Dynamic Calculus and Equations on Time Scales [1 ed.] 3111182894, 9783111182896

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Svetlin G. Georgiev (Ed.) Dynamic Calculus and Equations on Time Scales

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Dynamic Calculus and Equations on Time Scales �

Edited by Svetlin G. Georgiev

Mathematics Subject Classification 2020 Primary: 39A10, 34A60, 34B37; Secondary: 34C10, 34K20 Editor Prof. Dr. Svetlin G. Georgiev Department of Differential Equations Faculty of Mathematics and Informatics Kliment Ohridski University of Sofia 1126 Sofia Bulgaria [email protected]

ISBN 978-3-11-118289-6 e-ISBN (PDF) 978-3-11-118297-1 e-ISBN (EPUB) 978-3-11-118519-4 Library of Congress Control Number: 2023938954 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2023 Walter de Gruyter GmbH, Berlin/Boston Cover image: piranka / E+ / Getty Images Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface Time scale theory was first initiated by Stefan Hilger in 1988 in his PhD thesis to unify both approaches of dynamic modeling, namely difference and differential equations. Similar ideas have been used before and go back to the introduction of the Riemann– Stieltjes integral which unifies sums and integrals. Many results to differential equations carry over easily to corresponding results for difference equations, while other results seem to be totally different in nature. Because of these reasons, the theory of dynamic equations is an active area of research. The time scale calculus can be applied to any fields in which dynamic processes are described by discrete- or continuous-time models. So, the calculus of time scales has various applications involving discontinuous domains such as certain bug populations, phytoremediation of metals, wound healing, maximization problems in economics and traffic problems. This book presents some recent developments in the area of dynamic calculus and dynamic equations on time scales. The book contains ten chapters. Chapter 1 presents a projector analysis of dynamic systems on time scales. The linear time-varying dynamic systems are introduced and they are classified into those of the first, second, third, and fourth kind. The considered systems are investigated in the case when they are regular with tractability index 1. Jets of a function of one independent time scale variable and jets of a function of n independent real variables and one independent time scale variable are defined. Jet spaces are introduced and some of their properties are given. In the chapter, differentiable functions and total derivatives are defined. Nonlinear dynamic systems on arbitrary time scales are considered. Properly involved derivatives, constraints and consistent initial values for the considered equations are investigated. A linearization for nonlinear dynamic systems is introduced and the total derivative for regular linearized equations with tractability index 1 is investigated. Chapter 2 deals with the fundamental properties of the Muckenhoupt and Gehring weights on time scales. Some results related to the self-improving properties of the Muckenhoupt and Gehring classes and some higher integrability results for nonincreasing functions on time scales are presented. The main approach is based on proving some properties of integral operators with powers, the Hölder inequality, chain rules, as well as some connecting relations between Muckenhoupt and Gehring classes on time scales. In Chapter 3, a new definition of periodicity on isolated time scales introduced by Bohner, Mesquita, and Streipert is applied to the study of Ulam stability. If the graininess (step size) of an isolated time scale is bounded by a finite constant, then the linear 1- and 2-periodic dynamic equations are Ulam stable if and only if the exponential function has modulus different from unity. If the graininess increases at least linearly to infinity, the 1- and 2-periodic dynamic equations are not Ulam stable. Applying these results, several examples of the first-order linear 1- or 2-periodic dynamic equations on specific isolated time scales are given in the chapter, such as h-difference equations, q-difference equations, triangular equations, Fibonacci equations, and harmonic equations. In some cases the minimum Ulam stability https://doi.org/10.1515/9783111182971-201

VI � Preface constant is found. A multivalued logarithm on time scales recently introduced for deltadifferentiable functions that never vanish is covered in Chapter 4. This is accomplished using an extended definition of the cylinder transformation from which the definition of exponential functions on time scales arose. The definition of a logarithm function on arbitrary time scales with familiar and useful properties then follows. Chapter 5 deals with the existence and stability results for hybrid fuzzy differential equations with tempered Ξ-Hilfer fractional derivatives on time scales. Further, sufficient conditions for the existence and uniqueness of solutions by using hybrid fixed point theorem are obtained. In addition, it demonstrates Ulam-type stability. Finally, a suitable example to illustrate the main results is given. In Chapter 6, a new class of Ambartsumian equations of the fractional type with tempered Ξ-Hilfer fractional derivative with boundary conditions is examined. The provided problem is transformed into an equivalent fixed point problem, which is then resolved by using the Banach and Krasnosel’skii fixed point theorems. Ulam stability is investigated. An example is included to verify the theoretical results. In Chapter 7, a series solution method on general time scales is introduced. The derivation of the method and its application to dynamic and integral equations is discussed in detail. Several examples illustrating the method are presented. In Chapter 8, the dual results, delta and nabla inequalities, and their special cases, continuous and discrete inequalities, are unified into diamond alpha case, and new forms of such results as well as new diamond alpha Bennett–Leindler-type dynamic inequalities are established by developing a novel method, which does not require the integration by parts formula and the fundamental theorem of calculus. These theorems are standard arguments in the proofs of similar theorems in the delta and nabla approaches but do not follow naturally in the diamond-alpha calculus. In Chapter 9, a de la Vallée Poussin-type inequality for impulsive dynamic equations on time scales is derived. This inequality is often used in conjunction with disconjugacy and/or (non)oscillation. Hence, it appears to be a very useful tool for the qualitative study of dynamic equations. In this chapter, generalizing the classical de la Vallée Poussin inequality for impulsive dynamic equations on arbitrary time scales, disconjugacy criteria and some results on nonoscillation are obtained. In Chapter 10, the divided differences and σ-divided differences on time scales are introduced. The Newton and σ-Newton interpolation polynomial are constructed. In addition, the Hermite interpolation polynomial on time scales is constructed by using the divided differences table. Examples are presented to illustrate the theoretical results. This book is addressed to a wide audience of specialists such as mathematicians, physicists, engineers, and biologists. It can be used as a textbook at the graduate level and as a reference book for several disciplines. Paris, May 2023

Svetlin G. Georgiev

Contents Preface � V List of Contributing Authors � IX Svetlin G. Georgiev 1 Projector analysis of dynamic systems on time scales � 1 Samir H. Saker 2 Muckenhoupt and Gehring weights on time scales � 77 Douglas R. Anderson and Masakazu Onitsuka 3 Ulam stability and instability of first-order linear 1- and 2-periodic dynamic equations on isolated time scales � 147 Douglas R. Anderson, Martin Bohner, and Guo-Cheng Wu 4 A logarithm on time scales and its uses � 175 R. Vivek, S. Sivasundaram, D. Vivek, and K. Kanagarajan 5 Qualitative analysis for hybrid fuzzy differential equations involving tempered Ξ-Hilfer fractional derivative on time scales � 197 S. Manikandan, S. Sivasundaram, K. Kanagarajan, and D. Vivek 6 The dynamical analysis of nonlinear Ambartsumian equation via tempered Ξ-Hilfer fractional derivative on time scales � 219 Svetlin G. Georgiev and İnci M. Erhan 7 Series solution method on time scales and its applications � 239 Zeynep Kayar, Billur Kaymakçalan, and Neslihan Nesliye Pelen 8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities � 259 Sı̇ bel Doğru Akgöl and Abdullah Özbekler 9 De La Vallée Poussin-type inequality for impulsive dynamic equations on time scales � 295 Najlaa Jaddoa, Rezan Sevinik-Adıgüzel, and İnci M. Erhan 10 Divided and σ-divided differences on time scales � 305 Index � 325

List of Contributing Authors Douglas R. Anderson Department of Mathematics Concordia College Moorhead MN 56562 USA E-mail: [email protected]

K. Kanagarajan Department of Mathematics Sri Ramakrishna Mission Vidyalaya College of Arts and Science Coimbatore 641020 India E-mail: [email protected]

Martin Bohner Department of Mathematics and Statistics Missouri University of Science and Technology Rolla MO 65409-0020 USA E-mail: [email protected]

Zeynep Kayar Department of Mathematics Van Yüzüncü Yıl University Van Turkey E-mail: [email protected]

Sı̇ bel Doğru Akgöl Department of Mathematics Atilim University 06830 Incek Ankara Turkey E-mail: [email protected] İnci M. Erhan Department of Computer Engineering Aydın Adnan Menderes University 09010 Aydın Turkey E-mail: [email protected]

Billur Kaymakçalan Department of Mathematics Çankaya University 06810 Ankara Turkey E-mail: [email protected] S. Manikandan Department of Mathematics Sri Ramakrishna Mission Vidyalaya College of Arts and Science Coimbatore 641020 TamilnaduIndia E-mail: [email protected]

Svetlin G. Georgiev Department of Mathematics Sorbonne University Paris France E-mail: [email protected]

Masakazu Onitsuka Department of Applied Mathematics Okayama University of Science Okayama 700-0005 Japan E-mail: [email protected]

Najlaa Jaddoa Department of Mathematics Gazi University 06500 Beşevler Ankara Turkey E-mail: [email protected]

Abdullah Özbekler Department of Mathematics Atilim University 06830 Incek Ankara Turkey E-mail: [email protected]

X � List of Contributing Authors

Neslihan Nesliye Pelen Department of Mathematics Ondokuz Mayıs University Samsun Turkey E-mail: [email protected]

D. Vivek Department of Mathematics PSG College of Arts & Science Coimbatore 641014 India E-mail: [email protected]

Samir H. Saker Department of Mathematics Mansoura University Mansoura 35516 Egypt E-mail: [email protected]

R. Vivek Department of Mathematics Sri Ramakrishna Mission Vidyalaya College of Arts and Science Coimbatore 641020 India E-mail: [email protected]

Rezan Sevinik-Adıgüzel Department of Mathematics Atilim University 06830 Incek Ankara Turkey E-mail: [email protected] S. Sivasundaram Department of Mathematics College of Natural Science, Engineering and Mathematics Daytona Beach FL 32114 USA E-mail: [email protected]

Guo-Cheng Wu Data Recovery Key Laboratory of Sichuan Province College of Mathematics and Information Science, Neijiang Normal University Neijiang 641100 PR China E-mail: [email protected]

Svetlin G. Georgiev

1 Projector analysis of dynamic systems on time scales Abstract: This chapter presents a projector analysis of dynamic systems on time scales. We investigate the linear time-varying dynamic systems and classify them into those of the first, second, third, and fourth kind. The considered systems are investigated in the case when they are regular with tractability index 1. Then, we define jets of a function of one independent time scale variable and jets of a function of n independent real variables and one independent time scale variable. We introduce jet spaces and give some of their properties. In the chapter, we also define differentiable functions and total derivatives. We consider nonlinear dynamic systems on arbitrary time scales. We define properly involved derivatives, constraints, and consistent initial values for the considered equations. We introduce a linearization for nonlinear dynamic systems and investigate the total derivative for regular linearized equations with tractability index 1.

1.1 Linear time-varying dynamic-algebraic equations This chapter is devoted to linear time-varying dynamic-algebraic equations. We classify them into those of the first, second, third and fourth kind. We investigate them in the case when they are regular with tractability index 1. Suppose 𝕋 is a time scale with forward jump operator and delta differentiation operator σ and Δ, respectively. Let I ⊆ 𝕋.

1.1.1 Linear time-varying dynamic-algebraic equations of the first kind In this section, we will investigate the following linear time-varying dynamic-algebraic equation: Aσ (t)(Bx)Δ (t) = C σ (t)x σ (t) + f (t),

t ∈ I,

(1.1)

where A : I → Mn×m , B : I → Mm×n , C : I → Mn×n , and f : I → ℝn are given. Here, with Mp×q we denote the set of all p × q real matrices. Definition 1.1. Equation (1.1) is said to be a linear time-varying dynamic-algebraic equation of the first kind. Svetlin G. Georgiev, Department of Mathematics, Sorbonne University, Paris, France, e-mail: [email protected] https://doi.org/10.1515/9783111182971-001

2 � S. G. Georgiev We will consider the solutions of (1.1) within the space CB1 (I). Below, we remove the explicit dependence on t for the sake of notational simplicity.

1.1.1.1 A particular case Suppose that A, C : I → Mn×n . Consider the equation Aσ x Δ = C σ x σ + f .

(1.2)

We will show that equation (1.2) can be reduced to equation (1.1). Suppose that P is a

C 1 -projector along ker Aσ . Then

Aσ P = Aσ and Aσ x Δ = Aσ Px Δ = Aσ (Px)Δ − Aσ PΔ x σ . Hence, equation (1.2) takes the form Aσ (Px)Δ − Aσ PΔ x σ = C σ x σ + f , or Aσ (Px)Δ = (Aσ PΔ + C σ )x σ + f . Set C1σ = Aσ PΔ + C σ . Thus, (1.2) takes the form Aσ (Px)Δ = C1σ x σ + f ,

(1.3)

i. e., equation (1.2) is a particular case of equation (1.1). Example. Let 𝕋 = 2ℕ0 , We have

1 A(t) = ( 0 0

0 −t 0

0 1 ), 0

−t C(t) = ( 0 t

1 1 0

t 2t ) , 1

t ∈ 𝕋.

(1.4)

1 Projector analysis of dynamic systems on time scales

σ(t) = 2t,

� 3

t ∈ 𝕋,

and 1 Aσ (t) = ( 0 0

0 −2t 0

0 1 ), 0

−2t C σ (t) = ( 0 2t

1 1 0

2t 4t ) , 1

t ∈ 𝕋.

We will find a vector y1 (t) y(t) = ( y2 (t) ) , y3 (t)

t ∈ 𝕋,

so that 0 A (t)y(t) = ( 0 ) , 0 σ

t ∈ 𝕋.

We have 0 1 ( 0 )=( 0 0 0

0 −2t 0

0 y1 (t) y1 (t) 1 ) ( y2 (t) ) = ( −2ty2 (t) + y3 (t) ) , 0 y3 (t) 0

t ∈ 𝕋,

whereupon y1 (t) 0 ( y2 (t) ) = ( 1 ) , y3 (t) 2t

t ∈ 𝕋,

and the null projector to Aσ (t), t ∈ 𝕋, is 0 Q(t) = ( 0 0

0 0 0

0 1 ), 2t

t ∈ 𝕋.

Hence, 1 P(t) = I − Q(t) = ( 0 0

0 1 0

0 0 0 )−( 0 1 0

is a projector along ker Aσ . Note that

0 0 0

0 1 1 )=( 0 2t 0

0 1 0

0 −1 ) , 1 − 2t

t ∈ 𝕋,

4 � S. G. Georgiev 0 PΔ (t) = ( 0 0

0 0 0

0 0 ), −2

C1σ (t) = Aσ (t)PΔ (t) + C σ (t) 1 =( 0 0

0 −2t 0

0 =( 0 0

0 0 0

0 0 0

0 −2t −2 ) + ( 0 0 2t 1 1 0

−2t =( 0 2t

0 0 1 )( 0 0 0

0 −2t 0 )+( 0 −2 2t 1 1 0

1 1 0

2t 4t ) 1

2t 4t ) 1

2t 4t − 2 ) . 1

Equation (1.2) can be written as follows: 1 ( 0 0

x1Δ (t) 0 −2t 1 ) ( x2Δ (t) ) = ( 0 0 2t x3Δ (t)

0 −2t 0

1 1 0

x1σ (t) 2t f1 (t) 4t ) ( x2σ (t) ) + ( f2 (t) ) , 1 f3 (t) x3σ (t)

t ∈ 𝕋,

or x1Δ (t) = −2tx1σ (t) + x2σ (t) + 2tx3σ (t) + f1 (t),

−2tx2Δ (t) + x3Δ (t) = x2σ (t) + 4tx3σ (t) + f2 (t), 0 = 2tx1σ (t) + x3σ (t) + f3 (t),

t ∈ 𝕋.

This system, using (1.3), can be rewritten in the form 1 ( 0 0

0 −2t 0

−2t =( 0 2t

0 1 1 ) (( 0 0 0 1 1 0

0 1 0

Δ

0 x1 (t) −1 ) ( x2 (t) )) 1 − 2t x3 (t)

x1σ (t) 2t f1 (t) 4t − 2 ) ( x2σ (t) ) + ( f2 (t) ) , 1 f3 (t) x3σ (t)

t ∈ 𝕋,

or 1 ( 0 0

0 −2t 0

t ∈ 𝕋,

Δ

0 x1 (t) −2tx1σ (t) + x2σ (t) + 2tx3σ (t) + f1 (t) ), 1 ) ( x2 (t) − x3 (t) ) = ( x2σ (t) + (4t − 2)x3σ (t) + f2 (t) 0 (1 − 2t)x3 (t) 2tx1σ (t) + x3σ (t) + f3 (t)

1 Projector analysis of dynamic systems on time scales

� 5

or 1 (0 0

0 −2t 0

x1Δ (t) 0 −2tx1σ (t) + x2σ (t) + 2tx3σ (t) + f1 (t) Δ Δ ) = ( x2σ (t) + (4t − 2)x3σ (t) + f2 (t) ) , 1 )( x2 (t) − x3 (t) 0 2tx1σ (t) + x3σ (t) + f3 (t) (1 − 4t)x3Δ (t) − 2x3 (t)

t ∈ 𝕋, or x1Δ (t) = −2tx1σ (t) + x2σ (t) + 2tx3σ (t) + f1 (t),

−2t(x2Δ (t) − x3Δ (t)) + (1 − 4t)x3Δ (t) − 2x3 (t) = x2σ (t) + (4t − 2)x3σ (t) + f2 (t), 0 = −2tx1σ (t) + x3σ (t) + f3 (t),

t ∈ 𝕋,

or x1Δ (t) = −2tx1σ (t) + x2σ (t) +2 tx3σ (t) + f1 (t),

−2tx2Δ (t) + (1 − 2t)x3Δ (t) = x2σ (t) + (4t − 2)x3σ (t) + 2x3 (t) + f2 (t), 0 = 2tx1σ (t) + x3σ (t) + f3 (t),

t ∈ 𝕋.

1.1.1.2 Standard form index 1 problems In this section, we will investigate the equation Aσ (Px)Δ = C σ x σ + f , where ker A is a C 1 -space, C ∈ C (I), P is a C 1 -projector along ker A. Then AP = A. Assume in addition that Q=I −P and (B1) the matrix A1 = A + CQ is invertible. Definition 1.2. Equation (1.5) is said to be regular with tractability index 1.

(1.5)

6 � S. G. Georgiev We will start our investigations with the following useful lemma. Lemma 1.1. Suppose that (B1) holds. Then A−1 1 A=P and A−1 1 CQ = Q. Proof. We have A1 P = (A + CQ)P = AP + CQP = A. Since Q = I − P and ker P = ker A, we have im Q = ker A and AQ = 0. Then A1 Q = (A + CQ)Q = AQ + CQQ = CQ. This completes the proof. σ Now, we multiply equation (1.5) by (A−1 1 ) and get σ

σ

σ

σ Δ −1 σ σ −1 (A−1 1 ) A (Px) = (A1 ) C x + (A1 ) f .

Now, we employ the first equation of Lemma 1.1 and get σ

σ

σ σ −1 Pσ (Px)Δ = (A−1 1 ) C x + (A1 ) f .

(1.6)

We decompose x in the following way: x = Px + Qx. Then equation (1.6) takes the following form: σ

σ

σ σ σ σ σ −1 Pσ (Px)Δ = (A−1 1 ) C (P x + Q x ) + (A1 ) f σ

σ

σ

σ σ σ −1 σ σ σ −1 = (A−1 1 ) C P x + (A1 ) C Q x + (A1 ) f .

Using the second equation of Lemma 1.1, the latter equation can be rewritten as follows: σ

σ

σ σ σ σ σ −1 Pσ (Px)Δ = (A−1 1 ) C P x + Q x + (A1 ) f .

We multiply equation (1.7) with the projector Pσ and, using

(1.7)

1 Projector analysis of dynamic systems on time scales

PP = P,

� 7

PQ = 0,

we find σ

σ

σ σ σ σ σ σ σ −1 Pσ Pσ (Px)Δ = Pσ (A−1 1 ) C P x + P Q x + P (A1 ) f ,

or σ

σ

σ σ σ σ −1 Pσ (Px)Δ = Pσ (A−1 1 ) C P x + P (A1 ) f .

(1.8)

Note that Pσ (Px)Δ = (PPx)Δ − PΔ Px = (Px)Δ − PΔ Px. Hence using (1.8), we find σ

σ

σ

σ

σ σ σ σ −1 (Px)Δ − PΔ Px = Pσ (A−1 1 ) C P x + P (A1 ) f ,

or σ σ σ σ −1 (Px)Δ = PΔ Px + Pσ (A−1 1 ) C P x + P (A1 ) f .

(1.9)

Now, we multiply equation (1.7) by Qσ and find σ

σ −σ σ σ σ σ σ σ σ −1 Qσ Pσ (Px)Δ = Qσ (A−1 1 )C B B P x + Q Q x + Q (A1 ) f ,

or σ

σ

σ −σ σ σ σ σ σ σ −1 0 = Qσ (A−1 1 ) C B B P x + Q x + Q (A1 ) f .

(1.10)

Set u = Px,

v = Qx.

Then, by (1.9) and (1.10), we get the system σ

σ

σ −σ σ σ −1 uΔ = PΔ u + Pσ (A−1 1 ) C B u + P (A1 ) f , σ

σ

σ −σ σ σ −1 vσ = −Qσ (A−1 1 ) C B u − Q (A1 ) f .

(1.11)

We find u ∈ C 1 (I) from the first equation of the system (1.11), and then we find vσ ∈ C (I) from the second equation of the system (1.11). Hence, for the solution x of equation (1.5), we have the following representation: x σ = uσ + vσ = Pσ x σ + Qσ x σ .

8 � S. G. Georgiev Example. Let

𝕋 = ℕ, 0 C(t) = ( 0 0

0 −t 2

1 1 ), 1

−1 A(t) = ( 0 0

t+1 0 2t + 2

−1 0 ), −2

1 P(t) = ( 0 0

0 0 −(t + 1)

0 0 ), 1

(1.12) t ∈ 𝕋.

Here σ(t) = t + 1,

t ∈ 𝕋.

We have −1 A(t)P(t) = ( 0 0

t+1 0 2t + 2

−1 1 0 )( 0 −2 0

0 0 −(t + 1)

0 −1 0 )=( 0 1 0

t+1 0 2t + 2

−1 0 ), −2

t ∈ 𝕋. Therefore P is a projector along ker A. Next, Q(t) = I − P(t) 1 =( 0 0

0 0 −(t + 1)

0 =( 0 0

0 1 t+1

0 1 0 )−( 0 1 0

0 0 ), 0

0 0 −(t + 1)

0 0 ) 1

1 0 1 )( 0 1 0

0 1 t+1

t ∈ 𝕋,

and A1 (t) = A(t) + C(t)Q(t) −1 =( 0 0

t+1 0 2t + 2

−1 0 0 )+( 0 −2 0

0 −t 2

−1 =( 0 0

t+1 0 2t + 2

−1 0 0 )+( 0 −2 0

t+1 1 t+3

−1 =( 0 0

2(t + 1) 1 3t + 5

−1 0 ), −1

t ∈ 𝕋.

0 0 ) 0

0 0 ) 0

� 9

1 Projector analysis of dynamic systems on time scales

Note that det A1 (t) = 2 ≠ 0,

t ∈ 𝕋.

Thus, A1 is invertible. We will find its cofactors. We compute 󵄨󵄨 󵄨󵄨 󵄨󵄨 = 0, 󵄨󵄨 󵄨

󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨 1 0 󵄨󵄨󵄨 󵄨 0 󵄨 0 0 󵄨󵄨󵄨 󵄨 1 󵄨󵄨 = 0, a13 (t) = 󵄨󵄨󵄨 󵄨󵄨 = −2, a12 (t) = − 󵄨󵄨󵄨 a11 (t) = 󵄨󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 0 3t + 5 󵄨󵄨 0 −1 󵄨󵄨 󵄨󵄨 3t + 5 −2 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 2(t + 1) −1 󵄨󵄨 󵄨󵄨 = −(−4(t + 1) + 3t + 5) = −(−4t − 4 + 3t + 5) = t − 1, a21 (t) = − 󵄨󵄨󵄨󵄨 󵄨󵄨 3t + 5 −2 󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨 −1 −1 󵄨󵄨󵄨 󵄨 −1 2(t + 1) 󵄨󵄨󵄨 󵄨󵄨 = 2, a23 (t) = − 󵄨󵄨󵄨 󵄨󵄨 = 3t + 5, a22 (t) = 󵄨󵄨󵄨󵄨 󵄨󵄨 0 3t + 5 󵄨󵄨󵄨 󵄨󵄨 0 −2 󵄨󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨 2(t + 1) −1 󵄨󵄨󵄨 󵄨 −1 −1 󵄨󵄨󵄨 󵄨󵄨 = 1, a32 (t) = − 󵄨󵄨󵄨 󵄨󵄨 = 0, a31 (t) = 󵄨󵄨󵄨󵄨 󵄨󵄨 0 1 0 󵄨󵄨󵄨 0 󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨 −1 2(t + 1) 󵄨󵄨󵄨 󵄨󵄨 = −1, t ∈ 𝕋. a33 (t) = 󵄨󵄨󵄨󵄨 󵄨󵄨 1 󵄨󵄨 0 󵄨 Consequently,

A−1 1 (t)

−2 1 = ( 0 2 0

t−1 2 3t + 5

t−1 2

−1 1 0 )=( 0 −1 0

1 2

1

3t+5 2

t ∈ 𝕋.

0 ), − 21

Hence, −1 A (t) = ( 0 0 σ

t+2 0 2t + 4

0 C σ (t) = ( 0 0

0 −t − 1 2

0 Qσ (t) = ( 0 0

0 1 t+2

−1 0 ), −2 1 1 ), 1 0 0 ), 0

σ (A−1 1 ) (t)

−1 =( 0 0

1 2

t 2

1

3t+8 2

1 Pσ (t) = ( 0 0

0 0 −(t + 2)

0 PΔ (t) = ( 0 0

0 0 −1

0 ), − 21

0 0 ), 1

0 0 ), 0

t ∈ 𝕋.

Also, 1 P (t)(A ) (t) = ( 0 0 σ

−1 σ

0 1 0

−1 0 −1 ) ( 0 −t − 1 0

t 2

1

3t+8 2

1 2

−1 0 )=( 0 0 − 21

t 2

0

t+4 2

1 2

0 ), − 21

10 � S. G. Georgiev

P

σ

σ σ (t)(A−1 1 ) (t)C (t)

0 0 )( 0 − 21 0

0

t+4 2 (t+2)(t−1) − 2

0 =( 0 0 σ Qσ (t)(A−1 1 ) (t)

1 2

t 2

−1 =( 0 0

0 − (t+2)(t+3) 2

0 =( 0 0

0 0 0

0 =( 0

0

0

0 σ σ 0 Qσ (t)(A−1 ) (t)C (t) = ( 1

0

3t+8 2 (3t+8)(t+2) 2

0

0 =( 0

0

0 ),

t+3 2

0 − 21

− t+2 2 0 − 21

− t+2 2

− (3t+5)(t+2) 2 2 − (3t+5)(t+2) 2

0

1 1 ) 1

t−1 2

−1 0 1 )( 0 t+2 0

3t+8 2 (3t+8)(t+2) 2

0 −t − 1 2

t 2

1 2

1

3t+8 2

0 ) − 21

), 0 )( 0 0

0 −t − 1 2

1 1 ) 1

0

3(t+2) 2 (3t+7)(t+2) 2

),

t ∈ 𝕋.

Then, equation (1.5) can be rewritten as follows: −1 ( 0 0

t+2 0 2t + 4

0 =( 0 0

−1 1 0 ) (( 0 −2 0

0 −1 − t 2

Δ

0 0 −(t + 1)

0 x1 (t) 0 ) ( x2 (t) )) 1 x3 (t)

x1σ (t) 1 f1 (t) 1 ) ( x2σ (t) ) + ( f2 (t) ) , 1 f3 (t) x3σ (t)

t ∈ 𝕋,

or −1 ( 0 0

t+2 0 2t + 4

Δ

x1σ (t) −1 x1 (t) ) = ( −(t + 1)x2σ (t) + x3σ (t) ) 0 )( 0 −2 −(t + 1)x2 (t) + x3 (t) 2x2σ (t) + x3σ (t) f1 (t) + ( f2 (t) ) , f3 (t)

or

t ∈ 𝕋,

1 Projector analysis of dynamic systems on time scales

−1 ( 0 0

t+2 0 2t + 4

� 11

x1Δ (t) x1σ (t) −1 ) = ( −(t + 1)x2σ (t) + x3σ (t) ) 0 )( 0 σ Δ Δ −2 2x2σ (t) + x3σ (t) −x2 (t) − (t + 1)x2 (t) + x3 (t) f1 (t) + ( f2 (t) ) , f3 (t)

t ∈ 𝕋,

or x1σ (gt) + f1 (t) −x1Δ (t) + x2σ (t) + (t + 1)x2Δ (t) − x3Δ (t) ) = ( −(t + 1)x2σ (t) + x3σ (t) + f2 (t) ) , ( 0 Δ σ Δ 2x2σ (t) + x3σ (t) + f3 (t) 2x2 (t) + 2(t + 1)x2 (t) − 2x3 (t) t ∈ 𝕋, or −x1Δ (t) + x2σ (t) + (t + 1)x2Δ (t) − x3Δ (t) = x1σ (t) + f1 (t),

0 = −(t + 1)x2σ (t) + x3σ (t) + f2 (t),

2x2σ (t) + 2(t + 1)x2Δ (t) − 2x3Δ (t) = 2x2σ (t) + x3σ (t) + f3 (t),

t ∈ 𝕋,

or −x1Δ (t) + (t + 1)x2Δ (t) − x3Δ (t) = x1σ (t) − x2σ (t) + f1 (t),

0 = −(t + 1)x2σ (t) + x3σ (t) + f2 (t),

2(t + 1)x2Δ (t) − 2x3Δ (t) = x3σ (t) + f3 (t),

t ∈ 𝕋.

The system (1.11) can be rewritten as follows: u1Δ (t)

0 ( u2Δ (t) ) = ( 0 0 u3Δ (t)

0 0 −1

−1 +( 0 0 0 0 ( 0 ) = −( 0 vσ (t) 0 0 −( 0

0

0 u1 (t) 0 0 ) ( u2 (t ) + ( 0 0 u3 (t) 0 t 2

1 2

0

t+4 2

0

f1 (t) 0 ) ( f2 (t) ) , − 21 f3 (t) 0

− (3t+5)(t+1) 2

3(t+2) 2

− (3t+5)(t+2) 2

(3t+7)(t+2 2

2

0

− (t−1)(t+2) 2 0 − (t+2)(t+3) 2

3t+8 2 (3t+8)(t+2) 2

0 − 21

− t+2 2

u1σ (t) ) ( u2σ (t) ) u3σ (t)

f1 (t) ) ( f2 (t) ) , f3 (t)

t ∈ 𝕋,

u1σ (t) 0 ) ( u2σ (t) ) t+3 u3σ (t) 2 t−1 2

12 � S. G. Georgiev or u1Δ (t)

u2σ (t) + − (t−1)(t+2) 0 2 ( u2Δ (t) ) = ( )+( 0 0 σ −u2σ (t) − (t+2)(t+3) u u3Δ (t) 2 (t) + 2 −f1 (t) + 2t f2 (t) + 21 f3 (t) +( ), 0 t+4 1 f (t) − f (t) 2 2 2 3

t−1 σ u (t) 2 3 t+3 σ u (t) 2 3

)

0 0 (3t+5)(t+2) σ u2 (t) + 3(t+2) u3σ (t) ) ( 0 )=( − 2 2 2 vσ (t) − (3t+5)(t+2) u2σ (t) + (3t+7)(t+2) u3σ (t) 2 2 0

−(

3t+8 f (t) − 21 f3 (t) 2 2 (3t+8)(t+2) f2 (t) − t+2 f (t) 2 2 3

),

t ∈ 𝕋,

or u1Δ (t) = − u2Δ (t) = 0,

(t − 1)(t + 2) σ t−1 σ t 1 u2 (t) + u3 (t) − f1 (t) + f2 (t) + f3 (t), 2 2 2 2

u3Δ (t) = − u2 (t) − 0= − vσ (t) = − +

(t + 2)(t + 3) σ t+3 σ t+4 1 u2 (t) + u (t) + f (t) − f3 (t), 2 2 3 2 2 2

(3t + 5)(t + 2) σ 3(t + 2) σ 3t + 8 1 u2 (t) + u3 (t) + f (t) − f3 (t), 2 2 2 2 2 (3t + 7)(t + 2) σ (3t + 5)(t + 2)2 σ u2 (t) + u3 (t) 2 2

(3t + 8)(t + 2) t+2 f2 (t) − f (t), 2 2 3

t ∈ 𝕋.

1.1.2 Linear time-varying dynamic-algebraic equations of the second kind In this chapter we will investigate the following linear time-varying dynamic-algebraic equation: Aσ (t)(Bx)Δ (t) = C(t)x(t) + f (t),

t ∈ I,

(1.13)

where A, B, C : I → Mm×m and f : I → ℝm are given. Equation (1.13) will be said to be a linear time-varying dynamic-algebraic equation of the second kind. We will consider the solutions of (1.13) within the space CB1 (I). Below, we remove the explicit dependence on t for the sake of notational simplicity.

1 Projector analysis of dynamic systems on time scales

� 13

1.1.2.1 A particular case Consider the equation Aσ x Δ = Cx + f .

(1.14)

We will show that equation (1.14) can be reduced to equation (1.13). Suppose that P is a C 1 -projector along ker A. Then AP = A and Aσ x Δ = Aσ Pσ x Δ = Aσ (Px)Δ − Aσ PΔ x. Hence, equation (1.14) takes the form Aσ (Px)Δ − Aσ PΔ x = Cx + f , or Aσ (Px)Δ = (Aσ PΔ + C)x + f . Set C1 = Aσ PΔ + C. Thus, (1.14) takes the form Aσ (Px)Δ = C1 x + f ,

(1.15)

i. e., equation (1.14) is a particular case of equation (1.13). Example. Let 𝕋 = 2ℕ0 and the matrices A and C be as in (1.4). Then σ(t) = 2t,

t ∈ 𝕋,

and 1 Aσ (t) = ( 0 0 We will find a vector

0 −2t 0

0 1 ), 0

t ∈ 𝕋.

14 � S. G. Georgiev y1 (t) y(t) = ( y2 (t) ) , y3 (t)

t ∈ 𝕋,

0 A(t)y(t) = ( 0 ) , 0

t ∈ 𝕋.

so that

We have 0 1 ( 0 )=( 0 0 0

0 −t 0

0 y1 (t) y1 (t) 1 ) ( y2 (t) ) = ( −ty2 (t) + y3 (t) ) , 0 y3 (t) 0

t ∈ 𝕋,

whereupon y1 (t) 0 ( y2 (t) ) = ( 1 ) , y3 (t) t

t ∈ 𝕋,

and the null projector to A(t), t ∈ 𝕋, is 0 Q(t) = ( 0 0

0 0 0

0 1 ), t

t ∈ 𝕋.

Hence, 1 P(t) = I − Q(t) = ( 0 0

0 1 0

0 0 0 )−( 0 1 0

0 0 0

0 1 1 )=( 0 t 0

0 1 0

0 −1 ) , 1−t

t ∈ 𝕋,

is a projector along ker A. Observe that 0 P (t) = ( 0 0 Δ

0 0 0

0 0 ), −1

1 C1 (t) = Aσ (t)PΔ (t) + C(t) = ( 0 0 0 =( 0 0

0 0 0

0 −t −1 ) + ( 0 0 t

0 −2t 0 1 1 0

0 0 1 )( 0 0 0

0 0 0

t −t 2t ) = ( 0 1 t

0 −t 0 )+( 0 −1 t 1 1 0

t 2t − 1 ) . 1

1 1 0

t 2t ) 1

1 Projector analysis of dynamic systems on time scales

� 15

Equation (1.2) can be written as follows: 1 ( 0 0

0 −2t 0

x1Δ (t) 0 −t 1 ) ( x2Δ (t) ) = ( 0 0 t x3Δ (t)

1 1 0

t x1 (t) f1 (t) 2t ) ( x2 (t) ) + ( f2 (t) ) , 1 x3 (t) f3 (t)

t ∈ 𝕋,

or −2tx2Δ (t)

x1Δ (t) = −tx1 (t) + x2 (t) + tx3 (t) + f1 (t),

+ x3Δ (t) = x2 (t) + 2tx3 (t) + f2 (t), 0 = tx1 (t) + x3 (t) + f3 (t),

t ∈ 𝕋.

This system, using (1.15), can be rewritten in the form 1 ( 0 0

0 −2t 0

−t =( 0 t

0 1 1 ) (( 0 0 0 1 1 0

0 1 0

0 x1 (t) −1 ) ( x2 (t) )) 1−t x3 (t)

Δ

t x1 (t) f1 (t) 2t − 1 ) ( x2 (t) ) + ( f2 (t) ) , 1 x3 (t) f3 (t)

t ∈ 𝕋,

or 1 ( 0 0

0 −2t 0

Δ

0 x1 (t) −tx1 (t) + x2 (t) + tx3 (t) f1 (t) 1 ) ( x2 (t) ) = ( x2 (t) + (2t − 1)x3 (t) ) + ( f2 (t) ) , 0 x3 (t) tx1 (t) + x3 (t) f3 (t)

t ∈ 𝕋,

or 1 ( 0 0

0 −2t 0

x1Δ (t) −tx1 (t) + x2 (t) + tx3 (t) + f1 (t) 0 1 ) ( x2Δ (t) ) = ( x2 (t) + (2t − 1)x3 (t) + f2 (t) ) , 0 tx1 (t) + x3 (t) + f3 (t) x3Δ (t)

t ∈ 𝕋,

or x1Δ (t) = −tx1 (t) + x2 (t) + tx3 (t) + f1 (t),

−2t(x2Δ (t) − x3Δ (t)) + (1 − 2t)x3Δ (t) − x3 (t) = x2 (t) + (2t − 1)x3 (t) + f2 (t), 0 = tx1 (t) + x3 (t) + f3 (t),

or −2tx2Δ (t)

x1Δ (t) = −tx1 (t) + x2 (t) + tx3 (t) + f1 (t),

+ x3Δ (t) = x2 (t) + 2tx3 (t) + f2 (t), 0 = tx1 (t) + x3 (t) + f3 (t),

t ∈ 𝕋.

t ∈ 𝕋,

16 � S. G. Georgiev 1.1.2.2 Standard form index 1 problems In this section, we will investigate the equation Aσ (Px)Δ = Cx + f , where ker A is a C 1 -space, C ∈ C (I), P is a C 1 -projector along ker A. Then AP = A. Assume in addition that Q=I −P and (C1) the matrix Aσ1 = A + CQ is invertible. Definition 1.3. Equation (1.16) is said to be regular with tractability index 1. We will start our investigations with the following useful lemma. Lemma 1.2. Suppose that (C1) holds. Then Aσ−1 1 A=P and Aσ−1 1 CQ = Q. Proof. We have Aσ1 P = (A + CQ)P = AP + CQP = A. Since Q = I − P and ker P = ker A, we have im Q = ker A and AQ = 0. Then Aσ1 Q = (A + CQ)Q = AQ + CQQ = CQ. This completes the proof.

(1.16)

1 Projector analysis of dynamic systems on time scales

� 17

Now, we multiply equation (1.16) with Aσ−1 and get 1 σ Δ σ−1 σ−1 Aσ−1 1 A (Px) = A1 Cx + A1 f .

Now, we employ the second equation of Lemma 1.2 and get σ−1 P(Px)Δ = Aσ−1 1 Cx + A1 f .

(1.17)

We decompose x in the following way: x = Px + Qx. Then equation (1.17) takes the following form: σ−1 P(Px)Δ = Aσ−1 1 C(Px + Qx) + A1 f

σ−1 σ−1 = Aσ−1 1 CPx + A1 CQx + A1 f .

Using the second equation of Lemma 1.2, the latter equation can be rewritten as follows: σ−1 P(Px)Δ = Aσ−1 1 CPx + Qx + A1 f .

(1.18)

We multiply equation (1.18) with the projector P and, using PP = P,

PQ = 0,

we find σ−1 PP(Px)Δ = PAσ−1 1 CPx + PQx + PA1 f ,

or σ−1 P(Px)Δ = PAσ−1 1 CPx + PA1 f .

(1.19)

Note that P(Px)Δ = (PPx)Δ − PΔ Pσ x σ = (Px)Δ − PΔ Pσ x σ . Hence by (1.19), we find σ−1 (Px)Δ − PΔ Pσ x σ = PAσ−1 1 CPx + PA1 f ,

or σ−1 (Px)Δ = PΔ Pσ x σ + PAσ−1 1 CPx + PA1 f .

(1.20)

18 � S. G. Georgiev Now, we multiply equation (1.18) by Q and find σ−1 QP(Px)Δ = QAσ−1 1 CPx + QQx + QA1 f ,

or σ−1 0 = QAσ−1 1 CPx + Qx + QA1 f .

(1.21)

Set u = Px,

v = Qx.

Then, by (1.20) and (1.21), we get the system σ−1 uΔ = PΔ uσ + PAσ−1 1 Cu + PA1 f ,

(1.22)

σ−1 v = −QAσ−1 1 Cu − QA1 f .

We find u ∈ C 1 (I) from the second equation of the system (1.22), and then we find v ∈ C (I) from the second equation of the system (1.22). Hence the solution of the equation (1.16) is given by x = u + v = Px + Qx. Example. Let 𝕋 = ℕ and A, P and C be as in (1.12). Then, equation (1.16) can be rewritten as follows: −1 ( 0 0

t+2 0 2t + 4

0 =( 0 0

0 −t 2

−1 1 0 ) (( 0 −2 0

0 1 0

Δ

0 x1 (t) −1 ) ( x2 (t) )) −1 x3 (t)

1 x1 (t) f1 (t) 1 ) ( x2 (t) ) + ( f2 (t) ) , 1 x3 (t) f3 (t)

t ∈ 𝕋,

or −1 ( 0 0

Δ

t+2 0 2t + 4

−1 x1 (t) x1 (t) f1 (t) 0 ) ( x2 (t) ) = ( −tx2 (t) + x3 (t) ) + ( f2 (t) ) , −2 x3 (t) 2x2 (t) + x3 (t) f3 (t)

t ∈ 𝕋,

t+2 0 2t + 4

x1Δ (t) −1 x1 (t) f1 (t) Δ 0 ) ( x2 (t) ) = ( −tx2 (t) + x3 (t) ) + ( f2 (t) ) , −2 2x2 (t) + x3 (t) f3 (t) x3Δ (t)

t ∈ 𝕋,

or −1 ( 0 0

1 Projector analysis of dynamic systems on time scales

� 19

or −x1Δ (t) + (t + 2)(x2Δ (t) − x3Δ (t)) + x3σ (t) + x3Δ (t) = x1 (t) + f1 (t), (2t +

4)(x2Δ (t)

−

x3Δ (t))

+

0 = −tx2 (t) + x3 (t) + f2 (t),

2x3σ (t)

+

2x3Δ (t)

= 2x2 (t) + x3 (t) + f3 (t),

t ∈ 𝕋,

or −x1Δ (t) + (t + 2)x2Δ (t) − (t + 1)x3Δ (t) = x1 (t) − x3 (t) + f1 (t), (2t +

4)x2Δ (t)

− 2(t +

0 = −tx2 (t) + x3 (t) + f2 (t),

1)x3Δ (t)

= 2x2 (t) − x3 (t) + f3 (t),

t ∈ 𝕋.

Next, we will rewrite the system (1.22). We have 1 σ P(t)(A−1 (t)) = ( 0 1 0

0 1 0

−1

=( 0 0 σ

2t+1 2 2t+3 2(t+2)

1 2 1 2(t+2)

2t+1 2 2t+3 2(t+2)

1 2 1 2(t+2)

t

−1

P(t)(A−1 (t)) C(t) = ( 0 0

t

0

2t+1 2 1 − 2(t+2)

−1 0 −1 ) ( 0 −t 0

0

0

−1

0

0 )( 0 0

0 −t − 1 2

)

),

−t(t + 2) 2

2t+1 2 2t +7t+7 2(t+2) 2

=( 0 0

−2 − (2t+3)(t+1) 2(t+2) −t(t + 1)

0 =( 0 0

0 0 0

0 =( 0 0

0 −1 −(t + 1)

0 0 ), 0

0 σ Q(t)(A−1 (t)) C(t) = ( 0 0

0 −1 −(t + 1)

0 0 0 )( 0 0 0

0 =( 0 0

0 t+1 (t + 1)2

0 ), −1 −(t + 1)

σ Q(t)(A−1 1 (t))

1 2 1 2(t+1)

t

−1 0 1 )( 0 t+1 0

1 1 ) 1

),

2t+1 2 1 − 2(t+2)

1 2 1 2(t+2)

−1

0 −t − 1 2 t ∈ 𝕋.

0

1 1 ) 1

)

20 � S. G. Georgiev Hence, the system (1.22) takes the form u1Δ (t)

0 ( u2Δ (t) ) = ( 0 0 u3Δ (t)

0 0 0

0 u1σ (t) 0 σ 0 ) ( u2 (t) ) + ( 0 −1 u3σ (t) 0

−1 u1 (t) × ( u2 (t) ) + ( 0 u3 (t) 0 v1 (t) 0 ( v2 (t) ) = − ( 0 v3 (t) 0

0 t+1 (t + 1)2

2t+1 2 2t+3 2(t+2)

t

1 2 1 2(t+2)

0

−t(t + 2) 2

−2 − (2t+3)(t+1) 2(t+2) −t(t + 1)

t

)

f1 (t) ) ( f2 (t) ) , f3 (t)

0 u1 (t) 0 ) ( u2 (t) ) − ( 0 −1 −(t + 1) u3 (t) 0

f1 (t) × ( f2 (t) ) , f3 (t)

2t+1 2 2t 2 +7t+7 2(t+2)

0 −1 −(t + 1)

0 0 ) 0

t ∈ 𝕋,

or u1Δ (t)

−t(t + 2)u2 (t) + 2t+1 u3 (t) 0 2 2 2 Δ (2t+3)(t+1) −2 2t +7t+7 ( u2 (t) ) = ( )+( − 0 u2 (t) + 2(t+2) u3 (t) ) 2(t+2) σ Δ −u (t) −t(t + 1)u2 (t) + tu3 (t) 3 u3 (t) 2t+1 f (t) + 21 f2 (t) 2 2 2t+3 1 f (t) + 2(t+2) f3 (t) 2(t+2) 2

−f1 (t) + +(

tf2 (t)

),

v1 (t) 0 0 ( v2 (t) ) = − ( )−( ), (t + 1)u2 (t) − u3 (t) −f2 (t) v3 (t) (t + 1)2 u2 (t) − (t + 1)u3 (t) −(t + 1)f3 (t) or 2t + 1 2t + 1 1 u3 (t) − f1 (t) + f (t) + f2 (t), 2 2 2 2 (2t + 3)(t + 1)2 − 2 2t 2 + 7t + 7 u2Δ (t) = − u2 (t) + u (t) 2(t + 2) 2(t + 2) 3 u1Δ (t) = −t(t + 2)u2 (t) +

+

2t + 3 1 f (t) + f (t), 2(t + 2) 2 2(t + 2) 3

u3Δ (t) = −u3σ (t) − t(t + 1)u2 (t) + tu3 (t) + tf2 (t), v1 (t) = 0,

v2 (t) = −(t + 1)u2 (t) + u3 (t) + f2 (t),

v3 (t) = −(t + 1)2 u2 (t) + (t + 1)u3 (t) + (t + 1)f3 (t),

t ∈ 𝕋.

t ∈ 𝕋,

1 Projector analysis of dynamic systems on time scales

� 21

1.1.3 Linear time-varying dynamic-algebraic equations of the third kind In this section, we will investigate the following linear time-varying dynamic-algebraic equation: A(t)(Bx)Δ (t) = C(t)x σ (t) + f (t),

t ∈ I,

(1.23)

where A, B, C : I → Mm×m and f : I → ℝm are given. Definition 1.4. Equation (1.23) will be said to be third kind linear time-varying dynamicalgebraic equation. We will consider the solutions of (1.23) within the space CB1 (I). Without loss of generality, we remove the explicit dependence on t.

1.1.3.1 A particular case In this section, we will investigate the equation Ax Δ = Cx σ + f .

(1.24)

We will show that it can be reduced to equation (1.23). Let P be a C 1 -projector along ker A. Then AP = A and from here, Ax Δ = APx Δ

= A(Px Δ )

= A((Px)Δ − PΔ x σ )

= A(Px)Δ − APΔ x σ . Then, equation (1.24) can be rewritten in the following manner: A(Px)Δ − APΔ x σ = Cx σ + f , or A(Px)Δ = (APΔ + C)x σ + f . Denoting

22 � S. G. Georgiev C1 = APΔ + C, we find A(Px)Δ = C1 x σ + f ,

(1.25)

and therefore we can consider equation (1.24) as a particular case of equation (1.23). Example. Let 𝕋 = 2ℕ0 and the matrices A and C be as in (1.4). Then σ(t) = 2t,

t ∈ 𝕋,

and 1 P(t) = ( 0 0

0 1 0

0 −1 ) , 1−t

t ∈ 𝕋,

is a projector along ker A. Also, we have 0 PΔ (t) = ( 0 0

0 0 0

0 0 ). −1

Hence, C1 (t) = A(t)PΔ (t) + C(t) 1 =( 0 0

0 −t 0

0 =( 0 0

0 0 0

−t =( 0 t

1 1 0

0 0 1 )( 0 0 0

0 0 0

0 −t −1 ) + ( 0 0 t

0 −t 0 )+( 0 −1 t 1 1 0

1 1 0

t 2t ) 1

t 2t ) 1

t 2t − 1 ) . 1

Equation (1.24) can be written as follows: 1 ( 0 0 or

0 −t 0

x1Δ (t) 0 −t 1 ) ( x2Δ (t) ) = ( 0 0 t x3Δ (t)

1 1 0

t x1 (t) f1 (t) 2t ) ( x2 (t) ) + ( f2 (t) ) , 1 x3 (t) f3 (t)

t ∈ 𝕋,

1 Projector analysis of dynamic systems on time scales

� 23

x1Δ (t) = −tx1σ (t) + x2σ (t) + tx3σ (t) + f1 (t),

−tx2Δ (t) + x3Δ (t) = x2σ (t) + 2tx3σ (t) + f2 (t), 0 = tx1σ (t) + x3σ (t) + f3 (t),

t ∈ 𝕋.

This system, using (1.25), can be rewritten in the form 1 ( 0 0

0 −t 0

−t =( 0 t

0 1 1 ) (( 0 0 0 1 1 0

0 1 0

Δ

0 x1 (t) −1 ) ( x2 (t) )) 1−t x3 (t)

x1σ (t) t f1 (t) σ ) ( ) + ( 2t − 1 x2 (t) f2 (t) ) , σ 1 f3 (t) x3 (t)

t ∈ 𝕋,

or 1 ( 0 0

0 −t 0

x1Δ (t) 0 −tx1σ (t) + x2σ (t) + tx3σ (t) + f1 (t) Δ 1 ) ( x2 (t) ) = ( x2σ (t) + (2t − 1)x3σ (t) + f2 (t) ) , 0 tx1σ (t) + x3σ (t) + f3 (t) x3Δ (t)

t ∈ 𝕋,

or x1Δ (t) = −tx1σ (t) + x2σ (t) + tx3σ (t) + f1 (t),

−t(x2Δ (t) − x3Δ (t)) + (1 − 2t)x3Δ (t) − x3 (t) = x2σ (t) + (2t − 1)x3σ (t) + f2 (t), 0 = tx1σ (t) + x3σ (t),

t ∈ 𝕋,

or x1Δ (t) = −tx1σ (t) + x2σ (t) + tx3σ (t) + f1 (t),

−tx2Δ (t) + (1 − t)x3Δ (t) = x2σ (t) + (2t − 1)x3σ (t) + x3 (t) + f2 (t), 0 = tx1σ (t) + x2σ (t) + f3 (t),

t ∈ 𝕋.

1.1.3.2 Standard form index 1 problems Consider the equation A(Px)Δ = Cx σ + f ,

(1.26)

where ker A is a C 1 -space, C ∈ C (I), P is a projector so that Pσ is a C 1 -projector along ker A. Denote Q = I − P.

24 � S. G. Georgiev Then APσ = A,

AQσ = 0.

Assume that (C1) the matrix A1 = A + CQσ is invertible. Definition 1.5. Equation (1.26) is called regular with tractability index 1. We have the following result. Lemma 1.3. Let (C1) hold. Then we have the following equations: σ A−1 1 A=P ,

σ σ A−1 1 CQ = Q .

Proof. By the definition of A1 , we get A1 Pσ = (A + CQσ )Pσ

= APσ + CQσ Pσ

= APσ =A

and A1 Qσ = (A + CQσ )Qσ

= AQσ + CQσ Qσ

= AQσ , which completes the proof.

We multiply equation (1.26) by A−1 1 and, using the first equation and then the second equation of Lemma 1.3, we arrive at Δ −1 σ −1 A−1 1 A(Px) = A1 Cx + A1 f ,

or σ −1 Pσ (Px)Δ = A−1 1 Cx + A1 f

σ σ σ −1 = A−1 1 C(P + Q )x + A1 f

σ σ σ σ −1 = A−1 1 CP x + Q x + A1 f ,

1 Projector analysis of dynamic systems on time scales

� 25

i. e., σ σ σ σ −1 Pσ (Px)Δ = A−1 1 CP x + Q x + A1 f .

(1.27)

We multiply both sides of equation (1.27) by Pσ and, using Pσ Pσ = Pσ

and

Pσ Qσ = 0,

we obtain σ σ σ σ σ σ −1 Pσ Pσ (Px)Δ = Pσ A−1 1 CP x + P Q x + P A1 f ,

or σ σ σ −1 Pσ (Px)Δ = Pσ A−1 1 CP x + P A1 f .

(1.28)

Note that Pσ (Px)Δ = (PPx)Δ − PΔ Px = (Px)Δ − PΔ Px. Then, equation (1.28) can be rewritten in the form σ σ Δ σ −1 (Px)Δ = Pσ A−1 1 CP x + P Px + P A1 f .

Setting Px = u, we find σ Δ σ −1 uΔ = Pσ A−1 1 Cu + P u + P A1 f .

Now, we multiply equation (1.27) by Qσ and find σ σ σ σ σ σ −1 Qσ Pσ (Px)Δ = Qσ A−1 1 CP x + Q Q x + Q A1 f ,

or σ σ σ σ σ −1 0 = Qσ A−1 1 CP x + Q x + Q A1 f .

Denoting v = Qx, from the latter equation we find

(1.29)

26 � S. G. Georgiev σ σ σ −1 0 = Qσ A−1 1 Cu + v + Q A1 f ,

or σ σ −1 vσ = −Qσ A−1 1 Cu − Q A1 f .

Using the latter equation and (1.29), we obtain the system σ Δ σ −1 uΔ = Pσ A−1 1 Cu + P u + P A1 f ,

(1.30)

σ σ −1 vσ = −Qσ A−1 1 Cu − Q A1 f .

From the first equation of (1.30) we find u, and then we find v from the second equation of (1.30). Example. Let 𝕋 = ℕ and A, C, P be as in (1.12). Then 0 P (t) = ( 0 0

0 0 0

0 0 ), −1

0 Qσ (t) = ( 0 0

0 0 0

0 1 ), t+2

1 σ −1 P (t)A1 (t) = ( 0 0

0 1 0

−1 0 −1 ) ( 0 −t − 1 0

Δ

−1 =( 0 0 P

σ

(t)A−1 1 (t)C(t)

=( 0 0

Q

(t)A−1 1 (t)

1 2 1 2(t+1)

2t−1 2 2t+1 2(t+1)

1 2 1 2(t+1)

0

t+1

0

2+t−2t 2 2

0

σ

2t−1 2 2t+1 2(t+1)

t+1

−1

1 P (t) = ( 0 0 σ

2

2−t−2t 2(t+1)

=( 0 0

−t(t + 1)

0 =( 0 0

0 0 0

2t−1 2 1 − 2(t+1)

0 1 0

0 −1 ) , −t − 1

1 2 1 2(t+1)

0

−1

)

), 0 )( 0 0

0 −t 2

1 1 ) 1

t 1 ), t+1

−1 0 1 )( 0 t+2 0

2t−1 2 1 − 2(t+1)

−1

1 2 1 2(t+1)

0

0 )=( 0 0

0 −1 −t − 2

0 0 ), 0

1 Projector analysis of dynamic systems on time scales

0 Qσ (t)A−1 (t)C(t) = ( 0 1 0

0 −1 −t − 1

0 =( 0 0

0 t t(t + 2)

0 0 0 )( 0 0 0 0 −1 ) , −t − 2

0 −t 2

� 27

1 1 ) 1

t ∈ 𝕋.

Hence, the system (1.30) takes the form u1Δ (t) (

u2Δ (t) u3Δ (t)

2+t−2t 2 2 2−t−2t 2 2(t+1)

0 )= ( 0 0

−t(t + 1) −1

+( 0 0 vσ1 (t) 0 ( vσ2 (t) ) = ( 0 0 vσ3 (t)

2t−1 2 2t+1 2(t+1)

t+1

0 t t(t + 2)

0 −( 0 0

u1σ (t) 0 σ ) ( ) + ( u (t) 0 2 1 σ 0 u3 (t) t+1 t

0 −1 −t − 2

1 2 1 2(t+1)

0

0 0 0

0 u1 (t) 0 ) ( u2 (t) ) −1 u3 (t)

f1 (t) ) ( f2 (t) ) , f3 (t)

u1σ (t) 0 −1 ) ( u2σ (t) ) −t − 2 u3σ (t) 0 f1 (t) 0 ) ( f2 (t) ) , 0 f3 (t)

t ∈ 𝕋,

or u1Δ (t)

( u2Δ (t) ) = u3Δ (t)

(

vσ1 (t) vσ2 (t) vσ3 (t)

2+t−2t 2 σ u2 (t) + tu3σ (t) 2 2 2−t−2t ( )( uσ (t) + tu3σ (t) 2(t+1) 2 σ σ −t(t + 1)u2 (t) + (t + 1)u3 (t) f (t) + 21 f3 (t) −f1 (t) + 2t−1 2 2 1 2t+1 +( f (t) + 2(t+1) f3 (t) ) , 2(t+1) 2

0 ) 0 −u3 (t)

(t + 1)f2 (t)

) = −(

0 0 tu2σ (t) − u3σ (t) )−( ), −f2 (t) −(t + 2)f3 (t) t(t + 2)u2σ (t) − (t + 2)u3σ (t)

or u1Δ (t) =

2 + t − 2t 2 σ 2t − 1 1 u2 (t) + tu3σ (t) − f1 (t) + f2 (t) + f3 (t), 2 2 2

u2Δ (t) =

2 − t − 2t 2 σ 2t + 1 1 u (t) + tu3σ (t) + f (t) + f (t), 2(t + 1) 2 2(t + 1) 2 2(t + 1) 3

t ∈ 𝕋,

28 � S. G. Georgiev u3Δ (t) = −t(t + 1)u2σ (t) + (t + 1)u3σ (t) + (t + 1)f2 (t), vσ1 (t) = 0,

vσ2 (t) = −tu2σ (t) + u3σ (t) + f2 (t),

vσ3 (t) = −t(t + 2)u2σ (t) + (t + 2)u3σ (t) + (t + 2)f3 (t),

t ∈ 𝕋.

1.1.4 Linear time-varying dynamic-algebraic equations of the fourth kind In this section, we will investigate the following linear time-varying dynamic-algebraic equation: A(t)(Bx)Δ (t) = C(t)x(t) + f (t),

t ∈ I,

(1.31)

where A : I → Mn×m , B : I → Mm×n , C : I → Mn×n , and f : I → ℝn are given. Definition 1.6. Equation (1.31) will be said to be a linear time-varying dynamic-algebraic equation of the fourth kind. We will consider the solutions of (1.31) within the space CB1 (I). Below, we remove the explicit dependence on t for the sake of notational simplicity.

1.1.4.1 A particular case In this section, we will consider the equation Ax Δ = Cx + f . Suppose that P is a C 1 -projector so that Pσ is a C 1 -projector along ker A. Then APσ = A and, from equation (1.32), we find Cx + f = APσ x Δ = A(Px)Δ − APΔ x, whereupon A(Px)Δ = (C + APΔ )x + f . We set A1 = APΔ + C.

(1.32)

1 Projector analysis of dynamic systems on time scales

� 29

Then, we get the equation A(Px)Δ = A1 x + f .

(1.33)

Thus, equation (1.32) can be reduced to equation (1.31). Example. Let 𝕋 = 2ℕ0 and A, C be as in (1.4). We will search for a vector y1 (t) y(t) = ( y2 (t) ) ∈ ℝ3 , y3 (t)

t ∈ 𝕋,

so that A(t)y(t) = 0,

t ∈ 𝕋.

We have 0 1 ( 0 )=( 0 0 0

0 −t 0

0 y1 (t) y1 (t) 1 ) ( y2 (t) ) = ( −ty2 (t) + y3 (t) ) , 0 y3 (t) 0

whereupon y1 (t) = 0,

y3 (t) = ty2 (t),

t ∈ 𝕋,

and 0 Qσ (t) = ( 0 0

0 0 0

0 2 ), 2t

t ∈ 𝕋,

is such that A(t)Qσ (t) = 0,

t ∈ 𝕋.

We have 0 Q(t) = ( 0 0 and

0 0 0

0 2 ), t

t ∈ 𝕋,

t ∈ 𝕋,

30 � S. G. Georgiev 1 P(t) = I − Q(t) = ( 0 0

0 1 0

0 0 0 )−( 0 1 0

0 0 0

0 1 2 )=( 0 t 0

0 1 0

0 −2 ) , 1−t

t ∈ 𝕋,

and 1 Pσ (t) = ( 0 0

0 1 0

0 −2 ) , 1 − 2t

t ∈ 𝕋.

Next, 1 A(t)P (t) = ( 0 0 σ

0 −t 0

0 1 1 )( 0 0 0

0 1 0

0 1 −2 ) = ( 0 1 − 2t 0

0 −t 0

0 1 ) = A(t), 0

t ∈ 𝕋.

Therefore Pσ is a C 1 -projector along ker A. Moreover, 0 PΔ (t) = ( 0 0

0 0 0

0 0 ), −1

t ∈ 𝕋,

and A1 (t) = A(t)PΔ (t) + C(t) 1 =( 0 0

0 −t 0

0 =( 0 0

0 0 0

−t =( 0 t

1 1 0

0 0 1 )( 0 0 0

0 0 0

0 −t 0 )+( 0 −1 t

0 −t −1 ) + ( 0 0 t

1 1 0

t −1 + 2t ) , 1

t ∈ 𝕋.

1 1 0

t 2t ) 1

t 2t ) 1

Equation (1.31) takes the form 1 ( 0 0 or

0 −t 0

x1Δ (t) 0 −t 1 ) ( x2Δ (t) ) = ( 0 0 t x3Δ (t)

1 1 0

t x1 (t) f1 (t) 2t ) ( x2 (t) ) + ( f2 (t) ) , 1 x3 (t) f3 (t)

t ∈ 𝕋,

1 Projector analysis of dynamic systems on time scales

x1Δ (t)

(

−tx2Δ (t)

+

0

x3Δ (t)

−tx1 (t) + x2 (t) + tx3 (t) f1 (t) )=( ) + ( f2 (t) ) , x2 (t) + 2tx3 (t) tx1 (t) + x3 (t) f3 (t)

� 31

t ∈ 𝕋,

or x1Δ (t) = −tx1 (t) + x2 (t) + tx3 (t) + f1 (t),

−x2Δ (t) + x3Δ (t) = x2 (t) + 2tx3 (t) + f2 (t), 0 = tx1 (t) + x3 (t) + f3 (t),

t ∈ 𝕋.

Equation (1.33) can be rewritten in the form 1 ( 0 0

0 −t 0

−t =( 0 t

0 1 1 ) (( 0 0 0 1 1 0

0 1 0

Δ

0 x1 (t) −2 ) ( x2 (t) )) 1−t x3 (t)

t x1 (t) f1 (t) −1 + 2t ) ( x2 (t) ) + ( f2 (t) ) , 1 x3 (t) f3 (t)

t ∈ 𝕋,

or 1 ( 0 0

0 −t 0

Δ

0 x1 (t) −tx1 (t) + x2 (t) + tx3 (t) 1 ) ( x2 (t) − 2x3 (t) ) = ( x2 (t) + (−1 + 2t)x3 (t) ) 0 (1 − t)x3 (t) tx1 (t) + x3 (t) f1 (t) + ( f2 (t) ) , f3 (t)

t ∈ 𝕋,

or 1 ( 0 0

0 −t 0

x1Δ (t) 0 −tx1 (t) + x2 (t) + tx3 (t) + f1 (t) Δ ) = ( x2 (t) + (−1 + 2t)x3 (t) + f2 (t) ) , 1 )( x2 (t) − 2x3Δ (t) 0 tx1 (t) + x3 (t) + f3 (t) −x3 (t) + (1 − 2t)x3Δ (t)

t ∈ 𝕋, or x1Δ (t) = −tx1 (t) + x2 (t) + tx3 (t) + f1 (t),

−t(x2Δ (t) − 2x3Δ (t)) − x3 (t) + (1 − 2t)x3Δ (t) = x2 (t) + (−1 + 2t)x3 (t) + f2 (t), 0 = tx2 (t) + x3 (t) + f3 (t),

or

t ∈ 𝕋,

32 � S. G. Georgiev x1Δ (t) = −tx1 (t) + x2 (t) + tx3 (t) + f1 (t),

−tx2Δ (t) + x3Δ (t) = x2 (t) + 2tx3 (t) + f2 (t), 0 = tx2 (t) + x3 (t) + f3 (t),

t ∈ 𝕋.

1.1.5 Standard form index 1 problems In this section, we will investigate the equation A(Px)Δ = Cx + f ,

(1.34)

where P is a C 1 -projector along ker A. Set Q = I − P. Suppose that (D1) A1 = A + CQ is invertible. Definition 1.7. Equation (1.34) is called regular with tractability index 1. By Lemma 1.1, it follows that A−1 1 A = P,

A−1 1 CQ = Q.

We multiply (1.34) by A−1 1 and find Δ −1 −1 A−1 1 A(Px) = A1 Cx + A1 f ,

or −1 P(Px)Δ = A−1 1 Cx + A1 f −1 −1 = A−1 1 CPx + A1 CQx + A1 f −1 = A−1 1 CPx + Qx + A1 f ,

i. e., −1 P(Px)Δ = A−1 1 CPx + Qx + A1 f .

We multiply equation (1.35) by P, then, using PP = P, and setting

P(Px)Δ = (Px)Δ − PΔ Pσ x σ ,

(1.35)

1 Projector analysis of dynamic systems on time scales

� 33

Px = u, we arrive at the equation −1 (Px)Δ = PΔ Pσ x σ + PA−1 1 CPx + PA1 f ,

or −1 uΔ = PΔ uσ + PA−1 1 Cu + PA1 f .

(1.36)

Now, we multiply (1.35) by Q and, setting Qx = v, we find −1 0 = QA−1 1 CPx + Qx + QA1 f −1 = QA−1 1 Cu + v + QA1 f .

From the latter equation and (1.37), we obtain −1 uΔ = PΔ uσ + PA−1 1 Cu + PA1 f ,

(1.37)

−1 v = −QA−1 1 Cu − QA1 f .

Example. Let 𝕋 = ℕ and A, C be as in (1.12). Let also 1 P(t) = ( 0 0

0 1 0

0

1 − t+1

0

),

0 Q(t) = ( 0 0

0 0 0

0

1 t+1

1

),

t ∈ 𝕋.

0 1 0

0 0 ), 1

t ∈ 𝕋,

t+1 0 2(t + 1)

−1 0 ) −2

We have 1 P(t) + Q(t) = ( 0 0

0 1 0

0 0 1 )+( 0 − t+1 0 0

0 0 0

0

1 t+1

1 )=( 0 1 0

0 1 0

0 −1 1 )=( 0 − t+1 0 0

and −1 A(t)P(t) = ( 0 0 = A(t),

and

t+1 0 2(t + 1)

t ∈ 𝕋,

−1 1 0 )( 0 −2 0

34 � S. G. Georgiev 1 P(t)P(t) = ( 0 0 = P(t),

0 1 0

1 0 1 )( 0 − t+1 0 0

t ∈ 𝕋.

0 1 0

0 1 1 )=( 0 − t+1 0 0

0 1 0

0 1 ) − t+1 0

Thus, P is a projector along ker A. Next, A1 (t) = A(t) + C(t)Q(t) −1 =( 0 0

t+1 0 2(t + 1)

−1 0 0 )+( 0 −2 0

0 −t 2

1 0 1 )( 0 1 0

−1 =( 0 0

t+1 0 2(t + 1)

0 −1 0 )+( 0 −2 0

0

1

−1 =( 0 0

t+1 0

0

2(t + 1)

1 t+1 − t−1 t+1

),

0

0

1 t+1 t+3 t+1

0 0 0

0

1 t+1

1

)

)

t ∈ 𝕋.

Hence, det A(t) = 2 ≠ 0,

t ∈ 𝕋.

Therefore A1 is invertible. We will find the cofactors of A1 . We have 1 󵄨󵄨 󵄨󵄨 0 󵄨 t+1 󵄨󵄨󵄨 a11 (t) = 󵄨󵄨󵄨󵄨 󵄨󵄨 = −2, 󵄨 󵄨󵄨 2(t + 1) − t−1 t+1 󵄨 󵄨󵄨 󵄨󵄨 0 󵄨 0 󵄨󵄨 󵄨󵄨 = 0, a13 (t) = 󵄨󵄨󵄨󵄨 󵄨󵄨 0 2(t + 1) 󵄨󵄨󵄨 󵄨 󵄨󵄨 0 󵄨󵄨󵄨 󵄨 t+1 a21 (t) = − 󵄨󵄨󵄨󵄨 t−1 󵄨󵄨󵄨 = t − 1, 󵄨󵄨 󵄨󵄨 2(t + 1) − t+1 󵄨󵄨 󵄨󵄨 t + 1 󵄨󵄨 󵄨 −1 󵄨󵄨 = 2(t + 1), a23 (t) = − 󵄨󵄨󵄨󵄨 󵄨󵄨 0 2(t + 1) 󵄨󵄨󵄨 󵄨 󵄨󵄨 󵄨 t + 1 −1 󵄨󵄨󵄨 󵄨󵄨 = 0, a31 (t) = 󵄨󵄨󵄨󵄨 0 󵄨󵄨󵄨 󵄨󵄨 0 󵄨󵄨 󵄨 󵄨 −1 t + 1 󵄨󵄨󵄨 󵄨󵄨 = 0, t ∈ 𝕋. a33 (t) = 󵄨󵄨󵄨󵄨 0 󵄨󵄨󵄨 󵄨󵄨 0

Hence,

󵄨󵄨 0 󵄨 a12 (t) = − 󵄨󵄨󵄨󵄨 󵄨󵄨 0

1 t+1 − t−1 t+1

󵄨󵄨 󵄨 −1 a22 (t) = 󵄨󵄨󵄨󵄨 󵄨󵄨 0

0

󵄨󵄨 󵄨 −1 a32 (t) = − 󵄨󵄨󵄨󵄨 󵄨󵄨 0

− t−1 t+1

󵄨󵄨 󵄨󵄨 󵄨󵄨 = 0, 󵄨󵄨 󵄨

󵄨󵄨 󵄨󵄨 t − 1 󵄨󵄨 = 󵄨󵄨 t + 1 , 󵄨

󵄨 0 󵄨󵄨󵄨 1 , 1 󵄨󵄨󵄨 = t + 1 󵄨 t+1 󵄨

� 35

1 Projector analysis of dynamic systems on time scales

−2 1 −1 A1 (t) = ( 0 2 0

t−1

0

t−1 t+1

−1

1 t+1

2(t + 1)

)=( 0

0

0

t−1 2 t−1 2(t+1)

0

1 2(t+1)

t+1

0

t ∈ 𝕋,

),

and

P(t)A−1 1 (t)

1

=( 0

0

0 1

0

=( 0

0

0

=( 0

0

Q(t)A−1 1 (t)

0

0

0

1 t+1

1

t+1

0

1 2(t+1)

0

)

0

1 2(t+1)

0

0

1 2(t+1)

0

), 0 )( 0 0

t−3 2 t+2 − 2(t+1)

0

0

0 PΔ (t) = ( 0 0

0

0

0

0 Q(t)A−1 (t)C(t) = ( 0 1 0

0

− t(t−1) 2 t+2 2

0

=( 0

)( 0

0

=( 0 0

1 − t+1

t−1 2 t+3 − 2(t+1)

−1

t−1 2 t−1 2(t+1)

−1

t−1 2 t+3 − 2(t+1)

−1

P(t)A−1 1 (t)C(t)

0

0

0

0 1 t+1

0 0 0 )( 0 0 0

0 0 0

0

1 (t+1)(t+2)

0

),

1 1 ) 1

),

−1

)( 0

0 −t 2

t−1 2 t−1 2(t+1)

t+1

0 −t 2

0

1 2(t+1)

0

0 )=( 0 0

1 0 1 )=( 0 1 0

0 1 t+1

0 −t −t(t + 1)

0 0 ), 0

0 1 ), t+1

t ∈ 𝕋.

Then, the system (1.37) takes the form u1Δ (t)

0 Δ ( u2 (t) ) = ( 0 0 u3Δ (t)

0 0 0

0

1 (t+1)(t+2)

0

0 u1σ (t) σ 0 ) ( u2 (t) ) + ( u3σ (t) 0

−1 u1 (t) × ( u2 (t) ) + ( 0 u3 (t) 0

t−1 2 t+3 − 2(t+1)

0

0

1 2(t+1)

0

− t(t−1) 2 t+2 2

0

t−3 2 t+2 − 2(t+1)

f1 (t) ) ( f2 (t) ) , f3 (t)

0

)

36 � S. G. Georgiev 0 0 ( v0 (t) ) = − ( 0 v1 (t) 0

0 −t −t(t + 1)

f1 (t) × ( f2 (t) ) , f3 (t)

0 u1 (t) 0 1 ) ( u2 (t) ) − ( 0 t+1 u3 (t) 0

0 1 t+1

0 0 ) 0

t ∈ 𝕋,

or u1Δ (t)

( u2Δ (t) ) = ( u3Δ (t)

+(

t−3 u (t) 2 3 t+2 u (t) 2(t+1) 3

u2 (t) + − t(t−1) 2

0

1 uσ (t) (t+1)(t+2) 3

)+(

0

t+2 u (t) 2 2

−f1 (t) + t−1 f (t) 2 2 t+3 1 − 2(t+1) f2 (t) + 2(t+1) f3 (t) 0

−

0

)

),

0 0 0 ( v0 (t) ) = − ( )−( ), −tu2 (t) + u3 (t) f2 (t) v1 (t) −t(t + 1)u2 (t) + (t + 1)u3 (t) (t + 1)f2 (t)

t ∈ 𝕋,

or u1Δ (t) = − u2Δ (t) =

t(t − 1) t−3 t−1 u2 (t) + u (t) − f1 (t) + f (t), 2 2 3 2 2

t+2 t+2 1 uσ (t) + u (t) − u (t) (t + 1)(t + 2) 3 2 2 2(t + 1) 3 −

u3Δ (t) = 0,

t+3 1 f (t) + f (t), 2(t + 1) 2 2(t + 1) 3

v0 (t) = tu2 (t) − u3 (t) + f2 (t),

v1 (t) = t(t + 1)u2 (t) − (t + 1)u3 (t) − (t + 1)f2 (t),

t ∈ 𝕋.

1.2 Jets and jet spaces In this chapter, we introduce jets of one independent time scale variable, as well as jets of n independent real variables and one independent time scale variable. We deduct some of their properties and then we define jet spaces and total derivative in jet variables. Suppose that 𝕋 is a time scale with forward jump operator and delta differentiation operator σ and Δ, respectively.

1 Projector analysis of dynamic systems on time scales

� 37

1.2.1 Taylor’s formula for a function of one independent time scale variable We introduce the generalized monomials hk : 𝕋 × 𝕋 → ℝ, k ∈ ℕ0 , defined recursively by h0 (t, s) = 1, t

hk (t, s) = ∫ hk−1 (τ, s)Δτ,

k ∈ ℕ,

s

t, s ∈ 𝕋.

Then t

h1 (t, s) = ∫ Δτ = t − s, Δ hk t (t, s)

s

= hk−1 (t, s),

k ∈ ℕ,

t, s ∈ 𝕋.

(t − s)k , k!

k ∈ ℕ,

t, s ∈ 𝕋.

k ∈ ℕ,

t ∈ 𝕋.

Example. Let 𝕋 = ℝ. Then hk (t, s) = Example. Let 𝕋 = ℤ. We define t (0) = 1,

k−1

t (k) = ∏(t − j), j=0

Then h0 (t, s) = (t − s)(0) , t

t

h1 (t, s) = ∫ h0 (τ, s)Δτ = ∫ Δτ = t − s = s

s

(t − s)(1) . 1!

Assume that hk (t, s) =

(t − s)(k) , k!

t, s ∈ 𝕋,

for some k ∈ ℕ. We will prove that hk+1 (t, s) =

(t − s)(k+1) . (k + 1)!

(1.38)

38 � S. G. Georgiev Indeed, we have Δ

(

(t − s)(k+1) t (k+1) ) = (σ(t) − s) − (t − s)(k+1) (k + 1)! 1 = ((σ(t) − s)(σ(t) − s − 1) ⋅ ⋅ ⋅ (σ(t) − s − k) (k+1 )!

− (t − s)(t − s − 1) ⋅ ⋅ ⋅ (t − s − k)) 1 = ((t + 1 − s)(t + 1 − s − 1) ⋅ ⋅ ⋅ (t + 1 − s − k) (k + 1)! − (t − s)(t − s − 1) ⋅ ⋅ ⋅ (t − s − k)) 1 = ((t + 1 − s)(t − s) ⋅ ⋅ ⋅ (t − s − k + 1) (k + 1)! − (t − s)(t − s − 1) ⋅ ⋅ ⋅ (t − s − k)) 1 = (t − s) ⋅ ⋅ ⋅ (t − s − k + 1)(t + 1 − s − t + s + k) (k + 1)! = =

1 (t − s) ⋅ ⋅ ⋅ (t − s − k + 1)(k + 1) (k + 1)! 1 (t − s)(t − s − 1) ⋅ ⋅ ⋅ (t − s − k + 1) k!

(t − s)(k) k! = hk (t, s), t, s ∈ 𝕋.

=

Therefore (1.38) holds for any k ∈ ℕ. Theorem 1.1. We have t

hk+m+1 (t, t0 ) = ∫ hk (t, σ(s))hm (s, t0 )Δs, t0

t, t0 ∈ 𝕋,

k, m ∈ ℕ0 .

Proof. Let t

g(t) = ∫ hk (t, σ(s))hm (s, t0 )Δs. t0

Then, using the chain rule, we get t

g Δ (t) = hk (σ(t), σ(t))hm (σ(t), t0 ) + ∫ hk−1 (t, σ(s)), hm (s, t0 )Δs t0

1 Projector analysis of dynamic systems on time scales

� 39

t

= ∫ hk−1 (t, σ(s))hm (s, t0 )Δs, t0

t

2

g Δ (t) = hk−1 (σ(t), σ(t))hm (t, t0 ) + ∫ hk−2 (t, σ(s))hm (s, t0 )Δs t0

t

= ∫ hk−2 (t, σ(s))hm (s)Δs, .. . Δk

t0

t

g (t) = ∫ hm (s, t0 )Δs = hm+1 (t, t0 ), t0

k+1

g Δ (t) = hm+1 (t, t0 ), .. . k+m

gΔ

(t) = hk+m+1 (t, t0 ).

This completes the proof. We define t

g0 (t, s) = 1,

gk+1 (t, s) = ∫ gk (σ(τ), s)Δτ, s

k ∈ ℕ0 ,

t, s ∈ 𝕋.

Lemma 1.4. Let n ∈ ℕ. If f is n times differentiable and pk , 0 ≤ k ≤ n − 1, are differentiable at some t ∈ 𝕋 with pΔk+1 (t) = pσk (t)

for all 0 ≤ k ≤ n − 2, n ≥ 2,

then we have n−1

Δ

k Δk

n

( ∑ (−1) f (t)pk (t)) = (−1)n−1 f Δ (t)pσn−1 (t) + f (t)pΔ0 (t). k=0

Proof. We have n−1

k

Δ

n−1

k

( ∑ (−1)k f Δ (t)pk (t)) = ∑ (−1)k (f Δ (t)pk (t)) k=0

Δ

k=0 n−1

k+1

k

= ∑ (−1)k (f Δ (t)pσk (t) + f Δ (t)pΔk (t)) k=0

40 � S. G. Georgiev n−1

n−1

k+1

k

= ∑ (−1)k f Δ (t)pσk (t) + ∑ (−1)k f Δ (t)pΔk (t) k=0

k=0

n−2

k+1

n

= ∑ (−1)k f Δ (t)pσk (t) + (−1)n−1 f Δ (t)pσn−1 (t) k=0

n−1

k

0

+ ∑ (−1)k f Δ (t)pΔk (t) + f Δ (t)pΔ0 (t) k=0

n−2

k+1

n

= ∑ (−1)k f Δ (t)pΔk+1 (t) + (−1)n−1 f Δ (t)pσn−1 (t) k=0

n−2

k+1

+ ∑ (−1)k+1 f Δ (t)pΔk+1 (t) + f (t)pΔ0 (t) k=0

n

= (−1)n−1 f Δ (t)pσn−1 (t) + f (t)pΔ0 (t). This completes the proof. Lemma 1.5. The functions gn (t, s) satisfy, for all t ∈ 𝕋, the relationship gn (ρk (t), t) = 0 for all n ∈ ℕ and all 0 ≤ k ≤ n − 1. Proof. Let n ∈ ℕ be arbitrarily chosen. Then t

gn (ρ0 (t), t) = gn (t, t) = ∫ gn−1 (σ(τ), t)Δτ = 0. t

Assume that gn−1 (ρk (t), t) = 0

and

gn (ρk (t), t) = 0

for some 0 ≤ k < n − 1. We will prove that gn (ρk+1 (t), t) = 0. Case 1. ρk (t) is left-dense. Then ρk+1 (t) = ρ(ρk (t)) = ρk (t). Consequently, using the induction assumption, we have gn (ρk+1 (t), t) = gn (ρk (t), t) = 0.

1 Projector analysis of dynamic systems on time scales

� 41

Case 2. ρk (t) is left-scattered. Then ρ(ρk (t)) < ρk (t) and there is no s ∈ 𝕋 such that ρk+1 (t) < s < ρk (t). Hence, σ(ρk+1 (t)) = ρk (t). Therefore gn (σ(ρk+1 (t)), t) = gn (ρk+1 (t), t) + μ(ρk+1 (t))gnΔ (ρk+1 (t), t), or gn (ρk (t), t) = gn (ρk+1 (t), t) + μ(ρk+1 (t))gnΔ (ρk+1 (t), t), whereupon gn (ρk+1 (t), t) = gn (ρk (t), t) − μ(ρk+1 (t))gnΔ (ρk+1 (t), t)

= gn (ρk (t), t) − μ(ρk+1 (t))gn−1 (σ(ρk+1 (t)), t) = gn (ρk (t), t) − μ(ρk+1 (t))gn−1 (ρk (t), t) = 0.

This completes the proof. Lemma 1.6. Let n ∈ ℕ and suppose that f is (n − 1) times differentiable at ρn−1 (t). Then n−1

k

f (t) = ∑ (−1)k f Δ (ρn−1 (t))gk (ρn−1 (t), t). k=0

Proof. I. Let n = 1. Then 0

k

0

∑ (−1)k f Δ (ρ0 (t))gk (ρ0 (t), t) = (−1)0 f Δ (t)g0 (t, t) = f (t).

k=0

II. Assume that m−1

k

f (t) = ∑ (−1)k f Δ (ρm−1 (t))gk (ρm−1 (t), t) k=0

for some m ∈ ℕ. We will prove that m

k

f (t) = ∑ (−1)k f Δ (ρm (t))gk (ρm (t), t). k=0

42 � S. G. Georgiev Case 1. ρm−1 (t) is left-dense. Then ρm (t) = ρ(ρm−1 (t)) = ρm−1 (t). Hence by the induction assumption, we obtain m

k

∑ (−1)k f Δ (ρm (t))gk (ρm (t), t)

k=0

m−1

k

m

= ∑ (−1)k f Δ (ρm (t))gk (ρm (t), t) + (−1)m f Δ (ρm (t))gm (ρm (t), t) k=0

m−1

k

= ∑ (−1)k f Δ (ρm−1 (t))gk (ρm−1 (t), t) k=0

m

+ (−1)m f Δ (ρm−1 (t))gm (ρm−1 (t), t) (now we apply Lemma 1.5 (gm (ρm−1 (t), t) = 0)) m−1

k

= ∑ (−1)k f Δ (ρm−1 (t))gk (ρm−1 (t), t) k=0

(now we apply the induction assumption) = f (t). Case 2. ρm−1 (t) is left-scattered. Then ρm (t) = ρ(ρm−1 (t)) < ρm−1 (t) and there is no s ∈ 𝕋 such that ρm (t) < s < ρm−1 (t). Also, σ(ρm (t)) = ρm−1 (t). Hence, gk (σ(ρm (t)), t) = gk (ρm−1 (t), t). Therefore gk (ρm−1 (t), t) = gk (σ(ρm (t)), t) = gk (ρm (t), t) + μ(ρm (t))gkΔ (ρm (t), t)

1 Projector analysis of dynamic systems on time scales

� 43

= gk (ρm (t), t) + μ(ρm (t))gk−1 (σ(ρm (t)), t)

= gk (ρm (t), t) + μ(ρm (t))gk−1 (ρm−1 (t), t), whereupon

gk (ρm (t), t) = gk (ρm−1 (t), t) − μ(ρm (t))gk−1 (ρm−1 (t), t). Consequently, m

k

∑ (−1)k f Δ (ρm (t))gk (ρm (t), t)

k=0

m

k

k=1 m

k

= f (ρm (t)) + ∑ (−1)k f Δ (ρm (t))gk (ρm (t), t) = f (ρm (t)) + ∑ (−1)k f Δ (ρm (t))gk (ρm−1 (t), t) k=1

m

k

+ ∑ (−1)k−1 f Δ (ρm (t))μ(ρm (t))gk−1 (ρm−1 (t), t) k=1

m−1

k

= f (ρm (t)) + ∑ (−1)k f Δ (ρm (t))gk (ρm−1 (t), t) m Δm

+ (−1) f

k=1

(ρm (t))gm (ρm−1 (t), t)

m−1

k−1

+ ∑ (−1)k f Δ (ρm (t))μ(ρm (t))gk (ρm−1 (t), t) k=0

m−1

k

= ∑ (−1)k f Δ (ρm (t))gk (ρm−1 (t), t) k=0

m−1

k+1

+ ∑ (−1)k μ(ρm (t))f Δ (ρm (t))gk (ρm−1 (t), t) k=0

m−1

k

k

Δ

= ∑ (−1)k (f Δ (ρm (t)) + μ(ρm (t))(f Δ ) (ρm (t)))gk (ρm−1 (t), t) k=0

m−1

k

= ∑ (−1)k f Δ (σ(ρm (t)))gk (ρm−1 (t), t) k=0

m−1

k

= ∑ (−1)k f Δ (ρm−1 (t))gk (ρm−1 (t), t) k=0

= f (t). This completes the proof.

44 � S. G. Georgiev n

Theorem 1.2 (Taylor’s formula). Let n ∈ ℕ. Suppose f is n times differentiable on 𝕋κ . Let n−1 α ∈ 𝕋κ , t ∈ 𝕋. Then n−1

ρn−1 (t)

Δk

k

n

f (t) = ∑ (−1) gk (α, t)f (α) + ∫ (−1)n−1 gn−1 (σ(τ), t)f Δ (τ)Δτ. k=0

α

Proof. We note that, applying Lemma 1.4 with pk = gk , we have n−1

Δ

Δk

k

( ∑ (−1) gk (τ, t)f (τ)) k=0

τ

n−1 Δn

f (τ)gn−1 (σ(τ), t) + f (τ)g0Δ (τ, t)

= (−1)

n

= (−1)n−1 f Δ (τ)gn−1 (σ(τ), t)

n

for all τ ∈ 𝕋κ .

