JOURNAL OF APPLIED FUNCTIONAL ANALYSIS 10 (2015).pdf

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Volume 10, Numbers 1-2

January-April 2015

ISSN:1559-1948 (PRINT), 1559-1956 (ONLINE) EUDOXUS PRESS,LLC

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS

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SCOPE AND PRICES OF

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS A quartely international publication of EUDOXUS PRESS,LLC ISSN:1559-1948(PRINT),1559-1956(ONLINE) Editor in Chief: George Anastassiou Department of Mathematical Sciences The University of Memphis Memphis, TN 38152,USA E mail: [email protected] Assistant to the Editor: Dr.Razvan Mezei,Lenoir-Rhyne University,Hickory,NC 28601, USA. --------------------------------------------------------------------------------

The purpose of the "Journal of Applied Functional Analysis"(JAFA) is to publish high quality original research articles, survey articles and book reviews from all subareas of Applied Functional Analysis in the broadest form plus from its applications and its connections to other topics of Mathematical Sciences. A sample list of connected mathematical areas with this publication includes but is not restricted to: Approximation Theory, Inequalities, Probability in Analysis, Wavelet Theory, Neural Networks, Fractional Analysis, Applied Functional Analysis and Applications, Signal Theory, Computational Real and Complex Analysis and Measure Theory, Sampling Theory, Semigroups of Operators, Positive Operators, ODEs, PDEs, Difference Equations, Rearrangements, Numerical Functional Analysis, Integral equations, Optimization Theory of all kinds, Operator Theory, Control Theory, Banach Spaces, Evolution Equations, Information Theory, Numerical Analysis, Stochastics, Applied Fourier Analysis, Matrix Theory, Mathematical Physics, Mathematical Geophysics, Fluid Dynamics, Quantum Theory. Interpolation in all forms, Computer Aided Geometric Design, Algorithms, Fuzzyness, Learning Theory, Splines, Mathematical Biology, Nonlinear Functional Analysis, Variational Inequalities, Nonlinear Ergodic Theory, Functional Equations, Function Spaces, Harmonic Analysis, Extrapolation Theory, Fourier Analysis, Inverse Problems, Operator Equations, Image Processing, Nonlinear Operators, Stochastic Processes, Mathematical Finance and Economics, Special Functions, Quadrature, Orthogonal Polynomials, Asymptotics, Symbolic and Umbral Calculus, Integral and Discrete Transforms, Chaos and Bifurcation, Nonlinear Dynamics, Solid Mechanics, Functional Calculus, Chebyshev Systems. Also are included combinations of the above topics. Working with Applied Functional Analysis Methods has become a main trend in recent years, so we can understand better and deeper and solve important problems of our real and scientific world. JAFA is a peer-reviewed International Quarterly Journal published by Eudoxus Press,LLC. We are calling for high quality papers for possible publication. The contributor should submit the contribution to the EDITOR in CHIEF in TEX or LATEX double spaced and ten point type size, also in PDF format. Article should be sent ONLY by E-MAIL [See: Instructions to Contributors] Journal of Applied Functional Analysis(JAFA)

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Journal of Applied Functional Analysis Editorial Board Associate Editors Editor in-Chief: George A.Anastassiou Department of Mathematical Sciences The University of Memphis Memphis,TN 38152,USA 901-678-3144 office 901-678-2482 secretary 901-751-3553 home 901-678-2480 Fax [email protected] Approximation Theory,Inequalities,Probability, Wavelet,Neural Networks,Fractional Calculus

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J. APPLIED FUNCTIONAL ANALYSIS, VOL. 10, NO.'S 1-2,13-18 ,2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

SOME RESULTS CONCERNING SYSTEM OF MODULATES FOR L2 (R) A. Zothansanga1 and Suman Panwar2 1,2 Department of Mathematics, University of Delhi, Delhi 110 007. INDIA 1 e-mail:[email protected] 2 e-mail: [email protected]

Abstract: Frames by integer translates were studied by Kim and Kwon [6]. Christensen et.al [3] studied Riesz sequences of translates and their generalized duals. In this paper we consider a system of modulates for L2 (R). A necessary and sufficient condition for a system of modulates for L2 (R) to be a frame sequence (Riesz sequence,orthonormal sequence) is given. We also prove a result related to the frame operator of a frame sequence {Ek ϕ}k∈Z . Finally, it is proved that for n ≥ 1, the system of modulates {Ek Bn }k∈Z is a Riesz Fischer sequence. AMS Subject Classification 2010: 42C15, 42C30 Key Words: Frame, system of modulates, Bessel sequence, Riesz-Fischer sequence.

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Introduction A basic approach to the decomposition of a signal in terms of elementary signals

was given by Dennis Gabor [7] in 1946. While answering some deep problems in nonharmonic Fourier series, Duffin and Schaeffer [5] in 1952 abstracted Gabor’s method and defined frames for Hilbert spaces. Frames are widely used in mathematics, science and engineering. In particular they are used in filter banks, wireless sensor networks, multiple-antenna code design, signal processing, image processing and many more. For a nice and comprehensive survey on various types of frames, one may refer to [1,2,4,8,10]. Coherent frames {fk }k∈I are the frames for which the elements fk appear by action of some operators (belonging to a special class) on a single element f in a Hilbert space. This feature is necessary for applications because it simplifies calculations and makes it simpler to collect information about the frame. In this paper, we consider the case where the operators act by modulation. A necessary and sufficient condition for a system of modulates for L2 (R) to be a frame sequence(Riesz sequence,orthonormal) is given. We also prove a result related to the frame operator of a frame sequence {Ek ϕ}k∈Z . Finally, it is proved that for n ≥ 1, the system of modulates {Ek Bn }k∈Z is a Riesz Fischer sequence.

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A. ZOTHANSANGA AND SUMAN PANWAR

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Preliminaries

Definition 2.1. [4] Let H denote a Hilbert space, I denotes a countable index set and [fi ]i∈I denotes the closed linear span of the sequence {fi }i∈I . A family of vectors {fi }i∈I is called a frame for H if there exist constants A and B with 0 < A ≤ B < ∞ such that A∥f ∥2 ≤

(1)

∑

|⟨f, fi ⟩|2 ≤ B∥f ∥2 ,

for all f ∈ H.

i∈I

The positive constants A and B are called lower frame bound and upper frame bound for the family {fi }i∈I , respectively. The inequality (1) is called the frame inequality. If in (1) only the upper inequality holds, then {fi }i∈I is called a Bessel sequence. If the family of vectors {fi }i∈I is a frame for [fi ]i∈I , then it is called a frame sequence. Definition 2.2. [4]A family of vectors {fi }i∈I is called a Riesz basis for H if it is complete in H and if there exist constants A and B with 0 < A ≤ B < ∞ such that ∑ ∑ ∑ (2) A |ci |2 ≤ ∥ ci fi ∥2 ≤ B |ci |2 . i∈I

i∈I

i∈I

If the family of vectors {fi }i∈I satisfies (2) , then it is called a Riesz sequence and if it satisfies the left hand inequality of (2), then it is called a Riesz-Fischer sequence. For a ∈ R and ϕ ∈ L2 (R), the modulation operator Ea on L2 (R) is defined as Ea ϕ(x) = e2πiax ϕ(x) , x ∈ R and the dilation operator Da , a ̸= 0 on L2 (R) is 1 defined as Da ϕ(x) = ϕ( x ), x ∈ R. |a| a Definition 2.3. Let ϕ ∈ L2 (R) . The sequence {Ek ϕ}k∈Z is called a system of modulates for L2 (R). Further, (I) If {Ek ϕ}k∈Z is a frame for L2 (R), then it is called a frame of modulates. (II) If {Ek ϕ}k∈Z is a Bessel sequence for L2 (R), then it is called a Bessel sequence of modulates. For a system of modulates {Ek ϕ}k∈Z , the associated pre-frame operator is T : ∑ l (Z) → L2 (R), given by T {αk }k∈Z = αk Ek ϕ. The adjoint operator of T or 2

k∈Z

the analysis operator T ∗ : L2 (R) → l2 (Z), is given by T ∗ {f } = {⟨f, Ek ϕ⟩}k∈Z , f ∈ L2 (R. If the system of modulates {Ek ϕ}k∈Z is a Bessel sequence, then by composing T and T ∗ , we obtain the frame operator S : L2 (R) → L2 (R) defined as Sf = ∑ ⟨f, Ek ϕ⟩Ek ϕ, f ∈ L2 (R). k∈Z

If the system of modulates {Ek ϕ}k∈Z is a frame, then the frame operator S is

invertible.

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SOME RESULTS CONCERNING SYSTEM OF MODULATES FOR L2 (R)

Definition 2.4. For n ∈ N, the B-splines Bn are defined inductively as B1 = 1 ∫2 χ[− 12 , 12 ] and Bn+1 = Bn ∗ B1 , that is Bn+1 (x) = Bn (x − t)dt. − 12

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Main Results We give necessary and sufficient condition for the system of modulates {Ek ϕ}k∈Z , ϕ ∈

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L (R), to be a frame sequence, Riesz sequence or an orhonormal sequence in terms of a function Φ associated with ϕ. Theorem 3.1. For ϕ ∈ L2 (R) consider the function Φ on R defined as Φ(x) = ∑ |ϕ(x + k)|2 . Consider the set N = {x ∈ [0, 1], Φ(x) = 0}. Then for A, B > 0, k∈Z

the following characterizations hold: (1) {Ek ϕ}k∈Z is a frame sequence with bounds A and B if and only if A ≤ Φ(x) ≤ B, a.e x ∈ [0, 1] − N. (2) {Ek ϕ}k∈Z is a Riesz sequence with bounds A and B if and only if A ≤ Φ(x) ≤ B, a.e x ∈ [0, 1]. (3) {Ek ϕ}k∈Z is an orthonormal sequence if and only if Φ(x) = 1, a.e x ∈ [0, 1]. Proof. (1) Let {ck }k∈Z ∈ l2 (Z) and {Ek ϕ}k∈Z be a Bessel sequence with the preframe operator T . Using Theorem 3.2 in [9], we have ∫1 ∑ ∥T {ck }k∈Z ∥ = | ck e2πikx |2 Φ(x)dx. 2

(3)

0

k∈Z

Let the Kernal of T be denoted as Ker(T). Then using Equation (3) { } ∑ 2 2πikx Ker(T ) = {ck }k∈Z ∈ l (Z), ck e = 0, for all x ∈ [0, 1] − N. k∈Z ⊥

Let {ck }k∈Z ∈ Ker(T ) .Then ⟨{ck }k∈Z , {dk }k∈Z ⟩ = 0, for all{dk }k∈Z ∈ Ker(T ). ⊥

As {Ek }k∈Z is an orthonormal basis for L2 (0, 1), {ck }k∈Z ∈ Ker(T ) if and only if ∫1 ∑

ck e2πikx

0 k∈Z

Hence, ⊥

Ker(T ) =

∑

dk e2πikx dx = 0.

k∈Z

{ } ∑ {ck }k∈Z ∈ l2 (Z), ck e2πikx = 0, for all x ∈ N. k∈Z ⊥

⊥

Using the definition of Ker(T ) , for all {ck }k∈Z ∈ Ker(T ) , we have ∫ ∑ ∑ (4) |ck |2 = A | ck e2πikx |2 dx k∈Z

[0,1]−N

k∈Z

16

A. ZOTHANSANGA AND SUMAN PANWAR

Now using equation (3) and (4)and Lemma 5.4.5 in [4], {Ek ϕ}k∈Z is a frame sequence with lower bound A if and only if ∫ ∑ A | ck e2πikx |2 dx ≤ [0,1]−N

k∈Z

∫ | [0,1]−N

∑

ck e2πikx |2 ϕ(x)dx.

k∈Z

which is equivalent to A ≤ Φ(x) a.e x ∈ [0, 1] − N. Also,{Ek ϕ}k∈Z is a frame sequence with upper bound B if and only if ∫ ∫ ∑ ∑ 2πikx 2 B | ck e | dx ≥ | ck e2πikx |2 ϕ(x)dx. [0,1]−N

k∈Z

[0,1]−N

k∈Z

which is equivalent to B ≥ Φ(x) a.e x ∈ [0, 1] − N. Hence, we obtain A ≤ Φ(x) ≤ B a.e x ∈ [0, 1] − N. (2) Let {ck }k∈Z ∈ l2 (Z). Then using Theorem 3.2 in [9] , we have ∫1 ∑ ∥T {ck }k∈Z ∥ = | ck e2πikx |2 Φ(x)dx and 2

(5)

k∈Z

0

∑

(6)

∫1 ∑ ck e2πikx |2 dx. |ck | = | 2

k∈Z

0

k∈Z

Also, {Ek ϕ}k∈Z is a Riesz sequence with bounds A and B if and only if ∑ ∑ (7) |ck |2 . |ck |2 ≤ ∥T {ck }k∈Z ∥2 ≤ B A k∈Z

k∈Z

Using (5) and (6) in (7) we get that {Ek ϕ}k∈Z is a Riesz sequence with bounds A and B if and only if ∫1 ∑ ∫1 ∑ ∫1 ∑ 2πikx 2 2πikx 2 ck e | Φ(x)dx ≤ B | ck e2πikx |2 dx. ck e | dx ≤ | A | 0

k∈Z

0

k∈Z

0

k∈Z

which implies that A ≤ Φ(x) ≤ B, a.e x ∈ [0, 1]. (3) Let {ck }k∈Z ∈ l2 (Z). We know that {Ek ϕ}k∈Z is an orthonormal sequence if and only if (8)

∑

|ck |2 ≤ ∥T {ck }k∈Z ∥2 ≤

k∈Z

∑

|ck |2 .

k∈Z

Thus, {Ek ϕ}k∈Z is an orthonormal sequence with bounds A and B if and only if ∫1 ∑ ∫1 ∑ ∫1 ∑ 2πikx 2 2πikx 2 | ck e | dx ≤ | ck e | Φ(x)dx ≤ | ck e2πikx |2 dx. 0

k∈Z

This proves (3).

0

k∈Z

0

k∈Z



17

SOME RESULTS CONCERNING SYSTEM OF MODULATES FOR L2 (R)

From general frame theory we know that if for ϕ ∈ L2 (R), {Ek ϕ}k∈Z is a frame sequence. Then, for all f ∈ [Ek ϕ]k∈Z , we have ∑ f= ⟨f, S −1 Ek ϕ⟩Ek ϕ. k∈Z

In order to apply the above frame decomposition, we should be able to calculate S −1 Ek ϕ for all k ∈ Z. The following theorem helps to simplify this process by proving that instead of calculating S −1 Ek ϕ for all k ∈ Z, it is sufficient to calculate S −1 ϕ. Theorem 3.2. For ϕ ∈ L2 (R) assume that {Ek ϕ}k∈Z is a frame sequence. Then, for all f ∈ [Ek ϕ]k∈Z , (1) SEk f = Ek Sf , for all k ∈ Z. (2) S −1 Ek f = Ek S −1 f , for all k ∈ Z. where S denotes the frame operator for [Ek ϕ]k∈Z . Proof. (1) Given f ∈ [Ek ϕ]k∈Z and k ∈ Z, we have SEk f = =

∑

⟨Ek f, Ek′ ϕ⟩Ek′ ϕ.

k′ ∈Z

∑

k′ ∈Z

⟨f, Ek′ −k ϕ⟩Ek′ ϕ.

letting k ′ −→ k ′ + k, we finally obtain SEk f = Ek =

∑ k′ ∈Z

⟨f, Ek′ ϕ⟩Ek′ ϕ Ek Sf.

(2) Since {Ek ϕ}k∈Z is a frame sequence, it is a frame for [Ek ϕ]k∈Z . Therefore, the operator S is invertible. Therefore, the proof of (2) follows from (1).

4



Applications We now prove that the B-spline B1 is a Riesz sequence with Riesz bound 1.

Theorem 4.1. For B1 = χ[ −1 , 1 ] , {Ek B1 }k∈Z is a Riesz sequence with Riesz bound 2

2

1. 1

Proof. Note that for k ∈ Z, ∥Ek B1 ∥2 =

∫2

|e2πikx |2 dx.

− 12 1

For k ̸= j, ⟨Ek B1 , Ej B1 ⟩ =

∫2

e2πi(k−j)x dx.

− 12

Thus, {Ek B1 }k∈Z is an orthonormal sequence and therefore in view of Theorem 3.1, the result follows.  As an important outcome of Theorem 3.1, we now prove that the integer modulates of any B-spline form a Riesz-Fischer sequence.

18

A. ZOTHANSANGA AND SUMAN PANWAR

Theorem 4.2. For n ≥ 1, {Ek Bn }k∈Z is a Riesz Fischer sequence with lower bound 1. Proof. By Theorem 3.1, for n ≥ 1, {Ek Bn }k∈Z is a Riesz Fischer sequence with lower bound 1 if and only if 1 ≤ Φ(x), x ∈ [0, 1], where Φ(x) =

∑

|Bn (x + k)|2 .

k∈Z

By corollary 6.2.1 in [4], we have ∑ Bn (x + k) = 1, for all x ∈ R. k∈Z

Now, (9)

1=|

∑

Bn (x + k)|2 ≤

k∈Z

∑

|Bn (x + k)|2 , for all x ∈ R.

k∈Z

Since {Ek B1 }k∈Z is a Riesz sequence with Riesz bound 1, by Theorem 3.1 we have ∑ (10) |B1 (x + k)|2 = 1, a.e x ∈ [0, 1]. k∈Z

Using (9) and (10), we have 1≤

∑

|Bn (x + k)|2 , a.e x ∈ [0, 1].

k∈Z

Thus, for n ≥ 1, {Ek Bn }k∈Z is a Riesz Fischer sequence with lower bound 1.



Acknowledgement The research of the second author is supported by CSIR vide letter no. 09 \ 045(1140) \ 2011 -EMR I dated 16 \ 11 \ 2011.

References [1] P.G. Casazza, D. Han and D.R. Larson, Frames for Banach spaces, Contem. Math., 247 , 149 - 181, 1999. [2] P.G.Casazza, The art of frame theory, Taiwanese J. Math.4(2),129-201, 2000. [3] O. Christensen, O.H. Kim, R.Y. Kim and J.K. Lim, Riesz sequences of translates and their generalized duals, J. Geometruc Analysis. 16(4), 585-596, 2006. [4] O. Christensen, Frames and Bases: An Introductory Course, Birkhausher, Boston, 2008. [5] R.J. Duffin, A.C. Schaeffer, A class of non-harmonic Fourier series, Trans. Amer. Math. Soc. 72(2), 341-366, 1952. [6] J.M. Kim and K.H. Kwon, Frames by integer translates and their generalized duals, J. Korean society for Industrial and Applied Mathematics. 11(3), 1-5, 2007. [7] D.Gabor, Theory of communications, J.IEE(London), 93(3), 429-457, 1946. [8] C. Heil, A basis theory primer: Birkhausher, Boston, 2011. [9] Suman Panwar, On System of Modulates for L2 (R), Journal of Wavelet theory and Applications, 7(1), 69-75, 2013. [10] R.M. Young, An Introduction to Non-harmonic Fourier Series, Academic Press, New York, 1980 (Revised First Edition, 2001).

J. APPLIED FUNCTIONAL ANALYSIS, VOL. 10, NO.'S 1-2,19-30 ,2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

19

SOME DEFINITION OF A NEW INTEGRAL TRANSFORM BETWEEN ANALOGUE AND DISCRETE SYSTEMS S.K.Q. AL-OMARI Department of Applied Sciences, Faculty of Engineering Technology Al-Balqa Applied University, Amman 11134, Jordan [email protected] ABSTRACT The D transform is proclaimed as a new integral transform which states an equivalence between analogue and discrete systems [16]. In this article, we establish a convolution theorem of the D transform and give its de…nition in the space of generalized functions. The de…nition of the D transform of a distribution is a one to one and linear mapping. Over and above, we establish that the D transform of a Boehmian is a distribution and retain its classical properties. Keywords: D Transformation; Integral Transform; Analogue System; Discrete System; Distribution; Boehmian.

1

INTRODUCTION

The sampling techniques are based on the transformation x ! x; where x is an analogue function and x is the sequence x (n) = x (nT ) ;

(1)

T being a sampling period. Let x be an analogue signal which is casual; i.e. t < 0 ) x (t) = 0: Then the D transform of x is de…ned by [16] D (x ) (m) :=

Z1

x (t) e T

t

t T

m

m!

dt; m 2 N:

(2)

0

Given an analogue system which is characterized by an impulse response h then a discrete system is described by an impulse response D (h ) whose input signal is D (x ) ; where x is an input signal of the analogue system with output D (y ) given by D (x y ) = D (x ) D (x ) ; (3) where x

y (t) =

Zt

0

x ( ) y (t

)d :

(4)

20

2

S.K.Q. Al-Omari

It also clear that the cited transform extends possibilities concerning linear di¤erence and di¤erential equations as well. For, we recall the following example from the given citation. Example [16] Given a system which is described by the linear di¤erential equation L R

dy +y =x ; dt

where L and R are given as in Figure 2 of [16]: Then an equivalent discrete system under the D transform can be written as L R where

1 y (n)

=

y (n)

y (n T

1)

1y

+y =x

:

Hence, calculations then lead to the solution x (t) = (t) ; where represents a Dirac delta distribution. We enumerate certain properties of the D transform summarized as follows : (1) Linearity : D ( x + y ) = D (x ) + D (y ) : (2) Derivation : D

dn x dtn

=

n D (x n

(3) Integration : For f =

=

) ; where 1(

n 1 x)

:

dF we have dt

D (F) (m) = T

m X k=0

D (f ) (k) tT :F (0) :

(4) Sum and integral correspondence :

Z1

x (t) dt =

0

2

Pm

k=0 D (x

) (m) :

CONVOLUTION THEOREM OF D TRANSFORM

Denote by

the usual convolution product but extended in its upper limit. That is, (f

g) (t) =

Z1

f ( ) g (t

0

Then we establish the following theorem.

)d :

(5)

21

3

some de…nition of a new integral transform...

Theorem 1 ( Convolution Theorem) Let x and y be casual analogue signals. Then we have D (x

y ) (m) =

m X k=0

D (x ) (k) D (y ) (m

k) :

(6)

Proof Let x and y be arbitrary signals; then, by (5) and (1) ; we get that 0 1 Z1 Z1 t m t T @ A D (x y ) (m) = d : x ( ) y (t )d eT m! 0

0

Fubinitz theorem also gives D (x

y ) (m) =

Z1

x ( )

0

The change of variables, D (x

=t

y ) (m) =

Z1

Z1

y (t

x ( )

k

0 B @

Using the formula (m k )= D (x

dtd :

m!

Z1

y ( )e

+ T

+ T

m

d d :

m!

0

m m X ( ) k=0

m

; implies that

Hence, the binomial theorem, ( + )m = y ) (m) =

t T

0

0

D (x

)e

t T

m!

Z1

Pm

k! (m

m k=0 (k )

01 Z @ k)! x ( )e T 0 m k

y ( )e T

m k k;

T

(m

k)!

0

suggests 1 k

T

k!

1

C d A:

d A (7)

m! ; Equation (7) can be simpli…ed to mean (m k)!k!

y ) (m) =

m X k=0

D (x ) (k) D (y ) (m

k)

This completes the proof of the theorem.

3

GENERALIZED D TRANSFORM

In this section we discuss the D transform on a distribution space [3,10,13] of compact support and a quotient space of Boehmians [7]. To this end, reader is supposed to be aquainted with the concept of distribution spaces [17]. The minimal structure of Boehmians has been given in Section 3.2. of this note.

22

4

S.K.Q. Al-Omari

3.1

DISTRIBUTIONAL D TRANSFORM

Denote by E (R+ ) the space of smooth functions over R+ equipped with its usual topology [13,10] then for real values t; t > 0; we certainly get that eT

t

t T

m

m!

2 E R+ ;

(8)

T being the sampling period. 0 By (8), we de…ne the distributionl D transform of a distribution in E (R+ ) ; the space of distributions of compact support, as * + m V

D (x ) (m) =

x (t) ; e T

t

t T

m!

:

(9)

V

Reader can simply establish some properties of D transform which are listed in the following remark. 0

Remark 1 Let x 2 E (R+ ) be given; then the following are satis…ed in the distributional sense : V

(1) D is linear. V

(2) D is one to one. Proof is straightforward. Moreover, the generalized convolution of (5) can also be described by the inner product h(x y ) (t) ; ' (t)i = hx (t) ; hy ( ) ; ' (t + )ii ; V

where ' ( ) 2 E (R+ ) is arbitrary. Hence, the distributional D transform therefore acts on the convolution by the equation V

D (x

3.2

m V X V y ) (m) = D (x ) (k) D (y ) (m

k) :

(10)

k=0

GENERAL BOEHMIAN

The space of Boehmians as the youngest space of generalized functions consists of the following elements : A linear space Y and a subspace X of Y. To all pairs (f; ) ; (g; ), f; g 2 Y; ; 2 X; is assigned the products f ;g such that the following holds : (1) 2 X and = : (2) (f ) =f ( ): (3) (f + g) =f +g : (4) k (f ) = (kf ) = f (k ) ; k 2 R: Let be the collection of sequences from X satisfying : (5)If ( n ) 2 and f n = g n ,n = 1; 2; :::; then f = g;

23

5

some de…nition of a new integral transform...

(6) ( n ) ; ( n ) 2 ) ( n ; n) 2 Then elements of are called delta sequences or approximating identities. Let A be the class of pairs of sequences de…ned by n A = ((fn ) ; ( n )) : (fn )

o

YN ; ( n ) 2

;

8n 2 N:

Then the pair ((fn ) ; ( n )) 2 A is said to be quotient of sequences, fn

Quotients of sequences

fn

m

and

n

gn

n ; 8n; m

are equivalent,

m

= gm

gn

n

n

n ; 8n; m

; when

n

2 N:

fn

n

fn The operation classes.

= fm

fn

; if

2 N:

is an equivalent relation on A and hence, splits A into equivalence

The equivalence class containing

fn

fn

is denoted by

n

: These equivalence

n

classes are called Boehmians and the space of all Boehmians is denoted by B (Y; X; ; ) : The sum and multiplication by a scalar of two Boehmians are naturally de…ned in the respective ways : fn

+

gn

n

=

n

fn

n

+ gn

n

n

n

and fn

=

n

fn

;

being complex number:

n

The operations and Dk ; the k-th derivative, are de…ned by and Dk

fn n

=

Dk fn

fn

gn

n

n

=

fn gn n

n

:

n

Many a time, Y is equipped with a notion of convergence. The intrinsic relationship between the notion of convergence and the product are given by: (1) If fn ! f as n ! 1 in Y and, 2 X is any …xed element, then fn

!f

in Y as n ! 1:

(2) If fn ! f as n ! 1 in Y and ( n ) 2 ; then fn n ! f in Y as n ! 1: The operation is extended to B (Y; X; ; ) X by the following de…nition. More details of the construction of Boehmians are given in [1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15].

24

6

3.3

S.K.Q. Al-Omari

D TRANSFORM OF A BOEHMIAN

0

Denote by U the space of test functions of bounded support and U its conjugate space of distributions de…ned on R+ : By we mean the subset of U of all sequences of analogue signals ( i )1 0 in U such that : Z1 (i) i (t) dt = 1; 0

(ii) i (t) > 0; 8i 2 N; (iii) supp i (t) ! 0 as i ! 1: Then each sequence of such signals is called delta sequences which correspond to dirac delta function. In the next, let us be concerned with the Boehmian space B (E; U; ; ) where E is the group, U is the subgroup of E and, is the operation, then, with ; a typical element of B (E; U; ; ) is written in the form x

:=

i

;

i

where x i is an analogue sequence from E (R+ ) and i 2 : Sum, derivation, multiplication by scalars and the operation between Boehmians in B (E; U; ; ) are respectively de…ned in the natural way as : x

i

+

y

i

i i

dk x dxk x

i

x

=

i

and

x

=

+y

i i

2

i

i

;

i

3 dk 6 dxk x i 7 7; =6 4 5 i

i i

i

i

;

being complex number;

i

x

y

i i

i

=

x

i

y

i

i

i i

Theorem 2 Let (x i ) 2 E, ( i ) 2 be a pair of analogue sequences be given such x i 0 that = 2 B (E; U; ; ) : Then Dx i converges uniformly in U (R+ ) : i

Proof Let ' 2 U (R+ ) be arbitrary. Choose ( k0 ) 2 such that ( k0 ) (m k) > 0 x i on the support of '; 1 k m: Then the fact that represents a quotient i means x i (11) j =x j i ; i; j 2 N: Applying D transform to (11) yields m X k=0

Dx i (k) D

j

(m

k) =

m X k=0

Dx

j

(k) D

i (m

k) ;

(12)

25

7

some de…nition of a new integral transform...

from which we get Dx i (k) D

(m

j

k) = Dx

j

(k) D

i (m

k) :

(13)

Therefore, D D

Dx i (k) (') = Dx i (k) = Dx i D

(m k0 (m

(m

k0

k) ' k)

k0

k)

' k0 (m

D

:

k)

By (13) we write Dx i (k) (') = Dx

k0

= Dx

k0

(k) D D D

i (m

k)

i (m

k) k) k0 (m

But the fact that D

i (m

D

k) ! e T

t

t T

k!

' k0 (m

k)

:

(14)

;

(15)

k

as i ! 1; when inserted in (14) ; yields Dx i (k) (') ! where

:= e T

k

t T

t

' k0 (m

k! D

Once again, the fact that

k)

as i ! 1

:

is ofcourse smooth function with a property that supp

supp ' 0

implies that Dx i (k) de…nes a distribution in U (R+ ) : This completes the proof of the theorem. x

Theorem 3 Let

i

=

y

i

i

i

2 B (E; U; ; ) ; then Dx i (k) and Dy i (k) con-

0

verge to the same limit in U (R+ ). Proof Let

x

i

=

i

y

i

2 B (E; U; ; ) ; then de…ning a set A

i

(A )i = and B implies

A B

i i

=

x

i i

i

=

=

(

x

y

(

y

i i

i+1 2

i 2

i+1 2 i 2

:

; if i odd i ; if i even

i+1 2

2

; if i odd i ; if i even i+1 2

2

i

and B

i

by

26

8

S.K.Q. Al-Omari

Therefore DA

i

0

converges in U (R+ ) and lim DA

(2i 1) (')

i!1

= lim Dx i (') : i!1

Thus, DA i and Dx i converge to same limit. Similarly we proceed for DA i and Dy i . Hence the proof is completed. Now, by virtue of above, we introduce the D transform of the equivalence class x i 2 B (E; U; ; ) as i

x

D

i i

= lim Dx

(16)

i

i!1

on compact subsets of R+ : 0

Theorem 4 The transform D : B (E; U; ; ) ! U (R+ ) is well de…ned. x

Proof Considering the equation

i

y

=

i

we get, by the concept of quotients,

i

that x

i

i

j

=y

i:

i

(17)

Applying D transform to (17) then using the idea of (12) and (13) we get Dx i (k) D

j

(m

k) = Dy

(k) D

i (m

k) :

k) = Dy i (k) D

i (m

k) :

j

In particular, for i = j; we get Dx i (k) D

i (m

(18)

Considering limits as i ! 1; from (18) ; we …nd lim Dx i (k) = lim Dy i (k) :

i!1

i!1

Hence the theorem. 0

Theorem 5 The transform D : B (E; U; ; ) ! U (R+ ) is in…nitely smooth. Proof Let

x

i i

2 B (E; U; ; ) and K be an open bounded set of R+ then, there is

j 2 N such that

D

j

(m

k) > 0; 1

k

m; on K:

Hence, D

x

i

=

i

=

lim Dx i (k)

i!1

lim Dx i (k)

i!1

D D

(m j (m j

k) k)

27

9

some de…nition of a new integral transform...

By (13) we get x

D

i

=

lim

i!1

i

j

t T

k

t T

e

!

Dx

k!

(k) D i (m k) D j (m k) D

Dx j (k) on K: k) j (m

Last step follows from (15) : Since Dx j and Dx j are in E (R+ ) our result follows. This completes the proof of the theorem. Theorem 6 Let (x i ) ; (y i ) 2 E; ( i ) ; ( i ) 2 x i y i that ; 2 B (E; U; ; ) : Then i

be given sequences of signals such

i

x

D

y

i

i

i

=

e

i

t T

k

t T

k!

D

x

i

y

D

i

i

:

i

Proof of this theorem is a result of (15) and (16) : x

Theorem 7 Let

i

de…ne a Boehmian in B (E; U; ; ) then

i

x

D

i

(D

j ) (m

k) = e

i

t T

t T

k

Dx

k!

j

(k) ; 1

k

m:

Proof Let ' 2 U (R+ ) then (15) and (16) give x

D

i

(D

j ) (m

i

k) (') = lim Dx i (k) (D

i ) (m

i!1

k) (')

The equation in (13) then yields D

x

i

(D

j ) (m

k) (') =

i

lim Dx

j

t T

k

i!1

= e

t T

k!

(k) (D (Dx

j

i ) (m

k) (')

(k)) (') :

Hence the theorem. 0

Theorem 8 The transform D : B (E; U; ; ) ! U (R+ ) is continuous with respect to - convergence. Proof Let ( i ) 2 B (E; U; ; ) ; exists x

i;k

; (x k ) 2 E; (

2 B (E; U; ; ) be such that i)

2

; such that

n

=

x

i

!

i ;k i

: There can x k ; = i

and x i ;k ! x k for every k 2 N as i ! 1 in E: Continuity of D implies 0 D (x i;k ) ! D (x k ) as i ! 1 in U (R+ ) :

28

10

S.K.Q. Al-Omari

Thus, D i ! D as i ! 1: This completes the proof of the theorem. 0

Theorem 9 The transform D : B (E; U; ; ) ! U (R+ ) is linear. Proof Let k1 ; k2 2 R then we get x

D k1

i

y

+ k2

i

i

k1 x

=

i

+

k2 y

i

i

i i

That is x

D k1

i

+ k2

y

i

i

= k1 lim Dx i!1

i

+ k2 lim Dy

i

i!1

i

That is x

D k1

i

y

+ k2

i

i

x

= k1 D

i

i

+ k2 D

i

y

i i

This completes the proof of the theorem. 0

Theorem 10 The transform D : B (E; U; ; ) ! U (R+ ) is one - to - one. Proof Let

x

i

y

=

i

i

; then x

i

j

=y

i:

j

i

Applying D transform then

using (13) and considering limits for i = j, imply lim x Thus, D

x

i

=D

i

y

i

i

= lim y

as i ! 1.

i

:

i

Hence the proof. We state without proof the following theorems. Due simplicity of the proofs we prefer they be omitted. x

Theorem 11 D

i

i (m

i

x

Theorem 12 If D

i i

k)

= 0; then

=D x

i

x i

i i

;(

)i 2

.

= 0 in the sense of Boehmians of

i

B (E; U; ; ) : Theorem 13 If

n

is a sequence of Boehmians in B (E; U; ; ) such that

n ! 1; then D

n

0

n

!

as

! D as n ! 1 in U (R+ ) on compact subsets.

Con‡ict of Interests The author declare that there is no con‡ict of interests regarding the publication of this paper.

29

some de…nition of a new integral transform...

11

References [1] Mikusinski, P. (1987), Fourier transform for integrable Boehmians, Rocky Mountain Journal of Mathematics, 17(3), 577–582. [2] Al-Omari, S.K.Q., Loonker D., Banerji P.K. and Kalla, S.L. (2008). Fourier Sine(Cosine) Transform for Ultradistributions and their Extensions to Tempered and UltraBoehmian spaces, Integ. Trans. Spl. Funct. 19(6), 453 –462. [3] Banerji, P.K., Al-Omari, S.K.Q. and Debnath, L.(2006). Tempered Distributional Fourier Sine(Cosine)Transform, Integ. Trans. Spl. Funct.17(11),759-768. [4] S.K.Q.Al-Omari and Kilicman, A. (2012). Note on Boehmians for Class of Optical Fresnel Wavelet Transforms, Journal of Function Spaces and Applications, Volume 2012, Article ID 405368, doi:10.1155/2012/405368. [5] Boehme, T.K. (1973), The Support of Mikusinski Operators, Tran.Amer. Math. Soc.176,319-334. [6] Mikusinski, P.(1983). Convergence of Boehmians, Japan. J. Math. (N.S.) 9 , 159–179. [7] S.K.Q.Al-Omari and Kilicman, A. On Generalized Hartley-Hilbert and Fourier-Hilbert Transforms, Advances in Di¤erence Equations 2012, 2012:232 doi:10.1186/1687-1847-2012-232. [8] Mikusinski, P. (1995), Tempered Boehmians and Ultradistributions, Proc. Amer. Math. Soc. 123(3), 813-817. [9] Roopkumar, R.(2009). Mellin transform for Boehmians, Bull. Inst. Math., Academica Sinica, 4(1), pp. 75–96. [10] Zemanian, A.H. (1987). Generalized Integral Transformation, Dover Publications, Inc., New York. First published by interscience publishers, New York. [11] Al-Omari, S.K.Q.(2011). Generalized Functions for Double Sumudu Transformation, Intern.J. Algeb., 6(3), 139 - 146. [12] S. K. Q. Al-Omari and A. Kilicman (2012), On Di¤raction Fresnel Transforms for Boehmians, Abstract and Applied Analysis, Volume 2011, Article ID 712746. [13] Pathak, R. S. (1997). Integral transforms of Generalized Functions and their Applications, Gordon and Breach Science Publishers, Australia , Canada, India, Japan. [14] Roopkumar, R.(2007). Stieltjes Transform for Boehmians, Integral Transf. Spl. Funct.18(11), 845-853. [15] Al-Omari, S.K.Q. (2011). Notes for Hartley Transforms of Generalized Functions , Italian J. Pure Appl. Math. N.28, 21-30.

30

12

S.K.Q. Al-Omari

[16] J.Hekrdla (1988), New Integral Transform Between Analogue and Discrete Systems. Electronic Letters 24(10),620-623. [17] Zemanian, A.H. (1987). Distribution theory and transform analysis, Dover Publications, Inc., New York. First Published by McGraw-Hill, Inc. New York (1965).

J. APPLIED FUNCTIONAL ANALYSIS, VOL. 10, NO.'S 1-2,31-39 ,2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

SOME FIXED POINT THEOREMS FOR ITERATED CONTRACTION MAPS Santosh Kumar Department of Mathematics College of Natural and Applied Sciences University of Dar es salaam P.O.Box-35062 Tanzania. E. Mail: [email protected] ABSTRACT The study of iterated contraction was initiated by Rheinboldt in 1969. The concept of iterated contraction proves to be very useful in the study of certain iterative process and has wide applicability. In this paper a brief introduction of iterated contraction maps is given and some fixed point results are discussed. 1. INTRODUCTION In 1969, Rheinboldt [4] gave unified convergence theory for a class of iterative processes. He initiated the process of finding solution of the nonlinear operator equation F x = 0 using metric fixed point theory. Rheinboldt developed a general convergence theory for iterative processes of the form xk+1 = Gxk , k = 0, 1, ...; and founded on certain nonlinear estimates for the iteration function G as well as on a so called concept of majorizing sequences, His new approach reduces the study of the iterative process to that of a second order nonlinear difference equation. 0

2000 Mathematics Subject Classification: 47H10, 54H25. Key words and phrases: Nonexpansive mappings, Iteration process, Fixed points of nonexpansive map,Compact map, Iterated contraction map.

31

SOME FIXED POINT THEOREMS FOR ITERATED CONTRACTION MAPS Santosh Kumar

32

The purpose of this paper is to establish yet another fixed point theorem for iterated contractions maps in different settings. Our results extends several results contained in [1], [2], [6] etc. 2. PRELIMINARIES If f is a contractive mapping, then under certain conditions the sequence xn+1 = fxn tends to the unique fixed-point of f . Ortega, J. M. and Rheinboldt, W. C. [3] proved this type results. The following definitions are useful to our discussion. Definition 2.1 If f : X → X is map such that d(f x, ff x) ≤ k d(x, fx) for all x ∈ X, 0 ≤ k < 1, then f is said to be an iterated contraction map. In case d(f x, f f x) < d(x, f x), x = f x, then f is an iterated contractive map. See for further results [3,4]. Definition 2.2 A map f : X → X, satisfying d(f x, f y) ≤ kd(x, y), 0 ≤ k < 1, for all x, y ∈ X, is called a contraction map. A contraction map is continuous and is an iterated contraction. For example, if y = f x, then d(f x, f fx) ≤ kd(x, f x) is satisfied. However, converse is not true. If f : [−1/2, 1/2] → [−1/2, 1/2] is given by f x = x2 , then f is an iterated contraction but not a contraction map. If f : R → R, defined by f x = 0 for x ∈ [0, 1/2) and fx = 1 for x ∈ [1/2, 1], then f is not continuous at x = 1/2, and f is an iterated contraction. Remark 2.1 A contraction map has a unique fixed point. However, an iterated contraction map may have more than one fixed point. For example, the iterated contraction function fx = x2 on [0, 1] has f 0 = 0 and f1 = 1, two fixed points. Remark 2.2 If f : R → R is given by fx = 2x+1, then f is a continuous map but f is not an iterated contraction. It is easy to see that d(f x, f fx) ≤ kd(x, fx) is not satisfied for 0 ≤ k < 1. Remark 2.3 A discontinuous function need not always be an iterated contraction. Let X = [0, 1]. Define f : [0, 1] → [0, 1] by fx = 1/2, x ∈ [0, 1/2), and

2

SOME FIXED POINT THEOREMS FOR ITERATED CONTRACTION MAPS Santosh Kumar

33

f x = 0, x ∈ [1/2, 1]. Then f is discontinuous. If we take x = 1/4, then d(f x, f fx) ≤ kd(x, fx), 0 ≤ k < 1, is not satisfied and hence f is not an iterated contraction. If f : [0, 1] → [0, 1] defined by f x = 1/4, x ∈ [0, 1/2), and f x = 3/4, x ∈ [1/2, 1],then f is discontinuous at x = 1/2, f has fixed points when x = 1/4 and x = 3/4. If f : R → R defined by f x = x/3 + 1/3 for x ≤ 0, and f x = x/3 for x > 0, then f is an iterated contraction and f has no fixed point. If f : [0, 1] → [0, 1] is defined by f x = 0 for x ∈ [0, 1/2), and f x = 1/2 for x ∈ [1/2, 1], then f is discontinuous at x = 1/2 and f is an iterated contraction map and has a fixed point f (1/2) = 1/2. The following is a fixed point theorem for iterated contraction map. Theorem 2.1 If f : X → X is a continuous iterated contractive map and the sequence of iterates {xn } defined by xn+1 = f xn , n = 1, 2, ... for x ∈ X, has a subsequence converging to y ∈ X, then y = f y, that is,f has a fixed point. Proof: Let xn+1 = fxn , n = 1, 2, ... Then the sequence {d(xn+1 , xn )} is a non-increasing sequence of reals. It is bounded below by 0, and therefore has a limit. Since the subsequence converges to y and f is continuous on X, so f(xni ) converges to fy and ff (xni ) converges to f f y. Thus d(y, f y) = lim d(xni , xni+1 ) = lim d(xni+1 , xni+2 ) = d(f y, f fy). If y = f y, then d(f f y, fy) < d(f y, y), since f is an iterative contractive map. Consequently, d(f y, y) = d(f fy, f y) < d(f y, y), a contradiction and hence f y = y. We give the following example to show that if f is an iterated contraction that is not continuous, then f may not have a fixed point. Let f : R → R, given by f x = x/5 + 1/5 for x ≤ 0, and f x = x/5 for x > 0. Then f does not have a fixed point. Here f is discontinuous at x = 0. If f is continuous but X is not complete, then f need not have a fixed point. For example, if f : (0, 1] → (0, 1] given by f x = x/2, then f is an iterated contraction, but has no fixed point.

3

SOME FIXED POINT THEOREMS FOR ITERATED CONTRACTION MAPS Santosh Kumar

34

A continuous function f that is not an iterated contraction may not have a fixed point. The following example illustrates the fact: If f : R → R is a translation map f x = x + 2, then f has no fixed point but f is continuous. Note 2.1 If f is not contraction but some powers of f is contraction, then f has a unique fixed point on a complete metric space. For example, if f : R → R is defined by f x = 1 for x rational and fx = 0 for x irrational, then f ofx = f 2 x = 1 a constant map, is a contraction map, and f has a unique fixed point. Note 2.2 Continuity of an iterated contraction is sufficient but not necessary. Let f : [0, 1] → [0, 1] be defined by fx = 0 on [0, 1/2), and f x = 2/3, for x ∈ [1/2, 1]. Then f is an iterated contraction and f 0 = 0, f 2/3 = 2/3, and f is not continuous. As stated in Note 2.1 that if f is not contraction still f may have a unique fixed point when some powers of f is a contraction map. The same is true for iterated contraction map. Theorem 2.2 Let f : X → X be an iterated contraction map on a complete metric space X. If for some power of f , say f r is an iterated contraction, that is, d(f r x, f r f r x) ≤ k d(x, f r x) and f r is continuous at y, where y = lim(f r )nx, for any arbitrary x ∈ X. Then f has a fixed point. Proof: Since X is a complete metric space and f is an iterated contraction that is continuous at y, therefore y = f r y, where y = lim(f r )n x. It is easy to show that d(y, f y) ≤ k r d(y, f y). Since k r ≤ 1, therefore d(y, f y) = 0 and hence f has a fixed point. We give the following example to illustrate the theorem. Example 2.1 If f : [0, 1] → [0, 1] is given by f x = 1/4, x ∈ [0, 1/4], f(x) = 0, x ∈ (1/4, 1/2), and f x = 1/2, x ∈ [1/2, 1], then f is not continuous at x = 1/4. However, if we take iteration, then f r (1/4) = 1/4 on [0, 1/2) and f r (1/4) = 1/2 on [1/2, 1] for r = 2, 3, ... It is clear that

4

SOME FIXED POINT THEOREMS FOR ITERATED CONTRACTION MAPS Santosh Kumar

35

f (1/4) = 1/4. Note 2.3 If f is not an iterated contraction in Theorem 2.2, but f r is an iterated contraction with f r y = y, then f has a fixed point. The following example is worth mentioning. Example 2.2 Let f : [0, 1] → [0, 1] defined by f x = 1/4 for x ∈ [0, 1/4], fx = 0 for x ∈ (1/4, 1/2), f x = 1/2 for x ∈ [1/2, 3/4] and fx = 3/4 for x ∈ (3/4, 1]. Here f is not iterated contraction but f r is iterated contraction for r = 2, 3, ...f 2 (x) = 1/4 for x ∈ [0, 1/2) and f 2 (x) = 1/2 for x ∈ [1/2, 1]. In this example f has a fixed point at x = 1/4. 3. MAIN RESULTS In this section, we will prove some fixed point theorems for iterated maps under different settings. Our results will corollarize many existing results. The first theorem is as follows: Theorem 3.1 If f : X → X is an iterated contraction map, and X is a complete metric space, then the sequence of iterates xn converges to y ∈ X. In case, f is continuous at y, then y = fy, that is,f has a fixed point. Proof: Let xn+1 = f xn , n = 1, 2...; x1 ∈ X. It is easy to show that {xn } is a Cauchy sequence, since f is an iterated contraction. The Cauchy sequence {xn } converges to y ∈ X, since X is a complete metric space. Moreover, if f is continuous at y, then xn+1 = fxn converges to f y. It follows that y = f y. Note 3.1 A continuous iterated contraction map on a complete metric space has a unique fixed point. If an iterated contraction map is not continuous, even then it may have a fixed point. For example, if f : [0, 1] → [0, 1] is given by f x = 0 for x ∈ [0, 1/2) and f x = 2/3 for x ∈ [1/2, 1]. In this example, X = [0, 1] is complete but f is not continuous at x = 1/2. However, f is continuous at x = 0, f 0 = 0 and f is continuous at x = 2/3, f 2/3 = 2/3. Theorem 3.2 If f : C → C is an iterated contraction and is continuous, where C is closed subset of a metric space X, then f has a fixed point provided that f (C) is compact.

5

SOME FIXED POINT THEOREMS FOR ITERATED CONTRACTION MAPS Santosh Kumar

36

Proof: We show that the sequence {xn } has a convergent subsequence. Using iterated contraction and continuity of f we get a fixed point. Definition 3.1 Let X be a metric space and f : X → X. Then f is said to be an iterated nonexpansive map if d(fx, f f x) ≤ d(x, fx) for all x ∈ X. The following is a fixed point theorem for the iterated nonexpansive map. Theorem 3.3 Let X be a metric space and f : X → X an iterated nonexpensive map satisfying the following : (i) if x = f x, then d(f f x, fx) < d(fx, x), (ii) if for some x ∈ X, the sequence of iterates xn+1 = fxn has a convergent subsequence converging to y say and f is continuous at y. Then f has a fixed point. Proof: It is easy to show that the sequence {d(xn+1 , xn )} is a nonincreasing sequence of positive reals bounded below by 0. The sequence has a limit. Hence, d(fy, y) = lim d(xni , xni+1 ) = lim d(xni+1 , xni+2 ) = d(f fy, f y). This is a contradiction to (i). Therefore f has a fixed point, that is, f y = y. If C is a compact subset of a metric space X and f : C → C an iterated nonexpansive map, satisfying condition (i) of the above theorem, then f has a fixed point. Note 3.2 If C is compact, then condition (ii) of Theorem 3.3 is satisfied, and hence the result. If C is a closed subset of a metric space X and f : C → C an iterated contraction. If the sequence {xnn } converges to y, where f is continuous at y, then f y = y, that is, f has a fixed point. The following theorem deals with two metrics on X. Theorem 3.4 Let f : X → X satisfy the following: (i) X is complete with metric d and d(x, f x) ≤ δ(x, f x) for all x, f x ∈ X, (ii)f is iterated contraction with respect to δ, Then for x ∈ X, the sequence of iterates xn = f n x converges to y ∈ X. If f is continuous at y, then f has a fixed point, say f y = y. 6

SOME FIXED POINT THEOREMS FOR ITERATED CONTRACTION MAPS Santosh Kumar

37

Proof: It is easy to show that {xn } is a Cauchy sequence with respect to δ. Since d(x, f x) ≤ δ(x, f x), therefore {xn } is a Cauchy sequence with respect to d. The sequence {xn } converges to y in (X, d) since it is complete. The function f is continuous at y, therefore f y = f lim xn = lim fxn = lim xn+1 = y. Hence the result. Kannan [2] considered the following map. Let f : X → X satisfy d(f x, f y) ≤ k[d(x, f x) + d(y, fy)] for all x, y ∈ X and 0 ≤ k < 1/2. Then f is said to be a Kannan map. If y = fx, then we get d(fx, ff x) ≤ k[d(x, f x) + d(fx, f f x)]. This gives d(fx, f f x) ≤

k d(x, f x), (1−k)

where 0 ≤

k (1−k)

< 1, that is, f is an iterated

contraction. A Kannan map is an iterated contraction map. The following result due to Kannan [2] is valid for iterated contraction map. Theorem 3.5 Let f : X → X satisfy d(fx, fy) ≤ k[d(x, f x) + d(y, f y)] for all x, y ∈ X, 0 ≤ k < 1/2, f continuous on X and let the sequence of iterates {xn } have a subsequence {xni } converging to y. Then f has a fixed point. The following result deals with two mappings. Theorem 3.6 Let f : X → X and g : X → X, where X is a complete metric space, satisfy d(f x, gfx) ≤ kd(x, f x) and d(gx, fgx) ≤ kd(x, gx) for some k, 0 ≤ k < 1 and for each x1 ∈ X. If f and g are continuous on X, then f and g have a common fixed point. Proof: We consider the sequence of iterates as follows.

Let x2 =

f x1 , x3 = gx2 , x4 = f x2 , and so on. In this case it is easy to show that the sequence {xn } is a Cauchy sequence and converges in X since X is a complete metric space. Let lim xn = y. Then lim x2n = y and lim x2n−1 = y. Since f and g are continuous on X, therefore, f y = f lim x2n−1 = lim f x2n−1 = lim x2n = y. gy = g lim x2n = lim gx2n = lim x2n+1 = y. 7

SOME FIXED POINT THEOREMS FOR ITERATED CONTRACTION MAPS Santosh Kumar

38

Hence y = fy = gy and y is a common fixed point. Results of this type for topological vector spaces are given in [5]. We are giving the following: Theorem 3.7 Let f : X → X be a continuous iterated contraction map such that: (i) if x = f x, then d(f x, f fx) < d(x, fx), and (ii) the sequence xn+1 = f (xn ) has a convergent subsequence converging to y(say). Then the sequence {xn } converges to a fixed point of f. Proof: It is easy to see that the sequence {d(xn , xn+1 )} is a nonincreasing and bounded below by 0. Let {xni } be a subsequence of {xn } converging to y. Then, d(y, f y) = lim d(xni , xni+1 ) = d(xni+1 , xni+2 ) = d(f y, f fy) < d(y, f y). This is a contradiction so y = fy. Since d(xn+1 , y) < d(xn , y) for all n, so {xn } converges to y = f y. The result due to Cheney and Goldstein [1] follows as a corollary. Corollary 3.1 Let f be a map of a metric space X into itself such that (i) f is a nonexpansive map on X, that is, d(f x, f y) ≤ d(x, y) for all x, y ∈ X, (ii) if x = fx, then d(f x, f fx) < d(x, f x), (iii) the sequence xn+1 = f (xn ) has a convergent subsequence converging to y(say). Then the sequence {xn } converges to a fixed point of f. The following is well known for contractive maps. Note 3.3 If f : X → X is a contractive map and f (X) compact, then f has a unique fixed point. It is easy to see that the sequence of iterates {xn } converges to a unique fixed point of f. However, for nonexpansive map, a sequence of iterates need not converge to a fixed point of f. For example, if f x = −x, then the sequence of iterates {xn } does not converge to a fixed point of f (f 0 = 0). Note 3.4 If g = af x + (1 − a)x, 0 < a < 1, then the fixed point of f is the same as of g. Let f y = y. Then gy = af y + (1 − a)y, that is, gy = y since fy = y. 8

SOME FIXED POINT THEOREMS FOR ITERATED CONTRACTION MAPS Santosh Kumar

39

Let gy = y. Then we show that y = f y. Here gy = afy + (1 − a)y = y. Then f y = y, that is, f has a fixed point y [6]. In case the sequence xn+1 = gxn converges to y a fixed point of g, then y = f y. Acknowledgement: The author is extremely thankful to Late Prof. S. P. Singh of Memorial University, St.John’s, NL, Canada for his valuable suggestions in the preparation of this manuscript. REFERENCES 1. Cheney, E.W. and Goldstein, A.A., Proximity maps for convex sets, Proc. Amer. Math. Soc. 10 (1959), 448 - 450. 2. Kannan, R., Some results on fixed points -II, Amer. Math. Monthly, 76(1969),405 - 408. 3. Ortega, J. and Rheinboldt, W. C., On a class of approximate iterative process, Arch. Rat. Mech. Anal. 23 (1967) 352 - 365. 4. Rheinboldt, W.C., A unified convergence theory for a class of iterative process, SIAM J. Numerical Anal., 5 (1968) 42 - 63. 5. Singh, K. L. and Whitfield, J. H. M., A fixed point theorem in nonconvex domain, Bull. Polish Acad. of Sc., 32(1984) 95 - 99. 6. Singh, S. P., Watson, B. and Srivastava, P., Fixed Point theory and Best Approximation; The KKM-map Principle, Kluwer Academic Publishers, The Netherlands 1997. On Leave from: Department of Applied Mathematics Inderprastha Engineering College, 63, Site-IV, Surya Nagar Flyover Road, Sahibabad Ghaziabad, U.P-201010. India.

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J. APPLIED FUNCTIONAL ANALYSIS, VOL. 10, NO.'S 1-2,40-52 ,2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

Nonlinear Regularized Nonconvex Random Variational Inequalities with Fuzzy Event in q-uniformly Smooth Banach Spaces Salahuddin Department of Mathematics Jazan University, Jazan Kingdom of Saudi Arabia [email protected] Abstract The main purpose of this paper is to introduced and studied a new class of nonlinear regularized nonconvex random variational inequalities with fuzzy mappings in q-uniformly smooth Banach spaces. An existence theorem of random solutions for this kind of nonlinear regularized nonconvex random variational inequalities with fuzzy mappings is established and a new iterative random algorithm with random errors is suggested and discussed. The results presented in this paper generalized, improved and unified some recent works in this fields. Keywords: Nonlinear regularized nonconvex random variational inequalities, fuzzy mappings, uniformly r-prox regular sets, random iterative sequences, randomly relaxed (κt , ζt )-cocoercive mapping, randomly accretive mappings, convergence analysis, random mixed errors, q-uniformly smooth Banach spaces. AMS Mathematics Subject Classification: 49J40, 47H06.

1

Historical Backbone

Variational inequalities are an important and generalization of classical variational inequalities which have wide applications in many fields for example mechanics, physics, optimization and control theory, nonlinear programming, economics and engineering sciences and in face, the problems for random variational inequalities are just so. It is known that accretivity of the underlying operator plays indispensable roles in the theory of variational inequalities and its generalizations. In 2001, Huang and Fang [16] were first to introduced generalized m-accretive mapping and gave the definition of the resolvent operators for generalized m-accretive mapping in Banach spaces. The fuzzy set theory which was given by Zadeh [29] at university of California in 1965 has emerged as an interesting and fascinating branch of pure and applied sciences. The applications of the fuzzy set theory can be found in many branches of regional, physical, mathematical and engineering sciences. In 1989, Chang and Zhu [8] first introduced and studied a class of variational inequalities for fuzzy mappings. Since then several classes of variational inequalities, quasi variational inequalities and complementarity problems with fuzzy mappings were considered by Agarwal et al. [1], Anastassiou et al. [2], Chang and Huang [7], Ding et al. [11], Huang [15], Lee et al. [18, 19], Salahuddin [22, 23],and Zhang and Bi [30]. Note that most of results in this direction for variational inequalities have been done in the setting of Hilbert spaces.

1

40

Nonlinear Regularized Nonconvex Random Variational Inequalities with Fuzzy Event in q-uniformly Smooth Banach Spaces

41

2

Salahuddin

On the other hand, random variational inequality problems and random quasivariational inequality problems have been studied by Bharucha and Reid [4], Chang [5, 6], Chang and Huang [7], Huang [15], Khan and Salahuddin [17], Salahuddin [22], Tan [24] and Yuan [28], etc.. The projection method is an important tools for finding an approximate solutions of various types of variational inequalities and quasi variational inequalities. The idea of this techniques is to established the equivalence between the variational inequalities and fixed point problems using the concepts of projection. The most of results regarding the existence and the iterative approximation of solutions to variational inequality problems have been investigated and considered so far the case where the underlying set is a convex set. Recently the concepts of convex set has been generalized in many directions which has pointed and important applications in various fields. It is well known that the uniformly prox regular set are nonconvex and included the convex sets as special cases . This class of uniformly prox regular sets has played an important part in many nonconvex applications such as optimization, dynamic systems and differential inclusions, see [9, 10, 26]. Inspired by the recent works going on this fields, see [3, 13, 15, 16, 17, 18, 20, 21, 25], in this paper, we introduced and considered a nonlinear regularized nonconvex random variational inequalities with fuzzy mapping involving randomly relaxed cocoercive and randomly accretive mappings in q-uniformly smooth Banach spaces. We suggested a random perturbed projection iterative algorithm with random mixed errors for finding a random solutions of aforementioned problems for fuzzy mappings. We also proved the convergence of the defined random iterative sequences under some suitable assumptions. The results presented in this paper generalized, improve and unify some recent results in this paper.

