Understanding Integro-differential Equations

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MATHEMATICS RESEARCH DEVELOPMENTS

UNDERSTANDING INTEGRODIFFERENTIAL EQUATIONS

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MATHEMATICS RESEARCH DEVELOPMENTS

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MATHEMATICS RESEARCH DEVELOPMENTS

UNDERSTANDING INTEGRODIFFERENTIAL EQUATIONS

J. Vasundhara Devi, PhD Zahia Drici, PhD Farzana A. McRae, PhD EDITORS

Copyright © 2023 by Nova Science Publishers, Inc. DOI: https://doi.org/10.52305/NRLJ6556

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Published by Nova Science Publishers, Inc. † New York

Contents

Preface

…………………………………………………………. vii

Chapter 1

An Overview of Existence Results for Some Integro-Differential Equations ..…………..…………….. 1 CH. V. Sreedhar, I. S. N. R. G. Bharat and J. Vasundhara Devi

Chapter 2

A Brief Overview of the Stability Theory of Integro-Differential Equations ………………….…… 31 Z. Drici, F. A. McRae and J. Vasundhara Devi

Chapter 3

Advances in the Qualitative Theory of Integro–Differential Equations ……….…………………55 Osman Tunç, Seenith Sivasundaram and Cemil Tunç

Chapter 4

Mathematical Methods for Integro-Differential Equations and Their Applications …..…………………125 Gunvant A. Birajdar and N. Giribabu

Chapter 5

Operational Matrices for Solving Fractional Order Integral and Integro-Differential Equations.…..149 H. Tajadodi, R. M. Ganji, S. M. Narsale and H. Jafari

Chapter 6

On a Second Order Nonlinear Integro-Differential Equation of Fredholm Type in the Complex Plane……175 Lemita Samir, Selim Raja and Belahbib Zeineb

vi

Contents

Chapter 7

Linear and Nonlinear Partial Integro-Differential Equations Arising from Finance …….………………... 191 José Cruz, Maria Grossinho, Daniel Ševčovič and Cyril Udeani

Index

…..……………………………………………………….. 257

PREFACE This book presents an overview of the theory of Integro-Differential Equations (InDEs). Beginning with the history of the inception of InDEs, the existence and uniqueness results for the initial value problem of InDEs is given. Next, stability concepts are discussed, including Lyapunov stability and UlamRassias stability. Numerical techniques and operational matrix methods are also introduced. A specific study of second order InDEs of Fredholm InDes in complex plane is presented and an application to Finance is given.

In: Understanding Integro-Differential Equations ISBN: 979-8-89113-040-1 c 2023 Nova Science Publishers, Inc. Editors: J. Vasundhara Devi et al.

Chapter 1

A N OVERVIEW OF E XISTENCE R ESULTS FOR S OME I NTEGRO -D IFFERENTIAL E QUATIONS CH. V. Sreedhar ∗ I. S. N. R. G. Bharat and J. Vasundhara Devi Department of Mathematics, GVP-Prof. V. Lakshmikantham Institute for Advanced Studies, Gayatri Vidya Parishad College of Engineering(Autonomous), Visakhapatnam, AP, India

Abstract Starting with a brief sketch of the history of integro-differential equations, we proceed to give existence and uniqueness results for nonlinear Volterra integro-differential equations. Next we present monotone iterative technique for integro-differential equations with retardation and anticipation, periodic boundary-value problems of integro-differential equations, and higher order integro differential equations. Then, we give quasilinearization for periodic boundary-value problems of integro-differential equations, integro-differential equations with retardation and anticipation.

Keywords: integral equation, integro-differential equation, initial value problem, periodic boundary value problem, lower and upper solutions, existence ∗ Corresponding Author’s

Email: [email protected].

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CH. V. Sredhar, I. S. N. R. G. Bharat and J. Vasundhara Devi

and uniqueness, monotone iterative technique, quasilinearization AMS Subject Classification: 45D05, 45B05, 45J05, 45J99, 47G20

1.

Introduction

Modeling of physical phenomenon is necessary to understand and to predict the behavior of the phenomenon in future. Modeling involves developing mathematical equations which are obtained through the laws of nature. We illustrate the process through the following example. Electronic or magnetic polarization of substances depends on the electro magnetic field at the moment. Interestingly, for some other substances the electronic or magnetic polarization depends both on the present moment and also on the history of the electromagnetic state of the matter in all the past moments (this is called hysteresis). When these physical facts are translated into mathematical equations, the result is integral equation and if dynamics is also involved we get an integro-differential equation [1]. The theory of integro-differential equations (InDEs) can be traced to the works of Abel, Lotka, Fredholm, Malthus, Verhulst and Volterra. Abel’s work on the tautochrome problem is one of the first problems involving an InDE. Fredhlom also worked on InDEs and Volterra worked with a particular type of Fredholm integro-differential equation (FInDE). To solve an InDE, it is either transformed into a system of differential equations or integral equations and the solutions of these systems are solutions of InDE [2]. As the classification of InDEs is parallel to that of integral equations, we begin with the classification of integral equations and proceed with the classification of InDEs. In this chapter we concentrate on the theoretical aspects of solving nonlinear InDEs, particularly of Volterra type. Of the many types of existence results available, we focus on the constructive methods of monotone iterative technique (MIT) [3] and quasilinearization [4]. An obvious advantage of these techniques is that the solution is obtained in a sector, that is, a closed set. Further, the iterates are solutions of the corresponding linear integro-differential equations and are either increasing or decreasing in nature and these iterates converge to a solution of the considered problem. We begin with existence results obtained by using fixed point theorems. We

An Overview of Existence Results for Some Integro-Differential ...

3

first present the MIT for InDEs involving retardation and anticipation. This result includes the MIT of an InDE and MIT for an InDE involving retardation as special cases. Next, we present MIT developed for Periodic Boundary Value Problem (PBVP) from [5]. In this paper, the authors made a simple choice of initial value as the boundary value of the previous iterate, while constructing the iterates, which guaranteed the existence of the solutions in a sector. Thus one need not prove an existence result separately. Further, we develop MIT for fourth order nonlinear InDE. To develop the MIT, we need to use the maximum principle which is established in [6]. There are some papers involving the concept of lower and upper solutions with reverse inequalities. We introduce this concept from [7]. Next, we proceed to present results obtained using the method of quasilinearization for InDEs involving retardation and anticipation and describe the method of generalized quasilinearization for PBVP of InDE [8].

2.

Integral Equations

The term, integral equation, was first used by Du Bois-Reymond. The development of the theory of integral equations is one of the most powerful tools in both pure and applied mathematics. It has many applications in physical problems [9]. Definition 1. The general form of linear integral equation is given below ξ (z) h (z) = u (z) + λ

Zr(z)

H (z,t)h (t)dt,

(1)

q(z)

where q (z) and r (z) are the limits of integration, λ is a non-zero parameter, H (z,t) is the kernel of the integral equation and is a function of z, t ∈ [t0 , T ]. If the unknown function h(z) involved in the integral equation is nonlinear, then such an integral equation is called a nonlinear integral equation.

3.

Classification of Integral Equations

In this section we present various types of integral equations that are used frequently in applications. Integral equations can be broadly categorized under

4

CH. V. Sredhar, I. S. N. R. G. Bharat and J. Vasundhara Devi

two heads as Fredholm integral equations and Volterra integral equations. The classification mainly depends on the limits of integration. In Fredholm integral equation, the integral is over the whole interval of definition. If the upper limit of the integral is "t", then such an integral equation is called Volterra integral equation. The various types of integral equations are briefly given below.

3.1.

Fredholm Integral Equation

In the integral equation (1), if the limits of the integration q (z) and r (z) are constants, then such an equation is called a Fredholm integral equation. The general form of a Fredholm integral equation is ξ (z) h (z) = u (z) + λ

Zb

H (z,t) h (t) dt,

(2)

a

where a and b are the limits of the integration, λ is a non-zero parameter, H (z,t) is the kernel of the integral equation, and is a function of z,t ∈ [t0 , T ]. R1

Example 1. f (z) = ez + (zt) f (t)dt is a Fredholm integral equation. 0

3.1.1.

Fredholm Integral Equation of First Kind

In the Fredholm integral equation (2) if ξ (z) ≡ 0 and the unknown function h (z) appears only under the integral sign then such an equation is called an Fredholm integral equation of first kind. The general form of a Fredholm integral equation of first kind is u (z) + λ

Zb

H (z,t)h (t)dt = 0.

(3)

a

Example 2. In ez =

R1

(zt) u (t) dt, the unknown function u (t) appears only

0

inside the integral sign and both the limits of the integration are constants, hence (3) is an example of a Fredholm integral equation of first kind.

An Overview of Existence Results for Some Integro-Differential ... 3.1.2.

5

Fredholm Integral Equation of Second Kind

If ξ (z) ≡ 1 and the unknown function h (x) appears both inside and outside the integral sign in Fredholm integral equation (2) then such an equation is called a Fredholm integral equation of second kind. The general form of a Fredholm integral equation of second kind is h (z) = u (z) + λ

Zb

H (z,t) h (t)dt.

(4)

a

R1

Example 3. The equation u (z) = sin(z) + (zt) u (t) dt, is an example of a 0

Fredholm integral equation of second kind.

3.2.

Volterra Integral Equation

The Volterra integral equation is an equation which contains a fixed lower limit and a variable upper limit for the integral. The general form of a Volterra integral equation is given by ξ (z) h (z) = u (z) + λ

Zz

H (z,t) h (t) dt.

(5)

a

Rz

Example 4. The equation f (z) = ez + (zt) f (t)dt. It is an example of Volterra 0

integral equation (5). 3.2.1.

Volterra Integral Equation of First Kind

If ξ (z) ≡ 0, in the Volterra integral equation (5) then such an equation is called a Volterra integral equation of first kind. Example 5. An example of Volterra integral equation of first kind is given by Rz

ez + (zt) f (t) dt = 0. 0

6 3.2.2.

CH. V. Sredhar, I. S. N. R. G. Bharat and J. Vasundhara Devi Volterra Integral Equation of Second Kind

If ξ (z) ≡ 1, in Volterra integral equation (5), then we get a Volterra integral equation of second kind. The general form of Volterra integral equation of second kind is given by Rz

h (z) = u (z) + λ H (z,t)h (t)dt. a

Example 6. f (z) = sinz +

Rz 0

second kind.

4.


z2t f (t)dt, is a Volterra integral equation of

Integro-Differential Equations

An InDE is an equation in the form z

(n)

(t) = c (t, z (t)) +

Zr(t)

H (t, s) z (s)ds.

(6)

q(t)

Rt

Example 7. z00 = 5z + 6 4z (τ)sin(t)dτ. 0

4.1.

Classification of Integro-Differential Equations

In this section we discuss various types of InDEs that are frequently used as mathematical tools. Parallel to the classification of integral equations, InDEs are also divided into two types, namely, Fredholm integro-differential equations and Volterra integro-differential equations. These equations are further subdivided into various types, which are briefly given below.

4.1.1.

Fredholm Integro-Differential Equation

In InDE (6), if the limits of integration are constants, then such an equation is called a Fredholm integro-differential equation. The Fredholm integro-

An Overview of Existence Results for Some Integro-Differential ...

7

differential equation has the form z

(n)

(t) = c (t, z (t)) + λ

Zb

H (t, s)z (s)ds, n > 1.

(7)

a π

00

R2

0

Example 8. z (t) + z (t) = z − cos z − z (s) ds. 0

4.1.2.

Fredholm Integro-Differential Equation of Second Kind

If the kernel involved in the Fredholm InDE (7) is a separable kernel, that is, n

H (t, s) = ∑ hk (t)rk (s), then such an equation is called a Fredholm integrok=1

differential equation of second kind. 4.1.3.

Volterra Integro-Differential Equation

In InDE (6) if the upper limit of integration is variable, then such an equation is called a Volterra integro-differential equation. The Volterra integro-differential equation has the form z

(n)

(t) = c (t, z (t)) + λ

Zt

H (t, s)z (s)ds, n > 1.

(8)

a

Rt

Example 9. z000 (t) = 1 + tz (s) ds. 0

4.1.4.

Volterra-Fredhlom Integro-Differential Equation

The Volterra-Fredhlom integral equation is an equation which contains integrals with fixed limits and with variable limits. An example of Volterra-Fredholm integro-differential equation is z

(n)

(t) = u (t) + λ1

Zt a

H1 (z,t) h (t)dt + λ2

Zb a

H2 (z,t) h (t)dt.

(9)

8

5.

CH. V. Sredhar, I. S. N. R. G. Bharat and J. Vasundhara Devi

Existence Using Fixed Point Theorems

In this section, we state global existence and uniqueness results for nonlinear Volterra InDEs using fixed point theorems and the comparison method with a scalar integro-differential equation.

5.1.

Existence Result Using Tychonoff Fixed Point Theorem

Consider the IVP of InDE given by, 0

z (t) = c (t, z (t)) +

Zt

H (t, s, z (s)) ds,

(10)

t0

z (t0 ) = z0 .

(11)

We state the following theorem dealing with the existence of solutions using Tychonoff fixed point theorem, see [10]. Theorem 1. Assume that   (H1 ) c ∈ C [R+ × Rn , Rn ], l ∈ C R2+ , R+ , l (t, u) is monotone nondecreasing in u, for each t ∈ J, and |c (t, z)| 6 l (t, |z|) , (t, z) ∈ R+ × Rn ;     (H2 ) H ∈ C R2+ × Rn , Rn , G ∈ C R2+ , R+ , G (t, s, u) is monotone nondecreasing in u for each (t, s) ∈ R2+ ; and (H3 ) for each u0 > 0, the scalar integro-differential equation u0 (t) = l (t, u (t)) +

Rt

G (t, s, u (s))ds, u (t0 ) = u0 has a solution u (t)

t0

existing for t > t0 ;

(H4 )

Rt

|H (σ, s, x (s))| dσ 6 N for t, s ∈ R+, z ∈ C [R+ , Rn ] .

s

Then, for every z0 ∈ Rn such that |z0 | 6 u0 , there exists a solution z (t) of IVP of InDE (10) and (11).

An Overview of Existence Results for Some Integro-Differential ...

5.2.

9

Existence Using Schauder Fixed Point Theorem

The following theorem deals with the existence of solutions using Schauder fixed point theorem, for details see [10]. Theorem 2. Assume that (H1 ) c ∈ C [J × Rn , Rn ], H ∈ C [J × J × Rn , Rn ], and

Rt

|H (σ, s, z (s))| dσ 6 N,

s

for t0 6 s 6 t 6 t0 + a, z ∈ Ω = {φ ∈ C [J, Rn ] : φ (t0 ) = z0 , |φ (t) − z0 | 6 b} . Then, the IVP of InDE (10) and (11) possesses at least one solution z (t) on t0 6 s 6 t 6 t0 + α, for some 0 < α < a.

6.

Monotone Iterative Technique

The method of monotone iterative technique (MIT), together with the method of lower and upper solutions, is an important tools that yields sequences of monotone iterates that converge to the maximal and minimal solution of the considered problem. This method is a flexible mechanism and a constructive technique to obtain the existence of solutions in a closed sector. In this section we will consider some InDEs and develop MIT. Comparison results play a pivotal role in studying nonlinear InDEs. A complicated InDE can be studied through a much simpler scalar InDE using the comparison principle. We present the following comparison result using Dini derivatives [10, 11]. This theorem plays an important role in the qualitative theory of InDEs.     Theorem 3. Assume that l ∈ C R2+ , R , H ∈ C R3+ , R , H (t, s, u) is nondecreasing in u for each (t, s) and for t > t0 , D− m (t) > l (t, m (t)) +

Zt t0

H (t, s, m (s))ds,

10

CH. V. Sredhar, I. S. N. R. G. Bharat and J. Vasundhara Devi

where m ∈ C [R+ , R] and D− m (t) = lim inf− h−1 [m (t + h) − m (t)] . Suppose h→0

that γ (t) is the maximal solution of 0

u (t) = l (t, u (t)) +

Zt

H (t, s, u (s)) ds, u (t0 ) = u0 > 0,

t0

existing on [t0 , ∞) . Then, m (t) 6 γ (t), t > t0 , provided m (t0 ) 6 u0 . As finding the solutions of scalar integro-differential equations is also difficult, the above comparison theorems are not helpful in many situations. The previous theorems require knowledge of these solutions, which are not always. Thus, we present a comparison theorem which uses a scalar differential equation to estimate the nonlinear InDE. This result is an adaptation of results in [12] and is given in [10]. We first state a lemma that is essential to prove the comparison theorem.   Lemma 4. Let l0 , l ∈ C R2+ , R satisfy l0 (t, u) 6 l (t, u), (t, u) ∈ R2+ .

Then, the right maximal solution γ (t,t0, u0 ) of u0 = l (t, u),

u (t0 ) = u0 ,

and the left maximal solution η (t, T, v0 ) of u0 = l0 (t, u) , u (T ) = v0 > 0, satisfy the relation, γ (t,t0, u0 ) 6 η (t, T, v0 ), t ∈ [t0 , T ] , whenever γ (T,t0, u0 ) 6 v0 .     Theorem 5. Let m ∈ C [R+, R+ ], l ∈ C R2+ , R , G ∈ C R3+ , R and D− m (t) > l (t, m (t)) +

Zt

t0

G (t, s, m (s)) ds, t ∈ I0 ,

An Overview of Existence Results for Some Integro-Differential ...

11

where I0 = {t > t0 : m (s) 6 η (s,t, m (t)) , t0 6 s 6 t} , η (t, T, v0 ) being the left maximal solution of u0 = l0 (t, u) , u(T ) = v0 , existing on [t0 , T ] , assume that l0 (t, u) 6 F (t, u,t0) , where F (t, u,t0) = l (t, u) +

Zt

G (t, s, u (s))ds,

t0

and γ (t) is the maximal solution of u0 = F (t, u,t0) , u (t0 ) = u0 , existing on [t0 , ∞) , then, m (t0 ) 6 u0 implies m (t) 6 γ (t), t > t0 .

6.1.

InDE with Retardation and Anticipation

The past history or memory and anticipation play a vital role in physical phenomena. Hence, we present the MIT for InDEs with retardation and anticipation (InDE with R and A). Consider the nonlinear InDE given by, z0 = h(t, z, Bz, zt , zt ), t ∈ J = [t0, T ],

(12)

zt0 = φ0 , zT = ψ0 ,

(13)

where φ0 ∈ C1 , ψ0 ∈ C2 and h ∈ C[J × R × R ×C1 ×C2 , R] and Bz(t) =

Zt

H(t, s)z(s)ds,

t ∈ J,

t0

H ∈ C[J × J, R+ ], C1 = C[[−h1 , 0], R], C2 = C[[0, h2], R] and the term zt = xt (s) = z(t + s), s ∈ [−h1 , 0], represents retardation or delay and the term zt = zt (σ) = z(t + σ), σ ∈ [0, h2 ] represents anticipation. We begin by defining the natural lower and upper solutions for InDE with R and A (12) and (13).

12

CH. V. Sredhar, I. S. N. R. G. Bharat and J. Vasundhara Devi

Definition 2. (i) A function v0 ∈ C1 [[t0 − h1 , T + h2 ], R] is said to be a natural lower solution of InDE with R and A(12) and (13) if v00 ≤ h(t, v0 , Bv0 , v0t , vt0 ) where v0t0 = φ1 , vT0 = ψ1 , v0t0 ≤ φ0 and vT0 ≤ ψ0 . (ii) A function w0 ∈ C1 [[t0 − h1 , T + h2 ], R] is said to be a natural upper solution of InDE with R and A (12) and (13) if w00 ≥ h(t, w0, Bw0 , w0t , wt0 ) where w0t0 ≥ φ0 and wT0 ≥ ψ0 and w0t0 = φ2 , wT0 = ψ2 . Note that φ1 , φ2 ∈ C1 , ψ1 , ψ2 ∈ C2 and φ1 ≤ φ0 ≤ φ2 , ψ1 ≤ ψ0 ≤ ψ2 . Next, we introduce the concept of maximal and minimal solutions for InDE with R and A (12) and (13). Definition 3. (i) Let r1 (t) be a solution of InDE with R and A (12) and (13), then r1 (t) is said to be a maximal solution of (12) and (13) if for every solution z(t) of (12) and (13) existing on [t0 − h1 , T + h2 ], the inequality z(t) ≤ r1 (t) holds for t ∈ [t0 − h1 , T + h2 ]. (ii) Let ρ1 (t) be a solution of InDE with R and A (12) and (13), then ρ1 (t) is said to be a minimal solution of (12) and (13) if for every solution z(t) of (12) and (13) existing on [t0 − h1 , T + h2 ], the inequality ρ1 (t) ≤ z(t) holds for t ∈ [t0 − h1 , T + h2 ]. Definition 4. (i) A function v0 ∈ C1 [I, R] is said to be a coupled lower solution of InDE with R and A (12) and (13) if v00 ≤ h(t, v0, Bv0 , v0t , wt0 ), where v0t0 = φ1 , vT0 = ψ1 , v0t0 ≤ φ0 and vT0 ≤ ψ0 ; (ii) A function w0 ∈ C1 [I, R] is said to be a coupled upper solution of InDE with R and A (12) and (13) if w00 ≥ h(t, w0, Bw0 , w0t , vt0 ),

An Overview of Existence Results for Some Integro-Differential ...

13

where w0t0 = φ2 , wT0 = ψ2 , w0t0 ≥ φ0 and wT0 ≥ ψ0 . Note that φ1 , φ2 ∈ C1 , ψ1 , ψ2 ∈ C2 and φ1 ≤ φ0 ≤ φ2 , ψ1 ≤ ψ0 ≤ ψ2 . The following result is from [13] and is used to develop the monotone iterates corresponding to (12) and (13). Lemma 6. Let k ∈ C[[t0 − h1 , T ], R] C1 [J, R] be continuously differentiable on J = [t0 , T ] and T

0

k (t) ≤ −L1 k(t) − L2 Bk(t) − L3

Z0

kt (s)ds on J,

(14)

−h1

where

Rt

where L1 , L2 , L3 ≥ 0, Bk(t) = H(t, s)k(s)ds,

k1 = maxt,s∈J H(t, s).

t0

Suppose further that either

A) k(t0) ≤ kt0 (s) ≤ 0, s ∈ [−h1 , 0] and [L1 + L2 k1 (T − t0 ) + L3 h1 ](T − t0 ) ≤ 1, or B) kt0 (s) ≤ 0, s ∈ [−h1 , 0], k ∈ C1 [[t0 − h1 , T ], R] and k 0 (t) ≤

λ T −t0 +h1 ,

where t ∈ [t0 − h1 ,t0 ], min[t0 −h1 ,t0 ] k(s) = −λ, λ ≥ 0 and [L1 + L2 k1 (T − t0 ) + L3 h1 ](T − t0 + h1 ) ≤ 1.

(15)

Then, k(t) ≤ 0 on J. In order to develop MIT for InDE with R and A (12) and (13), we need the following assumptions relative to (12) and (13). (H1 ) Let v0 , w0 ∈ C1 [I, R], satisfying v00 ≤ h(t, v0 , Bv0 , vot , vt0 ), v0t0 = φ1 , vT0 = ψ1 , w00 ≥ h(t, w0 , Bw0 , wot , wt0 ),

14

CH. V. Sredhar, I. S. N. R. G. Bharat and J. Vasundhara Devi w0t0 = φ2 , wT0 = ψ2 , such that φ1 ≤ φ0 ≤ φ2 , ψ1 ≤ ψ0 ≤ ψ2 , v0 ≤ w0 on I = [t0 − h1 , T + h2 ], φ1 , φ2 ∈ C1 = C[[−h1 , 0], R] and ψ1 , ψ2 ∈ C2 = C[[0, h2], R];

(H2 ) h(t, x, ξ, φ, ψ) is nonincreasing in ψ for each (t, x, ξ, φ); (H3 ) h(t, x, ξ1, φ1 , ψ) − h(t, y, ξ2 , φ2 , ψ) ≥ −L1 (x − y) − L2 (ξ1 − ξ2 ) − L3

R0

(φ1 − φ2 )(s)ds

−h1

with [L1 + L2 k1 (T − t0 ) + L3 h1 ](T − t0 ) ≤ 1 and L(T − t0 ) > 12 , whenever v0 ≤ y ≤ x ≤ w0 , Bv0 ≤ ξ2 ≤ ξ1 ≤ Bw0 , k1 = maxt,s∈J , H(t, s) L1 , L2 , L3 ≥ 0, v0t ≤ φ2 ≤ φ1 ≤ wot , ψ ∈ C2 , φ1 , φ2 ∈ C1 ; (H4 ) v0t0 − φ0 , φ0 − w0t0 satisfy the assumptions of Lemma 6; (H5 ) h(t, x, ξ1, φ1 , ψ1 ) − h(t, y, ξ2, φ2 , ψ2 ) ≤ −O1 (x − y) + O2 (ξ1 − ξ2 ) + O3

R0

(φ1 − φ2 )(s)ds

−h1

Rh2

+ O4 (ψ1 − ψ2 )(s)ds, 0

where v0 ≤ y ≤ x ≤ w0 , Bv0 ≤ ξ2 ≤ ξ1 ≤ Bw0 , v0t0 ≤ φ2 ≤ φ1 ≤ w0t0 , vt0 ≤ ψ2 ≤ ψ1 ≤ wt0 , L1 , L2 , L3 , L4 ≥ 0 and [O2 k1 T + O3 h1 + O4 h2 ] < O1 . We now state the following theorem dealing with the monotone iterative technique for InDE involving retardation and anticipation. Theorem 7. Suppose that the assumptions (H1 ) to (H5 ) are satisfied. Then there exist monotone sequences {vn } and {wn } that converge uniformly to a solution of InDE with R and A (12) and (13) on I. The uniqueness of the solution for InDE with R and A (12) and (13) can be obtained by using the following lemma .

An Overview of Existence Results for Some Integro-Differential ...

15

Lemma 8. Let k ∈ C1 [[t0 − h1 , T + h2 ], R] be continuously differentiable on J = [t0 , T ] and k 0 (t) ≤ −L1 p(t) + L2 Bk(t) + L3

R0

−h1

on J, where

Rt

Bk(t) = H(t, s)k(s)ds,

Rh2

kt (s)ds + L4 kt (s)ds, 0

k1 = maxt,s∈J H(t, s),

t0

and L1 , L2 , L3 , L4 > 0 satisfying [L2 k1 (T − t0 ) + L3 h1 + L4 h2 ] < L1 , kt0 (s) ≤ 0, kT ≤ 0. Then, k(t) ≤ 0 on J. Now we state the criteria for obtaining the unique solution of InDE with R and A (12) and (13). Theorem 9. Suppose that (H1 ) to (H5 ) are satisfied. If in addition, (H6 ) h(t, x, ξ1 , φ1 , ψ2 ) − h(t, y, ξ2 , φ2 , ψ1 ) ≤ −L1 (x − y) + L2 (ξ1 − ξ2 ) + L3

R0

(φ1 − φ2 )(s)ds

−h1

Rh2

+ L4 (ψ1 − ψ2 )(s)ds, 0

where v0 ≤ y ≤ x ≤ w0 , Bv0 ≤ ξ2 ≤ ξ1 ≤ Bw0 , v0t0 ≤ φ2 ≤ φ1 ≤ w0t0 , vt0 ≤ ψ2 ≤ ψ1 ≤ wt0 , L1 , L2 , L3 , L4 ≥ 0, [L2 k1 T + L3 h1 + L4 h2 ] < L1 . Then, ρ = x = r is the unique solution for InDE with R and A (12) and (13) on J. Remark 1. Observe that if h is restricted to h ∈ C[J × R × R × C1, R] then the InDE with R and A (12) and (13) is reduced to an InDE with retardation. If h ∈ C[J × R × R, R] then the system InDE with R and A (12) and (13) is reduced to an ordinary InDE. In both the cases the above theorem holds with suitable modifications.

6.2.

PBVP of InDE

We present below MIT developed for PBVP of InDE using coupled upper and lower solutions of Type I. This result is interesting due to the following reason.

16

CH. V. Sredhar, I. S. N. R. G. Bharat and J. Vasundhara Devi

The authors in [5] considered the iterates of the linear InDEs as an IVP with a clever choice of the initial value as the boundary value of the previous iterate. This choice resulted in obtaining the existence of solution. Thus there was no need to prove the existence of solution separately. u0 (t) = h (t, u, Bu) + l (t, u, Bu) , t ∈ J = [0, T ] , u (0) = u (T ) , where h, l ∈ C [J × R × R, R], Bu (t) =

Rt

(16)

H (t, s) u (s)ds, and H ∈ C [J × R, R+]

0

To develop the MIT, the following definitions and comparison result are required. Definition 5. Functions v0 , w0 ∈ C1 [J, R] are said to be coupled lower and upper solutions of Type-I for PBVP of InDE (16) if v0 0 (t) 6 h (t, v0 , Bv0 ) + l (t, w0 , Bw0 ) , t ∈ J, v0 (0) 6 v0 (T ) ; and w0 0 (t) > h (t, w0 , Bw0 ) + l (t, v0 , Bv0 ) , t ∈ J, w0 (0) > w0 (T ) .

Definition 6. Two functions v0 , w0 ∈ C1 [J, R] are said to be coupled lower and upper solutions of Type-II for PBVP of InDE (16) if v0 0 (t) 6 h (t, w0 , Bw0 ) + l (t, v0 , Bv0 ) , t ∈ J, v0 (0) 6 v0 (T ) ; and w0 0 (t) > h (t, v0 , Bv0 ) + l (t, w0 , Bw0 ) , t ∈ J, w0 (0) > w0 (T ) .

Lemma 10. Let q ∈ C1 [J, R] be such that q0 6 −Mq + NBq and q (0) 6 q (T ), where constants M > 0, N > 0, and k0 = max {H (t, s) : (t, s) ∈ J × J} satisfy Nk0 T < M. Then, q (t) 6 0 for t ∈ J.

An Overview of Existence Results for Some Integro-Differential ...

17

Using the above definitions, the comparison result and the fact that the corresponding IVP of linear integro-differential equations have unique solutions. We now state the MIT in this setup. Theorem 11. Suppose that 1. v0 , w0 are coupled lower and upper solutions of Type-I relative to the PBVP of InDE (16) on J; 2. The forcing functions h(t, u, Bu) and l(t, u, Bu) are nondecreasing and nonincreasing respectively in the last two variables. Then, the iterative scheme vn 0 (t) = h (t, vn−1, Bvn−1 ) + l (t, wn−1 , Bwn−1 ), t ∈ J, vn (0) = vn−1 (T ) ; and wn 0 (t) = h (t, wn−1, Bwn−1 ) + l (t, vn−1 , Bvn−1 ), t ∈ J, vn (0) = vn−1 (T ) ; yields two monotone sequences {vn } and {wn } such that v0 6 v1 6 v2 6 ... 6 vk 6 ... 6 wk 6 ... 6 w2 6 w1 6 w0 on J. Further vn → ρ and wn → r, where ρ and r are coupled minimal and maximal solutions of the PBVP of InDE (16) on J with ρ ≤ r on J. Also, ρ0 (t) 6 h (t, ρ, Bρ) + l (t, r, Br) , t ∈ J, ρ (0) = ρ (T ); and r0 (t) > h (t, r, Br) + l (t, ρ, Bρ) , t ∈ J, r (0) = r (T ) .

The following theorem deals with the uniqueness of the solutions of PBVP of InDE (16) on J. Theorem 12. If, in addition to the assumptions of Theorem 11, the forcing functions h(t, u, Bu) and l(t, u, Bu) satisfy h (t, u1 , Bu1) − h (t, u2 , Bu2 ) 6 M1 (u1 − u2 ) + N1 B (u1 − u2 ), t ∈ J; l (t, u1, Bu1 ) − l (t, u2 , Bu2 ) 6 M2 (u1 − u2 ) + N2 B (u1 − u2 ), t ∈ J. whenever u1 6 u2 , Mi > 0, Ni > 0, for i = 1, 2, and M1 > M2 . Then, ρ = r on J. Consequently PBVP of InDE (16) has a unique solution on J.

18

CH. V. Sredhar, I. S. N. R. G. Bharat and J. Vasundhara Devi

In place of natural upper and lower solutions of PBVP of InDE (16), if coupled lower and upper solutions of Type-II are considered then the iterative scheme guarantees coupled, intertwining, alternating sequences that converge uniformly and monotonically to the coupled minimal and maximal solutions of PBVP of InDE (16) on J. We state this result from [5] Theorem 13. Let the hypothesis of Theorem 11, hold. Then, for any solution u (t) for PBVP of InDE (16) with v0 (t) 6 u (t) 6 w0 (t) on J, the iterative scheme vn 0 (t) = h (t, wn−1, Bwn−1 ) + l (t, vn−1 , Bvn−1 ), t ∈ J, vn (0) = vn−1 (T ) ; and wn 0 (t) = h (t, vn−1, Bvn−1 ) + l (t, wn−1 , Bwn−1 ), t ∈ J, vn (0) = vn−1 (T ) ; yields the intertwined alternating sequences {v2n , w2n+1 }, {w2n , v2n+1 } satisfying v0 6 w1 6 v2 6 ... 6 v2n 6 w2n+1 6 u 6 v2n+1 6 w2n ... 66 w2 6 v1 6 w0 on J, for every n ≥ 1. Moreover the sequences {v2n , w2n+1} , {w2n , v2n+1 } converge to ρ and r respectively on J, where ρ and r are coupled minimal and maximal solutions for PBVP of InDE (16), with ρ ≤ r on J. The same technique was extended for nonlinear boundary value problems with casual operators in [14].

6.3.

MIT for Higher Order Integro-Differential Equations

We consider the nonlinear fourth-order InDE given by y(4) (x) = C (x, y (x) , Ty (x)) ,

(17)

with the Navier boundary conditions y (0) = y (1) = y00 (0) = y00 (1) = 0,

(18)


R1 where C ∈ C [0, 1] × R2 , R , Ty (x) = h (x,t)y (t)dt, h ∈ C ([0, 1] × [0, 1], R). 0

The BVP of higher-order InDE (17) and (18) is a generalization of the linear fourth-order problem. y(4) (x) + λy (x) − µ

Z1 0

h (x,t)y (t)dt = p (x) , x ∈ (0, 1) ,

(19)

An Overview of Existence Results for Some Integro-Differential ... y (0) = y (1) = y00 (0) = y00 (1) = 0,

19 (20)

where λ and µ are constants, p ∈ C [0, 1] . It is known that for general second-order DEs with various types of boundary conditions, the existence of lower and upper solutions satisfying v0 ≤ w0 is sufficient for existence of a solution in a sector. But in [15] there is a counterexample to show that this result does not hold for fourth-order DEs. The application of the method lower and upper solutions for boundary value problems of fourth order DEs is highly dependent on the conclusion of the maximum principle for the corresponding linear operators. In [6], in view of this observation, they developed first the maximum principle and then the MIT for linear higher- order InDE (19), (20) and then the MIT has been developed. We proceed to present the established results from [6]. Consider the nonlocal nonhomogeneous boundary value problem (4)

y

(x) + λy (x) − µ

Z1

h (x,t)y (t)dt = p (x) , x ∈ (0, 1) ,

(21)

0

y (0) = M, y (1) = N, y00 (0) = P, y00 (1) = Q.

(22)

Theorem 14. Assume that (H1 ) 0 < λ 6 c1 ≈ 950.8843 and (H2 ) khk∞ = max {h (x,t)}
0, v000 (1) > 0,

(24)

R1

where T v0 (x) = h (x,t)v0 (t)dt 0

Definition 8. The function w0 ∈ C4 [0, 1] is said to be an upper solution for the BVP of higher-order InDE (17), (18) if (4)

w0 (x) > C (x, w0 (x) , T w0 (x)) , x ∈ (0, 1) ,

(25)

w0 (0) > 0, w0 (1) > 0, w0 (0) 6 0, w000 (1) 6 0,

(26)

R1

where Tw0 (x) = h (x,t)w0 (t)dt. 0

We now state the MIT from [6], Theorem 16. Let v0 , w0 be lower and upper solutions for the nonhomogeneous BVP of higher-order InDE
(17), (18) which satisfy v0 (x) 6 w0 (x) for x ∈ [0, 1] and C ∈ C [0, 1] × R2 , R , h > 0. If there exist two constants λ, µ > 0 satisfying 1 R1 such that λ < c1 ≈ 125.137 and khk∞ < |µ| max

0 G(x,s)ds x∈[0,1]

C (x, r1 , s1 ) −C (x, r2 , s2 ) > −λ (r1 − r2 ) + µ (s1 − s2 ),

(27)

for v0 (x) 6 r1 6 r2 6 w0 (x) and Z1 0

h (x,t) v0 (t) dt 6 s1 6 s2 6

Z1

h (x,t)w0 (t)dt, x ∈ [0, 1],

0

then, by the maximum principle the iterative sequences {vn } and {wn } with the initial functions y0 = v0 = w0 , satisfy vn−1 6 vn 6 wn 6 wn−1 and converge

An Overview of Existence Results for Some Integro-Differential ...

21

uniformly to the extremal solutions of BVP of higher-order InDE (17), (18) in [v0 , w0 ] . In [7], the authors using the method of upper and lower solutions together with the monotone iterative technique, developed the existence of maximal and minimal solutions under reversed ordering condition. Consider the class of integro-differential equations with integral boundary conditions x0 (t) = C (t, x (t), Hx (t) , Sx (t)), t ∈ J, x (0) + λ

ZT

(28)

x (s)ds = x (T ),

(29)

0

where


J = [0, T ], T > 0, C ∈ C J × R3 , R ,

H (t, s) ∈ C (D, R+),

Rt

Sx (t) = h (t, s) x (s)ds, where h (t, s) ∈ C (J × J, R+ ), R+ = [0, +∞), λ > 0.] 0 and D = (t, s) ∈ R2 , 0 6 s 6 t 6 T . Note that if λ = 0, in (29), then class of InDE (28), (29) is a PBVP of InDE. We state the following lemmas from [7]. Lemma 17. Assume that there exist P > 0, Q, R > 0 such that 0

x (t) > Px (t) + Q

Zt

k (t, s) x (s)ds + R

0

ZT

h (t, s) x (s)ds, t ∈ J,

0



ePT (Qk0 + Rh0 ) 1 − e−PT ePT − 1 < 1. x (0) > x (T ) P2 Then, x(t) ≤ 0, where k0 = max {k (t, s) : (t, s) ∈ D} , h0 = max {k (t, s); (t, s) ∈ J × J} . Lemma 18. Assume that P > 0, Q, R > 0, σ (t) ∈ C [J, R] and then, the equation x0 (t) − Px (t) − Q

Zt 0

R (t, s) x (s)ds − L

ZT 0

(Qk0 +Rh0 )T P

h (t, s)x (s) ds = σ (t), t ∈ J

< 1,

22

CH. V. Sredhar, I. S. N. R. G. Bharat and J. Vasundhara Devi x (0) + d = x (T ), d ∈ R,

has a unique solution. The following definitions deal with lower and upper solutions of class of InDE with integral boundary condition (28), (29). Definition 9. The functions v0 , w0 ∈ C0 [J, R] are said to be the lower and upper solutions of class of InDE with integral boundary condition (28), (29), respectively if v00 (t) 6 C (t, x (t), Hx (t), Sx (t)) , t ∈ J v0 (0) + λ

ZT

x (s) ds 6 v0 (T ),

0

w00 (t) > C (t, x (t) , Hx (t) , Sx (t)), w0 (0) + λ

ZT

t ∈ J,

x (s) ds > w0 (T ).

0

The following assumptions are necessary to develop the MIT for InDE with integral boundary condition (28), (29). (H1 ) v0 , w0 ∈ C1 [J, R] are lower and upper solutions of class of InDE with integral boundary conditions (28), (29), respectively then w0 (t) 6 v0 (t) for t ∈ J ;
(H2 ) C ∈ C J × R3 , R ; (H3 ) there exist P > 0, Q, R > 0 such that C (t, m2 , n2 , l2 ) −C (t, m1 , n1 , l1 ) 6 −P (m2 − m1 ) + Q (n2 − n1 ) + R (l2 − l1 ) if w0 6 m1 6 m2 6 v0 , and Kw0 6 Kn1 6 Kn2 6 v0 , Sw0 6 Sl1 6 Sl2 6 Sv0 , t ∈ J; (H4 ) (H5 )

ePT (Qk0 +Rh0 )(1−e−PT )(ePT −1) P2 (Qk0 +Rh0 )T P

61

< 1 where k0 = max {K (t, s) : (t, s) ∈ D},

An Overview of Existence Results for Some Integro-Differential ...