We integrate the latter relation from α to ρn−1 (t) and get ρn−1 (t)

n−1

Δ

Δk

k

∫ ( ∑ (−1) gk (τ, t)f (τ)) Δτ k=0

α

n−1

ρ

τ

(t)

n

= ∫ (−1)n−1 f Δ (τ)gn−1 (σ(τ), t)Δτ, α

or n−1

n−1

k

k

∑ (−1)k gk (ρn−1 (t), t)f Δ (ρn−1 (t)) − ∑ (−1)k gk (α, t)f Δ (α)

k=0

k=0

n−1

ρ

(t)

n

= ∫ (−1)n−1 f Δ (τ)gn−1 (σ(τ), t)Δτ. α

Hence, applying Lemma 1.6, n−1

k

ρn−1 (t)

n

f (t) − ∑ (−1)k gk (α, t)f Δ (α) = ∫ (−1)n−1 f Δ (τ)gn−1 (σ(τ), t)Δτ. k=0

α

This completes the proof. Theorem 1.3. The functions gn and hn satisfy the relationship hn (t, s) = (−1)n gn (s, t) n

for all t ∈ 𝕋 and all s ∈ 𝕋κ .

1 Projector analysis of dynamic systems on time scales

� 45

n

Proof. Let t ∈ 𝕋 and s ∈ 𝕋κ be arbitrarily chosen. We apply Theorem 1.2 for α = s and f (τ) = hn (τ, s). We observe that k

f Δ (τ) = hn−k (τ, s),

0 ≤ k ≤ n.

Hence, k

f Δ (s) = hn−k (s, s) = 0, Δn

f (s) = h0 (s, s) = 1,

0 ≤ k ≤ n − 1, f

Δn+1

(τ) = 0.

From here, using Taylor’s formula, we get f (t) = hn (t, s) n

Δk

k

ρn (t)

n+1

= ∑ (−1) gk (α, t)f (α) + ∫ (−1)n gn (σ(τ), t)f Δ (τ)Δτ k=0

α

n

k

ρn (t)

n+1

= ∑ (−1)k gk (s, t)f Δ (s) + ∫ (−1)n gn (σ(τ), t)f Δ (τ)Δτ k=0

s

n−1

k

n

= ∑ (−1)k gk (s, t)f Δ (s) + (−1)n gn (s, t)f Δ (s) k=0

n

= (−1)n gn (s, t)f Δ (s) = (−1)n gn (s, t),

i. e., hn (t, s) = (−1)n gn (s, t). This completes the proof. From Theorems 1.2 and 1.3, the following theorem, known as the Taylor formula of order n around α, follows. n

Theorem 1.4 (Taylor’s formula). Let n ∈ ℕ. Suppose f is n times differentiable on 𝕋κ . Let n−1 also, α ∈ 𝕋κ , t ∈ 𝕋. Then n−1

k

ρn−1 (t)

n

f (t) = ∑ hk (t, α)f Δ (α) + ∫ hn−1 (t, σ(τ))f Δ (τ)Δτ. k=0

α

Now we will formulate and prove another variant of Taylor’s formula. Theorem 1.5 (Taylor’s formula). Let n ∈ ℕ. Suppose that the function f is (n + 1) times n+1 n+1 differentiable on 𝕋κ . Let α ∈ 𝕋κ , t ∈ 𝕋, and t > α. Then

46 � S. G. Georgiev n

t

Δk

n+1

f (t) = ∑ hk (t, α)f (α) + ∫ hn (t, σ(τ))f Δ (τ)Δτ. k=0

(1.39)

α

Proof. Let n+1

g(t) = f Δ (t). Then f solves the problem n+1

xΔ

= g(t),

k

k

x Δ (α) = f Δ (α),

k ∈ {0, . . . , n}.

We have that y(t, s) = hn (t, σ(s)) n+1

is the Cauchy function for yΔ

= 0. Hence, it follows that t

f (t) = u(t) + ∫ y(t, σ(τ))g(τ)Δτ α

t

= u(t) + ∫ hn (t, σ(s))g(s)Δs,

(1.40)

α

where, u solves the initial value problem n+1

uΔ

m

m

uΔ (α) = f Δ (α),

= 0,

m ∈ {0, . . . , n}.

We set n

k

w(t) = ∑ hk (t, α)f Δ (α).

(1.41)

k=0

We have n

m

k

wΔ (t) = ∑ hk−m (t, α)f Δ (α),

m ∈ {0, . . . , n},

k=0

and hence m

n

k

m

wΔ (α) = ∑ hk−m (α, α)f Δ (α) = f Δ (α), k=0

m ∈ {0, . . . , n},

i. e., w solves (1.40). Consequently, w = u. Hence using (1.41), we obtain (1.39). This completes the proof.

1 Projector analysis of dynamic systems on time scales

� 47

Example. Let 𝕋 = 2ℕ0 and f (t) = t 4 + t,

t ∈ 𝕋.

(1.42)

We will apply Taylor’s formula of order 3 for f around α = 1. Here σ(t) = 2t,

t ∈ 𝕋,

and t

h0 (t, 1) = 1,

h1 (t, 1) = ∫ Δs = t − 1, 1

t

t

t

t

h2 (t, 1) = ∫ h1 (τ, 1)Δτ = ∫(τ − 1)Δτ = ∫ τΔτ − ∫ Δτ 1

=

1

2 󵄨󵄨τ=t

τ 3

1

2

2

1

1 t 2 t 󵄨󵄨 󵄨󵄨 − t + 1 = − − t + 1 = − t + , 󵄨󵄨τ=1 3 3 3 3 t

t

h2 (t, σ(τ)) = h2 (t, 2τ) = ∫ h1 (s, 2τ)Δs = ∫(s − 2τ)Δs 2τ

t

2τ

t

= ∫ sΔs − 2τ ∫ Δs = 2τ 2

2τ

2 󵄨s=t

s 󵄨󵄨󵄨 − 2τ(t − 2τ) 󵄨 3 󵄨󵄨󵄨s=2τ

4 t2 8 t = − τ 2 − 2τt + 4τ 2 = − 2τt + τ 2 , 3 3 3 3

t, τ ∈ 𝕋,

Next, f (1) = 2,

f Δ (t) = (23 + 22 + 2 + 1)t 3 + 1 = 15t 3 + 1,

f Δ (1) = 16, 2

f Δ (t) = 15(22 + 2 + 1)t 2 = 105t 2 , 2

f Δ (1) = 105, 3

f Δ (t) = 105(2 + 1)t = 315t,

t ∈ 𝕋.

Now, applying Taylor’s formula of order 3 around t = 1, we find Δ2

t

3

f (1) + h1 (t, 1)f (1) + h2 (t, 1)f (1) + ∫ h2 (t, σ(τ))f Δ (τ)Δτ Δ

1

τ < t.

48 � S. G. Georgiev t

t2 2 t2 8 = 2 + 16(t − 1) + 105( − t + ) + 315 ∫( − 2τt + τ 2 )τΔτ 3 3 3 3 1

t

t

t

1

1

1

= 2 + 16t − 16 + 35t 2 − 105t + 70 + 105t 2 ∫ τΔτ − 630t ∫ τ 2 Δτ + 840 ∫ τ 3 Δτ τ=t 󵄨󵄨τ=t 󵄨󵄨τ=t τ3 τ4 τ 2 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 = 35t 2 − 89t + 56 + 105t 2 󵄨󵄨󵄨 − 630t 2 󵄨󵄨 + 840 3 󵄨 3 󵄨󵄨τ=1 2 + 2 + 1 󵄨󵄨τ=1 2 + 22 + 2 + 1 󵄨󵄨󵄨τ=1

= 35t 2 − 89t + 56 + 35t 4 − 35t 2 − 90t 4 + 90t + 56t 4 − 56

= t4 + t = f (t),

t ∈ 𝕋.

Theorem 1.6. For any k ∈ ℕ0 , we have 0 ≤ hk (t, s) ≤

(t − s)k , k!

t ≥ s.

(1.43)

Proof. Let g(t) = (t − s)k+1 ,

t, s ∈ 𝕋,

k ∈ ℕ.

Then g Δ (t) = lim y→t

g(σ(t)) − g(y) σ(t) − y

= lim

(σ(t) − s)k+1 − (y − s)k+1 σ(t) − y

= lim

(σ(t) − y) ∑kν=0 (σ(t) − s)ν (y − s)k−ν σ(t) − y

y→t

y→t

k

= lim ∑ (σ(t) − s)(y − s)k−ν y→t

ν=0

k

ν

= ∑ (σ(t) − s) (t − s)k−ν , ν=0

t, s ∈ 𝕋,

k ∈ ℕ.

Note that the inequalities (1.43) are true for k = 0. Assume that they are true for some k ∈ ℕ. We will prove the inequalities (1.43) for k + 1. We have 0 ≤ hk+1 (t, s) t

= ∫ hk (τ.s)Δτ s

1 Projector analysis of dynamic systems on time scales

� 49

t

1 ≤ ∫(τ − s)k Δτ k! s

t

k 1 = ∫ ∑ (τ − s)k Δτ (k + 1)! ν=0 s

t

=

k 1 ∫ ∑ (τ − s)ν (τ − s)k−ν Δτ (k + 1)! ν=0

≤

k 1 ν ∫ ∑ (σ(τ) − s) (τ − s)k−ν Δτ (k + 1)! ν=0

s

t

s

t

1 = ∫ g Δ (τ)Δτ (k + 1)! s

1 = g(τ)|τ=t τ=s (k + 1)! =

1 󵄨τ=t (τ − s)k+1 󵄨󵄨󵄨τ=s (k + 1)!

=

(t − s)k+1 , (k + 1)!

t, s ∈ 𝕋,

t ≥ s.

By the principle of mathematical induction, it follows that (1.43) is true for any k ∈ ℕ. This completes the proof. Let ρn−1 (t)

n

Rn (t, α) = ∫ hn−1 (t, σ(τ))f Δ (τ)Δτ. α

Theorem 1.7. Let t ∈ 𝕋, t ≥ α and 󵄨 n 󵄨 Mn (t) = sup{󵄨󵄨󵄨f Δ (τ)󵄨󵄨󵄨 : τ ∈ [α, t]}. Then |Rn (t, α)| ≤ Mn (t)

(t − α)n . (n − 1)!

Proof. Let τ ∈ [α, t). Then α ≤ σ(τ) ≤ t and, applying (1.43), we get 0 ≤ hn−1 (t, σ(τ)) ≤ Hence,

(t − σ(τ))n−1 (t − τ)n−1 (t − α)n−1 ≤ ≤ . (n − 1)! (n − 1)! (n − 1)!

50 � S. G. Georgiev n−1

󵄨󵄨 󵄨󵄨 ρ (t) n 󵄨󵄨 󵄨󵄨 |Rn (t, α)| = 󵄨󵄨󵄨 ∫ hn−1 (t, σ(τ))f Δ (τ)Δτ 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 α t

󵄨 n 󵄨 ≤ ∫ hn−1 (t, σ(τ))󵄨󵄨󵄨f Δ (τ)󵄨󵄨󵄨Δτ α

t

≤ Mn (t) ∫ α

(t − α)n−1 Δτ (n − 1)!

(t − α)n = Mn (t) . (n − 1)! This completes the proof.

1.2.2 Taylor’s formula for a function of n independent real variables and one independent time scale variable Suppose that S is an open and convex set in ℝn , I ⊆ 𝕋, t0 ∈ I, f : S×I → ℝ, f ∈ C k+1 (S×I), (x10 , . . . , xn0 ), (x1 , . . . , xn ) ∈ S, t ∈ 𝕋, t ≥ t0 . Then, applying the classical Taylor’s formula for a function of n independent real variables and then the Taylor’s formula for a function of one independent time scale variable, we obtain f (x1 , . . . , xn , t) − f (x10 , . . . , xn0 , t0 )

= f (x1 , . . . , xn , t) − f (x10 , . . . , xn0 , t) + f (x10 , . . . , xn0 , t) − f (x10 , . . . , xn0 , t0 ) = ∑ |α|≤k

(x1 − x10 )α1 ⋅ ⋅ ⋅ (xn − xn0 )αn 𝜕α1 ⋅ ⋅ ⋅ 𝜕αn 0 0 α f (x1 , . . . , xn , t) α α1 ! ⋅ ⋅ ⋅ αn ! 𝜕x1 1 ⋅ ⋅ ⋅ 𝜕xn n

+ Rn,k (x1 − x10 , . . . , xn − xn0 , t) k

+ ∑ hl (t, t0 ) l=0

= ∑ |α|≤k

𝜕l f (x10 , . . . , xn0 , t0 ) + R1k (x10 , . . . , xn0 , t0 ) Δt l

(x1 − x10 )α1 ⋅ ⋅ ⋅ (xn − xn0 )αn 𝜕α1 ⋅ ⋅ ⋅ 𝜕αn k 𝜕l 0 0 αn ∑ hl (t, t0 ) l f (x1 , . . . , xn , t0 ) α1 α1 ! ⋅ ⋅ ⋅ αn ! Δt 𝜕x1 ⋅ ⋅ ⋅ 𝜕xn l=0

+ Rn,k (x1 − x10 , . . . , xn − xn0 , t) + R1k (x10 , . . . , xn0 , t, t0 ) k

+ ∑ hl (t, t0 ) l=0

𝜕l f (x10 , . . . , xn0 , t0 ) Δt l

(x1 − x10 )α1 ⋅ ⋅ ⋅ (xn − xn0 )αn 𝜕α1 ⋅ ⋅ ⋅ 𝜕αn 𝜕l hl (t, t0 ) α1 f (x10 , . . . , xn0 , t0 ) αn l α ! ⋅ ⋅ ⋅ α ! Δt 𝜕x ⋅ ⋅ ⋅ 𝜕x 1 n n |α|≤k l=0 1 k

= ∑ ∑ k

+ ∑ hl (t, t0 ) l=0

𝜕l f (x10 , . . . , xn0 , t0 ) Δt l

+ Rn,k (x1 − x10 , . . . , xn − xn0 , t) + R1k (x10 , . . . , xn0 , t, t0 ),

1 Projector analysis of dynamic systems on time scales

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where Rnk (⋅, . . . , ⋅, ⋅) is the remainder in the Taylor’s formula for a function of n independent real variables, R1k (⋅, . . . , ⋅, ⋅) is the remainder in the Taylor’s formula for a function of one independent time scale variable, α = (α1 , . . . , αn ) ∈ ℕn0 , |α| = α1 +⋅ ⋅ ⋅+αn . Therefore f (x1 , . . . , xn , t)

= f (x10 , . . . , xn0 , t0 )

(x1 − x10 )α1 ⋅ ⋅ ⋅ (xn − xn0 )αn 𝜕α1 ⋅ ⋅ ⋅ 𝜕αn 𝜕l hl (t, t0 ) α1 f (x10 , . . . , xn0 , t0 ) αn l α ! ⋅ ⋅ ⋅ α ! Δt 𝜕x ⋅ ⋅ ⋅ 𝜕x 1 n n |α|≤k l=0 1 k

+ ∑ ∑ k

+ ∑ hl (t, t0 ) l=0

𝜕l f (x10 , . . . , xn0 , t0 ) Δt l

+ Rn,k (x1 − x10 , . . . , xn − xn0 , t) + R1k (x10 , . . . , xn0 , t, t0 ).

(1.44)

Definition 1.8. Equation (1.44) is said to be the Taylor’s formula of order (k, k) for a function of n independent real variables and one independent time scale variable around (x10 , . . . , xn0 , t0 ). Let now f ∈ C k (S, C m (I)). As above, one can deduct the following formula: f (x1 , . . . , xn , t)

= f (x10 , . . . , xn0 , t0 )

(x1 − x10 )α1 ⋅ ⋅ ⋅ (xn − xn0 )αn 𝜕α1 ⋅ ⋅ ⋅ 𝜕αn 𝜕l hl (t, t0 ) α1 f (x10 , . . . , xn0 , t0 ) α α1 ! ⋅ ⋅ ⋅ αn ! 𝜕x1 ⋅ ⋅ ⋅ 𝜕xn n Δt l |α|≤k l=0 m

+ ∑ ∑ m

+ ∑ hl (t, t0 ) l=0

𝜕l f (x10 , . . . , xn0 , t0 ) Δt l

+ Rn,k (x1 − x10 , . . . , xn − xn0 , t) + R1m (x10 , . . . , xn0 , t, t0 ).

(1.45)

Definition 1.9. Equation (1.45) is said to be the Taylor’s formula of order (k, m) for a function of n independent real variables and one independent time scale variable around (x10 , . . . , xn0 , t0 ).

1.2.3 Jets of a function of one independent time scale variable Let t0 ∈ 𝕋. Definition 1.10. Let f be (k + 1) times delta differentiable in a neighborhood U of t0 . Then the k-jet of f at t0 is defined to be the function k

(Jtk0 f )(x) = f (t0 ) + h1 (z, t0 )f Δ (t0 ) + ⋅ ⋅ ⋅ + hk (z, t0 )f Δ (t0 ).

52 � S. G. Georgiev Example. Let 𝕋 = 2ℕ0 and f be as in (1.42). We will find (J12 f )(z). We have h1 (z, 1) = z − 1, h2 (z, 1) =

2 z2 −z+ . 3 3

Then 2

(J12 f )(z) = f (1) + h1 (z, 1)f Δ (1) + h2 (z, 1)f Δ (1) = 2 + 16(z − 1) + 105(

2 z2 −z+ ) 3 3

= 2 + 16z − 16 + 35z2 − 105z + 70 = 35z2 + 89z + 56.

Theorem 1.8. Let f and g be (k + 1) times delta differentiable in a neighborhood U of t0 . Then (Jtk0 (af + bg))(z) = a(Jtk0 f )(z) + b(Jtk0 g)(z) for any a, b ∈ ℝ. Proof. We have (Jtk0 (af + bg))(z)

k

= (af + bg)(t0 ) + h1 (z, t0 )(af + bg)Δ (t0 ) + ⋅ ⋅ ⋅ + hk (z, t0 )(af + bg)Δ (t0 ) k

= af (t0 ) + ah1 (z, t0 )f Δ (t0 ) + ⋅ ⋅ ⋅ + ahk (z, t0 )f Δ (t0 ) k

+ bg(t0 ) + bh1 (z, t0 )g Δ (t0 ) + ⋅ ⋅ ⋅ + bhk (z, t0 )g Δ (t0 )

= a(Jtk0 f )(z) + b(Jtk0 g)(z). This completes the proof.

Remark 1.1. Let f and g be (k + 1) times delta differentiable in a neighborhood U of t0 . If f Λ exists for any Λ ∈ Sl(k) , where Sl(k) is the set consisting of all possible strings of length k containing σ exactly l times and Δ exactly k − l times, then k

k

l

(fg)Δ = ∑( ∑ f Λ )g Δ . j=0 Λ∈S (k) j

Therefore it is impossible to deduce in the general case a representation for a k-jet (Jtk0 (fg))(z) containing jets of f and jets of g. The same situation is for a representation in the general case for a jet of the composition of f and g via jets of f and jets of g.

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1.2.4 Jets of a function of n independent real variables and one independent time scale variable Definition 1.11. Let S be open and connected set in ℝn that contains x 0 = (x10 , . . . , xn0 ) ∈ ℝn , t0 ∈ 𝕋, and U be a neighborhood of t0 . Suppose that f : S × U → ℝ, f ∈ C k+1 (S, C m+1 (U)). Then the (k, m)-jet of f at (x 0 , t0 ) is defined to be k,m (J(x 0 ,t ) f )(z1 , . . . , zn , zn+1 ) 0

= f (x10 , . . . , xn0 , t0 )

(z1 − x10 )α1 ⋅ ⋅ ⋅ (zn − xn0 )αn 𝜕α1 ⋅ ⋅ ⋅ 𝜕αn 𝜕l f (x10 , . . . , xn0 , t0 ) hl (zn+1 , t0 ) α1 αn l α ! ⋅ ⋅ ⋅ α ! Δt 𝜕x1 ⋅ ⋅ ⋅ 𝜕xn 1 n |α|≤k l=0 m

+ ∑ ∑ m

+ ∑ hl (zn+1 , t0 ) l=0

𝜕l f (x10 , . . . , xn0 , t0 ). Δt l

k When k = m, we will write (J(x 0 ,t ) f )(z1 , . . . , zn , zn+1 ). 0

Let f (x1 , . . . , xn , t) = f1 (x1 , . . . , xn )f2 (t), where f1 ∈ C k (S) and f2 ∈ C m (U). Then k,m (J(x 0 ,t ) f )(z1 , . . . , zn , zn+1 ) 0

= f (x10 , . . . , xn0 , t0 )

(z1 − x10 )α1 ⋅ ⋅ ⋅ (zn − xn0 )αn 𝜕α1 ⋅ ⋅ ⋅ 𝜕αn 𝜕l hl (zn+1 , t0 ) α1 f (x10 , . . . , xn0 , t0 ) αn l α ! ⋅ ⋅ ⋅ α ! Δt 𝜕x ⋅ ⋅ ⋅ 𝜕x 1 n n |α|≤k l=0 1 m

+ ∑ ∑ m

+ ∑ hl (zn+1 , t0 ) l=0

𝜕l f (x10 , . . . , xn0 , t0 ) Δt l

= f1 (x10 , . . . , xn0 )f2 (t0 )

(z1 − x10 )α1 ⋅ ⋅ ⋅ (zn − xn0 )αn hl (zn+1 , t0 ) α1 ! ⋅ ⋅ ⋅ αn ! |α|≤k l=0 m

+ ∑ ∑ ×

l 𝜕α1 ⋅ ⋅ ⋅ 𝜕αn 0 0 𝜕 αn f1 (x1 , . . . , xn ) l f2 (t0 ) α1 Δt 𝜕x1 ⋅ ⋅ ⋅ 𝜕xn m

+ f1 (x10 , . . . , xn0 ) ∑ hl (zn+1 , t0 ) l=0

= f1 (x10 , . . . , xn0 )f2 (t0 ) +( ∑ |α|≤k

𝜕l f2 (t0 ) Δt l

(z1 − x10 )α1 ⋅ ⋅ ⋅ (zn − xn0 )αn 𝜕α1 ⋅ ⋅ ⋅ 𝜕αn 0 0 α f1 (x1 , . . . , lxn )) α α1 ! ⋅ ⋅ ⋅ αn ! 𝜕x1 1 ⋅ ⋅ ⋅ 𝜕xn n

54 � S. G. Georgiev m

× (∑ hl (zn+1 , t0 ) l=0

𝜕l f2 (t0 )) + f1 (x10 , . . . , xn0 )(Jtm0 f ))(zn+1 ) Δt l

= f (x10 , . . . , xn0 , t0 ) + (Jxk0 f1 )(z1 , . . . , zn )(Jtm0 f2 )(zn+1 ) + f1 (x10 , . . . , xn0 )(Jtm0 f ))(zn+1 ).

Example. Let 𝕋 = 2ℕ0 and f1 (x) = x 4 + x 2 + x,

x ∈ ℝ,

as well as g(x, t) = f1 (x)f (t),

x ∈ ℝ,

t ∈ 𝕋,

where f is the function in (1.42). We will find 3,2 (J(1,1) f )(z1 , z2 ).

We have g(x, t) = (x 4 + x 2 + x)(t 4 + t),

(x, t) ∈ ℝ × 𝕋,

g(1, 1) = 3 ⋅ 2 = 6,

and

f1′ (x) f1′′ (x) f1′′′ (x)

3

= 4x + 2x + 1, = 12x 2 + 2, = 24x,

f1 (1) = 3, f1′ (1) = 7, f1′′ (1) = 14,

f1′′′ (1) = 24.

Then 1 1 (J13 f1 )(z1 ) = f1 (1) + f1′ (1)z1 + z21 f1′′ (1) + z31 f1′′′ (1) 2 6 = 3 + 7z1 + 7z21 + 4z31 . Now, we find 3,2 (J(1,1) g)(z1 , z2 ) = g(1, 1) + (J13 f1 )(z1 )(J12 f )(z2 ) + f1 (1)(J12 f2 )(z2 )

= 6 + (3 + 7z1 + 7z21 + 4z31 )(35z22 + 89z2 + 56) + 3(35z22 + 89z2 + 56)

= 6 + (6 + 7z1 + 7z21 + 4z31 )(35z22 + 89z2 + 56).

1 Projector analysis of dynamic systems on time scales

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1.2.5 Jet spaces Suppose that f : ℝp+1 × 𝕋 → ℝ, f = f (x1 , . . . , xp−1 , t) depends on p − 1 independent real variables and one independent time scale variable t. For convenience, we introduce the notation xp = t and will write 𝜕xp instead of Δt. The function f has pk = (

p+k−1 ) k

different kth partial derivatives 𝜕J f =

𝜕k f 𝜕xj1 ⋅ ⋅ ⋅ 𝜕xjk

indexed by unordered multiindices J = (j1 , . . . , jk ),

1 ≤ jk ≤ p,

of order k = #J. Thus, if we have q dependent variables (u1 , . . . , uq ), we will require qk = qpk different coordinates ujα = 𝜕j f α (x) of a function u = f (x). For the total space E = X × U ≃ ℝp−1 × 𝕋 × ℝq , the nth jet space J n = J n E = X × U (n) is the Euclidean space of dimension p + q(n) = p + q (

p+n ), n

56 � S. G. Georgiev whose coordinates consist of the p independent variables xi , the q dependent variables uα , and the derivative coordinates ujα , α = 1, . . . , q, of order 1 ≤ #J ≤ n. Definition 1.12. The space U (n) will be called the vertical space or the fiber. The points of the vertical space U (n) are denoted by u(n) and consist of all the dependent variables and their derivatives up to order n. Thus, the coordinates of a typical point z ∈ J n are denoted by (x, u(n) ). Because the derivative coordinates u(n) form a subset of the derivative coordinates u(n+k) , there is a projector πnn+k : J nk → J n on the jet space with πnn+k (x, u(n+k) ) = (x, u(n) ). In particular, we have that π0n (x, u(n) ) = (x, u) is the projector from J n to E = J 0 . If M ⊂ E is an open subset, then J n M = (π0n ) M ⊂ J n E −1

is the open subset of the nth jet space which projects back down to M.

1.2.6 Total derivatives Definition 1.13. A smooth real-valued function F : J n → ℝ, defined on an open subset of the nth jet space, is called a differentiable function. Definition 1.14. Let F(x, u(n) ) be a differentiable function of order n. The total derivative of F with respect to xi is the (n + 1)th order differential function Di F that satisfies Di F(x, f (n+1) (x)) = for any smooth function u = f (x).

𝜕 F(x, f (n) (x)) 𝜕xi

1 Projector analysis of dynamic systems on time scales

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Example. In the case of one independent time scale variable t and one dependent variable, the total derivative of a given function n

F(t, u, uΔ (t), . . . , uΔ (t)) is given by n

Dt F(t, u(t), uΔ (∗t), . . . , uΔ (t)) =

n n 𝜕 𝜕 F(c1 , u(c1 ), uΔ (c1 ), . . . , uΔ (c1 )) + uΔ F(σ(c2 ), u(c2 ), uΔ (c2 ), . . . , uΔ (c2 )) Δt 𝜕u n−1 n 𝜕 Δn+1 + ⋅ ⋅ ⋅ + u (t) Δn F(σ(cn+1 ), u(σ(cn+1 )), . . . , uΔ (σ(cn+1 )), uΔ (cn+1 )) 𝜕u

for some cj ∈ [t, σ(t)], j ∈ {1, . . . , n + 1}.

1.3 Nonlinear dynamic systems In this chapter, we will investigate the following nonlinear dynamic system: Δ

f ((g(x(t), t)) , x(t), t) = 0,

(1.46)

where f : ℝn × Df × If → ℝk , Df × If ⊆ ℝm × 𝕋 is continuous and has continuous classical derivatives fy , fx , as well as a continuous delta derivative ftΔ , g : Df × If → ℝn is continuous and has a continuous classical derivative gx and a continuous delta derivative gtΔ . A solution x of the nonlinear dynamic system (1.46) is a function defined on an interval I∗ ⊆ If so that x(t) ∈ Df for any t ∈ I∗ , the function g(x(⋅), ⋅) is continuously delta differentiable on I∗ and x satisfies equation (1.46) pointwise on I∗ .

1.3.1 Properly involved derivatives Define 1

2

1

F(y (t), y (t), x(t), t) = ∫ 0

1

G(x 1 (t), x 2 (t), t) = ∫ 0

where

𝜕 f (hy1 (t) + (1 − h)y2 (t), x(t), t)dh, 𝜕y 𝜕 g(hx 1 (t) + (1 − h)x 2 (t), t)dh, 𝜕x

58 � S. G. Georgiev 𝜕 f (hy1 (t) + (1 − h)y2 (t), x(t), t) 𝜕y =(

k,n

𝜕 f (y11 (t), . . . , y1j−1 (t), hy1j (t) + (1 − h)y2j (t), y2j+1 (t), . . . , y2n (t), x(t), t)) 𝜕yj i,j=1

and 𝜕 g(hx 1 (t) + (1 − h)x 2 (t), t) 𝜕x n,m 𝜕 1 2 2 = ( g(x11 (t), . . . , xj−1 (t), hxj1 (t) + (1 − h)xj2 (t), xj+1 (t), . . . , xm (t), σ(t))) , 𝜕xj i,j=1 and y1 (t), y2 (t) ∈ ℝn , x(t), x 1 (t), x 2 (t) ∈ Df , t ∈ If . Definition 1.15. The nonlinear dynamic equation (1.46) has a properly involved derivative, also called a properly stated leading term, if im G and ker F are C 1 -subspaces and the transversality condition ker F(y1 (t), y2 (t), x(t), t) ⊕ im G(x 1 (t), x 2 (t), t) = ℝn , y1 (t), y2 (t) ∈ ℝn ,

x(t), x 1 (t), x 2 (t) ∈ Df ,

t ∈ If ,

(1.47)

holds. Example. Let 𝕋 = 2ℕ0 . Consider the following nonlinear dynamic system: Δ

(x1 (t) − 2x2 (t)x3 (t)) + x2 (t) − q1 (t) = 0,

x1 (t) + x2 (t) − q2 (t) = 0,

x2 (t) − 2x3 (t) − q3 (t) = 0,

t ∈ 𝕋.

Here n = 1,

k = m = 3.

We have 1 x2 f (y1 , y2 , x, t) = ( 0 ) y + ( x1 + x2 ) − q(t), 0 x2 − 2x3

x ∈ ℝ3 ,

y1 , y2 ∈ ℝ,

and g(x 1 , x 2 , t) = x1 (t) − 2x2 (t)x3 (t), Then

x 1 , x 2 ∈ ℝ3 ,

t ∈ 𝕋.

t ∈ 𝕋,

1 Projector analysis of dynamic systems on time scales

1 1 1 F(y1 (t), y2 (t), x(t), t) = ∫ ( 0 ) dh = ( 0 ) , 0 0 0

x ∈ ℝ3 ,

y1 , y2 ∈ ℝ,

� 59

t ∈ 𝕋.

Hence, ker F(y1 (t), y2 (t), x(t), t) = {0},

y1 (t), y2 (t)ℝ,

x(t) ∈ ℝ3 ,

t ∈ 𝕋.

x 1 (t), x 2 (t) ∈ ℝ3 ,

t ∈ 𝕋.

Next, 1

G(x 1 (t), x 2 (t), t) = ∫(1, −2x32 (t), −2x21 (t))dh 0

= (1, −2x32 (t), −2x21 (t)), Then ker G(x 1 (t), x 2 (t), t) = {0},

x 1 (t), x 2 (t) ∈ ℝ3 ,

t ∈ 𝕋,

and im G(x 1 (t), x 2 (t), t) = ℝ,

x 1 (t), x 2 (t) ∈ ℝ3 ,

t ∈ 𝕋.

Consequently, ker F(y1 (t),y2 (t), x(t), t) ⊕ im G(x 1 (t), x 2 (t), t) = ℝ, y1 (t), y2 (t) ∈ 𝕋,

x(t), x 1 (t), x 2 (t) ∈ ℝ3 ,

t ∈ 𝕋.

Thus, the considered nonlinear dynamic system has a properly involved derivative. Note that the nonlinear dynamic system (1.46) covers equation (1.31) with f (y, x, t) = A(t)y − C(t)x − f (t), g(x, t) = B(t)x.

If equation (1.46) has a properly involved derivative, then F and G have constant rank on their definition domain.

1.3.2 Constraints and consistent initial values Consider equation (1.46) subject to the initial condition x(t0 ) = x0 , where t0 ∈ If and x0 ∈ Df .

(1.48)

60 � S. G. Georgiev Definition 1.16. For a given nonlinear dynamic system (1.46) and a given t0 ∈ If , the value x0 ∈ Df is said to be consistent if the IVP (1.46), (1.48) possesses a solution. Definition 1.17. For a nonlinear dynamic system (1.46) with a properly involved derivative, the set n

M0 (t) = {x(t) ∈ Df : ∃y(t) ∈ ℝ such that

y(t) − gtΔ (x(σ(t)), t) ∈ im G(x σ (t), x(t), t),

f (y(t), x(t), t) = 0}

is called obvious restriction set or obvious constraint of the equation (1.46) at t ∈ If . Example. Let 𝕋 = ℤ. Consider the nonlinear dynamic equation x1Δ (t) = 2x1 (t),

4

2

3(x1 (t)) + (x2 (t)) = 1 + q(t),

t ∈ 𝕋,

(1.49)

with two equations and two unknown functions on Df = {x ∈ ℝ2 : x2 > 0}, with If = 𝕋, q ∈ C (If ), q > −1 on If . This equation can be rewritten in the form (1.46) using f (y, x, t) = ( g(x, t) = x1 ,

3x14

y − 2x1 ), + x22 − 1 − q(t)

x ∈ Df ,

y ∈ ℝ,

t ∈ If .

We have F(y1 (t), y2 (t), x(t), t) = (

1 ), 0

G(x 1 (t), x 2 (t), t) = (1, 0),

y1 (t), y2 (t) ∈ ℝ,

x(t), x 1 (t), x 2 (t) ∈ Df ,

From here, ker F(y1 (t), y2 (t), x(t), t) = ker G(x 1 (t), x 2 (t), t) = {0}, y1 (t), y2 (t) ∈ ℝ,

x 1 (t), x 2 (t), x(t) ∈ Df ,

t ∈ If .

Therefore im G(x 1 (t), x 2 (t), t) = ℝ, and

x 1 (t), x 2 (t) ∈ Df ,

t ∈ If ,

t ∈ If .

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ker F(y1 (t), y2 (t), x(t), t) ⊕ im G(x 1 (t), x 2 (t), t) = ℝ, y1 (t), y2 (t) ∈ ℝ,

x(t), x 1 (t), x 2 (t) ∈ Df ,

t ∈ If .

Thus, equation (1.49) has a properly involved derivative. From the second equation of (1.49), we conclude that the solutions of (1.49) must lie in 4

2

M0 (t) = {x ∈ Df : 3x1 + x2 − 1 − q(t) = 0}.

Let x1 (0) = x10 . Then 4 . x20 = ±√1 + q(0) − 3x10

By the first equation of (1.49), we find 2

x1 (t) = e2 (t, 0)x10 = x10 e∫0 log(1+2)Δτ = x10 et log 3 = x10 3t ,

t ∈ 𝕋,

and then 4 34t , x2 (t) = √1 + q(t) − 3x10

t ∈ 𝕋.

It is clear that through each point of M0 (0) passes exactly one solution. Now, we consider equation (1.46). Suppose that x ∈ C 1 (If ) and set u(t) = g(x(t), t),

t ∈ If .

Using the generalized Pötzsche chain rule, we find 1

uΔ (t) = (∫ g(x(t) + hμ(t)x Δ (t), t)dh)x Δ (t) + gtΔ (x(σ(t)), t) 0

= G(x σ (t), x(t), t)x Δ (t) + gtΔ (x(σ(t)), t),

t ∈ If ,

or uΔ (t) − gtΔ (x(σ(t)), t) = G(x σ (t), x(t), t)x Δ (t),

t ∈ If .

From the inclusion uΔ (t) − gtΔ (x(σ(t)), t) ∈ im G(x σ (t), x(t), t),

t ∈ If ,

it follows that there is a w(t) ∈ im G(x σ (t), x(t), t), t ∈ If , such that

(1.50)

62 � S. G. Georgiev uΔ (t) = G(x σ (t), x(t), t)w(t) + gtΔ (x σ (t), t),

t ∈ If .

Note that the inclusion (1.50) holds trivially in the case when G(x(t), t) has full row rank for any t ∈ If . For any solution of equation (1.46), we have the following valid identities: f (uΔ (t), x(t), t) = 0,

t ∈ If ,

and f (G(x σ (t), x(t), t)w(t) + gtΔ (x σ (t), t)) = 0,

t ∈ If ,

and then the values x(t) belong to the set n

̃ M 0 (t) = {x ∈ Df : ∃y ∈ ℝ : f (y, x, t) = 0}. ̃ The sets M0 (t) and M 0 (t) are defined for any t ∈ If . We have ̃ M0 (t) ⊆ M 0 (t),

t ∈ If .

If G(x σ (t), x(t), t) has full row rank, we have ̃ M0 (t) = M 0 (t),

t ∈ If .

Theorem 1.9. Let equation (1.46) have a properly involved derivative. Then, for each t ∈ If and each x(t) ∈ M0 (t), there is a unique y(t) ∈ ℝn such that y(t) − gtΔ (x(σ(t)), t) ∈ im G(x σ (t), x(t), t) and f (y(t), x(t), t) = 0. Proof. Let t1 ∈ If and x 1 (t1 ) ∈ M0 (t1 ) be arbitrarily chosen. Suppose that there are y1 (t1 ), y2 (t1 ) ∈ ℝn so that y1 (t1 ) − gtΔ (x 1 (σ(t1 )), x 1 (t1 ), t1 ) ∈ G(x 1 (σ(t1 )), x 1 (t1 ), t1 ),

y2 (t1 ) − gtΔ (x 1 (σ(t1 )), x 1 (t1 ), t1 ) ∈ G(x 1 (σ(t1 )), x 1 (t1 ), t1 ). Let N = ker G(x 1 (σ(t1 )), x 1 (t1 ), t1 ) and w1 (t1 ) = G(x 1 (σ(t1 )), x 1 (t1 ), t1 ) (y1 (t1 ) − gtΔ (x 1 (t1 ), t1 )), +

(1.51)

1 Projector analysis of dynamic systems on time scales

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w2 (t1 ) = G(x 1 (σ(t1 )), x 1 (t1 ), t1 ) (y2 (t1 ) − gtΔ (x 1 (t1 ), t1 )). +

Here G(x 1 (σ(t1 )), x 1 (t1 ), t1 )+ is the Moore–Penrose inverse of G(x 1 (σ(t1 )), x 1 (t1 ), t1 ). Then w1 (t1 ), w2 (t1 ) ∈ N ⊥ and G(x 1 (σ(t1 )), x 1 (t1 ), t1 )w1 (t1 )

= G(x 1 (σ(t1 )), x 1 (t1 ), t1 )G(x 1 (σ(t1 )), x 1 (t1 ), t1 ) (y1 (t1 ) − gtΔ (x 1 (t1 ), t1 )), +

G(x 1 (σ(t1 )), x 1 (t1 ), t1 )w2 (t1 )

= G(x 1 (σ(t1 )), x 1 (t1 ), t1 )G(x 1 (σ(t1 )), x 1 (t1 ), t1 ) (y2 (t1 ) − gtΔ (x 1 (t1 ), t1 )). +

By (1.51), it follows that there are q1 (t1 ), q2 (t1 ) such that y1 (t1 ) − gtΔ (x 1 (σ(t1 )), x 1 (t1 ), t1 ) = G(x 1 (σ(t1 )), x 1 (t1 ), t1 )q1 (t1 ),

y2 (t1 ) − gtΔ (x 1 (σ(t1 )), x 1 (t1 ), t1 ) = G(x 1 (σ(t1 )), x 1 (t1 ), t1 )q2 (t1 ). Then G(x 1 (σ(t1 )), x 1 (t1 ), t1 )w1 (t1 )

= G(x 1 (σ(t1 )), x 1 (t1 ), t1 )G(x 1 (σ(t1 )), x 1 (t1 ), t1 ) G(x 1 (σ(t1 )), x 1 (t1 ), t1 )q1 (t1 ) +

= G(x 1 (σ(t1 )), x 1 (t1 ), t1 )q1 (t1 )

= y1 (t1 ) − gtΔ (x 1 (σ(t1 )), x 1 (t1 ), t1 ). As above, G(x 1 (σ(t1 )), x 1 (t1 ), t1 )w2 (t1 ) = y2 (t1 ) − gtΔ (x 1 (σ(t1 )), x 1 (t1 ), t1 ). Thus, y1 (t1 ) − y2 (t1 ) = G(x 1 (σ(t1 )), x 1 (t1 ), t1 )(w1 (t1 ) − w2 (t1 )) and f (gtΔ (x 1 (σ(t1 )), t1 ) + G(x 1 (σ(t1 )), x 1 (t1 ), t1 )w1 (t1 ), x 1 (t1 ), t1 ) = 0,

f (gtΔ (x 1 (σ(t1 )), t1 ) + G(x 1 (σ(t1 )), x 1 (t1 ), t1 )w2 (t1 ), x 1 (t1 ), t1 ) = 0. Hence, 0 = f (gtΔ (x 1 (σ(t1 )), t1 ) + G(x 1 (σ(t1 )), x 1 (t1 ), t1 )w1 (t1 ), x 1 (t1 ), t1 )

− f (gtΔ (x 1 (σ(t1 )), t1 ) + G(x 1 (σ(t1 )), x 1 (t1 ), t1 )w2 (t1 ), x 1 (t1 ), t1 ) 1

= (∫ 0

𝜕 f (s(gtΔ (x 1 (σ(t1 )), t1 ) + G(x 1 (σ(t1 )), x 1 (t1 ), t1 )w1 (t1 )) 𝜕y

64 � S. G. Georgiev

+ (1 − s)(gtΔ (x 1 (σ(t1 )), t1 ) + G(x 1 (σ(t1 )), x 1 (t1 ), t1 )w2 (t1 ))x 1 (t1 ), t1 )ds) × G(x 1 (σ(t1 )), x(t1 ), t1 )(w1 (t1 ) − w2 (t1 )). Since equation (1.46) has a properly involved derivative, we have 1

ker ((∫ 0

𝜕 f (s(gtΔ (x 1 (σ(t1 )), t1 ) + G(x 1 (σ(t1 )), x 1 (t1 ), t1 )w1 (t1 )) 𝜕y

+ (1 − s)(gtΔ (x 1 (σ(t1 )), t1 ) + G(x 1 (σ(t1 )), x 1 (t1 ), t1 )w2 (t1 ))x 1 (t1 ), t1 )ds) × G(x 1 (σ(t1 )), x 1 (t1 ), t1 )) = ker G(x 1 (σ(t1 )), x 1 (t1 ), t1 ). Therefore w1 (t1 ) − w2 (t1 ) ∈ N. Since w1 (t1 ), w2 (t2 ) ∈ N ⊥ , the latter relation is possible if and only if w1 (t1 ) = w2 (t1 ). Then y1 (t1 ) = y2 (t1 ). This completes the proof. Theorem 1.10. Let equation (1.46) have a properly involved derivative and let ker F(y1 (t), y2 (t), x(t), t),

y1 (t), y2 (t) ∈ ℝn , x(t) ∈ Df ,

t ∈ If ,

be independent of the choice of y1 and y2 . Suppose that there exists a projector R(x(t), t), x(t) ∈ Df , t ∈ If , such that im R(x(t), t) = im G(x 1 (t), x 2 (t), t),

ker R(x(t), t) = ker F(y1 (t), y2 (t), x(t), t), y1 (t), y2 (t) ∈ ℝn ,

Then, we have the following:

x(t), x 1 (t), x 2 (t) ∈ Df ,

t ∈ If .

1 Projector analysis of dynamic systems on time scales

� 65

1. f (y(t), x(t), t) = f (R(x(t), t)y(t), x(t), t), 2. 3.

y(t) ∈ ℝn ,

x(t) ∈ Dd ,

t ∈ If .

Projector R is continuously-differentiable on Df × If . ̃ M0 (t) = M 0 (t) for any t ∈ If .

Proof. 1. Let t ∈ If , x(t) ∈ Df , y(t) ∈ ℝn be arbitrarily chosen. Set η(t) = (I − R(x(t), t))y(t). Then f (y(t), x(t), t) − f (R(x(t), t)y(t), x(t), t) 1

=∫ 0

1

=∫ 0

𝜕 f (sy(t) + (1 − s)R(x(t), t)y(t), x(t), t)ds(I − R(x(t), t))y(t) 𝜕y 𝜕 f (sy(t) + (1 − s)R(x(t), t)y(t), x(t), t)dsη(t) 𝜕y

= F(y(t), R(x(t), t)y(t), x(t), t)η(t).

(1.52)

Note that η(t) ∈ im (I − R(x(t), t)) = ker F(y(t), R(x(t), t)y(t), x(t), t). Therefore F(y(t), R(x(t), t)y(t), x(t), t) = 0 and f (y(t), x(t), t) = f (R(x(t), t)y(t), x(t), t). 2. 3.

The function R is continuously differentiable because it is a projector defined on C 1 -subspaces. ̃0 (t) be arbitrarily chosen and fixed. Let also, y1 (t) ∈ ℝn be such Let t ∈ If , x(t) ∈ M that 0 = f (y1 (t), x(t), t) = f (R(x(t), t)y1 (t), x(t), t). Define

66 � S. G. Georgiev y(t) = R(x(t), t)y1 (t) + (I − R(x(t), t))gtΔ (x(σ(t)), t). Then y(t) − gtΔ (x(σ(t)), t) = R(x(t), t)(y1 (t) − gtΔ (x(σ(t)), t))

∈ im R(x(t), t) = im G(x 1 (t), x 2 (t), t).

By the definition of y(t), we find R(x(t), t)y(t) = R(x(t), t)R(x(t), t)y1 (t) + R(x(t), t)(I − R(x(t), t))gtΔ (x(σ(t)), t) = R(x(t), t)y1 (t).

From here and using claim 1, we arrive at f (y(t), x(t), t) = f (R(x(t), t)y(t), x(t), t) = f (R(x(t), t)y1 (t), x(t), t) = f (y1 (t), x(t), t) = 0.

̃ Consequently, x(t) ∈ M0 (t). Because x(t) ∈ M 0 (t), and since it is an element of M0 (t), we get the inclusion ̃ M 0 (t) ⊆ M0 (t). Hence, using ̃ M0 (t) ⊆ M 0 (t), we get ̃ M0 (t) = M 0 (t). This completes the proof.

1.3.3 Linearization Define 1

H(y(t), x 1 (t), x 2 (t), t) = ∫ 0

where

𝜕 f (y(σ(t)), sx 1 (t) + (1 − s)x 2 (t), t)dh, 𝜕y

1 Projector analysis of dynamic systems on time scales

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𝜕 f (y(t), hx 1 (t) + (1 − h)x 2 (t), t) 𝜕y =(

𝜕 1 f (y(σ(t), x11 (t), . . . , xj−1 (t), hxj1 (t) + (1 − h)xj2 (t), 𝜕xj 2 2 xj+1 (t), . . . , xm (t), t)))

k,m

i,j=1

,

with y(t) ∈ ℝn , x 1 (t), x 2 (t) ∈ Df , t ∈ If . Let I∗ ⊆ I and x∗ ∈ C 1 (I∗ ) be such that x∗ (t) ∈ Df , t ∈ If , and g(x∗ (⋅), ⋅) ∈ C 1 (I∗ ). Set Δσ

Δ

A(t) = F((g(x∗ (t), t)) , (g(x∗ (t), t)) , x∗ (t), t), B(t) = G(x∗σ (t), x∗ (t), t), Δ

C(t) = H((g(x∗ (t), t)) , x∗σ (t), x∗ (t), t),

t ∈ I∗ .

Consider the equation Δ

A(t)(B(t)x(t)) + C(t)x(t) = q(t),

t ∈ I∗ .

(1.53)

Definition 1.18. Equation (1.53) is said to be a linearization of equation (1.46) along the reference function x∗ . Note that the reference function x∗ is not necessarily a solution of equation (1.46). Example. Let 𝕋 = ℤ. Consider the following nonlinear dynamic system: 2

Δ

3((x1 (t)) + x2 (t)x3 (t)) = q1 (t),

2x1 (t) + x3 (t) = q2 (t),

x2 (t) + x3 (t) = q3 (t),

t ∈ 𝕋.

Here, n = 1, k = m = 3, and σ(t) = t + 1,

t ∈ 𝕋.

We have 3 0 f (y, x, t) = ( 0 ) y + ( 2x1 + x3 ) − q(t), 0 x2 + x3 g(x, t) = x12 + x2 x3 .

68 � S. G. Georgiev Then 1 3 3 F(y1 (t), y2 (t), x(t), t) = ∫ ( 0 ) dh = ( 0 ) , 0 0 0

y1 (t), y2 (t) ∈ ℝ,

x(t) ∈ ℝ3 ,

t ∈ If ,

and 1

G(x 1 (t), x 2 (t), t) = ∫(2hx11 (t) + 2(1 − h)x12 (t), x32 (t), x21 (t))dh 0

= (x11 (t) − x12 (t), x32 (t), x21 (t)),

x 1 (t), x 2 (t) ∈ ℝ3 ,

t ∈ If ,

as well as 0 H(y(t), x (t), x (t), t) = ∫ ( 2 0 0 1

y(t) ∈ ℝ,

2

1

x 1 (t), x 2 (t) ∈ ℝ3 ,

0 0 1

0 0 1 ) dh = ( 2 1 0

0 0 1

0 1 ), 1

t ∈ 𝕋.

Let x∗ ∈ C 1 (𝕋) be arbitrarily chosen. Then 3 F(yσ (t), y(t), x∗ (t), t) = ( 0 ) , 0

σ σ G(x∗σ (∗t), x∗ (t), t) = (x∗1 (t) − x∗1 (t), x∗3 (t), x∗1 (t))

= (x∗1 (t + 1) − x∗1 (t), x∗3 (t), x∗1 (t + 1)),

0 H(y(t), x∗σ (t), x∗ (t), t) = ( 2 0

0 0 1

0 1 ), 1

t ∈ 𝕋,

y(t) ∈ ℝ.

Next, Δ

(g(x∗ (t), t)) = G(x∗σ (t), x∗ (t), t)x∗Δ (t)

Δ x∗1 (t)

Δ = (x∗1 (t + 1) − x∗1 (t), x∗3 (t), x∗1 (t + 1)) ( x∗2 (t) ) Δ x∗3 (t)

Δ Δ Δ = (x∗1 (t + 1) − x∗1 (t))x∗1 (t) + x∗3 (t)x∗2 (t) + x∗1 (t + 1)x∗3 (t),

3 Δσ Δ A(t) = F((g(x∗ (t), t)) , (g(x∗ (t), t)) , x∗ (t), t) = ( 0 ) , 0

1 Projector analysis of dynamic systems on time scales

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B(t) = G(x∗σ (t), x∗ (t), t) = (x∗1 (t + 1) − x∗1 (t), x∗3 (t), x∗1 (t + 1)),

0 Δ C(t) = H((g(x∗ (t), t)) , x∗σ (t), x∗ (t)) = ( 2 0

0 0 1

0 1 ), 1

t ∈ 𝕋.

Therefore Δ

x1 (t) 3 ( 0 ) ((x∗1 (t + 1) − x∗1 (t), x∗3 (t), x∗1 (t + 1)) ( x2 (t) )) 0 x3 (t) 0 +( 2 0

0 0 1

0 x1 (t) 1 ) ( x2 (t) ) = q(t), 1 x3 (t)

t ∈ 𝕋,

or 3 Δ ( 0 ) ((x∗1 (t + 1) − x∗1 (t))x1 (t) + x∗3 (t)x2 (t) + x∗1 (t + 1)x3 (t)) 0 0 + ( 2x1 (t) + x3 (t) ) = q(t), x2 (t) + x3 (t)

t ∈ 𝕋,

or Δ

((x∗1 (t + 1) − x∗1 (t))x1 (t) + x∗3 (t)x2 (t) + x∗1 (t + 1)x3 (t)) = q1 (t),

2x1 (t) + x3 (t) = q2 (t),

x2 (t) + x3 (t) = q3 (t),

t ∈ 𝕋.

If equation (1.46) has a properly involved derivative, then the decomposition ker A(t) ⊕ im B(t) = ℝn ,

t ∈ I∗ ,

holds if ker A(t), t ∈ I∗ , and im B(t), t ∈ I∗ , are C -subspaces. If the subspace ker F(y1 (t), y2 (t), x(t), t), y1 (t), y2 (t) ∈ ℝn , x(t) ∈ Df , t ∈ I∗ , does not depend on y1 and y2 and x∗ ∈ C 1 (I∗ ), by Theorem 1.10 it follows that ker A(t), t ∈ I∗ , and im B(t), t ∈ I∗ , are C 1 -subspaces. We set B∗ (x(t), t) = G(x σ (t), x(t), t),

σ

A∗ (x 1 (t), x(t), t) = F((B∗ (x(t), t)x 1 (t) + gtΔ (x(σ(t)), t)) ,

B∗ (x(t), t)x 1 (t) + gtΔ (x(σ(t)), t), x(t), t),

C∗ (x 1 (t), x(t), t) = H(B∗ (x(t), t)x 1 (t) = gtΔ (x(σ(t)), t), x σ (t), x(t), t),

70 � S. G. Georgiev for x 1 (t), x(t) ∈ Df , t ∈ If . Then, we have the following: Δσ

Δ

A(t) = F((g(x∗ (t), t)) , (g(x∗ (t), t)) , x∗ (t), t)

σ

= F((G(x∗σ (t), x∗ (t), t)x∗Δ (t) + gtΔ (x∗ (σ(t)), t)) ,

G(x∗σ (t), x∗ (t), t)x∗Δ (t) + gtΔ (x∗ (σ(t)), t), x∗ (t), t) σ

= F((B∗ (x∗ (t), t)x∗Δ (t) + gtΔ (x∗ (σ(t)), t)) ,

B∗ (x∗ (t), t)x∗Δ (t) + gtΔ (x∗ (σ(t)), t), x∗ (t), t)

= A∗ (x∗Δ (t), x∗ (t), t),

B(t) = G(x∗σ (t), x∗ (t), t) = B∗ (x∗ (t), t), Δ

C(t) = H((g(x∗ (t), t)) , x∗σ (t), x∗ (t), t)

= H(G(x∗σ (t), x∗ (t), t)x∗Δ (t) + gtΔ (x∗ (σ(t)), t), x∗σ (t), x∗ (t), t) = H(B∗ (x∗ (t), t)x∗Δ (t) + gtΔ (x∗ (σ(t)), t), x∗σ (t), x∗ (t), t) = C∗ (x∗Δ (t), x∗ (t), t),

t ∈ I∗ .

Theorem 1.11. Let equation (1.46) have a properly involved derivative. Then the decomposition ker A∗ (x 1 (t), x(t), t) ⊕ im B∗ (x(t), t) = ℝn

(1.54)

holds for any x 1 (t) ∈ ℝm , x(t) ∈ Df , t ∈ If , and ker A∗ and im B∗ are C -subspaces. Proof. Since equation (1.46) has a properly involved derivative, we have ker F(y1 (t), y2 (t), x(t), t) ⊕ im G(x 1 (t), x 2 (t), t) for any y1 (t), y2 (t) ∈ ℝn , x 1 (t), x 2 (t) ∈ ℝm , x(t) ∈ Df , t ∈ If . For each triple (x(t), x(t), t) ∈ ℝm × Df × If , we set σ

y1 (t) = (B∗ (x(t), t)x(t) + gtΔ (x(σ(t)), t)) ,

y2 (t) = B∗ (x(t), t)x(t) + gtΔ (x(σ(t)), t),

x 1 (t) = x σ (t),

x 2 (t) = x(t). Then

F(y1 (t), y2 (t), x(t), t) = A∗ (x(t), x(t), t), G(x 1 (t), x 2 (t), t) = B∗ (x(t), t),

and ker A∗ (x(t), x(t), t) ⊕ im B∗ (x(t), t) = ℝn .

1 Projector analysis of dynamic systems on time scales

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Because A∗ and B∗ are continuous matrix functions with constant rank, we have that ker A∗ and im B∗ are C -subspaces. This completes the proof. Definition 1.19. Let equation (1.46) have a properly involved derivative. The projectorvalued function R defined by im R(x 1 (t), x(t), t) = im B∗ (x(t), t),

ker R(x 1 (t), x(t), t) = ker A∗ (x 1 (t), x(t), t), for x 1 (t) ∈ ℝm , x(t) ∈ Df , t ∈ If , is said to be a border projector function or border projector of equation (1.46). The basic assumption below is as follows: (E1) 1. The function f is classically continuously differentiable with respect to its first and second arguments and delta continuously differentiable with respect to its third argument on ℝn × Df × If . The functions F and H are continuous on ℝn ×ℝn ×Df ×If . The function g is classically continuously differentiable with respect to its first argument and delta continuously differentiable with respect to its second argument on Df × If . 2. Equation (1.46) has a properly involved derivative. 3. If ker F(y1 , y2 , x, t) depends on y1 and y2 , then suppose that g has a continuous classical second derivative with respect to its first argument and a continuous delta second derivative with respect to its second argument on Df × If . 4. The transversality conditions (1.47) and (1.54) are equivalent.

1.3.4 Regular linearized equations with tractability index 1 Suppose that (E1) holds and the matrices A∗ , B∗ , C∗ are defined as in the previous section. Then A∗ , B∗ , C∗ and the border projector are continuous matrix functions. Assume that the linearized equation (1.53) is regular with tractability index 1. Denote N0 (x(t), t) = ker B∗ (x(t), t),

x(t) ∈ Df ,

t ∈ If ,

x(t) ∈ Df ,

t ∈ If .

and let Q0 be a projector onto B∗ , P0 (x(t), t) = I − Q0 (x(t), t),

We can choose P0 and Q0 to be continuous. Let B∗−1 be the {1, 2}-inverse of B∗ defined by

72 � S. G. Georgiev B∗ (x(t), t)B∗− (x 1 (t), x(t), t)B∗ (x(t), t) = B∗ (x(t), t),

B∗− (x 1 (t), x(t), t)B∗ (x(t), t)B∗−1 (x 1 (t), x(t), t) = B∗− (x 1 (t), x(t), t), B∗ (x(t), t)B∗− (x 1 (t), x(t), t) = R(x 1 (t), x(t), t),

B∗− (x 1 (t), x(t), t)B∗ (x(t), t) = P0 (x 1 (t), x(t), t), x 1 (t) ∈ ℝm ,

x(t) ∈ Df ,

t ∈ If ,

(1.55)

where R(x 1 (t), x(t), t), x 1 (t) ∈ ℝm , x(t) ∈ Df , t ∈ If , is a continuous projector along ker A∗ (x 1 (t), x(t), t), x 1 (t) ∈ ℝm , x(t) ∈ Df , t ∈ If . Note that B∗− is uniquely determined by (1.55). Suppose that G0 (x 1 (t), x(t), t) = A∗ (x 1 (t), x(t), t)B∗ (x(t), t), 1

Π0 (x(t), t) = P0 (x(t), t),

C0 (x (t), x(t), t) = C∗ (x 1 (t), x(t), t),

G1 (x 1 (t), x(t), t) = G0 (x 1 (t), x(t), t) + C0 (x 1 (t), x(t), t)Q0 (x(t), t),

N1 (x 1 (t), x(t), t) = ker G1 (x 1 (t), x(t), t),

Π1 (x 1 (t), x(t), t) = Π0 (x(t), t)P1 (x 1 (t), x(t), t),

x 1 (t) ∈ ℝm ,

x(t) ∈ Df ,

t ∈ If ,

where Q1 (x 1 (t), x(t), t), x 1 (t) ∈ ℝm , x(t) ∈ Df , t ∈ If , is a continuous projector onto N1 (x 1 (t), x(t), t), x 1 (t) ∈ ℝm , x(t) ∈ Df , t ∈ If , and P1 (x 1 (t), x(t), t) = I − Q1 (x 1 (t), x(t), t),

x 1 (t) ∈ ℝm ,

x(t) ∈ Df ,

t ∈ If .

The total derivative of B∗ Π0 B∗− in jet variables we will denote as follows: Diff1 (x 2 (t), x 1 (t), x(t), t) = Dt (B∗ Π0 B∗− )(x 1 (t), x(t), t), x 1 (t), x 2 (t) ∈ ℝm ,

x(t) ∈ Df ,

t ∈ If .

2

The new jet variable x 2 (t) ∈ ℝm , t ∈ If , can be considered as a place holder for x Δ (t), t ∈ If . We have that there are c1 , c2 ∈ [t, σ(t)] such that Diff1 (x 2 (t), x Δ (t), x(t), t) =

𝜕 (B Π B− )(x Δ (t), x(t), t) Δt ∗ 0 ∗ 𝜕 + x Δ (t) (B∗ Π0 B∗− )(x Δ (c1 ), x(c1 ), σ(c1 )) 𝜕x 2 𝜕 + x Δ (t) 1 (B∗ Π0 B∗− )(x Δ (c2 ), x(σ(c2 )), σ(c2 )). 𝜕x

Example. Let 𝕋 = ℤ. Consider the following nonlinear dynamic system: 2

x1Δ (t) + 2x1 (t) = 0, 2

(x1 (t)) + (x2 (t)) − 2 = t 2

1 Projector analysis of dynamic systems on time scales

on Df = {x ∈ ℝ2 : x2 > 0},

If = 𝕋.

Here n = 1, m = k = 2, and f (y, x, t) = ( g(x, t) = x1 ,

y + x1 ), x12 + x22 − 2 − t 2 y ∈ ℝ,

x ∈ ℝ2 ,

t ∈ 𝕋.

Then fy (y, x, t) = (

1 ), 0

fx (y, x, t) = (

gx (x, t) = (1, 0),

gt (x, t) = 0,

1 2x1

0 ), 2x2

x ∈ ℝ2 ,

y ∈ ℝ,

t ∈ 𝕋.

Hence, 1

F(y1 (t), y2 (t), x(t), t) = ∫ ( 0

1

1 1 ) dh = ( ), 0 0

G(x 1 (t), x 2 (t), t) = ∫(1, 0)dh = (1, 0), 0

H(y(t), x 1 (t), x 2 (t), t) 1

= ∫( 0

1 2sx11 (t) + 2(1 − s)x12 (t)

=(

󵄨s=1 x11 (t)s2 󵄨󵄨󵄨s=0

=(

x11 (t) 2

1

−

1 + x12 (t)

x 1 (t), x (t) ∈ Df ,

x12 (t)(1 x21 (t)

−

0 ) ds 2sx21 (t) + 2(1 − s)x22 (t)

󵄨s=1 s)2 󵄨󵄨󵄨s=0

0 ), + x22 (t)

󵄨s=1 x21 (t)s2 󵄨󵄨󵄨s=0

y(t), y1 (t), y2 (t) ∈ ℝ,

0

󵄨s=1 ) − x22 (t)(1 − s)2 󵄨󵄨󵄨s=0

t ∈ 𝕋.

Therefore B∗ (x(t), t) = (1, 0), 1 A∗ (x (t), x(t), t) = ( ), 0 1

C∗ (x 1 (t), x(t), t) = H((1, 0)x 1 (t), x(t + 1), x(t), t) 1 0 =( ), x1 (t + 1) + x1 (t) x2 (t + 1) + x2 (t) x 1 (t) ∈ ℝ2 ,

x(t) ∈ Df ,

t ∈ If .

� 73

74 � S. G. Georgiev Next, G0 (x 1 (t), x(t), t) = A∗ (x 1 (t), x(t), t)B∗ (x(t), t) =(

1 ) (1, 0) 0

=(

1 0

0 ), 0

x 1 (t) ∈ ℝ2 ,

x(t) ∈ Df ,

t ∈ If .

Let z(t) = (

z1 (t) ) ∈ ℝ2 , z2 (t)

t ∈ 𝕋,

be such that G0 (x 1 (t), x(t), t)z(t) = 0,

x 1 (t) ∈ ℝ2 ,

x(t) ∈ Df ,

t ∈ 𝕋.

Then (

1 0

0 z (t) 0 )( 1 )=( ), 0 z2 (t) 0

t ∈ 𝕋,

or (

z1 (t) 0 )=( ), 0 0

t ∈ 𝕋,

whereupon z1 (t) = 0,

t ∈ 𝕋.

Let Q0 = (

0 0

0 ), p

where p ∈ ℝ, p ≠ 0, is chosen such that Q0 = Q0 Q0 . We have ( whereupon

0 0

0 0 )=( p 0

0 0 )( p 0

0 0 )=( p 0

0 ), p2

1 Projector analysis of dynamic systems on time scales

� 75

p2 = p and p = 1. Consequently, Q0 = (

0 0

0 ) 1

and G1 (x 1 (t), x(t), t) = G0 (x 1 (t), x(t), t) + C∗ (x 1 (t), x(t), t)Q0 =(

1 0

0 1 )+( 0 x1 (t + 1) + x1 (t)

=(

1 0

0 0 )+( 0 0

=(

1 0

0 ), x1 (t) + x2 (t)

0 0 )( x2 (t + 1) + x2 (t) 0

0 ) 1

0 ) x1 (t) + x2 (t) x 1 (t) ∈ ℝ2 ,

x(t) ∈ Df ,

t ∈ 𝕋.

Note that det G0 (x 1 (t), x(t), t) = x1 (t) + x2 (t) ≠ 0,

x 1 (t) ∈ ℝ2 ,

x(t) ∈ Df ,

t ∈ If .

Samir H. Saker

2 Muckenhoupt and Gehring weights on time scales Abstract: This chapter deals with the fundamental properties of the Muckenhoupt and Gehring weights on time scales. We also present some results related to the selfimproving properties of the Muckenhoupt and Gehring classes and some higher integrability results for nonincreasing functions on time scales. This chapter is organized as follows: In Section 2.1, we present some essential preliminaries on time scales that will act as prerequisites to the main results. Section 2.2 deals with the definitions and properties of the classical weights in the integral forms and the definitions and properties of the discrete weights, and then we write the definitions of Muckenhoupt and Gehring weights on time scales. Section 2.3 deals with the fundamental properties of the Muckenhoupt and Gehring classes on time scales. Section 2.4 deals with essential relations between the norms of these classes on time scales. Section 2.5 deals with the self-improving properties of Muckenhoupt and Gehring classes. In Section 2.6, we present some higher integrability results for nonincreasing functions on time scales. Our approach is based on proving some properties of integral operators with powers, Hölder inequality, chain rules, as well as some connecting relations between Muckenhoupt and Gehring classes on time scales. A science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life. Godfrey Harold Hardy (1877–1947).

2.1 Preliminaries on time scales We assume the reader has a good background in time scale calculus. For the reader who is not familiar with this calculus, we present to him, in this section, some preliminaries, definitions, concepts, and the basic dynamic inequalities on time scales that will be needed throughout the book. The results in this section will cover delta derivatives and integrals. For the notions used below, we refer the reader to the books [8, 9], and for more details about related inequalities on time scales, we refer the reader to the books [1, 2]. A time scale is an arbitrary nonempty closed subset of the real numbers. Throughout the chapter, we denote the time scale by the symbol 𝕋. For example, the real numbers ℝ, the integers ℤ, and the natural numbers ℕ are time scales. For t ∈ 𝕋, we define Samir H. Saker, Department of Mathematics, Mansoura University, Mansoura 35516, Egypt, e-mail: [email protected] https://doi.org/10.1515/9783111182971-002

78 � S. H. Saker the forward jump operator σ : 𝕋 → 𝕋 by σ(t) := inf{s ∈ 𝕋 : s > t}. A time scale 𝕋 equipped with the order topology is metrizable and is a Kσ -space, i. e., it is a union of at most countably many compact sets. The metric on 𝕋 which generates the order topology is given by d(r; s) := μ(r; s), where μ(⋅) = μ(⋅; τ) for a fixed τ ∈ 𝕋 is defined as follows. The mapping μ : 𝕋 → ℝ+ = [0, ∞) such that μ(t) := σ(t) − t is called the graininess function. When 𝕋 = ℝ, we see that σ(t) = t and μ(t) ≡ 0 for all t ∈ 𝕋, and when 𝕋 = ℕ, we have that σ(t) = t + 1 and μ(t) ≡ 1 for all t ∈ 𝕋. The backward jump operator ρ : 𝕋 → 𝕋 is defined by ρ(t) := sup{s ∈ 𝕋 : s < t}. The mapping ν : 𝕋 → ℝ+0 such that ν(t) = t − ρ(t) is called the backward graininess function. If σ(t) > t, we say that t is right-scattered, while if ρ(t) < t, we say that t is left-scattered. Also, if t < sup 𝕋 and σ(t) = t, then t is called right-dense, and if t > inf 𝕋 and ρ(t) = t, then t is called left-dense. If 𝕋 has a left-scattered maximum m, then 𝕋k = 𝕋 − {m}. Otherwise 𝕋k = 𝕋. In summary, 𝕋\(ρ sup 𝕋, sup ∞), if sup 𝕋 < ∞,

𝕋k = {

if sup 𝕋 = ∞.

𝕋,

Likewise, 𝕋k is defined as the set 𝕋k = 𝕋\[inf 𝕋, σ(inf 𝕋)] if | inf 𝕋| < ∞, and 𝕋k = 𝕋 if inf 𝕋 = −∞. For a weight f : 𝕋 → ℝ, we define the derivative f Δ at t ∈ 𝕋 as follows: If there exists a number α ∈ ℝ such that for all ε > 0 there exists a neighborhood U of t with 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨f (σ(t)) − f (s) − α(σ(t) − s)󵄨󵄨󵄨 ≤ ε󵄨󵄨󵄨σ(t) − s󵄨󵄨󵄨, for all s ∈ U, then f is said to be differentiable at t, and we call α the delta derivative of f at t and denote it by f Δ (t). For example, if 𝕋 = ℝ, then f Δ (t) = f ′ (t) = lim

Δt→0

f (t + Δt) − f (t) , Δt

for all t ∈ 𝕋.

If 𝕋 = ℕ, then f Δ (t) = f (t + 1) − f (t) for all t ∈ 𝕋. For a weight f : 𝕋 → ℝ, the (delta) derivative is defined by f Δ (t) =

f (σ(t)) − f (t) , σ(t) − t

if f is continuous at t and t is right-scattered. If t is not right-scattered then the derivative is defined by f Δ (t) = lim s→t

f (σ(t)) − f (s) f (t) − f (s) = lim , t→∞ t−s t−s

provided this limit exists. A useful formula is f σ = f + μf Δ

where f σ := f ∘ σ.

2 Muckenhoupt and Gehring weights on time scales

� 79

A weight f : [a, b] → ℝ is said to be right-dense continuous (rd-continuous) if it is right continuous at each right-dense point and there exists a finite left limit at all left-dense points, and f is said to be differentiable if its derivative exists. The space of rd-continuous functions is denoted by Crd (𝕋, ℝ). A time scale 𝕋 is said to be regular if the following two conditions are satisfied simultaneously: (a) For all t ∈ 𝕋, σ(ρ(t)) = t, (b) For all t ∈ 𝕋, ρ(σ(t)) = t. Remark 2.1. If 𝕋 is a regular time scale, then both operators ρ and σ are invertible with σ −1 = ρ and ρ−1 = σ. The following theorem gives the product and quotient rules for the derivative of the product fg and the quotient f /g (where gg σ ≠ 0) of two delta differentiable functions f and g. Theorem 2.1. Assume f ; g : 𝕋 → ℝ are delta differentiable at t ∈ 𝕋. Then (fg)Δ = f Δ g + f σ g Δ = fg Δ + f Δ g σ , Δ

Δ

(2.1)

Δ

f g − fg f . ( ) = g gg σ

(2.2)

By using the product rule, we see that the derivative of f (t) = (t − α)m for m ∈ ℕ, and α ∈ 𝕋 can be calculated as Δ

m−1

ν

f Δ (t) = ((t − α)m ) = ∑ (σ(t) − α) (t − α)m−ν−1 . ν=0

(2.3)

As a special case when α = 0, we see that the derivative of f (t) = t m for m ∈ ℕ can be calculated as Δ

m−1

(t m ) = ∑ σ γ (t)t m−γ−1 . γ=0

Note that when 𝕋 = ℝ, we have σ(t) = t,

μ(t) = 0,

f Δ (t) = f ′ (t).

When 𝕋 = ℤ, we have σ(t) = t + 1,

μ(t) = 1,

Δ

f (t) = Δf (t),

When 𝕋 =hℤ, h > 0, we have σ(t) = t + h, μ(t) = h,

b

b−1

∫ f (t)Δt = ∑ f (t). a

t=a

80 � S. H. Saker

f (t + h) − f (t) , f (t) = Δh f (t) = h Δ

b

b−a−h h

∫ f (t)Δt = ∑ f (a + kh)h. k=0

a

When 𝕋 = {t : t = qk , k ∈ ℕ0 , q > 1}, we have σ(t) = qt, μ(t) = (q − 1)t, (f (q t) − f (t)) f (t) = Δq f (t) = , (q − 1) t

∞

Δ

∞

∫ f (t)Δt = ∑ f (qk )μ(qk ). k=0

t0

When 𝕋 = ℕ20 = {t 2 : t ∈ ℕ}, we have σ(t) = (√t + 1)2 and f Δ (t) = Δ0 f (t) = (f ((√t + 1)2 ) − f (t))/1 + 2√t.

μ(t) = 1 + 2√t,

When 𝕋 = 𝕋n = {tn : n ∈ ℕ}, where (tn ) are the harmonic numbers that are defined by t0 = 0 and tn = ∑nk=1 k1 , n ∈ ℕ0 , we have σ(tn ) = tn+1 ,

μ(tn ) =

1 , n+1

f Δ (t) = Δ1 f (tn ) = (n + 1)f (tn ).

When 𝕋2 ={√n : n ∈ ℕ}, we have σ(t) = √t 2 + 1, μ(t) = √t 2 + 1 − t,

f Δ (t) = Δ2 f (t) =

(f (√t 2 + 1) − f (t)) . √t 2 + 1 − t

3

When 𝕋3 ={√3 n : n ∈ ℕ}, we have σ(t) = √t 3 + 1 and 3

μ(t) = √t 3 + 1 − t,

f Δ (t) = Δ3 f (t) =

3 (f (√t 3 + 1) − f (t)) . √3 t 3 + 1 − t

For a, b ∈ 𝕋, and a delta-differentiable weight f , the Cauchy integral of f Δ is defined by b

∫ f Δ (t)Δt = f (b) − f (a). a

Theorem 2.2. Let f , g ∈ Crd ([a, b], ℝ) be rd-continuous functions, a, b, c ∈ 𝕋, and α, β ∈ ℝ. Then, the following are true: b

b

b

1.

∫a [αf (t) + βg(t)]Δt = α ∫a f (t)Δt + β ∫a g(t)Δt,

2.

∫a f (t)Δt = − ∫b f (t)Δt,

3. 4.

b

a

c

b

c

∫a f (t)Δt = ∫a f (t)Δt + ∫b f (t)Δt, b

b

| ∫a f (t)Δt| ≤ ∫a |f (t)|Δt.

2 Muckenhoupt and Gehring weights on time scales

� 81

An integration by parts formula reads b

Δ

∫ f (t)g (t)Δt = a

b

f (t)g(t)|ba

− ∫ f Δ (t)g σ (t)Δt,

(2.4)

a

and improper integrals are defined as b

∞

∫ f (t)Δt = lim ∫ f (t)Δt. b→∞

a

a

Note that when 𝕋 = ℝ, we have b

b

∫ f (t)Δt = ∫ f (t)dt. a

a

When 𝕋 = ℤ, we have b

b−1

∫ f (t)Δt = ∑ f (t). t=a

a

When 𝕋 =hℤ, h > 0, we have b

b−a−h h

∫ f (t)Δt = ∑ f (a + kh)h. k=0

a

When 𝕋 = {t : t = qk , k ∈ ℕ0 , q > 1}, we have ∞

∞

∫ f (t)Δt = ∑ f (qk )μ(qk ). k=0

t0

Note that the integration formula on a discrete time scale is defined by b

∫ f (t)Δt = ∑ f (t)μ(t). a

t∈(a,b)

It is well known that rd-continuous functions possess antiderivatives. If f is rd-continuous and F Δ = f , then σ(t)

∫ f (s)Δs = F(σ(t)) − F(t) = μ(t)F Δ (t) = μ(t)f (t). t

82 � S. H. Saker Theorem 2.3. If a, b ∈ 𝕋 and f ∈ Crd (𝕋, ℝ) is such that f (t) ≥ 0 for all a ≤ t < b, then b

∫ f (t)Δt ≥ 0. a

1 ̃ = v(𝕋) be a time scale. If Lemma 2.1. Let v ∈ Crd (𝕋, ℝ) be strictly increasing and 𝕋 f ∈ Crd (𝕋, ℝ), then for a, b ∈ 𝕋, we have b

v(b)

a

v(a)

̃ ∫ f (x)vΔ (x)Δx = ∫ f (v−1 (y))Δy. Throughout the chapter, we will use the following results: ∞

∫

t0

Δs = ∞, sν

∞

if 0 ≤ ν ≤ 1,

and

∫

t0

Δs < ∞, sν

if ν > 1,

and, without loss of generality, we assume that sup 𝕋 = ∞, and define the time scale interval [a, b]𝕋 by [a, b]𝕋 := [a, b]∩𝕋. The two chain rules that we will use in this chapter are given in the next two lemmas. Lemma 2.2. Let f : ℝ → ℝ be continuously differentiable and suppose g : 𝕋 → ℝ is delta differentiable. Then f ∘ g : 𝕋 → ℝ is delta differentiable and f Δ (g(t)) = f ′ (g(ζ ))g Δ (t),

for ζ ∈ [t, σ(t)].

(2.5)

Lemma 2.3. Let f : ℝ → ℝ be continuously differentiable and suppose g : 𝕋 → ℝ is delta differentiable. Then f ∘ g : 𝕋 → ℝ is delta differentiable and the formula 1

(f ∘ g)Δ (t) = {∫ f ′ (g(t) + hμ(t)g Δ (t))dh}g Δ (t)

(2.6)

0

holds. In the following, we present the Jensen, Hölder, and Minkowski inequalities that will be used later in the proofs of the results in this book. For more details, we refer the reader to the book [1]. Theorem 2.4. Let a, b ∈ 𝕋 and c, d ∈ ℝ. Suppose that g ∈ Crd ([a, b], (c, d)) and h ∈ Crd ([a, b]𝕋 , ℝ) with b

󵄨 󵄨 ∫󵄨󵄨󵄨h(s)󵄨󵄨󵄨Δs > 0. a

2 Muckenhoupt and Gehring weights on time scales

� 83

If F ∈ C((c, d), ℝ) is convex, then b

b

F(

∫a |h(s)|g(s)Δs b

∫a |h(s)|Δs

)≤

∫a |h(s)|F(g(s))Δs b

∫a |h(s)|Δs

(2.7)

.

If F is strictly convex, then the inequality ≤ can be replaced by 1, and α

(

1 1 ∫ g(s)Δs) ≥ ∫ g α (s)Δs |I| |I| I

(2.9)

I

holds for α ∈ (0, 1). Theorem 2.5. Let a, b ∈ 𝕋. For f , g ∈ Crd (𝕀, ℝ), the Hölder inequality is given by b

b

1 p

󵄨p

b

󵄨q

1 q

󵄨 󵄨 󵄨 󵄨 ∫󵄨󵄨󵄨u(t)v(t)󵄨󵄨󵄨Δt ≤ [∫󵄨󵄨󵄨v(t)󵄨󵄨󵄨 Δt] [∫󵄨󵄨󵄨u(t)󵄨󵄨󵄨 Δt] , a

a

(2.10)

a

where p > 1 and p1 + q1 = 1, a, b ∈ 𝕋 and u, v ∈ Crd (𝕋, ℝ). This inequality is reversed if 0 < p < 1 and if p < 0 or q < 0. For example, if p = 1/γ < 1, then b

b

a

a

γ

b

−(γ−1)

󵄨 󵄨 󵄨 󵄨1/γ 󵄨 󵄨−1/(γ−1) Δt] ∫󵄨󵄨󵄨u(t)v(t)󵄨󵄨󵄨Δt ≥ [∫󵄨󵄨󵄨u(t)󵄨󵄨󵄨 Δt] [∫󵄨󵄨󵄨v(t)󵄨󵄨󵄨

(2.11)

.

a

Theorem 2.6. Let h, f , g ∈ Cr ([a, b]𝕋 , [0, ∞)). If 1/p + 1/q = 1, with p > 1, then b

b

1/p

∫ h(t)f (t)g(t)Δt ≤ (∫ h(t)f p (t)Δt) a

a

b

1/q

(∫ h(t)g q (t)Δt) a

.

(2.12)

The following theorems give the reverse Hölder-type inequality on time scales. Theorem 2.7. Let a, b ∈ 𝕋 with a < b and f and g be two positive functions defined on the interval [a, b]𝕋 such that 0 < m ≤ f p /g q ≤ M < ∞. Then for p > 1 and q > 1 with 1/p + 1/q = 1, we have

84 � S. H. Saker b

1/p

p

(∫ f (t)Δt) a

b

1/q

q

(∫ g (t)Δt)

1

b

M pq ≤ ( ) ∫ f (t)g(t)Δt. m

(2.13)

a

a

Next, we present the Hölder-type inequality in two dimensions on time scales. Theorem 2.8. Let a, b ∈ 𝕋 with a < b and f and g be two rd-continuous functions defined on the square [a, b]𝕋 × [a, b]𝕋 . Then b b

󵄨 󵄨 ∫ ∫󵄨󵄨󵄨f (x, y)g(x, y)󵄨󵄨󵄨ΔxΔy a a

b b

1/p

󵄨 󵄨p ≤ (∫ ∫󵄨󵄨󵄨f (x, y)󵄨󵄨󵄨 ΔxΔy) a a

b b

󵄨 󵄨q (∫ ∫󵄨󵄨󵄨g(x, y)󵄨󵄨󵄨 ΔxΔy)

1/q

,

a a

(2.14)

where p > 1 and q = p/(p − 1). In the following, we present the Minkowski inequality on time scales. Theorem 2.9. Let f , g, h ∈ Crd ([a, b]𝕋 , ℝ) and p > 1. Then b

󵄨 󵄨󵄨 󵄨p (∫󵄨󵄨󵄨h(x)󵄨󵄨󵄨󵄨󵄨󵄨f (x) + g(x)󵄨󵄨󵄨 Δx) a

b

1 p

1 p

b

1 p

󵄨 󵄨󵄨 󵄨p 󵄨 󵄨󵄨 󵄨p ≤ (∫󵄨󵄨󵄨h(x)󵄨󵄨󵄨󵄨󵄨󵄨f (x)󵄨󵄨󵄨 Δx) + (∫󵄨󵄨󵄨h(x)󵄨󵄨󵄨󵄨󵄨󵄨g(x)󵄨󵄨󵄨 Δx) . a

(2.15)

a

Now, we present the definition of Δ-measurable functions on time scales to define the Lebesgue Δ-measure on 𝕋. Let 𝕋 be a time scale. Denote by S the family of all leftclosed and right-open intervals of 𝕋 of the form [a, b) = {t ∈ 𝕋 : a ≤ t < b} with a, b ∈ 𝕋 and a ≤ b. The interval [a, a) is understood as the empty set. Obviously, the set weight m : S → [0, ∞), where S is a semiring of subsets of 𝕋, defined by m([a, b)) = b − a is a countably additive measure. An outer measure m∗ : P(𝕋) → [0, ∞] generated by m is defined by ∞

∞

n=1

n=1

m∗ (A) = inf{ ∑ m(An ) : An is a sequence of S with A ⊂ ⋃ An }. ∗ If there is no sequence (An ) of S such that A ⊂ ⋃∞ n=1 An , then we let m (A) = ∞. We ∗ ∗ define the family S(m ) of all m -measurable subsets of 𝕋, i. e.,

S(m∗ ) = {E ∈ 𝕋 : m∗ (A) = m∗ (A ∩ E) + m∗ (A ∩ E C ) for all A ⊂ 𝕋}. The collection S(m∗ ) of all m∗ -measurable sets is a σ-algebra and the restriction of m∗ to S(m∗ ) which we denote by mΔ is a countably additive measure on S(m∗ ). This measure

2 Muckenhoupt and Gehring weights on time scales

� 85

mΔ is a Carathéodory extension of the set weight m associated with the family S, and called the Lebesgue Δ-measure on 𝕋. We say that f : 𝕋 → ℝ is a measurable function if p f −1 (O) ∈ S(m∗ ) for every open subset of O of ℝ. We say that f belongs to LΔ (𝕋) provided that 1/p

‖f ‖Lp (𝕋) = (∫ |f |p Δs) Δ

< ∞,

if 1 ≤ p < ∞.

𝕋

Throughout this chapter, we will assume that the functions are nonnegative rd-continuous functions, Δ-differentiable, locally delta integrable, and the left hand sides of the inequalities exits if the right hand sides exist. We also assume that all the constants and boundaries of the integrals that will appear in the inequalities are positive real numbers.

2.2 Background on weights In this section, we introduce the necessary background on weights. We fix an interval I0 ⊂ ℝ+ = [0, ∞) and consider subinterval I of I0 of the form [0, s], for 0 < s < ∞ and denote by |I| the Lebesgue measure of I. A weight is a nonnegative locally integrable function defined on a bounded interval I ⊂ I0 with values in [0, +∞). The classical Muckenhoupt class of weights Ap has been introduced by Muckenhoupt [37] in connection with the boundedness of the Hardy and Littlewood maximal operator in the space Lpw (ℝ+ )

1 p

∞

󵄨 󵄨p = {f : ‖f ‖ = ( ∫ 󵄨󵄨󵄨f (x)󵄨󵄨󵄨 ω(x)dx) < ∞}. 0

A nonnegative weight ω is said to belong to the Muckenhoupt class Ap (C ) on the interval I0 for p > 1 and C > 1 (independent of p) if the inequality 1−p

1 1 1 ∫ ω(x)dx ≤ C ( ∫ ω 1−p (x)dx) |I| |I|

I

(2.16)

I

holds for every subinterval I ⊂ I0 . For p > 1, we define the Ap -norm of the weight ω by [Ap (ω)] := sup( I⊂I0

p−1

−1 1 1 ∫ ω(x)dx)( ∫ ω p−1 (x)dx) |I| |I|

I

< ∞.

I

The weight ω is said to belong to the Muckenhoupt class A1 (C ) on the interval I0 if the inequality 1 ∫ ω(x)dx ≤ C inf ω(x), x∈I |I| I

for C > 1,

86 � S. H. Saker holds for every subinterval I ⊂ I0 , and we define the A1 -norm by [A1 (ω)] := sup( I⊂I0

−1 1 ∫ ω(x)dx)(inf ω(x)) < ∞. x∈I |I| I

The weight ω is said to belong to the Muckenhoupt class A∞ (C ) on the interval I0 if the inequality (

1 1 1 dx) ≤ C , ∫ ω(x)dx)(exp ∫ log |I| |I| ω(x) I

for C > 1,

I

holds for every subinterval I ⊂ I0 , and we define the A∞ -norm by [A∞ (ω)] := sup( I⊂I0

1 1 1 dx) < ∞. ∫ ω(x)dx)(exp ∫ log |I| |I| ω(x) I

I

In [37] Muckenhoupt proved the following result. Theorem 2.10. If 1 < p < ∞ and ω satisfies the Ap -condition (2.16) on the interval I0 with constant C , then there exist constants q and C1 depending on p and C such that 1 < q < p and ω satisfies the Aq -condition q−1

(

1 1 − 1 ∫ ω(t)dt)( ∫ ω q−1 (t)dt) |I| |I| I

≤ C1 ,

(2.17)

I

for every subinterval I ⊂ I0 . In other words, the Muckenhoupt result (see also Coifman and Fefferman [15]) for self-improving property states that: if ω ∈ Ap (C ), then there exist a constant ϵ > 0 and a positive constant C1 such that ω ∈ Ap−ϵ (C1 ), and then Ap (C ) ⊂ Ap−ϵ (C1 ).