2

Conceptual Backbone

Throughout this paper, we suppose that (Ω, R, µ) is a complete σ-finite measurable space and X is a separable real Banach space endowed with dual space X ∗ , the norm k · k and an dual pairing h·, ·i between X and X ∗ . We denote by B(X ) the class of Borel σ-fields in X ; 2X and CB(X ) denote the family of all nonempty subsets of X , the family of all nonempty closed bounded subsets of X , respectively. The generalized duality mapping ∗ Jq : X → 2X is defined by Jq (x) = {f ∗ ∈ X ∗ : hx, f ∗ i = kxkq , kf ∗ k = kxkq−1 } ∀x ∈ X where q > 1 is a constant. In particular J2 is a usual normalized duality mapping. It is known that in general Jq (x) = kxkq−1 J2 (x) for all x 6= 0 and Jq is single valued if X ∗ is strictly convex. In the sequel, we always assume that X is a real Banach space such that Jq is a single valued. If X is a Hilbert space then Jq becomes the identity mapping on X . The modulus of smoothness of X is the function πX : [0, ∞) → [0, ∞) is defined by 1 πX (t) = sup{ (kx + yk + kx − yk) − 1 : kxk ≤ 1, kyk ≤ t}. 2

Nonlinear Regularized Nonconvex Random Variational Inequalities with Fuzzy Event in q-uniformly Smooth Banach Spaces

42

3

Salahuddin

A Banach space X is called uniformly smooth if lim

t→0

πX (t) = 0. t

X is called q-uniformly smooth if there exists a constant c > 0 such that πX (t) < ctq , q > 1. Note that Jq is a single valued if X is uniformly smooth. Concerned with the characteristic inequalities in q-uniformly smooth Banach spaces. Xu [27] proved the following results. Lemma 2.1 The real Banach space X is q-uniformly smooth if and only if there exists a constant cq > 0 such that for all x, y ∈ X kx + ykq ≤ kxkq + qhy, Jq (x)i + cq kykq . Let D(·, ·) represent the Hausdorff metric on CB(X ) defined by   D(A, B) = max sup d(a, B), sup d(A, b) for all A, B ∈ CB(X ), a∈A

b∈B

where d(a, B) = d(B, a) = inf ka − bk for a ∈ A. b∈B

Definition 2.2 A mapping x : Ω → X is said to be measurable if for any B ∈ B(X ), {t ∈ Ω, x(t) ∈ B ∈ R}. Definition 2.3 A mapping T : Ω × X → X is said to be a random mapping if for each fixed x ∈ X , T (t, x) = y(t) is measurable. A random mapping Tt is said to be continuous if for each fixed t ∈ Ω, f (t, ·) : Ω × X → X is continuous. Definition 2.4 A multivalued mapping T : Ω → 2X is said to be measurable if for any B ∈ B(X ), T −1 (B) = {t ∈ Ω : T (t) ∩ B 6= ∅} ∈ Σ. Definition 2.5 A mapping u : Ω → X is said to be a measurable selection of a measurable mapping T : Ω × X → 2X , if u is measurable and for any t ∈ Ω, u(t) ∈ Tt (x(t)). Definition 2.6 A mapping T : Ω × X → 2X is said to be a random multivalued mapping if for each fixed x ∈ X , T (·, x) : Ω × X → 2X is a measurable multivalued mapping. A random multivalued mapping T : Ω × X → CB(X ) is said to be D-continuous if for each fixed t ∈ Ω, T (t, ·) : Ω × X → 2X is a randomly continuous with respect to the Hausdorff metric on D. Definition 2.7 A multivalued mapping T : Ω×X → 2X is said to be a random multivalued mapping if for any x ∈ X , T (·, x) is measurable (denoted by Tt,x or Tt ).

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Let Ω be a set and F(X ) be a collection of fuzzy sets over X . A mapping F˜ : Ω×X → F(X ) is called a fuzzy mapping. For each x ∈ X , F˜ (x) (denote it by T˜x in the sequel) is a fuzzy mapping on X and F˜x (y) is the membership-grade of y in F˜x . Let A ∈ F(X ), α ∈ (0, 1], then the set Aα = {x ∈ X : A(x) ≥ α} is called an α-cut of A. Definition 2.8 A fuzzy mapping T˜ : Ω × X → F(X ) is called measurable, if for any α ∈ (0, 1], (T˜(·))α : Ω → 2X is a measurable multivalued mapping. Definition 2.9 A fuzzy mapping T˜ : Ω × X → F(X ) is a random fuzzy mapping if for any x ∈ X , T˜(·, x) : Ω × X → F(X ) is a measurable fuzzy mapping (denoted by T˜t,x short down T˜t ). Definition 2.10 Let K be a nonempty closed subsets of q-uniformly smooth Banach space X . The proximal normal cone of K at a point u ∈ X is given by NKP (u) = {ζ ∈ X : u ∈ PK (u + αζ) for some α > 0} where α > 0 is a constant and PK (u) = {v ∈ K : dK (u) = ku − vk}. Here dK (u) is the usual distance function to the subset K that is dK (u) = inf ku − vk. v∈K

Lemma 2.11 Let K be a nonempty closed subset in q-uniformly smooth Banach space X . Then ζ ∈ NKP (u) if and only if there exists a constant α = α(ζ, u) > 0 such that hζ, jq (v − u)i ≤ αkv − uk2 ∀v ∈ K. Lemma 2.12 Let K be a nonempty closed and convex subset in q-uniformly smooth Banach space X . Then ζ ∈ NKP (u) if and only if hζ, jq (v − u)i ≤ 0 ∀v ∈ K. Lemma 2.13 [5] Let G : Ω × X → CB(X ) be a D-continuous random multivalued mapping. Then for a measurable mapping u : Ω → X , a multivalued mapping G(·, u(·)) : Ω → CB(X ) is measurable. Lemma 2.14 [5] Let A, T : Ω → CB(X) be measurable multivalued mappings and u : Ω → X be a measurable selection of A. Then there exists a measurable selection v : Ω → X of T such that for all t ∈ Ω and  > 0, ku(t) − v(t)k ≤ (1 + ) D(A(t), T (t)).

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Definition 2.15 The tangent cone TK (x) to K at a point x ∈ K is defined as follows TK (x) = {v ∈ X : doK (x; v) = 0}. The normal cone of K at x by polarity with TK (x) defined as follows NK (x) = {ζ : hζ, jq (v)i ≤ 0 ∀v ∈ TK (x)}. The Clarke normal cone NKC (x) is given by NKC (x) = co[NKP (x)] where co(S) mean the closure of the convex hull of S. Clearly NKP (x) ⊆ NKC (x) but the converse is not true. Since NKP (x) is always closed and convex cone where NKC (x) is convex but may not be closed, see [9, 10, 21]. Definition 2.16 For any r ∈ (0, +∞], a subset Kr of X is called the normalized uniformly prox-regular (or uniformly r-prox-regular) if every nonzero proximal normal to Kr can be realized by an r-ball that is ∀u ∈ Kr and 0 6= ζ ∈ NKPr (u), kζk = 1 one has 1 kv − uk2 ∀v ∈ K. 2r Proposition 2.17 Let r > 0 and Kr be a nonempty closed and uniformly r-prox regular subset of q-uniformly smooth Banach space X . Set hζ, jq (v − u)i ≤

U(r) = {u ∈ X : 0 ≤ dKr (u) < r}. Then the following statements are hold: (a) for all x ∈ U(r), PKr (x) 6= ∅; (b) for all r0 ∈ (0, r), PKr is Lipschitz continuous mapping with constant 0

r r−r0

on

0

U(r ) = {u ∈ X : 0 ≤ dKr (u) < r }; (c) the proximal normal cone is closed as a set valued mapping. Let T˜ : Ω × X → F(X ) be a random fuzzy mapping satisfying the condition (∗) : (∗) : there exists a function a : X → (0, 1] such that for all (t, x) ∈ Ω × X , we have (T˜t )α(x) ∈ CB(X ) where CB(X ) denotes the family of all nonempty bounded closed subsets of X . By using the random fuzzy mapping T˜t , we can define a random multivalued mapping Tt : Ω × X → CB(X ) by Tt = (T˜t )α(x) for x ∈ X . In the sequel Tt is called the multivalued mapping induced by the random fuzzy mapping T˜t . Let α : X → (0, 1] be a function and g, h : Ω × X → X be the nonlinear random single valued mapping such that Kr ⊂ gt (X ). For a given measurable mapping η : Ω → (0, 1) we finding measurable mappings x, u : Ω → X such that 1 kgt (y(t)) − ht (x(t))k2 ∀ gt (y(t)) ∈ Kr , 2r (2.1) which is called a nonlinear regularized nonconvex random variational inequalities with fuzzy mappings. hηt u(t) + ht (x(t)) − gt (x(t)), gt (y(t)) − ht (x(t))i +

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Lemma 2.18 Let Kr be a uniformly r-prox regular set then the problem (2.1) is equivalent to finding x(t) ∈ X , u(t) ∈ Tt (x(t)) such that ht (x(t)) ∈ Kr and 0 ∈ ηt u(t) + ht (x(t)) − gt (x(t)) + NKPr (ht (x(t))),

(2.2)

where NKPr (s) denotes the P -normal cone of Kr at s in the sense of nonconvex analysis. Proof. Let (x(t), u(t)) with x(t) ∈ X , ht (x(t)) ∈ Kr and u(t) ∈ Tt (x(t)) be the random solution sets of the problem (2.2). If ηt u(t) + ht (x(t)) − gt (x(t)) = 0 because the vector zero always belongs to any normal cone, then 0 ∈ ηt u(t) + ht (x(t)) − gt (x(t)) + NKPr (ht (x(t))). If ηt u(t) + ht (x(t)) − gt (x(t)) 6= 0 then for all x(t) ∈ X with gt (x(t)) ∈ Kr one has h−(ηt u(t) + ht (x(t)) − gt (x(t))), gt (y(t)) − ht (x(t))i ≤

1 kgt (y(t)) − ht (x(t))k2 . 2r

(2.3)

From Lemma 2.11, we have −(ηt u(t) + ht (x(t)) − gt (x(t))) ∈ NKPr (ht (x(t))) and 0 ∈ ηt u(t) + ht (x(t)) − gt (x(t)) + NKPr (ht (x(t))).

(2.4)

Conversely if (x(t), u(t)) with x(t) ∈ X , ht (x(t)) ∈ Kr and u(t) ∈ Tt (x(t)) is a random solution sets of the problem (2.2) then from Definition 2.16, x(t) ∈ X and u(t) ∈ Tt (x(t)) with ht (x(t)) ∈ Kr are random solution sets of the problem (2.1). Lemma 2.19 Let T˜t : Ω×X → F(X ) be a random fuzzy mapping induced by a multivalued mapping T : Ω × X → CB(X ) and g, h : Ω × X → X be the random single valued mappings and η : Ω → (0, 1) be a measurable mapping, then (x(t), u(t)) with (x(t)) ∈ X , ht (x(t)) ∈ Kr , u(t) ∈ T˜t (x(t)) is a random solution sets of the problem (2.1) if and only if ht (x(t)) = PKr [gt (x(t)) − ηt u(t)],

(2.5)

where PKr is the projection of X onto the uniformly r-prox regular set Kr . Proof. Let (x(t), u(t)) with x(t) ∈ X , ht (x(t)) ∈ Kr , u(t) ∈ T˜t (x(t)) be a random solution sets of the problem (2.1). Then from Lemma 2.18, we have 0 ∈ ηt u(t) + ht (x(t)) − gt (x(t)) + NKPr (ht (x(t)))

(2.6)

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⇔ gt (x(t)) − ηt u(t) ∈ (I + NKPr )(ht (x(t)))

(2.7)

ht (x(t)) = PKr [gt (x(t)) − ηt u(t)],

(2.8)

⇔ where I is an identity mapping and PKr = (I +

NKPr )−1 .

Remark 2.20 The inequality (2.5) can be written as follows x(t) = x(t) − ht (x(t)) + PKr [gt (x(t)) − ηt u(t)],

(2.9)

where η : Ω → (0, 1) is a measurable mapping. Algorithm 2.21 Let T˜t : Ω × X → F(X ) be a random fuzzy mapping satisfying the condition (∗) and T : Ω × X → CB(X ) be a multivalued random mapping induced by the random fuzzy mapping Tt . Let g, h : Ω × X → X be the random single valued mappings. For any given x0 : Ω → X the random single valued mappings, Tt (x0 (t)) : Ω×X → CB(X ) is measurable by Lemma 2.13. We know that for any x0 (t) ∈ X , Tt (x0 (t)) is measurable and there exists a measurable selection u0 (t) ∈ Tt (x0 (t)), see [13, 14]. Set x1 (t) = (1 − λt )x0 (t) + λt [x0 (t) − ht (x0 (t)) + PKr [gt (x0 (t)) − ηt u0 (t)] + λt e0 (t) + r0 (t) where η, λ : Ω → (0, 1) are measurable mappings and 0 < λt < 1 is a random constant and e0 (t), r0 (t) : Ω → X is the random measurable mapping which is a random errors to take into account of a possible inexact computation for the proximal point. Then it is easy to see that x1 : Ω → X is a random measurable mapping. Since u0 (t) ∈ Tt (x0 (t)) then there exists a random measurable selection u1 (t) ∈ Tt (x1 (t)) for all t ∈ Ω 1 ku0 (t) − u1 (t)k ≤ (1 + ( )D(Tt (x0 (t)), Tt (x1 (t))). 1 By induction, we can define the measurable random sequences xn (t), un (t) : Ω → X inductively satisfying

xn+1 (t) = (1−λt )xn (t)+λt [xn (t)−ht (xn (t))+PKr (gt (xn (t))−ηt un (t))+λn (t)en (t)+rn (t), (2.10) −1 un (t) ∈ Tt (xn (t)); kun (t) − un+1 (t)k ≤ (1 + (1 + n) )D(Tt (xn (t)), Tt (xn+1 (t))) (2.11) ∞ where 0 < λt < 1 is a random constant and {en (t)}∞ n=0 , {rn (t)}n=0 are random sequences in X to take into account of a possible inexact computation for the resolvent operator satisfying the following conditions:

lim en (t) = lim rn (t) = 0,

n→∞ ∞ X n=1

n→∞

ken (t) − en−1 (t)k < ∞ and

∞ X n=1

krn (t) − rn−1 (t)k < ∞.

(2.12)

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Definition 2.22 Let X be a q-uniformly smooth Banach spaces. A random single valued mapping g : Ω × X → X is called (i) randomly accretive if hgt (x(t)) − gt (y(t)), jq (x(t) − y(t))i ≥ 0 ∀x(t), y(t) ∈ X , t ∈ Ω; (ii) randomly µt -strongly accretive if there exists a measurable mapping µ : Ω → (0, 1) such that hgt (x(t)) − gt (y(t)), jq (x(t) − y(t))i ≥ µt kx(t) − y(t)kq ∀x(t), y(t) ∈ X , t ∈ Ω; (iii) randomly relaxed µt -accretive mapping if there exists a measurable mapping µ : Ω → (0, 1) such that hgt (x(t)) − gt (y(t)), jq (x(t) − y(t))i ≥ −µt kx(t) − y(t)kq ∀x(t), y(t) ∈ X , t ∈ Ω; (iv) randomly αt -Lipschitz continuous mapping if there exists a measurable mapping α : Ω → (0, 1) such that kgt (x(t)) − gt (y(t))k ≤ αt kx(t) − y(t)k ∀x(t), y(t) ∈ X , t ∈ Ω. Definition 2.23 Let g : Ω × X → X be a random single valued mapping and T : Ω × X → CB(X ) a random multivalued mapping. Then Tt is said to be (i) randomly accretive if hu(t) − v(t), jq (x(t) − y(t))i ≥ 0, ∀x(t), y(t) ∈ X , u(t) ∈ Tt (x(t)), v ∈ Tt (y(t)), (ii) randomly (κt , ζt )-relaxed cocoercive mapping with respect to gt if there exist the measurable mappings κ, ζ : Ω → (0, 1) such that for all x(t), y(t) ∈ X , t ∈ Ω hu(t) − v(t), jq (gt (x(t)) − gt (y(t)))i ≥ −κt ku(t) − v(t)kq + ζt kgt (x(t)) − gt (y(t))kq ∀u(t) ∈ Tt (x(t)), v(t) ∈ Tt (y(t)), (iii) randomly α − D-Lipschitz continuous mapping if there exists a measurable mapping α : Ω → (0, 1) such that for all x(t), y(t) ∈ X , t ∈ Ω D(Tt (x(t)), Tt (y(t))) ≤ αt kx(t) − y(t)k where D is the Hausdorff pseudo metric defined in X .

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3

Main Backbone

In this section, we prove the existence and the convergence of problem (2.1). Theorem 3.1 Let X be a q-uniformly smooth real Banach space. Let T˜t : Ω × X → F(X ) be a random fuzzy mapping satisfying the condition (∗) and Tt : Ω × X → CB(X ) be a random multivalued mapping induced by random fuzzy mapping. Suppose that Tt is a randomly D-Lipschitz continuous with random measurable mapping α : Ω → (0, 1). Let Tt be the randomly (κt , ζt )-relaxed cocoercive with respect to the random mapping gt and g : Ω × X → X be the randomly Lipschitz continuous with a measurable mapping β : Ω → (0, 1). Let h : Ω × X → X be the randomly relaxed µt -accretive mapping with measurable mapping µ : Ω → (0, 1) and randomly Lipschitz continuous with the measurable mapping γ : Ω → (0, 1). If for any t ∈ Ω, y(t), z(t) ∈ X , we have kPKr (y(t)) − PKr (z(t)) ≤

r ky(t) − z(t)k. r − r0

(3.1)

Assume that the measurable mapping η : Ω → (0, 1) satisfying the following assumptions 0 < θt = 1 − λt (1 − pt ) < 1, q r q q βt − qηt (−κt αtq + ζt βtq ) + cq ηtq αtq < 1, Qt + r − r0 q Qt = q 1 + qµt + cq γtq Qt < 1,

0 < λt < 1,

pt < 1

and lim ken (t)k = lim krn (t)k = 0,

n→∞ ∞ X n=0

ken (t) − en−1 (t)k < ∞,

n→∞

∞ X

krn (t) − rn−1 (t)k < ∞,

(3.2)

n=0

where r0 ∈ (0, r), then the random mappings xn (t), un (t) : Ω × X → X generated by random algorithm converge strongly to the random mappings x∗ (t), u∗ (t) : Ω × X → X and x∗ (t), u∗ (t) ∈ X with ht (x(t)) ∈ Kr is a random solution sets of problem (2.1). Proof. From Algorithm 2.21 and (3.1), t ∈ Ω and 0 < λt < 1, we have kxn+1 (t)−xn (t)k ≤ (1−λt )kxn (t)−xn−1 (t)k+λt (kxn (t)−xn−1 (t)−(ht (xn (t))−ht (xn−1 (t)))k r kgt (xn (t))−gt (xn−1 (t))−ηt (un (t)−un−1 (t))k)+λt ken (t)−en−1 (t)k+krn (t)−rn−1 (t)k. r − r0 (3.3) Since ht is randomly relaxed µt -accretive mapping and randomly Lipschitz continuous mapping, we have +

kxn (t) − xn−1 (t) − (ht (xn (t)) − ht (xn−1 (t)))kq = kxn (t) − xn−1 (t)kq

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−qhht (xn (t)) − ht (xn−1 (t)), jq (xn (t) − xn−1 (t))i + cq kht (xn (t)) − ht (xn−1 (t))kq ≤ kxn (t) − xn−1 (t)kq + qµt kxn (t) − xn−1 (t)kq + γtq cq kxn (t) − xn−1 (t)kq ≤ (1 + qµt + γtq cq )kxn (t) − xn−1 (t)kq q ⇒ kxn (t)−xn−1 (t)−(ht (xn (t))−ht (xn−1 (t)))k ≤ q 1 + qµt + cq γtq kxn (t)−xn−1 (t)k. (3.4) Since Tt is randomly αt − D-Lipschitz continuous mapping, then we have 1 )D(Tt (xn (t)), Tt (xn−1 (t))) n 1 ≤ αt (1 + )kxn (t) − xn−1 (t)k. n

kun (t) − un−1 (t)k ≤ (1 +

(3.5)

Again Tt is randomly (κt , ζt )-relaxed cocoercive mapping with respect to gt and gt is randomly Lipschitz continuous mapping with measurable mapping β : Ω → (0, 1), we have kgt (xn (t)) − gt (xn−1 (t)) − ηt (un (t) − un−1 (t))kq ≤ kgt (xn (t)) − gt (xn−1 (t))kq −qηt hun (t) − un−1 (t), jq (gt (xn (t)) − gt (xn−1 (t))))i + cq ηtq kun (t) − un−1 (t)kq ≤ kgt (xn (t)) − gt (xn−1 (t))kq − qηt (−κt kun (t) − un−1 (t)kq + ζt kgt (xn (t)) − gt (xn−1 (t))kq ) +cq ηtq kun (t) − un−1 (t)kq 1 q ) kxn (t) − xn−1 (t)kq − qηt ζt βtq kxn (t) − xn−1 (t)kq n 1 +cq ηtq αtq (1 + )q kxn (t) − xn−1 (t)kq n 1 1 ≤ (βtq − qηt (−κt αtq (1 + )q + ζt βtq ) + cq ηtq αtq (1 + )q )kxn (t) − xn−1 (t)kq n n

≤ βtq kxn (t) − xn−1 (t)kq + qηt κt αtq (1 +

⇒ kgt (xn (t)) − gt (xn−1 (t)) − ηt (un (t) − un−1 (t))k r 1 1 q ≤ βtq − qηt (−κt αtq (1 + )q + ζt βtq ) + cq ηtq αtq (1 + )q kxn (t) − xn−1 (t)k. n n

(3.6)

From (3.3),(3.4) and (3.6), we get p kxn+1 (t) − xn (t)k ≤ (1 − λt )kxn (t) − xn−1 (t)k + λt ( q 1 + qµt + cq γtq r + r − r0

r q

βtq − qηt (−κt αtq (1 +

1 q 1 ) − ζt βtq ) + cq ηtq αtq (1 + )q )kxn (t) − xn−1 (t)k n n

+λt ken (t) − en−1 (t)k + krn (t) − rn−1 (t)k ≤ (1 − λt + λt pn,t )kxn (t) − xn−1 (t)k + λt ken (t) − en−1 (t)k + krn (t) − rn−1 (t)k ≤ θn,t kxn (t) − xn−1 (t)k + λt ken (t) − en−1 (t)k + krn (t) − rn−1 (t)k

(3.7)

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where θn,t = 1 − λt + λt pn,t pn,t

r = Qt + r − r0

r q

βtq − qηt (−κt αtq (1 +

1 q 1 ) + ζt βtq ) + cq ηtq αtq (1 + )q n n

p and Qt = q 1 + qµt + cq γtq . Let θt = 1 − λt + λt pt and pt = Qt +

r r − r0

q q

βtq − qηt (−κt αtq + ζt βtq ) + cq ηtq αtq .

We have pn,t → pt and θn,t → θt as n → ∞. It follows from (3.2), 0 < λt < 1 and θt∗ ∈ (0, 1) such that θn,t < θt∗ for all n ≥ N0 . Therefore from (3.7), we have kxn+1 (t) − xn (t)k ≤ θt∗ kxn (t) − xn−1 (t)k + λt ken (t) − en−1 (t)k + krn (t) − rn−1 (t)k, ∀n ≥ 1 without loss of generality, we may assume that kxn+1 (t) − xn (t)k ≤ θt∗ kxn (t) − xn−1 (t)k + λt ken (t) − en−1 (t)k + krn (t) − rn−1 (t)k, ∀n ≥ 1. Hence for any m > n > 0, we have kxn+1 (t) − xn (t)k ≤

m−1 X

kxi+1 (t) − xi (t)k ≤

i=n

+λt

m−1 i XX

θt∗,i−j kej (t) − ej−1 (t)k +

i=n j=1

m−1 X

θt∗,i kx1 (t) − x0 (t)k

i=n m−1 i XX

θt∗,i−j krj (t) − rj−1 (t)k, ∀n ≥ 1.

(3.8)

i=n j=1

It follows from condition (3.2) that kxm (t) − xn (t)k → 0 as n → ∞ and so {xn (t)} is a Cauchy sequence in X . Let xn (t) → x∗ (t) as n → ∞. By the random D-Lipschitz continuity of Tt (·) we have 1 )D(Tt (xn+1 (t)), Tt (xn (t))) 1+n 1 ≤ (1 + ))αt kxn+1 (t) − xn (t)k → 0, as n → ∞. 1+n

kun+1 (t) − un (t)k ≤ (1 +

(3.9)

It follows that {un (t)} is a Cauchy sequence in X . We can assume that un (t) → u∗ (t). Note that un (t) ∈ Tt (xn (t)) we have d(u∗ (t), Tt (x∗ (t))) ≤ ku∗ (t) − un (t)k + d(un (t), Tt (x∗ (t))) 1 ≤ ku∗ (t) − un (t)k + (1 + )D(Tt (xn (t)), Tt (x∗ (t))) n 1 ∗ ≤ ku (t) − un (t)k + (1 + )αt kxn (t) − x∗ (t)k → 0 as n → ∞. (3.10) n Hence d(u∗ (t), Tt (x∗ (t))) = 0 and therefore u∗ (t) ∈ Tt (x∗ (t)) ∈ X . By the random D-Lipschitz continuity of Tt (·), Lemma 2.19 and condition (3.2) and lim ken (t)k = lim krn (t)k = 0,

n→∞

n→∞

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we have x∗ (t) = (1 − λt )x∗ (t) + λt [x∗ (t) − ht (x∗ (t)) + PKr (gt (x∗ (t)) − ηt u∗ (t)).

(3.11)

By Lemma 2.19, we know that (x∗ (t), u∗ (t)) is a random solution sets of problem (2.1). This completes the proof.

References [1] R. P. Agarwal, M. F. Khan, D. O. Regan and Salahuddin, On generalized multivalued nonlinear variational inclusions with fuzzy mappings, Adv. Nonlinear Var. Inequal., 8(1) (2005), 41-55. [2] G. A. Anastassiou, M. K. Ahmad and Salahuddin, Fuzzified random generalized nonlinear variational inclusions, J. Concrete and Appl. Math., 10(3) (2012), 186-206. [3] J. Balooee, Y. J. Cho and M. K. Kang, Projection methods and a new system of extended general regularized nonconvex set valued variational inequalities, J. Applied Math., 2012, Article ID 690648, (2012), 18 pages. [4] A. T. Bharucha-Reid, Random Integral Equations, Academic Press, New York, 1972. [5] S. S. Chang, Fixed Point Theory with Applications, Chongging Publishing House, Chongging, 1984. [6] S. S. Chang, Variational Inequality and Complementarity Problem Theory with Applications, Shanghai Scientific and Tech. Literature Publ. House, Shanghai, 1991. [7] S. S. Chang and N. J. Huang, Generalized random quasi multivalued complementarity problems, Indian J. Math., 35 (1993), 305-320. [8] S. S. Chang, and Y. G. Zhu, On variational inequalities for fuzzy mappings, Fuzzy Sets and Systems, 32 (1989), 359-367. [9] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley Int. Science, New York, 1983. [10] F. H. Clarke, Y. S. Ledyaw, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York 1998. [11] X. P. Ding, M. K. Ahmad and Salahuddin, Fuzzy generalized vector variational inequalities and complementarity problems, J. Nonlinear Functional Analysis, 13 (2) (2008), 253-263. [12] Y. P. Fang and N. J. Huang, H-accretive operators and resolvent operator techniques for solving variational inclusions in Banach spaces, Appl. Math. Letts., 17(6)(2004), 647-653.

Nonlinear Regularized Nonconvex Random Variational Inequalities with Fuzzy Event in q-uniformly Smooth Banach Spaces

Salahuddin

52

13

[13] P. R. Halemous, Measure Theory, Springer Verlag, New York, 1974. [14] G. J. Himmelberg, Measurable relations, Fund. Math., 87 (1975), 53–72. [15] N. J. Huang, Random generalized nonlinear variational inclusions for random fuzzy mappings, Fuzzy Sets and Systems, 105(1999), 437-444. [16] N. J. Huang and Y. P. Fang, Generalized m-accretive mappings in Banach spaces, J. Sichuan University, 38(4)(2001), 591-592. [17] M. F. Khan and Salahuddin, Completely generalized nonlinear random variational inclusions, South East Asian Bulletin of Mathematics, 30(2) (2006), 261-276. [18] B. S. Lee, M. F. Khan and Salahuddin, Fuzzy nonlinear mixed random variational like inclusions, Pacific J. Optimization, 6(3)(2010), 573-590. [19] B. S. Lee, M. F. Khan and Salahuddin, Fuzzy nonlinear set valued variational inclusions, Comp. Math. Appl., 60(6)(2010), 1768-1775. [20] Jr. S. B. Nadler, Multivalued contraction mappings, Pacific J. Math., 30 (1969), 475– 485. [21] R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352(2000), 5231-5249. [22] Salahuddin, Some Aspects of Variational Inequalities, Ph.D. Thesis, A.M.U., Aligarh, India, 2000. [23] Salahuddin, Random hybrid proximal point algorithm for fuzzy nonlinear set valued inclusions, J. Concrete Appl. Math., 12(1-2)(2014), 47-62. [24] X. Z. Tan, Random quasi-variational inequality, Math. Nachr., 125 (1986), 319–328. [25] R. U. Verma, Generalized system for relaxed cocoercive variational inequalities and projection methods, J. Optim. Theo. Appl., 121(2004), 203-210. [26] X. Weng, Fixed point iteration for local strictly pseudo contractive mapping, Proceedings of American Mathematical Society, 113(3) (1991), 727-737. [27] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16(12) (1991) ,1127-1138. [28] X. Z. Yuan, Noncompact random generalized games and random quasi variational inequalities, J. Math. Stochastic. Analysis, 7(1994), 467-486. [29] L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338-353. [30] C. Zhang and Z. S. Bi, Random generalized nonlinear variational inclusions for random fuzzy mappings, J. Sichuan Univ., 6(2007), 499-502.

J. APPLIED FUNCTIONAL ANALYSIS, VOL. 10, NO.'S 1-2,53-69 ,2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

53

New White Noise Functional Solutions For Wick-type Stochastic Coupled KdV Equations Using F-expansion Method Hossam A. Ghany1,2 and M. Zakarya3 1 Department of Mathematics, Faculty of Science, Taif University, Taif, Saudi Arabia. 2 Department of Mathematics, Helwan University, Cairo, Egypt. [email protected] 3 Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt. mohammed [email protected] Abstract Wick-type stochastic coupled KdV equations are researched. By means of Hermite transformation, white noise theory and F-expansion method, three types of exact solutions for Wick-type stochastic coupled KdV equations are explicitly given. These solutions include the white noise functional solutions of Jacobi elliptic function (JEF) type, trigonometric type and hyperbolic type.

Keywords: Coupled KdV equations; F-expansion method; Hermite transform; Wicktype product; White noise theory. PACS No. : 05.40.±a, 02.30.Jr.

1

Introduction

In this paper, we shall explore exact solutions for the following variable coefficients coupled KdV equations.   ut + h1 (t)uux + h2 (t)vvx + h3 (t)uxxx = 0, , (t, x) ∈ R+ × R, (1.1)  vx + h4 (t)uvx + h3 (t)vxxx = 0,

where h1 (t), h2 (t), h3 (t) and h4 (t) are bounded measurable or integrable functions on R+ . Random wave is an important subject of stochastic partial differential equations 1

Wick-type Stochastic Coupled KdV Equations Using F-expansion Method, Hossam A. Ghany and M. Zakarya

54

(PDEs). Many authors have studied this subject. Wadati first introduced and studied the stochastic KdV equations and gave the diffusion of soliton of the KdV equation under Gaussian noise in [28, 30] and others [3–6, 23] also researched stochastic KdVtype equations. Xie first introduced Wick-type stochastic KdV equations on white noise space and showed the auto- Backlund transformation and the exact white noise functional solutions in [35]. Furthermore, Xie [36–39], Ghany et al. [12–14, 16–19] researched some Wick-type stochastic wave equations using white noise analysis. In this paper we use F-expansion method for finding new periodic wave solutions of nonlinear evolution equations in mathematical physics, and we obtain some new periodic wave solution for coupled KdV equations. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear (PDEs) arising in mathematical physics. The effort in finding exact solutions to nonlinear equations is important for the understanding of most nonlinear physical phenomena. For instance, the nonlinear wave phenomena observed in fluid dynamics, plasma, and optical fibers. Many effective methods have been presented, such as variational iteration method [7, 8], tanh-function method [9, 32, 40], homotopy perturbation method [11, 27, 33], homotopy analysis method [1], tanh-coth method [29, 31, 32], exp-function method [21, 22, 34, 41, 42] , Jacobi elliptic function expansion method [10, 24–26], the F-expansion method [43–46]. The main objective of this paper is using the F-expansion method to construct white noise functional solutions for wicktype stochastic coupled KdV equations via hermite transform, wick-type product, white noise theory. If equation (1.1) is considered in a random environment, we can get stochastic coupled KdV equations. In order to give the exact solutions of stochastic coupled KdV equations, we only consider this problem in white noise environment. We shall study the following Wick-type stochastic coupled KdV equations.  Ut + H1 (t)  U  Ux + H2 (t)  V  Vx + H3 (t)  Uxxx = 0, (1.2) Vx + H4 (t)  U  Vx + H3 (t)  Vxxx = 0, where “  ” is the Wick product on the Kondratiev distribution space (S−1 ) which was defined in [20], H1 (t), H2 (t), H3 (t) and H4 (t) are (S−1 )-valued functions.

2

Description of the F-expansion Method

In order to simultaneously obtain more periodic wave solutions expressed by various Jacobi elliptic functions to nonlinear wave equations, we introduce an F-expansion method which can be thought of as a succinctly over-all generalization of Jacobi elliptic function expansion. We briefly show what is F-expansion method and how to use it to obtain various periodic wave solutions to nonlinear wave equations. Suppose a nonlinear wave equation for u(t,x) is given by. p(u, ut , ux , uxx , uxxx , ...) = 0, 2

(2.1)

Wick-type Stochastic Coupled KdV Equations Using F-expansion Method, Hossam A. Ghany and M. Zakarya

55

where u = u(t, x) is an unknown function, p is a polynomial in u and its various partial derivatives in which the highest order derivatives and nonlinear terms are involved. In the following we give the main steps of a deformation F-expansion method. Step 1. Look for traveling wave solution of Eq.(2.1)by taking Z t u(t, x) = u(ξ) , ξ(t, x) = kx + s δ(τ )dτ + c, (2.2) 0

Hence, under the transformation (2.2). Eq.(2.1) can be transformed into ordinary differential equation (ODE) as following. O(u, sδu0 , ku0 , k 2 u00 , k 3 u000 , ...) = 0,

(2.3)

Step 2. Suppose that u(ξ) can be expressed by a finite power series of F (ξ) of the form. u(t, x) = u(ξ) =

N X

ai F i (ξ),

(2.4)

i=1

where a0 , a1 , ..., aN are constants to be determined later, while F 0 (ξ) in(2.4) satisfy [F 0 (ξ)]2 = P F 4 (ξ) + QF 2 (ξ) + R,

(2.5)

and hence holds for F (ξ)  0 00 F F = 2P F 3 F 0 + QF F 0 ,    00 F = 2P F 3 + QF, F 000 = 6P F 2 F 0 + QF 0 ,    ...

(2.6)

where P, Q, and R are constants. Step 3. The positive integer N can be determined by considering the homogeneous balance between the highest derivative term and the nonlinear terms appearing in (2.3). Therefore, we can get the value of N in (2.4). Step 4. Substituting (2.4) into (2.3) with the condition (2.5), we obtain polynomial in f i (ξ)[f 0 (ξ)]j , (i = 0 ± 1, ±2, ..., j = 0, 1). Setting each coefficient of this polynomial to be zero yields a set of algebraic equations for a0 , a1 , ..., aN , s and δ. Step 5. Solving the algebraic equations with the aid of Maple we have a0 , a1 , ..., aN , s and δ can be expressed by P, Q and R. Substituting these results into F-expansion (2.4), then a general form of traveling wave solution of Eq. (2.1) can be obtained. Step 6. Since the general solutions of (2.4) have been well known for us Choose properly (P, Q and R.) in ODE (2.5) such that the corresponding solution F (ξ) of it is one of Jacobi elliptic functions. (See Appendix A, B and C.)

3

Wick-type Stochastic Coupled KdV Equations Using F-expansion Method, Hossam A. Ghany and M. Zakarya

3

56

New Exact Traveling Wave Solutions of Eq. (1.2)

Taking the Hermite transform, white noise theory, and F-expansion method to explore new exact wave solutions for Eq.(1.2). Applying Hermite transform to Eq.(1.2), we get the deterministic equations:  U˜t (t, x, z) + H˜1 (t, z)U˜ (t, x, z)U˜x (t, x, z) + H˜2 (t, z)V˜ (t, x, z)V˜x (t, x, z)    +H˜3 (t, z)U˜xxx (t, x, z) = 0, (3.1)    ˜ Vt (t, x, z) + H˜4 U˜ (t, x, z)V˜x (t, x, z) + H˜3 V˜xxx (t, x, z) = 0,

where z = (z1 , z2 , ...) ∈ (CN ) is a vector parameter. To look for the travelling wave solution of Eq.(3.1), we make the transformations H˜1 (t, z) := h1 (t, z), H˜2 (t, z) := h2 (t, z), H˜3 (t, z) := h3 (t, z), H˜4 (t, z) := h4 (t, z), U˜ (t, x, z) =: u(t, x, z) = u(ξ(t, x, z)) and V˜ (t, x, z) =: v(t, x, z) = v(ξ(t, x, z)) with. Z t δ(τ, z)dτ + c, ξ(t, x, z) = kx + s 0

where k, s and c are arbitrary constants which satisfy sk 6= 0 , δ(τ ) is a nonzero function of the indicated variables to be determined later. So, Eq.(3.1) can be transformed into the following (ODE).  0 0 0 000 sδu + kh1 uu + kh2 vv + k 3 h3 u = 0, (3.2) 0 0 000 sδv + kh4 uv + k 3 h3 v = 0,

where the prime denote to the differential with respect to ξ. In view of F-expansion method, the solution of Eq. (3.1), can be expressed in the form.  i u(t, x, z) = u(ξ) = ΣN i=1 (ai F (ξ)), (3.3) i v(t, x, z) = v(ξ) = ΣM i=1 (bi F (ξ)), where ai and bi are constants to be determined later. considering homogeneous balance 000 0 0 000 0 between u and uu , vv and the order of v and uv in (3.2), then we can obtain N = M = 2, so (3.3) can be rewritten as following.  u(t, x, z) = a0 + a1 F (ξ) + a2 F 2 (ξ), (3.4) v(t, x, z) = b0 + b1 F (ξ) + b2 F 2 (ξ), where a0 , a1 , a2 , b0 , b1 and b2 are constants to be determined later. Substituting (3.4) with the conditions (2.5),(2.6) into (3.2) and collecting all terms with the same power

4

Wick-type Stochastic Coupled KdV Equations Using F-expansion Method, Hossam A. Ghany and M. Zakarya

of f i (ξ)[f 0 (ξ)]j , (i = 0 ± 1, ±2, ..., j = 0, 1). as following.  2 2 2 3 0 2k[12k a P h + b h + a h ]F F 2 3 2 1  2 2   +3k[2k 2 a P h + a a h + b b h ]F 2 F 0   1 3 1 2 1 1 2 2  0  2 3  +[2a2 sδ + kb1 h2 + 8k a2 Qh3 + 2ka0 a2 h1 + ka21 h1 + 2kb0 b2 h2 ]F F   0    +[sa1 h3 + ka0 a1 h1 + kb0 b1 h2 + k 3 a1 Qh3 ]F = 0, 0   2kb2 [12k 2 P h3 + a2 h4 ]F 3 F    0   +k[6k 2 b1 P h3 + 2a1 b2 h4 + a2 b1 h4 ]F 2 F   0   +[2sb2 δ + 2ka0 b2 h4 + ka1 b1 h4 + 8k 3 b2 Qh3 ]F F   0 +b1 [sδ + ka0 h4 + k 3 Qh3 ]F = 0.

57

(3.5)

Setting each coefficient of f i (ξ)[f 0 (ξ)]j to be zero, we get a system of algebraic equations which can be expressed by. 2k[12k 2 a2 P h3 + b22 h2 + a22 h1 ] = 0, 3k[2k 2 a1 P h3 + a1 a2 h1 + b1 b2 h2 ] = 0, 2a2 sδ + kb21 h2 + 8k 3 a2 Qh3 + 2ka0 a2 h1 + ka21 h1 + 2kb0 b2 h2 = 0, sa1 h3 + ka0 a1 h1 + kb0 b1 h2 + k 3 a1 Qh3 = 0, (3.6) 2

2kb2 [12k P h3 + a2 h4 ] = 0, k[6k 2 b1 P h3 + 2a1 b2 h4 + a2 b1 h4 ] = 0, 2sb2 δ + 2ka0 b2 h4 + ka1 b1 h4 + 8k 3 b2 Qh3 = 0, b1 [sδ + ka0 h4 + k 3 Qh3 ] = 0. with solving by Maple to get the following coefficient.  a1 = b1 = 0, a0 = arbitrary constant,    12k2 P h3 (t,z)   , a 2 = −  h4 (t,z)  q q  2 h3 (t,z) h1 (t,z)−h4 (t,z) h1 (t,z)−h4 (t,z) b2 = ±i 12kh4P(t,z) = ∓ia , 2 h2 (t,z) h2 (t,z)  2  [3k Qh (t,z)−a (h1 (t,z)−h4 (t,z))]  ,  b0 = ± √h3 (t,z)[h0(t,z)−h  2 1 4 (t,z)]   2 Qh (t,z)]  3 . δ = −k[a0 h4 (t,z)+k S

(3.7)

Substituting by coefficient (3.7) into (3.4) yields general form solutions of eq. (1.2). u(t, x, z) = a0 − [

12k 2 P h3 (t, z) 2 ]F (ξ(t, x, z)), h4 (t, z) 5

(3.8)

Wick-type Stochastic Coupled KdV Equations Using F-expansion Method, Hossam A. Ghany and M. Zakarya

58

2

√ 3 (t,z)+a0 (h1 (t,z)−h4 (t,z))] v(t, x, z) = ± [3k Qh h2 (t,z)[h1 (t,z)−h4 (t,z)]

2 h (t,z) 3 ± 12ikh4P(t,z)

with 

Z

ξ(t, x, z) = k x −

q

(3.9)

h1 (t,z)−h4 (t,z) 2 F (ξ(t, x, z)), h2 (t,z)

t



2



a0 h4 (τ, z) + k Qh3 (τ, z) dτ .

0

From Appendix C, we give the special cases as following: Case 1. if we take P = 1, Q = (2 − m2 ) and R = (1 − m2 ), we have F (ξ) → cs(ξ) , 12k 2 h3 (t, z) ] cs2 (ξ1 (t, x, z)), h4 (t, z)

u1 (t, x, z) = a0 − [ v1 (t, x, z) = ± [3k

2h

3 (t,z)(2−m

√

with ξ1 (t, x, z) = k x −

Z

0

2 )+a (h (t,z)−h (t,z))] 0 1 4

h2 (t,z)(h1 (t,z)−h4 (t,z))

2 h (t,z) 3 ± 12ikh4 (t,z)



(3.10)

q

(3.11) h1 (t,z)−h4 (t,z) h2 (t,z)

t

cs2 (ξ1 (t, x, z)),

2



2



a0 h4 (τ, z) + k (2 − m )h3 (τ, z) dτ .

In the limit case when m → o we have cs(ξ) → cot(ξ), thus (3.10),(3.11) become. u2 (t, x, z) = a0 − [

12k 2 h3 (t, z) ] cot2 (ξ2 (t, x, z)), h4 (t, z)

(3.12)

2

h3 (t,z)+a0 (h1 (t,z)−h4 (t,z))] v2 (t, x, z) = ± [6k √ h2 (t,z)[h1 (t,z)−h4 (t,z)]

2 h (t,z) 3 ± 12ikh4 (t,z)

with 

ξ2 (t, x, z) = k x −

Z

q

h1 (t,z)−h4 (t,z) h2 (t,z)

t

(3.13) 2

cot (ξ2 (t, x, z)),

2





a0 h4 (τ, z) + 2k h3 (τ, z) dτ .

0

In the limit case when m → 1 we have cs(ξ) → csch(ξ), thus (3.10).(3.11) become. u3 (t, x, z) = a0 − [

12k 2 h3 (t, z) ] csch2 (ξ3 (t, x, z)), h4 (t, z) 6

(3.14)

Wick-type Stochastic Coupled KdV Equations Using F-expansion Method, Hossam A. Ghany and M. Zakarya

59

2

h3 (t,z)+a0 (h1 (t,z)−h4 (t,z))] v3 (t, x, z) = ± [3k √ h2 (t,z)[h1 (t,z)−h4 (t,z)]

2 h (t,z) 3 ± 12ikh4 (t,z)

with



ξ3 (t, x, z) = k x −

q

Z

0

h1 (t,z)−h4 (t,z) h2 (t,z)

(3.15) 2

csch (ξ3 (t, x, z)),

 a0 h4 (τ, z) + k h3 (τ, z) dτ .

t

2

Case 2. if we take P = 1, Q = (2m2 − 1) and R = −m2 (1 − m2 ), then F (ξ) → ds(ξ) , 12k 2 h3 (t, z) ] ds2 (ξ4 (t, x, z)), h4 (t, z)

u4 (t, x, z) = a0 − [ v4 (t, x, z) = ± [3k

2 h (t,z)(2m2 −1)+a (h (t,z)−h (t,z))] 3 0 1 4

√

2 h (t,z) 3 ± 12ikh4 (t,z)

with



ξ4 (t, x, z) = k x −

Z

0

(3.16)

h2 (t,z)[h1 (t,z)−h4 (t,z)]

(3.17)

q

h1 (t,z)−h4 (t,z) h2 (t,z)

t

2

ds (ξ4 (t, x, z)),

2

2





a0 h4 (τ, z) + k (2m − 1)h3 (τ, z) dτ .

In the limit case when m → o we have ds(ξ) → csc(ξ), thus (3.16),(3.17) become. 12k 2 h3 (t, z) ] csc2 (ξ5 (t, x, z)), h4 (t, z)

u5 (t, x, z) = a0 − [

(3.18)

2

v5 (t, x, z) = ± [−3k√h3 (t,z)+a0 (h1 (t,z)−h4 (t,z))] h2 (t,z)[h1 (t,z)−h4 (t,z)]

q

2 h (t,z) 3 ± 12ikh4 (t,z)

with



ξ5 (t, x, z) = k x −

Z

0

h1 (t,z)−h4 (t,z) h2 (t,z)

(3.19)

csc2 (ξ5 (t, x, z)),

 a0 h4 (τ, z) − k h3 (τ, z) dτ .

t

2

Remark that. In the limit case when m → 1 we have ds(ξ) = cs(ξ) → csch(ξ), thus (3.16),(3.17) become the same solutions in case 1. Case 3. if we take P = 14 , Q =

1−2m2 2

and R = 14 , then F (ξ) → ns(ξ) ± cs(ξ), 2  3k 2 h3 (t, z) ns(ξ6 (t, x, z)) ± cs(ξ6 (t, x, z)) , u6 (t, x, z) = a0 − h4 (t, z) 7

(3.20)

Wick-type Stochastic Coupled KdV Equations Using F-expansion Method, Hossam A. Ghany and M. Zakarya

v6 (t, x, z) = ± [3k

2h

3 (t,z)(1−2m

2

2 3 (t,z) ± 3ikh4h(t,z)

with

√

2 )+2a (h (t,z)−h (t,z))] 0 1 4

h2 (t,z)[h1 (t,z)−h4 (t,z)]

q



60

1 ξ6 (t, x, z) = k x − 2

h1 (t,z)−h4 (t,z) h2 (t,z)

Z

0



2

(3.21)

ns(ξ6 (t, x, z)) ± cs(ξ6 (t, x, z)) ,

t

2



2



2a0 h4 (τ, z) + k (1 − 2m )h3 (τ, z) dτ .

In the limit case when m → o we have (ns(ξ) ± cs(ξ)) → (csc(ξ) ± cot(ξ)), thus (3.22),(3.23) become. 2  3k 2 h3 (t, z) u7 (t, x, z) = a0 − csc(ξ7 (t, x, z)) ± cot(ξ7 (t, x, z)) , h4 (t, z)

(3.22)

2

3 (t,z)+2a0 (h1 (t,z)−h4 (t,z))] v7 (t, x, z) = ± [3k h√

2

±

h2 (t,z)[h1 (t,z)−h4 (t,z)]

3ik2 h3 (t,z) h4 (t,z)

with

q

h1 (t,z)−h4 (t,z) h2 (t,z)



1 ξ7 (t, x, z) = k x − 2

Z



2

(3.23)

csc(ξ7 (t, x, z)) ± cot(ξ7 (t, x, z)) ,

t

2





2a0 h4 (τ, z) + k h3 (τ, z) dτ .

0

In the limit case when m → 1 we have (ns(ξ) ± cs(ξ)) → (coth(ξ) ± csch(ξ)), thus (3.22),(3.23) become.  2 3k 2 h3 (t, z) coth(ξ8 (t, x, z)) ± csch(ξ8 (t, x, z)) , u8 (t, x, z) = a0 − h4 (t, z)

(3.24)

2

h3 (t,z)+2a0 (h1 (t,z)−h4 (t,z))] v8 (t, x, z) = ± [−3k √ 2

2 3 (t,z) ± 3ikh4h(t,z)

with

h2 (t,z)[h1 (t,z)−h4 (t,z)]

q

2

h1 (t,z)−h4 (t,z) h2 (t,z)





 2a0 h4 (τ, z) − k h3 (τ, z) dτ .

1 ξ8 (t, x, z) = k x − 2

Z

0

(3.25)

coth(ξ8 (t, x, z)) ± csch(ξ8 (t, x, z)) ,

t

2

Remark that: there are another solutions for Eq.(1.2). These solutions come from setting different values for the cofficients P, Q and R.(see Appendex C.). The above mentioned cases are just to clarify how far our technique is applicable. 8

Wick-type Stochastic Coupled KdV Equations Using F-expansion Method, Hossam A. Ghany and M. Zakarya

4

61

White Noise Functional Solutions of Eq.(1.2)

In this section, we employ the results of the Section 3 by using Hermite transform to obtain exact white noise functional solutions for Wick-type stochastic coupled KdV equations (1.2). The properties of exponential and trigonometric functions yield that there exists a bounded open set D ⊂ R+ × R, ρ < ∞, λ > 0 such that the solution u(t, x, z) of Eq.(3.1) and all its partial derivatives which are involved in Eq. (3.1) are uniformly bounded for (t, x, z) ∈ D × Kρ (λ), continuous with respect to (t, x) ∈ D for all z ∈ Kρ (λ) and analytic with respect to z ∈ Kρ (λ), for all (t, x) ∈ D. From Theorem 4.1.1 in [20], there exists U (t, x, z) ∈ (S)−1 such that u(t, x, z) = U˜ (t, x)(z) for all (t, x, z) ∈ D × Kρ (λ) and U (t, x) solves Eq.(1.2) in (S)−1 . Hence, by applying the inverse Hermite transform to the results of Section 3, we get New exact white noise functional solutions of Eq.(1.2) as follows: • New Wick-type stochastic solutions of (JEF): U1 (t, x) = a0 − [ V1 (t, x) = ± [3k

2H

3 (t)(2−m

√



±

2 )+a (H (t)−H (t))] 0 1 4

q

(4.2) H1 (t)−H4 (t) H2 (t)

 cs2 (Ξ1 (t, x)),

12k 2 H3 (t) ]  ds2 (Ξ2 (t, x)), H4 (t)

U2 (t, x) = a0 − [ 2H

(4.1)

H2 (t)[H1 (t)−H4 (t)]

2 ± 12ikH4H(t)3 (t)

V2 (t, x) = ± [3k

12k 2 H3 (t) ]  cs2 (Ξ1 (t, x)), H4 (t)

3 (t)(2m

√

(4.3)

2 −1)+a (H (t)−H (t))] 0 1 4

H2 (t)[H1 (t)−H4 (t)]

12ik2 H3 (t) H4 (t)



q

(4.4) H1 (t)−H4 (t) H2 (t)

 ds2 (Ξ2 (t, x)),

 2 3k 2 H3 (t)    ns (Ξ3 (t, x)) ± cs (Ξ3 (t, x)) , U3 (t, x) = a0 − H4 (t) V3 (t, x) = ± [3k ±

2H

3ik2 H

3 (t)(1−2m

2

3 (t) H4 (t)

√ 

(4.5)

2 )+2a (H (t)−H (t))] 0 1 4

H2 (t)[H1 (t)−H4 (t)]

q

H1 (t)−H4 (t) H2 (t)



 ns (Ξ3 (t, x)) ± cs (Ξ3 (t, x)) 9

2

(4.6) ,

Wick-type Stochastic Coupled KdV Equations Using F-expansion Method, Hossam A. Ghany and M. Zakarya

62

with 

Z



Z

Ξ1 (t, x) = k x −

Ξ2 (t, x) = k x −

0

0



Z

1 Ξ3 (t, x) = k x − 2

t

2

a0 H4 (τ ) + k (2 − m )H3 (τ ) dτ ,

t

0





2

2





2

a0 H4 (τ ) + k (2m − 1)H3 (τ ) dτ ,

t

2

2





2a0 H4 (τ ) + k (1 − 2m )H3 (τ ) dτ .

• New Wick-type stochastic solutions of trigonometric functions: 12k 2 H3 (t)  cot2 (Ξ4 (t, x)), H4 (t)

U4 (t, x) = a0 −

(4.7)

2

V4 (t, x) = ± [6k√H3 (t)+a0 (H1 (t)−H4 (t))] H2 (t)(H1 (t)−H4 (t))

2 ± 12ikH4H(t)3 (t)

U5 (t, x) = a0 − [



q

(4.8)

H1 (t)−H4 (t) H2 (t)

2

 cot (Ξ4 (t, x)),

12k 2 H3 (t) ]  csc2 (Ξ5 (t, x)), H4 (t)

2

V5 (t, x) = ± [−3k√H3 (t)+a0 (H1 (t)−H4 (t))] ± H2 (t)[H1 (t)−H4 (t)]

q



H1 (t)−H4 (t) H2 (t)

12ik2 H3 (t) H4 (t)

(4.10)

 csc2 (Ξ5 (t, x)),

 2 3k 2 H3 (t)   U6 (t, x) = a0 −  csc (Ξ6 (t, x)) ± cot (Ξ6 (t, x)) , H4 (t) 2

H3 (t)+2a0 (H1 (t)−H4 (t))] V6 (t, x) = ± [3k √ ± 2

(4.9)

H2 (t)[H1 (t)−H4 (t)]

3ik2 H3 (t) H4 (t)

 2 q H1 (t)−H4 (t)     csc (Ξ6 (t, x)) ± cot (Ξ6 (t, x)) , H2 (t) 10

(4.11)

(4.12)

Wick-type Stochastic Coupled KdV Equations Using F-expansion Method, Hossam A. Ghany and M. Zakarya

63

with 

Z



Z

Ξ4 (t, x) = k x −

t

0



1 Ξ6 (t, x) = k x − 2



a0 H4 (τ ) + 2k H3 (τ ) dτ ,

0

Ξ5 (t, x) = k x −



2

Z

t



2



a0 H4 (τ ) − k H3 (τ ) dτ ,

t





2

2a0 H4 (τ ) + k H3 (τ ) dτ .

0

• New Wick-type stochastic solutions of hyperbolic functions:

U7 (t, x) = a0 − [

12k 2 H3 ]  csch2 (Ξ7 (t, x)), H4

2

V7 (t, x) = ± [3k√H3 (t)+a0 (H1 (t)−H4 (t))] ± H2 (t)[H1 (t)−H4 (t)]

q



H1 (t)−H4 (t) H2 (t)

(4.13)

12ik2 H3 (t) H4 (t)

(4.14)

2

 csch Ξ7 (t, x),

 2 3k 2 H3 (t)   U8 (t, x) = a0 −  coth (Ξ8 (t, x)) ± csch (Ξ8 (t, x)) , H4 (t) 2

V8 (t, x) = ± [−3k√H3 (t)+2a0 (H1 (t)−H4 (t))] ± 2

H2 (t)[H1 (t)−H4 (t)]

(4.15)

3ik2 H3 (t) H4 (t)

 2 q H1 (t)−H4 (t)     coth (Ξ8 (t, x)) ± csch (Ξ8 (t, x)) , H2 (t)

(4.16)

with 

Ξ7 (t, x) = k x − 

1 Ξ8 (t, x) = k x − 2

Z

t





2

a0 H4 (τ ) + k H3 (τ ) dτ ,

0

Z

0

t

2





2a0 H4 (τ ) − k H3 (τ ) dτ .

We observe that. For different forms of H1 , H2 , H3 and H4 , we can get different exact white noise functional solutions of Eq.(1.2) from Eqs.(4.1)-(4.16).