23

h0 = max {S (t, s) ; (t, s) ∈ J × J} The following lemma deals with lower and upper solutions satisfying reverse inequalities. Lemma 19. Assume that (H1 ) − (H5 ) hold if 0

y (t) − Py (t) − Q

Zt

ZT

K (t, s) y (s)ds − R

0

S (t, s) y (s)ds =

0

C (t, v0 (t) , Hv0 (t), Sv0 (t)) − Pv0 (t) − Q

Zt

K (t, s) v0 (s) ds

0

−R

ZT

S (t, s)v0 (s)ds, t ∈ J

0

y (0) + λ

ZT

v0 (s)ds = y (T )

0

and 0

z (t) − Pz (t) − Q

Zt

K (t, s)z (s)ds − R

0

ZT

S (t, s) z (s) ds =

0

C (t, w0 (t), Kw0 (t), Sw (t)) − Pw0 (t) − Q

Zt

K (t, s) w0 (s)ds

0

−R

ZT

S (t, s)w0 (s)ds, t ∈ J

0

z (0) + λ

ZT

w0 (s)ds = z (T )

0

then, w0 (t) 6 z (t) 6 y (t) 6 v0 (t), t ∈ J Using the reverse inequalities for lower and upper solutions, the MIT is developed for InDE with integral boundary conditions (28), (29). This result is stated below without proof [7].

24

CH. V. Sredhar, I. S. N. R. G. Bharat and J. Vasundhara Devi

Theorem 20. Suppose that (H1 ) − (H5 ) hold. Then, there exist monotone sequences {vn }, {wn } that converge uniformly and monotonically to ρ, and r respectively on J. Moreover, ρ and r are maximal and minimal solutions of InDE with integral boundary conditions (28), (29) in [w0 , v0 ] = {g ∈ C0 (J, R) : w0 6 g 6 v0 }

7.

Quasilinearization

Quasilinearization is another constructive iterative technique yielding quadratic convergence. This method coupled with the method of lower and upper solutions yields a unique solution of the considered problem. The quasilinearization technique was developed using coupled lower and upper solutions for an IVP of InDE and then this result was used to obtain existence and uniqueness of solutions for the corresponding PBVP of an InDE given by z0 = h1 (t, z, Bz) + h2(t, z, Bz), h(0) = h(T ).

(30) (31)

In order to develop the quasilinearization technique for the PBVP of InDE (30) and (31), the quasilinearization technique for the following initial value problem of an InDE was first obtained. z0 = h1 (t, z, Bz) + h2(t, z, Bz), h(0) = h0 .

(32) (33)

Rt

where h1 , h2 ∈ C[I ×Rn ×Rn , Rn ], Bx(t) = H(t, s)x(s)ds, with H ∈ C[I ×I, R+ ] 0

and I = [0, T ].

Before developing the quasiliearization technique for the IVP of an InDE (32) and (33), we present the following basic definitions. Definition 10. Let v0 , w0 ∈ C1 [I, Rn ]. Then, v0 , w0 are said to be (a) natural lower and upper solutions of PBVP of InDE (32) and (33) if  v0 ≤ h1 (t, v0, Bv0 ) + h2 (t, v0, Bv0 ), v0 (0) ≤ z0 , w00 ≥ h1 (t, w0 , Bw0 ) + h2 (t, w0 , Bw0 ), w0 (0) ≥ z0 , t ∈ I;

(34)

An Overview of Existence Results for Some Integro-Differential ...

25

(b) coupled lower and upper solutions of Type I for PBVP of InDE (32) and (33) if  v00 ≤ h1 (t, v0 , Bv0 ) + h2 (t, w0 , Bw0 ), v0 (0) ≤ z0 , (35) w00 ≥ h1 (t, w0 , Bw0 ) + h2 (t, v0 , Bv0 ), w0 (0) ≥ z0 , t ∈ I; (c) coupled lower and upper solutions of Type II for PBVP of InDE (32) and (33) if  v00 ≤ h1 (t, w0 , Bw0 ) + h2 (t, v0 , Bv0 ), v0 (0) ≤ z0 , w00 ≥ h1 (t, v0 , Bv0 ) + h2 (t, w0 , Bw0 ), w0 (0) ≥ z0 , t ∈ I;

(36)

(d) coupled lower and upper solutions of Type III for PBVP of InDE (32) and (33) if  v00 ≤ h1 (t, w0 , Bw0 ) + h2 (t, w0 , Bw0 ), v0 (0) ≤ z0 , (37) w00 ≥ h1 (t, v0, Bv0 ) + h2 (t, v0, Bv0 ), w0 (0) ≥ z0 , t ∈ I. We observe that the lower and upper solutions defined by (34) and (37) reduce to (35) and (36) when v(t) ≤ w(t), t ∈ I. This is due to the fact that h1 (t, z, ξ) is nondecreasing in z and y and h2 (t, z, ξ) is nonincreasing in z and y for each t ∈ I, and hence it is sufficient to investigate the cases (35) and (36). We begin with a comparison result, which will be useful in establishing quasilinearization. Lemma 21. Let q ∈ C1 [I, R], where I = [0, T ] is such that q 0 (t) ≤ −Mq(t) − NBq(t) on I, q(0) ≤ 0,

(38)

where M > 0, N ≥ 0 are constants such that Nk1 T (eMT − 1) ≤ M,

(39)

where k1 = maxt∈I H(t, s). Then, q(t) ≤ 0 on I. The following assumptions are useful for developing the quasilinearization technique for PBVP of InDE (30) and (31).

26

CH. V. Sredhar, I. S. N. R. G. Bharat and J. Vasundhara Devi

(H1 ) (i) The second-order Frechet derivatives of h1 (t, z, ξ), h2 (t, z, ξ) with respect to all variables exist and are bounded; (ii) h1 (t, z, ξ) is convex in z, ξ ; (iii) h1z (t, z, ξ) is nondecreasing in ξ for each (t, z); (iv) h1 is nondecreasing function in z, ξ for each t ∈ I and h2 is nonincreasing function in z, ξ for each t ∈ I. (H2 ) (i) −M1 ≤ h1z (t, z, ξ) ≤ −M, 0 < M < M1 ; (ii) −M2 ≤ h1ξ (t, z, ξ) ≤ −N, 0 < N < M2 ; (iii) Nk1 T < M; where M > 0, N ≥ 0. (H3 ) v0 , w0 are coupled lower and upper solutions IVP of InDE (32) and (33). (H4 ) (i) h1 (t, z, Bz) ≥ h1 (t, y, By) + f 1z(t, y, By)(z − y) + h1ξ(t, y, By)(Bx − By); (ii) |h1z(t, z, Bz) − h1y(t, y, By)| ≤ L1 (z − y) + M1 (Bz − By), L1 , M1 ≥ 0

Theorem 22. Suppose that the assumptions (H1 ) to (H4 ) are satisfied. Then, there exists a monotone sequence {vn }, such that vn → ρ, as n → ∞ uniformly and monotonically to the unique solution ρ of the IVP of InDE (32) and (33)on I and the convergence is quadratic. Next, an existence and uniqueness result is obtained for an PBVP of InDE. z0 = h1 (t, z, Bz) + h2(t, z, Bz), z(0) = z(T ),

(40) (41)

Rt

where h1 , h2 ∈ C[I ×Rn ×Rn , Rn ], Bx(t) = H(t, s)x(s)ds, and K ∈ C[I ×I, R+ ], 0

I = [0, T ].

The following definitions give the various types of lower and upper solutions corresponding to the PBVP of InDE (40) and (41).

An Overview of Existence Results for Some Integro-Differential ...

27

Definition 11. Let v0 , w0 ∈ C1[I, Rn]. Then, v0 , w0 are said to be (i) natural lower and upper solutions for PBVP of InDE (40) and (41) if v00 ≤ h1 (t, v0, Bv0 ) + h2 (t, v0, Bv0 ), v0 (0) ≤ v0 (T ), 0 w0 ≥ h1 (t, w0, Bw0 ) + h2 (t, w0, Bw0), w0 (0) ≥ w0 (T ), t ∈ I;



(ii) coupled lower and upper solutions of Type III for PBVP of InDE (40) and (41) if  v00 ≤ h1 (t, w0, Bw0) + h2 (t, w0, Bw0 ), v0 (0) ≤ v0 (T ), w00 ≥ h1 (t, v0 , Bv0 ) + h2 (t, v0, Bv0 ), w0 (0) ≥ w0 (T ), t ∈ I.

(42)

(43)

Now we present the quasilinearization technique to establish the existence of a unique solution of the PBVP of InDE (40) and (41). It is interesting to observe that the proof of the theorem follows the technique developed in [5]. Theorem 23. Suppose that the assumptions of Theorem 22 are satisfied. Then, there exists monotone sequence {vn }, such that vn → ρ, as n → ∞ uniformly and monotonically to the unique solution ρ for PBVP of InDE (40) and (41) on I and the convergence is quadratic. In [9] the quasilinearization is developed for the integro-differential equation with retardation and anticipation given by z0 = h(t, z, Bz, zt, zt ), t ∈ J = [0, T ], z0 = φ0 , zT = ψ0 ,

(44) (45)

where φ0 ∈ C1 , ψ0 ∈ C2 , h ∈ C[J × R × R ×C1 ×C2 , R]. and Bz(t) =

Zt

H(t, s)z(s)ds

t0

where H ∈ C[J × J, R+ ], C1 = C[[−h1 , 0], R] and C2 = C[[0, h2], R]. In order to state an existence and uniqueness result using quasilinearization we list the following assumptions relative to InDE with R and A (44) and (45) for convenience. (H1 )(i) All second order frechet derivatives of h(t, z, ξ, φ, ψ) exist and

28

CH. V. Sredhar, I. S. N. R. G. Bharat and J. Vasundhara Devi are bounded; (ii) h(t, z, ξ, φ, ψ) is convex in z, ξ, φ and is concave in ψ; (iii) hz (t, z, ξ, φ, ψ) is nondecreasing in ξ for each (t, z, φ, ψ) and is nondecreasing in φ for each (t, z, ξ, ψ) and nondecreasing in ψ for each (t, z, ξ, φ); (iv) hξ (t, z, ξ, φ, ψ) is nondecreasing in z for each (t, ξ, φ, ψ) and is nondecreasing in φ for each (t, z, ξ, ψ) and is nondecreasing in ψ for each (t, z, ξ, φ); (v) hφ (t, z, ξ, φ, ψ) is nondecreasing in z for each (t, ξ, φ, ψ) and is nondecreasing in ξ for each (t, z, φ, ψ) and is idependent of ψ for each (t, z, ξ, φ); (vi) hψ(t, z, ξ, φ, ψ) is nondecreasing in z for each (t, ξ, φ, ψ) and is nondecreasing in ξ for each (t, z, φ, ψ) and is idependent of φ for each (t, z, ξ, ψ);

(H2 ) (i) −M1 ≤ hz(t, z, ξ, φ, ψ) ≤ −N1 , (ii) −M2 ≤ hξ (t, z, ξ, φ, ψ) ≤ N2 , (iii) −M3

R0

0 < N1 < M1 ; 0 < N2 < M2 ;

η1 (s)ds ≤ hφ (t, z, ξ, φ, ψ)η1 ≤ N3

−h1

R0

η1 (s)ds, 0 < N3 < M3 ,

−h1

where η1 ∈ C1 ; Rh2

(iv) 0 ≤ hψ (t, z, ξ, φ, ψ)η2 ≤ N4 η2 (s)ds, where η2 ∈ C2 , N4 > 0; 0

(v) [N2 k1 T + N3 h1 + N4 h2 ] < N1 ; (H3 ) v0 , w0 are natural lower and upper solutions of InDE with R and A (44) and (45) ,that is, v0 , w0 ∈ C1 [J, R], where J = [0, T ] v00 ≤ h(t, v0, Bv0 , vot , v0t ) v00 = φ1 , v0 T = ψ1 , w00 ≥ h(t, w0, Bw0 , wot , w0 t ) w00 = φ2 , w0 T = ψ2 , where v0 ≤ w0 , Bv0 ≤ Bw0 , t ∈ J and φ1 , φ2 ∈ C1 , ψ1 , ψ2 ∈ C2 ,

An Overview of Existence Results for Some Integro-Differential ...

29

such that φ1 ≤ φ0 ≤ φ2 and ψ1 ≤ ψ0 ≤ ψ2 ; (H4 ) v00 − φ0 , φ0 − w00 satisfy the assumptions of Lemma (6). Theorem 24. Suppose that the assumptions (H1 ) to (H4 ) are satisfied. Then, there exist monotone sequences {vn }, {wn }, which converge uniformly on J to a unique solution of InDE with R and A (44), (45) and the convergence is quadratic.

Conclusion In this chapter we briefly described different types of integro-differential equations, and presented some criteria for the existence and uniqueness of the solution of a nonlinear Volterra integro-differential equation. Further we discussed the monotone iterative technique for integro-differential equations with retardation and anticipation, periodic boundary-value problems of integro-differential equations, and higher order integro differential equations. Next, quasilinearization for periodic boundary-value problems of integro-differential equations, and integro-differential equations with retardation and anticipation were given.

References [1] Vito Volterra, Theory of Functionals and of Integral and Integrodifferential equations , Dover Publications, New York, 1959. [2] Addul-Majid Wazwaz, Linear and Nonlinear Integral Equations Methods and Applications, Higher Education Press, Beijing and SpringerVerlag Berlin Heidelberg Dordrecht, London, New York, 2011. [3] G. S. Ladde, V. Lakshmikantham and A. S. Vatsala, Monotone Iterative Technique for Nonlinear Differential Equations, Pitman Publishing INC., 1985. [4] V. Lakshmikantham, A. S. Vatsala, Generalized Quasilinearization for Nonlinear Problems , Kluwer Academic Publishers, Netherlands, 1998.

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CH. V. Sredhar, I. S. N. R. G. Bharat and J. Vasundhara Devi

[5] S. G. Pandit, D. H. Dezerna, J. O. Adeyeye, Periodic boundary value problems for nonlinear integro differential equations, Proceedings of Neural, Parallel, and Scientific Computations , 4 (2010) 316-320. [6] Jinxiang Wang, Monotone iterative technique for nonlinear fourth order integro-differential equations, arXiv:2003.04697v1 [Math. CA], pg. 113, 2020. [7] Guoping Chen, Jianhua Shen,Monotone iterative technique for a class of integro-differential equations with integral boundary conditions, International Journal of Math. Analysis, Vol. 1, No.3, 2007, 113-122. [8] Ch. V. Sreedhar, J.Vasundhara Devi, Generalized Quasilinearization using coupled lower and upper solutions of periodic boundary value problem of an integro-differential equation, European Journal of Pure and Applied Mathematics, Vol. 12, No. 4, 2019, 1662-1675. [9] J.Vasundhara Devi, Ch. V. Sreedhar, Quasilinearization for integrodifferential equations with retardation and anticipation, Nonlinear Studies, Vol. 19, (2012), no. 1, 99-122. [10] V. Lakshmikantham, M. Rama Mohana Rao, Theory of integrodifferential equations, Gordon and Breach Science Publishers, S.A, 1995. [11] V. Lakshmikantham, S. Leela and M. Rama Mohana Rao, D Integral and Integro-differential Inequalities, Vol. I, Applicable Analysis, 34 (1987) 157-164. [12] V. Lakshmikantham, S. Leela, Differential and Integral Inequalities, Vol. I, Academic press, New York, 1969. [13] J. Vasundhara Devi, Ch. V. Sreedhar, S. Nagamani, Monotone iterative technique for integro-differential equations with retardation and anticipation, Communicaiton in Applied Analysis, 14 (2010), No.4, 325-336. [14] Wen-Li Wang, Jing-Feng Tian, Generalized monotone iterative method for nonlinear boundary value problems with casual operators, Boundary Value Problems, 2014, 1-12. [15] A. Cabada, J. A. Cid, L. Sanchez, Positivity and lower and upper solutions for fourth order boundary value problems, Nonlinear Analysis, Vol. 67, 2007, 1599-1612.

In: Understanding Integro-Differential Equations ISBN: 979-8-89113-040-1 c 2023 Nova Science Publishers, Inc. Editors: J. Vasundhara Devi et al.

Chapter 2

A B RIEF OVERVIEW OF THE S TABILITY T HEORY OF I NTEGRO -D IFFERENTIAL E QUATIONS Z. Drici1 F. A. McRae2 and J. Vasundhara Devi3 ∗ 1 Department of Mathematics, lllinois Wesleyan University, Bloomington, Illinois, USA 2 Department of Mathematics, Catholic University of America, Washington DC, USA 3 Department of Mathematics and GVP Prof. V. Lakshmikantham Institute for Advanced Studies Gayatri Vidya Parishad College of Engineering, Visakhapatnam, AP India

Abstract In this chapter an overview of the theory of Lyapunov stability of integrodifferential equations (InDEs) is presented. The chapter is divided into two parts. In the first part the stability theory of different types of linear InDEs is discussed and different approaches to study stability concepts are given. The second part covers the stability theory of nonlinear Volterra ∗ Corresponding Author’s

Email: [email protected].

32

Z. Drici, F. A. McRae and J. V. Devi InDEs using the comparison method. Various types of stability notions, such as, practical stability and stability in terms of two measures are some of the topics discussed.

Keywords: Lyapunov stability, comparison theorems, practical stability, stability in two measures AMS Subject Classification: 34D05, 34D20

1.

34A08, 34A12, 34A34, 34A37, 34A45,

Introduction

In this section we will briefly introduce the concepts leading to Lyapunov stability and then proceed to trace the developments in the stability theory in general. The word stability is derived from the Latin ’stare’ and is used in the sense of staying firm or durable. This word was first used in mechanics in the definition of a type of equilibrium of a material particle or a system. As this definition was inadequate to study many other setups, new definitions were introduced. Foremost among these definitions is Lyapunov stability[1], as it pertains to a general differential equation and not to any specific particle, and it applies to a solution of a differential equation and not only to an equilibrium or a critical point. In addition to the above features, the beauty of Lyapunov’s ideas is that the approach and the theoretical concepts involved do not require knowledge of the solution of the problem under study[1]. Further, they can be adapted to study other important aspects of system behaviour. Both theorems introduced by Lyapunov involve a function V (t, x) called Lyapunov function. The criteria satisfied by the Lyapunov function determine the types of stability possessed by the solution of the problem under consideration. Since its introduction, the usefulness of the Lyapunov function has increased substantially. It is employed to study various qualitative and quantitative properties of dynamic systems. It is also used as a vehicle to reduce the study of a complicated system to the study of a relatively simpler system. The properties of the complicated system are then assessed using the properties of the simple system. In many situations a system may be unstable in the sense of Lyapunov stability but for all practical purposes it is stable. An example of such a system is an aeroplane under turbulence. To accommodate such situations the notion

A Brief Overview of the Stability Theory ...

33

of practical stability [2] was introduced. It must be noted that this concept is neither weaker nor stronger than Lyapunov stability. The notions of Lyapunov stability enriched the stability theory by giving rise to many new concepts such as, eventual stability, partial stability, and relative stability. These concepts were also extended to practical stability and to boundedness of solutions, also known as Lagrange stability. As generalization of ideas is core to development in mathematics, it is natural to ask whether all the above mentioned concepts could be unified. This resulted in the notion of stability in terms of two measures [3], which provides a unified structure to study several nonlinear problems. The literature dealing with the stability theory of InDEs is vast [4] and varied in the sense that many types of InDEs were studied. InDEs are broadly classified as linear and nonlinear InDEs. The linear InDEs studied include both scalar InDEs and systems of InDEs (InDS). The development of the theory of linear InDSs and InDEs is similar to that of ordinary differential sytems and ordinary differential equations (ODEs). One approach is to consider the integral part of the InDE as a perturbation of the ODE and the InDE is studied using the theory of ODEs. The study of nonlinear InDEs is comparatively difficult and one way to study them is to consider the nonlinear term as a perturbation of the linear InDE. In this chapter, an attempt is made to showcase several types of linear InDEs and present the stability results pertaining to them. With respect to nonlinear InDEs the stability results presented here deal with Lyapunov theorems in this setup and their generalizations. These results are developed using comparison theorems and by reducing the study of a complicated system to the study of a simpler system using a Lyapunov function or a Lyapunov-like function [4].

2. 2.1.

Part I - Linear Integro-Differential Equations InDE of Convolution Type

A very simple type of integro-differential equations is the convolution type. Since the integral can be considered as a convolution, one can use Laplace transforms to obtain a solution of the InDE. We consider an InDS of convolution type given by, y0 (t) = Cy(t) +

Z t 0

H(t − s)y(s)ds

(1)

34

Z. Drici, F. A. McRae and J. V. Devi

where C is an nxn constant matrix and H(t) is an nxn continuous matrix on R+. Suppose y(t0 ) = φ, where φ(t) exists for 0 ≤ t ≤ t0 , t0 > 0. System (1) can be written as y(t) = Z(t − t0 )φ(t0) +

Z t t0

Z(t − s)[

Z t0 0

H(s − σ)φ(σ)dσ]ds

(2)

where Z(t) is a solution of the equation, Z 0 (t) = CZ(t) +

Z t 0

H(t − s)Z(s)ds, Z(0) = I.

(3)

Define p(t) = 0∞ | 0t Z(t − σ)H(σ + u)dσ|du and assume that it exists and is finite for all t ≥ 0. Before proceeding to define the stability concepts, we need the following notation. Let t ≥ 0 be fixed. Set B(t) = {φ : [0,t] → R, φ is continuous and |φ|t = max0≤s≤t |φ(s)| is bounded} We now present the definition of stability for InDS (1) and proceed to give some stability results from [4, 5] R

R

Definition 1. The trivial solution of InDS (1) is said to be stable if given ε > 0 and t0 ∈ R+ , there exists a δ = δ(t0 , ε) > 0 such that |φ|t0 = max0≤s≤t0 |φ(s)| < δ implies |y(t)| < ε. Definition 2. The trivial solution of InDS (1) is said to be uniformly stable if the δ in the above Definition 1 depends only on ε, t ≥ t0 . Definition 3. The trivial solution of InDS (1) is said to be quasi-equiasymptotically stable or attractive if given ε > 0 and t0 ∈ R+ , there exist a δ = δ(t0 ) and a T (t0, ε) such that |φ|t0 = max0≤s≤t0 |φ(s)| < δ implies |y(t)| < ε, for all t ≥ t0 + T . Definition 4. The trivial solution of InDS (1) is said to be quasi-uniformlyasymptotically stable or uniformly attractive, if the δ andT in the Definition 3 depend only on ε. Definition 5. The trivial solution of InDS (1) is asymptotically stable if it satisfies both Definitions 1 and 3. Definition 6. The trivial solution of InDS (1) is uniformly asymptotically stable if it satisfies both Definitions 2 and 4.

A Brief Overview of the Stability Theory ...

35

Remark 1. The definitions of various types of stability for both the InDEs and InDSs are the same except that the norm for InDE is absolute value. We now state the following stability result from [5]. Theorem 1. Suppose that H ∈ L1 (R+). Then, the equilibrium solution y ≡ 0 of InDS (1) is 1. Uniformly stable if and only if the two functions Z(t) and p(t) are uniformly bounded on R+ , and 2. Uniformly asymptotically stable if and only if it is uniformly stable and the functions Z(t) and p(t) → 0 as t → ∞.

2.2.

InDE of Convolution Type with Infinite Memory

We make a change in the lower limit of the integral in InDS (1) and consider InDS of the type y0 (t) = Cy(t) +

Z t

−∞

H(t − s)y(s)ds,

(4)

for t ≥ t0 ≥ 0 with y(t) = φ(t) on −∞ ≤ t ≤ t0 for some t0 > 0, where φ is a continuous function in Rn and the norm is suitably defined. Then, the various stability properties of the zero solution of InDS (4) can be defined in a similar fashion as that of InDS (1). We next present a result from [6] that establishes an equivalent criterion between InDS (1) and InDS (4). Theorem 2. Suppose that H ∈ L1 (R+ ). Then the following statements are equivalent: 1. The zero solution of InDS (1) is uniformly stable; 2. The zero solution of InDS (4) is uniformly stable; 3. The function Z(t) is bounded and for each φ ∈ C(R), the solution y(t, 0, φ) of equation ( 4) is bounded on R+ .

36

2.3.

Z. Drici, F. A. McRae and J. V. Devi

Integral as Perturbation

An InDE can be considered as a perturbation of an ODE. We now present two results in that setup from [7]. Consider a linear InDS of the type y0 (t) = C(t)y(t) +

Z t 0

H(t − s)y(s)ds

(5)

and its corresponding ordinary differential system x0 (t) = C(t)x(t).

(6)

Let X(t) be the fundamental matrix solution of the InDS (6). We now state the following theorems that deal with the stability and asymptotic stability of InDS (5). Theorem 3. Assume that |X(t)X −1(s)| ≤ L for t ≥ s ≥ 0 and R∞Rt 0 0 |H(t, s)|ds dt ≤ K. Then, the trivial solution of InDS (5) is uniformly stable. Theorem 4. Assume that 0t |X(t)X −1(s)|ds ≤ M for t ≥ 0 and R R supt≥0 0t |H(t, s)|ds ≤ M1 . Further, let lims→∞ 0t |H(s, u)|du = 0 for all t ≥ 0. Then, the trivial solution of InDS (5) is uniformly asymptotically stable. R

2.4.

Linear InDE with Nonlinear Perturbation

We now proceed to present a problem that is studied using an approach similar to the approach in the previous subsection, the difference being that the ODE is replaced by an InDE and the perturbation term is nonlinear. Consider the scalar linear Volterra InDE of the form 0

y (t) = f (t)y(t) +

Z t

H(at − s)y(s)ds

(7)

Z t

H(at − s)z(s)ds + g(t, z(t)),

(8)

0

and its perturbation 0

z (t) = f (t)z(t) +

0

where a is a constant with a > 1, f (t) is continuous on R+ and H : R+ → R+ is continuous in t and x, and g satisfies |g(t, z(t))| ≤ λ(t)|z(t)|,

(9)

A Brief Overview of the Stability Theory ...

37

where λ(t) is continuous on R+ . It is assumed that for each φ ∈ B(t0 ), t0 ≥ 0, there exists a unique solution y(t,t0, φ) of InDE (8) on an interval [t0, η) with y(s) = φ, 0 ≤ t ≤ t0 . We now proceed to give stability results from [8]. The stability results were obtained by constructing the Lyapunov functional V (t) and also using certain estimates given below. 1. Assume that for some constant c < 0, Gc (t) = e−ct and Gc(t) ∈ L1 [0, ∞); 2. Set a1 (t) =

f (t) a

+

R∞ t

H(u)ecudu exists

1 G (at − t); a2 c

3. Let a2 (t) = λ(t)[1 +

λ(t) ]; 2a2

4. Let the Lyapunov functional be defined by R R R∞ V (t) = 12 [y(t) + a1 0t Gc(at − s)y(s)ds]2 + p 0t at−s |Gc(u)|du y2(s)ds, where p is a positive constant to be suitably defined.

Theorem 5.

1. Let Gc (t) ∈ L1 [0, ∞) with

2. Assume that a a1 (t) + a2 (t) +

(a1 (t)−c)2 2

R 1 ∞ a 0 |Gc (u)|du

< 1.

+ a3 Q ≤ −β, for some β ≥ 0, 2

3. and also assume that Q = [ √1a 0∞ |Gc(u)|du]2 and a3 = 21 + L2 + |c| a . Then, the trivial solution of (8) is stable. R

Theorem 6. Suppose that the assumptions in Theorem 5 hold. Further assume R at R∞ that there exists a positive constant k such that (a−1)t v |Gc (u)|dudv < k for all t > 0. Then, the zero solution of (8 ) is uniformly stable. Theorem 7. Suppose that the assumptions in the Theorem 5 hold such that the R second condition is satisfied for β > 0. Further, assume that v∞ |Gc (u)|du ∈ L1 [0, ∞), then the zero solution of (8) is uniformly asymptotically stable.

38

Z. Drici, F. A. McRae and J. V. Devi

2.5.

Quasilinear InDS

In [9] the authors considered a quasilinear system obtained by adding two terms to InDS (1), as follows. y0 (t) = Cy(t) +

Z t 0

H(t − s)y(s)ds + f (t, y) +

Z t

G(t, s, y(s)ds,

(10)

0

where f and G are continuous for 0 ≤ s ≤ t < ∞, ||x|| ≤ r and f (t, 0) = G(t, s, 0) ≡ 0. Let Hˆ be the Laplace transform of H. ˆ Theorem 8. Let the matrix zI − C − H(z) be invertible for Rez ≥ 0 and let R supt || f (t, x)|| = o(||x||), x → 0 and supt 0t sup||x||≤r ||G(t, s, x)||ds = o(τ), τ → 0. Then, the trivial solution of quasilinear system (10) is stable . R If, in addition, limt→∞ 0T sup||x||≤r ||G(t, s, x)||ds = 0, for any T > 0 and τ ≤ r, then, the trivial solution of system (10) is asymptotically stable. Theorem 9. Assume that 1. for certain α > 0, qi (t), Qi(t, s), i = 1, 2 || f (t, x)|| ≤ q1 (t)o(||x||) + q2(t)||x||1+α

||G(t, s, x)|| ≤ Q1 (t, s)o(||x||) + Q2(t, s)||x||1+α 2. ||C(t)|| ≤ Mexp(−γt) for certain γ > 0, M > 0. Further, if the function 1t 0t [q1 (s) + 0s exp(γ(s − τ))Q1(s, τ)dτ]ds is bounded on R [0, ∞) and the function exp(−βt)[q2 (t) + 0t Q2 (t, s)exp(γ(1 + α)(t − s))ds] ∈ L1 [0, ∞) for some 0 < β < ∞, then the trivial solution of system (10) is asymptotically stable. R

R

The following theorem gives criteria for the instability of system (10) −1 ˆ Theorem 10. Let z0 be a pole of the matrix (zI −C − H(z)) with the maximal real part α = Rez0 > 0, and let (k + 1) be the maximal order of poles located on the line Rez = α. Let the assumptions (1) and (2) of the above theorem hold. Further assume that

A(t) = q1 (t)(t + 1)k +

Z t 0

Q1 (t, s, )exp(−α(t − s))(s + 1)k ds ∈ L1 [0, ∞)

A Brief Overview of the Stability Theory ... B(t) = q2 (t)(t + 1)k(1+α) +

Z t 0

39

Q2 (t, s, )exp(−α(t − s)(1 + α))(s + 1)k(1+α)ds

be bounded on [0, ∞]. Then, the trivial solution of quasilinear InDS (10) is unstable.

2.6.

Linear Barbashin Type InDE

We discuss the stability and boundedness of a linear InDE of Barbashin type from [10]. Consider the following linear InDE of Barbashin type given by ∂y(t, x) = c(t, x)y(t, x) + ∂t

Z 1 0

h(t, x, s)y(t, s)ds + f (t, x), t > 0, 0 ≤ x ≤ 1

(11)

where c(., .) : [0, ∞) × [0, 1] → R, h(., ., .) : [0, ∞) × [0, 1]2 → R and f (., .) : [0, ∞) × [0, 1] → R are given functions and y(., .) is unkown. This equation is transformed into an operator equation in a suitable space as follows. Let C(t) be such that (C(t)y(t))(x) ˆ = c(t, x)y(t, x) and H(t) be such that R (H(t)y(t))(x) ˆ = 01 h(t, x, s)y(t, s)ds and B(t) = C(t) + H(t). Then the operator equation u0 (t) = A(t)u(t), (t ≥ 0), is considered in a Hilbert space and apriori estimates are obtained on A and are then extended to B(t). In order to do so, the following assumptions are made. 1. For almost all x, s ∈ [0, 1], c(t, x) and h(t, x, s) have bounded measurable derivatives in t, ct0 (t, x) and ht0 (t, x, s). 2. The operators B(t) and B0 (t) are ∈ L2 and are defined as follows. For all x ∈ [0, 1] and w ∈ L2 , B(t)w(x) = c(t, x)w(x) +

Z 1

h(t, x, s)w(s)ds,

Z 1

ht0 (t, x, s)w(s)ds,

0

B0 (t)w(x) = ct0 (t, x)w(x) +

0

are assumed to be bounded uniformly in t, t ∈ [0, ∞). 3. The Frobenius norm or the Hilbert-Schmidt norm, Z Z 1

N2 (HI (t)) := (

0

1

(h(t, x, s) − h(t, s, x))2dsdx) 2 < ∞.

40

Z. Drici, F. A. McRae and J. V. Devi 4. supt≥0 α(B(t)) < 0, where α(B(t)) = sups∈σ(B(t)) Re s, and σ(B(t)) is the spectrum of B(t) and HI (t) =

H(t)−H ∗ (t) 2i

where H ∗ (t) is the adjoint operator of H(t).

The homogeneous equation corresponding to the linear Barbashin InDE (11) is given by, ∂y(t, x) = c(t, x)y(t, x) + ∂t

Z 1 0

h(t, x, s)y(t, s)ds (t > 0, 0 ≤ x ≤ 1),

(12)

with initial condition y(0, x) = y0 (x), 0 ≤ x ≤ 1 and y0 ∈ L2 is given. Definition 7. The equation (12) is said to be exponentially stable if there exist positive constants M and β such that any solution y(t, .) of (11) and (12) satisfies ||y(t, 0)||L2 ≤ Me−βt ||y0 ||L2 (t ≥ 0). Theorem 11. Let assumption (3) and the condition supt≥0 µ2 (t)||B0(t)||L2 < 2 hold, where j+k

∞

µ(t) =

N (HI (t))(k + j)! ∑ 2( j+k)/22|α(B(t))| j+k+1( j!k!)3/2 . j,k=1

(13)

Then, equation (12) is exponentially stable. Further, if || f (t, .)||L2 is bounded on [0, ∞), then any solution of (11) is bounded on [0, ∞).

2.7.

Impulsive InDS

Now we consider an impulsive InDE and proceed to present the stability results from [11]. The stability results are obtained by using the solution of the corresponding integral equation and the corresponding impulsive differential system. dy(t) = ay(t) + dt

Z ∞ 0

h(s)y(t − s)ds, t 6= τi

(14)

y(s) = φ(s), s ≤ 0,

(15)

y/t=τi = pi φ(s), 0 ≤ pi ≤ p, i = 1, 2, 3....

(16)

0 < τ1 < τ2 < ... < τi < ..., τi → ∞ ast → ∞, and τi+1 − τi ≥ θ > 0, i = 0, 1, 2, ... where φ is bounded and continuous on (−∞, 0] and a, b and p = max{pi }, i = 1, 2, 3... are real numbers. R The kernel h : [0,R∞) → [0, ∞) is bounded and continuous on [0, ∞) and is such that 0∞ h(s)ds < ∞ and 0∞ sh(s)ds < ∞.

A Brief Overview of the Stability Theory ...

41

The solution of system (14) - (16) can be obtained as the solution of the corresponding integral equation given by

y(t) = W (t, 0)φ(0) + b

Z t 0

Z ∞

W (t, s)(

0

h(u)y(s − u)du)ds,

(17)

where W (t, s) is the fundamental solution of the associated unperturbed differential system, that is, system (14) (16) - without the integral term. The following is a stability result in this set up. Theorem 12. Suppose the delay kernel h, the parameters a, b, θ and p are such that ∞ 1 h(s)ds, a − ln(1 + p) > |b|(1 + p) θ 0 then, the trivial solution of the impulsive InDS (14) -(16) is asymptotically stable.

Z

3. 3.1.

Part II - Nonlinear Integro-Differential Equations Introduction

In the second part of this chapter, stability properties of nonlinear integro-differential equations are presented. It is well known that Lyapunov’s second method is one of the basic tools for the study of the qualitative behavior of the solutions of nonlinear differential equations. In extending this method to Volterra integro-differential equations, one can either use Lyapunov functionals or Lyapunov functions. The use of Lyapunov functionals requires prior knowledge of the solutions, whereas the use of Lyapunov functions requires choosing appropriate minimal subsets of functions in C[R, Rn ] along which the derivative of the Lyapunov function admits a convenient estimate [12 - 14]. However, the Lyapunov functionals used in applications are usually a combination of a Lyapunov function and a Lyapunov functional, constructed so that the corresponding derivative can be estimated suitably without requiring the minimal classes of functions or prior knowledge of solutions [4]. The development of the method of Lyapunov functions on product spaces is a consequence of this fact. The stability and boundedness properties of solutions of InDEs were studied by many authors. Some of the early results developed using Lyapunov’s second method are those of Miller [5], Grossman and Miller [15], Burton and Mahfoud [16], Mahfoud [ 17], Rama Mohana Rao and Srinivas [18], Lakshmikantham and Rama Mohana Rao [19], Hara, Yoneyama, and Itoh [ 7], Elaydi and Sivasundaram [20], Elaydi [21], Zhang [ 22], and Vanualailai and Nakagiri [23]. Numerous references of more recent results can be found in C.Tunc and O. Tunc [24, 25], Andreev and Sedova [13] and Sedova [26]. The work we present in this section uses the comparison method introduced in [ 27].

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3.2.

Lyapunov Functionals

3.2.1.

Existence of Lyapunov Functionals

In this subsection we present results dealing with the existence of a Lyapunov functional for the following linear InDS y0 (t) = C(t)y(t) +

Z t t0

H(t, s)y(s)ds, y(t0 ) = y0 ,

(18)

and the nonlinear InDS 0

y (t) =

Z t t0

H(t, s)G(s, y(s)ds, y(t0 ) = y0 ,

(19)

where C(t), H(t, s) are n × n matrices defined on t0 ≤ t < ∞ and t0 ≤ s ≤ t < ∞, t0 ≥ 0. The following definition is necessary to introduce the Lyapunov function. Definition 8. A function p ∈ C[R+, R+] is said to belong to the class K if p(0) = 0 and p(t) is monotonically increasing in t. We now present the concept of exponential asymptotic stability Definition 9. The trivial solution of InDS (19) is generalized exponentially asymptotically stable (GEAS) if ||y(t, t0, y0 )|| ≤ L(t)||y0|| exp[P(t0 ) − P(t)] where L(t) > 0 and is continuous for t ∈ [t0 , ∞), P ∈ K and P(t) → ∞ as t → ∞. Remark 2. If L(t) = L > 0 and P(t) = a t, with a > 0, then, the trivial solution of (18) is exponentially stable. We next present the following theorem from [28] dealing with the existence of a Lyapunov function. Theorem 13. Assume that the trivial solution of (19) is GEAS. Further suppose that there exists a function P0 (t) which is continuous on J = [t0, ∞) (say). Then, there exists a function V (t, x) satisfying, 1. V ∈ C[J × Sρ , R] and |V (t, x) −V(t, y)| ≤ L(t)||x − y||, t ∈ J, x, y ∈ Sρ, 2. ||x|| ≤ V (t, x) ≤ L(t)||x||, (t, x) ∈ J × Sρ ,

3. D−V (t, x) ≤ −P0 (t)V (t, x), (t, x) ∈ J × Sρ .

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43

Theorem 14. Assume that the trivial solution of (19) is GEAS. Further suppose that, sup{ t≥0

Z t 0

||H(t, s)||ds : 0 ≤ s ≤ t < ∞)} ≤ M

and ||G(s, y) − G(s, x)|| ≤ K(t, r)||y − x||,t ∈ J, x, y ∈ Sρ , where K(t, r) > 0 such that isfying:

R∞ t0

K(s, r)ds = N. Then, there exists a function V (t, x) sat-

1. V ∈ C[J × Sρ , R] and |V (t, x) −V(t, y)| ≤ L(t)||x − y||, t ∈ J, x, y ∈ Sρ, 2. ||x|| ≤ V (t, x) ≤ L(t)||x||, (t, x) ∈ J × Sρ

3. D−V (t, x) ≤ −P0 (t)V (t, x), (t, x) ∈ J × Sρ .

3.2.2.

Lyapunov Functional for a Specific Scalar InDE

In this subsection, we will consider a scalar InDE corresponding to (18) and construct a Lyapunov functional following the work in [4]. Let y0 (t) = cy +

Z t t0

h(t − s)y(s)ds,

(20)

where c is a constant and h(t) is continuous for 0 ≤ t < ∞, with a strong sign condition. Remark 3. The definitions of stability concepts of InDS (1) are given in Section 2.1. Those definitions can be easily modified to define these concepts for scalar InDE (20). We now proceed to state the following results from [4, 16] and use a result involving a Lyapunov functional to prove stability results for (20). Theorem 15. Suppose that c < 0, h(t) > 0 and c + statements are equivalent.

R∞ 0

h(t)dt 6= 0. Then, the following

1. All the solutions of (20) tend to zero; 2. c +

R∞ 0

h(t)dt < 0;

3. Each solution of (20) is in L1 (R+ ); 4. The zero solution of (20) is uniformly asymptotically stable; 5. The zero solution of (20) is asymptotically stable.

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Proof. We shall only prove 2 =⇒ 3. Let y(t, t0, φ) be any solution of the InDE (20) with the initial condition φ(t) = 2 on [0, t0]. We choose a Lyapunov functional in this set up as V (t, y(.)) = |y| +

Z tZ ∞ 0

t

h(τ − s)dτ|y(s)|ds.