(2.18)

Further, Muckenhoupt proved the following result: Theorem 2.11. If 1 < p < ∞ and ω(x) ∈ Ap (C ) on the interval I with a constant C , then there exist constants r and C1 depending only on p and C such that ωr (x) ∈ Ap (C1 ) for r > 1. Despite a variety of ideas related to weighted inequalities that appeared with the birth of singular integrals, it was only in the 1970s that a better understanding of the subject was obtained and the full characterization of the weights w for which the Hardy– Littlewood maximal operator

2 Muckenhoupt and Gehring weights on time scales

M f (x) := sup x∈I

1 ∫ f (y)dy, |I|

� 87

(2.19)

I

p

is bounded on Lw (ℝ+ ) by means of the so-called Ap -condition was achieved by Muckenhoupt and published in 1972 (see [37]). Muckenhoupt’s result became a landmark in the theory of weighted inequalities because most of the previously known results for classical operators had been obtained for special classes of weights (like power weights) and has been extended to cover several operators like Hardy operator, Hilbert operator, Calderón–Zygmund singular integral operators, fractional integral operators, etc. A year later after the Muckenhoupt paper, a different class of weights satisfying the reverse Hölder inequality has been introduced and developed by Gehring [21, 22] in connection with the integrability properties of the gradient of quasiconformal mappings. A weight ω is said to belong to the Gehring class Gq (K ), 1 < q < ∞ on the interval I0 , if there exists a constant K > 1 such that the inequality 1

q 1 1 ( ∫ ωq (x)dx) ≤ K ( ∫ ω(x)dx) |I| |I|

I

(2.20)

I

holds for every subinterval I ⊂ I0 , and we define the Gq -norm by 1

q

q q−1 1 1 [Gq (ω)] := sup[( ∫ ωq (x)dx) ( ∫ ω(x)dx) ] . |I| |I| I⊂I0

−1

I

I

The weight ω is said to belong to the Gehring class G∞ (K ) if the inequality sup(sup ω(x))( I

x∈I

1 ∫ ω(x)dx) |I|

−1

≤K,

I

holds for every subinterval I ⊂ I0 . The weight ω is said to belong to the Gehring class G1 (K ) if the inequality exp(

1 ∫ |I| I

ω(x)

1 ∫ |I| I

ω(x)dx

log(

ω(x)

1 ∫ |I| I

ω(x)dx

)dx) ≤ K ,

holds for every interval I ⊂ I0 . Gehring proved that if (2.20) holds, then there exist a p > q and a positive constant K1 such that p

1 1 ∫ ωp (x)dx ≤ K1 ( ∫ ω(x)dx) . |I| |I| I

(2.21)

I

In other words, the Gehring result for self-improving property states that: if ω ∈ Gq (K ) then there exist ϵ > 0 and a positive constant K1 such that ω ∈ Gq+ϵ (K1 ) and

88 � S. H. Saker Gq (K ) ⊂ Gq+ϵ (K1 ).

(2.22)

The self-improving property of the Gehring class has applications in different fields, especially in studying the optimal regularity of solutions to some elliptic PDEs (see, for example, Kenig [28]) where the solution of the Dirichlet problem div A(σ)∇θ = 0 on the unit disc D, with θ|D = φ, can be expressed in terms of Gq conditions on the boundary 𝜕D for the harmonic measures associated to A(σ), with 1/p + 1/q = 1. We refer the reader to the book [28] for more applications of these classes on extrapolation theory, vectorvalued inequalities, and estimates for certain classes of nonlinear partial differential equations. The relation between Gehring and Muckenhoupt classes (inclusion properties) was given by Coifman and Fefferman in [15]. They proved that any Gehring class is contained in some Muckenhoupt class, and vice versa. In other words, they proved the following inclusions: Gq (K ) ⊂ Ap (K1 )

(2.23)

Ap1 (K1 ) ⊂ Gq1 (K ).

(2.24)

and

The sharp results for the reverse Hölder inequalities can be found in Martio–Sbordone [33]. The proof of Gehring’s inequality is based on the use of the Calderón–Zygmund decomposition and the scale structure of Lp -spaces. For further studies of the Muckenhoupt and Gehring classes, we refer the reader to [5, 11, 18–20, 25–27, 33–37, 40, 41, 47, 56, 57, 65–68]. In recent years the study of regularity and boundedness of discrete operator on p p lϑ (ℤ+ ) analogues for Lw (ℝ+ )-regularity and boundedness has been considered by some authors; see, for example, [7, 29, 31] and the references therein. One of the reasons for this surge of interest in discrete cases is due to the fact that the discrete operators may even behave differently from their continuous counterparts as is exhibited by the discrete spherical maximal operator [32]. In some special cases it is possible to translate or adapt almost straightforwardly the objects and results from the continuous setting to the discrete setting, or vice versa, however, in some other cases that is far from trivial [13, 14, 16]. For example, in the simplest cases of lp -bounds for discrete analogues of classical operators such as Calderón–Zygmund singular integral operators, fractional integral operators, and the maximal Hardy–Littlewood operator follow from known Lp -bounds for the original operators in the Euclidean setting, via elementary comparison arguments (see [42–44]). But lp -bounds for discrete analogues of more complicated operators are not implied by results in the continuous setting, and, moreover, the discrete analogues are resistant to conventional methods. The main challenge is that there are no general methods to study these questions. These methods have to be developed starting from

2 Muckenhoupt and Gehring weights on time scales

� 89

the basic definitions. In [6] the authors mentioned that the study of discrete inequalities is not an easy task and more difficult to analyze than its integral counterparts and discovered that the conditions do not correspond, in any natural way, with those that are obtained by discretizing the results for functions, but the converse is true. This means that what goes for sums goes, with the obvious modifications, for integrals which in fact proved the first part of the basic principle of Hardy, Littlewood, and Polya [24, p. 11]. Indeed, the proofs for series translate immediately and become much simpler when applied to integrals. This fact motivated a lot of authors to study the characterizations of discrete weights and use the new characterizations to formulate some conditions for the boundedness of the discrete operators and prove some embedding theorems for Lorentz spaces. For further studies of the discrete classes, we refer the reader to [12, 45, 46, 48, 50, 51, 53–55, 57, 58, 61–63]. In the following, for the sake of completeness, we present the background and the basic definitions for discrete weights. Throughout this chapter, ℤ+ stands for the set of nonnegative integers, i. e., ℤ+ = {1, 2, . . .}. By an interval 𝕁 we mean a finite subset of ℤ+ consisting of consecutive integers and, for J ⊂ 𝕁, the number |J| stands for its cardinality. A discrete weight on ℤ+ is a sequence ϑ = {ϑ(n)}∞ n=1 of nonnegative real numbers. The p space lϑ (ℤ+ ), for 1 ≤ p < ∞, is the space containing all real-valued sequences u defined on ℤ+ and satisfying the condition ∞

1/p

󵄨 󵄨p ‖u‖lp (ℤ+ ) := ( ∑ 󵄨󵄨󵄨u(n)󵄨󵄨󵄨 ϑ(n)) ϑ

n=1

< ∞,

where ϑ is a discrete weight. We shall denote by A = 2ℤ+ the power set of ℤ+ . A discrete weight ϑ belongs to the discrete Muckenhoupt class A1 (C ) on ℤ ⊂ ℤ+ for p > 1 and C > 1 if the inequality 1 ∑ ϑ(k) ≤ inf ϑ(k), k∈J |J| k∈J

for all k ∈ J,

(2.25)

holds for every subinterval J ⊂ 𝕁, with |J| being the cardinality of the set J. Sometimes it is convenient to consider a symmetric form of which it is equivalent to (2.25). A discrete weight ϑ belongs to the discrete Muckenhoupt class A2 (C ) on the interval 𝕁 ⊆ ℤ+ for p > 1 and C > 1 if the inequality ∑ ϑ(k) ∑ ϑ −1 (k) ≤ A|J|2 ,

k∈J

k∈J

(2.26)

holds for every subinterval J ⊂ 𝕁. This class has been used in harmonic analysis by some authors. For example, in [4], Arińo and Muckenhoupt proved that if ϑ is nonincreasing ∗ and satisfies (2.25), then the space Λ(ϑ −q /q , q∗ ) is the dual space of the discrete classical Lorentz space

90 � S. H. Saker

Λ(ϑ, q) = {x : ‖x‖ϑ,q

1/q

∞

󵄨q 󵄨 = ( ∑ 󵄨󵄨󵄨x ∗ (n)󵄨󵄨󵄨 ϑ(n))

< ∞},

n=1

where x ∗ (n) is the nonincreasing rearrangement of |x(n)| and q∗ is the conjugate of q. In [39] Pavlov gave a full description of all complete interpolating sequences on the real line by using the integral from of (2.26). In particular, he proved that a sequence λn of real numbers is a complete interpolating sequence if and only if the weight υ = |F(x + iy)|2 , x, y ∈ ℝ, satisfies the Muckenhoupt condition ∫ υ(x)dx ∫ υ−1 (x)dx ≤ C |J|2 , J

(2.27)

J

for some constant C > 0 and some y ≠ 0 for all intervals J ⊂ ℝ of finite length |J|, where F(z) = lim ∏ (1 − R→∞

|λn | 0 and all finite sets J of consecutive integers containing |J| elements. Checking the Muckenhoupt condition (2.27) for a weight F given by an infinite product (covering in the Cauchy principle value sense) is particularly quite hard. However, condition (2.28) is relatively easier to verify since it involves only countably many sets J instead of all finite intervals. A discrete nonnegative sequence ϑ belongs to the discrete Muckenhoupt class Ap (C ) on the interval 𝕁 ⊆ ℤ+ for p > 1 and C > 1 if the inequality p−1

(

−1 1 1 ∑ ϑ(k))( ∑ ϑ p−1 (k)) |J| k∈J |J| k∈J

(2.29)

≤C

holds for every subinterval J ⊂ 𝕁. For a given exponent p > 1, we define the A p -norm of the discrete weight ϑ by the following quantity: [A p (ϑ)] := sup( J⊂𝕁

−1 1 1 ∑ ϑ(k))( ∑ ϑ p−1 (k)) |J| k∈J |J| k∈J

p−1

< ∞,

(2.30)

where the supremum is taken over all intervals J ⊂ 𝕁. When we fix a constant C > 1, the pair of real numbers (p, C ) defines the Ap discrete Muchenhoupt class Ap (C ):

2 Muckenhoupt and Gehring weights on time scales

� 91

ϑ ∈ Ap (C ) ⇐⇒ [Ap (ϑ)] ≤ C , and we will refer to C as the Ap -constant of the class. The Hardy–Littlewood maximal sequence M f of the sequence f is defined by (M f )(n) := sup n∈J

1 ∑ f (k). |J| J

(2.31)

The operator M : f → M f is called discrete Hardy–Littlewood maximal operator. Observe that M is merely sublinear, rather than linear, and it is a contraction on ℓ∞ . The boundedness of discrete Hardy–Littlewood maximal operator has been characterized in [54], and it has been proved that M f (n) is bounded on ℓp (ϑ) if and only if ϑ ∈ Ap . In [61] the authors proved that if q > 1, C > 1, and ϑ is a nondecreasing weight belonging to Aq (C ), then ϑ ∈ Ap (C1 ) for p ∈ (p0 , q] where p0 is a unique solution of an algebraic inequality. This result proves that if ϑ ∈ Aq (C ) then there exist an ϵ > 0 and a constant C1 = A1 (p, C ) such that ϑ ∈ Aq−ϵ (C1 ), (self-improving property) and thus Aq (C ) ⊂ Aq−ϵ (C1 ).

(2.32)

In the following, we present some basic properties of discrete Muckenhoupt weights. Theorem 2.12 ([61]). Let ϑ be a nonnegative weight and p and q be positive real numbers. The following properties hold: ′ ′ ′ (1) ϑ ∈ Ap if and only if ϑ 1−p ∈ Ap′ , with Ap′ (ϑ 1−p ) = [Ap (ϑ)]p −1 where p′ is the conjugate of p; (2) Ap ⊂ Aq for all 1 < p ≤ q; (3) if ϑ ∈ Ap , then ϑ α ∈ Ap , for 0 ≤ α ≤ 1, with Ap (ϑ α ) = [Ap (ϑ)]α ; (4) A1 ⊂ Ap ⊂ A∞ , for all 1 < p < ∞; (5) A∞ = ⋃1

1 Ap ; (6) if ϑ1 , ϑ2 ∈ Ap , then ϑ1α ϑ21−α ∈ Ap , 0 ≤ α ≤ 1, with a constant Ap (ϑ1α ϑ21−α ) = [Ap (ϑ1 )]α [Ap (ϑ2 )]1−α . Theorem 2.13 ([61]). Let ϑ be a nonnegative weight and p and q be positive real numbers. The following properties hold: 1−p (1) ϑ ∈ Ap if and only if there exists ϑ1 , ϑ2 ∈ A1 such that ϑ = ϑ1 ϑ2 , 1 < p < ∞; τ (2) if ϑ ∈ Ap , then ϑ ∈ Ap for some τ > 1; (3) if ϑ ∈ Ap , p > 1, then ϑ ∈ Ap−ϵ , for some ϵ > 0; 1

(4) ϑ ∈ Ap if and only if ϑ and ϑ 1−p are in A∞ .

A discrete nonnegative weight ϑ belongs to the discrete Gehring class Gq (K ) for a given exponent q > 1 and a constant K > 1 (or satisfies the reverse Hölder inequality) on the interval 𝕁 ⊂ ℤ+ if, for every subinterval J ⊆ 𝕁, we have

92 � S. H. Saker 1

(

q 1 1 ∑ ϑ q (k)) ≤ K ∑ ϑ(k). |J| k∈J |J| k∈J

For a given exponent q > 1, we define the Gq -norm of ϑ as 1

q

q q−1 1 1 [G (ϑ)] := sup[( ∑ ϑ(k)) ( ∑ ϑ q (k)) ] , |J| k∈J |J| k∈J J⊂I

−1

q

where the supremum is taken over all intervals J ⊂ 𝕁 and represents the best constant for which the G q -condition holds true independently of the interval J ⊆ 𝕁. We say that ϑ is a discrete Gehring weight if its Gq -norm is finite, i. e., ϑ ∈ Gq ⇐⇒ [Gq (ϑ)] < ∞. When we fix a constant K > 1, the pair of real numbers (q, K ) defines the G p -discrete Gehring class G p (K ): ϑ ∈ Gq (K ) ⇐⇒ [Gq (ϑ)] ≤ K , and we will refer to K as the Gq - constant of the class. In [50] the authors proved that if q > 1 and Kq > 1, and ϑ is a nonincreasing sequence belonging to Gq (Kq ), then ϑ ∈ Gp (Kp ) for p ∈ [q, q∗ ) where q∗ is a unique solution of an algebraic inequality. In [53] it has been proved that if ϑ is a nonincreasing sequence and satisfies (2.25) for C > 1, then for p ∈ [1, C /(C − 1)) the inequality p

1 1 ∑ ϑ p (k) ≤ C1 ( ∑ ϑ(k)) , |J| k∈J |J| k∈J

for J ⊂ 𝕁,

(2.33)

holds for every subinterval J ⊂ 𝕁. This result proves that the Muckenhoupt A1 weight belongs to some Gehring class of weights satisfying reverse Hölder inequality (a transition property). A discrete nonnegative weight ϑ belongs to the discrete Gehring class G 1 (K ) for a constant K > 1 on the interval 𝕁 ⊂ ℤ+ if, for every subinterval J ⊆ 𝕁, we have [G1 (v)] = sup exp( J⊂𝕁

1 ϑ(k) ϑ(k) log )≤K, ∑ |J| k∈J ϑJ ϑJ

where ϑJ = (1/|J|) ∑k∈J ϑ(k) and the supremum is taken over all J ⊂ 𝕁. The class G∞ consists of all weights v defined on 𝕁 ⊂ ℤ+ such that G∞ (v)-norm is finite, where [G∞ (v)] = sup J⊂𝕁

ess supk∈J ϑ(k) |J| ∑k∈J ϑ(k)

.

In the following, we present some basic properties of discrete Gehring weights.

2 Muckenhoupt and Gehring weights on time scales

� 93

Theorem 2.14 ([61]). Let ϑ be a nonnegative weight and p and q be real positive numbers such that p, q > 1. The following properties hold: (1) Gq ⊂ Gp for all 1 < p ≤ q; (2) G∞ ⊂ Gq ⊂ G1 for all 1 < q ≤ ∞; (3) G1 = ⋃1 0; (5) G1 = A∞ = ⋃1 1, we denote by B p.q (C ) the class of all nonnegative weights v that satisfy a generalized reverse Hölder inequality 1/q

(

1 ∑ vq (n)) |J| n∈J

1/p

≤ C(

1 ∑ vp (n)) |J| n∈J

,

for all J ⊂ 𝕁.

(2.34)

By recalling the classical Hölder inequality, it is clear that the definition of B p.q is well posed only for C ≥ 1, where the equality prevails in case of constant sequences. The smallest constant, independent of the interval J, satisfying the inequality (2.34) is called the B p.q -norm of the weight v and will be denoted by B p,q (v); it is given by B

p,q

− p1

1 (v) := sup( ∑ vp (n)) J⊂𝕁 |J| n∈J

1

q 1 ( ∑ vq (n)) . |J| n∈J

(2.35)

We say that v is a B p.q -weight if its B p.q -norm is finite, i. e., v ∈ B p,q ⇐⇒ B p,q (v) < +∞. When we fix a constant C > 1, the triple of real numbers (p, q, C ) defines the B p.q discrete class: v ∈ B p,q (C ) ⇐⇒ B p,q (v) ≤ C , and we will refer to C as the B p.q -constant of the class. Moreover, v ∈ B p,q (C ) ⇐⇒ vp ∈ B 1,q/p (C p ) ⇐⇒ vq ∈ B p/q,1 (C q ), and the following properties hold: B B

p,q p,q

(C ) ⊂ B p,r (C ), r,q

(C ) ⊂ B (C ),

for p < r ≤ q,

for p ≤ r < q.

It is immediate to observe that the classes A p and G q are special cases of the B p.q (C ) of discrete weights as follows:

94 � S. H. Saker 1

(1) Ap (C ) = B 1−p (C ) ⇐⇒ A p−1 (C ) = B 1,p (C ), ,1

p

(2) Gq (C ) = B 1,q (C ).

The natural question that arises here is: Is it possible to prove the properties of general class of weights on time scales which are special cases contain the properties of the continuous and discrete Muckenhoupt and Gehring weights? The objective of this chapter is to provide an affirmative answer to this question and prove some properties and some relations between Muckenhoupt and Gehring classes in the context of time scales and use these properties to prove that self-improving properties hold and then extend the results to prove some higher integrability on time scales. Now, we present the definitions of the Muckenhoupt and Gehring weights on time scales. We assume that ω is a nonnegative locally Δ-integrable weight defined on 𝕀 = [0, ∞)𝕋 = [0, ∞) ∩ 𝕋 and p is a positive real number. The nonnegative weight ω is said to belong to the Muckenhoupt class 𝔸p (C ) on time scales on the interval 𝕀0 for p > 1 and C > 1 (independent of p) if the inequality 1−p

1 1 1 ∫ ω(s)Δs ≤ C ( ∫ ω 1−p (s)Δs) |I| |I|

I

(2.36)

I

holds for every subinterval I ⊂ 𝕀0 . For p > 1, we define the 𝔸p -norm on time scales by [𝔸p (ω)] := sup( I⊂𝕀

p−1

−1 1 1 ∫ ω(s)Δs)( ∫ ω p−1 (s)Δs) |I| |I|

I

< ∞.

I

The weight ω is said to belong to the Muckenhoupt class 𝔸1 (C ) on time scales on the interval 𝕀0 if the inequality 1 ∫ ω(s)Δs ≤ C inf ω(x), x∈I |I|

for C > 1,

I

holds for every subinterval I ⊂ 𝕀. We define the 𝔸1 -norm on time scales by [𝔸1 (ω)] := sup( I⊂𝕀0

−1 1 ∫ ω(x)Δx)(inf ω(x)) . x∈I |I| I

The weight ω is said to belong to the Muckenhoupt class 𝔸∞ (C ) on time scales on the interval 𝕀 if the inequality (

1 1 1 Δs)) ≤ C , ∫ ω(s)Δs)(exp( ∫ log |I| |I| ω(s) I

C > 1,

I

holds for every subinterval I ⊂ 𝕀, and we define the 𝔸∞ -norm on time scales by

2 Muckenhoupt and Gehring weights on time scales

� 95

1 1 1 Δs)). ∫ ω(s)Δs)(exp( ∫ log |I| |I| ω(s)

[𝔸∞ (ω)] := sup( I⊂I0

I

I

The weight ω is said to belong to the Gehring class 𝔾q (K) on time scales on the interval 𝕀 for q > 1 and K > 1 (independent of q) if the inequality 1

q 1 1 ( ∫ ωq (s)Δs) ≤ K ∫ ω(s)Δs |I| |I|

I

(2.37)

I

holds for every subinterval I ⊂ 𝕀, and the 𝔾q -norm is defined by 1

q

q q−1 1 1 [𝔾q (ω)] := sup[( ∫ ωq (s)Δs) ( ∫ ω(s)Δs) ] < ∞. |I| |I| I⊂I0

−1

I

I

The weight ω is said to belong to the Gehring class 𝔾∞ (K) on time scales on the interval 𝕀 if the inequality [𝔾∞ (ω)] = sup(sup ω(s)( I⊂𝕀0

s∈I

1 ∫ ω(s)Δs) ) ≤ K, |I| −1

for K > 0,

I

holds for every subinterval I ⊂ 𝕀. The weight ω is said to belong to the Gehring class 𝔾1 (K) on the interval 𝕀 if the inequality [𝔾1 (ω)] = exp(

1 ∫ |I| I

ω(s)

1 ∫ |I| I

ω(s)Δs

log(

ω(s)

1 ∫ |I| I

ω(s)Δs

)Δs) ≤ K,

holds for every interval I ⊂ 𝕀. We say that a nonnegative weight ω belongs to the class q Up (B) if it satisfies the reverse Hölder inequality 1

1

q p 1 1 [ ∫ ωq (s)Δs] ≤ B[ ∫ ωp (s)Δs] , |I| |I|

I

I

with a positive constant B > 1 for 1 ≤ p < q for every interval I ⊂ 𝕀. When we fix a q constant C > 1, the triple of real numbers (p, q, C ) defines the Up class: ω ∈ Upq (C ) ⇐⇒ [Upq (ω)] ≤ C , q

and we will refer to C as the Up -constant of the class. It is immediate to observe that q the classes 𝔸p and 𝔾q are special cases of the class Up of weights as follows: 𝔸p := U 11

1−p

and

q

𝔾q := U1 .

96 � S. H. Saker

2.3 Properties of Muckenhoupt and Gehring weights In this section, we state some basic properties of the Muckenhoupt 𝔸p -weights and Gehring 𝔾q -weights on time scales. The results are adapted from [49]. The results are particular cases when 𝕋 = ℝ and cover the results due to Cruz-Uribe [17], Johnson and Neugebauer [26], and Popoli [41]. Throughout this chapter, we assume that the functions in the statements of the theorems are nonnegative and rd-continuous, while the integrals considered are assumed to exist and be finite. Therefore, these conditions will be omitted, for brevity. Definition 2.1. We define the operator Mq ω : 𝕀 → ℝ+ by 1

q 1 q Mq ω := ( ∫ ω (s)Δs) , |I|

(2.38)

I

for any real number q ≠ 0 and every I ⊂ 𝕀. In the following lemma, we state some basic properties of the operator Mq which will be needed in the proof of the main results. The results are adapted from [49]. Lemma 2.4. Assume that p, q ≠ 0 are real numbers and Mq is defined as in (2.38). Then the following properties hold: (1) M−1 ω(s) ≤ M1 ω(s); (2) Mq ω(s) ≥ M1 ω(s) for all q ≥ 1; (3) Mq ω(s) ≤ M1 ω(s) for all q < 1; (4) Mp ω(s) ≤ Mq ω(s) for all p ≤ q. Lemma 2.5. Let q be a positive real numbers. If ω ∈ 𝔾q (K) for q > 1 and K > 1, then Mq ω ≤ K M1 ω and, consequently, Mq ω ≤ K Mp ω for all p ≥ 1. Lemma 2.6. If ω ∈ 𝔸p (C ) and p > 1, then M1 ω ≤ C exp(M1 log ω)

(2.39)

holds. The next lemma gives the inclusion of the Gehring classes of weights 𝔾q into the 𝔾1 -class. Lemma 2.7. Assume that q > 1 is a nonnegative number. If ω ∈ 𝔾q , then exp(

1 ∫ |I| I

holds for all I ⊂ 𝕀.

ω(s)

1 ∫ |I| I

ω(s)Δs

log(

ω(s)

1 ∫ |I| I

ω(s)Δs

)Δs) < ∞

(2.40)

2 Muckenhoupt and Gehring weights on time scales

� 97

In the following theorems, we present some basic inclusion properties of Muckenhoupt and Gehring classes on time scales. Theorem 2.15. Let p and q be positive real numbers. Then the following inclusion properties of Muckenhoupt classes hold: (1) 𝔸p ⊂ 𝔸q for all 1 < p ≤ q; (2) let 1 < p < ∞, then 𝔸1 ⊂ 𝔸p ⊂ 𝔸∞ ; (3) 𝔸∞ = ⋃1

1 𝔸p . Theorem 2.16. Let p and q be nonnegative real numbers. Then the following inclusion properties of Gehring classes hold: (1) 𝔾q ⊂ 𝔾p for all 1 ≤ p ≤ q; (2) 𝔾∞ ⊂ 𝔾q ⊂ 𝔾1 for all 1 ≤ p ≤ ∞; (3) 𝔾1 = ⋃1 1, that ω ∈ 𝔸p ⇐⇒ (

1−p

1 1 1 ∫ ω(s)Δs) ≤ C ( ∫ ω 1−p (s)Δs) |I| |I|

I

⇐⇒ (

1 ∫ ω(s)Δs) |I| I

1 1−p

I

≥C

1 1−p

1 1 ∫ ω 1−p (s)Δs |I|

I

1−p′

1 ′ ′ ′ 1 1 ⇐⇒ ∫ ω1−p (s)Δs ≤ C p −1 ( ∫ (ω1−p (s)) 1−p′ Δs) |I| |I|

I

⇐⇒ ω

1−p′

I

∈ 𝔸p′ ,

with [𝔸p′ (ω1−p )] = [𝔸p (ω)]p −1 . This is the desired result. (2) Let 1 ≤ p < ∞, 0 < ϵ < 1, and r = ϵp + 1 − ϵ. Then r − 1 = ϵ(p − 1) and, by applying Lemma 2.4 for ϵ < 1, we have ′

′

98 � S. H. Saker r−1

(

−1 1 1 ∫ ωϵ (s)Δs)( ∫(ωϵ (s)) r−1 Δs) |I| |I|

I

I

r−1

=(

−ϵ 1 1 ∫ ωϵ (s)Δs)( ∫ ω r−1 (s)Δs) |I| |I|

I

I

ϵ

≤(

ϵ(p−1)

−1 1 1 ∫ ω(s)Δs) ( ∫ ω p−1 (s)Δs) |I| |I|

I

≤ C ϵ,

I

whereupon ωϵ ∈ 𝔸ϵp+1−ϵ . This is the desired result, completing our proof. In the next theorem, we discuss the power rule for weights in the Muckenhoupt classes on time scales. Theorem 2.18. Assume that 1 < p < ∞ is a positive real number. Then the following properties hold: (1) If ω ∈ 𝔸p , then ωα ∈ 𝔸p for 0 ≤ α ≤ 1, with [𝔸p (ωα )] ≤ [𝔸p (ω)]α ; (2) If ω1 , ω2 ∈ 𝔸p , then ωα1 ω1−α ∈ 𝔸p for 0 ≤ α ≤ 1, with 2 α

[𝔸p (ωα1 ω1−α 2 )] ≤ [𝔸p (ω1 )] [𝔸p (ω2 )]

1−α

.

Proof. (1) For 0 ≤ α ≤ 1 and ω ∈ 𝔸p , we have 1/(p − 1) ≥ α/(p − 1) > 0 and, by Lemma 2.4, for α < 1 and for all I ⊂ I0 , we have p−1

(

−1 1 1 ∫ ωα (s)Δs)( ∫(ωα ) p−1 (s)Δs) |I| |I|

I

I

p−1

=(

−α 1 1 ∫ ωα (s)Δs)( ∫ ω p−1 (s)Δs) |I| |I|

I

I

α(p−1)

α

≤(

−1 1 1 ∫ ω(s)Δs) ( ∫ ω p−1 (s)Δs) |I| |I|

I

I

(p−1) α

= [(

−1 1 1 ∫ ω(s)Δs)( ∫ ω p−1 (s)Δs) |I| |I|

I

] ≤ C α,

I

that is, ωα ∈ 𝔸p , with [𝔸p (ωα )] ≤ [𝔸p (ω)]α . This is the desired result. (2) Since ω1 , ω2 ∈ 𝔸p , we get that p−1

−1 1 1 ∫ ω1 (s)Δs( ∫ ω1p−1 (s)Δs) |I| |I|

I

and

I

≤ C1

(2.41)

� 99

2 Muckenhoupt and Gehring weights on time scales p−1

−1 1 1 ∫ ω2 (s)Δs( ∫ ω2p−1 (s)Δs) |I| |I|

I

(2.42)

≤ C2 ,

I

where C1 , C2 > 1. By applying the Hölder inequality (note that 0 ≤ α ≤ 1) with 1/α > 1 and 1/(1 − α), and using (2.41) and (2.42), we have α

1−α

1 1 1 ∫ ω1 α (s)ω2 1−α (s)Δs ≤ ( ∫ ω1 (s)Δs) ( ∫ ω2 (s)Δs) |I| |I| |I| I

≤ ((

C1

|I|

1−p α

1 1−p

∫ ω1 (s)Δs)

I

I

) ((

I

= C1 α C2 1−α ((

C2

|I| α

1−p 1−α

1 1−p

∫ ω2 (s)Δs)

)

I

1−α 1−p

1 1 1 1 ∫ ω11−p (s)Δs) ( ∫ ω21−p (s)Δs) |I| |I|

I

)

(2.43)

.

I

By applying the Hölder inequality with 1/α and 1/(1 − α) on the term α 1−α 1 ∫ ω1 1−p (s)ω2 1−p (s)Δs, |I|

I

we have α

1−α

1 1 1−α α 1 1 1 ∫ ω1 1−p (s)ω2 1−p (s)Δs ≤ ( ∫ ω11−p (s)Δs) ( ∫ ω21−p (s)Δs) |I| |I| |I|

I

I

(2.44)

.

I

By substituting (2.44) into (2.43) and since 1 − p < 0, we have 1−α α 1 1 ∫ ω1 α (s)ω2 1−α (s)Δs ≤ C1 α C2 1−α [ ∫ ω1 1−p (s)ω2 1−p (s)Δs] |I| |I|

I

I

1−p

1−p

1 1 = C1 α C2 1−α [ ∫(ω1 α (s)ω2 1−α (s) ) 1−p Δs] |I|

.

I

This proves that ω1 , ω2 ∈ 𝔸p implies ωα1 ω1−α ∈ 𝔸p for 0 ≤ α ≤ 1, with 2 α

[𝔸p (ωα1 ω1−α 2 )] ≤ [𝔸p (ω1 )] [𝔸p (ω2 )]

1−α

.

The proof is complete. Theorem 2.19. Assume that p is a nonnegative real number. If ω ∈ 𝔸p , then

1 ω

∈ 𝔾p′ −1 .

Proof. Let ω ∈ 𝔸p . Then there exists a constant C > 1 such that the inequality 1−p

1 1 ∫ ω(s)Δs ≤ C ( ∫ ω1/(1−p) (s)Δs) |I| |I| I

I

(2.45)

100 � S. H. Saker holds for all I ⊂ I0 . From the property (1) in Lemma 2.4, we have 1 1 Δs) ∫ |I| ω(s)

≤

1/(p−1)

p−1

−1

(

I

1 ∫ ω(s)Δs, |I| I

and (2.45) becomes (

1 1 ∫( ) |I| ω

(s)Δs)

≤C

I

That is,

1 ω

1 1 Δs. ∫ |I| ω(s) I

∈ 𝔾p′ −1 . The proof is completed.

Theorem 2.20. Suppose that 1 < p1 < p2 < ∞, 0 < δ < 1, and ω1 , ω2 ∈ 𝔸p . Then the following properties hold: (1) If p = δp1 + (1 − δ)p2 , then δ

[𝔸p (ωδ1 ω1−δ 2 )] ≤ [𝔸p1 (ω1 )] [𝔸p2 (ω2 )] (2) If p = ( pδ + 1

1−δ −1 ) , p2

1−δ

;

then δp/p1

[𝔸p (ω1

(1−δ)p/p2

ω2

)] ≤ [𝔸p1 (ω1 )]

δp/p1

[𝔸p2 (ω2 )]

(1−δ)p/p2

.

Proof. (1) Since ω1 , ω2 ∈ 𝔸p , we have 𝔸p (ωδ1 ω1−δ 2 )=(

p−1

−1 1 1 p−1 Δs) ∫ ωδ1 (s)ω1−δ ∫[ωδ1 (s)ω1−δ 2 (s)Δs)( 2 (s)] |I| |I|

I

I

p−1

=(

−(1−δ) −δ 1 1 ∫ ωδ1 (s)ω1−δ ∫ ω1p−1 (s)ω2 p−1 (s)Δs) 2 (s)Δs)( |I| |I|

I

.

(2.46)

I

By applying the Hölder inequality with 1/δ > 1 and 1/(1 − δ), we obtain δ

1−δ

1 1 1 ∫ ωδ1 (s)ω1−δ ∫ ω1 (s)Δs) ( ∫ ω2 (s)Δs) 2 (s)Δs ≤ ( |I| |I| |I| I

I

I

Since 1 < p1 < p2 < ∞, 0 < δ < 1 (0 < 1 − δ < 1), we can easily see that p = δp1 + (1 − δ)p2 > δp1 + (1 − δ)p1 = p1 > 1, and, by using the fact that (1 − δ)p2 > (1 − δ), we have p = δp1 + (1 − δ)p2 > δp1 + 1 − δ = δ(p1 − 1) + 1,

.

(2.47)

2 Muckenhoupt and Gehring weights on time scales

� 101

and then (p − 1)/[δ(p1 − 1)] > 1.

(2.48)

From (2.48) and by applying the Hölder inequality with (p − 1)/[δ(p1 − 1)] > 1 and (p − 1)/[(1 − δ)(p2 − 1)], and taking into account that p = δp1 + (1 − δ)p2 , we obtain −δ −(1−δ) 1 ∫ ω1p−1 (s)ω2 p−1 (s)Δs |I|

I

1 p −1 ≤ ( ∫ ω1 1 (s)Δs) |I| −1

δ(p1 −1) p−1

I

1 p −1 ( ∫ ω2 2 (s)Δs) |I| −1

(1−δ)(p2 −1) p−1

(2.49)

.

I

By using (2.47) and (2.49), then (2.46) becomes 𝔸p (ωδ1 ω1−δ 2 ) ≤(

δ

1−δ

1 1 ∫ ω1 (s)Δs) ( ∫ ω2 (s)Δs) |I| |I| I

I

1 p −1 × (( ∫ ω1 1 (s)Δs) |I| −1

δ(p1 −1) p−1

I

= [(

1 p −1 ( ∫ ω2 2 (s)Δs) |I| I

1 1 ∫ ω1 (s)Δs)( ∫ ω1 |I| |I|

−1 p1 −1

I

−1

]

I

p2 −1 1−δ

1 1 p −1 ∫ ω2 (s)Δs)( ∫ ω2 2 (s)Δs) |I| |I| I

δ

1−δ

≤ [𝔸p1 (ω1 )] [𝔸p2 (ω2 )]

p−1

)

p1 −1 δ

(s)Δs) −1

× [(

(1−δ)(p2 −1) p−1

]

I

.

This is the desired result. The proof of (2) is similar to the proof of (1) and hence is omitted. The proof is completed. Theorem 2.21. Assume that 1 < p < ∞ is a positive real number. Then the following properties hold: (1−δ)/p (1) if ω1 , ω2 ∈ 𝔸p , then ωδ/r ∈ 𝔸p for p > 1, 0 < r < 1 with δ = (1 − p1 )/( r1 − p1 ), and 1 ω2 (1−δ)/p [𝔸p (ωδ/r )] ≤ 𝔸p [ω1 ]δ/r 𝔸p [ω2 ](1−δ)/p ; 1 ω2 1

(2) ω ∈ 𝔸p if and only if ω and ω 1−p are in 𝔸∞ .

102 � S. H. Saker Proof. (1) Assume that ω1 , ω2 ∈ 𝔸p , then 𝔸p (ωδ/r 1 ω2

(1−δ)/p

=(

)

p−1

−1 1 1 (1−δ)/p (1−δ)/p (s)ω2 (s)Δs)( ∫[ωδ/r (s)ω2 (s)] p−1 Δs) ∫ ωδ/r 1 1 |I| |I|

I

I

p−1

=(

−δ −(1−δ) 1 1 (1−δ)/p (s)ω2 (s)Δs)( ∫ ω1r(p−1) (s)ω2p(p−1) (s)Δs) ∫ ωδ/r 1 |I| |I|

I

(2.50)

.

I

Note that 0 < δ < 1 and δ/r + (1 − δ)/p = 1, so that, by letting γ = δ/r, we have 1 − γ = (1 − δ)/p, and (2.50) can be written as γ 1−γ

𝔸p (ω1 ω2 ) = (

p−1

−γ −(1−γ) 1 1 γ 1−γ ∫ ω1 (s)ω2 (s)Δs)( ∫ ω1p−1 (s)ω2 p−1 (s)Δs) |I| |I|

I

.

(2.51)

I

By applying the Hölder inequality with exponents 1/γ > 1 and 1/(1 − γ), we obtain γ

(

1−γ

1 1 1 γ 1−γ ∫ ω1 (s)ω2 (s)Δs) ≤ ( ∫ ω1 (s)Δs) ( ∫ ω2 (s)Δs) |I| |I| |I| I

I

(2.52)

I

and γ

1−γ

−γ −(1−γ) −1 −1 1 1 1 ∫ ω1p−1 (s)ω2 p−1 (s)Δs ≤ ( ∫ ω1p−1 (s)Δs) ( ∫ ω2p−1 (s)Δs) |I| |I| |I|

I

I

.

(2.53)

I

By substituting (2.52) and (2.53) into (2.51), we have γ 1−γ

𝔸p (ω1 ω2 ) ≤ (

γ

1−γ

1 1 ∫ ω1 (s)Δs) ( ∫ ω2 (s)Δs) |I| |I| I

× ((

I γ

1−γ p−1

−1 1 1 ∫ ω1 (s)Δs) ( ∫ ω2p−1 (s)Δs) |I| |I| −1 p−1

I

)

I

p−1 γ

= ((

−1 1 1 ∫ ω1 (s)Δs)( ∫ ω1p−1 (s)Δs) |I| |I|

I

× ((

)

I

p−1 1−γ

−1 1 1 ∫ ω2 (s)Δs)( ∫ ω2p−1 (s)Δs) |I| |I|

I γ

1−γ

≤ [𝔸p (ω1 )] [𝔸p (ω2 )]

I

δ/r

= [𝔸p (ω1 )]

)

[𝔸p (ω2 )]

(1−δ)/p

.

(2) Using property (3) in Theorem 2.15, since 𝔸∞ = ⋃1≤p 1, if and only if ω ∈ 𝔸∞ . Now, we have by property (1) in Theorem 2.17 that

2 Muckenhoupt and Gehring weights on time scales

� 103

ω ∈ 𝔸p if and only if ω1−p = ω1/(1−p) ∈ 𝔸p′ . That is, since 𝔸p′ ⊂ 𝔸∞ , ω ∈ 𝔸p if and only if ω1/(1−p) ∈ 𝔸∞ . The proof is complete. ′

2.4 Some fundamental relations In this section, we prove some fundamental relations connecting different Muckenhoupt and Gehring classes. The results are adapted from [49]. Theorem 2.22. Assume that ω is a positive weight and p is a positive real number. Then max{[𝔸∞ (ω)], [A∞ (ω1−p )] ′

p−1

} ≤ [𝔸p (ω)] ≤ [𝔸∞ (ω)][𝔸∞ (ω1−p )] ′

p−1

.

(2.54)

Proof. For p ≤ q, we have [𝔸p (ω)] ≥ [𝔸q (ω)], and thus, (2.55)

[𝔸∞ (ω)] ≤ [𝔸p (ω)]. Furthermore, for q < ∞, we have p−1

[𝔸q (ω1−p )] ′

= sup{( I⊂I0

I

= sup{( I⊂I0

=

q−1 p−1

′ ′ ′ 1 1 ∫ ω1−p (s)Δs)( ∫ ω(1−p )(1−q ) (s)Δs) |I| |I|

I

p−1

1 ∫ ω1−p (s)Δs) |I| ′

}

′ ′ 1 ∫ ω(1−p )(1−q ) (s)Δs) |I|

(q−1)(p−1)

(

I

}

I

( |I|1 ∫I

ω(s)Δs)( |I|1 ∫I ω p−1 (s)Δs)p−1 sup 1 ′ ′ ∫ ω(s)Δs)( |I|1 ∫I ω(1−p )(1−q ) (s)Δs)−(q−1)(p−1) I⊂I0 ( |I| I −1

≤ 𝔸p (ω).

(2.56)

Taking the limit in (2.56) as q tends to ∞, we have p−1

[𝔸∞ (ω1−p )] ′

≤ [𝔸p (ω)].

From (2.55) and (2.57), then max{[𝔸∞ (ω)], [𝔸∞ (ω1−p )] ′

p−1

Now, for the second inequality, we have p−1

′ 1 1 ∫ ω(s)Δs( ∫ ω1−p (s)Δs) |I| |I|

I

I

} ≤ [𝔸p (ω)].

(2.57)

104 � S. H. Saker q−1

=

′ 1 1 ∫ ω(s)Δs( ∫ ω1−q (s)Δs) |I| |I|

I

I

1−q

p−1

p−1 ′ ′ 1 1 × ( ∫ ω1−p (s)Δs( ∫ ω1−q (s)Δs) ) |I| |I|

I

(2.58)

.

I

Since 1 − q and 1 − q′ < 0, by Lemma 2.4 for 1 − q′ < q′ − 1, we have 1−q

p−1 ′ ′ 1 1 ∫ ω1−p (s)Δs( ∫ ω1−q (s)Δs) |I| |I|

I

I

q−1

p−1 ′ ′ 1 1 ≤ ∫ ω1−p (s)Δs( ∫ ωq −1 (s)Δs) |I| |I|

I

I

q−1

p−1 ′ ′ ′ ′ 1 1 = ∫ ω1−p (s)Δs( ∫ ω(1−p )(q −1)/(1−p ) (s)Δs) . |I| |I|

I

(2.59)

I

By setting r − 1 = (q − 1)/(p − 1), we have r′ − 1 =

p − 1 q′ − 1 1 = = . r − 1 q − 1 p′ − 1

Hence, from (2.58) and (2.59), we have p−1

′ 1 1 ∫ ω(s)Δs( ∫ ω1−p (s)Δs) |I| |I|

I

≤

I

q−1

′ 1 1 ∫ ω(s)Δs( ∫ ω1−q (s)Δs) |I| |I|

I

I

′ ′ 1 1 ∫ ω1−p (s)Δs( ∫ ω(1−p )(1−r ) (s)Δs) |I| |I| ′

×[

I

r−1 p−1

]

.

I

Taking supremum over all I ⊂ I0 , we have p−1

[𝔸p (ω)] ≤ [𝔸q (ω)][𝔸r (ω1−p )] ′

(2.60)

.

Now by taking the limit on the both sides of (2.60) as q tends to ∞, we get that p−1

[𝔸p (ω)] ≤ [𝔸∞ (ω)][𝔸∞ (ω1−p )] ′

The proof is complete.

.

2 Muckenhoupt and Gehring weights on time scales

� 105

Theorem 2.23. Assume that p, r > 1. Then, ω ∈ 𝔸p ∩ 𝔾r if and only if ωr ∈ 𝔸q for q = r(p − 1) + 1. Proof. First, assume that ω ∈ 𝔸p ∩ 𝔾r . Then ω ∈ 𝔸p and ω ∈ 𝔾r . In other words, there exists a constant C > 1 such that p−1

(

−1 1 1 ∫ ω(s)Δs)( ∫ ω p−1 (s)Δs) |I| |I|

I

(2.61)

≤ C,

I

and there exists a constant K > 1 such that 1/r

(

1 ∫ ωr (s)Δs) |I|

≤ K(

I

1 ∫ ω(s)Δs). |I|

(2.62)

I

From (2.62), we see that r

1 1 ∫ ωr (s)Δs ≤ K r ( ∫ ω(s)Δs) . |I| |I| I

(2.63)

I

If q = r(p − 1) + 1, then 1/(p − 1) = r/(q − 1), and from (2.61), we have −r 1 1 ( ∫ ω(s)Δs)( ∫ ω q−1 (s)Δs) |I| |I|

I

q−1 r

≤ C.

I

Thus q−1

(

−1 1 ∫(ωr ) q−1 (s)Δs) |I|

≤ C r(

I

1 ∫ ω(s)Δs) . |I| −r

(2.64)

I

From (2.63) and (2.64), we see that q−1

(

−1 1 1 ∫ ωr (s)Δs)( ∫(ωr ) q−1 (s)Δs) |I| |I|

I

I

r

1 1 ≤ C r K r ( ∫ ω(s)Δs) ( ∫ ω(s)Δs) |I| |I| r

r

I

−r

I

=C K , which implies that ωr ∈ 𝔸q . Conversely, if ωr ∈ 𝔸q for q = r(p − 1) + 1, then there exist C1 > 1 such that q−1

(

−1 1 1 ∫ ωr (s)Δs)( ∫(ωr ) q−1 (s)Δs) |I| |I|

I

I

≤ C1 .

106 � S. H. Saker Since q − 1 = r(p − 1), we obtain r(p−1)

(

−1 1 1 ∫ ωr (s)Δs)( ∫ ω p−1 (s)Δs) |I| |I|

I

≤ C1

I

and 1

p−1

r −1 1 1 ( ∫ ωr (s)Δs) ( ∫ ω p−1 (s)Δs) |I| |I|

I

I

1

≤ C1r .

(2.65)

From (2.65), by using Lemma 2.4, we get 1

r 1 1 ( ∫ ωr (s)Δs) ≥ ∫ ω(s)Δs. |I| |I|

I

(2.66)

I

From (2.65) and (2.66), we get p−1

(

−1 1 1 ∫ ω(s)Δs)( ∫ ω p−1 (s)Δs) |I| |I|

I

I

1

≤ C1r ,

(2.67)

which implies that ω ∈ 𝔸p , and, by using Lemma 2.4 with −1/(p − 1) < 0, we find −1

p−1 −1 1 1 ( ∫ ω(s)Δs) ≤ ∫ ω p−1 (s)Δs |I| |I|

I

I

and 1 ∫ ω(s)Δs) |I|

p−1

−1

(

≤(

I

−1 1 ∫ ω p−1 (s)Δs) |I|

.

I

From this and (2.65), we get that 1

r 1 1 1 ( ∫ ω(s)Δs) ( ∫ ωr (s)Δs) ≤ C r , |I| |I|

−1

I

I

or equivalently, 1

r 1 1 1 ( ∫ ωr (s)Δs) ≤ C r ( ∫ ω(s)Δs), |I| |I|

I

(2.68)

I

which implies that ω ∈ 𝔾r . From (2.67) and (2.68), we get ω ∈ 𝔸p ∩ 𝔾r . The proof is complete.

2 Muckenhoupt and Gehring weights on time scales

� 107

Theorem 2.24. Assume that p is a positive real number. Then the following properties hold: (1) If 1 < r < ∞, then [𝔸∞ (ωr )]1/r 1/r ≤ [𝔾r (ω)] ≤ [𝔸∞ (ωr )] ; [𝔸∞ (ω)] (2) If ω ∈ ⋂p>1 𝔸p , then 1/ω ∈ ⋂r1 𝔸p , then p−1

(

−1 1 ∫ ω p−1 (s)Δs) |I|

1 ∫ ω(s)Δs) |I|

−1

≤ C(

I

(2.71)

I

holds for all p > 1. From (2.71), by using Lemma 2.4, we get that p−1

(

1 1 1 ) p−1 Δs) ∫( |I| ω(s)

≤ C(

I

1 1 Δs). ∫ |I| ω(s) I

Hence, (1/ω) ∈ 𝔾r for all 0 < r = 1/(p − 1) < ∞, we have 1/ω ∈ ⋂r 0. In addition, by Lemma 2.9, we see that if ω is nonincreasing, then so is A ωq for q > 0.

Now, we state the Hardy inequality in a finite interval (see [2, Corollary 1.5.1]). Theorem 2.25. If q > 1 and ω is nonnegative and nonincreasing, then σ q

A [(A ω) ] ≤ (

q

q ) A ωq . q−1

(2.73)

We assume that there exists a constant λ ≥ 1 such that σ(t) ≤ λt,

for all t ∈ 𝕋.

(2.74)

110 � S. H. Saker We now apply the time scales chain rule to obtain some estimates that will be used later. Lemma 2.10. Let x(t) = t. If 0 < γ < 1, then 1−γ , σγ

(2.75)

(1 − γ)λγ . σγ

(2.76)

Δ

(x 1−γ ) ≥ and if γ > 1 and (2.74) holds, then Δ

(x 1−γ ) ≥ Proof. By the chain rule, we obtain 1

Δ

(x 1−γ ) (t) = (1 − γ)x Δ (t) ∫ 0

1

= (1 − γ) ∫ 0

dh (hx(σ(t)) + (1 − h)x(t))γ

dh . (hσ(t) + (1 − h)t)γ

Thus, if 0 < γ < 1, then 1

Δ

(x 1−γ ) (t) ≥ (1 − γ) ∫ 0

1−γ dh = , (hσ(t) + (1 − h)σ(t))γ (σ(t))γ

which is (2.75), and if γ > 1 and (2.74) holds, then 1

Δ

(x 1−γ ) (t) ≥ (1 − γ) ∫ 0

1 − γ (1 − γ)λγ dh = ≥ , (ht + (1 − h)t)γ tγ (σ(t))γ

which is (2.76). Lemma 2.11. If ω is nonnegative and nondecreasing and γ > 1, then Δ

(W γ ) ≥ γW Δ W γ−1 .

(2.77)

Proof. Again we apply the chain rule to see that 1

Δ

γ−1

(W γ ) = γW Δ ∫(hW σ + (1 − h)W )

dh

0 Δ

1

γ−1

≥ γW ∫(hW + (1 − h)W ) 0

which shows (2.77).

dh = γW Δ W γ−1 ,

2 Muckenhoupt and Gehring weights on time scales

� 111

Theorem 2.26. Assume that ω is a nonnegative and nonincreasing and p > 1. Then, for any q ∈ (0, p), we have (p − q)λp/q q p/q q σ p/q [A ωq ] + A [(A ω ) ] . p p

p

Aω ≤

(2.78)

Proof. From the Hardy inequality (see (2.73)), we see that the second integral on the right-hand side of (2.78) is finite. Now, we consider this integral. Then, for 0 < q < p, we put t

p γ= >1 q

and W (t) = ∫ ωq (s)Δs. 0

Using the notation from Lemma 2.10, we have t

σ(s)

0

0

p/q

(p − q)λp/q 1 ∫[ ∫ ωq (τ)Δτ] pt σ(s) =

t

σ(s)

0

0

Δs γ

(γ − 1)λγ 1 ∫[ ∫ ωq (τ)Δτ] Δs γt σ(s) t

≥−

1 Δ ∫ W γ (σ(s))(x 1−γ ) (s)Δs γt 0

t

1 W γ (s)x 1−γ (s) W γ (t)x 1−γ (t) Δ − + ∫(W γ ) (s)x 1−γ (s)Δs = lim+ γt γt γt s→0 0

t

=

γ

0

t

γ

1 γW Δ (s)W γ−1 (s) 1 W (t) ≥ Δs − ( ) ∫ γ−1 γt γ t s 0

t

=

1 1 γ−1 γ ∫ ωq (s)[A ωq (s)] Δs − [A ωq (t)] t γ 0

t

≥

γ

1 1 1 W (t) W (s) Δ lim [s( ) ]− ( ) ∫ s1−γ (W γ ) (s)Δs + γt γt s→0+ s γ t

1 1 γ−1 γ ∫ ωq (s)[ωq (s)] Δs − [A ωq (t)] t γ 0

t

q 1 1 γ γ p/q = ∫[ωq (s)] Δs − [A ωq (t)] = A ωp (t) − [A ωq (t)] , t γ p 0

112 � S. H. Saker from which (2.78) follows. Now, we are ready to state and prove our first time scales version of the Gehring self-improving property for monotone functions. Theorem 2.27 (Gehring inequality I). Assume that ω is a nonnegative and nonincreasing weight and q > 1 such that q

q

A ω ≤ κ[A ω]

for some κ > 0,

(2.79)

then p

p

A ω ≤ κ[̃ A ω] ,

(2.80)

where p > q and qκp/q p p > 0. p − (p − q)(λκ)p/q ( p−1 )

κ̃ :=

(2.81)

Proof. Assuming (2.79), we find t

1 ∫ ωp (s)Δs t 0

p/q

t

≤

q 1 [ ∫ ωq (s)Δs] p t 0

p

t

p/q

σ(s)

t

+

(p − q)λp/q 1 ∫[ ∫ ωq (τ)Δτ] pt σ(s) 0

0

t

σ(s)

Δs p

(p − q)λp/q q 1 1 ≤ κp/q [ ∫ ω(s)Δs] + ∫ κp/q [ ∫ ω(τ)Δτ] Δs p t pt σ(s) 0

≤

0

p

t

0

p t

p/q

q p/q 1 (p − q)(λκ) κ [ ∫ ω(s)Δs] + p t pt 0

(

p ) ∫ ωp (s)Δs p−1 0

so that, due to (2.81), t

t

0

0

p

1 1 ̃ ∫ ω(s)Δs] , ∫ ωp (s)Δs ≤ κ[ t t from which (2.80) follows. It is natural to ask what happens if in (2.21) we fix p > 1 and consider the improvement to this inequality that would result from lowering the exponent on the right-hand side. The following result gives an answer.

2 Muckenhoupt and Gehring weights on time scales

� 113

Theorem 2.28. Suppose that the assumptions of Theorem 2.27 hold and define κ̃ as in (2.81). Then, for all 0 < r < 1, we have r p/r

p

A ω ≤ κ[A ω ]

where κ := κ̃ 1/θ with θ :=

,

1− 1 r

−

1 p 1 p

(2.82)

.

Proof. Note first that θ ∈ (0, 1) and 1−θ θ + = 1. p r Then, by the Hölder inequality with exponents p/(1 − θ) and r/θ, we have 1/p

t

1 [ ∫ ωp (s)Δs] t 0

t

κ̃ 1/p ≤ ∫ ω(s)Δs t 0

t

κ̃ 1/p = ∫ ω1−θ (s)ωθ (s)Δs t 0

≤

t

(1−θ)/p t

κ̃ 1/p [∫ ωp (s)Δs] t 0

= κ̃

1/p

[∫ ωr (s)Δs)θ/r

0 (1−θ)/p

t

1 [ ∫ ωp (s)Δs] t 0

θ/r

t

1 [ ∫ ωr (s)Δs] t 0

so that, by dividing, we find θ/p

t

1 [ ∫ ωp (s)Δs] t 0

t

θ/r

1 ≤ κ̃ 1/p [ ∫ ωr (s)Δs] t

,

0

i. e., (2.82) is true. p

We say that ω : [0, ∞)𝕋 → ℝ belongs to 𝕃Δ ([0, ∞)𝕋 ) provided ∞

1 p

󵄨 󵄨p ( ∫ 󵄨󵄨󵄨ω(t)󵄨󵄨󵄨 Δt) < ∞. 0

By Theorem 2.28, under the assumptions of Theorem 2.27, if ω ∈ 𝕃rΔ [0, ∞)𝕋 for 0 < r < 1, p p then ω ∈ 𝕃Δ [0, ∞)𝕋 for p > 1. But in the general case when p ≠ r, 𝕃Δ [0, ∞)𝕋 neither includes nor is included in 𝕃rΔ [0, ∞)𝕋 . The following theorem gives some results for p 𝕃Δ [0, ∞)𝕋 -interpolation.

114 � S. H. Saker Theorem 2.29. Suppose that 0 < p0 < p1 < ∞ and that 0 < θ < 1. p p p (i) If p = (1 − θ)p0 + θp1 and ω ∈ 𝕃Δ0 [0, ∞)𝕋 ∩ 𝕃Δ1 [0, ∞)𝕋 , then ω ∈ 𝕃Δ [0, ∞)𝕋 and p

p

1−θ

A ω ≤ [A ω 0 ]

(ii) If p =

p

1

θ

[A ωp1 ] .

p

p

and ω ∈ 𝕃Δ0 [0, ∞)𝕋 ∩ 𝕃Δ1 [0, ∞)𝕋 , then ω ∈ 𝕃Δ [0, ∞)𝕋 and

1−θ + pθ p0 1

p

p

A ω ≤ [A ω 0 ]

(1−θ)p/p0

θp/p1

[A ωp1 ]

.

Proof. For (i), we apply the Hölder inequality with exponents 1/(1−θ) and 1/θ to see that t

t

0

0

1 1 ∫ ωp (s)Δs = ∫ ω(1−θ)p0 (s)ωθp1 (s)Δs t t 1−θ

t

1 ≤ [ ∫ ωp0 (s)Δs] t 0

θ

t

1 [ ∫ ωp1 (s)Δs] , t 0

which shows (i). For (ii), we apply the Hölder inequality with exponents 1/(1 − γ) and 1/γ, where θp p1

γ :=

so that 1 − γ =

(1 − θ)p , p0

to see that t

t

0

0

1 1 ∫ ωp (s)Δs = ∫ ω(1−θ)p (s)ωθp (s)Δs t t 1−γ

t

1 ≤ [ ∫ ω(1−θ)p/(1−γ) (s)Δs] t 0

t

(1−θ)p/p0

1 = [ ∫ ωp0 (s)Δs] t 0

γ

t

1 [ ∫ ωθp/γ (s)Δs] t t

0

θp/p1

1 [ ∫ ωp1 (s)Δs] t

,

0

which shows (ii). In the following, we give a new proof of the Gehring mean inequality on time scales proving that if the weight belongs to the 𝔸1 -Muckenhoupt class, then it will be a member of a Gehring class. The inequality will be proved by using a condition similar to that for the class A1 of Muckenhoupt. In fact, we do not assume that the reverse Hölder inequality holds.

2 Muckenhoupt and Gehring weights on time scales

� 115

Theorem 2.30 (Gehring inequality II). Assume (2.74). If ω is nonnegative and nonincreasing such that σ

A ω ≤ νω

for some ν > 1,

(2.83)

then p σ

σ p

A (ω ) ≤ ν[̃ A ω ]

and ν̃ :=

α > 0, α − p(α − 1)

(2.84)

for p ∈ [1, α/(α − 1)), where α = λν. Proof. For this proof, we put t

W (t) = ∫ ωσ (s)Δs,

l(t) = log(t),

L(t) = log(ω(t)).

0

By the chain rule, we get 1

1 dh 1 Δ l (t) = ∫ α λν hσ(t) + (1 − h)t 0

1

1 dh λ 1 1 ≤ ⋅ = ≤ ∫ λν hσ(t) + (1 − h) σ(t) λν σ(t) νσ(t) λ 0

1

≤

ω(σ(t)) W Δ (t) dh = = W Δ (t) ∫ W (σ(t)) W (σ(t)) hW (σ(t)) + (1 − h)W (σ(t)) 1

0

≤ W Δ (t) ∫ 0

dh = LΔ (t), hW (σ(t)) + (1 − h)W (t)

and hence, by integrating, log(

1/α

t ) σ(s)

=

1 1 W (t) l(t) − l(σ(s)) ≤ L(t) − L(σ(s)) = log( ) α α W (σ(s))

so that σ(s)

ω(σ(s)) ≤

1/α

1 W (σ(s)) σ(s) ≤( ) ∫ ω(σ(τ))Δτ = σ(s) σ(s) t 0

W (t) , σ(s)

and, by integrating again, putting γ = p(1 − 1/α) ∈ (0, 1), and using the notation from Lemma 2.10, we obtain

116 � S. H. Saker t

t

0

0

1 W p (t) Δs ∫ ωp (σ(s))Δs ≤ 1+p/α ∫ t t (σ(s))p(1−1/α) ≤

t

W p (t) Δ ∫(x 1−γ ) (s)Δs (1 − γ)t 1+p/α 0

p

t 1−γ W p (t) 1 W (t) = = ( ) , 1+p/α 1 − γ t (1 − γ)t proving (2.84). q

In the following, we consider the class Up (B) of all nonnegative weights ω that satisfy the reverse Hölder inequality 1 q

σ(x)

1 p

σ(x)

1 1 [ ∫ ωq (t)Δt] ≤ B[ ∫ ωp (t)Δt] , σ(x) − a σ(x) − a a

(2.85)

a

q

q

for 1 ≤ p < q where the constant B > 1. We say that ω is a Up -weight if its Up -norm is finite, i. e., ω ∈ Upq ⇐⇒ [Upq (ω)] < +∞. q

When we fix a constant C > 1, the triple of real numbers (p, q, C ) defines the Up class: ω ∈ Upq (C ) ⇐⇒ [Upq (ω)] ≤ C , q

and we will refer to C as the Up -constant of the class. It is immediate to observe that q the classes 𝔸p and 𝔾q are special cases of the class Up of weights as follows: 𝔸p := U 11

1−p

q

𝔾q := U1 .

and

In order to establish the main results, we need a new version of the refinement of Hardy inequality on a time scale. To prove this inequality, we will use the following elementary inequality: (ϑ + υ)p ≥ ϑ p + pϑ p−1 υ,

where p > 1 or p < 0.

(2.86)

Recall that this relation is a variant of the well-known Bernoulli inequality and it is valid for all ϑ ≥ 0 and ϑ + v ≥ 0, if p > 1, or for ϑ > 0 and ϑ + v > 0, if p < 0. The equality in (2.86) holds if and only if v = 0. We will assume that the forward jump operator is uniformly bounded from above by a linear function. More precisely, we suppose that there exists a real number m ≥ 1 such that σ(t) − a ≤ m(t − a),

for t > a.

(2.87)

2 Muckenhoupt and Gehring weights on time scales

� 117

It should be noticed here that the condition σ(t) ≤ λt may be removed if the graininess weight μ(t) on the time scale 𝕋 satisfies the relation μ(t) = O(t). Indeed, if μ(t) = O(t), then there exists λ > 1 such that 0 < μ(t)/t ≤ λ − 1, for all t ∈ 𝕋. Hence, 1 ≤ (t + μ(t))/t ≤ λ and therefore, 1 ≤ σ(t)/t ≤ λ, for all t ∈ 𝕋. Note also that if 𝕋 = ℝ, then σ(t) = t, while for 𝕋 = ℕ, we have σ(t) = t + 1. In the following, we prove a time scale version of an integral inequality due to Hardy, Littlewood, and Pólya [23] and the new refinement of Hardy-type inequality on time scales that will play important roles in the proof of the main results of higher integrability. Theorem 2.31. Assume that ϕ : [0, ∞) → ℝ is a differentiable convex function. If ω is a nonnegative decreasing function, then the inequality b

b

ϕ(0) + ∫ ϕ ((x − a)ω(x))ω(x)Δx ≤ ϕ(∫ ω(x)Δx) ′

a

(2.88)

a

holds. Proof. Let x ∈ [a, b]𝕋 . Since ω is a decreasing function, it follows that (x − a)ω(x) ≤ x ∫a ω(t)Δt. On the other hand, by defining x

W (x) = ∫ ω(t)Δt, a

we see that ωΔ (x) = ω(x) ≥ 0, which implies that ω is an increasing function. Now, due to convexity of the weight ϕ, we have x

ϕ ((x − a)ω(x))ω(x) ≤ ϕ (∫ ω(t)Δt)ω(x) = ϕ′ (W (x))ω(x). ′

′

(2.89)

a

Further, taking into account the chain rule, it follows that ϕΔ (W (x)) = ϕ′ (W (ζ ))W Δ (x), where ζ ∈ [x, σ(x)], and consequently, ϕΔ (W (x)) ≥ ϕ′ (W (x))W Δ (x) = ϕ′ (W (x))ω(x),

(2.90)

since ω is an increasing function. Now, considering the relations (2.89) and (2.90), we obtain the inequality ϕ′ ((x − a)ω(x))ω(x) ≤ ϕΔ (W (x)).

118 � S. H. Saker Finally, integrating the latter inequality from a to b yields the relation b

b

Δ

∫ ϕ′ ((x − a)ω(x))ω(x)Δx ≤ ∫(ϕ(W (x))) Δx = ϕ(W (b)) − ϕ(0), a

a

which proves our assertion. The proof is complete. In the sequel, we consider a special case of Theorem 2.31, when the weight ϕ : [0, ∞) → ℝ+ is defined by ϕ(ϑ) = ϑ p , for p ≥ 1. Clearly, this weight is differentiable and convex, so we have the following consequence. Corollary 2.2. Assume that ω is a nonnegative decreasing function. If p ≥ 1, then the inequality b

∫(x − a) a

p

b

1 ω (x)Δx ≤ (∫ ω(x)Δx) p

p−1 p

(2.91)

a

holds. Remark 2.3. Applying the Hölder inequality with exponents 1/p and (p − 1)/p to the right-hand side of (2.91), we obtain the following inequality: b

∫(x − a)p−1 ωp (x)Δx ≤ a

b

(b − a)p−1 ∫ ωp (x)Δx. p a

Our next intention is to rewrite the inequality (2.91) in a form which will be more suitable in our further discussion. First of all, it should be noticed here that if a nonnegative weight ω is a decreasing function, then the weight (x − a)γ−1 ω, where γ ≤ 1, is also decreasing. Therefore, considering the relation (2.91) with (x − a)γ−1 ω instead of ω, we obtain the following result. Corollary 2.3. Assume that ω is a decreasing function. If p ≥ 1 and γ ≤ 1, then the inequality b

∫(x − a)pγ−1 ωp (x)Δx ≤ a

b

1 (∫(x − a)γ−1 ω(x)Δx) p

p

(2.92)

a

holds. Remark 2.4. Let r and s be positive real numbers such that r ≤ s. Considering the relation (2.92) with the weight ωr instead of ω, and with parameters p = s/r ≥ 1, γ = 1/p ≤ 1, we see that the inequality

2 Muckenhoupt and Gehring weights on time scales

b

r s

b

r r (∫ ω (x)Δx) ≤ ∫(x − a) s −1 ωr (x)Δx s

s

a

� 119

(2.93)

a

holds. The above inequality will be an important relation that will be used later in the proof of our main higher integrability theorem. On the other hand, it is important in its own right, since it is the time scale version of an inequality due to Hardy, Littlewood, and Pólya (for more details, see [23]). Our next intention is to give a time scale extension and refinement of the famous Hardy inequality (2.73). In order to summarize our further discussion, we first define an operator H by x

H (x) :=

1 ∫ ω(t)Δt, x−a a

for all x ∈ [a, ∞)𝕋 ,

(2.94)

where ω : [a, ∞)𝕋 → ℝ is a nonnegative function. Next, we give several simple facts about the operator H following from its definition. Lemma 2.12. If ω is a nonnegative and decreasing function, then H (x) ≥ ω(x) for x ∈ [a, ∞)𝕋 . Proof. Since ω is decreasing, it follows that x

H (x) =

x

1 1 ∫ ω(t)Δt ≥ ∫ ω(x)Δt = ω(x), x−a x−a a

a

x ∈ [a, ∞)𝕋 ,

which completes the proof. It should be also noticed here that H inherits the decreasing nature of the weight ω. Lemma 2.13. If ω is a nonnegative decreasing function, then so is H . Proof. By utilizing the quotient rule, it follows that Δ

H (x) =

x

ω(x)(x − a) − ∫a ω(t)Δt (x − a)(σ(x) − a)

,

x ∈ [a, ∞)𝕋 .

Hence, by virtue of Lemma 2.12, we have (σ(x) − a)H Δ (x) = ω(x) − H (x) ≤ 0, for x ∈ [a, ∞)𝕋 , which proves our assertion. Now, we are ready to state and prove a refinement of the Hardy inequality on time scales. Theorem 2.32. Let 𝕋 be a time scale with a, b ∈ 𝕋, and let ω be a nonnegative decreasing function. Further, assume that (2.87) holds. If α ≤ 1 and β > 1, then the inequality

120 � S. H. Saker b

β

∫(x − a)α−1 (H σ (x)) Δx + ( a

β )(b − a)α H β (b) β−α

β b

≤(

β ) ∫(x − a)α−1 ωβ (x)Δx β−α

(2.95)

a

holds, where H is defined in (2.94). Proof. Taking into account the chain rule and utilizing the fact that H is decreasing on [a, b]𝕋 by Lemma 2.13, we obtain the inequality 1

Δ

(H β (x)) = β ∫[hH σ (x) + (1 − h)H (x)]

β−1

dhH Δ (x)

0

1

≤ β ∫[hH σ (x) + (1 − h)H σ (x)] 0

β−1

= βH Δ (x)(H σ (x))

β−1

dhH Δ (x)

.

x

On the other hand, since (x − a)H (x) = ∫a ω(t)Δt, the product rule implies the equality (x − a)H Δ (x) + H σ (x) = ω(x), and then we have (x − a)α H Δ (x) = (x − a)α−1 [ω(x) − H σ (x)]. Further, applying integration by parts formula with x

ϑ(x) = ∫(t − a)α−1 Δt

υ(x) = H β (x),

and

a

it follows that b

β

∫(x − a)α−1 (H σ (x)) Δx a

b

Δ

= ϑ(b)H β (b) − lim+ ϑ(x)H β (x) − ∫ ϑ(x)(H β (x)) Δx x→a

a

b

≥

(b − a)α H β (b) β β−1 − ∫(x − a)α H Δ (x)(H σ (x)) Δx α α − lim+ ϑ(x)H β (x) x→a

a

(2.96)

2 Muckenhoupt and Gehring weights on time scales

� 121

b

(b − a)α H β (b) β β−1 − ∫(x − a)α−1 [ω(x) − H σ (x)](H σ (x)) Δx = α α a

b

=

(b − a)α H β (b) β β−1 − ∫(x − a)α−1 ω(x)(H σ (x)) Δx α α a

b

+

β β ∫(x − a)α−1 (H σ (x)) Δx − lim+ ϑ(x)H β (x). α x→a a

Now, our intention is to show that limx→a+ ϑ(x)H β (x) = 0. Taking into account the definitions of the functions ϑ(x) and H β (x), we have x

β

α−1

ϑ(x)H (x) = ∫(t − a) a

x

1 Δt( ∫ ω(t)Δt) x−a a

β x

β

x

1 =( ) ∫(t − a)α−1 Δt(∫ ω(t)Δt) x−a a

≤ ωβ (a)(

β x

a

β

x

1 ) ∫(t − a)α−1 Δt(∫ Δt) x−a a

a

x

β

= ωβ (a) ∫(t − a)α−1 Δt a x

≤ ωβ (a) ∫ a

≤m

1−α

1−α

1 σ(t) − a ( ) t−a (σ(t) − a)1−α

β

x

ω (a) ∫ a

Δt

1 Δt. (σ(t) − a)1−α

(2.97)

Since α ≤ 1, another application of the chain rule yields the estimate 1

α Δ

α−1

((t − a) ) = α ∫[h(σ(t) − a) + (1 − h)(t − a)]

dh

0

1

α−1

≥ α ∫[h(σ(t) − a) + (1 − h)(σ(t) − a)] 0

= α(σ(t) − a) which implies the inequality

α−1

,

dh

122 � S. H. Saker x

α−1

∫(σ(t) − a) a

x

Δt ≤ ∫ a

((t − a)α )Δ (x − a)α Δt = . α α

Thus, utilizing the above inequality and (2.97), we obtain the estimate ϑ(x)H β (x) ≤ m1−α ωβ (a)

(x − a)α , α

and consequently, lim ϑ(x)H β (x) = 0.

x→a+

Clearly, from the above discussion, we obtain the inequality b

β−α (b − a)α H β (b) β ) ∫(x − a)α−1 (H σ (x)) Δx + ( α α a

b

≤

β β−1 ∫(x − a)α−1 ω(x)(H σ (x)) Δx. α a

Now, applying the Hölder inequality with exponents 1/β and (β − 1)/β to the right-hand side of the latter inequality yields the relation b

β−α (b − a)α H β (b) β ( ) ∫(x − a)α−1 (H σ (x)) Δx + α α a

1 β

b

b

β β ≤ {∫(x − a)α−1 ωβ (x)Δx} {∫(x − a)α−1 (H σ (x)) Δx} α a

β−1 β

a

,

which can be rewritten in the following form: β b

(

β ) ∫(x − a)α−1 ωβ (x)Δx β−α a

b

α−1

≥ [{∫(x − a) a

σ

β

1 β

(H (x)) Δx} +

b

{∫a (x − a)α−1 (H σ (x))β Δx}

Finally, applying the Bernoulli inequality (2.86), with b

β

ϑ = {∫(x − a)α−1 (H σ (x)) Δx} a

β

(b−a)α H β (b) (β−α)

1 β

and

β−1 β

] .

2 Muckenhoupt and Gehring weights on time scales

υ=

(b−a)α H β (b) (β−α) b

{∫a (x − a)α−1 (H σ (x))β Δx}

β−1 β

� 123

,

to the right-hand side of the latter inequality, we obtain β b

β ( ) ∫(x − a)α−1 ωβ (x)Δx β−α a

b

β

≥ ∫(x − a)α−1 (H σ (x)) Δx + a

β (b − a)α H β (b), β−α

which represents the desired inequality (2.95). The proof is now complete. The established inequality (2.95) is both a refinement and a time scale extension of the Hardy inequality (2.73) for a class of decreasing functions. To see this, let α = q/p and β = q, where p ≥ q > 1. In this setting, Theorem 2.32 reduces to the following corollary. Corollary 2.4. Let 𝕋 be a time scale with a, b ∈ 𝕋, let ω be a nonnegative decreasing function. Suppose that (2.87) holds. If p ≥ q > 1, then the inequality b

q

q

∫(x − a) p (H σ (x)) Δx + −1

a

q p (b − a) p H q (b) p−1

q b

≤(

q p −1 ) ∫(x − a) p ωq (x)Δx p−1

(2.98)

a

holds, where H is defined in (2.94). x

Clearly, if 𝕋 = ℝ, a = 0 and p = q, then σ(x) = x and H (x) = (1/x) ∫0 ω(t)dt. So the inequality (2.98) provides a refinement of the classical Hardy inequality (2.73) for a class of decreasing functions. In the next few remarks, we will compare our Theorem 2.32 with some results known from the literature. Remark 2.5. If α = 1, Theorem 2.32 provides the following refinement of the Hardy-type inequality: b

β b

β(b − a) β β H (b) ≤ ( ) ∫ ωβ (x)Δx. ∫(H (x)) Δx + β−1 β−1 a

σ

β

a

Remark 2.6. If 𝕋 = ℝ, our inequality (2.95) can be regarded as a refinement of the inequality

124 � S. H. Saker b

α−1

∫(x − a)

β b

β ) ∫(x − a)α−1 ωβ (x)dx, H (x)dx ≤ ( β−α β

a

a

established by Popoli [40, Theorem 2.1], for α ≤ 1 and β > 1. Clearly, H is defined here x by H (x) := (1/(x − a)) ∫a ω(t)dt. In the sequel, we first state and prove several lemmas, interesting in their own right, which will be utilized in establishing the higher integrability theorem. Lemma 2.14. Assume that φ, ψ are nonnegative functions. Then b

b

σ(t)

b

∫ φ(t)(∫ ψ(x)Δx)Δt = ∫ ψ(t)( ∫ φ(x)Δx)Δt. a

a

t

a

b

Proof. Define Ψ(t) = ∫t ψ(x)Δx. Applying the integration by parts formula to the term b

∫a φ(t)Ψ(t)Δt with ϑ(t) = Ψ(t) and υΔ (t) = φ(t), we get b

b

b

b

a

t

a

a

∫ φ(t)(∫ ψ(x)Δx)Δt = ∫ φ(t)Ψ(t)Δt = Ψ(t)υ(t)|ba − ∫ ΨΔ (t)υσ (t)Δt, t

where υ(t) = ∫a φ(x)Δx. Now, since υ(a) = 0 and Ψ(b) = 0, it follows that b

b

b

a

t

a

Δ

b

σ

∫ φ(t)(∫ ψ(x)Δx)Δt = ∫[−Ψ (t)]υ (t)Δt = ∫ ψ(t)υσ (t)Δt a

σ(t)

b

= ∫ ψ(t)( ∫ φ(x)Δx)Δt, a

a

which completes the proof. Lemma 2.15. Assume that w is a nonnegative increasing function. Further, suppose that there exists m ≥ 1 such that wσ (x) ≤ mw(x)

for all x ∈ [a, b]𝕋 .

(2.99)

If λ < 0, then the inequality b

∫ wλ (x)wΔ (x)H σ (x)Δx ≥ a

x

b

b

a

a

1 [wλ (b) ∫ φ(x)Δx − ∫ wλ (x)φ(x)Δx], λm

holds, where H (x) := (1/w(x)) ∫a φ(t)Δt.

(2.100)

2 Muckenhoupt and Gehring weights on time scales

� 125

Proof. Taking into account the definition of H , assumption (2.99), and Lemma 2.14, we obtain b

λ

Δ

b

σ

Δ

∫ w (x)w (x)H (x)Δx = ∫ w (x)w a

λ−1

a

a

σ(x)

b

≥

=

σ(x)

w (x)( σ ∫ φ(t)Δt)Δx w

1 ∫ wΔ (x)wλ−1 (x)( ∫ φ(t)Δt)Δx m a

a

b

b

a

x

1 ∫ φ(x)(∫ wΔ (t)wλ−1 (t)Δt)Δx. m

Further, since λ < 0 and wΔ (x) > 0, by applying the chain role, it follows that λ

1

Δ

λ−1

(w (t)) = λ ∫[hwσ (t) + (1 − h)w(t)]

dhwΔ (t)

0

1

λ−1

≥ λ ∫[hw(t) + (1 − h)w(t)]

dhwΔ (t) = λwΔ (t)wλ−1 (t),

0

which implies the estimate wΔ (t)wλ−1 (t) ≥ λ1 (wλ (t))Δ . Hence, we obtain b

b

b

1 Δ ∫ φ(x)(∫(wλ (t)) Δt)Δx ∫ w (x)w (x)H (x)Δx ≥ λm λ

Δ

σ

x

a

a

=

b

b

a

a

1 [wλ (b) ∫ φ(x)Δx − ∫ wλ (x)φ(x)Δt], λm

which proves our assertion. The proof is complete. We will also employ the following lemma which has been proved in [40, Lemma 2.2]. Lemma 2.16. Let C > 1, q > p > 0, and let L be defined by q

p q L(p, q, x, C) = 1 − C (1 − x)( ) , q − px

q

(2.101)

where x ∈ [0, 1]. Then, there exists a unique solution xq of the equation L(p, q, x, C) = 0. In addition, L(p, q, x, C) > 0 if and only if x ∈ (xq , 1]. Before we state and prove our main theorem, we first need the following auxiliary result.

126 � S. H. Saker Theorem 2.33. Let 0 < p < q, K > 1. Further, assume that ω is a nonnegative decreasing weight satisfying (2.85). If q

p q ) L (p, q, α, K) = 1 − K m(1 − α)( q − pα

q

(2.102)

and the condition (2.87) holds, then the inequality b

∫(x − a)α−1 ωq (x)Δx ≤ a

b

(b − a)α−1 ∫ ωq (x)Δx L (p, q, α, K)

(2.103)

a

holds for all α ∈ (αq , 1], where αq is the unique root of equation (2.101) with C = Km1/q . Proof. Without loss of generality, we can suppose that α ∈ (αq , 1) since for α = 1 the inequality (2.103) holds trivially. Now, since ω : [a, b]𝕋 → ℝ satisfies the reverse Hölder inequality, it follows that q

Wqσ (x) ≤ K q [Wpσ (x)] p , x

x

where Wp (x) = (1/(x − a)) ∫a ωp (t)Δt and Wq (x) = (1/(x − a)) ∫a ωq (t)Δt. By integrating, it follows that b

α−1

∫(x − a) a

Wqσ (x)Δx

b

q

≤ K ∫(x − a)α−1 (Wpσ (x)) p Δx. q

(2.104)

a

Further, it should be noticed that if w(x) = x − a, then the condition (2.99) reduces to (2.87). Therefore, applying Lemma 2.15 with λ = α − 1 < 0, w(x) = x − a and φ(x) = ωq (x) to the left-hand side of (2.104), it follows that b

∫(x − a)α−1 Wqσ (x)Δx a

b

b

a

a

1 ≥ [(b − a)α−1 ∫ ωq (x)Δx − ∫(x − a)α−1 ωq (x)Δx]. (α − 1)m On the other hand, applying Theorem 2.32 with β = q/p > 1 and ω = ωp to the right-hand side of (2.104), we have b

∫(x − a)

α−1

a

q

(Wpσ (x)) p Δx

q

b

p q ≤( ) ∫(x − a)α−1 ωq (x)Δx q − pα

a

b

q q α− −( )(b − a) p (∫ ωp (x)Δx) q − pα

a

q p

2 Muckenhoupt and Gehring weights on time scales

q

� 127

b

p q ) ∫(x − a)α−1 ωq (x)Δx. ≤( q − pα

a

Now, taking into account (2.104) and the previous two estimates, we obtain the inequality b

b

a

a

1 [(b − a)α−1 ∫ ωq (x)Δx − ∫(x − a)α−1 ωq (x)Δx] (α − 1)m ≤ K q(

b

q p

q ) ∫(x − a)α−1 ωq (x)Δx, q − pα a

which can be rewritten as b

(b − a)α−1 ∫ ωq (x)Δx a

q

b

p q ≥ [1 − K (1 − α)m( ) ] ∫(x − a)α−1 ωq (x)Δx q − pα

q

a

b

= L (p, q, α, K) ∫(x − a)α−1 ωq (x)Δx. a

Finally, by Lemma 2.16, there exists a unique αq ∈ (0, 1) such that L (p, q, α, K) > 0 for α ∈ (αq , 1]. This provides the inequality (2.103), and the proof is complete. Remark 2.7. If p = 1 and α = qr , provided that r ≥ q, the inequality (2.103) reduces to b

q

∫(x − a) r −1 ωq (x)Δx ≤ a

q

b

(b − a) r −1 ∫ ωq (x)Δx, L (1, q, q/r, K) a

where L (1, q, q/r, K) is defined by (2.102). In particular, if 𝕋 = ℝ this inequality provides a relation established by D’Apuzzo and Sbordone [18, Lemma 3.2]. Finally, we are able to state and prove the higher integrability theorem for decreasing functions on time scales. Theorem 2.34. Let 0 < p < q, K > 1. Further, suppose that ω is a nonnegative decreasing weight satisfying (2.85). If the condition (2.87) holds, then b

q s

b

q 1 1 ( ( ∫ ωs (x)Δx) ≤ ∫ ωq (x)Δx), q b−a sL (p, q, s , K) b − a a

a

for q ≤ s < q0 , where q0 is the unique solution of the equation

(2.105)

128 � S. H. Saker 1

(

1

q p 1 x x ) = Km q ( ) . x−q x−p

(2.106)

Proof. Utilizing Theorem 2.33 with α = q/s and αq = q/q0 , we obtain the inequality b

∫(x − a)

q −1 s

a

b

q

(b − a) s −1 ω (x)Δx ≤ ∫ ωq (x)Δx, q L (p, q, s , K) q

a

where q ≤ s < q0 . In addition, applying the inequality (2.93) with r = q to the left-hand side of the previous inequality, we have q s

b

(∫ ωs (x)Δx) ≤ a

b

q

q(b − a) s −1 ∫ ωq (x)Δx, sL (p, q, qs , K) a

for q ≤ s < q0 ,

and thus, b

(

q s

b

q 1 1 ( ∫ ωs (x)Δx) ≤ ∫ ωq (x)Δx), b−a sL (p, q, qs , K) b − a a

a

which provides the inequality (2.105). Clearly, q0 is the unique solution of the equation q x

L (p, q, , K) = 0,

which reduces to (2.106). The proof is now complete. Remark 2.8. Since ω is a decreasing function, we have ωσ (x) ≤ ω(x), so the relation (2.105) implies that the inequality b

q s

b

q 1 1 ( ( ∫ ωs (σ(x))Δx) ≤ ∫ ωq (x)Δx) q b−a b − a sL (p, q, s , K) a

(2.107)

a

holds for q ≤ s < q0 , where q0 is the unique solution of the equation (2.106) and L (p, q, q/s, K) is defined in (2.102). We conclude this section with a discrete version of the inequality (2.107) from the previous remark. Namely, if 𝕋 = ℕ, then σ(n) = n + 1, so the condition (2.87) is satisfied for m = 2. In this case, the constant L defined by (2.102) reduces to q

p q q s q L (p, q, , K) ≡ 1 − 2K (1 − )( ) , s s s−p

so we have the following result.

2 Muckenhoupt and Gehring weights on time scales

� 129

Corollary 2.5. Let 0 < p < q, K > 1, and let ω(n) be a nonnegative decreasing sequence such that 1 q

n n 1 1 ( ∑ ωq (k)) ≤ K( ∑ ωp (k)) n + 1 − a k=a n + 1 − a k=a

1 p

holds for n ∈ [a, b]ℕ . Then, ω ∈ ls [a, b]ℕ for q ≤ s < q0 and q s

q 1 b−1 q 1 b−1 s ( ( ∑ ω (n + 1)) ≤ ∑ ω (n)), q b − a n=a sL (p, q, s , K) b − a n=a where q0 is the unique solution of the equation 1

1

q p 1 x x ) = 2 q K( ) . ( x−q x−p

Now, we derive the self-improving properties of the two classes p

A := U

1

1 1−p

and

H

q

q

:= U1 .

Theorem 2.35. Let p > 1 and ω be any nonnegative and nondecreasing weight belonging to 𝔸p (B) for B > 1. Then ω ∈ 𝔸η (B2′′ ) for η ∈ (η− , p], where η− is the root of the equation 1 p−η (Bη) p−1 = 1. p−1

(2.108)

Proof. Since 𝔸p := U 11 , equation (2.106) becomes 1−p

p−1

(

(p − 1)x x )( ) x − 1 (p − 1)x + 1

1

= Bm q .

By applying the transform η → 1/(1 − x), we see that η− is determined from the equation 1 1 p−η (Bm q η) p−1 = 1, p−1

(2.109)

which is the desired equation (2.108). The proof is complete. Theorem 2.36. Let q > 1 and g be any nonnegative and nondecreasing weight belongs to 𝔾q (B) for B > 1. Then g ∈ 𝔾η (B1′′ ) for η ∈ [q, η+ ), where η+ is the root of the equation 1

q 1 x−1 x ( )( ) = Bm q , x x−q

(2.110)

130 � S. H. Saker q

Proof. Since 𝔾q := U1 , equation (2.106) becomes 1

q 1 x x ) ( ) = Bm q , x−1 x−q

−1

(

which is the desired equation (2.21). The proof is complete.

2.6 Higher integrability theorems In this section, we apply the self-improving properties of the Gehring weights to prove higher integrability theorems for monotone nonincreasing weights on time scales. In [38], Nania considered a new type of reverse inequalities of the form t

t

0

0

1 1 ∫ ωq (s)ds ≤ Cωq−1 (t) ∫ ω(s)ds, t t

for all t ∈ I,

(2.111)

and proved a higher integrability theorem for nonincreasing functions with the constants C > 1 and q > 1. In particular, Nania proved that if (2.111) holds, then for every p ∈ [q, q + ε], we have 1/p

(

1 ∫ ωp (t)dt) |I|

≤ K(

I

1 ∫ ω(s)ds), |I|

(2.112)

I

where ε = q/(α − 1), α = Cq(q − 1), as well as K =[

1/p

αr+1 ] α − r(α − 1)

and

r = p/q.

Nania proved (2.112) by employing the Hardy inequality p

t

p

p 1 1 1 ) ∫( ∫ ω(s)ds) dt ≤ ( ∫ ωp (t)dt, |I| t p − 1 |I| I

p > 1.