11

Wick-type Stochastic Coupled KdV Equations Using F-expansion Method, Hossam A. Ghany and M. Zakarya

5

64

Example

It is well known that Wick version of function is usually difficult to evaluate. So, in this section, we give non-Wick version of solutions of Eq.(1.2). Let Wt = B˙ t be the Gaussian white P noise, where Bt is the Brownian motion. We have the Hermite Rt 2 ∞ ˜ t (z) = transform W z η (s)ds [20]. Since exp (Bt ) = exp(Bt − t2 ), we have i=1 i 0 i 2 2 2 sin (Bt ) = sin(Bt − t2 ), cos (Bt ) = cos(Bt − t2 ), cot (Bt ) = cot(Bt − t2 ), csc (Bt ) = 2 2 2 csc(Bt − t2 ), coth (Bt ) = coth(Bt − t2 ) and csch (Bt ) = csch(Bt − t2 ). Suppose that H1 (t) = H2 (t) = λ1 H3 (t), H3 (t) = λ2 H4 (t) and H4 (t) = Γ(t) + λ3 Wt where λ1 , λ2 and λ3 are arbitrary constants and Γ(t) is integrable or bounded measurable function on R+ . Therefore, for H1 (t)H2 (t)H3 (t)H4 (t) 6= 0. thus exact white noise functional solutions of Eq.(1.2) are as follows:   a0 2 2 U9 (t, x) = 3k − 4λ2 cot (Φ1 (t, x)) , (5.1) 3k 2 V9 (t, x) = ±6k 2 λ2



1+

√

a0 6k2 λ2 (λ1 λ2 −1)

λ1 λ2 (λ1 λ2 −1)



+ (5.2)

 q (λ1 λ2 −1) 2 2i cot (Φ1 (t, x)) , λ 1 λ2 U10 (t, x) = 3k

2



 a0 2 − 4λ2 csc (Φ2 (t, x)) , 3k 2

V10 (t, x) = ±3k 2 λ2 4i

q



a0 −1 3k2 λ2 (λ1 λ2 −1)

√

(λ1 λ2 −1) λ1 λ 2

λ1 λ2 (λ1 λ2 −1)

+ (5.4)



csc2 (Φ2 (t, x)) ,



2



(5.3)

2

U11 (t, x) = a0 − 3k λ2 csc(Ξ(t, x)) ± cot(Φ3 (t, x)) ,

V11 (t, x) = ±3k 2 λ2



1+

2

2a0 3k2 λ2 (λ1 λ2 −1)

√



λ1 λ2 (λ1 λ2 −1)

+

 2  q (λ1 λ2 −1) i csc(Φ(t, x)) ± cot(Φ3 (t, x)) , λ1 λ2 12

(5.5)

(5.6)

Wick-type Stochastic Coupled KdV Equations Using F-expansion Method, Hossam A. Ghany and M. Zakarya

65

U12 (t, x) = a0 − 12k 2 λ2 csch2 (Φ4 (t, x)), V12 (t, x) = ±3k 2 λ2



1+

√

a0 3k2 λ2 (λ1 λ2 −1)

λ1 λ2 (λ1 λ2 −1)



(5.7)

+ (5.8)

 q (λ1 λ2 −1) 2 csch (Φ4 (t, x) , 4i λ 1 λ2 

2

2

U13 (t, x) = a0 − 3k λ2 coth(Φ5 (t, x)) ± csch(Φ5 (t, x)) ,

2

V13 (t, x) = ±3k λ2



2a0 −1 3k2 λ2 (λ1 λ2 −1)

2

√



λ1 λ2 (λ1 λ2 −1)

(5.9)

+ (5.10)

2   q (λ1 λ2 −1) 3i coth(Φ5 (t, x)) ± csch(Φ5 (t, x)) , λ 1 λ2 with 

2

Φ1 (t, x) = k x − (a0 + 2k λ2 ) 

2

Φ2 (t, x) = k x − (a0 − k λ2 )

Z

Z



(2a0 + k 2 λ2 ) Φ3 (t, x) = k x − 2 

2

Φ4 (t, x) = k x − (a0 + k λ2 )

t 0

t

t2 Γ(τ )dτ + λ3 [Bt − ] 2

0

Z

Z

t2 Γ(τ )dτ + λ3 [Bt − ] 2

t 0

t 0





t2 Γ(τ )dτ + λ3 [Bt − ] 2

t2 Γ(τ )dτ + λ3 [Bt − ] 2

,

,



,



and 

(2a0 − k 2 λ2 ) Φ5 (t, x) = k x − 2

Z

13

t 0

t2 Γ(τ )dτ + λ3 [Bt − ] 2



.

Wick-type Stochastic Coupled KdV Equations Using F-expansion Method, Hossam A. Ghany and M. Zakarya

6

66

Conclusion

We have discussed the solutions of stochastic (PDEs) driven by Gaussian white noise. There is a unitary mapping between the Gaussian white noise space and the Poisson white noise space. This connection was given by Benth and Gjerde [2]. We can see in the section 4.9 [20] clearly. Hence, by the aid of the connection, we can derive some stochastic exact soliton solutions if the coefficients , and are Poisson white noise functions in Eq. (1.2). In this paper, using Hermite transformation, white noise theory and F-expansion method, we study the white noise solutions of the Wick-type stochastic coupled KdV equations. This paper shows that the F-expansion method is sufficient to solve the stochastic nonlinear equations in mathematical physics. The method which we have proposed in this paper is powerful, direct and computerized method, which allows us to do complicated and tedious algebraic calculation. It is shown that the algorithm can be also applied to other NLPDEs in mathematical physics such as modified Hirota-Satsuma coupled KdV, (2+1)-dimensional coupled KdV, KdV-Burgers, schamel KdV, modified KdV Burgers, Sawada-Kotera, Zhiber-Shabat equations and Benjamin-Bona-Mahony equations. Since the equation (1.2) has other solutions if select other values of P, Q and R (see Appendix A, B and C). So there are many other of exact solutions for wick-type stochastic coupled KdV equations. Appendix A. the jacobi elliptic functions degenerate into trigonometric functions when m → 0. snξ → sin ξ, cnξ → cos ξ, dnξ → 1, scξ → tan ξ, sdξ → sin ξ, cdξ → cos ξ, nsξ → csc ξ, ncξ → sec ξ, ndξ → 1, csξ → cot ξ, dsξ → csc ξ, dcξ → sec ξ. Appendix B. the jacobi elliptic functions degenerate into hyperbolic functions when m → 1. snξ → tan ξ, cnξ → sech ξ, dnξ → sech ξ, scξ → sinh ξ, sdξ → sinh ξ, cdξ → 1, nsξ → coth ξ, ncξ → cosh ξ, ndξ → cosh, csξ → csch ξ, dsξ → csch ξ, dcξ → 1. Appendix C. The ODE and Jacobi Elliptic Functions Relation between values of (P , Q , R) and corresponding F (ξ) in ODE (F 0 )2 (ξ) = P F 4 (ξ) + QF 2 (ξ) + R,

14

Wick-type Stochastic Coupled KdV Equations Using F-expansion Method, Hossam A. Ghany and M. Zakarya

P m2 −m2 −1 1 1 − m2

m2 − 1 1 − m2

−m2 (1 − m2 )

Q −1 − m2

R 1

2m2 − 1

1 − m2

−1 − m2

m2

2 − m2

2m2 − 1 2 − m2 2 − m2

2m2 − 1

67

F (ξ) snξ, cdξ = cnξ

m2 − 1

dnξ ξ nsξ = sn1 ξ , dcξ = dn cnξ ncξ = cn1 ξ

−m2 −1

ndξ =

1

scξ =

1

sdξ =

1

2 − m2

1 − m2

csξ =

1

2m2 − 1

−m2 (1 − m2 )

dsξ =

m4 4

m2 −2 2

1 4

m2 4

m2 −2 2

m2 4

1 4

1−2m2 2

1 4

m2 −1 4

m2 +1 2

m2 −1 4

1−m2 4

m2 +1 2

1−m2 4

−1 4 1 4

m2 +1 2 m2 −2 2

−(1−m2 )2 4 m2 4

cnξ dnξ

1

dnξ snξ cnξ snξ dnξ cnξ snξ dnξ snξ cnξ

snξ , √ 1−m2 ±dnξ 1±dnξ snξ ± icnξ, nsξ ± csξ,

dnξ √ , m snξ i 1−m2 snξ±cnξ 1±dnξ cnξ √ , snξ , 1−m2 snξ±dnξ 1±cnξ dnξ 1±msnξ

ξ ncξ ± iscξ 1±cn snξ

mcnξ ± dnξ nsξ ± dsξ

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Wick-type Stochastic Coupled KdV Equations Using F-expansion Method, Hossam A. Ghany and M. Zakarya

68

[8] M. Dehghan and M. Tatari. Chaos Solitons and Fractals, 36 (2008): 157-166. [9] E. Fan. Phys. Lett. A, 277 (2000): 212-218. [10] Z. T. Fu, S. K. Liu, S. D. Liu, Q. Zhao. Phys. Lett. A, 290 (2001): 72-76. [11] D. D. Ganji and A. Sadighi. Int J Non Sci Numer Simul, 7 (2006): 411-418. [12] H. A. Ghany. Chin. J. Phys., 49 (2011): 926-940. [13] H. A. Ghany. International Journal of pure and applied mathematics, 78 (2012): 17-27. [14] H. A. Ghany and A. Hyder. International Review of Physics, 6 (2012): 153-157. [15] H. A. Ghany and A. Hyder. J. Comput. Anal. Appl., 15 (2013): 1332-1343. [16] H. A. Ghany and A. Hyder. Kuwait Journal of Science, (2013), Accepted. [17] H. A. Ghany and A. Hyder. IAENG International Journal of Applied Mathematics, (2013), Accepted. [18] H. A. Ghany and M. S. Mohammed. Chin. J. Phys., 50 (2012): 619-627. [19] H. A. Ghany, A. S. Okb El Bab, A. M. Zabal and A. Hyder. Chin. Phys. B, (2013), Accepted. [20] H. Holden, B. Øsendal, J. U b øe and T. Zhang. Stochastic partial differential equations, Bihk¨auser: Basel, (1996). [21] J .H. He and X. H. Wu. Chaos Solitons and Fractals, 30 (2006):700-708. [22] J. H. He and M. A. Abdou Chaos Solitons and Fractals, 34 (2007): 1421-1429. [23] V.V. Konotop, L. Vazquez. Nonlinear Random Waves, World Scientific, Singapore, (1994). [24] S. K. Liu, Z. T. Fu, S. D. Liu, Q. Zhao. Phys. Lett. A, 289 (2001): 69-74. [25] J. Liu, L. Yang, K. Yang Chaos, Solitons and Fractals, 20 (2004): 1157-1164. [26] E. J. Parkes, B. R. Duffy, P. C. Abbott Phys. Lett. A, 295 (2002): 280-286. [27] F. Shakeri and M. Dehghan. Math. Comput. Model, 48 (2008): 486-498. [28] M. Wadati. J. Phys. Soc. Jpn., 52 (1983): 2642-2648. [29] A. M. Wazwaz. Chaos Solitons Fractals, 188 (2007): 1930-1940. [30] M. Wadati and Y. Akutsu. J. Phys. Soc. Jpn., 53 (1984): 3342-3350. 16

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[31] A. M. Wazwaz. Appl. Math. Comput, 177 (2006): 755-760. [32] A. M. Wazwaz. Appl. Math. Comput, 169 (2005): 321-338. [33] X. Ma, L. Wei, and Z. Guo. J. Sound Vibration, 314 (2008): 217-227. [34] X. H. Wu and J.H. He. Chaos Solitons and Fractals, 38 (2008): 903-910. [35] Y. C. Xie . Phys. Lett. A, 310 (2003): 161-167. [36] Y. C. Xie. Solitons and Fractals, 21 (2004): 473-480. [37] Y. C. Xie. Solitons and Fractals, 20 (2004): 337-342. [38] Y. C. Xie. J. Phys. A: Math. Gen, 37 (2004): 5229-5236. [39] Y. C. Xie. Phys. Lett. A, 310 (2003): 161-167. [40] S. Zhang, T. C. Xia. Communications in Theoretical Physics, 46 (2006): 985-990. [41] S. D. Zhu. Int J Non Sci Numer Simul, 8 (2007): 461-464. [42] S. D. Zhu. Int J Non Sci Numer Simul, 8 (2007): 465-468. [43] Yubin Zhou. Mingliang Wang. Yueming Wang. Physics Letters A, 308 (2003): 3136. [44] Sheng Zhang. Tiecheng Xia. Applied Mathematics and Computation, 189 (2007): 836843. [45] Sheng Zhang. TieCheng Xia. Applied Mathematics and Computation, 183 (2006): 11901200. [46] Sheng Zhang. Tiecheng Xia. Communications in Nonlinear Science and Numerical Simulation, 13 (2008): 12941301.

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J. APPLIED FUNCTIONAL ANALYSIS, VOL. 10, NO.'S 1-2,70-77 ,2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

A NOTE ON n-BANACH LATTICES ¼ 1 AND N I·HAN GÜNGÖR 2 B I·RSEN SA GIR Abstract. In this paper we describe n-Banach lattices by Riesz space equipped with n-norm. We consider spaces of reguler operators and order bounded operators between n-Banach lattices and Banach lattices, we generate norm on these space by the aid of n-norm. Then we investigate properties of operators on these space with generated norm.

1. Introduction Riesz spaces, also called vector lattices, K-lineals which were …rst considered by F. Riesz, L. Kantorovic, and H. Freudenthal in the middle of nineteen thirties. L. Kantorovic and his school …rst recognized the importance of studying vector lattices in connection with Banach’s theory of normed vector spaces; they investigated normed vector lattices as well as order-related linear operators between such vector lattices. In the middle seventies the research on this subject was essentially in‡uenced by H.H. Schaefer. H. H. Schaefer present the theory of Banach lattices and positive operators as an inseparable part of the general Banach space and operator theory. In particular, deep results of the general theory and classical analysis were used to prove related properties in case of general Banach lattices. More recently other important study concerning this subject appeared CD. Aliprantis and O. Burkinshaw. Let a real vector space X be an ordered vector space equipped with an order relation . A vector x in an ordered vector space X is called positive whenever 0 x holds. The set of all positive vectors denoted by X + . An operator is a map between two vector space. An operator T : X ! Y between two ordered vector space is called positive if 0 T (x) for all x 2 X. A Riesz space is an ordered vector space X with property that for each pair vectors x; y 2 X the supremum and the in…mum of the set fx; yg both exists in X: For any vector x in a Riesz space X, de…ne x+ := x _ 0 = sup fx; 0g, x := x ^ 0 = sup f x; 0g, jxj := sup fx; xg. If T : X ! Y be an linear operator between two Riesz spaces such that sup fjT yj : jyj xg exists in Y for each x 2 X + , then the modulus of T exists and jT j x = sup fjT yj : jyj xg for each x 2 X + . Also jT (x)j jT j (jxj) holds for all x 2 X. A subset A of a Riesz space is said to be bounded above (bounded below) whenever there exists some x satisfying y x (x y) for all _ A subset in a Riesz space is called order bounded if it is bounded both above y 2 X. and below. A Riesz space is called Dedekind complete whenever every nonempty bounded above subset has supremum (whenever every nonempty bounded below subset has in…mum). An linear operator T : X ! Y between two Riesz spaces is 2000 Mathematics Subject Classi…cation. 46B42. Key words and phrases. Riesz spaces, n-norm, n-Banach lattices, reguler operators, order bounded operators. 1

70

71

¼ 1 AND N I·HAN GÜNGÖR 2 B I·RSEN SA GIR

2

said to be reguler if it can be written as a di¤erence of two positive operators. The vector space of all reguler operators form X to Y denoted by Lr (X; Y ). Of course this is equivalent to saying there exists a positive operator S : X ! Y satisfying T S. An linear operator T : X ! Y between two Riesz spaces is said to be order bounded if it maps an order bounded subsets of X to order bounded subsets of Y . The vector space of all order bounded operators from X to Y denoted by Lb (X; Y ). Every positive operator is order bounded. Therefore every reguler operator is also order bounded. Hence Lr (X; Y ) Lb (X; Y ). A norm k:k on a Riesz space is said to be a lattice norm whenever jxj jyj implies kxk kyk. For more detailed, it can be looked [1, 2, 6, 7]. 2. Operators on n-Banach Lattices Fistly, we will give the some required de…nitions. The notion of n-normed space as well as its properties was introduced by S. Gähler in [4] as a generalization of the normed space. Let X be a real linear space. A function k:; : : : ; :k : X n ! R is called a n-norm on X n if it satis…es following conditions, for all 2 R and x1 ; x2 ; :::; xn ; y 2 X; i) kx1 ; x2 ; :::; xn k = 0 , x1 ; x2 ; :::; xn are linearly dependent ii) kx1 ; x2 ; :::; xn k is invariant under permutations of x1 ; x2 ; :::; xn iii) k x1 ; x2 ; :::; xn k = j j kx1 ; x2 ; :::; xn k iv)kx1 + y; x2 ; :::; xn k kx1 ; x2 ; :::; xn k + ky; x2 ; :::; xn k. Then the pair (X; k:; : : : ; :k) is called n-normed space. A sequence fxk g of n-normed space X is said to be convergent if there exists an element x 2 X such that lim kxk

k!1

x; z1 ; :::; zn

1k

=0

for all z1 ; :::; zn 1 2 X [5]. Let (X; k:; : : : ; :k) and (Y; k:; : : : ; :k) be n-normed spaces. For an operator T : (X; k:; : : : ; :k) ! (Y; k:; : : : ; :k), de…ne [:]n by [T ]n := sup

kT (x1 ) ; T (x2 ) ; :::; T (xn )k : kx1 ; x2 ; :::; xn k = 6 0 . kx1 ; x2 ; :::; xn k

But, this natural de…nition of [:]n is not a norm [8] .For this reason, we need to take Y as normed space. Let X be n-normed space and Y be normed space. An operator T : X n ! Y be a n-linear operator on X if T is linear in each of variable. An n-linear operator is bounded if there exists a constant k > 0 such that , kT (x1 ; x2 ; :::; xn )k

k kx1 ; x2 ; :::; xn k

for all (x1 ; x2 ; :::; xn ) 2 X n . If T is a bounded n-linear operator, then the n-operator norm de…ned by jjjT jjj := sup fkT (x1 ; x2 ; :::; xn )k : kx1 ; x2 ; :::; xn k

1g

and the space of all bounded n-linear operators from X n to Y denoted by L(X n ; Y ). If Y is a Banach space, then L(X n ; Y ) is a Banach space with n-operator norm. Also it’s known that when T : X n ! Y is a n-linear operator, T is bounded if and only if T is continuous [8].

72

A NOTE ON n-BANACH LATTICES

3

Let X be a Riesz space. The cartesian product X n is a Riesz space under the ordering x = (x1 ; x2 ; :::; xn ) z = (z1 ; z2 ; :::; zn ) , xi zi holds in X for each i = 1; 2; ::; n. If x = (x1 ; x2 ; :::; xn ) and z = (z1 ; z2 ; :::; zn ) are vectors of X n , then x _ y = (x1 _ z1 ; x2 _ z2 ; :::; xn _ zn ) and x ^ y = (x1 ^ z1 ; x2 ^ z2 ; :::; xn ^ zn ) [2].Hence we can see jxj = j(x1 ; x2 ; :::; xn )j = (jx1 j ; jx2 j ; :::; jxn j) for all x = (x1 ; x2 ; :::; xn ) 2 X n . Also, it’s clearly that +

+ + x+ = (x1 ; x2 ; :::; xn ) = x+ 1 ; x2 ; :::; xn

and x = (x1 ; x2 ; :::; xn ) = x1 ; x2 ; :::; xn : +

+

We demonstrate the space of positive vectors x+ = (x1 ; x2 ; :::; xn ) by X n . Through this paper X n will be equipped with this ordering: In [2, 10], the space of reguler operators and order bounded operators between Banach lattices is de…ned and equipped with reguler norm and bound norm, respectively. Also some properties of these operators are showed. In this study, we de…ne n-Banach lattice with the aid of Riesz space equipped with n-norm and then we generalize some of these properties by using n-normed. De…nition 1. Let X be a Riesz space. A n-norm k ; : : : ; k on X is said to be a n-lattices norm whenever jxj 4 jyj implies kx; z1 ; :::; zn 1 k ky; z1 ; :::; zn 1 k for all z1 ; :::; zn 1 2 X. When X equipped with a n-lattice norm, it’s de…ned as a n-normed Riesz space. If n-normed Riesz space X is complete, then it’s called as n-Banach lattice. Since jxj j jxj j and j jxj j jxj for all x element of n-normed Riesz space X, we can see easily that the following equality kx; z1 ; :::; zn

1k

= kjxj ; z1 ; :::; zn

1k

for all z1 ; :::; zn 1 2 X. Also kx+ y + ; z1 ; :::; zn 1 k kx y; z1 ; :::; zn kjxj jyj ; z1 ; :::; zn 1 k kx y; z1 ; :::; zn 1 k for all z1 ; :::; zn 1 2 X. It’s known that the space lp with 1 p < 1 is n-normed space with 3 p1 2 X X 1 p ::: jdet (xijk )j 5 kx1 ; x2 ; :::; xn kp := 4 n! j j 1

1k

and

n

, i = 1; :::; n [5].If lp equipped with an order relation such that jxj jyj , jxk j jyk j whenever x = (xk ), y = (yk ) for all k 2 N , then lp is a Riesz space according

to this relation and k:; :kp is a n-lattice norm. So lp ; k:; :kp is an example of n-normed Riesz space. The following lemma will used in the proof of the future theorem. We prove the theorem by using the method which is well-known as in [3, 9]. Lemma 1. n-normed space X is a n-Banach space if and only if every convergent series is absolutely convergent.

73

¼ 1 AND N I·HAN GÜNGÖR 2 B I·RSEN SA GIR

4

Proof. Firstly, let X be a n-Banach space.

1 P

Assume that

k=1

kxk ; z1 ; :::; zn

1k

is convergent for all z1 ; :::; zn 1 2 X. From convergent principle of real series 1 m P P kxk ; z1 ; :::; zn 1 k ! 0 (m ! 1). If we take Sm = xk , then

k=m+1

k=1

kSl

Sm ; z1 ; :::; zn

1k

= kxm+1 + xm+2 + + xl ; z1 ; :::; zn 1 k kxm+1 ; z1 ; :::; zn 1 k + kxm+2 ; z1 ; :::; zn + + kxl ; z1 ; :::; zn 1 k l X

k=m+1

So that kSl

Sm ; z1 ; :::; zn

kxk ; z1 ; :::; zn

1k

+

1k .

1k

! 0 (m; l ! 1) and then we see that (Sm ) is a 1 P Cauchy sequence. Since X is n-Banach space, xk is convergent. k=1

For the converse, every convergent series is absolutely convergent. Let fxk g be a Cauchy sequence of X, so kxk xm ; z1 ; :::; zn 1 k ! 0 (k; m ! 1). Then there exists strictly increasing sequence fkm g of natural numbers such that kxkm +1 for all m 2 N. Hence we obtain

xkm ; z1 ; :::; zn

1 P

m=1

kxkm +1

1k


−β|γ| n f (z) RDλ,γ n o P n (n+j−1)! n z− ∞ aj z j z(RDλ,γ f (z))0 j=2 j α [1 + (j − 1) λ] + (1 − α) n!(j−1)! n o = P∞ n f (z) n RDλ,γ z − j=2 α [1 + (j − 1) λ] + (1 − α) (n+j−1)! aj z j n!(j−1)! n o ⎧ P ⎫ n (n+j−1)! j⎬ ⎨− ∞ (j − 1) α [1 + (j − 1) λ] + (1 − α) z a j j=2 n!(j−1)! n o > −β |γ| Re ⎩ z − P∞ α [1 + (j − 1) λ]n + (1 − α) (n+j−1)! a z j ⎭ j j=2 n!(j−1)!

Re{

By chossing the valkues of z on the real ais and letting z → 1− through real values, the above inequality immediately yields the required condition (2.1). Conversely, by applying the hypothesis (2.1) and letting |z| = 1, we obtain n o ¯ ¯ ¯¯ P∞ ¯ ¯ ¯ j=2 (j − 1) α [1 + (j − 1) λ]n + (1 − α) (n+j−1)! ¯ z(RDn f (z))0 aj z j ¯¯ n!(j−1)! ¯ ¯ ¯ λ,γ ¯ n o − 1¯ = ¯ ¯ n f (z) ¯ ¯ z − P∞ α [1 + (j − 1) λ]n + (1 − α) (n+j−1)! a z j ¯¯ ¯ RDλ,γ j j=2 n!(j−1)! n o ´ ³ P n (n+j−1)! + (1 − α) α [1 + (j − 1) λ] aj β |γ| 1 − ∞ j=2 n!(j−1)! o ≤ β|γ|. ≤ P∞ n n 1 − j=2 α [1 + (j − 1) λ] + (1 − α) (n+j−1)! aj n!(j−1)! 2

On a certain subclass of analytic functions, Alina Alb Lupas and Loriana Andrei

91

Hence, by maximum modulus theorem, we have f ∈ T S nλ,α (β, γ), which evidently completes the proof of theorem. Finally, the result is sharp with the extremal functions fj be in the class T S nλ,α (β, γ) given by fj (z) = z −

β |γ| n o z j , for j ≥ 2. (j − 1 + β|γ|) α [1 + (j − 1) λ]n + (1 − α) (n+j−1)! n!(j−1)!

(2.2)

Corollary 2.2 Let the function f ∈ T be in the class T S nλ,α (β, γ). Then, we have β|γ| n o , for j ≥ 2. (j − 1 + β|γ|) α [1 + (j − 1) λ]n + (1 − α) (n+j−1)! n!(j−1)!

aj ≤

The equality is attained for the functions f given by (2.2).

3

Growth and distortion theorems

Theorem 3.1 Let the function f ∈ T be in the class T S nλ,α (β, γ). Then for |z| = r, we have r−

β|γ| β|γ| r2 ≤ |f (z)| ≤ r + r2 . n n (1 + β|γ|) [α (1 + λ) + (1 − α) (n + 1)] (1 + β|γ|) [α (1 + λ) + (1 − α) (n + 1)]

with equality for f (z) = z −

β|γ| z2 . n (1 + β|γ|) [α (1 + λ) + (1 − α) (n + 1)]

Proof. In view of Theorem 2.1, we have (1 + β|γ|) [α (1 + λ)n + (1 − α) (n + 1)]

∞ X j=2

aj ≤

∞ X n (j − 1 + β|γ|) [α (1 + λ) + (1 − α) (n + 1)] aj ≤ β|γ|. j=2

Hence |f (z)| ≤ r +

∞ X

aj rj ≤ r + r2

∞ X

aj ≤ r +

β|γ| r2 (1 + β|γ|) [α (1 + λ)n + (1 − α) (n + 1)]

|f (z)| ≥ r −

∞ X

aj rj ≥ r − r2

∞ X

aj ≥ r −

β|γ| r2 . (1 + β|γ|) [α (1 + λ)n + (1 − α) (n + 1)]

and

j=2

j=2

This completes the proof.

j=2

j=2

Theorem 3.2 Let the function f ∈ T be in the class T S nλ,α (β, γ). Then, for |z| = r we have r−

2β|γ| 2β|γ| r2 ≤ |f 0 (z)| ≤ r + r2 n n (1 + β|γ|) [α (1 + λ) + (1 − α) (n + 1)] (1 + β|γ|) [α (1 + λ) + (1 − α) (n + 1)]

with equality for f (z) = z −

β|γ| z2 . n (1 + β|γ|) [α (1 + λ) + (1 − α) (n + 1)]

Proof. We have |f 0 (z)| ≤ r +

∞ X

jaj rj ≤ r + 2r2

∞ X

aj ≤ r +

2β|γ| r2 (1 + β|γ|) [α (1 + λ)n + (1 − α) (n + 1)]

|f 0 (z)| ≥ r −

∞ X

jaj rj ≥ r − 2r2

∞ X

aj ≥ r −

2β|γ| r2 , (1 + β|γ|) [α (1 + λ)n + (1 − α) (n + 1)]

and

j=2

j=2

which completes the proof.

j=2

j=2

3

(2.3)

On a certain subclass of analytic functions, Alina Alb Lupas and Loriana Andrei

4

92

Extreme points The extreme points of the class T S nλ,α (β, γ) will be now determined. β|γ| j (n+j−1)! z , j ≥ 2. (j−1+β|γ|){α[1+(j−1)λ]n +(1−α) n!(j−1)! } T S nλ,α (β, γ) if and only if it can be expressed in

Theorem 4.1 Let f1 (z) = z and fj (z) = z − Assume that f is analytic in U . Then f ∈

f (z) =

∞ X

the form

µj fj (z),

j=1

where µj ≥ 0 and

P∞

µj = 1.

j=1

Proof. Suppose that f (z) =

P∞

j=1

µj fj (z) with µj ≥ 0 and

f (z) =

∞ X

µj fj (z) = µ1 f1 (z) +

j=1

µ1 z +

∞ X j=2

z−

⎛

µj ⎝z − ∞ X

µj

j=2

Then ∞ X j=2

µj

P∞

j=1

µj = 1. Then

∞ X

µj fj (z) =

j=2

⎞

β|γ| n o zj ⎠ = n (n+j−1)! (j − 1 + β|γ|) α [1 + (j − 1) λ] + (1 − α) n!(j−1)!

β|γ| n o zj . n (n+j−1)! (j − 1 + β|γ|) α [1 + (j − 1) λ] + (1 − α) n!(j−1)!

β|γ| n o (j − 1 + β|γ|) α [1 + (j − 1) λ]n + (1 − α) (n+j−1)! n!(j−1)! ∞ X

µj =

j=2

∞ X j=1

n o n (j − 1 + β|γ|) α [1 + (j − 1) λ] + (1 − α) (n+j−1)! n!(j−1)! β|γ|

µj − µ1 = 1 − µ1 ≤ 1.

Thus f ∈ T S nλ,α (β, γ) by Theorem 2.1. Conversely, suppose that f ∈ T S nλ,α (β, γ). By using (2.3) we may set and µj = P∞

for j ≥ 2 and µ1 = 1 − Then

j=2

f (z) = z −

∞ X j=1

n o n (j − 1 + β|γ|) α [1 + (j − 1) λ] + (1 − α) (n+j−1)! n!(j−1)! β|γ|

µj .

aj z j = z −

∞ X j=2

µj

β|γ| n o zj = n (j − 1 + β|γ|) α [1 + (j − 1) λ] + (1 − α) (n+j−1)! n!(j−1)!

µ1 f1 (z) +

∞ X

µj fj (z) =

j=2

with µj ≥ 0 and

P∞

j=1

aj

∞ X

µj fj (z) ,

j=1

µj = 1, which completes the proof.

Corollary 4.2 The extreme points of T S nλ,α (β, γ) are the functions f1 (z) = z and fj (z) =

β|γ| n o z j , for j ≥ 2. n (n+j−1)! (j − 1 + β|γ|) α [1 + (j − 1) λ] + (1 − α) n!(j−1)! 4

=

On a certain subclass of analytic functions, Alina Alb Lupas and Loriana Andrei

5

93

Partial sums

We investigate in this section the partial sums of functions in the class T S nλ,α (β, γ). We shall obtain sharp lower bounds for the real part of its ratios. We shall follow similar works done by Silverman [15] and Khairnar and Moreon [12] about the partial sums of analytic functions. In what follows, we will use the well known result that for an analytic function ω in U , ¶ µ 1 + ω (z) > 0, z ∈ U, Re 1 − ω (z)

if and only if the inequality |ω (z)| < 1 is satisfied. P∞ Theorem 5.1 Let f (z) = z − j=2 aj z j ∈ T S nλ,α (β, γ) and define its partial sums by f1 (z) = z and fm (z) = Pm z − j=2 aj z j , m ≥ 2. Then ¶ µ 1 f (z) ≥1− Re , z ∈ U , m ∈ N, (5.1) fm (z) cm+1 and

Re where cj =

µ

fm (z) f (z)

¶

≥

cm+1 , 1 + cm+1

z ∈ U , m ∈ N,

(5.2)

n o (j + β|γ|) α [1 + (j − 1) λ]n + (1 − α) (n+j−1)! n!(j−1)! β|γ|

(5.3)

This estimates in (5.1) and (5.2) are sharp. Proof. To prove (5.1), it suffices to show that µ µ ¶¶ 1 f (z) 1+z cm+1 − 1− , z ∈ U. ≺ fm (z) cm+1 1−z By the subordination property, we can write P P j−1 j−1 1− m − cm+1 ∞ 1 + ω (z) j=2 aj z j=m+1 aj z Pm = 1 − ω (z) 1 − j=2 aj z j−1

for certain ω analytic in U with |ω (z)| ≤ 1. Notice that ω (0) = 0 and P∞ cm+1 j=m+1 aj P P∞ |ω (z)| ≤ . 2−2 m j=2 aj − cm+1 j=m+1 aj Now |ω (z)| ≤ 1 if and only if

m X

aj + cm+1

j=2

∞ X

j=m+1

aj ≤

∞ X j=2

cj aj ≤ 1.

The above inequality holds because cj is a non-decreasing sequence. This completes the proof of (5.1). Finally, it πi is observed that equality in (5.1) is attained for the function given by (2.3) when z = re2 m as r → 1− . Similarly, we take P P ¶ µ j−1 j−1 + cm+1 ∞ 1− m cm+1 fm (z) 1 + ω (z) j=2 aj z j=m+1 aj z Pm − , = = (1 + cm+1 ) f (z) 1 + cm+1 1 − ω (z) 1 − j=2 aj z j−1 where

|ω (z)| ≤ Now |ω (z)| ≤ 1 if and only if

m X j=2

(1 + cm+1 )

2−2

Pm

P∞

aj + cm+1

∞ X

j=m+1

aj ≤

aj P∞

j=m+1

j=2 aj + (1 − cm+1 ) ∞ X j=2

j=m+1

aj

.

cj aj ≤ 1.

This immediately leads to assertion (5.2) of Theorem 5.1. This completes the proof. Using a similar method, we can prove the following theorem. 5

On a certain subclass of analytic functions, Alina Alb Lupas and Loriana Andrei

Theorem 5.2 If f ∈ T S nλ,α (β, γ) and define the partial sums by f1 (z) = z and fm (z) = z − Re

µ

f 0 (z) 0 (z) fm

and Re

µ

¶

≥1−

0 (z) fm f (z)

¶

m+1 , cm+1

≥

94

Pm

j=2

aj z j . Then

z ∈ U , m ∈ N,

cm+1 , m + 1 + cm+1

where cj is given by (5.3). The result is sharp for every m, with extremal functions given by (2.2).

References [1] A. Abubaker, M. Darus, On a certain subclass of analytic functions involving differential operators, TJMM 3 (2011), No. 1, 1-8. [2] A. Alb Lupa¸s, On special differential subordinations using Sa˘la˘gean and Ruscheweyh operators, Mathematical Inequalities and Applications, Volume 12, Issue 4, 2009, 781-790. [3] A. Alb Lupa¸s, On a certain subclass of analytic functions defined by Salagean and Ruscheweyh operators, Journal of Mathematics and Applications, No. 31, 2009, 67-76. [4] A. Alb Lupa¸s, On special differential subordinations using a generalized Sa˘la˘gean operator and Ruscheweyh derivative, Journal of Computational Analysis and Applications, Vol. 13, No.1, 2011, 98-107. [5] A. Alb Lupa¸s, On a certain subclass of analytic functions defined by a generalized Sa˘la˘gean operator and Ruscheweyh derivative, Carpathian Journal of Mathematics, 28 (2012), No. 2, 183-190. [6] A. Alb Lupa¸s, D. Breaz, On special differential superordinations using Sa˘la˘gean and Ruscheweyh operators, Geometric Function Theory and Applications’ 2010 (Proc. of International Symposium, Sofia, 27-31 August 2010), 98-103. [7] A. Alb Lupa¸s, On special differential superordinations using a generalized Sa˘la˘gean operator and Ruscheweyh derivative, Computers and Mathematics with Applications 61, 2011, 1048-1058, doi:10.1016/j.camwa.2010.12.055. [8] A. Alb Lupa¸s, Certain special differential superordinations using a generalized Sa˘la˘gean operator and Ruscheweyh derivative, Analele Universitatii Oradea, Fasc. Matematica, Tom XVIII, 2011, 167-178. [9] F.M. Al-Oboudi, On univalent functions defined by a generalized Sa˘la˘gean operator, Ind. J. Math. Math. Sci., 27 (2004), 1429-1436. [10] L. Andrei, Differential subordinations using Ruscheweyh derivative and generalized Sa˘la˘gean operator, Advances in Difference Equation, 2013, 2013:252, DOI: 10.1186/1687-1847-2013-252. [11] L. Andrei, V. Ionescu, Some differential superordinations using Ruscheweyh derivative and generalized Sa˘la˘gean operator, Journal of Computational Analysis and Applications, Vol. 17, No. 3, 2014, 437-444 [12] S.M. Khairnar, M. Moore, On a certain subclass of analytic functions involving the Al-Oboudi differential operator, J. Ineq. Pure Appl. Math., 10 (2009), 1-22. [13] St. Ruscheweyh, New criteria for univalent functions, Proc. Amet. Math. Soc., 49(1975), 109-115. [14] G. St. S˘ al˘ agean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, 1013(1983), 362-372. [15] H. Silverman, Partial sums of starlike and convex functions, J. Math. Anal. Appl., 209 (1997), 221-227.

6

J. APPLIED FUNCTIONAL ANALYSIS, VOL. 10, NO.'S 1-2,95-100 ,2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

ON THE (r; s) CONVEXITY AND SOME HADAMARD-TYPE INEQUALITIES H M.

EMIN ÖZDEMIR,

F ERHAN

SET, AND

H AHMET

OCAK AKDEMIR

Abstract. In this paper, we de…ne a new class of convex functions which is called (r; s) convex functions. We also prove some Hadamard’s type inequalities related to this new class of functions.

1. INTRODUCTION Let f : I R ! R be a convex function de…ned on the interval I of real numbers and a; b 2 I with a < b: Then, the following double inequality is well known in literature as Hermite-Hadamard inequality holds for convex functions: f

a+b 2

1 b

a

Zb

f (a) + f (b) : 2

f (x)dx

a

In [1], Pearce et al. generalized this inequality to r convex positive function f which is de…ned on an interval [a; b]; for all x; y 2 [a; b] and 2 [0; 1]; ( r r 1 ( [f (x)] + (1 ) [f (y)] ) r ; if r 6= 0 f ( x + (1 )y) : 1 [f (x)] [f (y)] ; if r = 0 Obviously, one can see that 0 convex functions are simply log convex functions and 1 convex functions are ordinary convex functions. Another inequality which well known in the literature as Minkowski Inequality is given as following; Zb Zb Let p 1; 0 < f (x)p dx < 1; and 0 < g(x)p dx < 1: Then a

(1.1)

a

0 b 1 p1 Z @ (f (x) + g(x))p dxA a

0 b 1 p1 0 b 1 p1 Z Z @ f (x)p dxA + @ g(x)p dxA : a

a

De…nition 1. [See [4]] A function f : I ! [0; 1) is said to be log convex or multiplicatively convex if log f is convex, or, equivalently, if for all x; y 2 I and t 2 [0; 1] one has the inequality: (1.2)

f (tx + (1

t

t)y)

1 t

[f (x)] [f (y)]

We note that a log convex function is convex, but the converse may not necessarily be true. 2000 Mathematics Subject Classi…cation. Primary 26D15, 26A07. Key words and phrases. Hadamard’s inequality, r convex, s convex. 1

95

96

H

2

M . EM IN ÖZDEM IR,

F

ERHAN SET, AND

H

AHM ET OCAK AKDEM IR

In [6], Gill et al. proved following inequality for r convex functions. Theorem 1. Suppose f is a positive r convex function on [a; b]. Then Z b 1 (1.3) f (t)dt Lr (f (a); f (b)) : b a a If f is a positive r concave function, then the inequality is reversed. For recent results on r convexity see the papers [2], [5] and [6]. De…nition 2. [See [3]] Let s 2 (0; 1] : A function f : [0; 1) ! [0; 1) is said to be s convex in the second sense if (1.4)

f (tx + (1

t) y)

ts f (x) + (1

s

t) f (y)

for all x; y 2 R+ and t 2 [0; 1] : In [10], s convexity introduced by Breckner as a generalization of convex functions. Also, Breckner proved the fact that the set valued map is s convex only if the associated support function is s convex function in [11]. Several properties of s convexity in the …rst sense are discussed in the paper [3]. Obviously, s convexity means just convexity when s = 1. Theorem 2. [See [8]] Suppose that f : [0; 1) ! [0; 1) is an s convex function in the second sense, where s 2 (0; 1] and let a; b 2 [0; 1) ; a < b: If f 2 L1 [0; 1] ; then the following inequalities hold: Z b a+b 1 f (a) + f (b) s 1 (1.5) 2 f : f (x) dx 2 b a a s+1

1 The constant k = s+1 is the best possible in the second inequality in (1.5). The above inequalities are sharp.

Some new Hermite-Hadamard type inequalities based on concave functions and s convexity established by K¬rmac¬et al. in [9]. For related results see the papers [7], [8] and [9]. The main purpose of this paper is to give de…nition of (r; s) convexity and to prove some Hadamard-type inequalities for (r; s) convex functions. 2. MAIN RESULTS De…nition 3. A positive function f is (r; s) convex on [a; b] x; y 2 [a; b] ; s 2 (0; 1] and 2 [0; 1] f ( x + (1

)y)

( s f r (x) + (1 f (x)f 1 (y)

[0; 1) if for all

1

)s f r (y)) r ; if r 6= 0 : ; if r = 0

This de…nition of (r; s) convexity naturally complements the concept of (r; s) concave functions in which the inequality is reversed. Remark 1. We have that (0; s) convex functions are simply log convex functions and (1; 1) convex functions are ordinary convex functions. Remark 2. We have that (r; 1) convex functions are r convex functions. Remark 3. We have that (1; s) convex functions are s convex functions. Now, we will prove inequalities based on above de…nition.

97

HADAM ARD-TYPE INEQUALITIES

3

Theorem 3. Let f : [a; b] [0; 1) ! (0; 1) be (r; s) convex function on [a; b] with a < b: Then the following inequality holds; Z b 1 1 r (2.1) f (x)dx [f r (a) + f r (b)] r b a a r+s for r; s 2 (0; 1]: Proof. Since f is (r; s) convex function and r > 0; we can write f (ta + (1

t)b)

r

1

r

(ts [f (a)] + (1

t)s [f (b)] ) r

for all t 2 [0; 1] and s 2 (0; 1]. It is easy to observe that 1 b

a

Z

b

f (x)dx =

a

Z1

f (ta + (1

Z1

(ts [f (a)] + (1

t)b)dt

0

r

r

1

t)s [f (b)] ) r dt:

0

Using the inequality (1.1), we get Z1

r

r

(ts [f (a)] + (1

20 1 1r 0 1 Z Z 4@ t rs f (a)dtA + @ (1

1 r

t)s [f (b)] ) dt

0

0

0

r

= = Thus 1 b

a

Z

b

a

f (x)dx

r (f r (a) + f r (b)) r+s 1 r [f r (a) + f r (b)] r : r+s

r r+s

s r

1 r

1r 3 r1

t) f (b)dtA 5

1

[f r (a) + f r (b)] r

which completes the proof. Corollary 1. In Theorem 3, if we choose (1; s) convex function on [a; b] with a < b: Then, we have the following inequality; Z b 1 1 f (x)dx [f (a) + f (b)] : b a a s+1 Corollary 2. In Theorem 3, if we choose (r; 1) convex function on [a; b] with a < b: Then, we have the following inequality; Z b 1 1 r [f r (a) + f r (b)] r : f (x)dx b a a r+1 Remark 4. In Theorem 3, if we choose (1; 1) convex function on [a; b] with a < b: Then, we have the right hand side of Hadamard’s inequality.

[0; 1)

98

H

4

M . EM IN ÖZDEM IR,

F

ERHAN SET, AND

H

AHM ET OCAK AKDEM IR

Theorem 4. Let f; g : [a; b] [0; 1) ! (0; 1) be (r1 ; s) convex and (r2 ; s) convex function on [a; b] with a < b: Then the following inequality holds; (2.2) Z b 1 1 1 1 r r r r f (x)g(x)dx ([f (a)] 1 + [f (b)] 1 ) r1 ([g(a)] 2 + [g (b)] 2 ) r2 b a a s+1 1 r1

for r1 > 1 and

1 r2

+

= 1:

Proof. Since f is (r1 ; s) convex function and g is (r2 ; s) convex function, we have r

1

r

1

r1

+ (1

t)s [f (b)] 1 ) r1

r2

+ (1

t)s [g(b)] 2 ) r2

f (ta + (1

t)b)

(ts [f (a)]

g(ta + (1

t)b)

(ts [g(a)]

and for all t 2 [0; 1] and r1 ; r2 ; s 2 (0; 1]: Since f and g are non-negative functions, hence f (ta + (1

t)b)g(ta + (1

r1

s

(t [f (a)]

t)b) r1

s

1

r2

t) [f (b)] ) r1 (ts [g(a)]

+ (1

r

1

t)s [g(b)] 2 ) r2 :

+ (1

Integrating both sides of the above inequality over [0; 1] with respect to t; we obtain Z1

[f (ta + (1

t)b)g(ta + (1

t)b)] dt

0

Z1 h

r1

(ts [f (a)]

1

r

r2

t)s [f (b)] 1 ) r1 (ts [g(a)]

+ (1

i 1 r t)s [g(b)] 2 ) r2 dt:

+ (1

0

By applying the Hölder’s inequality, we have Z1 h

r1

(ts [f (a)]

+ (1

r

1

r2

t)s [f (b)] 1 ) r1 (ts [g(a)]

i 1 r t)s [g(b)] 2 ) r2 dt

+ (1

0

2 1 Z 4 (ts [f (a)]r1 + (1 0

=

1 s+1

r1

([f (a)]

3 r1 2 1 1 Z r r1 s 5 4 (ts [g(a)] 2 + (1 t) [f (b)] dt) 0

r1

+ [f (b)] )

1 r1

r2

([g(a)]

r

1

+ [g (b)] 2 ) r2 :

3 r1

2

r2

t) [g(b)] dt)5 s

By using the fact that 1 b

a

Z

b

f (x)g(x)dx =

a

Z1

[f (ta + (1

t)b)g(ta + (1

t)b)] dt:

0

We obtain the desired result. Corollary 3. In Theorem 4, if we choose s = 1; r1 = r2 = 2 and f (x) = g(x); we have the following inequality; Z b 2 2 1 [f (a)] + [f (b)] f (x)2 dx : b a a 2

99

HADAM ARD-TYPE INEQUALITIES

5

Corollary 4. In Theorem 4, if we choose s = 1 and r1 = r2 = 2; we have the following inequality; s s Z b 2 2 2 2 [f (a)] + [f (b)] [g(a)] + [g (b)] 1 f (x)g(x)dx : b a a 2 2 Theorem 5. Let f : [a; b] [0; 1) ! (0; 1) be a (r; s) convex function on [a; b] with r; s 2 (0; 1]: If f 2 L1 [a; b]; then one has the following inequalities; Z b s a+b 1 f r (a) + f r (b) f r (x)dx (2.3) 2 r 1f : 2 b a a s+1 Proof. By the (r; s) convexity of f , we have that f

x+y 2

1 r r s [f (x) + f (y)] 2r

for all x; y 2 [a; b] : If we take x = ta + (1

t)b and y = (1

a+b 2

t)b) + f r ((1

f

1 r s [f (ta + (1 2r

t)a + tb; we obtain t)a + tb)]

for all t 2 [0; 1] : Integrating the above inequality over [0; 1] with respect to t; we get 2 1 3 Z Z1 a+b 1 4 (2.4) f f r (ta + (1 t)b)dt + f r ((1 t)a + tb) dt5 : s 2 2r 0

0

From the facts that

Z1

f r (ta + (1

t)b)dt =

1 b

a

0

Z

b

f r (x)dx

a

and Z1

r

f ((1

t)a + tb) dt =

1 b

a

0

Z

b

f r (x) dx

a

we obtain the left hand side of the (2.3). By the (r; s) convexity of f , we also have that (2.5)

f (ta + (1 t)b) + f (tb + (1 t)a) ts f r (a) + (1 t)s f r (b) + ts f r (b) + (1

t)s f r (a)

for all t 2 [0; 1] : Integrating the inequality (2.5) over [0; 1] with respect to t; we get Z b f r (a) + f r (b) 1 f r (x)dx : b a a s+1 which completes the proof. Remark 5. In Theorem 5, if we choose r = 1; we have the inequality (1.5). Remark 6. In Theorem 5, if we choose s = r = 1; we have the Hadamard’s inequality.

100

6

H

M . EM IN ÖZDEM IR,

F

ERHAN SET, AND

H

AHM ET OCAK AKDEM IR

References [1] C.E.M. Pearce, J. Peµcari´c, V. Simi´c, Stolarsky Means and Hadamard’s Inequality, Journal Math. Analysis Appl., 220, 1998, 99-109. [2] G.S. Yang, D.Y. Hwang, Re…nements of Hadamard’s inequality for r convex functions, Indian Journal Pure Appl. Math., 32 (10), 2001, 1571-1579. [3] H. Hudzik, L. Maligranda, Some remarks on s convex functions, Aequationes Math., 48 (1994) 100–111. [4] J. Peµcari´c, F. Proschan and Y.L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Inc., 1992. [5] N.P.G. Ngoc, N.V. Vinh and P.T.T. Hien, Integral inequalities of Hadamard-type for r convex functions, International Mathematical Forum, 4, 2009, 1723-1728. [6] P.M. Gill, C.E.M. Pearce and J. Peµcari´c, Hadamard’s inequality for r convex functions, Journal of Math. Analysis and Appl., 215, 1997, 461-470. [7] S. Hussain, M.I. Bhatti and M. Iqbal, Hadamard-type inequalities for s convex functions, Punjab University, Journal of Mathematics, 41 (2009) 51-60. [8] S.S. Dragomir and S. Fitzpatrick, The Hadamard’s inequality for s convex functions in the second sense, Demonstratio Math., 32 (4) (1999) 687-696. [9] U. S. K¬rmac¬, M. K. Bakula, M. E. Özdemir and J. Peµcari´c, Hadamard-type inequalities for s convex functions, Applied Mathematics and Computation, 193 (2007) 26-35. [10] W.W. Breckner, Stetigkeitsaussagen f¨ ur eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen Raumen, Pupl. Inst. Math., 23 (1978) 13–20. [11] W.W. Breckner, Continuity of generalized convex and generalized concave set-valued functions, Rev Anal. Num´ er. Thkor. Approx., 22 (1993) 39–51. H Ataturk

University, K. K. Education Faculty, Department of Mathematics, 25640, Kampus, Erzurum, Turkey E-mail address : [email protected] F Department of Mathematics, Faculty of Science and Arts, Ordu University, Ordu / Turkey E-mail address : [email protected] H AG ·brahim Çeçen University, Faculty of Science and Arts, Department of Math¼ r¬ I ematics, 04100, AG¼ r¬, Turkey E-mail address : [email protected]

J. APPLIED FUNCTIONAL ANALYSIS, VOL. 10, NO.'S 1-2,101-108 ,2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

COMPACT COMPOSITION OPERATORS ON WEIGHTED HILBERT SPACES WALEED AL-RAWASHDEH

Abstract. Let ϕ be an analytic self-map of open unit disk D. A composition operator is defined as (Cϕ f )(z) = f (ϕ(z)), for z ∈ D and f analytic on D. Given an admissible weight ω, the weighted Hilbert Z space Hω consists of all analytic functions f such that kf k2Hω = |f (0)|2 +

|f 0 (z)|2 w(z)dA(z) is

D

finite. In this paper, we study composition operators acting between weighted Bergman space A2α and the weighted Hilbert space Hω . Using generalized Nevalinna counting functions associated with ω, we characterize the boundedness and compactness of these composition operators.

1. Introduction Let D be the unit disk {z ∈ D : |z| < 1} in the complex plane. Suppose ϕ is an analytic function maps D into itself and ψ is an analytic function on D, the weighted composition operator Wψ,ϕ is defined on the space H(D) of all analytic functions on D by (Wψ,ϕ f )(z) = ψ(z)Cϕ f (z) = ψ(z)f (ϕ(z)), for all f ∈ H(D) and z ∈ D. The composition operator Cϕ is a weighted composition operator with the weight function ψ identically equal to 1. It is well known that the composition operator Cϕ f = f ◦ ϕ defines a linear operator Cϕ which acts boundedly on various spaces of analytic or harmonic functions on D. These operators have been studied on many spaces of analytic functions. During the past few decades much effort has been devoted to the study of these operators with the goal of explaining the operatortheoretic properties of Wψ,ϕ in terms of the function-theoretic properties of the induced maps ϕ and ψ. We refer to the monographs by Cowen and MacCluer [1], Duren and Schuster [2], Hedenmalm, Korenblum, and Zhu [3], Shapiro [6], and Zhu ([9], [10]) for the overview of the field as of the early 1990s. For α > −1, we define the measure dAα on D by dAα (z) = (− log |z|)α dA(z), where dA is the normalized Lebesgue measure on the unit disk D. The weighted Bergman space A2α consists of all functions f analytic on D that 2010 Mathematics Subject Classification. Primary 47B33, 47B38, 30H10, 30H20; Secondary 47B37, 30H05, 32C15. Key words and phrases. Composition operators, weighted Bergman spaces, Hardy spaces, weighted Hilbert spaces, compact operators, generalized Nevalinna counting function. 1

101

102

2

AL-RAWASHDEH

satisfy kf k2α

Z

|f (z)|2 dAα (z) < ∞.

= D

In this definition, the measure dAα can be replaced by the measure (1 − |z|)α dA(z) and this results an equivalent norm; see [5] for example. Shapiro [8] showed that f ∈ A2α if and only if f 0 ∈ A2α+2 . In particular, he proved that kf k2α = |f (0)|2 + kf 0 k2α+2 , this formula is just the Bergman space version of the Little-Paley identity. The Hardy space H 2 (D) consists of functions f analytic on D that satisfy Z 2 kf k = sup |f (rζ)|p dσ(ζ) < ∞, 0 −1, is the weighted Hilbert space Hα+2 . Recently, Kellay and Lef´evre [4] characterized the compactness of composition operators acting on the weighted Hilbert space Hω with the weight function ω that satisfies the following conditions: (1) ω is non-decreasing. (2) ω(r)(1 − r)−(1+δ) is non-decreasing for some δ > 0. (3) lim ω(r) = 0. r→1−

(4) One of the two properties of convexity is fulfilled: ω is convex and lim ω 0 (r) = 0; or ω is concave. r→1

In this paper, we consider the admissible weight ω that only satisfies two conditions; namely we say ω is admissible weight if it is convex and satisfies

103

COMPACT COMPOSITION OPERATORS ON WEIGHTED HILBERT SPACES

3

lim ω(r) = lim ω 0 (r) = 0. We characterize boundedness and compactness

r→1−

r→1

of composition operators act between weighted bergman space A2α and the weighted Hilbert space Hω . The results we obtain about composition operators will be given in terms of a generalized Nevalinna counting functions, which we define next. The Nevalinna counting functions play an essential role in study of composition operators on weighted Hilbert spaces. The classical Nevalinna function of the inducing map ϕ is defined by   X 1 log Nϕ (z) = , z ∈ D\{ϕ(0)}, |a| −1 a∈ϕ

{z}

ϕ−1 {z}

where as usual a ∈ is repeated according to the multiplicity of the zero of ϕ − z at a. The Nevalinna counting function Nϕ was first used by Shapiro in [8] to study composition operators on Hardy spaces H 2 . In the same paper Shapiro also defined the generalized counting functions Nϕ,γ for γ > 0 by X   1 γ Nϕ,γ (z) = log , z ∈ D\{ϕ(0)}, |a| −1 a∈ϕ

{z}

and used them to study composition operators from a weighted Bergman space to itself. For more information about the Nevalinna counting functions; see [8], [6], and [1], for example. Kellay and Lef`evre [4] defined the generalized Nevalinna function associated to the admissible weight ω by X Nϕ,ω (z) = ω(a), z ∈ D\{ϕ(0)}, a∈ϕ−1 {z}

and used them to study composition operators from a weighted Hilbert space to itself. Note that if ω(r) = (log(1/r))γ , then Nϕ,ω is the generalized Nevalinna counting function Nϕ,γ . The next lemma explains how the generalized Nenalinna counting function Nϕ,ω arise naturally in the study of composition operators on weighted Hilbert spaces, and its proof is a modification of that of (Theorem 2.32, [1]). This lemma is a generalization of the change of variables formula [8]. Shapiro’s formula is a special case of Stanton’s formula for integral means of analytic functions [7] that extends the Little-Paley identity. The utility of the change of variables formula in the study of composition operators on Hardy and Bergman spaces comes form Stanton’s formula. Lemma 1.1. Let g and w be positive measurable functions, and ϕ is a nonconstant analytic self-map of D. Then Z Z 0 2 g (ϕ(z)) |ϕ (z)| ω(z)dA(z) = g(λ)Nϕ,ω (λ)dA(λ). D

ϕ(D)

104

4

AL-RAWASHDEH

Proof. Suppose ϕ is a nonconstant analytic self-map of D. Since ϕ0 (z) is not identically zero, ϕ is locally univalent on D except at the points at which the derivative equal to zero. Since the zeros of ϕ at most countably many, we can find a countable collection of disjoint open sets {Rn } such that the area of D\ ∪ Rn is zero, and such that ϕ is univalent on each Rn . Let ψn be the inverse of ϕ : Rn → ϕ(Rn ). Then the usual change of variables applied on Rn , with z = ψn (λ) and dA(λ) = |ϕ0 (z)|2 dA(z), gives Z Z 0 2 g (ϕ(z)) |ϕ (z)| ω(z)dA(z) = g(λ) ω (ψn (λ)) dA(λ). Rn

ϕ(Rn )

Let χn denotes the characteristic function of the set ϕ(Rn ). Since the zeros of ϕ0 (z) is countable, we have Z ∞ Z X g (ϕ(z)) |ϕ0 (z)|2 ω(z)dA(z) = g(λ) ω (ψn (λ)) dA(λ) n=1 ϕ(Rn )

D

Z g(λ)

= ϕ(D)

Z =

g(λ) ϕ(D)

∞ X n=1 ∞ X

! χn (λ) ω (ψn (λ)) dA(λ) ! ω (zn (λ)) dA(λ),

n=1

where {zn (λ)} denotes the sequence of zeros of ϕ(z) − λ, repeated according to multiplicity. This completes the proof, since the sum in the last equation is Nϕ,ω (λ).  Recall that an operator is said to be compact if it takes bounded sets to sets with compact closure. The following lemma characterizes the compactness of a composition operator, and its proof is just a modification of that of (Proposition 3.11, [1]). The proof is a standard technique of Montel’s Theorem and definition of a composition operator Cϕ , so we omit the proof’s details. Lemma 1.2. Let ϕ be an analytic self-map of D. Then Cϕ : A2α → Hω is compact if and only if whenever {fn } is a bounded sequence in A2α that converges to 0 uniformly on compact subsets of D then lim kCϕ fn kHω = 0. n→∞

The next lemma shows that the generalized Nevalinna functions Nϕ,ω , while not subharmonic themselves, satisfy a subharmonic mean value property. This lemma is a modification of (Corollary 6.7, [8]) and its proof similar to that of (Corollary 4.6, [8]), so we omit the proof’s details. Lemma 1.3. Let ω be an admissible weight. Let ϕ be an analytic self-map of D and ϕ(0) 6= 0. Then for every 0 < r < |ϕ(0)| we have Z 1 Nϕ,ω (0) ≤ 2 Nϕ,ω (ζ)dA(ζ). r rD

105

COMPACT COMPOSITION OPERATORS ON WEIGHTED HILBERT SPACES

5

2. Compactness of a composition operator In this section we characterize the boundedness and compactness of a composition operator Cϕ mapping weighted Bergman space A2α into weighted Hilbert space Hω associated with the admissible weight ω. The notation A ≈ B means that there are two positive constants c1 and c2 independent of A and B such that c1 A ≤ B ≤ c2 A. Theorem 2.1. let ϕ be an analytic self-map of D. Then Cϕ : A2α → Hω is bounded if and only if
Nϕ,ω (λ) = O (− log |λ|)2+α , as |λ| → 1. Proof. We first show that Nϕ,ω satisfies the given growth condition. For λ ∈ D, consider the test function
(2+α)/2 1 − |λ|2 fλ (z) = ¯ 2+α . (1 − λz) Let ψλ (z) be the M¨obius self-map of D that interchanges 0 and λ, that is ψλ (z) =

λ−z ¯ . 1 − λz

It is well-known that kfλ k2A2 ≈ 1. Then we have α Z kCϕ (fλ )k2Hω = |fλ (ϕ(0))|2 + |fλ0 (ϕ(z))|2 |ϕ0 (z)|2 ω(z)dA(z) ZD = |fλ (ϕ(0))|2 + |fλ0 (z)|2 Nϕ,ω (z)dA(z) ϕ(D) Z = |fλ (ϕ(0))|2 + |fλ0 (z)|2 Nϕ,ω (z)dA(z) ϕ(D)

= |fλ (ϕ(0))|2 + (2 + α)2 |λ|2 (1 − |α|2 )2+α

Z D

= |fλ (ϕ(0))|2 + (2 + α)2 |λ|2 (1 − |α|2 )α

Z

= |fλ (ϕ(0))|2 + (2 + α)2 |λ|2 (1 − |α|2 )α

Z

D

D

Nϕ,ω (z) dA(z) ¯ 6+2α 1 − λz

Nϕ,ω (z) |ψ 0 (z)|2 dA(z) ¯ 2+2α λ 1 − λz Nϕ,ω (ψλ (z)) dA(z) ¯ λ (z) 2+2α 1 − λψ

Now for any |z| ≤ 21 , we get kCϕ (fλ )k2Hω

(2 + α)2 |λ|2 ≥ 1+α 4 (1 − |λ|2 )2+α ≥

Z 1 D 2

Nϕ,ω (ψλ (z)) dA(λ)

(2 + α)2 |λ|2 Nϕ,ω (λ), − |λ|2 )2+α

42+α (1

106

6

AL-RAWASHDEH

where the last inequality can be seen by using the sub-mean property Lemma 1.3. If |λ| is sufficiently close to 1, we get for some positive constant C Nϕ,ω (λ) ≤ C(1 − |λ|2 )2+α kCϕ (fλ )k2Hω . 1 Since kfλ k2A2 ≈ 1 and (1 − |λ|2 ) is comparable to log( |λ| ), we get the desired α growth condition.