Let y(t) be a solution of the scalar InDE (20) and y(t) not equal to the zero solution, then V 0 (t, y(.)) ≤ c|y| + = [c +

Z t

Z

0 ∞

h(t − s)|y(s)|ds +

Z ∞ t

h(τ − t)dτ]|y| ≤ [c +

t

h(τ − t)dτ |y| −

Z ∞ 0

Z t 0

h(t − s)|y(s)|ds

h(τ)dτ]|y| = −β|y|,

for some constant β > 0. This yields 0 ≤ V (t, y(.)) ≤ V (t0 , φ(.)) − β

Z t t0

|y(s)|ds,

which implies that t∞ |y(s)|ds < ∞, meaning that y ∈ L1 (R+ ), completing the proof. 0 If we relax the condition on h(t), we can modify the Lyapunov functional suitably to get asymptotic stability of the trivial solution of (20). We state this result. For a proof see [4, 16]. R

Theorem 16. Suppose that in Theorem 15, c + 0∞ |h(t)|dt < 0. Then, the trivial solution of scalar InDE (20) is uniformly asymptotically stable. R

To prove this theorem, choose the Lyapunov functional as V (t, y(.)) = y2 +

Z tZ ∞ 0

t

|h(τ − s)|dτ|y(s)|ds,

and obtain an estimate for the Lyapunov functional as in the earlier theorem.

3.2.3.

Lyapunov Functional for a General Scalar InDE

Next, we shall consider a more general scalar InDE [4, 16] and discuss the stability concepts using a Lyapunov functional. To that end, let y0 (t) = c(t)y +

Z t t0

h(t, s)y(s)ds, t ∈ R+

Theorem 17. Assume that 1. c : R+ → R+ and h are continuous for 0 ≤ s ≤ t < ∞;

(21)

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45

2. the integral t∞ |h(τ, t)|dτ is defined and finite for all t ≥ 0, and suppose there exists a β ∈ R+ such that R

Z t 0

|h(t, s)|ds +

Z ∞ t

|h(τ, t)|dτ − 2|c(t)| ≤ −β.

Then, the trivial solution of the InDE (21) is stable if and only if c(t) < 0. Proof. Suppose that c(t) < 0 and consider the Lyapunov functional given by V (t, y(.)) = y2 +

Z tZ ∞ 0

t

|h(τ, s)|dτ y2ds.

Then, we obtain that V 0 (t, y(.)) ≤ −β y2 . Observing that V is positive definite and that V 0 (t, y(.)) ≤ 0, we conclude that the solution of InDE (21) is stable. For the second part of the proof refer to [4, 16].

3.3.

Method of Lyapunov Functions

In this section, we discuss the stability properties of a general class of InDEs using the method of Lyapunov functions. A Lyapunov function can be viewed as a transformation which reduces the study of a given complicated differential system to the study of a scalar differential equation. In the case of InDEs, this method relies on carefully selecting appropriate minimal subsets of functions in C[R, Rn ], along which the derivative of the Lyapunov function admits a convenient estimate.

3.3.1.

Stability Using Minimal Classes

Consider the InDS given by, y0 (t) = G(t, y(t), (Ty)(t)), t ≥ t0 , y(t0 ) = y0 ,

(22)

where (Ty)(t) = tt0 g(t, s, y(s))ds, G ∈ C[R+ × S(ρ) × Rn , Rn ] and g ∈ C[R+ × R+ × S(ρ), Rn ]. Assume that G(t, 0, 0) ≡ 0 and g(t, s, 0) ≡ 0, so that the InDS (22) admits the trivial solution. As mentioned earlier, the estimate of the derivative of the Lyapunov function along continuous functions in the minimal class is used to construct the scalar differential equation mentioned above [4]. Then, the study of the behavior of solutions of the InDE reduces to the study of the behavior of the solutions of this scalar differential equation. For this purpose, let R

V ∈ C[R+ × S(ρ), R+ ], and for y ∈ C[R+, Rn ], 1 D−V (t, y(t)) = lim inf [V (t + h, y(t) + hG(t, y(t), (Ty)(t))) − V (t, y(t)). h h→0−

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Z. Drici, F. A. McRae and J. V. Devi

Now we proceed to introduce three subsets of C[R, Rn ] along which we estimate D−V (t, y(t)) subject to the demands we impose on the solutions of the InDS (22). 1. Eα = {y ∈ C[R+, Rn ] : V (s, y(s))α(s) ≤ V (t, y(t))α(t), t0 ≤ s ≤ t};

2. E1 = {y ∈ C[R+, Rn ] : V (s, y(s)) ≤ V (t, y(t)), t0 ≤ s ≤ t};

3. E0 = {y ∈ C[R+, Rn ] : V (s, y(s)) ≤ f (V (t, y(t))), t1 ≤ s ≤ t, t1 ≥ t0 }; where α(t) > 0 is a continuous function on R+ and f (r) is continuous on R+ , nondecreasing in r and f (r) > r for r > 0. We next present comparison theorems that are useful in establishing the results stated later [4, 19]. Theorem 18. Let V ∈ C[R+ × S(ρ), R+ ] and V (t, y) be Lipschitzian in y. Assume that for t ≥ t0 and y ∈ E1 , D−V (t, y(t)) ≤ p(t,V(t, y(t)),

(23)

where p ∈ C[R+ × R+ , R+ ]. Let r(t) = r(t, t0, u0) be the maximal solution of the scalar ordinary differential equation: u0 = p(t, u), u(t0 ) = u0 ≥ 0,

(24)

existing on t0 ≤ t < ∞. Let y(t) = y(t, t0, y0 ) be any solution of the InDS (22) such that y(t) ∈ S(ρ) for t ∈ [t0, t1] and satisfying V (t0 , y(t0 )) ≤ u0 .

(25)

V (t, y(t)) ≤ r(t) for all t ∈ [t0, t1].

(26)

Then,

Proof. We briefly sketch the proof of the theorem. Let y(t) = y(t, t0, y0 ) be any solution of the InDS (22) such that y(t) ∈ S(ρ) for t ∈ [t0, t1]. Define m(t) = V (t, y(t)). For sufficiently small ε > 0, consider ε increment - scalar ODE, u0 = p(t, u) + ε, u(t0 ) = u0 + ε,

(27)

whose solutions u(t, ε) exist as far as r(t) exists to the right of t0 . Using the hypothesis on p, we note that the solutions of the scalar ODE (24) and the solutions, u(t, ε), of the ε increment - scalar ODE (27), are monotonically increasing functions of t. We establish, using m(t) and u(t, ε), that y(t) ∈ E1 for t0 ≤ t ≤ t2 . Using the fact that V (t, y) is Lipschitzian in y and the relation between V and p, we conclude that V (t, y(t)) ≤ r(t), completing the proof. For a detailed proof of the theorem and the following results see [4, 29]. The following theorem deals with the solutions of InDS (22) in the set E0 .

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47

Theorem 19. Let V ∈ C[R+ × S(ρ), R+ ] and V (t, y) be Lipschitzian in y. Assume that for t ≥ t0 and y ∈ E0 , D−V (t, y(t)) ≤ 0 Let y(t) = y(t, t0, y0 ) be any solution of the InDS (22) such that y(t) ∈ S(ρ) for t ∈ [t0, t1]. Then, V (t, y(t)) ≤ V (t0 , y(t0 )), for all t ∈ [t0 , t1].

(28)

Proof. Proceeding as in Theorem 18 with p ≡ 0, we arrive at the inequality V (s, y(s)) ≤ V (t2 , y(t2)), t2 ∈ (t0 , t1 ). Since V (t2 , y(t2)) ≤ V (t0 , y(t0)) + ε + ε(t2 − t0 ), from the assumption on f (r) we get V (s, y(s)) ≤ f (V (t2 , y(t2))), which means that y(t) ∈ E0 for t0 ≤ s, t ≤ t2 . The remaining proof follows the same pattern as in Theorem 18. The next result is more general than Theorem 18 and is useful to establish the asymptotic stability of the trivial solution of InDS (22). The proof of the theorem is given below to draw attention to a subtle point in the proof. Theorem 20. Assume that the hypotheses of Theorem 18 hold except that the inequality (23) is replaced by α(t)D−V (t, y(t)) + V (t, y(t))D−α(t) ≤ p(t,V(t, y(t)α(t)),

(29)

for t > 0 and y ∈ Eα , where α(t) > 0, is continuous on R+ and 1 D− α(t) = lim inf [α(t + h) − α(t)]. − h h→0 Then, α(t0 )V (t0 , y0 ) ≤ u0 implies that α(t)V (t, y(t)) ≤ r(t), t ≥ t0 . Proof. Let L(t, y(t)) = α(t)V (t, y(t)), then, L(t + h, y(t) + hG(t, y(t), (Ty)(t))) − L(t, y(t)) = V (t + h, y(t) +hG(t, y(t), (Ty)(t)))(α(t + h) − α(t)[V (t + h, y(t) + hG(t, y(t), (Ty)(t))) −V(t, y(t))]. Now using the relation (28), we obtain, D− L(t, y(t)) ≤ p(t, L(t, y(t)), for t ∈ [t0 , t1], y ∈ E1 .

(30)

Observe that L replaces V in E1 , and also that L is Lipschitzian in y. Hence, all the assumptions of Theorem 18 are satisfied and the conclusion of the theorem follows from the proof of Theorem 18. The following results provide sufficient conditions for the stability and asymptotic stability of the trivial solution of InDS (22). Theorem 21. Assume that there exist functions V (t, y) and p(t, y) satisfying the following conditions:

48

Z. Drici, F. A. McRae and J. V. Devi 1. V ∈ C[R+ × S(ρ), R+ ], and for y ∈ C[R+, Rn ], V (t, 0) ≡ 0, V (t, y) is positive definite and locally Lipschitzian in y; 2. p ∈ C[R+ × R+ , R+ ] and p(t, 0) ≡ 0 3. α(t) > 0 is continuous for t ∈ R+ and α(t) → ∞ as t → ∞ 4.

α(t)D−V (t, y(t)) + V (t, y(t))D−α(t) ≤ p(t,V(t, y(t)α(t)), for t > t0 and y ∈ Eα .

(31)

Then, stability of the trivial solution of scalar ODE (24) implies the asymptotic stability of the trivial solution of InDS (22). The following theorem deals with the uniform asymptotic stability of the trivial solution of InDS (22) Theorem 22. Let V ∈ C[R+ × S(ρ), R+ ] and V (t, y) be Lipschitzian in y. Assume that t ≥ t0 , y ∈ E0 , and D−V (t, y(t)) ≤ −φ(y(t)), where, φ ∈ K.

(32)

Then, the trivial solution of InDS (22) is uniform asymptotically stable. Remark 4. All the results that were established in this section deal with InDS (22) with the initial condition as a value, that is, y(t0 ) = y0 . However, the InDS (22) can be considered with the initial value replaced with a continuous function, representing the past history of the system. Thus, we have y(t0 ) = φ, for t ≥ 0 and φ : [0, t0] → Rn is continuous function. All the above theorems can also be proved in this case with suitable modifications.

3.4.

Lyapunov Functions on Product Spaces

As mentioned in the introduction, the Lyapunov functionals used in applications are usually a combination of a Lyapunov function and a Lyapunov functional, constructed so that the corresponding derivative can be estimated suitably without requiring the minimal class of functions or prior knowledge of solutions. The development of the method of Lyapunov functions on product spaces [4, 30] is a consequence of this fact. Let yt (.) ∈ C[[0, t], Rn ]. Suppose y ∈ C[R+, Rn ] = C(R+ ), then for each t ∈ R+ , yt (.) is the restriction of y(s) given by yt (s) = y(s), 0 ≤ s ≤ t, and |yt (.)| is the norm defined by |yt (.)| = sup |y(s)|. 0≤s≤t

Observe that |y(t)| ≤ |yt (.)|.

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49

Consider a system of the form. y0 (t) = G(t, y(t), yt (.)), t ∈ R+,

(33)

where G ∈ C[R+ × Rn × C(R+), Rn ]. Assume that G(t, 0, 0) ≡ 0 and let y(t) = y(t, t0, yt0 (.)) be a solution of the system (32) with initial values (t0 , yt0 (.)). A special case of the system (32) is the Volterra InDE given below: y0 (t) = c(t, y(t)) +

Z t t0

h(t, s, y(s))ds, t ∈ R+ ,

(34)

where c ∈ C[R+ × Rn , Rn ], h ∈ C[R+ × R+ ×C(R+ ), Rn ], c(t, 0) ≡ 0 and h(t, s, 0) ≡ 0. We now proceed to employ Lyapunov functions on the product space Rn ×C(R+ ) and present some results pertaining to the stability of the system (32). Consider V ∈ C[R+ ×Rn ×C(R+ ), Rn ], and suppose that V (t, y) is positive definite and locally Lipschitzian in y. Let y(t) = y(t, t0, yt0 (.)) be a solution of the system (32). Now define, 1 D+V (t, y, yt (.)) = lim sup [V (t + h, y + hG(t, y, yt (.)), yt+h (.)) − V (t, y, yt (.))]. + h h→0 We shall now give sufficient conditions guaranteeing asymptotic stability of the trivial solution of the system (32). Theorem 23. Assume that there exists a function V (t, y, yt (.)) ∈ C[R+ × S(ρ) × C(R+ ), R+ ], such that 1. a(|y(t)|) ≤ V (t, y, yt (.)) ≤ b(|yt (.)|), where a, b ∈ K;

2. D+V (t, y, yt (.)) ≤ 0

Then, the trivial solution of system (32) is uniformly stable. Theorem 24. Assume that there exists a function V (t, y, yt (.)) ∈ C[R+ × S(ρ) × C(R+ ), R+ ], such that 1. a(|y(t)|) ≤ V (t, y, yt (.)) ≤ b(|yt (.)|), where a, b ∈ K;

2. D+V (t, y, yt (.)) ≤ −c(|yt (.)|), where c ∈ K;

Then, the trivial solution of system (32) is uniformly asymptotically stable.

3.4.1.

Stability in Two Measures

The theory of stability in terms of two measures [31, 32] unifies a variety of known stability concepts in a single set up. The next theorem gives a stability result using this approach. We begin with the following definitions.

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Z. Drici, F. A. McRae and J. V. Devi

Definition 10.

ˆ y) = 0 for each t ∈ R+ }; 1. Γ = {hˆ ∈ C[R+ × Rn , R+] : infy∈Rn h(t,

ˆ h0 ∈ Γ and yr (.) ∈ C(R+ )), let 2. For h, ˜ yt (.)) = sup0≥s≥t h(s, ˆ y(s)). h0 (t, yt (.)) = sup0≤s≤t h0 (s, y(s)) and h(t, ˜ yt (.)) ≥ h(s, ˆ y(s)). It is clear that h0 (s, y(s)) ≤ h0 (t, yt (.)) and h(t, ˆ h0 ∈ Γ and let h, ˜ h0 be as defined above. Then, h0 is said to be finer Definition 11. Let h, ˜ yt (.)) ≤ than h˜ if there exist a λ > 0 and φ ∈ K such that h0 (t, yt (.)) ≤ λ implies h(t, φ(h0 (t, yt (.))). Next, we define the criteria for V to be positive definite and decrescent in this setup. Definition 12. Consider V ∈ C[R+ × Rn ×C(R+ ), Rn ]. Then, V is said to be

ˆ ˆ y) < ρ implies 1. h-positive definite if there exist ρ > 0 and a ∈ K such that h(t, ˆ y)) ≤ V (t, y, yt (.)); a(h(t, 2. h0 -decrescent if there exist ρ0 > 0 and b ∈ K such that h0 (t, y) < ρ0 implies V (t, y, yt (.)) ≤ b(h0 (t, yt (.))).

We now proceed to define the stability concepts in the set up of two measures. We present the stability definition. All other concepts of stability and boundedness can be defined in a similar fashion. ˆ uniformly Definition 13. The trivial solution of system (32) is said to be(h0 h − h) stable, if given ε > 0 and t0 ∈ R+, there exists a δ = δ(ε) > 0 such that t0 ≥ 0 and h0 (t0 , yt0 (.)) < δ implies h(t, y(t)) < ε, for all t ≥ t0 . To illustrate the generality of this concept, we give a couple of examples. For more details see [4,32]. ˆ y) = |y|, which implies h0 (t, yt(.)) = |yt (.)|, where |y| is the 1. If h0 (t, y) = h(t, usual Euclidean norm and |yt (.)| = sup0≤s≤t |y(s)|, then we obtain the well known uniform stability result for the trivial solution of system (32); ˆ y) = |y − 2. Let z(t) = z(t, t0, zt0 ) be any prescribed solution of system (32). If h(t, z(t)|, andh0 (t, yt(.)) = |yt (.) − zt (.)|, where |y| is the usual Euclidean norm and |yt (.)| = sup0≤s≤t |y(s)|, then we obtain the stability of the prescribed solution of system (32). As stability is a local concept, we describe the ρ-ball in this set up. ˆ ρ) = {(t, y) : h(t, ˆ y) < ρ} and for h∗ ∈ For any h ∈ Γ, define S(h, ∗ ∗ C[R+ × C(R+ ), R+ ], let S(h , ρ) = {(t, yt (.)) : h (t, yt (.)) < ρ}. We state below a generalization of Lyapunov’s second theorem in terms of two measures.

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51

Theorem 25. Assume that ˆ h0 ∈ Γ and h0 is finer than h; ˜ 1. h,

ˆ ρ)×C(R+), R+ ] that is locally 2. there exists a function V (t, y, yt (.)) ∈ C[R+ ×S(h, ˆ Lipschitzian in y, h− positive definite and h0 − decrescent; ˆ ρ) × S(h0, ρ0) 3. D+V (t, y, yt (.)) ≤ −η(h0 (t, yt (.))), for all (t, y, yt (.) ∈ R+ × S(h,

ˆ uniformly asymptotically stable. Then, the trivial solution of the system (32) is (ho , h)−

3.4.2.

Practical Stability for InDE

Consider the following initial value problem [33] y0 (t) = G(t, y(t), (Ty)(t)), t ≥ t0 , y(t0 ) = y0 ,

(35)

Rt

where (Ty)(t) = t0 g(t, s, y(s))ds, G ∈ C[R+ × Rn × Rn , Rn ] and g ∈ C[R+ × R+ × Rn , Rn ]. In this subsection, we present a practical stability result using the comparison system [33]. In this case, the Lyapunov function V will serve as a vehicle to reduce the complex system to a simpler one. Then, the practical stability of the complex system is obtained through the practical stability of the corresponding scalar ODE. We assume the following hypotheses: 1. V ∈ C[R+ × Rn , R+ ], V (t, y) is locally Lipschitzian in y and a(|y(t)|) ≤ V (t, y) ≤ b(|y(t)|), where a, b ∈ K; 2. f 0 , f ∈ C[R+ × R+ , R], f 0 (t, w) ≤ f (t, w), r(t, t0, w0) is the right maximal solution of w0 = g(t, w), w(t0 ) = w0 ≥ 0,

existing on [t0, ∞) and µ(t, t 0, u0) is the right maximal solution of u0 = g0 (t, u), u(t 0) = u0 ≥ 0, t 0 > t0 , existing on t0 ≤ t ≤ t 0 ;

(36)

3. D−V (t, y(t)) = limh→0− inf h1 [V (t + h, y(t) + hG(t, y(t), (Ty)(t))) − V (t, y(t)), such that, D+V (t, y) ≤ f (t,V(t, y)), (t, y) ∈ Ω, where Ω = {y ∈ C[R+, Rn ] : V (s, y(s)) ≤ µ(s, t,V(t, y(t)), t0 ≤ s ≤ t}.

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Z. Drici, F. A. McRae and J. V. Devi

We now state a general comparison theorem used in establishing the practical stability results of the system (35). Theorem 26. Assume that all the forementioned hypotheses hold. Let y(t, t0, y0 ) be any solution of system (35), such that V (t0 , y0 ) ≤ w0 . Then, V (t, y(t, t0, y0 ) ≤ r(t, t0, w0), t ≥ t0 . Having the comparison theorem at our disposal, we are in a position to state a very general result that establishes criteria for various types of practical + concepts. Theorem 27. Assume that all the forementioned hypotheses hold. Then, the practical stability properties of system (36) imply the corresponding practical stability properties of (35) .

Conclusion This chapter dealt with the stability of various types of linear and nonlinear integrodifferential equations. First, several approaches to the study the stability of various types of linear integro-differential equations were given. Next, stability results for nonlinear Volterra integro-differential equations, obtained using the comparison method, were presented. In the comparison method Lyapunov or Lyapunov-like functions and comparison theorems are used to reduce the study of a complicated system to the study of a simpler system. Different types of stability notions such as practical stability and stability in terms of two measures were also discussed.

References [1] Rouche, N., Habets, P. and Laloy, M. Stability Theory by Liapunov’s Direct Method, Springer-Verlag, New York, 1977. [2] Lakshmikantham, V., Leela, S. and Martynyuk, A. A., Practical Stability of Nonlinear Systems, World Scientific, Singapore,1990. [3] Lakshmikantham, V. and Liu, X., Stability Analysis in Terms of Two Measures, World Scientific, Singapore,1993. [4] Lakshmikantham, V. and Rama Mohana Rao, M., Theory of Integro-differential Equations, Gordon and Breach Publishers, Switzerland, 1995. [5] Miller R.K., Asymptotic stability properties of linear Volterra integro-differential equations, Journal of Differential Equations, 10, 485-506, 1971. [6] Hara,T., Yoneyama,T. and Itoh,T., Characterization of stability concepts of Volterra integro-differential equations, Journal of Mathematical Analysis and Applications Vol. 142, 558-572, 1989.

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[7] Hara,T., Yoneyama,T. and Itoh,T., Asymptotic stability criteria for nonlinear Volterra integro-differential equations, Funkcial. Ekvac. Vol. 33, 39-57, 1990. [8] M. N. Islam and Y. N. Raffoul, Stability in linear Volterra integrodifferential equations with nonlinear perturbation, Journal of Integral Equations and Applications Vol. 17, No. 3, 259-276, 2005. [9] O.Yu. Khvorost and Z.B.Tsalyuk, Stability and Unstability of quasilinear systems of integrodifferential equations, Russian Mathematics, Vol. 51, No.11, 7679, 2007. [10] Michael Gil, On stability of linear Barbashin type integrodiffferential equations, Mathematical Problems in Engineering 2015, Article ID 962565. http//dx.doi.org/10.1155/2015/962565. [11] Sariyasa, Stability of integrodifferential Equations with Impulses, J. of Indonesian Mathematical Society, Vol. 13, No. 1,17-26, 2007. [12] V. Lakshmikantham and Leela, S., Differential and Integral Inequalities, Vol. 2, Academic Press, New York, 1969. [13] A. S. Andreev and N. O. Sedova, The method of Lyapunov-Razumikhin functions in stability analysis of systems with delay, Automation and Remote control, Vol. 80, No. 7, 1185-1229, 2019. [14] V. S. Sergeev, Stability of solutions of Volterra integrodifferential equations, Mathematical and Computer Modelling, Vol. 45, 1376-1394, 2007. [15] Grossman S.I. and Miller R.K. Perturbation theory for Volterra integro-differential equations, J. Differential Equations 8, 457-474, 1970 [16] Burton, T. A. and Mahfoud,W.E. Stability Criteria for Volterra Equations, Transactions of American Mathemtical Society, 279, 143-174,1983. [17] Mahfoud,W.E. Boundedness properties in Volterra integro-differential systems. Proceedings in American Mathematical Society, 100, 37-45, 1987 . [18] Rama Mohana Rao, M. and Srinivas, P. Asymptotic behavior of solutions of Volterra integro-differential equations, Proceedings of the American Mathematical Society, 94, 55-60,1985. [19] V. Lakshmikantham and Rama Mohana Rao, M., Stability in variation for nonlinear integro-differential equations, Applicable Analysis, 24, 165-173,1987. [20] Elaydi, S and Sivasundaram, S., A unified approach to stability in integrodifferential equations via Lyapunov functions, J. Math. Anal. Appl. 144, 503 531, 1989. [21] Elaydi, S., Stability of integrodifferential systems of nonconvolution type, Math. Inequal. Appl. 1, 423-430, 1998.

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[22] Zhang, B., Construction of Liapunov functionals for linear Volterra integrodifferential equations and stability of delay systems, Electron. J. Qual. Theory Differ. Equ. 30,1-17, 2000. [23] Vanualailal, J. and Nakagiri, S., Stability of a system of Volterra integrodifferential equations, Journal of Mathematical Analysis and Applications, 281,602-619, 2003. [24] Cemil Tunc and Osman Tunc, New qualitative criteria for solutions of Volterra integro-differential equations, Arab Journal of Basic and Applied Sciences, 25:3, 158-165,DOI: 10.1080/25765299.2018.1509554, 2018. [25] Cemil Tunc and Osman Tunc, A note on the qualitative analysis of Volterra integro-differential equations, Journal of Talibah University for Science, 13:1, 490-496, 2019 [26] Natalia Sedova, On uniform asymptotic stability for nonlinear integro-differential equations of Volterra type, Cybernetics and Physics, Vol. 8, No. 3, 161-166, 2019. [27] Lakshmikantham, V. and Leela, S., Differential and Integral Inequalities, Vol. 1, Academic Press, New York,1969. [28] Kandil, F. M., A note on stability of integrodifferential equations, International Mathematical Forum, Vol. 5, No. 67, pp 3343-3348, 2010. [29] Lakshmikantham, V. and Rama Mohana Rao, M. Integro-differential equations and extensions of Lyapunovs method, Journal of Mathematical Analysis, 30, 435447, 1970. [30] Rama Mohana Rao, M. and Sanjay, K. S., Lyapunov functions on product spaces and stability of integro-differential equations, Dynamical Systems Applications 1, 93-102, 1992. [31] Leela, S. and Rama Mohana Rao, M. (h0 , h, M0)-stability for integro-differential equations, Journal of Mathematical Analysis and Applications, 130, 460-468, 1988. [32] Lakshmikantham, V. and Liu, X. Z., Stability Analysis in terms of two measures, World Scientific Publishing Co. Pte. Ltd. NJ, USA 1993. [33] Lakshmikantham, V., Leela, S. and Martynyuk, A.A., Practical Stability of Nonlinear Systems, World Scientific Publishing Co. Pte. Ltd. Teaneck, NJ, USA 1993.

In: Understanding Integro-Differential Equations ISBN: 979-8-89113-040-1 Editors: J. Vasundhara Devi et al. ©2023 Nova Science Publishers, Inc.

Chapter 3

Advances in the Qualitative Theory of Integro–Differential Equations Osman Tunç1,* Seenith Sivasundaram2,† and Cemil Tunç3,‡ 1

Department of Computer Programing, Baskale Vocational School, Van Yuzuncu Yil University, Van, Turkey 2 Department of Mathematics, College of Science Engineering and Mathematics, Daytona Beach, Florida, USA 3 Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, Campus, Van, Turkey Abstract This work investigates the advances from the past until now in the qualitative properties of solutions of linear and nonlinear integro – differential equations (IDEs). Here, we present an extensive literature on the qualitative properties of solutions, including asymptotic stability, uniform stability, instability and global uniform asymptotic stability of the zero solution, as well as boundedness, square integrability and existence of solutions to various linear and non-linear Volterra IDEs, without delay and with delay. We also present some applications of

Corresponding Author’s E-mail: [email protected] Corresponding Author’s E-mail: [email protected] ‡ Corresponding Author’s E-mail: [email protected] * †

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Osman Tunç, Seenith Sivasundaram and Cemil Tunç such equations in sciences and engineering. Some examples are given to illustrate the results of this work and show their applications.

Keywords: qualitative theory, integro–differential equation, fractional integro-differential equation, Lyapunov function, Lyapunov–Krasovskii functional (LKF), Lyapunov-Razumikhin method, fixed point method, stabilty, Ulam stability, Ulam-Hyers stability, Ulam-Hyers-Rassias stability 202 k0 (MOS) Subject Classifications: 34D05, 34K06, 34K20, 34K37, 45J05

1. Introduction Vito Volterra (1860 –1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations. He is one of the founders of functional analysis. Volterra’s work on elasticity was the origin of his theory of IDEs: He found that for certain substances, the electric or magnetic polarization depends not only on the electromagnetic field at that moment, but also on the history of the electromagnetic state of the matter at all previous instants. These physical facts are modeled by IDEs. Indeed, from the qualitative theory of IDEs, it can be seen that IDEs occur very often as mathematical models in various disciplines, see, the sources [1120]. The origins of the investigations on integral and integro-differential equations are based on the works of Abel, Lotka, Fredholm, Malthus, Verhulst and Volterra on problems in mathematical biology, elasticity, mechanics, electromagnetics, economics (see Burton [16], Lakshmikantham and Rama Mohana Rao [48], Volterra [96], Wazwaz [99] and the references therein). In addition, the work of Volterra [96] on the problems of competing species and electromagnetics is of fundamental importance in the development of mathematical modeling of various real-world problems. Also, Volterra assumed linear “heredity”, i.e., that the strain is a linear functional of the stress. In this case, the fundamental equations are systems of linear IDEs. He proved that the strain in a definite interval of time can be determined, given the forces in the body and the stress and strain on its surface for this time-interval (see, for example, Hatamzadeh et al. [40], Kyselka [46], Volterra [96] and the references therein).

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From those beginnings, the theory and applications of IDEs with bounded delays, unbounded delays and without delay have emerged as new areas of investigation. Over the past few decades, numerous research papers and books on the qualitative properties of IDEs and functional differential equations (FDEs) were published (see [1-39], [41-45], [47-110] and the references therein). Indeed, mathematical models called IDEs, integral equations, integrodelay differential equations (IDDEs), ordinary differential equations, FDEs are very attractive and interesting equations in the literature due to their very effective roles in numerous scientific fields and applications. In the past and in recent years, an extensive literature such as the numerous books by Burton [16], Hale [37], Hale and Verduyn Lunel [38], Kress [47], Lakshmikantham and Rama Mohana Rao [48], Rahman [71], Reissig et al. [75] and the related papers in the references of this work have been devoted to discussions of stability, uniform stability, uniform asymptotically stability, exponential stability, instability, integrability, boundedness, existence of solutions, etc., of these kinds of mathematical models. IDEs and numerous kind of FDEs are extensively used for modeling various problems from mechanics, power systems, control theory, physics, electricity, biology, population ecology, neural networks, medicine, economics, etc., and their fundamental properties were intensively investigated theoretically (Burton [16], Hale [37], Hale and Verduyn Lunel [38], Hatamzadeh et al.[40], Kyselka [46], Lakshmikantham and Rama Mohana Rao [48], Volterra [96]). For example, as a specific example in physics, an RLC, standard closed electric circuit, can be mathematically modeled by the below linear IDE: 𝐿

𝑑 1 𝑡 𝐼(𝑡) + 𝑅𝐼(𝑡) + ∫ 𝐼(𝑠)𝑑𝑠 = 𝐸(𝑡), 𝑑𝑡 𝐶 0

where 𝐼(𝑡) is the electric current, 𝑅 is the resistance, 𝐿 is the inductance, 𝐶is the capacitance and the impressed voltage 𝐸(𝑡) (Bohner et al. [15]).

2. Basic Results We consider the following system of delay differential equations (DDEs) of the form:

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dx = F (t , xt ), 𝑥𝑡 (𝜃) = 𝑥(𝑡 + 𝜃), −𝜏 ≤ 𝜃 ≤ 0. dt

(1)

We suppose 𝐹: (−∞, ∞) × 𝐶 → 𝑅 𝑛 , where 𝐶 is a set of continuous functions 𝜙: [−𝜏, 0] → 𝑅 𝑛 , 𝜏 > 0. Here, 𝐹 is continuous and takes closed bounded sets into bounded sets, and 𝐹(𝑡, 0) = 0. Since 𝐹(𝑡, 0) = 0, the DDE (1) includes the solution 𝑥(𝑡) ≡ 0, with zero initial function 𝜙 ≡ 0. Recall that for 𝑡 ∈ 𝑅, 𝐶(𝑡) = {𝜙: [−𝜏, 0] → 𝑅 𝑛 |𝜙 is continous}. If 𝜙 ∈ 𝐶(𝑡), then ‖𝜙‖𝑡0 =

𝑠𝑢𝑝 |𝜙(𝑠)|, where |. | is a convenient 𝑡0 −𝜏≤𝑠≤𝑡0

norm on 𝑅 𝑛 (see, Burton [16]). For any 𝜙 ∈ 𝐶([−𝜏, 0], 𝑅 𝑛 ), we refer the usual Euclidean norm ‖. ‖, which is defined by ‖𝜙‖ = 𝑠𝑢𝑝 |𝜙(𝑠)|. −𝜏≤𝑠≤0

We mean a continuously differentiable function 𝑥(𝑠) on 0 ≤ 𝑠 < 𝑇 ≤ ∞ such that (1) is satisfied on 0 ≤ 𝑡 < 𝑇 for 𝑥𝑡 (. ) a segment of 𝑥(𝑠). If 𝑇 < ∞, and there is no solution 𝑦(𝑠) of (1) on 0 ≤ 𝑠 < 𝑇1 with 𝑇1 > 𝑇 such that 𝑥(𝑠) = 𝑦(𝑠) on 0 ≤ 𝑠 < 𝑇, we say that 𝑥(𝑠) is noncontinuable on 0 ≤ 𝑠 < 𝑇. If 𝑇 = ∞, 𝑥(𝑠) is said to be noncontinuable i.e., defined for all 𝑠 ≥ 0. We assume that: If 𝑥(𝑠) is a noncontinuaable solution of (1) 0 ≤ 𝑠 < 𝑇 ≤ ∞, then there exists a sequence {𝑡𝑗 } such that 𝑡𝑗 → 𝑇 and |𝑥(𝑡𝑗 )| → ∞ as 𝑗 → ∞. Let 𝑅 denote the set of real numbers, and 𝑅 𝑛 the set of real 𝑛 -tuples. If 𝑥 and 𝑦 are elements of 𝑅 𝑛 with 𝑥 = (𝑥1 , . . . , 𝑥𝑛 ), 𝑦 = (𝑦1 , . . . 𝑦𝑛 ), then (𝑥, 𝑦) = ∑𝑛𝑗=1 𝑥𝑗 𝑦𝑗 and |𝑥| = (𝑥, 𝑥)1/2 . 𝑥𝑡 (. )denotes a function continuous on the interval 0 ≤ 𝑠 ≤ 𝑡 to 𝑅 𝑛 and by 𝑆𝑡 the set {𝑥𝑡 (. )}of all such functions. If 𝑥(𝑠) is a function defined and continuous on 0 ≤ 𝑠 < 𝑇 ≤ ∞ to 𝑅 𝑛 , then for each fixed 𝑡, 0 ≤ 𝑡 < 𝑇, this function defines a member 𝑥𝑡 (. )of 𝑆𝑡 given by 𝑥(𝑠), 0 ≤ 𝑠 ≤ 𝑡. The function 𝑥𝑡 (. ) is called a segment of 𝑥(𝑠). Definition 1 (Seifert [78]). The point 𝑥 = 0 is stable for (1) if given 𝜀 > 0, there exists a 𝛿(𝜀) > 0 such that 𝑖𝑓 |𝑥0 | < 𝛿(𝜀), then every solution 𝑥(𝑡, 𝑥0 ) of (1) is defined for 𝑡 ≥ 0 and satisfies |𝑥(𝑡, 𝑥0 )| < 𝜀 for 𝑡 ≥ 0.

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Definition 2 (Seifert [78]). The point𝑥 = 0 is said to be asymptotically stable for (1) if it is stable and if there exists a 𝑟0 > 0 such that |𝑥0 | < 𝑟0 implies 𝑥(𝑡, 𝑥0 ) → 0 as 𝑡 → +∞. Definition 3 (Seifert [78]). The solutions of (1) are bounded if given 𝑥0 in 𝑅 𝑛 , every solution 𝑥(𝑡, 𝑥0 ) is defined for 𝑡 ≥ 0 and there exists a 𝐵(𝑥0 ) such that |𝑥(𝑡, 𝑥0 )| ≤ 𝐵(𝑥0 ) for 𝑡 ≥ 0. Theorem 1 (Burton [16]). Let 𝑉(𝑡, 𝑥𝑡 ) be a differentiable scalar functional defined when 𝑥: [𝛼, 𝑡] → 𝑅 𝑛 is continuous and bounded by some 𝐷 ≤ ∞. Assume the following conditions hold: (𝐴1)𝑉(𝑡, 0) = 0, 𝑊1 (|𝑥(𝑡)|) ≤ 𝑉(𝑡, 𝑥𝑡 ), where 𝑊1 (𝑟) is a wedge; ( A2) 𝑉̇ (𝑡, 𝑥𝑡 ) ≤ 0. Then, the zero solution of the system of DDEs (1) is stable. Next, consider a system of DDEs of the form: 𝑥̇ (𝑡) = 𝐺(𝑥𝑡 ),

(2)

which is a particular case of system of DDEs (1). We assume that 𝐶 = 𝐶([−𝜏, 0], 𝑅 𝑛 ) is the space of continuous functions from [−𝜏, 0] into 𝑅 𝑛 and 𝐺: 𝐶 → 𝐺 𝑛 is continuous, and 𝐺(0) = 0, (see Yoshizawa [104]). If 𝐻 is a given positive constant, we use the notation 𝐶𝐻 for the set {𝜙 in 𝐶: ‖𝜙(𝑡)‖ < 𝐻}; that is, 𝐶𝐻 is the open ball in 𝐶 of radius 𝐻. Lemma 1 (Sinha [81, Lemma 1]). Suppose 𝐺(0) = 0. Let 𝑉 be a continuous functional defined on 𝐶𝐻 with 𝑉(0) = 0 and let 𝑢(𝑠) be a function, non-negative and continuous for 0 ≤ 𝑠 < ∞, 𝑢(𝑠) → ∞ as 𝑠 → ∞ with 𝑢(0) = 0. If for all 𝜑 in 𝐶, 𝒖(𝝓(𝟎)) ≤ 𝑽(𝝓), V ( )  0, 𝑉 ′ (𝜑) ≤ 0, then the zero solution of (2) is stable. Let ℝ ⊂ 𝐶𝐻 be a set of all functions 𝜙 ∈ 𝐶𝐻 where 𝑉 ′ (𝜑) = 0. If {0} is the largest invariant set in ℝ, then the solution 𝑥(𝑡) = 0 of (2) is asymptotically stable. Let 𝑉(𝑡, 𝑥) denote a function continuous on 𝑅 × 𝑅 𝑛 to 𝑅. Throughout this work, 𝑥 represents 𝑥(𝑡).