(2.113)

I

0

Alzer [3] improved the Nania result and proved that if ω is a nonnegative and nonincreasing weight on I which satisfies (2.111) for all t ∈ I, then 1/p

1 ( ∫ ωp (t)dt) |I| I

1/I

1 ≤ K1 ( ∫ ωI (s)ds) , |I|

(2.114)

I

holds with a new constant K1 smaller than K and for all p ∈ [I, I + δ] and δ > ε. Alzer proved his results by employing the Shum inequality (see [64])

2 Muckenhoupt and Gehring weights on time scales p

t

p

p

p p 1 1 1 1 ( ∫ ω(t)dt) ≤ ( ) ∫( ∫ ω(s)ds) dt + ∫ ωp (t)dt. |I| t p − 1 |I| p − 1 |I| I

I

0

� 131

(2.115)

I

Definition 2.3. For any weight ω : (0, T]𝕋 → ℝ+ which is a nonnegative and nonincreasing function, we define A ω : (0, T]𝕋 → ℝ+ by t

Aω=

1 ∫ ω(s)Δs t 0

for all t ∈ (0, T]𝕋 ,

(2.116)

as well as σ(t)

1 A ω= ∫ ω(s)Δs, σ(t) σ

for all t ∈ (0, T]𝕋 ,

0

and σ

σ(t)

σ

A ω (t) =

1 ∫ ωσ (s)Δs, σ(t) 0

and assume that there exists a constant λ > 1 such that σ(t) ≤ λt. We mentioned here that the condition σ(t) ≤ λt may be removed if the graininess weight μ(t) on the time scale 𝕋 satisfies μ(t) = O(t). Indeed, if μ(t) = O(t), then there exists λ > 1 such that 0 < μ(t)/t ≤ λ − 1 for all t ∈ 𝕋. Hence 1 ≤ (t + μ(t))/t ≤ λ, and thus 1 ≤ σ(t)/t ≤ λ for all t ∈ 𝕋. Note that when 𝕋 = ℝ we have σ(t) = t, and when 𝕋 = ℕ, we have σ(t) = t + 1. Theorem 2.37. Assume that ω be a nonnegative and nonincreasing. If there exists a constant A > 1 such that σ

σ

σ

A ω ≤ Aω ,

for all t ∈ (0, ∞)𝕋 ,

(2.117)

then σ

σ p

A (ω ) ≤

M M p p (A σ ωσ ) ≤ (A σ ω) , M − p(M − 1) M − p(M − 1)

(2.118)

for p ∈ [1, M/(M − 1)], where M := Aλ > 1. t

Proof. Let W (t) := ∫0 ωσ (s)Δs. From this and (2.117), we get that (note that ω is decreasing) 1 ωσ (t) ωΔ (t) ≤ σ(t) = σ . Aσ(t) ∫ ωσ (s)Δs ω (t) 0

132 � S. H. Saker Since σ(t) ≤ λt for some constant λ ≥ 1, we have that W Δ (t) 1 ≤ σ , Mt W (t)

where M = Aλ > 1.

From the chain rule, we see that Δ

1

(log W (t)) = {∫ 0

hωσ

1 dh}W Δ (t). + (1 − h)ω

(2.119)

Now, since ωΔ (t) = ωσ (t) > 0, we obtain ωσ ≥ ω(t) and this implies that 1 1 1 ≥ = . (1 − h)W (t) + hW σ (1 − h)W σ + hW σ W σ This and (2.119) yield 1

1 W Δ (t) Δ ≤ {∫ dh}W Δ (t) = (log W (t)) . Wσ hW σ + (1 − h)W

(2.120)

0

Also, we have (since σ(t) ≥ t) that Δ

1

(log t) = {∫ 0

1 1 dh} ≤ . hσ + (1 − h)t t

(2.121)

Combining (2.120) and (2.121), we deduce 1 Δ (log t)Δ ≤ (log W (t)) . M

(2.122)

Integrating the latter inequality from σ(y) to σ(t) (where 0 < σ(y) < σ(t)), we get that 1

log(

σ(t) M W σ (t) ) ≤ log σ . σ(y) W (y)

Hence, σ(y)

1

∫ ωσ (s)Δs ≤ ( 0

σ(t)

σ(y) M ) ∫ ωσ (s)Δs. σ(t) 0

Since ω is nonincreasing, we see that ωσ (y) ≤ ωσ (s), where σ(y) ≥ σ(s), and then σ(y)

1

σ(t)

σ(y) M 1 1 ω (y) ≤ ( ) ∫ ω(s)Δs. ∫ ωσ (s)Δs ≤ σ(y) σ(y) σ(t) σ

0

0

2 Muckenhoupt and Gehring weights on time scales

� 133

Thus, p

(ωσ (y)) ≤ (

p

σ(t)

p

p

σ(y) M 1 ) ( ) ( ∫ ω(s)Δs) . σ(y) σ(t) 0

Integrating from 0 to σ(t), we see that σ(t)

σ(t)

p

σ

p

σ

(2.123)

∫ (ω (y)) Δy ≤ Θ( ∫ ω (s)Δs) , 0

0

where σ(t)

p

p

p

σ(t)

σ(y) M 1 1 M 1 Θ= ∫( ) ( ) Δy = ( ) ∫ γ Δy, σ(y) σ(t) σ(t) σ (y) 0

0

with γ = p(M − 1)/M < 1. From the chain rule, we see that 1

Δ

(s1−γ ) = (1 − γ) ∫[hσ(s) + (1 − h)s] dh −γ

0

1

= (1 − γ) ∫ 0

1

≥ (1 − γ) ∫ 0

dh [hσ(s) + (1 − h)s]γ (1 − γ) dh = γ . [hσ + (1 − h)σ(s)]γ σ (s)

This implies that p

σ(t)

1 1 M ) ∫ γ Δy Θ=( σ(t) σ (y) 0

p

σ(t)

1 1 M Δ ≤ ( ) ∫ (s1−γ ) Δs (1 − γ) σ(t) p M

=

0

p(M−1) 1 1 M 1− ( ) (σ(t)) M = σ 1−p (t). (1 − γ) σ(t) M − p(M − 1)

Substituting into (2.123) yields σ(t)

σ(t)

p

M σ 1−p (t)( ∫ ωσ (s)Δs) , ∫ (ω ) Δy ≤ M − p(M − 1) 0

σ p

0

134 � S. H. Saker and hence (since ω is nonincreasing), we get that σ(t)

σ(t)

1 M 1 p ( ∫ (ωσ (y)) Δy ≤ ∫ ωσ (s)Δs) σ(t) M − p(M − 1) σ(t) 0

0

p

p

σ(t)

M 1 ≤ ( ∫ ω(s)Δs) , M − p(M − 1) σ(t) 0

which is the desired inequality (2.118). The proof is complete. q

Notice that for all nonnegative and nonincreasing functions ω ∈ 𝕃Δ [0, ∞)𝕋 with q > 1, we always have t

t

t

0

0

0

1 ωq−1 (t) 1 q q−1 A ω (t) = ∫ ω (s)Δs = ∫ ω (s)ω(s)Δs ≥ ∫ ω(s)Δs. t t t q

(2.124) q

Let us now consider the class of nonnegative and nonincreasing functions ω ∈ 𝕃Δ [0, ∞)𝕋 that satisfy the reverse of (2.134), namely q

q−1

A ω ≤ ηω

A ω,

for some η > 1.

(2.125)

Theorem 2.38. Assume that ω is a nonnegative and nonincreasing weight such that (2.125) holds for q > 1. Then p σ

q p/q

A (ω ) ≤ η[̃ A ω ]

η̃ :=

,

ληq 1+p/q

ληq − pq (ληq − 1)

ηq =

,

ηq , q−1

(2.126)

for p ∈ [q, q + c], c ∈ (q, η). Proof. In this proof, we write W = A ωq for brevity. By using the Hölder inequality with exponents q/(q − 1) and q, we obtain t

1 ∫ W (σ(s))Δs t 0

σ(s)

t

≤

η 1 q−1 ∫(ω(σ(s))) ⋅ ∫ ω(τ)ΔτΔs t σ(s) 0

0 (q−1)/q

t

η q ≤ [∫(ω(σ(s))) Δs] t 0

t

0

1/q

1 [∫( ∫ ω(τ)Δτ) Δs] σ(s)

0 (q−1)/q

ηq q ≤ [∫(ω(s)) Δs] (q − 1)t

q

σ(s)

t

t

0

q

[∫(ω(s)) Δs]

1/q

0

2 Muckenhoupt and Gehring weights on time scales

=

ηq t

� 135

t

∫ ωq (s)Δs = ηq W (t), 0

i. e., σ

A W ≤ ηq W .

(2.127)

Since ω is also nonnegative and nonincreasing, it satisfies the assumptions of Theorem 2.30 and thus, t

t

0

0

r

1 1 r ∫[W (σ(s))] Δs ≤ η̃ q [ ∫ W (σ(s))Δs] , t t

(2.128)

with η̃ q =

αq

αq − r(αq − 1)

,

αq = ληq , r =

αq p ∈ [1, ). q αq − 1

Noting that t

1 W (t) = ∫ ωq (s)Δs ≥ ωq (t), t

(2.129)

0

we obtain t

t

0

0

t

1 1 1 p r r ∫(ω(σ(s))) Δs = ∫(ωq (σ(s))) Δs ≤ ∫(W (σ(s))) Δs t t t t

1 ≤ η̃ q ( ∫ W σ (s)Δs) t

0

r

0

t

p/q

1 r r ̃ (t)] = η[ ̃ ∫ ωq (s)Δs] ≤ η̃ q ηq r [W (t)] = η[W t

,

0

proving (2.136). In Theorem 2.38, if 𝕋 = ℝ, then we have that σ(t) = t, αq = ηq , and we get the following result. Corollary 2.6. Let η > 1 and q > 1 and ω is a nonnegative nonincreasing weight satisfying t

q

q−1

∫ ω (x)dx ≤ ηω 0

t

(t) ∫ ω(x)dx, 0

136 � S. H. Saker then t

t

0

0

1 1 ̃ ∫ ωp (x)dx ≤ η( ∫ ωq (x)dx) t t

p/q

,

for p ∈ [q, q + c], c ∈ (q, η), and p

η̃ :=

ηq q−1

ηq q ( q−1 )

+1

ηq − pq ( q−1 − 1)

.

In the following, we prove a new extension of the Hardy inequality on time scales with reiteration term on a finite interval. The results are adapted from [59]. The proof depends on the applications of the Hölder inequality, the chain rules on time scales, and the algebraic inequality (ϑ + v)p ≥ ϑ p + pϑ p−1 v

if p < 0 or p > 1.

(2.130)

This inequality is valid for all ϑ ≥ 0 and ϑ + v ≥ 0 (if p > 0), or ϑ > 0 and ϑ + v > 0 (if p < 0) and equality holds if only if v = 0. Theorem 2.39. Assume that ω is nonincreasing. Then for q > 1, we have 󵄨󵄨 q 󵄨󵄨 󵄨 󵄨

q

σ

q

q

q

q

󵄨󵄨(A ω) ≤ ( A (A ω) + 󵄨󵄨󵄨 ) A (ω ). 󵄨󵄨 q − 1 󵄨󵄨󵄨 q−1

(2.131)

Proof. In this proof, we write W = A ω for brevity. By the chain rule, we obtain 1

Δ

(W q ) = q(W )Δ ∫(hW σ + (1 − h)W ) 1

q−1

dh,

0

≤ qW Δ ∫(hW σ + (1 − h)W σ )

q−1

q−1

dh = qW Δ (W σ )

.

0 t

Moreover, since t W (t) = ∫0 ω(s)Δs, the product rule in (2.12) yields that t W Δ (t) + W σ (t) = ω(t),

where W σ = A σ ω.

Now, putting ϑ = t and v = W q , and using integration by parts, we find that t

t

σ q

∫(W ) Δs = tv(t) − lim+ sv(s) − ∫ svΔ (s)Δs. 0

s→0

0

By applying the Hölder inequality with exponents q and q/(q − 1), we see that

� 137

2 Muckenhoupt and Gehring weights on time scales

tv(t) =

1

q

t

t q−1

(∫ ω(s)Δs) ≤ 0

1

t q−1

(t

q−1 q

t

1/q

q

(∫ ω (s)Δs)

q

t

) = ∫ ωq (s)Δs,

0

0

implying (note that tv(t) ≥ 0) limt→0+ tv(t) = 0. This and the fact that W is decreasing imply that t

σ q

t

q

t

0

q

q−1

∫(W ) Δs = t W (t) − ∫ sv (s)Δs ≥ t W (t) − q ∫ sW Δ (s)(W σ (s)) 0

Δ

Δs

0

t

q−1

= t W q (t) − q ∫[ω(s) − W σ (s)](W σ (s)) 0

t

q−1

= t W q (t) − q ∫(W σ (s))

Δs

t

q

ω(s)Δs + q ∫(W σ (s)) Δs. 0

0

By applying the Hölder inequality with exponents q and q/(q − 1) on the term t

q−1

∫(W σ (s))

ω(s)Δs,

0

we get t

q

(q − 1) ∫(W σ (s)) Δs + t W q (t) 0

t

q−1

σ

≤ q ∫(W (s))

t

q

ω(s)Δs ≤ q[∫ ω (s)Δs]

1/q

0

0

t

q

σ

(q−1)/q

[∫(W (s)) Δs]

,

0

i. e., q t

q ( ) ∫ ωq Δs q−1 t

0

q

σ

1/q

≥ [(∫(W (s)) Δs) t

q

1 t W q (t) + ] t (q − 1) [∫ (W σ (s))q Δs](q−1)/q 0

0

t

(q−1)/q

q t W q (t) q ≥ ∫(W (s)) Δs + [∫(W σ (s)) Δs] t (q − 1) [∫ (W σ (s))q Δs](q−1)/q 0

This implies

σ

q

0

0

.

138 � S. H. Saker t

t

q

q q 1 1 q q W (t) ≤ ( ) ∫ ωq (s)Δs, ∫(W σ (s)) Δs + t (q − 1) q−1 t 0

0

which is the desired inequality (2.131). The proof is complete. As a special case of Theorem 2.39, if 𝕋 = ℝ, we obtain the following Shum’s inequality [64]. Corollary 2.7. If 1 < q, then for 0 < T < ∞, we have T

t

0

0

q

q

T

q 1 1 1 ( ∫ ω(t)dt) ∫( ∫ ω(s)ds) Δt + T t q−1 T q

≤(

0

T

q 1 ) ∫ ωq (t)dt. q−1 T 0

Remark 2.9. From the inequality (2.131), we get a Hardy-type inequality for decreasing functions on a finite time scales interval, namely q

σ(t)

T

q

T

q 1 1 1 ) ∫ ωq (t)Δt. ∫( ∫ ω(s)Δs) Δt ≤ ( T σ(t) q−1 T 0

(2.132)

0

0

In Theorem 2.39 if 𝕋 = ℕ, we have that σ(n) = n + 1, and then we get the following result. Corollary 2.8. If 1 < q then for 0 < N < ∞, we have q

q

q q 1 n 1 N−1 1 N−1 1 N−1 q ( ∑ ω(k)) ≤ ( ) ∑( ∑ ω(k)) + ∑ ω (n). N n=0 n + 1 k=0 (q − 1) N k=0 q − 1 N n=0 From the proof of Theorem 2.39, we see that the inequality (2.131) is still satisfied if we replace A by A σ , and we thus have the following result. Corollary 2.9. Assume that ω is a nonnegative and nonincreasing weight. Then for q > 1, we have σ

σ

q

q 󵄨󵄨 q 󵄨󵄨 q q 󵄨 󵄨󵄨 σ σ q ( A ω(t)) ≤ ( ) A ω (t). 󵄨 󵄨󵄨 q − 1 󵄨󵄨󵄨 q−1

A (A ω) (t) + 󵄨󵄨󵄨

(2.133)

In the following, we prove a higher integrability result for monotone decreasing functions by employing the inequalities (2.118) and (2.133). The results are adapted from [52]. For all nonnegative and nonincreasing functions ω, the following inequality holds: ωq−1 (σ(t))A σ ω(t) ≤ A σ ωq (t).

(2.134)

2 Muckenhoupt and Gehring weights on time scales

� 139

For C > 1 and q > 1, let us consider the class Lq (0, ∞)𝕋 of nonnegative nonincreasing functions which satisfy σ

q

q−1

σ

A ω (t) ≤ C(ω (t))

σ

A ω(t).

(2.135)

Notice that the inequality (2.125) is the reverse of the inequality (2.134). Theorem 2.40. If ω is a positive decreasing weight satisfying (2.135) for C > 1, then p 1/p

(A σ (ωσ ) )

1/q

≤ R(A σ ωq )

(2.136)

,

for all p ∈ [q, q + δ], where R := [

βλr+1 ] βλ − r(βλ − 1)

1/p

,

p , q

r=

δ=

q , (β − 1)

βλ = λβ,

and β := [C q (

q

qC q−1 q ) − ] q−1 q−1

1/q

> 1.

Proof. Since ω is decreasing and σ(t) ≥ t, we see that (ωσ (t))q−1 ≤ ωq−1 (t). From this and (2.125), we get that σ(t)

σ(t)

1 1 ∫ ωq (s)Δs ≤ Cωq−1 (t) ∫ ω(s)Δs. σ(t) σ(t) 0

(2.137)

0

Let us integrate (2.137) between 0 and σ(y) ∈ (0, ∞)𝕋 , to get σ(y)

σ(t)

σ(y)

σ(t)

0

0

0

0

1 1 ∫ ωq (s)ΔsΔt ≤ C ∫ ωq−1 (t) ∫ ω(s)ΔsΔt. ∫ σ(t) σ(t)

Applying the Hölder inequality with indices q and q/(q − 1), we obtain σ(y)

(∫ 0

σ(t)

σ(y)

σ(t)

0

0

0

1 1 ∫ ωq (s)ΔsΔt) ≤ C ∫ ωq−1 (t) ∫ ω(s)ΔsΔt σ(t) σ(t) σ(y)

≤ C( ∫ (ω

q−1

q/(q−1)

(t))

Δt)

q−1 q

0 σ(y)

σ(t)

0

0

q

1 q

1 ×( ∫ ( ∫ ω(s)Δs) Δt) . σ(t)

(2.138)

140 � S. H. Saker This implies that σ(y)

σ(t)

1 (∫ ∫ ωq (s)ΔsΔt) σ(t) 0

0

σ(y)

q

q

q−1 σ(y)

q

q

σ(t)

1 ∫( ∫ ω(s)Δs) Δt. σ(t)

≤ C ( ∫ ω (t)Δt) 0

0

(2.139)

0

Applying (2.133) on the second term of the right-hand side, we see that σ(y)

σ(t)

0

0

q

1 ∫ ω(s)Δs) Δt ∫( σ(t) σ(y)

σ(y)

q q q q (∫0 ω(t)Δt) ≤( ) ∫ ωq (t)Δt − , q−1 q−1 σ q−1 (y) 0

This and (2.139) imply that σ(y)

σ(t)

0

0

q

1 1 ( ∫ ∫ ωq (s)ΔsΔt) σ(y) σ(t) q

σ(y)

q−1

q

≤ C ( ∫ ω (t)Δt) 0

σ(y)

q

σ(y)

q

σ(y)

q q q (∫0 ω(t)Δt) q ) ∫ ωq (t)Δt − [( ] q−1 q−1 σ q−1 (y) 0

q−1 ( 1 ∫σ(y) ω(t)Δt)q σ(y) 0 ]. σ(y) q 1 (∫ ω (t)Δt) σ(y) 0

q−1 Cq 1 ) ( ) =( ∫ ωq (t)Δt) [1 − ( q−1 σ(y) q 0

(2.140)

Setting t

1 Λ(t) := ∫ ωq (s)Δs, t

(2.141)

0

we get from (2.140) that σ(y)

q

1 ( ∫ Λσ (t)Δt) σ(y) 0

q

q−1 ( 1 ∫σ(y) ω(t)Δt)q σ(y) 0 ]. σ(y) q 1 ω (t)Δt ∫ σ(y) 0

Cq σ q−1 ≤( Λ (y)) [1 − ( ) q−1 q

(2.142)

2 Muckenhoupt and Gehring weights on time scales

� 141

Since ω is nonincreasing, we have σ(y)

1 ω (y) ≤ ∫ ω(t)Δt. σ(y) σ

(2.143)

0

From (2.125) it follows that σ(y)

1 ( σ(y) ∫0

ω(t)Δt)q

σ(y) q 1 ω (t)Δt ∫ σ(y) 0

σ(y)

≥

=

1 ( σ(y) ∫0

ω(t)Δt)q σ(y)

1 Cωq−1 (σ(y)) σ(y) ∫0

1 ( C

ω(s)Δs

σ(y) 1 ω(t)Δt q ∫ σ(y) 0 ) . ωσ (y)

This and (2.143) imply q

σ(y)

σ(y)

1 1 1 ( ∫ ω(t)Δt) ≥ ∫ ωq (t)Δt. σ(y) C σ(y) 0

0

Substituting into (2.142), we have that q

σ(y)

q−1

1 1 q−1 ( ) ∫ Λσ (t)Δt) ≤ [1 − ( σ(y) C q

q

Cq σ Λ (y)) . q−1

](

0

This leads to σ(y)

1 ∫ Λσ (t)Δt ≤ βΛσ (y), σ(y)

(2.144)

0

with β = [C q (

q

1/q

q q ) − C q−1 ( )] q−1 q−1

.

From, the inequality x q − yq ≥ qyq−1 (x − y) for q ≥ 1, x > y > 0, we see that x q − 1 > q(x − 1) for q > 1. This implies, by setting x = (q/(q − 1)), that q

(

q q ) −( )>1 q−1 q−1

for q > 1.

Since the weight G(t) := t q (

q

q q ) − t q−1 ( ) q−1 q−1

142 � S. H. Saker is strictly increasing on [1, ∞), we conclude for C > 1 that β = [C q (

q

1/q

q q ) − C q−1 ( )] q−1 q−1

> 1.

Since ωq is decreasing, it follows that Λ(t) is also decreasing (see Lemma 2.15) and satisfies (2.144). So we can apply the inequality (2.118) with Λ instead of ω and βλ instead of A to get for all r ∈ [1, βλ /(βλ − 1)) that σ r

σ

A (Λ ) ≤

βλ r (A σ (Λσ )) , βλ − r(βλ − 1)

where βλ = λβ.

By using the fact that σ

(ω (s))

qr

r

σ(s)

1 r ≤( ∫ ωq (ϑ)Δϑ) = (Λσ (s)) , σ(s) 0

we obtain σ(t)

σ(t)

0

0

1 1 r ∫ ωrq (σ(s))Δs ≤ ∫ (Λσ ) Δs σ(t) σ(t) σ(t)

r

βλr+1 1 ( ≤ ∫ Λσ (s)Δs) . βλ − r(βλ − 1) σ(t) 0

From this and (2.144), we see that σ rq

σ

A (ω )

≤

βλr+1 r (A σ (ωq )) . βλ − r(βλ − 1)

Putting r = p/q, we get the desired inequality (2.136). The proof is complete. In Theorem 2.38 if 𝕋 = ℝ, we have that σ(t) = t and λ = 1 and then we get the following result due to Alzer. Corollary 2.10. Let C > 1, q > 1, and let ω be a positive decreasing weight satisfying t

t

0

0

1 1 ∫ ωq (s)ds ≤ Cωq−1 (t) ∫ ω(s)ds. t t Then T

T

0

0

1 1 ∫ ωp (s)ds ≤ M ∗ ( ∫ ωq (s)ds) T T

p/q

2 Muckenhoupt and Gehring weights on time scales

� 143

for all p ∈ [q, q + δ], where δ = q/(β − 1), M∗ =

βr+1 , β − r(β − 1)

and

β := (C q (

q

1/q

q q ) − C q−1 ) q−1 q−1

.

In Theorem 2.38 if 𝕋 = ℕ, we have that σ(n) = n + 1 and, by considering λ = 2, we get the following. Corollary 2.11. Let C > 1, q > 1, and let a(n) be a positive decreasing sequence which satisfies 1 n q 1 n ∑ a (i) ≤ Caq−1 (n + 1) ∑ a(i), n + 1 i=0 n + 1 i=0 then 1 N p 1 N q ∑ a (i + 1) ≤ M∗ ( ∑ a (i)) N + 1 i=1 N + 1 i=0

p/q

,

for all p ∈ [q, q + δ], where δ = q/(β∗ − 1), M∗ =

β∗r+1 , β∗ − r(β∗ − 1)

and

β∗ := 2(C q (

q

1/q

q q ) − C q−1 ) q−1 q−1

.

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2 Muckenhoupt and Gehring weights on time scales

� 145

[39] B. S. Pavlov, Basicity of an exponential system and Muckenhoupt’s condition, Dokl. Akad. Nauk SSSR, 247(1) (1979), 37–40; English translation in Sov. Math. Dokl. 20(4) (1979), 655–659. q [40] A. Popoli, Optimal integrability in Bp classes, Le Matematiche, LII (1997), 157–168. [41] A. Popoli, Sharp integrability exponents and constants for Muckenhoupt and Gehring weights as solution to a unique equation, Ann. Acad. Sci. Fenn., Math., 43 (2018), 785–805. [42] E. M. Stein and S. Wainger, Discrete analogues in harmonic analysis I: l 2 -estimates for singular Radon transforms, Am. J. Math., 121 (1999), 1291–1336. [43] E. M. Stein and S. Wainger, Discrete analogues in harmonic analysis II: Fractional integration, J. Anal. Math., 80 (2000), 335–355. [44] E. M. Stein and S. Wainger, Two discrete fractional integral operators revisited, J. Anal. Math., 87 (2002), 451–479. [45] S. H. Saker and R. P. Agarwal, Discrete Hardy’s type inequalities and structure of discrete class of weights satisfy reverse Hölder’s inequality, Math. Inequal. Appl., 24(2) (2021), 521–541. [46] S. H. Saker and R. P. Agarwal, Theory of discrete Muckenhoupt weights and discrete Rubio de Francia extrapolation theorems, Appl. Anal. Discrete Math., 15 (2021), 295–316. [47] S. H. Saker, J. Alzabut, D. O’Regan and R. P. Agarwal, Self-improving properties of weighted Gehring classes with applications to partial differential equations, Adv. Differ. Equ., 2021 (2021), 1–20. [48] S. H. Saker, D. Baleanu and M. Krnić, Some further properties of discrete Muckenhoupt and Gehring weights, J. Math. Inequal., 16 (2022), 1–18. [49] S. H. Saker, M. A. Darwish, H. A. Elshamy and R. P. Agarwal, Fundamental properties of Muckenhoupt and Gehring weights on time scales, J. Anal. (submitted). [50] S. H. Saker and M. Krnić, The weighted discrete Gehring classes, Muckenhoupt classes and their basic properties, Proc. Am. Math. Soc., 149 (2021), 231–243. [51] S. H. Saker, M. Krnić and J. Pečarić, Higher summability theorems from the weighted reverse discrete inequalities, Appl. Anal. Discrete Math., 13 (2019), 423–439. [52] S. H. Saker and I. Kubiaczyk, Reverse dynamic inequalities and higher integrability theorems, J. Math. Anal. Appl., 471(1–2) (2019), 671–686. [53] S. H. Saker and I. Kubiaczyk, Higher summability and discrete weighted Muckenhoupt and Gehring type inequalities, Proc. Edinb. Math. Soc., 62 (2019), 949–973. [54] S. H. Saker and R. R. Mahmoud, Boundedness of both discrete Hardy and Hardy–Littlewood maximal operators via Muckenhoupt weights, Rocky Mt. J. Math., 51(2) (2021), 733–746. [55] S. H. Saker, D. O’Regan and R. P. Agarwal, Self-improving properties of discrete Muckenhoupt weights, Analysis, 41 (2021), 169–178. [56] S. H. Saker, D. O’Regan and R. P. Agarwal, A higher integrability theorem from a reverse weighted inequality, Bull. Lond. Math. Soc., 51(6) (2019), 967–977. [57] S. H. Saker, D. O’Regan and R. P. Agarwal, Self-improving properties of a generalized Muckenhoupt class, Acta Math. Hungr., 164 (2021), 113–134. [58] S. H. Saker, S. S. Rabie, G. AlNemer and M. Zakarya, On structure of discrete Muckenhoupt and discrete Gehring classes, J. Inequal. Appl., 2020 (2020), 1–18. [59] S. H. Saker, D. O’Regan, M. M. Osman and R. P. Agarwal, New dynamic inequalities for decreasing functions and theorems of higher integrability, Ann. Appl. Math., 34(2) (2018), 165–177. [60] S. H. Saker, M. M. Osman and M. Krnić, Higher integrability theorems on time scales from reverse Hölder’s inequalities, Appl. Anal. Discrete Math., 13 (2019), 819–838. [61] S. H. Saker, S. S. Rabie, J. Alzabut, D. O’Regan and R. P. Agarwal, Some basic properties and fundamental relations for discrete Muckenhoupt and Gehring classes, Adv. Differ. Equ., 2021(1) (2021), 1–22. [62] S. H. Saker, S. S. Rabie and M. Krnić, Sharp self-improving property of discrete Muckenhoupt and Gehring weights, Rev. R. Acad. Cienc. Exactas, 115 (2021), 119. [63] S. H. Saker, S. S. Rabie and R. P. Agarwal, Properties of a generalized class of weights satisfying reverse Hölder’s inequality, J. Funct. Spaces, 2021 (2021), 1–16.

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Douglas R. Anderson and Masakazu Onitsuka

3 Ulam stability and instability of first-order linear 1- and 2-periodic dynamic equations on isolated time scales Abstract: We apply a new definition of periodicity on isolated time scales introduced by Bohner, Mesquita, and Streipert to the study of Ulam stability. If the graininess (step size) of an isolated time scale is bounded by a finite constant, then the linear 1- and 2-periodic dynamic equations are Ulam stable if and only if the exponential function has modulus different from unity. If the graininess increases at least linearly to infinity, the 1- and 2-periodic dynamic equations are not Ulam stable. Applying these results, we give several interesting examples of first-order linear 1- or 2-periodic dynamic equations on specific isolated time scales such as h-difference equations, q-difference equations, triangular equations, Fibonacci equations, and harmonic equations. In some cases the minimum Ulam stability constant is found.

3.1 Introduction Bohner, Mesquita, and Streipert [10] recently introduced the idea of periodicity of functions on isolated time scales by focusing on repeated area under the curve rather than repeated function values; see also the earlier paper by Bohner and Chieochan [9], which establishes this concept first for q-difference equations. The nature of this new notion of periodicity is distinct once the graininess (step size) of the isolated time scale is nonconstant, making even 1-periodic functions new and interesting. In this work we explore the Ulam stability and instability of first-order linear dynamic equations over isolated time scales, where the coefficient function is 1- or 2-periodic in the sense of [10]. Stability analysis that variously goes by the names Ulam stability [30], Hyers–Ulam stability [17], Hyers–Ulam–Rassias stability [27] is an area of interest that differs from Lyapunov stability analysis. Brillouët-Belluot, Brzdek, and Ciepliński [12] give an overview on some recent developments in Ulam-type stability, while Brzdek, Popa, Raşa, and Xu [13] have a book on the Ulam stability of operators. Acknowledgement: The second author was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI (grant number JP20K03668). Douglas R. Anderson, Department of Mathematics, Concordia College, Moorhead, MN 56562, USA, e-mail: [email protected] Masakazu Onitsuka, Department of Applied Mathematics, Okayama University of Science, Okayama, 700-0005, Japan, e-mail: [email protected] https://doi.org/10.1515/9783111182971-003

148 � D. R. Anderson and M. Onitsuka Brzdek and Wójcik [14] write on approximate solutions of some difference equations. Popa [24, 25] studies Hyers–Ulam and Hyers–Ulam–Rassias stability of the linear recurrence, while Rasouli, Abbaszadeh, and Eshaghi [26] study approximately linear recurrences. Jung and Nam [18] are interested in the Hyers–Ulam stability of the Pielou logistic difference equation. Nam [19–21] has explored the Hyers–Ulam stability of hyperbolic, elliptic, and loxodromic Möbius difference equations, respectively. Fukutaka and Onitsuka [15] and Onitsuka [22, 23] have studied Hyers–Ulam stability for first-order homogeneous linear differential equations with a periodic coefficient, first-order homogeneous linear difference equations, and second-order nonhomogeneous linear difference equations, respectively. Anderson and Onitsuka [5, 6] investigate Hyers–Ulam stability for Cayley quantum equations and its application to h-difference equations, while Anderson, Onitsuka, and Rassias [7] find the best constant for Ulam stability of first-order h-difference equations with periodic coefficient. András and Mészáros [8] study Ulam– Hyers stability of dynamic equations on time scales via Picard operators. Hua, Li, and Feng [16] investigate Hyers–Ulam stability of dynamic integral equation on time scales. Shen [28] has done work on the Ulam stability of first order linear dynamic equations on time scales, while Shen and Li [29] examine Hyers–Ulam stability of first order nonhomogeneous linear dynamic equations on time scales. The outline of the paper is the following. In Section 3.2, a brief review of time scales and the corresponding notation is provided. In Section 3.3, we investigate the stability and instability of first-order dynamic equations on isolated time scales with a 1-periodic coefficient. In Section 3.4, we investigate the stability and instability of first-order dynamic equations on isolated time scales with a 2-periodic coefficient. In Section 3.5, we consider a certain dynamic equation with two variable coefficient functions, and show through a change of variables a connection to the earlier cases, establishing new Ulam stability results in the process for this equation. In Section 3.6, we provide examples of isolated time scales with 1- or 2-periodic coefficient functions and apply the results from Sections 3.3, 3.4 and 3.5 to these cases. In Section 3.7, we offer a conclusion, together with possible future directions.

3.2 Brief review of time scales A time scale 𝕋 is any closed subset of the real line. For t ∈ 𝕋, define the forward and backward jump operators σ, ρ : 𝕋 → 𝕋 by σ(t) := inf{s ∈ 𝕋 : s > t} and

ρ(t) := sup{s ∈ 𝕋 : s < t},

respectively, and the graininess function μ : 𝕋 → [0, ∞) by μ(t) := σ(t) − t.

3 Ulam stability and instability of first-order linear 1- and 2-periodic dynamic equations

� 149

A time scale 𝕋 is isolated if σ(t) > t and ρ(t) < t both hold for all t ∈ 𝕋. For f : 𝕋 → ℂ, (t) . For an isolated time scale 𝕋 = {tn }∞ the derivative of f at t ∈ 𝕋 is f Δ (t) = f (σ(t))−f n=0 μ(t) that is unbounded above, the derivative can also be written as f Δ (tn ) =

f (tn+1 ) − f (tn ) . tn+1 − tn

Throughout this work, we will assume 𝕋 is an isolated time scale that is unbounded above.

3.3 Investigating stability of first-order dynamic equations with a 1-periodic coefficient Consider an arbitrary isolated time scale 𝕋 = {tn }∞ n=0 . In this section we consider on 𝕋 the Ulam stability of the first-order linear homogeneous periodic dynamic equation on isolated time scales, with a 1-periodic coefficient, represented by zΔ (t) − p(t)z(t) = 0,

t ∈ 𝕋,

(3.1)

for all t ∈ 𝕋.

(3.2)

where p : 𝕋 → ℂ is 1-periodic [9, Definition 3.1] if p(t) = σ Δ (t)p(σ(t)),

−1 Let p(t0 ) = p0 for p0 ∈ ℂ\{ μ(t }. Through iteration of (3.2), one sees that p is given by ) 0

p(t) :=

μ(t0 )p(t0 ) μ(t0 )p0 = , μ(t) μ(t)

t ∈ 𝕋.

(3.3)

−1 Remark 3.1 (Exponential function). Let p(0) = p0 for p0 ∈ ℂ\{ μ(t }. For t = tn ∈ 𝕋 with 0) n ∈ ℕ0 , set n−1

ep (tn ) := ∏[1 + μ(tj )p(tj )], j=0

−1

where ∏[1 + μ(tj )p(tj )] ≡ 1. j=0

(3.4)

By the 1-periodic nature of p given in (3.2) and (3.3), we have (t = tn and n ∈ ℕ0 ) n

ep (tn ) = [1 + μ(t0 )p0 ] ,

n ∈ ℕ0 .

(3.5)

−1 Note that the condition p(t0 ) = p0 ∈ ℂ\{ μ(t } is the regressive condition, since p(t0 ) = ) 0

−1 p0 ≠ μ(t is equivalent to 1 + μ(t0 )p(t0 ) ≠ 0. For regressive conditions on general time 0) scales, see [11, Definition 2.25]. Throughout the remainder of this section, for notational convenience we define the base of the exponential function to be

150 � D. R. Anderson and M. Onitsuka P := 1 + μ(t0 )p0 ,

ep (tn ) = Pn ,

n ∈ ℕ0 .

(3.6)

Definition 3.1 (Ulam stability). Let η : 𝕋 → ℂ be a perturbation. Equation (3.1) has Ulam stability if and only if there exists a constant L > 0 such that, for arbitrary ε > 0, if a function ψ : 𝕋 → ℂ satisfies ψΔ (t) − p(t)ψ(t) = η(t),

󵄨󵄨 󵄨 󵄨󵄨η(t)󵄨󵄨󵄨 ≤ ε,

t ∈ 𝕋,

(3.7)

then there exists a solution z : 𝕋 → ℂ of (3.1) such that |ψ(t) − z(t)| ≤ Lε for all t ∈ 𝕋. If (3.1) is Ulam stable, the constant L is called an Ulam stability constant for (3.1) on 𝕋. Remark 3.2. According to [10, Theorem 5.1], the role of constant functions in the classical sense is assumed by 1-periodic functions on isolated time scales, where a function f c is 1-periodic on 𝕋 if and only if there exists a real constant c such that f (t) = μ(t) for all ∞ t ∈ 𝕋 = {tn }n=0 , as in this paper. Theorem 3.1. Let 𝕋 = {tn }∞ n=0 be an isolated time scale with graininess function μ > 0 μ0 p0 such that limn→∞ tn = ∞. Let p(t) = μ(t) for all t = tn ∈ 𝕋, i. e., p is 1-periodic, where p0 ∈ ℂ\{ −1 }. If there exists a positive constant μmax ∈ (0, ∞) such that μ 0

μ(tn ) ≤ μmax ,

n ∈ ℕ0 ,

(3.8)

then (3.1) is Ulam stable if and only if ρ = |1 + μ0 p0 | ≠ 1, with Ulam constant L =

μmax . |1−ρ|

Proof. The proof is a simpler case of the proof given later for the 2-periodic coefficient case in Theorem 3.3, and thus is omitted. μ p

0 0 Theorem 3.2. Let 𝕋 = {tn }∞ n=0 be an isolated time scale. Let p(t) = μ(t) for all t = tn ∈ 𝕋, i. e., p is 1-periodic. If there exist real constants a > 0 and b ≥ 0 such that

μ(tn ) ≥ an + b,

n ∈ ℕ0 ,

(3.9)

then (3.1) is not Ulam stable. Proof. Let ε > 0 be a fixed arbitrary constant throughout the proof. By the definition of Ulam stability given in Definition 3.1, we need to find a function ψ : 𝕋 → ℂ fulfilling (3.7), such that there is no solution z of (3.1) satisfying |ψ(t) − z(t)| ≤ Lε for all t ∈ 𝕋, for any constant L ∈ ℝ. Suppose a function ψ fulfills (3.7) for all t ∈ 𝕋, and takes the form (for t = tn and n ∈ ℕ0 ) n−1

ψ(tn ) = ψ(t0 )ep (tn ) + ep (tn ) ∑

j=0

Take ψ(t0 ) = 0, and let

μ(tj )η(tj ) ep (tj+1 )

−1

,

assuming ∑

j=0

μ(tj )η(tj ) ep (tj+1 )

≡ 0.

(3.10)

3 Ulam stability and instability of first-order linear 1- and 2-periodic dynamic equations

η(tj ) =

ε(aj + b)(1 + μ0 p0 )j+1 , μ(tj )|1 + μ0 p0 |j+1

󵄨 󵄨󵄨 󵄨󵄨η(tj )󵄨󵄨󵄨 ≤ ε,

� 151

j ∈ ℕ0 ,

while recalling that (aj + b) ≤ μ(tj ) for tj ∈ 𝕋 and j ∈ ℕ0 , from (3.9). Moreover, for ρ > 0 and θ ∈ [0, 2π), express (1 + μ0 p0 ) = ρeiθ , and thus ep (tn ) = (1 + μ0 p0 )n = ρn einθ ,

n ∈ ℕ0 .

Then we have from (3.10) that for n ∈ ℕ0 , n−1

ψ(tn ) = ερn einθ ∑

j=0

(aj + b) ρj+1

εeinθ { (1−ρ)2 (a(n

− nρ − 1 + ρn ) + b(1 − ρ)(1 − ρn )) = { inθ εne { 2 (a(n − 1) + 2b)

if ρ ≠ 1, if ρ = 1.

Then, ψ satisfies (3.7) for all t = tn ∈ 𝕋 and n ∈ ℕ0 , but ε

| (1−ρ)2 (a(n − nρ − 1) + b(1 − ρ)) { { { { { 󵄨󵄨 󵄨 󵄨󵄨ψ(tn ) − z(tn )󵄨󵄨󵄨 = { + ρn ( ε(a−b(1−ρ)) − z0 )| if ρ ≠ 1, (1−ρ)2 { { { { εn if ρ = 1 {| 2 (a(n − 1) + 2b) − z0 | →∞ as n → ∞ for any choice of z0 ∈ ℂ, since z(tn ) = z0 (1 + μ0 p0 )n = z0 ρn einθ for n ∈ ℕ0 is the general solution of (3.1). Therefore, equation (3.1) is not Ulam stable on isolated time scales with graininess satisfying (3.9), and a 1-periodic coefficient function. Remark 3.3. The linear growth condition on the graininess μ in (3.9) can be significantly weakened to the condition μ(tn ) ≥ anγ + b,

(3.11)

for any real γ > 0.

3.4 Investigating stability of first-order dynamic equations with a 2-periodic coefficient In this section we consider the Ulam stability of the first-order linear homogeneous dynamic equation (3.1) on 𝕋 = {tn }∞ n=0 for all n ∈ ℕ0 with a 2-periodic coefficient p : 𝕋 → ℂ. Here, p is 2-periodic [9, Definition 3.1] if

152 � D. R. Anderson and M. Onitsuka p(tn ) =

μ(tn+2 )p(tn+2 ) , μ(tn )

∀t = tn ∈ 𝕋.

(3.12)

Let p(tk ) = pk , μ(tk ) = μk , and pk ∈ ℂ\{ −1 } for k ∈ {0, 1}. Through iteration of (3.12), one μk sees that p is given by p(tn ) :=

μ0 p0 1 { μ(tn ) μ1 p1

if n ≡ 0 (mod 2),

if n ≡ 1 (mod 2),

n ∈ ℕ0 .

(3.13)

} for k = Remark 3.4 (Exponential function). Let p(tk ) = pk , μ(tk ) = μk , and pk ∈ ℂ\{ −1 μk 0, 1. By the 2-periodic nature of p given in (3.12) and (3.13), the exponential function (3.4) takes the form n

n

{[1 + μ0 p0 ] 2 [1 + μ1 p1 ] 2 ep (tn ) = { n+1 n−1 2 2 {[1 + μ0 p0 ] [1 + μ1 p1 ]

if n ≡ 0 (mod 2), if n ≡ 1 (mod 2).

(3.14)

Note that the conditions pk ∈ ℂ\{ −1 } for k ∈ {0, 1} are really regressive conditions, since μ k

p(tk ) = pk ≠ −1 is equivalent to 1 + μk pk ≠ 0, for k = 0, 1. μk For notational convenience throughout the remainder of this section, define the key constants as Ek := 1 + μk pk ,

k = 0, 1.

(3.15)

It is left to the reader to check that ep indeed satisfies (3.1), with ep (t0 ) = 1. Theorem 3.3. Let 𝕋 = {tn }∞ n=0 be an isolated time scale with graininess function μ > 0 such that limn→∞ tn = ∞. Let p(tk ) = pk , μ(tk ) = μk , and pk ∈ ℂ\{ −1 } for k = 0, 1. μk If μ satisfies (3.8) and p is a 2-periodic function, then (3.1) is Ulam stable if and only if |E0 E1 | = |(1 + μ0 p0 )(1 + μ1 p1 )| ≠ 1, with Ulam constant L = max{|E0 | + 1, |E1 | + 1}

μmax . ||E0 E1 | − 1|

Proof. Recall the criteria for Ulam stability given in Definition 3.1. Since p is 2-periodic in this case, p satisfies (3.12) and has the form (3.13). Let Ek take the form (3.15) for k = 0, 1. (i) First, suppose |E0 E1 | = 1, and let ep be given by (3.14). It follows in this case that |ep (tn )| = |E0 |n (mod 2) , n ∈ ℕ0 . Given arbitrary ε > 0, set m := min{1, |E0 |}, and let ψ(tn ) =

εtn e (t ), M p n

M := max{1, |E0 |},

n ∈ ℕ0 . Then, ψ satisfies

󵄨 ε󵄨 󵄨 󵄨󵄨 Δ 󵄨󵄨ψ (tn ) − p(tn )ψ(tn )󵄨󵄨󵄨 = 󵄨󵄨󵄨ep (σ(tn ))󵄨󵄨󵄨 ≤ ε M

3 Ulam stability and instability of first-order linear 1- and 2-periodic dynamic equations

� 153

for all tn ∈ 𝕋 and n ∈ ℕ0 , so that (3.7) holds, but 󵄨 󵄨󵄨 εt 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 εt 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 εt 󵄨 󵄨 󵄨󵄨 󵄨󵄨ψ(tn ) − z(t)󵄨󵄨󵄨 = 󵄨󵄨󵄨 n ep (tn ) − z0 ep (tn )󵄨󵄨󵄨 = 󵄨󵄨󵄨ep (tn )󵄨󵄨󵄨󵄨󵄨󵄨 n − z0 󵄨󵄨󵄨 ≥ m󵄨󵄨󵄨 n − z0 󵄨󵄨󵄨 → ∞ 󵄨󵄨 M 󵄨󵄨 M 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 M as n → ∞, making (3.1) unstable in the Ulam sense. (ii) Second, suppose |E0 E1 | > 1, where E0 E1 = (1 + μ0 p0 )(1 + μ1 p1 ). For arbitrary ε > 0, suppose a function ψ fulfills (3.7) for all t ∈ 𝕋. Moreover, by (3.14), the exponential function has the form ep (t2m ) = (E0 E1 )m ,

ep (t2m+1 ) = E0 (E0 E1 )m ,

m ∈ ℕ0 ,

and from (3.10) we have that ψ has the form n−1

ψ(tn ) = ψ0 ep (tn ) + ep (tn ) ∑

j=0

μ(tj )η(tj ) ep (tj+1 )

󵄨󵄨 󵄨 󵄨󵄨η(tn )󵄨󵄨󵄨 ≤ ε

,

(3.16)

for all t = tn ∈ 𝕋 and n ∈ ℕ0 , where ψ(t0 ) = ψ0 . From this expression for ψ(tn ), we have n−1 󵄨󵄨 ψ(t ) 󵄨󵄨 1 󵄨󵄨 n 󵄨󵄨 , 󵄨󵄨 󵄨󵄨 ≤ |ψ0 | + εμmax ∑ 󵄨󵄨 ep (tn ) 󵄨󵄨 |e (t )| j=0 p j+1

n ∈ ℕ0 ,

as |η(tj )| ≤ ε, and μ satisfies (3.8) for all t = tj ∈ 𝕋 and j ∈ ℕ0 in this case. These facts allow us to rewrite ψ as ∞

ψ(tn ) = [ψ0 + ∑

j=0

μ(tj )η(tj ) ep (tj+1 )

∞

]ep (tn ) − ep (tn ) ∑

μ(tj )η(tj )

j=n

ep (tj+1 )

,

n ∈ ℕ0 ,

(3.17)

where ∞

z0 := ψ0 + ∑

j=0

μ(tj )η(tj ) ep (tj+1 )

∈ℂ

exists and is a finite number due to ep (tn ), n ∈ ℕ0 , given in (3.14), the assumption |E0 E1 | > 1, and the absolute convergence of the infinite series. It is straightforward to see that z(tn ) := z(t0 )ep (tn ) for tn ∈ 𝕋 is a solution of (3.1), and ∞ μ(t )η(t ) ψ(tn ) j j = ψ0 + ∑ = z0 n→∞ e (t ) e (t ) p n j=0 p j+1

lim

exists. Consequently, z(tn ) = ( lim

n→∞

ψ(tn ) )e (t ), ep (tn ) p n

n ∈ ℕ0 ,

154 � D. R. Anderson and M. Onitsuka and 󵄨󵄨 ∞ μ(t )η(t ) 󵄨󵄨󵄨 j j 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨ψ(tn ) − z(tn )󵄨󵄨 = 󵄨󵄨−ep (tn ) ∑ 󵄨󵄨 󵄨 e (t ) j=n p j+1 󵄨󵄨 󵄨 ∞

1 |e (t )| j=n p j+1

󵄨 󵄨 ≤ εμmax 󵄨󵄨󵄨ep (tn )󵄨󵄨󵄨 ∑

=

(|E1 | + 1) if n ≡ 0 (mod 2), εμmax { |E0 E1 | − 1 (|E0 | + 1) if n ≡ 1 (mod 2)

holds for all t = tn ∈ 𝕋. Therefore, (3.1) is Ulam stable with Ulam constant L = max{|E0 | + 1, |E1 | + 1}

μmax |E0 E1 | − 1

for E0 E1 = (1 + μ0 p0 )(1 + μ1 p1 ) and |E0 E1 | > 1. (iii) Third, suppose 0 < |E0 E1 | < 1, where E0 E1 = (1 + μ0 p0 )(1 + μ1 p1 ). As before, we know that a function ψ that satisfies (3.7) takes the form (3.16) by (3.10). Let z = z(tn ) = ψ0 ep (tn ) for t = tn ∈ 𝕋 be the solution of (3.1) such that z(t0 ) = ψ0 . Then we have n−1

ψ(tn ) − z(tn ) = ep (tn ) ∑

μ(tj )η(tj )

j=0

ep (tj+1 )

,

and thus n−1 1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨ψ(tn ) − z(tn )󵄨󵄨󵄨 ≤ εμmax 󵄨󵄨󵄨ep (tn )󵄨󵄨󵄨 ∑ |e (t )| j=0 p j+1

≤

(|E1 | + 1) if n ≡ 0 (mod 2), εμmax { 1 − |E0 E1 | (|E0 | + 1) if n ≡ 1 (mod 2)

for all t = tn ∈ 𝕋. Therefore, (3.1) is Ulam stable with Ulam constant L = max{|E0 | + 1, |E1 | + 1}

μmax 1 − |E0 E1 |

for 0 < |E0 E1 | < 1, where E0 E1 = (1 + μ0 p0 )(1 + μ1 p1 ). Overall, we have shown that (3.1) is Ulam stable if and only if |E0 E1 | = |(1 + μ0 p0 )(1 + μ1 p1 )| ≠ 1, with Ulam constant μ L = max{|E0 | + 1, |E1 | + 1} |1−|EmaxE || , completing the proof. 0 1

Theorem 3.4. Let 𝕋 = {tn }∞ n=0 be an isolated time scale. Assume p satisfies (3.12) and takes the form (3.13) for all t = tn ∈ 𝕋, i. e., p is 2-periodic. If there exist real constants a > 0 and b ≥ 0 such that (3.9) holds, then (3.1) is not Ulam stable.

3 Ulam stability and instability of first-order linear 1- and 2-periodic dynamic equations

� 155

Proof. Let ε > 0 be a fixed arbitrary constant throughout the proof. Let p(tk ) = pk , } for k = 0, 1. By the definition of Ulam stability given in μ(tk ) = μk , and pk ∈ ℂ\{ −1 μk Definition 3.1, we need to find a function ψ : 𝕋 → ℂ fulfilling (3.7), such that there is no solution z of (3.1) satisfying |ψ(t) − z(t)| ≤ Lε for all t ∈ 𝕋, for any constant L ∈ ℝ. Suppose a function ψ fulfills (3.7) for all t ∈ 𝕋, and has the form (3.10). Take ψ(t0 ) = 0, and η(tj ) =

ε(aj + b)ep (tj+1 ) μ(tj )|ep (tj+1 )|

,

󵄨 󵄨󵄨 󵄨󵄨η(tj )󵄨󵄨󵄨 ≤ ε,

while recalling that (aj + b) ≤ μ(tj ) for tj ∈ 𝕋 from (3.9). Suppose n = 2m and |E0 E1 | = 1. We then have from (3.10) that 2m−1

ψ(t2m ) = εep (t2m ) ∑

j=0

=

(aj + b) |ep (tj+1 )|

εma(E0 E1 )m 1 (m|E0 | + m − 1) + εmb(E0 E1 )m (1 + ). |E0 | |E0 |

Then, ψ satisfies (3.7) for all t = t2m ∈ 𝕋, but 󵄨󵄨 󵄨 󵄨󵄨ψ(t2m ) − z(t2m )󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 εma(E0 E1 )m 1 󵄨 = 󵄨󵄨󵄨 (m|E0 | + m − 1) + εmb(E0 E1 )m (1 + ) − z0 (E0 E1 )m 󵄨󵄨󵄨 󵄨󵄨 |E0 | |E0 | 󵄨󵄨 󵄨 󵄨󵄨 εma 󵄨󵄨 1 󵄨 (m|E0 | + m − 1) + εmb(1 + ) − z0 󵄨󵄨󵄨 = 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 |E0 | |E0 | →∞ as m → ∞ for any choice of z0 ∈ ℂ, since |E0 E1 | = 1, z(t2m ) = z0 (E0 E1 )m is the general solution of (3.1), and a > 0 with b ≥ 0. Suppose n = 2m, m ∈ ℕ, and |E0 E1 | ≠ 1. We then have from (3.10) that 2m−1

ψ(t2m ) = εep (t2m ) ∑

j=0

=

(aj + b) |ep (tj+1 )|

εa(E0 E1 )m (|E0 E1 | − 1)2 |E0 E1 |m × (2m − 1 + |E0 E1 |m + |E1 |(2m − 2 − |E0 |(2m|E1 | + 2m + 1)) + (2 + |E0 |)|E0 E1 |m ) +

εb(E0 E1 )m (1 + |E1 |)(|E0 E1 |m − 1) . (|E0 E1 | − 1)|E0 E1 |m

156 � D. R. Anderson and M. Onitsuka Then, ψ satisfies (3.7) for all t = t2m ∈ 𝕋, but 󵄨 󵄨󵄨 󵄨󵄨ψ(t2m ) − z(t2m )󵄨󵄨󵄨 󵄨󵄨 εa 󵄨 = 󵄨󵄨󵄨 (2m − 1 + |E0 E1 |m + |E1 |(2m − 2 − |E0 |(2m|E1 | + 2m + 1)) 󵄨󵄨 (|E0 E1 | − 1)2 󵄨󵄨m εb(1 + |E1 |)(|E0 E1 |m − 1) 󵄨 + (2 + |E0 |)|E0 E1 |m ) + − z0 |E0 E1 󵄨󵄨󵄨 | 󵄨󵄨 (|E0 E1 | − 1) →∞ as m → ∞ for any choice of z0 ∈ ℂ, since |E0 E1 | ≠ 1 and the coefficient of m is 2(1 + |E1 |)(1 − |E0 E1 |) ≠ 0, where z(t2m ) = z0 (E0 E1 )m is the general solution of (3.1), and a > 0 with b ≥ 0. Suppose n = 2m + 1, m ∈ ℕ, and |E0 E1 | = 1. We then have 󵄨󵄨 󵄨󵄨 ε(am + b)(1 + m + m|E |) 󵄨󵄨 󵄨 󵄨 󵄨 0 − z0 󵄨󵄨󵄨 → ∞ 󵄨󵄨ψ(t2m+1 ) − z(t2m+1 )󵄨󵄨󵄨 = |E0 |󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 |E0 | as m → ∞ for any choice of z0 ∈ ℂ, since |E0 E1 | = 1, z(t2m+1 ) = z0 (E0 E1 )m E0 is the general solution of (3.1), and a > 0 with b ≥ 0. Suppose n = 2m + 1, m ∈ ℕ, and |E0 E1 | ≠ 1. We then have 󵄨󵄨 󵄨 󵄨󵄨ψ(t2m+1 ) − z(t2m+1 )󵄨󵄨󵄨 󵄨󵄨 ε 󵄨 = 󵄨󵄨󵄨 [2am + b + |E0 |(a(2m − 1) + b − |E1 |(2a(m + 1) 󵄨󵄨 (|E0 E1 | − 1)2 + b + (a + b + 2am)|E0 |)

󵄨󵄨m+1 󵄨 + (a − b + |E1 |(2a − b + |E0 |(a + b + b|E1 |)))|E0 E1 |m )] − z0 |E0 󵄨󵄨󵄨 |E1 |m | 󵄨󵄨 →∞ as m → ∞ for any choice of z0 ∈ ℂ, since |E0 E1 | ≠ 1 and the coefficient of m is 2a(1 + |E0 |)(1 − |E0 E1 |) ≠ 0, where z(t2m+1 ) = z0 (E0 E1 )m E0 is the general solution of (3.1), and a > 0 with b ≥ 0. Therefore, equation (3.1) is not Ulam stable on isolated time scales with graininess satisfying (3.9), and a 2-periodic coefficient function. Remark 3.5. The previous theorem can be made stronger by weakening condition (3.9) to (3.11).

3 Ulam stability and instability of first-order linear 1- and 2-periodic dynamic equations

� 157

3.5 Applications to two variable-coefficient equations Let 𝕊 = {sn }∞ n=0 be an arbitrary isolated time scale. In this section, we consider the two variable-coefficient equation α(s)yΔ (s) − β(s)y(s) = 0,

s ∈ 𝕊,

(3.18)

where α : 𝕊 → ℝ is a nonzero real-valued function, but β : 𝕊 → ℂ is a complex-valued function. Definition 3.2 (Ulam stability). Let φ : 𝕊 → ℂ be a perturbation. Equation (3.18) has Ulam stability if and only if there exists a constant L > 0 such that, for arbitrary ε > 0, if a function ζ : 𝕊 → ℂ satisfies α(s)ζ Δ (s) − β(s)ζ (s) = φ(s),

󵄨󵄨 󵄨 󵄨󵄨φ(s)󵄨󵄨󵄨 ≤ ε,

s ∈ 𝕊,

(3.19)

then there exists a solution y : 𝕊 → ℂ of (3.18) such that |ζ (s) − y(s)| ≤ Lε for all s ∈ 𝕊. If (3.18) is Ulam stable, the constant L is called an Ulam stability constant for (3.18) on 𝕊. Theorem 3.5. Let 𝕊 = {sn }∞ n=0 be an isolated time scale with graininess function μ(sn ) > 0. Suppose that there exists an increasing function ϕ : 𝕊 → ℝ such that limn→∞ ϕ(sn ) = ∞ and α(s)ϕΔ (s) = K,

s ∈ 𝕊,

where K is a positive constant. Suppose also that β α

i. e., is 1-periodic, where such that

β(s0 ) α(s0 )

∈

−1 ℂ\{ μ(s }. 0)

β(s) α(s)

=

(3.20) β(s0 )μ(s0 ) α(s0 )μ(s)

holds for all s = sn ∈ 𝕊,

If there exists a positive constant M ∈ (0, ∞)

μ(sn ) ≤ M, α(sn ) then (3.18) is Ulam stable if and only if ρ = |1+

sn ∈ 𝕊, β(s0 )μ(s0 ) | α(s0 )

≠ 1, with Ulam constant Ls =

(3.21) M . |1−ρ|

Proof. Let 𝕊 = {sn }∞ n=0 be an isolated time scale with μ(sn ) > 0 for all n ∈ ℕ0 . Suppose that there exists an increasing function ϕ such that limn→∞ ϕ(sn ) = ∞ and (3.20) holds. Define tn := ϕ(sn ) for all n ∈ ℕ0 . Then 𝕋 = {tn }∞ n=0 is also an isolated time scale with μ(tn ) > 0, and satisfies limn→∞ tn = ∞. Let y be a solution of (3.18). Define z(t) := Ky(ϕ−1 (t)) and p(t) := β(ϕK (t)) for t = ϕ(s) ∈ 𝕋, where ϕ−1 is the inverse function of ϕ. Then, by (3.20), we have −1

zΔ (t) = K

y(ϕ−1 (tn+1 )) − y(ϕ−1 (tn )) tn+1 − tn

158 � D. R. Anderson and M. Onitsuka =K

y(sn+1 ) − y(sn ) ϕ(sn+1 ) − ϕ(sn )

= α(sn )

y(sn+1 ) − y(sn ) sn+1 − sn

= α(s)yΔ (s), and so that

zΔ (t) − p(t)z(t) = α(s)yΔ (s) − β(s)y(s) = 0 for t ∈ 𝕋, that is, (3.18) is transformed into (3.1) with p(t) = and (3.20) hold, we see that p(t) =

β(ϕ−1 (t)) . Since β(s) K α(s)

=

β(s0 )μ(s0 ) α(s0 )μ(s)

β(s0 )μ(s0 )α(sn ) β(s0 )μ(s0 ) p(t0 )μ(t0 ) β(s) = = = K Kα(s0 )(sn+1 − sn ) α(s0 )(ϕ(sn+1 ) − ϕ(sn )) μ(t)

is satisfied for all t ∈ 𝕋. Moreover, by p(t0 ) =

β(s0 ) α(s0 )

−1 ∈ ℂ\{ μ(s } and (3.20), we have ) 0

β(s0 ) −α(s0 ) −1 ≠ = , K Kμ(s0 ) μ(t0 )

−1 and thus, p(t0 ) = p0 ∈ ℂ\{ μ(t }. If there exists a positive constant M ∈ (0, ∞) such 0) that (3.21) holds, then

μ(tn ) = ϕ(sn+1 ) − ϕ(sn ) =

Kμ(sn ) ≤ KM = μmax , α(sn )

Hence, by Theorem 3.1, we can conclude that (3.1) with p(t) = and only if

n ∈ ℕ0 . β(ϕ−1 (t)) K

is Ulam stable if

󵄨 β(s0 )μ(s0 ) 󵄨󵄨󵄨󵄨 󵄨 󵄨 󵄨󵄨 ρ = 󵄨󵄨󵄨1 + μ(t0 )p(t0 )󵄨󵄨󵄨 = 󵄨󵄨󵄨1 + 󵄨 ≠ 1, 󵄨󵄨 α(s0 ) 󵄨󵄨󵄨 with Ulam constant L =

μmax |1−ρ|

=

KM . |1−ρ|

Next we will prove that (3.1) with p(t) = β(ϕK (t)) is Ulam stable, with an Ulam conKM M stant |1−ρ| if and only if (3.18) is Ulam stable, with an Ulam constant |1−ρ| . Assume that Ulam stability for (3.1), with an Ulam constant

−1

KM . |1−ρ|

Let ε > 0 and let ζ : 𝕊 → ℂ sat-

isfy (3.19). Define ψ(t) := Kζ (ϕ (t)) for t = ϕ(s) ∈ 𝕋. Then, by (3.20), we obtain −1

ψΔ (t) = K and thus,

ζ (ϕ−1 (tn+1 )) − ζ (ϕ−1 (tn )) ζ (s ) − ζ (sn ) = α(sn ) n+1 = α(s)ζ Δ (s), tn+1 − tn sn+1 − sn

3 Ulam stability and instability of first-order linear 1- and 2-periodic dynamic equations

η(t) := ψΔ (t) − p(t)ψ(t) = α(s)ζ Δ (s) − β(s)ζ (s) = φ(s)

� 159

(3.22)

for t ∈ 𝕋. Since |η(t)| = |φ(s)| ≤ ε holds for t = ϕ(s) ∈ 𝕋, and (3.1) is Ulam stable on 𝕋, KM there exists a solution z(t) of (3.1) such that |ψ(t)−z(t)| ≤ |1−ρ| ε for t ∈ 𝕋. Let y(s) := z(ϕ(s)) K for s ∈ 𝕊. Then y(s) is a solution of (3.18). Moreover, we obtain KM 󵄨 󵄨 󵄨 󵄨 K 󵄨󵄨󵄨ζ (s) − y(s)󵄨󵄨󵄨 = 󵄨󵄨󵄨ψ(t) − z(t)󵄨󵄨󵄨 ≤ ε. |1 − ρ| M . This says that (3.18) is Ulam stable on 𝕊, with an Ulam constant |1−ρ| Conversely, we assume that Ulam stability holds for (3.18), with an Ulam constant M . Let ε > 0 and suppose ψ : 𝕋 → ℂ satisfies (3.7). Define ζ (s) := ψ(ϕ(s)) for s = |1−ρ| K

ϕ−1 (t) ∈ 𝕊. Then, by (3.20), we obtain (3.22). Since |φ(s)| = |η(t)| ≤ ε holds for s = ϕ−1 (t) ∈ 𝕊, and (3.18) is Ulam stable on 𝕊, there exists a solution y of (3.18) such that M |ζ (s) − y(s)| ≤ |1−ρ| ε for s ∈ 𝕊. Let z(t) := Ky(ϕ−1 (t)) for t ∈ 𝕋. Then z is a solution of (3.1). Moreover, we obtain M 1 󵄨󵄨 󵄨 󵄨 󵄨 ε, 󵄨ψ(t) − z(t)󵄨󵄨󵄨 = 󵄨󵄨󵄨ζ (s) − y(s)󵄨󵄨󵄨 ≤ K󵄨 |1 − ρ|

s = ϕ−1 (t) ∈ 𝕊.

This says that (3.1) is Ulam stable on 𝕋, with an Ulam constant |1 +

KM . |1−ρ|

Summarizing the above, we conclude that (3.18) is Ulam stable if and only if ρ = β(s0 )μ(s0 ) M | ≠ 1, with Ulam constant Ls = |1−ρ| . α(s ) 0

Theorem 3.6. Let 𝕊 = {sn }∞ n=0 be an isolated time scale with graininess function μ(sn ) > 0. Suppose that there exists an increasing function ϕ : 𝕊 → ℝ such that limn→∞ ϕ(sn ) = ∞ and (3.20) holds, where K is a positive constant. Suppose also that β(s0 )μ(s0 )

β(sn ) 1 { α(s0 ) = α(sn ) μ(sn ) { β(s1 )μ(s1 ) { α(s1 )

if n ≡ 0 (mod 2), if n ≡ 1 (mod 2)

(3.23)

β(s )

−1 holds for all sn ∈ 𝕊, i. e., αβ is 2-periodic, where α(sk ) ∈ ℂ\{ μ(s } for k ∈ {0, 1}. If there exists k k) a positive constant M ∈ (0, ∞) such that (3.21) holds, then (3.18) is Ulam stable if and only if |Ẽ 0 Ẽ 1 | ≠ 1, with Ulam constant

Ls = max{|Ẽ 0 | + 1, |Ẽ 1 | + 1}

M , ||Ẽ 0 Ẽ 1 | − 1|

where β(sk )μ(sk ) Ẽ k := 1 + , α(sk )

k = 0, 1.

Proof. Let 𝕊 = {sn }∞ n=0 be an isolated time scale with μ(sn ) > 0 for all n ∈ ℕ0 , and let ϕ be an increasing function such that limn→∞ ϕ(sn ) = ∞ and (3.20) holds. Put tn := ϕ(sn )

160 � D. R. Anderson and M. Onitsuka for all n ∈ ℕ0 . Then tn satisfies limn→∞ tn = ∞, and 𝕋 = {tn }∞ n=0 is also an isolated time

β(ϕ−1 (t)) for K β(ϕ−1 (t)) can be seen that (3.18) is transformed into (3.1) with p(t) = K

scale with μ(tn ) > 0. Set z(t) := Ky(ϕ−1 (t)) and p(t) :=

Theorem 3.5. Using (3.20) and (3.23), we see that

t = ϕ(s) ∈ 𝕋. Then, it in the same way as in

β(sn ) β(s0 )μ(s0 )α(sn ) β(s0 )μ(s0 ) p(t0 )μ(t0 ) = = = K Kα(s0 )(sn+1 − sn ) α(s0 )(ϕ(sn+1 ) − ϕ(sn )) μ(tn )

p(tn ) =

if n ≡ 0 (mod 2), and p(tn ) =

β(s1 )μ(s1 )α(sn ) β(sn ) β(s1 )μ(s1 ) p(t1 )μ(t1 ) = = = K Kα(s1 )(sn+1 − sn ) α(s1 )(ϕ(sn+1 ) − ϕ(sn )) μ(tn )

if n ≡ 1 (mod 2). Moreover, by

β(sk ) α(sk )

p(tk ) =

−1 ∈ ℂ\{ μ(s } and (3.20), we have ) k

β(sk ) −α(sk ) −1 ≠ = , K Kμ(sk ) μ(tk )

−1 for k ∈ {0, 1}. From these facts, p(t) is 2-periodic, where p(tk ) ∈ ℂ\{ μ(t } for k ∈ {0, 1}. If k) there exists a positive constant M ∈ (0, ∞) such that (3.21) holds, then

μ(tn ) = ϕ(sn+1 ) − ϕ(sn ) =

Kμ(sn ) ≤ KM = μmax , α(sn )

Hence, by Theorem 3.3, we can conclude that (3.1) with p(t) = and only if

n ∈ ℕ0 . β(ϕ−1 (t)) K

is Ulam stable if

󵄨 󵄨 |E0 E1 | = 󵄨󵄨󵄨(1 + μ(t0 )p(t0 ))(1 + μ(t1 )p(t1 ))󵄨󵄨󵄨 󵄨󵄨 β(s0 )μ(s0 ) β(s1 )μ(s1 ) 󵄨󵄨󵄨󵄨 󵄨 )(1 + )󵄨󵄨 = 󵄨󵄨󵄨(1 + 󵄨󵄨 󵄨󵄨 α(s0 ) α(s1 ) = |Ẽ 0 Ẽ 1 | ≠ 1,

with Ulam constant L = max{|Ẽ 0 | + 1, |Ẽ 1 | + 1}

μmax KM = max{|Ẽ 0 | + 1, |Ẽ 1 | + 1} . ̃ ̃ ̃ ||E0 E1 | − 1| ||E0 Ẽ 1 | − 1|

By the same technique as Theorem 3.5, we can conclude that (3.1) with p(t) = is Ulam stable, with an Ulam constant max{|Ẽ 0 | + 1, |Ẽ 1 | + 1}

KM ̃ ||E0 Ẽ 1 | − 1|

if and only if (3.18) is Ulam stable, with an Ulam constant

β(ϕ−1 (t)) K

3 Ulam stability and instability of first-order linear 1- and 2-periodic dynamic equations

max{|Ẽ 0 | + 1, |Ẽ 1 | + 1}

� 161

M . ̃ ||E0 Ẽ 1 | − 1|

So, the proof is complete.

3.6 Examples of isolated time scales with 1- or 2-periodic coefficient functions In this section, we present several specific isolated time scales of interest, and relate their Ulam stability or lack thereof to theorems in the previous sections. Example (h-difference equation). Let h > 0, and set 𝕋 = {0, h, 2h, 3h, . . .}. If p is a 1-periodic function on 𝕋, then p(t + h) = p(t) for all t ∈ 𝕋, so that p(t) ≡ p is a constant, due to the constant graininess μ(t) ≡ h. It is known [2, Theorem 2.6] that (3.1) is Ulam stable for p ∈ ℂ\{− h1 } if and only if |1 + hp| ≠ 1, with best Ulam constant L=

1 h = . |1 − |1 + hp|| | Reh (p)|

This result also follows directly from Theorem 3.1 and (3.8), where h = μmax = μ0 . If p is 2-periodic, then p(t + 2h) = p(t) for all t ∈ 𝕋, so p0 p(tn ) = p(nh) := { p1

if n ≡ 0 (mod 2), if n ≡ 1 (mod 2),

for pk ∈ ℂ\{− h1 }, k = 0, 1. By Theorem 3.3, (3.1) is also Ulam stable in this case, if 1 ≠ |(1 + hp0 )(1 + hp1 )|. Example (Two step sizes). Let η, τ > 0 be two step sizes, and let the isolated time scale 𝕋 constructed with them be denoted by 𝕋η,τ := {0, η, η + τ, (η + τ) + η, 2(η + τ), 2(η + τ) + η, . . . }. Then the derivative is z(t+η)−z(t)

{ η zΔ (t) := { z(t+τ)−z(t) { τ

if

t η+τ

∈ ℤ,

if

t−η η+τ

∈ ℤ,

for t ∈ 𝕋η,τ . This time scale has been studied in [1, 4], but only for the constant coefficient case. For the periodic coefficient case, μ(tn ) ≤ μmax := max{η, τ}, whence (3.8) holds. If p is 1-periodic, then

162 � D. R. Anderson and M. Onitsuka if n ≡ 0 (mod 2),

1 p(tn ) = p0 { η

if n ≡ 1 (mod 2),

τ

for p0 ∈ ℂ\{− η1 }. By Theorem 3.1, we have the new result that (3.1) is Ulam stable if and

only if ρ = |1 + ηp0 | ≠ 1, with Ulam constant given by L = p0

p(tn ) = {

p1

max{η,τ} . |1−ρ|

If p is 2-periodic, then

if n ≡ 0 (mod 2), if n ≡ 1 (mod 2),

for p0 ∈ ℂ\{− η1 } and p1 ∈ ℂ\{− τ1 }. By Theorem 3.3, we have the new result that (3.1) is Ulam stable if and only if |(1 + ηp0 )(1 + τp1 )| ≠ 1, with Ulam constant given by L = max{|1 + ηp0 | + 1, |1 + τp1 | + 1}

max{η, τ} . ||(1 + ηp0 )(1 + τp1 )| − 1|

Example (q-difference equation). Let q > 1 and set 𝕋 = {1, q, q2 , q3 , . . .}. If p is a 1-periodic function on 𝕋, then qp(qt) = p(t) for all t ∈ 𝕋. Then p takes the form p(t) =

p(1) , t

t ∈ 𝕋,

−1 for p(1) ∈ ℂ\{ q−1 }. The instability result follows directly from Theorem 3.2 and (3.9), n where μ(q ) ≥ an for a = (q − 1)e ln q for all n ∈ ℕ0 . If p is a 2-periodic function on 𝕋, then q2 p(q2 t) = p(t) for all t ∈ 𝕋. Then p takes the form

p(t) :=

1 p1 { t qpq

if logq t ≡ 0 (mod 2), if logq t ≡ 1 (mod 2),

−1 −1 } and pq ∈ ℂ\{ q(q−1) }. Likewise, that (3.1) is not Ulam stable for any choice for p1 ∈ ℂ\{ q−1

of p1 or pq follows from Theorem 3.4.

Example (Triangular equation). Let ℕ0 = ℕ ∪ {0} = {0, 1, 2, 3, . . .}, and let ∞

𝕋={

n(n + 1) } = {0, 1, 3, 6, 10, . . .} 2 n=0

be the set of triangular numbers. It follows that tn+1 − tn = μ(tn ) = n + 1,

n ∈ ℕ0 .

If p is a 1-periodic function on 𝕋, then ( n+2 )p(tn+1 ) = p(tn ) for all t = tn ∈ 𝕋. Then p takes n+1 the form p(t) =

p0 p 󳨐⇒ p(tn ) = 0 , μ(t) n+1

t ∈ 𝕋,

3 Ulam stability and instability of first-order linear 1- and 2-periodic dynamic equations

� 163

for p0 ∈ ℂ\{−1}. The exponential function is ep (tn ) = (1 + p0 )n for p0 ∈ ℂ\{−1}. Then, (3.1) is not Ulam stable over the triangular numbers by Theorem 3.2 and (3.9), since μ(tn ) = an + b for a = 1 = b for all n ∈ ℕ0 . Example (Fibonacci equation). Let ∞

𝕋={

(1 + √5)n+1 − (1 − √5)n+1 } = {1, 2, 3, 5, 8, 13, . . .} 2n+1 √5 n=1 = {t1 , t2 , t3 , t4 , t5 , t6 , . . .},

the set of Fibonacci numbers, where we have omitted the first 1 to avoid the redundancy of two consecutive 1s. It follows that tn+1 − tn = μ(tn ) = tn−1 ,

n ∈ {2, 3, 4, . . .},

μ(t1 ) = 1.

t

If p is a 1-periodic function on 𝕋, then ( t n )p(tn+1 ) = p(tn ) for all t = tn ∈ 𝕋. This p takes n−1 the form p(t) =

p1 p 󳨐⇒ p(tn ) = 1 , μ(t) tn−1

t ∈ 𝕋,

t0 = 1,

for p1 ∈ ℂ\{−1}. The exponential function is ep (tn ) = (1+p1 )n−1 for p1 ∈ ℂ\{−1}. Then, (3.1) is not Ulam stable over the Fibonacci numbers by Theorem 3.2 and (3.9), since μ(tn ) = tn−1 ≥ 21 n for all n ∈ ℕ0 . Example (Harmonic equation). Let ℕ0 = ℕ ∪ {0} = {0, 1, 2, 3, . . .}, and let n

∞

3 11 25 1 = {0, 1, , , , . . .}, 𝕋 = {Hn = ∑ } j 2 6 12 j=1 n=0

n ∈ ℕ0 ,

be the set of harmonic numbers. It follows that Hn+1 − Hn = μ(Hn ) =

1 ≤ 1, n+1

n ∈ ℕ0 .

n+1 If p is a 1-periodic function on 𝕋, then ( n+2 )p(Hn+1 ) = p(Hn ) for all t = Hn ∈ 𝕋. This p takes the form

p(t) =

p0 󳨐⇒ p(Hn ) = (n + 1)p0 , μ(t)

t ∈ 𝕋,

for p0 ∈ ℂ\{−1}. The exponential function is ep (Hn ) = (1 + p0 )n for p0 ∈ ℂ\{−1}. By Theorem 3.1 and condition (3.8), equation (3.1) is Ulam stable if and only if |1 + p0 | ≠ 1. ρ Additional work shows that (3.1) is Ulam stable with Ulam constant L = ln( ρ−1 ) for ρ > 1,

1 + p0 = ρeiθ , and ep (Hn ) = ρn einθ , and (3.1) is Ulam stable with Ulam constant

164 � D. R. Anderson and M. Onitsuka 1−ρ 1 {max{ln( ρ ), ρ + 2 } L={ ρ 1 1 {max{ln( 1−ρ ), ρ + 2 , 3 +

if 0 < ρ ≤ 21 , ρ 2

+ ρ2 }

if

1 2

≤ ρ < 1,

for 0 < ρ < 1, 1 + p0 = ρeiθ , and ep (Hn ) = ρn einθ . Proof. (i) Suppose 1 + p0 = ρeiθ for any θ ∈ [0, 2π) and any real ρ > 1. For arbitrary ε > 0, suppose ψ satisfies (3.7), with |η(Hn )| ≤ ε for all t = Hn ∈ 𝕋. It follows that ep (Hn ) = ρn einθ , and from (3.10) that ψ has the form η(Hj )

n−1

ψ(Hn ) = ψ0 ρn einθ + ρn einθ ∑

j=0

(j + 1)ρj+1 e

−1

, i(j+1)θ

assuming ∑ f (j) ≡ 0. j=0

From this expression for ψ(Hn ), we have n−1 󵄨󵄨 ψ(H ) 󵄨󵄨 1 󵄨󵄨 n 󵄨󵄨 , 󵄨󵄨 n inθ 󵄨󵄨 ≤ |ψ0 | + ε ∑ j+1 󵄨󵄨 ρ e 󵄨󵄨 j=0 (j + 1)ρ

as |η(Hj )| ≤ ε for all t = Hj ∈ 𝕋. Note that ∞

ρ 1 = ln( ) j+1 ρ −1 j=0 (j + 1)ρ ∑

converges, as in this case ρ > 1. Thus, ψ can be expressed anew as η(Hj )

∞

ψ(Hn ) = [ψ0 + ∑

j=0

(j +

1)ρj+1 e

∞

]ep (Hn ) − ep (Hn ) ∑ i(j+1)θ

j=n

η(Hj )

(j + 1)ρj+1 ei(j+1)θ

,

(3.24)

where ∞

z0 := ψ0 + ∑

j=0

η(Hj )

(j + 1)ρj+1 ei(j+1)θ

∈ℂ

exists and is a finite number due to ep (Hn ) = ρn einθ , and the assumption ρ > 1. It is straightforward to see that z(Hn ) := z(H0 )ep (Hn ) = z0 ρn einθ ,

Hn ∈ 𝕋

is a solution of (3.1), and ∞ η(Hj ) ψ(Hn ) = ψ0 + ∑ = z0 j+1 i(j+1)θ n→∞ e (H ) p n j=0 (j + 1)ρ e

lim

exists. Consequently,

3 Ulam stability and instability of first-order linear 1- and 2-periodic dynamic equations

z(Hn ) = ( lim

n→∞

� 165

ψ(Hn ) )e (H ), ep (Hn ) p n

and 󵄨󵄨 󵄨󵄨 ∞ η(Hj ) 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨ψ(Hn ) − z(Hn )󵄨󵄨󵄨 = 󵄨󵄨󵄨−ep (Hn ) ∑ j+1 ei(j+1)θ 󵄨󵄨 (j + 1)ρ 󵄨󵄨󵄨 󵄨󵄨 j=n ∞

ρ 1 ε 1 = Φ( , 1, n + 1) ≤ ε ln( ) j+1 ρ ρ ρ −1 (j + 1)ρ j=n

≤ ερn ∑

holds for all t ∈ 𝕋, where Φ = Φ(⋅, ⋅, ⋅) is the Lerch transcendent function. Therefore, (3.1) ρ is Ulam stable with Ulam constant L = ln( ρ−1 ) for ρ > 1, 1+p0 = ρeiθ , and ep (Hn ) = ρn einθ .

(ii) Now assume 1 + p0 = ρeiθ for 0 < ρ < 1. As before, we know that a function ψ that satisfies (3.7) takes the form η(Hj )

n−1

ψ(Hn ) = ψ0 ep (Hn ) + ep (Hn ) ∑

j=0

(j + 1)ep (Hj+1 )

by (3.10). Let z = z(Hn ) = ψ0 ep (Hn ) for t = Hn ∈ 𝕋 be the solution of (3.1) such that z0 = ψ0 . Then we have n−1

ψ(Hn ) − z(Hn ) = ep (Hn ) ∑

j=0

η(Hj )

(j + 1)ep (Hj+1 )

,

and thus (using a computer algebra system and the Lerch transcendent Φ) n−1 1 󵄨󵄨 󵄨 n 󵄨󵄨ψ(Hn ) − z(Hn )󵄨󵄨󵄨 ≤ ερ ∑ j+1 (j + 1)ρ j=0

ρ−1 ε 1 ) = − Φ( , 1, n + 1) − ερn ln( ρ ρ ρ 1−ρ 1 {max{ln( ρ ), ρ + 2 } ≤ ε{ ρ 1 1 {max{ln( 1−ρ ), ρ + 2 , 3 +

if 0 < ρ ≤ 21 , ρ 2

+ ρ2 } if

1 2

≤ ρ < 1,

for all t = Hn ∈ 𝕋. Therefore, (3.1) is Ulam stable with Ulam constant 1−ρ 1 {max{ln( ρ ), ρ + 2 } L={ ρ 1 1 {max{ln( 1−ρ ), ρ + 2 , 3 +

if 0 < ρ ≤ 21 , ρ 2

+ ρ2 }

if

1 2

≤ ρ < 1,

for 0 < ρ < 1, 1 + p0 = ρeiθ , and ep (Hn ) = ρn einθ . Overall, we have shown that (3.1) is Ulam stable if |1 + p0 | = ρ ≠ 1, completing the proof.

166 � D. R. Anderson and M. Onitsuka Example. Let ℕ0 = ℕ ∪ {0} = {0, 1, 2, 3, . . .}, and let ∞

n

1 7 29 73 𝕋 = {tn := n + Hn = n + ∑ } = {0, 2, , , , . . .}, j 2 6 12 j=1 n=0

n ∈ ℕ0 ,

where Hn is the nth harmonic number. It follows that tn+1 − tn = μ(tn ) = 1 +

1 n+2 = ≤ 2, n+1 n+1

n ∈ ℕ0 .

)p(tn+1 ) = p(tn ) for all t = tn ∈ 𝕋. This p If p is a 1-periodic function on 𝕋, then ( (n+3)(n+1) (n+2)2 takes the form p(t) =

2p0 2(n + 1)p0 󳨐⇒ p(tn ) = , μ(t) n+2

t ∈ 𝕋,

for p0 ∈ ℂ\{− 21 }. The exponential function is ep (tn ) = (1 + 2p0 )n for p0 ∈ ℂ\{− 21 }. By Theorem 3.2, equation (3.1) is Ulam stable if and only if |1 + 2p0 | ≠ 1. Additional work ρ 1 ) + ρ−1 ) for ρ > 1, 1 + 2p0 = shows that (3.1) is Ulam stable with Ulam constant L = (ln( ρ−1

ρeiθ , and ep (tn ) = ρn einθ , and (3.1) is Ulam stable with Ulam constant 󵄨󵄨 ρ 󵄨󵄨󵄨󵄨 1 󵄨 L = (󵄨󵄨󵄨Log( )󵄨󵄨 + ) 󵄨󵄨 󵄨 ρ−1 󵄨 1−ρ for 0 < ρ < 1, 1 + 2p0 = ρeiθ , and ep (tn ) = ρn einθ .

Proof. (i) Suppose 1+2p0 = ρeiθ for any θ ∈ [0, 2π) and any real ρ > 1. For arbitrary ε > 0, suppose ψ satisfies (3.7), with |η(tn )| ≤ ε for all t = tn ∈ 𝕋. It follows that ep (tn ) = ρn einθ , and from (3.10) that ψ has the form n−1

ψ(tn ) = ψ0 ρn einθ + ρn einθ ∑

j=0

(j + 2)η(tj )

(j + 1)ρj+1 e

, i(j+1)θ

−1

assuming ∑ f (j) ≡ 0.

From this expression for ψ(tn ), we have n−1 |ψ(tn )| 󵄨󵄨󵄨󵄨 ψ(tn ) 󵄨󵄨󵄨󵄨 (j + 2) , = ≤ |ψ | + ε ∑ 󵄨 󵄨 0 󵄨 󵄨 n j+1 n inθ 󵄨 󵄨 ρ 󵄨ρ e 󵄨 j=0 (j + 1)ρ

as |η(tj )| ≤ ε for all t = tj ∈ 𝕋. Note that ∞

(j + 2) ρ 1 = ln( )+ j+1 ρ−1 ρ−1 j=0 (j + 1)ρ ∑

converges, as in this case ρ > 1. Thus, ψ can be expressed anew as

j=0

3 Ulam stability and instability of first-order linear 1- and 2-periodic dynamic equations

(j + 2)η(tj )

∞

ψ(tn ) = [ψ0 + ∑

j=0

(j +

1)ρj+1 e

∞

]ep (tn ) − ep (tn ) ∑ i(j+1)θ

j=n

(j + 2)η(tj )

(j + 1)ρj+1 ei(j+1)θ

,

� 167

(3.25)

where ∞

z0 := ψ0 + ∑

j=0

(j + 2)η(tj )

(j + 1)ρj+1 ei(j+1)θ

∈ℂ

exists and is a finite number due to ep (tn ) = ρn einθ , the assumption ρ > 1, and the absolute convergence of the infinite series. It is straightforward to see that z(tn ) := z(t0 )ep (tn ) = z0 ρn einθ ,

tn ∈ 𝕋

is a solution of (3.1), and lim (

n→∞

∞ (j + 2)η(tj ) ψ(tn ) = z0 ) = ψ0 + ∑ ep (tn ) (j + 1)ρj+1 ei(j+1)θ j=0

exists. Consequently, z(tn ) = ( lim

n→∞

ψ(tn ) )e (t ), ep (tn ) p n

and 󵄨 󵄨󵄨 ∞ (j + 2)η(tj ) 󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨ψ(tn ) − z(tn )󵄨󵄨󵄨 = 󵄨󵄨󵄨−ep (tn ) ∑ j+1 ei(j+1)θ 󵄨󵄨 󵄨󵄨 (j + 1)ρ 󵄨󵄨 j=n 󵄨 ∞

(j + 2) j+1 j=n (j + 1)ρ

≤ ερn ∑

1 ε(n + 2) 1 = ερn B( , n + 2, −1) + Φ( , 1, n + 1) ρ ρ ρ ≤ ε(ln(

ρ 1 )+ ) ρ−1 ρ−1

holds for all t ∈ 𝕋, where B = B(⋅, ⋅, ⋅) is the incomplete Beta function, and Φ = Φ(⋅, ⋅, ⋅) is the Hurwitz–Lerch transcendent function. Therefore, (3.1) is Ulam stable with Ulam ρ 1 constant L = (ln( ρ−1 ) + ρ−1 ) for ρ > 1, 1 + 2p0 = ρeiθ , and ep (tn ) = ρn einθ . (ii) Now assume 1 + 2p0 = ρeiθ for 0 < ρ < 1. As before, we know that a function ψ that satisfies (3.7) takes the form n−1

ψ(tn ) = ψ0 ep (tn ) + ep (tn ) ∑

j=0

(j + 2)η(tj )

(j + 1)ep (tj+1 )

168 � D. R. Anderson and M. Onitsuka by (3.10). Let z = z(tn ) = ψ0 ep (tn ) for t = tn ∈ 𝕋 be the solution of (3.1) such that z0 = ψ0 . Then we have n−1

ψ(tn ) − z(tn ) = ep (tn ) ∑

j=0

(j + 2)η(tj )

(j + 1)ep (tj+1 )

,

and thus (using a computer algebra system, the Hurwitz–Lerch transcendent Φ, and the principal logarithm Log) n−1

(j + 2) 󵄨󵄨 󵄨 n 󵄨󵄨ψ(tn ) − z(tn )󵄨󵄨󵄨 ≤ ερ ∑ (j + 1)ρj+1 j=0 = ε(

−1 + ρn 1 ρ−1 1 − Φ( , 1, n + 1) − ρn Log( )) −1 + ρ ρ ρ ρ

󵄨󵄨 ρ 󵄨󵄨󵄨󵄨 1 󵄨 )󵄨 + ) ≤ ε(󵄨󵄨󵄨Log( 󵄨󵄨 ρ − 1 󵄨󵄨󵄨 1 − ρ for all t = tn ∈ 𝕋. Therefore, (3.1) is Ulam stable with Ulam constant 󵄨󵄨 ρ 󵄨󵄨󵄨󵄨 1 󵄨 L = (󵄨󵄨󵄨Log( )󵄨 + ) 󵄨󵄨 ρ − 1 󵄨󵄨󵄨 1 − ρ for 0 < ρ < 1, 1 + 2p0 = ρeiθ , and ep (tn ) = ρn einθ . Overall, we have shown that (3.1) is Ulam stable if |1 + 2p0 | = ρ ≠ 1, completing the proof. Example (Euler type q-difference equation). Let q > 1, and set 𝕊 = qℕ0 = {1, q, q2 , q3 , . . .}. When α(s) = s and β(s) ≡ β ∈ ℂ, equation (3.18) becomes the first-order linear homogeneous dynamic equation of Euler type, represented by syΔ (s) − βy(s) = 0.