For the converse, let f ∈ A2α . Then, by the change of variables formula Lemma 1.1, we get Z 2 2 |f 0 (ϕ(z))|2 |ϕ0 (z)|2 ω(z)dA(z) kCϕ (f )kHω = |f (ϕ(0))| + ZD 2 |f 0 (z)|2 Nϕ,ω (z)dA(z). = |f (ϕ(0))| + D

By our hypothesis, there is a finite positive constant C and 0 < r < 1 such that   2+α 1 Nϕ,ω (z) ≤ C log , for z ∈ D\rD. |z| Therefore, we get Z 2 |f 0 (z)|2 dAα+2 (z) kCϕ (f )kHω ≤ C ≤

D\rD Ckf k2A2α .

By using the Closed Graph Theorem, we get the boundedness of Cϕ .



In the next theorem we are following operator-theoretic wisdom: If a “big-oh” condition determines when an operator is bounded, then the corresponding “little-oh” condition determines when it is compact. Theorem 2.2. let ϕ be an analytic self-map of D. Then Cϕ : A2α → Hω is compact if and only if
Nϕ,ω (λ) = o (− log |λ|)2+α , as |λ| → 1. Proof. Suppose that Cϕ is compact and the given grow condition does not hold. Then there exists a sequence {λn } in D with |λn | → 1 as n → ∞ such that Nϕ,ω (λn ) ≥ β (− log |λn |)2+α , for some β > 0. Now consider the function
(2+α)/2 1 − |λn |2 fn (z) =
2+α . 1 − λ¯n z It is clear that kfn kA2α ≈ 1 and {fn } converges uniformly to 0 on compact subsets of D. So, by compactness of Cϕ and Lemma 1.2, we have lim kCϕ (fn )k2Hω = 0. n→∞

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COMPACT COMPOSITION OPERATORS ON WEIGHTED HILBERT SPACES

7

On the other hand, using similar argument as that in the proof of Theorem 2.1, there are positive constants C and C1 such that Z 2 |fn0 (ϕ(z))|2 |ϕ0 (z)|2 ω(z)dA(z) kCϕ (fn )kHω ≥ D

Nϕ,ω (λn )

≥C

(1 − |λn |2 )2+α Nϕ,ω (λn ) ≥ CC1 (− log |λn |)2+α ≥ CC1 β. Which contradicts the compactness of Cϕ , because C and C1 are constants independent of n. For the converse, Let {fn } be a sequence in the unit ball of A2α converges to 0 uniformly on compact subsets of D. Then Cauchy’s estimate gives fn0 converges uniformly to 0 on compact subsets of D. Let  > 0, by our hypothesis there exists δ ∈ (0, 1) such that Nϕ,ω (z) ≤  (− log |z|)2+α ,

for δ < |z| < 1.

To end the proof, by Lemma 1.2, it is enough to show lim kCϕ (fn )k2Hω = 0. n→∞

kCϕ (fn )k2Hω = |fn (ϕ(0))|2 + = |fn (ϕ(0))|2 +

Z

|fn0 (ϕ(z))|2 |ϕ0 (z)|2 ω(z)dA(z)

ZD

|fn0 (z)|2 Nϕ,ω (z)dA(z)

D 2

Z

= |fn (ϕ(0))| + |z|≤δ 2

Z

= |fn (ϕ(0))| + |z|≤δ 2

Z

= |fn (ϕ(0))| + |z|≤δ

|fn0 (z)|2 Nϕ,ω (z)dA(z) |fn0 (z)|2 Nϕ,ω (z)dA(z)

Z + |z|>δ

|fn0 (z)|2 Nϕ,ω (z)dA(z)

Z + |z|>δ

|fn0 (z)|2 (− log |z|)2+α dA(z)

|fn0 (z)|2 Nϕ,ω (z)dA(z) + kfn k2α .

Since  > 0 is arbitrary, {fn } and {fn0 } converge to 0 uniformly on compact subsets of D, we can make the right hand side of the last inequality as small as we wish by choosing n sufficiently large. This completes the proof.  References [1] C.C. Cowen and B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC press, Boca Raton, 1995. [2] P. Duren and A. Schuster, Bergman Spaces, American Mathematical Society, 2004 [3] H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman Spaces, Springer-Verlag, New York, 2000. [4] K. Kellay and P. Lef` evre, Compact composition operators on weighted Hilbert spaces of analytic functions, J. Math. Anal. Appl. 386(2012), 718-727.

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[5] B.D. MacCluer and J.H. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38(1986), 878-906. [6] J.H. Shapiro, Composition Operators and Classical Function Theory. Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993. [7] C.S. Stanton, Counting funcions and majorization for jensen measures, Pacific J. Math. 125(1986), 459-468. [8] J.H. Shapiro, The essential norm of a composition operators, Annals of Mathematics 125(1987), 375-404. [9] K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer-Verlag, New York, 2005. [10] K. Zhu, Operator Theory in Function Spaces. Monograghs and Textbooks in Pure and Applied Mathematics, 139. Marcel Dekker, Inc.,New York, 1990. (Waleed Al-Rawashdeh) Department of Mathematical Sciences, Montana Tech of The University of Montana, Butte, Montana 59701, USA E-mail address: [email protected]

J. APPLIED FUNCTIONAL ANALYSIS, VOL. 10, NO.'S 1-2,109-116 ,2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

A NEW DOUBLE CESÀRO SEQUENCE SPACE DEFINED BY MODULUS FUNCTIONS ¼ ¼ O GUZ O GUR Abstract. The object of this paper is to introduce a new double Cesàro sequence spaces Ces(2) (F; p) de…ned by a double sequence of modulus functions. We study some topologic properties of this space and obtain some inclusion relations.

1. Introduction As usual, N, R and R+ denote the sets of positive integers, real numbers and nonnegative real numbers, respectively. A double sequence on a normed linear space X is a function x from N N into X and brie‡y denoted by x = (x(i; j)). Throughout this work , w and w2 denote the spaces of all single real sequences and double real sequences, respectively. First of all, let us recall preliminary de…nitions and notations. If for every " > 0 there exists n" 2 N such that kxk;l akX < " whenever k; l > n" then a double sequence fxk;l g is said to be converge (in terms of Pringsheim) to a 2 X [10]. A double sequence fxk;l g is called a Cauchy sequence if and only if for every " > 0 there exists n0 = n0 (") such jxk;l xp;q j < " for all k; l; p; q n0 . Pthat 1 A double series is in…nity sum k;l=1 xk;l and its convergence implies the converPn Pm gence by k:kX of partial sums sequence fSn;m g, where Sn;m = k=1 l=1 xk;l (see [2],[3]). If each double Cauchy sequence in X converges an element of X according to norm of X, then X is said to be a double complete space. A normed double complete space is said to be a double Banach space [2]. Let X be a linear space. A function p : X ! R is called paranorm, if i) p(0) = 0; ii) p(x) 0 for all x 2 X; iii) p( x) = p(x) for all x 2 X; iv) p(x + y) p(x) + p(y) for all x; y 2 X; v) if ( n ) is a sequence of scalars with n ! as n ! 1 and (xn ) is a sequence of vectors with p (xn x) ! 0 as n ! 1, then p ( n xn x) ! 0 as n ! 1 ( continuity of multiplication by scalars). A paranorm p for which p(x) = 0 implies x = 0 is called total [7]. A double sequence space E is said to be solid (or normal) if (ykl ) 2 E whenever jykl j jxkl j for all k; l 2 N and (xkl ) 2 E. 2000 Mathematics Subject Classi…cation. 40A05, 40A25, 40A30. Key words and phrases. Double sequence, modulus function, paranorm, Cesàro sequence space, solidity. 1

109

110

¼ ¼ O GUZ O GUR

2

A function f : [0; 1) ! [0; 1) is said to be a modulus function if it satis…es the following: 1) f (x) = 0 if and only if x = 0; 2) f (x + y) f (x) + f (y) for all x; y 2 [0; 1) ; 3) f is increasing; 4) f is continuous from right at 0: It follows that f is continuous on [0; 1) : The modulus function may be bounded or unbounded. For example, if we take f (x) = x (x + 1), then f (x) is bounded. But, for 0 < p < 1, f (x) = xp is not bounded [9]. By the condition 2); we have f (nx) n:f (x) for all n 2 N and so f (x) nf nx ; hence f nx n1 x 1 for all n 2 N: f (x) f n n The FK-spaces L (f ), introduced by Ruckle in [11], is in the form ) ( 1 X f (jxk j) < 1 L(f ) = x 2 w : k=1

where f is a modulus function. This space is closely related to the space `1 which is an L (f ) space with f (x) = x for all real x 0: Later on, this space was investigated by many authors (see [1; 4; 5; 8; 13]) : For 1 p < 1, the Cesàro sequence space is de…ned by 8 9 !p j 1 = < X 1X Cesp = x 2 w : jx(i)j 0:

Since x 2 Ces(2) (F; p); then there exist n0 ; m0 2 N such that 2 0 13pn;m 2 0 13pn;m (s) (s) n;m n;m n0 1 1 1 X X X X X X 4fn;m @ 4fn;m @ jxi;j jA5 + jxi;j jA5 n:m i;j=1 n:m i;j=1 n=n0 m=m0 n=1 m=m0 +1 2 0 13pn;m (s) n;m m0 1 X X X 4fn;m @ + jxi;j jA5 n:m n=n0 +1 m=1 i;j=1 2 0 13pn;m 13pn;m 2 0 n;m n;m n0 1 1 1 X X X X X X T T 4fn;m @ 4fn;m @ + jxi;j jA5 jxi;j jA5 n:m i;j=1 n:m i;j=1 n=n0 m=m0 n=1 m=m0 +1 2 0 13pn;m n;m m0 1 X X X T 4fn;m @ + jxi;j jA5 n:m i;j=1 n=n +1 m=1 0

" < 2 for all s 2 N: Also, we have 2 0 n0 X m0 X 4fn;m @ n=1 m=1

as s ! 1: Consequently, 2 0 1 1 X X 4fn;m @ n=1 m=1

as s ! 1: Since g(

(s) (s)

x

n:m

(s)

n:m

0

H

x)

(s)

(K) M g(x(s) x)+@

n;m X

i;j=1

n;m X

i;j=1

1 1 X X

n=1 m=1

we get g(

(s) (s)

x

x) ! 0 as s ! 1:

13pn;m

jxi;j jA5

13pn;m

jxi;j jA5

2

0

4fn;m @


0 there exists N 2 N such that 13pn;m 2 0 n;m 1 X 1 X X 1 (s) (t) 4fn;m @ < "M xi;j xi;j A5 n:m n=1 m=1 i;j=1 for s; t > N and so by taking t ! 1 we have 2 0 n;m 1 X 1 X X (s) 4fn;m @ 1 xi;j n:m n=1 m=1 i;j=1

13pn;m

xi;j A5

< "M

for all s N: Thus we get g(x(s) x) ! 0 for all s N: Finally we must show that x 2 Ces(2) (F; p): By de…nition of a modulus function, for all s N we get 2 0 0 13pn;m 1 M1 n;m 1 1 X X X 1 4fn;m @ @ A jxi;j jA5 n:m i;j=1 n=1 m=1 0

2

0

2

0

n;m X 4fn;m @ 1 = @ xi;j n:m i;j=1 n=1 m=1

0

1 1 X X

n;m X 4fn;m @ 1 @ xi;j n:m i;j=1 n=1 m=1 1 1 X X

2

(s) (s) xi;j + xi;j A5

1 1 X X

A

13pn;m 1 M1

(s) xi;j A5

13pn;m 1 M1 n;m X 1 (s) 4fn;m @ A xi;j A5 +@ n:m n=1 m=1 i;j=1 0

0

13pn;m 1 M1 A

" + g(x(s) );

which implies that x 2 Ces(2) (F; p): Then Ces(2) (F; p) is complete. Theorem 4. Let F = (fn;m ) and H = (hn;m ) be two modulus functions. Then f

(t)

(i) lim sup hn;m < 1 for all n; m 2 N implies Ces(2) (H; p) n;m (t)

(ii) Ces(2) (H; p) \ Ces(2) (F; p)

Ces(2) (F; p);

Ces(2) (F + H; p):

Proof. (i) By the hypothesis there exists K > 0 such that fn;m (t) K:hn;m (t) for all n; m 2 N: Let x 2 Ces(2) (H; p); then we have 2 0 13pn;m 2 0 13pn;m n;m n;m 1 X 1 1 X 1 X X X X 1 1 4fn;m @ 4K:hn;m @ jxi;j jA5 jxi;j jA5 n:m n:m n=1 m=1 n=1 m=1 i;j=1 i;j=1 2 0 13pn;m n;m 1 X 1 X X 1 4hn;m @ = CH jxi;j jA5 n:m i;j=1 n=1 m=1 < 1;

where C = max (1; K) : Hence we get x 2 Ces(2) (F; p):

115

A NEW DOUBLE CESÀRO SEQUENCE SPACE DEFINED BY M ODULUS FUNCTIONS

7

(ii) Let x 2 Ces(2) (H; p) \ Ces(2) (F; p): Hence we have 13pn;m 2 0 n;m 1 X 1 X X 4(fn;m + hn;m ) @ 1 jxi;j jA5 n:m i;j=1 n=1 m=1 2 0 1 0 13pn;m n;m n;m 1 X 1 X X X 1 4fn;m @ 1 = jxi;j jA + hn;m @ jxi;j jA5 n:m n:m n=1 m=1 i;j=1 i;j=1 8 13pn;m 13pn;m 9 2 0 2 0 n;m n;m 1 X 1 1 X 1 = 0, z ∈ U , the class of normalized convex functions in U . Let the functions f and g be analytic in U . We say that the function f is subordinate to g, written f ≺ g, if there exists a Schwarz function w, analytic in U , with w(0) = 0 and |w(z)| < 1, for all z ∈ U, such that f(z) = g(w(z)), for all z ∈ U . In particular, if the function g is univalent in U , the above subordination is equivalent to f (0) = g(0) and f (U ) ⊂ g(U ). Let ψ : C3 × U → C and h be an univalent function in U . If p is analytic in U and satisfies the second order differential subordination ψ(p(z), zp0 (z), z 2 p00 (z); z) ≺ h(z),

for z ∈ U,

(1.1)

then p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, or more simply a dominant, if p ≺ q for all p satisfying (1.1). A dominant qe that satisfies qe ≺ q for all dominants q of (1.1) is said to be the best dominant of (1.1). The best dominant is unique up to¡a rotation of U . ¢ Let ψ : C2 × U → C and h analytic in U . If p and ψ p (z) , zp0 (z) , z 2 p00 (z) ; z are univalent and if p satisfies the second order differential superordination h(z) ≺ ψ(p(z), zp0 (z), z 2 p00 (z) ; z),

z ∈ U,

(1.2)

then p is a solution of the differential superordination (1.2) (if f is subordinate to F , then F is called to be superordinate to f ). An analytic function q is called a subordinant if q ≺ p for all p 1

A differential sandwich-type result for Ruscheweyh operator Andrei Loriana

126

satisfying (1.2). An univalent subordinant qe that satisfies q ≺ qe for all subordinants q of (1.2) is said to be the best subordinant. Miller and Mocanu [17] obtained conditions h, q and ψ for which the following implication holds h(z) ≺ ψ(p(z), zp0 (z), z 2 p00 (z) ; z) ⇒ q (z) ≺ p (z) . P P∞ j j For two functions f (z) = z + ∞ j=2 aj z and g(z) = z + j=2 bj z analytic in the open unit disc U , the Hadamard product (or convolution product) of f (z) and g (z), written as (f ∗ g) (z), is defined by ∞ P f (z) ∗ g (z) = (f ∗ g) (z) = z + aj bj z j . j=2

Definition 1.1 (Al Oboudi [7]) For f ∈ A, λ ≥ 0 and n ∈ N, the operator Dλm is defined by Dλm : A → A, Dλ0 f (z) = f (z) Dλ1 f (z) = (1 − λ) f (z) + λzf 0 (z) = Dλ f (z) ...

¡ ¢ = (1 − λ) Dλm−1 f (z) + λz (Dλm f (z))0 = Dλ Dλm−1 f (z) , for z ∈ U. P P∞ m j m j Remark 1.1 If f ∈ A and f (z) = z + ∞ j=2 aj z , then Dλ f (z) = z + j=2 [1 + (j − 1) λ] aj z , for z ∈ U . Dλm f (z)

Remark 1.2 For λ = 1 in the above definition we obtain the Sa ˘la˘gean differential operator [20]. Definition 1.2 (Ruscheweyh [19]) For f ∈ A and n ∈ N, the operator Rn is defined by Rn : A → A, R0 f (z) = f (z) R1 f (z) = zf 0 (z) ... f (z) = z (Rn f (z))0 + nRn f (z) , z ∈ U. P∞ (n+j−1)! P j n j Remark 1.3 If f ∈ A, f (z) = z + ∞ j=2 aj z , then R f (z) = z + j=2 n!(j−1)! aj z for z ∈ U . n+1

(n + 1) R

The purpose of this paper is to derive the several subordination and superordination results involving a differential operator. Furthermore, we studied the results of M. Darus, K. Al-Shaqs [15], Shanmugam, Ramachandran, Darus and Sivasubramanian [21], R.W. Ibrahim, M. Darus [16]. In order to prove our subordination and superordination results, we make use of the following known results. Definition 1.3 [18] Denote by Q the set of all functions f that are analytic and injective on U \E (f), where E (f ) = {ζ ∈ ∂U : lim f (z) = ∞}, and are such that f 0 (ζ) 6= 0 for ζ ∈ ∂U \E (f ). z→ζ

Lemma 1.1 [18] Let the function q be univalent in the unit disc U and θ and φ be analytic in a domain D containing q (U ) with φ (w) 6= 0 when w ∈ q (U ). Set Q (z) = zq 0 (z) φ (q (z)) and h (z) = θ (q (z)) + Q (z). Suppose that 1. Q is ³ starlike ´ univalent in U and zh0 (z) 2. Re Q(z) > 0 for z ∈ U . If p is analytic with p (0) = q (0), p (U ) ⊆ D and θ (p (z)) + zp0 (z) φ (p (z)) ≺ θ (q (z)) + zq0 (z) φ (q (z)) , then p (z) ≺ q (z) and q is the best dominant. 2

A differential sandwich-type result for Ruscheweyh operator Andrei Loriana

127

Lemma 1.2 [14] Let the function q be convex univalent in the open unit disc U and ν and φ be analytic in³ a domain ´ D containing q (U ). Suppose that ν 0 (q(z)) 1. Re φ(q(z)) > 0 for z ∈ U and 2. ψ (z) = zq 0 (z) φ (q (z)) is starlike univalent in U . If p (z) ∈ H [q (0) , 1] ∩ Q, with p (U ) ⊆ D and ν (p (z)) + zp0 (z) φ (p (z)) is univalent in U and ν (q (z)) + zq 0 (z) φ (q (z)) ≺ ν (p (z)) + zp0 (z) φ (p (z)) , then q (z) ≺ p (z) and q is the best subordinant.

2

Main results

Definition 2.1 Let λ ≥ 0 and n, m ∈ N. Denote by DRλm,n : A → A the operator given by the Hadamard product of the generalized Sa˘la˘gean operator Dλm and the Ruscheweyh operator Rn , DRλm,n f (z) = (Dλm ∗ Rn ) f (z) , for any z ∈ U and each nonnegative integers m, n. P j Remark 2.1 If f ∈ A and f (z) = z + ∞ j=2 aj z , then P m (n+j−1)! 2 j DRλm,n f (z) = z + ∞ j=2 [1 + (j − 1) λ] n!(j−1)! aj z , for z ∈ U . This operator was studied in [12]. Remark 2.2 For λ = 1, m = n, we obtain the Hadamard product SRn [1] of the Sa ˘la˘gean operator S n and Ruscheweyh derivative Rn , which was studied in [2], [3]. Remark 2.3 For m = n we obtain the Hadamard product DRλn [4] of the generalized Sa˘la˘gean operator Dλn and Ruscheweyh derivative Rn , which was studied in [5], [6], [8], [9], [10], [11]. Using simple computation one obtains the next result. Proposition 2.1 [12] For m, n ∈ N and λ ≥ 0 we have ¢0 ¡ z DRλm,n f (z) = (n + 1) DRλm,n+1 f (z) − nDRλm,n f (z) .

(2.1)

We begin with the following

0

m,n z (DRλ f (z)) ∈ H (U ) , z ∈ U , f ∈ A, m, n ∈ N, λ ≥ 0 and let the function q (z) Theorem 2.2 Let DRm,n f (z) λ be convex and univalent in U such that q (0) = 1. Assume that ¶ µ zq 0 (z) zq 00 (z) α + 0 > 0, z ∈ U, (2.2) Re 1 + q (z) − β q (z) q (z)

for α, β ∈ C,β 6= 0, z ∈ U, and

ψ m,n (α, β; z) := λ ´0 ³ ¡ ¢0 DRλm,n+1 f (z) z DRλm,n f (z) − βn. β (n + 1) ¡ ¢0 + (α − .β) DRλm,n f (z) DRλm,n f (z)

(2.3)

If q satisfies the following subordination

ψ m,n (α, β, µ; z) ≺ αq (z) + λ 3

βzq 0 (z) , q (z)

(2.4)

A differential sandwich-type result for Ruscheweyh operator Andrei Loriana

for, α, β ∈ C, β 6= 0 then

DRλm+1,n f (z) ≺ q (z) , DRλm,n f (z)

z ∈ U,

128

(2.5)

and q is the best dominant. Proof. Our aim is to apply Lemma 1.1. Let the function p be defined by p (z) := z ∈ U , z 6= 0, f ∈ A. The function p is analytic in U and p (0) = 1 Differentiating this function, with respect to z,we get ¸ ∙ 0 00 0 2 m,n m,n m,n z (DRλ f (z)) z 2 (DRλ f (z)) z (DRλ f (z)) 0 zp (z) = DRm,n f (z) + DRm,n f (z) − DRm,n f (z) λ

λ

0

z (DRm,n f (z)) λ m,n DRλ f (z)

,

λ

By using the identity (2.1), we obtain

³ ´0 DRλm,n+1 f (z) = (n + 1) ¡ ¢0 − p (z) DRλm,n f (z)

zp0 (z)

¢0 ¡ z DRλm,n f (z) −n DRλm,n f (z)

(2.6)

β , α, β ∈ C, β 6= 0 it can be easily verified that θ is analytic By setting θ (w) := αw and φ (w) := w in C, φ is analytic in C\{0} and that φ (w) 6= 0, w ∈ C\{0}. 0 (z) Also, by letting Q (z) = zq0 (z) φ (q (z)) = βzq q(z) ,we find that Q (z) is starlike univalent in U.

Let h (z) = θ (q (z)) + Q (z) = αq (z) +

βzq 0 (z) q(z) ,

z∈U

³ 0 ´ (z) If we derive the function Q, with respect to z, perform calculations, we have Re zh = Q(z) ´ ³ 0 (z) 00 (z) + zqq0 (z) > 0. Re 1 + αβ q (z) − zqq(z) By using (2.6), we obtain ∙ ¸ 0 0 0 m,n z (DRm,n f (z)) z (DRλ f (z)) (DRλm,n+1 f (z)) zp0 (z) λ αp (z) + β p(z) = α DRm,n f (z) + .β (n + 1) −n = m,n 0 − DRλ f (z) (DRλm,n f (z)) λ 0 0 m,n z (DRλ f (z)) (DRλm,n+1 f (z)) − βn. β (n + 1) m,n 0 + (α − .β) m,n DR f (z) (DRλ f (z)) λ 0 (z) 0 (z) ≺ αq (z) + β zqq(z) . By using (2.4), we have αp (z) + β zpp(z) 0 m,n z (DRλ f (z)) Therefore, the conditions of Lemma 1.1 are met, so we have p (z) ≺ q (z), z ∈ U, i.e. DRm,n f (z) λ ≺ q (z), z ∈ U, and q is the best dominant. Corollary 2.3 Let q (z) = holds. If f ∈ A and

1+Az 1+Bz ,

−1 ≤ B < A ≤ 1, m, n ∈ N, λ ≥ 0, z ∈ U. Assume that (2.2)

(α, β; z) ≺ α ψ m,n λ

1 + Az (A − B) z +β , 1 + Bz (1 + Az) (1 + Bz)

is defined in (2.3), then for α, β ∈ C, β 6= 0, −1 ≤ B < A ≤ 1, where ψ m,n λ ¡ ¢0 z DRλm,n f (z) 1 + Az ≺ m,n DRλ f (z) 1 + Bz

and

1+Az 1+Bz

is the best dominant.

Proof. For q (z) =

1+Az 1+Bz ,

−1 ≤ B < A ≤ 1, in Theorem 2.2 we get the corollary. 4

A differential sandwich-type result for Ruscheweyh operator Andrei Loriana

Corollary 2.4 Let q (z) =

1+z 1−z , m, n

129

∈ N, λ ≥ 0, z ∈ U. Assume that (2.2) holds. If f ∈ A and

(α, β, µ; z) ≺ α ψ m,n λ

1+z 2z , +β 1−z 1 − z2

for α, β ∈ C, β 6= 0, where ψ m,n is defined in (2.3), then λ

¢0 ¡ z DRλm,n f (z) 1+z , ≺ m,n 1−z DRλ f (z)

and

1+z 1−z

is the best dominant.

Proof. Corollary follows by using Theorem 2.2 for q (z) =

1+z 1−z

Theorem 2.5 Let q be convex and univalent in U, such that q (0) = 1, m, n ∈ N, λ ≥ 0. Assume that ¶ µ α q (z) > 0, for α, β ∈ C, µ 6= 0, z ∈ U. (2.7) Re β 0

z (DRm,n f (z)) λ If f ∈ A, DRm,n ∈ H [q (0) , 1] ∩ Q and ψ m,n (α, β, µ; z) is univalent in U , where ψ m,n (α, β; z) λ λ f (z) λ is as defined in (2.3), then

αq (z) + β implies

zq 0 (z) ≺ ψ m,n (α, β, µ; z) , λ p (z)

¢0 ¡ z DRλm,n f (z) q (z) ≺ , DRλm,n f (z)

z ∈ U,

z ∈ U,

(2.8)

(2.9)

and q is the best subordinant.

Proof. Let the function p be defined by p (z) := β w

0

m,n z (DRλ f (z)) m,n DRλ f (z)

, z ∈ U , z 6= 0, f ∈ A.

it can be easily verified that ν is analytic in C, φ is By setting ν (w) := αw and φ (w) := analytic in C\{0} and that φ (w) 6= 0, w ∈ C\{0}. ³ 0 ´ ´ ³ (q(z)) Since q is convex and univalent function, it follows that Re νφ(q(z)) = Re αβ q (z) > 0, for α, β ∈ C, β 6= 0. By using (2.8) we obtain αq (z) + β

zq 0 (z) zp0 (z) ≺ αp (z) + β . q (z) p (z)

Using Lemma 1.2, we have ¢0 ¡ z DRλm,n f (z) q (z) ≺ p (z) = , DRλm,n f (z) and q is the best subordinant.

5

z ∈ U,

A differential sandwich-type result for Ruscheweyh operator Andrei Loriana

130

1+Az Corollary 2.6 Let q (z) = 1+Bz , −1 ≤ B < A ≤ 1, m, n ∈ N, λ ≥ 0. Assume that (2.7) holds. If 0 m,n z (DRλ f (z)) f ∈ A, DRm,n f (z) ∈ H [q (0) , 1] ∩ Q and λ

α

(A − B) z 1 + Az (α, β; z) , +β ≺ ψ m,n λ 1 + Bz (1 + Az) (1 + Bz)

for α, β ∈ C, β 6= 0, −1 ≤ B < A ≤ 1, where ψ m,n is defined in (2.3), then λ ¢0 ¡ z DRλm,n f (z) 1 + Az ≺ 1 + Bz DRλm,n f (z)

and

1+Az 1+Bz

is the best subordinant.

Proof. For q (z) =

1+Az 1+Bz ,

Corollary 2.7 Let q (z) = ∈ H [q (0) , 1] ∩ Q and

−1 ≤ B < A ≤ 1 in Theorem 2.5 we get the corollary.

1+z 1−z , m, n

α

∈ N, λ ≥ 0. Assume that (2.7) holds. If f ∈ A,

0

m,n z (DRλ f (z)) m,n DRλ f (z)

1+z 2z ≺ ψ m,n (α, β; z) , +β λ 1−z 1 − z2

for α, β ∈ C, β 6= 0, where ψ m,n is defined in (2.3), then λ

¢0 ¡ z DRλm,n f (z) 1+z ≺ 1−z DRλm,n f (z)

and

1+z 1−z

is the best subordinant.

Proof. Corollary follows by using Theorem 2.5 for q (z) = 1+z 1−z . Combining Theorem 2.2 and Theorem 2.5, we state the following sandwich theorem. Theorem 2.8 Let q1 and q2 be analytic and univalent in U such that q1 (z) 6= 0 and q2 (z) 6= 0, for all z ∈ U , with zq10 (z) and zq20 (z) being starlike univalent. Suppose that q1 satisfies (2.2) and 0 z (DRm,n f (z)) λ q2 satisfies (2.7). If f ∈ A, DRm,n ∈ H [q (0) , 1] ∩ Q and ψ m,n (α, β; z) is as defined in (2.3) λ f (z) λ univalent in U , then αq1 (z) + for α, β ∈ C, β 6= 0, implies

βzq10 (z) βzq20 (z) (α, β; z) ≺ αq2 (z) + ≺ ψ m,n , λ q1 (z) q2 (z) ¡ ¢0 z DRλm,n f (z) q1 (z) ≺ ≺ q2 (z) , DRλm,n f (z)

and q1 and q2 are respectively the best subordinant and the best dominant. For q1 (z) = corollary.

1+A1 z 1+B1 z ,

q2 (z) =

1+A2 z 1+B2 z ,

where −1 ≤ B2 < B1 < A1 < A2 ≤ 1, we have the following

6

A differential sandwich-type result for Ruscheweyh operator Andrei Loriana

Corollary 2.9 Let m, n ∈ N, λ ≥ 0. Assume that (2.2) and (2.7) hold for q1 (z) = 0 m,n z (DRλ f (z)) 2z , respectively. If f ∈ A, ∈ H [q (0) , 1] ∩ Q and q2 (z) = 1+A m,n 1+B2 z DR f (z)

131

1+A1 z 1+B1 z

and

λ

α

(A1 − B1 ) z 1 + A1 z (α, β; z) +β ≺ ψ m,n λ 1 + B1 z (1 + A1 z) (1 + B1 z) ≺α

(A2 − B2 ) z 1 + A2 z +β , 1 + B2 z (1 + A2 z) (1 + B2 z)

is defined in (2.3), then for α, β ∈ C, β 6= 0, −1 ≤ B2 ≤ B1 < A1 ≤ A2 ≤ 1, where ψ m,n λ ¡ ¢ 0 z DRλm,n f (z) 1 + A2 z 1 + A1 z ≺ ≺ , m,n 1 + B1 z DRλ f (z) 1 + B2 z hence

1+A1 z 1+B1 z

and

1+A2 z 1+B2 z

are the best subordinant and the best dominant, respectively.

References [1] A. Alb Lupas, Certain differential subordinations using Sa ˘la˘gean and Ruscheweyh operators, Acta Universitatis Apulensis, No. 29/2012, 125-129. [2] A. Alb Lupas, A note on differential subordinations using Sa˘la ˘gean and Ruscheweyh operators, Romai Journal, vol. 6, nr. 1(2010), 1—4. [3] A. Alb Lupas, Certain differential superordinations using Sa ˘la˘gean and Ruscheweyh operators, Analele Universita˘¸tii din Oradea, Fascicola Matematica, Tom XVII, Issue no. 2, 2010, 209-216. [4] A. Alb Lupas, Certain differential subordinations using a generalized Sa˘la˘gean operator and Ruscheweyh operator I, Journal of Mathematics and Applications, No. 33 (2010), 67-72. [5] A. Alb Lupas, Certain differential subordinations using a generalized Sa ˘la˘gean operator and Ruscheweyh operator II, Fractional Calculus and Applied Analysis, Vol 13, No. 4 (2010), 355360. [6] A. Alb Lupas, Certain differential superordinations using a generalized Sa˘la˘gean and Ruscheweyh operators, Acta Universitatis Apulensis nr. 25/2011, 31-40. [7] F.M. Al-Oboudi, On univalent functions defined by a generalized Sa˘la˘gean operator, Ind. J. Math. Math. Sci., 27 (2004), 1429-1436. [8] L. Andrei, Differential subordination results using a generalized Sa˘la˘gean operator and Ruscheweyh operator, Acta Universitatis Apulensis (to appear). [9] L. Andrei, Some differential subordination results using a generalized Sa˘la˘gean operator and Ruscheweyh operator, Jokull Journal (to appear). [10] L. Andrei, Differential superordination results using a generalized Sa˘la˘gean operator and Ruscheweyh operator, Analele Universitatii Oradea (to appear). [11] L. Andrei, Some differential superordination results using a generalized Sa ˘la˘gean operator and Ruscheweyh operator, Stud. Univ. Babes-Bolyai Math (to appear). [12] L. Andrei, Differential Sandwich Theorems using a generalized Sa˘la˘gean operator and Ruscheweyh operator, submitted Journal of Inequalities and Applications. 7

A differential sandwich-type result for Ruscheweyh operator Andrei Loriana

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[13] L. Andrei, On some differential Sandwich Theorems using a generalized Sa˘la˘gean operator and Ruscheweyh operator, Journal of Computational Analysis and Applications (to appear). [14] T. Bulboaca˘, Classes of first order differential superordinations, Demonstratio Math., Vol. 35, No. 2, 287-292. [15] M. Darus, K. Al-Shaqsi, Differential sandwich theorems with generalised derivative operator, Advanced Technologies, October, Kankesu Jayanthakumaran (Ed), ISBN:978-953-307-009-4 2009 [16] R.W. Ibrahim, M. Darus, On sandwich theorems of analytic functions involving Noor integral operator, African Journal of Mathematics and Computer Science Research, vol.2, pp.132-137, 2009 [17] S.S. Miller, P.T. Mocanu, Subordinants of Differential Superordinations, Complex Variables, vol. 48, no. 10, 815-826, October, 2003. [18] S.S. Miller, P.T. Mocanu, Differential Subordinations: Theory and Applications, Marcel Dekker Inc., New York, 2000. [19] St. Ruscheweyh, New criteria for univalent functions, Proc. Amet. Math. Soc., 49(1975), 109115. [20] G. St. Sa˘la˘gean, Subclasses of univalent functions, Lecture Notes in Math., Springer Verlag, Berlin, 1013(1983), 362-372. [21] T.N. Shanmugan, C. Ramachandran, M. Darus, S. Sivasubramanian, Differential sandwich theorems for some subclasses of analytic functions involving a linear operator, Acta Math. Univ. Comenianae, 16 (2007), no. 2, 287-294.

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J. APPLIED FUNCTIONAL ANALYSIS, VOL. 10, NO.'S 1-2,133-142 ,2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

Predictor Corrector type method for solving certain non convex equilibrium problems Wasim Ul-Haq1;2 , Manzoor Hussain2 1

Department of Mathematics, College of Science in Al-Zul…, Majmaah University, Kingdom of Saudi Arabia 2

Department of Mathematics, Abdul Wali Khan University Mardan, Pakistan Email: [email protected], [email protected]

Abstract: In this paper, we propose and analyze a new predictor corrector type method for solving uniformly regular invex equilibrium problems by using the auxiliary principle technique. The convergence criteria of this new method under some mild restrictions is also studied. New and known methods for solving variational-like inequalities and equilibrium problems appear as special consequences of our work.

Key Words: Equilibrium problem, Invex function, Prox regular set. MSC: 49J40, 91B50

1

Introduction

A variety of problems arising in optimization, economics, …nance, networking, transportation, structural analysis and elasticity can be studied in the framework of Equilibrium problems. The theory of equilibrium problems provides productive and innovative techniques to investigate a wide range of such problems. Blum and Oettli [1] and Noor and Oettli [2] has shown that variational inequalities and mathematical programming problems can be seen as a special case of the abstract equilibrium problems. There are a many numerical methods including the projection technique and its variant forms, Weiner-Hopf equations, the auxiliary principle technique and resolvent equations method for solving variational inequalities. However, it is known that projection, Weiner-Hopf equations, and proximal and resolvent equations techniques cannot be extended and generalized to suggest and analyze similar iterative methods for solving uniformly regular invex equilibrium problems and variational-like inequalities due to the presence of function (:; :). To overcome this shortcoming, one may use auxiliary principle technique. Several interesting generalizations and extensions of classical convexity have been studied and investigated in recent years. Hanson [3] introduced the invex functions as a generalization of convex functions. Later, subsequent works inspired from Hanson’s result have greatly found the role

1

133

Predictor Corrector type method, Wasim Ul-Haq, Manzoor Hussain

and applications of invexity in non linear optimization and other branches of pure and applied sciences. The basic properties and role of preinvex functions in optimization, equilibrium problems and variational inequalities was studied by Noor [2,4,5] and Wier and Mond [8]. Motivated from the above and recent work of Noor et al. [47], we use the auxiliary principle technique which is mainly due to Glowinski et al. [12] to propose and analyze a new predictor-corrector type method for solving the uniformly regular invex equilibrium problems. We also study the convergence of this new method under some suitable conditions.

2

Preliminaries

Let h:; :i and k:k denote the inner product and norm respectively in a real Hilbert space H. In the sequel let f : K ! H and (:; :) : K K ! H be a continuous function, where K is a nonempty closed convex set in H. Now, we recall some basic de…nitions and concepts of non-smooth analysis, for details see [48,49]. P (u) of the set K at u is De…nition 1.The proximal normal cone denoted by NK de…ned by P NK (u) = f 2 H : u 2 PK [u + ]g

where > 0 is a constant and PK [:] is the projection of H onto the set K and de…ned as PK [u] = fu 2 K : dK = kv u kg Here dK (:) is the usual distance function to the subset K; and de…ned as dK (u) = inf kv v2K

uk .

The proximal normal cone has the following characterization. Lemma 1. Let K be a closed subset in H: Then exists a constant such that h ;v

ui

kv

P (u) if and only if there 2 NK

uk2 ; 8v 2 K.

C (u) is de…ned as De…nition 2. The Clarke normal cone, denoted by NK P C (u)]; NK (u) = co[NK

8u 2 K

P (u) C (u); but the where co means the closure of the convex hull. Clearly NK NK C P (u) converse is not true. Note that NK (u) is always closed and convex, whereas NK is convex but may not be closed, see [49].

Poliquin et al. [49] and Clarke et al. [48] have introduced and studied a new class of nonconvex sets, which are also called uniformly prox-regular sets. This class of uniformly prox-regular sets has played an important role in many nonconvex

2

134

Predictor Corrector type method, Wasim Ul-Haq, Manzoor Hussain

135

applications such as optimization, dynamical systems and di¤erential inclusion. In particular, we have De…nition 3. For a given r 2 (0; 1]; a subset Kr is said to be uniformly r proximal regular if and only if for each u 2 Kr and all proximal normal vectors 0 6= 2 N PK (u), k k

; (v; u)

1 k (v; u)k2 ; 8v 2 Kr : 2r

The class of uniformly prox-regular sets contains the class of convex sets, p-convex sets, C 1;1 submanifolds (possibly with boundary) of H; the images under a C 1;1 di¤eomorphism of convex sets and many other nonconvex sets, see [48,49]. It is clearsimple to note that if r = 1 then uniformly r proximal regular set K reduces to convex set. This observation plays a key role in this work. It is known that if K P (u) is closed is a uniformly r prox-regular set, then the proximal normal cone NK C P as a set-valued mapping. Thus, we have NK (u) = NK (u): From now onward the set Kr is considered to be uniformly proximal regular invex set. For a given continuous bifunction F (:; :) : Kr Kr ! H; consider the problem of …nding u 2 Kr such that F (u; v) +

k 2r

k (v; u)k2

where k is a positive constant. By letting the above problem becomes

=

F (u; v) + k (v; u)k2

0; k 2r

0;

8v 2 Kr ;

such that

= 0 for r = 1; then

8v 2 Kr :

(2.1)

This type of problem is called uniformly regular invex equilibrium problem. Note that for (v; u) = v u; then the uniformly regular invex equilibrium problem reduces to uniformly regular equilibrium problem introduced and studied by Noor [26; 3 5] and Noor and Noor [30] and for r = 1 uniformly regular invex equilibrium problem reduces to classical equilibrium problem introduced and studied by Blum and Oettli [1] and Noor and Oettli [2]. If F (u; v) = hT u; (v; u)i ; where T : H ! H is a nonlinear continuous operator, then problem (2:1) is equivalent to …nding u 2Kr such that D E T u + k (v; u)k2 ; (v; u) 0; 8v 2 Kr : (2.2)

which is a variant form of varitainol-like inequality problem. Note that for r = 1 problem (2:2) reduces to varitainol-like inequality problem studied by various authors in recent years, see Noor [19] and Yang and Chen [43]: If (v; u) = v u and r = 1 uniformly r proximal regular set reduces to convex set K.and problem (2:2) is equivalent to …nding u 2 K such that hT u; v

ui

0;

8 v 2 K:

(2.3)

which is known as the classical variational inequality introduced and studied by Stampacchia [40]. For the recent applications, numerical methods and formulation of variational inequalities, variational-like inequalities and equilibrium problems, see [1-51] and the references therein. De…nition 4. The function F (:; :) : K

K ! H is said to be:

3

Predictor Corrector type method, Wasim Ul-Haq, Manzoor Hussain

(i).

136

pseudomonotone

F (u; v)

0 =)

F (v; u)

0;

8u; v 2 K:

(ii). partially relaxed strongly monotone, if there exist a constant that F (u; v) + F (v; z) k (z; u)k2 ; 8u; v; z 2 K;

> 0 such

Note that for z = u; (z; u) = 0, 8 u 2 K; thus partially relaxed strongly monotonicity reduces to F (u; v) + F (v; u) 0; 8u; v; 2 K, which is known as the monotonicity of F (:; :) : It is obvious that implies pseudomonotonicity, but the converse is not true.

3

monotonicity

Iterative methods and convergence analysis

In this section, we introduce some iterative scheme for solving the uniformly regular invex equilibrium problem (2:1) by using the auxiliary principle technique, which is due to Glowinski et al. [12]. For a given u 2 Kr ; consider the auxiliary uniformly regular invex equilibrium problem of …nding w 2 Kr such that F (u; v) +

k (v; u)k2 + E 0 (w)

E 0 (wn ); (v; un+1 )

0;

8 v 2 Kr , (3.1)

where is a positive constant and E 0 (u) is the di¤erential of a strongly preinvex function E at u 2 K. From the strong preinvexity of the di¤erentiable function E(u); it follows that problem (3:1) has a unique solution. Note that if w = u; then w is a solution of uniformly regular invex equilibrium problem (2:1). This observation leads us to propose and analyze the following iterative scheme for solving the uniformly regular invex equilibrium problem (2:1). Algorithm 1. Given u0 2 H; calculate un+1 from the following iterative scheme F (wn ; v)+ F (un ; v) + where ;

k (v; wn )k2 + E 0 (un+1 )

E 0 (wn ); (v; un+1 )

k (v; un )k2 + E 0 (wn )

E 0 (un ); (v; wn )

0; 8 v 2 Kr (3.2) 0; 8 v 2 Kr

(3.3)

are both positive constants.

If F (u; v) = hT u; (v; u)i ; then we have Algorithm 2. Given u0 2 H; calculate un+1 through iterative scheme D E T wn + k (v; wn )k2 + E 0 (un+1 ) E 0 (wn ); (v; un+1 ) 0; 8 v 2 Kr

4

Predictor Corrector type method, Wasim Ul-Haq, Manzoor Hussain

D

where ;

T un +

E E 0 (un ); (v; un )

k (v; un )k2 + E 0 (wn )

are both positive constants.

137

0; 8 v 2 Kr

When r = +1; then = 0; so the uniformly regular invex equilibrium problem reduces to equilibrium problem. i.e. F (u; v)

0; 8 v 2 K

and the uniformly prox-regular set K becomes convex set K: Thus, we have Algorithm 3 [47]. For u0 2 H we calculate un+1 through the following iterative scheme F (wn ; v) + E 0 (un+1 ) E 0 (wn ); v un+1 0; 8 v 2 K F (un ; v) + E 0 (wn )

where ;

E 0 (un ); v

wn

0; 8 v 2 K

are both positive constants.

Now if, F (u; v) = hT u; vi ; then we have Algorithm 4[47]. For u0 2 H we calculate un+1 through the following iterative scheme T wn + E 0 (un+1 ) E 0 (wn ); v un+1 0; 8 v 2 K T un + E 0 (wn )

where ;

E 0 (un ); v

wn

0; 8 v 2 K

are both positive constants.

We now study the convergence criteria of Algorithm 1: For this purpose, we need the following condition. Assumption 1[47]. 8 u; v; z 2 K; the function tion

(:; :) satis…es the following condi-

(u; v) = (u; z) + (z; v) :

(3.4)

Assumption has been used to study the existence of a solution of a variational-like inequalities. Note that (u; v) = 0 if and only if u = v; 8u; v 2 K: Theorem 1. Suppose the function F (:; :) is partially relaxed strongly monotone with constant > 0 and suppose E(u) be a strongly preinvex function with > 0: If 0 < < + ; 0 < < + and the above assumption holds, then the approximate solution obtained from Algorithm 1 converges to a solution u 2 K of the uniform regular invex equilibrium problem (2:1). Proof Let u 2Kr be a solution of (2:1); see [1,19]. Then F (u; v) +

k (v; u)k2

0; 8v 2 Kr

(3.5)

F (u; v) +

k (v; u)k2

0; 8v 2 Kr ;

(3.6)

5

Predictor Corrector type method, Wasim Ul-Haq, Manzoor Hussain

138

where and are both positive constants.Taking v = un+1 in (3:5) and v = u in (3:2); we have k (un+1 ; u)k2

F (u; un+1 ) + F (wn ; u) +

D 0 k (u; wn )k2 + E (un+1 )

0; 8v 2 Kr ;

E 0 E (wn ); (u; un+1 )

(3.7) 0; 8v 2 Kr : (3.8)

Now we consider the generalized Bergman function as

B(u; z) = E(u)

D

E(z)

E 0 E (z); (u; z)

k (u; z)k2 ;

where we have used the fact that the function E(u) is strongly preinvex. Combining (3:7)

(3:9) and (3:4); we have D

B(u; wn ) B(u; un+1 ) = E(un+1 ) E(wn )

= E(un+1 )

E(wn )

D

0

E (wn )

D 0 k (un+1 ; wn )k2 + E (un+1 ) k (un+1 ; wn )k2

f

E D 0 E 0 E (wn ); (u; wn ) + E (un+1 ); (u; un+1 )

E 0 E (un+1 ); (u; un+1 ) E 0 E (wn ); (u; un+1 )

fF (u; un+1 ) + F (wn ; u)g

D

E 0 E (wn ); (un+1 ; wn )

fk (un+1 ; u)k2 + k (u; wn )k2 g

( + )g k (un+1 ; wn )k2 :

since F (:; :) is partially relaxed strongly monotone with a constant

> 0:

In a similar way, we have B(u; un ) B(u; wn ) k (wn ; u)k2 g f

k (un ; wn )k2

fF (u; wn ) + F (un ; u)g

fk (u; un )k2 +

( + )g k (un ; wn )k2 :

since F (:; :) is partially relaxed strongly monotone with a constant

> 0:

If un+1 = wn = un ; then clearly un is a solution of (2:1): Otherwise, for 0 < < + and 0 < < + ; the sequences B(u; wn ) B(u; un+1 ) and B(u; un ) B(u; wn ) are nonnegative and we must have limn!1 (k (un+1 ; wn )k) = 0 and limn!1 (k (wn ; un )k) = 0: Thus limn!1 k (un+1 ; un )k = limn!1 (k (un+1 ; wn )k) + limn!1 (k (wn ; un )k) = 0:

6

Predictor Corrector type method, Wasim Ul-Haq, Manzoor Hussain

_

It follows that the sequence fun g is bounded. Let u be a cluster point of the subse_ quence funj g; and let funj g be a subsequence converging towards u: Now using the technique of Zhu and Marcotte [46], it can be shown that the entire sequence fun g _ converges to the cluster point u satisfying the uniformly regular invex equilibrium problem (2:1).

References

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J. APPLIED FUNCTIONAL ANALYSIS, VOL. 10, NO.'S 1-2,143-163 ,2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

ON SOME ZWEIER I-CONVERGENT DIFFERENCE SEQUENCE SPACES KULDIP RAJ AND SURUCHI PANDOH

Abstract. In the present paper we introduce some Zweier ideal convergent difference sequence spaces defined by a sequence of Orlicz functions. We study some topological properties and inclusion relations between these spaces. We also make an effort to study these sequence spaces over n-normed spaces.

1. Introduction and Preliminaries An ideal convergence is a generalization of statistical convergence. The ideal convergence plays a vital role not only in pure mathematics but also in other branches of science especially in computer science, information theory, biological science, dynamical systems, geographic information systems, and motion planning in robotics. Kostyrko et al. [14] introduced the notion of I−convergence based on the structure of admissible ideal I of subsets of natural number N. For more details about ideal convergence see ([10], [15], [24], [27]). Let X be a non empty set. Then a family of sets I ⊆ 2X (Power set of X) is said to be an ideal if I is additive, that is, A, B ∈ I ⇒ A ∪ B ∈ I and A ∈ I, B ⊆ A ⇒ B ∈ I. A non empty family of sets £(I) ⊆ 2X is said to be filter on X if and only if Φ ∈ £(I), for A, B ∈ £(I) we have A ∩ B ∈ £(I) and for each A ∈ £(I) and A ⊆ B implies B ∈ £(I). An ideal I ⊆ 2X is called non-trivial if I 6= 2X . A non-trivial ideal I ⊆ 2X is called admissible if {{x} : x ∈ X} ⊆ I. A non-trivial ideal I is maximal if there cannot exists any non-trivial ideal J 6= I containing I as a subset. n 1X A subset A of N is said to have asymptotic density δ(A) if δ(A) = lim χA (k) exists, n→∞ n k=1 where χA is the characteristic function of A. If we take I = If = {A ⊆ N : A is a finite subset}, then If is a non trivial admissible ideal of N and the corresponding convergence coincide with the usual convergence. If we take I = Iδ = {A ⊆ N : δ(A) = 0} where δ(A) denotes the asymptotic density of the set A, then Iδ is a non trivial admissible ideal of N and the corresponding convergence coincide with the statistical convergence. Recently, some new sequence spaces by mean of the matrix domain have been discussed by Malkowsky [17] and many others. Sengonul [25] defined the sequence y = (yi ) which is frequently used as the Z p −transformation of the sequence x = (xi ), that is, yi = pxi + (1 − p)xi−1 where x−1 = 0, 1 < p < ∞ and Z p denotes the matrix Z p = (zik ) defined by  (i = k);  p, 1 − p, (i − 1 = k)(i, k ∈ N); zik =  0, otherwise. 2010 Mathematics Subject Classification. 40A05, 40D25, 40G15. Key words and phrases. Ideal, I-convergence, Zweier sequence, Orlicz function, n-normed space. 1

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Sengonul [25] introduced the Zweier sequence spaces Z and Z0 as follows: Z = {x = (xk ) ∈ w : Z p x ∈ c} and Z0 = {x = (xk ) ∈ w : Z p x ∈ c0 }, where w denotes the space of all real or complex sequences x = (xk ). By l∞ , c, c0 we denote the classes of all bounded, convergent and null sequence spaces. The notion of difference sequence spaces was introduced by Kızmaz [12] who studied the difference sequence spaces l∞ (∆), c(∆) and c0 (∆). The notion was further generalized by Et and C ¸ olak [3] by introducing the spaces l∞ (∆n ), c(∆n ) and c0 (∆n ). Let n be a nonnegative integer, then for z = c, c0 and l∞ , we have sequence spaces z(∆n ) = {x = (xk ) ∈ w : (∆n xk ) ∈ z}, where ∆n x = (∆n xk ) = (∆n−1 xk − ∆n−1 xk+1 ) and ∆0 xk = xk for all k ∈ N, which is equivalent to the following binomial representation   n X n ∆n xk = (−1)v xk+v . v v=0

Taking n = 1, we get the spaces studied by Et and C ¸ olak [3]. For more details about sequence spaces see ([19], [21], [22]) and references therein. Let X be a linear metric space. A function p : X → R is called paranorm, if (1) p(x) ≥ 0 for all x ∈ X, (2) p(−x) = p(x) for all x ∈ X, (3) p(x + y) ≤ p(x) + p(y) for all x, y ∈ X, (4) if (λn ) is a sequence of scalars with λn → λ as n → ∞ and (xn ) is a sequence of vectors with p(xn − x) → 0 as n → ∞, then p(λn xn − λx) → 0 as n → ∞. A paranorm p for which p(x) = 0 implies x = 0 is called total paranorm and the pair (X, p) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [28], Theorem 10.4.2, pp. 183). An Orlicz function M : [0, ∞) → [0, ∞) is a continuous, non-decreasing and convex such that M (0) = 0, M (x) > 0 for x > 0. Lindenstrauss and Tzafriri [16] used the idea of Orlicz function to define the following sequence space. Orlicz sequence space is defined as ∞ o n  |x |  X k < ∞, for some ρ > 0 . `M = x ∈ w : M ρ k=1

The space `M is a Banach space with the norm ∞ n  |x |  o X k ||x|| = inf ρ > 0 : M ≤1 . ρ k=1

It is shown by Lindenstrauss and Tzafriri [16] that every Orlicz sequence space `M contains a subspace isomorphic to `p (p ≥ 1) which is an Orlicz sequence space with M (t) = tp for 1 ≤ p < ∞. An Orlicz function M satisfies ∆2 −condition if and only if for any constant L > 1 there exists a constant K(L) such that M (Lu) ≤ K(L)M (u) for all values of u ≥ 0. Later on, Orlicz sequence spaces were investigated by Parashar and Chaudhary [20], Esi [4], Tripathy et al. [26], Bhardwaj and Singh [1], Et [2], Esi and Et [5] and many others. Also if M is an Orlicz function, then we may write M (λx) ≤ λM (x) for all λ with 0 < λ < 1.

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ON SOME ZWEIER I-CONVERGENT DIFFERENCE SEQUENCE SPACES

3

Definition 1.1. A sequence (xk ) ∈ w is said to be I-convergent to a number L if for every  > 0, the set {k ∈ N : |xk − L| ≥ } ∈ I. In this case we write I − lim xk = L. Definition 1.2. A sequence (xk ) ∈ w is said to be I-null if L = 0. In this case we write I − lim xk = 0. Definition 1.3. A sequence (xk ) ∈ w is said to be I-Cauchy if for every  > 0, there exist a number m = m() such that {k ∈ N : |xk − xm | ≥ } ∈ I. Definition 1.4. A sequence (xk ) ∈ w is said to be I-bounded if there exist M > 0 such that {k ∈ N : |xk | ≥ M } ∈ I. Definition 1.5. A sequence space E is said to be solid (or normal) if (αk xk ) ∈ E whenever (xk ) ∈ E and for all sequence (αk ) of scalars with |αk | < 1 for all k ∈ N. Definition 1.6. A sequence space E is said to be symmetric if (xk ) ∈ E implies (xπ(k) ) ∈ E, where π is a permutation of N. Definition 1.7. A sequence space E is said to be sequence algebra if (xk yk ) ∈ E whenever (xk ), (yk ) ∈ E. Definition 1.8. A sequence space E is said to be convergence free if (yk ) ∈ E whenever (xk ) ∈ E and xk = 0 implies yk = 0. Definition 1.9. Let K = {k1 < k2 < ...} ⊂ N and let E be a sequence space. A K−step space of E is a sequence space λE K = {(xkn ) ∈ w : (xk ) ∈ E}. Definition 1.10. A canonical preimage of a sequence (xkn ) ∈ λE K is a sequence (yk ) ∈ w defined by  xk , if k ∈ K yk = 0 , otherwise. A canonical preimage of a step space λE K is a set of canonical preimages of all the elements E in λE K , that is, y is in the canonical preimage of λK if and only if y is a canonical preimage E of some x ∈ λK . Definition 1.11. A sequence space E is said to be monotone if it contain the canonical preimages of its step spaces. I Throughout the article Z I , Z0I , Z∞ , mIZ and mIZ0 represents Zweier I-convergent, Zweier I-null, Zweier bounded I-convergent and Zweier bounded I-null sequence spaces, respectively.