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Lemma 2 (Hale and Verduyn Lunel [38, Theorem 4.2, pp.152]). The system of DDEs (1) is globally uniformly asymptotically stable if there exists a continuous function 𝑉(𝑡, 𝑥), positive definite functions 𝒖, 𝒗, 𝝎 and a continuous non-decreasing function 𝑞(𝑠) > 𝑠 for 𝑠 > 0 such that the following conditions hold: 𝑢(|𝑥|) ≤ 𝑉(𝑡, 𝑥) ≤ 𝑣(|𝑥|) ,∀𝑡 ∈ 𝑅 + , ∀𝑥 ∈ ℝ𝑛 , 𝑑 𝑑𝑡

𝑉(𝑡, 𝑥) ≤ −𝜔(|𝑥|)

if 𝑉(𝑡 + 𝑠, 𝑥(𝑡 + 𝑠)) < 𝑞(𝑉(𝑡, 𝑥(𝑡))), ∀𝑠 ∈ [−𝜏, 0]. Theorem 2 (Seifert [78]). Let there exist functions 𝑢(𝑠), 𝑣(𝑠) and 𝑓(𝑠) continuous for 𝑠 ≥ 0 and such that 𝑢(0) = 𝑣(0) = 0, f(s) is increasing, 𝑓(𝑠) > 𝑠 for 𝑠 > 0, and suppose 𝑉(𝑡, 𝑥) is a real-valued function continuous in (𝑡, 𝑥) for 𝑡 ≥ 0 and 𝑥 in 𝐷, 𝐷 is an open subset of 𝑅 𝑛 containing the zero vector. Let 𝑉 satisfy: (𝐻2 ) 𝑢(|𝑥|) ≤ 𝑉(𝑡, 𝑥) ≤ 𝑣(|𝑥|) for 𝑡 ≥ 0, 𝑥 in𝐷, (𝐻3 ) 𝑉̇ (𝑡, 𝑥(𝑡)) ≤ 0 for any solution 𝑥(𝑡) of (1) for which 𝑥(𝑠) is in 𝐷 and 𝑓(𝑉(𝑡, 𝑥(𝑡)) > 𝑉(𝑠, 𝑥(𝑠)) for 0 ≤ 𝑠 ≤ 𝑡; here and henceforth 𝑉̇ (𝑡, 𝑥(𝑡)) = 𝑙𝑖𝑚 𝑠𝑢𝑝[ 𝑉(𝑡 + ℎ), 𝑥(𝑡 + ℎ) − 𝑉(𝑡, 𝑥(𝑡))]/ℎ. ℎ→0+

Then, the point 𝑥 = 0 is stable for the system of DDEs (1). Theorem 3. Let there exist a function 𝑉 satisfying the hypotheses of Theorem 2 except that now 𝐷 = 𝑅 𝑛 . Then, the solutions of the system of DDEs (1) are bounded, (see, Seifert [78]). We now assume that the following conditions hold. (a) There exist real-valued functions 𝑢(𝑠) and 𝑣(𝑠), continuous and increasing for 𝑠 ≥ 0 such that 𝑢(0) = 𝑣(0) = 0, and 𝑢(|𝑥|) ≤ 𝑉(𝑡, 𝑥) ≤ 𝑣(|𝑥|) for 𝑡 ≥ 0, 𝑥 ∈ 𝑅 𝑛 ,

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(b) Let 𝑓(𝑠) be a function continuous for 𝑠 ≥ 0 such that 𝑓(𝑠) > 𝑠 for 𝑠 > 0. Let 𝑥(𝑡) be a solution of (1) on [0, 𝑇), 𝑇 ≤ ∞. Then, there exists a number 𝑟 > 0, and a function 𝑤(𝑠) > 0 for 𝑠 > 0, such that the condition (𝐢) 𝑉(𝑠, 𝑥(𝑠)) < 𝑓(𝑉(𝑡, 𝑥(𝑡))) for 𝑠 ∈ [𝑡0 , 𝑡], 𝑡 > 0, where 𝑡0 = 𝑚𝑎𝑥[ 0, 𝑡 − 𝑟], implies. (ii) 𝑉̇ (𝑡, 𝑥(𝑡)) ≤ −𝑤(|𝑥(𝑡)|), where 𝑤(𝑠) and 𝑟 may depend on the solution 𝑥(𝑡), as well as on 𝑉 and 𝑓, and 𝑉̇ (𝑡, 𝑥(𝑡)) = 𝑙𝑖𝑚 𝑠𝑢𝑝[ 𝑉(𝑡 + ℎ, 𝑥(𝑡 + ℎ)) − 𝑉(𝑡, 𝑥(𝑡))]/ℎ. ℎ→0

Let 𝑉1 : 𝑅 + × 𝐶𝐻 → 𝑅 + , 𝑅 + = [0, ∞), be a continuous LKF with 𝑉1 (𝑡, 0) = 0. Further, let

𝑑 𝑉 (𝑡, 𝑥) 𝑑𝑡 1

denote the

derivative of 𝑉1 (𝑡, 𝑥) on the right along any solution of (1). Theorem 4 (Burton [16, Theorem 4. 2.9]). Assume that (𝐴1) The Lyapunov –Krasovskii functional (LKF) 𝑉1 (𝑡, 𝑥) satisfies the locally Lipschitz condition in 𝑥, i.e., for every compact 𝑆 ⊂ ℝ𝑛 and 𝛾 > 𝑡0 , there is a 𝐾𝛾𝑠 ∈ 𝑅 with 𝐾𝛾𝑠 > 0 such that |𝑉1 (𝑡, 𝑥𝑡 ) − 𝑉1 (𝑡, 𝑦𝑡 )| ≤ 𝐾𝛾𝑠 ‖𝑥 − 𝑦‖[𝑡0 −𝜏,𝑡] for all 𝑡 ∈ [𝑡0 , 𝛾] and 𝑥, 𝑦 ∈ 𝐶([𝑡0 − 𝜏, 𝑡0 ], 𝑆). (𝐴2) Let 𝑍: 𝑅 + × 𝐶𝐻 → 𝑅 + be a continuous functional such that it satisfies the one –sided locally Lipschitz condition in 𝑡, i.e., 𝑍( 𝑡2 , 𝜙) − 𝑍( 𝑡1 , 𝜙 ≤ 𝐾(𝑡2 − 𝑡1 ),0 < 𝑡1 < 𝑡2 < ∞, 𝐾 > 0, 𝐾 ∈ 𝑅, 𝜙 ∈ 𝐶𝐻 . (𝐴3) There are four strictly increasing functions 𝜔, 𝜔1 , 𝜔2 , 𝜔3 : 𝑅 + → 𝑅 with value 0 at 0 such that +

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𝜔(‖𝜙(0)‖) + 𝑍( 𝑡, 𝜙) ≤ 𝑉1 (𝑡, 𝜙) ≤ 𝜔1 (‖𝜙(0)‖) + 𝑍(𝑡, 𝜙), 𝑍( 𝑡, 𝜙) ≤ 𝜔2 (‖𝜙‖𝐶 ), and 𝑑 𝑉 (𝑡, 𝑥) ≤ −𝜔3(‖𝑥(𝑡)‖) 𝑑𝑡 1 whenever 𝑡 ∈ ℝ+and 𝑥 ∈ 𝐶𝐻 . Then, the solution 𝑥(𝑡) = 0 of (1) is uniformly asymptotically stable.

3. Lyapunov and Lyapunov –Krasovskii Qualitative Results of IDEs The qualitative properties of solutions, such as stability, instability, and boundedness of solutions, are very important for IDEs and different kinds of FDEs. These concepts and various others have attracted the attention of many researchers during the last decades. Hence, numerous results related to the qualitative properties of certain IDEs and FDEs have been obtained in the literature (see, [1-39], [41-45], [47-110]). Next, we summarize some of the work from the references cited in this chapter dealing with the qualitative proprieties of IDEs and systems of IDEs. In the book by Burton [16], a reference book on integral equations and IDEs, stability, uniform stability, asymptotic stability, uniform asymptotic stability of the zero solution, as well as, integrability and boundedness of non-zero solutions when 𝐹(𝑡) ≠ 0, are discussed for the following systems of IDEs using Lyapunov’s second method 𝑡

𝑥 ′ = 𝐴(𝑡)𝑥 + ∫ 𝐶(𝑡, 𝑠)𝑥(𝑠)𝑑𝑠, 0

𝑡

′

𝑥 = 𝐴𝑥 + ∫ 𝐶(𝑡, 𝑠)𝑥(𝑠)𝑑𝑠 + 𝐹(𝑡), 0

𝑡

𝑥 ′ = 𝐴𝑥 + ∫ 𝐵(𝑡 − 𝑠)𝑥(𝑠)𝑑𝑠, 𝑡

0

𝑥 ′ = 𝐴𝑥 + ∫ 𝐷(𝑡 − 𝑠)𝑥(𝑠)𝑑𝑠 + 𝐹(𝑡), 0

Advances in the Qualitative Theory of Integro–Differential Equations 𝑡

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𝑡

𝑥 ′ = 𝐴(𝑡)𝑥 + ∫ 𝐶1 (𝑡, 𝑠)𝑥(𝑠)𝑑𝑠 + ∫ 𝐶2(𝑡, 𝑠)𝑥(𝑠)𝑑𝑠, 0

𝑡

0

𝑥 ′ = 𝐴𝑥 + 𝑓(𝑡, 𝑥) + ∫ 𝐶(𝑡, 𝑠)𝑥(𝑠)𝑑𝑠. 0

In the book by Lakshmikantham and Rama Mohana Rao [48], a reference book on the theory of IDEs, various qualitative behaviors of solutions such as stability, uniform stability, asymptotic stability, uniform asymptotic stability of the zero solution, as well as, integrability and boundedness of non-zero solutions when 𝑓(𝑡, 𝑥) ≠ 0 and𝑔(𝑡, 𝑦) ≠ 0, are investigated using Lyapunov’s second method. Very interesting results are obtained for the following scalar and systems of IDEs : 𝑡

𝑢′ = 𝛼𝑢 + ∫ 𝑎(𝑡 − 𝑠)𝑢(𝑠)𝑑𝑠, 0

𝑡

𝑢′ = 𝛼(𝑡)𝑢 + ∫ 𝑎(𝑡, 𝑠)𝑢(𝑠)𝑑𝑠, 0 𝑡

𝑥 ′ = 𝐴(𝑡)𝑥 + ∫ 𝐾(𝑡, 𝑠)𝑥(𝑠)𝑑𝑠, 0 𝑡

′

𝑥 = 𝐴𝑥 + ∫ 𝐾(𝑡, 𝑠)𝑥(𝑠)𝑑𝑠, 0

𝑡

𝑥 ′ = 𝐴(𝑡)𝑥 + ∫ 𝐾(𝑡, 𝑠)𝑥(𝑠)𝑑𝑠 + 𝑓(𝑡, 𝑥), −∞

𝑡

𝑥 ′ = 𝐴𝑥 + ∫ 𝐶(𝑡 − 𝑠)𝑥(𝑠)𝑑𝑠, ′

𝑡

0

𝑦 = 𝐴(𝑡)𝑦 + ∫ 𝐶(𝑡 − 𝑠)𝑦(𝑠)𝑑𝑠 + 𝑔(𝑡, 𝑦). 0

In [17], Burton first considered the following class of scalar IDEs: 𝑡

𝑥 ′ (𝑡) = 𝐴(𝑡)𝑓(𝑥(𝑡)) + ∫ 𝐵(𝑡, 𝑠)𝑔(𝑥(𝑠))𝑑𝑠, 𝑡

0

𝑥 ′ (𝑡) = − ∫ 𝐶(𝑡, 𝑠)ℎ(𝑥(𝑠))𝑑𝑠 0

and

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𝑥 ′ (𝑡) = 𝐴(𝑡)𝑓(𝑥(𝑡)) + ∫ [𝐵(𝑡, 𝑠) − 𝐶(𝑡, 𝑠) − 𝐷(𝑡, 𝑠)]𝑓(𝑥(𝑠)𝑑𝑠. 0

He discussed the stability of the zero solution, the boundedness and convergence of the bounded solutions to the zero solution of the first equation by a Lyapunov–Krasovskii functional [17, Theorem 1]. He also discussed the stability of the zero solution, boundedness of solutions, the square integrability of 𝑥 ′ (𝑡), the convergence of the solution 𝑥(𝑡) to the zero solution as 𝑡 → ∞ of the second equation via a Lyapunov –Krasovskii functional [17, Theorem 2], as well as, the stability of the zero solution, the ∞ boundedness of solutions and showed that ∫0 𝑓 2 (𝑥(𝑡))𝑑𝑡 < ∞ by a Lyapunov –Krasovskii functional [17, Theorem 4]. Later, Burton [17] considered the following system of IDEs and some of its modified versions: 𝑡

𝑥 ′ (𝑡) = 𝐴(𝑡)𝑓(𝑥(𝑡)) + ∫ 𝐵(𝑡, 𝑠)𝐸(𝑥(𝑠)𝑥(𝑠)𝑑𝑠. 0

In Burton [17], some sufficient conditions are established for this equation and its modified versions such that under those conditions the zero solution is stable or uniformly stable, the solutions are bounded, and all bounded solutions tend to zero for the considered equations [17, Theorems 5-8]. As for some recent works on the qualitative properties of solutions of IDEs, Matsunaga, Hashimoto [56] dealt with the problem of asymptotic stability for the following linear system of IDEs with distributed delay in the diagonal terms: 𝑡

𝑥 ′ = 𝐴𝑥 − 𝑎 ∫ 𝑥(𝑠)𝑑𝑠. 𝑡−𝜏

They also established some explicit conditions for the asymptotic stability of the zero solution of the above system. In particular, as the delay parameter increases monotonically under certain conditions, the zero solution switches in finite times from stability to instability to stability, and becomes eventually unstable. Raffoul and Rai [70] used LKFs and discussed exponential stability of the following IDE with finite delay:

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𝑡

𝑥 ′ = 𝑃𝑥 + ∫ 𝐶(𝑡, 𝑠)𝑔(𝑥(𝑠))𝑑𝑠. −∞

In [70], the authors used modified versions of the Lyapunov –Krasovskii functional that were used previously to obtain a criterion for the stability of the zero solution of the infinite delay nonlinear Volterra IDE. By using spectral properties of Metzler matrices and the comparison principle, Ngoc and Anh [59] established some new explicit criteria for uniform asymptotic stability and exponential stability of the following nonlinear IDE of the form: 𝑡

𝑥 ′ = ℎ(𝑡, 𝑥) + ∫0 𝑞(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠. In this paper, Ngoc and Anh presented a novel approach to problems of stability of Volterra IDEs. The approach of [59] is simple and relies upon the spectral properties of Metzler matrices and the comparison principle. Consequently, they obtained some explicit criteria for the uniform asymptotic stability and the exponential stability of the above nonlinear Volterra IDE. Some examples are given to illustrate the obtained results. In Berezansky et al. [12], criteria for uniform exponential stability are derived for the linear IDEs with several point delays and several distributed delays of the form: 𝑚

𝑘

′

𝑡

𝑥 = ∑ 𝐴𝑘 (𝑡)𝑥(ℎ𝑘 (𝑡)) + ∑ ∫

𝑃𝑘 (𝑡, 𝑠)𝑥(𝑠))𝑑𝑠.

𝑘=1 𝑔𝑘 (𝑡)

𝑘=1

In Berezansky et al. [12], the main technique of the proofs is splitting the linear expressions in the equation (both with point and with distributed delays) into a “dominant” and a “remainder”' part, which can be done in a number of different ways, thus providing a number of different criteria. The next important ingredient is the use of a Bohl-Perron type result stating that a linear equation is exponentially stable if all of the solutions of the inhomogeneous counterpart of that equation are bounded. Seifert [79] considered the following IDEs of Volterra type: 𝑡

𝑥̇ (𝑡) = 𝐺(𝑡, 𝑥(𝑡)) + ∫0 𝐾(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠.

(3)

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As an application, Seifert [79] considered the following particular case of IDE (3) 𝑡

𝑥̇ (𝑡) = 𝐴𝑥(𝑡) + ∫0 𝐾(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠,

(4)

where 𝐴 is a real constant 𝑛 × 𝑛 matrix, the eigenvalues of which all have negative real parts, 𝐾 is continuous in (𝑡, 𝑠, 𝑥) for 0 ≤ 𝑠 ≤ 𝑡 < ∞ and 𝑥 in 𝑅 𝑛 , and is such that there exist constants 𝜇 and 𝜌 for which 𝑡

|∫0 𝐾(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠| ≤ 𝜇 𝑠𝑢𝑝 |𝑥(𝑠)|

(5)

0≤𝑠≤𝑡

for any function 𝑥(𝑠) continuous on 0 ≤ 𝑠 ≤ 𝑡 with |𝑥(𝑠)| ≤ 𝜌 on this interval. It is well known from results in matrix theory that there exists a positive definite symmetric real matrix 𝐵 such that 𝐵𝐴 + 𝐴𝑇 𝐵 = −𝐼, where 𝐼 denotes the identity matrix and 𝐴𝑇 the transpose of 𝐴. If 𝜆 is the least eigenvalue of 𝐵, then 𝜆 > 0 and (𝑥, 𝐵𝑥) ≥ 𝜆2 (𝑥, 𝑥) for all 𝑥 in 𝑅 𝑛 . Choose a Lyapunov function as 𝑉(𝑡, 𝑥) = 𝑉(𝑥) = (𝑥, 𝐵𝑥). Clearly (𝐻2 ) holds for the Lyapunov function 𝑉 since we may take 𝑣(𝑠) = 𝜆𝑠 2 and 𝑢(𝑠) = 𝛬𝑠 2 , where 𝜆 and 𝛬 are, respectively, the least and greatest eigenvalues of 𝐵. The time derivative of the Lyapunov function 𝑉 along (4) yields 𝑡 𝑉̇ (𝑥(𝑡)) = (𝑥(𝑡), (𝐴𝑇 𝐵 + 𝐵𝐴)𝑥(𝑡)) + 2(∫0 𝐾(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠, 𝐵𝑥(𝑡)) 𝑡

= −|𝑥(𝑡)|2 + 2(∫0 𝐾(𝑡, 𝑠, 𝑥(𝑠)), 𝐵𝑥(𝑡)).

(6)

Let 𝑓(𝑠) = 𝑞2 𝑠 for 𝑞 > 1, 𝑡 > 0 and 𝑥(𝑠) be a solution of (4) such that |𝑥(𝑠)| < 𝜌 for 0 ≤ 𝑠 ≤ 𝑡. Suppose that 𝑞2 (𝑥(𝑡), 𝐵𝑥(𝑡)) > (𝑥(𝑠), 𝐵𝑥(𝑠)) for 0 ≤ 𝑠 ≤ 𝑡. This inequality implies 𝑞2 (𝑥(𝑡), 𝐵𝑥(𝑡)) > 𝜆2 |𝑥𝑡 |2 ,

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where we define |𝑥𝑡 | = 𝑠𝑢𝑝 |𝑥(𝑠)|. 0≤𝑠≤𝑡

2 If 𝐵 = (𝑏𝑖𝑗 ) and |𝐵| = √𝛴𝑖,𝑗 𝑏𝑖,𝑗 , using (5), it follows that

𝑡

𝑡

(∫0 𝐾(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠, 𝐵𝑥(𝑡)) ≤ |𝐵| |∫0 𝐾(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠| |𝑥(𝑡)| ≤ |𝐵|𝜇|𝑥𝑡 ||𝑥(𝑡)| < |𝐵|𝜇𝑞(𝑥(𝑡), 𝐵𝑥(𝑡))1/2 |𝑥(𝑡)|𝜆 ≤ |𝐵|𝜇𝑞|𝑥(𝑡)|2𝛬/𝜆. If 2|𝐵|𝜇𝛬/𝜆 < 1, then there obviously exists a 𝑞 > 1 such that 2|𝐵|𝜇𝛬/ 𝜆 < 1. For this choice of 𝑞, it follows from (6) that 𝑉̇ (𝑥(𝑡)) ≤ −(1 − 2|𝐵|𝜇𝑞𝛬/𝜆)|𝑥(𝑡)|2 ≤ 0; thus, (𝐻3 ) holds for 𝐷 the set of 𝑥 in 𝑅 𝑛 such that |𝑥| < 𝜌. As a consequence of the above result, we have, therefore, the following result. Seifert [79] proved the following theorem . Theorem 5. We assume that

2|𝐵|𝜇𝛬 𝜆

< 1 and the conditions above hold.

Then the trivial solution of IDE (4) is stable. In [79], Seifert used Lyapunov functions of Razumkhim type to give conditions sufficient for the stability of the zero state of a system of ordinary differential equations involving an interval of delay which becomes unbounded as 𝑡 → +∞. An example of such a system is an IDE of Volterra type: 𝑡

𝑥̇ (𝑡) = 𝐺(𝑡, 𝑥(𝑡)) + ∫0 𝐾(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠, where 𝑥, 𝐺 and 𝐾 denote functions with values in 𝑅 𝑛 . As an example, Seifert [79] also considered the following system of IDEs: 𝑡

𝑥̇ (𝑡) = 𝐴𝑥(𝑡) + ℎ(𝑡, 𝑥𝑡 ) + ∫0 𝑔(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠,

(7)

where 𝐴 is a real stable 𝑛 × 𝑛 -matrix, ℎ is a function continuous on 𝑅 × 𝐶𝑟 , (𝐶𝑟 is the space (with the usual supremum norm) of continuous functions from [−𝑟, 0] to 𝑅 𝑛 ), and satisfying

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|ℎ(𝑡, 𝜙)| ≤ 𝜇‖𝜙‖, where ‖𝜙‖ = 𝑠𝑢𝑝 ≥ |𝜙(𝑠)|, 𝑠∈[−𝑟,0]

and 𝑔(𝑡, 𝑠, 𝑥) is continuous on 𝑅 × 𝑅 × 𝑅 𝑛 , 𝑡 ≥ 𝑠 ≥ 0, and satisfies |𝑔(𝑡, 𝑠, 𝑥)| ≤ 𝐾(𝑡, 𝑠)|𝑥| with 𝑡

∫0 𝐾(𝑡, 𝑠)𝑑𝑠 → 0 as𝑡 → +∞. In Seifert [79], it is shown that for 𝜇 sufficiently small, each bounded solution of (7) tends to zero as 𝑡 → +∞. Since 𝐴 is a stable matrix, by a wellknown result, there exists a positive definite symmetric matrix 𝐵 such that 𝐵𝐴 + 𝐴𝑇 𝐵 = −𝐼, where 𝐴𝑇 denotes the transpose of 𝐴 and 𝐼 the identity matrix. Define the Lyapunov function 𝑉 = (𝐵𝑥, 𝑥). Then, there exist positive numbers 𝜆 and 𝛬 such that 𝜆2 |𝑥|2 ≤ 𝐵𝑥. 𝑥 ≤ 𝛬2 |𝑥|2 for all 𝑥 ∈ 𝑅 𝑛 . If 𝜇 < 𝜆/(2|𝐵|𝛬), 𝑛 where |𝐵| = 𝛴𝑖,𝑗=1 | 𝑏𝑖𝑖 |, 𝐵 = (𝑏𝑖𝑗 ). Then, there exists a 𝑞 > 1 such that

𝜇 < 𝜆/(2𝑞|𝐵|𝛬). Let 𝑓(𝑠) = 𝑞2 𝑠. Then, for any positive integer 𝑘 and any solution 𝑥(𝑡) of (7) such that (𝐵𝑥(𝑠), 𝑥(𝑠)) < 𝑞2 (𝐵𝑥(𝑡), 𝑥(𝑡)) for 𝑠 ∈ [𝑡 − 𝑘𝑟, 𝑡], 𝑡 ≥ 𝑘𝑟, it is clear that 𝑞2 𝛬2|𝑥(𝑡)|2 > 𝜆2 ‖𝑥𝑡 ‖2 , 𝑡 ≥ 𝑘𝑟.

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Thus, |(𝐵𝑥(𝑡), ℎ(𝑡, 𝑥𝑡 ))| ≤ |𝐵| |𝑥(𝑡)|𝜇‖𝑥(𝑡)‖ ≤ 𝜇|𝐵|𝑞𝛬|𝑥(𝑡)|2 /𝜆, 𝑡 ≥ 𝑘𝑟. Fix 𝜇1 > 0 such that (2|𝐵|𝑞𝛬/𝜆)(𝜇 + 𝜇1 ) < 1. Then, there exists a 𝑘 > 0 such that for 𝑡 ≥ 𝑘𝑟, 𝑡

∫0 𝐾(𝑡, 𝑠)𝑑𝑠 ≤ 𝜇1 . Since 𝑡

𝑡

|∫𝑡−𝑘𝑟 𝑔(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠| ≤ ∫𝑡−𝑘𝑟 𝐾(𝑡, 𝑠)|𝑥(𝑠)| 𝑑𝑠 𝑡

≤

𝑠𝑢𝑝 |𝑥(𝜃)| ∫ 𝐾(𝑡, 𝑠)𝑑𝑠, 𝑡 ≥ 𝑘𝑟, 𝜃∈[𝑡−𝑘𝑟,𝑡]

0

and 𝑥(𝑡) is a solution of (7) satisfying (𝐵𝑥(𝑠), 𝑥(𝑠)) < 𝑞2 (𝐵𝑥(𝑡), 𝑥(𝑡)), it follows that for such 𝑘, 𝑡 and 𝑥(𝑡), 𝑡

|2(𝐵𝑥(𝑡), ∫𝑡−𝑘𝑟 𝑔(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠)| ≤ 2𝑞𝛬𝜇1 |𝐵||𝑥(𝑡)|2 /𝜆. If we define 𝛼 = 1 − 2|𝐵|𝑞𝛬(𝜇 + 𝜇1 )/𝜆, and 𝑤(𝑠) = 𝛼𝑠 2 , then (2|𝐵|𝑞𝛬/𝜆)(𝜇 + 𝜇1 ) < 1, which implies that 𝛼 > 0, and 𝑡−𝑘𝑟

|∫0

𝑡

𝑔(𝑡, 𝑠, 𝑥)𝑑𝑠| ≤ |𝑥| ∫0 𝐾(𝑡, 𝑠)𝑑𝑠 → 0 as 𝑡 → +∞,

Hence, it can be concluded that each bounded solution of (7) tends to zero as 𝑡 → +∞, (see, Seifert [79, Theorem 2]). In [95], Vanualailai and Nakagiri considered the IDE of the form: 𝑑 𝑑𝑡

𝑡

[𝑥(𝑡)] = 𝐴(𝑡)𝑓(𝑥(𝑡)) + ∫0 𝐵(𝑡, 𝑠)𝑔(𝑥(𝑠))𝑑𝑠,

(8)

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in which 𝐴(𝑡) is an 𝑛 × 𝑛 - matrix function continuous on [0, ∞), 𝐵(𝑡, 𝑠) is an 𝑛 × 𝑛 matrix continuous for 0 ≤ 𝑠 ≤ 𝑡 < ∞, and 𝑓 and 𝑔 are 𝑛 × 1 vector functions continuous on (−∞, ∞). In Vanualailai and Nakagiri [95], it is assumed that 𝐴(𝑡) is continuous 𝑡 for 0 ≤ 𝑡 < ∞, 𝐵(𝑡, 𝑠) is continuous for 0 ≤ 𝑠 ≤ 𝑡 < ∞, ∫0 |𝐵(𝑢, 𝑠)|𝑑𝑢 is defined and continuous for 0 ≤ 𝑠 ≤ 𝑡 < ∞, 𝑓(𝑥) and 𝑔(𝑥) are continuous on (−∞, ∞), 𝑥𝑓(𝑥) > 0, ∀𝑥 ≠ 0, a𝑓(0) = 𝑔(0) = 0. Theorem 6 (Vanualailai and Nakagiri [95]). Let the conditions above hold and suppose there are constants 𝑚 > 0 and 𝑀 > 0 such that, 𝑔2 (𝑥) ≤ 𝑚2𝑓 2 (𝑥) if |𝑥| ≤ 𝑀. Define ¥

1

𝑡

𝛽(𝑡, 𝑘) = 𝐴(𝑡) + 𝑘 ∫𝑡 |𝐵(𝑢, 𝑡)|𝑑𝑢 + 2 ∫0 |𝐵(𝑡, 𝑠)|𝑑𝑠. If there exists 𝑘 > 0 with 𝑚2 < 2𝑘 and 𝛽(𝑡, 𝑘) ≤ 0 for 𝑡 ≥ 0, then the zero solution of IDE (8) is stable. Theorem 7 (Vanualailai and Nakagiri [95]). Let the condition above hold, with 𝐴(𝑡) < 0, and suppose there are constants 𝑚 > 0 and 𝑀 > 0 such that 𝑔2 (𝑥) ≤ 𝑚2 𝑓 2 (𝑥) if |𝑥| ≤ 𝑀, 𝛼 > 4; and 𝑁 > 0 such that 4𝑥 2 ≤ (𝛼 − 1 𝑡 1 4)𝑓 2 (𝑥); if |𝑥| ≤ 𝑁, and 𝐽 ≥ 1 such that − ∫0 |𝐵(𝑡, 𝑠)|𝑑𝑠 < for every 4𝐴(𝑡)

𝑡 ≥ 0. Suppose further there is some constant 𝑘 > 0 such that ∞ 𝑘 ∫𝑡 |𝐵(𝑢, 𝑡)|𝑑𝑢

and 𝐴(𝑡) + (8) is stable.

𝐽 (1+𝛼) 𝑚2 𝐽

1 such that when 𝑡 ∈ 𝑅 we have 𝑡

𝜇𝐴(𝑡) + 𝐾 [∫0 ‖𝐶(𝑡, 𝑠)‖𝑑𝑠 + 𝑏(𝑡)] ≤ 0,. Then, the zero solution of (9) is uniformly stable. Example 1 [22]. Consider a functional differential equation such that 𝜆∗𝑀 (𝑡) = −4𝑒 −2𝑡 , ‖𝐶(𝑡, 𝑠)‖ = 𝑒 −𝑡−𝑠 , ‖𝐶(𝑢, 𝑡)‖ = 𝑒 −𝑢−𝑠 . Then, 1

1 ∗ 𝜆 (𝑡)𝜆2𝑚 (𝐵) 2 𝑀 𝑡

1

+∞

+ 𝜆2𝑀 (𝐵) ∫0 ‖𝐶(𝑢, 𝑡)‖𝑑𝑢 = (−2 + √2)𝑒 −2𝑡 < 0

+∞

1

and when ∫0 ∫𝑡 𝐶(𝑢, 𝑠)𝑑𝑢𝑑𝑠 = 1 − 2 𝑒 −2𝑡 , 𝑡 ≥ 0. This is obviously bounded, and then from Theorem 10 , the zero solution of the corresponding IDE is uniformly stable. Example 2 [22]. Consider the integro-differential equation 2𝑡 𝑥̇ (𝑡) = (−𝑒 0

0 ) 𝑥(𝑡) + 𝑡 (𝑒 −𝑡+2𝑠 ∫0 −𝑒 3𝑡 0 𝑡

0 ) 𝑥(𝑠)𝑑𝑠. 𝑒 −𝑡+𝑠 1

Obviously, it follows that 𝜇(𝐴(𝑡)) = 𝑒 −2𝑡 , ∫0 ‖𝐶(𝑡, 𝑠)‖𝑑𝑠 = (𝑒 −𝑡 − 𝑒 −𝑡 ). 2 1

Let 𝐾 = 2. Then, we have 𝜇(𝐴(𝑡)) + 2 [2 (𝑒 𝑡 − 𝑒 −𝑡 )] = −𝑒 2𝑡 + 𝑒 𝑡 + 𝑒 −𝑡 ≤ 0, 𝑡 ∈ 𝑅. According to Theorem 10, the zero solution of this equation is uniformly 𝑡 +∞ stable, but because ∫0 ∫𝑡 ‖𝐶(𝑢, 𝑠)‖𝑑𝑢𝑑𝑠 = 𝑒 𝑡 − 1 when 𝑡 ≥ 0 is unbounded, from Theorem 9 we cannot determine its uniform stability. In a recent and very interesting paper, Sedova [77] considered the following IDE: 𝑡

𝑥̇ (𝑡) = 𝐺(𝑡, 𝑥(𝑡)) + ∫0 𝐻(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠,

(10)

Advances in the Qualitative Theory of Integro–Differential Equations

where

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𝑥(𝑡) ∈ 𝑅𝑛 , 𝑡 ∈ 𝑅+ , 𝑅 + = [0, ∞), 𝐺 ∈ 𝐶(𝑅 + × 𝑅𝑛 , 𝑅𝑛 ), 𝐻 ∈ 𝐶(𝐷 ×

𝑅 , 𝑅 ). Also, for a continuous function 𝑥: (−∞, 𝐴] → 𝑅 𝑛 , 0 ≤ 𝐴 ≤ +∞, for 𝑛

𝑛

each 𝑡 ≤ 𝐴, the function

xt : R − → R n , 𝑅− = (−∞, 0],

is defined as

𝑥𝑡 (𝑠) = 𝑥(𝑡 + 𝑠). It is also assumed that 𝐺(𝑡, 0) = 0, 𝐻(𝑡, 𝑠, 0) = 0, so IDE (10) admits the zero solution. If the sequence {𝐺(𝑡 + 𝑡𝑘 , 𝑥)} converges to 𝐺 ∗ (𝑡, 𝑥), and the sequence {𝐻(𝑡 + 𝑡𝑘 , 𝑡 + 𝑡𝑘 + 𝑠, 𝑥)} converges to 𝐻 ∗ (𝑡, 𝑡 + 𝑠, 𝑥) as 𝑡𝑘 → +∞ uniformly on the corresponding compact subsets, then a limiting equation for IDE (10) has the form: 𝑡

𝑥̇ (𝑡) = 𝐺 ∗ (𝑡, 𝑥(𝑡)) + ∫−∞ 𝐻 ∗ (𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠.

(11)

It is clear that equation (11) can be considered in the same phase space as the orginal one, and the limiting functions satisfy the same constraints as the original ones. In [77], the specific applications of Razumikhin technique to the stability analysis of this IDE are considered and new sufficient conditions for uniform asymptotic stability of the zero solution are given using the phase space of a special construction and constraints on the right side of IDE (11). Theorem 11 (Sedova [77]). Suppose that there exists a positive definite continuously differentiable function 𝑉(𝑥) such that one of the following conditions is satisfied: 1) the derivative of 𝑉(𝑥) by virtue of (10) is negative definite for 𝜑 ∈ 𝐶𝑔 satisfying 𝑠𝑢𝑝𝑠≤0 𝑉 (𝜑(𝑠)/𝑔(𝑠)) ≤ 𝑉(𝜑(0)) and for sufficiently large 𝑡. 2) the derivative

𝑑 𝑑𝑡

𝑉(𝑥(𝑡)) by virtue of every limiting equation (11)

does not exceed a function 𝑊(𝑥(𝑡)) for 𝜑 ∈ 𝐶𝑔 , (𝜑 ∈ 𝐶𝑔 ⇒ |𝜑|𝑔 ≤ ℎ, ℎ > 0), satisfying 𝑠𝑢𝑝𝑠≤0 𝑉 (𝜑(𝑠)/𝑔(𝑠)) ≤ 𝑉(𝜑(0)), and 𝑊(𝑥) is negative definite. Then, the zero solution of IDE (10) is uniformly asymptotically stable. Example 3 (Sedova [77]). Consider the following scalar IDE: 𝑡

𝑥̇ (𝑡) = −ℎ(𝑡)𝑥(𝑡) − 𝑏(𝑡)𝑥 3(𝑡) + ∫0 𝑐(𝑎𝑡 − 𝑠)𝑥(𝑠)𝑑𝑠,

(12)

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Osman Tunç, Seenith Sivasundaram and Cemil Tunç

where 𝑐 ∈ 𝐿1 [0, +∞], 𝑎 > 1, ℎ(𝑡) ≥ 0 and 𝑏(𝑡) ≥ 𝑏0 > 0 are continuous functions. The following conditions for uniform asymptotic stability are obtained via an LKF: ∞

2ℎ(𝑡) ≥ [1 + (1/𝑎)] ∫

|𝑐(𝑣)|𝑑𝑣,

(𝑎−1)𝑡

∞

∫ |𝑐(𝑢)|𝑑𝑢 ∈ 𝐿1 0, +∞). 𝑡 0

∞

We denote 𝑀(𝑡) = ∫−∞|𝑐((𝑎 − 1)𝑡 − 𝑠)|𝑑𝑠 = ∫(𝑎−1)𝑡|𝑐(𝑣)|𝑑𝑣 (this function is bounded). Then, for an arbitrarily small 𝛿 > 0 there exists an appropriate function 𝑔 such that 0

∫−∞|𝑐((𝑎 − 1)𝑡 − 𝑠|𝑔(𝑠)𝑑𝑠 < 𝑀(𝑡) + 𝛿. Using the Lyapunov function 𝑉(𝑥) = 𝑥 2 , we obtain that the zero solution of (12) is uniformly asymptotically stable if the functions ℎ(𝑡) and 𝑏(𝑡) are bounded and satisfy the inequality ∞

ℎ(𝑡) ≥ ∫

|𝑐(𝑣)|𝑑𝑣 + 𝜀

(𝑎−1)𝑡

for some 𝜀 > 0. Example 4 (Sedova [77]). Consider the following scalar IDE: 𝑡

𝑥̇ (𝑡) = −𝑝(𝑡)𝑥(𝑡) + 𝑞(𝑡, 𝑥(𝑡)) + ∫ 𝐶(𝑡 − 𝑠)ℎ(𝑥(𝑠))𝑑𝑠, 0

Where ℎ(0) = 0, |ℎ(𝑥)| ≤ 𝛾|𝑥| for 𝑥 ≠ 0, ∞ 𝑝(𝑡) − 𝛾 ∫𝑡 |𝐶(𝑢 − 𝑡)|𝑑𝑢 > 𝛿 > 0 for 𝑡 ∈ 𝑅 + , |𝑞(𝑡, 𝑥)| ≤ 𝑟(𝑡)|𝑥| for some 𝑟(𝑡) ∈ 𝐿1 .

(13)

By using an LKF, it is proved that under the given conditions the zero solution of IDE (13) is asymptotically stable. Next, the inequality 𝑝(𝑡) −

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0

𝛾 ∫𝑡 |𝐶(𝑢 − 𝑡)|𝑑𝑢 > 𝛿 > 0 for 𝑡 ∈ 𝑅 + implies that ∫−∞|𝐶(−𝑠)|𝑑𝑠 < 𝜀 = (𝑖𝑛𝑓𝑡∈𝑅+ 𝑝 (𝑡) − 𝛿1 )/𝛾 for some 𝛿1 ∈ (0, 𝛿). Then, there exists an 0 appropriate function 𝑔 such that ∫−∞|𝐶(−𝑠)|𝑔(𝑠)𝑑𝑠 < 𝜀. Using the function

𝑉(𝑥) = 𝑥 2 , it can now be obtained that the time derivative 𝑉 ′ is negative definite for 𝜑 ∈ 𝐶𝑔 satisfying 𝑠𝑢𝑝𝑠≤0 𝑉 (𝜑(𝑠)/𝑔(𝑠)) ≤ 𝑉(𝜑(0)) and for sufficiently large 𝑡. Then, these results imply the uniform asymptotic stability of the zero solution of the above IDE. Nieto and Tunç [64] considered the non-linear VIDE as follows: 𝑑𝑥 𝑑𝑡

𝑡

= −𝐹(𝑡, 𝑥)𝑥 + ∫0 𝐾(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠, ∀𝑡 > 𝑡0 ≥ 0,

(14)

where 𝑡 ∈ 𝑅 + , 𝑥 ∈ 𝑅 𝑛 , 𝐹(𝑡, 𝑥) ∈ 𝐶(𝑅+ × 𝑅 𝑛 , 𝑅 𝑛 × 𝑅 𝑛 ), 𝐷 = {(𝑢, 𝑣) ∈ 𝑅 2 : 0 ≤ 𝑣 ≤ 𝑢 < ∞},and 𝐾(𝑢, 𝑣, 𝑥) ∈ 𝐶(𝐷 × 𝑅 𝑛 , 𝑅 𝑛 ) and 𝐾(𝑢, 𝑣, 𝑥) = 0 ⇔ 𝑥 = 0. Initially, they investigated the asymptotic and uniform stability of the trivial solution of IDE (14) using the LKF approach, under the following conditions: (𝐶1) 𝐹(𝑡, 𝑥) ∈ 𝐶(𝑅 + × 𝑅 𝑛 , 𝑅 𝑛 × 𝑅 𝑛 ) and is positive definite such that 𝑠𝑢𝑝 ‖𝐹(𝑡, 𝑥)‖ < ∞, (𝑡,𝑥)∈ℜ+ ×ℜ𝑛

and the eigenvalues of 𝐹(𝑡, 𝑥) satisfy,

f1i  i ( F (t , x))  f0i , 𝑓0𝑖 > 0 +

t  R , 𝑅 = [0, ∞), ∀𝑥 ∈ 𝑅 𝑛 , (𝑖 = 1, . . , 𝑛). +

(𝐶2) Let 𝛽 > 0 be a positive constant such that ‖𝐾(𝑡, 𝑠, 𝑥(𝑠))‖ ≤ ‖𝐷(𝑡, 𝑠)‖ ‖𝑓(𝑥(𝑠))‖, ‖𝑓(𝑥(𝑠))‖ ≤ 𝛽‖𝑥(𝑠)‖, 𝑡 ∞ ∫0 ‖𝐷(𝑡, 𝑠)‖𝑑𝑠 ≤ 𝛼1 (𝑡), ∫𝑡 ‖𝐷(𝑢, 𝑡)‖𝑑𝑢 ≤ 𝛼2 (𝑡),

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where 𝛼1 , 𝛼2 ∈ 𝐶(𝑅 + , 𝑅 + ) and are bounded for ∀𝑡 ∈ 𝑅, and 1

𝑡

1

∞

𝛼(𝑡) = 𝑓0 − 2 ∫0 ‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠 − 2 𝛽 2 ∫𝑡 ‖𝐷(𝑢, 𝑡)‖𝑑𝑢 ≥ 𝑐 > 0, ∀𝑡 ∈ 𝑅, where 𝑐 ∈ 𝑅, 𝑓0 = 𝑚𝑖𝑛{ 𝑓01 , 𝑓02 , . . . , 𝑓0𝑛 ). Theorem 12 (Nieto and Tunç [64]). The zero solution of IDE (14) is asymptotic stable if assumptions(𝐶1) and (𝐶2) hold. Proof. To prove the theorem, we use the LKF method. Define the 𝑡 ∞ 1 functional: 𝑊 = 𝑊(𝑡, 𝑥𝑡 ) = 2 ⟨𝑥(𝑡), 𝑥(𝑡)⟩ + 𝜎 ∫0 ∫𝑡 ‖𝐷(𝑢, 𝑠)‖ ‖𝑓(𝑥(𝑠))‖2 𝑑𝑢𝑑𝑠, where 𝜎 > 0, 𝜎 ∈ 𝑅, the constant 𝜎 is chosen later in the proof. For the first step, from the given functional, we derive 𝑊(𝑡, 0) = 0 and 𝑊(𝑡, 𝑥𝑡 ) ≥ 1 ‖𝑥(𝑡)‖2 . Thus, clearly, we see that 𝑊is positive definite and has a lower 2

bound. Next, differentiating 𝑊 gives: 𝑑 1 1 𝑊(𝑡, 𝑥𝑡 ) = ⟨𝑥 ′ (𝑡), 𝑥(𝑡)⟩ + ⟨𝑥(𝑡), 𝑥 ′ (𝑡)⟩ 𝑑𝑡 2 2 ∞

+ 𝜎 ∫ ‖𝐷(𝑢, 𝑡)‖ ‖𝑓(𝑥(𝑡))‖2𝑑𝑢 𝑡

𝑡

−𝜎 ∫ ‖𝐷(𝑡, 𝑠)‖ ‖𝑓(𝑥(𝑠))‖2 𝑑𝑠 1

0

1

= − 2 ⟨𝐹(𝑡, 𝑥(𝑡))𝑥(𝑡), 𝑥(𝑡)⟩ − 2 ⟨𝑥(𝑡), 𝐹(𝑡, 𝑥(𝑡))𝑥(𝑡)⟩ + 1

𝑡 1 𝑡 ⟨𝑥(𝑡), ∫0 𝐾(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠⟩ + 2 ⟨∫0 𝐾(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠, 𝑥(𝑡)⟩ 2 ∞ 𝑡 +𝜎 ∫𝑡 ‖𝐷(𝑢, 𝑡)‖ ‖𝑓(𝑥(𝑡))‖2 𝑑𝑢 − 𝜎 ∫0 ‖𝐷(𝑡, 𝑠)‖ ‖𝑓(𝑥(𝑠))‖2 𝑑𝑠.