(3.26)

Note from Example 3.6 that simply dividing this equation by s is not Ulam stable. Howq−1 −1 ever, (3.26) is Ulam stable if and only if β ∈ ℂ\{ q−1 }, with Ulam constant Ls = |1−|1+β(q−1)|| . It is known [3, Theorem 1] that the Ulam constant Ls is the best Ulam constant for (3.26) on qℤ . n ∞ Proof. Let ϕ(s) = logq s. Since 𝕊 = {sn }∞ n=0 = {q }n=0 is an isolated time scale with the n n graininess function μ(sn ) = μ(q ) = (q − 1)q > 0 for n ∈ ℕ0 , we see that

lim ϕ(sn ) = lim n = ∞,

n→∞

n→∞

and α(sn )ϕΔ (sn ) = qn

logq qn+1 − logq qn qn+1 − qn

=

1 =K >0 q−1

3 Ulam stability and instability of first-order linear 1- and 2-periodic dynamic equations

� 169

for all sn = s ∈ 𝕊; that is, (3.20) holds. Moreover, β(sn ) β(s0 )μ(s0 ) β = = α(sn ) qn α(s0 )μ(sn ) for all sn = s ∈ 𝕊 means that μ(s )

β(s) α(s)

−1 is 1-periodic. β ∈ ℂ\{ q−1 } says that

β(s0 ) α(s0 )

−1 ∈ ℂ\{ μ(s }. ) 0

From α(sn ) = q − 1 = M for all sn = s ∈ 𝕊, we obtain (3.21). Hence, by Theorem 3.5, we can n conclude that (3.26) is Ulam stable if and only if 󵄨󵄨 β(s0 )μ(s0 ) 󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨 󵄨 ρ = 󵄨󵄨󵄨1 + 󵄨 = 󵄨1 + β(q − 1)󵄨󵄨󵄨 ≠ 1, 󵄨󵄨 α(s0 ) 󵄨󵄨󵄨 󵄨 −1 }, with Ulam constant Ls = that is, β ∈ ℂ\{ q−1

M |1−ρ|

=

q−1 . |1−|1+β(q−1)||

Example (Euler type q-difference equation with variable coefficient β(s)). Let q > 1, and set 𝕊 = qℕ0 = {1, q, q2 , q3 , . . .}. Consider the first-order linear homogeneous dynamic equation syΔ (s) − β(s)y(s) = 0,

(3.27)

where β0

β(s) := {

β1

if logq s ≡ 0 (mod 2), if logq s ≡ 1 (mod 2).

Note from Example 3.6 that dividing this equation by s is not Ulam stable. However, (3.27) −1 } for k ∈ {0, 1}, with Ulam constant is Ulam stable if and only if βk ∈ ℂ\{ q−1 󵄨 󵄨 󵄨 󵄨 Ls = max{󵄨󵄨󵄨(1 + β0 (q − 1))󵄨󵄨󵄨 + 1, 󵄨󵄨󵄨(1 + β1 (q − 1))󵄨󵄨󵄨 + 1} q−1 . × ||(1 + β0 (q − 1))(1 + β1 (q − 1))| − 1| Proof. Let ϕ(s) = logq s. Then, as in the previous example, we get the following facts: 𝕊 is an isolated time scale with the graininess function μ(sn ) = (q − 1)qn > 0 for n ∈ ℕ0 , 1 limn→∞ ϕ(sn ) = ∞, (3.20) and (3.21) hold with K = q−1 and M = q − 1. Moreover, we have {(q − 1)β0 β(sn ) 1 = α(sn ) (q − 1)qn {(q − 1)β 1 { β(s0 )μ(s0 )

1 { α(s0 ) = μ(sn ) { β(s1 )μ(s1 ) { α(s1 )

if n ≡ 0 (mod 2), if n ≡ 1 (mod 2)

if n ≡ 0 (mod 2), if n ≡ 1 (mod 2).

170 � D. R. Anderson and M. Onitsuka This says that (3.23) holds, and β(sk ) α(sk )

β(s) α(s)

−1 is 2-periodic; βk ∈ ℂ\{ q−1 } for k ∈ {0, 1} implies that

−1 ∈ ℂ\{ μ(s } for k ∈ {0, 1}. Using Theorem 3.6, we can conclude that (3.27) is Ulam ) k

stable if and only if

󵄨 󵄨󵄨 󵄨󵄨(1 + β0 (q − 1))(1 + β1 (q − 1))󵄨󵄨󵄨 ≠ 1; −1 that is, βk ∈ ℂ\{ q−1 } for k ∈ {0, 1}, with Ulam constant

󵄨 󵄨 󵄨 󵄨 Ls = max{󵄨󵄨󵄨1 + β0 (q − 1)󵄨󵄨󵄨 + 1, 󵄨󵄨󵄨1 + β1 (q − 1)󵄨󵄨󵄨 + 1} q−1 . × ||(1 + β0 (q − 1))(1 + β1 (q − 1))| − 1| This completes the proof. Example. Set 𝕊 = {1, 22 , 32 , 42 , . . .}. When α(s) = 2√s + 1, s ∈ 𝕊, and β(s) ≡ β ∈ ℂ, s ∈ 𝕊, equation (3.18) becomes the first-order linear homogeneous dynamic equation (2√s + 1)yΔ (s) − βy(s) = 0,

s ∈ 𝕊.

(3.28)

Then (3.28) is Ulam stable if and only if β ∈ ℂ\{−1}, with Ulam constant Ls =

1 . |1−|1+β||

the other hand, if we divide both sides of this equation by α(s), we get the equation yΔ (s) −

β y(s) = 0, 2√s + 1

s ∈ 𝕊.

On

(3.29)

Then (3.29) is not Ulam stable for any β ∈ ℂ. 2 ∞ Proof. Let ϕ(s) = √s, s ∈ 𝕊. Since 𝕊 = {sn }∞ n=0 = {(n + 1) }n=0 is an isolated time scale with the graininess function

μ(sn ) = μ((n + 1)2 ) = 2n + 3 > 0 for n ∈ ℕ0 , we have limn→∞ ϕ(sn ) = limn→∞ n = ∞ and μ(sn ) = α(sn ). So, α(sn )ϕΔ (sn ) = 1 = K > 0 for all sn = s ∈ 𝕊. Thus, (3.20) is satisfied. In addition, β(s0 )μ(s0 ) β(sn ) β = = α(sn ) μ(sn ) α(s0 )μ(sn ) holds for all sn = s ∈ 𝕊. Thus, From

μ(sn ) α(sn )

β(s) α(s)

is 1-periodic, and β ∈ ℂ\{−1} implies

β(s0 ) α(s0 )

−1 ∈ ℂ\{ μ(s }. ) 0

= 1 = M for all sn = s ∈ 𝕊, we get (3.21). Hence, by Theorem 3.5, we can

conclude that (3.28) is Ulam stable if and only if ρ = |1 + β ∈ ℂ\{−1}, with Ulam constant Ls =

M |1−ρ|

=

1 . |1−|1+β||

β(s0 )μ(s0 ) | α(s0 )

= |1 + β| ≠ 1, that is,

3 Ulam stability and instability of first-order linear 1- and 2-periodic dynamic equations

� 171

Next we will show that (3.29) is unstable in the Ulam sense. From μ(sn ) = α(sn ), it follows that p(s) =

β β β = , = 2√s + 1 α(s) μ(s)

s ∈ 𝕊,

that is, p is 1-periodic. Since we have μ(sn ) = 2n + 3, (3.9) holds. By using Theorem 3.2, we see that (3.29) is not Ulam stable, completing the proof. Example. Set 𝕊 = {1, 22 , 32 , 42 , . . .}. Consider the first-order linear homogeneous dynamic equations (2√s + 1)yΔ (s) − β(s)y(s) = 0,

s ∈ 𝕊,

(3.30)

and yΔ (s) −

β(s) y(s) = 0, 2√s + 1

s ∈ 𝕊,

(3.31)

where β0 β(s) := { β1

if √s ≡ 0 (mod 2),

if √s ≡ 1 (mod 2).

Then (3.30) is Ulam stable if and only if βk ∈ ℂ\{−1} for k ∈ {0, 1}, with Ulam constant Ls =

max{|1 + β0 | + 1, |1 + β1 | + 1} . ||(1 + β0 )(1 + β1 )| − 1|

On the other hand, (3.31) is not Ulam stable. Proof. It can be easily shown by using the proofs of Examples 3.6 and 3.6, so we omit it.

3.7 Conclusion and future directions In this paper, we investigate the stability and instability of first-order dynamic equations on isolated time scales with 1- and 2-periodic coefficients. Additionally, we provide examples of isolated time scales with 1- or 2-periodic coefficient functions, and apply the results to these cases, including for h-difference equations, q-difference equations, triangular equations, Fibonacci equations, harmonic equations, and Euler type q-difference equations. In the future, we will extend the results to include the stability and instability of first-order linear dynamic equations on isolated time scales with n-periodic coefficients, and also to higher-order linear dynamic equations on isolated time scales.

172 � D. R. Anderson and M. Onitsuka

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[23] M. Onitsuka, Hyers–Ulam stability of second-order nonhomogeneous linear difference equations with a constant step size, J. Comput. Anal. Appl., 28(1) (2020), 152–165. [24] D. Popa, Hyers–Ulam stability of the linear recurrence with constant coefficients, Adv. Differ. Equ., 2005 (2005), 407076. [25] D. Popa, Hyers–Ulam–Rassias stability of a linear recurrence, J. Math. Anal. Appl., 309 (2005), 591–597. [26] H. Rasouli, S. Abbaszadeh and M. Eshaghi, Approximately linear recurrences, J. Appl. Anal., 24(1) (2018), 81–85. [27] Th. M. Rassias, On the stability of linear mapping in Banach spaces, Proc. Am. Math. Soc., 72 (1978), 297–300. [28] Y. H. Shen, The Ulam stability of first order linear dynamic equations on time scales, Results Math., 72(4) (2017), 1881–1895. [29] Y. H. Shen and Y. J. Li, Hyers–Ulam stability of first order nonhomogeneous linear dynamic equations on time scales, Commun. Math. Res., 35(2) (2019), 139–148. https://doi.org/10.13447/j.16745647.2019.02.05. [30] S. M. Ulam, A Collection of the Mathematical Problems, Interscience, New York, 1960.

Douglas R. Anderson, Martin Bohner, and Guo-Cheng Wu

4 A logarithm on time scales and its uses Abstract: A multivalued logarithm on time scales recently introduced for delta-differentiable functions that never vanish is covered in this chapter. This is accomplished using an extended definition of the cylinder transformation from which the definition of exponential functions on time scales arose. The definition of a logarithm function on arbitrary time scales with familiar and useful properties then follows.

4.1 Prelude to the time scales logarithm The new multivalued logarithm recapitulated here fills a gap for time scales and dynamic equations, namely how to define and represent a logarithm function on time scales with properties that extend the well-known properties of the logarithm function for the continuous case. The purpose of what follows below is to present how the novel multivalued logarithm arises naturally from the cylinder transformation employed in definitions of exponential functions for dynamic equations, and the properties that follow from this extension. The logarithm on general time scales and its development as summarized in this chapter will proceed as follows. The definition of the traditional single-valued cylinder transformation extended to a multivalued cylinder transformation is given in Section 4.2. Useful properties across the circle plus (⊕) and circle dot (⊙) operations are preserved by this transformation, at least for nonvanishing delta-differentiable functions. In Section 4.3, familiar and desired properties of this new logarithm are shown to hold in this more general setting. A similar logarithm for the nabla case is established in Section 4.4. The Cayley cylinder transformation is also considered, in Section 4.5, and is shown to lead to the same exact logarithm. In the literature, this is not the only approach to the logarithm question, so in Section 4.6 we give a summary of the proposed logarithm functions on time scales to date. To conclude, a numerical comparison of the various logarithms on a specific time scale is given in Section 4.7, and several examples on various time scales illustrating the properties of the new definition are also provided. These results may be found in Anderson and Bohner [1]. Douglas R. Anderson, Department of Mathematics, Concordia College, Moorhead, MN 56562, USA, e-mail: [email protected] Martin Bohner, Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA, e-mail: [email protected] Guo-Cheng Wu, Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, PR China, e-mail: [email protected] https://doi.org/10.1515/9783111182971-004

176 � D. R. Anderson et al.

4.2 A new logarithm on time scales The presentation of a new definition of a logarithm for dynamic equations on general time scales starts with some motivation provided by the base definition of the cylinder transformation upon which the exponential functions for dynamic equations are built. The following definition [4, Definition 2.21] (see also Hilger [10, Section 7]) is the original cylinder transformation; the proposed modified cylinder transformation is also presented, in Section 4.5. Any closed subset of the real line may serve as a time scale. Such a subset induces the forward jump operator σ : 𝕋 → 𝕋 given by σ(t) = inf{s ∈ 𝕋 : s > t}, and the graininess function μ(t) := σ(t) − t; see [2–5] for more details. Definition 4.1 (Single-valued cylinder transformation). Fix h > 0, and define the cylinder transformation ξh : ℂh → ℤh by 1

ξh (z) = { h z

Log(1 + zh)

for h ≠ 0, for h = 0,

(4.1)

where ℂ is the set of complex numbers, 1 ℂh = {z ∈ ℂ : z ≠ − }, h

ℤh = {z ∈ ℂ : −

π π < Im(z) ≤ }, h h

(4.2)

and Log represents the principal complex logarithm function. The following definition is [4, Definition 2.25]. Definition 4.2 (Regressive function). A function p : 𝕋 → ℝ is regressive granted 1 + μ(τ)p(τ) ≠ 0

for each τ ∈ 𝕋κ

holds. We will denote via R the set of all rd-continuous and regressive functions p : 𝕋 → ℝ. The following definition is [4, Definition 2.30]. This definition sets the foundation for offering the extended formulation of logarithms to time scales. The new definition is merely the extension to a multivalued function, which requires only a modification of the single-valued cylinder function given above in (4.1). Definition 4.3 (Exponential function). For functions p ∈ R , the time scales exponential function is formulated via t

ep (t, s) = exp(∫ ξμ(τ) (p(τ))Δτ) for s, t ∈ 𝕋; s

here, ξh (z) is the cylinder transformation given in (4.1).

4 A logarithm on time scales and its uses

� 177

Definition 4.4 (Multivalued cylinder transformation). Fix h > 0, and define the multivalued cylinder transformation ζh : ℂh → ℂ by 1

ζh (z) = { h z

log(1 + zh)

for h ≠ 0,

(4.3)

for h = 0,

where the set of complex numbers is ℂ, the set ℂh is given in (4.2), and log represents the multivalued complex logarithm function. Lemma 4.1. Let f , g : 𝕋 → ℂ be Δ-differentiable functions with f , g ≠ 0 on 𝕋, and let the multivalued cylinder transformation ζ be given by (4.3). Then, for fixed τ ∈ 𝕋κ , ζμ(τ) ((

f Δ (τ) g Δ (τ) f Δ gΔ ⊕ )(τ)) = ζμ(τ) ( ) + ζμ(τ) ( ). f g f (τ) g(τ)

Proof. First, note that the simple useful formula f σ = μf Δ + f (suppressing the variable) implies f ΔgΔ f Δ gΔ (fg)Δ f σ g Δ + f Δ g (f + μf Δ )g Δ f Δ f Δ g Δ = = + = + +μ = ⊕ , fg fg fg f f g fg f g by the definition [4, Theorem 2.7] of ⊕. From this, we observe that for fixed τ ∈ 𝕋κ , ζμ(τ) ((

f Δ gΔ ⊕ )(τ)) f g

= ζμ(τ) (

(fg)Δ (τ) ) (fg)(τ)

Δ

(fg) (τ) 1 { μ(τ) log(1 + μ(τ) (fg)(τ) ) ={ Δ (fg) (τ) { (fg)(τ)

for μ(τ) ≠ 0, for μ(τ) = 0

σ

(fg) (τ) 1 { μ(τ) log( (fg)(τ) ) for μ(τ) ≠ 0, ={ Δ f gΔ for μ(τ) = 0 {( f ⊕ g )(τ) σ

f (τ) 1 { μ(τ) log( f (τ) ) + ={ Δ f gΔ {( f + g )(τ) 1

Δ

{ μ(τ) log( f (τ) ={ Δ f (τ) g Δ (τ) { f (τ) + g(τ) = ζμ(τ) ( The proof is complete.

1 μ(τ)

(f +μf )(τ)

σ

(τ) log( gg(τ) )

for μ(τ) ≠ 0, for μ(τ) = 0

)+

1 μ(τ)

Δ

)(τ) log( (g+μg ) for μ(τ) ≠ 0, g(τ)

f Δ (τ) g Δ (τ) ) + ζμ(τ) ( ). f (τ) g(τ)

for μ(τ) = 0

178 � D. R. Anderson et al. Lemma 4.2. Let α ∈ ℝ, and let p : 𝕋 → ℂ be a Δ-differentiable function with p ≠ 0 on 𝕋. For the multivalued cylinder transformation ζ given by (4.3) and for fixed τ ∈ 𝕋κ , ζμ(τ) ((α ⊙

pΔ (τ) pΔ )(τ)) = αζμ(τ) ( ). p p(τ)

Proof. Let α ∈ ℝ, and let p : 𝕋 → ℂ be a Δ-differentiable function with p ≠ 0 on 𝕋. Then [5, Theorem 2.43] yields 1 + μ(α ⊙ f ) = (1 + μf )α on 𝕋κ for f =

pΔ . p

It follows that for fixed τ ∈ 𝕋κ , ζμ(τ) ((α ⊙

pΔ )(τ)) p

1

{ μ(τ) log(1 + μ(τ)(α ⊙ ={ pΔ {(α ⊙ p )(τ)

pΔ )(τ)) p

Δ

p (τ) α 1 { μ(τ) log(1 + μ(τ) p(τ) ) ={ Δ p (τ) {α p(τ)

for μ(τ) ≠ 0, for μ(τ) = 0

for μ(τ) ≠ 0, for μ(τ) = 0

Δ

p (τ) 1 { μ(τ) log(1 + μ(τ) p(τ) ) for μ(τ) ≠ 0, = α{ Δ p (τ) for μ(τ) = 0 { p(τ)

= αζμ(τ) (

pΔ (τ) ). p(τ)

The proof is complete. Definition 4.5 (Logarithm function). Given a Δ-differentiable function p : 𝕋 → ℂ with p ≠ 0 on 𝕋, the multivalued logarithm function on time scales is given by t

ℓp (t, s) = ∫ ζμ(τ) ( s

pΔ (τ) )Δτ p(τ)

for s, t ∈ 𝕋,

where ζh (z) is the multivalued cylinder transformation given in (4.3). Define the principal logarithm on time scales to be t

Lp (t, s) = ∫ ξμ(τ) ( s

pΔ (τ) )Δτ p(τ)

for s, t ∈ 𝕋,

where ξh (z) is the single-valued cylinder transformation given in (4.1).

4 A logarithm on time scales and its uses

� 179

Remark 4.1. According to this definition, if p ≡ constant, then ℓp (t, s) = 0 for each t, s ∈ 𝕋. Thus, this logarithm does not distinguish between either constants or constant multiples of functions. We, moreover, note here that even when we restrict the time scale to 𝕋 = ℝ, the dynamics along the negative and positive real line necessitate the existence of a logarithm with principal and multiple values, making a multivalued logarithm on general time scales both natural and expected, though heretofore unexplored.

4.3 Properties of the logarithm Using the definition of the multivalued logarithm on time scales given above in Definition 2.7, we establish the following properties. Theorem 4.1. If p : 𝕋 → ℂ is a Δ-differentiable function with p ≠ 0 on 𝕋, then exp(Lp (t, s)) = e pΔ (t, s),

t, s ∈ 𝕋.

exp(Lep (t, s)) = ep (t, s),

t, s ∈ 𝕋.

p

In particular, if p ∈ R , then

Proof. Presuming p : 𝕋 → ℂ is a Δ-differentiable function with p ≠ 0 on 𝕋, t

Lp (t, s) = ∫ ξμ(τ) ( s

pΔ (τ) )Δτ. p(τ)

Now, exponentiate both sides and use the definition of ep (t, s), the exponential function. Theorem 4.2 (Logarithm of product, quotient, and power). Presume f , g, p : 𝕋 → ℂ are Δ-differentiable functions with f , g, p ≠ 0 on 𝕋. Then, for s, t ∈ 𝕋 and α ∈ ℝ, we have the following: 1. ℓfg (t, s) = ℓf (t, s) + ℓg (t, s), 2. ℓ f (t, s) = ℓf (t, s) − ℓg (t, s), 3.

g

ℓpα (t, s) = αℓp (t, s).

Proof. Presume f , g, p : 𝕋 → ℂ are Δ-differentiable functions with f , g, p ≠ 0 on 𝕋. Then, for s, t ∈ 𝕋, we have via Lemma 4.1 and its proof that t

ℓfg (t, s) = ∫ ζμ(τ) ( s

(fg)Δ (τ) )Δτ (fg)(τ)

180 � D. R. Anderson et al. t

= ∫ ζμ(τ) (( s

f Δ gΔ ⊕ )(τ))Δτ f g

t

t

s

s

g Δ (τ) f Δ (τ) )Δτ + ∫ ζμ ( )Δτ = ∫ ζμ ( f (τ) g(τ) = ℓf (t, s) + ℓg (t, s). In a similar manner, t

ℓ f (t, s) = ∫ ζμ(τ) ( g

s

t

( gf )Δ (τ) ( gf )(τ)

= ∫ ζμ(τ) (( s

t

)Δτ

f Δ gΔ ⊖ )(τ))Δτ f g t

= ∫ ζμ ( s

f Δ (τ) g Δ (τ) )Δτ − ∫ ζμ ( )Δτ f (τ) g(τ)

= ℓf (t, s) − ℓg (t, s).

s

Let α ∈ ℝ. For the multivalued cylinder transformation ζ given by (4.3) and for fixed τ ∈ 𝕋κ , ζμ(τ) ((α ⊙

pΔ (τ) pΔ )(τ)) = αζμ(τ) ( ) p p(τ)

using Lemma 4.2. Moreover, by [5, Theorem 2.37], we have (pα )Δ pΔ = α ⊙ . pα p Consequently, t

ℓpα (t, s) = ∫ ζμ(τ) ( s

(pα )Δ (τ) )Δτ pα (τ)

t

= ∫ ζμ(τ) ((α ⊙ s

t

= ∫ αζμ(τ) ( s

= αℓp (t, s). The proof is complete.

pΔ )(τ))Δτ p

pΔ (τ) )Δτ p(τ)

4 A logarithm on time scales and its uses

� 181

Theorem 4.3. Let p : 𝕋 → ℝ be a Δ-differentiable function with p ≠ 0 on 𝕋. Then, for s, t ∈ 𝕋, we have σ

p (t) 1 { μ(t) log( p(t) ) for μ(t) ≠ 0, Δ ℓp (t, s) = { Δ p (t) for μ(t) = 0, { p(t)

where Δ-differentiation is with respect to t. Proof. Using the definition of the logarithm and Δ-differentiating with respect to t, ℓpΔ (t, s) = ζμ(t) (

pΔ (t) ) p(t)

Δ

p (t) 1 { μ(t) log(1 + μ(t) p(t) ) ={ Δ p (t) { p(t)

for μ(t) ≠ 0, for μ(t) = 0.

Now substitute μpΔ = pσ − p. The argument is finished.

4.4 The nabla case An analogous logarithm may likewise be defined for the nabla case. Definition 4.6 (Cylinder transformation). For h > 0, define the single-valued cylinder ̂ h → ℤh by transformation ξ̂h : ℂ −1 Log(1 − zh) for h ≠ 0, ξ̂h (z) = { h z for h = 0,

(4.4)

̂ h → ℂ by and the multivalued cylinder transformation ζ̂h : ℂ −1 log(1 − zh) for h ≠ 0, ζ̂h (z) = { h z for h = 0.

Here ℂ is the set of complex numbers, ℤh is in (4.2), 1 Ĉh = {z ∈ ℂ : z ≠ }, h and as before Log represents the principal complex logarithm function. The following definition is [5, Definition 3.4].

(4.5)

182 � D. R. Anderson et al. Definition 4.7 (Regressive function). A function p : 𝕋 → ℝ is ν-regressive granted 1 − ν(t)p(t) ≠ 0

for each t ∈ 𝕋κ

̂ signify the set of all ld-continuous and ν-regressive functions p : 𝕋 → ℝ. holds. Let R The following definition is [5, Definition 3.10]. ̂, the time scales Definition 4.8 (Exponential function). Let t, s ∈ 𝕋. For function p ∈ R nabla exponential function is formulated via t

̂ep (t, s) = exp(∫ ξ̂ν(τ) (p(τ))∇τ), s

where ξ̂h (z) is the single-valued cylinder transformation given in (4.4). We now offer a new definition of logarithms for the nabla case on time scales. Definition 4.9 (Logarithm function). Given a ∇-differentiable function p : 𝕋 → ℝ with p ≠ 0 on 𝕋, the multivalued nabla logarithm function on time scales is given by t

p (τ) ℓ̂p (t, s) = ∫ ζ̂ν(τ) ( )∇τ p(τ) ∇

s

for s, t ∈ 𝕋,

where ζ̂h (z) is the multivalued cylinder transformation given in (4.5), while the principal nabla logarithm is given by t

̂ p (t, s) = ∫ ξ̂ν(τ) ( p (τ) )∇τ L p(τ) ∇

s

for s, t ∈ 𝕋,

where ξ̂h (z) is the single-valued nabla cylinder transformation given in (4.4). Analogous properties to those given in the delta case earlier may be established for the nabla case in a similar manner.

4.5 Logarithms for Cayley-exponential functions Cieśliński introduces another time scales exponential function, named the Cayleyexponential function in [7], defined by t

Ep (t, s) = exp(∫ Ψμ(τ) (p(τ))Δτ), s

(4.6)

4 A logarithm on time scales and its uses

� 183

where p : 𝕋 → ℂ is rd-continuous and satisfies the regressivity condition μ(τ)p(τ) ≠ ±2 for all τ ∈ 𝕋κ , and the modified cylinder transformation Ψ is given by 1 + 21 zh 1 Log( ), h 1 − 21 zh

Ψh (z) =

Ψ0 (z) = z,

(4.7)

for h > 0. Once more, Log represents the principal complex logarithm. Now consider the multivalued function version of (4.7) denoted, i. e., 1 + 21 zh 1 log( ), h 1 − 21 zh

ψh (z) =

ψ0 (z) = z,

(4.8)

where log represents the multivalued complex logarithm. The following Cayley-logarithm function on time scales may then be formulated. Definition 4.10. For a Δ-differentiable function p : 𝕋 → ℂ with p ≠ 0 on 𝕋, the multivalued Cayley-logarithm function on time scales is given by t

caylogp (t, s) = ∫ ψμ(τ) ( s

2pΔ (τ) )Δτ p(τ) + pσ (τ)

for s, t ∈ 𝕋,

where ψh (z) is the multivalued cylinder transformation given in (4.8). Define the principal Cayley-logarithm on time scales to be t

CayLogp (t, s) = ∫ Ψμ(τ) ( s

2pΔ (τ) )Δτ p(τ) + pσ (τ)

for s, t ∈ 𝕋,

where Ψh (z) is the single-valued cylinder transformation given in (4.7). Lemma 4.3. The Cayley-logarithm functions are well-defined functions. Proof. For a Δ-differentiable function p : 𝕋 → ℂ with p ≠ 0 on 𝕋, we need to show that μ(τ)

2pΔ (τ) ≠ ±2, p(τ) + pσ (τ)

in other words, that the regressivity condition holds. The following are equivalent: 2μ(τ)pΔ (τ) pσ (τ) − p(τ) = ±2 ⇐⇒ = ±1, σ p(τ) + p (τ) p(τ) + pσ (τ)

pσ (τ) − p(τ) = ±(p(τ) + pσ (τ)) ⇐⇒ pσ (τ) ∓ pσ (τ) = p(τ) ± p(τ), so that we have either 0 = 2p(τ) or 2pσ (τ) = 0, both contradictions.

184 � D. R. Anderson et al. Theorem 4.4. For a Δ-differentiable function p : 𝕋 → ℂ with p ≠ 0 on 𝕋, caylogp (t, s) = ℓp (t, s)

and

CayLogp (t, s) = Lp (t, s)

for all s, t ∈ 𝕋.

Proof. Consider (4.8). For fixed τ ∈ 𝕋κ with μ(τ) ≠ 0, observe that 1+ 2pΔ (τ) 1 )= log( ψμ(τ) ( σ p(τ) + p (τ) μ(τ) 1−

Δ 1 2p (τ) μ(τ) 2 p(τ)+pσ (τ)

1 2pΔ (τ) μ(τ) 2 p(τ)+pσ (τ)

1+ 1 log( = μ(τ) 1−

μ(τ)pΔ (τ) p(τ)+pσ (τ)

1+ 1 = log( μ(τ) 1−

pσ (τ)−p(τ) p(τ)+pσ (τ) pσ (τ)−p(τ) p(τ)+pσ (τ)

= =

μ(τ)pΔ (τ) p(τ)+pσ (τ)

)

)

)

pσ (τ) 1 log( ) μ(τ) p(τ)

p(τ) + μ(τ)pΔ (τ) 1 log( ) μ(τ) p(τ)

= ζμ(τ) (

pΔ (τ) ) p(τ)

for ζh defined in (4.3). For fixed τ ∈ 𝕋κ with μ(τ) = 0, we have τ = σ(τ) and 2pΔ (τ) pΔ (τ) = . p(τ) + pσ (τ) p(τ) Consequently, ψμ(τ) (

2pΔ (τ) 2pΔ (τ) pΔ (τ) pΔ (τ) ) = = = ζ ( ). μ(τ) p(τ) + pσ (τ) p(τ) + pσ (τ) p(τ) p(τ)

Thus, in either case, we have ψμ(τ) (

pΔ (τ) 2pΔ (τ) ) = ζ ( ). μ(τ) p(τ) + pσ (τ) p(τ)

It follows that t

t

s

s

2pΔ (τ) pΔ (τ) caylogp (t, s) = ∫ ψμ(τ) ( )Δτ = ζ ( )Δτ = ℓp (t, s). ∫ μ(τ) p(τ) + pσ (τ) p(τ) Similarly, we have

(4.9)

4 A logarithm on time scales and its uses

� 185

CayLogp (t, s) = Lp (t, s), completing the proof. Remark 4.2. The previous theorem and proof may be generalized, as we will now show. Let θ ∈ [0, 1], and set ψθh (z) =

1 + (1 − θ)zh 1 log( ), h 1 − θzh

ψθ0 (z) = z.

(4.10)

Then, for a Δ-differentiable function p : 𝕋 → ℂ with p ≠ 0 on 𝕋, and for all τ ∈ 𝕋κ , we have ψθμ(τ) (

pΔ (τ) ) (1 − θ)p(τ) + θpσ (τ) pΔ (τ)

1 + (1 − θ)μ(τ) (1−θ)p(τ)+θpσ (τ) 1 ) = log( pΔ (τ) μ(τ) 1 − θμ(τ) σ (1−θ)p(τ)+θp (τ)

(1 − θ)p(τ) + θpσ (τ) + (1 − θ)μ(τ)pΔ (τ) 1 = log( ) μ(τ) (1 − θ)p(τ) + θpσ (τ) − θμ(τ)pΔ (τ) =

(1 − θ)p(τ) + θpσ (τ) + (1 − θ)(pσ (τ) − p(τ)) 1 log( ) μ(τ) (1 − θ)p(τ) + θpσ (τ) − θ(pσ (τ) − p(τ))

=

pσ (τ) 1 log( ) μ(τ) p(τ)

=

p(τ) + μ(τ)pΔ (τ) 1 log( ) μ(τ) p(τ)

= ζμ(τ) (

pΔ (τ) ) p(τ)

for ζh defined in (4.3). For fixed τ ∈ 𝕋κ with μ(τ) = 0, we have τ = σ(τ) and pΔ (τ) pΔ (τ) = . (1 − θ)p(τ) + θpσ (τ) p(τ) As a result, ψθ0 (

pΔ (τ) pΔ (τ) pΔ (τ) pΔ (τ) )= = = ζ0 ( ). σ σ (1 − θ)p(τ) + θp (τ) (1 − θ)p(τ) + θp (τ) p(τ) p(τ)

Thus, in either case, we have ψθμ(τ) (

pΔ (τ) pΔ (τ) ) = ζμ(τ) ( ) σ (1 − θ)p(τ) + θp (τ) p(τ)

186 � D. R. Anderson et al. for all θ ∈ [0, 1]. Consequently, t

logθp (t, s) := ∫ ψθμ(τ) ( s

t

pΔ (τ) pΔ (τ) )Δτ = ∫ ζμ(τ) ( )Δτ = ℓp (t, s). σ (1 − θ)p(τ) + θp (τ) p(τ) s

This ends the remark.

4.6 Alternative logarithms on time scales As shown in the previous sections of this chapter, the key to arriving at useful logarithm properties is to allow for a multivalued logarithm, as for the 𝕋 = ℝ case. Here, we present the previous definitions of logarithms on time scales, noting that they are all single-valued functions. Moreover, only Definition 4.5 leads to results as given in Theorems 4.1, 4.2, 4.4, and Remark 4.2, justifying this new approach, and emphasizing the advantages of having a function satisfying familiar properties, while ensconced in the more general time scales context. The first logarithm on time scales [11] interprets the integral t

∫

t0

2 Δτ τ + σ(τ)

as a time scales analogue of ln t. This is understandable because if 𝕋 = ℝ, then τ = σ(τ) and t

∫

t0

t

2 2 Δτ = ∫ dτ = ln t − ln t0 . τ + σ(τ) 2τ t0

A recent paper [12] applies iterates of this logarithm to Riemann–Weber-type equations; see also [14]. A second approach [6, Section 3] is to view the slightly different integral t

∫

t0

1 Δτ τ + 2μ(τ)

as the time scales version of ln t, due to the same fact that it reduces to ln t − ln t0 on 𝕋 = ℝ, and as it is part of a solution form to a certain Euler–Cauchy dynamic equation whose differential equation analogue involves the natural logarithm. A third approach [6, Section 4] could be to define a logarithm via

4 A logarithm on time scales and its uses t

Lp (t, t0 ) = ∫ t0

� 187

pΔ (τ) Δτ p(τ)

for Δ-differentiable functions p : 𝕋 → ℝ. Clearly, if p(τ) = τ, then this is t

t

t0

t0

pΔ (τ) 1 Δτ = ∫ Δτ, Lp (t, t0 ) = ∫ p(τ) τ a form that is similar to its continuous analogue for 𝕋 = ℝ. A fourth approach [13] is to take the logarithm to be given by log𝕋 p(t) =

pΔ (t) p(t)

for Δ-differentiable functions p : 𝕋 → ℝ, where the time scale logarithm on ℝ does not play the role of the logarithm, clearly, but rather its derivative. The motivation here is to maintain some attractive algebraic properties of logarithms, and to serve in some sense as an inverse to the exponential function. A fifth approach [15], only for time scales such that 1 ∈ 𝕋, is to define the natural logarithm via t

1 L𝕋 (t) = ∫ Δτ, τ 1

which hearkens back to [6, Section 4]. Here the motivation is clearly that Lℝ (t) = ln t,

L𝕋 (1) = 0,

1 LΔ𝕋 (t) = . t

Other possibilities may be possible but have yet to be explored.

4.7 Examples of logarithms and numerical comparisons Each of the definitions given in the previous section has advantages and drawbacks, and each satisfies some of what one might wish for in a logarithm function. As shown earlier in this work, however, a multivalued logarithm on time scales with a definition based on cylinder transformations is a natural move that leads to nice properties, and has not been introduced until the foundational paper for this chapter. We now consider the following examples.

188 � D. R. Anderson et al. Example. In this example, we compare the values of the various logarithms on the time scale 𝕋 := (−∞, −k] ∪ {−k + 1, −k + 2, . . . , −1, 0, 1, . . . , k − 2, k − 1} ∪ [k, ∞),

k ∈ ℕ.

For p(t) = t on [1, k + 3]𝕋 , we have the following plot and table of comparison for the logarithms on time scales mentioned in the literature to date. As can be seen in Table 4.1, the new definition presented in [1, Definition 2.7] and restated in Definition 2.7 of this chapter leads to a unique and accurate value for this time scale. The comparison of graphs on [1, 8]𝕋 = {1, 2, 3, 4} ∪ [5, 8] is given in Figure 4.1. Table 4.1: Comparison chart of proposed logarithm functions on time scales with the new definition in Definition 2.7. Citation

Logarithm

[11]

∑k−� j=�

[6, Section 3] [6, Section 4] [13] [15] Definition 2.7

∑k−� j=� ∑k−� j=� ∑k−� j=� ∑k−� j=� ∑k−� j=�

� + ln( kt ) �j+� � + ln( kt ) j+� � + ln( kt ) j � + ln( kt ) j � + ln( kt ) j j+� ln( j ) + ln( kt )

Value at t = �

Figure 1 color

1.75692

blue

1.13232

orange

2.26565

green

2.26565

green

2.26565

green

1.79176

red

Figure 4.1: Comparison plot of various logarithms on [1, k + 3]𝕋 for k = 5.

In the rest of this section, we provide numerous examples of the new logarithm from Definition 2.7, for various time scales.

4 A logarithm on time scales and its uses

� 189

Example. For 𝕋 = ℝ, t

t

s

s

p′ (τ) p(t) pΔ (τ) )Δτ = ∫ dτ = log( ), ℓp (t, s) = ∫ ζμ(τ) ( p(τ) p(τ) p(s) where log represents the multivalued complex logarithm function. For 𝕋 = hℤ, ΛΔ (τ) = Δh Λ(τ) :=

Λ(h + τ) − Λ(τ) h

and t

t

h Δ p(jh) pΔ (τ) ℓp (t, s) = ∫ ζμ(τ) ( )Δτ = ∑ ζh ( h )h p(τ) p(jh) s j=

s

−1 h

t −1 h

= ∑ j= hs

hΔ p(jh) 1 log(1 + h )h h p(jh)

t −1 h

t

h p(jh + h) p((j + 1)h) p(t) = ∑ log( ) = log( ∏ ) = log( ). p(jh) p(jh) p(s) s s j= j= h

−1 h

For 𝕋 = qℕ0 , f Δ (τ) = Dq f (τ) :=

f (qτ) − f (τ) (q − 1)τ

and t

ℓp (t, s) = ∫ ζμ(τ) ( s

pΔ (τ) pΔ (τ) )Δτ = ∑ ζ(q−1)τ ( )(q − 1)τ p(τ) p(τ) τ∈[s,t)

(q − 1)τpΔ (τ) 1 log(1 + )(q − 1)τ (q − 1)τ p(τ) τ∈[s,t)

= ∑

= ∑ log( τ∈[s,t)

p(t) p(qτ) ) = log( ). p(τ) p(s)

This ends the example. Example. For real numbers a, b, c, d with a < b < c < d, set 𝕋 = [a, b] ∪ [c, d]. Assume p : 𝕋 → ℂ is differentiable with p ≠ 0 on 𝕋. If s, t ∈ [a, b) or s, t ∈ [c, d], then μ(τ) ≡ 0 for τ ∈ [s, t], so that by the definition of the multivalued cylinder function (4.3),

190 � D. R. Anderson et al. t

ℓp (t, s) = ∫ s

p′ (τ) p(t) dτ = log( ). p(τ) p(s)

Presume without loss of generality that s ∈ [a, b] and t ∈ [c, d]. Then c = σ(b), and t

ℓp (t, s) = ∫ ζμ(τ) ( s

pΔ (τ) )Δτ p(τ)

σ(b)

b

t

= (∫ + ∫ + ∫ )ζμ(τ) ( s

b

σ(b)

pΔ (τ) )Δτ p(τ) σ(b)

p(b) pΔ (τ) p(t) = log( ) + log( ) + ∫ ζμ(τ) ( )Δτ p(s) p(σ(b)) p(τ) b

= log(

p(b) p(t) pΔ (b) ) + log( ) + μ(b)ζμ(b) ( ) p(s) p(c) p(b)

= log(

p(b) p(t) μ(b)pΔ (b) 1 ) + log( ) + μ(b)[ log(1 + )] p(s) p(c) μ(b) p(b)

= log(

p(t) pσ (b) p(b) ) + log( ) + log( ) p(s) p(c) p(b)

= log(

p(b) p(t) p(c) ) + log( ) + log( ) p(s) p(c) p(b)

= log(

p(t) ). p(s)

p(t) Consequently, in all cases, we see that ℓp (t, s) = log( p(s) ) on this time scale as well.

Example. Let 𝕋 = (−∞, −4] ∪ [2, ∞), and p(t) = t 3 . Let t ≥ 2 and s = −5. Then μ(−4) = σ(−4) − (−4) = 2 − (−4) = 6, and the principal logarithm on this time scale is t

Lp (t, s) = Lp (t, −5) = ∫ ξμ(τ) ( −4

−5

2

t

= ( ∫ + ∫ + ∫)ξμ(τ) ( −5

−4

−4

t

−5

2

2

= 3( ∫ + ∫)

(τ 3 )Δ )Δτ τ3 σ(τ)2 + τσ(τ) + τ 2 )Δτ τ3

dτ 22 − 4(2) + (−4)2 + μ(−4)ξμ(−4) ( ) τ (−4)3

4 A logarithm on time scales and its uses

= 3(Log[−4] − Log[−5] + Log[t] − Log[2]) + Log(1 + 6

� 191

12 ) −64

t = 3 ln( ) + iπ, 5 where ln is the natural logarithm and Log represents the principal complex logarithm. Again for sake of comparison, the logarithms in [11] and [6, Section 3] do not apply as they are defined exclusively in terms of p(t) = t, and [15] does not apply as that logarithm requires 1 ∈ 𝕋. If we use the logarithm in [6, Section 4] or [13], we get 3 ln( 2t5 ) − 89 , a real-valued function, as opposed to our principal value of 3 ln( 5t ) + iπ, a complex-valued function. This example justifies our approach. Example. Here is an example of Theorem 4.3. Let t ∈ 𝕋 with t ≠ 0, and set p(t) = t. For s ∈ 𝕋, we have 1

{ μ(t) log( ℓpΔ (t, s) = { 1 {t

σ(t) ) t

for μ(t) ≠ 0, for μ(t) = 0,

where Δ-differentiation is with respect to t. Thus, 1 { { t { { {1 Δ ℓp (t, s) = { h log(1 + ht ) { { { { log(q) { (q−1)t

for 𝕋 = ℝ, for 𝕋 = hℤ, for 𝕋 = qℕ0 ,

where h > 0 and q > 1. See Figure 4.2 for h = 1 and 𝕋 = ℤ. This ends the example. Example. Construct a discrete time scale with two step sizes that alternate, that is, for the two alternating step sizes α, β > 0 with α ≠ β, let 𝕋 := 𝕋α,β = {0, α, (α + β), (α + β) + α, 2(α + β), 2(α + β) + α, 3(α + β), . . . }. Then, for t ∈ 𝕋 and k ∈ ℕ0 = {0, 1, 2, . . .}, we have α

μ(t) = {

β

for t = k(α + β),

for t = k(α + β) + α.

Set p(t) = t. We claim that for t ∈ 𝕋α,β with t ≠ 0, ℓpΔ (t, s)

1 α { α log(1 + t ) ={ β 1 { β log(1 + t )

for t = k(α + β), for t = k(α + β) + α.

192 � D. R. Anderson et al.

Figure 4.2: A plot of the function

1 t

versus Log(1 + 1t ) for Example 4.7.

To verify this, note that ℓpΔ (t, s) =

1 σ(t) log( ) μ(t) t

k(α+β)+α 1 { α log( k(α+β) ) ={ (k+1)(α+β) 1 { β log( k(α+β)+α ) 1 α { α log(1 + t ) ={ β 1 { β log(1 + t )

for t = k(α + β), for t = k(α + β) + α

for t = k(α + β), for t = k(α + β) + α.

This ends the example. Remark. As given above, the first three examples suggest that this new logarithm may be a kind of exact discretization. In other words, by definition it yields the usual logarithm function restricted to the given time scale. This remains an open question for more intricate and general time scales.

4.8 Uses of this logarithm on time scales For trends on time scales generally, see the recent monographs [2, 3, 9] and the paper [8]. For developments associated with the logarithm introduced in [1] and summarized in the earlier sections of this chapter, we highlight the following. Song, Wu, and Wei [16] use the multivalued logarithm on time scales to formulate the Hadamard fractional calculus on time scales, and Wu, Song, and Wang [17] the Caputo–Hadamard fractional

4 A logarithm on time scales and its uses

� 193

differential equations on time scales, including exploring a numerical scheme, asymptotic stability, and chaos. The famous Hadamard derivative [18] is defined by H α a It x(t)

t

=

ds 1 α−1 ∫(log(t) − log(s)) x(s) , Γ(α) s a

t ∈ [a, b].

(4.11)

A “nice” log(t) is a key step to define Hadamard fractional calculus on time scales. The newly proposed logarithm function [1], summarized in this chapter, is strictly monotonous and invertible. Consider a set {t0 , t1 , . . . , tN } satisfying 1 ≤ a = t0 < t1 < ⋅ ⋅ ⋅ < tN−1 < tN = b. The logarithm function can be reduced to the standard one in ℝ. Motivated by the nonequidistant partition idea in numerical methods of general fractional differential equations [19, 20], ℓ (b,s)−ℓ (a,s) we take h = p N p and p(t) = t. The logarithm function and its inverse are reduced to t ℓp (t, s) = log( ) s

and

ℓp−1 (t, s) = set ,

respectively. Then the set becomes tj = ℓp−1 (ℓp (a, s) + jh, s), j = 0, 1, . . . , N, and tj ∈ {a, aeh , ae2h , . . . , ae(N−1)h , aeNh = b}.

(4.12)

So the backward and forward jump operators are ρ(t) = te−h and σ(t) = teh . We denote the isolated time scale by e(hℕ)log a for N → ∞. We now use the falling factorial function α

th := hα

Γ( ht + 1)

Γ( ht + 1 − α)

,

t ∈ (hℕ)αh ,

α∈ℝ

(4.13)

to give Hadamard fractional sum on the time scale. Definition 4.11 ([16]). Suppose f : e(hℕ)log a → ℝ and α > 0. The Hadamard fractional sum of order α is defined by ρα−1 (t)

α−1 ̃ −α f (t) = 1 Δ ∫ (ℓp (t, r) − ℓp (σ(s), r))h ℓpΔ (s, r)f (s)Δs, a Γ(α) a

with t ∈ e(hℕ)log a+αh . Remark. Let f : e(hℕ)log a → ℝ and M − 1 < α ≤ M. If t = aeαh ⋅ e(N−1)h and s = aejh , j = 0, 1, . . . , N − 1, then the fractional sum’s discrete numerical scheme reads

194 � D. R. Anderson et al. α N−1 ̃ −α f (t) = h ∑ Γ(N + α − j − 1) f (aejh ). Δ a Γ(α) j=0 Γ(N − j)

Then the Caputo and Riemann–Liouville Hadamard differences can be defined, respectively. Definition 4.12 ([16]). Suppose f : e(hℕ)log a → ℝ, M ∈ ℕ1 and M − 1 < α ≤ M. The left Hadamard fractional difference of order α is given by ̃ α f (t) = 1 Δ a Γ(−α)

ρ−α−1 (t)

−α−1 Δ ℓp (s, r)f (s)Δs,

∫ (ℓp (t, r) − ℓp (σ(s), r))h a

(4.14)

with t ∈ e(hℕ)log a+(M−α)h . The logarithm function is used as ℓp (t, s) = log( st ) from [1]. The Δ- and ∇-derivatives 1 1 t of ℓp (t, s) with respect to t are ℓpΔ (t, s) = μ(t) log( σ(t) ) and ℓp∇ (t, s) = ν(t) log( ρ(t) ), respect tively. Definition 4.13 ([17]). Let f : e(hℕ)log a → ℝ, M ∈ ℕ1 and M − 1 < α ≤ M. The Caputo– Hadamard fractional difference of order α is given by C ̃α Δa f (t)

1 = Γ(M − α)

ρM−α−1 (t)

M−α−1 Δ ̃ M f (s)Δs, ℓp (s, r)𝕋 Δ

(ℓp (t, r) − ℓp (σ(s), r))h

∫ a

(4.15)

where t ∈ e(hℕ)log a+(M−α)h . The initial value problem C ̃α Δa x(t)

= λx(teαh ),

x(a) = C, 0 < α ≤ 1, t ∈ e(hℕ)log a+(1−α)h ,

(4.16)

has a solution α

x(t) = Cεα (λ, (t − σ(a)) ) if we define a discrete Mittag-Leffler function of Hadamard type α

∞

εα (λ, (t − σ(a)) ) := ∑ λk k=0

kα

(ℓp (t, r) − ℓp (ae(1−kα)h , r))h Γ(kα + 1)

,

t ∈ e(hℕ)log a+h .

(4.17)

The fractional-order sum and difference satisfy many useful properties such as the composition rule and Leibniz sum laws which are similar to those of the continuous case [18]. We call them exact discretization operators. More details and the right-hand side Hadamard fractional difference can be found in [21].

4 A logarithm on time scales and its uses

� 195

Definition 4.14 ([21]). Suppose f : elog b (hℕ) → ℝ, M ∈ ℕ1 , and M − 1 < α ≤ M. The right Hadamard fractional difference of order α is given by ̃ α f (t) =

bΔ

b 1 −α−1 ∑ (ℓp (ρ(s), r) − ℓp (t, r))h ℓp∇ (s, r)f (s)ν(s), Γ(−α) −αh

(4.18)

s=te

with t ∈ elog b−(M−α)h (hℕ) . Using Definition 4.14, we can formulate and solve the following problem. If we consider a terminal value problem of the right fractional Hadamard difference equation ̃ α x(t) = λx(te(1−α)h ),

bΔ

{

̃ α−1 x(be(α−1)h ) = C,

t ∈ elog b−(1−α)h (hℕ) , 0 < α ≤ 1,

(4.19)

bΔ

we will obtain a right Mittag-Leffler solution expressed as α

x(t) = Cεα,α (λ, (ρ(b) − t) ),

t ∈ elog b−h (hℕ) ,

where εα,α (λ, (ρ(b) − t)α ) is defined by α

∞

εα,α (λ, (ρ(b) − t) ) = ∑ λ

m (ℓp (be

m=0

(m+1)(α−1)h

, r) − ℓp (t, r))h

Γ(mα + α)

(m+1)α−1

,

(4.20)

with t ∈ elog b−h (hℕ) . This ends a brief introduction to one of the applications of the logarithm on time scales.

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[10] S. Hilger, Analysis on measure chains – a unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18–56. [11] S. Huff, G. Olumolode, N. Pennington and A. Peterson, Oscillation of an Euler–Cauchy dynamic equation, Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, Discrete Contin. Dyn. Syst. 2003 (2003), 423–431. https://doi.org/10.3934/proc.2003.2003. 423. [12] B. Ito, P. R̆ehák and N. Yamaoka, Applications of iterated logarithm functions on time scales to Riemann–Weber-type equations, Proc. Am. Math. Soc., 148 (2020), 1611–1624. https://doi.org/10.1090/ proc/14812. [13] B. Jackson, The time scale logarithm, Appl. Math. Lett., 21 (2008), 215–221. [14] J. Jekl, Closed-form solutions of second-order linear difference equations close to the self-adjoint Euler type, Math. Methods Appl. Sci., 46 (2022), 5314–5327. https://doi.org/10.1002/mma.8836. [15] D. Mozyrska and D. F. M. Torres, The natural logarithm on time scales, J. Dyn. Syst. Geom. Theories, 7 (2009), 41–48. [16] T. T. Song, G. C. Wu and J. L. Wei, Hadamard fractional calculus on time scales, Fractals, 30 (2022), 2250145. https://doi.org/10.1142/S0218348X22501456. [17] G. C. Wu, T. T. Song and S. Wang, Caputo–Hadamard fractional differential equations on time scales: Numerical scheme, asymptotic stability, and chaos, Chaos, 32 (2022), 093143. https://doi.org/10.1063/5.0098375. [18] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North Holland Mathematical Studies, Amsterdam, 2006. [19] S. D. Zeng, D. Baleanu, Y. R. Bai and G. C. Wu, Fractional differential equations of Caputo–Katugampola type and numerical solutions, Appl. Math. Comput., 315 (2017), 549–554. https://doi.org/10.1016/j.amc.2017.07.003. [20] G. C. Wu, H. Kong, M. Luo, H. Fu and L. L. Huang, Unified predictor–corrector method for fractional differential equations with general kernel functions, Fract. Calc. Appl. Anal., 25 (2022), 648–667. https://doi.org/10.1007/s13540-022-00029-z. [21] J. L. Wei, G. C. Wu and R. Lozi, Right Hadamard fractional differences and integration by parts, Rocky Mt. J. Math. (2022, accepted).

R. Vivek, S. Sivasundaram, D. Vivek, and K. Kanagarajan

5 Qualitative analysis for hybrid fuzzy differential equations involving tempered Ξ-Hilfer fractional derivative on time scales Abstract: This chapter deals with the existence and stability results for hybrid fuzzy differential equations with tempered Ξ-Hilfer fractional derivatives on time scales. Further, we obtain sufficient conditions for the existence and uniqueness of solutions by using a hybrid fixed point theorem. In addition, it demonstrates Ulam-type stability. Finally, we give a suitable example to illustrate our main results.

5.1 Introduction Hilger in 1990 introduced time scales to unify and extend the theory of differential equations, difference equations, and other differential systems defined over nonempty closed subset of real line. It is more realistic to model a phenomenon by a dynamical system that incorporates both continuous and discrete times, namely, as an arbitrary closed set of reals known as a time scale. In addition, the continuous and discrete processes are seen in option-pricing and stock dynamics in finance, robust 3D tracking in shape and motion estimation, frequency of markets and duration of market trading in economics, large-scale models of DNA dynamics. For a basic discussion on fuzzy-valued functions and fuzzy differential equations (FDEs), we refer to [12, 21, 27, 28, 38]. For more details on time scales, we refer to [4, 7, 16, 31, 35, 40]. The theory of fractional differential equations received considerable interest in recent years both in pure mathematics and applications, see [23, 33]. There are several kinds of fractional derivatives such as the Riemann–Liouville (RL), Caputo, Hilfer, Hadamard, and others. A new type of derivative, called the Hilfer fractional derivative (HFD), was defined by R. Hilfer [19]. In 2018, Sousa and Oliveira [24, 36] presented new HFD – the so-called Ξ-HFD was defined. Tempered fractional calculus can be considered as the extension to traditional fractional calculus concepts, multiplying fractional derivatives and integrals by an exponential factor leads to tempered fractional derivatives and integrals. Some properties R. Vivek, K. Kanagarajan, Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore 641020, India, e-mails: [email protected], [email protected] S. Sivasundaram, College of Natural Science, Engineering and Mathematics, Daytona Beach, FL 32114, USA, e-mail: [email protected] D. Vivek, Department of Mathematics, PSG College of Arts & Science, Coimbatore 641014, India, e-mail: [email protected] https://doi.org/10.1515/9783111182971-005

198 � R. Vivek et al. and applications of the tempered fractional calculus are given in [25, 29, 30, 32] and the references therein. In [5, 18, 20, 22, 26, 34], the issue of fuzzy fractional calculus and fractional FDEs has emerged as a significant subject and this new theory became very attractive to many scientists. Significant results from the theory of fuzzy differential equations and their applications have appeared. The concept of Hukuhara differentiability has been correlated with fuzzy RL-differentiability by employing the Hausdorff measure of noncompactness in [6, 10]. The stability analysis of integral and differential equations is important in many applications and for basic results and recent development on Ulam stability of integral and differential equations. Different types of stability such as Ulam–Hyers (UH), generalized UH, Ulam–Hyers–Rassias (UHR), and generalized UHR stability have been given much attention for fuzzy fractional differential equations which involve different types of operators; we refer to a series of papers [1, 14, 37, 41]. Furthermore, hybrid differential equations arise from a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross-section, a three-layer beam, electromagnetic waves or gravity driven flows, and so on, see [8, 15]. One more attractive class of problems involves hybrid fractional differential equations. For some works on this topic, one can refer to [2, 3, 9, 11, 13, 39]. In [17], the following first-order hybrid FDE is discussed: d

v(t)

{ dt ( f (t,v(t)) ) = G(t, v(t)), { d {v(t0 ) = v0 ∈ E ,

t = [0, b] ∈ J,

where f ∈ C(J × E d , E d − {0}) and G ∈ C(J × E d , E d ) are continuous fuzzy functions. They established some fundamental hybrid fuzzy differential inequalities that are useful for establishing the existence of extremal solutions. Motivated by these works, we consider the following hybrid FDE involving tempered Ξ-Hilfer fractional derivative on time scale of the form: p,q,λ

v(t) TH𝕋 {0+ ΔΞ ( f (t,v(t)) ) = G(t, v(t)), 0 < λ < 1, {T𝕋 1−γ,λ v(0) {0+ IΞ ( f (0,v(0)) ) = v0 , γ = p + q − pq, p,q,λ

t ∈ [0, b] = J ⊆ 𝕋,

(5.1)

where TH𝕋 is the tempered Ξ-Hilfer fractional derivative on 𝕋, p ∈ (0, 1), q ∈ [0, 1], 0+ ΔΞ λ ∈ ℝ, E d denotes the space of fuzzy sets on ℝ, v0 ∈ E d is given. Let 𝕋 be a time scale 1−γ,λ and suppose that T𝕋 is the tempered Ξ-RL fractional integral operator of order 1 − γ 0+ IΞ defined on 𝕋, while f ∈ C(𝕋 × E d , E d − {0}) and G ∈ C(𝕋 × E d , E d ) are continuous fuzzy functions.

5 Qualitative analysis for hybrid fuzzy differential equations

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5.2 Preliminaries In this section, we recall some notations, definitions, and results related to time scale calculus, as well as fuzzy calculus, that are used throughout this paper.

5.2.1 Time scale calculus Let 𝕋 be the time scale, i. e., an arbitrary nonempty closed subset of ℝ. Since a time scale 𝕋 is not connected, we need the concept of jump operators. Definition 5.1 ([12]). The mappings ξ, τ : 𝕋 → 𝕋 defined as ξ(t) = inf{s ∈ 𝕋 | s > t}

and τ(t) = sup{s ∈ 𝕋 | s < t},

are called jump operators. Definition 5.2 ([12]). A nonmaximal element t ∈ 𝕋 is said to be right-scattered (rs), if ξ(t) > t and right-dense (rd), if ξ(t) = t. A nonminimal element t ∈ 𝕋 is called leftscattered (ls), if τ(t) < t and left-dense (ld), if τ(t) = t. If 𝕋 has an ls maximum m, then 𝕋k = 𝕋 − {m}. Otherwise 𝕋k = 𝕋. Definition 5.3 ([12]). The mapping μ : 𝕋 → ℝ+ defined by μ(t) = ξ(t) − t is called graininess. When 𝕋 = ℤ, μ(t) = 1, and when 𝕋 = ℝ, μ(t) = 0. Definition 5.4 ([12]). The mapping g : 𝕋 → E d , where E d is a fuzzy space, is called rdcontinuous if (i) it is continuous at each right-dense (rd)t ∈ 𝕋, (ii) at each left-dense point, the left-sided limit g(t − ) exists. Definition 5.5 ([12]). A fuzzy function g : 𝕋 → E d is said to be differentiable at t ∈ 𝕋k , if there exists a δ > 0 such that for any ε > 0, there exists a neighborhood N𝕋 (t, δ) satisfying 󵄨 󵄨 D[g(ξ(t)) ⊖gH g(s), g Δ (t)(ξ(t) − s)] ≤ ε󵄨󵄨󵄨ξ(t) − s󵄨󵄨󵄨,

for all s ∈ N𝕋 (t, δ),

g(ξ(t))⊖ g(t)

gH . In this case, g Δ (t) is called the delta generalized Hukuhara where g Δ (t) = μ(t) derivative (ΔgH -derivative) of g at t. Moreover, g is said to be delta generalized Hukuhara differentiable (ΔgH -differentiable) on 𝕋k , if g Δ (t) ∈ E d exists for all t ∈ 𝕋k . The fuzzy function g Δ : 𝕋k → E d is then called the ΔgH -derivative of g on 𝕋k .

Definition 5.6 ([12]). A fuzzy function g : 𝕋 → E d is called regulated provided its rightsided limits exist (finite) at all right-dense (rd) points in 𝕋 and its left-sided limits exist (finite) at all left-dense points in 𝕋, and g is said to be rd-continuous if it continuous at

200 � R. Vivek et al. all right-dense points in 𝕋 and its left-sided limits exists (finite) at all left-dense points in 𝕋. We denote the set of rd-continuous functions from 𝕋 to E d by Crd [𝕋, E d ]. Definition 5.7 ([12]). Assume that g : 𝕋 → E d is a continuous fuzzy function on 𝕋 and ΔgH -differentiable on 𝕋k . If g Δ (t) ≥ 0 for all t ∈ 𝕋k , then g is called nondecreasing on 𝕋. If g ΔgH (t0 ) ≤ 0 for all t ∈ 𝕋k then g is called nonincreasing on 𝕋. Theorem 5.1 ([21]). Let g : 𝕋 → E d be a fuzzy function and t ∈ 𝕋k . Then we have the following: (i) If g is ΔgH -differentiable at t, then g is continuous at t. (ii) If g is continuous at t and t is rs, then g is ΔgH -differentiable at t with g Δ (t) =

g(ξ(t)) ⊖gH g(t) μ(t)

.

(iii) If t is rd, then g is ΔgH -differentiable at t if and only if the limits lim+

g(t + h) ⊖gH g(t) h

h→0

lim+

,

g(t) ⊖gH g(t − h) h

h→0

exist. In this case lim+

g(t + h) ⊖gH g(t) h

h→0

= lim+

g(t) ⊖gH g(t − h) h

h→0

= g Δ (t).

(iv) If g is ΔgH -differentiable at t, then g(ξ(t)) ⊖gH g(t) = μ(t)g Δ (t). Theorem 5.2. Let g : [a, b]𝕋 → E d be a continuous fuzzy function on [a, b]𝕋 . If g is ΔgH differentiable on [a, b)𝕋 such that g Δ is continuous on [a, b)𝕋 , then b

∫ g Δ (t)Δ(t) = g(b) ⊖gH g(a). a

Remark 5.1. If g is nondecreasing on [a, b]𝕋 , then Eq. (5.2) is equivalent to b

g(b) = g(a) + ∫ g Δ (t)Δt, a

and if g is nonincreasing, then Eq. (5.2) is equivalent to b

g(b) = g(a) ⊖gH (−1) ∫ g Δ (t)Δt. a

(5.2)

5 Qualitative analysis for hybrid fuzzy differential equations

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Theorem 5.3. Suppose a, b ∈ 𝕋, a < b, and g(t) is continuous on [a, b]. Then b

b

Δ

∫ g (t)Δt = [ξ(a) − a]g(a) + ∫ g(t)Δt. a

ξ(a)

Theorem 5.4. Suppose 𝕋 is a time scale and g is an increasing continuous function on [a, b]𝕋 . If g is the extension of g1 to the real interval [a, b] given by g1 (s),

g(s) = {

g1 (t),

s ∈ 𝕋, s ∈ (t, ξ(t)) ∉ 𝕋,

then b

b

∫ g1 (t)Δt ≤ ∫ g(t)dt. a

(5.3)

a

5.2.2 Fuzzy calculus Definition 5.8 ([34]). Let Pk (ℝ) denote the family of all nonempty compact convex subsets of ℝ, then the Hausdorff metric dH is defined by dH (A, B) = max{sup inf ‖y − z‖, sup inf ‖y − z‖}, y∈A z∈B

z∈B y∈A

A, B ∈ Pk (ℝ),

where ‖ ⋅ ‖ denotes the usual Euclidean norm on ℝ. Definition 5.9 ([34]). Let E d denote the class of fuzzy subsets of the real axis v : ℝ → [0, 1], which satisfy the following conditions: (1) v is normal, i. e., there exists y0 ∈ ℝ with v(y0 ) = 1; (2) v is a convex fuzzy set, i. e., v(ωy + (1 − ω)z) ≥ min{v(y), v(z)}, for all ω ∈ [0, 1] and y, z ∈ ℝ; (3) v is upper semicontinuous on ℝ; (4) the closure of {y ∈ ℝ : v(y) > 0}, denoted by [v]0 , is compact. We call E d the space of fuzzy numbers or a fuzzy set; obviously, E d ⊂ Pk (ℝ). Definition 5.10 ([34]). Let E d represent the fuzzy set v : ℝ → [0, 1] such that [v]α ∈ Pk (ℝ) for all α ∈ [0, 1], where [v]α = {y ∈ ℝ | v(y) ≥ α} for 0 < α ≤ 1, and [v]0 = cl{y ∈ ℝ : v(y) > 0}. The notation [v]α = [vαl , vαr ] denotes the α-level set of v. We refer to vl , vr as the left and right branches of v.

202 � R. Vivek et al. Definition 5.11 ([26]). Let v1 and v2 be two fuzzy numbers defined on E d and ω ∈ ℝ. Due to Zadeh’s extension principle, v1 + v2 and ωv1 are in E d and are defined as [v1 + v2 ]α = [v1 ]α + [v2 ]α , [ωv1 ]α = ω[v1 ]α ,

for all α ∈ [0, 1],

where [v1 ]α + [v2 ]α is the usual addition of two intervals of ℝ and ω[v1 ]α is the usual scalar product of a number and a subset of ℝ. Definition 5.12 ([26]). The Hausdorff distance between fuzzy numbers v1 , v2 ∈ E d is given as D[v1 , v2 ] = sup {dH ([v1 ]α , [v2 ]α )}, α∈[0,1]

where dH ([v1 ]α , [v2 ]α ) is the Hausdorff distance between two sets [v1 ]α , [v2 ]α of E d . Definition 5.13 ([22]). Let g : [a, b] → E d be measurable and bounded. The integral of g b

over [a, b], denoted by ∫a g(t)dt, is defined level-wise by the expression b

α

b

α

[∫ g(t)dt] = ∫[g(t)] dt a

a b

α

̄ | ḡ : [a, b] → ℝ is a measurable selection of [g(⋅)] }. = {∫ g(t)dt a

Definition 5.14 ([10]). The generalized Hukuhara difference (gH-difference) of two fuzzy numbers v1 , v2 ∈ E d is v3 ∈ E d , defined as v1 = v2 + v3 ,

or

v1 ⊖gH v2 = v3 ⇐⇒ {

v2 = v1 + (−1)v3 .

(5.4)

Definition 5.15. The delta generalized Hukuhara derivative (ΔgH -derivative) of a fuzzy function g : [a, b] → E d at t is defined as g Δ (t) = lim

h→0

g(t + h) ⊖gH g(t) h

.

If g Δ (t) ∈ E d , then g is said to be ΔgH -differentiable at t. Also, g is [(i)-ΔgH ]-differentiable at t if (i) g Δ (t; α) = [(glα )Δ (t, α), (grα )Δ (t, α)] and g is [(ii)-ΔgH ]-differentiable at t if (ii) g Δ (t; α) = [(grα )Δ (t, α), (glα )Δ (t, α)], where α ∈ (0, 1). Let [0, b] ⊂ ℝ be an interval and suppose Ξ ∈ C 1 ([0, b], ℝ+ ) is such that Ξ is increasing and positive-defined, as well as ΞΔ (t) ≠ 0 for every t ∈ [0, b]. We consider the weighted space C1−γ,Ξ ([0, b], E d ) of fuzzy functions g on (0, b] defined by

5 Qualitative analysis for hybrid fuzzy differential equations

1−γ

C1−γ,Ξ ([0, b], E d ) = {g : (0, b] → E d : (Ξ(t) − Ξ(0))

� 203

g(t) ∈ C([0, b], E d )},

with 0 < γ ≤ 1 and the norm 1−γ

̂ = max D[(Ξ(t) − Ξ(0)) ‖g‖C1−γ,Ξ ([0,b],Ed ) = D[g(t), 0] t∈[0,b]

̂ g(t), 0].

We denote by C([0, b], E d ) the space of all continuous fuzzy functions on [0, b]. Definition 5.16. Suppose 𝕋 is a time scale and [0, b] an interval of 𝕋. Let g be an integrable fuzzy-valued function defined on [0, b]. Then the tempered Ξ-RL fuzzy fractional integral on time scales of order p of the fuzzy-valued function g with respect to Ξ is defined by p,λ

(T𝕋 0+ IΞ g)(t) =

t

1 p−1 ∫ ΞΔ (s)(Ξ(t) − Ξ(s)) e−λ(Ξ(t)−Ξ(s)) g(s)Δs. Γ(p)

(5.5)

0

Definition 5.17. Suppose 𝕋 is a time scale and [0, b] an interval of 𝕋. Then, the tempered Ξ-Hilfer fuzzy fractional derivative on time scales of order p ∈ (0, 1) and type q ∈ [0, 1] of the fuzzy-valued function g ∈ C 1 ([0, b], E d ) is defined by p,q,λ

(TH𝕋 0+ ΔΞ where 𝕋 Δ =

q(1−p),Ξ

g)(t) = (T0+ IΞ

)(

Δ

ΞΔ (t)

+ λ)(T𝕋 0+ IΞ

(1−q)(1−p),λ

g)(t),

d . dt

Lemma 5.1. Let 𝕋 be a time scale. Let also p ∈ (0, ∞), λ ∈ ℝ, and ν ∈ (−1, ∞). Then, we have T𝕋 p,λ 0+ IΞ (Ξ(t)

ν

ν+p

− Ξ(0)) = e−λ(Ξ(t)−Ξ(0)) (Ξ(t) − Ξ(0))

Γ(ν + 1) Γ(ν + p + 1)

× 1 G1 (ν + 1; ν + p + 1; λ(Ξ(t) − Ξ(0))), TR𝕋 p,λ 0+ ΔΞ (Ξ(t)

ν

ν−p

− Ξ(0)) = e−λ(Ξ(t)−Ξ(0)) (Ξ(t) − Ξ(0))

Γ(ν + 1) Γ(ν − p + 1)

× 1 G1 (ν + 1; ν − p + 1; λ(Ξ(t) − Ξ(0))). Proof. Using the expression for the Ξ-tempered integral as a conjugation of the Ξ-RL ν integral, together with the functions of the form (Ξ(t)−Ξ(0)) , we have Γ(ν+1) T𝕋 p,λ 0+ IΞ (Ξ(t)

ν

− Ξ(0))

ν

p,λ

λ(Ξ(t)−Ξ(0)) = e−λ(Ξ(t)−Ξ(0))𝕋 (Ξ(t) − Ξ(0)) ) 0+ IΞ (e p,λ

(λ(Ξ(t)) − Ξ(0))l ν (Ξ(t) − Ξ(0)) ) l! l=0 ∞

= e−λ(Ξ(t)−Ξ(0))𝕋 0+ IΞ (∑

204 � R. Vivek et al. λl 𝕋 p,λ ν+l 0+ IΞ (Ξ(t) − Ξ(0)) l! l=0 ∞

= e−λ(Ξ(t)−Ξ(0)) ∑

λl Γ(ν + l + 1) ν+p+l (Ξ(t) − Ξ(0)) l! Γ(ν + l + p + 1) l=0 ∞

= e−λ(Ξ(t)−Ξ(0)) ∑

ν+p

= e−λ(Ξ(t)−Ξ(0)) (Ξ(t) − Ξ(0)) ×{ =

Γ(ν + 1) Γ(ν + p + 1)

Γ(ν + p + 1) ∞ Γ(ν + 1 + l) (λ(Ξ(t) − Ξ(0))l ) } ∑ Γ(ν + 1) l=0 Γ(ν + p + 1 + l) l!

Γ(ν + 1) ν+p (Ξ(t) − Ξ(0)) e−λ(Ξ(t)−Ξ(0)) Γ(ν + p + 1) × 1 G1 (ν + 1; ν + p + 1; λ(Ξ(t) − Ξ(0))).

This proves the first of the stated identities. The identity for the Ξ-tempered derivative of RL type can be proved similarly by direct calculation. Lemma 5.2. Let g ∈ C 1 ([0, b], E d ), 0 < p < 1, and 0 ≤ q ≤ 1. Then, we have:

(i)

T𝕋 p,λ TH𝕋 p,q,λ 0+ IΞ 0+ ΔΞ g(t)

= g(t) − e−λ(Ξ(t))

(ii)

(Ξ(t) − Ξ(0))γ−1 T𝕋 1−γ,λ λΞ(t) [0+ IΞ (e g(t))]t=0 , Γ(γ)

with γ = p + q(1 − p). TH𝕋 p,q,λ T𝕋 p,λ 0+ ΔΞ 0+ IΞ g(t)

= g(t).

Theorem 5.5 (Gronwall’s inequality, [30]). Let p > 0, λ ∈ ℝ and let X (t), Y (t) be two integrable functions and Z (t) be a continuous function on [0, b]. Assume that X (t), Y (t) are nonnegative, and Z (t) is nonnegative and nondecreasing. If t

′

X (t) ≤ Y (t) + Z (t) ∫ Ξ (s)(Ξ(t) − Ξ(s))

p−1 −λ(Ξ(t)−Ξ(s))

e

X (s)ds,

t ∈ [0, b],

0

then t ∞

[Z(t)Γ(p)]k ′ pk−1 −λ(Ξ(t)−Ξ(s)) Ξ (s)(Ξ(t) − Ξ(s)) e Y (s)ds. Γ(pk) k=1

X (t) ≤ Y (t) + ∫ ∑ 0

Let Y be a nondecreasing function. Then p

X (t) ≤ Y (t)Ep {Z (t)Γ(p)(Ξ(t) − Ξ(0)) }.

Lemma 5.3 (Hybrid fixed point theorem, [15]). Let S be a nonempty closed convex and bounded subset of a Banach algebra Y and let P : Y → Y , Q : S → Y be two operators

5 Qualitative analysis for hybrid fuzzy differential equations

� 205

such that (i) P is Lipschitzian with Lipschitz constant β; (ii) Q is completely continuous; (iii) v = PvQv̄ ↔ v ∈ S for every v̄ ∈ S; (iv) Lβ < 1, where L = sup{‖Q(s)‖ : v̄ ∈ S}. Then the operator equation v = PvQv has a solution in S.

5.3 Main results Lemma 5.4. Assume that J = [0, b] ⊆ 𝕋. Let γ = p + q − pq, where p ∈ (0, 1) and q ∈ [0, 1]. Let also G : J × E d → E d be a function such that G(⋅, v(⋅)) ∈ C1−γ,Ξ (J, E d ) for any v ∈ C1−γ,Ξ (J, E d ). A fuzzy function v is a solution of the problem (5.1) if and only if v satisfies the following integral equation: v(t) = f (t, v(t))(

v0 e−λ(Ξ(t)−Ξ(0)) γ−1 (Ξ(t) − Ξ(0)) Γ(γ) t

⊖ (−1)

1 p−1 ∫ ΞΔ (s)(Ξ(t) − Ξ(s)) e−λ(Ξ(t)−Ξ(s)) G(s, v(s))Δs). Γ(p)

(5.6)

0

Proof. Let v ∈ C1−γ,Ξ (J, E d ) be a solution of the problem (5.1). We will prove that v is p,λ

also a solution of the integral equation (5.6). Applying T𝕋 0+ IΞ on both sides of the first equation of the problem and using Lemma 5.2, we obtain e−λ(Ξ(t)−Ξ(0)) (Ξ(t) − Ξ(0))γ−1 T𝕋 1−γ,λ v(t) v(t) − [0+ IΞ ( )] f (t, v(t)) Γ(γ) f (t, v(t)) t=0 p,λ = T𝕋 0+ IΞ G(t, v(t)).

(5.7)

Hence, we get the integral equation (5.6). In the latter equation we have used that T𝕋 1−γ,λ ( 0+ IΞ

v(0) ) = v0 . f (0, v(0))

(5.8)

Thus v is a solution of the integral equation (5.6). Conversely, assume that v ∈ C1−γ,Ξ (J, E d ) is a solution of Eq. (5.6). Then Eq. (5.7) becomes v e−λ(Ξ(t)−Ξ(0)) v(t) γ−1 =( 0 (Ξ(t) − Ξ(0)) f (t, v(t)) Γ(γ) t

1 p−1 ⊖ (−1) ∫ ΞΔ (s)(Ξ(t) − Ξ(s)) e−λ(Ξ(t)−Ξ(s)) G(s, v(s))Δs). Γ(p) 0

(5.9)

206 � R. Vivek et al. p,q,λ

Applying 0TH𝕋 ΔΞ on both sides of the above equation and using + Ξ(0))γ−1 = 0, together with Lemma 5.2, we obtain TH𝕋 p,q,λ 0+ ΔΞ [

TH𝕋 p,q,λ 0+ ΔΞ (Ξ(t)

−

v e−λ(Ξ(t)−Ξ(0)) TH𝕋 p,q,λ v(t) γ−1 ]=( 0 0+ ΔΞ (Ξ(t) − Ξ(0)) f (t, v(t)) Γ(γ) p,q,λ T𝕋 p,λ 0+ JΞ G(t, v(t)))

⊖ (−1)TH𝕋 0+ ΔΞ = G(t, v(t)).

1−γ,λ

Finally, we need to verify that the initial condition T𝕋 0+ IΞ (5.1) also is satisfied. From Eq. (5.9), we have T𝕋 1−γ,λ [ 0+ IΞ

v(0) ( f (0,v(0)) ) = v0 in the problem

v v(t) 1−γ+p,λ ] = ( 0 ⊖ (−1)T𝕋 G(t, v(t))). 0+ IΞ f (t, v(t)) Γ(γ)

For this purpose, we put t = 0 and then the above equation reduces to T𝕋 1−γ,λ [ 0+ IΞ

v(0) ] = v0 , f (t, v(0))

which is the initial condition of (5.1). This completes the proof. For establishing the main results, we will impose the following hypotheses on f and G: (H1) The functions f : J × E d → E d − {0} and G : J × E d → E d are continuous and there exists a bounded function p1 : J → ℝ+ such that ̄ ̄ D[f (t, v(t)), f (t, v(t))] ≤ p1 (t)D[v(t), v(t)], ̂ for v, v̄ ∈ E d for every t ∈ J. with p∗1 = supt∈J D[p1 (t), 0], (H2) There exist a function q1 ∈ C(J, ℝ+ ) and a continuous nondecreasing function U : [0, ∞) → [0, ∞) such that ̂ ≤ q1 (t)U(D[v, 0]), ̂ D[G(t, v(t)), 0] ̂ for v ∈ E d , for every t ∈ J. with q1∗ = supt∈J D[q1 (t), 0], (H3) There exists a constant L1 > 0 such that ̄ ̄ ≤ L1 D[v, v], D[f (t, v(t)), f (t, v(t))]

for every v, v̄ ∈ E d ,

̂ ≤ M1 . and we put D[f (t, v(t)), 0] (H4) There exists a constant L2 > 0 such that ̄ ̄ D[G(t, v(t)), G(t, v(t))] ≤ L2 D[v, v],

for every v, v̄ ∈ E d ,

5 Qualitative analysis for hybrid fuzzy differential equations

� 207

̂ ≤ M2 . and we put D[G(t, v(t)), 0] + (H5) Let φ ∈ C1−γ,Ξ (J, ℝ ) be a nondecreasing function. Then there exists K > 0 such that T𝕋 p,λ 0+ JΞ φ(t)

≤ Kφ(t).

Theorem 5.6. Assume that (H1) and (H2) are satisfied. Then, the problem (5.1) has a solution v ∈ C1−γ,Ξ (J, E d ) provided p∗1 (

v0 (Ξ(b) − Ξ(0))1−γ+p −λ(Ξ(b)−Ξ(0)) ∗ + e q1 U(r)) < 1. Γ(γ) Γ(p + 1)

(5.10)

Proof. Define a subset S of C1−γ,Ξ (J, E d ) by S = {v ∈ C1−γ,Ξ (J, E d ) | ‖v‖ ≤ r},

(5.11)

where r=K(

v0 (Ξ(b) − Ξ(0))1−γ+p −λ(Ξ(b)−Ξ(0)) ∗ + e q1 U(r)), Γ(γ) Γ(p + 1)

and K is bounded on f . Obviously, S is a closed, convex, and bounded subset of C1−γ,Ξ (J, E d ). By Lemma 5.4, the problem (5.1) is equivalent to the integral equation (5.6). Let us consider two operators P : C1−γ,Ξ (J, E d ) → C1−γ,Ξ (J, E d ) defined by Pv(t) = f (t, v(t)),

t ∈ J,

and Q : S → C1−γ,Ξ (J, E d ) by Qv(t) =

v0 e−λ(Ξ(t)−Ξ(0)) γ−1 (Ξ(t) − Ξ(0)) Γ(γ) t

1 p−1 ⊖ (−1) ∫ ΞΔ (s)(Ξ(t) − Ξ(s)) e−λ(Ξ(t)−Ξ(s)) G(s, v(s))Δs. Γ(p) 0

Then, v = PvQv. We shall show that the operators P and Q fulfill all the assumptions of Lemma 5.3. We divide our proof into several steps. Step 1: P is Lipschitzian on C1−γ,Ξ (J, E d ). Let v, v̄ ∈ C1−γ,Ξ (J, E d ). Then by (H1) we have 1−γ

D[(Ξ(t) − Ξ(0))

1−γ

Pv(t), (Ξ(t) − Ξ(0)) 1−γ

≤ D[(Ξ(t) − Ξ(0))

̄ Pv(t)] 1−γ

f (t, v(t)), (Ξ(t) − Ξ(0))

≤ p1 (t)D[(Ξ(t) − Ξ(0))

1−γ

1−γ

v(t), (Ξ(t) − Ξ(0))

̄ f (t, v(t))]

̄ v(t)].

208 � R. Vivek et al. This gives ̄ D[Pv − Pv]̄ ≤ p∗1 D[v − v].

(5.12)

Thus, the operator P is Lipschitzian on C1−γ,Ξ (J, E d ) with Lipschitz constant p∗1 . Step 2: Q is completely continuous on S. Let us take a sequence {vn } ⊂ S and v ∈ S such that D[vn , v] → 0 as n → ∞. Then the Lebesgue dominated convergence theorem gives that 1−γ

lim (Ξ(t) − Ξ(0))

n→∞

= lim ( n→∞

Qvn (t)

v0 e−λ(Ξ(t)−Ξ(0)) (Ξ(t) − Ξ(0))1−γ ⊖ (−1) Γ(γ) Γ(p)

t

× ∫ ΞΔ (s)(Ξ(t) − Ξ(s))

p−1 −λ(Ξ(t)−Ξ(s))

e

̂ D[G(s, vn (s)), 0]Δs)

0

=

v0 e−λ(Ξ(t)−Ξ(0)) (Ξ(t) − Ξ(0))1−γ ⊖ (−1) Γ(γ) Γ(p) t

× ∫ ΞΔ (s)(Ξ(t) − Ξ(s))

p−1 −λ(Ξ(t)−Ξ(s))

e

0

=

̂ lim D[G(s, vn (s)), 0]Δs

n→∞

v0 e−λ(Ξ(t)−Ξ(0)) (Ξ(t) − Ξ(0))1−γ ⊖ (−1) Γ(γ) Γ(p) t

× ∫ ΞΔ (s)(Ξ(t) − Ξ(s)) 0

1−γ

= (Ξ(t) − Ξ(0))

p−1 −λ(Ξ(t)−Ξ(s))

e

Qv(t),

̂ D[G(s, v(s)), 0]ds

for every t ∈ J.

This proves that Q is continuous on S. Step 3: Q is uniformly bounded in S. For each t ∈ J, v ∈ S and by (H2), one has 1−γ

D[(Ξ(t) − Ξ(0)) ≤

̂ Qv(t), 0]

v0 e−λ(Ξ(t)−Ξ(0)) (Ξ(t) − Ξ(0))1−γ ⊖ (−1) Γ(γ) Γ(p) t

p−1 −λ(Ξ(t)−Ξ(s))

× ∫ ΞΔ (s)(Ξ(t) − Ξ(s))

e

̂ D[G(s, v(s)), 0]Δs

0

≤

v0 e−λ(Ξ(t)−Ξ(0)) (Ξ(b) − Ξ(0))1−γ+p −λ(Ξ(b)−Ξ(0)) + q1 (t)U(r) e . Γ(γ) Γ(p + 1)

Taking the supremum over the interval [0, b], the above inequality becomes

5 Qualitative analysis for hybrid fuzzy differential equations

v0 (Ξ(b) − Ξ(0))1−γ+p −λ(Ξ(b)−Ξ(0)) + q1∗ U(r) e , Γ(γ) Γ(p + 1)

‖Qv‖ ≤

� 209

(5.13)

for every v ∈ S. This proves that Q is uniformly bounded on S. Step 4: Q(s) is an equicontinuous set in C1−γ,Ξ (J, E d ). Indeed, let v ∈ S and 0 ≤ t1 < t2 ≤ b with t1 < t2 . Then one has 1−γ

D[(Ξ(t2 ) − Ξ(0)) ≤K(

1−γ

Qv(t2 ), (Ξ(t1 ) − Ξ(0))

Qv(t1 )]

v0 −λ(Ξ(t1 )−Ξ(0)) (Ξ(t2 ) − Ξ(0))1−γ −λΞ(t2 ) (e − e−λ(Ξ(t2 )−Ξ(0)) ) + e Γ(γ) Γ(p)

t2

p−1 λΞ(s)

× ∫ ΞΔ (s)(Ξ(t2 ) − Ξ(0))

e

̂ D[G(s, v(s)), 0]Δs

0

+

(Ξ(t1 ) − Ξ(0))1−γ −λΞ(t1 ) e Γ(p) t1

p−1 λΞ(s)

× ∫ ΞΔ (s)(Ξ(t1 ) − Ξ(0))

e

̂ D[G(s, v(s)), 0]Δs).