Lemma 1.12. The sequence space E is solid implies that E is monotone (See [13]). Let M = (Mn ) be a sequence of Orlicz functions, u = (un ) be a sequence of strictly positive real numbers and p = (pn ) be a bounded sequence of positive real numbers. In the present paper we define the following sequence spaces:

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KULDIP RAJ AND SURUCHI PANDOH

Z I (M, ∆r , p, u) = ( ( x = (xk ) :

n∈N:

∞ X

" Mn

n=1

( Z0I (M, ∆r , p, u)

=

x = (xk ) :

|un (Z p (∆r x))n − L| ρ ( n∈N:

∞ X n=1

I Z∞ (M, ∆r , p, u) = ( (

x = (xk ) :

n ∈ N : ∃ K > 0,

∞ X n=1

" Mn

" Mn

# pn

) ≥

) ∈I

for some L ∈ C.

|un (Z p (∆r x))n | ρ

|un (Z p (∆r x))n | ρ

# pn

#pn

) ≥K

) ≥

) ∈I .

) ∈I .

I (M, ∆r , p, u) and mIZ0 (M, ∆r , p, u) = Also, mIZ (M, ∆r , p, u) = Z I (M, ∆r , p, u) ∩ Z∞ r I r I Z0 (M, ∆ , p, u) ∩ Z∞ (M, ∆ , p, u).

If we take (pk ) = 1 for all k, then we have I (M, ∆r , p, u) = Z I (M, ∆r , p, u) = Z I (M, ∆r , u), Z0I (M, ∆r , p, u) = Z0I (M, ∆r , u), Z∞ r I r I r I r I I Z∞ (M, ∆ , u), mZ (M, ∆ , p, u) = Z (M, ∆ , u)∩Z∞ (M, ∆ , u) and mZ0 (M, ∆r , p, u) = I Z0I (M, ∆r , u) ∩ Z∞ (M, ∆r , u). If we take (uk ) = 1 for all k, then we have I (M, ∆r , p, u) = Z I (M, ∆r , p, u) = Z I (M, ∆r , p), Z0I (M, ∆r , p, u) = Z0I (M, ∆r , p), Z∞ r I r I r I r I I Z∞ (M, ∆ , p), mZ (M, ∆ , p, u) = Z (M, ∆ , p) ∩ Z∞ (M, ∆ , p) and mZ0 (M, ∆r , p, u) = I Z0I (M, ∆r , p) ∩ Z∞ (M, ∆r , p). If we take (Mn ) = M , (pn ) = 1, (un ) = 1 for all n ∈ N and r = 0, then we get the sequence spaces defined by Hazarika et al. [11]. The main purpose of this paper is to introduce the sequence spaces Z I (M, ∆r , p, u), I Z0I (M, ∆r , p, u) and Z∞ (M, ∆r , p, u). We study some topological properties and inclusion relations between these sequence spaces. An effort is made to study these sequence spaces over n-normed spaces in the section third of this paper. 2. Main Results Theorem 2.1. Let M = (Mn ) be a sequence of Orlicz functions, u = (un ) be a sequence of strictly positive real numbers and p = (pn ) be a bounded sequence of positive real numbers. I (M, ∆r , p, u) are linear over the Then the spaces Z I (M, ∆r , p, u), Z0I (M, ∆r , p, u) and Z∞ complex field C. Proof. We shall prove the result for the space Z I (M, ∆r , p, u). Let x = (xk ), y = (yk ) ∈ Z I (M, ∆r , p, u) and α, β be scalars. Then there exist positive numbers ρ1 and ρ2 such that ( " # pn ) ∞ X |un (Z p (∆r x))n − L1 | n∈N: Mn ≥  ∈ I for some L1 ∈ C. ρ1 n=1 ( " # pn ) ∞ X |un (Z p (∆r y))n − L2 | n∈N: Mn ≥  ∈ I for some L2 ∈ C. ρ2 n=1

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ON SOME ZWEIER I-CONVERGENT DIFFERENCE SEQUENCE SPACES

that is, ( (2.1)

A1 =

n∈N: (

(2.2)

A2 =

n∈N:

∞ X n=1 ∞ X

" Mn " Mn

n=1

|un (Z p (∆r x))n − L1 | ρ1

# pn

|un (Z p (∆r y))n − L2 | ρ2

# pn

 ≥ 2

)

 ≥ 2

)

5

∈ I. ∈ I.

Let ρ3 = max{2|α|ρ1 , 2|β|ρ2 }. Since M = (Mn ) is a nondecreasing and convex function, we have # pn "
| αun (Z p (∆r x))n + βun (Z p (∆r y))n − (αL1 + βL2 )| Mn ρ3 " # pn |α||un (Z p (∆r x))n − L1 | |β||un (Z p (∆r y))n − L2 | ≤ Mn + ρ3 ρ3 " #pn #pn " |un (Z p (∆r x))n − L1 | |un (Z p (∆r y))n − L2 | ≤ Mn + Mn . ρ1 ρ2 Now from equations (2.1) and (2.2), we have # pn ( " )
∞ X | αun (Z p (∆r x))n + βun (Z p (∆r y))n − (αL1 + βL2 )| n∈N: Mn ≥ ρ3 n=1 ⊂ A1 ∪ A2 ∈ I. Therefore, αx + βy ∈ Z I (M, ∆r , p, u). Hence Z I (M, ∆r , p, u) is a linear space. SimiI (M, ∆r , p, u) are linear spaces.  larly, we can prove that Z0I (M, ∆r , p, u) and Z∞ Theorem 2.2. The spaces mIZ (M, ∆r , p, u) and mIZ0 (M, ∆r , p, u) are paranormed spaces, with the paranorm g defined by ( # pHn ) " pn |un (Z p (∆r x))n | g(x) = inf ρ H : sup Mn ≤ 1, for some ρ > 0 , ρ n where H = max{1, sup pn }. n

Proof. Clearly g(−x) = g(x) and g(0) = 0. Let x = (xk ) and y = (yk ) ∈ mIZ (M, ∆r , p, u). Now for ρ1 , ρ2 > 0 we denote ( " #pn ) |un (Z p (∆r x))n | A3 = ρ1 : sup Mn ≤1 ρ1 n and ( A4 =

"

ρ2 : sup Mn n

|un (Z p (∆r y))n | ρ2

#pn

) ≤1 .

If ρ = ρ1 + ρ2 . Then by using Minkowski’s inequality, we obtain #pn " #pn " |un (Z p (∆r (x + y)))n | ρ1 |un (Z p (∆r x))n | Mn ≤ Mn ρ ρ1 + ρ2 ρ1 " #pn ρ2 |un (Z p (∆r y))n | + Mn . ρ1 + ρ2 ρ2

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KULDIP RAJ AND SURUCHI PANDOH

Therefore, " sup Mn n

|un (Z p (∆r (x + y)))n | ρ

#pn ≤1

and # pHn ) |un (Z p (∆r (x + y)))n | g(x + y) = inf (ρ1 + ρ2 ) : Mn ≤ 1, ρ1 ∈ A3 , ρ2 ∈ A4 ρ ( " # pHn ) pn |un (Z p (∆r x))n | ≤ inf (ρ1 ) H : Mn ≤ 1, ρ1 ∈ A3 ρ1 " # pHn ( ) pn |un (Z p (∆r y))n | ≤ 1, ρ2 ∈ A4 + inf (ρ2 ) H : Mn ρ2 "

(

pn H

= g(x) + g(y). Let tm → L and let g(∆r xm −∆r x) → 0 as m → ∞. To prove that g(tm ∆r xm −L∆r x) → 0 as m → ∞. We put # pn ) ( " |un (Z p (∆r xm ))n | ≤1 A5 = ρm > 0 : sup Mn ρm n and ( A6 =

"

ρs > 0 : sup Mn n

|un (Z p (∆r (xm − x)))n | ρs

#pn

) ≤1

by the continuity of M = (Mn ), we observe that # pn " #pn " |un (Z p (tm ∆r xm − L∆r xm ))n | |un (Z p (tm ∆r xm − L∆r x))n | ≤ Mn Mn |tm − L|ρm + |L|ρs |tm − L|ρm + |L|ρs " #pn |un (Z p (L∆r xm − L∆r x))n | + Mn |tm − L|ρm + |L|ρs #pn " |tm − L|ρm |un (Z p (∆r xm )n | ≤ Mn |tm − L|ρm + |L|ρs ρm " # pn |un (Z p (∆r xm − ∆r x))n | |L|ρs Mn + . |tm − L|ρm + |L|ρs ρs From the above inequality it follows that " #pn |un (Z p (tm ∆r xm − L∆r x))n | ≤1 sup Mn |tm − L|ρm + |L|ρs n and consequently (2.3)

g(tm ∆r xm − L∆r x)

pn

=

inf{[|tm − L|ρm + |L|ρs ] H : ρm ∈ A5 , ρs ∈ A6 }

≤

|tm − L| H inf{(ρm ) H : ρm ∈ A5 }

pn

pn

pn

pn

+ |L| H inf{(ρs ) H : ρs ∈ A6 } pn

pn

≤ max{1, |tm − L| H }g(∆r xm ) + max{1, |L| H } g(∆r xm − ∆r x).

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ON SOME ZWEIER I-CONVERGENT DIFFERENCE SEQUENCE SPACES

7

It is obvious that g(∆r xm ) ≤ g(∆r x) + g(∆r xm − ∆r x), for all m ∈ N. Clearly, the right hand side of the relation (2.3) tends to 0 as m → ∞ and the result follows. This completes the proof.  Theorem 2.3. Let M1 and M2 be Orlicz functions satisfying ∆2 −condition. Then (i) W (M2 , ∆r , p, u) ⊆ W (M1 oM2 , ∆r , p, u). I (ii) W (M1 , ∆r , p, u) ∩ W (M2 , ∆r , p, u) = W (M1 + M2 , ∆r , p, u) for W = Z I , Z0I , Z∞ . Proof. (i) Let x = (xk ) ∈ Z I (M2 , ∆r , p, u). Let  > 0, be given. For some ρ > 0, we have #pn ) ) ( ( " ∞ X |un (Z p (∆r x))n − L| ≥ ∈I . (2.4) x = (xk ) : n ∈ N : M2 ρ n=1 Let  > 0 and choose 0 < δ < 1 such that M1 (t) ≤  for 0 ≤ t ≤ δ. We define # " |un (Z p (∆r x))n − L| yn = M2 ρ and consider

lim

n∈N, 0≤yn ≤δ

[M1 (yn )]pn =

lim

n∈N, yn ≤δ

[M1 (yn )]pn +

lim

n∈N, yn >δ

[M1 (yn )]pn .

We have (2.5)

lim

n∈N, yn ≤δ

[M1 (yn )]pn ≤ [M1 (2)]G +

lim

n∈N, yn ≤δ

[(yn )]pn , G = sup pn . n yn δ .

yn δ

< 1+ Since M1 is nondeFor second summation (i.e. yn > δ), we have yn < creasing and convex, it follows that   2y  yn  1 1 n M1 (yn ) < M1 1 + ≤ M1 (2)+ ≤ M1 . δ 2 2 δ Since M1 satisfies the ∆2 −condition, we can write 1 yn yn 1 yn K M1 (2) + K M1 (2) = K M1 (2). 2 δ 2 δ δ We get the following estimates: M1 (yn )
δ

[M1 (yn )]pn ≤ max{1, (Kδ −1 M1 (2))H } +

lim

n∈N, yn >δ

[yn ]pn .

From equations (2.4), (2.5) and (2.6), it follows that ( ( " # pn ! ) ) ∞ X |un (Z p (∆r x))n − L| ≥ ∈I . x = (xk ) : n ∈ N : M1 M2 ρ n=1 Hence Z I (M2 , ∆r , p, u) ⊆ Z I (M1 oM2 , ∆r , p, u). (ii) Let x = (xk ) ∈ Z I (M1 , ∆r , p, u) ∩ Z I (M2 , ∆r , p, u). Let  > 0, be given. Then there exists ρ > 0 such that ( ( " #pn ) ) ∞ X |un (Z p (∆r x))n − L| x = (xk ) : n ∈ N : M1 ≥ ∈I ρ n=1 and ( x = (xk ) :

( n∈N:

∞ X n=1

" M2

|un (Z p (∆r x))n − L| ρ

#pn

) ≥

) ∈I .

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KULDIP RAJ AND SURUCHI PANDOH

The rest of the proof follows from the following relations: ( #pn ) " ∞ X |un (Z p (∆r x))n − L| n∈N: ≥ (M1 + M2 ) ρ n=1 ( #pn ) " ∞ X |un (Z p (∆r x))n − L| ⊆ n∈N: ≥ M1 ρ n=1 ( #pn " ) ∞ X |un (Z p (∆r x))n − L| ∪ n∈N: M2 ≥ ρ n=1 Taking M2 (x) = x and M1 (x) = M (x) for all x ∈ [0, ∞), we have the following result.  Corollary 2.4. W ⊆ W (M, ∆r , p, u), where W = Z I , Z0I . Theorem 2.5. The spaces Z0I (M, ∆r , p, u) and mIZ0 (M, ∆r , p, u) are solid and monotone. Proof. We shall prove the result for Z0I (M, ∆r , p, u). For mIZ0 (M, ∆r , p, u), the result can be proved similarly. Let x = (xk ) ∈ Z0I (M, ∆r , p, u), then there exists ρ > 0 such that #pn ) ( " ∞ X |un (Z p (∆r x))n | ≥  ∈ I. (2.7) n∈N: Mn ρ n=1 Let (αk ) be a sequence scalar with |αk | ≤ 1 for all k ∈ N. Then the result follows from (2.7) and the following inequality # pn " # pn " |un (Z p (∆r x))n | |αk un (Z p (∆r x))n | ≤ |αk |Mn Mn ρ ρ " # pn |un (Z p (∆r x))n | ≤ Mn ρ for all n ∈ N. The space Z0I (M, ∆r , p, u) is monotone follows from the Lemma (1.12).  Theorem 2.6. The spaces Z I (M, ∆r , p, u) and mIZ (M, ∆r , p, u) are neither monotone nor solid in general. Proof. The proof of this result follows from the following example. Let I = If , Mn (x) = x for all x ∈ [0, ∞), p = (pn ) = 1, u = (un ) = 1, for all n and r = 0. Consider the K-step space Tk of T defined as follows. Let (xk ) ∈ T and (yk ) ∈ Tk be such that  xk , if k is odd. yk = 0, otherwise. Consider the sequence (xk ) defined by xk = 21 for all k ∈ N. Then (xk ) ∈ Z I (M, ∆r , p, u) but its K- step space preimage does not belongs to Z I (M, ∆r , p, u). Thus Z I (M, ∆r , p, u) are not monotone. Hence, Z I (M, ∆r , p, u) is not solid by Lemma (1.12).  Theorem 2.7. The spaces Z I (M, ∆r , p, u) and Z0I (M, ∆r , p, u) are sequence algebra. Proof. We prove that Z0I (M, ∆r , p, u) is sequence algebra. For the space Z I (M, ∆r , p, u), the result can be proved similarly. Let x = (xk ), y = (yk ) ∈ Z0I (M, ∆r , p, u). Then ( " # pn ) ∞ X |un (Z p (∆r x))n | n∈N: Mn ≥  ∈ I, for some ρ1 > 0. ρ1 n=1

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ON SOME ZWEIER I-CONVERGENT DIFFERENCE SEQUENCE SPACES

and ( n∈N:

∞ X

" Mn

n=1

|un (Z p (∆r y))n | ρ2

# pn

9

) ≥

∈ I, for some ρ2 > 0.

Let ρ = ρ1 ρ2 > 0. Then we can show that ) ( "
#pn ∞ X |un (Z p (∆r xy))n | n∈N: Mn ≥  ∈ I. ρ n=1 Thus, (xk yk ) ∈ Z0I (M, ∆r , p, u). Hence Z0I (M, ∆r , p, u) is sequence algebra.



Theorem 2.8. Let M = (Mk ) be a sequence of Orlicz functions. Then Z0I (M, ∆r , p, u) ⊂ I Z I (M, ∆r , p, u) ⊂ Z∞ (M, ∆r , p, u) and the inclusions are proper. Proof. Let x = (xk ) ∈ Z I (M, ∆r , p, u). Then there exists L ∈ C and ρ > 0 such that #pn ) ( " ∞ X |un (Z p (∆r x))n − L| ≥  ∈ I. n∈N: Mn ρ n=1 We have " Mn

|un (Z p (∆r x))n | 2ρ

#pn

" # pn " #pn 1 |un (Z p (∆r x))n − L| 1 |L| ≤ Mn + Mn . 2 ρ 2 ρ

I Taking supremum over n on both sides we get x = (xk ) ∈ Z∞ (M, ∆r , p, u). The inclusion I r I r Z0 (M, ∆ , p, u) ⊂ Z (M, ∆ , p, u) is obvious. The inclusion is proper follows from the following example. Let I = Id , Mn (x) = x2 for all x ∈ [0, ∞), u = (un ) = 1, p = (pn ) = 1 and r = 0 for all n ∈ N. (a) Consider the sequence (xk ) defined by xk = 1 for all k ∈ N. Then (xk ) ∈ Z I (M, ∆r , p, u) but (xk ) ∈ / Z0I (M, ∆r , p, u). (b) Consider the sequence (yk ) defined as  2, if k is even yk = 0, otherwise. I (M, ∆r , p, u) but (yk ) ∈ / Z I (M, ∆r , p, u). Then (yk ) ∈ Z∞



Theorem 2.9. The spaces Z I (M, ∆r , p, u) and Z0I (M, ∆r , p, u) are not convergence free in general. Proof. The proof of this result follows from the following example. Let I = If , Mn (x) = x2 for all x ∈ [0, ∞), u = (un ) = 1, p = (pn ) = 1 and r = 0 for all n ∈ N. Consider the sequence (xk ) and (yk ) defined by xk = k12 and yk = k 2 for all k ∈ N. Then (xk ) belongs to Z I (M, ∆r , p, u) and Z0I (M, ∆r , p, u), but (yk ) does not belongs to both Z I (M, ∆r , p, u) and Z0I (M, ∆r , p, u). Hence, the spaces are not convergence free.  Theorem 2.10. mIZ (M, ∆r , p, u) is a closed subspace of l∞ (M, ∆r , p, u). (i)

Proof. Let (xk ) be a Cauchy sequence in mIZ (M, ∆r , p, u) such that x(i) → x. We show (i) that x ∈ mIZ (M, ∆r , p, u). Since (xk ) ∈ mIZ (M, ∆r , p, u), then there exists ai such that ( " #pn ) ∞ X |un (Z p (∆r x(i) ))n − ai | n∈N: Mn ≥  ∈ I. ρ n=1 We need to show that (i) (ai ) converges to a.

152

10

KULDIP RAJ AND SURUCHI PANDOH

( (ii) If U =

( n∈N:

x = (xk ) :

∞ X

" Mn

n=1

|un (Z p (∆r x))n − L| ρ

# pn

)) , then U c ∈ I.

≥

(i)

(i) Since (xk ) be a Cauchy sequence in mIZ (M, ∆r , p, u), then for a given  > 0, there exists k0 ∈ N such that "
# pn |un (Z p (∆r x(i) ))n − (Z p (∆r x(j) ))n |  sup Mn < for all i, j ≥ k0 . ρ 3 n For a given  > 0, we have ( " )
# pn ∞ X |un (Z p (∆r x(i) ))n − (Z p (∆r x(j) ))n |  Bij = n ∈ N : Mn < ρ 3 n=1 ( n∈N:

Bi =

∞ X

|un (Z p (∆r x(i) − ai ))n | ρ

"

|un (Z p (∆r x(j) − aj ))n | ρ

Mn

n=1

( n∈N:

Bj =

∞ X

Mn

n=1 c B c = Bji

#pn

"

 < 3

# pn

)

 < 3

)

∪ Bic ∪ Bjc , where ( " #pn ) ∞ X |un (ai − aj )| B= n∈N: Mn 0 : sup Mn n

# )

u (Z p (∆r (xm − x)))

pn

n

n , z1 , · · · , zn−1 ≤1

ρs

by the continuity of M = (Mn ), we observe that # "

pn

u (Z p (tm ∆r xm − L∆r x))

n n , z1 , · · · , zn−1 Mn |tm − L|ρm + |L|ρs " #

u (Z p (tm ∆r xm − L∆r xm ))

pn

n

n ≤ Mn , z1 , · · · , zn−1 |tm − L|ρm + |L|ρs # "

pn

u (Z p (L∆r xm − L∆r x))

n n +Mn , z1 , · · · , zn−1 |tm − L|ρm + |L|ρs " #

u (Z p (∆r xm ))

pn |tm − L|ρm

n

n Mn , z1 , · · · , zn−1 ≤ m |t − L|ρm + |L|ρs ρm " #

u (Z p (∆r (xm − x)))

pn |L|ρs

n

n Mn , z1 , · · · , zn−1 + m . |t − L|ρm + |L|ρs ρs From the above inequality it follows that # "

pn

u (Z p (tm ∆r xm − L∆r x))

n n , z1 , · · · , zn−1 ≤1 sup Mn |tm − L|ρm + |L|ρs n and consequently (3.3)

g(tm ∆r xm − L∆r x)

pn

=

inf{[|tm − L|ρm + |L|ρs ] H : ρm ∈ D5 , ρs ∈ D6 }

≤

|tm − L| H inf{(ρm ) H : ρm ∈ D5 }

pn

pn

pn

pn

+ |L| H inf{(ρs ) H : ρs ∈ D6 } pn

pn

≤ max{1, |tm − L| H }g(∆r xm ) + max{1, |L| H } g(∆r xm − ∆r x). It is obvious that g(∆r xm ) ≤ g(∆r x) + g(∆r xm − ∆r x), for all m ∈ N. Clearly, the right hand side of the relation (3.3) tends to 0 as m → ∞ and the result follows. This completes the proof. 

158

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KULDIP RAJ AND SURUCHI PANDOH

Theorem 3.3. Let M1 and M2 be Orlicz functions that satisfies ∆2 −condition. Then (i) W (M2 , ∆r , p, u, k·, · · · , ·k) ⊆ W (M1 oM2 , , ∆r p, u, k·, · · · , ·k). (ii) W (M1 , ∆r , p, u, k·, · · · , ·k) ∩ W (M2 , ∆r , p, u, k·, · · · , ·k) = W (M1 + M2 , ∆r , p, u, I k·, · · · , ·k) for W = Z I , Z0I , Z∞ . Proof. (i) Let x = (xk ) ∈ Z I (M2 , ∆r , p, u, k·, · · · , ·k). Let  > 0, be given. For some ρ > 0, we have # ) ) ( ( " ∞

pn

u (Z p (∆r x)) − L X

n n ≥ ∈I . , z1 , · · · , zn−1 (3.4) x = (xk ) : n ∈ N : M2 ρ n=1 Let  > 0 and choose 0 < δ < 1 such that M1 (t) ≤  for 0 ≤ t ≤ δ. We define # "

u (Z p (∆r x)) − L

n n , z1 , · · · , zn−1 yn = M2 ρ and consider

lim

n∈N, 0≤yn ≤δ

[M1 (yn )]pn =

lim

n∈N, yn ≤δ

[M1 (yn )]pn +

lim

n∈N, yn >δ

[M1 (yn )]pn .

We have (3.5)

lim

n∈N, yn ≤δ

[M1 (yn )]pn ≤ [M1 (2)]G +

lim

n∈N, yn ≤δ

[(yn )]pn , G = sup pn .

yn δ

n yn δ .

For second summation (i.e. yn > δ), we have yn < < 1+ Since M1 is nondecreasing and convex, it follows that   2y  1 yn  1 n M1 (yn ) < M1 1 + ≤ M1 (2)+ ≤ M1 . δ 2 2 δ Since M1 satisfies the ∆2 −condition, we can write M1 (yn )
δ

[M1 (yn )]pn ≤ max{1, (Kδ −1 M1 (2))H } +

lim

n∈N, yn >δ

[yn ]pn .

From equations (3.4), (3.5) and (3.6), it follows that # ! ) ) ( ( " ∞

pn

u (Z p (∆r x)) − L X

n , z1 , · · · , zn−1 ≥ ∈I . x = (xk ) : n ∈ N : M1 M2 ρ n=1 Hence Z I (M2 , ∆r , p, u, k·, · · · , ·k) ⊆ Z I (M1 oM2 , ∆r , p, u, k·, · · · , ·k). (ii) Let x = (xk ) ∈ Z I (M1 , ∆r , p, u, k·, · · · , ·k) ∩ Z I (M2 , ∆r , p, u, k·, · · · , ·k). Let  > 0, be given. Then there exists ρ > 0 such that ( ( " # ) ) ∞

u (Z p (∆r x)) − L

pn X

n n x = (xk ) : n ∈ N : M1 , z1 , · · · , zn−1 ≥ ∈I ρ n=1 and ( x = (xk ) :

( n∈N:

∞ X n=1

" M2

# ) )

u (Z p (∆r x)) − L

pn

n

n , z1 , · · · , zn−1 ≥ ∈I .

ρ

159

ON SOME ZWEIER I-CONVERGENT DIFFERENCE SEQUENCE SPACES

17

The rest of the proof follows from the following relations: ( # ) " ∞

pn

u (Z p (∆r x)) − L X

n n ≥ n∈N: , z1 , · · · , zn−1 (M1 + M2 ) ρ n=1 ( ) # " ∞

u (Z p (∆r x)) − L

pn X

n

n ⊆ n∈N: ≥ M1 , z1 , · · · , zn−1 ρ n=1 ( ) # " ∞

u (Z p (∆r x)) − L

pn X

n

n ∪ n∈N: ≥ M2 , z1 , · · · , zn−1 ρ n=1 Taking M2 (x) = x and M1 (x) = M (x) for all x ∈ [0, ∞), we have the following result.  Corollary 3.4. W ⊆ W (M, ∆r , p, u, k·, · · · , ·k), where W = Z I , Z0I . Theorem 3.5. The spaces Z0I (M, ∆r , p, u, k·, · · · , ·k) and mIZ0 (M, ∆r , p, u, k·, · · · , ·k) are solid and monotone. Proof. We shall prove the result for Z0I (M, ∆r , p, u, k·, · · · , ·k). For mIZ0 (M, ∆r , p, u, k·, · · · , ·k), the result can be proved similarly. Let x = (xk ) ∈ Z0I (M, ∆r , p, u, k·, · · · , ·k), then there exists ρ > 0 such that ( " # ) ∞

u (Z p (∆r x))

pn X

n

n (3.7) n∈N: Mn , z1 , · · · , zn−1 ≥  ∈ I. ρ n=1 Let (αk ) be a sequence scalar with |αk | ≤ 1 for all k ∈ N, we have # # " "

pn

pn

α u (Z p (∆r x))

u (Z p (∆r x))

k n

n n n , z1 , · · · , zn−1 , z1 , · · · , zn−1 ≤ |αk |Mn Mn ρ ρ # "

pn

u (Z p (∆r x))

n n , z1 , · · · , zn−1 ≤ Mn ρ for all n ∈ N. The space Z0I (M, ∆r , p, u, k·, · · · , ·k) is monotone follows from the Lemma (1.12).  Theorem 3.6. The spaces Z I (M, ∆r , p, u, k·, · · · , ·k) and mIZ (M, ∆r , p, u, k·, · · · , ·k) are neither monotone nor solid in general. Proof. The proof of result follows from the Theorem (2.6).



Theorem 3.7. The spaces Z I (M, ∆r , p, u, k·, · · · , ·k) and Z0I (M, ∆r , p, u, k·, · · · , ·k) are sequence algebra. Proof. We prove that Z0I (M, ∆r , p, u, k·, · · · , ·k) is sequence algebra. For the space Z I (M, ∆r , p, u, k·, · · · , ·k), the result can be proved similarly. Let x = (xk ), y = (yk ) ∈ Z0I (M, ∆r , p, u, k·, · · · , ·k). Then ( " # ) ∞

u (Z p (∆r x))

pn X

n

n n∈N: Mn , z1 , · · · , zn−1 ≥  ∈ I, for some ρ1 > 0. ρ1 n=1 and " # )

u (Z p (∆r y))

pn

n n n∈N: Mn , z1 , · · · , zn−1 ≥  ∈ I, for some ρ2 > 0. ρ2 n=1

(

∞ X

160

18

KULDIP RAJ AND SURUCHI PANDOH

Let ρ = ρ1 ρ2 > 0. Then we can show that ( # " ) ∞

u (Z p (∆r xy))


pn X n

n

n∈N: Mn , z1 , · · · , zn−1 ≥  ∈ I. ρ n=1 Thus, (xk yk ) ∈ Z0I (M, ∆r , p, u, k·, · · · , ·k). Hence Z0I (M, ∆r , p, u, k·, · · · , ·k) is sequence algebra.  Theorem 3.8. Let M = (Mn ) be a sequence of Orlicz function. Then Z0I (M, ∆r , p, u, k·, · · · , ·k) I ⊂ Z I (M, ∆r , p, u, k·, · · · , ·k) ⊂ Z∞ (M, ∆r , p, u, k·, · · · , ·k) and the inclusions are proper. Proof. Let x = (xk ) ∈ Z I (M, ∆r , p, u, k·, · · · , ·k). Then there exists L ∈ R and ρ > 0 such that # ) ( " ∞

pn

u (Z p (∆r x)) − L X

k n , z1 , · · · , zn−1 ≥  ∈ I. n∈N: Mn ρ n=1 We have # "

pn

|u (Z p (∆r x)) |

n n , z1 , · · · , zn−1 Mn 2ρ

" #

u (Z p (∆r x)) − L

pn 1

n

n Mn , z1 , · · · , zn−1 ≤ 2 ρ " #

pn 1

L

+ Mn , z1 , · · · , zn−1 . 2 ρ

I (M, ∆r , p, u, k·, · · · , ·k). The Taking supremum over n on both sides we get x = (xk ) ∈ Z∞ I r I r inclusion Z0 (M, ∆ , p, u, k·, · · · , ·k) ⊂ Z (M, ∆ , p, u, k·, · · · , ·k) is obvious. The inclusion is proper follows from the following example. Let I = Id , Mn (x) = x2 for all x ∈ [0, ∞), u = (un ) = 1, p = (pn ) = 1 and r = 0 for all n ∈ N. (a) Consider the sequence (xk ) defined by xk = 1 for all k ∈ N. Then (xk ) ∈ Z I (M, ∆r , p, u, k·, · · · , ·k) but (xk ) ∈ / Z0I (M, ∆r , p, u, k·, · · · , ·k). (b) Consider the sequence (yk ) defined as  2, if k is even yk = 0, otherwise. I (M, ∆r , p, u, k·, · · · , ·k) but (yk ) ∈ / Z I (M, ∆r , p, u, k·, · · · , ·k). Then (yk ) ∈ Z∞



Theorem 3.9. The spaces Z I (M, ∆r , p, u, k·, · · · , ·k) and Z0I (M, ∆r , p, u, k·, · · · , ·k) are not convergence free in general. Proof. The proof follows from Theorem (2.9). Theorem 3.10.

mIZ (M, ∆r , p, u, k·, · · ·

 r

, ·k) is a closed subspace of l∞ (M, ∆ , p, u, k·, · · · , ·k).

(i)

Proof. Let (xk ) be a Cauchy sequence in mIZ (M, ∆r , p, u, k·, · · · , ·k) such that x(i) → x. (i) We show that x ∈ mIZ (M, ∆r , p, u, k·, · · · , ·k). Since (xk ) ∈ mIZ (M, ∆r , p, u, k·, · · · , ·k), then there exists ai such that # ) ( " ∞

pn

u (Z p (∆r x(i) )) − a X

n

n i n∈N: Mn , z1 , · · · , zn−1 ≥  ∈ I. ρ n=1 We need to show that (i) (ai ) converges to a. ( ( (ii) If V =

x = (xk ) :

" # ))

u (Z p (∆r x)) − L

pn

n n , z1 , · · · , zn−1 ≥ , n∈N: Mn ρ n=1 ∞ X

161

ON SOME ZWEIER I-CONVERGENT DIFFERENCE SEQUENCE SPACES

19

then V c ∈ I. (i)

(i) Since (xk ) be a Cauchy sequence in mIZ (M, ∆r , p, u, k·, · · · , ·k), then for a given  > 0, there exists k0 ∈ N such that # "

u (Z p (∆r x(i) )) − (Z p (∆r x(j) ))


pn  n n

n

sup Mn , z1 , · · · , zn−1 < for all i, j ≥ k0 ρ 3 n For a given  > 0, we have ( # ) " ∞

u (Z p (∆r x(i) )) − (Z p (∆r x(j) ))


pn X  n n

n

< Fij = n ∈ N : Mn , z1 , · · · , zn−1 ρ 3 n=1 # ) "

pn

u (Z p (∆r x(i) − a )) 

n i n Fi = n ∈ N : Mn , z1 , · · · , zn−1 < ρ 3 n=1 ( " # ) ∞

u (Z p (∆r x(j) − a ))

pn X 

n

j n Mn Fj = n ∈ N : , z1 , · · · , zn−1 < ρ 3 n=1 (

∞ X

c Then Fij , Fi , Fj ∈ I. Let F c = Fji ∪ Fic ∪ Fjc , where # ) ( " ∞

pn

u (a − a ) X

n i j 0

Pn Thus, h(n) ≥ h(10) > 0. We have proven i=1 a2i < 22n−1 /n for all n ≥ 10. Finally, we can easily check the inequality is also true for all 1 ≤ n ≤ 9. The case of n = 1 is trivial since S1 (z) = z. For the cases of 2 ≤ n ≤ 9, denote the difference of P n 2 2n−1 /n by αn . Thus, i=1 ai − 2 n X

22n−1 n i=1 Z 2π n−1 X  2n − 1   1 + cos(ξ/2) n−1−i  1 − cos(ξ/2) i 1 22n−1 = dξ − i 2π 0 i=0 2 2 n

αn =

a2i −

Using Mathematica, we have the following table n 2 3 4 5 6 7 20223 20687 1343063 71 αn -2 − 12 -19 − 320 − 96 − 1792

8

9

− 84763 32

− 1398580967 147456

As for n from 2 to 9, all the αn are negative, inequality (3.3) is completely proven, which indicates that the scaling function φn (x) is in L2 (R) and also completes the proof of Theorem 2.1

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Spline Type Orthogonal Scaling Functions and Wavelets

As examples, we consider the cases of n = 1 and 2. It is easy to find that S1 (z) = z and φn (x) is the Haar function. For n = 2, √ √ 1− 3 2 1+ 3 z+ z S2 (z) = 2 2 and the corresponding φ2 (x) is the Daubechies scaling function.

3.2

Refinement relation and corresponding wavelets

P2n Let Mn (z) := Pn (z)Sn (z) = 12 k=0 ck z k . Then Mn (z) is the mask of φn (x) and thus we have the refinement equation X φn (x) = ck φn (2x − k) (3.5) ∀k

We also have the corresponding wavelets X ψn (x) = (−1)k c1−k φn (2x − k)

(3.6)

∀k

One natural question at this point is from the refinement equation to determine explicitly the scaling function φn (x) and thus the wavelet functions ψn (x). We propose a method by the iterative procedure described in Theorem 5.23 of [1]. (1) (0) (0) We start with φn (x) = B1 (x). From φn (x) we can construct φn (x) by using the refinement equation (3.5) φ(1) n (x) =

X

ck φ(0) n (2x − k)

∀k (1)

(2)

Now from φn (x) we can continue to construct φn (x) again by the refinement equation. Continuing the process for a certain number of times, we can obtain an approximation of φn (x). Simple as it seems, this iteration method is not a very good way to construct φ as we do not know how many times should we repeat the process to find a good enough approximation for φ. For this reason, more direct method is developed and will be addressed in future paper. From Theorem 2 of [11], we have the following result on the regularities of spline type wavelets ψn (x). Theorem 3.1 Let φn (x) be constructed as the previous section with the mask shown in Theorem 2.1. Then the corresponding wavelets ψn (x) established in (3.6) are in C βn , where βn are greater than   n X 1 n − log2 n a2j  . 2 j=1

199

T. Nguyen and T.-X. He

3.3

11

Decomposition and reconstruction formula

Finally, upon obtaining the new scaling function and corresponding wavelet function, we may establish the corresponding decomposition and reconstruction formulas by using a routine argument. Roughly speaking consider a function f ∈ L2 (R) with fi being an approximation in Vj , a space generated by the j th order scaling function. Then fj =

X

< f, φjk > φjk

(3.7)

k∈Z

Because we have the orthogonal direct sum decomposition Vj = Vj−1 + Wj−1 , we can express fj using bases of Vj−1 and bases of Wj−1 as follows fj = fj−1 + gj−1 =

X

X

< f, φj−1,k > φj−1,k +

k∈Z

< f, ψj−1,k > ψj−1,k

(3.8)

k∈Z

And now we have the following theorem regarding the relationship of these coefficients. Let {Vj : j P ∈ Z} be an MRA with scaling function φ satisfying the scaling relation φ(x) = k∈Z pk φ(2x − k), and let Wj be the orthogonal complement of Vj in Vj+1 with wavelet function ψ. Then the coefficients relative to the different bases in (3.7) and (3.8) satisfy the following decomposition formula

< f, φj−1,l > = 2−1/2

X

pk−2l < f, φjk >

k∈Z

< f, ψj−1,l > = 2−1/2

X

(−1)k p1−k+2l < f, φjk >

k∈Z

and the reconstruction formula

< f, φjl >= 2−1/2

X k∈Z

4

pl−2k < f, φj−1,k > +2−1/2

X

(−1)k p1−l+2k < f, φj−1,k >

k∈Z

Examples for n = 3 and n = 4

Besides the examples of the cases of n = 1 and 2 shown at the end of Section 3.1, in this section we give an examples on the cases of n = 3 and n = 4. According to the previous sections, we will construct an orthogonal scaling function φ3 (x) from the third order B-spline function B3 (x). In order to construct the function φ3 (x), we start with its mask P3 (z)S3 (z), where 3 2 3 P3 (z) = ( 1+z 2 ) is the mask of the third order B-spline. Let S3 (z) = a1 z+a2 z +a3 z ,

200

12

Spline Type Orthogonal Scaling Functions and Wavelets

then by Lemma 2.2, we have Q3 (x) = |S3 (z)|2 = (a21 + a22 + a23 ) + 2(a1 a2 + a2 a3 )cos(ξ/2) + 2a1 a3 cos(ξ) = (a21 + a22 + a23 ) + 2(a1 a2 + a2 a3 )cos(ξ/2) + 2a1 a3 (2cos2 (ξ/2) − 1) = (a21 + a22 + a23 − 2a1 a3 ) + 2(a1 a2 + a2 a3 )cos(ξ/2) + 4a1 a3 cos2 (ξ/2) = (a21 + a22 + a23 − 2a1 a3 ) + 2(a1 a2 + a2 a3 )x + 4a1 a3 x2 where x = cos(ξ/2) and z = e−iξ/2 . On the other hand, by equation (2.7) we have 2−i  i  2  X 1−x 1+x 5 Q3 (x) = i 2 2 i=0   2    2 1+x 1+x 1−x 1−x +5 = + 10 2 2 2 2 3 2 9 = x − x+4 2 2 Thus, from the above equations, we have the following system of equations  2 2 2  a1 + a2 + a3 − 2a1 a3 = 4 2(a1 a2 + a2 a3 ) = − 29   4a1 a3 = 32

(4.1)

Simplify (4.1) we get  19 2 2 2  a1 + a2 + a3 = 4 a1 a2 + a2 a3 = − 49   a1 a3 = 38

(4.2)

From this system, we have (a1 + a2 + a3 )2 = a21 + a22 + a23 + 2(a1 a2 + a2 a3 + a1 a3 ) = 1. Without loss of generality, consider the case a1 + a2 + a3 = 1. Combining this with a1 a2 + a2 a3 = − 49 and a1 a3 = 38 we have the following solution   p √  √ 1  a = 10 − 5 + 2 10 1 + 1  4  √
1 (4.3) a2 = 2 1 − 10  p   a = 1 1 + √10 + 5 + 2√10 3

4

We verify the condition for φ3 to be in L2 (R) 19 32 22·3−1 < = 4 3 3 Thus φ3 is indeed in L2 (R). Now we will attempt to construct φ3 explicitly. It has the mask  3 1+z P3 (z)S3 (z) = (a1 z + a2 z 2 + a3 z 3 ) 2 a21 + a22 + a23 =

= 0.0249x − 0.0604x2 − 0.095x3 + 0.325x4 + 0.571x5 + 0.2352x6

201

T. Nguyen and T.-X. He

13

By (1.1), (1.2) and (1.3) we have the refinement equation φ3 (x) = 0.0498φ3 (2x − 1) − 0.121φ3 (2x − 2) − 0.191φ3 (2x − 3) + 0.650φ3 (2x − 4) + 1.141φ3 (2x − 5) + 0.4705φ3 (2x − 6)

(4.4)

The corresponding spline type wavelet ψ3 (x) can be expressed as

ψ3 (x) = 0.0498φ3 (2x) + 0.121φ3 (2x + 1) − 0.191φ3 (2x + 2) − 0.650φ3 (2x + 3) + 1.141φ3 (2x + 4) − 0.4705φ3 (2x + 5), which is in C β3 and β3 > 4 −

1 log2 (57). 2

We now consider the case n = 4. Again, we construct an orthogonal scaling 4 function φ4 (x) from the forth order B-spine B4 (x) with the mask P4 (z) = ( 1+z 2 ) . We examine the mask P4 (z)S4 (z) of φ4 (x), where S4 (z) = a1 z + a2 z 2 + a3 z 3 + a4 z 4 . By Lemma 2.2, we have Q4 (x) = |S4 (z)|2 = (a21 + a22 + a23 + a24 ) + 2(a1 a2 + a2 a3 + a3 a4 )cos(ξ/2) + 2(a1 a3 + a2 a4 )cos(ξ) + 2a1 a4 cos(3ξ/2) = (a21 + a22 + a23 + a24 ) + 2(a1 a2 + a2 a3 + a3 a4 )cos(ξ/2) + 2(a1 a3 + a2 a4 )(2cos2 (ξ/2) − 1) + 2a1 a4 (4cos3 (ξ/2) − 3cos(ξ/2)) = (a21 + a22 + a23 + a24 − 2a1 a3 − 2a2 a4 ) + (2a1 a2 + 2a2 a3 + 2a3 a4 − 6a1 a4 )cos(ξ/2) + (4a1 a3 + 4a2 a4 )cos2 (ξ/2) + 8a1 a4 cos3 (ξ/2) = (a21 + a22 + a23 + a24 − 2a1 a3 − 2a2 a4 ) + (2a1 a2 + 2a2 a3 + 2a3 a4 − 6a1 a4 )x + (4a1 a3 + 4a2 a4 )x2 + 8a1 a4 x3 where x = cos(ξ/2) and z = e−iξ/2 . Using equation (2.7) we get another expression for Q4 (x) 3−i  i  3  X 1−x 1+x 7 i 2 2 i=0  3  2     2  3 1+x 1+x 1−x 1+x 1−x 1−x = +7 + 21 + 35 2 2 2 2 2 2 29 5 = 8 − x + 10x2 − x3 2 2

Q4 (x) =

202

14

Spline Type Orthogonal Scaling Functions and Wavelets

Thus, from the above equations, we have the following system of equations  2 a + a22 + a23 + a24 − 2a1 a3 − 2a2 a4 = 8    1  2a1 a2 + 2a2 a3 + 2a3 a4 − 6a1 a4 = − 29 2  4a1 a3 + 4a2 a4 = 10    8a1 a4 = − 25 Simplify (5.1) we have the following system  2 a1 + a22 + a23 + a24 = 13    a a + a a + a a = − 131 1 2 2 3 3 4 16 a1 a3 + a2 a4 = 52    5 a1 a4 = − 16

(4.5)

(4.6)

Solving for this system of equations yields 8 solutions. One of the numerical solutions is  a1 = 2.6064    a = −2.3381 2 (4.7)  a 3 = 0.8516    a4 = −0.1199 We verify the condition for φ4 to be in L2 (R) a21 + a22 + a23 + a24 = 13 < 32 =

22·4−1 4

Thus φ4 is indeed in L2 (R). Now we will attempt to construct φ4 explicitly. It has the mask  4 1+z P4 (z)S4 (z) = (a1 z + a2 z 2 + a3 z 3 + a4 z 4 ) 2 = 0.1629z + 0.5055z 2 + 0.4461z 3 − 0.0198z 4 − 0.1323z 5 + 0.0218z 6 + 0.0233z 7 − 0.0075z 8 By (1.1), (1.2) and (1.3) we have the refinement equation φ4 (x) = 0.3258φ4 (2x − 1) + 1.011φ4 (2x − 2) + 0.8922φ4 (2x − 3) − 0.0396φ4 (2x − 4) − 0.2646φ4 (2x − 5) + 0.0436φ4 (2x − 6) + 0.0466φ4 (2x − 7) − 0.015φ4 (2x − 8) (4.8) The corresponding spline type wavelet ψ3 (x) can be expressed as ψ4 (x) = 0.3258φ4 (2x) − 1.011φ4 (2x + 1) + 0.8922φ4 (2x + 2) + 0.0396φ4 (2x + 3) − 0.2646φ4 (2x + 4) − 0.0436φ4 (2x + 5) + 0.0466φ4 (2x + 6) + 0.015φ4 (2x + 7) which is in C β4 and β4 > 4 −

1 log2 (52). 2

203

T. Nguyen and T.-X. He

15

References [1] A. Boggess and F. J. Narcowich, A first course in wavelets with Fourier analysis, Second edition, John Wiley & Sons, Inc., Hoboken, NJ, 2009. [2] C. de Boor, B(asic)-Spline Basics, http://www.cs.unc.edu/ dm/UNC/COMP 258/Papers/bsplbasic.pdf. [3] C. de Boor, A Practical Guide to Splines, Springer-Verlag, New York, 1978. [4] C. K. Chui, An Introduction to Wavelets, Wavelet Analysis and Its Applications, Vol. 1, Academic Press, INC., New York, 1992. [5] O. Christensen, An Introduction to Frames and Riesz Bases, Birkh¨auser, Boston, 2002. [6] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, Second Edition, MIT Press and McGraw-Hill, 2001. [7] I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. [8] T. Erd´elyi and J. Szabados, On polynomials with positive coefficients. J. Approx. Theory 54 (1988), no. 1, 107–122. [9] E. Hern´ andez and G. Weiss, A first Course on Wavelets, With a foreword by Yves Meyer, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996. [10] D. Han, K. Kornelson, D. Larson, and E. Weber, Frames for Undergraduates, AMS, Providence, Rhode Island, 2007. [11] T.-X. He, Biorthogonal wavelets with certain regularities. Appl. Comput. Harmon. Anal. 11 (2001), no. 2, 227–242. [12] T.-X. He, Biorthogonal spline type wavelets. Comput. Math. Appl. 48 (2004), no. 9, 1319–1334. [13] T.-X. He, Construction of biorthogonal B-spline type wavelet sequences with certain regularities. J. Appl. Funct. Anal. 2 (2007), no. 4, 339–360. [14] S. Jukna, Extremal Combinatorics, With Applications in Computer Science, Texts in Theoretical Computer Science, An EATCS Series, Springer-Verlag, Berlin, 2001. [15] G. G. Lorentz, The degree of approximation by polynomials with positive coefficients. Math. Ann. 151 1963 239251. [16] http://en.wikipedia.org/wiki/Extended-Euclidean-algorithm.

J. APPLIED FUNCTIONAL ANALYSIS, VOL. 10, NO.'S 3-4,204-210 ,2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

Fixed point results for commuting mappings in modular metric spaces Emine K¬l¬nç1 , Cihangir Alaca2; 1

2

Department of Mathematics, Institute of Natural and Applied Sciences, Celal Bayar

University, 45140 Manisa, Turkey Department of Mathematics, Faculty of Science and Arts, Celal Bayar University, 45140 Manisa, Turkey

Abstract: The purpose of this paper is to prove that two main …xed point theorems for commuting mappings in modular metric spaces. Our main results are more general than from those of Jungck [17] and Fisher [16]. 2010 Mathematics Subject Classi…cation: 46A80, 47H10, 54E35. Key words and phrases: Modular metric spaces, …xed point, commuting mappings.

1

Introduction

The Banach Contraction Principle which published in 1922 [5] and then the beginning of …xed point theory in metric spaces is related to which in particular situation was already obtained by Liouville, Picard and Goursat. A number of authors have de…ned and studied contractive type mappings on a complate metric space X which are generalizations of well-known Banach contraction (see [18], [19],[26], [11]). The notion of modular space was introduced by Nakano [24] and was intensively developed by Koshi, Shimogaki, Yamamuro (see [21, 27]) and others. A lot of mathematicians are interested …xed point of modular space. In 2008, Chistyakov introduced the notion of modular metric space generated by F-modular and developed the theory of this space [7], on the same idea was de…nedthe notion of a modular on an arbitrary set ad developed the theory of metric space generated by modular such that called the modular metric spaces in 2010 [8]. Afrah A. N. Abdou [1] studied and proved some new …xed points theorems for pointwise and asymptotic pointwise contraction mappings in modular metric spaces. Azadifer et. al. [2] introduced the notion of modular G-metric spaces and proved some …xed point theorems of contractive and Azadifer et. al. [4] proved the existence and uniqueness of a common …xed point of compatible mappings of integral type in this Corresponding author. Tel.: +90 236 2013209; fax: +90 236 2412158. E-mail addresses: [email protected] (E. K¬l¬nç), [email protected] (C. Alaca).

1

204

Fixed point results for commuting mappings, Emine Kilinc, Cihangir Alaca

205

space. K¬l¬nç and Alaca [20] de…ned ("; k) uniformly locally contractive mappings and -chainable concept and proved a …xed point theorem for these concepts in a complete modular metric spaces. Recently, many authors [3, 6, 9, 10, 14, 23] studied on di¤erent …xed point results for modular metric spaces. In this paper, we give modular metric versions of the …xed point theorem which was given by Jungck [17] and a generalization of this theorem which was given by Fisher [16].

2

Preliminaries

In this section, we will give some basic concepts and de…nitions about modular metric spaces. De…nition 2.1 [[8], De…nition 2.1] Let X be a nonempty set, a function w : (0; 1) X X ! [0; 1] is said to be a metric modular on X if satisfying, for all x; y; z 2 X the following condition holds: (i) w (x; y) = 0 for all

> 0 , x = y;

(ii) w (x; y) = w (y; x) for all (iii) w

+

(x; y)

> 0;

w (x; z) +w (z; y) for all ;

> 0:

If instead of (i), we have only the condition i w (x; x) = 0 for all

> 0, then w is said to be a (metric) pseudomodular on X.

The main property of a metric modular [[8]] w on a set X is the following: given x; y 2 X, the function 0 < 7! w (x; y) 2 [0; 1] is nonincreasing on (0; 1). In fact, if 0 < < , then (iii), i and (ii) imply w (x; y)

w

(x; x) + w (x; y) = w (x; y) :

It follows that at each point

> 0 the right limit w

left limit w

"

0

(x; y) = lim w "!+0

hold: w

+0

+0

(x; y) = lim w (x; y) and the ! +0

(x; y) exist in [0; 1] and the following two inequalities

(x; y)

w (x; y)

w

0

(x; y) :

(2.1)

De…nition 2.2 [[23]] Let Xw be a complete modular metric space. A self-mapping T on Xw is said to be a contraction if there exists 0 k < 1 such that w (T x; T y) for all x; y 2 Xw and

kw (x; y)

(2.2)

> 0:

Theorem 2.1 [[23]] Let Xw be a complete modular metric space and T a contraction on Xw . Then, the sequence (T n x)n2N converges to the unique …xed point of T in Xw for any initial x 2 Xw . De…nition 2.3 [[23]], De…nition 2.4] Let Xw be a modular metric space. Then following de…nitions exist: 2

Fixed point results for commuting mappings, Emine Kilinc, Cihangir Alaca

206

(1) The sequence (xn )n2N in Xw is said to be convergent to x 2 Xw if w (xn ; x) ! 0, as n ! 1 for all > 0: (2) The sequence (xn )n2N in Xw is said to be Cauchy if w (xm ; xn ) ! 0, as m; n ! 1 for all > 0: (3) A subset C of Xw is said to be closed if the limit of a convergent sequence of C always belong to C. (4) A subset C of Xw is said to be complete if any Cauchy sequence in C is a convergent sequence and its limit is in C.

3

Main Results

In this section we will give two main theorems for commuting mappings in modular metric spaces, the …xed point theorem which was given by Jungck in 1976 [17] and a generalization of this theorem which was given by Fisher [16] in 1981 for metric spaces. Theorem 3.1 Let I and T be commuting mappings of complete modular metric space Xw into itsef satis…ying the inequality w (T x; T y)

(3:1)

k:w (Ix; Iy)

for all x; y 2 Xw , where 0 < k < 1:If the range of I contains the range of T ,I is continuous and w (x; Ix) < 1 then T and I have a common unique …xed point. Proof. Let x0 2 Xw be arbitrary. Then T x0 and Ix0 are well de…ned. Since T x0 2 I (Xw ) ; there is some x1 2 Xw such that Ix1 = T x0 : Then choose x2 2 Xw such that Ix2 = T x1 : In general if xn is choosen, then we choose a point xn+1 in Xw such that Ixn+1 = T xn : We show that (xn ) is Cauchy sequence . From (3:1) we have w (Ixn ; Ixn+1 ) = w (T xn 1 ; T xn )

k:w (Ixn 1 ; Ixn )

Then with induction we have w (Ixn ; Ixn+1 )

k:w (Ixn 1 ; Ixn )

k n w (Ix0 ; Ix1 )

:::

If we take the limit as n ! 1, we get w (Ixn ; Ixn+1 ) ! 0 Without loss of generality we can assume that for w

m n

(Ixn ; Ixn+1 )

m n

> 0 there exists

" m n

such that

" m

n

For m; n 2 N and m > n we have w (Ixn ; Ixm )

(Ixn ; Ixn+1 ) + w (Ixn+1 ; Ixn+2 ) + ::: + w m n m " " " + + ::: + m n m n m n " w

m n

3

n

(Ixm 1 ; Ixm )

Fixed point results for commuting mappings, Emine Kilinc, Cihangir Alaca

207

Thus (xn )isa Cauchy sequence. Since Xw is complete, there is some u 2 Xw such that lim Ixn = lim T xn

n!1

n!1

1

=u

Since I is continuous and T and I commute we get Iu = I Iu = I

lim Ixn = lim I 2 xn

n!1

n!1

lim T xn = lim IT xn = lim T Ixn

n!1

n!1

n!1

From (3:1) we get w (T Ixn ; T u)

k:w

I 2 xn ; Iu

Taking the limit as n ! 1 we obtain w (Iu; T u)

k:w (Iu; Iu)

Hence we get Iu = T u. Again from (3:1) w (T xn ; T u)

k:w (Ixn ; Iu)

Letting n tend to 1 we get w (u; T u)

k:w (u; Iu) = k:w (u; T u)

Hence T u = u: To show the uniqueness of u, assume that there exists v 2 Xw such that T v = Iv = v, then w (u; v) = w (Iu; T v) = w (T u; T v)

k:w (Iu; Iv)

Then u = v: This completes the proof. Theorem 3.2 Let S and T be commuting mappings and I and J be commuting mappings of a complete modular metric spaceXw into itself satisfying w (Sx; T y)

k:w (Ix; Jy)

for all x; y 2 Xw , where 0 k < 1. If I andJ are continuous and w (Ix0 ; Jx1 ) < 1, then all S; T; I and J have a unique common …xed point. Proof. Let x0 2 Xw be arbitrary. Since Sx0 2 J (Xw ), let x1 2 Xw be such that Jx1 = Sx0 and also, as T x1 2 I (Xw ) ; let x2 2 Xw such that Ix2 = T x1 : In general x2n+1 2 Xw is choosen such that Jx2n+1 = Sx2n and x2n+2 2 Xw suh that Ix2n+2 = T x2n+1 ; n = 0; 1; ::: Denote y2n = Jx2n+1 = Sx2n y2n+1 = Ix2n+2 = T x2n+1 ;

4

n

0

Fixed point results for commuting mappings, Emine Kilinc, Cihangir Alaca

208

Then, it can be proved that (yn ) isa Cauchy sequence. w (Jx2n+1 ; Ix2n+2 ) = w (Sx2n ; T x2n+1 ) k:w (Ix2n ; Jx2n+1 ) k:w (T x2n 1 ; Sx2n ) k:k:w (Jx2n 1 ; Ix2n ) k 2 w (Jx2n 1 ; Ix2n ) k 3 w (Jx2n 3 ; Ix2n 2 ) ::: k 2n w (Jx3 ; Ix2 ) k 2n+2 w (Jx1 ; Ix0 ) Then taking n ! 1 we have w (Jx2n+1 ; Ix2n+2 ) ! 0 Then there exists " > 0 such that w (Jx2n+1 ; Ix2n+2 ) Without loss of generality we can assume that for that (Jx2n+1 ; Ix2n+2 ) w

" m n

< 0, there exists

" m n

> 0 such

"

m n Hence choosing m = 2k + 1; n = 2l + 2; k > l and using triangle inequality we get m n

w (ym ; yn ) = w (y2k+1 ; y2l+2 ) = w (Ix2k+2 ; Jx2l+1 ) w (Ix2k+2 ; Jx2k+1 ) + w (Jx2k+1 ; Ix2k ) + ::: + w m n m n m " " " + + ::: + m n m n m n "

n

(Ix2l+2 ; Jx2l+1 )

Hence (yn ) is a Cauchy sequence. Let u 2 Xw be such that lim Sx2n = lim Jx2n+1 = lim T x2n+1 = lim Ix2n+2 = u

n!1

n!1

n!1

n!1

Since I is continuous and I and S commute it follows that lim I 2 x2n+2 = Iu;

n!1

lim SIx2n = lim ISx2n = Iu = u

n!1

n!1

From (3:2) w (SIx2n ; T x2n+1 )

k:w

I 2 x2n ; Jx2n+1

Taking the limit as n ! 1 we get w (Iu; u)

k:w (Iu; u)

Thence Iu = u: Similarly we obtain Ju = u, as J is continuous and T and J are commute. 5

Fixed point results for commuting mappings, Emine Kilinc, Cihangir Alaca

From (3:2) we have, as Iu = u; w (Su; T x2n+ )

k:w (u; Jx2n+1 )

Taking the limit as n ! 1 we get w (Su; u) = 0 Hence Su = u: Again from (3:2) considering the results above, we have w (Su; T u) = 0 Hence T u = Su:Thus we proved that Su = T u = Iu = Ju = u Clearly u is the unique common …xed point of all S; T; I and J. This completes the proof.