Since 𝑓1𝑖 ≥ 𝜆𝑖 (𝐹(𝑡, 𝑥(𝑡))) ≥ 𝑓0𝑖 , 𝑓0𝑖 > 0, we can assume that 𝑓0 = 𝑚𝑖𝑛{ 𝑓01 , 𝑓02 , . . . , 𝑓0𝑛 ). Then, by assumptions (𝐶1), (𝐶2) and an elementary inequality, we have 𝑑 𝑡 𝑊(𝑡, 𝑥𝑡 ) ≤ −𝑓0 ‖𝑥(𝑡)‖2 + ∫0 ‖𝐾(𝑡, 𝑠, 𝑥(𝑠))‖𝑑𝑠 ‖𝑥(𝑡)‖ 𝑑𝑡 ∞ 𝑡 +𝜎 ∫𝑡 ‖𝑓(𝑥(𝑡))‖2‖𝐷(𝑢, 𝑡)‖ 𝑑𝑢 − 𝜎 ∫0 ‖𝑓(𝑥(𝑠))‖2‖𝐷(𝑡, 𝑠)‖ 𝑡 ≤ −𝑓0 ‖𝑥(𝑡)‖2 + ∫ ‖𝑓(𝑥(𝑠))‖‖𝐷(𝑡, 𝑠)‖ ‖𝑥(𝑡)‖𝑑𝑠 0

𝑑𝑠

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𝑡

+𝜎 ∫𝑡 ‖𝐷(𝑢, 𝑡)‖ ‖𝑓(𝑥(𝑡))‖2 𝑑𝑢 − 𝜎 ∫0 ‖𝐷(𝑡, 𝑠)‖ ‖𝑓(𝑥(𝑠))‖2 𝑑𝑠 1 𝑡 ≤ −𝑓0 ‖𝑥(𝑡)‖2 + ∫ ‖𝐷(𝑡, 𝑠)‖ [‖𝑓(𝑥(𝑠))‖2 + ‖𝑥(𝑡)‖2 ]𝑑𝑠 2 0 ∞

𝑡

+𝜎 ∫ ‖𝑓(𝑥(𝑡))‖2 ‖𝐷(𝑢, 𝑡)‖𝑑𝑢 − 𝜎 ∫ ‖𝑓(𝑥(𝑠))‖2‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠 𝑡

0

1 𝑡 1 𝑡 = −𝑓0 ‖𝑥(𝑡)‖2 + ∫ ‖𝑓(𝑥(𝑠))‖2 ‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠 + ∫ ‖𝐷(𝑡, 𝑠)‖ ‖𝑥(𝑡)‖2 𝑑𝑠 2 0 2 0 ∞

𝑡

+𝜎 ∫ ‖𝑓(𝑥(𝑡))‖2 ‖𝐷(𝑢, 𝑡)‖𝑑𝑢 − 𝜎 ∫ ‖𝑓(𝑥(𝑠))‖2‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠 𝑡

0

1 𝑡 1 𝑡 ≤ −𝑓0 ‖𝑥(𝑡)‖2 + ∫ ‖𝑓(𝑥(𝑠))‖2 ‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠 + ∫ ‖𝐷(𝑡, 𝑠)‖ ‖𝑥(𝑡)‖2 𝑑𝑠 2 0 2 0 ∞

𝑡

+𝜎𝛽 2 ∫ ‖𝐷(𝑢, 𝑡)‖ ‖𝑥(𝑡)‖2 𝑑𝑢 − 𝜎 ∫ ‖𝐷(𝑡, 𝑠)‖ ‖𝑓(𝑥(𝑠))‖2 𝑑𝑠. 𝑡

0

1

Let 𝜎 = . Then, we rearrange this inequality as follows: 2

𝑑 1 𝑡 𝑊(𝑡, 𝑥𝑡 ) ≤ −𝑓0 ‖𝑥(𝑡)‖2 + ∫ ‖𝐷(𝑡, 𝑠)‖ ‖𝑥(𝑡)‖2 𝑑𝑠 𝑑𝑡 2 0 1 2 ∞ + 𝛽 ∫ ‖𝐷(𝑢, 𝑡)‖ ‖𝑥(𝑡)‖2𝑑𝑢 2 𝑡 1 𝑡 1 2 ∞ = − [𝑓0 − ∫ ‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠 − 𝛽 ∫ ‖𝐷(𝑢, 𝑡)‖𝑑𝑢]‖𝑥(𝑡)‖2 ≤ 𝑐‖𝑥(𝑡)‖2 2 0 2 𝑡 ≤0 by (𝐶2). This result together with the discussion above imply that the trivial solution of IDE (14) is stable. As for the last step of the proof, consider the set: 𝐼 ≡ {(𝑡, 𝑥(𝑡)):

𝑑 𝑊(𝑡, 𝑥𝑡 ) = 0}. 𝑑𝑡

We now observe that if (𝑡, 𝑥(𝑡)) ∈ 𝐼, then 𝛼(𝑡) = 0 or ‖𝑥(𝑡)‖2 = 0. Since 𝛼(𝑡) ≥ 𝑐 > 0, then (𝑡, 𝑥(𝑡)) ∈ 𝐼 implies that ‖𝑥(𝑡)‖2 = 0. Hence, we have ‖𝑥(𝑡)‖2 = 0 ⇔ (𝑥1 (𝑡), 𝑥2 (𝑡), . . . , 𝑥𝑛 (𝑡)) = (0,0, . . . ,0). This equality

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and IDE (14), together, imply that 𝑥(𝑡) = 0 since 𝐾(𝑡, 𝑠, 𝑥(𝑡)) = 0 ⇔ 𝑥(𝑡) = 0. Thus, the largest invariant set contained in 𝐼 is (𝑡, 0) ∈ 𝐼. As a consequence, the trivial solution of IDE (14) is asymptotically stable (see the LaSalle’s invariance principle (Reissig et al. [75] and Sinha [81, Lemma 1]). Hence, we complete the proof of Theorem 12. Next, they discuss the uniform stability of the trivial solution of IDE (14). Before proceeding further, it is necessary to introduce additional assumptions given below. (𝐶3) Let 𝛽 > 0, 𝛥 > 0 and 𝛾 > 0 with 𝛽, 𝛥, 𝛾 ∈ 𝑅 such that ‖𝐾(𝑡, 𝑠, 𝑥(𝑠))‖ ≤ ‖𝐷(𝑡, 𝑠)‖ ‖𝑓(𝑥(𝑠))‖, ‖𝑓(𝑥(𝑠))‖ ≤ 𝛽‖𝑥(𝑠)‖, 𝑡 ∫0 ‖𝐷(𝑡, 𝑠)‖𝑑𝑠 ≤ 𝛼1 (𝑡), ∀𝑡 ∈ 𝑅, 𝑡0

∞

∫ ∫ ‖𝐷(𝑢, 𝑠)‖ 𝑑𝑢𝑑𝑠 = 𝛥 < ∞, 0

𝑡0

where 𝛼1 ∈ 𝐶(𝑅 + , 𝑅 + ), 𝛼1 is also bounded for ∀𝑡 ∈ 𝑅, and 1

𝑡

1

∞

𝛼(𝑡) = 𝑓0 − 2 ∫0 ‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠 − 2 𝛽 2 ∫𝑡 ‖𝐷(𝑢, 𝑡)‖𝑑𝑢 ≥ 0, ∀𝑡 ∈ 𝑅. Theorem 13 (Nieto and Tunç [64]). The zero solution of IDE (14) is uniformly stable if assumptions (𝐶1) and (𝐶3) hold. Proof. Let 𝑥 ∈ 𝑅 𝑛 and |𝑥| be any norm, 𝐶 denote the Banach space of continuous functions 𝜙: [𝑡0 − 𝜏, 𝑡0 ] → 𝑅 𝑛 , 𝜏 > 0, with ‖𝜙‖𝑡0 = 𝑠𝑢𝑝 |𝜙(𝑠)|. 𝑡0 −𝜏≤𝑠≤𝑡0

In the proof of this theorem, the main tool is the LKF 𝑊 = 𝑊(𝑡, 𝑥𝑡 ), which is used in the proof of Theorem 12. By the time derivative of the functional 𝑊 and the conditions (𝐶1) and (𝐶3), it is easily derived that 𝑑 𝑑𝑡

𝑊(𝑡, 𝑥𝑡 ) ≤ 0. Hence, it is clear that the functional 𝑊(𝑡, 𝑥𝑡 ) is decreasing.

In view of this information, we can write that 1 ‖𝑥(𝑡)‖2 ≤ 𝑊(𝑡, 𝑥𝑡 ) ≤ 𝑊(𝑡0 , 𝜙(𝑡0 )), 𝑡 ≥ 𝑡0 . 2

Advances in the Qualitative Theory of Integro–Differential Equations

81

From this point, we have 𝑡0 ∞ 1 𝑊(𝑡0 , 𝜙(𝑡0)) = ‖𝜙(𝑡0)‖2 + 𝜎 ∫ ∫ ‖𝐷(𝑢, 𝑠)‖ ‖𝑓(𝜙(𝑠))‖2 𝑑𝑢𝑑𝑠 2 0 𝑡0 𝑡0 ∞ 1 ≤ ‖𝜙(𝑡0 )‖2 + 𝜎𝛽 2 ∫ ∫ ‖𝐷(𝑢, 𝑠)‖ ‖𝜙(𝑠)‖2 𝑑𝑢𝑑𝑠 2 0 𝑡0 𝑡0 ∞ 1 ≤ [1 + 2𝜎𝛽 2 ]‖𝜙‖2𝑡0 ∫ ∫ ‖𝐷(𝑢, 𝑠)‖ 𝑑𝑢𝑑𝑠 2 0 𝑡0 1

= 2 [1 + 2𝜎𝛽 2 𝛥]‖𝜙‖2𝑡0 . Hence, we obtain that ‖𝑥(𝑡)‖2 ≤ [1 + 2𝜎𝛽 2 𝛥]‖𝜙‖2𝑡0 . Now, by the definition of the stability, for each 𝜀 > 0, choose a constant 𝛿=(

1

)

𝜀

√1+2𝜎𝛽2 𝛥 2

such

that

‖𝜙(𝑡)‖ < 𝛿,

if

∀𝑡 ∈ [−𝜏, 𝑡0 ],

then

‖𝑥(𝑡, 𝑡0 , 𝜙)‖ ≤ (√1 + 2𝜎𝛽 2 𝛥)𝛿 < 𝜀, ∀𝑡 ≥ 𝑡0 . Then, since 𝛿does not depend on the constant 𝒕𝟎 , the solution 𝑥(𝑡) ≡ 0 of IDE (14) is uniformly stable. This competes the proof. Next, Nieto and Tunç [64] considered the following IDDE: 𝑑𝑥 𝑑𝑡

𝑡

= −𝐹(𝑡, 𝑥)𝑥 + ∫𝑡−𝜏 𝐾(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠,

(15)

with the initial condition 𝑥(𝑡) = 𝜙(𝑡), 𝑡 ∈ [−𝜏, 0], 𝜙 ∈ 𝐶([−𝜏, 0], 𝑅 𝑛 ), where 𝑡 ∈ [−𝜏, ∞), 𝜏 is a given positive constant, i.e., constant delay, 𝑥 ∈ 𝑅 𝑛 , 𝐹 is defined as in IDE (14) and 𝐾(𝑡, 𝑠, 𝑥) ∈ 𝐶(𝑅 × 𝑅 × 𝑅 𝑛 , 𝑅 𝑛 ) with −𝜏 ≤ 𝑠 ≤ 𝑡 < ∞ and 𝐾(𝑡, 𝑠, 𝑥) = 0 iff 𝑥 = 0. Finally, for the last result of Nieto and Tunç [64], we require an additional assumption as follows. (𝐶4) There exist positive constants 𝛽, 𝑓0 and 𝜎1 such that ‖𝐾(𝑡, 𝑠, 𝑥(𝑠))‖ ≤ ‖𝐷(𝑡, 𝑠)‖ ‖𝑓(𝑥(𝑠))‖, ‖𝑓(𝑥(𝑠))‖ ≤ 𝛽‖𝑥(𝑠)‖, 𝑡 𝑡 ∫𝑡−𝜏‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠 < +∞, ∫0 ‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠 < +∞, 𝑡

and 𝜗(𝑡) = 𝑓0 − 𝛽 ∫𝑡−𝜏‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠 ≥ 𝜎1 .

82

Osman Tunç, Seenith Sivasundaram and Cemil Tunç

Theorem 14 (Nieto and Tunç [64]). The trivial solution of IDDE (15) is globally asymptotically stable if assumptions (𝐶1) and (𝐶4) hold. Proof. The proof is by means of the Lyapunov- Razumikhin method (see, Hale ([37] and Zhou and Egorov [109]). For the first step, we choose a Lyapunov function 𝑊1 (𝑥(𝑡)) = 𝑊1 (𝑡, 𝑥(𝑡)) as follows: 1

1

𝑊1 (𝑡, 𝑥(𝑡)) = 2 ⟨𝑥(𝑡), 𝑥(𝑡)⟩ = 2 ‖𝑥(𝑡)‖2. Next, from the Lyapunov function 𝑊1 (𝑡, 𝑥(𝑡)), it follows that 𝑊1 (𝑡, 0) = 0. Let 𝑘1 ∈ (0,1) and

k2  1, 𝑘1, 𝑘2 ∈ 𝑅 . Then, it is clear that

the Lyapunov function 𝑊1 satisfies the inequality: 𝑘1 ‖𝑥(𝑡)‖2 ≤ 𝑊1 (𝑡, 𝑥) ≤ 𝑘2 ‖𝑥(𝑡)‖2. We now consider an arbitrary initial data (𝑡0 , 𝜙) ∈ 𝑅 + × 𝐶([−𝜏, 0], 𝑅) and a point 𝑡 > 𝑡0 such that the Razumikhin condition 𝑊1 (𝑡 + 𝜃, 𝑥(𝑡 + 1 1 𝑠)) < 𝑊1 (𝑡, 𝑥(𝑡)), 𝑠 ∈ [−𝜏, 0], holds, i.e., ‖𝑥(𝑡 + 𝑠)‖2 < ‖𝑥(𝑡)‖2holds 2

2

for 𝑠 ∈ [−𝜏, 0]. Let 𝑥(𝑡) = 𝑥(𝑡, 𝑡0 , 𝜙) denote the solution of IDDE (15) such that 𝑥(𝑡0+ + 𝜃) = 𝜙(𝜃) for 𝜃 ∈ [−𝜏, 0]. Differentiating 𝑊1 (𝑡, 𝑥(𝑡)), we find: 𝑑 1 1 𝑊1 (𝑡, 𝑥(𝑡)) = − ⟨𝐹(𝑡, 𝑥(𝑡))𝑥(𝑡), 𝑥(𝑡)⟩ − ⟨𝑥(𝑡), 𝐹(𝑡, 𝑥(𝑡))𝑥(𝑡)⟩ 𝑑𝑡 2 2 𝑡 𝑡 1 1 + ⟨𝑥(𝑡), ∫ 𝐾(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠⟩ + ⟨𝑥(𝑡), ∫ 𝐾(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠⟩. 2 2 𝑡−𝜏 𝑡−𝜏 Next, assumptions (𝐶1), (𝐶4) and an elementary inequality give that 𝑑

𝑡

𝑑𝑡

𝑊1 (𝑡, 𝑥(𝑡)) ≤ −𝑓0 ‖𝑥(𝑡)‖2 + ‖𝑥(𝑡)‖ ∫𝑡−𝜏‖𝐾(𝑡, 𝑠, 𝑥(𝑠))‖𝑑𝑠 𝑡

≤ −𝑓0 ‖𝑥(𝑡)‖2 + ‖𝑥(𝑡)‖ ∫ ‖𝑓(𝑥(𝑠))‖ ‖𝐷(𝑡, 𝑠)‖𝑑𝑠 𝑡−𝜏

𝑡

≤ −𝑓0 ‖𝑥(𝑡)‖2 + 𝛽‖𝑥(𝑡)‖ ∫ ‖𝑥(𝑠)‖ ‖𝐷(𝑡, 𝑠)‖𝑑𝑠 𝑡−𝜏

≤ −𝑓0

‖𝑥(𝑡)‖2

𝛽 𝑡 + ∫ ‖𝐷(𝑡, 𝑠)‖ [‖𝑥(𝑠)‖2 + ‖𝑥(𝑡)‖2 ]𝑑𝑠 2 𝑡−𝜏

Advances in the Qualitative Theory of Integro–Differential Equations

= −𝑓0 ‖𝑥(𝑡)‖2 +

83

𝛽 𝑡 ∫ ‖𝑥(𝑠)‖2 ‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠 2 𝑡−𝜏

𝛽 𝑡 + ∫ ‖𝑥(𝑡)‖2 ‖𝐷(𝑡, 𝑠)‖𝑑𝑠 2 𝑡−𝜏 𝛽 𝑡 ≤ −𝑓0 ‖𝑥(𝑡)‖2 + ∫ ‖𝑥(𝑠)‖2 ‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠 2 𝑡−𝜏 𝛽

𝑡

+ ‖𝑥(𝑡)‖2 ∫𝑡−𝜏‖𝐷(𝑡, 𝑠)‖𝑑𝑠. 2 We note that the term:

𝛽

𝑡 ∫ ‖𝑥(𝑠)‖2 ‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠, 2 𝑡−𝜏

(16) is included in (16).

We now apply the transformation 𝑠 − 𝑡 = 𝜉. Then, it follows that 𝑑𝑠 = 𝑑𝜉. Hence, if 𝑠 = 𝑡 − 𝜏, then 𝜉 = −𝜏. Similarly, if 𝑠 = 𝑡, then 𝜉 = 0. From this point, we have 𝛽 𝑡 𝛽 0 ∫ ‖𝑥(𝑠)‖2 ‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠 = ∫ ‖𝑥(𝑡 + 𝜉)‖2 ‖𝐷(𝑡, 𝑡 + 𝜉)‖ 𝑑𝜉 2 𝑡−𝜏 2 −𝜏 0 𝛽 ≤ ∫ ‖𝑥(𝑡)‖2 ‖𝐷(𝑡, 𝑡 + 𝜉)‖ 𝑑𝜉 2 −𝜏 0 𝛽 = ‖𝑥(𝑡)‖2 ∫ ‖𝐷(𝑡, 𝑡 + 𝜉)‖ 𝑑𝜉 2 −𝜏 𝛽

𝑡

= ‖𝑥(𝑡)‖2 ∫𝑡−𝜏‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠. 2

(17)

Substituting the inequality (17) into the inequality (16) and using the given condition, we get 𝑡 𝑑 𝛽 𝑊1 (𝑡, 𝑥(𝑡)) ≤ −𝑓0 ‖𝑥(𝑡)‖2 + ‖𝑥(𝑡)‖2 ∫ ‖𝐷(𝑡, 𝑠)‖𝑑𝑠 𝑑𝑡 2 𝑡−𝜏 𝑡 𝛽 + ‖𝑥(𝑡)‖2 ∫ ‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠. 2 𝑡−𝜏 𝑡

= − [𝑓0 − 𝛽 ∫𝑡−𝜏‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠]‖𝑥(𝑡)‖2 ≤ −𝜎1 ‖𝑥(𝑡)‖2 ≤ 0. Thus, as consequence, the zero solution of IDDE (15) is globally asymptotically stable. This inequality yields 𝑑 𝑊 (𝑡, 𝑥(𝑡)) ≤ 0. 𝑑𝑡 1

84

Osman Tunç, Seenith Sivasundaram and Cemil Tunç

From this point of view, since 𝑊1 (𝑡, 𝑥(𝑡)) is decreasing and positive definite, it is clear that 𝑊1 (𝑡, 𝑥(𝑡)) ≤ 𝑊1 (0, 𝑥(0)) = 𝐿0 , 𝐿0 > 0. Next, by this discussion, we arrive at 𝑘1 ‖𝑥(𝑡)‖2 ≤ 𝑊1 (𝑡, 𝑥(𝑡)) ≤ 𝑊1 (0, 𝑥(0)) = 𝐿0 , and ‖𝑥(𝑡)‖ ≤ √𝑘1−1 𝐿0 , ∀𝑡 ≥ 𝑡0 . Finally, by this inequality, it is clear that the solutions of IDDE (15) are bounded as 𝑡 → ∞. This result completes the proof. Example 5 (Nieto and Tunç [64]). We now have the IDE: 𝑥′ ( 1′ ) 𝑥2 9+ =−

1 1 + 𝑒𝑥𝑝( 𝑡) + 𝑥12 + 𝑥22 1

( 𝑡

𝑒𝑥𝑝( − 2𝑡 + 𝑠)

1

𝑥 ( 1) 1 𝑥2 9+ 1 + 𝑒𝑥𝑝( 𝑡) + 𝑥12 + 𝑥22 )

√𝑥12 (𝑠)+𝑥22 (𝑠)

+ ∫0

2 𝑒𝑥𝑝( − 2𝑡 + 𝑠) (

1+𝑥12 (𝑠)

𝑑𝑠,

(18)

𝑠𝑖𝑛 √𝑥12 (𝑠)+𝑥22 (𝑠) 1+𝑥22 (𝑠)

)

𝑥 (𝑡) 𝑥 where 𝑡 ≥ 0 and ( 1 ) = ( 1 ) = 𝑥(𝑡) = 𝑥 ∈ ℝ2 . Compare IDE (18) with 𝑥2 𝑥2 (𝑡) IDE (14). Then, we have: 9+ 𝐹(𝑡, 𝑥1 , 𝑥2 )=

1 1 + 𝑒𝑥𝑝( 𝑡) + 𝑥12 + 𝑥22 1

( and

1 9+

1 1 + 𝑒𝑥𝑝( 𝑡) + 𝑥12 + 𝑥22 )

Advances in the Qualitative Theory of Integro–Differential Equations

𝑒𝑥𝑝( − 2𝑡 + 𝑠) 𝐾(𝑡, 𝑠, 𝑥(𝑠)) =

√𝑥12 (𝑠) + 𝑥22 (𝑠) 1 + 𝑥12 (𝑠)

. 𝑠𝑖𝑛 √𝑥12 (𝑠) + 𝑥22 (𝑠) 2 𝑒𝑥𝑝( − 2𝑡 + 𝑠) 1 + 𝑥22 (𝑠) ( )

Now, some simple calculations give the following relations: ‖𝐹(𝑡, 𝑥1 , 𝑥2 )‖= 9+

‖ ‖

1 1 + 𝑒𝑥𝑝( 𝑡) + 𝑥12 + 𝑥22 1

( 𝑠𝑢𝑝

(𝑡,𝑥)∈ℝ+ ×ℝ2

1

‖, ‖ 1 9+ 2 2 1 + 𝑒𝑥𝑝( 𝑡) + 𝑥1 + 𝑥2 )

‖𝐹(𝑡, 𝑥1 , 𝑥2 )‖ ≤ 22 < ∞,

1 , 1 + 𝑒𝑥𝑝( 𝑡) + 𝑥12 + 𝑥22 1 𝜆2 (𝐹(𝑡, 𝑥1 , 𝑥2 )) = 10 + , 1 + 𝑒𝑥𝑝( 𝑡) + 𝑥12 + 𝑥22 1 𝑓0𝑖 = 8 ≤ 8 + ≤ 𝜆𝑖 (𝐹(𝑡, 𝑥1 , 𝑥2 )), 1 + 𝑒𝑥𝑝( 𝑡) + 𝑥12 + 𝑥22 1 𝜆𝑖 (𝐹(𝑡, 𝑥1 , 𝑥2 )) = 10 + ≤ 11 = 𝑓1𝑖 , 1 + 𝑒𝑥𝑝( 𝑡) + 𝑥12 + 𝑥22 𝑓0 = 8, 8 ≤ 𝑓0𝑖 ≤ 𝜆𝑖 (𝐹(𝑡, 𝑥)) ≤ 𝑓1𝑖 ≤ 11, (𝑖 = 1,2). 𝜆1 (𝐹(𝑡, 𝑥1 , 𝑥2 )) = 8 +

‖𝐾(𝑡, 𝑠, 𝑥(𝑠))‖ =

‖ ‖

𝑒𝑥𝑝( − 2𝑡 + 𝑠) 2 𝑒𝑥𝑝( − 2𝑡 + (

√𝑥12 (𝑠) + 𝑥22 (𝑠) 1 + 𝑥12 (𝑠)

‖

𝑠𝑖𝑛 √𝑥12 (𝑠) + 𝑥22 (𝑠) ‖ 𝑠) 1 + 𝑥22 (𝑠) )

≤ 𝑒𝑥𝑝( − 2𝑡 + 𝑠)√𝑥12 (𝑠) + 𝑥22 (𝑠) + 2 𝑒𝑥𝑝( − 2𝑡 + 𝑠) |𝑠𝑖𝑛 √𝑥12 (𝑠) + 𝑥22 (𝑠)| ≤ 𝑒𝑥𝑝( − 2𝑡 + 𝑠)√𝑥12 (𝑠) + 𝑥22 (𝑠) + 2 𝑒𝑥𝑝( − 2𝑡 + 𝑠)√𝑥12 (𝑠) + 𝑥22 (𝑠) ≤ 3 𝑒𝑥𝑝( − 2𝑡 + 𝑠)√𝑥12 (𝑠) + 𝑥22 (𝑠) = ‖𝐷(𝑡, 𝑠)‖ ‖𝑓(𝑥(𝑠))‖,

85

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Osman Tunç, Seenith Sivasundaram and Cemil Tunç

where ‖𝐷(𝑡, 𝑠)‖ = 3 𝑒𝑥𝑝( − 2𝑡 + 𝑠), 0 ≤ 𝑠 ≤ 𝑡, ‖𝐷(𝑢, 𝑡)‖ = 3 𝑒𝑥𝑝( − 2𝑢 + 𝑡), 0 ≤ 𝑡 ≤ 𝑢, ‖𝑓(𝑥1 (𝑠), 𝑥2 (𝑠))‖ = √𝑥12 (𝑠) + 𝑥22 (𝑠) = ‖𝑥(𝑠)‖, 𝛽 = 1. Then, 𝑡

𝑡

∫ ‖𝐷(𝑡, 𝑠)‖𝑑𝑠 = 3 ∫ 𝑒𝑥𝑝( − 2𝑡 + 𝑠)𝑑𝑠 = 3 [𝑒𝑥𝑝( − 𝑡) − 𝑒𝑥𝑝( − 2𝑡)] 0

0

= 𝛼1 (𝑡), ∞ ∞ 3 ∫ ‖𝐷(𝑢, 𝑡)‖𝑑𝑢 = 3 ∫ 𝑒𝑥𝑝( − 2𝑢 + 𝑡)𝑑𝑢 = 𝑒𝑥𝑝( − 𝑡) = 𝛼2 (𝑡). 2 𝑡 𝑡

After these estimates, ∞ 1 𝑡 1 𝛼(𝑡) = 𝑓0 − ∫ ‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠 − 𝛽 2 ∫ ‖𝐷(𝑢, 𝑡)‖𝑑𝑢 2 0 2 𝑡 3 3 3 = 8 − 𝑒𝑥𝑝( − 𝑡) + 𝑒𝑥𝑝( − 2𝑡) − 𝑒𝑥𝑝( − 𝑡) 2 2 2 3 3 ≥ 8 − − = 5 = 𝜌 > 0. 2 2

Thus, assumptions (𝐶1) and (𝐶2) hold, which leads that the trivial solution of IDE (18) is asymptotic stable. For the next step, we derive the following results: ‖𝐷(𝑢, 𝑠)‖ = 3 𝑒𝑥𝑝( − 2𝑢 + 𝑠), 0 ≤ 𝑠 ≤ 𝑢, t0 

 0 t0

t 

0 3 D(u, s) duds = 3  exp(−2u + s)duds = [exp(3t0 ) − exp(2t0 )] =   , 2 0 t0

𝑡0 ≥ 0, 𝑡

𝜔(𝑡) = 𝑓0 − 𝛽 2 ∫ ‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠 0

= 8 − 3 𝑒𝑥𝑝( − 𝑡) + 3 𝑒𝑥𝑝( − 2𝑡) ≥ 8 − 3 𝑒𝑥𝑝( − 𝑡) ≥ 5 = 𝛾.

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Thus, assumption (𝐶3) of Theorem 13 is fulfilled. As a consequence of this fact, the trivial solution of IDE (18) is uniformly stable. Example 6. We now consider the IDDE: 𝑥′ ( 1′ ) 𝑥2 9+ =−

1 1 + 𝑒𝑥𝑝( 𝑡) + 𝑥12 + 𝑥22 1

( 𝑡

𝑒𝑥𝑝( − 2𝑡 + 𝑠)

1

𝑥 ( 1) 1 𝑥2 9+ 1 + 𝑒𝑥𝑝( 𝑡) + 𝑥12 + 𝑥22 )

√𝑥12 (𝑠)+𝑥22 (𝑠)

+ ∫𝑡−1 2 𝑒𝑥𝑝( − 2𝑡 + 𝑠) (

1+𝑥12 (𝑠)

𝑑𝑠,

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𝑠𝑖𝑛 √𝑥12 (𝑠)+𝑥22 (𝑠) 1+𝑥22 (𝑠)

)

𝑥 where 𝑡 − 1 ≥ 0, 𝜏 = 1 is fixed constant delay and ( 1 ) = 𝑥(𝑡) = 𝑥 ∈ ℝ2 . 𝑥2 Compare IDDE (19) and IDDE (15): 𝐹(𝑡, 𝑥) is the same as in Example 5 and satisfies assumption (𝐶1) of Theorem 14. Next, we have

t



t −

𝑡 K (t , s, x(s))ds = ∫𝑡−1

𝑒𝑥𝑝( − 2𝑡 + 𝑠)

√𝑥12 (𝑠)+𝑥22 (𝑠)

2 𝑒𝑥𝑝( − 2𝑡 + 𝑠) (

1+𝑥12 (𝑠)

𝑑𝑠.

𝑠𝑖𝑛 √𝑥12 (𝑠)+𝑥22 (𝑠) 1+𝑥22 (𝑠)

)

Then 𝑒𝑥𝑝( − 2𝑡 + 𝑠) 𝐾(𝑡, 𝑠, 𝑥(𝑠)) =

√𝑥12 (𝑠) + 𝑥22 (𝑠) 1 + 𝑥12 (𝑠)

. 𝑠𝑖𝑛 √𝑥12 (𝑠) + 𝑥22 (𝑠) 2 𝑒𝑥𝑝( − 2𝑡 + 𝑠) 1 + 𝑥22 (𝑠) ( )

Similarly, as in Example 4, we obtain the following: ‖𝐷(𝑡, 𝑠)‖ = 3 𝑒𝑥𝑝( − 2𝑡 + 𝑠), 0 ≤ 𝑠 ≤ 𝑡,

88

Osman Tunç, Seenith Sivasundaram and Cemil Tunç 𝑡

𝑡

∫ ‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠 = 3 ∫ 𝑒𝑥𝑝( − 2𝑡 + 𝑠) 𝑑𝑠 𝑡−𝜏

𝑡−1

= 3[𝑒𝑥𝑝( − 𝑡) − 𝑒𝑥𝑝( − 𝑡 + 1)] < +∞, ‖𝑓(𝑥1(𝑠), 𝑥2 (𝑠))‖ = ‖𝑥(𝑠)‖, 𝛽 = 1. 𝑡 1 𝜗(𝑡) = 𝑓0 − (𝛽 2 + 1) ∫ ‖𝐷(𝑡, 𝑠)‖ 𝑑𝑠 2 𝑡−1 = 8 − 3[𝑒𝑥𝑝( − 𝑡) − 𝑒𝑥𝑝( − 𝑡 + 1)] ≥ 5 = 𝜎. Subject to the previous discussion, condition (𝐶4) of Theorem 14 is fulfilled. Then, we conclude that the trivial solution of IDDE (19) is globally uniformly asymptotic stable. Furthermore, the solutions of IDDE (19) are square integrable and bounded as 𝑡 → ∞. Example 7 (Bohner et al. [15]). We now consider the standard electric RLC circuit governed by the scalar IDDE with constant delay: 𝑑

1

𝑡

𝐿 𝑑𝑡 𝐼(𝑡) + 𝑅𝐼(𝑡) + 𝐶 ∫𝑡−1 𝐼(𝑠)𝑑𝑠 = 𝐸(𝑡),

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where 𝐼, 𝑅, 𝐿, 𝐶 and 𝐸are defined just as in the introduction. Equation (20) can be written as 𝑑𝐼 𝑑𝑡

𝑅

1

𝑡

1

= − 𝐿 𝐼(𝑡) − 𝐶𝐿 ∫𝑡−1 𝐼(𝑠)𝑑𝑠 + 𝐿 𝐸(𝑡). 1

Define a Lyapunov function by 𝜈(𝑡) = 𝐼 2 (𝑡). Differentiating the 2

Lyapunov function 𝝂(𝒕) and considering IDDE (20), we have 𝑡 𝑅 1 𝐼(𝑡) 𝜈 ′ = − 𝐼 2 (𝑡) − 𝐼(𝑡) ∫ 𝐼(𝑠)𝑑𝑠 + 𝐸(𝑡) 𝐿 𝐶𝐿 𝐿 𝑡−1 𝑡 𝑅 1 𝐼(𝑡) ≤ − 𝐼 2 (𝑡) + 𝐸(𝑡) ∫ [𝐼 2 (𝑡) + 𝐼 2 (𝑠)]𝑑𝑠 + 𝐿 2𝐶𝐿 𝑡−1 𝐿 𝑅

1

1

𝑡

≤ − 𝐿 𝐼 2 (𝑡) + 2𝐶𝐿 𝐼 2 (𝑡) + 2𝐶𝐿 ∫𝑡−1 𝐼 2 (𝑠)𝑑𝑠 + 𝑡

𝐼(𝑡) 𝐿

𝐸(𝑡).

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Note that the term: ∫𝑡−1 𝐼 2 (𝑠) 𝑑𝑠, is included in (21). We now consider an arbitrary initial data (𝑡0 , 𝜙) ∈ ℝ+ × 𝐶([−1,0], 𝑅) and a point 𝑡 > 𝑡0 such that 𝐼 2 (𝑡 + 𝑠) < 𝐼 2 (𝑡), 𝑠 ∈ [−1,0]. Then, we apply

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the transformation 𝑠 − 𝑡 = 𝜉. In this case, it follows that 𝑑𝑠 = 𝑑𝜉. Hence, if 𝑠 = 𝑡 − 1, then 𝜉 = −1. As before, if 𝑠 = 𝑡, then 𝜉 = 0., Thus, 𝑡

0

0

∫ 𝐼 2 (𝑠) 𝑑𝑠 = ∫ 𝐼 2 (𝑡 + 𝜉) 𝑑𝜉 ≤ ∫ 𝐼 2 (𝑡) 𝑑𝜉 = 𝐼 2 (𝑡). 𝑡−1

−1

−1

Substitution of this inequality into (21) implies 𝑅 1 2 𝐼(𝑡) 𝑣 ′ ≤ − 𝐼 2 (𝑡) + 𝐼 (𝑡) + 𝐸(𝑡). 𝐿 𝐶𝐿 𝐿 1 Let 𝐸(𝑡) ≤ 𝐼(𝑡). Then, 2𝐶 𝑅 1 2 1 2 𝑅 3 2 𝑣 ′ ≤ − 𝐼 2 (𝑡) + 𝐼 (𝑡) + 𝐼 (𝑡) = − [ − ]𝐼 (𝑡). 𝐿 𝐶𝐿 2𝐿𝐶 𝐿 2𝐶𝐿 3

We now have for a positive constant 𝛽 that 𝑣 ′ ≤ −𝛽𝐼 2 (𝑡) if 𝑅𝐶 > . 2

Thus, we proved that that the solutions of IDDE (20) are globally uniformly asymptotically stable. Tunç [93] discussed the system of IIDEs of the form: 𝑑𝑥 𝑑𝑡

𝑡

= −𝑓(𝑡, 𝑥(𝑡)) + 𝑔(𝑡, 𝑥(𝑡), 𝑥(𝑡 − 𝜏)) + ∫𝑡−𝜏 ℎ(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠 + 𝑞(𝑡, 𝑥(𝑡), 𝑥(𝑡 − 𝑟(𝑡)))

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with 𝑥(𝑡) = 𝜙(𝑡), 𝑡 ∈ [𝑡0 − 𝜏, 𝑡0 ], where 𝜙 is a continuous initial function, 𝑥 = (𝑥1 , . . . , 𝑥𝑛 )𝑇 ∈ 𝑅 𝑛 , 𝑡 ∈ 𝑅, 𝑠 ∈ [−𝜏, ∞), 𝜏 is a positive constant, 𝑟(𝑡), 𝑟 ∈ 𝐶([0, ∞), [0, 𝜏]), is the variable delay, 𝑓 ∈ 𝐶(ℝ × ℝ𝑛 , ℝ𝑛 ), 𝑔 ∈ 𝐶(𝑅 × 𝑅 𝑛 × 𝐶𝐻 , 𝑅 𝑛 ), ℎ ∈ 𝐶(𝑅 × [−𝜏, ∞) × 𝐶𝐻 , 𝑅 𝑛 ) and 𝑞 ∈ 𝐶(𝑅 × 𝑅 𝑛 × 𝐶𝐻 , 𝑅 𝑛 ). we assume that 𝑓(𝑡, 0) = 0, 𝑔(𝑡, 0,0) = 0 and ℎ(𝑡, 𝑠, 0) = 0. Then, the system of IDDEs (22) includes the zero solution when 𝑞(𝑡, 𝑥(𝑡), 𝑥(𝑡 − 𝑟(𝑡))) = 0. Now, consider an unperturbed system of IDDEs, which is obtained from the system of IDDEs (22), when 𝑞(. ) ≡ 0: 𝑑𝑥 𝑑𝑡

𝑡

= −𝑓(𝑡, 𝑥(𝑡)) + 𝑔(𝑡, 𝑥(𝑡), 𝑥(𝑡 − 𝜏)) + ∫𝑡−𝜏 ℎ(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠.

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Next, new sufficient conditions are given, which are needed in the proofs of Theorems 15 and 16.