0

Thus, as t2 → t1 , the right-hand side of the above inequality tends to zero. As a consequence, the Arzela–Ascoli theorem gives that Q is a completely continuous operator on S. Step 5: For v ∈ C1−γ,Ξ (J, E d ), v = PvQv̄ ↔ v ∈ S for every v̄ ∈ S. Let v ∈ C1−γ,Ξ (J, E d ) and v̄ ∈ S be such that v = PvQv.̄ Then, one obtains that 1−γ

D[(Ξ(t) − Ξ(0))

̂ v(t), 0] 1−γ

= D[(Ξ(t) − Ξ(0))

̂ ≤ D[g(t, v(t)), 0]{ t

̂ ̄ 0] Pv(t)Qv(t),

v0 e−λ(Ξ(t)−Ξ(0)) (Ξ(t) − Ξ(0))1−γ ⊖ (−1) Γ(γ) Γ(p) p−1 −λ(Ξ(t)−Ξ(s))

× ∫ ΞΔ (s)(Ξ(t) − Ξ(s)) 0

≤K{

e

̂ D[G(s, v(s)), 0]Δs}

v0 e−λ(Ξ(t)−Ξ(0)) (Ξ(b) − Ξ(0))1−γ+p −λ(Ξ(b)−Ξ(0)) + q1 (t)U(r) e }. Γ(γ) Γ(p + 1)

Taking the supremum for t ∈ J, we obtain ‖v‖ ≤ K {

v0 (Ξ(b) − Ξ(0))1−γ+p −λ(Ξ(b)−Ξ(0)) + q1∗ U(r) e } = r. Γ(γ) Γ(p + 1)

This implies that v ∈ S. Step 6: We will prove that βL < 1, where β = p∗1 and L = sup{‖Qv‖ : v ∈ S}.

210 � R. Vivek et al. From inequality (5.13), we have v0 (Ξ(b) − Ξ(0))1−γ+p −λ(Ξ(b)−Ξ(0)) + q1∗ U(r) e . Γ(γ) Γ(p + 1)

L = sup{‖Qv‖ : v ∈ S} ≤

From the inequality (5.12), we have β = L. Hence, by using Eq. (5.10), we have βL ≤ p∗1 (

v0 (Ξ(b) − Ξ(0))1−γ+p −λ(Ξ(b)−Ξ(0)) + q1∗ U(r) e ) < 1. Γ(γ) Γ(p + 1)

Thus, all the conditions of Lemma 5.3 hold true and hence the operator equation v = PvQv has a solution in S. Consequently, the problem (5.1) has a solution J. This completes the proof. Theorem 5.7. Assume that (H3)–(H4) are satisfied. If (Ξ(b) − Ξ(0))p −λ(Ξ(b)−Ξ(0)) e (L1 M2 + M1 L2 ) < 1, Γ(p + 1)

(5.14)

then the problem (5.1) has a unique solution on J. Proof. Consider the operator Π : C1−γ,Ξ (J, E d ) → C1−γ,Ξ (J, E d ) defined as follows: (Πv)(t) = f (t, v(t))(

v0 e−λ(Ξ(t)−Ξ(0)) γ−1 (Ξ(t) − Ξ(0)) Γ(γ) t

⊖ (−1)

1 p−1 ∫ ΞΔ (s)(Ξ(t) − Ξ(s)) e−λ(Ξ(t)−Ξ(s)) G(s, v(s))Δs). Γ(p)

(5.15)

0

By Eq. (5.15), finding a solution of Eq. (5.1) in C1−γ,Ξ (J, E d ) is equivalent to finding a fixed point of the operator Π. For any v, v̄ ∈ C1−γ,Ξ (J, E d ) and each t ∈ J, we have ̄ D[(Πv)(t), (Πv)(t)] t

=

1 p−1 D[f (t, v(t)) ∫ ΞΔ (s)(Ξ(t) − Ξ(s)) e−λ(Ξ(t)−Ξ(s)) G(s, v(s))Δs, Γ(p) 0 t

̄ f (t, v(t)) ∫ ΞΔ (s)(Ξ(t) − Ξ(s))

p−1 −λ(Ξ(t)−Ξ(s))

e

̄ G(s, v(s))Δs]

0

=

1 ̄ [D[f (t, v(t)), f (t, v(t))] Γ(p) t

p−1 −λ(Ξ(t)−Ξ(s))

× ∫ ΞΔ (s)(Ξ(t) − Ξ(s)) 0

e

̂ D[G(s, v(s)), 0]Δs

5 Qualitative analysis for hybrid fuzzy differential equations

� 211

̂ ̄ + D[f (t, v(t)), 0] t

p−1 −λ(Ξ(t)−Ξ(s))

× ∫ ΞΔ (s)(Ξ(t) − Ξ(s)) 0

̄ ̄ D[G(s, v(s)), G(s, v(s))]Δs]

t

≤ max[ t∈J

e

1 p−1 {(L1 M2 + M1 L2 ) ∫ ΞΔ (s)(Ξ(t) − Ξ(s)) e−λ(Ξ(t)−Ξ(s)) Δs}] Γ(p) 0

× D[v, v]̄ ≤

(Ξ(b) − Ξ(0))p −λ(Ξ(b)−Ξ(0)) ̄ e (L1 M2 + M1 L2 )D[v, v]. Γ(p + 1)

This further implies that (Ξ(b) − Ξ(0))p −λ(Ξ(b)−Ξ(0)) e (L1 M2 + M1 L2 )D[v, v]̄ ≥ 0. Γ(p + 1)

(5.16)

̄ By the condition (5.14), the relation (5.16) is true whenever D[(Πv)(t), (Πv)(t)] = 0, which ̄ further implies that Πv(t) = Πv(t). Ultimately, the problem (5.1) has a unique solution on J. This completes the proof.

5.4 Stability theory Our aim in this section is to prove that the problem (5.1) is UH stable and UHR stable. In the fuzzy setting, we have the following definitions. Definition 5.18. Equation (5.1) is called UH stable if there exists a constant CG > 0 such that for each ε > 0 and for each solution v̄ ∈ C1−γ,Ξ (J, E d ) of the inequality p,q,λ

D[TH𝕋 0+ ΔΞ

(

̄ v(t) ̄ ), G(t, v(t))] ≤ ε, ̄ f (t, v(t))

t ∈ J,

(5.17)

there exists a solution v ∈ C1−γ,Ξ (J, E d ) satisfying (5.1) with ̄ v(t)] ≤ CG ε, D[v(t),

t ∈ J.

Definition 5.19. Equation (5.1) is called generalized UH stable if there exists χG ∈ C(ℝ+ , ℝ+ ), χG (0) = 0 such that, for each solution v̄ ∈ C1−γ,Ξ (J, E d ) of the inequality (5.17), there exists a solution v ∈ C1−γ,Ξ (J, E d ) satisfying (5.1) with ̄ v(t)] ≤ χG ε, D[v(t),

t ∈ J.

212 � R. Vivek et al. Definition 5.20. Equation (5.1) is UHR stable with respect to φ ∈ C1−γ,Ξ (J, E d ) if there exists a constant CG,φ > 0 such that, for each ε > 0 and for each solution v̄ ∈ C1−γ,Ξ (J, E d ) of the inequality p,q,λ

D[TH𝕋 0+ ΔΞ

(

̄ v(t) ̄ ), G(t, v(t))] ≤ εφ(t), ̄ f (t, v(t))

t ∈ J,

(5.18)

there exists a solution v ∈ C1−γ,Ξ (J, E d ) satisfying (5.1) with 1−γ

D[(Ξ(t) − Ξ(0))

1−γ

̄ (Ξ(t) − Ξ(0)) v(t),

v(t)] ≤ εCG,φ φ(t),

t ∈ J.

Definition 5.21. Equation (5.1) is called generalized UHR stable with respect to φ ∈ C1−γ,Ξ (J, E d ) if there exists a constant CG,φ > 0 such that for each solution v̄ ∈ C1−γ,Ξ (J, E d ) of the inequality p,q,λ

D[TH𝕋 0+ ΔΞ

(

̄ v(t) ̄ ≤ φ(t), ), G(t, v(t))] ̄ f (t, v(t))

t ∈ J,

(5.19)

there exists a solution v ∈ C1−γ,Ξ (J, E d ) satisfying Eq. (5.1) with 1−γ

D[(Ξ(t) − Ξ(0))

1−γ

̄ (Ξ(t) − Ξ(0)) v(t),

v(t)] ≤ CG,φ φ(t),

t ∈ J.

Remark 5.2. A fuzzy function v̄ ∈ C1−γ,Ξ (J, E d ) is a solution of the inequality p,q,λ

D[TH𝕋 0+ ΔΞ

(

̄ v(t) ̄ ), G(t, v(t))] ≤ ε, ̄ f (t, v(t))

t ∈ J,

if and only if there exists a h ∈ C1−γ,Ξ (J, E d ) such that ̂ ≤ εφ(t), t ∈ J, {(i) D[h(t), 0] { ̄ v(t) TH𝕋 p,q,λ ̄ ) = G(t, v(t)) + h(t), ̄ {(ii) 0+ ΔΞ ( f (t,v(t))

t ∈ J.

(5.20)

Theorem 5.8. Assume that (H1), (H2), (H4) and (5.14) are satisfied. Then, the problem (5.1) is UHR stable. Proof. Fix ε > 0 and let v̄ ∈ C1−γ,Ξ (J, E d ) be a solution of the inequality p,q,λ

D[TH𝕋 0+ ΔΞ

(

̄ v(t) ̄ ), G(t, v(t))] ≤ εφ(t), ̄ f (t, v(t))

t ∈ J.

Let v ∈ C1−γ,Ξ (J, E d ) be the unique solution of Eq. (5.1). By Lemma 5.4, v(t) = f (t, v(t))(

v0 e−λ(Ξ(t)−Ξ(0)) γ−1 (Ξ(t) − Ξ(0)) Γ(γ)

(5.21)

5 Qualitative analysis for hybrid fuzzy differential equations

� 213

t

1 p−1 ⊖ (−1) ∫ ΞΔ (s)(Ξ(t) − Ξ(s)) e−λ(Ξ(t)−Ξ(s)) G(s, v(s))Δs), Γ(p) 0

while the solution of Eq. (5.20) is given by v̄0 e ̄ { (Ξ(t) − Ξ(0))γ−1 f (t, v(t))( { Γ(γ) { { { ̄ = { ⊖ (−1) 1 ∫t ΞΔ (s)(Ξ(t) − Ξ(s))p−1 e−λ(Ξ(t)−Ξ(s)) G(s, v(s))Δs) v(t) ̄ Γ(p) 0 { { { { T𝕋 p,λ { + 0+ IΞ h(t). −λ(Ξ(t)−Ξ(0))

(5.22)

From Eq. (5.22) we obtain ̄ f (t, v(t))( ̄ D[v(t),

v̄0 e−λ(Ξ(t)−Ξ(0)) γ−1 (Ξ(t) − Ξ(0)) Γ(γ)

t

1 p−1 ̄ ⊖ (−1) ∫ ΞΔ (s)(Ξ(t) − Ξ(s)) e−λ(Ξ(t)−Ξ(s)) G(s, v(s))Δs)] Γ(p)

≤

0 T𝕋 p,λ ̂ 0+ IΞ D[h(t), 0]

p,λ

≤ T𝕋 0+ IΞ φ(t) ≤ εKφ(t).

On the other hand, we have t

̄ v(t)] ≤ εKφ(t) + D[v(t),

1 p−1 ̄ D[f (t, v(t)) ∫ ΞΔ (s)(Ξ(t) − Ξ(s)) Γ(p) 0

t

p−1

̄ × e−λ(Ξ(t)−Ξ(s)) G(s, v(s))Δs, f (t, v(t)) ∫ ΞΔ (s)(Ξ(t) − Ξ(s)) 0

× e−λ(Ξ(t)−Ξ(s)) G(s, v(s))Δs] t

1 p−1 ̄ [D[f (t, v(t)), f (t, v(t))] ∫ ΞΔ (s)(Ξ(t) − Ξ(s)) ≤ εKφ(t) + Γ(p) 0

t

̂ ̂ ∫ ΞΔ (s)(Ξ(t) − Ξ(s))p−1 ̄ × e−λ(Ξ(t)−Ξ(s)) D[G(s, v(s)), 0]Δs + D[f (t, v(t)), 0] 0

̄ × e−λ(Ξ(t)−Ξ(s)) D[G(s, v(s)), G(s, v(s))]Δs] t

1 p−1 ≤ εKφ(t) + {(L1 M2 + M1 L2 ) ∫ ΞΔ (s)(Ξ(t) − Ξ(s)) Γ(p) 0

̄ × e−λ(Ξ(t)−Ξ(s)) D[v(s), v(s)]Δs}.

214 � R. Vivek et al. ̄ v(t)], Y (t) = εKφ(t), and Applying Gronwall’s inequality given that X (t) = D[v(t), (L1 M2 +M1 L2 ) , we obtain Γ(p)

Z (t) =

p

̄ v(t)] ≤ εKφ(t)Ep ((L1 M2 + M1 L2 )(Ξ(t) − Ξ(0)) ). D[v(t), Therefore, 1−γ

D[(Ξ(t) − Ξ(0))

1−γ

̄ (Ξ(t) − Ξ(0)) v(t), 1−γ

≤ εKφ(t)(Ξ(t) − Ξ(0)) ≤ εCG,φ φ(t),

v(t)] p

Ep ((L1 M2 + M1 L2 )(Ξ(t) − Ξ(0)) )

(5.23)

where CG,φ = K(Ξ(t) − Ξ(0))1−γ Ep ((L1 M2 + M1 L2 )(Ξ(t) − Ξ(0))p ). This proves that Eq. (5.1) is UHR stable. Further, in the same fashion, it is easy to check that the solution of the problem (5.1) is generalized UHR stable, which follows by taking ε = 1 in the inequality (5.23). Corollary 5.1. (i) The proof that Eq. (5.1) is UH stable follows by taking φ(t) = 1 in the inequality (5.23). (ii) The proof that Eq. (5.1) is generalized UH stable follows by taking φ(t) = 1 and χG (ε) = εCG,1 in the inequality (5.23).

5.5 An example Consider the following hybrid FDE involving tempered Ξ-Hilfer fractional derivative on time scales 1 1

, ,1

v(t) TH𝕋 2 3 { {0+ Δt2 ( f (t,v(t)) ) = G(t, v(t)), t ∈ [0, 1] ⊆ 𝕋, { {T𝕋 1− 32 ,λ v(0) 1 1 1 2 {0+ It2 ( f (0,v(0)) ) = 1, γ = 2 + 3 − 6 = 3 .

(5.24)

v(t) Here, p = 21 , q = 31 , λ = 1, b = 1, v0 = 1 and Ξ(t) = t 2 . Set f (t, v(t)) = e18+e t and t G(t, v(t)) = 4 cos v(t). It is observed that the functions f and G are continuous. Let v, v̄ ∈ E d and t ∈ [0, 1]. Then, we get −t

̄ D[f (t, v(t)), f (t, v(t))] =

e−t e−t ̄ ̄ D[v(t), v(t)] ≤ D[v, v]. t 18 + e 18 + et

Thus, hypothesis (H1) holds true with p1 (t) =

e−t 18+et

and p∗ =

1 . 19

5 Qualitative analysis for hybrid fuzzy differential equations

� 215

Moreover, for v ∈ E d and t ∈ [0, 1], we get ̂ = D[ t cos v(t), 0]. ̂ D[G(t, v(t)), 0] 4 This shows that hypothesis (H2) holds true with q1 (t) = 4t , q1∗ = 41 , and U(r) = 1. Hence, all the conditions of Theorem 5.6 are satisfied, along with condition that p∗1 (

v0 (Ξ(b) − Ξ(0))1−γ+p −λ(Ξ(b)−Ξ(0)) ∗ + e q1 U(r)) Γ(γ) Γ(p + 1)

=

1 1 1 1 ( + ( )(0.36787)) 19 1.35412 0.88623 4

= 0.04432 < 1.

It follows from Theorem 5.7 that the problem (5.24) has a unique solution on [0, 1]. Moreover, Theorem 5.8 ensures that the problem (5.24) is UHR stable.

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[14] R. Chaharpashlou, R. Saadati and A. Atangana, Ulam–Hyers–Rassias stability for nonlinear ψ-Hilfer stochastic fractional differential equation with uncertainty, Adv. Differ. Equ., 10 (2020), 339. [15] B. C. Dhage, A fixed point theorem in Banach algebras with applications to functional integral equations, Kyungpook Math. J., 44 (2004), 145–155. [16] M. Hannabou and K. Hilal, Existence results for a system of coupled hybrid differential equations with fractional order, Int. J. Differ. Equ., 2020 (2020), 1–8. [17] A. Harir, S. Melliani and L. S. Chadli, Hybrid fuzzy differential equations, AIMS Math., 5(1) (2020), 273–285. [18] A. Harir, S. Melliani and L. S. Chadli, An algorithm for the solution of fuzzy fractional differential equations, J. Univers. Math., 3(1) (2020), 11–20. [19] R. Hilfer, Applications of fractional Calculus in Physics, World Scientific, Singapore, 1999. [20] N. V. Hoa, V. Lupulescu and D. O’Regan, A note on initial value problems for fractional fuzzy differential equations, Fuzzy Sets Syst., 347 (2018), 54–69. [21] S. Hong, Differentiability of multivalued functions on time scales and applications to multivalued dynamic equations, Nonlinear Anal., 71 (2009), 3622–3637. [22] O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst., 24 (1987), 301–317. [23] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of the fractional differential equations, North-Holland Mathematics Studies, Amsterdam, Netherlands, 2006. [24] K. D. Kucche, A. D. Mali and J. V. D. C. Sousa, On the nonlinear ψ-Hilfer fractional differential equations, Comput. Appl. Math., 38(2) (2019), 73. [25] K. K. Kucche, A. D. Mali, A. Fernandez and H. M. Fahad, On tempered Hilfer fractional derivatives with respect to functions and the associated fractional differential equations, Chaos Solitons Fractals, 163 (2022), 112547. [26] V. Lakshmikantham and R. N. Mohapatra, Theory of Fuzzy Differential Equations and Applications, London, 2003. [27] V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, Dynamic systems on measure chains, Kluwer Academic Publishers, Dordrecht, 1996. [28] V. Lakshmikantham and A. S. Vatsala, Hybrid systems on time scales, J. Comput. Appl. Math., 141 (2002), 227–335. [29] C. Li, W. Deng and L. Zhao, Well-posedness and numerical approximation of tempered fractional ordinary differential equations, Discrete Contin. Dyn. Syst., 20(5) (2017), 1239–1262. [30] A. D. Mali, K. D. Kucche, A. Fernandez and H. M. Fahad, On tempered fractional calculus with respect to functions and associated fractional differential equations, Math. Methods Appl. Sci., 45(17) (2022), 11134–11157. [31] S. Manikandan, S. Sivasundaram, D. Vivek and K. Kanagarajan, Controllability and qualitative property results for Ambartsumian equation via Ξ-Hilfer generalized proportional fractional derivative on time scales, Nonlinear Stud., 29(4) (2022), 1–23. [32] M. D. Obeidat and D. E. Bentil, New theories and applications of tempered fractional differential equations, Nonlinear Dyn., 105 (2021), 1689–1702. [33] I. Podlubny, Fractional differential equations, Academic Press, New York, 1999. [34] A. Rivaz, O. S. Fard and T. A. Bidgoli, On the existence and uniqueness of solutions for fuzzy fractional differential equations, Tbil. Math. J., 10 (2017), 197–205. [35] J. V. C. Sousa, D. S. Oliveira, G. S. F. Frederico and D. F. M. Torres, Existence, uniqueness and controllability for g-Hilfer differential equations on time scales, hal-03726777, 2022. [36] J. V. C. Sousa and E. C. Oliveira, On the ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72–91. [37] J. V. C. Sousa and E. C. Oliveira, On the Ulam–Hyers–Rassias stability for nonlinear fractional differential equations using the ψ-Hilfer operator, J. Fixed Point Theory Appl., 20 (2018), 1–21. [38] Ch. Vasavi, G. Suresh Kumar and M. S. N. Murty, Fuzzy Hukuhara delta differential and applications to fuzzy dynamic equations on time scales, J. Uncertain Syst., 10(3) (2016), 163–180.

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S. Manikandan, S. Sivasundaram, K. Kanagarajan, and D. Vivek

6 The dynamical analysis of nonlinear Ambartsumian equation via tempered Ξ-Hilfer fractional derivative on time scales Abstract: In this chapter, we examine a new class of Ambartsumian equations of the fractional type with tempered Ξ-Hilfer fractional derivative with boundary conditions. The provided problem is transformed into an equivalent fixed point problem, which is then solved by using the Banach and Krasnosel’skii fixed point theorems. Ulam stability is investigated. An example is included to verify the theoretical results.

6.1 Introduction In comparison to differential equations of classical order, fractional-order differential equations more correctly model a variety of real-world phenomena. Recently, fractional differential equations (FDEs) have been used in a wide range of engineering, mathematics, physics, bioengineering, and applied sciences fields; we refer to [27–29] for some key findings in the theory of fractional calculus and its applications. The literature contains numerous definitions of fractional integrals and derivatives. We refer to the references given therein for more information on the various uses of FDEs employing a Hilfer derivative. There are actual events in the real world that have unusual dynamics, such as the atmospheric diffusion of pollution, the transmission of signals through powerful magnetic fields, the impact of stock market profitability theory, the theoretical simulation of dielectric relaxation in glass-forming materials, network traffic, and so on. The presence of initial and boundary value problem solutions for FDEs with the Hilfer fractional derivative has received a lot of attention recently, see [8, 10– 12, 20, 21, 24]. Several authors [1, 7, 9, 17, 23] used particular versions of the proportional derivatives, called modified conformable derivatives, to present the fractional counterpart proportional derivatives and integrals. Later, authors [25] generalized proportional derivatives and used them to generate more general classes of nonlocal fractional integrals and derivatives. S. Manikandan, K. Kanagarajan, Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore 641020, India, e-mails: [email protected], [email protected] S. Sivasundaram, Department of Mathematics, College of Natural Science, Engineering and Mathematics, Daytona Beach, FL 32114, USA, e-mail: [email protected] D. Vivek, Department of Mathematics, PSG College of Arts & Science, Coimbatore 641014, India, e-mail: [email protected] https://doi.org/10.1515/9783111182971-006

220 � S. Manikandan et al. In recent years, many fractional models with tempered fractional derivatives have been widely applied in many fields of science and technology, and a lot of research results have been obtained. For more details, see [5, 14, 16, 19]. In [3, 6, 18, 26], the authors worked on time scales. Ulam brought up the topic of functional equation stability first, and Hyers followed suit. Ulam–Hyers stability is the name given to this sort of stability nowadays. By taking variables into account, Rassias accomplished a surprising expansion of the Ulam–Hyers stability of maps. When we swap out the functional equation with an inequality that perturbs the equation, the idea of stability for a functional equation emerges. The study of Ulam–Hyers and Ulam–Hyers–Rassias stability for various functional equations has received significant attention. The prime target of this work is to investigate a fractional type of the Ambartsumian equation. This equation is very useful to describe the surface brightness of the Milky Way. In [2, 15, 22], the authors discussed about Ambartsumian equations in different aspects. In this work, we discuss the existence theory and stability result for the following problem: t p,q,λ { (TH𝕋 A )(t) = ℚ(t, A (t), A ( )), t ∈ J = [a, b], { a+ △Ξ { { η { { { { A (a) = 0, { { { m n r { { TH𝕋 ϕi ,λ TH𝕋 ωk ,λ { { {A (b) = ∑ αj A (ζj ) + ∑ βi a+ IΞ A (θi ) + ∑ ϱk a+ △Ξ A (μk ), j=1 i=1 k=1 {

(6.1)

p,q,λ

where ℚ(t, A (t), A ( ηt )) = η1 A ( ηt ) − A (t), η > 1, TH𝕋 is the tempered Ξ-Hilfer fraca+ △Ξ tional derivative on time scales of order p > 0 and type q > 0 and αj , βi , ϱk ∈ ℝ, λ ∈ ℝ are given constants, ℚ : J × ℝ × ℝ → ℝ is a given continuous function, ϕi , ωk > 0 and ζj , θi , μk ∈ J, j = 1, 2, . . . , m, i = 1, 2, . . . , n, k = 1, 2, . . . , r.

6.2 Prerequisites Let 0 ≤ a < b < ∞, J be a finite interval, and ϑ a parameter such that n − 1 ≤ ϑ < n. Let X = C(J, ℝ) be the Banach space of the continuous functions ℚ on J with the norm defined by 󵄨 󵄨 ‖ℚ‖C(J,ℝ) = max󵄨󵄨󵄨ℚ(t)󵄨󵄨󵄨. t∈J

Definition 6.1 ([26]). A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers. For t ∈ 𝕋, one defines the forward jump operator σ : 𝕋 → 𝕋 by

6 The dynamical analysis of nonlinear Ambartsumian equation

� 221

σ(t) = inf{s ∈ 𝕋 : s > t}, while the backward jump operator ρ : 𝕋 → 𝕋 is defined by ρ(t) = sup{s ∈ 𝕋 : s > t}. If max 𝕋 is finite and there exists a finite min 𝕋 in addition, we put σ(max 𝕋) = max 𝕋 and ρ(min 𝕋) = min 𝕋. If σ(t) > t, then we say that t is right-scattered, while if ρ(t) < t, then we say that t is left-scattered and also if t < max 𝕋, and σ(t) = t then t is called right-dense, and if t > min 𝕋 and ρ(t) = t, then t is called right-dense. The derivative makes use of the set 𝕋κ , which is derived from the time scale 𝕋 as follows: if 𝕋 has left-scattered maximum M, then 𝕋κ := 𝕋\{M}; otherwise, 𝕋κ = 𝕋. Definition 6.2 (Delta derivative, [26]). Suppose that ℚ : 𝕋 → ℝ and let t ∈ 𝕋κ . The delta derivative is defined by ℚ(σ(s)) − ℚ(t) , σ(s) − t

ℚ△ (t) = lim s→t

t ≠ σ(s).

Definition 6.3 ([26]). Let J denote a closed bounded interval in 𝕋. A function F : J → ℝ is called a delta antiderivative of a function f : [a, b) → ℝ provided F is continuous on J, delta differentiable on [a, b), and F △ (t) = f (t) for all t ∈ [a, b). Then we define the △-integral by b

∫ f (t) △ (t) = F(b) − F(a). a

Proposition 6.1 ([26]). Suppose a, b ∈ 𝕋, a < b and ℚ is continuous on J. Then, b

b

∫ ℚ(t) △ t = [σ(a) − a]ℚ(a) + ∫ ℚ(t) △ t. a

σ(a)

Proposition 6.2 ([26]). Let 𝕋 be a time scale and ℚ be an increasing continuous function on J. If ℚ is the extension of ℚ to the real interval J given by ℚ(s), s ∈ 𝕋,

ℚ(s) = {

ℚ(t),

s ∈ (t, σ(t)) ⊄ 𝕋,

then b

b

∫ ℚ(t) △ t ≤ ∫ ℚ(t)dt. a

a

222 � S. Manikandan et al. Definition 6.4. Suppose 𝕋 is a time scale, ℚ is an absolutely Ξ integrable function defined on [a, b]. Then the Ξ-tempered fractional integral of order p ∈ (0, 2) and the index λ ∈ ℝ of the function ℚ is defined by T𝕋 p,λ a+ IΞ ℚ(t)

t

=

1 p−1 ∫ Ξ△ (s)(Ξ(t) − Ξ(s)) e−λ(Ξ(t))−Ξ(s) ℚ(s) △ s. Γ(p) a+

Definition 6.5. Suppose 𝕋 is a time scale, λ ∈ ℝ, Ξ is an increasing real C n function on J such that Ξ△ > 0 on J. Then the tempered (nonfractional) tempered derivative with respect to Ξ is defined by T𝕋 1,λ a+ △Ξ

𝕋

△ + λ, Ξ△ (t)

=

which can be taken to the nth power (n ∈ ℕ) to get T𝕋 n,λ a+ △Ξ

where 𝕋 △ =

n

𝕋

=(

△ + λ) , Ξ△ (t)

d . dt

Definition 6.6. Let p, λ ∈ ℝ with p > 0 and Ξ be an increasing real C n function on J such that Ξ△ > 0 on J. Then, the tempered fractional integrals of order p and index λ with respect to Ξ, of Riemann–Liouville and Caputo type, respectively, are defined as follows, applied to a function ℚ ∈ ACΞn J: TR𝕋 a+

TC𝕋 a+

p,λ

n−p,λ

n,λ T𝕋 △Ξ ℚ(t) = T𝕋 a+ △Ξ a+ IΞ p,λ △Ξ

ℚ(t) =

T𝕋 n−p,λ T𝕋 a + IΞ a+

ℚ(t),

△n,λ Ξ

ℚ(t).

Definition 6.7. Let n − 1 < p < n with n ∈ ℕ, J be the interval such that −∞ ≤ a ≤ b ≤ ∞ and ℚ, Ξ ∈ C n (J, ℝ) two functions such that Ξ is an increasing function and Ξ△ (t) ≠ 0, for all t ∈ J. The tempered Ξ-Hilfer fractional derivative of a function ℚ of order p and type q is defined by TH𝕋 a+

p,q,λ

△Ξ

q(n−p),λ

ℚ(t) = T𝕋 a+ IΞ

𝕋

(

△ (1−q)(n−p),λ + λ)T𝕋 ℚ(t). a+ △Ξ Ξ△ (t)

Lemma 6.1. Let p, q > 0. Then we have the following semigroup property: T𝕋 p,λ T𝕋 q,λ a+ IΞ a+ IΞ ℚ(t)

p+q,λ

= T𝕋 a + IΞ

ℚ(t),

t > a.

Lemma 6.2. Let ℚ ∈ C n (J, ℝ), 0 ≤ a ≤ b ≤ ∞, p, q > 0, and ϑ = p + q(n − p). If ℚ ∈ p,q,λ n L1 (J), (T𝕋 a+ △Ξ ℚ)(t) ∈ ACΞ J, then

6 The dynamical analysis of nonlinear Ambartsumian equation

� 223

T𝕋 p,λ T𝕋 p,q,λ a+ IΞ (a+ △Ξ ℚ)(t)

= ℚ(t) n

[n−k]

e−λ(Ξ(t)−Ξ(a)) (Ξ(t) − Ξ(a))ϑ−k 𝕋 △ ( △ ) Γ(ϑ − k + 1) Ξ (t) Ξ k=1

−∑

T𝕋 (1−q)(n−p),λ ℚ(a). a + IΞ

(6.2)

Lemma 6.3 (Banach fixed point theorem, [4]). Let X be a Banach space, D ⊂ X closed and F : D → D a strict contraction, i. e., |Fx − Fy| ≤ k|x − y| for some k ∈ (0, 1) and all x, y ∈ D. Then F has a fixed point in D. Lemma 6.4 (Krasnoselskii’s fixed point theorem, [13]). Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be operators such that – Ax + By ∈ M whenever x, y ∈ M, – A is compact and continuous, – B is a contraction mapping. Then there exists z ∈ M such that z = Az + Bz. Theorem 6.1 (Gronwall’s inequality, [16]). Let p ∈ ℝ+ , λ ∈ ℝ, u, v ∈ L1 (J, dΞ) be nonnegative functions, and let w : J → [0, ∞) be a continuous, nonnegative and nondecreasing function. If for all t ∈ J we have t

p−1 −λ(Ξ(t)−Ξ(s))

u(t) = v(t) + w(t) ∫ Ξ′ (s)(Ξ(t) − Ξ(s))

e

u(s)ds,

a+

then for all t ∈ J we have t ∞

|w(t)Γ(p)|k ′ pk−1 −λ(Ξ(t)−Ξ(s)) Ξ (s)(Ξ(t) − Ξ(s)) e v(s)ds. Γ(pk) k=1

u(t) ≤ v(t) + ∫ ∑ a+

Further, if v is a nondecreasing function, then for all t ∈ J we have p

u(t) ≤ v(t)Ep (w(t)Γ(p)(Ξ(t) − Ξ(a)) ).

6.3 Existence theory The problem (6.1) will be converted into an equivalent fixed point problem using the auxiliary lemma in this part. Standard fixed point theorems are used to study the main results.

224 � S. Manikandan et al. Lemma 6.5. Let ℚ ∈ C(J, ℝ) and ∧ = e−λ(Ξ(b)−Ξ(a)) (Ξ(b) − Ξ(a))

ϑ−1

m

ϑ−1

− ∑ αj e−λ(Ξ(ζj )−Ξ(a)) (Ξ(ζj ) − Ξ(a)) j=1

n

ϑ ϑ+ϕ −1 e−λ(Ξ(θi )−Ξ(a)) (Ξ(θi ) − Ξ(a)) i − ∑ βi Γ(ϑ + ϕ ) i i=1 r

− ∑ ϱk k=1

ϑ ϑ−ω −1 e−λ(Ξ(μk )−Ξ(a)) (Ξ(μk ) − Ξ(a)) k ≠ 0. Γ(ϑ − ωk )

(6.3)

Then, A is a solution of our proposed problem (6.1) if and only if it satisfies the integral equation e−λ(Ξ(t)−Ξ(a)) (Ξ(t) − Ξ(a))ϑ−1 ∧ m ζ j p,λ × ( ∑ αj T𝕋 a+ IΞ ℚ(ζj , A (ζj ), A ( )) η j=1 t η

p,λ

T𝕋

A (t) = a+ IΞ ℚ(t, A (t), A ( )) +

n

p,λ

− ∑ βi T𝕋 a+ IΞ ℚ(θi , A (θi ), A ( i=1 r

p−ωk ,λ

− ∑ ϱk T𝕋 a + IΞ k=1

θi )) η

ℚ(μk , A (μk ), A (

μk )) η

b p,λ − T𝕋 a+ IΞ ℚ(b, A (b), A ( ))). η

(6.4)

Proof. Applying the tempered fractional integral operator of order p on both sides of Eq. (6.4) and using Lemma 6.2, we find that 2−ϑ,λ δ(T𝕋 A )(a) −λ(Ξ(t)−Ξ(a)) ϑ−1 a + IΞ e (Ξ(t) − Ξ(a)) Γ(ϑ) 2−ϑ,λ δ(T𝕋 A )(a) −λ(Ξ(t)−Ξ(a)) + I ϑ−2 − a Ξ e (Ξ(t) − Ξ(a)) Γ(ϑ − 1) t p,λ = T𝕋 a IΞ ℚ(t, A (t), A ( )), η

A (t) =

(6.5)

which can be rewritten as follows: A (t) = c0 e

−λ(Ξ(t)−Ξ(a)) t

ϑ−1

(Ξ(t) − Ξ(a))

+ c1 e−λ(Ξ(t)−Ξ(a)) (Ξ(t) − Ξ(a))

ϑ−2

1 s p−1 + ∫ Ξ△ (s)e−λ(Ξ(t)−Ξ(s)) (Ξ(t) − Ξ(s)) ℚ(s, A (s), A ( )) △ s, Γ(p) η

(6.6)

a

where c0 , c1 are arbitrary constants. The first boundary condition A (a) = 0 in Eq. (6.6) gives c1 = 0, since ϑ ∈ [p, 2]. In consequence, Eq. (6.6) takes the following form:

6 The dynamical analysis of nonlinear Ambartsumian equation

A (t) = c0 e

−λ(Ξ(t)−Ξ(a))

(Ξ(t) − Ξ(a))

� 225

ϑ−1

t

1 s p−1 + ∫ Ξ△ (s)e−λ(Ξ(t)−Ξ(s)) (Ξ(t) − Ξ(s)) ℚ(s, A (s), A ( )) △ s. Γ(p) η

(6.7)

a

Now, employing the second boundary condition, m

n

j=1

i=1

TH𝕋

ϕ ,λ

r

TH𝕋

A (b) = ∑ αj A (ζj ) + ∑ βi a+ IΞ i A (θi ) + ∑ ϱk a+ k=1

ω ,λ

△Ξ k A (μk ),

and using Eq. (6.5), we obtain the following: c0 =

ζj 1 m T𝕋 p,λ { ∑ αj a+ IΞ ℚ(ζj , A (ζj ), A ( )) ∧ j=1 η n

p,λ

− ∑ βi T𝕋 a+ IΞ ℚ(θi , A (θi ), A ( i=1 r

p−ωk ,λ

− ∑ ϱk T𝕋 a+ IΞ k=1

θi )) η

ℚ(μk , A (μk ), A (

μk )) η

b p,λ − T𝕋 a+ IΞ ℚ(b, A (b), A ( ))}. η After substituting the value of c0 into Eq. (6.7), we get the solution. Hence A satisfies our proposed problem (6.1). By direct computation, one can obtain the converse of the lemma. The proof is completed. Next, we define the operator T : X → X associated with the problem (6.1) as follows: t

T A (t) =

1 z p−1 ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) ℚ(z, A (z), A ( )) △ z Γ(p) η +

e

a −λ(Ξ(t)−Ξ(z))

m

× (∑ j=1

αj

Γ(p)

ζj

(Ξ(t) − Ξ(z))ϑ−1 ∧ p−1

∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) a

z × ℚ(z, A (z), A ( )) △ z η n

θi

βi p+ϕ −1 −∑ ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) i Γ(p + ϕ ) i i=1 a

z × ℚ(z, A (z), A ( )) △ z η

226 � S. Manikandan et al. μk

r

ϱk p−ω −1 −∑ ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) k Γ(p − ω ) k k=1 a

z × ℚ(z, A (z), A ( )) △ z η b

1 p−1 − ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) Γ(p) a

z × ℚ(z, A (z), A ( )) △ z), η

t ∈ J.

(6.8)

In the sequel, we use the following notation: Ω=

e−λ(Ξ(b)−Ξ(a)) (Ξ(b) − Ξ(a))p e−λ(Ξ(b)−Ξ(a)) (Ξ(b) − Ξ(a))ϑ−1 + Γ(p + 1) |∧| m

× {∑

|αj |e−λ(Ξ(ζj )−Ξ(a)) (Ξ(ζj ) − Ξ(a))p

j=1

n

Γ(p + 1)

|βi |e−λ(Ξ(θi )−Ξ(a)) (Ξ(θi ) − Ξ(a))p+ϕi Γ(p + ϕi + 1)

+∑ i=1 r

|ϱk |e−λ(Ξ(μk )−Ξ(a)) (Ξ(μk ) − Ξ(a))p−ωk Γ(p − ωk + 1) k=1

+∑

+

e−λ(Ξ(b)−Ξ(a)) (Ξ(b) − Ξ(a))p }. Γ(p + 1)

(6.9)

The following result is discussed by using the Banach fixed point theorem. Theorem 6.2. Suppose that the following condition holds: (H1 ) There exits a constant l > 0 such that for all t ∈ J and u, v, u, v ∈ ℝ, 󵄨󵄨 󵄨 󵄨󵄨ℚ(t, u, v) − ℚ(t, u, v)󵄨󵄨󵄨 ≤ l{|u − u| + |v − v|}. Then, the problem (6.1) has a unique solution on J, if Ω < 1, where Ω is defined by (6.9). Proof. We will verify that the operator T satisfies the hypotheses of the Banach contraction mapping principle. Fixing N = maxt∈J |ℚ(t, 0, 0)| < ∞ and using assumption (H1 ), we obtain the following: 󵄨󵄨 t 󵄨󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨ℚ(t, A (t), A ( ))󵄨󵄨󵄨 ≤ l󵄨󵄨󵄨A (t)󵄨󵄨󵄨 + 󵄨󵄨󵄨ℚ(t, 0, 0)󵄨󵄨󵄨 ≤ l‖A ‖ + N. 󵄨󵄨 η 󵄨󵄨 The proof is divided into two steps.

(6.10)

6 The dynamical analysis of nonlinear Ambartsumian equation

� 227

Step I. We will show that T (Br ) ⊂ Br , where (Br ) = {A ∈ X : ‖A ‖ < r}

with r ≥

NΩ . (1 − lΩ)

Let A ∈ Br . Then, we have the following: t 󵄨 z 󵄨󵄨󵄨 1 p−1 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) 󵄨󵄨󵄨ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z 󵄨󵄨T A (t)󵄨󵄨 = 󵄨󵄨 Γ(p) η 󵄨󵄨

+

e

a −λ(Ξ(t)−Ξ(z))

m

× {∑ j=1

|αj |

Γ(p)

ζj

(Ξ(t) − Ξ(z))ϑ−1 ∧ p−1

∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) a

󵄨󵄨 z 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z η 󵄨󵄨 󵄨󵄨 θi

n

|βi | p+ϕ −1 −∑ ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) i Γ(p + ϕi ) i=1 a

󵄨󵄨 z 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z 󵄨󵄨 η 󵄨󵄨 μk

r

|ϱk | p−ω −1 −∑ ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) k Γ(p − ω ) k k=1 a

󵄨󵄨 z 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z 󵄨󵄨 η 󵄨󵄨 b

−

1 p−1 ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) Γ(p) a

󵄨󵄨 z 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z} 󵄨󵄨 η 󵄨󵄨 =

e−λ(Ξ(b)−Ξ(a)) (Ξ(b) − Ξ(a))ϑ−1 e−λ(Ξ(a)−Ξ(b)) (Ξ(a) − Ξ(b))p (l‖A ‖ + N) + Γ(p + 1) |∧| m

× {∑ j=1

n

+∑ i=1 r

|αj |e−λ(Ξ(ζj )−Ξ(a)) (Ξ(ζj ) − Ξ(a))p Γ(p + 1)

|βi |e−λ(Ξ(θi )−Ξ(a)) (Ξ(θi ) − Ξ(a))p+ϕi Γ(p + ϕi + 1)

|ϱk |e−λ(Ξ(μk )−Ξ(a)) (Ξ(μk ) − Ξ(a))p−ωk Γ(p − ωk + 1) k=1

+∑

228 � S. Manikandan et al.

+ ≤[

e−λ(Ξ(b)−Ξ(a)) (Ξ(b) − Ξ(a))p }(l‖A ‖ + N) Γ(p + 1)

e−λ(Ξ(a)−Ξ(b)) (Ξ(a) − Ξ(b))p e−λ(Ξ(b)−Ξ(a)) (Ξ(b) − Ξ(a))ϑ−1 + Γ(p + 1) |∧| m

× {∑

|αj |e−λ(Ξ(ζj )−Ξ(a)) (Ξ(ζj ) − Ξ(a))p Γ(p + 1)

j=1

n

+∑ i=1 r

|βi |e−λ(Ξ(θi )−Ξ(a)) (Ξ(θi ) − Ξ(a))p+ϕi Γ(p + ϕi + 1)

|ϱk |e−λ(Ξ(μk )−Ξ(a)) (Ξ(μk ) − Ξ(a))p−ωk Γ(p − ωk + 1) k=1

+∑

+

e−λ(Ξ(b)−Ξ(a)) (Ξ(b) − Ξ(a))p }](lr + N) Γ(p + 1)

= Ω(lr + N) ≤ r. Thus the following is the case: 󵄩󵄩 󵄩 󵄨 󵄨 󵄩󵄩T (A )󵄩󵄩󵄩 = max󵄨󵄨󵄨T u(t)󵄨󵄨󵄨 ≤ r, t∈J which means that T (Br ) ⊂ Br . Step II. We will show that the operator A is a contraction. Let A , A ∈ X. Then for any t ∈ J, we have the following: t

|T A − T A | ≤

1 p−1 ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) Γ(p) a

󵄨󵄨 z z 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(z, A (z), A ( )) − ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z η η 󵄨󵄨 󵄨󵄨 −λ(Ξ(t)−Ξ(z)) ϑ−1 e (Ξ(t) − Ξ(z)) + ∧ m

× {∑ j=1

|αj |

Γ(p)

ζj

p−1

∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) a

󵄨󵄨 z z 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(z, A (z), A ( )) − ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z 󵄨󵄨 η η 󵄨󵄨 n

θi

|βi | p+ϕ −1 −∑ ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) i Γ(p + ϕ ) i i=1 a

󵄨󵄨 z z 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(z, A (z), A ( )) − ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z η η 󵄨󵄨 󵄨󵄨

6 The dynamical analysis of nonlinear Ambartsumian equation

� 229

μk

r

|ϱk | p−ω −1 −∑ ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) k Γ(p − ω ) k k=1 a

󵄨󵄨 z z 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(z, A (z), A ( )) − ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z 󵄨󵄨 η η 󵄨󵄨 b

1 p−1 − ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) Γ(p) a

󵄨󵄨 z z 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(z, A (z), A ( )) − ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z} 󵄨󵄨 η η 󵄨󵄨 ≤[

e−λ(Ξ(a)−Ξ(b)) (Ξ(a) − Ξ(b))p e−λ(Ξ(b)−Ξ(a)) (Ξ(b) − Ξ(a))ϑ−1 + Γ(p + 1) |∧| m

× {∑

|αj |e−λ(Ξ(ζj )−Ξ(a)) (Ξ(ζj ) − Ξ(a))p Γ(p + 1)

j=1

n

+∑ i=1 r

|βi |e−λ(Ξ(θi )−Ξ(a)) (Ξ(θi ) − Ξ(a))p+ϕi Γ(p + ϕi + 1)

|ϱk |e−λ(Ξ(μk )−Ξ(a)) (Ξ(μk ) − Ξ(a))p−ωk Γ(p − ωk + 1) k=1

+∑

+

e−λ(Ξ(b)−Ξ(a)) (Ξ(b) − Ξ(a))p }]l‖A − A ‖. Γ(p + 1)

Thus, the following is the case: ‖T A − T A ‖ = max |T A − T A | ≤ lΩ‖A − A ‖, t∈J

which, in view of lΩ < 1, shows that the operator T is a contraction. Hence, the operator T has a unique fixed point by the Banach contraction mapping principle. Therefore, our proposed problem (6.1) has a unique solution on J. The proof is completed. Now the existence of solution for our proposed problem is discussed by using Krasnoselskii’s fixed point theorem. Theorem 6.3. Let ℚ : J × ℝ × ℝ → ℝ be a continuous function satisfying (H1 ). In addition, we assume that the following condition is satisfied: (H2 ) There exists a continuous function ϕ ∈ X such that 󵄨󵄨 t 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨ℚ(t, A (t), A ( ))󵄨󵄨󵄨 ≤ ϕ(t). 󵄨󵄨 η 󵄨󵄨 Then, our proposed problem (6.1) has at least one solution on J, provided the following condition holds:

230 � S. Manikandan et al. e−λ(Ξ(b)−Ξ(a)) (Ξ(b) − Ξ(a))p e−λ(Ξ(b)−Ξ(a)) (Ξ(b) − Ξ(a))ϑ−1 + Γ(p + 1) |∧| m

× {∑

|αj |e−λ(Ξ(ζj )−Ξ(a)) (Ξ(ζj ) − Ξ(a))p Γ(p + 1)

j=1

n

+∑ i=1 r

|βi |e−λ(Ξ(θi )−Ξ(a)) (Ξ(θi ) − Ξ(a))p+ϕi Γ(p + ϕi + 1)

|ϱk |e−λ(Ξ(μk )−Ξ(a)) (Ξ(μk ) − Ξ(a))p−ωk Γ(p − ωk + 1) k=1

+∑

+

e−λ(Ξ(b)−Ξ(a)) (Ξ(b) − Ξ(a))p }l < 1. Γ(p + 1)

(6.11)

Proof. By assumption (H2 ), we can fix ρ ≥ Ω‖ϕ‖ and consider a closed ball Bρ = {A ∈ C(J, ℝ) : ‖A ‖ ≤ ρ}, where ‖ϕ‖ = supt∈J |ϕ(t)|. We verify the hypotheses of Krasnoselskii’s fixed point theorem by splitting the operator T as T = G + H , where G , H are defined by the following: G A (t) = H A (t) =

t 󵄨 z 󵄨󵄨󵄨 1 p−1 󵄨󵄨 ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) 󵄨󵄨󵄨ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z, 󵄨󵄨 Γ(p) η 󵄨󵄨

e

a −λ(Ξ(t)−Ξ(z))

m

× {∑ j=1

|αj |

(Ξ(t) − Ξ(z))ϑ−1 ∧

Γ(p)

ζj

p−1

∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) a

󵄨󵄨 z 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z 󵄨󵄨 η 󵄨󵄨 n

−∑ i=1

θi

|βi | p+ϕ −1 ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) i Γ(p + ϕi ) a

󵄨󵄨 z 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z η 󵄨󵄨 󵄨󵄨 μk

r

|ϱk | p−ω −1 −∑ ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) k Γ(p − ω ) k k=1 a

󵄨󵄨 z 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z 󵄨󵄨 η 󵄨󵄨 b

1 p−1 − ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) Γ(p) a

󵄨󵄨 z 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z}. 󵄨󵄨 η 󵄨󵄨

6 The dynamical analysis of nonlinear Ambartsumian equation

� 231

For any A , A ∈ Bρ , we have the following: 󵄨󵄨 󵄨 󵄨󵄨G A (t) + H A (t)󵄨󵄨󵄨 =

t 󵄨 1 z 󵄨󵄨󵄨 p−1 󵄨󵄨 ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) 󵄨󵄨󵄨ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z 󵄨󵄨 Γ(p) η 󵄨󵄨

+

e

a −λ(Ξ(t)−Ξ(z))

m

× {∑ j=1

|αj |

Γ(p)

ζj

(Ξ(t) − Ξ(z))ϑ−1 ∧ p−1

∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) a

󵄨󵄨 z 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z 󵄨󵄨 η 󵄨󵄨 θi

n

|βi | p+ϕ −1 −∑ ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) i Γ(p + ϕi ) i=1 a

󵄨󵄨 z 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z 󵄨󵄨 η 󵄨󵄨 μk

r

|ϱk | p−ω −1 −∑ ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) k Γ(p − ω ) k k=1 a

󵄨󵄨 z 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z 󵄨󵄨 η 󵄨󵄨 b

−

1 p−1 ∫ Ξ△ (z)e−λ(Ξ(t)−Ξ(z)) (Ξ(t) − Ξ(z)) Γ(p) a

󵄨󵄨 z 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z} 󵄨󵄨 η 󵄨󵄨 ≤[

e−λ(Ξ(a)−Ξ(b)) (Ξ(a) − Ξ(b))p e−λ(Ξ(b)−Ξ(a)) (Ξ(b) − Ξ(a))ϑ−1 + Γ(p + 1) |∧| m

× {∑

|αj |e−λ(Ξ(ζj )−Ξ(a)) (Ξ(ζj ) − Ξ(a))p

j=1

n

+∑ i=1 r

Γ(p + 1)

|βi |e−λ(Ξ(θi )−Ξ(a)) (Ξ(θi ) − Ξ(a))p+ϕi Γ(p + ϕi + 1)

|ϱk |e−λ(Ξ(μk )−Ξ(a)) (Ξ(μk ) − Ξ(a))p−ωk Γ(p − ωk + 1) k=1

+∑

+

e−λ(Ξ(b)−Ξ(a)) (Ξ(b) − Ξ(a))p }]‖ϕ‖ Γ(p + 1)

= Ω‖ϕ‖ ≤ ρ.

232 � S. Manikandan et al. Hence ‖G A (t) + H A (t)‖ ≤ ρ, which shows that G A (t) + H A (t) ∈ Bρ . Now, it is easy to prove that the operator H is a contraction mapping. The operator G is continuous by the continuity of ℚ. Moreover, G is uniformly bounded on Bρ since ‖G A ‖ ≤

e−λ(Ξ(b)−Ξ(a)) (Ξ(b) − Ξ(a))p ‖ϕ‖. Γ(p + 1)

Finally, we prove the compactness of the operator G . For t1 , t2 ∈ J, t1 < t2 , we have the following: 󵄨󵄨 󵄨 󵄨󵄨G A (t2 ) − G A (t1 )󵄨󵄨󵄨 t1

≤

1 p−1 ∫ Ξ△ (z)[e−λ(Ξ(t2 )−Ξ(z)) (Ξ(t2 ) − Ξ(z)) Γ(p) a

p−1

− e−λ(Ξ(t1 )−Ξ(z)) (Ξ(t1 ) − Ξ(z))

󵄨󵄨 z 󵄨󵄨󵄨 󵄨 ]󵄨󵄨󵄨ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z 󵄨󵄨 η 󵄨󵄨

t

2 󵄨 1 z 󵄨󵄨󵄨 p−1 󵄨󵄨 + ∫ Ξ△ (z)e−λ(Ξ(t2 )−Ξ(z)) (Ξ(t2 ) − Ξ(z)) 󵄨󵄨󵄨ℚ(z, A (z), A ( ))󵄨󵄨󵄨 △ z 󵄨 Γ(p) η 󵄨󵄨 󵄨

t1

≤

‖ϕ‖ p [2(e−λ(Ξ(t2 )−Ξ(t1 )) (Ξ(t2 ) − Ξ(t1 )) ) Γ(p + 1) p󵄨 󵄨 + 󵄨󵄨󵄨e−λ(Ξ(t2 )−Ξ(t1 )) (Ξ(t2 ) − Ξ(t1 )) 󵄨󵄨󵄨],

so that |G A (t2 ) − G A (t1 )| → 0, as t2 → t1 . Thus, G is equicontinuous, By applying the Arzela–Ascoli theorem, we deduce that the operator G is compact on Bρ . Thus, the hypotheses of Krasnoselskii’s fixed point theorem hold true. In consequence, there exists at least one solution for our proposed problem (6.1) on J. This completes the proof.

6.4 Stability theory Definition 6.8. Let p > 0, λ ∈ ℝ, and ℚ ∈ C(J, ℝ). Equation (6.1) is said to be Ulam– Hyers (UH) stable if there exists a constant cℚ > 0 such that for each ϵ > 0 and for each z ∈ AC n (J, ℝ) the following inequality holds true: 󵄨󵄨 t 󵄨󵄨󵄨 󵄨󵄨 TH𝕋 p,q,λ 󵄨󵄨(a+ △Ξ z)(t) − ℚ(t, z(t), z( ))󵄨󵄨󵄨 ≤ ϵ, 󵄨󵄨 η 󵄨󵄨 Then there exists a solution A ∈ AC n (J, ℝ) of Eq. (6.1) with |z − A |AC n (J,ℝ) ≤ cℚ ϵ.

t ∈ [a, b].

(6.12)

6 The dynamical analysis of nonlinear Ambartsumian equation

� 233

Equation (6.1) is said to be generalized UH stable if there exits a function cℚ ∈ C[ℝ+0 , ℝ+0 ] with cℚ (0) = 0 such that for each ϵ > 0 and for each zAC n (J,ℝ) satisfying the inequality (6.12), there exits a solution A ∈ AC n [J, ℝ] of Eq. (6.1) with |z − A |AC n ([a,b],ℝ) ≤ cℚ ϵ. Definition 6.9. Let p > 0, λ ∈ ℝ, and ℚ ∈ C(J × ℝ × ℝ, ℝ). Equation (6.1) is said to be Ulam–Hyers–Rassias (UHR) stable with respect to a given function ν ∈ C(J, ℝ+ ) if there exits a constant ϕℚ,ν > 0 such that, for each ϵ > 0 and for each z ∈ AC n (J, ℝ) satisfying the inequality 󵄨󵄨 t 󵄨󵄨󵄨 󵄨󵄨 TH𝕋 p,q,λ 󵄨󵄨(a+ △Ξ A )(t) − ℚ(t, z(t), z( ))󵄨󵄨󵄨 ≤ ϵν(t), 󵄨󵄨 η 󵄨󵄨

t ∈ J,

(6.13)

t ∈ J,

(6.14)

there exists a solution A ∈ AC n (J, ℝ) of Eq. (6.1) with ‖z − A ‖AC n (J,ℝ) ≤ ϵϕℚ,ν ν(t),

t ∈ J,

and if 󵄨󵄨 t 󵄨󵄨󵄨 󵄨󵄨 TH𝕋 p,q,λ 󵄨󵄨(a+ △Ξ A )(t) − ℚ(t, z(t), z( ))󵄨󵄨󵄨 ≤ ν(t), 󵄨󵄨 η 󵄨󵄨 then there exists a solution A ∈ AC n (J, ℝ) of Eq. (6.1) with |z − A |AC n (J,ℝ) ≤ ϕℚ,ν ν(t),

t ∈ J.

Theorem 6.4. Let p > 0, λ ∈ ℝ, and ℚ ∈ C([a, b] × ℝ × ℝ, ℝ). If ν ∈ C(J, ℝ+ ) is a nondecreasing function and K > 0 is a constant such that T𝕋 p,λ a+ IΞ ν(t)

≤ Kν(t),

t ∈ J,

(6.15)

then the differential equation (6.1) is UHR stable with respect to ν. Proof. Fix ϵ > 0 and let z ∈ AC n (J, ℝ) be any solution of the inequality 󵄨󵄨 t 󵄨󵄨󵄨 󵄨󵄨 TH𝕋 p,q,λ 󵄨󵄨(a+ △Ξ z)(t) − ℚ(t, z(t), z( ))󵄨󵄨󵄨 ≤ ϵν(t), 󵄨󵄨 η 󵄨󵄨

t ∈ J.

(6.16)

t ∈ J,

(6.17)

Then, we can define w ∈ AC n (J, ℝ) such that p,q,λ

(aTH𝕋 △Ξ +

t )(t)z(t) = ℚ(t, z(t), z( )) + w(t), η

and |w(t)| ≤ ϵν(t), for all t ∈ (a, b]. Now let A ∈ AC n (J, ℝ) be a solution of the following boundary value problem:

234 � S. Manikandan et al. t p,q,λ { (aTH𝕋 △Ξ A )(t) = ℚ(t, A (t), A ( )), t ∈ J, { + { { η { { { { A (a) = 0, { { { m n r { { TH𝕋 ϕi ,λ TH𝕋 ωk ,λ { { {A (b) = ∑ αj A (ζj ) + ∑ βi a+ IΞ A (θi ) + ∑ ϱk a+ △Ξ A (μk ). j=1 i=1 k=1 {

(6.18)

The solution of Eq. (6.18) is p,q,λ

T𝕋

A (t) = a+ IΞ

t e−λ(Ξ(t)−Ξ(a)) (Ξ(t) − Ξ(a))ϑ−1 ℚ(t, A (t), A ( )) + η ∧

m

p,q,λ

× { ∑ αj T𝕋 a+ IΞ j=1

n

p,q,λ

− ∑ βi T𝕋 a+ IΞ i=1 r

− ∑ ϱk k=1

ζj ℚ(ζj , A (ζj ), A ( )) η

ℚ(θi , A (θi ), A (

θi )) η

μ T𝕋 p−ωk ,λ ℚ(μk , A (μk ), A ( k )) a + IΞ η

b p,λ − T𝕋 a+ IΞ ℚ(b, A (b), A ( ))}, η

(6.19)

while the solution of Eq. (6.16) is given by t e−λ(Ξ(t)−Ξ(a)) (Ξ(t) − Ξ(a))ϑ−1 p,λ z(t) = T𝕋 I ℚ(t, z(t), z( )) + + a Ξ η ∧ m ζj p,λ × { ∑ αj T𝕋 a+ IΞ ℚ(ζj , z(ζj ), z( )) η j=1 n

p,q,λ

− ∑ βi T𝕋 a+ IΞ i=1 r

ℚ(θi , z(θi ), z(

p−ωk ,q,λ

− ∑ ϱk T𝕋 a + IΞ k=1

p,q,λ

− T𝕋 a + IΞ

θi )) η

ℚ(μk , z(μk ), z(

μk )) η

b ℚ(b, z(b), z( ))}, η

where h1 (⋅) = ℚ(⋅, z(⋅), z( η⋅ )). From Eq. (6.20) and inequality (6.15), we have 󵄨󵄨 󵄨󵄨 t e−λ(Ξ(t)−Ξ(a)) (Ξ(t) − Ξ(a))ϑ−1 p,λ 󵄨󵄨z(t) − [T𝕋 I ℚ(t, z(t), z( )) + + a Ξ 󵄨󵄨󵄨 η ∧ 󵄨 m n ζj θi p,λ T𝕋 p,λ × { ∑ αj T𝕋 a+ IΞ ℚ(ζj , z(ζj ), z( )) − ∑ βi a+ IΞ ℚ(θi , z(θi ), z( )) η η j=1 i=1

(6.20)

6 The dynamical analysis of nonlinear Ambartsumian equation

r

p−ωk ,λ

− ∑ ϱk T𝕋 a+ IΞ k=1

ℚ(μk , z(μk ), z(

� 235

󵄨󵄨 󵄨󵄨 μk b p,λ 󵄨󵄨 )) − T𝕋 ))}] I ℚ(b, z(b), z( + a 󵄨󵄨 Ξ η η 󵄨󵄨

p,λ 󵄨󵄨 󵄨 ≤ 󵄨󵄨󵄨T𝕋 a+ IΞ w(t)󵄨󵄨 p,λ

≤ ϵT𝕋 a+ IΞ ν(t) ≤ ϵKν(t). Using this together with Eq. (6.19) and Lipschitz condition for each t ∈ J, we have 󵄨󵄨 t e−λ(Ξ(t)−Ξ(a)) (Ξ(t) − Ξ(a))ϑ−1 󵄨󵄨 󵄨 󵄨󵄨 T𝕋 p,λ 󵄨󵄨z(t) − A (t)󵄨󵄨󵄨 = 󵄨󵄨󵄨z(t) − a+ IΞ ℚ(t, A (t), A ( )) + η ∧ 󵄨󵄨󵄨 m ζj p,λ × { ∑ αj T𝕋 a+ IΞ ℚ(ζj , A (ζj ), A ( )) η j=1 n

p,λ

− ∑ βi T𝕋 a+ IΞ ℚ(θi , A (θi ), A ( i=1 r

p−ωk ,λ

− ∑ ϱk T𝕋 a + IΞ k=1

θi )) η

ℚ(μk , A (μk ), A (

μk )) η

󵄨󵄨

−

󵄨󵄨 b T𝕋 p,λ 󵄨 a+ IΞ ℚ(b, A (b), A ( ))}󵄨󵄨󵄨 η

󵄨󵄨

t

≤ ϵKν(t) +

1 p−1 ∫ Ξ′ (s)(Ξ(t) − Ξ(s)) e−λ(Ξ(t)−Ξ(s)) Γ(p) a

󵄨󵄨 s s 󵄨󵄨󵄨 󵄨 × 󵄨󵄨󵄨ℚ(s, z(s), z( )) − ℚ(s, A (s), A ( ))󵄨󵄨󵄨 △ s η η 󵄨󵄨 󵄨󵄨 t

L p−1 ≤ ϵKν(t) + ∫ Ξ′ (s)(Ξ(t) − Ξ(s)) e−λ(Ξ(t)−Ξ(s)) Γ(p) a

󵄨 s 󵄨󵄨󵄨 󵄨 󵄨 󵄨󵄨 s × {󵄨󵄨󵄨z(s) − A (s)󵄨󵄨󵄨 + 󵄨󵄨󵄨z( ) − A ( )󵄨󵄨󵄨} △ s. 󵄨󵄨 η η 󵄨󵄨 Applying Gronwall’s inequality, with u(t) = |z(t) − A (t)|, v(t) = ϵKν(t), and w(t) = we obtain

L , Γ(p)

p 󵄨󵄨 󵄨 󵄨󵄨z(t) − A (t)󵄨󵄨󵄨 ≤ ϵKν(t)Ep (L(Ξ(b) − Ξ(a)) ),

≤ ϵcℚ,ν ν(t),

t ∈ [a, b],

(6.21)

where cℚ,ν := KEp (L(Ξ(b) − Ξ(a))p ). Thus our proposed problem (6.1) is UHR stable with respect to ν.

236 � S. Manikandan et al.

6.5 An example Consider the following tempered Ξ-Hilfer type Ambartsumian boundary value problem on time scales: t p,q,λ { (aTH𝕋 △Ξ A )(t) = ℚ(t, A (t), A ( )), t ∈ [0, 5], { + { { η { { { { A (a) = 0, { { { m n r { { TH𝕋 ωk ,λ TH𝕋 ϕi ,λ { { {A (b) = ∑ αj A (ζj ) + ∑ βi a+ IΞ A (θi ) + ∑ ϱk a+ △Ξ A (μk ), j=1 i=1 k=1 { 1 , q = 41 , m = 4, n = 3, r = 2, α1 = 151 , α2 = 101 , α3 = 152 , α4 3 2 = = 5 , ζ4 = 2, β1 = 181 , β2 = 112 , β3 = 72 , ϕ1 = 21 , ϕ2 = 31 , ϕ3 = 52 , θ1 = 1, μ1 = 3, μ2 = 72 , ω1 = 41 , ω2 = 31 , ϱ1 = 281 , ϱ2 = 141 , λ = 111 , ϑ = 21 , η = 1.175.

(6.22)

with Ξ(t) = 1 + t 2 , p =

=

ζ1 =

=

θ2 =

1 ,ζ 4 2 3 , θ 4 3

1 ,ζ 2 3

2 , 7 1 , 2

From the above values we get ∧ ≈ −1.10200512 and Ω ≈ 0.3631064. Also 1 t t A( ) − A (t), ℚ(t, A (t), A ( )) = η 1.175 1.175 1 ℚ(t, u, v) = (u − v), u, v ∈ ℝ. 1.175

t ∈ [0, 5],

(6.23)

Clearly, ℚ is continuous. For any u, v, u, v ∈ ℝ and t ∈ [0, 5], 1 󵄨󵄨 󵄨 |u − v − u + v| 󵄨󵄨ℚ(t, u, v) − ℚ(t, u, v)󵄨󵄨󵄨 = 1.175 1 ≤ {|u − u| + |v − v|}. 1.175

(6.24)

1 Hence, assumption (H1 ) holds for the nonlinear function ℚ(t, A (t), A ( ηt )) with l = 1.175 and also the condition lΩ ≈ 0.3090267234 < 1 holds. Hence our proposed problem (6.22) with (6.23) has a unique solution on [0, 5]. Theorem 6.4 implies that (6.22) is UHR stable.

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Svetlin G. Georgiev and İnci M. Erhan

7 Series solution method on time scales and its applications Abstract: In this chapter, a series solution method on general time scales is introduced. The derivation of the method and its application to dynamic and integral equations is discussed in details. Several examples illustrating the method are presented.

7.1 Introduction In this chapter, we introduce the series solution method on a general time scale. The series solution method on time scales has been discussed for some particular cases in recent studies [4, 7–9]. The series solution method for certain integral equations has been used in recent books [3, 6]. Also, the series solution method has been applied to initial value problems associated with first-order dynamic equations and fractional dynamic equations in [5]. We present the series solution method on an arbitrary time scale and its application to linear dynamic equations and to nonlinear Volterra integral equations in which the nonlinear term is at most a quadratic function. We consider Volterra integral equations of both the first and second kind. Moreover, the method is applicable in cases when the graininess function is constant or nonconstant. The chapter is organized as follows. In the next section, we recall some preliminary concepts such as the Taylor series and Cauchy product of power series on time scales, and derive other preliminary results. The series solution method for dynamic equations and its application to particular examples is discussed in Section 7.3. Finally, we introduce an application of the series solution method to a certain type nonlinear Volterra integral equations of the first and second kind and give specific examples in Sections 7.4 and 7.5.

7.2 Preliminary results We first recall the Taylor series on general time scales. Let 𝕋 be a time scale with forward jump operator and delta differentiation operator, σ and Δ, respectively. Let s, t ∈ 𝕋. The monomials hk (t, s) are defined recursively as follows [2]: Svetlin G. Georgiev, Sorbonne University, Paris, France, e-mail: [email protected] İnci M. Erhan, Department of Computer Engineering, Aydın Adnan Menderes University, Aydın, Turkey, e-mail: [email protected] https://doi.org/10.1515/9783111182971-007

240 � S. G. Georgiev and İ. M. Erhan h0 (t, s) = 1, t

hk+1 (t, s) = ∫ hk (τ, s)Δτ,

k = 0, 1, 2, . . .

s

Then it is obvious that t

t

h1 (t, s) = ∫ h0 (τ, s)Δτ = ∫ Δτ = t − s, s

s

t

t

h2 (t, s) = ∫ h1 (τ, s)Δτ = ∫(τ − s)Δτ, s

s

and so on. Note that hkΔ (t, s) = hk−1 (t, s),

k ∈ ℕ.

Theorem 7.1 (Taylor formula, [1, 3]). Let n ∈ ℕ. Suppose that f is n times differentiable n n−1 on 𝕋κ . Let also α ∈ 𝕋κ , t ∈ 𝕋. Then n−1

ρn−1 (t)

Δk

n

f (t) = ∑ hk (t, α)f (α) + ∫ hn−1 (t, σ(τ))f Δ (τ)Δτ. k=0

α

Theorem 7.2 (Leibnitz formula, [1, 3]). Let Sk(n) be the set consisting of all possible strings of length n, containing exactly k times σ and n − k times Δ. If fΛ

exists for all Λ ∈ Sk(n) ,

then n

n

k

(fg)Δ = ∑ ( ∑ f Λ )g Δ . k=0 Λ∈S (n) k

The next theorem is an important result which is crucial in the application of the series solution method. Theorem 7.3 ([3]). For every m, n ∈ ℕ0 , we have m+n

Λ

hn (t, α)hm (t, α) = ∑ ( ∑ hn l,m (α, α))hl (t, α) (l) l=m Λl,m ∈Sm

for every t, α ∈ 𝕋. Proof. If m = 0 or n = 0, the assertion is evident. Suppose that m ≠ 0 and n ≠ 0. By the Taylor formula, we have

7 Series solution method on time scales and its applications

∞

Δl 󵄨 hn (t, α)hm (t, α) = ∑(hn (t, α)hm (t, α)) 󵄨󵄨󵄨t=α hl (t, α),

t, α ∈ 𝕋.

l=0

By the Leibnitz formula, we have l

Δl

Λ

k

Δ (hn (t, α)hm (t, α)) = ∑ ( ∑ hn l,k (t, α))hm (t, α),

t, α ∈ 𝕋.

k=0 Λl,k ∈S (l) k

If l < m, then l

Δl

Λ

(hn (t, α)hm (t, α)) = ∑ ( ∑ hn l,k (t, α))hm−k (t, α),

t, α ∈ 𝕋.

k=0 Λl,k ∈S (l) k

From here, for l < m, we have hm−k (α, α) = 0 and Δl 󵄨 (hn (t, α)hm (t, α)) 󵄨󵄨󵄨t=α = 0,

For l ≥ m, using that h0 (t, α) = 1,

t, α ∈ 𝕋.

t, α ∈ 𝕋, we get m−1

Δl 󵄨 Λ (hn (t, α)hm (t, α)) 󵄨󵄨󵄨t=α = ∑ ( ∑ hn l,k (t, α))hm−k (t, α)|t=α k=0 Λl,k ∈S (l) k

+ =

Λ

∑ hn l,m (t, α)|t=α

(l) Λl,m ∈Sm

Λ

∑ hn l,m (α, α),

(l) Λl,m ∈Sm

t, α ∈ 𝕋.

Hence, using that Λl,m contains m times σ and l − m times Δ, and fσ = f

or

f σ = f + μf Δ ,

f σσ = f

or

f σσ = f + μf Δ + μσ (f Δ + μf Δ ),

2

and so on, we obtain ∞

Δl 󵄨 hn (t, α)hm (t, α) = ∑ (hn (t, α)hm (t, α)) 󵄨󵄨󵄨t=α hl (t, α) l=m ∞

Λ

= ∑ ( ∑ hn l,m (α, α))hl (t, α) (l) l=m Λl,m ∈Sm

m+n

Λ

= ∑ ( ∑ hn l,m (α, α))hl (t, α), (l) l=m Λl,m ∈Sm

which completes the proof.

t, α ∈ 𝕋,

� 241

242 � S. G. Georgiev and İ. M. Erhan Theorem 7.4. We have ∞

∞

∞

i=0

j=0

r=0

(∑ Ai hi (t, α))(∑ Bj hj (t, α)) = ∑ Cr hr (t, α),

t, α ∈ 𝕋,

where ∞

k

Cr = ∑ ( ∑ Al Bk−l ( k=r

l=k−r

Λ

∑

(r) Λr,k−l ∈Sk−l

hl r,k−l (α, α))),

(7.1)

with Ai , Bi , i ∈ ℕ0 , being constants. Proof. Using the Cauchy product of two infinite series and Theorem 7.3, we get ∞

∞

(∑ Ai hi (t, α))(∑ Bj hj (t, α)) i=0

j=0

k

∞

= ∑ (∑ Al hl (t, α)Bk−l hk−l (t, α)) k=0 l=0 k

∞

k

= ∑ (∑ Al Bk−l ( ∑ ( k=0 l=0

r=k−l

∑

(r) Λr,k−l ∈Sk−l

Λ

t, α ∈ 𝕋.

(7.2)

Λ

t, α ∈ 𝕋.

(7.3)

hl r,k−l (α, α))hr (t, α))),

Then the double sum in (7.2) becomes ∞

∞

(∑ Ai hi (t, α))(∑ Bj hj (t, α)) i=0

j=0

k

∞

k

= ∑ (∑ Al Bk−l ( ∑ ( k=0 l=0

r=k−l

∑

(r) Λr,k−l ∈Sk−l

hl r,k−l (α, α))hr (t, α))),

k k ∞ ∞ k If we reorder the triple sum ∑∞ k=0 ∑l=0 ∑r=k−l in (7.3) as ∑r=0 ∑k=r ∑l=k−r and denote ∞

k

Cr = ∑ ( ∑ Al Bk−l ( k=r

l=k−r

∑

(r) Λr,k−l ∈Sk−l

Λ

hl r,k−l (α, α))),

r ∈ ℕ0 ,

α∈𝕋

for the sake of brevity, we conclude ∞

∞

∞

i=0

j=0

r=0

(∑ Ai hi (t, α))(∑ Bj hj (t, α)) = ∑ Cr hr (t, α), which completes the proof.

t, α ∈ 𝕋,

(7.4)

7 Series solution method on time scales and its applications

� 243

In order to avoid the complicated structure of the constants Cr involved in the Cauchy product of infinite series, we define the constants Dn,m,l as Dn,m,l =

Λ

∑ hn l,m (α, α)

(7.5)

(l) Λl,m ∈Sm

for α ∈ 𝕋, n, m ∈ ℕ0 and l ∈ {m − n, . . . , m}. Then we have m+n

hn (t, α)hm (t, α) = ∑ Dn,m,l hl (t, α), l=m

t, α ∈ 𝕋,

and also ∞

∞

∞ k

k

i=0

j=0

k=0 l=0

r=k−l

(∑ Ai hi (t, α))(∑ Bj hj (t, α)) = ∑ ∑ Al Bk−l ( ∑ Dl,k−l,r )hr (t, α),

t, α ∈ 𝕋.

The constants Dn,m,l defined in (7.5) have some properties which we discuss below. 1. For all m ∈ ℕ0 and l = m, . . . , m + n, we have D0,m,l = Dm,0,l = {

0 1

if l ≠ m, if l = m.

(7.6)

Proof. This statement easily follows from the fact that h0 (t, α)hm (t, α) = hm (t, α)h0 (t, α) = hm (t, α),

t, α ∈ 𝕋.

Indeed, m

hm (t, α) = h0 (t, α)hm (t, α) = ∑ D0,m,l hl (t, α) = D0,m,m hm (t, α), l=m

t, α ∈ 𝕋

implies D0,m,m = 1, and m

hm (t, α) = hm (t, α)h0 (t, α) = ∑ Dm,0,l hl (t, α), l=0

t, α ∈ 𝕋

implies Dm,0,m = 1 and Dm,0,l = 0 for l = 0, . . . , m − 1. 2.

For all n, m ∈ ℕ0 and l = m, . . . , n + m we have Dn,m,l = Dm,n,l

for n ≥ m and l ≥ n,

Dn,m,l = Dm,n,l = 0

for n ≥ m and m < l < n.

Proof. Since hn (t, α)hm (t, α) = hm (t, α)hn (t, α),

t, α ∈ 𝕋,

(7.7)

244 � S. G. Georgiev and İ. M. Erhan we have m+n

m+n

l=m

l=n

∑ Dn,m,l hl (t, α) = ∑ Dm,n,l hl (t, α),

t, α ∈ 𝕋,

from which it follows that if n ≥ m we should have Dn,m,m = Dn,m,m+1 = ⋅ ⋅ ⋅ = Dn,m,n−1 = 0 and also Dn,m,l = Dm,n,l

for l = n, n + 1, . . . , n + m.

We next give the computation of the constants Dn,m,l for small values of the subscripts on arbitrary time scales. For larger values, the computation becomes long and complicated. Let 𝕋 be a time scale with the forward jump operator σ, delta differentiation operator Δ, and the graininess function μ. Let hl (t, α), t, α ∈ 𝕋, l = 0, 1, 2, 3 be the monomials on 𝕋. We are using the following relations in the computations of the constants Dn,m,l : hkΔ (t, α) = hk−1 (t, α),

hkσ (t, α) = hk (t, α) + μ(t)hkΔ (t, α) = hk (t, α) + μ(t)hk−1 (t, α),

σ hkΔσ (t, α) = hk−1 (t, α) = hk−1 (t, α) + μ(t)hk−2 (t, α),

hkσΔ (t, α) = (1 + μΔ (t))hkΔσ (t, α),

2

hkσσ (t, α) = hk (t, α) + μ(t)hkΔ (t, α) + μσ (t)[hkΔ (t, α) + μ(t)hkΔ (t, α)]

= hk (t, α) + (μ(t) + μσ (t))hk−1 (t, α) + μσ (t)μ(t)hk−2 (t, α),

hkΔΔ (t, α) = hk−2 (t, α),

hkσσσ (t, α) = (hk (t, α) + μ(t)hk−1 (t, α))

σσ

= hk (t, α)

+ [μ(t) + (μ(t) + μσ (t))(1 + μΔ (t)) + μ(t)μσ (t)μΔΔ (t)]hk−1 (t, α)

+ [μσ (t)(μ(t) + μσ (t)) + μ(t)μσ (t)(1 + μΔσ (t) + μσΔ (t))]hk−2 (t, α)

+ μσσ (t)μσ (t)μ(t)hk−3 (t, α),

t, α ∈ 𝕋.

Then we compute the following: 1. Let l = 0. Then from the property (7.6) we have D0,0,0 = 1. 2.

Let l = 1. Then from the property (7.6) we have D1,0,1 = D0,1,1 = 1,

7 Series solution method on time scales and its applications

D1,1,1 = h1σ (α, α) = h1 (α, α) + μ(α)h0 (α, α) = μ(α), 3.

� 245

α ∈ 𝕋.

Let l = 2. Then we have D2,0,2 = D0,2,2 = 1,

D1,1,2 = h1σΔ (α, α) + h1Δσ (α, α) = (2 + μΔ (α))h1Δσ (α, α) = (2 + μΔ (α))h0σ (α, α) = 2 + μΔ (α),

D2,1,2 = D1,2,2 = h1σσ (α, α)

= h1 (α, α) + (μ(α) + μσ (α))h0 (α, α) = μ(α) + μσ (α),

α ∈ 𝕋,

and D2,2,2 = h2σσ (α, α)

= h2 (α, α) + (μ(α) + μσ (α))h1 (αα) + μ(α)μσ (α)h0 (α, α) = μ(α)μσ (α),

4.

α ∈ 𝕋.

Let l = 3. Then we compute D3,0,3 = D0,3,3 = 1,

D1,1,3 = h1ΔΔσ (α, α) + h1ΔσΔ (α, α) + h1σΔΔ (α, α) = μΔΔ (α),

D1,2,3 = D2,1,3 (α) = h2ΔΔσ (α, α) + h2ΔσΔ (α, α) + h2σΔΔ (α, α) = h0σ (α, α) + h1σΔ (α, α) + (h2 (α, α) + μ(α)h1 (α, α))

ΔΔ

= 3 + μΔ (α) + μΔσ (α) + μσΔ (α),

D2,2,3 = h2Δσσ (α, α) + h2σΔσ (α, α) + h2σσΔ (α, α) σ

= 2(μ(α) + μσ (α)) + (μ(α) + μσ (α)) Δ

+ (μ(α)μσ (α)) + μΔσ (α)(2 + μΔ (α)),

D3,1,3 = D1,3,3 = h1σσσ (α, α) = h1 (α, α)

+ [μ(α) + (μ(α) + μσ (α))(1 + μΔ (α)) + μ(α)μσ (α)μΔΔ (α)]h0 (α, α)

= μ(α) + (μ(α) + μσ (α))(1 + μΔ (α)) + μ(α)μσ (α)μΔΔ (α),

D3,2,3 = D2,3,3 = h2σσσ (α, α) = h2 (α, α)

+ [μ(α) + (μ(α) + μσ (α))(1 + μΔ (α)) + μ(α)μσ (α)μΔΔ (α)]h1 (α, α)

+ [μσ (α)(μ(α) + μσ (α)) + μ(α)μσ (α)(1 + μΔσ (α) + μσΔ (α))]h0 (α, α)

= μσ (α)(μ(α) + μσ (α)) + μ(α)μσ (α)(1 + μΔσ (α) + μσΔ (α)),

246 � S. G. Georgiev and İ. M. Erhan D3,3,3 = h3σσσ (α, α) = h3 (α, α) + [μ(α) + (μ(α) + μσ (α))(1 + μΔ (α)) + μ(α)μσ (α)μΔΔ (α)]h2 (α, α) + [μσ (α)(μ(α) + μσ (α)) + μ(α)μσ (α)(1 + μΔσ (α) + μσΔ (α))]h1 (α, α) + μ(α)μσ (α)μσσ (α)h0 (α, α) = μ(α)μσ (α)μσσ (α),

α ∈ 𝕋.

As seen above, the computation of the constants Dn,m,l becomes hard for large values of the subscripts and depends solely on the forward jump operator and the graininess function of the time scale under consideration.

7.3 Series solution method for dynamic equations In this section, we use the series solution method to solve linear dynamic equations. Let 𝕋 be a time scale with forward jump operator and delta differentiation operator, σ and Δ, respectively. Suppose that the graininess function μ is differentiable. Consider the dynamic equation n

n−1

yΔ (t) + a1 (t)yΔ (t) + ⋅ ⋅ ⋅ + an (t)y(t) = b(t),

t ∈ 𝕋,

(7.8)

where ∞

am (t) = ∑ aim hi (t, α), i=0

∞

b(t) = ∑ bi hi (t, α), i=0

t, α ∈ 𝕋,

and aim , bi , i ∈ ℕ0 , m ∈ {1, . . . , n}, are given constants. We will search for a solution y of equation (7.8) in the form ∞

y(t) = ∑ ci hi (t, α), i=0

(7.9)

t, α ∈ 𝕋,

where ci , i ∈ ℕ0 , are constants that will be determined below. We have r

∞

r

∞

∞

i=r

i=0

yΔ (t) = ∑ ci hiΔ (t, α) = ∑ ci hi−r (t, α) = ∑ ci+r hi (t, α), i=0

By Theorem 7.4, we obtain

t, α ∈ 𝕋,

r ∈ ℕ0 .

7 Series solution method on time scales and its applications

n−m

am (t)yΔ

∞

� 247

∞

(t) = (∑ aim hi (t, α))(∑ cj+n−m hj (t, α)) i=0

∞

j=0

k

= ∑ (∑ alm ck−l+n−m hl (t, α)hk−l (t, α)) k=0 l=0 ∞

k

k

= ∑ (∑ alm ck−l+n−m ( ∑ ( k=0 l=0 ∞

∑

r=k−l Λr,k−l ∈S (r) k−l

k

Λ

hl r,k−l (α, α))hr (t, α)))

k

= ∑ (∑ alm ck−l+n−m ( ∑ Dl,k−l,r hr (t, α))), k=0 l=0

r=k−l

t, α ∈ 𝕋,

(7.10)

where Dl,k−l,r can be computed using (7.5) for k = 0, 1, . . . , l = 0, 1, . . . , k, and r = k − l, . . . , k. Note that the double sum ∑kl=0 ∑kr=k−l can be reordered as ∑kr=0 ∑kl=k−r . In addition, k ∞ ∞ the sum ∑∞ k=0 ∑r=0 can be reordered as ∑r=0 ∑k=r using the Fubini theorem. Therefore, ∞ k k ∞ k we rewrite the triple sum ∑k=0 ∑l=0 ∑r=k−l in (7.10) as ∑∞ r=0 ∑k=r ∑l=k−r , which yields n−m

am (t)yΔ

∞

k

∞

(t) = ∑ ( ∑ ∑ alm ck−l+n−m Dl,k−l,r )hr (t, α), r=0 k=r l=k−r

t, α ∈ 𝕋.

(7.11)

Then, equation (7.8) implies n

n

n−m

yΔ (t) + ∑ am (t)yΔ m=1

󳨐⇒

(t) = b(t) n

r

∞

∞

∞

r=0

r=0 m=1 k=r l=k−r

∞

∑ cr+n hr (t, α) + ∑ ( ∑ ∑ ∑ alm ck−l+n−m Dl,k−l,r )hr (t, α) = ∑ br hr (t, α), r=0

for t, α ∈ 𝕋. Hence, we deduce the following recurrence relation for the computation of the coefficients cj , j ≥ 0: n

∞

r

cr+n = − ∑ ∑ ∑ alm ck−l+n−m Dl,k−l,r + br . m=1 k=r l=k−r

(7.12)

Solving this recurrence relation will lead to a complete determination of the coefficients cj , j ≥ 0. Having determined the coefficients cj , j ≥ 0, the series solution follows immediately upon substituting the derived coefficients into (7.9). The exact solution may be obtained if such an exact solution exists. If an exact solution does not exist, then the infinite series can be used for numerical purposes. In this case, the more terms we determine, the higher accuracy level we achieve. Our first example is a dynamic equation with constant coefficients. It provides an opportunity for testing the series solution method presented above.

248 � S. G. Georgiev and İ. M. Erhan Example. Consider the second-order constant-coefficient dynamic equation yΔΔ − 3yΔ + 2y = 0,

t ∈ 𝕋 = 2ℕ0 .

(7.13)

Its characteristic equation is λ2 − 3λ + 2 = 0, with the solutions λ = 1, λ = 2. Then the linearly independent solutions of the dynamic equation for α = 1 are y1 (t) = e1 (t, 1), y2 (t) = e2 (t, 1), t ∈ 𝕋, [1]. We will apply the series solution method to the equation (7.13). Take α = 1 and propose a solution in the form ∞

t ∈ 𝕋.

y(t) = ∑ cn hn (t, 1), n=0

Then we have ∞

yΔ (t) = ∑ cn+1 hn (t, 1), n=0

∞

yΔΔ (t) = ∑ cn+2 hn (t, 1), n=0

t ∈ 𝕋.

Put y(t), yΔ (t), and yΔΔ (t) into the equation which gives ∞

∞

∞

n=0

n=0

n=0

∑ cn+2 hn (t, 1) − 3 ∑ cn+1 hn (t, 1) + 2 ∑ cn hn (t, 1) = 0,

and upon combining the series we get ∞

∑ (cn+2 − 3cn+1 + 2cn )hn (t, 1) = 0,

n=0

t ∈ 𝕋.

Then we have the following recurrence relation: cn+2 = 3cn+1 − 2cn ,

for n ≥ 0.

We compute first few terms, for arbitrary c0 and c1 , as follows: n = 0,

c2 = 3c1 − 2c0 ,

n = 1,

c3 = 3c2 − 2c1 = 7c1 − 6c0 ,

n = 3,

c5 = 3c4 − 2c3 = 31c1 − 30c0 ,

n = 2,

n = 4,

c4 = 3c3 − 2c2 = 15c1 − 14c0 ,

c6 = 3c5 − 2c4 = 63c1 − 62c0 ,

so that the recurrence relation can be generalized as

t ∈ 𝕋,

7 Series solution method on time scales and its applications

� 249

cn = (2n − 1)c1 − (2n − 2)c0 , for arbitrary c0 , c1 and n ≥ 2. Then the solution y(t) becomes ∞

y(t) = ∑ cn hn (t, 1) n=0

∞

= c0 h0 (t, 1) + c1 h1 (t, 1) + ∑ cn hn (t, 1) n=2 ∞

= c0 h0 (t, 1) + c1 h1 (t, 1) + ∑ [(2n − 1)c1 − (2n − 2)c0 ]hn (t, 1), n=2

t∈𝕋

and can be arranged as ∞

∞

n=0

n=0

y(t) = (c1 − c0 ) ∑ 2n hn (t, 1) + (2c0 − c1 ) ∑ hn (t, 1),

t ∈ 𝕋.

n Recalling that eλ (t, 1) = ∑∞ n=0 λ hn (t, 1), t ∈ 𝕋, we conclude

y(t) = d1 e2 (t, 1) + d2 e1 (t, 1),

t ∈ 𝕋,

d1 = c1 − c0 ,

d2 = 2c0 − c1 ,

which is the general solution of the dynamic equation. We now apply the series solution method to a dynamic equation with nonconstant coefficients. Example. Consider the second-order dynamic equation 2

[1 − (σ(t)) ]yΔΔ (t) − (σ(t) + t)yΔ (t) + k(k + 1)y(t) = 0,

t ∈ 𝕋,

(7.14)

on some time scale 𝕋 with differentiable graininess function, where k is a real constant. We assume that ∞

2

1 − (σ(t)) = ∑ am hm (t, α), m=0 ∞

−(σ(t) + t) = ∑ bm hm (t, α),

(7.15)

m=0

for some t, α ∈ 𝕋 and propose the following series expansion for the solution y: ∞

y(t) = ∑ cn hn (t, α). n=0

(7.16)

As we did in the previous example, we substitute the series (7.15) and (7.16) into the dynamic equation (7.14) and, using the Cauchy product for infinite series, we get

250 � S. G. Georgiev and İ. M. Erhan ∞

n

∞

∑ ∑ (am cn−m+2 + bm cn−m+1 )hm (t, α)hn−m (t, α) + k(k + 1) ∑ cn hn (t, α) = 0,

n=0 m=0

t, α ∈ 𝕋.

n=0

Note that the product hm (t, α)hn−m (t, α), t, α ∈ 𝕋 involved in the last relation can be written as n

hm (t, α)hn−m (t, α) = ∑ Dm,n−m,l hl (t, α), l=n−m

t, α ∈ 𝕋,

(7.17)

for all n = 0, 1, . . . , and m = 0, 1, . . . , n where the constants Dm,n−m,l are defined in (7.5). Employing (7.17) and (7.5), we obtain ∞

n

n

∑ ∑ ∑ [(am cn−m+2 + bm cn−m+1 )Dm,n−m,l ]hl (t, α)

n=0 m=0 l=n−m

∞

+ k(k + 1) ∑ cn hn (t, α) = 0, n=0

t, α ∈ 𝕋.