References [1] Afrah A.N. Abdou, On asymptotic pointwise contractions in modular metric spaces, Abstract and Applied Analysis Volume 2013, Article ID 501631, 7 pages, http://dx.doi.org/10.1155/2013/501631. [2] B. Azadifar, M. Maramaei, Gh. Sadeghi, On the modular G-metric spaces and …xed point theorems, J. Nonlinear Sci. Appl. 6 (2013) 293-304. [3] B. Azadifar, M. Maramaei, Gh. Sadeghi, Common …xed point theorems in modular G-metric spaces, Journal Nonlinear Analysis and Application 2013 (2013) 1-9. [4] B. Azadifar, Gh. Sadeghi, R. Saadati, C. Park, Integral type contractions in modular metric spaces, Journal of Inequalities and Applications 2013 (2013):483. doi:10.1186/1029-242X-2013-483. [5] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922) 133-181. [6] P. Chaipunya, Y.J. Cho, P. Kumam, Geraghty-type theorems in modular metric spaces with an application to partial di¤erential equation, Advances in Di¤erence Equations 2012 (2012):83. doi:10.1186/1687-1847-2012-83. [7] V.V. Chistyakov, Modular metric spaces generated by F-modulars, Folia Math. 14 (2008) 3-25. [8] V.V. Chistyakov, Modular metric spaces I. basic conceps, Nonlinear Anal. 72 (2010) 1-14. [9] V.V. Chistyakov, Fixed points of modular contractive maps, Doklady Mathematics 86(1) (2012) 515-518. 6

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Fixed point results for commuting mappings, Emine Kilinc, Cihangir Alaca

[10] Y.J. Cho, R. Saadati, Gh. Sadeghi, Quasi-contractive mappings in modular metric spaces, Journal of Applied Mathematics Volume 2012, Article ID 907951, 5 pages, doi:10.1155/2012/907951. ´ c, Generalized contractions and …xed point theorems, Publ. Inst. Math. 12 [11] Lj.B. Ciri´ (1971) 19-26. ´ c, On a familiy of contractive maps and and …xed points, Publ. Inst. Math. [12] Lj.B. Ciri´ 17 (1974) 52-58. ´ c, A new …xed point theorem forcontractive mappings, Publ. Inst. Math. [13] Lj.B. Ciri´ 30(44) (1981) 25-27. [14] H. Dehghan, M. Eshaghi Gordji, A. Ebadian, Comment on ’Fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory and Applications 2012 (2012):144. doi:10.1186/1687-1812-2011-93. [15] M. Edelstein, An extension of Banach’s contraction principle, Proc. Amer. Math. Soc. 12(1) (1961) 7-10. [16] B. Fisher, Four mappings with a common …xed point, (Arabic summary), J. Univ. Kuwait Sci. 8 (1981) 131-139. [17] G. Jungck, Commuting maps and …xed points, Amer. Math. Monthly 83 (1976) 261-263. [18] R. Kannan, Some results on …xed points, Bull. Calcutta Math. Soc. 60 (1968) 71-76. [19] R. Kannan, On certain sets and …xed point theorems, Roum. Math. Pure Appl. 14 (1969) 51-54. [20] E. K¬l¬nç, C. Alaca, A …xed point theorem in modular metric spaces, Advances in Fixed Point Theory 4(2) (2014) 199-206. [21] S. Koshi, T. Shimogaki, On F-norms of quasi-modular space, J. Fac. Sci. Hokkaido Univ. Ser 1. 15(3-4) (1961) 202-218. [22] A. Meir, E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969) 326-329. [23] C. Mongkolkeha, W. Sintunavarat, P. Kumam, Fixed point theorems for contraction mappings in modular metric spaces, Fixed Point Theory and Applications (2011). doi:10.1186/1687-1812-2011-93. [24] H. Nakano, Modulared semi-ordered linearspace, In Tokyo Math. Book Ser, vol.1, Maruzen Co, Tokyo (1950). [25] E. Rakotch, A note on contractive mappings, Amer. Math. Soc. 3 (1962) 459-465. [26] S. Reich, Kannan’s …xed point theorem, Boll. Un. Mat. Ital. 4 (1971) 1-11. [27] S. Yamammuro, On conjugate space of Nakano space, Trans. Amer. Math. Soc. 90 (1959) 291-311. 7

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J. APPLIED FUNCTIONAL ANALYSIS, VOL. 10, NO.'S 3-4,211-218 ,2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

211

A general common fixed point theorem for weakly compatible mappings Kristaq Kikina, Luljeta Kikina and Sofokli Vasili Department of Mathematics and Computer Sciences, University of Gjirokastra, Gjirokastra, Albania. [email protected], [email protected] Abstract. The purpose of this paper is to complete and generalize many existing results in the literature for weakly compatible mappings and to perform the proof without the Hausdorff’s condition. Mathematics Subject Classification: 47H10, 54H25 Keywords: Cauchy sequence, fixed point, generalized metric space, generalized quasi-metric space, quasi-contraction.

1. Introduction and preliminaries It is known that common fixed point theorems are generalizations of fixed point theorems. Over the past few decades, there have been many researchers who have interested in generalizing fixed point theorems to coincidence point theorems and common fixed point theorems. The concept of metric space, as an ambient space in fixed point theory, has been generalized in several directions. Some of such generalizations are: the quasi-metric spaces, the generalized metric spaces and the generalized quasi-metric spaces. The notion of quasi-metric, also known as b-metric, is in line of metric in which the triangular inequality d ( x, y) d ( x, z ) + d ( z, y) is replaced by quasi-triangular inequality d ( x, y ) k[d ( x, z ) + d ( z , y )], k 1 (see [1], [6], [7], [8], [11], [12], [25], [30], [34], [35]). In 2000 Branciari [8] introduced the concept of generalized metric space (gms), by replacing the triangle inequality with tetrahedral inequality. Starting with the paper of Branciari, many fixed point and common fixed point results have been established in this interesting space. For further information, the reader can refer to [2, 13-16, 20, 22-24, 28-29, 32-33, 36]. Recently L. Kikina and K. Kikina [26] introduced the concept of generalized quasi-metric space (gqms) by replacing the tetrahedral inequality d ( x, y ) d ( x, z ) + d ( z , w) + d ( w, y ) with a more general inequality, namely quasi-tetrahedral inequality d ( x, y ) k[d ( x, z ) + d ( z, w) + d ( w, y )] , k 1 . The generalized metric space is a special case of generalized quasi-metric space (for k = 1 ). Also, every quasi-metric space is a gqms, while the converse is not true [26]. Further aspects may be found in ([21], [26], [27]). By using this generalized quasi-metric space successfully defined an “open” ball and hence a topology. On the other hand this topology fails to provide some useful topological properties: a “open” ball in generalized quasi-metric space needs not to be open set; the generalized quasi-metric needs not to be continuous; a convergent sequence in generalized quasi-metric space needs not to be

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Cauchy; the generalized metric space needs not to be Hausdorff and hence the uniqueness of limits cannot be guaranteed. The above properties of generalized metric spaces (spesial case of gqms for k = 1 ) that do not hold for metric spaces were first observed by Das and Dey [13], [14] and also these facts were observed independently by Samet [32] and also by Sarma, Rao and Rao [33]. Initially these were considered to be true, implying non correct proofs of several theorems. This made some of the previous results to be reconsidered and to be corrected. Under these circumstances, not every fixed point theorem from metric spaces can be extended in gms or in gqms. Even in the case this extension may be done, the proof of such theorem is technically more complicated than in the usual setting. Because of those difficulties, some authors have taken the Hausdorffity as additional condition in their theorems, but this is not always necessary. For example, the assertion in [33] that the space needs to be Hausdorff is superfluous, a fact first noted by Kikina and Kikina [23]. See also [36]. The aim of this paper is to present a general theorem, from which a lot of current results on common fixed points in different kind of spaces (metric, quasi metric, gms, gqms, etc.) are taken as corollaries. The following definition was given by Kikina in 2012. R+ is called a generalized quasiDefinition 1.1 [26] Let X be a set. A function d : X × X metric in X if and only if there exists a constant k 1 such that for all x, y X and for all distinct points z , w X , each of them different from x and y the following conditions hold: (i ) d ( x, y ) = 0 x = y; (ii) d ( x, y ) = d ( y, x); (iii ) d ( x, y) k[d ( x, z ) + d ( z, w) + d ( w, y )] (quasi-tetrahedral inequality) Inequality (3) is often called quasi-tetrahedral inequality and k is often called the coefficient of d . A pair ( X , d ) is called a generalized quasi-metric space (gqms) if X is a set and d is a generalized quasi-metric in X. The set B( a, r ) = {x X : d ( x, a) < r} is called “open” ball with center a X and radius r > 0. The family = {Q X : a Q, r > 0, B(a, r ) Q} is a topology on X and it is called induced topology by the generalized quasi-metric d . Definition 1.2 Let ( X , d ) be a gqms. (1) A sequence {xn } in X is said to be convergent to a point x lim d ( xn , x ) = 0 . We denote this by xn n

X , if and only if

x.

(2) A sequence {xn } is a Cauchy sequence if and only if for each

> 0 there exists

a natural number n( ) such that d ( xn , xm ) < for all m > n > n( ). (3) ( X , d ) is called complete if every Cauchy sequence is convergent in X. Definition 1.3 Let ( X , d ) be a gqms. The generalized quasi-metric d is said to be continuous if and only if for any sequences { xn },{ yn } X , such that lim xn = x and lim y n = y , we have n

lim d ( xn , y n ) = d ( x , y ) . n

n

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The following example illustrates the existence of the generalized quasi-metric space for an arbitrary coefficient k 1 : 1 R as follow: Example 1.4 [26] Let X = 1 : n = 1, 2,... {1, 2} , Define d : X × X n 0 for x = y 1 1 1 for x {1, 2} and y = 1 or y {1, 2} and x = 1 ,x y d ( x, y ) = n n n 3k for x, y {1, 2}, x y 1 otherwise Then it is easy to see that ( X , d ) is a generalized quasi-metric space and is not a generalized metric space (for k > 1 ). Note that the sequence {xn } = {1 1n } converges to both 1 and 2 and it is not a Cauchy

sequence: d ( xn , xm ) = d (1 1n ,1 m1 ) = 1, n, m N The above example shows us some of the properties of metric spaces which do not hold in gqms (see [26]). Kikina and Kikina in [27] extended the well-known Ciric's quasi-contraction principle [10] in a generalized quasi-metric space. We restate Theorem 2.6[27] as follows: Theorem 1.5 Let (X, d) be a complete gqms with the coefficient k 1 and let T : X self-mapping satisfying: (1) d (Tx, Ty ) h max{d ( x, y ), d ( x, Tx), d ( y, Ty ), d ( x, Ty ), d ( y, Tx)}

for all x, y

X be a

X , where 0 h < 1k . Then T has a unique fixed point in X,

By this Theorem, in special case k = 1 , we obtain the Ciric's quasi-contraction principle in a generalized metric space which is presented in [15] and [22]. Let F be the set of functions :[0, ) [0, ) satisfying the condition (t ) = 0 if and only if t =0. We denote by F such that is continuous and non-decreasing. the set of functions We reserve ! for the set of functions " F such that + is continuous. Finally, by # we denote the set of functions $ F satisfying the following condition: , is lower semi-continuous. Lakzian and Samet in [28] established the following fixed point theorem. X be a Theorem 1.6. ([28]). Let ( X , d ) be a Hausdorff and complete gms and let T : X self-mapping satisfying ( d (Tx , Ty ) ( d ( x , y )) % ( d ( x , y )) . For all x , y X , where and % ! . Then T has a unique fixed point in X. In [5], it is given a slightly improved version of Theorem 1.6, obtained by replacing the continuity condition of % with a lower semi-continuity. Definition 1.7 ([19]) Let X be a non-empty set and T , F : X X . The mappings T and F are said to be weakly compatible if they commute at their coincidence points (i.e. TFx = FTx whenever Tx = Fx ). A point y X is called point of coincidence of T and F if there exists a point x X such that y = Tx = Fx .

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Di Bari and Vetro [16] extended Theorem 1.6 of Lakzian and Samet to the following common fixed point theorem: Theorem 1.8 ([16]) Let ( X , d ) be a Hausdorff gms and let T and F be self-mappings on X such that TX FX . Assume that ( FX , d ) is a complete gms and that the following condition holds: ( d (Tx , Ty ) ( d ( Fx , Fy )) % ( d ( Fx , Fy )) for all x , y X , where and % # . Then T and F have a unique point of coincidence in X. Moreover, if T and F are weakly compatible, then T and F have a unique common fixed point. In [9], Choudury and Kundu established the (0,+,,)-weakly contraction principle in partially ordered metric spaces. Recently, Isik and Turkoglu [18], and Bilgili et al [5] extended the results in [9] to the set of generalized metric spaces: X be a Theorem 1.9 ([5]). Let ( X , d ) be a Hausdorff and complete gms, and let T : X self-mapping such that ( d (Tx , Ty ) " ( d ( x , y )) $ ( d ( x , y ) for all x , y X , where , " ! , $ # and these mappings satisfy condition (t ) " (t ) + $ (t ) > 0 , for all t > 0 . Then T has a unique fixed point in X. Theorem 1.10 (Isik and Turkoglu [18]) Let ( X , d ) be a Hausdorff and complete g.m.s. and let T , F : X X be self-mappings such that TX FX , and FX is a closed subspace of X, and that the following condition holds: ( d (Tx , Ty ) " ( d ( Fx , Fy )) $ ( d ( Fx , Fy )) for all x , y X , where , " ! , $ # and these mappings satisfy condition (t ) " (t ) + $ (t ) > 0 , for all t > 0 Then T and F have a unique coincidence point in X. Moreover, if T and F are weakly compatible, then T and F have a unique common fixed point. The following lemma will play an important role in obtaining the main results of this paper. Lemma 1.11 [17] Let X be a nonempty set and f : X 1 X a function. Then there exists a subset E X such that f (E) = f (X) and f: E 1 X is one-to-one.

2. Main results In this section, we state and prove our main results. Firstly, we prove an important proposition that makes unnecessary the Hausdorffity condition for the most of well known theorems on gms and gqms. Proposition 2.1 If { yn } is a Cauchy sequence in a generalized quasi-metric space ( X , d ) with the coefficient k 1 and lim yn = u , then the limit u is unique. n

Proof. Assume that u ' u is also lim yn . n

A general common fixed point theorem for weakly compatible mappings, Kristaq Kikina et al

If { yn } is a sequence of distinct points, yn

ym for all n

215

m , there exists n0

N such that u

and u ' are different from yn for all n > n0 . For n > n0 , by rectangular property of Definition 1.1 we obtain d (u , u ') k[ d (u , yn ) + d ( yn , yn +1 ) + d ( yn +1 , u ')] Letting n tend to infinity we get d (u, u ') = 0 and so u = u ' , a contradiction. If { yn } is not a sequence of distinct points, then we have the following two cases: a) The sequence { yn } contains a subsequence { yn } of distinct points, and in this case, we can p

use the same reasoning as above. b) The sequence { yn } contains a constant subsequence { yn } : y n = c; p p

p

N . Then

c = lim yn = u = u ' , a contradiction. This completes the proof of the Proposition. n

Remark 2.2 From the above proposition it follows that even in the cases of gms and gqms, the including of the Hausdorffity condition in any theorem, in order to implies the uniqueness of limit of Cauchy sequence, is unnecessary The following results are inspired by the techniques and ideas of [17]. The main Theorem A natural way to unify and prove in a simple manner several fixed point and common fixed point theorems is by considering ”functional” contractive condition like F (d (Tx, Ty), d ( x, y), d ( x, Tx), d ( y, Ty), d ( x, Ty), d ( y, Tx)) 0, x, y X 6 where F : R+ R is an appropriate function. For more details about the possible choices of F we refer to the 1977 paper by Rhoades [31]; see also Turinici [37], Berinde and Vetro [3], Berinde [4], etc. Let (X, d) be a space (metric, quasi-metric, generalized metric, generalized quasi-metric, etc.) in which the following theorem has been proved: X is a self-mapping satisfying Theorem *: If (X, d) is a complete space and T : X F (d (Tx, Ty), d ( x, y), d ( x, Tx), d ( y, Ty), d ( x, Ty), d ( y, Tx)) 0, x, y X (4) for all x, y X . Then T has a unique fixed point in X. We will prove here the main theorem: X be selfTheorem 2.3 Let (X, d) be a space on which the Theorem * holds and T , f : X mappings such that T ( X ) f ( X ) . Assume that ( fX , d ) is a complete space and that the following condition holds: (5) F (d (Tx, Ty), d ( fx, fy), d ( fx, Tx), d ( fy, Ty), d ( fx, Ty), d ( fy, Tx)) 0 for all x, y X . Then T and f have a unique coincidence point in X. Moreover if T and f are weakly compatible, then T and f have a unique common fixed point. X is one-to-one. Proof . By Lemma 2.1, there exists E X such that f ( E ) = f ( X ) and f : E Now, define a map h : f ( E ) f ( E ) by h( fx) = Tx . Since f is one-to-one on E, h is well-defined. Note that, F (d (h( fx), h( fy)), d ( fx, fy ), d ( fx, h( fx)), d ( fy, h( fy)), d ( fx, h( fy )), d ( fy, h( fx))) 0

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f ( E ) . Since f ( E ) = f ( X ) is complete, by using Theorem *, there exists x0 X such that h( fx0 ) = fx0 . Hence, T and f have a point of coincidence, which is also unique. It is clear that T and f have a unique common fixed point whenever T and f are weakly compatibles. for all fx, fy

Corollary 2.4 (A generalization of the Ciric's quasi-contraction principle [10]) Let (X, d) be a gqms with the coefficient k 1 and let T , f : X X be self-mappings such that T ( X ) f ( X ) . Assume that ( fX , d ) is a complete gqms and that the following condition holds: d (Tx, Ty ) h max{d ( fx, fy), d ( fx, Tx), d ( fy, Ty ), d ( fx, Ty ), d ( fy, Tx)} (1) 1 for all x, y X , where 0 h < k . Then T and f have a unique coincidence point in X. Moreover if T and f are weakly compatible, then T and f have a unique common fixed point. Proof. Corollary 2.4 is taken from Theorem 2.3 in case when Theorem* is the Theorem 1.5 and F (d (Tx, Ty ), d ( x, y ), d ( x, Tx), d ( y, Ty ), d ( x, Ty ), d ( y, Tx))

= d (Tx, Ty ) h max{d ( x, y ), d ( x, Tx), d ( y, Ty ), d ( x, Ty ), d ( y, Tx)} where 0 h
0 such that

hT x b).

T y; x

yi

2

kx

yk ; 8y 2 H; > 0 such that

relaxed monotone, if there exists a constant

hT x

T y; x

yi

(

) kx

2

yk ; 8y 2 H:

Note that for = 0 the De…nition 1.1(a) reduces to monotonicity of T: c). Lipschitz continuous, if there exist a constant > 0 such that

kT x

T yk

kx

yk ; 8y 2 H:

De…nition 1.2. A mapping S : K ! K is called k strictly pseudocontractive in the sense of [13], if there exists a constant 0 k < 1 such that kSx

Syk

2

kx

2

yk + k(I

S) x

(I

2

S) yk ; 8x; y 2 K:

where I is the identity map. Observe that k = 0 implies that S is the usual nonexpansive map. Lemma 1.2 [4] Let the sequence fan g of non-negative real numbers satisfying the following

an+1

(1

n ) an

+ bn ; 8n

n0

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3 P where n0 is some non-negative integer, f n g is a sequence in (0; 1) such that n = 1; bn = o( n ); then limn!1 an = 0: Lemma 1.3 [3, 5] Let K(x) be a closed convex set in H: Then, for a given z 2 H; x 2 K(x) satis…es the inequality hx

z; y

xi

0; 8y 2 K(x);

if and only if

x = PK(x) z; where PK(x) is the projection of H onto the closed convex-valued set K(x) in

H: Lemma 1.4 [3, 5] The function x 2 H : h(x) 2 K(x) is a solution of the extended general quasi variational inequality if and only if x 2 H : h(x) 2 K(x) satis…es the relation h(x) = PK(x) fg(x)

T xg;

where PK(x) is the projection mapping and > 0 is a positive constant. Let S : K ! K be a k strict pseudo-contractive mapping with a …xed point. Let us denote by fSk gn k=1 ; the …nite family of nonexpansive mappings de…ned as:

Sk x = kx + (1

k)Sx; 8x 2 K:

(2.1)

with the set of …xed points F (S k ) (i.e., F (S k ) = fx 2 K : x = T xg): Setting 1 z(S) = \n=1 F (S k ) then the following assertions hold: Lemma 1.6 [11] Let K be a nonempty closed convex subset of a real Hilbert space H and let S : K ! K be a k strict pseudo-contraction with a …xed point. Then F (S) is closed and convex. De…ne a mapping Sk : K ! K by Sk x = kx + (1 k)Sx; 8x 2 K: Then Sk is nonexpansive such that

F (Sk ) = F (S): Lemma 1.7 [12] Let K be a nonempty closed convex subset of strictly convex Banach space X: Let fTn : n 2 N g be a sequence of nonexpansive 1 mappings on K: Suppose P1 \n=1 F (Tn ) 6= ;: Let f n g be a sequence of positive numbers such that n=1 n = 1: Then a mapping S on K can be de…ned by Sx =

1 X

n Tn x

(2.2)

n=1

for each x 2 K is well de…ned, nonexpansive and F (S) = \1 n=1 F (Tn ) holds.

3. Iterative methods and convergence analysis Let T; g; h : H ! H be certain nonlinear mappings and K(x) is a nonempty closed convex-valued set in H . Consider the problem of …nding an element x 2 H;

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4 h(x) 2 K(x) such that hT x; g(y)

h(x)i

0; 8y 2 H; g(y) 2 K(x)

(3.1)

such type of inequality known as extended general quasi variational inequalities introduced and studied by Noor et al. [3]. We denote by the set of solutions of variational inequality (3:1) : Variational inequality (3:1) entails extended general variational inequality, general variational inequality, variational inequality and other numerous probelms as special cases as qouted in [3]. We also faces the probelm of solving the Wiener-Hopf equations which observe a close reltions to variational inequalities as proved by Shi [2]. A huge amount of interest has been shown by many reaschers in this regard, see e.g. [3, 4, 10]. To be more speci…c, let us denote by QK(x) = I Sk gh 1 PK(x) ; where PK(x) is the implicit projection mapping, Sk is the …nite family of nonexpansive mappings de…ned as by (2:1) ; I is identity map and assume that h 1 exist. Let us consider the problem of …nding z 2 H such that

Th

1

Sk PK(x) z +

1

QK(x) z = 0:

(3.2)

where T; g; h : H ! H are given nonlinear mappings and is a positive constant. Equation (3:2) is known as implicit general Wiener-Hopf equations containing the …nite family of nonexpansive mappings Sk . Now we suggest the iterative method as follow: Algorithm 1. For a given iterate z0 2 H; compute zn+1 via the following scheme:

h(xn ) = (1 z = (1 where f If 1

ng

n ) PK(xn ) zn

+ n Sk PK(xn ) zn ; T xn ] ; n ) zn + n [g(xn )

and f

n g are two sequences in [0; 1] and Sk is de…ned by (2:1) : = 0; then Algorithm 1 becomes: n Algorithm 2. For a given iterate z0 2 H; compute zn+1 via the following

scheme:

h(xn ) = Sk PK(xn ) zn ; z = (1 n ) zn +

n

[g(xn )

T xn ] ;

where f n g is the sequence in [0; 1] and Sk is de…ned by (2:1) : If Sk = I; then Algorithm 1 reduces to: Algorithm 3. For a given iterate z0 2 H; compute zn+1 via the following scheme:

h(xn ) = (1 z = (1

n ) PK(xn ) zn

+ n PK(xn ) zn ; T xn ] ; n ) zn + n [g(xn )

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5 where f n g and f n g are two sequences in [0; 1] : If Sk = I and 1 n = 0; then Algorithm 1 collapses with the same one as suggested by Noor et al. [3]. i.e. Algorithm 4. For a given iterate z0 2 H; compute zn+1 via the following scheme:

h(xn ) = PK(xn ) zn ; z = (1 n ) zn +

n

[g(xn )

T xn ] ;

where f n g is the sequence in [0; 1] : In short, suitable and appropriate choices of mappings may leads one to obtain certain new and known algorithms for solving various classes of WienerHopf equations and variational inequality problems. Now we study the convergence of our proposed algorithm which is the main objective of this section. Since the implicit projection PK(:) is not nonexpansive but satisfy some Lipschitz type continuity as quoted by Noor et al. [3]. Also they pointed out that this mapping satisfy an inequality stated as below. Assumption 1 [3] The implicit projection PK(:) mapping satis…es the following inequality

PK(x) z

PK(y) z

kx

yk

where is a positive constant. This assumption plays an important part in studying the existence and convergence of iterative algorithms, see [3, 5]. We also need the following: Lemma 1.5 [3, 5] The solution x 2 H : h(x) 2 K(x) satis…es the extended general quasi variational inequality if and only if, z 2 H is a solution of the extended general implicit Wiener-Hopf equation, where

h(x) = Sk PK(x) z z = fg(x) T xg; > 0 is a positive constant. Theorem 3.1. Let T; g; h : H ! H be relaxed monotone mappings with constants 1 ; 2 ; and 3 and Lipschitz continuous with constants 1 ; 2 and 3 respectively. Let Sk be de…ned by (2:1) such that = z(S) \ 6= ?: Let fzn g and fxn g be two sequences generated by Algorithm 2.1. Let f n g and f n g are two sequences P1 in [0; 1] satisfying the following conditions: (i) n=0 n = 1: p where

(ii)

with

1

p 1

1

>

1


H = E(XY ), for every X, Y in H. If values of a random process are in a Hilbert space then one can find the best linear prediction based on projections on the past values. When the second moment is infinite the classical method (prediction based on the Hilbert space H) for prediction fail. Several researchers have studied prediction of infinite variance processes. Cline and Brockwell (1985) used minimum dispersion criterion for the prediction of causal ARMA processes with i.i.d. innovations. They assumed that innovations have regularly varying tails. In the case of α-stable random processes, minimum dispersion criterion focuses tails of errors. This criterion is used in Kokoszka (1996) for predicting FARIMA processes with innovations in the domain of attraction of an α-stable distribution with 1 < α ≤ 2. Generally, the predictor using minimum dispersion criterion is not unique and, except some cases, the calculations for finding predictors are not routine. We refer the reader interested in the prediction of infinite variance processes to Cambanis and Soltani (1984), Miamee and Pourahmadi (1988) and Soltani and Moeanaddin (1994). In Mohammadi and Mohammadpour (2009), using integral representation of α-stable random variables a Hilbert space, is defined: the following Hilbert space {∫ } f (x)dM (x) f ∈ Lα (E, E, p) , E

where M is an α-stable, 0 < α ≤ 2, random measure on (E, E) with finite control measure p. The prediction method is based on the introduced Hilbert space with an inner product known as stable covariation. In Karchera et al. (2013) three methods for predicting α-stable random fields are investigated.

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3

Non-causal ARMA processes with finite or infinite variance innovations are important process that the above described methods cannot be applied to. As we know, there is not any model based method for predicting such processes. It should be noted that there is a best linear predictor for non-causal ARMA processes with finite variance innovations, but, in practice we can not compute predictor. In this article, our aim is to find the best linear prediction of the linear process {Xt |t ∈ Z} of the following form Xt =

+∞ ∑

cj Wt−j ,

t ∈ Z,

(1)

j=−∞

where {Wt } is an arbitrary sequence of random variables which satisfies many weak assumptions (see Theorem 2.1). Consider the process {Xt |t ∈ Z} which satisfies the following system Φ(B)Xt = Θ(B)Zt , where {Zt } is an arbitrary sequence, Φ(B) = 1−ϕ1 B 1 −· · ·−ϕp B p , Θ(B) = 1+θ1 B 1 +· · ·+ θq B q , p, q ∈ N and B is a backward shift operator. Obviously, two processes {Φ(B)Xt } and {Θ(B)Zt } satisfy (1). Section 2 is devoted to the construction of Hilbert spaces based on linear processes of the form (1). In Section 3, we explain the prediction method. Finally, in Section 4, we explain the application of the new prediction method for a financial time series which a non-causal ARMA(2, 0) is an appropriate model for that.

2

Hilbert space generated by a linear process

In this section, under mild conditions we present a Hilbert space such that it contains the linear process of the form (1). We can rewrite the linear process of the form (1) as Xt =

+∞ ∑ j=−∞

ct+j W−j .

(2)

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2 HILBERT SPACE GENERATED BY A LINEAR PROCESS

4

In the sequel, we consider sequence of random variables {Wt } such that for every sequence {aj } ⊂ C, +∞ ∑

at+j W−j = 0 if and only if aj = 0, for every j ∈ Z.

(3)

j=−∞

Also, we consider linear processes of the form (2) such that +∞ ∑

+∞ ∑ |cj | < ∞ and ct+j W−j < ∞, a.s. 2

j=−∞

(4)

j=−∞

Theorem 2.1. Let {Wt } be a sequence of random variables satisfying (3). Let A =

} ct+j W−j + µ t ∈ Z, µ ∈ C, and {cj } satisfies 4 .

+∞ { ∑ j=−∞

For every two elements X =

∑+∞

j=−∞ ct1 +j W−j

+ µ1 and Y =

∑+∞ j=−∞

dt2 +j W−j + µ2 in A,

we assign the following value to them, < X, Y >A =

+∞ ∑

ct1 +j d¯t2 +j + µ1 µ ¯2 .

j=−∞

¯ containing A such that Then, there is a Hilbert space (denoted by A) < X, Y >A =< X, Y >A¯ ,

(5)

for every X, Y ∈ A. Proof. It is obvious that A is a vector space with field C. It is enough we show that A is an inner product space. Condition (3) ensures that the function < ., . >A is well defined and < ., . >A satisfies properties of the inner product. Therefore, A is an inner product space. Now, the completion of A is a Hilbert space satisfying (5). ¯ the Hilbert space generated by the process {Wt }. Definition 2.2. We call the space A, Remark 2.3. If {Wt } is an i.i.d. sequence of random variables such that V ar(Wt ) < +∞ and E(Wt ) = 0, for every t ∈ Z then A¯ is a subset of the Hilbert space H generated from all finite variance random variables. In this case, we have < X, Y >H = V ar(W1 ) < X, Y >A , for every X, Y ∈ A.

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5

Remark 2.4. The inner product of the space A is not necessary based on distribution function. For example, even if the distributions W1 and W2 in A are different, it is possible to have ||W1 ||A = ||W2 ||A , where ||.||A = (< ., . >A )1/2 . Conversely, consider two non-zero random variables W1 and aW2 in A such that they have same distribution for a ̸= 1. However, it is possible to have ||W1 ||A ̸= ||aW2 ||A . Example 2.5. Assume that the process {Xt } satisfies the system Xt =

+∞ ∑

cj Wt−j , t ∈ Z,

j=−∞

where {cj } and {Wj } satisfy (3) and (4). From Theorem (2.1), the process {Wj } takes its values in the space A. On the other hand, by assuming that {Xt } satisfies (3), the process {Xt } takes its values in the following space A1 =

+∞ { ∑

} dt+j X−j t ∈ Z, {dj } ⊂ C, and {dj } satisfies 4 .

j=−∞

Two functions ||.||A and ||.||A1 are not necessary equal. This result follows from the fact that the introduced inner product is not based on distribution function. By a similar argument as in the proof of Theorem 2.1, we can generalize the space A to a larger space such that sum of linear processes takes its values in a Hilbert space. Theorem 2.6. Let the following two sets { } { } {Wtν |t ∈ Z}, ν = 1, . . . , I and {cνt |t ∈ Z}, ν = 1, . . . , I satisfy I +∞ ∑ ∑

ν cνtν +j W−j = 0 if and only if cνj = 0, for all j ∈ Z and ν = 1, . . . , I,

(6)

ν=1 j=−∞

where I ∈ N. Moreover, assume that +∞ ∑ j=−∞

+∞ ∑ ν < ∞, a.s. cνtν +j W−j |cνj |2 < ∞ and j=−∞

(7)

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3 PREDICTION METHOD

6

for every ν = 1, . . . , I. Consider the space B=

I +∞ {∑ ∑

} ν cνtν +j W−j + µ tν ∈ Z, µ ∈ C, and {cνj } satisfies 6 and 7 .

ν=1 j=−∞

For every two elements X=

I +∞ ∑ ∑

ν cνtν +j W−j

+ µ1 and Y =

ν=1 j=−∞

I +∞ ∑ ∑

ν dνsν +j W−j + µ2

ν=1 j=−∞

in B assign the following value to them < X, Y >B =

I +∞ ∑ ∑

cνtν +j d¯νsν +j + µ1 µ ¯2 .

ν=1 j=−∞

¯ containing B such that Then, there is a Hilbert space (denoted by B) < X, Y >B =< X, Y >B¯ , for every X, Y ∈ B.

3

Prediction method

In the previous section, under conditions (3) and (4) we showed that every linear process {Xt } in the form of (2) takes its values in a Hilbert space. Therefore, using the projection theorem (see Brockwell and Davis (1991) chapter 2), one can find the best linear prediction of Xn+h in terms of some past values X1 , . . . , Xn . In other words, using the projection theorem, we can find the coefficient vector an = (a1n , . . . , ann )′ such that n

∑

ain Xn+1−i − Xn+h =

¯ A

i=1

inf



K − Xn+h ,

K∈span{X1 ,...,Xn }

¯ A

where A¯ is the Hilbert space generated by the linear process {Wt }. The coefficient vector an satisfies the following equations
A¯ = 0, j = 1, . . . , n.

(8)

On the Best Linear Prediction of Linear Processes, Mohammad Mohammadia and Adel Mohammadpour

3 PREDICTION METHOD

260

7

Set γ(h) :=< X0 , Xh >A¯ , h ∈ Z. Then, equations (8) can be rewritten as Γ n an = γ n ,

(9)

where Γn = [γ(i − j)]i,j=1,...,n and γn = (γ(h), . . . , γ(n + h − 1))′ . If Γn is non-singular then we have the unique solution, an = Γ−1 n γn . Remark 3.1. Similarly to Proposition 5.1.1 in Brockwell and Davis (1991), it can be shown that if γ(0) > 0 and γ(h) → 0, as h → +∞ then Γn is non-singular. For the linear process {Xt } of the form (2) with conditions (3) and (4) we have γ(h) → 0, as h → +∞. Therefore, Γn in (9) is non-singular.

3.1

Prediction method for a class of linear processes with infinite variance

Consider the real valued process {Xt =

∑+∞

j=−∞ cj Wt−j

t ∈ Z}, with conditions (3) and

(4) and {Wt } is an i.i.d. sequence of random variables such that P (|Wt | > x) = x−α L(x),

P (Wt < −x) P (Wt > x) → p and → q, P (|Wt | > x) P (|Wt | > x)

(10)

as x → +∞, where L(.) is a slowly varying function at infinity, α ∈ (0, 2), p ∈ [0, 1] and q = 1 − p. In the sequel, for the process {Xt } we describe a reasonable method for estimating the coefficient vector an in (9). Assume that ∑+∞ ∑n k=−∞ ck ck+h t=1 Xt Xt+h ∑ , h ∈ Z. ρ(h) = ∑+∞ and ρ ˆ (h) = n n 2 2 t=1 Xt k=−∞ ck In Davis and Resnick (1985) it is shown that (

( ) ) ρˆn (1), . . . , ρˆn (H) →P ρ(1), . . . , ρ(H) ,

H>0

(11)

On the Best Linear Prediction of Linear Processes, Mohammad Mohammadia and Adel Mohammadpour

4 APPLICATION IN FINANCE

261

8

as n → +∞. The notation “ →P ” denotes convergence in probability. By dividing two ∑ 2 sides (9) on +∞ k=−∞ |ck | we obtain Γ∗n an = γn∗ ,

(12)

where Γ∗n = [ρ(i − j)]i,j=1,...,n and γn∗ = (ρ(h), . . . , ρ(n + h − 1))′ . Using (11), we can estimate Γ∗n and γn∗ . We summarize the obtained results in the following lemma. In this lemma a reasonable estimator for an is presented. Lemma 3.2. Let X1 , . . . , XN be a path of length N from the linear process {Xt = ∑+∞ j=−∞ cj Wt−j } with conditions (3) and (4) and {Wt } is an i.i.d. sequence of random variables satisfying (10). Let ˆ ∗n,N = [ˆ Γ ρN (|i − j|)]i,j=1,...,n , −1 ∗ ˆn,N = Γˆ∗ n,N γˆ∗ n,N . Then, be non-singular and γˆn,N = (ˆ ρN (h), . . . , ρˆN (n + h − 1))′ . Let a

ˆn,N →P an , as N → +∞. a

4

Application in finance

In this section, we utilize the new prediction method for predicting a set of financial data. Andrews et al. (2009) showed that the natural logarithms of the volumes of WalMart stock traded daily on the New York Stock Exchange between December 1, 2003 and December 31, 2004, can be fitted by the following non-causal ARMA(2,0) process Xt + 2.0766Xt−1 − 2.0772Xt−2 = Zt , where {Zt } is an i.i.d. sequence of α-stable random variables with α = 1.8335. As we know, there is not any model base method for predicting this process. For the values ∑274 {xt }274 t=1 we found t=1 xt /274 = 16.08896. Our aim is to predict Xn+1 in terms of some past values. Firstly, we find the best linear prediction for the process {Xt∗ = ˆ ∗ + 16.08896 as a predictor of Xn+1 . Xt − 16.08896|t ∈ Z}. Then, we consider X n+1

On the Best Linear Prediction of Linear Processes, Mohammad Mohammadia and Adel Mohammadpour

4 APPLICATION IN FINANCE

262

9

∗ ∗ We have Xt∗ + 2.0766Xt−1 − 2.0772Xt−2 = Zt∗ , where Zt∗ = Zt − 16.07931. We can ∗ ∗ +2.0772Xt−2 . Using the introduced space in Theorem rewrite Xt∗ as Xt∗ = Zt∗ −2.0766Xt−1 ∗ ∗ 2.6, the values of the process {Zt∗ −2.0766Xt−1 +2.0772Xt−2 |t ∈ Z} are in a Hilbert space.

Therefore, for the process {Xt∗ } we have    1 + 2.07662 + 2.07722 h = 0   γ(h) = . −2.0766 × 2.0772 h=1     0 otherwise Figure 1 shows the value of Xn+1 and its prediction in terms of 20 past values, for n = 20, . . . , 273. We calculated the mean square error of prediction and true values: 273 1 ∑ (ˆ xn+1 − xn+1 )2 = 0.2435914. 253 n=21

The procedure described in this section can be used for predicting of causal and noncausal ARMA processes.

On the Best Linear Prediction of Linear Processes, Mohammad Mohammadia and Adel Mohammadpour

263

10

15.5

16.0

16.5

17.0

4 APPLICATION IN FINANCE

0

50

100

150

200

250

Figure 1: Dotted line displays the natural logarithms of the volumes of Wal-Mart stock traded daily on the New York Stock Exchange between December 21, 2003 and December 31, 2004. Smooth line displays the predicted values for Xn+1 in terms of 20 past values, where n = 20, . . . , 273.

On the Best Linear Prediction of Linear Processes, Mohammad Mohammadia and Adel Mohammadpour

REFERENCES

264

11

References [1] B. Andrews, M. Calder, R.A. Davis, Maximum likelihood estimation for α-stable autoregressive processes, Annals of Statistics, 37, 1946-1982 (2009). [2] P.J. Brockwell, R.A. Davis, Time Series: Theory and Methods (2nd ed.), SpringerVerlage, New York, 1991. [3] S. Cambanis, A.R. Soltani, Prediction of stable processes: Spectral and moving average representation, Probability Theory and Related Fields, 66, 593-612 (1984). [4] D.B.H. Cline, P.J. Brockwell, Linear prediction of ARMA processes with infinite variance, Stochastic Processes and their Applications, 19, 181-296 (1985). [5] R.A. Davis, S. Resnick, Limit theory for moving averages of random variables with regularly varying tail probabilities, Annals of Probability, 13, 179-195 (1985). [6] J.R. Gallardo, Fractional Stable Noise Processes and Their Application to Traffic Models and Fast Simulation of Broadband Telecommunications Networks, Ph.D. dissertation, Department of Electrical and Computer Science, George Washington University, 2000. [7] A. Kabaˇsinskas, S.T. Rachev, L. Sakalauskas, W. Sun, I. Belovas, Alpha-stable paradigm in financial markets, Journal of Computational Analysis and Applications, 11(4), 641-668 (2009). [8] W. Karchera, E. Shmilevab, E. Spodareva, Extrapolation of stable random fields, Journal of Multivariate Analysis, 115, 516-536 (2013). [9] P.S. Kokoszka, Prediction of infinite variance fractional ARIMA, Probability and Mathematical Statistics, 16, 65-83 (1996). [10] A.G. Miamee, M. Pourahmadi, Wold decomposition, prediction and parameterization of stationary processes with infinite variance, Probability Theory and Related Fields, 79, 145-164 (1988).

On the Best Linear Prediction of Linear Processes, Mohammad Mohammadia and Adel Mohammadpour

REFERENCES

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12

[11] M. Mohammadi, A. Mohammadpour, Best linear prediction for α-stable random processes, Statistics and Probability Letters, 79, 2266-2272 (2009). [12] J.P. Nolan, Modeling financial distributions with stable distributions, Chapter 3 of Handbook of Heavy Tailed Distributions in Finance, Elsevier Science, Amsterdam (2003). [13] S.T. Rachev, S. Mittnik, Stable Paretian Models in Finance, Wiley, New York, 2000. [14] G. Samorodnitsky, M. Taqqu, Stable non-Gaussian Random Processes, ChapmanHill, New York, 1994. [15] A.R. Soltani, R. Moeanaddin, On dispersion of stable random vectors and its application in the prediction of multivariate stable processes, Journal of Applied Probability, 31, 3, 691-699 (1994). [16] R.S. Tsay, Analysis of Financial Time Series, Wiley, New York, 2002.

J. APPLIED FUNCTIONAL ANALYSIS, VOL. 10, NO.'S 3-4,266-280 ,2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

266

SOME ESTIMATE FOR THE EXTENDED FRESNEL TRANSFORM AND ITS PROPERTIES IN A CLASS OF BOEHMIANS S.K.Q. Al-Omari Department of Applied Sciences, Faculty of Engineering Technology Al-Balqa Applied University, Amman 11134, Jordan E-mail: [email protected] Abstract The construction of Boehmians is similar to the construction of …eld of quotients and in some cases, it just gives the …eld of quotients. Strong Boehmians are introduced as a subclass of generalized functions to generalize distributions [4]. In this article, we continue the analysis obtained in [20] and [21]. We discuss the Fresnel transform on a certain space of strong Boehmians. The Fresnel transform and its inverse are extended to a context of Boehmians and are well-de…ned, linear, analytic and one-to-one mappings. New convergence is also de…ned. Keywords: Fresnel Transform; Generalized Fresnel Transform; Tempered Distribution; Boehmian; Strong Boehmian.

1

INTRODUCTION

In wave optics theory, the generalized Fresnel transform is frequently used to describe beam propagation in paraxial approximation and Fresnel di¤raction of light . The generalized Fresnel transform (the di¤raction Fresnel transform) of an arbitrary function f (x) is de…ned as [16; 9] Z F df ( ) = E ( 1 ; 1 ; 2 ; 2 ; x; ) f (x) dx: (1) R

where E(

1;

1;

2;

2 ; x;

) := p

1 2 i

exp 1

i 2

2 1x

2x +

2

2

1

is the transform kernel parameterized by a real matrix M being restricted by 1 2 1 2 = 1: Since di¤raction Fresnel transforms are related to a wide class of optical transforms; their various properties and applications have brought great interests for physicists recently. In this article, we pay attention to a particular case of the di¤raction Fresnel transform, namely, the fresnel transform 1 = 2 : The reason we choose this simpli…ed form of the di¤raction Fresnel transform ”Fresnel transform” lies in two aspects: the …rst is that the Fresnel transform can be analyzed more conveniently which provide a favourable performance; the second reason is related

267

2

S.K.Q. Al-Omari

to the complexity of …nding a strong Boehmian space that can handle the di¤raction Fresnel transform entangled with four parameters. However, our results are new even for usual Boehmian spaces. We consider certain space of Boehmians named as "strong Boehmian space”for the distributional Fresnel transform. Enlightened by this space, the extended Fresnel transform is shown to constitute a usual Boehmian. Further properties are also discussed. We spread the paper into …ve sections: The Fresnel transform is reviewed in Section 1. Section 2 presents a general construction of usual Boehmian spaces. The strong Boehmian space is obtained in Section 3. The image space of Boehmians is described in Section 4. Properties of the extended Fresnel transform and its inverse are discussed in Section 5. By considering the case, 1 = 2 ; the reduction of Equ.(1) is the so-called Fresnel transform whose integral equation given by [8; p.p:207] Z T f ( ) := F ( ) := f (x) exp i ( x)2 dx; (2) R

0

where f (x) is a single valued function over R. Let E (R) be the dual space of E (R) of all in…nitely smooth complex-valued functions (x) ; over R( the set of real numbers), such that k (3) k ( ) = sup D x (x) < 1; x2K

where k 2 N (k = 0; 1; 2; :::) and K run through compact subsets of R. Elements of 0 E (R) are called distributions of compact support. The topology of E (R) is generated by the sequence ( k ( ))1 0 of multinorms which makes E (R) locally convex Hausdör¤ topological vector space, D (R) E (R) ; D (R) is the test function space of compact support. The topology of D (R) is, therefore, stronger than that induced on D (R) 0 0 by E (R) ; and the restriction of any element of E (R) to D (R) is in D (R) ; the dual space of Schwartz distributions. 0 The classical theory of the Fresnel transform is extended to distributions in E (R) by the formula D E T f ( ) := F ( ) = f (x) ; exp i (

x)2 ;

where 2 R is arbitrary but …xed. We state without proof the following theorem. 0

Theorem 1.1(Analyticity of T ). Let f 2 E (R) ; then T is di¤erentiable and D Dk T f ( ) = f (x) ; Dk exp i (

for every nonnegative integer k: Theorem 1.2. The mapping T is linear and one-one . Proof is straightforward.

E x)2 ;

(4)

268

3

some estimate for the extended Fresnel transform and ... 0

De…nition 1.3. Let f and g 2 E (R) : The generalized convolution product f of f and g is de…ned as h(f f g) (x) ; (x)i = hf (x) ; hg ( ) ; (x + )ii ; for every

2 E (R) : 0

Theorem 1.4. For every f 2 E (R) ; and satis…es the relation

for all k 2 N:

(x) = hf ( ) ; (x + )i is in…nitely smooth

D E Dkx (x) = f ( ) ; Dkx (x + ) ;

Proof (see [16] for more details).

2

THE STRONG SPACE OF BOEHMIANS B

Let Y=N R; R = ( 1; 1) ; then f (n; x) 2 E (Y) if and only if f (n; x) is in…nitely smooth and k (5) k (f (n; x)) = sup D x f (n; x) < 1; x2K

for every k 2 N; K being compact subset of R: Let f (n; x) 2 E (Y) and ! 2 D (R) : The usual convolution product of f (n; x) and ! with respect to x is given by Z (f f !) (n; x) = f (n; y) ! (x y) dy; (6) R

n 2 N: Let E (R) (

D (R)) be the set of all test functions ! such that Z ! (x) dx = 1: R

The pair (f; !) ; or

f (n; x) ; is said to be a quotient of functions if and only if ! f (n; x) f dm ! (x) = f (m; x) f dn ! (x) ;

n; m 2 N and dr ! (x) = r! (rx) : Two quotients of functions are equivalent,

f (n; x) !

g (m; x)

f (n; x) f dm (x) = g (m; x) f dn ! (x) :

; if and only if (7)

f (n; x) : f (n; x) 2 E (Y) ; ! 2 E (R) ; then the equivalence class ! f (n; x) f (n; x) containing is said to be a strong Boehmian. The space of all ! ! such Boehmians is denoted by B : Let Q =

269

4

S.K.Q. Al-Omari

Addition, scalar multiplication, di¤erentiation and convolution in B are de…ned by

f (n; x) g (m; x) f (n; x) f + g (m; x) f ! + = ; ! !f k

f (n; x) kf (n; x) f (n; x) Dkx f (n; x) = ; Dkx = ! ! ! !

and

f (n; x) f !

=

f (n; x) f !

;

respectively. E

E convergence ! f (n; x) = ; ! 0 E (Y) :

0

in B if for some f (n; x) ; f (n; x) 2 E (Y) such that f (n; x) = we have f (n; x) ! f (n; x) as ! 1 in !

Theorem 2.1. The mapping f (n; x) !

f (n; x) f dn ! is continuous and imbed!

0

ding from E (Y) ! B : Proof see [7, Theorem 3.1] for detailed proof.

3

STRUCTURE OF USUAL BOEHMIANS

One of most youngest generalizations of functions, and more particularly of distributions, is the theory of Boehmians. The idea of construction of Boehmians was initiated by the concept of regular operators introduced by Boehme [6]. Regular operators form a subalgebra of the …eld of Mikusinski operators and they include only such functions whose support is bounded from the left. In a concrete case, the space of Boehmians contains all regular operators, all distributions and some objects which are neither operators nor distributions. The construction of Boehmians is similar to the construction of the …eld of quotients and in some cases, it gives just the …eld of quotients. On the other hand, the construction is possible where there are zero divisors, such as space C (the space of continous functions) with the operations of pointwise additions and convolution: Let G be a linear space and S be a subspace of G: We assume to each pair of elements f 2 G and ! 2 S; is assigned the product f g such that the following conditions are satis…ed: (1) If !; 2 S; then ! ? 2 S and ! ? = ? !: (2) If f 2 G and !; 2 S; then (f !) = f (! ? ) : (3) If f; g 2 G; ! 2 S and 2 R; then (f + g) ! = f Let

! + g ! and

(f

be a family of sequences from S; such that:

!) = ( f ) !:

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5

some estimate for the extended Fresnel transform and ...

If f; g 2 G; ( n ) 2 and f n =g n (n = 1; 2; :::) ; then f = g; 8n: If (! n ) ; ( n ) 2 ; then (! n ? n ) 2 : Elements of will be called delta sequences. Consider the class A of pair of sequences de…ned by 1 2

A = ((fn ) ; (! n )) : (fn )

G N ; (! n ) 2

;

for each n 2 N: An element ((fn ) ; (! n )) 2 A is called a quotient of sequences, denoted by;

f !n

if fn ! m = fm ! n ; 8n; m 2 N: fn gn fn gn and are said to be equivalent, ; if Two quotients of sequences !n !n n n fn m = gm ! n ; 8n; m 2 N: The relation is an equivalent relation on A and hence, splits A into equivfn fn alence classes. The equivalence class containing is denoted by : These !n !n equivalence classes are called Boehmians; or usual Boehmians; and the space of all Boehmians is denoted by B: The sum of two Boehmians and multiplication by a scalar can be de…ned in a natfn fn fn gn fn fn n + gn ! n ural way = and + = = ; 2 !n !n !n !n !n n n C, space of complex numbers. gn fn fn gn The operation and the di¤ erentiation are de…ned by = !n !n n n D fn fn = : and D !n !n Many a time, G is equipped with a notion of convergence. The relationship between the notion of convergence and are given by: (4) If fn ! f as n ! 1 in G and, ! 2 S is any …xed element, then fn ! ! f

! in G (as n ! 1) :

(5) If fn ! f as n ! 1 in G and (! n ) 2

; then

fn ! n ! f in G (as n ! 1) : The operation

can be extended to B fn !n

In B; two types of convergence, follows:

S by : If !=

fn ! : !n

convergence and

convergence: A sequence of Boehmians ( a Boehmian such that

in B; denoted by (

n

fn 2 B and ! 2 S, then !n

!n) ; (

n

n)

convergence, are de…ned as

in B is said to be

convergent to

! ; if there exists a delta sequence (! n ) ! n ) 2 G; 8k; n 2 N;

271

6

S.K.Q. Al-Omari

and (

!k ) ! (

n

! k ) as n ! 1; in G; for every k 2 N:

The following is equivalent for the statement of

convergence:

(n ! 1) in B if and only if there is fn;k ; fk 2 G and fn;k fk ; = and for each k 2 N; fn;k ! fk as (! k ) 2 such that n = !k !k n ! 1 in G:

Theorem 3.1.

n

!

convergence: A sequence of Boehmians (

n)

in B is said to be

convergent

to a Boehmian in B; denoted by n ! ; if there exists a (! n ) 2 such that ( n ) ! n 2 G; 8n 2 N; and ( n ) ! n ! 0 as n ! 1 in G: See; [1 7; 15 17; 19] ; for further investigation.

4

THE IMAGE SPACE OF BOEHMIANS 0

The function ` (n; ) is a member in L (Y) ; Y = N R; if there is f (n; x) 2 E (Y) such that ` (n; ) := (T f ) (n; ) := F (n; ) : A sequence (` (n; ))1=1 ! ` (n; ) as ! 1; in L (Y) ; if and only if there are 0 f (n; x) ; f (n; x) in E (Y) ; such that f (n; x) ! f (n; x) as

! 1;

where, ` (n; ) = (T f ) (n; ) ; ` (n; ) = (T f ) (n; ) : To L (Y) and E (R) ; the usual convolution product g is interpreted to mean Z (` g !) (n; y) = ` (n; y) dm ! (y) dy; n; m 2 N; (8) R

where ` = T f; ! 2 E (R) : Let ~ (R) = f(dn !) : ! 2 E (R) ; supp dn ! ! 0 as n ! 1g E ~ (R) is a family of delta sequences. then E ~ g of general Boehmians. Following results constitute the space B L; (E; f) ; E; 0

Lemma 4.1. Let f 2 E (Y) and ! 2 E (R) ; then the following holds T (f (n; x) f dm ! (x)) (n; ) = (` g !) (n; ) ; where ` = T f . 0

Proof. Let f 2 E (Y) and ! 2 E (R) ; then by using Equ.(4) we get D T (f (n; x) f dm !) (n; ) = (f (n; x) f dm !) (x) ; exp i (

E x)2 :

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some estimate for the extended Fresnel transform and ...

De…nition 1.3 implies D

D f (n; x) ; dm ! (y) ; exp i ( D D = f (n; x) ; dm ! (y) ; exp i ((

T (f (n; x) f dm !) (n; ) =

That is,

T (f (n; x) f dm !) (n; ) =

Z

R

D

f (n; x) ; exp i ((

y)

(x + y))2 y)

x)2

EE

EE

:

E x)2 dm ! (y) dy:

Equ.(8) gives T (f (n; x) f dm !) (n; ) =

Z

y) dm ! (y) dy = (` (n; ) g !) (n; ) ;

` (n; R

where ` = T f: This completes the proof of the lemma. Lemma 4.2. Let ` 2 L (Y) and ! 2 E (R) ; then ` g ! 2 L (Y) : 0

Proof. Let f (n; x) 2 E (Y) be such that ` (n; ) = T f (n; ) : From Lemma 4.1. we see that (` g !) (n; ) = T (f (n; x) f dm !) (n; ) ; m; n 2 N: The fact that ! 2 E (R) implies (dm !) 2 E (R) ; by [7, p.p.886,(2.4)] : Hence, 0 f (n; x) f dm ! 2 E (Y) :

Thus, ` g ! 2 L (Y) :

This completes the proof of the lemma. Lemma 4.3. Let ` (n; ) 2 L (Y) ; !;

2 E (R) ; then ` g (! f ) = (` g !) f :

Proof. The hypothesis of the lemma and Liebniz Theorem yield Z (` g (! f )) (n; ) = ` (n; y) dm (! f ) (y) dy Z R Z = ` (n; y) m! (my t) dy R

(t) dt:

R

The substitution my

t = x implies Z Z (` g (! f )) (n; ) = ` n; R

R

The substitution x = mz implies Z Z (` g (! f )) (n; ) = ` n; R

t+x m

R

z

t m

m! (x) dx

(t) dt

dm ! (z) dz m (t) dt

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S.K.Q. Al-Omari

Once again, the substitution t = mr togather with Liebniz Theorem imply Z Z ` (n; z r) dm ! (z) dz dm (r) dr (` g (! f )) (n; ) = R

R

Hence, by Equ.(8); we get that Z

(` g (! f )) (n; ) =

R

(` g !) (

r) dm (r) dr

= ((` g !) f ) (n; ) The lemma is therefore completely proved. Lemma 4.4.

Let `1 ; `2 2 L (Y) ; ! 2 E (R) ; then

(i) (`1 + `2 ) g ! = `1 g ! + `2 g !; (ii) k (`1 g !) = (k`1 ) g ! = `1 g (k!) : Proof is straightforward from properties of integral operators. ~ (R) and `1 g ! = `2 g !; then `1 = `2 Lemma. 4.5. Let `1 ; `2 2 L (Y) ; (d !) 2 E in L (Y) : Proof. For every `1 (n; ) ; `2 (m; ) 2 L (Y), there are the corresponding dis0 tributions f1 (n; x) ; f2 (m; x) 2 E (Y) such that `1 = T f1 and `2 = T f2 : Hence, the hypothesis of the lemma and linearity of the distributional Fresnel transform, imply T (f1 (n; x) f2 (m; x)) g ! = 0; 8n; m 2 N: Therefore T (f1 (n; x) f2 (m; x)) = 0; n; m 2 N; x 2 R, in L (Y) : Thus `1 (n; ) = `2 (m; ) in L (Y) ; 8n; m 2 N; 2 R: Hence the lemma. ! 1 and ! 2 E (R) ; then ` g ! ! ` g !

Lemma 4.6. Let ` ! ` in L (Y) as in L (Y) as ! 1:

Proof is a straightforward consequence from de…nitions. ~ (R) ; then ` g ! ! ` ! 1 and (d !) 2 E

Lemma 4.7. Let ` ! ` in L (Y) as as ! 1:

Proof. We have ` (n; ) g ! ` (n; ) = T f (n; ) g ! 0 f (n; x) ; f (n; x) 2 E (R) : Hence, Lemma 4.1. implies ` (n; ) g !

` (n; ) = T (f (n; x) f dr !) (n; )

T f (n; ) ; for some

T f (n; ) ; for some r 2 N:

Once again, linearity of T and the fact that (dr !) is delta sequence yield ` (n; ) g !

` (n; )

=

T (f (n; x) f dr !