90

Osman Tunç, Seenith Sivasundaram and Cemil Tunç

We suppose that the following conditions hold: (𝐻1)𝑓(𝑡, 0) = 0, 𝑔(𝑡, 0,0) = 0, ℎ(𝑡, 𝑠, 0) = 0, 𝑥𝑖 (𝑡)𝑓𝑖 (𝑡, 𝑥(𝑡)) > 0 for all

xi (t )  0, 𝑥𝑖 (𝑡) ∈ 𝑅; ∞

(𝐻2)𝐻(𝑡, 𝑠, 0) = 0, 𝐻(𝑡, 𝑠, 𝑥) ≡ ∫𝑡 ‖ℎ(𝑢, 𝑠, 𝑥(𝑠))‖𝑑𝑢 ≤ 𝐿̃ < ∞ for 𝑡0 ≤ 𝑠 ≤ 𝑡, 𝑥 ∈ 𝑅 𝑛 , where L  0, 𝐿̃ ∈ 𝑅; (𝐻3) the function 𝐻 satisfies the locally Lipschitz condition in 𝑥 and ∞

‖𝑓(𝑡, 𝑥(𝑡))‖ − ‖𝑔(𝑡, 𝑥(𝑡), 𝑥(𝑡 − 𝜏))‖ − ∫𝑡 ‖ℎ(𝑢, 𝑡, 𝑥(𝑡))‖𝑑𝑢 ≥ 𝐾‖𝑥(𝑡)‖ for 𝑡0 ≤ 𝑠 ≤ 𝑡, 𝑥 ∈ 𝑅 𝑛 , where 𝐾 > 0, 𝐾 ∈ 𝑅 . Theorem 15 (Tunç [93]). The solution 𝑥(𝑡) = 0 of IDDEs (23) is uniformly asymptotically stable if conditions(𝐻1), (𝐻2) and (𝐻3) hold. Proof. We introduce an LKF 𝛤: = 𝛤(𝑡, 𝑥) defined by 𝑡

∞

𝛤(𝑡, 𝑥): = ‖𝑥‖ + 𝜆 ∫𝑡−𝜏 ∫𝑡 ‖ℎ(𝑢, 𝑠, 𝑥(𝑠))‖ 𝑑𝑢𝑑𝑠 𝑡

∞

= |𝑥1 |+. . . +|𝑥𝑛 | + 𝜆 ∫𝑡−𝜏 ∫𝑡 ‖ℎ(𝑢, 𝑠, 𝑥(𝑠))‖ 𝑑𝑢𝑑𝑠,

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in which 𝜆 > 0, 𝜆 ∈ 𝑅, 𝜆is an arbitrary constant. From this point of view, the LKF 𝛤 satisfies that 𝛤(𝑡, 0) = 0, 𝛤(𝑡, 𝑥) ≥ ‖𝑥‖. Next, by some simple calculations, we obtain that |𝛤(𝑡, 𝑥) − 𝛤(𝑡, 𝑦)| ≤ | ‖𝑥‖ − ‖𝑦‖ | 𝑡

∞

𝑡

∞

+𝜆 |∫ ∫ ‖ℎ(𝑢, 𝑠, 𝑥(𝑠))‖ 𝑑𝑢𝑑𝑠 − ∫ ∫ ‖ℎ(𝑢, 𝑠, 𝑦(𝑠))‖ 𝑑𝑢𝑑𝑠| 𝑡−𝜏 𝑡

𝑡−𝜏 𝑡 𝑡

𝑡

= | ‖𝑥‖ − ‖𝑦‖ | + 𝜆 |∫ 𝐻(𝑡, 𝑠, 𝑥(𝑠)) 𝑑𝑠 − ∫ 𝐻(𝑡, 𝑠, 𝑦(𝑠)) 𝑑𝑠| 𝑡−𝜏 𝑡

𝑡−𝜏

≤ ‖𝑥 − 𝑦‖ + 𝜆 ∫ |𝐻(𝑡, 𝑠, 𝑥(𝑠)) − 𝐻(𝑡, 𝑠, 𝑦(𝑠))| 𝑑𝑠 𝑡−𝜏 𝑡

≤ ‖𝑥 − 𝑦‖ + 𝜆𝐿𝐻 ∫ ‖𝑥(𝑠) − 𝑦(𝑠)‖ 𝑑𝑠 𝑡−𝜏

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91

≤ ‖𝑥(𝑡) − 𝑦(𝑡)‖ + 𝜆𝜏𝐿𝐻 𝑠𝑢𝑝 ‖𝑥(𝑠) − 𝑦(𝑠)‖ 𝑡−𝜏≤𝑠≤𝑡

= 𝐾0 𝑠𝑢𝑝 ‖𝑥(𝑠)) − 𝑦(𝑠)‖, 𝑡−𝜏≤𝑠≤𝑡

where 𝐿𝐻 > 0is the Lipschitz constant and 𝐾0 : = 1 + 𝜆𝜏𝐿𝐻 . Hence, it is seen that the LKF 𝛤 satisfies the locally Lipschitz condition in 𝑥. For the following step, let ∞

𝑡

∫ ‖ℎ(𝑢, 𝑠, 𝑥(𝑠))‖ 𝑑𝑢𝑑𝑠.

𝑍(𝑡, 𝑥): = 𝜆 ∫

𝑡−𝜏 𝑡

Then, using condition (𝐻2), it follows that 𝑡

∞

𝑡

𝑍(𝑡, 𝑥) = 𝜆 ∫ ∫ ‖ℎ(𝑢, 𝑠, 𝑥(𝑠))‖ 𝑑𝑢𝑑𝑠 = 𝜆 ∫ 𝐻(𝑡, 𝑠, 𝑥(𝑠)) 𝑑𝑠 𝑡−𝜏 𝑡

𝑡−𝜏

𝑡

= 𝜆 ∫ |𝐻(𝑡, 𝑠, 𝑥(𝑠)) − 𝐻(𝑡, 𝑠, 0)| 𝑑𝑠 𝑡−𝜏

𝑡

≤ 𝜆𝜏𝐿𝐻

≤ 𝜆𝐿𝐻 ∫𝑡−𝜏‖𝑥(𝑠) − 0‖ 𝑑𝑠 𝑠𝑢𝑝 ‖𝑥(𝑠)‖ = 𝜆𝜏𝐿𝐻 ‖𝑥(𝑠)‖[𝑡−𝜏,𝑡] ,

𝑡−𝜏≤𝑠≤𝑡

where 𝐿𝐻 > 0 is the Lipschitz constant. From this point of view, for any two

t1  0, 𝑡2 > 0

with

𝑡1 < 𝑡2 < ∞,

adding and subtracting the term

𝑡 −𝜏

𝜆 ∫𝑡 2−𝜏 𝐻(𝑡, 𝑠, 𝑥(𝑠)) 𝑑𝑠, it follows that 1

𝑡2

𝑍(𝑡2 , 𝑥) − 𝑍(𝑡1 , 𝑥) = 𝜆 ∫

𝑡1

𝐻(𝑡, 𝑠, 𝑥(𝑠)) 𝑑𝑠 − 𝜆 ∫

𝑡2 −𝜏 𝑡2

𝑡2 −𝜏

= 𝜆 ∫ 𝐻(𝑡, 𝑠, 𝑥(𝑠)) 𝑑𝑠 − 𝜆 ∫ 𝑡1 𝑡2

𝐻(𝑡, 𝑠, 𝑥(𝑠)) 𝑑𝑠

𝑡1 −𝜏

𝐻(𝑡, 𝑠, 𝑥(𝑠)) 𝑑𝑠

𝑡1 −𝜏 𝑡2

≤ 𝜆 ∫ 𝐻(𝑡, 𝑠, 𝑥(𝑠)) 𝑑𝑠 ≤ 𝜆 ∫ 𝐿̃ 𝑑𝑠 = 𝜆𝐿̃(𝑡2 − 𝑡1 ). 𝑡1

𝑡1

Differentiating the LKF 𝛤 in (24) along the solution of IDDEs (23) and using condition (𝐻1), we derive

92

Osman Tunç, Seenith Sivasundaram and Cemil Tunç 𝑛

∞ 𝑑 + 𝛤 (𝑡, 𝑥(𝑡)) = ∑ 𝑥𝑖′ (𝑡) 𝑠𝑔𝑛 𝑥𝑖 (𝑡 + 0) + 𝜆 ∫ ‖ℎ(𝑢, 𝑡, 𝑥(𝑡))‖𝑑𝑢 𝑑𝑡 𝑡 ∞

𝑛

𝑖=1

𝑡

−𝜆 ∫𝑡 ‖ℎ(𝑢, 𝑡 − 𝜏, 𝑥(𝑡 − 𝜏))‖𝑑𝑢 − 𝜆 ∫𝑡−𝜏‖ℎ(𝑡, 𝑠, 𝑥(𝑠))‖ 𝑑𝑠 𝑡

≤ ∑[−|𝑓𝑖 (𝑡, 𝑥(𝑡))| + |𝑔𝑖 (𝑡, 𝑥(𝑡), 𝑥(𝑡 − 𝜏))| + ∫ |ℎ𝑖 (𝑡, 𝑠, 𝑥(𝑠))|𝑑𝑠] 𝑡−𝜏

𝑖=1

∞

∞

+𝜆 ∫ ‖ℎ(𝑢, 𝑡, 𝑥(𝑡))‖𝑑𝑢 − 𝜆 ∫ ‖ℎ(𝑢, 𝑡 − 𝜏, 𝑥(𝑡 − 𝜏))‖𝑑𝑢 𝑡

𝑡

𝑡

−𝜆 ∫𝑡−𝜏‖ℎ(𝑡, 𝑠, 𝑥(𝑠))‖ 𝑑𝑠

𝑡

≤ −‖𝑓(𝑡, 𝑥(𝑡))‖ + ‖𝑔(𝑡, 𝑥(𝑡), 𝑥(𝑡 − 𝜏))‖ + ∫ ‖ℎ(𝑡, 𝑠, 𝑥(𝑠))‖𝑑𝑠 ∞ +𝜆 ∫𝑡 ‖ℎ(𝑢, 𝑡, 𝑥(𝑡))‖𝑑𝑢

−

𝑡−𝜏 𝑡 𝜆 ∫𝑡−𝜏‖ℎ(𝑡, 𝑠, 𝑥(𝑠))‖ 𝑑𝑠.

Choose 𝜆 = 1. Then, we have 𝑑 + 𝛤 (𝑡, 𝑥(𝑡)) ≤ −‖𝑓(𝑡, 𝑥(𝑡))‖ + ‖𝑔(𝑡, 𝑥(𝑡), 𝑥(𝑡 − 𝜏))‖ 𝑑𝑡 ∞ + ∫𝑡 ‖ℎ(𝑢, 𝑡, 𝑥(𝑡))‖𝑑𝑢.

(25)

Using condition (𝐻3), from (25) we conclude that 𝑑 𝑑𝑡

𝛤 + (𝑡, 𝑥(𝑡)) ≤ −[‖𝑓(𝑡, 𝑥(𝑡))‖ − ‖𝑔(𝑡, 𝑥(𝑡), 𝑥(𝑡 − 𝜏))‖ − ∞

∫𝑡 ‖ℎ(𝑢, 𝑡, 𝑥(𝑡))‖𝑑𝑢] ≤ −𝐾‖𝑥(𝑡)‖, Thus, 𝑑 𝑑𝑡

𝛤 + (𝑡, 𝑥(𝑡)) ≤ −𝐾‖𝑥(𝑡)‖.

Let

W = W1 = x , 𝑊2 = 𝜆𝜏𝐿𝐻 ‖𝑥(𝑠)‖[𝑡−𝜏,𝑡]

(26) and 𝑊3 = 𝐾‖𝑥‖. Hence,

the proof is completed (see, Burton [16, Theorem 4. 2.9]). The following result is on the integrability of the unperturbed system of IDDEs (23).

Advances in the Qualitative Theory of Integro–Differential Equations

93

Theorem 16 (Tunç [93]). The norms of solutions of the system of IDDEs (23) are integrable in the sense of Lebesgue on 𝑅 + , 𝑅 + = [𝑡0 , ∞), if conditions (𝐻1), (𝐻2) and (𝐻3) hold. Proof. We again consider the L.K.F. 𝛤, which is defined by (24). By conditions (𝐻1) − (𝐻3), we have the inequality (26). Then, integrating the inequality (26), we obtain 𝑡

𝛤(𝑡, 𝑥) − 𝛤(𝑡0 , 𝜙(𝑡0)) ≤ −𝐾 ∫ ‖𝑥(𝑠)‖ 𝑑𝑠, 𝑡 ≥ 𝑡0 . 𝑡0

Since the LKF 𝛤 is decreasing, it is clear that. 𝑡

𝐾 ∫ ‖𝑥(𝑠)‖ 𝑑 ≤ 𝛤(𝑡0 , 𝜙(𝑡0 )) − 𝛤(𝑡, 𝑥) ≤ 𝛤(𝑡0 , 𝜙(𝑡0 )). 𝑡0

Following this inequality, it is therefore seen that 𝑡

∫ ‖𝑥(𝑠)‖ 𝑑𝑠 ≤ 𝐾 −1 [𝛤(𝑡, 𝑥(𝑡) − 𝛤(𝑡0 , 𝜙(𝑡0 ))] ≤ 𝐾 −1 𝛤(𝑡0 , 𝜙(𝑡0 )). 𝑡0

Let 𝐾 −1 𝛤(𝑡0 , 𝜙(𝑡0 )) = 𝐾0 , 𝐾0 > 0, 𝐾0 ∈ 𝑅. Let 𝑡 → +∞. Hence, we obtain that, ∞

∫ ‖𝑥(𝑠)‖ 𝑑𝑠 ≤ 𝐾0 . 𝑡0

This result shows that the solutions of Volterra IDDEs (23) are integrable on

R + , 𝑅+ = [𝑡0, ∞).

We now consider the perturbed system of IDDEs (22). We give the following additional condition to obtain the next boundedness theorem. (𝐻4) There is a continuous nonnegative function 𝑞0 (𝑡) such that ∞

‖𝑞(𝑡, 𝑥, 𝑥(𝑡 − 𝑟(𝑡)))‖ ≤ |𝑞0 (𝑡)| ‖𝑥(𝑡)‖ with∫𝑡 |𝑞0 (𝑠)|𝑑𝑠 < ∞. 0

94

Osman Tunç, Seenith Sivasundaram and Cemil Tunç

Theorem 17 (Tunç [93]). The solutions of the perturbed system IDDEs (22) are bounded as 𝑡 → +∞ if conditions (𝐻1), (𝐻2), (𝐻3) and (𝐻4) hold. Proof. Consider the LKF 𝛤, which is given by (24). Differentiating the LKF (24) and using conditions (𝐻1) − (𝐻4), we obtain that, 𝑑 + 𝛤 (𝑡, 𝑥(𝑡)) ≤ ‖𝑞(𝑡, 𝑥, 𝑥(𝑡 − 𝑟(𝑡)))‖ ≤ |𝑞0 (𝑡)|‖𝑥‖ ≤ |𝑞0 (𝑡)|𝛤(𝑡, 𝑥). 𝑑𝑡

Integration of this ∞ |𝑞 𝛤(𝑡0 , 𝜙(𝑡0 )) 𝑒𝑥𝑝( ∫𝑡 0 (𝑠)|)𝑑𝑠. 0

inequality

yields

𝛤(𝑡, 𝑥(𝑡)) ≤

∞

Hence, by the condition ∫𝑡 |𝑞0 (𝑠)|𝑑𝑠 < ∞, we can obtain for a positive 0

constant 𝐾1 that ‖𝑥(𝑡)‖ ≤ 𝐾1 . Calculating the limit of this inequality as 𝑡 → ∞, we have 𝑙𝑖𝑚‖𝑥(𝑡)‖ ≤ 𝑙𝑖𝑚𝐾1 = 𝐾1 . 𝑡→∞

𝑡→∞

Clearly, we obtain that the solution 𝑥(𝑡) is bounded at infinity. Hence, this completes the proof of Theorem 17. We now give a result on the instability of the unperturbed system of Volterra IDDEs (23). We suppose that the following conditions hold. (𝐻5) ℎ(𝑡, 𝑠, 0) = 0, 𝑔(𝑡, 0,0) = 0, 𝑓(𝑡, 0) = 0,

xi (t )  0, 𝑥𝑖 (𝑡) ∈ 𝑅;

𝑥𝑖 (𝑡)𝑓𝑖 (𝑡, 𝑥(𝑡)) < 0 for all (𝐻6)

h(u, t , x(t ))  h0 K (u, t )

x(t ) , ∫𝑡∞‖𝐾(𝑢, 𝑡)‖𝑑𝑢 ≤ 𝐾0 < ∞

for 𝑡0 ≤ 𝑢 ≤ 𝑡, 𝑥 ∈ ℝ𝑛 , where

h0 , K0  0, ℎ0, 𝐾0 ∈ 𝑅;

(𝐻7) ‖𝑓(𝑡, 𝑥(𝑡))‖ − ‖𝑔(𝑡, 𝑥(𝑡), 𝑥(𝑡 − 𝜏))‖ − ℎ0 𝐾0 ‖𝑥(𝑡)‖ ≥ 𝐾1 ‖𝑥(𝑡)‖ for 𝑡0 ≤ 𝑡, 𝑥 ∈ ℝ𝑛 , where K1  0, 𝑲𝟏 ∈ ℝ . Theorem 18 (Tunç [93]). The solution 𝑥(𝑡) = 0 of the system of Volterra IDDEs (23) is unstable if the following conditions (𝐻5) − (𝐻7) and 𝜏 < ℎ

1

0 𝐾0

hold.

Proof. Let us define a new LKF 𝛤1 : = 𝛤1 (𝑡, 𝑥) by 𝑡

∞

𝛤1 (𝑡, 𝑥): = ‖𝑥‖ − ∫𝑡−𝜏 ∫𝑡 ‖ℎ(𝑢, 𝑠, 𝑥(𝑠))‖ 𝑑𝑢𝑑𝑠. Then, using (27) and (𝐻6),

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Advances in the Qualitative Theory of Integro–Differential Equations 𝑡

𝛤1 (𝑡, 𝑥) = ‖𝑥‖ − ∫

95

∞

∫ ‖ℎ(𝑢, 𝑠, 𝑥(𝑠))‖ 𝑑𝑢𝑑𝑠

𝑡−𝜏 𝑡 ∞

𝑡

≥ ‖𝑥‖ − ℎ0 ∫

∫ ‖𝐾(𝑢, 𝑠)‖ ‖𝑥(𝑠)‖𝑑𝑢𝑑𝑠

𝑡−𝜏 𝑡

𝑡

≥ ‖𝑥‖ − ℎ0 𝐾0 ∫ ‖𝑥(𝑠)‖ 𝑑𝑠 𝑡−𝜏

 x − h0 K0 inf x(s) t −  s t

 (1 − h0 K0 )inf x(s)  0 if 𝜏
0 is delay , 𝜙 ∈ 𝐶([−ℎ, 0], 𝑅 𝑛 ) is an initial function, 𝐴(𝑡) = (𝑎𝑖𝑗 (𝑡)), 𝐴𝑑 = (𝑎𝑑 𝑖𝑗 (𝑡)), 𝐴(𝑡), 𝐴𝑑 (𝑡) ∈ 𝐶[𝑅 + , 𝑅 𝑛×𝑛 ], 1,2, . . . , 𝑛,

R + = [0, ), 𝐴𝐷 = (𝑎𝐷𝑖𝑗 (𝑠)), 𝑖, 𝑗 =

−h  s  t  , 𝐴𝐷 ∈ 𝐶([−ℎ, ∞), 𝑅𝑛×𝑛 ), 𝐹, 𝐺 ∈ 𝐶[𝑅𝑛 , 𝑅𝑛 ], 𝐹(0) =

𝐺(0) = 0 and 𝐻 ∈ 𝐶(𝑅 + × 𝑅 𝑛 × 𝑅 𝑛 , 𝑅 𝑛 ). Consider now the following perturbed system 𝑡

𝑥̇ (𝑡) = 𝐴(𝑡)𝑥(𝑡) + 𝐴𝑑 (𝑡)𝐹(𝑥(𝑡 − ℎ)) + ∫𝑡−ℎ 𝐴𝐷 (𝑠)𝐺(𝑥(𝑠))𝑑𝑠,

(29)

which is obtained from system (28). Tunç et al. [94] improved and generalized an asymptotic stability result which is available in the literature. They obtained an asymptotic stability result under less conservative conditions and gave additional new results for the system of IDDEs (29). The technique used in the proofs is based on the LKF approach. The following conditions are needed in the next sections. (𝐴1) The functions 𝐹, 𝐺 ∈ 𝐶[𝑅 𝑛 , 𝑅 𝑛 ] satisfy ‖𝐹(𝑥)‖ ≤ 𝐹0 ‖𝑥‖ for all 𝑥 ∈ 𝑅 𝑛 with 𝐹(0) = 0, ‖𝐺(𝑥)‖ ≤ 𝐺0 ‖𝑥‖ for all 𝑥 ∈ 𝑅 𝑛 with 𝐺(0) = 0 where 𝐹0 , 𝐺0 and ℎ are positive constants. (𝐴2) There are constants 𝐹0 , 𝐺0 , ℎ from (𝐴1) and 𝐴0 > 0 such that

Advances in the Qualitative Theory of Integro–Differential Equations

97

𝑎𝑖𝑖 (𝑡) + ∑𝑛𝑗=1,𝑗≠𝑖|𝑎𝑗𝑖 (𝑡)| ≤ −𝐴0 for all 𝑡 ∈ 𝑅 + and 𝐴0 − ℎ𝐺0 ‖𝐴𝐷 (𝑡)‖ − 𝐹0 ‖𝐴𝑑 (𝑡 + ℎ)‖ ≥ 0 for all 𝑡 ∈ [−ℎ, ∞), (𝐴3) There are constants 𝐴0 , 𝐺0 , 𝐹0 , ℎ > 0 from (𝐴2) and𝐾0 > 0 such that 𝑎𝑖𝑖 (𝑡) + ∑𝑛𝑗=1,𝑗≠𝑖|𝑎𝑗𝑖 (𝑡)| ≤ −𝐴0 for all 𝑡 ∈ 𝑅 + and 𝐴0 − ℎ𝐺0 ‖𝐴𝐷 (𝑡)‖ − 𝐹0 ‖𝐴𝑑 (𝑡 + ℎ)‖ ≥ 𝐾0 for all 𝑡 ∈ [−ℎ, ∞). The next theorem is the first main result in [94]. Theorem 19 (Tunç et al. [94]). We assume that conditions (𝐴1) and (𝐴2) are satisfied. Then, the solution𝑥(𝑡) ≡ 0 of system (29) is uniformly stable. Proof. Consider a new LKF 𝛱: = 𝛱(𝑡, 𝑥𝑡 ) given by 𝑡

0

𝛱(𝑡, 𝑥𝑡 ): = ‖𝑥(𝑡)‖ + ∫𝑡−ℎ‖𝐴𝑑 (𝑠 + ℎ)‖ ‖𝐹(𝑥(𝑠))‖𝑑𝑠 + 𝑡

∫−ℎ ∫𝑡+𝜂‖𝐴𝐷 (𝑠)‖ ‖𝐺(𝑥(𝑠))‖𝑑𝑠𝑑𝜂,

(30)

where −ℎ ≤ 𝜂 ≤ 0. Next, note that the KLF 𝛱 satisfies 𝛱(𝑡, 0) = 0, 𝛱(𝑡, 𝑥𝑡 ) ≥ ‖𝑥‖. Differentiating the LKF𝛱 in (30) along the solution of system (29), we find 𝑛

𝑑 𝛱(𝑡, 𝑥𝑡 ) = ∑ 𝑥𝑖′ (𝑡) 𝑠𝑔𝑛 𝑥𝑖 (𝑡 + 0) 𝑑𝑡 𝑖=1

+‖𝐴𝑑 (𝑡 + ℎ)‖ ‖𝐹(𝑥(𝑡))‖ − ‖𝐴𝑑 (𝑡)‖ ‖𝐹(𝑥(𝑡 − ℎ))‖ 0

+ ∫ [‖𝐴𝐷 (𝑡)‖ ‖𝐺(𝑥(𝑡))‖ − ‖𝐴𝐷 (𝑡 + 𝜂)‖ ‖𝐺(𝑥(𝑡 + 𝜂))‖]𝑑𝜂 −ℎ

𝑛

= ∑ 𝑥𝑖′ (𝑡) 𝑠𝑔𝑛 𝑥𝑖 (𝑡 + 0) 𝑖=1

𝑡

+ℎ‖𝐴𝐷 (𝑡)‖ ‖𝐺(𝑥(𝑡))‖ − ∫𝑡−ℎ‖𝐴𝐷 (𝑠)‖ ‖𝐺(𝑥(𝑠))‖𝑑𝑠 +‖𝐴𝑑 (𝑡 + ℎ)‖ ‖𝐹(𝑥(𝑡))‖ − ‖𝐴𝑑 (𝑡)‖ ‖𝐹(𝑥(𝑡 − ℎ))‖ 𝑛

≤ ∑ 𝑥𝑖′ (𝑡) 𝑠𝑔𝑛 𝑥𝑖 (𝑡 + 0) + ℎ𝐺0 ‖𝐴𝐷 (𝑡)‖ ‖𝑥(𝑡)‖ 𝑖=1

98

Osman Tunç, Seenith Sivasundaram and Cemil Tunç 𝑡

− ∫ ‖𝐴𝐷 (𝑠)‖ ‖𝐺(𝑥(𝑠))‖𝑑𝑠 𝑡−ℎ

+‖𝐴𝑑 (𝑡 + ℎ)‖ ‖𝐹(𝑥(𝑡))‖ − ‖𝐴𝑑 (𝑡)‖ ‖𝐹(𝑥(𝑡 − ℎ))‖.

(31)

Using conditions(𝐴1) and(𝐴2), we obtain that 𝑛

𝑛

∑ 𝑠𝑔𝑛 𝑥𝑖 (𝑡

+ 0)𝑥𝑖′ (𝑡)

𝑖=1

𝑛

≤ ∑ 𝑎𝑖𝑖 (𝑡) |𝑥𝑖 (𝑡)| + ∑ ∑ |𝑎𝑗𝑖 (𝑡)| |𝑥𝑖 (𝑡)| 𝑖=1

𝑛

𝑛

𝑖=1 𝑗=1,𝑗≠𝑖

𝑛

𝑡

+ ∑ ∑|𝑎𝑑𝑖𝑗 (𝑡)| |𝐹𝑗 (𝑥(𝑡 − ℎ))| + ∫ ‖𝐴𝐷 (𝑠)‖ ‖𝐺(𝑥(𝑠))‖𝑑𝑠 𝑡−ℎ

𝑖=1 𝑗=1 𝑛

𝑛

≤ ∑ (𝑎𝑖𝑖 (𝑡) + ∑ |𝑎𝑗𝑖 (𝑡)|) |𝑥𝑖 (𝑡)| 𝑖=1

𝑗=1,𝑗≠𝑖 𝑡

+‖𝐴𝑑 (𝑡)‖ ‖𝐹(𝑥(𝑡 − ℎ))‖ + ∫ ‖𝐴𝐷 (𝑠)‖ ‖𝐺(𝑥(𝑠))‖𝑑𝑠 𝑡−ℎ

𝑡

≤ −𝐴0‖𝑥(𝑡)‖ + ‖𝐴𝑑 (𝑡)‖ ‖𝐹(𝑥(𝑡 − ℎ))‖ + ∫𝑡−ℎ‖𝐴𝐷 (𝑠)‖ ‖𝐺(𝑥(𝑠))‖𝑑𝑠 Combining this inequality and inequality (31) yields 𝑑 𝛱(𝑡, 𝑥𝑡 ) ≤ −𝐴0 ‖𝑥(𝑡)‖ + ‖𝐴𝑑 (𝑡)‖ ‖𝐹(𝑥(𝑡 − ℎ))‖ 𝑑𝑡 𝑡

+ ∫ ‖𝐴𝐷 (𝑠)‖ ‖𝐺(𝑥(𝑠))‖𝑑𝑠 𝑡−ℎ

𝑡

+ℎ𝐺0 ‖𝐴𝐷 (𝑡)‖ ‖𝑥(𝑡)‖ − ∫ ‖𝐴𝐷 (𝑠)‖ ‖𝐺(𝑥(𝑠))‖𝑑𝑠 𝑡−ℎ

+‖𝐴𝑑 (𝑡 + ℎ)‖ ‖𝐹(𝑥(𝑡))‖ − ‖𝐴𝑑 (𝑡)‖ ‖𝐹(𝑥(𝑡 − ℎ))‖ ≤ −𝐴0‖𝑥(𝑡)‖ + ℎ𝐺0 ‖𝐴𝐷 (𝑡)‖ ‖𝑥(𝑡)‖ + 𝐹0 ‖𝐴𝑑 (𝑡 + ℎ)‖ ‖𝑥(𝑡)‖. Hence, we obtain 𝑑 𝑑𝑡

𝛱(𝑡, 𝑥𝑡 ) ≤ − [𝐴0 − ℎ𝐺0 ‖𝐴𝐷 (𝑡)‖ − 𝐹0 ‖𝐴𝑑 (𝑡 + ℎ)‖ ]‖𝑥(𝑡)‖.

Using condition (𝐴2), from (32) we obtain

𝑑 𝑑𝑡

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𝛱(𝑡, 𝑥𝑡 ) ≤ 0. Thus, it is

proved that the zero solution of system (29) is uniformly stable. The next theorem, Theorem 20, is the second main result of [94].

Advances in the Qualitative Theory of Integro–Differential Equations

99

Theorem 20 (Tunç et al. [94]). We assume that conditions (𝐴1) and (𝐴3) are satisfied. Then, the solution 𝑥(𝑡) ≡ 0 of system (29) is uniformly asymptotically stable. Proof. Using condition(𝐴1) and revising the inequality (8) according condition (𝐴3), we have, 𝑡

𝛱(𝑡, 𝑥𝑡 ): = ‖𝑥(𝑡)‖ + ∫ ‖𝐴𝑑 (𝑠 + ℎ)‖ ‖𝐹(𝑥(𝑠))‖𝑑𝑠 0

𝑡−ℎ 𝑡

+ ∫ ∫ ‖𝐴𝐷 (𝑠)‖ ‖𝐺(𝑥(𝑠))‖𝑑𝑠𝑑𝜂 −ℎ 𝑡+𝜂 𝑡

≤ ‖𝑥(𝑡)‖ + 𝐹0 ∫ ‖𝐴𝑑 (𝑠 + ℎ)‖ ‖𝑥(𝑠)‖𝑑𝑠 𝑡−ℎ

0

𝑡

+ 𝐺0 ∫ ∫ ‖𝐴𝐷 (𝑠)‖ ‖𝑥(𝑠)‖𝑑𝑠𝑑𝜂 −ℎ 𝑡+𝜂

≤ ‖𝑥(𝑡)‖ + ℎ𝐹0 𝑠𝑢𝑝 ‖𝐴𝑑 (𝑠 + ℎ)‖ ‖𝑥(𝑠)‖ − 𝑡−ℎ≤𝑠≤𝑡

ℎ𝜂𝐺0 𝑠𝑢𝑝 ‖𝐴𝐷 (𝑠)‖ ‖𝑥(𝑠)‖, −ℎ ≤ 𝜂 ≤ 0, 𝛱(𝑡, 𝑥𝑡 ) ≥ ‖𝑥(𝑡)‖, and

𝑡+𝜂≤𝑠≤𝑡 𝑑 𝛱(𝑡, 𝑥𝑡 ) 𝑑𝑡

≤ − [𝐴0 − ℎ𝐺0 𝐴𝐷 (𝑡) − 𝐹0 ‖𝐴𝑑 (𝑡 + ℎ)‖]‖𝑥(𝑡)‖

 − K0 x(t )  0, ‖𝑥(𝑡)‖ ≠ 0. Hence, conditions (𝐴1) and (𝐴3) are satisfied. Thus, the solution 𝑥(𝑡) ≡ 0 of (29) is uniformly asymptotically stable. This is the end of the proof. The integrability of solutions is given in the following theorem. Theorem 21 (Tunç et al. [94]). We assume that conditions (𝐴1) and (𝐴3) are satisfied. Then, the norms of the solutions of system (29) are integrable on ℝ+ in sense of Lebesgue. Proof. Using the LKF 𝛱: = 𝛱(𝑡, 𝑥𝑡 ), it follows that (see the proof of Theorem 2) 𝑑 𝛱(𝑡, 𝑥𝑡 ) ≤ −𝐾0 ‖𝑥(𝑡)‖. 𝑑𝑡

100

Osman Tunç, Seenith Sivasundaram and Cemil Tunç

From this inequality, an integration gives that 𝛱(𝑡, 𝑥𝑡 ) − 𝛱(𝑡0 , 𝜙(𝑡0 )) ≤ 𝑡 ∞ −𝐾0 ∫𝑡 ‖𝑥(𝑠)‖ 𝑑𝑠. Hence, as the next step, we obtain ∫𝑡 ‖𝑥(𝑠)‖ 𝑑𝑠 ≤ 0

0

𝐾0−1 𝛱(𝑡0 , 𝜙(𝑡0 ) < +∞. The proof is complete. Example 8 (Tunç et al. [92]). We consider the nonlinear delay system 𝑥 ′ (𝑡) −51 + 𝑐𝑜𝑠 𝑡 ( 1′ ) = ( 1 𝑥2 (𝑡)

𝑥 (𝑡) 1 )( 1 ) −51 + 𝑐𝑜𝑠 𝑡 𝑥2 (𝑡)

𝑥1 (𝑡) 1 + 𝑥12 (𝑡 − 5−1) 𝑒𝑥𝑝( − 𝑡) 1 +( ) 𝑥2 (𝑡) 1 𝑒𝑥𝑝( − 𝑡) 2 −1 (1 + 𝑥2 (𝑡 − 5 )) 1

1+𝑒𝑥𝑝(𝑠) 𝑡 + ∫𝑡−1 ( 5

2

𝑥1 (𝑠)

1 )

1

1+𝑥12 (𝑠) ( 𝑥 (𝑠) ) 𝑑𝑠, 2

(33)

1+𝑥22 (𝑠)

1+𝑒𝑥𝑝(𝑠)

where 𝑡 ≥ 5−1 . Comparing systems (33) and (29), we obtain the following relations: 𝑎 (𝑡) 𝑎12 (𝑡) −51 + 𝑐𝑜𝑠 𝑡 1 𝐴(𝑡) = ( ), ) = ( 11 𝑎 1 −51 + 𝑐𝑜𝑠 𝑡 21 (𝑡) 𝑎22 (𝑡) 𝑒𝑥𝑝( − 𝑡) 1 𝑎 (𝑡) 𝑎𝑑12 (𝑡) 𝐴𝑑 (𝑡) = ( ) = ( 𝑑11 ), 1 𝑒𝑥𝑝( − 𝑡) 𝑎𝑑21 (𝑡) 𝑎𝑑22 (𝑡) 𝑥1 (𝑡) 2 −1 1 + 𝑥 1 (𝑡 − 5 ) 𝐹(𝑥(𝑡 − ℎ)) = 𝐹(𝑥1 (𝑡 − 5−1 ), 𝑥2 (𝑡 − 5−1 )) = , 𝑥2 (𝑡) 2 −1 (1 + 𝑥2 (𝑡 − 5 )) 𝑥1

G(𝑥) = 𝐺(𝑥1 , 𝑥2 ) =

1+𝑥 2 ( 𝑥2 1 ), 1+𝑥22 1

1+𝑒𝑥𝑝(𝑡)

(

2

𝑥 where 𝑥 = ( 1 ) , 𝐴𝐷 (𝑡) = 𝑥2 1 1

).

1+𝑒𝑥𝑝(𝑡)

Considering the above relations, we have the following estimates:

Advances in the Qualitative Theory of Integro–Differential Equations 101 𝑥1

𝐹(0) = 0, ‖𝐹(𝑥)‖ = ‖

1+𝑥 2 (𝑡−5−1 ) ( 1 𝑥2 )‖ 1+𝑥22 (𝑡−5−1 )

=

x1 −1

x2

+

1 + x (t − 5 ) 1 + x (t − 5−1 ) 2 1

2 2

 F0 x , 𝐹0 = 1,

𝑥1

𝐺(0) = 0, ‖𝐺(𝑥)‖ = ‖

1+𝑥 2 ( 𝑥2 1 ) ‖ 1+𝑥22

=

|𝑥1 | 1+𝑥12

+

|𝑥2 | 1+𝑥22

≤ 𝐺0 ‖𝑥‖, 𝐺0 = 1,

𝑎𝑖𝑖 (𝑡) + ∑𝑛𝑗=1,𝑗≠𝑖|𝑎𝑗𝑖 (𝑡)| ≤ −49 = −𝐴0 Since 𝑎11 (𝑡) + |𝑎21 (𝑡)| = −51 + 𝑐𝑜𝑠 𝑡 + 1 ≤ − 49 = −𝐴0 and 𝑎22 (𝑡) + |𝑎12 (𝑡)| = −51 + 𝑐𝑜𝑠 𝑡 + 1 ≤ − 49 = −𝐴0 , 2

𝑒𝑥𝑝( − 𝑡 − ℎ) 1 ‖𝐴𝑑 (𝑡 + ℎ)‖ = ‖ ( ) ‖ 𝑚𝑎𝑥 ∑ |𝑎𝑑 𝑖𝑗 (𝑡 + ℎ)| 1 𝑒𝑥𝑝( − 𝑡 + ℎ) 1≤𝑗≤2 𝑖=1

= 𝑚𝑎𝑥{𝑒𝑥𝑝( − 𝑡 − ℎ) + 1,1 + 𝑒𝑥𝑝( − 𝑡 − ℎ)} 1 = 1 + 𝑒𝑥𝑝( − 𝑡 − ℎ) ≤ 2 = 𝑑0 , ℎ = , 𝑡 ≥ 5−1 , 5

1 2 1 1 + 𝑒𝑥𝑝( 𝑡) ‖ ‖ ‖𝐴𝐷 (𝑡)‖ = ‖ 𝑚𝑎𝑥 ∑ |𝑎𝐷 𝑖𝑗 (𝑡)| ‖ = 1≤𝑗≤2 1 𝑖=1 2 1 + 𝑒𝑥𝑝( 𝑡) ( ) 1 1 1 = 𝑚𝑎𝑥 { + 2, + 1} = + 2 ≤ 3 = 𝐷0 . 1 + 𝑒𝑥𝑝( 𝑡) 1 + 𝑒𝑥𝑝( 𝑡) 1 + 𝑒𝑥𝑝( 𝑡) As for the last step, from these estimates, we arrive at 𝐴0 − ℎ𝐺0 ‖𝐴𝐷 (𝑡)‖ − 𝐹0 ‖𝐴𝑑 (𝑡 + ℎ)‖ ≥ 49 − 5−1 × 3 − 1 − 𝑒𝑥𝑝( − 𝑡 − 5−1 ) > 46 = 𝐾0 . Thus, conditions (𝐴1) − (𝐴3) hold, so the solution 𝑥(𝑡) ≡ 0 of system (33) is uniformly stable, uniformly asymptotically stable and the norms of the solutions of the system are integrable. For the next boundedness result of this paper, the following conditions are needed.