∞ n Upon reordering the triple sum as ∑∞ l=0 ∑n=l ∑m=n−l , we rewrite the above equation as n

∞

∞

l=0

n=l m=n−l

∑[(∑ ∑ (am cn−m+2 + bm cn−m+1 )Dm,n−m,l ) + k(k + 1)cl ]hl (t, α) = 0,

t, α ∈ 𝕋,

which implies the following recurrence relation for the computation of the coefficients cl , l = 0, 1, 2, . . . : ∞

n

∑ ∑ ((am cn−m+2 + bm cn−m+1 )Dm,n−m,l ) + k(k + 1)cl = 0,

n=l m=n−l

l = 0, 1, 2, . . .

(7.18)

For computational purposes, we consider the particular case of 𝕋 = sℕ0 , s > 0 and choose α = 0. Then we have σ(t) = t + s,

μ(t) = s,

t ∈ 𝕋,

and h0 (t, 0) = 1,

h1 (t, 0) = t,

h2 (t, 0) =

t(t − s) , 2

t ∈ 𝕋.

The dynamic equation (7.14) can be written as (1 − s2 + 2st − t 2 )yΔΔ − (2t + s)yΔ + k(k + 1)y = 0,

t ∈ 𝕋,

(7.19)

where 2

1 − (σ(t)) = 1 − s2 + 2st − t 2 = (1 − s2 )h0 (t, 0) + sh1 (t, 0) − 2h2 (t, 0), −t − σ(t) = −2t − s = −sh0 (t, 0) − 2h1 (t, 0),

t ∈ 𝕋,

(7.20)

7 Series solution method on time scales and its applications

� 251

that is, a0 = 1 − s2 , a1 = s, a2 = −2, b0 = −s, and b1 = −2. For arbitrary c0 and c1 , the recurrence relation (7.18) becomes l+2

2

∑ ∑ [(am cn−m+2 + bm cn−m+1 )Dm,n−m,l ] + k(k + 1)cl = 0,

n=l m=n−l

l = 0, 1, 2, . . . ,

which yields (a0 c2 + b0 c1 )D0,0,0 + k(k + 1)c0 = 0,

(a0 c3 + b0 c2 )D0,1,1 + (a1 c2 + b1 c1 )D1,0,1

for l = 0,

+ (a1 c3 + b1 c2 )D1,1,1 + k(k + 1)c1 = 0,

for l = 1,

and (a0 cl+2 + b0 cl+1 )D0,l,l + (a1 cl+1 + b1 cl )D1,l−1,l + a2 cl D2,l−2,l + (a1 cl+2 + b1 cl+1 )D1,l,l + a2 cl+1 D2,l−1,l + a2 cl+2 D2,l,l

+ k(k + 1)cl = 0,

for l ≥ 2. Therefore, for s ≠ 1 we compute s k(k + 1) c1 − c0 , 2 1−s 1 − s2 c3 = 2sc2 − (k(k + 1) − 2)c1 , c2 =

and cl+2 = −

sD1,l−1,l − 2D2,l−1,l − sD0,l,l − 2D1,l,l

(1 − s2 )D0,l,l + sD1,l,l − 2D2,l,l k(k + 1) − 2D2,l−2,l − 2D1,l−1,l − cl (1 − s2 )D0,l,l + sD1,l,l − 2D2,l,l

cl+1

for l ≥ 2. On the other hand, if s = 1, for arbitrary c0 and c2 we have c1 = −k(k + 1)c0 ,

c3 = 2c2 − (k(k + 1) − 2)c1 , and for l = 2, 3, 4, . . . we obtain D1,l−1,l − 2D2,l−1,l − D0,l,l − 2D1,l,l cl+1 D1,l,l − 2D2,l,l k(k + 1) − 2D2,l−2,l − 2D1,l−1,l − cl . D1,l,l − 2D2,l,l

cl+2 = −

(7.21)

252 � S. G. Georgiev and İ. M. Erhan Clearly, a generalization of the recurrence relation is not possible for this example. On the other hand, one can get a good approximate solution by computing many terms of the sequence of coefficients.

7.4 Volterra integral equations of the first kind In this section we develop the series solution method for the nonlinear Volterra integral equation of the first kind x

u(x) = ∫ K(x, t, σ(x), σ(t))F(φ(t))Δt,

x ∈ 𝕋,

a

(7.22)

where K : 𝕋 × 𝕋 × 𝕋 × 𝕋 → ℝ, u : 𝕋 → ℝ, and F : ℝ → ℝ are given functions and φ : 𝕋 → ℝ is unknown. We suppose that the nonlinear function F is a quadratic function in φ, that is, 2

F(φ(x)) = α(φ(x)) + βφ(x) + γ,

x ∈ 𝕋.

Assume that the unknown function φ(x) has a Taylor series expansion about x = a of the form ∞

φ(x) = ∑ fn hn (x, a), n=0

x ∈ 𝕋,

(7.23)

where the coefficients fn will be determined from the equation. Then we have 2

∞ ∞

(φ(x)) = ∑ ∑ fn fm hn (x, a)hm (x, a), n=0 m=0

x ∈ 𝕋.

By Theorem 7.4, the double series in (7.24) can be written as 2

∞

(φ(x)) = ∑ gr hr (x, a), r=0

x ∈ 𝕋,

where ∞

k

gr = ∑ ∑ fl fk−l Dr,k,l , k=r l=k−r

and Dr,k,l =

∑

(r) Λr,k−l ∈Sk−l

Λ

hl r,k−l (a, a),

a ∈ 𝕋.

(7.24)

7 Series solution method on time scales and its applications

� 253

Note that in the above relation, the definition of the constants Dr,k,l is slightly different from the definition given in (7.5), however, this is only a notational difference which does not affect the properties and computation of the constants. Then we get 2

F(φ(x)) = α(φ(x)) + βφ(x) + γh0 (x, a) ∞

= ∑ (αgr + βfr + γδr0 )hr (x, a) r=0 ∞

= ∑ Qr hr (x, a),

x ∈ 𝕋,

r=0

(7.25)

where Qr = αgr + βfr + γδr0 for r ∈ ℕ0 , and δr0 is the Kronecker delta. We also assume that the kernel K(x, t, σ(x), σ(t)), x, t ∈ 𝕋, has a series representation of the form ∞

∞

m=0

i=0

K(x, t, σ(x), σ(t)) = ∑ am hm (x, a) ∑ bi hi (t, a),

x, t ∈ 𝕋,

(7.26)

where am , m ∈ ℕ0 , and bi , i ∈ ℕ0 , are given constants. Finally, we suppose that the Taylor series of the function u(x) is also known to be ∞

u(x) = ∑ un hn (x, a), n=0

x ∈ 𝕋.

(7.27)

If we substitute the series in (7.25), (7.26), and (7.27) into the nonlinear integral equation (7.22), we obtain ∞

x ∞

∞

∞

n=0

a m=0

i=0

r=0

∑ un hn (x, a) = ∫ ∑ am hm (x, a) ∑ bi hi (t, a) ∑ Qr hr (t, a)Δt,

x ∈ 𝕋.

(7.28)

Now, we take the x-dependent series out of the integral sign and use Theorem 7.4 to write the product of the two series inside the integral as a single one, which gives ∞

∞

x ∞

n=0

m=0

a p=0

∑ un hn (x, a) = ∑ am hm (x, a) ∫ ∑ yp hp (t, a)Δt,

x ∈ 𝕋,

and upon integrating, ∞

∞

∞

n=0

m=0

p=0

∑ un hn (x, a) = ∑ am hm (x, a) ∑ yp hp+1 (x, a),

where

x ∈ 𝕋,

(7.29)

254 � S. G. Georgiev and İ. M. Erhan ∞

k

yp = ∑ ∑ bl Qk−l Dp,k,l . k=p l=k−p

Finally, we employ again Theorem 7.4 to write the product on the right-hand side as a single series. This yields ∞

∞

n=0

n=1

∑ un hn (x, a) = ∑ wn hn (x, a),

x ∈ 𝕋,

(7.30)

where ∞

wn = ∑

k+1

∑

k=n−1 r=k−n+1

ar yk−r Dn,k+1,r−1

and we set

y−1 ≡ 0.

We equate the coefficients of hn (x, a), x ∈ 𝕋, of both sides. This will result in a nonlinear recurrence relation for the computation of the coefficients fn of the Taylor series of the unknown function φ. We present a specific example to illustrate the method. Example. Consider the equation x

2

u(x) = ∫ h2 (x, 0)(ϕ(t)) Δt,

x ∈ 𝕋,

(7.31)

0

on any time scale 𝕋 containing 0. We will search for a solution of equation (7.31) in the form ∞

ϕ(x) = ∑ fi hi (x, 0), i=0

x ∈ 𝕋,

where fi are constants to be determined from the equation. We assume that the function u(x) has the form ∞

u(x) = ∑ un hn (x, 0), n=1

x ∈ 𝕋.

Observe that 2

∞

(ϕ(x)) = ∑ gr hr (x, 0), r=0

x ∈ 𝕋,

where ∞

k

gr = ∑ ∑ fl fk−l Dr,k,l . k=r l=k−r

7 Series solution method on time scales and its applications

� 255

Then, equation (7.31) can be written as ∞

x ∞

n=1

0 r=0

∑ un hn (x, 0) = h2 (x, 0) ∫ ∑ gr hr (t, 0)Δt,

x ∈ 𝕋,

and upon integration, ∞

∞

n=1

r=0

∑ un hn (x, 0) = ∑ gr h2 (x, 0)hr+1 (x, 0),

x ∈ 𝕋.

(7.32)

We notice that according to Theorems 7.3 and 7.4, r+3

h2 (x, 0)hr+1 (x, 0) = ∑ Dn,r+3,2 hn (x, 0), n=r+1

x ∈ 𝕋.

Hence, we have r+3

∞

∞

r=0

r=0 n=r+1 ∞

∑ gr h2 (x, 0)hr+1 (x, 0) = ∑ ∑ gr Dn,r+3,2 hn (x, 0) ∞

= ∑ gr Dr+1,r+3,2 hr+1 (x, 0) + ∑ gr Dr+2,r+3,2 hr+2 (x, 0) r=0 ∞

r=0

+ ∑ gr Dr+3,r+3,2 hr+3 (x, 0) r=0 ∞

∞

= ∑ gr−1 Dr,r+2,2 hr (x, 0) + ∑ gr−2 Dr,r+1,2 hr (x, 0) r=1 ∞

r=2

+ ∑ gr−3 Dr,r,2 hr (x, 0) r=3

= g0 D1,3,2 h1 (x, 0) + (g1 D2,4,2 + g0 D2,3,2 )h2 (x, 0) ∞

+ ∑ (gr−1 Dr,r+2,2 + gr−2 Dr,r+1,2 + gr−3 Dr,r,2 )hr (x, 0), r=3

x ∈ 𝕋.

Thus, equation (7.32) yields ∞

∑ un hn (x, 0) = g0 D1,3,2 h1 (x, 0) + (g1 D2,4,2 + g0 D2,3,2 )h2 (x, 0)

n=1

∞

+ ∑ (gr−1 Dr,r+2,2 + gr−2 Dr,r+1,2 + gr−3 Dr,r,2 )hr (x, 0), r=3

and we obtain u1 = g0 D1,3,2 ,

x ∈ 𝕋,

(7.33)

256 � S. G. Georgiev and İ. M. Erhan u2 = g0 D2,3,2 + g1 D2,4,2 ,

ur = (gr−1 Dr,r+2,2 + gr−2 Dr,r+1,2 + gr−3 Dr,r,2 ),

r ≥ 3.

7.5 Volterra integral equations of the second kind In this section we consider the Volterra integral equation of the second kind x

φ(x) = u(x) + ∫ K(x, t, σ(x), σ(t))F(φ(t))Δt,

x ∈ 𝕋,

a

(7.34)

on an arbitrary time scale, where K : 𝕋 × 𝕋 × 𝕋 × 𝕋 → ℝ, u : 𝕋 → ℝ, and F : ℝ → ℝ are given functions and φ : 𝕋 → ℝ is unknown. We make the same assumption about the nonlinear function F inside the integral, that is, let F(φ(x)) = αφ2 (x) + βφ(x) + γ,

x ∈ 𝕋.

As in the case of Volterra integral equations of the first kind, we assume that the unknown function φ has a Taylor series expansion about x = a of the form ∞

φ(x) = ∑ fn hn (x, a),

x ∈ 𝕋,

n=0

(7.35)

where the coefficients fn will be determined from the equation. We assume that the kernel K(x, t, σ(x), σ(t)), x ∈ 𝕋, and the given function u have the Taylor series representations given in (7.26) and (7.27). We substitute these series and (7.25) into the nonlinear integral equation (7.34), which results in ∞

∞

x ∞

∞

∞

n=0

n=0

a m=0

i=0

r=0

∑ fn hn (x, a) = ∑ un hn (x, a) + ∫ ∑ am hm (x, a) ∑ bi hi (t, a) ∑ Qr hr (t, a)Δt,

x, a ∈ 𝕋. (7.36)

Now, we arrange the equation as ∞

∞

∞

x ∞

n=0

n=0

m=0

a p=0

∑ fn hn (x, a) = ∑ un hn (x, a) + ∑ am hm (x, a) ∫ ∑ yp hp (t, a)Δt,

x, a ∈ 𝕋,

and upon integrating, ∞

∞

∞

∞

n=0

n=0

m=0

p=0

∑ fn hn (x, a) = ∑ un hn (x, a) + ∑ am hm (x, a) ∑ yp hp+1 (x, a),

x, a ∈ 𝕋,

(7.37)

7 Series solution method on time scales and its applications

� 257

where ∞

k

yp = ∑ ∑ bl Qk−l Dp,k,l . k=p l=k−p

Finally, we combine the series on the right-hand side as a single one to get ∞

∞

n=0

n=0

∑ fn hn (x, a) = ∑ [un + wn ]hn (x, a),

x, a ∈ 𝕋,

(7.38)

where ∞

wn = ∑

k

∑

k=n−1 r=k−n+1

ar yk−r Dn,k,r−1 .

We equate the coefficients of hn (x, a), x, a ∈ 𝕋 of both sides. This will result in a nonlinear recurrence relation for the computation of the coefficients fn of the Taylor series of the unknown function φ. Remark 7.1. Notice that the series solution method simplifies considerably when applied to linear integral equations. In this case, the recurrence relation and the constants involved in this relation have a simpler form and their computation is easier.

Bibliography [1] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2003. [2] M. Bohner and R. P. Agarwal, Basic Calculus on Time Scales and some of its Applications, Results Math., 35 (1999), 3–22. [3] S. Georgiev, Integral Equations on Time Scales, Atlantis Press, 2016. [4] S. Georgiev, Fractional Dynamic Calculus and Fractional Dynamic Equations on Time Scales, Springer, 2017. [5] S. Georgiev and İ. M. Erhan, Series Solution Method for Cauchy Problems with Fractional Δ-derivative on Time Scales, Fract. Differ. Calc., 9 (2019), 243–261. [6] S. Georgiev and İ. M. Erhan, Nonlinear Integral Equations on Time Scales, Nova Science Publ., 2019. [7] D. Mozyrska and E. Pawluszewicz, Hermite’s Equations on Time Scales, Appl. Math. Lett., 22 (2009), 1217–1219. [8] H.-K. Liu, Developing a Series Solution Method of q-difference Equation, J. Appl. Math., 2013 (2013), 743973. [9] B. D. Haile and L. M. Hall, Polynomial and Series Solutions of Dynamic Equations on Time Scales, Dyn. Syst. Appl., 12 (2003), 149–157. https://web.mst.edu/~bohner/bimadsa/hall.pdf

Zeynep Kayar, Billur Kaymakçalan, and Neslihan Nesliye Pelen

8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities Abstract: The dual results; delta and nabla inequalities and their special cases; continuous and discrete inequalities are unified into diamond alpha case and new forms of such results as well as new diamond alpha Bennett–Leindler-type dynamic inequalities are established by developing a novel method, which does not require the Integration by Parts Formula and the Fundamental Theorem of Calculus. These theorems are standard arguments in the proofs of Bennett–Leindler-type dynamic inequalities in the delta and nabla approaches but do not follow naturally in the diamond alpha calculus.

8.1 Introduction Since Hardy’s inequality is one of those inequalities which turns information about derivatives of functions into information about the size of the function, it is essential part of all areas of mathematics and useful in various applications. In 1920, when Hardy tried to find a simple and elementary proof of Hilbert’s inequality [39], 1/2

1/2

∞ ∞ a c 2 ) ( ∑ cn2 ) , ∑ ∑ m n ≤ π( ∑ am m+n n=1 m=1 n=1 m=1 ∞ ∞

∞ 2 2 where am , cn ≥ 0 and ∑∞ m=1 am and ∑n=1 cn are convergent, he showed the following pioneering discrete inequality [24]: ζ

ζ

∞ 1 m ζ ) ∑ cζ (j), ∑( ∑ c(i)) ≤ ( j i=1 ζ − 1 j=1 j=1 ∞

c(j) ≥ 0, ζ > 1,

(8.1)

and a pioneering continuous inequality [24] for a nonnegative function ϕ and for a real constant ζ > 1 stated as follows: ∞

t

0

0

ζ

ζ ∞

1 ζ ) ∫ ϕζ (t)dt, ∫ ( ∫ ϕ(s)ds) dt ≤ ( t ζ −1

(8.2)

0

Zeynep Kayar, Department of Mathematics, Van Yüzüncü Yıl University, Van, Turkey, e-mail: [email protected] Billur Kaymakçalan, Department of Mathematics, Çankaya University, 06810 Ankara, Turkey, e-mail: [email protected] Neslihan Nesliye Pelen, Department of Mathematics, Ondokuz Mayıs University, Samsun, Turkey, e-mail: [email protected] https://doi.org/10.1515/9783111182971-008

260 � Z. Kayar et al. ∞

where ∫0 ϕζ (t)dt < ∞. In fact, Hardy only stated inequality (8.2) in [24] but did not prove it. In 1925, the proof of inequality (8.2), which depends on the calculus of variations, was shown by Hardy in [25]. The constant ( ζ ζ−1 )ζ that is seen in the above inequalities also has been found as the best possible constant, because if we replace it by a smaller constant, then inequalities (8.1) and (8.2) are no longer satisfied for the involved sequences and functions, respectively. Later in [27, Theorem 330], Hardy et al. generalized inequality (8.2) to ∞

󵄨󵄨 ζ 󵄨󵄨ζ ∞ ϕζ (t) Φζ (t) 󵄨󵄨 󵄨 ∫ θ dt ≤ 󵄨󵄨󵄨 󵄨 ∫ θ−ζ dt, 󵄨󵄨 θ − 1 󵄨󵄨󵄨 t t 0

ζ > 1,

(8.3)

0

t

∞

where ϕ is a nonnegative function and Φ(t) = ∫0 ϕ(s)ds, if θ > 1 and Φ(t) = ∫t ϕ(s)ds, if θ < 1. The discrete and continuous Hardy inequalities have been improved and generalized in many directions and used in many applications; see the books [9, 27, 39, 40, 44] and the references therein. The investigation of the reverse Hardy–Copson inequalities, which are called Bennett–Leindler inequalities, started almost at the same time with the original inequalities. The first reverse discrete Hardy–Copson inequalities were obtained by Hardy and Littlewood [26] in 1927 for 0 < ζ < 1 without finding the best possible constants. Then Copson [16], Bennett [10], and Leindler [41] established discrete Bennett–Leindler inequalities by means of the following: Assume that the sequences z and h are nonnegative. If 0 < ζ < 1, then ∞

∑

z(m)

m=1

ζ

∞

[G(m)]θ

∞

( ∑ h(j)z(j)) ≥ ζ ζ ∑

m=1

j=m

z(m)hζ (m) [G(m)]θ−ζ

,

0 ≤ θ < 1,

where G(m) = ∑m j=1 z(j) and ∞

∑

m=1

and for 0 < L ≤ ∞

∑

m=1

z(m)

[G(m)]θ

∞

ζ

∞ ζ z(m)hζ (m) ) ∑ , 1 − θ m=1 [G(m)]θ−ζ

ζ

∞ Lζ z(m)hζ (m) ) ∑ , θ − 1 m=1 [G(m)]θ−ζ

( ∑ h(j)z(j)) ≥ ( j=m

ζ

θ < 0,

(8.4)

θ > 1.

(8.5)

z(m) , z(m+1)

z(m)

[G(m)]θ

m

(∑ h(j)z(j)) ≥ ( j=1

ζ

There are some results in [42] about the reverse discrete Hardy–Copson inequalities different from those above and in [20] about finding conditions on the sequence z(m) for 0 < ζ < 1 to obtain the best possible constant. The first continuous Bennett–Leindler inequality, which is the reverse version of the continuous Hardy–Copson inequality (8.3), when θ = ζ , was established in [27, The-

8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities

� 261

∞

orem 337] for 0 < ζ < 1 and for H(t) = ∫t h(s)ds as ∞

∫ 0

ζ ∞

ζ

ζ H (t) dt ≥ ( ) ∫ hζ (t)dt, 1−ζ tζ

h(t) ≥ 0.

0

Later Copson derived continuous analogues of the discrete Bennett–Leindler inequalities (8.4) and (8.5), which are called continuous Bennett–Leindler inequalities, in t [17, Theorems 4 and 2], respectively, for z(t) ≥ 0 and h(t) ≥ 0, and G(t) = ∫0 z(s)ds, H(t) = t

∞

∫0 z(s)h(s)ds, H(t) = ∫t z(s)h(s)ds in the following manner: If 0 < ζ ≤ 1, θ < 1 then b

z(t)

ζ b

ζ ζ −θ [H(t)] dt ≥ ( ) ∫ z(t)[G(t)] hζ (t)dt, ∫ θ 1 − θ [G(t)] ζ

0

0 < b ≤ ∞.

0

If 0 < ζ ≤ 1 < θ, a > 0, then ∞

z(t)

ζ ∞

ζ ζ −θ [H(t)] dt ≥ ( ) ∫ z(t)[G(t)] hζ (t)dt. ∫ θ θ−1 [G(t)] a a ζ

Following the development of the time scale concept [8, 14, 15, 21, 22], the analysis of dynamic inequalities has become a popular research area and most classical inequalities have been extended to an arbitrary time scale. The surveys [1, 51] and the monograph [3] can be used to see these extended dynamic inequalities for the delta approach. Although the nabla dynamic inequalities are less attractive compared to the delta ones, some of the nabla dynamic inequalities can be found in [6, 12, 23, 47, 48]. The growing interest to Hardy–Copson type inequalities take place in the time scale calculus as well, and delta unifications of these inequalities are established in the book [4] and in the articles [2, 19, 49, 52, 54–58], whereas their nabla counterparts and extensions can be seen in [30–32]. In the delta time scale calculus, the reverse Hardy–Copson type inequalities, which are called delta Bennett–Leindler inequalities, can be found in [18, 53, 58, 59] for 0 < ζ < 1. These results are unifications of discrete and continuous Bennett–Leindler inequalities mentioned above. In addition to the delta calculus, the above discrete and continuous Bennett–Leindler inequalities can be unified in the nabla time scale calculus for 0 < ζ < 1, see [29, 37]. Some delta and nabla Bennett–Leindler-type inequalities can be given as below. Let us define the following functions and the constant in the delta calculus: t

∞

G(t) = ∫ z(s)Δs,

a

t

t

∞

G(t) = ∫ z(s)Δs, a

∃M1 > 0

H(t) = ∫ z(s)h(s)Δs, H(t) = ∫ z(s)h(s)Δs, t

G(t) such that σ ≤ M1 for t ∈ (a, ∞)𝕋 , G (t)

(8.6)

262 � Z. Kayar et al. where z, h ≥ 0 are rd-continuous, Δ-differentiable, and locally delta integrable functions, and 0 < a ∈ 𝕋.

Theorem 8.1 ([34, 37, 59]). For the functions G(t) and H(t) and the constant M1 defined in (8.6),

(i) ([34, 59]) if θ ≤ 0 < ζ < 1, then we have ∞

∞

a

a

ζ z(t)h(t)[H σ (t)]ζ −1 z(t)[H σ (t)]ζ Δt ≥ Δt, ∫ ∫ 1−θ [G(t)]θ [G(t)]θ−1 ∞

(8.7)

ζ ∞

z(t)[H σ (t)]ζ ζ z(t)hζ (t) Δt ≥ [ Δt. ] ∫ ∫ 1−θ [G(t)]θ [G(t)]θ−ζ a

(8.8)

a

(ii) ([34]) if 0 < ζ < 1 < θ, then we have ∞

∫ a

∞

ζ z(t)[H σ (t)]ζ z(t)h(t)[H σ (t)]ζ −1 Δt ≥ Δt, ∫ θ θ−1 [G(t)] [G(t)]θ−1 ∞

∫ a

a

(8.9)

ζ ∞

z(t)[H σ (t)]ζ ζ z(t)hζ (t) Δt ≥ [ ] ∫ Δt. θ θ−1 [G(t)] [G(t)]θ−ζ

(8.10)

a

(iii) ([37]) if ζ > 1, η ≥ 0, η + θ ≤ 0, then we have ∞

∞

a

a

η+ζ z(t)[H(t)]η+ζ z(t)h(t)[H(t)]η+ζ −1 Δt ≥ Δt, ∫ ∫ 1−η−θ [G(t)]η+θ [G(t)]η+θ−1 ∞

η+θ−1

∞

(8.11)

M1 (η + ζ ) 1/ζ z(t)h1/ζ (t)[H(t)]η+ζ − ζ z(t)[H(t)]η+ζ ] ∫ Δt. Δt ≥ [ ∫ 1 1−η−θ [G(t)]η+θ [G(t)]η+θ− ζ a a 1

(8.12)

(iv) ([37]) if ζ > 1, η ≥ 0, 0 ≤ η + θ < 1, then we have ∞

∫ ∞

∫ a

a

∞

η+ζ z(t)[H(t)]η+ζ z(t)h(t)[H(t)]η+ζ −1 Δt ≥ Δt, ∫ σ η+θ 1−η−θ [G (t)] [Gσ (t)]η+θ−1 a 1/ζ ∞

η+ζ z(t)[H(t)]η+ζ Δt ≥ [ ] 1−η−θ [Gσ (t)]η+θ

∫ a

(8.13)

1

z(t)h1/ζ (t)[H(t)]η+ζ − ζ 1

[Gσ (t)]η+θ− ζ

Δt.

(8.14)

Similar to the delta case, let us define the following functions and the constant in

the nabla calculus:

8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities

� 263

t

∞

G(t) = ∫ z(s)∇s,

H(t) = ∫ z(s)h(s)∇s, a

t

t

∞

G(t) = ∫ z(s)∇s, a

∃L1 > 0

(8.15)

H(t) = ∫ z(s)h(s)∇s,

such that

t

G(t) ρ

G (t)

≤ L1 for t ∈ (a, ∞)𝕋 ,

where z, h ≥ 0 are ld-continuous, ∇-differentiable, and locally nabla integrable functions, and 0 < a ∈ 𝕋. Theorem 8.2 ([29, 34, 37]). For the functions G(t) and H(t) and the constant L1 defined in (8.15), (i) ([29]) if θ ≤ 0 < ζ < 1, then we have ∞

∫

ρ

z(t)[H (t)]ζ [G(t)]θ

a

∞

∫

∞

ρ

z(t)[H (t)]ζ [G(t)]θ

a

ρ

z(t)h(t)[H (t)]ζ −1 ζ ∇t ≥ ∇t, ∫ 1−θ [G(t)]θ−1 a

(8.16)

ζ ∞

ζ z(t)hζ (t) ∇t ≥ [ ] ∫ ∇t. 1−θ [G(t)]θ−ζ a

(8.17)

(ii) ([34]) if 0 < ζ < 1 < θ, then we have ∞

∫ a

∞

ρ

z(t)[H (t)]ζ [G(t)]θ

∞

∫

∇t ≥

ρ

z(t)[H (t)]ζ

a

[G(t)]θ

ρ

ζ z(t)h(t)[H (t)]ζ −1 ∇t, ∫ θ−1 [G(t)]θ−1 a

(8.18)

ζ z(t)hζ (t) ∇t. ] ∫ θ−1 [G(t)]θ−ζ a

(8.19)

ζ ∞

∇t ≥ [

(iii) ([37]) if ζ > 1, η ≥ 0, η + θ ≤ 0, then we have ∞

∫ ∞

∫

z(t)[H(t)]η+ζ [G(t)]η+θ

a

z(t)[H(t)]η+ζ [G(t)]η+θ

a

∞

η+ζ z(t)h(t)[H(t)]η+ζ −1 ∇t ≥ ∇t, ∫ ρ 1−η−θ [G (t)]η+θ−1 a

(8.20)

(η + ζ ) 1/ζ z(t)h1/ζ (t)[H(t)]η+ζ − ζ ] ∫ ∇t. 1 1−η−θ [G(t)]η+θ− ζ a

(8.21)

∞

η+θ−1

∇t ≥ [

L1

1

(iv) ([37]) if ζ > 1, η ≥ 0, 0 ≤ η + θ < 1, then we have ∞

∫ ∞

∫ a

a

z(t)[H(t)]η+ζ ρ

[G (t)]η+θ

z(t)[H(t)]η+ζ ρ

[G (t)]η+θ

∞

η+ζ z(t)h(t)[H(t)]η+ζ −1 ∇t ≥ ∇t, ∫ ρ 1−η−θ [G (t)]η+θ−1 a 1/ζ ∞

η+ζ ∇t ≥ [ ] 1−η−θ

∫ a

(8.22)

1

z(t)h1/ζ (t)[H(t)]η+ζ − ζ ρ

1

[G (t)]η+θ− ζ

∇t.

(8.23)

264 � Z. Kayar et al. The diamond alpha time scale unifications of the classical Bennett–Leindler-type inequalities (0 < ζ < 1) on an arbitrary time scale are given in [34] while their complements are obtained in [38] for ζ > 1, some of which are given in the next two theorems. Let us define the following functions in the diamond alpha calculus: t

∞

G(t) = ∫ z(s)♦α s, t

H(t) = ∫ z(s)h(s)♦α s, a

t

(8.24)

∞

G(t) = ∫ z(s)♦α s, a

H(t) = ∫ z(s)h(s)♦α s, t

where z, h ≥ 0 are diamond alpha differentiable and locally diamond alpha integrable functions, and 0 < a ∈ 𝕋. Theorem 8.3 ([34]). For the functions G(t), G(t) and H(t), H(t) defined in (8.24), (i) if θ ≤ 0 < ζ < 1, then we have ∞

∫ a

ζ ∞

αζ z(t)[H σ (t)]ζ z(t)hζ (t) ♦α t ≥ [ ♦α t, ] ∫ θ 1−θ [G(t)] [G(t)]θ−ζ

(8.25)

a

where z is a nonincreasing function on [a, ∞)𝕋 and ∞

∫

ρ

z(t)[H (t)]ζ [G(t)]θ

a

ζ ∞

(1 − α)ζ z(t)hζ (t) ♦α t ≥ [ ] ∫ ♦α t, 1−θ [G(t)]θ−ζ a

(8.26)

where z is a nondecreasing function on [a, ∞)𝕋 . (ii) if 0 < ζ < 1 < θ, then we have ∞

ζ ∞

a

a

αζ z(t)[H σ (t)]ζ z(t)hζ (t) ♦ t ≥ [ ] ♦α t, ∫ ∫ α θ−1 [G(t)]θ [G(t)]θ−ζ

(8.27)

where z is a nondecreasing function on [a, ∞)𝕋 and ∞

∫ a

ρ

z(t)[H (t)]ζ [G(t)]θ

ζ ∞

z(t)hζ (t) (1 − α)ζ ] ∫ ♦α t ≥ [ ♦α t, θ−1 [G(t)]θ−ζ a

(8.28)

where z is a nonincreasing function on [a, ∞)𝕋 . Theorem 8.4 ([38]). For the functions G(t), G(t) and H(t), H(t) defined in (8.24), assume G(t) that there exist M1 , L1 > 0 such that GG(t) ≤ L1 for t ∈ (a, ∞)𝕋 . Let ρ σ (t) ≤ M1 and G (t)

ζ > 1, η ≥ 0 be real constants. (i) If η + θ ≤ 0, then we have ∞

η+θ−1 1/ζ ∞

α(η + ζ )M1 z(t)[H(t)]η+ζ ♦α t ≥ [ ∫ η+θ 1−η−θ [G(t)] a

]

∫ a

1

z(t)h1/ζ (t)[H(t)]η+ζ − ζ 1

[G(t)]η+θ− ζ

♦α t,

(8.29)

� 265

8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities

where z is a nonincreasing function on [a, ∞)𝕋 and ∞

∞

η+θ−1

(1 − α)(η + ζ ) 1/ζ z(t)h1/ζ (t)[H(t)]η+ζ − ζ ] ∫ ♦α t, 1 1−η−θ [G(t)]η+θ [G(t)]η+θ− ζ a a (8.30) where z is a nondecreasing function on [a, ∞)𝕋 . (ii) If 0 ≤ η + θ < 1, then we have ∫

z(t)[H(t)]η+ζ

♦α t ≥ [

L1

∞

1/ζ ∞

α(η + ζ ) z(t)[H(t)]η+ζ ♦α t ≥ [ ] ∫ σ η+θ 1 −η−θ [G (t)]

∫

a

1

1

z(t)h1/ζ (t)[H(t)]η+ζ − ζ 1

[Gσ (t)]η+θ− ζ

a

♦α t,

(8.31)

where z is a nonincreasing function on [a, ∞)𝕋 and ∞

∫ a

z(t)[H(t)]η+ζ ρ

[G (t)]η+θ

♦α t ≥ [

1/ζ ∞

(1 − α)(η + ζ ) ] 1−η−θ

∫

1

z(t)h1/ζ (t)[H(t)]η+ζ − ζ ρ

1

[G (t)]η+θ− ζ

a

♦α t,

(8.32)

where z is a nondecreasing function on [a, ∞)𝕋 . (iii) If η + θ > 1, then we have ∞

∫ a

η+θ−1 1/ζ ∞

α(η + ζ )M1 z(t)[H(t)]η+ζ ♦α t ≥ [ η+θ η+θ−1 [G(t)]

]

∫

1

z(t)h1/ζ (t)[H(t)]η+ζ − ζ 1

[G(t)]η+θ− ζ

a

♦α t,

(8.33)

where z is a nondecreasing function on [a, ∞)𝕋 and ∞

η+θ−1

∞

(1 − α)(η + ζ ) 1/ζ z(t)h1/ζ (t)[H(t)]+ζ + ζ ♦α t ≥ [ ] ∫ ♦α t, ∫ 1 η+θ−1 [G(t)]η+θ [G(t)]η+θ− ζ a a (8.34) where z is a nonincreasing function on [a, ∞)𝕋 . z(t)[H(t)]η+ζ

L1

1

Since there is a gap in the literature for diamond alpha Bennett–Leindler-type inequalities for 0 < ζ < 1, η ≥ 0, our aim is to fill this gap by obtaining new diamond alpha Bennett–Leindler-type inequalities to generalize the results in [34] and to complement the current literature given in [38]. Another significant contribution of this chapter is to employ a new method, which depends on using the diamond alpha calculus rather than algebra in contrast to the literature, for establishing new diamond alpha Bennett– Leindler-type inequalities. By this new method, we do not need Integration by Parts Formula and the Fundamental Theorem of Calculus, which are standard arguments in the proofs of Bennett–Leindler-type dynamic inequalities in the delta and nabla approaches but do not follow through in the diamond alpha calculus. Moreover, since both sides of the diamond alpha Bennett–Leindler-type inequalities include only single diamond alpha integrals, these inequalities have more compact forms. In addition, we notice that special cases of η + θ ≤ 0 and η + θ > 1 corresponding to η = 0 were considered in [34], while the case 0 ≤ η + θ < 1 was not investigated therein. By taking account a constant η ≥ 0, we not only generalize the diamond alpha Bennett–Leindler-type in-

266 � Z. Kayar et al. equalities presented in [34] for η ≥ 0, but also obtain complement inequalities of those in [38] established for ζ > 1. As a result, our novel technique allows us to generalize the results in [34], to unify the foregoing delta and nabla Bennett–Leindler-type inequalities and derive new ones, and to obtain complementary inequalities for the existing diamond alpha Bennett– Leindler-type inequalities in [38].

8.2 Preliminaries This section is devoted to the main definitions and theorems of delta, nabla, and diamond alpha calculi. We refer the reader to [8, 14, 15, 21, 22] for the concept of time scale calculus in the delta and nabla sense. A nonempty closed subset of ℝ is called a time scale which is denoted by 𝕋. Since the delta and nabla time scale calculi are very well known [8, 14, 15], we skip the details of them and consider only main properties which will be used in the sequel. Theorem 8.5 ([14]). Suppose that Λ, Γ : 𝕋 → ℝ and s ∈ 𝕋κ . For μ(s) = σ(s) − s, we have the following statements: 1. If Λ is delta differentiable at s, then Λ(σ(s)) = Λσ (s) = Λ(s) + μ(s)ΛΔ (s). 2. The product ΛΓ : 𝕋 → ℝ is differentiable at s with (ΛΓ)Δ (s) = ΛΔ (s)Γ(s) + Λ(σ(s))Γ Δ (s) = Λ(s)Γ Δ (s) + ΛΔ (s)Γ(σ(s)).

(8.35)

Lemma 8.1 (Chain rule for the delta derivative, [14]). If Γ ∈ C 1 (ℝ, ℝ) and Λ ∈ C(𝕋, ℝ) is delta differentiable on 𝕋κ , then Γ ∘ Λ is delta differentiable and (i) one can find c ∈ [s, σ(s)] with (Γ ∘ Λ)Δ (s) = Γ ′ (Λ(c))ΛΔ (s);

(8.36)

(ii) the following relation holds: 1

(Γ ∘ Λ)Δ (s) = ΛΔ (s) ∫ Γ ′ (Λ(s) + wμ(s)ΛΔ (s))dw.

(8.37)

0

Theorem 8.6 ([14]). Suppose that Γ : 𝕋 → ℝ and s ∈ 𝕋κ . For ν(s) = s − ρ(s), we have the following statements: 1. If Γ is nabla differentiable at s, then Γ(ρ(s)) = Γ ρ (s) = Γ(s) − ν(s)Γ ∇ (s). 2. The product ΛΓ : 𝕋 → ℝ is differentiable at s with (ΛΓ)∇ (s) = Λ∇ (s)Γ(s) + Λ(ρ(s))Γ ∇ (s) = Λ(s)Γ ∇ (s) + Λ∇ (s)Γ(ρ(s)).

(8.38)

Lemma 8.2 (Chain rule for the nabla derivative, [23]). If Γ ∈ C 1 (ℝ, ℝ) and Λ ∈ C(𝕋, ℝ) is nabla differentiable on 𝕋κ , then Γ ∘ Λ is nabla differentiable and the following holds:

8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities

∇

∇

� 267

1

(Γ ∘ Λ) (s) = Λ (s) ∫ Γ ′ (Λ(ρ(s)) + wν(s)Λ∇ (s))dw.

(8.39)

0

The next lemmas play crucial roles in the proofs of the main theorems. Lemma 8.3 ([8, 14]). If Γ is continuous, then t

Δ

Δ

t

(i) (∫ Γ(s)Δs) = Γ(t) and (∫ Γ(s)∇s) = Γ(σ(t)), t1 t

t1

∇

t

∇

(ii) (∫ Γ(s)∇s) = Γ(t) and (∫ Γ(s)Δs) = Γ(ρ(t)). t1

t1

Lemma 8.4 ([22]). If Γ is continuous for all t1 , t2 ∈ 𝕋 with t1 < t2 , then t2

t2

t2

∫ Γ(t)Δt = ∫ Γ(ρ(t))∇t t1

and

t1

t2

∫ Γ(t)∇t = ∫ Γ(σ(t))Δt. t1

t1

The diamond alpha time scale calculus has been introduced by Sheng et al. in [60]. This calculus deals with diamond alpha, denoted by ♦α , differentiable and diamond alpha integrable functions which are convex linear combinations of delta and nabla differentiable and integrable functions, respectively. For some developments of this calculus and for some dynamic inequalities in this calculus, we refer to [5, 7, 11–13, 28, 33– 36, 38, 43, 45–47, 50] and the references therein. Definition 8.1 ([60]). The diamond alpha derivative and integral have been introduced as follows. (i) Let ρ(s) − τ = asτ and σ(s) − τ = bsτ . Then the ♦α -derivative of Λ : 𝕋 → ℝ at the point s ∈ 𝕋κκ denoted by Λ♦α (s) is the number enjoying the property that for all ϵ > 0, there exists a neighborhood V ⊂ 𝕋 of s ∈ 𝕋κκ such that for any τ ∈ V , 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ♦ 󵄨 󵄨󵄨α|Λ(σ(s)) − Λ(τ)󵄨󵄨󵄨|asτ | + (1 − α)󵄨󵄨󵄨Λ(ρ(s)) − Λ(τ)󵄨󵄨󵄨|bsτ 󵄨󵄨󵄨−Λ α (s)|asτ ||bsτ |󵄨󵄨󵄨 ≤ ϵ|asτ ||bsτ |. (ii) Λ : 𝕋 → ℝ is ♦α -differentiable at s ∈ 𝕋κκ provided it is both delta and nabla differentiable at s. Moreover, for 0 ≤ α ≤ 1, such a function satisfies Λ♦α (s) = αΛΔ (s) + (1 − α)Λ∇ (s). (iii) Λ : 𝕋 → ℝ is diamond alpha integrable provided it is continuous. Moreover, for 0 ≤ α ≤ 1, we have s2

s2

s2

∫ Λ(s)♦α s = α ∫ Λ(s)Δs + (1 − α) ∫ Λ(s)∇s.

s1

s1

s1

268 � Z. Kayar et al. Lemma 8.5 ([22, 46]). For all s ∈ 𝕋, a time scale 𝕋 is said to be regular provided σ(ρ(s)) = ρ(σ(s)) = s. A regular time scale 𝕋 satisfies 𝕋κκ = 𝕋κ = 𝕋κ = 𝕋. Moreover, σ(𝕋) = ρ(𝕋) = 𝕋 in such a time scale. Lemma 8.6 (Diamond alpha Hölder’s inequality, [5, 47]). Let s1 , s2 ∈ 𝕋 and λ1 , λ2 > 1 be s given satisfying λ1 + λ1 = 1. For ϕ, ψ ∈ C([a, b]𝕋 , [0, ∞)) with ∫s 2 ψλ2 (s)♦α s > 0, the 1 2 1 diamond alpha Hölder’s inequality s2

s2

1/λ1

λ1

∫ ϕ(s)ψ(s)♦α s ≤ (∫ ϕ (s)♦α s)

s1

holds. If λ1 < 0 or 0 < λ1 < 1 with inequality s2

+ s2

1 λ2

1/λ2

s1

= 1, then the diamond alpha reverse Hölder’s 1/λ1

λ1

∫ ϕ(s)ψ(s)♦α s ≥ (∫ ϕ (s)♦α s)

s1

λ2

(∫ ψ (s)♦α s)

s1

1 λ1

s2

s1

s2

λ2

1/λ2

(∫ ψ (s)♦α s)

(8.40)

s1

is satisfied.

8.3 Diamond alpha Bennett–Leindler type dynamic inequalities This section is devoted to deriving new diamond alpha Bennett–Leindler-type integral inequalities, which are established using the properties of the delta, nabla and diamond alpha derivatives and integrals. Let 𝕋 be a regular time scale and a ∈ [0, ∞)𝕋 . The following theorem asserts not only novel diamond alpha and delta Bennett– Leindler-type inequalities when 0 < ζ < 1, η ≥ 0, and η+θ ≤ 0, but also complements the diamond alpha Bennett–Leindler type inequalities given in [38, Theorem 9] established for ζ > 1, η ≥ 0, and η + θ ≤ 0 and generalizations of the diamond alpha Bennett– Leindler-type inequalities given in [34, Theorem 15] proven for 0 < ζ < 1, η = 0, and θ ≤ 0. This novelty is caused by the condition η + ζ ≥ 1, which has not been considered until this current contribution to the literature. Theorem 8.7. Suppose that z is a nonincreasing function on [a, ∞)𝕋 . For the functions G(t) and H(t) defined in (8.24), assume that there exists M1 > 0 such that GG(t) σ (t) ≤ M1 for t ∈ (a, ∞)𝕋 . Let 0 < ζ < 1, η ≥ 0, and η + θ ≤ 0 be real constants. (i) If 0 < η + ζ ≤ 1, then ∞

∫ a

∞

α(η + ζ ) z(t)h(t)[H σ (t)]η+ζ −1 z(t)[H σ (t)]η+ζ ♦ t ≥ ♦α t ∫ α 1−η−θ [G(t)]η+θ [G(t)]η+θ−1 a

(8.41)

8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities

� 269

and ∞

ζ ∞

a

a

α(η + ζ ) z(t)[H σ (t)]η+ζ z(t)hζ (t)[H σ (t)]η ♦ t ≥ [ ♦α t. ] ∫ ∫ α 1−η−θ [G(t)]η+θ [G(t)]η+θ−ζ

(8.42)

(ii) If η + ζ ≥ 1, then ∞

∞

a

a

α(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 z(t)[H(t)]η+ζ ♦ t ≥ ♦α t ∫ ∫ α 1−η−θ [G(t)]η+θ [Gσ (t)]η+θ−1

(8.43)

and ∞

η+θ−1 ζ ∞

α(η + ζ )M1 z(t)[H(t)]η+ζ ♦α t ≥ [ ∫ η+θ 1−η−θ [G(t)]

] ∫

a

a

z(t)hζ (t)[H(t)]η ♦α t. [G(t)]η+θ−ζ

(8.44)

Proof. First we obtain some inequalities which will be used in the sequel. By utilizing Lemma 8.3, one can obtain Δ

t

Δ

H (t) = [∫ z(s)h(s)♦α s] a

Δ

t

Δ

t

= α[∫ z(s)h(s)Δs] + (1 − α)[∫ z(s)h(s)∇s] a

(8.45)

a

= αz(t)h(t) + (1 − α)zσ (t)hσ (t) ≥ 0. By taking into account equation (8.45) and by employing formula (8.36), one can observe that for t ≤ c ≤ σ(t), Δ

[H η+ζ (t)] = (η + ζ )H Δ (t)H η+ζ −1 (c) = (η + ζ )[αz(t)h(t) + (1 − α)zσ (t)hσ (t)]H η+ζ −1 (c) holds. Then two estimates can be obtained depending on whether 0 < η + ζ ≤ 1 or η + ζ ≥ 1 for the function [H η+ζ (t)]Δ as follows: (i) If 0 < η + ζ ≤ 1, then Δ

η+ζ −1

[H η+ζ (t)] ≥ (η + ζ )αz(t)h(t)[H σ (t)]

;

(8.46)

(ii) If η + ζ ≥ 1, then Δ

η+ζ −1

[H η+ζ (t)] ≥ (η + ζ )αz(t)h(t)[H(t)] where H(t) ≤ H(c) ≤ H σ (t) has been used for t ≤ c ≤ σ(t).

,

(8.47)

270 � Z. Kayar et al. By Lemma 8.3, note that Δ

∞

Δ

∞

Δ

∞

GΔ (t) = [ ∫ z(s)♦α s] = α[ ∫ z(s)Δs] + (1 − α)[ ∫ z(s)∇s] t

t

(8.48)

t

= −αz(t) − (1 − α)zσ (t) ≤ 0.

It follows from (8.48) and formula (8.36) that for η + θ ≤ 0 we have 1

Δ

[G1−η−θ (t)] = (1 − η − θ)GΔ (t) ∫ 0

1

=∫ 0

dw [(1 − w)G(t) + wGσ (t)]η+θ

(1 − η − θ)[−αz(t) − (1 − α)zσ (t)] dw [(1 − w)G(t) + wGσ (t)]η+θ

≥ −(1 − η − θ)

(8.49)

z(t) , [G(t)]η+θ

where Gσ (t) ≤ G(t) and the nonincreasingness property of z have been used. (i) Let us define u(t) = [H(t)]η+ζ [G(t)]1−η−θ for t ∈ [a, ∞). If we take the delta derivative of the function u using formula (8.35), we get Δ

η+ζ

uΔ (t) = [H η+ζ (t)] [G1−η−θ (t)] + [H σ (t)]

Δ

[G1−η−θ (t)] .

(8.50)

By making use of inequalities (8.46) and (8.49) in equation (8.50), we obtain uΔ (t) ≥ −(1 − η − θ)

α(η + ζ )z(t)h(t)[H σ (t)]η+ζ −1 z(t) η+ζ σ [H (t)] + , [G(t)]η+θ [G(t)]η+θ−1

or ∞

∞

∞

α(η + ζ ) z(t)h(t)[H σ (t)]η+ζ −1 1 z(t)[H σ (t)]η+ζ ♦α t ≥ ♦α t − ∫ ∫ uΔ (t)♦α t. η+θ 1−η−θ 1−η−θ [G(t)] [G(t)]η+θ−1 a a a (8.51) The definition of u implies u(∞) = u(a) = 0 and, by employing Lemma 8.4, we obtain ∫

∞

Δ

∞

∞

∫ u (t)♦α t = α ∫ u (t)Δt + (1 − α) ∫ uΔ (t)∇t a

a

Δ

a

(8.52)

= α[u(∞) − u(a)] + (1 − α)[u(σ(∞)) − u(σ(a))] ≤ 0, where we have imposed that (−1)η+θ = 1. Therefore we can infer that inequality (8.51) becomes ∞

∫ a

∞

α(η + ζ ) z(t)h(t)[H σ (t)]η+ζ −1 z(t)[H σ (t)]η+ζ ♦α t ≥ ♦α t, ∫ η+θ 1−η−θ [G(t)] [G(t)]η+θ−1 a

8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities

� 271

which is the desired inequality (8.41). Since ∞

∫ a

∞

η

1

z(t)h(t)[H σ (t)]η+ζ −1 z ζ (t)h(t)[H σ (t)] ζ z(t)[H σ (t)]η+ζ ♦α t = ∫ [ ] η+θ−ζ η+θ−1 [G(t)] [Gσ (t)]η+θ [G(t)] ζ

ζ −1 ζ

a

♦α t,

applying the reverse Hölder’s inequality (8.40) with the constants 0 < ζ < 1 and ζ /(ζ − 1) < 0 to the right-hand side of the above equation leads to inequality (8.42). (ii) Let us define u(t) = [H(t)]η+ζ [G(t)]1−η−θ for t ∈ [a, ∞). If we take the delta derivative of the function u using formula (8.35), we get Δ

Δ

uΔ (t) = [H η+ζ (t)][G1−η−θ (t)] + [H η+ζ (t)] [Gσ (t)]

1−η−θ

(8.53)

.

Using inequalities (8.47) and (8.49) in equation (8.53) yields uΔ (t) ≥ −(1 − η − θ)

α(η + ζ )z(t)h(t)[H(t)]η+ζ −1 z(t) η+ζ [H(t)] + , [G(t)]η+θ [Gσ (t)]η+θ−1

or ∞

∫ a

∞

∞

a

a

α(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 z(t)[H(t)]η+ζ 1 ♦α t ≥ ♦α t − ∫ ∫ uΔ (t)♦α t. η+θ 1−η−θ 1−η−θ [G(t)] [Gσ (t)]η+θ−1

If we employ inequality (8.52), we obtain from the above inequality that ∞

∞

a

a

α(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 z(t)[H(t)]η+ζ ♦ t ≥ ♦α t, ∫ ∫ α 1−η−θ [G(t)]η+θ [Gσ (t)]η+θ−1 which is the desired inequality (8.43). After using GG(t) σ (t) ≤ M1 on the right-hand side of inequality (8.43) and applying the reverse Hölder’s inequality (8.40) with the constants 0 < ζ < 1 and ζ /(ζ − 1) < 0 to the resulting integral, one can obtain inequality (8.44). Remark 8.1. The diamond alpha Bennett–Leindler-type inequalities (8.41)–(8.42) obtained for 0 < ζ < 1, η ≥ 0, and η + θ ≤ 0 are generalizations of the diamond alpha Bennett–Leindler-type inequality (8.25) established for 0 < ζ < 1, η = 0, and θ ≤ 0 given in [34, Theorem 15]. The diamond alpha Bennett–Leindler-type inequalities (8.43)–(8.44) obtained for 0 < ζ < 1, η ≥ 0, and η + θ ≤ 0 are complements of the diamond alpha Bennett–Leindler-type inequality (8.29) established for ζ > 1, η ≥ 0, and η + θ ≤ 0 given in [38, Theorem 9]. Remark 8.2. Although the special case of the condition η + ζ ≤ 1 is automatically satisfied in [34, Theorem 15], the other case, η + ζ ≥ 1 with 0 < ζ < 1, has not appeared in the literature previously even for the special cases. This is one of the gaps in the literature that this theorem aims to fill. By the novel diamond alpha Bennett–Leindler inequali-

272 � Z. Kayar et al. ties (8.43)–(8.44) obtained for 0 < ζ < 1, η ≥ 0, η + θ ≤ 0, and η + ζ ≥ 1 for the first time, this aim is achieved. Remark 8.3. Special cases of the diamond alpha Bennett–Leindler-type inequalities (8.41)–(8.44) can be seen below. (i) Expressing inequalities (8.41)–(8.42) in terms of delta and nabla integrals for 0 < ζ < 1, η ≥ 0, η + θ ≤ 0, and η + ζ ≤ 1 and then choosing α = 1 in those inequalities yields ∞

∞

a

a

η+ζ z(t)h(t)[H σ (t)]η+ζ −1 z(t)[H σ (t)]η+ζ Δt ≥ Δt ∫ ∫ 1−η−θ [G(t)]η+θ [G(t)]η+θ−1

(8.54)

and ∞

ζ ∞

a

a

η+ζ z(t)[H σ (t)]η+ζ z(t)hζ (t)[H σ (t)]η Δt ≥ [ ] Δt, ∫ ∫ 1−η−θ [G(t)]η+θ [G(t)]η+θ−ζ

(8.55)

respectively, where H(t) and G(t) are defined in (8.6). Inequalities (8.54)–(8.55) generalize delta Bennett–Leindler-type inequalities (8.7)–(8.8) established for 0 < ζ < 1, η = 0, and θ ≤ 0 given in [59, Theorem 2.1] and [34, Remark 5] and complement delta Bennett–Leindler-type inequalities (8.11)–(8.12) in [37, Corollary 1] and [38, Remark 3] established for ζ > 1, η ≥ 0, and η + θ ≤ 0. We can conclude that (i) of Theorem 8.7 is a diamond alpha unification of Theorem 8.1 which is given in [59, Theorem 2.1] and [34, Remark 5] and is a generalization of the diamond alpha Bennett– Leindler-type inequality (8.25) established for 0 < ζ < 1, η = 0, and θ ≤ 0 given in [34, Theorem 15] and is a completion of the diamond alpha Bennett–Leindler-type inequality (8.29) established for ζ > 1, η ≥ 0, and η + θ ≤ 0 given in [38, Theorem 9]. (ii) Expressing inequalities (8.43)–(8.44) in terms of delta and nabla integrals for 0 < ζ < 1, η ≥ 0, η + θ ≤ 0, and η + ζ ≥ 1, and then choosing α = 1 in those inequalities yields ∞

∞

a

a

η+ζ z(t)[H(t)]η+ζ z(t)h(t)[H(t)]η+ζ −1 Δt ≥ Δt ∫ ∫ 1−η−θ [G(t)]η+θ [Gσ (t)]η+θ−1 and

∞

∫ a

η+θ−1 ζ ∞

(η + ζ )M1 z(t)[H(t)]η+ζ Δt ≥ [ η+θ 1−η−θ [G(t)]

] ∫ a

z(t)hζ (t)[H(t)]η Δt, [G(t)]η+θ−ζ

(8.56)

(8.57)

respectively, where H(t), G(t) and M1 are defined in (8.6). Inequalities (8.56)–(8.57) are novel even in the delta calculus. These novel inequalities generalize delta Bennett–Leindler-type inequalities (8.7)–(8.8) given in [59, Theorem 2.1] and [34, Remark 5] established for 0 < ζ < 1, η = 0, and θ ≤ 0 and complement delta Bennett– Leindler-type inequalities (8.11)–(8.12) given in [37, Corollary 1] and [38, Remark 3] established for ζ > 1, η ≥ 0 and η + θ ≤ 0. We can conclude that (ii) of Theorem 8.7 is a diamond alpha unification of Theorem 8.1 which is given in [59, Theorem 2.1] and

8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities

� 273

[34, Remark 5] and is a generalization of the diamond alpha Bennett–Leindler-type inequality (8.25) established for 0 < ζ < 1, η = 0, and θ ≤ 0 given in [34, Theorem 15] and is a completion of the diamond alpha Bennett–Leindler-type inequality (8.29) established for ζ > 1, η ≥ 0, and η + θ ≤ 0 given in [38, Theorem 9]. The following theorem asserts not only novel diamond alpha and delta Bennett– Leindler-type inequalities when 0 < ζ < 1, η ≥ 0, and 0 ≤ η + θ < 1, but also complements the diamond alpha Bennett–Leindler-type inequalities given in [38, Theorem 9] established for ζ > 1, η ≥ 0, and 0 ≤ η + θ < 1, and generalizations of the diamond alpha Bennett–Leindler-type inequalities given in [34, Theorem 15] established for 0 < ζ < 1, η = 0, and θ ≤ 0. This novelty is caused by the condition η + ζ ≥ 1, which has not been considered so far. Theorem 8.8. Suppose that z is a nonincreasing function on [a, ∞)𝕋 . For the functions G(t) and H(t) defined in (8.24), let 0 < ζ < 1, η ≥ 0, and 0 ≤ η + θ < 1 be real constants. (i) If 0 < η + ζ ≤ 1, then ∞

∫ a

and

∞

α(η + ζ ) z(t)h(t)[H σ (t)]η+ζ −1 z(t)[H σ (t)]η+ζ ♦α t ≥ ♦α t ∫ σ η+θ 1−η−θ [G (t)] [G(t)]η+θ−1

(8.58)

a

∞

ζ ∞

a

a

α(η + ζ ) z(t)[H σ (t)]η+ζ z(t)hζ (t)[H σ (t)]η ♦ t ≥ [ ♦α t. ] ∫ ∫ α 1−η−θ [Gσ (t)]η+θ [Gσ (t)]η+θ−ζ

(8.59)

(ii) If η + ζ ≥ 1, then ∞

∫ a

∞

α(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 z(t)[H(t)]η+ζ ♦α t ≥ ♦α t ∫ σ η+θ 1−η−θ [G (t)] [Gσ (t)]η+θ−1

(8.60)

a

and ∞

∫ a

ζ ∞

α(η + ζ ) z(t)[H(t)]η+ζ z(t)hζ (t)[H(t)]η ♦ t ≥ [ ] ♦α t. ∫ α 1−η−θ [Gσ (t)]η+θ [Gσ (t)]η+θ−ζ

(8.61)

a

Proof. It follows from (8.48) and formula (8.37) for 0 ≤ η + θ < 1 that we have [G

1−η−θ

Δ

Δ

1

(t)] = (1 − η − θ)G (t) ∫ 1

=∫ 0

0

dw [(1 − w)G(t) + wGσ (t)]η+θ

(1 − η − θ)[−αz(t) − (1 − α)zσ (t)] dw [(1 − w)G(t) + wGσ (t)]η+θ

≥ −(1 − η − θ)

z(t)

[Gσ (t)]η+θ

,

where Gσ (t) ≤ G(t) and the nonincreasingness property of z have been used.

(8.62)

274 � Z. Kayar et al. (i) Using inequalities (8.46) and (8.62) in equation (8.50) yields uΔ (t) ≥ − (1 − η − θ)

α(η + ζ )z(t)h(t)[H σ (t)]η+ζ −1 z(t) η+ζ σ [H (t)] + , [Gσ (t)]η+θ [G(t)]η+θ−1

or ∞

∞

α(η + ζ ) z(t)h(t)[H σ (t)]η+ζ −1 z(t)[H σ (t)]η+ζ ♦ t ≥ ♦α t ∫ α 1−η−θ [Gσ (t)]η+θ [G(t)]η+θ−1

∫ a

a

−

∞

1 ∫ uΔ (t)♦α t. 1−η−θ a

If we employ inequality (8.52), we obtain from the above inequality that ∞

∫

∞

α(η + ζ ) z(t)h(t)[H σ (t)]η+ζ −1 z(t)[H σ (t)]η+ζ ♦α t ≥ ♦α t, ∫ σ η+θ 1−η−θ [G (t)] [G(t)]η+θ−1 a

a

which is the desired inequality (8.58). Since ∞

∫ a

∞

α(η + ζ ) z(t)h(t)[H σ (t)]η+ζ −1 z(t)[H σ (t)]η+ζ ♦ t ≥ ♦α t ∫ α 1−η−θ [Gσ (t)]η+θ [G(t)]η+θ−1 a ∞

≥

α(η + ζ ) z(t)h(t)[H σ (t)]η+ζ −1 ♦α t, ∫ 1−η−θ [Gσ (t)]η+θ−1 a

applying the reverse Hölder’s inequality (8.40) with the constants 0 < ζ < 1 and ζ /(ζ − 1) < 0 to the right-hand side of the above inequality leads to inequality (8.59). (ii) Using inequalities (8.47) and (8.62) in equation (8.53) yields uΔ (t) ≥ −(1 − η − θ)

α(η + ζ )z(t)h(t)[H(t)]η+ζ −1 z(t) η+ζ [H(t)] + , [Gσ (t)]η+θ [Gσ (t)]η+θ−1

or ∞

∞

a

a

α(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 z(t)[H(t)]η+ζ ♦ t ≥ ♦α t ∫ ∫ α 1−η−θ [Gσ (t)]η+θ [Gσ (t)]η+θ−1 ∞

1 − ∫ uΔ (t)♦α t. 1−η−θ a

If we employ inequality (8.52), we obtain from the above inequality that ∞

∫ a

∞

α(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 z(t)[H(t)]η+ζ ♦α t ≥ ♦α t, ∫ σ η+θ 1−η−θ [G (t)] [Gσ (t)]η+θ−1 a

8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities

� 275

which is the desired inequality (8.60). Applying the reverse Hölder’s inequality (8.40) with the constants 0 < ζ < 1 and ζ /(ζ − 1) < 0 to the right-hand side of inequality (8.60), one can obtain inequality (8.61). Remark 8.4. The diamond alpha Bennett–Leindler-type inequalities (8.58)–(8.61) obtained for 0 < ζ < 1, η ≥ 0, and 0 ≤ η + θ < 1 are derived for the first time due to the condition 0 ≤ η + θ < 1. Moreover, these inequalities generalize the diamond alpha Bennett–Leindler-type inequality (8.25) given in [34, Theorem 15] established for 0 < ζ < 1, η = 0, and θ ≤ 0 and complement the diamond alpha Bennett–Leindler-type inequality (8.31) given in [38, Theorem 9] established for ζ > 1, η ≥ 0, and 0 ≤ η + θ < 1. Remark 8.5. The condition 0 ≤ η+θ < 1 with 0 < ζ < 1 has not appeared in the literature before even in the special cases. This is one of the gaps in the literature that this theorem aims to fill. By the novel diamond alpha Bennett–Leindler inequalities (8.58)–(8.61) obtained for 0 < ζ < 1, η ≥ 0, and 0 ≤ η + θ < 1 for the first time, this aim is achieved. Remark 8.6. Special cases of the diamond alpha Bennett–Leindler-type inequalities (8.58)–(8.61) can be seen below. (i) Expressing inequalities (8.58)–(8.59) in terms of delta and nabla integrals for 0 < ζ < 1, η ≥ 0, 0 ≤ η + θ < 1, and η + ζ ≤ 1, and then choosing α = 1 in those inequalities yields ∞

∫ a

∞

η+ζ z(t)h(t)[H σ (t)]η+ζ −1 z(t)[H σ (t)]η+ζ Δt ≥ Δt ∫ 1−η−θ [Gσ (t)]η+θ [G(t)]η+θ−1

(8.63)

a

and ∞

ζ ∞

a

a

η+ζ z(t)[H σ (t)]η+ζ z(t)hζ (t)[H σ (t)]η Δt ≥ [ ] Δt, ∫ ∫ 1−η−θ [Gσ (t)]η+θ [Gσ (t)]η+θ−ζ

(8.64)

respectively, where H(t) and G(t) are defined in (8.6). Inequalities (8.63)–(8.64) are novel even in the delta calculus. These novel inequalities generalize delta Bennett– Leindler-type inequalities (8.7)–(8.8) given in [59, Theorem 2.1] and [34, Remark 5] established for 0 < ζ < 1, η = 0, and θ ≤ 0 and complement delta Bennett–Leindlertype inequalities (8.13)–(8.14) given in [37, Corollary 2] and [38, Remark 3] established for ζ > 1, η ≥ 0 and 0 ≤ η + θ < 1. We can conclude that (i) of Theorem 8.8 is a diamond alpha unification of Theorem 8.1 which is given in [59, Theorem 2.1] and [34, Remark 5] and is a generalization of the diamond alpha Bennett–Leindler-type inequality (8.25) established for 0 < ζ < 1, η = 0, and θ ≤ 0 given in [34, Theorem 15] and is a completion of the diamond alpha Bennett–Leindler-type inequality (8.31) established for ζ > 1, η ≥ 0, and 0 ≤ η + θ < 1 given in [38, Theorem 9].

276 � Z. Kayar et al. (ii) Expressing inequalities (8.60)–(8.61) in terms of delta and nabla integrals for 0 < ζ < 1, η ≥ 0, 0 ≤ η + θ < 1, and η + ζ ≥ 1, and then choosing α = 1 in those inequalities yields ∞

∫ a

∞

η+ζ z(t)[H(t)]η+ζ z(t)h(t)[H(t)]η+ζ −1 Δt ≥ Δt ∫ σ η+θ 1−η−θ [G (t)] [Gσ (t)]η+θ−1

(8.65)

a

and ∞

ζ ∞

a

a

η+ζ z(t)hζ (t)[H(t)]η z(t)[H(t)]η+ζ Δt ≥ [ ] Δt, ∫ ∫ 1−η−θ [Gσ (t)]η+θ [Gσ (t)]η+θ−ζ

(8.66)

respectively, where H(t), G(t) are defined in (8.6). Inequalities (8.65)–(8.66) are novel even in the delta calculus. These novel inequalities generalize delta Bennett– Leindler-type inequalities (8.7)–(8.8) given in [59, Theorem 2.1] and [34, Remark 5] established for 0 < ζ < 1, η = 0, and θ ≤ 0 and complement delta Bennett–Leindlertype inequalities (8.13)–(8.14) given in [37, Corollary 2] and [38, Remark 3] established for ζ > 1, η ≥ 0 and 0 ≤ η + θ < 1. We can conclude that (ii) of Theorem 8.8 is a diamond alpha unification of Theorem 8.1 which is given in [59, Theorem 2.1] and [34, Remark 5] and is a generalization of the diamond alpha Bennett–Leindler-type inequality (8.25) established for 0 < ζ < 1, η = 0, and θ ≤ 0 given in [34, Theorem 15] and is a completion of the diamond alpha Bennett–Leindler-type inequality (8.31) established for ζ > 1, η ≥ 0, and 0 ≤ η + θ < 1 given in [38, Theorem 9]. The following theorem asserts not only novel diamond alpha and delta Bennett– Leindler-type inequalities when 0 < ζ < 1, η ≥ 0, and η + θ > 1, but also complements the diamond alpha Bennett–Leindler-type inequalities given in [38, Theorem 10] established for ζ > 1, η ≥ 0, and η + θ > 1, and generalizations of the diamond alpha Bennett– Leindler-type inequalities given in [34, Theorem 17] established for 0 < ζ < 1, η = 0, and θ > 1. This novelty is caused by the condition η + ζ ≥ 1, which has not been considered so far. Theorem 8.9. Suppose that z is a nondecreasing function on [a, ∞)𝕋 . For the functions G(t) and H(t) defined in (8.24), let 0 < ζ < 1, η ≥ 0, and η + θ > 1. (i) If 0 ≤ η + ζ < 1, then ∞

∞

a

a

α(η + ζ ) z(t)h(t)[H σ (t)]η+ζ −1 z(t)[H σ (t)]η+ζ ♦ t ≥ ♦α t ∫ ∫ α η+θ−1 [G(t)]η+θ [G(t)]η+θ−1

(8.67)

and ∞

∫ a

ζ ∞

α(η + ζ ) z(t)[H σ (t)]η+ζ z(t)hζ (t)[H σ (t)]η ♦ t ≥ [ ] ♦α t. ∫ α η+θ−1 [G(t)]η+θ [G(t)]η+θ−ζ a

(8.68)

8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities

� 277

(ii) If η + ζ > 1, then ∞

∫ a

∞

α(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 z(t)[H(t)]η+ζ ♦α t ≥ ♦α t ∫ η+θ η+θ−1 [G(t)] [Gσ (t)]η+θ−1

(8.69)

a

and ∞

∫

ζ ∞

α(η + ζ ) z(t)hζ (t)[H(t)]η z(t)[H(t)]η+ζ ♦ t ≥ [ ♦α t. ] ∫ α η+θ−1 [G(t)]η+θ [G(t)]η+θ−ζ

(8.70)

a

a

Proof. It follows from (8.48) and the formula (8.36) for η + θ > 1 that

[G

1−η−θ

Δ

Δ

1

(t)] = (1 − η − θ)G (t) ∫ 1

=∫ 0

0

dw [(1 − w)G(t) + wGσ (t)]η+θ

(1 − η − θ)[−αz(t) − (1 − α)zσ (t)] dw [(1 − w)G(t) + wGσ (t)]η+θ

≥ (η + θ − 1)

(8.71)

z(t) , [G(t)]η+θ

where Gσ (t) ≤ G(t) and the nondecreasingness property of z have been used. (i) Using inequalities (8.46) and (8.71) in equation (8.50) yields uΔ (t) ≥ (η + θ − 1)

α(η + ζ )z(t)h(t)[H σ (t)]η+ζ −1 z(t) η+ζ σ [H (t)] + [G(t)]η+θ [G(t)]η+θ−1

or ∞

∫ a

∞

α(η + ζ ) z(t)h(t)[H σ (t)]η+ζ −1 z(t)[H σ (t)]η+ζ ♦ t ≥ ♦α t ∫ α 1−η−θ [G(t)]η+θ [G(t)]η+θ−1 a

−

∞

1 ∫ uΔ (t)♦α t. 1−η−θ a

If we employ inequality (8.52), we obtain from the above inequality that ∞

∫ a

∞

α(η + ζ ) z(t)h(t)[H σ (t)]η+ζ −1 z(t)[H σ (t)]η+ζ ♦α t ≥ ♦α t ∫ η+θ 1−η−θ [G(t)] [G(t)]η+θ−1 a

which is the desired inequality (8.67). Applying the reverse Hölder inequality (8.40) with the constants 0 < ζ < 1 and ζ / (ζ −1) < 0 to the right hand side of inequality (8.67), one can obtain inequality (8.68).

278 � Z. Kayar et al. (ii) Using inequalities (8.47) and (8.71) in equation (8.53) yields uΔ (t) ≥ (η + θ − 1)

α(η + ζ )z(t)h(t)[H(t)]η+ζ −1 z(t) η+ζ [H(t)] + η+θ [G(t)] [Gσ (t)]η+θ−1

≥ −(η + θ − 1)

α(η + ζ )z(t)h(t)[H(t)]η+ζ −1 z(t) η+ζ [H(t)] + η+θ [G(t)] [Gσ (t)]η+θ−1

or ∞

∞

a

a

α(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 z(t)[H(t)]η+ζ ♦ t ≥ ♦α t ∫ ∫ α η+θ−1 [G(t)]η+θ [Gσ (t)]η+θ−1 ∞

1 − ∫ uΔ (t)♦α t. 1−η−θ a

If we employ inequality (8.52), we obtain from the above inequality that ∞

∞

a

a

α(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 z(t)[H(t)]η+ζ ♦ t ≥ ♦α t, ∫ ∫ α η+θ−1 [G(t)]η+θ [Gσ (t)]η+θ−1 which is the desired inequality (8.69). Since ∞

∫ a

∞

α(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 z(t)[H(t)]η+ζ ♦α t ≥ ♦α t ∫ η+θ η+θ−1 [G(t)] [Gσ (t)]η+θ−1 a ∞

≥

α(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 ♦α t, ∫ η+θ−1 [G(t)]η+θ−1 a

applying the reverse Hölder inequality (8.40) with the constants 0 < ζ < 1 and ζ / (ζ −1) < 0 to the right hand side of the above inequality leads to inequality (8.70). Remark 8.7. The diamond alpha Bennett–Leindler-type inequalities (8.67)–(8.70) obtained for 0 < ζ < 1, η ≥ 0, and η + θ > 1 are generalizations of the diamond alpha Bennett–Leindler-type inequality (8.27) established for 0 < ζ < 1, η = 0, and θ > 1 given in [34, Theorem 17] and complements of the diamond alpha Bennett–Leindler-type inequality (8.33) established for ζ > 1, η ≥ 0, and η + θ > 1 given in [38, Theorem 10]. Remark 8.8. Although the special case of the condition η + ζ ≤ 1 is automatically satisfied in [34, Theorem 17], the other case, η + ζ ≥ 1 with 0 < ζ < 1, has not appeared in the literature before even in the special cases. This is one of the gaps in the literature that this theorem aims to fill. By the novel diamond alpha Bennett–Leindler inequalities (8.69)–(8.70) obtained for 0 < ζ < 1, η ≥ 0, η + θ > 1, and η + ζ ≥ 1 for the first time, this aim is achieved.

8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities

� 279

Remark 8.9. Special cases of the diamond alpha Bennett–Leindler-type inequalities (8.67)–(8.70) can be seen below. (i) Expressing inequalities (8.67)–(8.68) in terms of delta and nabla integrals for 0 < ζ < 1, η ≥ 0, η + θ > 1, and 0 ≤ η + ζ ≤ 1, and then choosing α = 1 in those inequalities yields ∞

∫

∞

η+ζ z(t)h(t)[H σ (t)]η+ζ −1 z(t)[H σ (t)]η+ζ Δt ≥ Δt ∫ η+θ η+θ−1 [G(t)] [G(t)]η+θ−1

(8.72)

a

a

and ∞

ζ ∞

a

a

η+ζ z(t)hζ (t)[H σ (t)]η z(t)[H σ (t)]η+ζ Δt, Δt ≥ [ ] ∫ ∫ η+θ−1 [G(t)]η+θ [G(t)]η+θ−ζ

(8.73)

respectively, where H(t), G(t) are defined in (8.6). Inequalities (8.72)–(8.73) generalize delta Bennett–Leindler-type inequalities (8.9)–(8.10) established for 0 < ζ < 1, η = 0, and θ > 1 given in [34, Remark 7] and complement delta Bennett–Leindlertype inequalities in [38, Remark 5] established for ζ > 1, η ≥ 0, and η + θ > 1. We can conclude that (i) of Theorem 8.9 is a diamond alpha unification of Theorem 8.1 given in [34, Remark 7] and a generalization of the diamond alpha Bennett–Leindler-type inequality (8.27) established for 0 < ζ < 1, η = 0, and θ > 1 given in [34, Theorem 17] and a completion of the diamond alpha Bennett–Leindler-type inequality (8.33) established for ζ > 1, η ≥ 0, and η + θ > 1 given in [38, Theorem 10]. (ii) Expressing inequalities (8.69)–(8.70) in terms of delta and nabla integrals for 0 < ζ < 1, η ≥ 0, η + θ > 1, and η + ζ ≥ 1, and then choosing α = 1 in those inequalities yields ∞

∫ a

∞

η+ζ z(t)[H(t)]η+ζ z(t)h(t)[H(t)]η+ζ −1 Δt ≥ Δt ∫ η+θ−1 [G(t)]η+θ [Gσ (t)]η+θ−1

(8.74)

a

and ∞

∫ a

ζ ∞

η+ζ z(t)[H(t)]η+ζ z(t)hζ (t)[H(t)]η ] Δt ≥ [ Δt, ∫ η+θ−1 [G(t)]η+θ [G(t)]η+θ−ζ

(8.75)

a

respectively, where H(t), G(t) are defined in (8.6). Inequalities (8.74)–(8.75) are novel even in the delta calculus. These novel inequalities generalize delta Bennett– Leindler-type inequalities (8.9)–(8.10) given in [34, Remark 7] established for 0 < ζ < 1, η = 0, and θ > 1 and complement delta Bennett–Leindler-type inequalities [38, Remark 5] established for ζ > 1, η ≥ 0, and η + θ > 1. We can conclude that (ii) of Theorem 8.9 is a diamond alpha unification of Theorem 8.1 which is given in [34, Remark 5] and is a generalization of the diamond alpha Bennett–Leindler-type inequality (8.27) established for 0 < ζ < 1, η = 0, and θ > 1 given in [34, Theorem 17] and is a completion of the diamond alpha Bennett–Leindler-type inequality (8.33) established for ζ > 1, η ≥ 0, and η + θ > 1 given in [38, Theorem 10].

280 � Z. Kayar et al. The following theorem asserts not only novel diamond alpha and nabla Bennett– Leindler-type inequalities when 0 < ζ < 1, η ≥ 0, and η+θ ≤ 0, but also complements the diamond alpha Bennett–Leindler-type inequalities given in [38, Theorem 11] established for ζ > 1, η ≥ 0, and η + θ ≤ 0, and generalizations of the diamond alpha Bennett– Leindler-type inequalities given in [34, Theorem 16] established for 0 < ζ < 1, η = 0, and θ ≤ 0. This novelty is caused by the condition η + ζ ≥ 1, which has not been considered so far. Theorem 8.10. Suppose that z is a nondecreasing function on [a, ∞)𝕋 . For the functions G(t) and H(t) defined in (8.24), assume that there exists L1 > 0 such that G(t) ≤ L1 for ρ G (t)

t ∈ (a, ∞)𝕋 . Let 0 < ζ < 1, η ≥ 0, and η + θ ≤ 0 be real constants. (i) If 0 ≤ η + ζ ≤ 1, then ∞

∫

ρ

z(t)[H (t)]η+ζ [G(t)]η+θ

a

∞

ρ

(1 − α)(η + ζ ) z(t)h(t)[H (t)]η+ζ −1 ♦α t ≥ ♦α t ∫ 1−η−θ [G(t)]η+θ−1 a

(8.76)

and ∞

∫

ρ

z(t)[H (t)]η+ζ [G(t)]η+θ

a

ζ ∞

ρ

(1 − α)(η + ζ ) z(t)hζ (t)[H (t)]η ♦α t ≥ [ ] ∫ ♦α t. 1−η−θ [G(t)]η+θ−ζ a

(8.77)

(ii) If η + ζ ≥ 1, then ∞

∫ a

z(t)[H(t)]η+ζ [G(t)]η+θ

∞

♦α t ≥

(1 − α)(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 ♦α t ∫ ρ 1−η−θ [G (t)]η+θ−1 a

(8.78)

and ∞

∫ a

z(t)[H(t)]η+ζ [G(t)]η+θ

♦α t ≥ [

∞

η+θ−1

L1

(1 − α)(η + ζ ) ζ z(t)hζ (t)[H(t)]η ] ∫ ♦α t. 1−η−θ [G(t)]η+θ−ζ a

(8.79)

Proof. First we obtain some inequalities which will be used in the sequel. By utilizing Lemma 8.3, one can obtain ∇

∇

∞

H (t) = [ ∫ z(s)h(s)♦α s] t

∞

∇

∇

∞

= α[ ∫ z(s)h(s)Δs] + (1 − α)[ ∫ z(s)h(s)∇s] t

t

= −αzρ (t)hρ (t) − (1 − α)z(t)h(t) ≤ 0. By taking account of (8.80) and employing formula (8.39), one can observe that

(8.80)

8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities

[H

η+ζ

1

∇

∇

ρ

(t)] = (η + ζ )H (t) ∫[wH(t) + (1 − w)H (t)]

η+ζ −1

� 281

dw

0

= (η + ζ )[−αzρ (t)hρ (t) − (1 − α)z(t)h(t)] 1

ρ

× ∫[wH(t) + (1 − w)H (t)]

η+ζ −1

dw.

0

Then two estimates can be obtained depending on whether η + ζ ≤ 1 or η + ζ ≥ 1 for the η+ζ

function [H (t)]∇ as follows: (i) If η + ζ ≤ 1, then [H

η+ζ

ρ

∇

(t)] ≤ −(1 − α)(η + ζ )z(t)h(t)[H (t)]

η+ζ −1

(8.81)

;

(ii) If η + ζ ≥ 1, then [H

η+ζ

∇

≤ −(1 − α)(η + ζ )z(t)h(t)[H(t)]

(t)]

η+ζ −1

(8.82)

,

ρ

where H (t) ≥ H(t) has been used. Similarly, by Lemma 8.3, we note that t

∇

t

∇

t

∇

∇

G (t) = [∫ z(s)♦α s] = α[∫ z(s)Δs] + (1 − α)[∫ z(s)∇s] a

a ρ

a

(8.83)

= αz (t) + (1 − α)z(t) ≥ 0. It follows from (8.83) and formula (8.39) for η + θ ≤ 0 that we have [G

1−η−θ

1

∇

(t)] = ∫ 0

1

=∫

∇

(1 − η − θ)G (t)dw ρ

[wG(t) + (1 − w)G (t)]η+θ (1 − η − θ)[αzρ (t) + (1 − α)z(t)]dw [wG(t) + (1 − w)G

0

≤

(1 − η − θ)z(t) [G(t)]η+θ

ρ

(8.84)

(t)]η+θ

,

where G(t) ≥ G(ρ(t)) and the nondecreasingness property of z have been used. (i) Let us define u(t) = [H(t)]η+ζ [G(t)]1−η−θ for t ∈ [a, ∞). If we take the nabla derivative of the function u using formula (8.38), we get u∇ (t) = [H

η+ζ

∇

(t)] [G

1−η−θ

ρ

(t)] + [H (t)]

η+ζ

[G

1−η−θ

Using inequalities (8.81) and (8.84) in equation (8.85) yields

∇

(t)] .

(8.85)

282 � Z. Kayar et al.

u∇ (t) ≤

(1 − η − θ)z(t) [G(t)]η+θ

ρ

η+ζ

[H (t)]

ρ

−

(1 − α)(η + ζ )z(t)h(t)[H (t)]η+ζ −1 [G(t)]η+θ−1

,

or ∞

∫

ρ

z(t)[H (t)]η+ζ [G(t)]η+θ

a

∞

♦α t ≥

ρ

(1 − α)(η + ζ ) z(t)h(t)[H (t)]η+ζ −1 ♦α t ∫ 1−η−θ [G(t)]η+θ−1 a

(8.86)

∞

1 + ∫ u∇ (t)♦α t. 1−η−θ a

The definition of u implies u(∞) = u(a) = 0 and, by using Lemma 8.4, we obtain ∞

∞ ∇

∞ ∇

∫ u (t)♦α t = α ∫ u (t)Δt + (1 − α) ∫ u∇ (t)∇t a

a

(8.87)

a

= α[u(ρ(∞)) − u(ρ(a))] + (1 − α)[u(∞) − u(a)] ≥ 0, where we have imposed that (−1)η+θ = 1. Therefore we can infer that inequality (8.86) becomes ∞

∫

ρ

z(t)[H (t)]η+ζ [G(t)]η+θ

a

∞

♦α t ≥

ρ

(1 − α)(η + ζ ) z(t)h(t)[H (t)]η+ζ −1 ♦α t, ∫ 1−η−θ [G(t)]η+θ−1 a

which is the desired inequality (8.76). Inequality (8.77) can be obtained by applying the reverse Hölder’s inequality (8.40) with the constants ζ < 1 and ζ /(ζ − 1) < 0 to the right-hand side of inequality (8.76). 1−η−θ

η+ζ

(t)] for t ∈ [a, ∞). If we take the nabla deriva(ii) Let us define u(t) = [H (t)][G tive of the function u by using the formula (8.38), we get u∇ (t) = [H

η+ζ

(t)][G

1−η−θ

∇

(t)] + [H

η+ζ

∇

ρ

(t)] [G (t)]

1−η−θ

(8.88)

.

Using inequalities (8.82) and (8.84) in equation (8.88) yields u∇ (t) ≤

(1 − η − θ)z(t)[H(t)]η+ζ [G(t)]η+θ

−

(1 − α)(η + ζ )z(t)h(t)[H(t)]η+ζ −1 ρ

[G (t)]η+θ−1

or ∞

∫ a

z(t)[H(t)]η+ζ [G(t)]η+θ

∞

(1 − α)(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 ♦α t ≥ ♦α t ∫ ρ 1−η−θ [G (t)]η+θ−1 a ∞

1 + ∫ u∇ (t)♦α t. 1−η−θ a

,

8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities

� 283

If we employ inequality (8.87), we obtain from the above inequality that ∞

∫

z(t)[H(t)]η+ζ [G(t)]η+θ

a

∞

(1 − α)(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 ♦α t, ♦α t ≥ ∫ ρ 1−η−θ [G (t)]η+θ−1 a

which is the desired inequality (8.78). Using G(t) ≤ L1 on the right-hand side of inequality (8.78) and applying the reverse ρ G (t)

Hölder inequality (8.40) with the constants ζ < 1 and ζ /(ζ − 1) < 0 to the resulting integral, one can obtain inequality (8.79).