! T (f (n; x)

f (n; x)) (n; )

f (n; x)) (n; ) as r ! 1:

Hence ` (n; ) g ! The proof is therefore completed.

` (n; ) ! 0 as

! 1:

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some estimate for the extended Fresnel transform and ...

~ g is thus constructed. The Boehmian space B L; (E; f) ; E;

~ g is denoted by A typical element in B L; (E; f) ; E; ` (n; ) : dn ! Two Boehmians

`2 (m; ) `1 (n; ) ~ g and are said to be equal in B L; (E; f) ; E; dn ! dm

if and only if `1 (n; ) g

= `2 (m; ) g !:

(9)

~ g , we de…ne, addition, scalar multiplication, di¤erentiation and In B L; (E; f) ; E; convolution between Boehmians by `1 (n; ) `2 (m; ) `1 (n; ) g + `2 (m; ) g ! + = ; dn ! dm !g

k

` (n; ) k` (n; ) ` (n; ) Dk ` (n; ) = ; Dk = dn ! dn ! dn ! dn !

and ` (n; ) g dn !

=

` (n; ) g dn !

;

respectively. ~ g in B L; (E; f) ; E; ~ (R) such that gent if there exists a delta sequence (d !) 2 E

De…nition 4.8. ( convergence):

(

!

is

conver-

g !) ; ( g !) 2 L; 8 ; n 2 N;

and ( Theorem 4.9.

g !) ! ( g !) as !

! 1; in L; for every

~ g ( ! 1) in B L; (E; f) ; E;

~ (R) such that ` (n; ) ; ` (n; ) 2 L and (dn !) 2 E ` (n; ) and for each dn !

2 N; ` (n; ) ! ` (n; ) as

2 N:

if and only if there are =

` (n; ) ; dn !

=

! 1 in L:

~ g is convergent; convergence): ( ) ! in B L; (E; f) ; E; ~ (R) such that ( if there exists a (d !) 2 E ) g ! 2 L; 8 2 N; and ( ) g ! ! 0 as ! 1 in L:

De…nition 4.10.(

275

10

5

S.K.Q. Al-Omari

FRESNEL TRANSFORMS TO BOEHMIANS

This section is devoted for extending the Fresnel transform to Boehmians in B : f (n; x) f (n; x) Let 2 B then, the extended Fresnel transform of is viewed ! ! as f (n; x) ` (n; ) ~ g ; T~ = 2 B L; (E; f) ; E; (10) ! dn ! where ` = T f: ~ g is well-de…ned. Theorem 5.1. The mapping T~ : B ! B L; (E; f) ; E; f (n; x) g (m; x) = 2 B ; then, using Equ:(7) yields !

Proof. Let

f (n; x) f dm

= g (m; x) f dn !:

(11)

Applying the Fresnel transform on both sides of Equ.(11), and applying Lemma 0 4.1. imply `1 (n; ) g = `2 (m; ) g ! where, `1 = T f; `2 = T g; f; g 2 E (R) : Hence, from Equ.(9) ; we obtain `1 (n; ) `2 (m; ) = dn ! dm ~ g . in the space B L; (E; f) ; E; The proof is completed. ~ g is linear. Theorem 5.2. The mapping T~ : B ! B L; (E; f) ; E; f (n; x) g (m; x) ; 2 B ; then !

Proof Let

f (n; x) g (m; x) f (n; x) f dm + g (m; x) f dn ! + = : ! !f

(12)

Employing the Fresnel transform for Equ.(12) yields T~

f (n; x) g (m; x) + !

= =

`1 (n; ) g

+ `2 (m; ) g ! !g `1 (n; ) `2 (m; ) + : dn ! dm

By using Lemma 4.1., we get T~

f (n; x) g (m; x) + !

where `1 = T f; `2 = T g:

= T~

f (n; x) !

+ T~

g (n; x)

;

276

11

some estimate for the extended Fresnel transform and ...

Further, for each k 2 R, we have k T~

f (n; x) !

f (n; x) = T~ k !

kf (n; x) !

= T~

=

k`1 (n; ) ; dn !

`1 = T f: Hence the theorem is proved. ~ g is one-one mapping. Theorem 5.3. T~ : B ! B L; (E; f) ; E; Proof Let

f (n; x) g (m; x) ; 2 B be such that ! g (m; x) f (n; x) = T~ T~ !

~ g : in B L; (E; f) ; E;

Equ.(10) then yields `1 (n; ) `2 (m; ) = ; dn ! dm where `1 (n; ) = T f (n; x) ; `2 (m; ) = T g (m; x) : From Equ.(9) we get, `1 (n; ) g

= `2 (m; ) g !: Applying Lemma 4.1. leads

to f (n; x) f dm

= g (m; x) f dn !:

Hence, upon considering Equ.(7); proof of our theorem is completed. ` (n; ) dn ! Fresnel transform by

~ g ; then we de…ne its inverse 2 B L; (E; f) ; E;

De…nition 5.4. Let

T~

` (n; ) dn !

1

=

f (n; x) ; !

where ` = T f: Theorem 5.5. The mapping T~

1

~ g ! B is well de…ned. : B L; (E; f) ; E;

`1 (n; ) `2 (m; ) ~ g ; then `1 (n; ) g = 2 B L; (E; f) ; E; dn ! dm `2 (m; ) g !: By virtue of Lemma 4.1. we get

Proof. Let

T (f (n; x) f dm ) = T (g (m; x) f dn !) ; 0

for some f and g in E (R) ; where `1 = T f; `2 = T g:

=

277

12

S.K.Q. Al-Omari

The property that T is one-one results in f (n; x) f dm

= g (m; x) f dn !:

Therefore, f (n; x) g (m; x) = ! in B : The proof is therefore completed. Theorem 5.6. The mapping T~

1

~ g ! B is linear. : B L; (E; f) ; E;

Proof of this theorem is analogous to that of Theorem 5.2. Details are, thus, avoided. Theorem 5.7. T~ convergence. Proof

Let

~ g : B L; (E; f) ; E;

1

! B is continuous with respect to

~ g : Then, using Theorem 3.1, there are in B L; (E; f) ; E;

!

~ g ` (n; ) ; ` (n; ) 2 B L; (E; f) ; E;

~ (R) such that and (d !) 2 E

=

` (n; ) ` (n; ) ; = and ` (n; ) ! ` (n; ) as ! 1 in L: Hence, dn ! dn ! 0 for some f (n; x) ; f (n; x) 2 E (Y) ; where ` (n; ) = T f (n; ) ; ` (n; ) = 0 T f (n; ) ; we have f (n; x) ! f (n; x) as ! 1 in E (Y) : This implies f (n; x) f (n; x) ! ! !

in B : That is, T~

! T~

1

1

as

! 1:

This completes the proof of the theorem. Theorem 5.8. T~ convergence. Proof

Let

~ g ! B is continuous with respect to : B L; (E; f) ; E;

1

!

~ g ; then we …nd ` (n; ) 2 ! 1 in B L; (E; f) ; E;

as

` (n; ) g ! ! 0 and ` (n; ) ! 0 as ! 1; ` (n; ) = T f (n; ) ; for some f (n; x) 2 E (Y) : By aid of Lemma 4.1. we get

~ g ; (d !) 2 E ~ (R) such that ( B L; (E; f) ; E;

( Employing T~ T~ as

1 1

)g! =

)g! =

T (f (n; x) f dm !) : !

yields

((

) g !) =

f (n; x) f dm ! := f (n; x) ! 0; !

! 1, by [7, Theorem 3.1]: Thus T~

1

! T~

1

in B :

278

13

some estimate for the extended Fresnel transform and ...

The theorem is completely proved. ~ g is continuous with respect to E conTheorem 5.9. T~ : B ! B L; (E; f) ; E; vergence. Proof. Let

E

0

! in B ; then for some f (n; x) ; f (n; x) 2 E (Y) we have f (n; x) f (n; x) = ; = and f (n; x) ! f (n; x) as ! 1: Hence, ! ! applying the Fresnel transform implies ` (n; ) ! ` (n; ) as ! 1: Therefore ` (n; ) g ! ! ` (n; ) g ! as ! 1 , or equivalently, ` (n; ) ` (n; ) ! dn ! dn !

as

! 1: Thus T~ (

E ) ! T~ ( ) as

! 1: The proof is, therefore, completed.

It can further be observed that : ~ g Theorem 5.10. T~ : B ! B L; (E; f) ; E; convergence. Proof Let

!

is continuous with respect to 0

in B as

! 1: Then, we …nd f (n; x) 2 E (Y) ; ! 2 f (n; x) f dm ! E (R) such that ( ) f dm ! = and f (n; x) ! 0 as ! T (f (n; x) f dm !) n ! 1: Hence, we have T~ (( ) f dm !) = = ! ` (n; ) g ! 0 := ` (n; ) ! 0 as ! 1 in E (Y) : Therefore ! T~ (( Hence, T~

! T~ as

) f dm !) = T~

T~

g ! ! 0 as

! 1:

~ g : ! 1 in B L; (E; f) ; E;

The theorem is proved. Con‡ict of Interests The author declares there is no con‡ict of interests regarding the publication of the paper.

References [1] Al-Omari, S.K.Q. ,Loonker D. , Banerji P. K. and Kalla, S. L. (2008). Fourier sine(cosine) transform for ultradistributions and their extensions to tempered and ultraBoehmian spaces, Integ.Trans. Spl.Funct. 19(6), 453 –462. [2] Al-Omari, S. K. Q. and Adam Kilicman, On the Generalized Hartley-Hilbert and Fourier-Hilbert Transforms, Advances in Di¤erence Equations, 2012, 2012:232, doi:10.1186/1687-1847-2012-232.

279

14

S.K.Q. Al-Omari

[3] Al-Omari, S. K. Q. (2011). On the Distributional Mellin Transformation and its Extension to Boehmian Spaces, Int. J. Contemp. Math. Sciences, 6(17), 801-810. [4] E. R. Dill and P. Mikusinski(1993), Strong Boehmians, Proc. Amer. Math. Soc. 119 , 885–888. [5] Banerji, P.K., Al-Omari, S.K. and Debnath, L.(2006). Tempered Distributional Fourier Sine(Cosine)Transform, Integ.Trans. Spl.Funct.17(11),759-768. [6] Boehme, T.K. (1973),The Support of Mikusinski Operators, Tran.Amer. Math. Soc.176,319-334. [7] Ellin R.Dill and Piotr Mikusinski (1993), Strong Boehmians, Proc. Amer. Math. Soc. 119(3). 885-888. [8] Daniel F.V. James, Girish S. Agatwal (1996), The generalized Fresnel transform and its application to optics, Optics Communications 126, 207-212. [9] Hong-Yi Fan,Hai-Liang Lu (2006), Wave-function Transformations by general SU(1,1) Single-Mode Squeezing and Analogy to Fresnel Transformations in Wave Optics, Optics Commun. 258, 51-58. [10] J. N. Pandy(1972), On the Stieltjes Transform of Generalized Functions , Proc. Cambridge Phil. Soc. 71, 85-96. [11] Mikusinski,P.(1987), Fourier Transform for Integrable Boehmians, Rocky Mountain J. Math. 17(3), 577-582. [12] Mikusinski,P.(1983), Convergence of Boehmians, Japan.J.Math.,9, 159-179. [13] Pathak, R.S.(1997), Integral transforms of generalized functions and their applications, Gordon and Breach Science Publishers, Australia, Canada, India, Japan. [14] R. S. Pathak(1976), Distributional Stieltjes Transformation ,Proc. Edinburgh Math. Soc. Ser., 20 (1), 15-22. [15] Roopkumar, R.(2007), Stieltjes Transform for Boehmians, Integral Transf. Spl. Funct.18(11),845-853. [16] S. K. Q. Al-Omari and A. K¬l¬cman (2012), On Di¤ raction Fresnel Transforms for Boehmians, Abstract and Applied Analysis, Volume 2011. Article ID 712746. [17] V.Karunakaran and R.Roopkumar (2000), Boehmians and Their Hilbert Transforms, Integ. Trans. Spl. Funct. ,13,131-141. [18] Zemanian, A.H. (1987), Generalized Integral transformation, Dover Publications, Inc., New York. First published by interscience publishers, New York(1968).

280

some estimate for the extended Fresnel transform and ...

15

[19] S. K. Q. Al-Omari1 and A. K¬l¬cman (2012), Note on Boehmians for Class of Optical Fresnel Wavelet Transforms, Journal of Function Spaces and Applications,Volume 2012, Article ID 405368, doi:10.1155/2012/405368. [20] S. K. Q. Al-Omari , Hartley Transforms on Certain Space of Generalized Functions, Georgian Mathematical Journal, 20(3),415–426, (2013). [21] S. K. Q. Al-Omari , On the Sumudu Transform and Its Extension to a Class of Boehmians, ISRN Mathematical Analysis, Volume 2014, Article ID 463901, 1-6.

J. APPLIED FUNCTIONAL ANALYSIS, VOL. 10, NO.'S 3-4,281-295 ,2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

281

Resonance Problems for Nonlinear Elliptic Equations with Nonlinear Boundary Conditions N. Mavinga Department of Mathematics & Statistics, Swarthmore College, Swarthmore, PA 19081-1390 E-mail: [email protected]

and M. N. Nkashama Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170 E-mail: [email protected]

Abstract We study the solvability of nonlinear second order elliptic partial differential equations with nonlinear boundary conditions where we impose asymptotic conditions on both nonlinearities in the differential equation and on the boundary in such a way that resonance occurs at a generalized eigenvalue; which is an eigenvalue of the linear problem in which the spectral parameter is both in the differential equation and on the boundary. The proofs are based on some variational techniques and topological degree arguments. Keywords: nonlinear elliptic equations, nonlinear boundary conditions, weighted RobinNeumann-Steklov eigenproblem, resonance conditions.

1

Introduction

In this paper we prove the existence of (weak) solutions to nonlinear second order elliptic partial differential equations with (possibly) nonlinear boundary conditions  −∆u + c(x)u = f (x, u) in Ω, (1)  ∂u + σ(x)u = g(x, u) on ∂Ω, ∂ν where the nonlinear reaction-function f (x, u) and the nonlinear boundary function g(x, u) interact, in some sense, with the generalized spectrum of the following linear problem (with possibly singular (m, ρ)-weights)  −∆u + c(x)u = µm(x)u in Ω, (2)  ∂u + σ(x)u = µρ(x)u on ∂Ω. ∂ν 1

N. Mavinga and M. N. Nkashama

Resonance Problems for Nonlinear Elliptic Equations

282

2

Notice that the eigenproblem (2) includes as special cases the weighted Steklov eigenproblem (when m ≡ 0 and ρ 6≡ 0) which was considered in [3, 4, 6, 17, 24] as well as the weighted Robin-Neumann eigenproblem (when ρ ≡ 0 and m 6≡ 0); the latter is also referred to in the literature as Neumann or regular oblique derivative boundary condition (see e.g. [1, 16] and references therein). When m 6≡ 0 and ρ 6≡ 0, we have then the eigenparameter µ both in the differential equation and on the boundary condition, we refer for instance to [5, 7, 8, 18]. Unlike previous results in the literature, what sets our results apart is that we compare both the reaction nonlinearity f in the differential and the boundary nonlinearity g in Eq.(1) with higher eigenvalues of the spectrum of problem (2), where the spectral parameter is both in the differential equation and on the boundary (with weights). The nonlinear problem (1) has received much attention in recent years. Such problem (and its parabolic analog) has been studied in [9, 22] as a model for heat conduction in a body where cooling and heating appear inside and at the boundary at a rate proportional to a power of u. Problem (1) has also been considerably studied by many authors in the framework of sub and super-solutions method. We refer e.g. to [1, 2, 21], and references therein. Since it is based on (the so-called) comparison techniques, the (ordered) sub-super solutions method does not apply when the nonlinearities are compared with higher eigenvalues. After Landesman-Lazer [14], much work has been devoted to the study of the solvability of elliptic boundary value problems (with linear homogeneous boundary conditions) where the reaction nonlinearity in the differential equation interacts with the eigenvalues of the corresponding linear differential equation with linear homogeneous boundary conditions (resonance and nonresonance problems). For some recent results in this direction we refer e.g. to [11, 12, 13, 19, 20, 23], and references therein. A few results on a disk (n = 2) were obtained in the case of linear elliptic equations with nonlinear boundary conditions, where the nonlinearity on the boundary was compared with the first Steklov eigenvalue (that is, m ≡ 0 in Eq.(2)). We refer to Cushing [10] and Klingelh¨ofer [15] (the results in [15] were significantly generalized to higher dimensions in [1] in the framework of sub and super-solutions method as aforementioned). In [3, 17] the resonance problem for elliptic equations with nonlinear boundary conditions was analyzed using bifurcation theory (see Remark 3.6 herein). More recently, the authors in [19] proved nonresonance results for problem (1) in which the nonlinearities interact, in some sense, only with either the Steklov or the Neumann spectrum. In a very recent paper of one of the authors [18], nonresonance results for problem (1) were proved in which both nonlinearities in the differential equation and on the boundary interact, in some sense, with the generalized spectrum of problem (2). It is our purpose in this paper to establish existence results for problem (1) by imposing asymptotic conditions on both nonlinearities in the differential equation and on the boundary in such a way that resonance occurs at a generalized eigenvalue of problem (2). Our results generalize earlier ones in [1, 2, 12, 23], and in some instances some of those in [3, 11, 17] which were obtained in a different setting. The content of this paper is organized as follows. In Section 2, we state and prove some preliminary results that we shall need in the sequel. Section 3 is devoted to the statement and proof of existence results for Eq.(1) at resonance. The proof of the main result is based on variational and topological degree arguments. We conclude the paper with some remarks which show (among other) how our result can be extended to problems with variable

N. Mavinga and M. N. Nkashama

283

Resonance Problems for Nonlinear Elliptic Equations

3

coefficients. Throughout this paper we assume that Ω is a bounded domain in Rn (n ≥ 2 ) with boundary ∂Ω of class C 0,1 , ∂/∂ν is the (unit) outer normal derivative. By a weak solution of Eq.(1) we mean a function u ∈ H 1 (Ω) such that Z Z I Z I ∇u∇v + c(x)uv + σ(x)uv = f (x, u)v + g(x, u)v for all v ∈ H 1 (Ω), (3) Z where

I denotes the (volume) integral on Ω and

denotes the (surface) integral on ∂Ω.

Let us mention that H 1 (Ω) denotes the usual real Sobolev space of functions on Ω endowed with the (c, σ)-inner product defined by Z Z I (u, v)(c,σ) = ∇u∇v + c(x)uv + σ(x)uv (4) with the associated norm denoted by kuk(c,σ) . (The conditions on c and σ which imply that (4) is an inner product are given below.) This norm is equivalent to the standard H 1 (Ω)-norm. Besides the Sobolev spaces, we shall make use, in what follows, of the real Lebesgue spaces Lq (∂Ω), 1 ≤ q ≤ ∞, and the compactness of the trace operator Γ : H 1 (Ω) → Lq (∂Ω) for 2(n − 1) 1≤q< . (Sometimes we will just use u in place of Γu when considering the trace n−2 of a function on ∂Ω.) The functions c : Ω → R, σ : ∂Ω → R, f : Ω × R → R and g : ∂Ω × R → R satisfy the following conditions. (C1) c ∈ Lp (Ω) with p ≥ n/2 when n ≥ 3 (p > 1 when n = 2) and σ ∈ Lq (∂Ω) with q ≥ n − 1 when n ≥ 3 (q > 1 when n = 2) with (c, σ) > 0; that is, c(x) ≥ 0 a.e. on Ω and σ(x) ≥ 0 a.e. on ∂Ω such that Z I c(x) dx + σ(x) dx 6= 0. (C2) f : Ω × R → R and g : ∂Ω × R → R are Carath´eodory functions (i.e., measurable in x for each u, and continuous in u for a.e. x). (C3) There exist constants a1 , a2 > 0 such that for a.e. x ∈ ∂Ω and all u ∈ R, |g(x, u)| ≤ a1 + a2 |u|s with 0 ≤ s
0 such that for a.e. x ∈ Ω and all u ∈ R, |f (x, u)| ≤ b1 + b2 |u|s with 0 ≤ s
0; that is, Z I m(x) ≥ 0 a.e. on Ω and ρ(x) ≥ 0 a.e. on ∂Ω with m(x) dx + ρ(x) dx 6= 0.

(5)

(6)

(We stress the fact that the weight-functions m and ρ may vanish on subsets of positive measure.) The (generalized) eigenproblem is to find a pair (µ, ϕ) ∈ R × H 1 (Ω) with ϕ 6≡ 0 such that Z  Z Z I I ∇ϕ∇v + c(x)ϕv + σ(x)ϕv = µ m(x)ϕv + ρ(x)ϕv for all v ∈ H 1 (Ω). (7) Picking v = ϕ Z I it follows that, if there is such an eigenpair, then one has that µ > 0 and 2 m(x)ϕ + ρ(x)ϕ2 > 0. Therefore, one can split the Hilbert space H 1 (Ω) as a direct (c, σ)-orthogonal sum in the following way, 1 H 1 (Ω) = V(m,ρ) (Ω) ⊕ H(m,ρ) (Ω),

 where V(m,ρ) (Ω) :=

1

u ∈ H (Ω) :

Z

2

I

m(x)u +

(8)

  ⊥ 1 ρ(x)u = 0 and H(m,ρ) (Ω) = V(m,ρ) (Ω) . 2

1 On H(m,ρ) (Ω) ⊂ H 1 (Ω), we will also consider the inner product defined by

Z (u, v)(m,ρ) :=

I m(x)uv +

ρ(x)uv,

(9)

with corresponding norm denoted by || · ||(m,ρ) (see e.g. [18] for details). Assuming that the above assumptions are satisfied, one of the authors [18] proved that, for n ≥ 2, the eigenproblem (5) has a sequence of real eigenvalues 0 < µ1 < µ2 ≤ . . . ≤ µj ≤ . . . → ∞, as j → ∞, each eigenvalue has a finite-dimensional eigenspace. The eigenfunctions ϕj corresponding to 1 these eigenvalues form a complete orthonormal family in the (proper) subspace H(m,ρ) (Ω). Moreover, the first eigenvalue µ1 is simple, and its associated eigenfunction ϕ1 is strictly positive (or strictly negative) in Ω and the following inequalities hold. (i) For all u ∈ H 1 (Ω), Z  Z I Z I 2 2 2 2 µ1 m(x)u + ρ(x)u ≤ |∇u| + c(x)u + σ(x)u2 ,

(10)

where µ1 > 0 is the least eigenvalue for Eq.(5). If equality holds in (10), then u is a multiple of an eigenfunction of Eq.(5) corresponding to µ1 .

N. Mavinga and M. N. Nkashama

Resonance Problems for Nonlinear Elliptic Equations

285

5

(ii) For every v ∈ ⊕i≤j E(µi ), and w ∈ ⊕i≥j+1 E(µi ), we have that ||v||2(c,σ) ≤ µj ||v||2(m,ρ)

and

||w||2(c,σ) ≥ µj+1 ||w||2(m,ρ) ,

(11)

where E(µi ) is the µi -eigenspace, ⊕i≤j E(µi ) is the span of eigenfunctions associated with eigenvalues below and up to µj , and || · ||(m,ρ) is the norm induced by (9). The next theorem concerns an existence result for the nonlinear problem (1) in the case of weighted nonresonance. We refer to [18] for the proof of this result. Theorem 2.1 (Weighted nonresonance between consecutive generalized eigenvalues) Suppose that the assumptions (C1)-(C3’) and (6) are met, and that the following conditions hold. (C4) There exist constants a, b, α, β ∈ R such that αm(x) ≤ lim inf

f (x, u) f (x, u) ≤ lim sup ≤ βm(x) u u |u|→∞

aρ(x) ≤ lim inf

g(x, u) g(x, u) ≤ lim sup ≤ bρ(x), u u |u|→∞

|u|→∞

and |u|→∞

uniformly for a.e. x ∈ Ω, respectively for a.e. x ∈ ∂Ω, where µj < min(a, α) ≤ max(b, β) < µj+1 .

(12)

Then, Eq.(1) has at least one (weak) solution u ∈ H 1 (Ω). Remark 2.2 The result in Theorem 2.1 remains valid if one replaces the functions f (x, u) and g(x, u) with f (x, u) + A(x) and g(x, u) + B(x) respectively, where A ∈ L2 (Ω) and B ∈ L2 (∂Ω).

3

Main Result

In this section, we prove an existence result for problem (1) at resonance which includes both the Steklov as well as Neumann and Robin problems with appropriate choices of the weights m and ρ as aforementioned. Notice that the nonlinearity in the boundary condition is at resonance as well. We require that the nonlinearities satisfy some sign conditions and a Landesman-Lazer type condition (possibly at a generalized higher eigenvalue). Consider the following nonlinear problem  −∆u + c(x)u = µj m(x)u + f (x, u) in Ω, (13)  ∂u + σ(x)u = µj ρ(x)u + g(x, u) on ∂Ω, ∂ν where µj is a generalized eigenvalue of the (weighted) problem (5). Theorem 3.1 (Resonance at any generalized eigenvalue) Suppose that the assumptions (C1)–(C3’) and (6) are met, and that the following conditions hold.

N. Mavinga and M. N. Nkashama

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Resonance Problems for Nonlinear Elliptic Equations

6

(C5) There exists a constant β such that 0 ≤ lim inf

f (x, u) f (x, u) ≤ lim sup ≤ βm(x) u u |u|→∞

0 ≤ lim inf

g(x, u) g(x, u) ≤ lim sup ≤ βρ(x), u u |u|→∞

|u|→∞

and |u|→∞

uniformly for a.e. x ∈ Ω, respectively for a.e. x ∈ ∂Ω, where β < (µj+1 − µj ). (C6) Sign conditions: There exist functions a, A ∈ L2 (Ω), b, B ∈ L2 (∂Ω), and constants r < 0 < R such that f (x, u) ≥ A(x)

and

g(x, u) ≥ B(x)

for a.e. x ∈ Ω, respectively for a.e. x ∈ ∂Ω and all u ≥ R, f (x, u) ≤ a(x)

and

g(x, u) ≤ b(x)

for a.e. x ∈ Ω, respectively for a.e. x ∈ ∂Ω and all u ≤ r. Then, Eq. (13) has at least one (weak) solution u ∈ H 1 (Ω) provided that the following Landesman-Lazer type condition holds: Z Z I I f+ ϕ + f− ϕ + g+ ϕ + g− ϕ > 0 for all ϕ ∈ E(µj ) \ {0}, (14) ϕ>0

ϕ0

ϕ0

ϕ>0

on the sets {x ∈ Ω : ϕ(x) > 0} and {x ∈ ∂Ω : ϕ(x) > 0} respectively. Unlike previous results in the literature, what sets our results apart is that we compare both the reaction nonlinearity f in the differential equation and the boundary nonlinearity g with higher eigenvalues of the spectrum of problem (2), where the spectral parameter is both in the differential equation and on the boundary (with possibly singular weights). It should also be noted that the presence of the boundary nonlinearity extends the range of allowable ‘forcing’ terms in the condition (14). Our results generalize earlier ones in [1, 2, 3, 12, 17, 23] (see Remark 3.6 for details). We will use variational and topological degree techniques combined with some duality arguments. Before giving a proof of our main result, we first prove several lemmas that are relevant in order to obtain a priori estimates. (For some of these lemmas, we borrow some techniques of proof from [12, 13].) For u ∈ H 1 (Ω), we shall write u = u0 + u ¯+u ˜ + v, where u0 ∈ E 0 := E(µj ), u ¯ ∈ ⊕i≤j−1 E(µi ), u ˜ ∈ ⊕i≥j+1 E(µi ), and v ∈ V(m,ρ) . Moreover, we shall set w := u ˜+v−u ¯ − u0 and u⊥ := u ˜+v+u ¯.

N. Mavinga and M. N. Nkashama

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7

Resonance Problems for Nonlinear Elliptic Equations

Lemma 3.2 Let β > 0 be as in Theorem 3.1. Then there exists δ = δ(β) > 0 such that for all u ∈ H 1 (Ω) Z  Z Z I I

2

∇u∇w + c(x)uw + σ(x)uw − (µj + β) . m(x)uw + ρ(x)uw ≥ δ u⊥ (c,σ)

Proof. Let u ∈ H 1 (Ω), define Dβ (u) by Z  Z Z I I Dβ (u) := ∇u∇w + c(x)uw + σ(x)uw − (µj + β) m(x)uw + ρ(x)uw . Taking into account the (c, σ)-orthogonality of u ˜, v, u ¯ and u0 in H 1 (Ω) and the fact that 0 0 v ∈ V(m,ρ) and u ∈ E , one has that

2 Dβ (u) = k˜ uk2(c,σ) + kvk2(c,σ) − k¯ uk2(c,σ) − (µj + β) k˜ uk2(m,ρ) + (µj + β) k¯ uk2(m,ρ) + β u0 (m,ρ)     (µj + β) µj 2 2 2 2 2 ≥ k˜ uk(c,σ) − k˜ uk(c,σ) + kvk(c,σ) + k¯ uk(c,σ) − k¯ uk(c,σ) µj+1 µj−1

2  

2 2 2 ≥ δ k˜ uk(c,σ) + kvk(c,σ) + k¯ uk(c,σ) = δ u⊥ , (c,σ)

 where δ = min 1, 1 −

µj + β µj , µj+1 µj−1

 − 1 . The proof is complete. 2

Lemma 3.3 Let β > 0 be as in Theorem 3.1, δ > 0 be associated with β by Lemma 3.2, and  > 0. Then for all τ¯ ∈ Lp (Ω) and τ˜ ∈ Lq (∂Ω) satisfying τ¯(x) ≤ βm(x) + , τ˜(x) ≤ βρ(x) +  respectively, and all u ∈ H 1 (Ω, ) one has Z  Z I I

2

(u, w)(c,σ) − µj m(x)uw + ρ(x)uw − τ¯(x)uw + τ˜(x)uw ≥ Cδ u⊥ , (c,σ)

where Cδ > 0 is a constant depending on δ and , provided that  > 0 is sufficiently small. Z  Z I I Proof. Let Dτ (u) := (u, w)(c,σ) − µj m(x)uw + ρ(x)uw − τ¯(x)uw + τ˜(x)uw. Using the computations in the proof of Lemma 3.2 we obtain that

2

2

2

2



˜ ˜ Dτ (u) ≥ δ u⊥ − K = (δ − K) = Cδ u⊥

u⊥

u⊥ (c,σ)

(c,σ)

(c,σ)

(c,σ)

,

˜ is a constant. If  is sufficiently small then we get that Cδ > 0. The proof is complete where K 2 Lemma 3.4 Assume (C1)–(C3’) and (6) are met, and that (in addition) f and g satisfy the sign-condition (C6). Then, for each real number K > 0 there are decompositions f (x, u) = pK (x, u) + fK (x, u)

and

g(x, u) = qK (x, u) + gK (x, u)

(15)

and

0 ≤ u qK (x, u)

(16)

of f and g such that 0 ≤ u pK (x, u)

N. Mavinga and M. N. Nkashama

Resonance Problems for Nonlinear Elliptic Equations

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8

for a.e. x ∈ Ω, respectively for a.e. x ∈ ∂Ω, and all u ∈ R. Moreover, there exist functions ω ¯ ∈ L2 (Ω) and ω ˜ ∈ L2 (∂Ω) depending on a, A, f and b, B, g respectively such that |fK (x, u)| ≤ ω ¯ (x)

and

|gK (x, u)| ≤ ω ˜ (x)

(17)

for a.e. x ∈ Ω, respectively for a.e. x ∈ ∂Ω, and all u ∈ R. ( inf{g(x, u), K} if u ≥ 1, Proof. Given K > 0, define gˆK (x, u) := sup{g(x, u), K} if u ≤ −1, and qˆK (x, u) := g(x, u) − gˆK (x, u) for x ∈ Ω and |u| ≥ 1.   if |u| ≥ 1, qˆK (x, u) Set qK (x, u) := u qˆK (x, u/|u|) if 0 < |u| ≤ 1,   0 if u = 0. Finally, define gK := g − qK . By an easy calculation, one can check that all the conditions of the lemma are satisfied with ω ˜ = C˜ + max{|b(x)|, |B(x)|, K}, where the constant C˜ > 0 depends on R, −r, 1, b1 , b2 . Similar arguments are used in the case of the function f. The proof is complete. 2 Lemma 3.5 Assume (C1)–(C3’) and (6) are met, and that in (addition) f and g satisfy (C5) and (C6). Then, for each real number K > 0, the functions pK and qK provided by Lemma 3.4 satisfy the following additional conditions |pK (x, u)| ≤ (β m(x) + )|u| − K

and

|qK (x, u)| ≤ (β ρ(x) + )|u| − K

(18)

for a.e. x ∈ Ω, respectively for a.e. x ∈ ∂Ω, and all u ∈ R with |u| ≥ max{1, }. Proof. It follows from (C5) that for all  > 0 there exists κ = κ() > 0 such that |f (x, u)| ≤ (β m(x) + )|u|

and

|g(x, u)| ≤ (β ρ(x) + )|u|

for a.e. x ∈ Ω, respectively for a.e. x ∈ ∂Ω,  and u ∈ R with |u| ≥ κ.  g(x, u) if u ≥ 1 and g(x, u) ≤ K,    K if u ≥ 1 and g(x, u) ≥ K, Let u ∈ R with |u| ≥ 1. Then gˆK (x, u) := g(x, u) if u ≤ −1 and g(x, u) ≥ −K,    K if u ≤ −1 and g(x, u) ≤ −K.   0    g(x, u) − K It follows that qK (x, u) =  0    g(x, u) + K

if u ≥ 1 if u ≥ 1 if u ≤ −1 if u ≤ −1

and g(x, u) ≤ K, and g(x, u) ≥ K, and g(x, u) ≥ −K, and g(x, u) ≤ −K.

By (19) we get that 0 ≤ qK (x, u) ≤ (β ρ(x) + ) u − K

if u ≥ max{1, κ}

(19)

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Resonance Problems for Nonlinear Elliptic Equations

9

and 0 ≥ qK (x, u) ≥ (β ρ(x) + ) u + K

if u ≤ − max{1, κ}.

Therefore |qK (x, u)| ≤ (β ρ(x) + ) |u| − K for a.e. x ∈ ∂Ω and all u ∈ R with |u| ≥ max{1, κ}. Similar arguments are used in the case of f. The proof is complete. 2 Proof of Theorem 3.1. The proof is divided into four steps. Step 1. Let δ be associated to the constant β by Lemma 3.2. Then, by assumption (C5), there exists κ ≡ κδ > 0 such that ˜ |f (x, u)| ≤ (β m(x) + d)|u|

and

˜ |g(x, u)| ≤ (β ρ(x) + d)|u|

(20)

for a.e. x ∈ Ω, respectively for a.e. x ∈ ∂Ω, and all u ∈ R with |u| ≥ κ, where d˜ is a sufficiently small constant such that 0 < d˜ 0 such that kukH 1 < C

(28)

for any (possible) weak solution u ∈ H 1 (Ω) of (26) (C is independent of u and λ). If we assume that the claim does not hold, then there exist sequences (λn ) in the interval (0, 1] and (un ) in H 1 (Ω) with kun kH 1 → ∞ such that un is a (weak) solution of the following problem  −∆u + c(x)u − µj m(x)u − (1 − λn )d m(x))u − λn f (x, u) = 0 in Ω, (29)  ∂u + σ(x)u − µj ρ(x)u − (1 − λn )d ρ(x)u − λn g(x, u) = 0 on ∂Ω. ∂ν That is, Z 0 = (un , v)(c,σ) −µj (un , v)(m,ρ) −(1−λn ) d (un , v)(m,ρ) −λn

I f (x, un )v −λn

g(x, un )v (30)

for every v ∈ H 1 (Ω). From (27), it follows that

0

un → ∞ and (c,σ) Therefore,

un 0 kun k(c,σ)

⊥

un (c,σ) ku0n k(c,σ)

→ 0.

(31)

is bounded in H 1 (Ω). By the reflexivity of H 1 (Ω), the compact embed-

ding of H 1 (Ω) into L2 (Ω) and the compactness of the trace operator, one can assume (taking a subsequence if it is necessary) that un 0 kun k(c,σ)

* w in H 1 (Ω);

un 0 kun k(c,σ)

→ w in L2 (Ω) (also in L2 (∂Ω));

u0n → w in L2 (Ω) (also in L2 (∂Ω)). ku0n k(c,σ) Set vn =

u0n . 0 kun k(c,σ)

Substituting v in (30) by (vn /λn ), and taking into account the orthogo-

nality and the fact that vn ∈ E 0 , we get

0 −1 0 2



0 ≤ (1 − λn )λ−1 n d un (c,σ) un (m,ρ) = −

Z

I f (x, un )vn −

By taking the liminf as n → ∞, we have that Z  I lim inf f (x, un )vn + g(x, un )vn ≤ 0. n→∞

g(x, un )vn

(32)

(33)

N. Mavinga and M. N. Nkashama

292

12

Resonance Problems for Nonlinear Elliptic Equations

Let I + = {x ∈ Ω : w(x) > 0} and I − = {x ∈ Ω : w(x) < 0}. Then for a.e. x ∈ I + there exists ν(x) ∈ N such that for all n ≥ ν(x), one has (passing to a subsequence if necessary),  −1 1

⊥ < w(x) un (x) u0n (c,σ) 4 and

−1 0  0 1 un (x) un − w(x) < w(x). (c,σ) 4

Therefore, for all n ≥ ν(x) one has  −1

0

0 1 −1 −1



un (x) u0n (c,σ) = (u0n (x)+u⊥ ≥ (u0n (x)−|u⊥ ≥ w(x). n (x))( un (c,σ) ) n (x)|)( un (c,σ) ) 2 It follows that for a.e. x ∈ I + , there exists an integer ν(x) ∈ N such that for all n ≥ ν(x), vn (x) > 0

and un (x) ≥



1 w(x)( u0n (c,σ) ) → ∞ (since u0n (c,σ) ) → ∞). 2

On the other hand, for a.e. x ∈ I − there exists ϑ(x) ∈ N such that for all n ≥ ϑ(x),  −1

0

0 1 −1 −1



un (x) u0n (c,σ) = (u0n (x)+u⊥ ≤ (u0n (x)+|u⊥ ≤ w(x). n (x))( un (c,σ) ) n (x)|)( un (c,σ) ) 2

1 Therefore, for n ≥ ϑ(x), un (x) ≤ w(x)( u0n (c,σ) ) → −∞. 2 In order to apply Fatou’s Lemma we need to find n0 ∈ N such that for n ≥ n0 , f (x, un ) vn ≥ ¯l(x) a.e.

and

g(x, un ) vn ≥ ˜l(x) a.e.,

(34)

for some ¯l ∈ L1 (Ω) and ˜l ∈ L1 (∂Ω). Indeed, from (27) one gets



2  −1 −1 

⊥

⊥ 0 0

+ 2a. ≤ 2a u u u u

n

n n (c,σ) n (c,σ) (c,σ)

(c,σ)

−1

2  0

u ≤ C, where C is a Therefore, by (31) one has that for n ≥ n0 , u⊥ n (c,σ) n (c,σ) constant independent of n. Since γ¯ (x, un (x)) ≥ 0, one has that for n ≥ n0 ,  −1 γ¯ (x, un (x))un (x)vn (x) = γ¯ (x, un (x))un (x)u0n (x) u0n (c,σ)  −1 1 = γ¯ (x, un (x)) u0n (c,σ) [(un (x))2 + (u0n (x))2 − (un (x) − u0n (x))2 ] 2  −1 1 2 0 ≥ − γ¯ (x, un (x))(u⊥ un (c,σ) ≥ −C¯ γ¯ (x, un (x)) l1 (x), n (x)) 2 where l1 ∈ L1 (Ω), and C¯ > 0 are independent of n. Therefore, for n ≥ n0 , ¯ m(x) + d) l1 (x). γ¯ (x, un (x))un (x)vn (x) ≥ −C(β Now, using the decomposition of f , one has that for n ≥ n0 , ¯ un (x))vn (x) f (x, un (x))vn (x) = γ¯ (x, un (x))un (x)vn (x) + h(x, ¯ m(x) + d) l1 (x) − K1 l2 (x) = ¯l(x), ≥ −C(β

N. Mavinga and M. N. Nkashama

293

Resonance Problems for Nonlinear Elliptic Equations

13

˜l in (34). Notice that it where l2 ∈ L1 (Ω). We use similar arguments to obtain the function Z I follows from (32) and (34) that sup f (x, un )vn < ∞ and sup g(x, un )vn < ∞. Therefore, by Fatou’s Lemma and the properties of lim inf, one has I I Z Z g+ w ≤ lim inf f (x, un )vn ; f+ w ≤ lim inf n→∞

w>0

Z

Z

I

f− w ≤ lim inf n→∞

w0

w 0 such that for all ξ ∈ Rn , hA(x)ξ, ξi ≥ γ|ξ|2

a.e. on Ω.

References [1] H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions, Proc. 2nd Schrveningen Conf. Diff. Eq., North-Holland Math. Studies 21 (1976), pp. 43-64. [2] P. Amster, M. C. Mariani and O. Mendez Nonlinear boundary conditions for elliptic equations, Electron. J. Diff. Equations (2005), No. 144, pp. 1-8. [3] J. M. Arrieta, R. Pardo and A. Rodr´ıguez-Bernal, Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity, Proceedings Royal Society Edinburgh 137 A (2007), 225-252. [4] G. Auchmuty, Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Numer. Funct. Anal. Optim. 25, Nos. 3&4 (2004), 321-348. [5] G. Auchmuty, Bases and Comparison Results for Linear Elliptic Eigenproblems, J.Math. Analysis and Applications 390 (2012), 394-406. [6] C. Bandle, Isoperimetric Inequalities and Applications, Pittman, London, 1980. [7] C. Bandle, A Rayleigh-Faber-Krahn Inequality and Some Monotonicity Properties for Eigenvalue Problems with Mixed Boundary Conditions, Internat. Ser. Numer. Math., vol. 157, Birkhuser, Basel, 2009, 3-12. [8] B. Belinskiy, Eigenvalue problems for elliptic type partial differential operators with spectral parameter contained linearly in boundary conditions, in: Partial Differential and Integral Equations, H.G.W Begehr, et al., eds., , Kluwer Academic Publ. (1999), 319-333. [9] M. Chipot, M. Fila, and P. Quittner, Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions, Acta Math. Univ. Comenianae, Vol. LX, 1 (1991), 35-103.

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15

[10] J. M. Cushing, Nonlinear Steklov problems on the unit circle, J. Math. Anal. Appl. 38 (1972), 766-783. [11] P. Dr´abek and S. B. Robinson, Resonance problems for the p-Laplacian, J. Functional Analysis 169 (1999), 189 - 200. [12] D. G. de Figueiredo, Semilinear elliptic equations at resonance: Higher eigenvalues and unbounded nonlinearities, in “Recent Advances in Differential Equations” (R. Conti, Ed.), pp. 89-99, Academic Press, London, 1981. [13] R. Iannacci and M. Nkashama, Unbounded perturbations of forced second order ordinary differential equations at resonance, J. Differential Equations 69 (1986), 289–309. [14] E. M. Landesman and A. C. Lazer, Nonlinear pertubations of linear elliptic boundary value problems I, J. Mech. Anal. 19 (1970), 609-623. [15] K. Klingelh¨ofer, Nonlinear harmonic boundary value problems I, Arch. Rat. Mech. Anal. 31 (1968), 364-371. [16] A. Manes and A. M. Micheletti, Un’estensione della teoria variazionale classica degli autovalori per operatri ellitici del secondo ordine, Bolletino U. M. I. Vol. 7 (1973), 285301. [17] S. Mart´ınez and J. D. Rossi,Weak solutions for the p-Laplacian with a nonlinear boundary condition at resonance, Electron. J. Diff. Equations (2003) 2003, No. 27, pp. 1-14. [18] N. Mavinga, Generalized eigenproblem and nonlinear elliptic equations with nonlinear boundary conditions, Proc. Royal Soc. Edinburgh 142A (2012), 137-153. [19] N. Mavinga and M. N. Nkashama, Steklov-Neumann eigenproblems and nonlinear elliptic equations with nonlinear boundary conditions, J. Differential Equations 248 (2010), 12121229. [20] J. Mawhin, Landesman-Lazer conditions for boundary value problems: A nonlinear version of resonance, Bol. de la Sociedad Espanola de Mat. Aplicada 16 (2000), 45-65. [21] J. Mawhin and K. Schmitt, Corrigendum:Upper and lower solutions and semilinear second order elliptic equations with non-linear boundary conditions, Proceedings Royal Society Edinburgh 100A (1985), 361. [22] K. Pfluger, On indefinite nonlinear Neumann Problems, in: Partial Differential and Integral Equations, H.G.W Begehr, et al., eds., , Kluwer Academic Publ. (1999), 335-346. [23] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Application to Differential Equations, CBMS Regional Conf. Series in Math., no. 65, Amer. Math. Soc., Providence, RI, 1986. [24] M. W. Steklov, Sur les probl`emes fundamentaux de la physique math´ematique, Ann. Sci. Ecole Norm. Sup. 19 (1902), 455-490.

J. APPLIED FUNCTIONAL ANALYSIS, VOL. 10, NO.'S 3-4,296-345 ,2015, COPYRIGHT 2015 EUDOXUS PRESS, LLC

Approximation by Complex Generalized Discrete Singular Operators George A. Anastassiou & Merve Kester

Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, U.S.A. [email protected] [email protected] Abstract. In this article, we work on the general complex-valued discrete singular operators over the real line regarding their convergence to the unit operator with rates in the Lp norm for 1 p 1: The related established inequalities contain the higher order Lp modulus of smoothness of the engaged function or its higher order derivative. Also we study the complex-valued fractional generalized discrete singular operators on the real line, regarding their convergence to the unit operator with rates in the uniform norm. The related established inequalities involve the higher order moduli of smoothness of the associated right and left Caputo fractional derivatives of the related function. AMS 2010 Mathematics Subject Classi…cation: 26A15, 26A33, 26D15, 41A17, 41A25, 41A28, 41A35. Key Words and Phrases: general singular integral, fractional singular integral, discrete singular operator, modulus of smoothness, fractional derivative, complex valued function.

1

Background

In [5], Chapter 19, the authors studied the approximation propeties of the general complex- valued singular integral operators r; (f ; x) de…ned as follows: They considered the complex p valued Borel measurable functions f : R ! C such that f = f1 + if2 ; i := 1: Here f1 ; f2 : R ! R are implied to be real valued Borel measurable functions.

1

296

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

be a Borel probability measure on R: For r 2 N and n 2 Z+

Let > 0 and they put j

as

297

=

(

r j

r ( 1) j n; Pr j r 1 i=1 ( 1)

i r i

n

i

j = 1; : : : ; r; j = 0:

;

(1)

Then, they de…ned the general complex-valued singular integral operators 0 1 Z 1 X r @ A (2) r; (f ; x) := j f (x + jt) d (t); 8 > 0: 1

j=0

Clearly by the de…nition of R.H.S.(2) they had r;

(f ; x) =

r;

(f1 ; x) + i

r;

(f2 ; x):

(3)

They supposed that r; (fj ; x) 2 R; 8x 2 R; j = 1; 2: In [5], Chapter 19, the authors de…ned the modulus of smoothness …nite as follows: Let f1 ; f2 2 C n (R); n 2 Z+ with the rth modulus of smoothness …nite, that is r (n) (x)k1;x t fe j

(n)

! r (fej ; h) := sup k jtj h

h > 0; where

r (n) (x) t fe j

e j = 1; 2:

:=

r X

( 1)r

j

j=0

< 1;

(4)

r (n) f (x + jt); j ej

(5)

In [5], Chapter 19, The authors also de…ned the rth Lp -modulus of smooth(n) (n) ness for f1 ; f2 2 C n (R) with f1 ; f2 2 Lp (R); 1 p < 1 as (n)

! r (fej ; h)p := sup k jtj h

r (n) (x)kp;x ; t fe j

h > 0; with e j = 1; 2:

(6)

(n) There they supposed that ! r (fej ; h)p < 1; h > 0; e j = 1; 2: They denoted r X k k := j j ; k = 1; : : : ; n 2 N: j=1

They assumed that the integrals

ck; :=

Z

1

tk d

(t)

1

are …nite, k = 1; : : : ; n: They noticed the inequalities j

r;

(f ; x)

f (x)j

j

r;

(f1 ; x) 2

f1 (x)j + j

r;

(f2 ; x)

f2 (x)j ;

(7)

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

k

r;

(f ; x)

f (x)k1;x

(f ; x)

f (x)kp;x

k

(f1 ; x)

r;

298

f1 (x)k1;x + k

r;

(f2 ; x)

f2 (x)k1;x ; (8)

and k

r;

k

r;

(f1 ; x)

f1 (x)kp;x +k

r;

(f2 ; x)

f2 (x)kp;x ; p

1: (9)

Furthermore, they stated that the equality (k)

(k)

f (k) (x) = f1 (x) + if2 (x);

(10)

holds for all k = 1; : : : ; n: The authors stated Theorem 1 ([5] p:365) Let f : R ! C such that f = f1 + if2 : Here j = 1; 2: Let fj 2 C n (R) ; n 2 Z+ : Assume that Z

1

jtjn 1 +

1

r

jtj

d

(t) < 1:

(n)

Suppose also that ! r (fj ; h) < 1; 8h > 0: Then r;

1 n!

(f ; x)

Z

1 1

f (x)

jtjn 1 +

jtj

n X f (k) (x)

k!

k=1 r

d

(t)

k ck;

(11) 1;x

(n) ! r (f1 ;

(n)

) + ! r (f2 ; ) :

When n = 0 the sum in L.H.S. (11 ) collapses. They gave their results for the n = 0 case as follows. Corollary 2 ([5] p:366)Let f : R ! C : f = f1 + if2 : Here j = 1; 2: Let fj 2 C (R) : Assume Z 1 r jtj 1+ d (t) < 1: 1

Suppose also ! r (fj ; h) < 1; 8h > 0: Then k

r;

(f )

f k1

Z

1

1+

jtj

r

d

1

Next they stated

3

(t) (! r (f1 ; ) + ! r (f2 ; )) :

(12)

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

299

Theorem 3 ([5] p:366) Let g 2 C n 1 (R), such that g (n) exists, n; r 2 N. e thermore assume that for each x 2 R the function g (j) (x + jt) 2 L1 (R; a function of t; for all e j = 0; 1; : : : ; n 1; j = 1; : : : ; r: Suppose that there e 0, j = 1; : : : ; n;j = 1; : : : ; r; with ej;j 2 L1 (R; ) such that for e j;j x 2 R we have e jg (j) (x + jt)j e j;j (t);

Fur) as exist each (13)

e for almost all t 2 R, all e j = 1; : : : ; n; j = 1; 2; : : : ; r: Then g (j) (x+jt) de…nes a integrable function with respect to t for each x 2 R, all e j = 1; : : : ; n; j = 1; : : : ; r, and e (e j) ( r; (g; x)) = r; g (j) ; x ; (14)

for all x 2 R, all e j = 1; : : : ; n:

They presented the following approximation result

Theorem 4 ([5] p:366)Let f : R ! C; such that f = f1 +if2 : Here j = 1; 2: Let (n+e j) fj 2 C n+ (R) ; n; 2 Z+ ; and ! r (fj ; h) < 1; 8h > 0; for e j = 0; 1; : : : ; : Assume Z 1 r jtj jtjn 1 + d (t) < 1: 1

We consider the assumptions of Theorem 3 valid regarding f1 ; f2 for n = : Then ( 1 n!

r;

Z

e j (f ; x))( ) 1 1

jtjn 1 +

e

f (j ) (x) jtj

e n X f (k+j ) (x)

k!

k=1 r

d

k ck; 1;x

(n+e j) ! r (f1 ;

(t)

(15) (n+e j)

) + ! r (f2

; ) ;

for all e j = 0; 1; : : : ; : When n = 0 the sum in L.H.S.(15 ) collapses.

They started to present their Lp results with the following theorem

Theorem 5 ([5] p:367)Let f : R ! C such that f = f1 + if2 : Here j = 1; 2: Let (n) fj 2 C n (R) with fj 2 Lp (R) ; n 2 N; p; q > 1 : p1 + 1q = 1: Suppose that Z

1 1

"

1+

jtj

#

rp+1

np 1

1 jtj

4

d

(t) < 1;

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

300

and ck; 2 R; k = 1; : : : ; n: Then r;

(f ; x)

n X f (k) (x)

f (x)

k ck;

k!

k=1

"Z

1 ((n 1 p

1

1

1)!)(q(n

1) + 1) q (rp + 1) p

(n)

(16) p;x 1

1+

rp+1

jtj

!

np 1

1 jtj

1

d

(n)

! r (f1 ; )p + ! r (f2 ; )p :

For the case of p = 1; they obtained

Theorem 6 ([5] p:368)Let f : R ! C such that f = f1 + if2 : Here j = 1; 2: (n) Let fj 2 C n (R) with fj 2 L1 (R) ; n 2 N: Suppose that Z

1 1

"

1+

#

r+1

jtj

n 1

1 jtj

d

(t) < 1;

and ck; 2 R; k = 1; : : : ; n: Then r;

(n

(f ; x)

n X f (k) (x)

f (x)

1 1)! (r + 1)

"Z

k ck;

k!

k=1 1

1+

r+1

jtj

!

n 1

1 jtj

1

(n)

(17) 1;x

d

#

(t)

(n)

! r (f1 ; )1 + ! r (f2 ; )1 :

For n = 0; they showed Proposition 7 ([5] p:368) Let f : R ! C such that f = f1 + if2 ; where f1 ; f2 2 (C (R) \ Lp (R)) ; p; q > 1 : p1 + 1q = 1: Suppose that Z

1

1+

jtj

rp

d

1

(t) < 1:

Then k

r;

(f )

f kp

Z

1

1+

jtj

1 p

rp

d

(t)

(! r (f1 ; )p + ! r (f2 ; )p ) : (18)

1

Finally, they gave the case of n = 0; p = 1 as 5

# p1

(t)

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

301

Proposition 8 ([5] p:368)Let f : R ! C such that f = f1 + if2 ; where f1 ; f2 2 (C (R) \ L1 (R)) : Suppose Z 1 r jtj d (t) < 1: 1+ 1

Then k

r;

(f )

Z

f k1

1

r

jtj

1+

d

(t) (! r (f1 ; )1 + ! r (f2 ; )1 ) :

(19)

1

Next, the authors demonstrated their simultenous Lp approximation results. They started with Theorem 9 ([5] p:369)Let f : R ! C; such that f = f1 + if2 : Here j = 1; 2: (n+e j) Let fj 2 C n+ (R) ; n 2 N; 2 Z+ ; with fj 2 Lp (R) ; e j = 0; 1; : : : ; : Let rp+1 R 1 np 1 p; q > 1 : p1 + 1q = 1: Suppose that 1 1 + jtj 1 jtj d (t) < 1; and ck; 2 R; k = 1; : : : ; n: We consider the assumptions of Theorem 3 valid regarding f1 ; f2 for n = : Then (

r;

(e j)

(f ; x))

f

(e j)

n e X f (k+j) (x)

(x)

k=1

1 (q(n

(n+e j)

1) + 1) (rp + 1) "Z rp+1 1 jtj 1+

!

np 1

d

# p1

(t)

1 p

(20)

1)!)

(n+e j)

; )p + ! r (f2

1 jtj

1

((n

p;x

! r (f1

1 p

1 q

1

k ck;

k!

; )p

;

for all e j = 0; 1; : : : ; : They had

(e j)

(e j)

Proposition 10 ([5] p:369)Let f : R ! C; such that f = f1 +if2 : Let f1 ; f2 (C (R) \ Lp (R)) ; e j = 0; 1; : : : ; 2 Z+ ; p; q > 1 : p1 + 1q = 1: Suppose that Z

1

1+

2

rp

jtj

d

1

(t) < 1:

We consider the assumptions of Theorem 3 valid regarding f1 ; f2 for n = : Then ( Z e j = 0; 1; : : : ; :

r; 1

e j (f ))( )

1+

jtj

e

f (j )

(21)

p 1 p

rp

d

(t)

1

6

(e j)

(e j)

! r (f1 ; )p + ! r (f2 ; )p ;

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

302

Next they gave the related L1 result as Theorem 11 ([5] p:369) Let f : R ! C; such that f = f1 + if2 ; and j = 1; 2: (n+e j) Let f1 ; f2 2 C n+ (R) ; with fj 2 L1 (R) ; n 2 N; e j = 0; 1; : : : ; 2 Z+ : Suppose that ! Z 1 r+1 jtj 1+ 1 jtjn 1 d (t) < 1; 1

and ck; 2 R; k = 1; : : : ; n: We consider the assumptions of Theorem 3 valid regarding f1 ; f2 for n = : Then (

r;

(n

e j (f ; x))( )

1 1)! (r + 1) (n+e j)

! r (f1

e n X f (k+j ) (x)

e

f (j ) (x) "Z

k!

k=1

1

1+

jtj

!

r+1

(n+e j)

; )1 + ! r (f2

1;x n 1

1 jtj

1

(22)

k ck;

d

#

(t)

; )1 ;

for all e j = 0; 1; : : : ; :

Their last simultaneous approximation result follows

Proposition 12 ([5] p:370)Let f : R ! C; such that f = f1 + if2 : Here (j) (j) f1 ; f2 2 (C (R) \ L1 (R)) ; j = 0; 1; : : : ; 2 Z+ : Suppose Z

1

1+

r

jtj

d

1

(t) < 1:

We consider the assumptions of Theorem 3 valid regarding f1 ; f2 for n = : Then Z 1 r jtj (j) (j) (j) (j) ( r; (f )) f 1+ d (t) ! r (f1 ; )1 + ! r (f2 ; )1 ; 1

1

(23)

for all j = 0; 1; : : : ; : Next in [5], Chapter 19, the authors de…ned the generalized complex fractional integral operators as follows: Let > 0; x; x0 2 R; f : R ! C Borel measurable, such that f = f1 + if2 : Sup(m) (m) pose f1 ; f2 2 C m (R) ; with f1 < 1; f2 < 1: Let probability 1

1

7

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

Borel measure on R; 8 > 0: Then 0 1 Z 1 X r @ A : = r; (f ; x) j f (x + jt) d 1

=

Z

=

1

1 r;

where j

for r 2 N and

=

j=0

0 r X @

j f1 (x

j=0

(f1 ; x) + i (

> 0:

r;

1

+ jt)A d

(t)

(24)

(t) + i

Z

1 1

(f2 ; x);

r j

r ( 1) j ; Pr j r 1 i=1 ( 1)

i r i

303

i

0 r X @

j f2 (x

j=0

j = 1; : : : ; r; j = 0:

;

1

+ jt)A d

(25)

Additionally, they denoted k

=

r X

jj

k

; k = 1; : : : ; m

1;

(26)

j=1

where m = d e ; d e is the ceiling of a number : They supposed here that r; (f1 ; x) ; r; (f2 ; x)R2 R; 8x 2 R: 1 The authors also assumed existence of ck; := 1 tk d (t); k = 1; : : : ; m 1; R1 +k and the existence of 1 jtj d (t); k = 0; 1; : : : ; r: Finally, they gave their fractional result as

Theorem 13 ([5] p:371)It holds r;

"

(f; )

f()

m X1 k=1

r X

k=0

(r

r! k)! ( + k + 1)

sup max ! r Dx f1 ; x2R

+ sup max ! r Dx f2 ; x2R

f (k) ( ) k! Z 1 k

1

k ck;

(27) 1

jtj

+k

d

#

(t)

; ! r (D x f1 ; ) ; ! r (D x f2 ; )

:

If m = 1 the sum disappears in L.H.S.(27): Let j = 1; 2: Above Dx0 fj is the right Caputo fractional derivative of order > 0 is given by m Z x0 ( 1) m 1 (m) Dx0 fj (x) := ( x) fj ( )d ; (28) (m ) x 8

(t)

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

8x

304

x0 2 R …xed : They supposed Dx0 f (x) = 0; 8x > x0 : Also D

x0 fj

is the left Caputo fractional derivative of order

> 0 is given

by D 8x

x0 fj (x) :=

1 (m

)

Z

x

m

(x

1

t)

x0

R1 x0 2 R …xed, where ( ) = 0 e t t 1 dt; They assumed D x0 fj (x) = 0; for x < x0 :

(m)

fj

(t)dt;

> 0:

On the other hand, in [1], the authors de…ned important special cases of operators for discrete probability measures as follows : Let f 2 C n (R), n 2 Z+ ; 0 < i) When

(29)

r;

1, x 2 R. j j

e 1 P

( )=

j j

e

,

(30)

= 1

they de…ned the generalized discrete Picard operators as ! 1 r P P j f (x + j ) e Pr; (f ; x) :=

= 1

j=0

1 P

j j

e

j j

.