102

Osman Tunç, Seenith Sivasundaram and Cemil Tunç

(𝐴6) The function 𝐻 ∈ 𝐶(𝑅 + × 𝑅 𝑛 × 𝑅 𝑛 , 𝑅 𝑛 ) satisfies, ‖𝐻(𝑡, 𝑥(𝑡), 𝑥(𝑡 − ℎ))‖ ≤ |𝐻0 (𝑡)| ‖𝑥(𝑡)‖ for all 𝑡 ∈ 𝑅 + and for all 𝑥, 𝑥(𝑡 − ℎ) ∈ 𝑅 𝑛 , where 𝐻0 ∈ 𝐶(𝑅 + , 𝑅), the function 𝐻0 is bounded and there are constants 𝐹0 , 𝐺0 , 𝐴0 > 0 and ℎ > 0 from (𝐴2) such that 𝑎𝑖𝑖 (𝑡) + ∑𝑛𝑗=1,𝑗≠𝑖|𝑎𝑗𝑖 (𝑡)| ≤ −𝐴0 for all 𝑡 ∈ 𝑅 +, and 𝐴0 − ℎ𝐺0 ‖𝐴𝐷 (𝑡)‖ − 𝐹0 ‖𝐴𝑑 (𝑡 + ℎ)‖ − |𝐻0 (𝑡)| ≥ 0 for all 𝑡 ∈ [−ℎ, ∞), The following theorem, Theorem 22, is the boundedness result of [94]. Theorem 22 (Tunç et al. [94]). The solutions of perturbed system (28) are bounded at infinity if conditions(𝐴1) and(𝐴6) are satisfied. Proof. In this proof, we use again the LKF given by (30). Clearly, using conditions (𝐴1) and (𝐴6), one can find that, 𝑑 𝛱(𝑡, 𝑥𝑡 ) ≤ −[𝐴0 − 𝐹0 ‖𝐴𝑑 (𝑡 + ℎ)‖ − ℎ𝐺0 ‖𝐴𝐷 (𝑡)‖ ]‖𝑥(𝑡)‖ 𝑑𝑡 + ‖𝐻(𝑡, 𝑥(𝑡), 𝑥(𝑡 − ℎ))‖ ≤ −[𝐴0 − 𝐹0 ‖𝐴𝑑 (𝑡 + ℎ)‖ − ℎ𝐺0 ‖𝐴𝐷 (𝑡)‖ − |𝐻0 (𝑡)| ]‖𝑥(𝑡)‖ ≤ 0. As a result of this inequality, we find 𝛱(𝑡, 𝑥𝑡 ) ≤ 𝛱(𝑡0, 𝜙(𝑡0 )), 𝑡 ≥ 𝑡0 . Hence, it is notable from the definition of LKF (30) that ‖𝑥(𝑡)‖ ≤ 𝛱(𝑡0 , 𝜙(𝑡0 )). If 𝑡 → +∞, then 𝑙𝑖𝑚 ‖|𝑥(𝑡)|‖ ≤ 𝑙𝑖𝑚 𝛱(𝑡0, 𝜙(𝑡0 )) = 𝑡→+∞

𝑡→+∞

𝛱(𝑡0 , 𝜙(𝑡0 )) ≡ a positive constant. Thus, the solutions of system (28) are bounded as 𝑡 → +∞. The proof of Theorem 22 is complete. Example 9 (Tunç et al. [94]). We now take into consideration the following nonlinear delay system:

Advances in the Qualitative Theory of Integro–Differential Equations 103

𝑥 ′ (𝑡) −51 + 𝑐𝑜𝑠 𝑡 ( 1′ ) = ( 1 𝑥2 (𝑡)

𝑥 (𝑡) 1 )( 1 ) −51 + 𝑐𝑜𝑠 𝑡 𝑥2 (𝑡)

𝑥1 (𝑡) 1 + 𝑥12 (𝑡 − 5−1 ) 𝑒𝑥𝑝( − 𝑡) 1 +( ) 𝑥2 (𝑡) 1 𝑒𝑥𝑝( − 𝑡) 2 −1 1 + 𝑥 2 (𝑡 − 5 )) ( 1

𝑥1 (𝑠)

1

1+𝑒𝑥𝑝(𝑠) 𝑡 + ∫𝑡−1 ( 5

1

2

)

1+𝑒𝑥𝑝(𝑠)

1+𝑥12 (𝑠) ( 𝑥2 (𝑠) ) 𝑑𝑠, 1+𝑥22 (𝑠)

𝑠𝑖𝑛 𝑥1 (𝑡) 1+𝑒𝑥𝑝(𝑡)+𝑒𝑥𝑝(|𝑥1 (𝑡−5−1 )|) +( ). 𝑠𝑖𝑛 𝑥2 (𝑡) 1+𝑒𝑥𝑝(𝑡)+𝑒𝑥𝑝(|𝑥2

(34)

(𝑡−5−1 )|)

Comparing systems (34) and (28), it follows that 𝐻(𝑡, 𝑥1 , 𝑥2 , 𝑥1 (𝑡 − 5−1 ), 𝑥2 (𝑡 − 5−1)) 𝑠𝑖𝑛 𝑥1 1 + 𝑒𝑥𝑝( 𝑡) + 𝑒𝑥𝑝( |𝑥1 (𝑡 − 5−1)|) = . 𝑠𝑖𝑛 𝑥2 (1 + 𝑒𝑥𝑝( 𝑡) + 𝑒𝑥𝑝( |𝑥2 (𝑡 − 5−1)|)) ‖𝐻(𝑡, 𝑥1 , 𝑥2 , 𝑥1 (𝑡 − 5−1 ), 𝑥2 (𝑡 − 5−1 ))‖ |𝑠𝑖𝑛 𝑥1 | ≤ 1 + 𝑒𝑥𝑝( 𝑡) + 𝑒𝑥𝑝( |𝑥1 (𝑡 − 5−1)|) |𝑠𝑖𝑛 𝑥2 | |𝑠𝑖𝑛 𝑥1 | + |𝑠𝑖𝑛 𝑥2 | |𝑥1 | + |𝑥2 | + ≤ ≤ −1 1 + 𝑒𝑥𝑝( 𝑡) + 𝑒𝑥𝑝( |𝑥2 (𝑡 − 5 )|) 1 + 𝑒𝑥𝑝( 𝑡) 1 + 𝑒𝑥𝑝( 𝑡) 1 ‖𝑥‖, = 1 + 𝑒𝑥𝑝( 𝑡) 1

where |𝐻0 (𝑡) =| 1+𝑒𝑥𝑝(𝑡). Next, the matrices A(t ), 𝐴𝑑 (𝑡), 𝐴𝐷 (𝑡), the functions G(𝑥1 , 𝑥2 ) and 𝐹(𝑥1 (𝑡 − 5−1 ), 𝑥2 (𝑡 − 5−1 )) are the same as those in Example 1. Hence, it is seen that, 𝐴0 − ℎ𝐺0 ‖𝐴𝐷 (𝑡)‖ − 𝐹0 ‖𝐴𝑑 (𝑡 + ℎ)‖ − |𝐻0 (𝑡)| = 49 − 5−1 × 3 − 2 − 1 > 0. 1+𝑒𝑥𝑝(𝑡)

104

Osman Tunç, Seenith Sivasundaram and Cemil Tunç

Then, it is obvious that (

𝑥1 (𝑡) ) of system (34) are bounded. 𝑥2 (𝑡)

Recently, Ngoc and Hieu [61] considered the vector IDE of the form: 𝑡

𝑥̇ (𝑡) = 𝐹(𝑡, 𝑥(𝑡), ∫0 𝐺(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠), 𝑡 ≥ 𝑡0 ,

(35)

𝑥(𝑡) = 𝜑(𝑡), 𝑡 ∈ [0, 𝑡0 ],

(36)

where 𝐹: 𝑅+ × 𝑅 𝑛 × 𝑅 𝑛 → 𝑅 𝑛 , 𝑅+ = [0, ∞), and 𝐺: 𝛥 × 𝑅 𝑛 → 𝑅 𝑛 (𝛥: = {(𝑡, 𝑠) ∈ 𝑅 2 : 𝑡 ≥ 𝑠 ≥ 0}) are given matrix-valued continuous functions. In [61], using a novel approach, the authors presented some new scalar criteria for the uniform asymptotic stability of general nonlinear the vector IDE above. Discussion and illustrative examples of the obtained results are also given. Vector IDE (35) encompasses a large class of Volterra integrodifferential equations. In particular, it includes nonlinear IDEs of the form 𝑡

𝑥̇ (𝑡) = 𝑓(𝑡, 𝑥(𝑡)) + ∫ 𝑔(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠 0

𝑡

and 𝑥̇ (𝑡) = 𝐴(𝑡)𝑓(𝑥(𝑡)) + ∫0 𝐵(𝑡, 𝑠)𝑔(𝑥(𝑠))𝑑𝑠. Let 𝑅 𝑛 be endowed with the Euclidean norm ‖. ‖. For given 𝛿 > 0, let 𝐵𝛿 (𝑅 𝑛 ): = {𝑥 ∈ 𝑅 𝑛 : ‖𝑥‖ ≤ 𝛿}. Denote by 𝐶𝑡0 : = 𝐶([0, 𝑡0 ], 𝑅 𝑛 ), the Banach space of all continuous functions on [0, 𝑡0 ], endowed with the norm ‖𝜗‖𝑡0 : = 𝑠𝑢𝑝𝑠∈[0,𝑡0 ]‖𝜗(𝑠)‖, for 𝜗 ∈ 𝐶𝑡0 . Consider the nonlinear IDE (35). For fixed 𝑡0 ∈ 𝑅 + and given 𝜗 ∈ 𝐶𝑡0 , an initial value condition to IDE (35) is defined by (36). To ensure existence and uniqueness of solutions of the initial-value problem (35)–(36), it is assumed throughout that the function 𝐹 is (locally) Lipschitz continuous with respect to second and third arguments on each compact subset of 𝑅 + × 𝑅 𝑛 × 𝑅 𝑛 and the function 𝐺 is (locally) Lipschitz continuous with respect to the third argument on each compact subset of 𝛥 × 𝑅 𝑛 . Definition 4 (Ngoc and Hieu [61]).

Advances in the Qualitative Theory of Integro–Differential Equations 105

(i) The zero solution of IDE (35) is said to be uniformly stable if for any there exists 𝑟1 = 𝑟1 (𝜀) > 0 such that 𝑡0 ≥ 0, 𝜗 ∈ 𝐶𝑡0 , ‖𝜗‖𝑡0 ≤ 𝑟1 ⇒ ‖𝑥(𝑡, 𝑡0 , 𝜗)‖ < 𝜀, ∀𝑡 ≥ 𝑡0 . (ii) The zero solution of IDE (35) is said to be uniformly asymptotically stable if it is uniformly stable and there exists 𝑟2 > 0 with the property that, for each 𝜀 > 0 there is a number 𝑇 = 𝑇(𝜀) > 0 such that 𝑡 ≥ 𝑡0 , 𝜗 ∈ 𝐶𝑡0 , ‖𝜗‖𝑡0 ≤ 𝑟2 ⇒ ‖𝑥(𝑡, 𝑡0 , 𝜗)‖ < 𝜀, ∀𝑡 ≥ 𝑡0 + 𝑇. We are now in the position to state the main result in this setup. Theorem 23 (Ngoc and Hieu [61]). Suppose there exists 𝛿 > 0 and continuous functions 𝛼, 𝛽: 𝑅 + → 𝑅 + and 𝛾: 𝛥 → 𝑅 + such that 𝑡

𝛼(𝑡) + 𝛽(𝑡) ∫0 𝛾(𝑡, 𝜃)𝑑𝜃 ≤ 0, 𝑡 ∈ 𝑅 + , 𝑡

𝑠𝑢𝑝 ∫0 𝛾(𝑡, 𝑠)𝑑𝑠 < ∞,

𝑡∈𝑅+

𝑥 𝑇 𝐹(𝑡, 𝑥, 𝑦) ≤ −𝛼(𝑡)‖𝑥‖2 + 𝛽(𝑡)‖𝑥‖‖𝑦‖, 𝑡 ≥ 0, 𝑥, 𝑦 ∈ 𝐵𝛿 (𝑅 𝑛 ), and ‖𝐺(𝑡, 𝑠, 𝑧)‖ ≤ 𝛾(𝑡, 𝑠)‖𝑧‖, (𝑡, 𝑠) ∈ 𝛥, 𝑧 ∈ 𝐵𝛿 (𝑅𝑛 ). If the the following linear IDE 𝑡

𝑦̇ (𝑡) = −𝛼(𝑡)𝑦(𝑡) + ∫0 𝛽(𝑡, 𝑠)𝑦(𝑠)𝑑𝑠, 𝑡 ≥ 𝑡0 , is uniformly asymptotically stable, then the zero solution of IDE (35) is uniformly asymptotically stable. Therorem 24 (Ngoc and Hieu [61]). Suppose there exist positive numbers 𝛿, 𝛼, 𝛽 and a continuous function 𝛾: 𝑅 + → 𝑅 + such that, 𝑥 𝑇 𝐹(𝑡, 𝑥, 𝑦) ≤ −𝛼‖𝑥‖2 + 𝛽‖𝑥‖ ‖𝑦‖ and

106

Osman Tunç, Seenith Sivasundaram and Cemil Tunç

‖𝐺(𝑡, 𝑠, 𝑧)‖ ≤ 𝛾(𝑡 − 𝑠)‖𝑧‖, ∞

for any (𝑡, 𝑠) ∈ 𝛥 and for any 𝑥, 𝑦, 𝑧 ∈ 𝐵𝛿 (𝑅𝑛 ). If−𝛼 + 𝛽 ∫0 𝛾(𝜃)𝑑𝜃 < 0, then the zero solution of IDE (35) is uniformly asymptotically stable.

4. Ulam Stability, Ulam- Hyers Stability, Ulam -Hyers-Rassias Stability of IDEs Kucche and Shikhare [118] investigated the existence and uniqueness of a solution and Ulam- type stabilities for Volterra delay integro-differential equations on a finite interval. The analysis of [118] is based on the Pachpatte’s inequality and Picard operator theory. Examples are provided to illustrate the stability results obtained in the case of a finite interval. Also, an example is given to illustrate that the Volterra delay integro-differential equations are not Ulam–Hyers stable on the infinite interval. Indeed, in [118], the authors consider the following nonlinear Volterra delay integro-differential equation: 𝑡

𝑥 ′ (𝑡) = 𝑓 (𝑡, 𝑥(𝑡), 𝑥(𝑔(𝑡)), ∫0 ℎ (𝑡, 𝑠, 𝑥(𝑠), 𝑥(𝑔(𝑠))) 𝑑𝑠) , 𝑡 ∈ 𝐼,

(37)

𝑥(𝑡) = 𝜙(𝑡), 𝑡 ∈ [−𝑟, 0], where 𝜙 ∈ 𝐶([−𝑟, 0], 𝑅). For a given 𝜀 > 0 and a positive nondecreasing continuous function 𝜓 ∈ 𝐶([−𝑟, 𝑏], 𝑅+ ), consider the following inequalities: 𝑡

|𝑦 ′ (𝑡) − 𝑓 (𝑡, 𝑦(𝑡), 𝑦(𝑔(𝑡)), ∫0 ℎ (𝑡, 𝑠, 𝑦(𝑠), 𝑦(𝑔(𝑠))) 𝑑𝑠)| ≤ 𝜀, 𝑡 ∈ 𝐼, (38) 𝑡

|𝑦 ′ (𝑡) − 𝑓 (𝑡, 𝑦(𝑡), 𝑦(𝑔(𝑡)), ∫0 ℎ (𝑡, 𝑠, 𝑦(𝑠), 𝑦(𝑔(𝑠))) 𝑑𝑠)| ≤ 𝜓(𝑡), 𝑡 ∈ 𝐼,

(39) 𝑡

|𝑦 ′ (𝑡) − 𝑓 (𝑡, 𝑦(𝑡), 𝑦(𝑔(𝑡)), ∫0 ℎ (𝑡, 𝑠, 𝑦(𝑠), 𝑦(𝑔(𝑠))) 𝑑𝑠)| ≤ 𝜀𝜓(𝑡), 𝑡 ∈ 𝐼.

(40)

Advances in the Qualitative Theory of Integro–Differential Equations 107

Definition 5 (Kucche and Shikhare [118]). Equation (37) is said to be Ulam-Hyers stable if there exists a real number 𝐶 > 0such that for each 𝜀 > 0 and for each solution 𝑦 ∈ 𝐶 1 ([−𝑟, 𝑏], 𝑅) of (38) there exists a solution 𝑥 ∈ 𝐶 1 ([−𝑟, 𝑏], 𝑅) of (37) with |𝑦(𝑡) − 𝑥(𝑡)| ≤ 𝐶𝜀 for 𝑡 ∈ [−𝑟, 𝑏] . Definition 6 (Kucche and Shikhare [118]). Equation (37) is said to be generalized Ulam-Hyers stable if there exists 𝜃𝑓 ∈ 𝐶(𝑅+ , 𝑅+ ), 𝜃𝑓 (0) = 0 such that for each solution 𝑦 ∈ 𝐶 ′ ([−𝑟, 𝑏], 𝑅) of (39) there exists a solution 𝑥 ∈ 𝐶 1 ([−𝑟, 𝑏], 𝑅) of (37) with |𝑦(𝑡) − 𝑥(𝑡)| ≤ 𝜃𝑓 (𝜀) for 𝑡 ∈ [−𝑟, 𝑏] . Definition 7 (Kucche and Shikhare [118]). Equation (37) is said to be Ulam-Hyers-Rassias stable with respect to the positive nondecreasing continuous function 𝜓: [−𝑟, 𝑏] → 𝑅+ if there exists 𝐶𝜓 > 0 such that for each 𝜀 > 0 and for each solution 𝑦 ∈ 𝐶 1 ([−𝑟, 𝑏], 𝑅) of (40) there exists a solution 𝑥 ∈ 𝐶 1 ([−𝑟, 𝑏], ℝ) of (37) with |𝑦(𝑡) − 𝑥(𝑡)| ≤ 𝐶𝜓 𝜀𝜓(𝑡) for 𝑡 ∈ [−𝑟, 𝑏]. Definition 8 (Kucche and Shikhare [118]). Equation (37) is said to be generalized Ulam-Hyers-Rassias stable with respect to the positive nondecreasing continuous function 𝜓: [−𝑟, 𝑏] → ℝ+ if there exists 𝐶𝜓 > 0 such that for each solution 𝑦 ∈ 𝐶 ′ ([−𝑟, 𝑏], ℝ) of (40) there exists a solution 𝑥 ∈ 𝐶 ′ ([−𝑟, 𝑏], 𝑅) of (37) with |𝑦(𝑡) − 𝑥(𝑡)| ≤ 𝐶𝜓 𝜓(𝑡) for 𝑡 ∈ [−𝑟, 𝑏]. The following assumptions are needed to state and prove the main results of Kucche and Shikhare [118]. (H1) (i) Let 𝑓 ∈ 𝐶([0, 𝑏] × 𝑅 3 , 𝑅), ℎ ∈ 𝐶([0, 𝑏] × [0, 𝑏] × 𝑅 2 , 𝑅) and 𝑔 ∈ 𝐶([0, 𝑏], [−𝑟, 𝑏]) be such that 𝑔(𝑡) ≤ 𝑡. (ii) There exist constants 𝐿𝑓 , 𝐿ℎ > 0 such that |𝑓(𝑡, 𝑢1 , 𝑢2 , 𝑢3 ) − 𝑓(𝑡, 𝑣1 , 𝑣2 , 𝑣3 )| ≤ 𝐿𝑓 (|𝑢1 − 𝑣1 | + |𝑢2 − 𝑣2 | + |𝑢3 − 𝑣3 |); |ℎ(𝑡, 𝑠, 𝑢1 , 𝑢2 , ) − ℎ(𝑡, 𝑠, 𝑣1 , 𝑣2 , )| ≤ 𝐿ℎ (|𝑢1 − 𝑣1 | + |𝑢2 − 𝑣2 |) for all 𝑡, 𝑠 ∈ 𝐼, 𝑢𝑖 , 𝑣𝑖 ∈ 𝑅, (𝑖 = 1,2,3).

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(H2) The function 𝜓: [−𝑟, 𝑏] → 𝑅+ is positive, nondecreasing and 𝑡 continuous and there exists 𝜆 > 0 such that ∫0 𝜓(𝑠) 𝑑𝑠 ≤ 𝜆𝜓(𝑡), 𝑡 ∈ [0, 𝑏]. Theorem 25 (Kucche and Shikhare [118]). Let the functions 𝑓 and ℎ in (37) satisfy (H1) and assume that (H2) holds. If 𝑏𝐿𝑓 |2 + 𝐿ℎ 𝑏| < 1, then the following assertions hold: The initial value problem of (37) has a unique solution, 𝑥 ∈ 𝐶([−𝑟, 𝑏], 𝑅) ∩ 𝐶 1 ([0, 𝑏], 𝑅); The equation (37) is Ulam-Hyers-Rassias stable with respect to the function 𝜓. Corollary 1 Let the functions 𝑓 and ℎ in (37) satisfy (H1) and assume that (H2) holds. If 𝑏𝐿𝑓 |2 + 𝐿ℎ 𝑏| < 1, then the initial-value problem of (37) has a unique solution and the equation (37) is generalized Ulam-HyersRassias stable with respect to the function 𝜓. Proof. By taking 𝜀 = 1 in the proof of Theorem 24, we obtain |𝑦(𝑡) − 𝑥(𝑡)| ≤ 𝐶𝜓 𝜓(𝑡) , 𝑡 ∈ [−𝑟, 𝑏], showing that (37) is generalized Ulam-Hyers-Rassias stable with respect to the function 𝜓. Using arguments similar to those applied in the proof of Theorem 24, one can prove Ulam-Hyers stability of equation (37). Observing that for 𝜓(𝑡) = 1, ∀𝑡 ∈ [−𝑟, 𝑏] , the assumption (H2) holds, we can state the following corollary of Theorem 24. Corollary 2 Let the functions 𝑓 and ℎ in (37) satisfy the hypothesis (H1). If 𝑏𝐿𝑓 |2 + 𝐿ℎ 𝑏| < 1, then the initial-value problem of (37) has a unique solution and the equation (37) is Ulam-Hyers stable. Proof. By taking 𝜓(𝑡) = 1, ∀𝑡 ∈ [−𝑟, 𝑏] in the proof of Theorem 24, we obtain |𝑦(𝑡) − 𝑥(𝑡)| ≤ 𝐶𝜓 𝜀 , 𝑡 ∈ [−𝑟, 𝑏] , and the result follows.

Advances in the Qualitative Theory of Integro–Differential Equations 109

Corollary 3 Let the functions 𝑓 and ℎ in (37) satisfy the hypothesis (H1). If 𝑏𝐿𝑓 |2 + 𝐿ℎ 𝑏| < 1, then the intial-value problem of (37) has a unique solution and the equation (37) is generalized Ulam-Hyers stable. Proof. The result follows from Corollary 2, by taking 𝜃𝑓 (𝜀) = 𝐶𝜀. Applications (Kucche and Shikhare [118]). Consider some important special cases of the initial-value problem (37). Fix any 𝑟 > 0, and define 𝑔1 (𝑡) = 𝑡 − 𝑟, 𝑡 ∈ [0, 𝑏]. Then, we get the following special case of the initial value problem of (37): 𝑡

𝑥 ′ (𝑡) = 𝑓1 (𝑡, 𝑥(𝑡), 𝑥(𝑡 − 𝑟), ∫0 ℎ1 (𝑡, 𝑠, 𝑥(𝑠), 𝑥(𝑠 − 𝑟))𝑑𝑠) , 𝑡 ∈ [0, 𝑏], (41) 𝑥(𝑡) = 𝜙(𝑡), 𝑡 ∈ [−𝑟, 0],

(42)

which is an initial-value problem for a nonlinear Volterra integro-differential difference equation. Consider the following inequality: 𝑡

|𝑦 ′ (𝑡) − 𝑓1 (𝑡, 𝑦(𝑡), 𝑦(𝑡 − 𝑟), ∫ ℎ1 (𝑡, 𝑠, 𝑦(𝑠), 𝑦(𝑠 − 𝑟))𝑑𝑠)| ≤ 𝜀𝜓(𝑡), 0

𝑡 ∈ [0, 𝑏], where 𝜀, 𝜓 and 𝜙 are as specified before. As an application, we have the following theorem for the problem (41), (42). Theorem 24 (Kucche and Shikhare [118]). Suppose that the following assumptions are fulfilled: (A1) (i) 𝑓1 ∈ 𝐶([0, 𝑏] × 𝑅 3 , 𝑅), ℎ1 ∈ 𝐶([0, 𝑏] × [0, 𝑏] × 𝑅 2 , 𝑅) and 𝑔1 ∈ 𝐶([0, 𝑏], [−𝑟, 𝑏]) be such that 𝑔1 (𝑡) ≤ 𝑡; (ii) there exist constants 𝐿𝑓1 , 𝐿ℎ1 > 0 such that |𝑓1 (𝑡, 𝑢1 , 𝑢2 , 𝑢3 ) − 𝑓1 (𝑡, 𝑣1 , 𝑣2 , 𝑣3 )| ≤ 𝐿𝑓1 (|𝑢1 − 𝑣1 | + |𝑢2 − 𝑣2 | + |𝑢3 − 𝑣3 |); |ℎ1 (𝑡, 𝑠, 𝑢1 , 𝑢2 , ) − ℎ1 (𝑡, 𝑠, 𝑣1 , 𝑣2 , )| ≤ 𝐿ℎ1 (|𝑢1 − 𝑣1 | + |𝑢2 − 𝑣2 |); for all 𝑡, 𝑠 ∈ [0, 𝑏], 𝑢𝑖 , 𝑣𝑖 ∈ ℝ, (𝑖 = 1,2,3);

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(A2) the function 𝜓: [−𝑟, 𝑏] → 𝑅+ is positive, nondecreasing and 𝑡

continuous, and there exists𝜆 > 0such that ∫0 𝑡 ∈ [0, 𝑏];

𝜓(𝑠)𝑑𝑠 ≤ 𝜆𝜓(𝑡),

(A3) 𝑏𝐿𝑓1 |2 + 𝐿ℎ2 𝑏| < 1. Then, the problem (40), (41) has a unique solution𝑥 ∈ 𝐶([−𝑟, 𝑏], 𝑅) ∩ 𝐶 1 ([0, 𝑏], 𝑅), and the equation (40) is Ulam-Hyers-Rassias stable with respect to the function 𝜓. Theorem 25 (Kucche and Shikhare [118]). Suppose that the following assumptions are fulfilled: (B1) (i) 𝑓2 ∈ 𝐶([0,1] × 𝑅 3 , 𝑅), ℎ2 ∈ 𝐶([0,1] × [0,1] × 𝑅 2 , 𝑅) and 𝑔2 ∈ 𝐶([0,1], [−𝑟, 1]) be such that 𝑔2 (𝑡) ≤ 𝑡; (ii) there exists constants 𝐿𝑓2 , 𝐿ℎ2 > 0such that |𝑓2 (𝑡, 𝑢1 , 𝑢2 , 𝑢3 ) − 𝑓2 (𝑡, 𝑣1 , 𝑣2 , 𝑣3 )| ≤ 𝐿𝑓2 (|𝑢1 − 𝑣1 | + |𝑢2 − 𝑣2 | + |𝑢3 − 𝑣3 |); |ℎ2 (𝑡, 𝑠, 𝑢1 , 𝑢2 , ) − ℎ2 (𝑡, 𝑠, 𝑣1 , 𝑣2 , )| ≤ 𝐿ℎ2 (|𝑢1 − 𝑣1 | + |𝑢2 − 𝑣2 |); for all 𝑡, 𝑠 ∈ [0,1], 𝑢𝑖 , 𝑣𝑖 ∈ 𝑅, (𝑖 = 1,2,3); (B2) the function 𝜓: [−𝑟, 1] → 𝑅+ is positive, nondecreasing and 𝑡 continuous, and there exists𝜆 > 0such that ∫0 𝜓 (𝑠)𝑑𝑠 ≤ 𝜆𝜓(𝑡), 𝑡 ∈ [0,1]; (B3) 𝑏𝐿𝑓2 |2 + 𝐿ℎ2 | < 1. Then, the problem (41), (42) has a unique solution 𝑥 ∈ 𝐶([−𝑟, 1], 𝑅) ∩ 𝐶 ([0,1], 𝑅), and the equation (41) is Ulam-Hyers-Rassias stable with respect to the function 𝜓. 1

Example 10 (Kucche and Shikhare [118]). Consider the following nonlinear delay Volterra integro-differential equation:

Advances in the Qualitative Theory of Integro–Differential Equations 111

𝑡 𝑐𝑜𝑠( 𝑥(𝑡)) 3𝑥(𝑡) 𝑡 𝑐𝑜𝑠( 𝑥(𝑔(𝑡)) − + 140 140 70 𝑡 {𝑠𝑖𝑛( 𝑥(𝑠) − 𝑠𝑖𝑛 (𝑥(𝑔(𝑠)))} 𝑑𝑠, 𝑡 ∈ [0,5], 70

𝑥 ′ (𝑡) = 1 + +

1 𝑡 ∫ 20 0

𝑥(𝑡) = 0, 𝑡 ∈ [−1,0],

(43) (44)

𝑡

where 𝑔(𝑡) = , 𝑡 ∈ [0,5]. Clearly, we have 𝑔(𝑡) ≤ 𝑡, 𝑡 ∈ [0,5]. 2

(i) Define ℎ: [0,5] × [0,5] × 𝑅 × 𝑅 → 𝑅 by ℎ(𝑡, 𝑠, 𝑥(𝑠), 𝑥(𝑔(𝑠))) =

𝑡 [𝑠𝑖𝑛( 𝑥(𝑠)) − 𝑠𝑖𝑛( 𝑥(𝑔(𝑠)))], 𝑡, 𝑠 ∈ [0,5]. 70

Then, for any 𝑡, 𝑠 ∈ [0,5] and 𝑥1 , 𝑥2 , 𝑦1 , 𝑦2 ∈ 𝑅, we have |ℎ(𝑡, 𝑠, 𝑥1 , 𝑥2 ) − ℎ(𝑡, 𝑠, 𝑦1 , 𝑦2 )| ≤ ≤

𝑡 {|sin𝑥1 − sin𝑦1 | + |sin𝑥2 − sin𝑦2 |} 70

5 {|𝑥 − 𝑦1 | + |𝑥2 − 𝑦2 |}. 70 1

(ii) Define 𝑓: [0,5] × 𝑅 × 𝑅 × 𝑅 → 𝑅 by 𝑡

𝑓 (𝑡, 𝑥(𝑡), 𝑥(𝑔(𝑡)), ∫ ℎ (𝑡, 𝑠, 𝑥(𝑠), 𝑥(𝑔(𝑠))))𝑑𝑠) 0

𝑡 𝑐𝑜𝑠(𝑥(𝑡)) 3𝑥(𝑡) 𝑡 𝑐𝑜𝑠( 𝑥(𝑔(𝑡)) =1+ − + 140 140 70 1 𝑡 𝑡 + ∫ [𝑠𝑖𝑛( 𝑥(𝑠) − 𝑠𝑖𝑛 (𝑥(𝑔(𝑠)))] 𝑑𝑠 20 0 70 𝑡 𝑐𝑜𝑠(𝑥(𝑡)) 3𝑥(𝑡) 𝑡 𝑐𝑜𝑠( 𝑥(𝑔(𝑡)) =1+ − + 140 140 70 𝑡 1 + ∫ ℎ(𝑡, 𝑠, 𝑥(𝑠), 𝑥(𝑔(𝑠)))𝑑𝑠, 𝑡 ∈ [0,5]. 20 0 Then, for any 𝑡 ∈ [0,5] and 𝑥1 , 𝑥2 , 𝑥3 , 𝑦1 , 𝑦2 , 𝑦3 ∈ 𝑅, we have |𝑓(𝑡, 𝑥1 , 𝑥2 , 𝑥3 ) − 𝑓(𝑡, 𝑦1 , 𝑦2 , 𝑦3 )|

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𝑡 3 𝑡 |𝑐𝑜𝑠 𝑥1 − 𝑐𝑜𝑠 𝑦1 | + |𝑥 − 𝑦1 |} + |𝑐𝑜𝑠 𝑥2 − 𝑐𝑜𝑠 𝑦2 | 140 140 1 70 1 + |𝑥3 − 𝑦3 |. 20 Next, for any 𝑥, 𝑦 ∈ 𝑅 with 𝑥 < 𝑦, by the Mean Value Theorem, there exists 𝑝, 𝑥 < 𝑝 < 𝑦 such that ≤{

𝑐𝑜𝑠 𝑥−𝑐𝑜𝑠 𝑦 𝑥−𝑦

= − 𝑠𝑖𝑛 𝑝 ⇒ |𝑐𝑜𝑠 𝑥 − 𝑐𝑜𝑠 𝑦| ≤ |𝑥 − 𝑦|.

Therefore, we have 5 |𝑥 140 1 5 1 |𝑥 − 𝑦2 | + |𝑥3 − 70 2 20

|𝑓(𝑡, 𝑥1 , 𝑥2 , 𝑥3 ) − 𝑓(𝑡, 𝑦1 , 𝑦2 , 𝑦3 )| ≤ {

≤

5 70

3

− 𝑦1 | + 140 |𝑥1 − 𝑦1 |} + 𝑦3 |

{|𝑥1 − 𝑦1 | + |𝑥2 − 𝑦2 | + |𝑥3 − 𝑦3 |}.

Hence, the above defined functions 𝑓 and ℎ verify the assumptions (H1) and (H2) with 5

5

𝐿𝑓 = 70 , 𝐿ℎ = 70 , 𝑏 = 5. Further, we see that 𝑏𝐿𝑓 (2 + 𝑏𝐿ℎ ) = 5

5 70

[2 +

5 70

5] = 0.84183673
0 independent of 𝑦 and 𝑦0 such that|𝑦(𝑥) − 𝑦0 (𝑥)| ≤ 𝐶𝜃, for all 𝑥 ∈ [𝑎, 𝑏], then we say thay the integro-differential equation (45) has the Hyers-Ulam stability. If for each function 𝑦 satisfying 𝑥

|𝑦 ′ (𝑥) − 𝑓(𝑥, 𝑦(𝑥), ∫𝑎 𝑘(𝑥, 𝜏, 𝑦(𝜏), 𝑦(𝛼(𝜏)))𝑑𝜏)| ≤ 𝜎(𝑥), 𝑥 ∈ [𝑎, 𝑏], where 𝜎 is a non-negative function, there is a solution 𝑦0 of the integrodifferential equation and a constant 𝐶 > 0 independent of 𝑦 and 𝑦0 such that|𝑦(𝑥) − 𝑦0 (𝑥)| ≤ 𝐶𝜎(𝑥), for all 𝑥 ∈ [𝑎, 𝑏], then we say they the integrodifferential equation (45) has the Hyers-Ulam Rassias stability. Theorem 26 Castro et al. [111]. Let 𝛼: [𝑎, 𝑏] → [𝑎, 𝑏] be a continuous delay function with 𝛼(𝑡) ≤ 𝑡 for all 𝑡 ∈ [𝑎, 𝑏] and 𝜎: [𝑎, 𝑏] → (0, ∞) a nondecreasing continuous function. In addition, suppose that there is𝛽 ∈ [0,1) 𝑥 such that ∫𝑎 𝜎(𝜏)𝑑𝜏 ≤ 𝛽𝜎(𝑥),for all 𝑥 ∈ [𝑎, 𝑏]. Moreover, suppose that 𝑓: [𝑎, 𝑏] × ℂ × ℂ → ℂ is a continuous function satisfying the Lipchitz condition |𝑓(𝑥, 𝑢(𝑥), 𝑔(𝑥)) − 𝑓(𝑥, 𝑣(𝑥), ℎ(𝑥)| ≤ 𝑀(|𝑢(𝑥) − 𝑣(𝑥)| + |𝑔(𝑥) − ℎ(𝑥)|) with 𝑀 > 0 and the kernel 𝑘: [𝑎, 𝑏] × [𝑎, 𝑏] × ℂ × ℂ → ℂ is a continuous function satisfying the Lipschitz condition |𝑘(𝑥, 𝑡, 𝑢(𝑡), 𝑢(𝛼(𝑡))) − 𝑘(𝑥, 𝑡, 𝑣(𝑡), 𝑣(𝛼(𝑡)))| ≤ 𝐿|𝑢(𝛼(𝑡)) − 𝑣(𝛼(𝑡))| with 𝐿 > 0. If 𝑦 ∈ 𝐶 1 ([𝑎, 𝑏]) is such that 𝑥

|𝑦 ′ (𝑥) − 𝑓(𝑥, 𝑦(𝑥), ∫ 𝑘(𝑥, 𝜏, 𝑦(𝜏), 𝑦(𝛼(𝜏)))𝑑𝜏)| ≤ 𝜎(𝑥), 𝑥 ∈ [𝑎, 𝑏], 𝑎

and 𝑀(𝛽 + 𝐿𝛽 2 ) < 1, then there is a unique function 𝑦0 ∈ 𝐶 1 ([𝑎, 𝑏]) such that

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𝑦0′ (𝑥) = 𝑓(𝑥, 𝑦0 (𝑥), ∫ 𝑘(𝑥, 𝜏, 𝑦0 (𝜏), 𝑦0 (𝛼(𝜏)))𝑑𝜏 𝑎

and|𝑦(𝑥) − 𝑦0 (𝑥)| ≤

𝛽 1−𝑀(𝛽+𝐿𝛽2 )

𝜎(𝑥) for all 𝑥 ∈ [𝑎, 𝑏]. This means that

under the above conditions, the integro-differential equation (1.1) has the Hyers-Ulam-Rassias stability. Theorem 27 (Castro et al. [111]). Let 𝛼: [𝑎, 𝑏] → [𝑎, 𝑏] be a continuous delay function with 𝛼(𝑡) ≤ 𝑡 for all 𝑡 ∈ [𝑎, 𝑏] and 𝜎: [𝑎, 𝑏] → (0, ∞) a nondecreasing continuous function. In addition, suppose that there is 𝛽 ∈ [0,1) 𝑥 such that∫𝑎 𝜎(𝜏)𝑑𝜏 ≤ 𝛽𝜎(𝑥), for all 𝑥 ∈ [𝑎, 𝑏]. Moreover, suppose that 𝑓: [𝑎, 𝑏] × ℂ × ℂ → ℂ is a continuous function satisfying the Lipschitz condition |𝑓(𝑥, 𝑢(𝑥), 𝑔(𝑥)) − 𝑓(𝑥, 𝑣(𝑥), ℎ(𝑥)| ≤ 𝑀(|𝑢(𝑥) − 𝑣(𝑥)| + |𝑔(𝑥) − ℎ(𝑥)|) with 𝑀 > 0 and 𝑘: [𝑎, 𝑏] × [𝑎, 𝑏] × ℂ × ℂ → ℂ is a continuous kernel function satisfying the Lipschitz condition |𝑘(𝑥, 𝑡, 𝑢(𝑡), 𝑢(𝛼(𝑡))) − 𝑘(𝑥, 𝑡, 𝑣(𝑡), 𝑣(𝛼(𝑡)))| ≤ 𝐿|𝑢(𝛼(𝑡)) − 𝑣(𝛼(𝑡))| with 𝐿 > 0. If 𝑦 ∈ 𝐶 1 ([𝑎, 𝑏]) is such that 𝑥

|𝑦 ′ (𝑥) − 𝑓(𝑥, 𝑦(𝑥), ∫ 𝑘(𝑥, 𝜏, 𝑦(𝜏), 𝑦(𝛼(𝜏)))𝑑𝜏)| ≤ 𝜃, 𝑥 ∈ [𝑎, 𝑏], 𝑎

where 𝜃 > 0 and 𝑀(𝛽 + 𝐿 𝛽 2 ) < 1, then there is a unique function 𝑦0 ∈ 𝐶 1 ([𝑎, 𝑏]) such that 𝑥

𝑦0′ (𝑥) = 𝑓(𝑥, 𝑦0 (𝑥), ∫ 𝑘(𝑥, 𝜏, 𝑦0 (𝑡), 𝑦0 (𝛼(𝑡)))𝑑𝑡) 𝑎 (𝑏−𝑎)𝜎(𝑏)

and |𝑦(𝑥) − 𝑦0 (𝑥)| ≤ [1−𝑀(𝛽+𝐿𝛽2 )]𝜎(𝑎) 𝜃 for all 𝑥 ∈ [𝑎, 𝑏]. Finally, some interesting results on Ulam stability, Ulam- Hyers stability, Ulam -Hyers-Rassias stability of IDEs can be outlined as follows:

Advances in the Qualitative Theory of Integro–Differential Equations 115

Castro and Guerra [112] address the problem of Hyers-Ulam-Rassias stability for nonlinear Volterra integral equations with delay and obtain conditions using the Banach fixed point theorem in a suitable complete metric space. Some illustrative examples are also given in [112]. Castro and Simões [113] discuss different types of stability of general nonlinear integro-differential equations. They introduce a new type of stability, in between the two notions of Hyers-Ulam-Rassias stability and Hyers-Ulam stability, namely the so-called semi-Hyers-Ulam-Rassias stability, in the framework of an appropriate metric space. At the end they introduce some examples to illustrate their results. Castro and Simões [114] analyse different types of Hyers-Ulam-Rassias stabilities for a class of integro-differential equations and Hyers-Ulam stabilities of certain integral equations with point and distributed delays of Fredholm and Volterra types. Finite and infinite intervals are considered as integration domains. A fixed point argument within the framework of the Bielecki metric is used, and the results are illustrated by concrete examples. Castro and Simões [115] obtain results regarding the Hyers-Ulam and the Hyers-Ulam-Rassias stabilities for the following class of Volterra integrodifferential equations: 𝑥

𝑦 ′ (𝑡) = 𝑓 (𝑡, 𝑦(𝑥), ∫ 𝑘(𝑡, 𝜏, 𝑦(𝑠𝜏), 𝑥(𝛼(𝜏)))𝑑𝑠) , 𝑦(𝑎) = 𝑐 ∈ 𝑅, 𝑎

with 𝑦 ∈ 𝐶[𝑎, 𝑏]. de Oliveira et al. [116] consider the problems generated by some fractional integro-differential equations. Using the Banach fixed point theorem, the authors introduce some stability theorems of Ulam-Hyers, Ulam Hyers-Rassias and semi-Ulam-Hyers-Rassias types. In Kucche [117], Pachpatte’s inequality is employed to discuss the Ulam-Hyers stabilities for Volterra integro-differential equations and Volterra delay integro-differential equations in Banach spaces on both finite and infinite intervals. Examples are given to show the applicability of our obtained results. In Vu and Hoa [119], the Ulam-Hyers stability and the Ulam-HyersRassias stability for the nonlinear Volterra integro-differential equations are established by employing the method of successive approximations. Some simple examples are given to illustrate the main results. In Shah et al. [120], the stability in terms of Bielecki-Ulam-Hyers and stability in terms of Bielecki-Ulam-Hyers-Rassias of non-linear impulsive

116

Osman Tunç, Seenith Sivasundaram and Cemil Tunç

Hammerstein integro-dynamic system with delay and non-linear impulsive mixed integro-dynamic system on time scales are achieved by utilizing a fixed point approach along with a Lipschitz condition and Gronwall’s inequality. Examples are also provided for the illustration of the results.