Remark 8.10. The diamond alpha Bennett–Leindler-type inequalities (8.76)–(8.79) obtained for 0 < ζ < 1, η ≥ 0, and η + θ ≤ 0 are generalizations of the diamond alpha Bennett–Leindler-type inequality (8.26) given in [34, Theorem 16] established for 0 < ζ < 1, η = 0, and θ ≤ 0, and complements of the diamond alpha Bennett–Leindlertype inequality (8.30) given in [38, Theorem 11] established for ζ > 1, η ≥ 0, and η+θ ≤ 0. Remark 8.11. Although a special case of the condition η+ζ ≤ 1 is automatically satisfied in [34, Theorem 16], the other case, η+ζ ≥ 1 with 0 < ζ < 1, has not appeared in the literature before even for the special cases. This is one of the gaps in the literature that this theorem aims to fill. By the novel diamond alpha Bennett–Leindler inequalities (8.78)–(8.79) obtained for 0 < ζ < 1, η ≥ 0, η+θ ≤ 0, and η+ζ ≥ 1 for the first time, this aim is achieved. Remark 8.12. Special cases of the diamond alpha Bennett–Leindler-type inequalities (8.76)–(8.79) can be seen below. (i) Expressing inequalities (8.76)–(8.77) in terms of delta and nabla integrals for 0 < ζ < 1, η ≥ 0, η + θ ≤ 0, and η + ζ ≤ 1, and then choosing α = 0 in those inequalities yields ∞

∫ a

and

∞

∫ a

∞

ρ

z(t)[H (t)]η+ζ [G(t)]η+θ

∇t ≥

ζ ∞

ρ

z(t)[H (t)]η+ζ [G(t)]η+θ

ρ

η+ζ z(t)h(t)[H (t)]η+ζ −1 ∇t ∫ 1−η−θ [G(t)]η+θ−1 a

∇t ≥ [

ρ

η+ζ z(t)hζ (t)[H (t)]η ] ∫ ∇t, η+θ−ζ 1−η−θ [G(t)] a

(8.89)

(8.90)

respectively, where H(t) and G(t) are defined in (8.24). Inequalities (8.89)–(8.90) generalize nabla Bennett–Leindler-type inequalities (8.16)–(8.17) established for 0 < ζ < 1, η = 0, and θ ≤ 0 given in [29, Theorem 3.9] and [34, Remark 6] and complement nabla Bennett–Leindler-type inequalities (8.20)–(8.21) established for ζ > 1, η ≥ 0, and η + θ ≤ 0 given in [37, Theorem 7] and [38, Remark 7]. We can conclude that (i) of Theorem 8.10 is a diamond alpha unification of Theorem 8.2 which is given in [29, Theorem 3.9] and [34, Remark 6] and is a generalization of the diamond alpha Bennett–Leindler-type inequality (8.26) established for 0 < ζ < 1, η = 0, and θ ≤ 0 given in [34, Theorem 16] and is a completion of the diamond alpha Bennett–

284 � Z. Kayar et al. Leindler-type inequality (8.30) established for ζ > 1, η ≥ 0, and η + θ ≤ 0 given in [38, Theorem 11]. (ii) Expressing inequalities (8.78)–(8.79) in terms of delta and nabla integrals for 0 < ζ < 1, η ≥ 0, η + θ ≤ 0, and η + ζ ≥ 1, and then choosing α = 0 in those inequalities yields ∞

∫ a

z(t)[H(t)]η+ζ [G(t)]η+θ

∞

η+ζ z(t)h(t)[H(t)]η+ζ −1 ∇t ≥ ∇t ∫ ρ 1−η−θ [G (t)]η+θ−1 a

(8.91)

and ∞

∫

z(t)[H(t)]η+ζ [G(t)]η+θ

a

∞

η+θ−1

∇t ≥ [

η + ζ ζ z(t)hζ (t)[H(t)]η L1 ] ∫ ∇t, 1−η−θ [G(t)]η+θ−ζ a

(8.92)

respectively, where H(t) and G(t) and L1 are defined in (8.24). Inequalities (8.91)– (8.92) are novel even in the nabla calculus. These novel inequalities generalize nabla Bennett–Leindler-type inequalities (8.16)–(8.17) in [29, Theorem 3.9] and [34, Remark 6] established for 0 < ζ < 1, η = 0 and θ ≤ 0 and complement nabla Bennett– Leindler-type inequalities (8.20)–(8.21) in [37, Theorem 7] and [38, Remark 7] established for ζ > 1, η ≥ 0, and η + θ ≤ 0. We can conclude that (ii) of Theorem 8.10 is a diamond alpha unification of Theorem 8.2 which is given in [29, Theorem 3.9] and [34, Remark 6] and is a generalization of the diamond alpha Bennett–Leindler-type inequality (8.26) established for 0 < ζ < 1, η = 0, and θ ≤ 0 given in [34, Theorem 16] and is a completion of the diamond alpha Bennett–Leindler-type inequality (8.30) established for ζ > 1, η ≥ 0, and η + θ ≤ 0 given in [38, Theorem 11]. The following theorem asserts not only novel diamond alpha and nabla Bennett– Leindler-type inequalities when 0 < ζ < 1, η ≥ 0, and 0 ≤ η + θ < 1, but also complements the diamond alpha Bennett–Leindler-type inequalities given in [38, Theorem 11] established for ζ > 1, η ≥ 0, and 0 ≤ η + θ < 1, and generalization of the diamond alpha Bennett–Leindler-type inequalities given in [34, Theorem 16] established for 0 < ζ < 1, η = 0, and θ ≤ 0. This novelty is caused by the condition η + ζ ≥ 1, which has not been considered so far. Theorem 8.11. Suppose that z is a nondecreasing function on [a, ∞)𝕋 . For the functions G(t) and H(t) defined in (8.24), let 0 < ζ < 1, η ≥ 0 and 0 ≤ η + θ < 1 be real constants. (i) If 0 ≤ η + ζ ≤ 1, then ∞

∫ a

ρ

z(t)[H (t)]η+ζ ρ

[G (t)]η+θ

∞

♦α t ≥

ρ

(1 − α)(η + ζ ) z(t)h(t)[H (t)]η+ζ −1 ♦α t ∫ η+θ−1 1−η−θ [G(t)] a

(8.93)

and ∞

∫ a

ρ

z(t)[H (t)]η+ζ ρ

[G (t)]η+θ

ζ ∞

ρ

(1 − α)(η + ζ ) z(t)hζ (t)[H (t)]η ♦α t ≥ [ ] ∫ ♦α t. ρ 1−η−θ [G (t)]η+θ−ζ a

(8.94)

8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities

� 285

(ii) If η + ζ ≥ 1, then ∞

∫

z(t)[H(t)]η+ζ ρ

[G (t)]η+θ

a

∞

(1 − α)(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 ♦α t ≥ ♦α t ∫ ρ 1−η−θ [G (t)]η+θ−1 a

(8.95)

and ∞

∫

z(t)[H(t)]η+ζ ρ

[G (t)]η+θ

a

ζ ∞

(1 − α)(η + ζ ) z(t)hζ (t)[H(t)]η ♦α t ≥ [ ♦α t. ] ∫ ρ 1−η−θ [G (t)]η+θ−ζ a

(8.96)

Proof. It follows from (8.83) and formula (8.39) for 0 ≤ η + θ < 1 that we have

[G

1−η−θ

1

∇

(t)] = ∫ 0

1

=∫

∇

(1 − η − θ)G (t)dw ρ

[wG(t) + (1 − w)G (t)]η+θ (1 − η − θ)[αzρ (t) + (1 − α)z(t)]dw [wG(t) + (1 − w)G

0

≤

(1 − η − θ)z(t) ρ

[G (t)]η+θ

ρ

(8.97)

(t)]η+θ

,

where G(t) ≥ G(ρ(t)) and the nondecreasingness property of z have been used. (i) Using inequalities (8.81) and (8.97) in equation (8.85) yields u∇ (t) ≤

(1 − η − θ)z(t) ρ

[G (t)]η+θ

ρ

η+ζ

[H (t)]

ρ

−

(1 − α)(η + ζ )z(t)h(t)[H (t)]η+ζ −1 [G(t)]η+θ−1

or ∞

∫ a

ρ

z(t)[H (t)]η+ζ ρ

[G (t)]η+θ

∞

ρ

(1 − α)(η + ζ ) z(t)h(t)[H (t)]η+ζ −1 ♦α t ≥ ♦α t ∫ 1−η−θ [G(t)]η+θ−1 a ∞

1 + ∫ u∇ (t)♦α t. 1−η−θ a

If we employ inequality (8.87), we obtain from the above inequality that ∞

∫ a

ρ

z(t)[H (t)]η+ζ ρ

[G (t)]η+θ

∞

♦α t ≥

ρ

(1 − α)(η + ζ ) z(t)h(t)[H (t)]η+ζ −1 ♦α t, ∫ 1−η−θ [G(t)]η+θ−1 a

which is the desired inequality (8.93).

,

286 � Z. Kayar et al. Since ∞

ρ

z(t)[H (t)]η+ζ

∫

ρ

[G (t)]η+θ

a

∞

ρ

∞

ρ

♦α t ≥

(1 − α)(η + ζ ) z(t)h(t)[H (t)]η+ζ −1 ♦α t ∫ η+θ−1 1−η−θ [G(t)] a

≥

(1 − α)(η + ζ ) z(t)h(t)[H (t)]η+ζ −1 ♦α t, ∫ ρ η+θ−1 1−η−θ [G (t)] a

inequality (8.94) can be obtained by applying the reverse Hölder’s inequality (8.40) with the constants ζ < 1 and ζ /(ζ − 1) < 0 to the right-hand side of inequality (8.93). (ii) Using inequalities (8.82) and (8.97) in equation (8.88) yields u∇ (t) ≤

(1 − η − θ)z(t)[H(t)]η+ζ ρ

[G (t)]η+θ

−

(1 − α)(η + ζ )z(t)h(t)[H(t)]η+ζ −1 ρ

[G (t)]η+θ−1

,

or ∞

∫ a

z(t)[H(t)]η+ζ ρ

[G (t)]η+θ

∞

(1 − α)(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 ♦α t ≥ ♦α t ∫ ρ 1−η−θ [G (t)]η+θ−1 a ∞

1 + ∫ u∇ (t)♦α t. 1−η−θ a

If we employ inequality (8.87), we obtain from the above inequality that ∞

∫ a

z(t)[H(t)]η+ζ ρ

[G (t)]η+θ

∞

(1 − α)(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 ♦α t ≥ ♦α t, ∫ ρ 1−η−θ [G (t)]η+θ−1 a

which is the desired inequality (8.95). Applying the reverse Hölder’s inequality (8.40) with the constants ζ < 1 and ζ / (ζ − 1) < 0 to the right-hand side of inequality (8.95), one can obtain inequality (8.96). Remark 8.13. The diamond alpha Bennett–Leindler-type inequalities (8.93)–(8.96) obtained for 0 < ζ < 1, η ≥ 0, and 0 ≤ η + θ < 1 are derived for the first time due to the condition 0 ≤ η + θ < 1. Moreover, these inequalities generalize the diamond alpha Bennett–Leindler-type inequality (8.26) given in [34, Theorem 16] established for 0 < ζ < 1, η = 0, and θ ≤ 0 and complement the diamond alpha Bennett–Leindler-type inequality (8.32) given in [38, Theorem 11] established for ζ > 1, η ≥ 0, and 0 ≤ η + θ < 1. Remark 8.14. The condition 0 ≤ η + θ < 1 with 0 < ζ < 1 has not appeared in the literature before even in the special cases. This is one of the gaps in the literature that this theorem aims to fill. By the novel diamond alpha Bennett–Leindler inequalities (8.93)–(8.96) obtained for 0 < ζ < 1, η ≥ 0, and 0 ≤ η + θ < 1 for the first time, this aim is achieved.

8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities

� 287

Remark 8.15. Special cases of the diamond alpha Bennett–Leindler-type inequalities (8.93)–(8.96) can be seen below. (i) Expressing inequalities (8.93)–(8.94) in terms of delta and nabla integrals for 0 < ζ < 1, η ≥ 0, 0 ≤ η + θ < 1, and η + ζ ≤ 1, and then choosing α = 0 in those inequalities yields ∞

∫

ρ

z(t)[H (t)]η+ζ ρ

[G (t)]η+θ

a

∞

ρ

η+ζ z(t)h(t)[H (t)]η+ζ −1 ∇t ≥ ∇t ∫ 1−η−θ [G(t)]η+θ−1 a

(8.98)

and ∞

∫

ζ ∞

ρ

z(t)[H (t)]η+ζ ρ

[G (t)]η+θ

a

∇t ≥ [

ρ

η+ζ z(t)hζ (t)[H (t)]η ] ∫ ∇t, ρ 1−η−θ [G (t)]η+θ−ζ a

(8.99)

respectively, where H(t) and G(t) are defined in (8.15). Inequalities (8.98)–(8.99) are novel even in the nabla calculus. These novel inequalities generalize nabla Bennett– Leindler-type inequalities (8.16)–(8.17) in [29, Theorem 3.9] and [34, Remark 6] established for 0 < ζ < 1, η = 0 and θ ≤ 0 and complement nabla Bennett–Leindlertype inequalities (8.22)–(8.23) in [37, Theorem 8] and [38, Remark 7] established for ζ > 1, η ≥ 0, and 0 ≤ η + θ < 1. We can conclude that (i) of Theorem 8.11 is a diamond alpha unification of Theorem 8.2 which is given in [29, Theorem 3.9] and [34, Remark 6] and is a generalization of the diamond alpha Bennett–Leindler-type inequality (8.26) established for 0 < ζ < 1, η = 0, and θ ≤ 0 given in [34, Theorem 16] and is a completion of the diamond alpha Bennett–Leindler-type inequality (8.32) established for ζ > 1, η ≥ 0, and 0 ≤ η + θ < 1 given in [38, Theorem 11]. (ii) Expressing inequalities (8.95)–(8.96) in terms of delta and nabla integrals for 0 < ζ < 1, η ≥ 0, η + θ ≤ 0, and η + ζ ≥ 1, and then choosing α = 0 in those inequalities yields ∞

∫ a

z(t)[H(t)]η+ζ ρ

[G (t)]η+θ

∞

∇t ≥

η+ζ z(t)h(t)[H(t)]η+ζ −1 ∇t ∫ ρ η+θ−1 1−η−θ [G (t)] a

(8.100)

and ∞

∫ a

z(t)[H(t)]η+ζ ρ

[G (t)]η+θ

ζ ∞

η+ζ z(t)hζ (t)[H(t)]η ] ∫ ∇t ≥ [ ∇t, ρ 1−η−θ [G (t)]η+θ−ζ a

(8.101)

respectively, where H(t) and G(t) are defined in (8.15). Inequalities (8.100)–(8.101) are novel even in the nabla calculus. These novel inequalities generalize nabla Bennett–Leindler-type inequalities (8.16)–(8.17) in [29, Theorem 3.9] and [34, Remark 6] established for 0 < ζ < 1, η = 0 and θ ≤ 0 and complement nabla Bennett– Leindler-type inequalities (8.22)–(8.23) in [37, Theorem 8] and [38, Remark 7] established for ζ > 1, η ≥ 0, and 0 ≤ η + θ < 1. We can conclude that (ii) of Theorem 8.11 is a diamond alpha unification of Theorem 8.2 which is given in [29, Theorem 3.9] and

288 � Z. Kayar et al. [34, Remark 6] and is a generalization of the diamond alpha Bennett–Leindler-type inequality (8.26) established for 0 < ζ < 1, η = 0, and θ ≤ 0 given in [34, Theorem 16] and is a completion of the diamond alpha Bennett–Leindler-type inequality (8.32) established for ζ > 1, η ≥ 0, and 0 ≤ η + θ < 1 given in [38, Theorem 11]. The following theorem asserts not only novel diamond alpha and nabla Bennett– Leindler-type inequalities when 0 < ζ < 1, η ≥ 0, and η + θ > 1, but also complements the diamond alpha Bennett–Leindler-type inequalities given in [38, Theorem 12] established for ζ > 1, η ≥ 0, and η + θ > 1, and generalizations of the diamond alpha Bennett– Leindler-type inequalities given in [34, Theorem 18] established for 0 < ζ < 1, η = 0, and θ > 1. This novelty is caused by the condition η + ζ ≥ 1, which has not been considered so far. Theorem 8.12. Suppose that z is a nonincreasing function on [a, ∞)𝕋 . For the functions G(t) and H(t) defined in (8.24), let 0 < ζ < 1, η ≥ 0 and η + θ > 1 be real constants. (i) If 0 ≤ η + ζ < 1, then ∞

∫

ρ

z(t)[H (t)]η+ζ [G(t)]η+θ

a

∞

♦α t ≥

ρ

(1 − α)(η + ζ ) z(t)h(t)[H (t)]η+ζ −1 ♦α t ∫ η+θ−1 [G(t)]η+θ−1 a

(8.102)

and ∞

∫

ρ

z(t)[H (t)]η+ζ [G(t)]η+θ

a

♦α t ≥ [

ζ ∞

ρ

(1 − α)(η + ζ ) z(t)hζ (t)[H (t)]η ] ∫ ♦α t. η+θ−ζ η+θ−1 [G(t)] a

(8.103)

(ii) If η + ζ ≥ 1, then ∞

∫ a

z(t)[H(t)]η+ζ [G(t)]η+θ

∞

♦α t ≥

(1 − α)(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 ♦α t ∫ ρ η+θ−1 [G (t)]η+θ−1 a

(8.104)

and ∞

∫ a

z(t)[H(t)]η+ζ [G(t)]η+θ

♦α t ≥ [

ζ ∞

(1 − α)(η + ζ ) z(t)hζ (t)[H(t)]η ] ∫ ♦α t. η+θ−ζ η+θ−1 [G(t)] a

(8.105)

Proof. It follows from (8.83) and formula (8.39) for η + θ > 1 that we have [G

1−η−θ

1

∇

(t)] = ∫ 0

1

=∫ 0

≤

∇

(1 − η − θ)G (t)dw ρ

[wG(t) + (1 − w)G (t)]η+θ (1 − η − θ)[αzρ (t) + (1 − α)z(t)]dw [wG(t) + (1 − w)G

(η + θ − 1)z(t) [G(t)]η+θ

,

ρ

(t)]η+θ

(8.106)

8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities

� 289

where G(t) ≥ G(ρ(t)) and the nonincreasingness property of z have been used. (i) Using inequalities (8.81) and (8.106) in equation (8.85) yields u∇ (t) ≤ − ≤

(η + θ − 1)z(t) [G(t)]η+θ

(η + θ − 1)z(t) [G(t)]η+θ

ρ

ρ

η+ζ

[H (t)] ρ

η+ζ

[H (t)]

−

−

(1 − α)(η + ζ )z(t)h(t)[H (t)]η+ζ −1 [G(t)]η+θ−1

ρ

(1 − α)(η + ζ )z(t)h(t)[H (t)]η+ζ −1 [G(t)]η+θ−1

,

or ∞

ρ

z(t)[H (t)]η+ζ

∫

[G(t)]η+θ

a

∞

ρ

(1 − α)(η + ζ ) z(t)h(t)[H (t)]η+ζ −1 ♦α t ≥ ♦α t ∫ η+θ−1 [G(t)]η+θ−1 a ∞

1 + ∫ u∇ (t)♦α t. η+θ−1 a

If we employ inequality (8.87), we obtain from the above inequality that ∞

∫

ρ

z(t)[H (t)]η+ζ [G(t)]η+θ

a

∞

ρ

(1 − α)(η + ζ ) z(t)h(t)[H (t)]η+ζ −1 ♦α t ≥ ♦α t, ∫ η+θ−1 [G(t)]η+θ−1 a

which is the desired inequality (8.102). Inequality (8.103) can be obtained by applying the reverse Hölder’s inequality (8.40) with the constants ζ < 1 and ζ /(ζ − 1) < 0 to the right-hand side of inequality (8.93). (ii) Using inequalities (8.82) and (8.106) in equation (8.88) yields u∇ (t) ≤ −

(η + θ − 1)z(t)[H(t)]η+ζ [G(t)]η+θ

−

(1 − α)(η + ζ )z(t)h(t)[H(t)]η+ζ −1 ρ

[G (t)]η+θ−1

or ∞

∫ a

z(t)[H(t)]η+ζ [G(t)]η+θ

∞

♦α t ≥

(1 − α)(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 ♦α t ∫ ρ η+θ−1 [G (t)]η+θ−1 a ∞

+

1 ∫ u∇ (t)♦α t. η+θ−1 a

If we employ inequality (8.87), we obtain from the above inequality that ∞

∫ a

z(t)[H(t)]η+ζ [G(t)]η+θ

∞

(1 − α)(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 ♦α t ≥ ♦α t, ∫ ρ η+θ−1 [G (t)]η+θ−1 a

which is the desired inequality (8.104).

,

290 � Z. Kayar et al. Since ∞

∫ a

z(t)[H(t)]η+ζ [G(t)]η+θ

∞

(1 − α)(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 ♦α t ≥ ♦α t ∫ ρ η+θ−1 [G (t)]η+θ−1 a ∞

(1 − α)(η + ζ ) z(t)h(t)[H(t)]η+ζ −1 ≥ ♦α t, ∫ η+θ−1 [G(t)]η+θ−1 a applying the reverse Hölder’s inequality (8.40) with the constants ζ < 1 and ζ / (ζ − 1) < 0 to the right-hand side of the above inequality, one can obtain inequality (8.105). Remark 8.16. The diamond alpha Bennett–Leindler-type inequalities (8.102)–(8.105) obtained for 0 < ζ < 1, η ≥ 0, and η + θ > 1 are generalizations of the diamond alpha Bennett–Leindler-type inequality (8.28) given in [34, Theorem 18] for 0 < ζ < 1, η = 0, and θ > 1 and complements of the diamond alpha Bennett–Leindler-type inequality (8.34) given in [38, Theorem 12] for ζ > 1, η ≥ 0, and η + θ > 1. Remark 8.17. Although the special case of the condition η + ζ ≤ 1 is automatically satisfied in [34, Theorem 18], the other case, η + ζ ≥ 1 with 0 < ζ < 1, has not appeared in the literature before even for the special case. This is one of the gaps in the literature that this theorem aims to fill. By the novel diamond alpha Bennett–Leindler-type inequalities (8.104)–(8.105) obtained for 0 < ζ < 1, η ≥ 0, η + θ > 1, and η + ζ ≥ 1 for the first time, this aim is achieved. Remark 8.18. Special cases of the diamond alpha Bennett–Leindler-type inequalities (8.102)–(8.105) can be seen below. (i) Expressing inequalities (8.102)–(8.103) in terms of delta and nabla integrals for 0 < ζ < 1, η ≥ 0, η + θ > 1, and η + ζ ≤ 1, and then choosing α = 0 in those inequalities yields ∞

∫ a

ρ

z(t)[H (t)]η+ζ [G(t)]η+θ

∞

ρ

η+ζ z(t)h(t)[H (t)]η+ζ −1 ∇t ≥ ∇t ∫ η+θ−1 [G(t)]η+θ−1 a

(8.107)

and ∞

∫ a

ρ

z(t)[H (t)]η+ζ [G(t)]η+θ

ζ ∞

ρ

η+ζ z(t)hζ (t)[H (t)]η ∇t ≥ [ ] ∫ ∇t, η+θ−1 [G(t)]η+θ−ζ a

(8.108)

respectively, where H(t) and G(t) are defined in (8.15). Inequalities (8.107)–(8.108) generalize nabla Bennett–Leindler-type inequalities (8.18)–(8.19) established for 0 < ζ < 1, η = 0, and θ > 1 given in [34, Remark 8] and complement nabla Bennett– Leindler-type inequalities given in [38, Remark 9] established for ζ > 1, η ≥ 0, and

8 Generalized diamond alpha Bennett–Leindler-type dynamic inequalities

� 291

η + θ > 1. We can conclude that (i) of Theorem 8.12 is a diamond alpha unification of Theorem 8.2 given in [34, Remark 8] and a generalization of the diamond alpha Bennett–Leindler-type inequality (8.28) established for 0 < ζ < 1, η = 0, and θ > 1 given in [34, Theorem 18] and a completion of the diamond alpha Bennett– Leindler-type inequality (8.34) established for ζ > 1, η ≥ 0, and η + θ > 1 given in [38, Theorem 12]. (ii) Expressing inequalities (8.104)–(8.105) in terms of delta and nabla integrals for 0 < ζ < 1, η ≥ 0, η + θ > 1, and η + ζ ≥ 1, and then choosing α = 0 in those inequalities yields ∞

∫

z(t)[H(t)]η+ζ

a

[G(t)]η+θ

∞

∇t ≥

η+ζ z(t)h(t)[H(t)]η+ζ −1 ∇t ∫ ρ η+θ−1 [G (t)]η+θ−1 a

(8.109)

and ∞

∫ a

z(t)[H(t)]η+ζ [G(t)]η+θ

ζ ∞

η+ζ z(t)h1/ζ (t)[H(t)]ηζ ∇t ≥ [ ] ∫ ∇t, η+θ−1 [G(t)]η+θ−ζ a

(8.110)

respectively, where H(t) and G(t) are defined in (8.15). Inequalities (8.109)–(8.110) are novel even in the nabla calculus. These novel inequalities generalize nabla Bennett–Leindler-type inequalities (8.18)–(8.19) established for 0 < ζ < 1, η = 0, and θ > 1 given in [34, Remark 8] and complement nabla Bennett–Leindler-type inequalities given in [38, Remark 9] established for ζ > 1, η ≥ 0, and η + θ > 1. We can conclude that (ii) of Theorem 8.12 is a diamond alpha unification of Theorem 8.2 given in [34, Remark 8] and a generalization of the diamond alpha Bennett– Leindler-type inequality (8.28) established for 0 < ζ < 1, η = 0, and θ > 1 given in [34, Theorem 18] and a completion of the diamond alpha Bennett–Leindler-type inequality (8.34) established for ζ > 1, η ≥ 0, and η + θ > 1 given in [38, Theorem 12].

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Sı̇ bel Doğru Akgöl and Abdullah Özbekler

9 De La Vallée Poussin-type inequality for impulsive dynamic equations on time scales Abstract: We derive a de La Vallée Poussin-type inequality for impulsive dynamic equations on time scales. This inequality is often used in conjunction with disconjugacy and/or (non)oscillation. Hence, it appears to be a very useful tool for the qualitative study of dynamic equations. In this work, generalizing the classical de La Vallée Poussin inequality for impulsive dynamic equations on arbitrary time scales, we obtain a disconjugacy criterion and some results on nonoscillation. We also present illustrative examples that support our findings.

9.1 Introduction The celebrated de La Vallée Poussin inequality was given for the second-order linear homogeneous differential equation x ′′ + p(t)x ′ + q(t)x = 0 with the Dirichlet boundary conditions x(a) = x(b) = 0 as in the following theorem. Theorem 9.1 ([11, 14]). If x(t) ≠ 0 in (a, b), then q0 (b − a)2 + p0 (b − a) > 1 2 holds, where 󵄨 󵄨 p0 = max 󵄨󵄨󵄨p(t)󵄨󵄨󵄨 t∈[a,b]

and

󵄨 󵄨 q0 = max 󵄨󵄨󵄨q(t)󵄨󵄨󵄨. t∈[a,b]

Later, Hartman and Wintner [11] obtained the less restrictive inequality b

b

b

a

a

a

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 max{∫(s − a)󵄨󵄨󵄨p(s)󵄨󵄨󵄨ds, ∫(b − s)󵄨󵄨󵄨p(s)󵄨󵄨󵄨ds} + ∫(s − a)(b − s)󵄨󵄨󵄨q(s)󵄨󵄨󵄨ds > b − a,

Sı̇ bel Doğru Akgöl, Abdullah Özbekler, Department of Mathematics, Atilim University, 06830 Incek, Ankara, Turkey, e-mails: [email protected], [email protected] https://doi.org/10.1515/9783111182971-009

(9.1)

296 � S. Doğru Akgöl and A. Özbekler which implies that q0 p (b − a)3 + 0 (b − a)2 > b − a, 6 2 or equivalently that q0 p (b − a)2 + 0 (b − a) > 1. 6 2 In 2018, some extensions of the de La Vallée Poussin-type inequalities were obtained for dynamic equations on time scales [8], fractional differential equations [9], and partial differential equations [1]. Recently, the inequality (9.1) has also been extended for impulsive differential equations, see [3]. In this work, we aim to obtain a de La Vallée Poussin-type inequality for impulsive dynamic equations on time scales. First, we recall some basic concepts regarding time scale calculus. A time scale 𝕋 is a nonempty arbitrary closed subset of ℝ. The set of real numbers ℝ, the set of integers ℤ, the Cantor set C , the set qℕ0 := {qn : n ∈ ℕ0 }, where q > 1 and hℤ := {hz : z ∈ ℤ}, where h > 0, are some of the well-known examples of time scales. The necessary definitions and elements of time scale calculus can be found in [2], and comprehensive and detailed information about time scales is given in the books [4–7, 10, 12, 13]. We let PLCrd [a, b] be the set of functions φ such that φ(t) is rd-continuous on (tk , tk+1 ]𝕋 := (tk , tk+1 ] ∩ 𝕋 and φ(tk+ ) exists for each k for which tk ∈ [a, b)𝕋 , k = 1, 2, . . . . Let 𝕋 be an arbitrary time scale and suppose a and b (a < b) are two consecutive zeros of the linear impulsive dynamic equation x ΔΔ + p(t)x Δ + q(t)x σ = 0,

{

Δ

Δ

σ

△x + pk x + qk x = 0,

t ≠ tk ;

t = tk

(9.2)

for t ∈ [a, b]𝕋 , k = 1, 2, . . ., where p, q ∈ Crd [a, b], {pk } and {qk } are real sequences, {tk } is a strictly increasing sequence of real numbers such that limk→∞ tk = ∞, and the impulse operator is defined by △φ(tk ) = φ(tk+ ) − φ(tk− ) with φ(tk± ) = limt→t± φ(t). Throughout this k study, we assume that σ(tk ) = tk , k = 1, 2, . . . , i. e., each impulse point is right dense, and we use the notations φ+ = max{φ, 0}, n(t) := inf{k : tk ≥ t} and n(t) := sup{k : tk < t}; also, for brevity, we shall denote the intervals as [a, b] instead of [a, b]𝕋 .

9.2 Main results In this section, we derive a de La Vallée Poussin-type inequality for Eq. (9.2). Below is our main result.

9 De La Vallée Poussin-type inequality for impulsive dynamic equations on time scales

� 297

Theorem 9.2. Let x ∈ Crd [a, ρ(b)] be a nontrivial solution of Eq. (9.2) satisfying the boundary conditions x(a) = x(b) = 0.

(9.3)

If x(t) ≠ 0 for t ∈ (a, b), then the inequality ρ(b)

n(b)

∫ (σ(s) − a)(b − σ(s))q+ (s)Δs + ∑ (tk − a)(b − tk )qk+ k=n(a)

a

ρ(b)

ρ(b)

+ max{ ∫ (σ(s) − a)p+ (s)Δs, ∫ (b − σ(s))p+ (s)Δs} a

a

n(b)

n(b)

k=n(a)

k=n(a)

+ max{ ∑ (tk − a)p+k , ∑ (b − tk )p+k } ≥ b − a

(9.4)

holds. Proof. For clarity, we define f (t) = p(t)x Δ + q(t)x σ and fk = pk x Δ (tk ) + qk x(tk ). Thus, Eq. (9.2) turns into x ΔΔ (t) = −f (t), { Δ △x (t) = −fk ,

t ≠ tk ; t = tk ,

k = 1, 2, . . .

(9.5)

Any solution of (9.5) satisfying the Dirichlet boundary conditions (9.3) is represented by ρ(b)

n(b)

x(t) = ∫ G(t, σ(s))f (s)Δs + ∑ G(t, tk )fk , k=n(a)

a

(9.6)

where G(t, τ) =

(τ − a)(b − t), 1 { b − a (t − a)(b − τ),

a ≤ τ < t; t≤τ≤b

is the Green function for Eq. (9.5) with the boundary conditions (9.3). Indeed, observe that (9.6) can be expanded as t

b

1 x(t) = {∫(b − t)(σ(s) − a)f (s)Δs − ∫(t − a)(b − σ(s))f (s)Δs b−a a

t

n(t)

n(b)

k=n(a)

k=n(t)

+ ∑ (tk − a)fk − ∑ (b − tk )fk }.

(9.7)

298 � S. Doğru Akgöl and A. Özbekler The delta differentiation of both sides of (9.7) with respect to t gives x Δ (t) = −

t

1 {∫(σ(s) − a)f (s)Δs b−a a

ρ(b)

n(t)

n(b)

k=n(a)

k=n(t)

− ∫ (b − σ(s))f (s)Δs + ∑ (tk − a)fk − ∑ (b − tk )fk }. t

If we denote t

ρ(b)

a

t

󵄨 󵄨 󵄨 󵄨 I(t) := ∫(σ(s) − a)󵄨󵄨󵄨f (s)󵄨󵄨󵄨Δs + ∫ (b − σ(s))󵄨󵄨󵄨f (s)󵄨󵄨󵄨Δs,

t ∈ [a, ρ(b)],

and n(t)

n(b)

k=n(a)

k=n(t)

S(t) := ∑ (tk − a)|fk | + ∑ (b − tk )|fk |,

t ∈ [a, ρ(b)],

then we can write 1 󵄨󵄨 Δ 󵄨󵄨 {I(t) + S(t)}. 󵄨󵄨x (t)󵄨󵄨 ≤ b−a By taking the delta derivative of I(t), one has 󵄨 󵄨 I Δ (t) = (2σ(t) − a − b)󵄨󵄨󵄨f (t)󵄨󵄨󵄨, which shows that the maximum occurs either at t = a or at t = ρ(b). Thus, we can write ρ(b)

ρ(b)

󵄨 󵄨 󵄨 󵄨 I(t) ≤ max{ ∫ (σ(s) − a)󵄨󵄨󵄨f (s)󵄨󵄨󵄨Δs, ∫ (b − σ(s))󵄨󵄨󵄨f (s)󵄨󵄨󵄨Δs}. a

a

On the other hand, we have n(b)

n(b)

k=n(a)

k=n(t)

n(b)

n(t)

k=n(a)

k=n(a)

S(t) − ∑ (tk − a)|fk | = ∑ (a + b − 2tk )|fk | and S(t) − ∑ (b − tk )|fk | = ∑ (2tk − a − b)|fk | for S(t). If tk < (a + b)/2 for tk ∈ [t, b) and tk > (a + b)/2 for tk ∈ [a, t), then both

(9.8)

9 De La Vallée Poussin-type inequality for impulsive dynamic equations on time scales

n(b)

∑ (a + b − 2tk )|fk |

� 299

(9.9)

k=n(a)

and n(b)

∑ (2tk − a − b)|fk |

(9.10)

k=n(a)

would be positive, but this is impossible since {tk } is increasing. Thus, at least one of (9.9) or (9.10) must be nonpositive. This yields n(b)

n(b)

k=n(a)

k=n(a)

S(t) ≤ max{ ∑ (tk − a)|fk |, ∑ (b − tk )|fk |}.

(9.11)

Now, observe that 󵄨 󵄨 󵄨󵄨 󵄨 󵄨 σ + 󵄨 Δ 󵄨 + 󵄨 Δ 󵄨󵄨f (t)󵄨󵄨󵄨 = 󵄨󵄨󵄨p(t)x (t) + q(t)x (t)󵄨󵄨󵄨 ≤ p (t)󵄨󵄨󵄨x (t)󵄨󵄨󵄨 + q (t)󵄨󵄨󵄨x(t)󵄨󵄨󵄨 and 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 |fk | = 󵄨󵄨󵄨pk x Δ (tk ) + qk x(tk )󵄨󵄨󵄨 ≤ p+k 󵄨󵄨󵄨x Δ (tk )󵄨󵄨󵄨 + qk+ 󵄨󵄨󵄨x(tk )󵄨󵄨󵄨. At this point, we need the following lemma: Lemma 9.1 (Mean value theorem [6]). Let φ be continuous on [a, b] and delta-differentiable on [a, b). Then, there exist some constants c, d ∈ [a, b) such that φΔ (c) ≤

φ(b) − φ(a) ≤ φΔ (d). b−a

Since x Δ ∈ PLCrd [a, b], there exists α ∈ [a, b] such that 󵄨 󵄨 󵄨 󵄨 max 󵄨󵄨󵄨x Δ (τ)󵄨󵄨󵄨 = 󵄨󵄨󵄨x Δ (α)󵄨󵄨󵄨.

τ∈[a,b]

On the other hand, since x ∈ Crd [a, b] and x Δ exists for all x ∈ [a, ρ(b)], Lemma 9.1 applies. Hence, 󵄨󵄨 󵄨 󵄨 Δ 󵄨 󵄨󵄨x(τ)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨x (α)󵄨󵄨󵄨(τ − a)

󵄨 󵄨 󵄨 󵄨 and 󵄨󵄨󵄨x(τ)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨x Δ (α)󵄨󵄨󵄨(b − τ).

Let m(τ) := min{(τ − a), (b − τ)}, then clearly |x(τ)| ≤ m(τ)|x Δ (α)|. If we define ρ(b)

ρ(b)

a

a

1 (max{ ∫ (σ(s) − a)p+ (s)Δs, ∫ (b − σ(s))p+ (s)Δs} M1 := b−a

300 � S. Doğru Akgöl and A. Özbekler n(b)

n(b)

k=n(a)

k=n(a)

+ max{ ∑ (tk − a)p+k , ∑ (b − tk )p+k }) and ρ(b)

M2 :=

ρ(b)

1 (max{ ∫ (σ(s) − a)m(s)q+ (s)Δs, ∫ (b − σ(s))m(s)q+ (s)Δs} b−a a

a

n(b)

n(b)

k=n(a)

k=n(a)

+ max{ ∑ (tk − a)m(tk )qk+ , ∑ (b − tk )m(tk )qk+ }), then it follows from (9.8) and (9.11) that 1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 Δ 󵄨󵄨 󵄨 {󵄨I(t)󵄨 + 󵄨S(t)󵄨󵄨󵄨} ≤ M1 󵄨󵄨󵄨x Δ (α)󵄨󵄨󵄨 + M2 󵄨󵄨󵄨x Δ (α)󵄨󵄨󵄨, 󵄨󵄨x (α)󵄨󵄨 ≤ b−a 󵄨 󵄨 󵄨 or equivalently 1 ≤ M1 + M2 .

(9.12)

Since (b − τ)m(τ) ≤ (τ − a)(b − τ) and (τ − a)m(τ) ≤ (τ − a)(b − τ) for τ ∈ [a, b], we can write ρ(b)

n(b)

1 M2 ≤ ( ∫ (σ(s) − a)(b − σ(s))q+ (s)Δs + ∑ (tk − a)(b − tk )qk+ ). b−a k=n(a)

(9.13)

a

From (9.12) and (9.13), we obtain the dynamic de La Vallée Poussin-type inequality given by (9.4). To express some corollaries of the above theorem, we introduce the following definition. Definition 9.1. Equation (9.2) is said to be disconjugate on (a, b) if every nontrivial solution of it has at most one zero on (a, b). Corollary 9.1. Suppose, on the contrary to (9.4), that the inequality ρ(b)

n(b)

∫ (σ(s) − a)(b − σ(s))q+ (s)Δs + ∑ (tk − a)(b − tk )qk+ k=n(a)

a

ρ(b)

ρ(b)

+ max{ ∫ (σ(s) − a)p+ (s)Δs, ∫ (b − σ(s))p+ (s)Δs} a

a

n(b)

n(b)

k=n(a)

k=n(a)

+ max{ ∑ (tk − a)p+k , ∑ (b − tk )p+k } < b − a

9 De La Vallée Poussin-type inequality for impulsive dynamic equations on time scales

� 301

holds. Then, Eq. (9.2) is disconjugate on (a, b). Remark 9.1. If we take △x Δ = 0, Eq. (9.2) turns into the nonimpulsive dynamic equation x ΔΔ + p(t)x Δ + q(t)x σ = 0,

(9.14)

and hence the associated de La Vallée Poussin inequality reduces to ρ(b)

∫ (σ(s) − a)(b − σ(s))q+ (s)Δs a ρ(b)

ρ(b)

+ max{ ∫ (σ(s) − a)p+ (s)Δs, ∫ (b − σ(s))p+ (s)Δs} ≥ b − a. a

(9.15)

a

Since p+ (s) ≤ |p(s)| and q+ (s) ≤ |q(s)|, inequality (9.15) is better than the inequality given in [8]. Thus, (9.15) is the best possible de La Vallée Poussin inequality for the dynamic equations of the form (9.14).

9.3 Examples In this section, we provide some examples that support the above results. Example. Let 𝕋 = ℙ1,1 = ⋃∞ n=0 [2n, 2n+1], and consider the following impulsive dynamic equation on the time scale ℙ1,1 : πt Δ ΔΔ σ {x + cos( 2 )x − tx = 0, { Δ x Δ {△x − x + (3t)3 = 0,

t ≠ 2k; t = 2k,

k = 1, 2, . . . , 5,

(9.16)

with the boundary conditions x(0) = x(9) = 0. Since p(t) = cos(πt/2) is positive on the intervals (0, 1), (3, 5), (7, 9) and negative on (1, 3), (5, 7), we have p+ (t) = cos(πt/2) for t ∈ (0, 1) ∪ (3, 5) ∪ (7, 9) and p+ (t) = 0 for t ∈ (1, 3) ∪ (5, 7). For q(s), pk , and qk , it is easy to see that q+ (s) = 0 = p+k and qk+ = 1/(6k)3 . Thus, ρ(b)

I1 := ∫ (σ(s) − a)p+ (s)Δs 1

a

5

9

= ∫ s cos(πs/2)Δs + ∫ s cos(πs/2)Δs + ∫ s cos(πs/2)Δs 0

3

7

302 � S. Doğru Akgöl and A. Özbekler 1

9

5

= ∫ s cos(πs/2)ds + ∫ s cos(πs/2)ds + ∫ s cos(πs/2)ds 4

0

30 12 = − 2 < 8.4. π π

8

Similarly, ρ(b)

I2 := ∫ (b − σ(s))p+ (s)Δs 1

a

9

5

= ∫(9 − s) cos(πs/2)ds + ∫(9 − s) cos(πs/2)ds + ∫(9 − s) cos(πs/2)ds 4

0

24 12 = + < 8.9 π π2

8

and n(b)

4

9 − 2k 1245 < 0.1. = 2 15552 108k k=1

S1 := ∑ (tk − a)(b − tk )qk+ = ∑ k=n(a)

Since max{I1 , I2 } + S1 < 9 = b − a, and the remaining terms on the left-hand side of (9.4) are all zero, the inequality in Corollary 9.1 is satisfied, as opposed to Theorem 9.2. Thus, (9.16) is disconjugate on (0, 9). Example. Consider the impulsive dynamic equation x ΔΔ + 13 x Δ − x σ = 0, { Δ s3 Δ △x − t x + sin( πt4 ) = 0,

t ≠ 4k; t = 4k,

k∈ℕ

(9.17)

on 𝕋 = ℙ2,2 = ⋃∞ n=0 [4n, 4n + 2] satisfying the Dirichlet boundary conditions x(0) = x(b) = 0, where b > 0. Observe that p+ (s) = 1/s3 and q+ (s) = p+k = qk+ = 0. Let j be the largest integer satisfying ρ(b) ≥ 4j + 4. Then, ρ(b)

j

4n+2

ρ(b)

j

1 1 1 J1 := ∫ (σ(s) − a)p (s)Δs = ∑ ∫ 2 ds + ∑ + ∫ 2 ds 2 s (4n + 2) s n=0 n=0 +

a

4n

j

= ∑[ n=0

and

4j+4

j

1 1 1 1 ]+ ∑ + − 2 2n(4n + 2) 4j + 4 ρ(b) (4n + 2) n=0

9 De La Vallée Poussin-type inequality for impulsive dynamic equations on time scales

� 303

ρ(b)

J2 := ∫ (b − σ(s))p+ (s)Δs a

4n+2

j

ρ(b)

j

b−s b − (4n + 2) b−s ds + ∑ + ∫ ds = ∑ ∫ 3 3 s (4n + 2) s3 n=0 n=0 4n

j

j

= ∑[ n=0

+

4j+4

b(4n + 1) 1 b − (4n + 2) − ]+ ∑ 3 4n2 (4n + 2)2 2n(4n + 2) n=0 (4n + 2)

b b 1 1 − − + . (4j + 4)2 (ρ(b))2 4j + 4 ρ(b)

Letting b → ∞ and thus, j → ∞, it is clearly seen that limt→∞ J1 < ∞ and limt→∞ J2 < ∞ which imply that max{J1 , J2 } < ∞. Thus, by Corollary 9.1, the dynamic impulsive Eq. (9.17) is disconjugate on (0, ∞). In other words, (9.17) is nonoscillatory. Example. Consider the impulsive dynamic equation x ΔΔ − 2x Δ + x σ = 0, t ≠ k; { Δ △x + 2(1 + cosh 1)x = 0, t = k,

k∈ℕ

(9.18)

on 𝕋 = ℝ with the boundary conditions x(

e e ) = x(1 + ) = 0. e+1 e+1

(9.19)

By direct computation, it can be shown that xk (t) = (−1)k et−k+1 [(1 + e)(k − t) − 1],

t ∈ (k − 1, k]

is a solution of (9.18) with the boundary conditions (9.19). Moreover, it is not hard to see that xk (t) > 0 if x ∈ (e/(e + 1), 1 + e/(e + 1)) which means that (9.18) is disconjugate on (e/(e + 1), 1 + e/(e + 1)). Thus, Theorem 9.2 applies, i. e., the inequality (9.4) must hold. Since p+ (t) = 0, q+ (t) = 1, p+k = 0 and qk+ = 2(1 + cosh 1), we have ρ(b)

∫ (σ(s) − a)(b − σ(s))q+ (σ(s))Δs a

1+e/(e+1)

=

∫ e/(e+1)

= and

1 6

(σ(s) −

e e )(1 + − σ(s))ds e+1 e+1

304 � S. Doğru Akgöl and A. Özbekler n(b)

1

k=n(a)

k=1

∑ (tk − a)(b − tk )qk+ = 2(1 + cosh 1) ∑ (tk −

e e )(1 + − t ) = 1. e+1 e+1 k

Thus, ρ(b)

n(b)

∫ (σ(s) − a)(b − σ(s))q+ (s)Δs + ∑ (tk − a)(b − tk )qk+ = a

k=n(a)

7 > b − a = 1. 6

Hence, the de La Vallée Poussin-type inequality (9.4) is indeed valid.

Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

R. P. Agarwal, M. Jleli and B. Samet, On de La Vallée Poussin-type inequalities in higher dimension and applications, Appl. Math. Lett., 86 (2018), 264–269. S. D. Akgöl, Asymptotic equivalence of impulsive dynamic equations on time scales, Hacet. J. Math. Stat., 52(2) (2023), 277–291. S. D. Akgöl and A. Özbekler, De La Vallée Poussin inequality for impulsive differential equations, Math. Slovaca, 71(4) (2021), 881–888. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Basel, 2001. M. Bohner and S. G. Georgiev, Multivariable Dynamic Calculus on Time Scales, Springer, Switzerland, 2016. İ. M. Erhan and S. G. Georgiev, Nonlinear Integral Equations on Time Scales, Nova Science Publishers, 2019. R. A. C. Ferreira, A de La Vallée Poussin Type Inequality on Time Scales, Results Math., 73(3) (2018), 88. R. A. C. Ferreira, Fract. Vallée Poussin Inequal., 22(3) (2019), 917–930. S. G. Georgiev, Integral Equations on Time Scales, Atlantis Studies in Dynamical Systems, Springer, New York, 2016. P. Hartman and A. Wintner, On an oscillation criterion of de La Vallée Poussin, Q. Appl. Math., 13 (1955), 330–332. S. Hilger, Analysis on Measure Chains – A Unified Approach to Continuous and Discrete Calculus, Results Math., 18 (1990), 18–56. B. Kaymakçalan, V. Lakshmikantham and S. Sivasundaram, Dynamic Systems on Measure Chains, Kluwer, Dordrecht, 1996. C. de La Vallée Poussin, Sur l’équation différentielle linéaire du second order. Détermination d’une intégrale par deux valuers assignés. Extension aux équasions d’ordre n, J. Math. Pures Appl., 8 (1929), 125–144.

Najlaa Jaddoa, Rezan Sevinik-Adıgüzel, and İnci M. Erhan

10 Divided and σ-divided differences on time scales Abstract: In this chapter, the divided differences and σ-divided differences on time scales are introduced. The Newton and σ-Newton interpolation polynomial are constructed. In addition, the Hermite interpolation polynomial on time scales is constructed by using the divided differences table. Examples are presented to illustrate the theoretical results.

10.1 Introduction The theory of time scales was commenced by Hilger in his PhD thesis [9]. He unified the continuous and discrete analysis [1, 10]. Afterwards it has been studied by many researchers (see [2, 4, 5] and the references therein). The development of the theory of time scales is still in progress. In particular, numerical analysis and numerical methods on time scales is one of the very recent subjects of interest [3, 6–8]. Polynomial interpolation is a very practical method to represent a discrete set of data by a polynomial. Lagrange interpolation is one of the most widely used techniques to construct a polynomial which interpolates a given function, or a given discrete data set of nodes and function values. Another approach to construct the interpolation polynomial is the Newton interpolation which is based on using divided differences. On the other hand, the polynomial interpolation problem extends further to constructing a polynomial which takes not only the given function values but also the derivative values at the interpolation points. This problem uses the Hermite interpolation polynomial. In a recent book [6], the authors investigated the interpolation problem on arbitrary time scales and defined Lagrange and σ-Lagrange interpolation polynomials, as well as Hermite and σ-Hermite interpolation polynomials, on time scales. In this study we intend to give the construction of Newton and σ-Newton interpolation polynomial by introducing the divided and σ-divided differences on arbitrary time scales. This approach provides an alternative way to compute the interpolation polynomial for a given set of data. Moreover, we propose an alternative and simpler way to

Najlaa Jaddoa, Department of Mathematics, Gazi University, 06500 Beşevler, Ankara, Turkey, e-mail: [email protected] Rezan Sevinik-Adıgüzel, Department of Mathematics, Atilim University, 06830 Incek, Ankara, Turkey, e-mail: [email protected] İnci M. Erhan, Department of Computer Engineering, Aydın Adnan Menderes University, 09010 Aydın, Turkey, e-mail: [email protected] https://doi.org/10.1515/9783111182971-010

306 � N. Jaddoa et al. write the Hermite interpolation polynomial using modified divided differences for an arbitrary time scale. The chapter is organized as follows. In Section 10.2, we propose the construction of divided differences on an arbitrary time scale and define the Newton interpolation polynomial via the divided difference table. In Section 10.3, we construct the σ-divided differences on an arbitrary time scale and the σ-Newton interpolation polynomial by using them. Section 10.4 contains the construction of the Hermite interpolation polynomial using modified divided differences on an arbitrary time scale.

10.2 Divided differences Let 𝕋 be a time scale with the forward jump operator σ and graininess function μ. Suppose that xi ∈ 𝕋, i ∈ {0, 1, . . . , n} are distinct points in the time scale and yi = f (xi ), i ∈ {0, 1, . . . , n} are given real numbers. The Newton interpolation polynomial is constructed in the following way. Using the notation f [x0 ] = f (x0 ) = y0 , f [x1 ] = f (x1 ) = y1 , . . ., f [xn ] = f (xn ) = yn , one can generate the polynomials Pn (x) recursively by taking P0 (x) = f (x0 ) = f [x0 ] and defining P1 (x) = P0 (x) + f [x0 , x1 ](x − x0 ) = f [x0 ] + f [x0 , x1 ](x − x0 ).

(10.1)

The next polynomial has the form P2 (x) = P1 (x) + f [x0 , x1 , x2 ](x − x0 )(x − x1 )

= f [x0 ] + f [x0 , x1 ](x − x0 ) + f [x0 , x1 , x2 ](x − x0 )(x − x1 ).

(10.2)

The nth degree polynomial is generalized as n

i−1

i=0

j=0

Pn (x) = ∑ f [x0 , . . . , xi ] ∏(x − xj ),

(10.3)

and is called the Newton interpolation polynomial, where f [x0 , . . . , xi ] represents the divided difference of order i. Here we take ∏−1 j=0 (x − xj ) = 1. In order to evaluate the divided differences f [x0 , . . . , xi ], for i = 0, 1, . . . , n, we first employ (10.1) and the interpolation conditions Pn (xi ) = f (xi ), leading to

for any n ∈ ℕ0 and i = 0, 1, . . . , n,

10 Divided and σ-divided differences on time scales

f [x0 , x1 ] =

� 307

f (x1 ) − f (x0 ) . x1 − x0

(10.4)

Next by using (10.2), we derive f [x0 , x1 , x2 ] =

f (x0 ) f (x1 ) f (x2 ) + + . (x0 − x1 )(x0 − x2 ) (x1 − x0 )(x1 − x2 ) (x2 − x0 )(x2 − x1 )

Substituting the identity f (x1 ) f (x1 ) f (x1 ) = − , (x1 − x0 )(x1 − x2 ) (x1 − x0 )(x0 − x2 ) (x1 − x2 )(x0 − x2 ) we deduce f [x0 , x1 , x2 ] =

(x1 ) − { f (xx2 )−f −x 2

1

f (x1 )−f (x0 ) } x1 −x0

x2 − x0

=

f [x1 , x2 ] − f [x0 , x1 ] . x2 − x0

(10.5)

Notice here that the form of f [x0 , x1 , x2 ] obtained in terms of f [x1 , x2 ] and f [x0 , x1 ] provides an easy way of generating divided differences recursively. Then, the general nth divided difference can be derived recursively as follows: f [x0 , . . . , xn ] =

f [x1 , . . . , xn ] − f [x0 , . . . , xn−1 ] . xn − x0

(10.6)

The formula (10.6) allows us to generate all the divided differences needed for the Newton interpolation polynomial in a simple manner by using a divided difference table. We illustrate such a table for the case n = 4 (see Table 10.1). Table 10.1: Divided difference table. xk

f (xk )

x�

f (x� )

x�

f (x� )

x�

f (x� )

x�

f (x� )

x�

f (x� )

f [xk , xk+� ] f [x� , x� ] f [x� , x� ] f [x� , x� ] f [x� , x� ]

f [xk , xk+� , xk+� ]

f [x� , x� , x� ] f [x� , x� , x� ] f [x� , x� , x� ]

f [xk , xk+� , xk+� , xk+� ]

f [x� , x� , x� , x� ] f [x� , x� , x� , x� ]

f [xk , xk+� , xk+� , xk+� , xk+� ]

f [x� , x� , x� , x� , x� ]

The divided differences in the table are calculated columnwise by using the formula f [xi , . . . , xi+k ] =

f [xi+1 , . . . , xi+k ] − f [xi , . . . , xi+k−1 ] , xi+k − xi

(10.7)

308 � N. Jaddoa et al. for i, depending on n and k. The coefficients used in the Newton interpolation polynomial are the first entries in each column. In the next theorem we prove that the Newton interpolation polynomial defined in (10.3) interpolates the given set of points (xi , yi ) for i = 0, 1, . . . , n. In the following, we use the notation y[x0 , x1 , . . . , xk ] instead of f [x0 , x1 , . . . , xk ], k = 0, 1, . . . , n, for the divided differences. Theorem 10.1. Consider the polynomial Pn (x) = y[x0 ] + y[x0 , x1 ](x − x0 ) + ⋅ ⋅ ⋅ + y[x0 , x1 , . . . , xn ](x − x0 ) ⋅ ⋅ ⋅ (x − xn−1 ), where y[x0 , . . . , xk ] =

y[x1 , . . . , xk ] − y[x0 , . . . , xk−1 ] , xk − x0

k = 1, . . . , n.

Then Pn (x) interpolates the points (xi , yi ) for i = 0, 1, . . . , n. Proof. We will prove the theorem by using induction. For n = 1, the Newton polynomial is P1 (x) = y[x0 ] + y[x0 , x1 ](x − x0 ). Clearly, P1 (x0 ) = y[x0 ] = y0 and P1 (x1 ) = y[x0 ] + y[x0 , x1 ](x1 − x0 ) = y[x1 ] = y1 gives y[x0 , x1 ] =

y[x1 ] − y[x0 ] , x1 − x0

which indeed is the form of y[x0 , x1 ]. Assume that the hypothesis holds for n = k, that is, Pk (x) = y[x0 ] + ⋅ ⋅ ⋅ + y[x0 , . . . , xk ](x − x0 ) ⋅ ⋅ ⋅ (x − xk−1 ) interpolates the points (xi , yi ) for i = 0, 1, . . . , k. Consider the polynomial qk (x) = y[x1 ] + y[x1 , x2 ](x − x1 ) + ⋅ ⋅ ⋅ + y[x1 , . . . , xk+1 ](x − x1 ) ⋅ ⋅ ⋅ (x − xk ). It is obvious that the polynomial qk interpolates the points (xi , yi ) for i = 1, . . . , k + 1. Define the (k + 1)th degree polynomial

10 Divided and σ-divided differences on time scales

r(x) =

� 309

(x − x0 )qk (x) + (xk+1 − x)Pk (x) . xk+1 − x0

Then for xi , i = 1, 2, . . . , k, we have (xi − x0 )qk (xi ) + (xk+1 − xi )Pk (xi ) xk+1 − x0 (xi − x0 )yi + (xk+1 − xi )yi = xk+1 − x0 (x − x0 )yi = k+1 xk+1 − x0

r(xi ) =

= yi .

For i = 0, we have r(x0 ) =

0.qk (x0 ) + (xk+1 − x0 )Pk (x0 ) = Pk (x0 ) = y0 , xk+1 − x0

and, for i = k + 1, r(xk+1 ) = (xk+1 − x0 )qk (xk+1 ) = qk (xk+1 ) = yk+1 . Thus, rk is a polynomial of degree k + 1 which interpolates the points (xi , yi ) for i = 0, 1, . . . , k + 1 and hence, rk = Pk+1 . The leading coefficient, that is, the coefficient of the highest degree term x k+1 of Pk+1 , is then y[x1 , . . . , xk+1 ] − y[x0 , . . . , xk ] , xk+1 − x0 which completes the proof.

10.3 σ-Divided differences In order to define the σ-divided differences, we need to give the definition of σ-distinct points on a time scale. Definition 10.1 ([6]). Let n ∈ ℕ0 . The points xj ∈ [a, b), j ∈ {0, 1, . . . , n}, will be called σ-distinct if σ(xn ) ≤ b and σ(x0 ) < σ(x1 ) < ⋅ ⋅ ⋅ < σ(xn ). Suppose that xi ∈ 𝕋, i ∈ {0, 1, . . . , n} are σ-distinct points on a time scale and yi = f (xi ), i ∈ {0, 1, . . . , n} are given real numbers. We will define the σ-Newton interpolation polynomial via σ-divided differences.

310 � N. Jaddoa et al. Using the notation fσ [x0 ] = f (x0 ) = y0 , fσ [x1 ] = f (x1 ) = y1 , . . ., fσ [xn ] = f (xn ) = yn , the polynomials Pσn (x) can be generated recursively as follows. Let Pσ0 (x) = f (x0 ) = fσ [x0 ], and define Pσ1 (x) = Pσ0 (x) + fσ [x0 , x1 ](σ(x) − σ(x0 )) = fσ [x0 ] + fσ [x0 , x1 ](σ(x) − σ(x0 )).

(10.8)

Then, Pσ2 (x) = Pσ1 (x) + fσ [x0 , x1 , x2 ](σ(x) − σ(x0 ))(σ(x) − σ(x1 )) = fσ [x0 ] + fσ [x0 , x1 ](σ(x) − σ(x0 ))

+ fσ [x0 , x1 , x2 ](σ(x) − σ(x0 ))(σ(x) − σ(x1 )).

(10.9)

The general formula for Pσn (x) is generalized as n

i−1

i=0

j=0

Pσn (x) = ∑ fσ [x0 , . . . , xi ] ∏(σ(x) − σ(xj )),

(10.10)

where we take ∏−1 j=0 (σ(x) − σ(xj )) = 1. The notation fσ [x0 , . . . , xi ] will be referred to as σ-divided difference of order i. Definition 10.2. The polynomial given by the formula (10.10) will be called σ-Newton interpolation polynomial. Remark 10.1. Note that if, on a given time scale the forward jump operator σ is a nonlinear function, the σ-Newton interpolation polynomial may not be a polynomial of degree n. Therefore, in [6] such functions are called σ-polynomials, since they are polynomials in σ(x) but not in x. The σ-divided differences fσ [x0 , . . . , xi ] used in the formula (10.10) are evaluated recursively below. From (10.8) we have fσ [x0 , x1 ] =

f (x0 ) f (x1 ) − f (x0 ) f (x1 ) + = , σ(x0 ) − σ(x1 ) σ(x1 ) − σ(x0 ) σ(x1 ) − σ(x0 )

and from (10.9) we can calculate fσ [x0 , x1 , x2 ] =

f (x2 )−f (x1 ) σ(x2 )−σ(x1 )

−

f (x1 )−f (x0 ) σ(x1 )−σ(x0 )

σ(x2 ) − σ(x0 )

=

fσ [x1 , x2 ] − fσ [x0 , x1 ] . σ(x2 ) − σ(x0 )

The form of fσ [x0 , x1 , x2 ] indicates an easy way of generating σ-divided differences recursively. The general form can be derived by the following formula:

10 Divided and σ-divided differences on time scales

fσ [x0 , . . . , xk ] =

� 311

fσ [x1 , . . . , xk ] − fσ [x0 , . . . , xk−1 ] , σ(xk ) − σ(x0 )

for k = 1, 2, . . . , n. This formula allows us to generate all the σ-divided differences needed for the σ-Newton interpolation polynomial in a simple manner by using a divided difference table. In Table 10.2, we show the σ-divided differences for n = 4. The calculation is done columnwise using the recursion formula fσ [xi , . . . , xi+k ] =

fσ [xi+1 , . . . , xi+k ] − fσ [xi , . . . , xi+k−1 ] , σ(xi+k ) − σ(xi )

for i depending on n, and k. The coefficients of the σ-Newton interpolation polynomial appear at the top of each column. Table 10.2: σ-Divided difference table. σ(xk )

f (xk )

σ(x� )

f (x� )

σ(x� )

f (x� )

σ(x� )

f (x� )

σ(x� )

f (x� )

σ(x� )

f (x� )

fσ [xk , xk+� ] fσ [x� , x� ] fσ [x� , x� ] fσ [x� , x� ] fσ [x� , x� ]

fσ [xk , xk+� , xk+� ]

fσ [x� , x� , x� ]

fσ [xk , xk+� , xk+� , xk+� ]

fσ [x� , x� , x� , x� ]

fσ [x� , x� , x� ]

fσ [x� , x� , x� , x� ]

fσ [x� , x� , x� ]

fσ [xk , xk+� , xk+� , xk+� , xk+� ]

fσ [x� , x� , x� , x� , x� ]

Theorem 10.2. Consider the σ- polynomial Pσn (x) = yσ [x0 ] + yσ [x0 , x1 ](σ(x) − σ(x0 ))

+ ⋅ ⋅ ⋅ + yσ [x0 , x1 , . . . , xn ](σ(x) − σ(x0 )) ⋅ ⋅ ⋅ (σ(x) − σ(xn−1 )),

(10.11)

where yσ [x0 , . . . , xk ] =

yσ [x1 , . . . , xk ] − yσ [x0 , . . . , xk−1 ] . σ(xk ) − σ(x0 )

Then Pσn (x) interpolates the σ-distinct points {x0 , . . . , xn }, {y0 , . . . , yn }. Proof. We consider the proof by induction. For n = 1, we have Pσ1 (x0 ) = yσ [x0 ] + yσ [x0 , x1 ](σ(x) − σ(x0 )) = yσ [x0 ] and

(10.12)

312 � N. Jaddoa et al. Pσ1 (x1 ) = yσ [x0 ] + yσ [x0 , x1 ](σ(x1 ) − σ(x0 )) = yσ [x1 ], provided that yσ [x0 , x1 ] =

yσ [x1 ] − yσ [x0 ] . σ(x1 ) − σ(x0 )

Let n = k and assume that Pσk (x) = yσ [x0 ] + ⋅ ⋅ ⋅ + yσ [x0 , . . . , xk ](σ(x) − σ(x0 )) ⋅ ⋅ ⋅ (σ(x) − σ(xk−1 )) interpolates the points {x0 , . . . , xk }, {y0 , . . . , yk }. Let qσk (x) = yσ [x1 ] + yσ [x1 , x2 ](σ(x) − σ(x1 ))

+ ⋅ ⋅ ⋅ + yσ [x1 , . . . , xk+1 ](σ(x) − σ(x1 )) ⋅ ⋅ ⋅ (σ(x) − σ(xk )).

(10.13)

Clearly, qσk interpolates the points {x1 , . . . , xk+1 }, {y1 , . . . , yk+1 }. Define rσ (x) =

(σ(x) − σ(x0 ))qσk (x) + (σ(xk+1 ) − σ(x))pσk (x) σ(xk+1 ) − σ(x0 )

.

It can be easily verified that rσ interpolates the σ-distinct points {x1 , . . . , xk+1 }, {y1 , . . . , yk+1 }. Indeed, we have rσ (xi ) =

(σ(xi ) − σ(x0 ))qσk (xi ) + (σ(xk+1 ) − σ(xi ))pσk (xi )

σ(xk+1 ) − σ(x0 ) (σ(xi ) − σ(x0 ))yi + (σ(xk+1 ) − σ(xi ))yi = σ(xk+1 ) − σ(x0 ) = yi .

For i = 0, we get rσ (x0 ) =

0.qσk (x0 ) + (σ(xk+1 ) − σ(x0 ))Pσk (x0 ) σ(xn+1 ) − σ(x0 )

= Pσk (x0 ) = y0 .

For i = k + 1, we have rσ (xk+1 ) = (σ(xk+1 ) − σ(x0 ))qσk (xk+1 ) = qσk (xk+1 ) = yk+1 . Then rσ is a σ-polynomial of degree (k + 1) which interpolates the σ-distinct points {x0 , . . . , xk+1 }, {y0 , . . . , yk+1 }. Hence, rσ = Pσ,k+1 and its leading coefficient is yσ [x1 , . . . , xk+1 ] − yσ [x0 , . . . , xk ] , σ(xk+1 ) − σ(x0 ) which completes the proof.

10 Divided and σ-divided differences on time scales

� 313

Next, we present some examples of σ-divided differences on different time scales and the resulting σ-Newton interpolation polynomial. We also compute the σ-Lagrange interpolation polynomial as described in [8]. Example. Let 𝕋 = {√2n + 1 : n = 0, 1, 2, . . .} and x0 = 1,

y0 = −2,

x1 = √5,

x2 = √11,

y1 = 0,

y2 = 4.

We will compute both σ-Lagrange and σ-Newton interpolation polynomials and verify that they are equal. On the given time scale 𝕋 = {√1, √3, √5, . . .} we have σ(x) = √x 2 + 2, so that σ(x0 ) = σ(1) = √3, σ(x1 ) = σ(√5) = √7,

σ(x2 ) = σ(√11) = √13.

The σ-divided differences are computed in the following Table 10.3. Table 10.3: The divided difference table for the first example. x

σ(x)

f (x)

�

√�

−�

√� √��

√�

�

√��

f [xk , xk+� ] �−(−�) √�−√� �−� √��−√�

�

=

=

� √�−√�

f [xk , xk+� , xk+� ]

� − � √��−√� √�−√�

√��−√�

� √��−√�

Then the σ-Newton interpolation polynomial can be derived as Nσ2 (x) = f (x) + f [x0 , x1 ](σ(x) − σ(x0 ))

+ f [x0 , x1 , x2 ](σ(x) − σ(x0 ))(σ(x) − σ(x1 ))

= −2+

2 (σ(x) − √3) + √7 − √3

4 √13−√7

−

2 √7−√3

√13 − √3

(σ(x) − √3)(σ(x) − √7),

which is simplified as Nσ2 (x) = 1.056 σ 2 (x) − 2.435 σ(x) − 0.951.

(10.14)

To verify the uniqueness of an interpolation polynomial, we compute the σ-Lagrange interpolation polynomial which is defined in [8]. Following the construction procedure presented in [8], we obtain

314 � N. Jaddoa et al. (σ(x) − σ(x1 ))(σ(x) − σ(x2 )) (σ(x0 ) − σ(x1 ))(σ(x0 ) − σ(x2 )) (σ(x) − √7)(σ(x) − √13) = (√3 − √7)(√3 − √13)

Lσ0 (x) =

σ 2 (x) + (−√13 − √7)σ(x) + √91 , 3 − √39 − √21 + √91 (σ(x) − σ(x0 ))(σ(x) − σ(x2 )) Lσ1 (x) = (σ(x1 ) − σ(x0 ))(σ(x1 ) − σ(x2 )) (σ(x) − √3)(σ(x) − √13) = (√7 − √3)(√7 − √13) =

σ 2 (x) + (−√13 − √3)σ(x) + √39 , 7 − √91 − √21 + √39 (σ(x) − σ(x0 ))(σ(x) − σ(x1 )) Lσ2 (x) = (σ(x2 ) − σ(x0 ))(σ(x2 ) − σ(x1 )) (σ(x) − √3)(σ(x) − √7) = (√13 − √3)(√13 − √7) =

=

σ 2 (x) + (−√7 − √3)σ(x) + √21 . 13 − √91 − √39 + √21

Then, the σ-Lagrange interpolation polynomial has the form Pσ2 (x) = y0 Lσ0 (x) + y1 Lσ1 (x) + y2 Lσ2 (x) = − 2Lσ0 (x) + 4Lσ2 (x)

σ 2 (x) + (−√13 − √7)σ(x) + √91 3 − √39 − √21 + √91 σ 2 (x) + (−√7 − √3)σ(x) + √21 +4 13 − √91 − √39 + √21

= −2

and simplifies as Pσ2 (x) = 1.056σ 2 (x) − 2.435 σ(x) − 0.951.

(10.15)

Observe that the σ-Lagrange polynomial (10.15) and σ-Newton polynomial (10.14) are equal. The next example presents a computation of a third-degree σ-interpolation polynomial. 1 , . . . , 31 , 21 , 1} and Example. Let 𝕋 = { n1 : n ∈ ℕ} = {. . . , n1 , n−1

1 , 16 1 y0 = , 272

x0 =

1 , 12 1 y1 = , 156

x1 =

1 , 10 1 y2 = , 110

x2 =

1 x3 = , 7 1 y3 = . 56

10 Divided and σ-divided differences on time scales

� 315

We will find the σ-Lagrange and the σ-Newton interpolating polynomials for the given points (xi , yi ), i = 0, 1, 2, 3. On the given time scale, the forward jump operator is σ(x) =

x , 1−x

so that we obtain 1 1 )= , 16 15 1 1 σ(x1 ) = σ( ) = , 12 11 1 1 σ(x2 ) = σ( ) = , 10 9 1 1 σ(x3 ) = σ( ) = . 7 6

σ(x0 ) = σ(

The σ-divided differences are computed in Table 10.4. Table 10.4: The divided difference table for the second example. x

σ(x)

f (x)

� ��

� ��

� ���

� ��

� ��

� �

� ��

� �

� �

f [xk , xk+� ] � � ��� − ��� � � − �� ��

� ���

� � ��� − ��� � � � − ��

� ���

� � �� − ��� � � �−�

� ��

=

=

=

f [xk , xk+� , xk+� ] ��� ����

�� ���

��� ����

�� ��� ��� − ���� � � − � ��

��� �� ���� − ��� � � − � ��

=

=

���� ����

��� ����

f [xk , xk+� , xk+� , xk+� ]

��� ���� ���� − ���� � � � − ��

=

−��� ���

From Table 10.4, the σ-Newton polynomial follows easily as Nσ3 (x) = f (x) + f [x0 , x1 ](σ(x) − σ(x0 ))

+ f [x0 , x1 , x2 ](σ(x) − σ(x0 ))(σ(x) − σ(x1 ))

+ f [x0 , x1 , x2 , x3 ](σ(x) − σ(x0 ))(σ(x) − σ(x1 ))(σ(x) − σ(x2 ))

=

1 257 1 2531 1 1 + (σ(x) − ) + (σ(x) − )(σ(x) − ) 272 2279 15 5646 15 11 −986 1 1 1 + (σ(x) − )(σ(x) − )(σ(x) − ), 843 15 11 9

316 � N. Jaddoa et al. which becomes Nσ3 (x) =

−986 3 1284 2 314 7 σ (x) + σ (x) + σ(x) − . 843 1745 12951 16371

For the σ-Lagrange interpolation polynomial, we first compute the σ-Lagrange basis polynomials as in [6, 8]: Lσ0 (x) = = Lσ1 (x) = = Lσ2 (x) = = Lσ3 (x) = =

(σ(x) − σ(x1 ))(σ(x) − σ(x2 ))(σ(x) − σ(x3 )) (σ(x0 ) − σ(x1 ))(σ(x0 ) − σ(x2 ))(σ(x0 ) − σ(x3 )) 1 )(σ(x) − 91 )(σ(x) − 61 ) 11 , ( 151 − 111 )( 151 − 91 )( 151 − 61 )

(σ(x) −

(σ(x) − σ(x0 ))(σ(x) − σ(x2 ))(σ(x) − σ(x3 )) (σ(x1 ) − σ(x0 ))(σ(x1 ) − σ(x2 ))(σ(x1 ) − σ(x3 )) 1 )(σ(x) − 91 )(σ(x) − 61 ) 15 , ( 111 − 151 )( 111 − 91 )( 111 − 61 )

(σ(x) −

(σ(x) − σ(x0 ))(σ(x) − σ(x1 ))(σ(x) − σ(x3 )) (σ(x2 ) − σ(x0 ))(σ(x2 ) − σ(x1 ))(σ(x2 ) − σ(x3 )) 1 )(σ(x) − 111 )(σ(x) − 61 ) 15 , ( 91 − 151 )( 91 − 111 )( 91 − 61 )

(σ(x) −

(σ(x) − σ(x0 ))(σ(x) − σ(x1 ))(σ(x) − σ(x2 )) (σ(x3 ) − σ(x0 ))(σ(x3 ) − σ(x1 ))(σ(x3 ) − σ(x2 )) 1 )(σ(x) − 111 )(σ(x) − 91 ) 15 . ( 61 − 151 )( 61 − 111 )( 61 − 61 )

(σ(x) −

This results in the σ-Lagrange polynomial Pσ3 (x) = f (x0 )Lσ0 (x) + f (x1 )Lσ1 (x) + f (x2 )Lσ2 (x) + f (x3 )Lσ3 (x) =

1 1 1 1 (σ(x) − 11 )(σ(x) − 9 )(σ(x) − 6 ) [ ] 272 ( 151 − 111 )( 151 − 91 )( 151 − 61 )

+ + +

1 1 1 1 (σ(x) − 15 )(σ(x) − 9 )(σ(x) − 6 ) [ ] 156 ( 111 − 151 )( 111 − 91 )( 111 − 61 )

1 1 1 1 (σ(x) − 15 )(σ(x) − 11 )(σ(x) − 6 ) [ ] 110 ( 91 − 151 )( 91 − 111 )( 91 − 61 ) 1 1 1 1 (σ(x) − 15 )(σ(x) − 11 )(σ(x) − 9 ) [ ], 56 ( 61 − 151 )( 61 − 111 )( 61 − 91 )

simplifying into Pσ3 (x) =

−986 3 1284 2 314 7 σ (x) + σ (x) + σ(x) − . 843 1745 12951 16371

10 Divided and σ-divided differences on time scales

� 317

The equality of the two σ-polynomials is obvious, which demonstrates the consistency with the theoretical results.

10.4 Hermite interpolation polynomial via divided differences on time scales In this section, the construction of a modified divided difference table is discussed, which simplifies the computation of the Hermite interpolation polynomial considerably. Let 𝕋 be a time scale with the forward jump operator σ, delta derivative Δ, and graininess function μ. Let x0 , x1 , . . . , xn be given σ-distinct points such that xj ≠ σ(xi ) for i, j ∈ {0, . . . , n}. Let y0 , y1 , . . . , yn and z0 , z1 , . . . , zn be the points such that yi = f (xi ), zi = f Δ (xi ) for some f . The idea is based on doubling the number of points in the following way. Define t2k = xk ,

t2k+1 = σ(xk ),

k = 0, . . . , n.

Then y(t2k ) = yk ,

y(t2k+1 ) = y(σ(xk )),

k = 0, . . . , n,

where y(σ(xi )) are in general unknown. The first divided differences for t0 , . . . , t2n will be defined as yk ,

y[ti ] = {

wk ,

i = 2k,

i = 2k + 1,

where wk is not given if the function itself is unknown. The second divided differences are defined as y[tj , tj+1 ] =

y(tj+1 ) − y(tj ) tj+1 − tj

,

j = 0, . . . , 2n.

Now for j = 2k we have y(t2k+1 ) − y(t2k ) t2k+1 − t2k y(σ(xk )) − y(xk ) = = yΔ (xk ) = zk σ(xk ) − xk

y[t2k , t2k+1 ] =

and y[t2k−1 , t2k ] =

y(xk ) − y(σ(xk−1 )) xk − σ(xk−1 )

318 � N. Jaddoa et al. y(xk ) − y(xk−1 ) − (y(σ(xk−1 )) − y(xk−1 )) xk − σ(xk−1 ) y(xk ) − y(xk−1 ) y(σ(xk−1 )) − y(xk−1 ) σ(xk−1 ) − xk−1 = − xk − σ(xk−1 ) σ(xk−1 ) − xk−1 xk − σ(xk−1 ) σ(xk−1 ) − xk−1 y(xk ) − y(xk−1 ) − zk−1 . = xk − σ(xk−1 ) xk − σ(xk−1 ) =

The other divided differences are computed in the usual way. Then, Table 10.5 enables us to construct the Hermite interpolation polynomial P2k+1 (x) = y[t0 ] + y[t0 , t1 ](x − x0 ) + y[t0 , t1 , t2 ](x − x0 )(x − σ(x0 )) + ⋅ ⋅ ⋅ + y[t0 , . . . , t2n+1 ](x − x0 )(x − σ(x0 )) ⋅ ⋅ ⋅ (x − xn ).

(10.16)

Since P2k+1 (x) interpolates the points (tj , f (tj )) for j = 0, 1, . . . , 2k, we have P2k+1 (xi ) = P2k+1 (t2i ) = yi ,

P2k+1 (σ(xi )) = P2k+1 (t2i+1 ) = wi , so that Δ P2k+1 (xi ) =

P2k+1 (σ(xi )) − P2k+1 (xi ) wi − yi f (σ(xi )) − f (xi ) = = = f Δ (xi ). σ(xi ) − xi σ(xi ) − xi σ(xi ) − xi

Therefore, the values of the polynomial in (10.16) and its Δ-derivative coincide with the values of the function and its delta derivative at the given points. In the examples, the divided difference table is employed in the construction of the Hermite interpolation polynomial. Table 10.5: Divided difference table for Hermite interpolation polynomial. tk

y[tk ]

t� = x�

y�

t� = σ(x� )

w�

t� = x�

y�

y[tk , tk+� ] z� y[t� , t� ]

y[tk , tk+� , tk+� ]

y[t� , t� , t� ] y[t� , t� , t� ]

z� t� = σ(x� )

t� = x� .. . t�k+� = σ(xk )

w�

y� .. . wk

y[t� , t� ] .. .

y[t� , t� , t� ] .. .

y[tk , tk+� , tk+� , tk+� ]

y[t� , t� , t� , t� ] y[t� , t� , t� , t� ] .. .

y[tk , tk+� , tk+� , tk+� , tk+� ]

y[t� , t� , t� , t� , t� ] .. .

10 Divided and σ-divided differences on time scales

� 319

1 , and x0 = 1, x1 = 4 be given points on 𝕋. We will Example. Let 𝕋 = ℤ, f (x) = x+5 compute the Hermite polynomial in two different ways and compare the results. Note that on this time scale we have σ(x) = x + 1 and hence,

f Δ (x) =

f (x + 1) − f (x) −1 = . 1 (x + 5)(x + 6)

Then, it follows that x0 = 1,

σ(x0 ) = 2,

x1 = 4,

1 y0 = , 6 −1 z0 = , 42

σ(x1 ) = 5,

1 , 9 −1 z1 = . 90

y1 =

We first compute the divided difference table for the given points (see Table 10.6). Table 10.6: Divided difference table for the first example. tk

y(tk )

t� = �

� �

t� = σ(x� ) = �

� � � �

t� = � t� = σ(x� ) = �

� ��

y[tk , tk+� ] � � �−�

�−�

� � �−�

�−�

� � �� − �

�−�

=

−� ��

y[tk , tk+� , tk+� ]

−� � �� + ��

�−�

=

−� ��

−� � �� + ��

�−�

=

= =

� ��� � ���

y[tk , tk+� , tk+� , tk+� ]

� � ��� − ���

�−�

=

−� ���

�

=

−� ����

−� ��

This enables us to write easily the Hermite interpolation polynomial as 1 1 1 1 − (x − 1) + (x − 1)(x − 2) − (x − 1)(x − 2)(x − 4) 6 42 378 3780 1 1 1 1 1 = − x+ + (x 2 − 3x + 2) − (x 3 − 7x 2 − 10x − 8) 6 42 42 378 3780 1 1 1 1 2 3 2 1 3 7 2 10 8 = − x+ + x − x+ − x + x + x+ 6 42 42 378 378 378 3780 3780 3780 3780 −1 3 17 2 67 187 = x + x − x+ . 3780 3780 1890 945

P3 =

The Hermite interpolation polynomial can also be computed using the formulation given in [6]. First, we compute

320 � N. Jaddoa et al. x − x1 x − 4 −1 = = (x − 4), x0 − x1 1−4 3 x − x0 x−1 1 = = (x − 1), L1 (x) = x1 − x0 4 − 1 3 x − σ(x1 ) x − 5 −1 M0 (x) = = = (x − 5), x0 − σ(x1 ) 1 − 5 4 x − σ(x0 ) x−2 1 = = (x − 2), M1 (x) = x1 − σ(x0 ) 4 − 2 2 −1 LΔ0 (x0 ) = , 3 1 LΔ1 (x1 ) = , 3 −1 M0Δ (x0 ) = , 4 1 M1Δ (x1 ) = , 2 −1 2 L0 (σ(x0 )) = L0 (2) = (2 − 4) = , 3 3 1 4 L1 (σ(x1 )) = L1 (5) = (5 − 1) = , 3 3 −1 3 M0 (σ(x0 )) = M0 (2) = (2 − 5) = , 4 4 1 3 M1 (σ(x1 )) = M1 (5) = (5 − 2) = . 2 2 L0 (x) =

Then, the Hermite interpolation polynomial has the form P3 = [(1 −

−1 4

+ [(1 −

+ 43 . −1 3

3 2 . 4 3 1 + 32 . 31 2 3 4 . 2 3

1 −1 −1 (x − 1)) + 3422 (x − 1)] (x − 4) (x − 5) 6 3 4 . 4 3 −1

1 1 1 (x − 4)) + 3904 (x − 4)] (x − 1) (x − 2) 9 3 2 . 2 3 −1

1 −1 −1 1 = [(1 + (x − 1)) + (x − 1)] (x − 1) (x − 2) 6 21 3 2 1 1 −1 1 1 + [(1 − (x − 4)) + (x − 4)] (x − 1) (x − 2) 2 9 180 3 2 −1 3 17 2 67 187 = x + x − x+ . 3780 3780 1890 945

Notice that the polynomials obtained in the two different ways are identical. Example. Let 𝕋 = 2ℕ , f (x) = x1 , and x0 = 1, x1 = 4. The Hermite interpolation polynomial is constructed by the two methods and the results are compared. First, note that on the given time scale the forward jump operator is σ(x) = 2x and we evaluate

10 Divided and σ-divided differences on time scales

f Δ (x) =

� 321

f (2x) − f (x) −1 = 2. x 2x

Then x0 = 1,

x1 = 4,

1 , 4 −1 z1 = , 32 σ(4) = 8.

y0 = 1,

y1 =

−1 , 2 σ(1) = 2, z0 =

The modified divided difference table is derived as follows (see Table 10.7). Table 10.7: Divided difference table for the second example. tk

y(tk )

t� = �

�

t� = σ(x� ) = �

� �

t� = �

� �

t� = σ(x� ) = �

� �

y[tk , tk+� ] � � −� �−� � � �−�

�−�

� � �−�

�−�

=

−� �

y[tk , tk+� , tk+� ]

−� � � +�

�−�

=

−� �

−� � �� + �

�−�

=

= =

� � � ��

y[tk , tk+� , tk+� , tk+� ]

� � �� − �

�−�

=

−� ��

�

=

−� ��

−� ��

Using this table, we can get the Hermite polynomial 1 1 1 P3 = 1 − (x − 1) + (x − 1)(x − 2) − (x − 1)(x − 2)(x − 4) 2 8 64 1 1 1 2 1 3 = 1 − x + + (x − 3x + 2) − (x − 7x 2 − 10x − 8) 2 2 8 64 1 1 1 2 3 1 1 7 10 1 = 1 − x + + x − x + − x3 + x2 + x + 2 2 8 8 4 64 64 64 8 15 −1 3 15 2 23 = x + x − x+ . 64 64 32 8 Considering the general form of the Hermite interpolation polynomial defined in [6], we also compute x − x1 x − 4 −1 = = (x − 4), x0 − x1 1−4 3 x − x0 x−1 1 L1 (x) = = = (x − 1), x1 − x0 4 − 1 3

L0 (x) =

322 � N. Jaddoa et al. x − σ(x1 ) x − 8 −1 = = (x − 8), x0 − σ(x1 ) 1 − 8 7 x − σ(x0 ) x−2 1 M1 (x) = = = (x − 2), x1 − σ(x0 ) 4 − 2 2 −1 LΔ0 (x0 ) = , 3 1 LΔ1 (x1 ) = , 3 −1 M0Δ (x0 ) = , 7 1 M1Δ (x1 ) = , 2 −1 −1 2 L0 (σ(x0 )) = L0 (2) = (2 − 4) = (−2) = , 3 3 3 1 7 L1 (σ(x1 )) = L1 (8) = (8 − 1) = , 3 3 −1 6 M0 (σ(x0 )) = M0 (2) = (2 − 8) = , 7 7 1 M1 (σ(x1 )) = M1 (8) = (8 − 2) = 3, 2 M0 (x) =

which yields the Hermite polynomial as P3 = [(1 −

−1 7

+ [(1 − = [(1 −

−3 7 4 7

+ 67 . −1 3

6 2 . 7 3 1 + 3. 31 2 3. 73

(x − 1)).1 + 1 (x − 4)) + 4

(x − 1)) + 3 2

−1 −1 −1 2 (x − 1)] (x − 4) (x − 8) 6 2 3 7 . 7 3 −1 1 1 32 (x − 4)] (x − 1) (x − 2) 3 2 3. 73

−7 −1 −1 (x − 1)] (x − 4) (x − 8) 8 3 7

−1 1 1 (x − 4)] (x − 1) (x − 2) 7 224 3 2 −1 3 15 2 23 15 = x + x − x+ . 64 64 32 8 + [(1 −

(x − 4)) +

Clearly, both polynomials are the same.

10.5 Conclusion In this chapter, we introduced the divided and σ-divided differences on time scales. They provide an alternative form for the interpolation polynomial for a given data set. Moreover, the modified divided differences make the construction of the Hermite interpolation polynomial very simple. As a further study, one can consider the deriva-

10 Divided and σ-divided differences on time scales

� 323

tion of a modified σ-divided differences table to be used to simplify the construction of the σ-Hermite interpolation polynomials, which is also complicated in its present form given in [6].

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Index σ-distinct points 309 σ-divided differences 310 σ-Newton interpolation polynomial 310 backward jump operator 78 Bennett–Leindler inequalities 260 Bennett–Leindler-type 259 border projector 71 Cauchy integral 80 chain rule 82 consistent initial values 60 continuous Bennett–Leindler inequalities 260, 261 de La Vallée Poussin inequality 295 delta antiderivative 221 delta Bennett–Leindler inequalities 261 delta derivative 78, 221 diamond alpha 259 diamond alpha Bennett–Leindler inequalities 264 diamond alpha Bennett–Leindler type dynamic inequalities 268 diamond alpha derivative 267 diamond alpha Hölder’s inequality 268 diamond alpha integral 267 diamond alpha reverse Hölder’s inequality 268 diamond alpha time scale calculus 267 differentiable function 56 Dirichlet boundary conditions 295 disconjugate 300 discrete Bennett–Leindler inequalities 260 discrete Gehring weights 91, 93 discrete Muckenhoupt class 89 discrete Muckenhoupt weights 91 discrete time scale 81 divided difference 307 dynamic de La Vallée Poussin-type inequality 300 fiber 56 forward jump operator 78 fractional sum 193 Gehring inequality 112, 115 Gehring’s weights 87 Green function 297 Hadamard difference 194, 195 Hardy–Littlewood maximal operator 91 https://doi.org/10.1515/9783111182971-011

Hardy’s inequality 119, 259 Hardy’s inequality on time scales 136 harmonic numbers 80 Hermite interpolation polynomial 318 higher integrability theorems 130 Hilbert’s inequality 259 Hölder’s inequality 82, 83 Hölder’s inequality in two dimensional 84 hybrid fixed point theorem 204 integration by parts formula 81 Jensen’s inequalities 82 (k, m)-jet of a function of n independent real variables and one independent time scale variable 53 k-jet of a function of one independent time scale variable 51 left-dense 78 left-scattered 78 Leibnitz formula 240 linearization 67 logarithm function 178 Lorentz’s space 89 mean value theorem 299 Minkowski inequality 82, 84 Muckenhoupt and Gehring weights on time scales 94 Muckenhoupt weights on time scales 97 Muckenhoupt’s weights 85 nabla Bennett–Leindler inequalities 261 Newton interpolation polynomial 306 nonoscillatory 303 product and quotient rules for the derivative 79 properly involved derivative 58 properly stated leading term 58 reverse Hardy–Copson inequalities 260 reverse Hölder’s inequality 83 right-dense 78 right-dense continuous 79 right-scattered 78 self-improving properties 108

326 � Index

Taylor formula 44, 45, 240 Taylor formula of order (k, k) for a function of n independent real variables and one independent time scale variable 51 Taylor formula of order (k, m) for a function of n independent real variables and one independent time scale variable 51 TH3a 138

time scale 77, 83, 220 time scale monomials 239 Ulam stability 150, 157 vertical space 56 Volterra integral equation of the first kind 252 Volterra integral equation of the second kind 256