(31)

= 1

ii) When

2

e 1 P

( )=

2

,

(32)

e

= 1

they de…ned the generalized discrete Gauss-Weierstrass operators as ! 2 1 r P P j f (x + j ) e Wr; (f ; x) :=

= 1

j=0

1 P

.

2

(33)

e

= 1

iii) Let

2 N, and

>

1

: When 2

( )=

1 P

= 1

9

+

2

2

+

; 2

(34)

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

305

they de…ned the generalized discrete Poisson-Cauchy operators as ! 1 r P P 2 + 2 j f (x + j ) Qr; (f ; x) :=

= 1

j=0

1 P

2

.

(35)

2

+

= 1

They observed that for c constant they have Pr; (c; x) = Wr; (c; x) = Qr; (c; x) = c.

(36)

They assumed that the operators Pr; (f ; x), Wr; (f ; x), and Qr; (f ; x) 2 R; for x 2 R: This is the case when kf k1;R < 1: iv) When ( ) :=

= 1

Pr; (f ; x) :=

;P (

) :=

j j

1 ; 1+2 e they de…ned the generalized discrete non-unitary Picard operators as ! j j 1 r P P j f (x + j ) e

Here

;P (

e

j=0

1+2 e

:

1

(37)

(38)

) has mass m

;P

:=

1 P

j j

e

= 1

1+2 e

1

:

(39)

They observed that ;P (

m

)

j j

e 1 P

=

;P

e

j j

;

(40)

= 1

which is the probability measure (30) de…ning the operators Pr; : v) When 2

( ) :=

with erf(x) =

p2

Rx

e

t2

;W (

e

) := p

1

erf

;

p1

(41)

+1

dt; erf(1) = 1; they de…ned the generalized discrete

0

non-unitary Gauss-Weierstrass operators as

Wr; (f ; x) :=

1 P

= 1

p

r P

!

j f (x

+j ) e

erf

p1

j=0

1

10

+1

2

:

(42)

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

Here

;W (

306

) has mass

m

;W

1 P

2

e

= 1

:= p

1

:

p1

erf

(43)

+1

They observed that 2

;W (

m

)

e 1 P

=

;W

2

;

(44)

e

= 1

which is the probability measure (32) de…ning the operators Wr; : The authors observed that Pr; (f ; x), Wr; (f ; x) 2 R; for x 2 R. In [1], for k = 1; :::; n, the authors de…ned the ratios of sums 1 P

k

= 1 1 P

ck; :=

j j

e

j j

e

;

(45)

;

(46)

= 1

1 P

2

k

= 1 1 P

pk; :=

e 2

e

= 1

and for

2 N;

>

n+r+1 ; 2

they introduced

qk; :=

1 P

k

2

= 1 1 P

2

+

: 2

+

(47)

2

= 1

Furthermore, they proved that these ratios of sums ck; ; pk; , and qk; are …nite for all 2 (0; 1]. In [1], the authors also proved

m

;P

=

;W

= 1+

j j

e

= 1

1+2 e

and

m

1 X

p

1 X

1

! 1 as

(48)

2

e

= 1

1

! 0+

erf 11

p1

! 1 as

! 0+ :

(49)

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

307

The authors introduced also k

:=

r X

jj

k

k = 1; : : : ; n 2 N:

;

j=1

(50)

Additionally, in [1], the authors de…ned the following error quantities: E0;P (f; x)

:

= Pr; (f ; x) = 1

=

E0;W (f; x)

:

f (x)

r P

1 P

j f (x

r P

= 1

=

f (x)

1

(52)

!

2

j f (x

+j ) e

erf

p1

j=0

p

f (x);

1

1+2 e

1 P

j j

+j ) e

j=0

= Wr; (f ; x)

(51)

!

f (x):

+1

Furthermore, they introduced the errors (n 2 N):

En;P (f; x) := Pr; (f ; x)

n X

f (x)

k=1

f (k) (x) k!

k

1 X

k

j j

e

= 1

(53)

1

1+2 e

and

En;W (f; x) := Wr; (f ; x)

f (x)

n X

k=1

f (k) (x) k!

kp

1 X

k

2

e

= 1

1

: (54)

p1

erf

+1

Next, they obtained the inequalities jE0;P (f; x)j

m

jE0;W (f; x)j

m

;P

Pr; (f ; x)

f (x) + jf (x)j jm

;W

Wr; (f ; x)

f (x) + jf (x)j jm

;P

1j ;

;W

(55)

1j ;

(56)

and jEn;P (f; x)j m

;P

Pr; (f ; x)

(57) f (x)

n X

k=1

f (k) (x) k!

12

k ck;

+ jf (x)j jm

;P

1j ;

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

308

with jEn;W (f; x)j m

;W

(58)

Wr; (f ; x)

f (x)

n X f (k) (x)

k=1

k!

k pk;

+ jf (x)j jm

;W

1j :

In [1], they …rst gave the following simultaneous approximation results for unitary operators. They showed Theorem 14 Let f 2 C n (R) with f (n) 2 Cu (R) ( uniformly continuous functions on R): i) For n 2 N; Pr; (f ; x)

f (x)

n X f (k) (x)

k=1

k!

0 P 1 j jn 1 + ! r (f (n) ; ) B = 1 B 1 @ P n! e

j j

k ck; r

(59) j j

e

Wr; (f ; x)

f (x)

n X f (k) (x)

k=1

k!

0 P 1 j jn 1 + (n) B ! r (f ; ) B = 1 1 @ P n! e

j j

k pk;

(60) 2

r

e

Pr; (f ; x)

f (x)

1;x

0 P 1 1+ B = 1 ! r (f; ) B 1 @ P

1;x

1

C C: A

2

= 1

ii) For n = 0;

1

C C; A

j j

= 1

and

1;x

r

j j

e

e

j j

Wr; (f ; x)

f (x)

1;x

0 P 1 1+ B = 1 B ! r (f; ) @ 1 P

= 1

j j

e 2

e

2

r

(61)

1

(62)

C C; A

j j

= 1

and

1

C C: A

In the above inequalities (59) - (62) ; the ratios of sums in their right hand sides (R.H.S.) are uniformly bounded with respect to 2 (0; 1] : 13

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

309

In [1], they had also Theorem 15 Let f 2 C n (R) with f (n) 2 Cu (R); n 2 N, and i) For n 2 N; Qr; (f ; x)

f (x)

n X f (k) (x)

k!

k=1

0 P 1 j jn 1 + (n) ! r (f ; ) B B = 1 1 @ P n!

j j 2

>

k qk; 1;x

r

+

(63)

2

1

2

+

C C: A

2

= 1

ii) For n = 0;

Qr; (f ; x)

f (x)

1;x

0 P 1 1+ B = 1 B ! r (f; ) @ 1 P

n+r+1 : 2

j j 2

r

2

+

+

1

2

C C : (64) A

2

= 1

In the above inequalities (63) - (64) ; the ratios of sums in their R.H.S. are uniformly bounded with respect to 2 (0; 1] : Next, they stated their results in [1] for the errors E0;P ; E0;W ; En;P ; and En;W : They had Corollary 16 Let f 2 Cu (R): Then i) jE0;P (f; x)j ii)

jE0;W (f; x)j

0

0 P 1 ! (f; j j)e B = 1 r B 1 @ 1+2 e

B B @p

1 P

= 1

1

j j

2

p1

;P

1

;W

C C + jf (x)j jm A

! r (f; j j)e erf

1

+1

C C + jf (x)j jm A

In [1], for En;P and En;W , the authors presented

14

1j ;

(65)

1j :

(66)

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

310

Theorem 17 Let f 2 C n (R) with f (n) 2 Cu (R), n 2 N;and kf k1;R < 1: Then i) kEn;P (f; x)k1;x 0 P 1 j jn 1 + (n) ! r (f ; ) B = 1 B @ n! 1+2 e

(67) r

j j

1

j j

e

C C + kf k 1;R jm A

1

;P

1j ;

ii) kEn;W (f; x)k1;x 0 P 1 ! r (f (n) ; ) B B = @ p n!

1

(68) j jn 1 + 1

2

r

j j

e

p1

erf

+1

1

C C + kf k 1;R jm A

;W

1j :

In the above inequalities (67) - (68) ; the ratios of sums in their R.H.S. are uniformly bounded with respect to 2 (0; 1] : In [2], the authors represented simultaneous Lp approximation results. They started with Theorem 18 i) Let f 2 C n (R); with f (n) 2 Lp (R) ; n 2 N; p; q > 1 : p1 + 1q = 1, and rest as above in this section:Then Pr; (f ; x)

n X f (k) (x)

f (x)

k!

k=1

1 ((n

1 q

1)!)(q(n

where

Mp; :=

k ck;

1) + 1) (rp + 1)

1 X

1+

= 1

1 p

Mp;

rp+1

j j

(69) p

np 1

1 j j 1 X

1 p

1 p

e

! r (f (n) ; )p

j j

(70) e

j j

= 1

which is uniformly bounded for all 2 (0; 1] : Additionally, as ! 0+ we obtain that R:H:S: of (69) goes to zero. ii) When p = 1; let f 2 C n (R), f (n) 2 L1 (R); and n 2 N f1g: Then Pr; (f ; x)

f (x)

n X f (k) (x)

k=1

(n

k!

1 M ! r (f (n) ; )1 1)! (r + 1) 1; 15

k ck;

(71) 1

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

311

holds where M1; is de…ned as in (70). Hence, as ! 0+ ; we obtain that R:H:S: of (71) goes to zero. iii) When n = 0, let f 2 (C (R) \ Lp (R)) ; p; q > 1 such that p1 + 1q = 1 and the rest as above in this section. Then Pr; (f ; x)

f (x)

where

Mp; :=

1=p

Mp;

p

1 X

1+

= 1

1 X

rp

j j

e

! r (f; )p

(72)

j j

(73) e

j j

= 1

which is uniformly bounded for all 2 (0; 1] : Hence, as ! 0+ ; we obtain that Pr; ! unit operator I in the Lp norm for p > 1. iv) When n = 0 and p = 1; let f 2 (C (R) \ L1 (R)) and the rest as above in this section. Then the inequality Pr; (f ; x)

f (x)

M1; ! r (f; )1

1

(74)

holds where M1; is de…ned as in (73). Furthermore, we get Pr; ! I in the L1 norm as ! 0+ : Next, the authors presented their quantitative results for the Gauss-Weierstrass operators, see [2]. They started with Theorem 19 i) Let f 2 C n (R); with f (n) 2 Lp (R) ; n 2 N; p; q > 1 : p1 + 1q = 1, and the rest as above in this section. Then Wr; (f ; x)

f (x)

n X f (k) (x)

k!

k=1

1 ((n

1)!)(q(n

where

Np; :=

1 X

= 1

1 q

1) + 1) (rp + 1)

1+

k pk;

1 p

rp+1

j j

(75) p

Np;

np 1

1 j j 1 X

1 p

1 p

! r (f (n) ; )p

2

e (76)

2

e

= 1

which is uniformly bounded for all 2 (0; 1] : Additionally, as ! 0+ we obtain that R:H:S: of (75) goes to zero.

16

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

312

ii) For p = 1, let f 2 C n (R), f (n) 2 L1 (R); and n 2 N Wr; (f ; x)

n X f (k) (x)

f (x)

k!

k=1

(n

f1g: Then k pk;

(77) 1

1 N ! r (f (n) ; )1 1)! (r + 1) 1;

holds where N1; is de…ned as in (76). Hence, as ! 0+ ; we obtain that R:H:S: of (77) goes to zero. iii) For n = 0, let f 2 (C (R) \ Lp (R)) ; p; q > 1 such that p1 + 1q = 1 and the rest as above in this section. Then Wr; (f ; x)

f (x)

where

1 X

1+

= 1

Np; :=

1=p

Np;

p

j j

! r (f; )p

rp

(78)

2

e (79)

1 X

2

e

= 1

which is uniformly bounded for all 2 (0; 1] : Hence, as ! 0+ ; we obtain that Wr; ! unit operator I in the Lp norm for p > 1. iv) For n = 0 and p = 1, let f 2 (C (R) \ L1 (R)) and the rest as above in this section. Then the inequality Wr; (f ; x)

f (x)

N1; ! r (f; )1

1

(80)

holds where N1; is de…ned as in (79). Furthermore, we get Wr; ! I in the L1 norm as ! 0+ : For the Possion-Cauchy operators, in [2], the authors showed Theorem 20 i) Let f 2 C n (R); with f (n) 2 Lp (R) ; n 2 N; p; q > 1 : p1 + 1q = 1, >

p(r+n)+1 ; 2

2 N; and the rest as above in this section. Then

Qr; (f ; x)

n X f (k) (x)

f (x)

k=1

1 ((n where

Sp; :=

1)!)(q(n 1 X

= 1

k!

1 q

1) + 1) (rp + 1)

1+

j j

k qk;

(81) p

Sp;

1 p

rp+1

np 1

1 j j

1 p

1 p

2

+

! r (f (n) ; )p

2

(82)

1 X

2

= 1

17

+

2

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

313

is uniformly bounded for all 2 (0; 1] : Additionally, as ! 0+ ; we obtain that R:H:S: of (81) goes to zero. ii) When p = 1, let f 2 C n (R), f (n) 2 L1 (R); > r+n+1 ; and n 2 N 2 Then Qr; (f ; x)

n X f (k) (x)

f (x)

k!

k=1

(n

f1g: (83)

k qk; 1

1 S ! r (f (n) ; )1 1)! (r + 1) 1;

holds where S1; is de…ned as in (82). Hence, as ! 0+ ; we obtain that R:H:S: of (83) goes to zero. iii)When n = 0, let f 2 (C (R) \ Lp (R)) ; p; q > 1 such that p1 + 1q = 1; >

p(r+2)+1 ; 2

and the rest as above in this section. Then Qr; (f ; x)

where

Sp; :=

1 X

f (x)

= 1

rp

j j

1+

1=p

Sp;

p

2

! r (f; )p

+

(84)

2

(85)

1 X

2

+

2

= 1

which is uniformly bounded for all 2 (0; 1] : Hence, as ! 0+ ; we obtain that Qr; ! unit operator I in the Lp norm for p > 1. and the rest as iv) When n = 0 and p = 1, let f 2 (C (R) \ L1 (R)) ; > r+3 2 above in this section. The inequality Qr; (f ; x)

f (x)

S1; ! r (f; )1

1

(86)

holds where S1; is de…ned as in (85). Furthermore, we get Qr; ! I in the L1 norm as ! 0+ : Next in [2], they stated their results for the errors E0;P ; E0;W ; En;P ; and En;W as follows Theorem 21 i) Let p; q > 1 such that

18

1 p

+

1 q

= 1, n 2 N such that np 6= 1,

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

314

f 2 Lp (R); and the rest as above in this section. Then kEn;P (f; x)kp 1 p

(87) 1 P

! r (f (n) ; )p

1 q

j j

e

= 1

((n 2 6 6 6 6 6 6 4

1)!)(q(n 1 X

1

1

1) + 1) q (rp + 1) p j j

1+

rp+1

np 1

1 j j

= 1

1+2 e

+ kf (x)kp jm

e

j j

1

1j

;P

! p1 3 7 7 7 7 7 7 5

holds. Additionally, as ! 0+ ; we obtain that R:H:S: of (87) goes to zero. ii) When p = 1; let f 2 C n (R); f 2 L1 (R), f (n) 2 L1 (R); and n 2 N f1g: Then kEn;P (f; x)k1

(88)

(n)

! r (f ; )1 (n 1)! (r + 1) 2 1 X r+1 n 1+ j j 1 j j 6 6 = 1 6 1 6 1+2 e 4

+ kf (x)k1 jm

;P

1

e

1j

j j

3 7 7 7 7 5

holds. Additionally, as ! 0+ ; we obtain that R:H:S: of (88) goes to zero. iii) When n = 0; let p; q > 1 such that p1 + 1q = 1; f 2 Lp (R), and the rest as above in this section. Then ! q1 1 X j j kE0;P (f; x)kp ! r (f; )p e (89) 2 6 6 6 6 6 4

= 1

1 X

1+

= 1

1+2 e

+ kf (x)kp jm holds. Hence, as

j j

;P

1j

rp

e 1

j j

! 1=p 3 7 7 7 7 7 5

! 0+ ; we obtain that R:H:S: of (89) goes to zero. 19

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

iv) When n = 0 and p = 1; the inequality 0 1 X 1+ j j B B = 1 B kE0;P (f; x)k1 B 1+2 e @ + kf (x)k1 jm

holds. Hence, as

r

j j

e 1

1j

;P

315

1

C C C ! r (f; )1 C A

(90)

! 0+ ; we obtain that R:H:S: of (90) goes to zero.

Next in [2], the authors gave quantitative results for En;W (f; x) Theorem 22 i) Let p; q > 1 such that p1 + 1q = 1, n 2 N such that np 6= 1, f 2 Lp (R); and the rest as above in this section. Then kEn;W (f; x)kp 1 p

(91) 1 X

! r (f (n) ; )p

2

e

= 1

((n 2 6 6 6 6 6 6 4

1)!)(q(n 1 X

+ kf (x)kp jm

1

1

1) + 1) q (rp + 1) p 1+

= 1

! q1

j j

p

1

;W

1j

rp+1

np 1

1 j j erf

p1

2

e

+1

! p1 3 7 7 7 7 7 7 5

holds. Additionally, as ! 0+ ; we obtain that R:H:S: of (91) goes to zero. ii) For p = 1; let f 2 C n (R); f 2 L1 (R), f (n) 2 L1 (R); and n 2 N f1g: Then kEn;W (f; x)k1

(92)

(n)

! r (f ; )1 (n 1)! (r + 1) 2 1 X 1+ j 6 6 = 1 6 p 6 4 1

+ kf (x)k1 jm

;W

j

r+1

n 1

1 j j erf

1j

p1

+1

2

e

3 7 7 7 7 5

holds. Additionally, as ! 0+ ; we obtain that R:H:S: of (92) goes to zero. iii) For n = 0; let p; q > 1 such that p1 + 1q = 1; f 2 Lp (R), and the rest as above 20

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

316

in this section. Then kE0;W (f; x)kp

1 X

! r (f; )p

2

e

= 1

2

1 X

6 6 6 = 6 6 p 6 4

1+

! q1

rp

j j

1

1

p1

erf

+ kf (x)kp jm

1j

;W

(93)

! p1 3 7 e 7 7 7 7 +1 7 5 2

holds. Hence, as ! 0+ ; we obtain that R:H:S: of (93) goes to zero. iv) For n = 0 and p = 1; the inequality 0

1 X

B B = B Bp @

kE0;W (f; x)k1

1+

2

e

1

p1

erf

1j

;W

1

C C C ! r (f; )1 C + 1A

1

+ kf (x)k1 jm

holds. Hence, as

r

j j

(94)

! 0+ ; we obtain that R:H:S: of (94) goes to zero.

Furthermore in [3], the authors gave their fractional approximation results as follows Theorem 23 Let f 2 C m (R) ; m = d e ; 1:Then i) Pr; (f ) "

f

m X1

> 0; with f (m)

f (k)

r X

k=0

(r

k ck;

r! k)! ( + k + 1)

sup max ! r Dx f; where S 1;k :=

1 P

j j

= 1 1 P

= 1

21

1 k

S

1 ;k

#

; ! r (D x f; )

x2R

+k

e

e j j

< 1; 0
0; with f (m)

1

< 1; 0


1

(116)

p

f1 (x)

p

+ Wr; (f2 ; x)

f2 (x)

:

p

: When 2

( )=

+

2

2

+

1 P

,

(117)

2

= 1

we de…ne the complex generalized discrete Poisson-Cauchy operators as ! 1 r P P 2 + 2 j f (x + j ) Qr; (f ; x) :=

= 1

j=0

1 P

.

2

+

(118)

2

= 1

Here we observe that

Qr; (f ; x) = Qr; (f1 ; x) + iQr; (f2 ; x) ; Qr; (f ; x)

f (x)

Qr; (f1 ; x) Qr; (f ; x)

and for p

(119) (120)

f1 (x) + Qr; (f2 ; x)

f (x)

(121)

1

Qr; (f1 ; x)

f1 (x)

Qr; (f ; x)

f (x)

f2 (x) ;

1

+ Qr; (f2 ; x)

f2 (x)

1

;

1;

Qr; (f1 ; x)

(122)

p

f1 (x)

p

25

+ Qr; (f2 ; x)

f2 (x)

p

:

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

321

Observe that for c 2 C constant we have Pr; (c; x) = Wr; (c; x) = Qr; (c; x) = c.

(123)

We assume that for x 2 R; the operators Pr; (fj ; x), Wr; (fj ; x), and Qr; (fj ; x) 2 R where j = 1; 2: This is the case when kfj k1;R < 1 for j = 1; 2: iv) When ( ) :=

;P (

j j

e

) :=

1+2 e

1

;

(124)

we de…ne the complex generalized discrete non-unitary Picard operators as ! j j 1 r P P j f (x + j ) e = 1

Pr; (f ; x) :=

Here

;P (

j=0

1+2 e

:

1

(125)

) has mass m

;P

1 P

= 1

:=

m

1

1+2 e

We observe that ;P (

j j

e

)

=

(126)

j j

e 1 P

;P

:

j j

e

;

(127)

= 1

which is the probability measure (105) de…ning the operators Pr; : v) When 2

( ) :=

;W (

e

) := p

1

erf

;

p1

(128)

+1

we de…ne the complex generalized discrete non-unitary Gauss-Weierstrass operators as ! 2 1 r P P j f (x + j ) e Wr; (f ; x) :=

Here

;W (

= 1

j=0

p

1

p1

erf

:

(129)

+1

) has mass

m

;W

:= p

1 P

2

e

= 1

1

26

erf

p1

: +1

(130)

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

We observe that

322

2

;W (

m

)

=

;W

e 1 P

2

;

(131)

e

= 1

which is the probability measure (111) de…ning the operators Wr; : We state our …rst result as follows Theorem 25 Let f : R ! C such that f = f1 + if2 : Here fj 2 C n (R) ; (n) fj 2 Cu (R) ; j = 1; 2; and n 2 N: Then i) Pr; (f ; x)

f (x)

n X f (k) (x)

k ck; k! 1;x 0 P 1 ! j jn 1 + (n) (n) ! r (f1 ; ) + ! r (f2 ; ) B = 1 B 1 @ P n! e

(132)

k=1

r

j j

e

j j

C C; A

j j

= 1

ii)

Wr; (f ; x)

f (x)

n X f (k) (x)

k pk; k! 1;x 0 P 1 ! j jn 1 + (n) (n) ! r (f1 ; ) + ! r (f2 ; ) B = 1 B 1 @ P n! e

1

(133)

k=1

2

r

j j

e

C C; A

2

= 1

iii) for

2 N and

>

n+r+1 ; 2

n X f (k) (x)

k qk; k! 1;x 0 P 1 ! j jn 1 + (n) (n) ! r (f1 ; ) + ! r (f2 ; ) B = 1 B 1 @ P n!

Qr; (f ; x)

1

f (x)

(134)

k=1

= 1

Proof. By Theorems 1, 14, 15.

For the case of n = 0; we give the following result 27

j j 2

r

+

2

2

+

2

1

C C: A

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

323

Corollary 26 Let f : R ! C such that f = f1 + if2 : Here fj 2 Cu (R) for j = 1; 2: Then i) Pr; (f ; x)

f (x)

(135)

1;x

0 P 1 1+ B = 1 B (! r (f1 ; ) + ! r (f2 ; )) @ 1 P

r

j j

e

e j j

j

j 1

C C; A

= 1

ii)

Wr; (f ; x)

f (x)

(136)

1;x

0 P 1 1+ B = 1 B (! r (f1 ; ) + ! r (f2 ; )) @ 1 P

2

r

j j

e

C C; A

2

e

= 1

iii) for

2 N and

>

r+1 2 ;

Qr; (f ; x)

f (x)

1

(137)

1;x

0 P 1 1+ B = 1 B (! r (f1 ; ) + ! r (f2 ; )) @ 1 P

j j

= 1

Proof. By Corollary 2, and the Theorems 14, 15.

2

r

2

+

2

+

2

1

C C: A

In [4], the authors gave Theorem 27 Let g 2 C n 1 (R), such that g (n) exists, n; r 2 N; 0 < 1: (~ j) Additionally, suppose that for each x 2 R the function g (x + j ) 2 L1 (R; ) as a function of ; for all ~j = 0; 1; : : : ; n 1; j = 1; : : : ; r: Assume that there exist ~j;j 0, ~j = 1; : : : ; n; j = 1; : : : ; r; with ~j;j 2 L1 (R; ) such that for each x 2 R we have jg (i) (x + j )j (138) ~ j;j ( ); for almost all 2 R, all i = 1; : : : ; n; j = 1; 2; : : : ; r: Then, g (i) (x + j ) de…nes a integrable function with respect to for each x 2 R, all ~j = 1; : : : ; n; j = 1; : : : ; r. i) When ( )=

e 1 P

j j

= 1

28

e

j j

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

324

we get Pr; (f ; x)

(~ j)

~

f (j) ; x ;

= Pr;

(139)

for all x 2 R, and for all ~j = 1; : : : ; n. ii) When 2

e 1 P

( )=

2

e

= 1

we have Wr; (f ; x)

(~ j)

~

f (j) ; x ;

= Wr;

(140)

for all x 2 R, and for all ~j = 1; : : : ; n. iii) Let

2 N, and

>

1

: When 2

( )=

1 P

+

2

2

+

, 2

= 1

we obtain Qr; (f ; x)

(~ j)

= Qr;

~

f (j) ; x ;

(141)

for all x 2 R, and for all ~j = 1; : : : ; n. Now, we present following simultaneous results for the operators Pr; ; Wr; ; and Qr; : Theorem 28 Let f : R ! C; such that f = f1 + if2 and 0 < 1: Here n+e j) n+ + ( e fj 2 C ; n 2 N; 2 Z ; j = 1; 2; and f 2 Cu (R) where j = 0; 1; : : : ; : We consider the assumptions of the Theorem 27 valid for n = there. Then i) Pr; (f ; x) 0 @

!r

(e j)

(n+e j) f1 ;

e

f (j) (x)

+ !r n!

n e X f (j+k) (x)

k=1 (n+e j) f2 ;

k ck; k! 1;x 0 P 1 1 n j j 1+ B AB = 1 1 @ P e = 1

29

(142) j j j j

r

e

j j

1

C C; A

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

325

ii) Wr; (f ; x) 0 @

!r

(e j)

e

f (j) (x)

(n+e j) f1 ;

n e X f (j+k) (x)

k pk; k! 1;x 0 P 1 1 n j j 1+ B = 1 AB 1 @ P e

k=1 (n+e j) f2 ;

+ !r n!

Qr; (f ; x)

@ where

!r

j j

r

2

e

(e j)

f

(n+e j) f1 ;

(e j)

(x)

n e X f (j+k) (x)

k qk; k! 1;x 0 P 1 1 n j j 1+ B AB = 1 1 @ P

k=1 (n+e j) f2 ;

+ !r n!

1

C C; A

2

= 1

iii)

0

(143)

(144) r

j j 2

+

2

+

2

C C; A

2

= 1

>

n+r+1 ; 2

2 N:

Proof. By Theorems 25, 27. Next, we state our Lp results for the operators Pr; ; Wr; ; and Qr; : We start with Theorem 29 Let f : R ! C such that f = f1 + if2 and 0 < 1: (n) i)Here fj 2 C n (R) with fj 2 Lp (R) ; n 2 N; j = 1; 2; and p; q > 1 such that 1 1 p + q = 1: Then Pr; (f ; x)

f (x)

n X f (k) (x)

k=1 (n)

k!

k ck;

(145) p

(n)

! r (f1 ; )p + ! r (f2 ; )p

1 p

((n where

Mp; :=

1 X

1)!)(q(n

= 1

1+

1 q

1) + 1) (rp + 1) rp+1

j j

np 1

1 j j 1 X

= 1

30

1 p

!

e

Mp;

1 p

j j

(146) e

j j

1

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

326

which is uniformly bounded for all 2 (0; 1] : (n) ii) When p = 1; let fj 2 C n (R), fj 2 L1 (R); j = 1; 2; and n 2 N Pr; (f ; x)

n X f (k) (x)

f (x)

k!

k=1 (n)

(n)

! r (f1 ; )1 + ! r (f2 ; )1 (n 1)! (r + 1)

!

f1g: Then (147)

k ck; 1

M1;

holds where M1; is de…ned as in (146). iii) When n = 0, let fj 2 (C (R) \ Lp (R)) ; j = 1; 2; and p; q > 1 such that 1 1 p + q = 1. Then Pr; (f ; x)

f (x)

(! r (f1 ; )p + ! r (f2 ; )p ) Mp;

p

where

Mp; :=

1 X

1+

= 1

1 X

rp

j j

e

1=p

(148)

j j

(149) e

j j

= 1

which is uniformly bounded for all 2 (0; 1] : iv) When n = 0 and p = 1; let fj 2 (C (R) \ L1 (R)) ; j = 1; 2. Then the inequality Pr; (f ; x)

f (x)

1

(! r (f1 ; )1 + ! r (f2 ; )1 ) M1;

(150)

holds where M1; is de…ned as in (149). Proof. By Theorems 5, 6, 18, and Propositions 7, 8. For the operators Wr; ; we obtain Theorem 30 Let f : R ! C such that f = f1 + if2 and 0 < 1: (n) i) Here fj 2 C n (R) with fj 2 Lp (R) ; n 2 N; j = 1; 2; p; q > 1 such that 1 1 p + q = 1: Then Wr; (f ; x)

n X f (k) (x)

f (x)

k!

k=1 (n)

k pk;

(151) p

(n)

! r (f1 ; )p + ! r (f2 ; )p

1 p

((n where

Np; :=

1 X

1 q

1) + 1) (rp + 1)

1)!)(q(n

= 1

1+

rp+1

j j

np 1

1 j j 1 X

= 1

31

1 p

!

Np;

1 p

2

e (152)

2

e

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

327

which is uniformly bounded for all 2 (0; 1] : (n) ii)When p = 1; let fj 2 C n (R), fj 2 L1 (R); j = 1; 2; and n 2 N Wr; (f ; x)

n X f (k) (x)

f (x)

f1g: Then

k pk;

k! ! (n) (n) ! r (f1 ; )1 + ! r (f2 ; )1 N1; (n 1)! (r + 1) k=1

(153) 1

holds where N1; is de…ned as in (152). iii) When n = 0, let fj 2 (C (R) \ Lp (R)) ; j = 1; 2; and p; q > 1 such that 1 1 p + q = 1. Then Wr; (f ; x)

f (x)

(! r (f1 ; )p + ! r (f2 ; )p ) Np;

p

where

1 X

1+

= 1

Np; :=

1 X

rp

j j

1=p

(154)

2

e (155) 2

e

= 1

which is uniformly bounded for all 2 (0; 1] : iv) When n = 0 and p = 1; let fj 2 (C (R) \ L1 (R)) ; j = 1; 2. Then the inequality Wr; (f ; x)

f (x)

(! r (f1 ; )1 + ! r (f2 ; )1 ) N1;

1

(156)

holds where N1; is de…ned as in (155). Proof. By Theorems 5, 6, 19, and Propositions 7, 8. For the operators Qr; ; we get Theorem 31 Let f : R ! C such that f = f1 + if2 and 0 < 1: (n) n i) Let fj 2 C (R) with fj 2 Lp (R) ; n 2 N; j = 1; 2; p; q > 1 such that 1 p

+

1 q

= 1,

>

p(r+n)+1 ; 2

2 N. Then

Qr; (f ; x)

n X f (k) (x)

f (x)

k=1 (n)

((n

Sp; :=

k qk;

(157) p

(n)

! r (f1 ; )p + ! r (f2 ; )p

1 p

where

k!

1 X

= 1

1 q

1)!)(q(n 1+

j j

1) + 1) (rp + 1)

rp+1

np 1

1 j j

1 p

2

! +

Sp;

1 p

2

(158)

1 X

2

= 1

32

+

2

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

328

is uniformly bounded for all 2 (0; 1] : (n) ii) When p = 1; let fj 2 C n (R), fj 2 L1 (R); j = 1; 2; n 2 N > r+n+1 : Then 2 Qr; (f ; x)

n X f (k) (x)

f (x)

f1g; and

(159)

k qk;

k! ! (n) (n) ! r (f1 ; )1 + ! r (f2 ; )1 S1; (n 1)! (r + 1) k=1

1

holds where S1; is de…ned as in (158). iii)When n = 0, let fj 2 (C (R) \ Lp (R)) ; j = 1; 2; p; q > 1 such that and

>

p(r+2)+1 . 2

1 p

+ 1q = 1;

Then

Qr; (f ; x)

f (x)

where

(! r (f1 ; )p + ! r (f2 ; )p ) Sp;

p

1 X

= 1

Sp; :=

rp

j j

1+

2

+

1=p

(160)

2

(161)

1 X

2

2

+

= 1

which is uniformly bounded for all 2 (0; 1]. iv) When n = 0 and p = 1; let fj 2 (C (R) \ L1 (R)) ; j = 1; 2; inequality Qr; (f ; x)

f (x)

>

(! r (f1 ; )1 + ! r (f2 ; )1 ) S1;

1

r+3 2 .

The

(162)

holds where S1; is de…ned as in (161). Proof. By Theorems 5, 6, 20, and Propositions 7, 8. Now, we give our simultaneous results for Lp norm. For the case of p > 1; we have Theorem 32 Here f : R ! C such that f = f1 + if2 and 0 < 1: Let (n+~ j) n+ + ~ fj 2 C (R); with fj 2 Lp (R) ; n 2 N; j = 0; 1; : : : ; 2 Z : Let p; q > 1 : p1 + 1q = 1: We consider the assumptions of Theorem 27 as valid for n = there. Then i) Pr; (f ; x)

(~ j)

f

(~ j)

(x)

n ~ X f (j+k) (x)

k=1 1 p

(n+~ j)

! r (f1 ((n

(n+~ j)

; )p + ! r (f2

1)!) (q (n

1 q

k!

; )p

1) + 1) (rp + 1) 33

1 p

k ck;

!

(163) p;x

Mp;

1 p

;

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

329

ii) (~ j)

Wr; (f ; x)

~

f (j) (x)

n ~ X f (j+k) (x)

(n+~ j)

! r (f1

1 p

((n iii) for

>

p(n+r)+1 ; 2

(n+~ j)

; )p + ! r (f2

; )p

1 q

1) + 1) (rp + 1)

1)!) (q (n

k pk;

k!

k=1

1 p

!

(164) p;x

Np;

1 p

;

2N

Qr; (f ; x)

(~ j)

~

f (j) (x)

n ~ X f (j+k) (x)

k=1 (n+~ j) ! r (f1 ;

1 p

((n

)p +

1)!) (q (n

(n+~ j) ! r (f2 ; 1 q

k!

)p

1) + 1) (rp + 1)

1 p

k qk;

!

(165) p;x

Sp;

1 p

:

Proof. By Theorems 9, 27, 29-31. Now, we give our results for the special case of n = 0: Proposition 33 Here f : R ! C such that f = f1 + if2 and 0 < 1: (~ j) Let fj 2 (C (R) \ Lp (R)) ; j = 1; 2; ~j = 0; 1; : : : ; 2 Z+ ; p; q > 1 such that 1 1 there. p + q = 1: We consider the assumptions of Theorem 27 as valid for n = Then for all ~j = 0; 1; : : : ; ; we have i) Pr; (f ; x)

(~ j)

~

f (j) (x)

p;x

(~j ) (~j ) ! r (f1 ; )p + ! r (f2 ; )p

Mp;

(~j ) (~j ) ! r (f1 ; )p + ! r (f2 ; )p

Np;

(~j ) (~j ) ! r (f1 ; )p + ! r (f2 ; )p

Sp;

1 p

; (166)

ii) Wr; (f ; x) iii) for

>

(~ j)

~

f (j) (x)

p(r+2)+1 ; 2

Qr; (f ; x)

(~ j)

p;x

1 p

; (167)

2N ~

f (j) (x)

p;x

1 p

: (168)

Proof. By Theorems 27, 29-31 and Proposition 10. For the special case of p = 1; we obtain Theorem 34 Here f : R ! C such that f = f1 + if2 and 0 < 1: Let (n+~ j) fj 2 C n+ (R); with fj 2 L1 (R) ; n 2 N f1g ; ~j = 0; 1; : : : ; 2 Z+ : We consider the assumptions of Theorem 27 as valid for n = there. Then for all

34

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

330

~j = 0; 1; : : : ; , we have i) Pr; (f ; x)

(~ j)

~

f (j) (x)

n ~ X f (j+k) (x)

k=1 (n+~ j)

! r (f1

(n+~ j)

; )1 + ! r (f2 (n 1)! (r + 1)

; )1

k! !

k ck;

(169) 1;x

M1; ;

ii) Wr; (f ; x)

(~ j)

~

f (j) (x)

n ~ X f (j+k) (x)

k=1 (n+~ j) ! r (f1 ;

(n iii) for

>

n+r+1 ; 2

(n+~ j) ! r (f2 ;

)1 + 1)! (r + 1)

)1

k! !

k pk;

(170) 1;x

N1; ;

2N Qr; (f ; x)

(~ j)

~

f (j) (x)

n ~ X f (j+k) (x)

k=1 (n+~ j) ! r (f1 ;

(n

(n+~ j) ! r (f2 ;

)1 + 1)! (r + 1)

)1

k! !

k qk;

(171) 1;x

S1; :

Proof. By Theorems 11, 27, 29-31. For p = 1 and n = 0; we give Proposition 35 Here f : R ! C such that f = f1 + if2 and 0 < 1: Let ~ f (j) 2 (C (R) \ L1 (R)) ; ~j = 0; 1; : : : ; 2 Z+ : We consider the assumptions of Theorem 27 as valid for n = there. Then for all ~j = 0; 1; : : : ; ; we have i) Pr; (f ; x)

(~ j)

~

f (j) (x)

1;x

(~j ) (~j ) ! r (f1 ; )1 + ! r (f2 ; )1 M1; ;

(172)

(~j ) (~j ) ! r (f1 ; )1 + ! r (f2 ; )1 N1; ;

(173)

(~j ) (~j ) ! r (f1 ; )1 + ! r (f2 ; )1 S1; :

(174)

ii) Wr; (f ; x)

(~j )

~

f (j) (x) 1;x

iii) for

>

r+3 2 ;

Qr; (f ; x)

2N (~ j)

~

f (j) (x)

1;x

35

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

331

Proof. By Theorems 27, 29-31 and Proposition 12. Next, we give the fractional approximation results for the operators Pr; ; Wr; ; and Qr; : We start with Theorem 36 Here f : R ! C such that f = f1 + if2 and 0 < (m) fj 2 C m (R) ; m = d e ; > 0; with fj < 1; j = 1; 2:Then

1: Let

1

i)

Pr; (f ) "

f

m X1

f (k)

(r k=0

(175)

k!

k=1

r X

k ck;

r! k)! ( + k + 1)

1 k

S 1;k

#

sup max ! r Dx f1 ;

; ! r (D x f1 ; )

+ sup max ! r Dx f2 ;

; ! r (D x f2 ; )

x2R

x2R

;

where S 1;k is de…ned as in (96) : ii) Wr; (f ) "

f

m X1

f (k)

k=1

r X

k=0

(r

k pk;

(176)

k!

r! k)! ( + k + 1)

1 k

S

2 ;k

#

sup max ! r Dx f1 ;

; ! r (D x f1 ; )

+ sup max ! r Dx f2 ;

; ! r (D x f2 ; )

x2R

x2R

where S 2;k is de…ned as in (98) :

36

;

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

iii)For

>

+r+1 ; 2

332

2 N; Qr; (f ) "

m X1

f

f (k)

k=0

1

r! k)! ( + k + 1)

(r

(177)

k!

k=1

r X

k qk;

k

S

3 ;k

#

sup max ! r Dx f1 ;

; ! r (D x f1 ; )

+ sup max ! r Dx f2 ;

; ! r (D x f2 ; )

x2R

x2R

;

where S 3;k is de…ned as in (100) : Proof. By Theorems 13, 23. Let f : R ! C such that f = f1 +if2 :We de…ne the following error quantities: E0;P (f; x)

:

=

E0;W (f; x)

:

= Pr; (f ; x) 1 P

= 1

f (x)

r P

j f (x

=

1 P

= 1

f (x)

1

(179)

!

2

j f (x

+j ) e

erf

p1

j=0

p

f (x);

1

1+2 e

r P

j j

+j ) e

j=0

= Wr; (f ; x)

(178)

!

f (x):

+1

Additionally, we introduce the errors (n 2 N):

En;P (f; x) := Pr; (f ; x)

f (x)

n X

f

k=1

(k)

(x) k!

k

1 X

k

e

= 1

1+2 e

j j

(180)

1

and

En;W (f; x) := Wr; (f ; x) f (x)

n X f (k) (x)

k=1

k!

37

kp

1 X

k

2

e

= 1

1

erf

p1

: (181) +1

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

333

We observe that jE0;P (f; x)j jE0;P (f1 ; x)j + jE0;P (f2 ; x)j ;

(182)

jE0;W (f; x)j jE0;W (f1 ; x)j + jE0;W (f2 ; x)j :

(183)

Next, we give following results for errors E0;P (f; x) and E0;W (f; x): Corollary 37 Here f : R ! C such that f = f1 + if2 :Let fj 2 Cu (R) for j = 1; 2: Then i)

jE0;P (f; x)j

0 P 1 (! (f ; j j) + ! r (f2 ; j j)) e B = 1 r 1 B 1 @ 1+2 e + (jf1 (x)j + jf2 (x)j) jm

;P

j j

1

(184)

1

(185)

C C A

1j ;

ii)

jE0;W (f; x)j

0 P 1 (! (f ; j j) + ! r (f2 ; j j)) e B = 1 r 1 B p @ 1 erf p1 +1

+ (jf1 (x)j + jf2 (x)j) jm

;W

1j :

2

C C A

Proof. By Corollary 16, (182) ; and (183) : We also observe that kEn;P (f; x)k1 kEn;P (f1 ; x)k1 + kEn;P (f2 ; x)k1

(186)

kEn;W (f; x)k1 kEn;W (f1 ; x)k1 + kEn;W (f2 ; x)k1 :

(187)

and

For En;P and En;W , we present Theorem 38 Here f : R ! C such that f = f1 + if2 : Let fj 2 C n (R) with (n) fj 2 Cu (R), j = 1; 2; n 2 N;and kfj k1;R < 1: Then 38

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

334

i) kEn;P (f; x)k1;x

(188)

0 P 1 n (n) (n) ! r (f1 ; ) + ! r (f2 ; ) B = 1 j j 1 + B @ n! 1+2 e

+ kf1 k1;R + kf2 k1;R jm

;P

j j

r

e

j j

1 C C A

1

1j ;

ii) kEn;W (f; x)k1;x

(189)

0 P 1 j j n (n) (n) ! r (f1 ; ) + ! r (f2 ; ) B = 1 j j 1 + B @ p n! 1 erf p1

+ kf1 k1;R + kf2 k1;R jm

1j :

;W

2

r

e +1

1 C C A

In the above inequalities (188) - (189) ; the ratios of sums in their R.H.S. are uniformly bounded with respect to 2 (0; 1] : Proof. By Theorem 17, (186) ; and (187) : Furthermore, for p

1; we obtain that kEn;P (f; x)kp

(190)

kEn;W (f; x)kp

(191)

kEn;P (f1 ; x)kp + kEn;P (f2 ; x)kp

and

kEn;W (f1 ; x)kp + kEn;W (f2 ; x)kp :

Next, we state our Lp approximation results for the errors E0;P ; E0;W ; En;P ; and En;W : We start with Theorem 39 i) Let p; q > 1 such that

39

1 p

+

1 q

= 1, n 2 N such that np 6= 1,

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

335

fj 2 Lp (R); j = 1; 2; and the rest as above in this section. Then kEn;P (f; x)kp (n) ! r (f1 ;

)p +

0

B B B B @ ((n 2 6 6 6 6 6 6 4

(192) (n) ! r (f2 ;

1 p

1 P

)p j j

1

1 q

C C C 1 C 1 1) + 1) q (rp + 1) p A e

= 1

1)!)(q(n

1 X

1+

rp+1

j j

np 1

1 j j

= 1

1+2 e

+ kf1 (x)kp + kf2 (x)kp jm

e

1

1j

;P

j

! p1 3 j 7 7 7 7 7 7 5

holds. (n) ii) When p = 1; let fj 2 C n (R); fj 2 L1 (R), fj 2 L1 (R); and n 2 N Then kEn;P (f; x)k1 (n) ! r (f1 ;

2 6 6 6 6 4

(n 1 X

)1 +

(193) (n) ! r (f2 ;

)1

1)! (r + 1) 1+

f1g:

j j

r+1

n 1

1 j j

= 1

1+2 e

1

+ (kf1 (x)k1 + kf2 (x)k1 ) jm holds. iii) When n = 0; let p; q > 1 such that

1 p

40

+

1 q

;P

1j

e

j j

3 7 7 7 7 5

= 1; fj 2 Lp (R), and the rest as

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

336

above in this section. Then kE0;P (f; x)kp

(! r (f1 ; )p + ! r (f2 ; )p ) 2 6 6 6 6 6 4

1 X

e

j j

= 1 1 X

1+

j j

= 1

1+2 e

rp

e

j j

! 1=p 3

(194)

7 7 7 7 7 5

1

+ kf1 (x)kp + kf2 (x)kp jm

! q1

1j

;P

holds. iv) When n = 0 and p = 1; the inequality kE0;P (f; x)k1

(! r (f1 ; )1 + ! r (f2 ; )1 ) 0 1 X r j 1+ j j e B B = 1 B 1 B 1+2 e @

j

(195)

1 C C C C A

+ (kf1 (x)k1 + kf2 (x)k1 ) jm

;P

1j

holds. Proof. By Theorem 21 and (190). We have also Theorem 40 i) Let p; q > 1 such that

41

1 p

+

1 q

= 1, n 2 N such that np 6= 1,

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

337

fj 2 Lp (R); j = 1; 2; and the rest as above in this section. Then kEn;W (f; x)kp (n) ! r (f1 ;

)p +

0

B B B B B B ((n @ 2 6 6 6 6 6 6 4

(196) (n) ! r (f2 ;

1 p

1 X

)p 2

C C C C 1 C 1 1) + 1) q (rp + 1) p C A e

= 1

1)!)(q(n

1 X

= 1

1+ p

rp+1

j j

np 1

1 j j

1

erf

p1

+1

;W

1j

+ kf1 (x)kp + kf2 (x)kp jm holds. ii) For p = 1; let n 2 N f1g: Then

1

! q1

(n)

fj 2 C n (R); fj 2 L1 (R), fj kEn;W (f; x)k1 (n) ! r (f1 ;

2 6 6 6 6 4

(n 1 X

= 1

)1 +

2

e

2 L1 (R); j = 1; 2; and (197)

(n) ! r (f2 ;

)1

1)! (r + 1) 1+ p

r+1

j j

n 1

1 j j

1

erf

p1

+ (kf1 (x)k1 + kf2 (x)k1 ) jm holds. iii) For n = 0; let p; q > 1 such that

! p1 3 7 7 7 7 7 7 5

1 p

+

42

1 q

+1 ;W

1j

2

e

3 7 7 7 7 5

= 1; fj 2 Lp (R), and the rest as

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

338

above in this section. Then kE0;W (f; x)kp

(! r (f1 ; )p + ! r (f2 ; )p )

1 X

2

e

= 1

2

1 X

6 6 6 = 6 6 p 6 4

1+

rp

j j

1

1

erf

p1

! p1 3 7 e 7 7 7 7 +1 7 5

! q1

(198)

2

+ kf1 (x)kp + kf2 (x)kp jm

;W

1j

holds. iv) For n = 0 and p = 1; the inequality (n)

kE0;W (f; x)k1

(n)

! r (f1 ; )1 + ! r (f2 ; )1 0 1 1 2 X r j j 1 + e B C B = 1 C B C Bp C @ 1 erf p1 + 1A

+ kf1 (x)kp + kf2 (x)kp jm

;W

(199)

1j

holds. Proof. By Theorem 22, and (191). Next, we demonstrate our simultaneous approximation results for the derivatives of the errors E0;P ; E0;W ; En;P ; and En;W : We start with (~ j) Corollary 41 Let fj 2 Cu (R); ~j = 0; 1; : : : ; ; 2 Z+ ; and 0 < 1: We consider the assumptions of Theorem 27 as valid for n = there. Then for all ~j = 0; 1; : : : ; ; we have

i)

~ j (E0;P (f; x))( )

0 P 1 (~j ) (~j ) ! r (f1 ; j j) + ! r (f2 ; j j) e B = 1 B 1 @ 1+2 e +

~

~

f (j ) (x) + f (j ) (x)

43

jm

;P

1j ;

j j

1

C C (200) A

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

339

ii) 0 P 1 (~j ) (~j ) ! r (f1 ; j j) + ! r (f2 ; j j) e B = 1 B p @ +1 1 erf p1

~ j (E0;W (f; x))( )

~

~

f (j ) (x) + f (j ) (x)

+

jm

;W

2

1

C C(201) A

1j :

Proof. By Corollary 37, and Theorem 27. We have also (n+~ j) Theorem 42 Let fj 2 C n+ (R); n 2 N; 2 Z+ and fj 2 Cu (R); ~j = ~ 0; 1; : : : ; ; 0 < 1, and f (j ) < 1: We consider the assumptions of 1;R

Theorem 27 valid for n = : Then for all ~j = 0; 1; : : : ; ; we have i) ~ j (En;P (f; x))( ) (n+~ j) ! r (f1 ;

)+

(202)

1;x (n+~ j) ! r (f2 ;

n!

+

(~j )

(~j )

f1

+ f2 1;R

1;R

0 P 1 n ) B = 1j j 1 + B @ 1+2 e

!

jm

;P

j j

r

e

j j

1 C C A

1

1j ;

and ii) ~ j (En;W (f; x))( ) (n+~ j) ! r (f1 ;

)+

(203)

1;x (n+~ j) ! r (f2 ;

n!

+

(~j )

(~j )

+ f2

f1

1;R

1;R

0 P 1 j j n ) B = 1j j 1 + B @ p 1 erf p1

!

jm

;W

1j :

Proof. By Theorems 27, 38: Next, we give following Lp approximation results. We obtain

44

2

r

e +1

1 C C A

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

340

(n+~ j) Theorem 43 i) Let fj 2 C n+ (R); with fj 2 Lp (R) ; n 2 N; ~j = 0; 1; : : : ; 2 1 1 + Z : Let p; q > 1 : p + q = 1; np 6= 1: We consider the assumptions of Theorem 27 as valid for n = there. Then for all j = 0; 1; : : : ; ; ~ j (En;P (f; x))( ) (n+~ j)

! r (f1 0

2

1 P

; ) 1

1 q

j j

C C C 1 C 1 1) + 1) q (rp + 1) p A e

= 1

1)!)(q(n

1 X

6 6 6 6 6 6 4 +

(n+~ j)

; ) + ! r (f2 1 p

B B xB B @ ((n

(204)

p

1+

rp+1

j j

np 1

1 j j

= 1

1

1+2 e

(~j )

(~j )

f1 (x)

+ f2

(x)

p

p

e

j j

!

jm

! p1 3 7 7 7 7 7 7 5

1j

;P

holds. (n+~ j) ii) Let fj 2 C n+ (R); with fj 2 L1 (R) ; n 2 N f1g ; ~j = 0; 1; : : : ; 2 Z+ : We consider the assumptions of Theorem 27 as valid for n = there. Then for all ~j = 0; 1; : : : ; , ~ j (En;P (f; x))( ) (n+~ j) ; ! r (f1

2

+

6 6 6 6 4

)1 +

(n 1 X

(205)

1 (n+~ j) ! r (f2 ;

)1

1)! (r + 1) 1+

j j

r+1

= 1

1+2 e

(~j ) f1 (x)

n 1

1 j j

j j

1

(~j ) + f2 (x) 1

e

1

jm

;P

1j

3 7 7 7 7 5

holds. (~ j) iii) Let fj 2 (C (R) \ Lp (R)) ; ~j = 0; 1; : : : ; 2 Z+ ; p; q > 1 such that 1 1 p + q = 1: We consider the assumptions of Theorem 27 as valid for n =

45

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

341

there. Then for all ~j = 0; 1; : : : ; ; (~j )

(~j )

(E0;P (f; x))

1 X

(~j )

! r (f1 ; )p + ! r (f2 ; )p

p

2

+

6 6 6 6 6 4

e

= 1

1 X

1+

= 1

1+2 e

j j

e

! 1=p 3

1

(~ j)

(~ j)

f1 (x)

rp

j j

p

+ f2 (x)

p

jm

7 7 7 7 7 5

j j

! q1

(206)

1j

;P

holds. (~ j) iv) Let fj 2 (C (R) \ L1 (R)) ; ~j = 0; 1; : : : ; 2 Z+ : We consider the assumptions of Theorem 27 as valid for n = there. Then for all ~j = 0; 1; : : : ; ; ~ j (E0;P (f; x))( )

1

(~j ) (~j ) ! r (f1 ; )1 + ! r (f2 ; )1 0

+

B B B B @

1 X

1+

j j

= 1

1+2 e

(~ j)

f1 (x)

r

e 1

(~ j)

1

j j

+ f2 (x)

(207)

1 C C C C A 1

jm

;P

1j

holds. Proof. By Theorems 27, 39: For En;W (f; x); we have (n+~ j)

Theorem 44 i) Let fj 2 C n+ (R); with fj 2 Lp (R) ; n 2 N; ~j = 0; 1; : : : ; 2 1 1 + Z : Let p; q > 1 : p + q = 1; np 6= 1: We consider the assumptions of Theorem

46

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

there. Then for all ~j = 0; 1; : : : ; ;

27 as valid for n =

~ j (En;W (f; x))( ) (n+~ j)

! r (f1 0

B B B B B B ((n @

2

(208)

p (n+~ j)

; ) + ! r (f2 1 p

1 X

1)!)(q(n

1+

= 1

p

e

rp+1

j j

np 1

1 j j

1

p1

erf

+ f2

f1 (x)

1

2

(~j )

(~j )

; ) ! q1

C C C 1 C 1 C 1 1) + 1) q (rp + 1) p C A

=

1 X

6 6 6 6 6 6 4 +

342

(x) p

p

2

e

+1

!

jm

;W

! p1 3 7 7 7 7 7 7 5

1j

holds. ~ ii) Let f 2 C n+ (R); with f (n+j) 2 L1 (R) ; n 2 N f1g ; ~j = 0; 1; : : : ; 2 Z+ : We consider the assumptions of Theorem 27 as valid for n = there. Then for all ~j = 0; 1; : : : ; , we have ~ j (En;W (f; x))( ) (n+~ j)

! r (f1 2

+

6 6 6 6 4

(n+~ j)

; )1 + ! r (f2

(n 1 X

= 1

; )1

1)! (r + 1) 1+

p

j j

1

(~ j)

f1 (x)

(209)

1

r+1

n 1

1 j j erf

p1

(~ j)

1

+ f2 (x)

1

2

e

+1 jm

;W

1j

3 7 7 7 7 5

holds. ~ iii) Let f (j) 2 (C (R) \ Lp (R)) ; ~j = 0; 1; : : : ; 2 Z+ ; p; q > 1 such that 1 1 p + q = 1: We consider the assumptions of Theorem 27 as valid for n =

47

Approximation by Complex Generalized Discrete Singular Operators, George A. Anastassiou & Merve Kester

343

there. Then for all ~j = 0; 1; : : : ; ; (~j )

(~j )

(E0;W (f; x))

1 X

(~j )

! r (f1 ; )p + ! r (f2 ; )p

p

e

= 1

2

+

2

1 X

6 6 6 = 6 6 p 6 4

1+

rp

j j

1

1

(~ j)

f1 (x)

p1

erf

(~ j)

p

! p1 3 7 e 7 7 7 7 +1 7 5

! q1

(210)

2

+ f2 (x)

jm

p

1j

;W

holds. ~ iv) Let f (j) 2 (C (R) \ L1 (R)) ; ~j = 0; 1; : : : ; 2 Z+ : We consider the assumptions of Theorem 27 as valid for n = there. Then for all ~j = 0; 1; : : : ; ; ~ j (E0;W (f; x))( )

(~j ) (~j ) ! r (f1 ; )1 + ! r (f2 ; )1

1

0

+

1 X

B B = B Bp @

1+

j j

r

2

e

1

1

erf

(~ j)

f1 (x)

p1 (~ j)

1

+ f2 (x)

(211) 1

C C C C + 1A 1

jm

;W

1j

holds. Proof. By Theorems 27, 40:

Finally, we give the fractional results for the error terms as Theorem 45 Here f : R ! C such that f = f1 + if2 : Let fj 2 C m (R) ;

m = d e;

(m)

> 0; with fj

1

< 1; 0