Conclusion This chapter provides numerous qualitative results with examples in relation to the scalar and vector linear and nonlinear integro –differential equations with and without delays. The given qualitative results can found in the related literature. The qualitative results of these chapter are called stability, instability, uniformly stability asymptotically stability, integrability and boundedness of solutions, etc., of IDEs, IDDEs, and Ulam stability, UlamHyers stability, Ulam -Hyers-Rassias stability, etc., of IDEs, IDDEs. Here, it can be seen that there is an extensive literature on the mentioned qualitative concepts. According to the given results, it can be seen that most of the qualitative results have been obtained using Lyapunov-Krasovskii functional method. Meanwhile a few of these qualitative results have been proved depending upon Lyapunov-Razumikhin technique. On the other hand, the qualitative results of Ulam stability, Ulam- Hyers stability, Ulam -HyersRassias stability, etc., have been proved using the fixed point method. Today, researchers are working effectively on the mentioned all the concepts. By this chapter, our aim is to provides researchers a bundle of scientific works on qualitative theory of IDEs, IDDEs, etc., which may be useful for researchers.

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Advances in the Qualitative Theory of Integro–Differential Equations 123 [108] Zhang, Z. D., Asymptotic stability of Volterra integro-differential equations. (Chinese) J. Harbin Inst. Tech. 1990, no. 4, 11–19. [109] Zhou, B., Egorov, A. V., Razumikhin and Krasovskiĭ stability theorems for timevarying time-delay systems. Automatica J. IFAC 71 (2016), 281–291. [110] Zhuang, W., Existence and uniqueness of solutions of nonlinear integrodifferential equations of Volterra type in a Banach space. Appl. Anal. 22 (1986), no. 2, 157–166. [111] Castro, L. P., Simões, A. M., Hyers-Ulam and Hyers-Ulam-Rassias stability for a class of integro-differential equations. Mathematical methods in engineering, 81– 94, Nonlinear Syst. Complex., 23, Springer, Cham, 2019. [112] Castro, L. P.,Guerra, R. C., Hyers-Ulam-Rassias stability of Volterra integral equations within weighted spaces. Lib. Math. (N.S.) 33 (2013), no. 2, 21–35. [113] Castro, L. P., Simões, A. M., Different types of Hyers-Ulam-Rassias stabilities for a class of integro-differential equations. Filomat 31 (2017), no. 17, 5379–5390. [114] Castro, L. P., Simões, A. M., Hyers-Ulam-Rassias stability of nonlinear integral equations through the Bielecki metric. Math. Methods Appl. Sci. 41 (2018), no. 17, 7367–7383. [115] Castro, L. P., Simões, A. M., Hyers-Ulam and Hyers-Ulam-Rassias stability for a class of integro-differential equations. Mathematical methods in engineering, 81– 94, Nonlinear Syst. Complex., 23, Springer, Cham, 2019. [116] de Oliveira, E. Capelas, Sousa, J. Vanterler da C. Ulam-Hyers-Rassias stability for a class of fractional integro-differential equations. Results Math. 73 (2018), no. 3, Paper No. 111, 16 pp. [117] Kucche, K. D., Shikhare, Pallavi U., Ulam-Hyers stability of integrodifferential equations in Banach spaces via Pachpatte's inequality. Asian-Eur. J. Math. 11 (2018), no. 4, 1850062, 19 pp. [118] Kucche, K. D., Shikhare, P. U., Ulam stabilities for nonlinear Volterra delay integro-differential equations. Izv. Nats. Akad. Nauk Armenii Mat. 54 (2019), no. 5, 27–43; reprinted in J. Contemp. Math. Anal. 54 (2019), no. 5, 276–287. [119] Vu, Ho, Hoa, N. V., Ulam-Hyers stability for a nonlinear Volterra integrodifferential equation. Hacet. J. Math. Stat. 49 (2020), no. 4, 1261–1269. [120] Shah, Syed Omar, Tunç, Cemil, Rizwan, Rizwan, Zada, Akbar, Khan, Qayyum Ullah, Ullah, Iftikhar, Ullah, I., Bielecki-Ulam's types stability analysis of Hammerstein and mixed integro-dynamic systems of non-linear form with instantaneous impulses on time scales. Qual. Theory Dyn. Syst. 21 (2022), no. 4, Paper No. 107, 21 pp.

In: Understanding Integro-Differential Equations ISBN: 979-8-89113-040-1 c 2023 Nova Science Publishers, Inc. Editors: J. Vasundhara Devi et al.

Chapter 4

M ATHEMATICAL M ETHODS FOR I NTEGRO -D IFFERENTIAL E QUATIONS AND T HEIR A PPLICATIONS Gunvant A. Birajdar1∗ and N. Giribabu2 1 School of Rural Development, Tata Institute of Social Sciences Tuljapur, Osmanabad, India 2 Department of Mathematics, Gayatri Vidya Parishad College of Engineering, Visakhapatnam, India

Abstract In this chapter, we have concentrated on powerful methods for solving intero-differential equations. Methods like the differential transform method, Laplace transform method, and Adomian decomposition method to obtain the solution of initial value problems of integro-differential equations. ∗ Corresponding Author’s

Email: [email protected].

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Keywords: differential transform method, adomian decomposition method, Laplace transform method, integro-differential equations AMS Subject Classification: 53D, 37C, 65P

1.

Introduction

Several scientific and engineering applications are usually described by integral equations or integro differential equations, standard or singular. We have seen a large class of initial and boundary value problems that can be converted to Volterra or Fredholm integral equations. Integral equations and integrodifferential equations arise in the potential theory more than any other field. Integro-differential equations arise also in diffraction problems, conformal mapping, water waves, scattering in quantum mechanics, and Volterra’s population growth model. The electrostatic, electro magnetic scattering problems and propagation of acoustical and elastical waves are scientific fields where integral and integro-differential equations appear. We have presented in this text a variety of traditional methods and some methods to handle integro-differential equations, Fredholm or Volterra type. Our concern in this text is the determination of the exact solutions in an easy computable fashion. Moreover, our aim is to present these selected methods to facilitate the computational work, and we avoided the abstract theorems that can be found in many other texts. In this chapter we focus on some powerful method, like Differential transform method (DTM), Laplace transform method (LTM) and, Adomian decomposition method (ADM) for initial value problems for integro-differential equations. Their has been lot of work has done in the area of numerical techniques integro-differential equations parallel to ordinary differential equations have been established for integro-differential equations also. Avudainayagam and Vani [1] developed the numerical technique as Wavelet-Galerkin method (WGM) for integro-differential equations. Rashed [2] employed the Lagrange interpolation method to obtain solutions differential integral and integrodifferential equations. We observe that Hosseini and Shahmorad [3] obtained the solution of Fredholm integro-differential equations using arbitrary polynomial bases through Tau method. The Taylor series method is used by Kanwal & Liu [4] and Maleknejad & Mahmoudi [5] for obtaining the solution of integrodifferential equations as well as higher order nonlinear integro-differential equa-

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tions respectively. Maleknejad et. al. [6] come up with rationalized Haar functions method to solve the system of integro-differential equations. The references ( [7–11]) also presented the use differential transform methods for various examples. In the second part we have discussed about the Adomian decomposition method. In 1980, Adomian proposed decomposition method popularly known as Adomian decomposition method (ADM) [12–14] has been intensively applied in research to obtain solution of varieties of initial boundary value problems of integral and integro-differential equation. Wazwaz used Adomian decomposition method very extensively in his research [15–17]. The Taylor’s method discussed by the researchers in [10, 18, 19]

2.

Differential Transform Method

The Differential Transform Method (DTM) is a numerical method for solving differential as well as integral equations. The concept of the differential transform method was first proposed by Zho [20].This method construct, for differential equations, an analytical solution in the form of a polynomial. The transformation of the kth derivative of a function in one variable is as follows:   1 d k p(x) P(k) = (1) k! dxk x=x0 and the inverse transformation is defined by ∞

p(x) =

∑ P(k)(x − x0)k k=0

The following theorems that can deducted from (1) and (2) are given as: Theorem 1. 1. If u(x) = v(x) ± w(x),then U(k) = V (k) ±W (k). 2. If u(x) = cv(x), then U(k) = cV (k), where c is a constant. 3. If u(x) =

d n v(x) (k + n)! then U(k) = V (k + n). n dx k!

4. If u(x) = v(x)w(x), then U(k) = ∑kk1 =0 V (k1 )W(k − k1 ).  1, k=n; n 5. If u(x) = x then U(k) = δ(k − n) where, δ(k − n) = 0, k 6= n.

(2)

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V (k − 1) , where k ≥ 1. k R 1 7. If u(x) = v(x) xx0 h(t)dt then U(k) = ∑kk1 =1 V (k − k1 )H(k1 − 1), where k ≥ k1 1. 6. If u(x) =

Rx

x0 v(t)dt

then, U(k) =

Example 1. The Volterra integral equation u(iv) (x) = x(1 + ex ) + 3ex + u(x) −

Z x

u(t)dt

(3)

0

with boundary conditions u(0) = 1,

0

u (0) = 1,

u(1) = 1 + e,

0

u (1) = 2e. (4)

Solution We can obtained the another boundary condition (BCs) by using the equation (4) by putting x = 0, in equation (3), we get u(iv) (0) = 4 We know that the differential transform for the exponential function is ex is k!1 . Also, using the results in the theorem, we obtain U(k + 4) =

3 1 δ(k − 1) + ∑kk1 =0 δ(k1 − 1) (k−k + k! +U(k) − 1k U(k − 1) 1 )!

(k + 1)(k + 2)(k + 3)(k + 4)

(5)

After simplifying equation (5), we get U(k + 1) =

1 [k!δ(k − 1) + 3 + k + k!U(k) − (k − 1)!U(k − 1)] (6) (k + 4)!

By using equations (1), (4) and uiv (0) = 4 the following transformed BCs at x = 0 can be obtained 1 U(0) = 1, U(1) = 1, U(2) = a, U(3) = b, U(4) = (7) 6 00

u (0)

000

u (0)

where, according to equation (1) , a = 2! and b = 3! . Applying the recurrence relation in equation (5) and transformed the boundary conditions in equation (6), U(k) for k ≥ 5 are easily obtained and the using equation (2), we get the terms up to N = 10: x4 x5 1 a 6 + +( + )x 6 24 180 360 1 a b 7 11 b x9 +( − + )x + ( − )x8 + 840 2520 840 40320 6720 40320 1 a 10 11 +( + )x + O(x ) 453600 1814400

u(x) = 1 + x + ax2 + bx3 +

(8)

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The constants a and b are evaluated from boundary conditions given in equation (3), for x = 1, by taking N = 10, as follows: a = 0.999998,

b = 0.500003.

For N = 20, these values are obtained as a = 1.000000000000001,

1 2.

b = 0.4999999999999991.

It is clear that in the limit case N → ∞, a converges to 1 and b converges to With these values of a and b, equation (8) becomes: u(x) = 1 + x + x2 +

x3 x4 x5 x6 x7 x8 x9 x10 + + + + + + + + ... 2! 3! 4! 5! 6! 7! 8! 9!

(9)

which can be written in the closed from as follows: u(x) = 1 + xex . The Fig. 1 shows the graphical representation and its comparison with exact solution. 4 DTM Solution Exact Solution

3.5

3

2.5

2

1.5

1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 1. Graphical solution of boundary value problem (3).

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Example 2. In this example we will consider the following nonlinear integrodifferential equation of initial value type: 0

u (x) = −1 +

Z x

u2 (t)dt

(10)

0

with the initial boundary condition u(0) = 0.

(11)

The given initial value problem is a nonlinear in nature. At the point x = 0, equations (10) and (11) gives out the following boundary condition 0

u (0) = −1.

(12)

Applying theorem and equation (10), we get the following recurrence relation; ! 1 k−1 1 −δ(k) + ∑ U(k1 )U(k − k1 − 1) . (13) U(k + 1) = k+1 k k1 =0 δ(k) = 0 and equation (13) can be simplified to following form: ! k−1 1 U(k1 )U(k − k1 − 1) . (14) U(k + 1) = k(k + 1) k∑ 1 =0

As k ≥ 1,

The differential transformation of the initial boundary conditions in equations (11) and (12) are U(0) = 0 U(1) = −1.

(15)

By using equations (14) and (15) and (2), the following series solution is obtained: x4 x7 x10 x13 37x16 − + + + 12 252 6048 157248 158505984 25x19 73x22 449x25 + + − 3011613696 252975550464 45535599083520 16531x28 + − O(x31 ). 49724874199203840

u(x) = −x +

(16) (17) (18)

The authors El-Sayed et al. [10] a series solution was obtained for the integrodifferential equations (11) by using the ADM [15] which is the same with one we presented in equation (16) up to O(x13 ). However, higher ordered terms are slightly different from the ones we obtained as shown in the following Table 1.

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Table 1. Comparison of WGM, ADM, and DTM method X 0 0.0312 0.0625 0.0938 0.1562 0.2188 0.2812 0.3125 0.3750 0.4062 0.4688 0.5000 0.5938 0.6250 0.6875 0.7188 0.7512 0.8125 0.8750 0.9062 0.9688 1.0000

2.1.

WGM 0 -0.0312 0.0625 0.0937 0.1562 0.2186 0.2807 0.3117 0.3734 0.4040 0.4648 0.4948 0.5835 0.6124 0.6692 0.6969 0.7509 0.7771 0.8277 0.8520 0.8984 0.9205

ADM 0 -0.0311999 0.0624987 0.0937935 0.1561500 0.2186090 0.2806800 0.3117060 0.3733560 0.4039390 0.4647950 0.4948230 0.5835420 0.6124310 0.6691670 0.6969410 0.7508550 0.7770900 0.8276670 0.8519340 0.8984520 0.9204760

DTM 0 -0.0311999210 0.0624987284 0.0937935492 0.1561504020 0.2186091064 0.2806795005 0.3117064249 0.3733561800 0.4039385134 0.4647946327 0.4948225080 0.5835419298 0.6124306816 0.6691672304 0.6969414638 0.7251900069 0.7770901037 0.8276674429 0.8519341746 0.8984522567 0.9204757107

Fredholm Integro-Differential Equations

In this section we apply the differential transformed method for Fredholm integro-differential equations. Consider the Fredholm integro-differential equation 0

u (x) + f (x)u(x) +

Z b

k(x,t)u(t)dt = g(x),

(19)

a

with boundary conditions u(a) = u0 ,

(20)

where u(x) is an unknown function, the function f (x), g(x) and k(x,t) are given real functions of a, b and the equation (19) and (20) called initial boundary value problem for Fredholm integro-differential equations.

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One-dimensional differential transform Here we will define the basic one-dimensional differential transformation are introduced as follows: Definition 1. If f (x) is analytic in the time domain T, then ∂k f (x) ∀x ∈ T φ(x, k) = t=t , ∂xk i

(21)

where k belongs to the set of non-negative integer, denoted as the K-domain. Therefore, equation (21) can be written as   1 1 ∂k f (x) F(k) = φ(xi , k) = (22) t=ti , ∀k ∈ K, k! k! ∂xk

where F(k) is called the spectrum of f (x) at t = ti in the K-domain. If f (x) can be represented by Taylor’s series, expansion as follows: ∞

f (x) =

∑ (x − xi )k F(k) ≡ D−1 F(k).

(23)

k=0

The equation (23) is known as inverse of f (x) with the symbol D which denotes the differential transformation process. The particular case of equation (23) when xi = 0 is referred to as the Maclaurin series of f (x) and is expressed as ∞

f (x) =

∑ (x)k F(k) ≡ D−1F(k).

(24)

k=0

Using this transformation a differential equation in the domain of interest can be transformed to an algebraic equation in K- domain and f (x) can be obtained by finite term Taylor series expansion and remainder as ∞

f (x) =

∑ (x − xi )k F(k) + Rn+1(x).

(25)

k=0

Lemma 1. If w(t) = u1 (t)u2 (t)...un−1(t)un(t), then k

W (k) =

ln−1

∑ ∑

ln−1 =0 ln−2 =0

3

... ∑

2

∑ U1(l1)U2(l2 − l1 )...Un(k − ln−1 ).

l2=0 l1 =0

(26)

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133

The fundamental operations for one-dimensional DTM are represented in the following Table 2. Table 2. The Fundamental operations of one-dimensional DTM Time Function w(t) = αu(t) ± βv(t) ∂m u(t) w(t) = ∂t m w(t) = u(t)v(t) w(t) = t n

Transformed Function W (k) = αU(K) ± βV (k) (k+m)! k! U(k + m) k ∑l=0 U(l)V  (k − l)

W (k) = W (k) =

W (k) = δ(k − m) =

1, k =m; 0, k = 6 m.

k k! k W (k) = ωk! sin( kπ 2 + α) k W (k) = ωk! cos( kπ 2 + α)

w(t) = exp(t)

W (k) =

w(t) = sin(ωt + α) w(t) = cos(ωt + α)

Proof. By using Definition (21), we have W (0) =

1 [u1 (t), u2(t), ..., un−1(t), un(t)] t=t0 0!

=U1 (0)U2 (0)...Un−1(0)Un (0), W (1) =

1 ∂ [u1 (t), u2(t), ..., un−1(t), un(t)] t=t0 1! ∂t 0

0

=[u1 (t)u2 (t), ..., un−1(t), un(t) + u1 (t), u2(t), ..., un−1(t), un(t) + ... 0 0 = + u1 (t), u2(t), ..., un−1 (t), un(t) + u1 (t), u2(t), ..., un−1(t), un(t)] t=t0

=U1 (1)U2 (0)...Un−1(0)Un (0) +U1 (0)U2 (1)...Un−1(0)Un (0) + ... +U1 (0)U2 (0)...Un−1(0)Un (0) +U1 (0)U2 (0)...Un−1(0)Un (1), W (2) =U1 (1)U2 (1)U3 (0)...Un−1(0)Un (0) +U1 (0)U2 (1)U3 (1)... ×Un−2 (0)Un−1 (0)Un (0) + ... +U1(0)U2 (0)U3 (0)... ...Un−2(1)Un−1 (1)Un (0) +U1 (0)U2 (0)U3 (0)...Un−2(0)Un−1 (1)Un (1). In the general form we have k

W (k) =

ln−1

∑ ∑

ln−1 =0 ln−2 =0

3

...

2

∑ ∑ U1(l1)U2(l2 − l1)...Un(k − ln−1 ).

l2=0 l1 =0

(27)

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Gunvant A. Birajdar and N. Giribabu

Two-dimensional differential transform The two dimensional differential transformation is introduced as Definition 2. W (k, h) =

1 ∂k+h x=0,y=0 w(x, y) k h k!h ∂x ∂y

(28)

where ∞

∞

 k+h  1 ∂ W (x, y) = ∑ ∑ w(x, y) x=0,y=0, k h k=0 h=0 k!h ∂x ∂y

xk yh ≡ D−1 [W (k, h)]

(29)

Definition 3. If w(x, y) = u(x, y)v(x, y),u(x, y) = D−1 [U(k, h)], v(x, y) = D−1 [V (k, h)] and ⊗ denote the convolution, then D[w(x, y)] = D[u(x, y)v(x, y)] = U(k.h) ⊗V (k, h). In the following Table 3 the operations for two-dimensional DTM are presented. Table 3. The fundamental operations of two-dimensional DTM Time Function w(x, y) = αu(x, y) ± βv(x, y) w(x, y) =

∂a+b u(x,y) ∂xa yb

W (k, h) = (k + 1)(k + 2)....(k + a)(h + 1)(h + 2)... (a+b)U(k+a,h+b) W (k, h) = ∑ka=0 ∑hb=0 U(a, h − a)V(k − b, b) W (k, h) = δ(k − m, h − n) = δ(k − m)δ(h − n)

w(x,y) = u(x,y)v(x,y) w(x, y) = xm yn

2.2.

Transformed Function W (k, h) = αU(k, h) ±V (k, h)

Higher-Order Integro-Differential Equations

This section will focus on higher order integro-differential equations by using differential transformed method. Consider the integro-differential equation n

un (t) + ∑ f i (t)yn−i (t) + i=1

Z b

w(t, x)y(x)dx = g(t)

(30)

a

We convert a equation (30) of nth order integro-differential equation with the initial conditions

Mathematical Methods for Integro-Differential Equations ... 0

135

00

u (a) = α2 , u (a) = α3 ,...,un−1 (a) = αn ,

u(a) = α1 ,

into a system of first-order integro-differential equations, let y1 (t) = u(t), y2 (t) = 0 u (t), ..., yn(t) = un−1 (t). By using this notation, we obtain the first-order system 0

y1 (t) = y2 (t), 0

y2 (t) = y3 (t), 0

y3 (t) = y4 (t), . .

(31)

0

yn−1 (t) = yn (t) 0

n

yn (t) = g(t) − ∑ f i (t)yn+1−i (t) − i=1

Z b a

w(t, x)y1 (t)dt

with initial conditions y1 (a) = α1 , y2 (a) = α2 ,..., yn (t) = αn (t). Now, using the differential transformation of system (31) and using the relations, we get 1 Y2 (k), k+1 1 Y2 (k + 1) = Y3 (k), k+1 1 Y3 (k + 1) = Y4 (k), k+1 . . 1 Yn−1 (n − 1) = Yn (k), k+1 " # n h 1 Yn (k + 1) = G(k) − ∑ F1 (k) ⊗Yn+1−i (k) + (b − a) ∑ W (k, h − l)Y1(k) k+1 i=1 i=0 Y1 (k + 1) =

Using the initial conditions we have, Y1 (0) = α1 Y2 (0) = α2 , Y3 (0) = α3 , ..., Yn (0) = αn .

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2.3.

Numerical Approach of Differential Transform Method

Here, we developed the numerical version of differential transform method. In the first sub-domain y(t) can be described by y0 (t) can be expressed by using Taylor’s series expansion near a = x0 as n

y0 (t) =

∑ Y10 ( j)(x − a) j ,

j=0

where the subscript "0" denotes that the Taylor’s polynomial near a. y(t1 ) ∼ = y0 (t1 ) =

n

∑ Y10 ( j)(h) j ,

h = x1 − x0 .

j=0

The final value y0 (x1 ) = U11 (0) y(t2 ) ∼ = y1 (t2 ) =

n

∑ Y11 ( j)(h) j ,

h = x2 − x1 ,

j=0

where a = x1 . In the same manner, y(t3 ), can be represented as y(t3 ) ∼ = y2 (t3 ) =

n

∑ Y12 ( j)(h) j ,

h = x2 − x1 .

j=0

Hence, the solution on the grid points ti+1 can be obtained from y(ti+1 ) ∼ = yi (ti+1 ) =

n

∑ Y1i ( j)(h) j , h = xi+1 − xi , i = 0, 1, 2, ..., N − 1.

(32)

j=0

3.

Numerical Results

As an application of differential transform method, we present the couple problems are given below. Example 3. Consider the integro-differential equations t y (t) = 1 − + 3 u(0) = 0. 0

Z 1

xty(x)dx,

(33)

0

(34)

Let h = 0.1 and m = 10. The integro-differential equation of the system between ti and ti+1 can be present as 0

y (t ∗ ) = 1 −

ti t ∗ − + 3 3

Z 1 0

(y∗ + yi )xy(x)dx,

(35)

Mathematical Methods for Integro-Differential Equations ...

137

where t ∗ = t − ti . Taking the differential transformation, we obtain  ti  1 (k +1)Yi (k +1) = 1 − δ(k)− δ(k −1)+ti δ(k −1)⊗Y (k)+δ(k −1, h −1)⊗Y (h) 3 3 Set k = h and by using tables, we obtain " # k 1 ti 1 Yi (k + 1) = (1 − )δ(k) − δ(k − 1) + ti ∑ Yi ( j)δ(k − j − 1) (k + 1) 3 3 j=0 k

+ ∑ Yi ( j)δ(k − j − 1)δ(k − 1).

(36)

j=0

With Y0 (0) = 0, the approximation of y(t) at the grid points can be obtained. The following table shows the comparison of numerical solution yi and exact solution y(ti ). The following Table 4 exhibits the error analysis. Table 4. Represent the solution with its error t 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.9 1.0

yi 0.0 0.09833333 0.19390611 0.28679821 0.37708594 0.46484216 0.55013645 0.71360162 0.79189609 0.86797601

y(ti ) 0.0 0.10000000 0.20000000 0.30000000 0.40000000 0.50000000 0.60000000 0.80000000 0.90000000 1.00000000

ky(ti ) − yi k 0.0 1.66666667e-03 6.09388620e-03 1.32017875e-02 2.29140636e-02 3.51578404e-02 6.69648304e-02 8.63983845e-02 1.08103910e-01 1.32023989e-01

In the following Fig. 2 we have given the graphical representation of solution of boundary value problem (33).

138

Gunvant A. Birajdar and N. Giribabu 1 0.9 0.8 0.7 0.6 0.5 DTM Solution Exact Solution

0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2. Graphical solution of boundary value problem. (33) Example 4. Consider the integro-differential equation 0

y (t) = tet + et − t +

Z t

ty(t)dt

y(0) = 0.

(37)

0

having exact solution y(t) = tet . Let h = 0.1 and n = 10. The integro-differential equation of the system between ti and ti+1 can be represented as 0

y (t ∗ ) = (x∗ + ti )et

∗ +t

i

− (t ∗ + ti ) +

Z 1 0

t ∗ + ti y(x)dx,

(38)

where t ∗ = t − ti . applying the differential transformation of equation (39), we get " # k 1 1 Yi (k + 1) = (ti + 1)eti − ti δ(k) − δ(k − 1) + tiYi (k) + ∑ Yi ( j)δ(k − j − 1) (k + 1) k! j=0 k

+∑

j=0

1 δ(k − j − 1). j!

(39)

By using Y0 (0) = 0, the approximation of y(t) on the grid points can be obtained. The following Table 5 exhibits the errors between the exact solution and solution obtained by DTM.

Mathematical Methods for Integro-Differential Equations ...

139

Table 5. Represent the solution along with its error. t 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

yi 0.0 0.00000000 0.10050526 0.21641542 0.35408455 0.52119424 0.72717178 0.98371957 1.30549366 1.71098147 2.22364623 2.87342954

y(ti ) 0.0 0.000000000 0.11051709 0.24428055 0.40495764 0.59672988 0.82436064 1.09327128 1.40962690 1.78043274 2.21364280 2.71828183

ky(ti ) − yi k 0.0 0.00000000 1.00118319e-02 2.78651355e-02 5.08730892e-02 7.55356316e-02 9.71888592e-02 1.09551714e-01 1.04133232e-01 0.94512700e-02 1.00034260e-02 1.55147712e-01

The following Fig. 3 shows the solution obtained by differential transform method. 3 DTM Solution Exact Solution

2.5

2

1.5

1

0.5

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3. Graphical solution of boundary value problem (37).

4.

The Adomain Decomposition Method

Consider a Volterra intergro-differential equation of the second kind given by g00 (x) = h(x) +

Z x 0

k(x, y)g(y)dy, g(0) = g0 , g0 (0) = g1 → (1)

140

Gunvant A. Birajdar and N. Giribabu

Integrating w.r.t. 0 x0 from 0 to x, we get  Z x Z x Z x 0 0 g (x) − g (0) = h(y) dy + k(x, y)g(y)dy dx 0

0

0

0 0

Again integrating w.r.t. x from 0 to x, we get g(x) − g0 − g1 x =

RxRx 0 0

⇒ g(x) = g0 + g1 x +

h(y) dydy +

RxRx 0 0

RxRx Rx 0 0

h(x) dxdx +

[

0

k(x, y)g(y)dy] dx dx.

RxRx Rx 0 0

[

0

k(x, y)g(y)dy] dx dx. → (2)

Now using the decomposition series, ∞

g(x) =

∑ gn(x) = g0(x) + g1 (x) + g2(x) + ....

n=0

Substituting g(x) as a series in (2), we get ∑∞ n=0 gnR(x) R= g0 +Rg1Rx +R 0x 0x h(x)dxdx+ x x x ∞ 0 0 [ 0 k(x, y) (∑n=0 gn (y)) dy] dx dx ⇒ g0 (x) + g1 (x) + g2 (x) + ... = g0 + g1 x + 0x 0x h(x)dx d+  Z x Z x Z x [k(x, y)(g0 (y) + g1 (y) + g2 (y) + ....)]dy dx dx. R R

0

0

0

Z xZ x

...

h(x)dxdx. ⇒ g0 (x) = g0 + g1 (x) + 0 0   Z xZ x Z x k(x, y) g0 (y)dy dx dx. g1 (x) = 0 0 0  Z x Z x Z x k(x, y) g1 (y)dy dx dx. g2 (x) = 0

0

0

The components g0 (x), g1 (x), g2 (x), .... of the solution g(x) =

∞

∑ gn(x) is

n=0

determined from the following recurrence relations. Z xZ x

h(x)dxdx. g0 (x) = g0 + g1 (x) + 0 0  Z x Z x Z x k(x, y) gn (y)dy dx dx, n = 0, 1, 2, 3, ... gn+1 (x) = 0

0

0

Example 5. Consider the following Voltera integro- differential equation

Mathematical Methods for Integro-Differential Equations ... Z x

g00 (x) = 1 +

141

(x + t)g(t) dt, g(0) = 1, g0 (0) = 2.

0

By comparing it with the standard form of Volterra integro-differential equation. g00 (x)

= h(x) +

Z x 0

h(x)

K(x, y) g(y))dy, g(0) = g0 , g0 (0) = g1 , we get

= 1, k(x, t) = x + t, g0 = 1, g1 = 2.

(40) (41)

∞

Let g(x) =

∑ gn(x) be a solution of (1).

n=0

Let us find g0 (x), g1 (x), ...bn by Adomain-Decomposition method. g0 (x)

= =

g0 + g1 (x) + 1 + 2x + 1 + 2x +

h(x) dx dx

Z x Z 0x 0

l dx dx = 1 + 2x +

0

=

Z xZ x

Z x

0

x = dx = 1 + 2x +

0

Z x 0

(x)x0 dx

x2 . 2

We obtain the recurrence formula as fallow  Z x Z x Z x gn+1 (x) = K(x, y)gn (y)dy dxdx, n = 0, 1, 2, ... 0 0 0    Z x Z x Z x y2 g1 (x) = (x + y) 1 + 2y + dy dxdx, 2 0 0 0 =

x4 x5 7 6 + + x , 8 24 720

and so on. So the solution in series form is g(x) = 1 + 2x +

5.

x2 x4 1 7 6 + + x5 + x + ... 2 8 24 720

The Laplace Decomposition Method

Consider the nonlinear integro- differential equation of the type U 0 (x) = f (x) +

Z x 0

K(t,U(t),U 0(t)) dt, → (1)

U(x0 ) = α, 0 ≤ x ≤ 1, f (x) is the source term, K(t,U(t),U 0(t)) is a linear or non-linear

142

Gunvant A. Birajdar and N. Giribabu

function. Applying the Laplace transform to (1), we get Z x   0 0 L U = L{f }+L K(t,U(t),U (t))dt . 0  Z x ⇒ s L {U} −U(0) = F(s) + L K(t,U(t),U 0(t))dt . Z x 0  ¯ − α = F(s) + L ⇒ s U(s) K(t,U(t),U 0(t))dt . 0

In the Laplce decomposition method, we assume the solution as an infinite series as follows: ∞

U=

∑ Un , where the terms Un are to be determined recursively.

n=0

Also the nonlinear term K(t,U(t),U 0(t)) is decomposed as an infinite series of Adomain polynomials as follows: ∞

K(t,U(t),U 0(t)) =

∑ An, where An = An (U0 ,U1,U2...,Un) are to be determined

n=0

by recursive relation: " " 1 dn K An = n! d λn

n

∑ Ui

λ

i=0

i

!## Z

.

λ=0 x

0

Then S L {U} = α + F(s) + L K(t,U(t),U (t))dt 0 "Z " # # ∞ x α 1 1 L {U} = + F(s) + L ∑ An dt s s s 0 n=0 ( ) Z ∞ α 1 1 x ∞ ⇒ L ∑ Un = + F(s) + ∑ L[An ]dt s s s 0 n=0 n=0 Define the following iterative algorithm: α + L { f (x)} ; s Z x  1 L {U1 } = + L A0 dt . s 0

L {U0 } =

In general,



becomes

Mathematical Methods for Integro-Differential Equations ... 1 L {Un+1 } = L s

Z

0

x

143



An dt , n ≥ 1.

with these, the components U0 ,U1,U2 ..., are identified and the series solution is entirely determined. In many cases, the exact solution in closed form may also be obtained. From numerical point ! of view, the approximation n−1

∑ Uk (x)

U(x) = lim

n→∞

5.1.

.

k=0

Laplace Transform Method

Consider a Volterra -integro differential equation of the form g(n) (x) = h(x) + λ

Z x

K(x − t) g(t) dt,

0

(n−1)

g(0) = g0 , g0 (0) = g1 , ...., g(0)

= gn−1 .

Applying Laplace transform, and using the given initial conditions, find L{g(x)} in terms of s, by using convolution theorem. Then by inverse Laplace transform, find g(x). Example 6. ZConsider the following Volterra integro- differential equation x

U 0 (x) = x −

U(t)dt,U(0) = 0.

0

Applying Laplace transform, and using the initial condition U(0) = 0, we get

¯ −0 = S U(s)

1 1 ¯ − .U(s). s2 s



 1 1 ⇒ s+ U(s) = 2 s s ⇒

s2 + 1 1 1 U(s) = 2 ⇒ U(s) = 2 s s s(s + 1)

⇒ U(s) =

1 s − s s2 + 1

144

Gunvant A. Birajdar and N. Giribabu Applying inverse Laplace transform, we get U(x) = 1 − cos x, is the solution.

6.

The Variational Iteration Method

Consider the jth order integro -differential equation (i)

g(x) = h(x) + (i)

Where g(x) = ( j−1)

g(0)

Z x

K(x, y)g(y)dy,

0

d ig , g(0) = g0 , g0(0) = g1 , ....., dx j

= g( j−1), are the initial conditions.

The correction functional for the integro differential equation is   Z x Z ς (x) (i) gn+1 = gn (x) + λ(ς) gn (ς) − h(ς) − K(ς, r)F(gn (r))dr dς, 0

0

where λ is a Lagrange multiplier. For λ : g0

+

h(g(ς), g0(ς)) = 0, λ = −1.

g00

+

h(g(ς), g0(ς), g00 (ς)) = 0, λ = ς − x.

g000

+

g(n)

+

λ =

1 h(g(ς), g0(ς), g00 (ς), g000 (ς)) = 0, λ = − , (ς − x)2 . 2 h(g(ς), g0(ς), g00 (ς), ...., g(n)(ς)) = 0, 1 (ς − x)n−1 . (−1)n (n − 1)!

Example 7. Consider the Volterra integro-differential equation g0 (x) = −1 +

Z x

g(t)dt = 1.

0

The correction functional for this equation is given by  Z x Z (ς) (x) gn (r)dr dς gn+1 = gn (x) − g0n (ς) + 1 − 0

0

Mathematical Methods for Integro-Differential Equations ...

145

where λ = −1 for first order integro -differential equation. Use initial condition to select g0 (x) = g0 (0) = 1. Now the successive approximations : g0 (x)

=

1

g1 (x)

=

g0 (x) −

=

Z x

(0 + 1 −

Z x

(1 − ςdr)dς) = 1 − x +

1−

Z x 0

(g00 (ς) + 1 −

0

=

1−

Z ς

Z ς 0

g0 (r)dr)dς

dr)dς

0

0

g2 (x)

= =

= =

g1 (x) − 1−x+

Z x

0 2 x

2

−

= =

Z x

Z ς 0

g1 (r)dr)dς

(−1 + ς + 1 −

0

Z ς

(1 − r +

0

r2 )dr)dς 2

x x2 ς2 ς3 − (ς − (ς − + )dς 2 2 6 0 x2 x3 x4 1−x+ − + −−−−−− 2 6 24 Z Z

Z

1−x+

x

g3 (x)

(g01 (ς) + 1 −

x2 2

g2 (x) −

1−x+

0 x2

2

(g01 (ς) + 1 −

−

ς

0

g2 (r)dr)dς

x3 x4 + − ...... 3! 4!

and so on. Now by VIM, U(x) = lim Un (x) = e−x is the exact solution. n→oo

Conclusion The DTM, ADM, LDM and Variational iteration methods are successfully applied to find the solutions of linear as well as nonlinear integro-differential equations. The efficiency and accuracy of the proposed methods are demonstrated by test problems. These methods may be successfully used to solve other linear as well as nonlinear integrodifferential equations.

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Gunvant A. Birajdar and N. Giribabu

References [1] A. Avudainayagam, C. Vani, Wavelet-Galerkin method for integro-differential equations, Appl. Numer. Math. 32 (2000) 247-254. [2] M. T. Rashed, Lagrange interpolation to compute the numerical solutions of differential,integral and integro-differential equations, Appl. Math. Comput. 151(3) (2004) 869-878. [3] S. M. Hosseini, S. Shahmorad, Tau numerical solution of Fredholm integrodifferential equations with arbitrary polynomial bases, Appl. Math. Model. 27 (2003) 145-154. [4] R. P. Kanwal, K. C. Liu, A Taylor expansion approach for solving integral equations, Int. J. Edu. Sci. Technol. 3 (1989) 411-414. [5] K. Maleknejad, Y. Mahmoudi, Taylor polynomial solution of high-order nonlinear Volterra-Fredholm integro-differential equations, Appl. Math. Comput. 145 (2003) 641-653. [6] K. Maleknejad, F. Mirzaee, S. Abbasbandy, Solving linear integro-differential equations system by using rationalized Haar functions method, Appl. Math. Comput. 155 (2), (2004) 317-328. [7] I. H. Abdel-Halim Hassan, Differential transformation technique for solving higher-order initial value problems, Appl. Math. Comput. 154 (2004)299-311. [8] C. K. Chen, S. H. Ho, Solving partial differential equations by two-dimensional differential transform method, Appl. Math. Comput. 106 (1999) 171-179. [9] P. Darania, Ali Ebadian, A method for the numerical solution of the integrodifferential equations, Appl. Math. Comput. 188 (1), (2007), 657-668. [10] S. M. El-Sayed, M. R. Abdel-Aziz, A Comparison of Adomian decompositon method and Wavelet-Galerkin method for solving integro-differential equations, Appl. Math. Comput. 136 (2003) 151-159. [11] M. J. Jang, C. L. Chen, Y. C. Liu, Two-dimensional differential transform for partial differential equations, Appl. Math. Comput. 121 (2001) 261-271. [12] G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, San Diego, (1986). [13] G. Adomian, Solving Frontier Problems of Physics, The Decomposition Method, Kluwer, Boston, (1994). [14] G. Adomian and R. Rach, Noise terms in decomposition series solution, Comput. Math. Appl., 24 (1992) 61-64.

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[15] A.-M. Wazwaz, A reliable algorithm for solving boundary value problems for higher-order integro-differential equations, Appl. Math. Comput. 118 (2001) 327-342. [16] A. M. Wazwaz, A First Course in Integral Equations, World Scientific, Singapore, (1997), 212. [17] A. M. Wazwaz, The variational iteration method; a reliable tool for solving linear and nonlinear wave equations, Comput. Math. Appl., 54 (20007) 926-932. [18] M. Sezer, Taylor polynomial solution of Volterra integral equations, Int. J. Math. Educ. Sci. Technol. 5 (1994) 625-633. [19] S. Yalcinbas, Taylor polynomial solution of nonlinear Volterra-Fredholm integral equations, Appl. Math. Comput. 127 (2002) 195-206. [20] J. K. Zhou, Differential Transformation and Its Application for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986.

In: Understanding Integro-Differential Equations ISBN: 979-8-89113-040-1 c 2023 Nova Science Publishers, Inc. Editors: J. Vasundhara Devi et al.

Chapter 5

O PERATIONAL M ATRICES FOR S OLVING F RACTIONAL O RDER I NTEGRAL AND I NTEGRO -D IFFERENTIAL E QUATIONS H. Tajadodi1 R. M. Ganji2 S. M. Narsale3 and H. Jafari4∗ 1 Department of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran 2 Department of Applied Mathematics, University of Mazandaran, Babolsar, Iran 3 Symbiosis Institute of Technology, Symbiosis International (Deemed University), Pune, India 4 Department of Mathematical Sciences, University of South Africa, South Africa

Abstract In this chapter a numerical method is presented to obtain numerical solutions of fractional-order differential equations, as well as fractional ∗ Corresponding Author’s

Email: [email protected]

150

H. Tajadodi, R. M. Ganji, S. M. Narsale et al. integro-differential equations. In this method, operational matrices are computed and used to convert the main equations into a system of algebraic equations. After solving the algebraic equations, the values are substituted in the approximate function to obtain a solution of the governing equation. The operational matrices are obtained by using Legendre polynomials as basic functions. Further the existence, uniqueness, stability and convergence of solutions are discussed.

Keywords: fractional-order differential equations, operational matrices, fractional integro-differential equations, Legendre polynomials AMS Subject Classification: 53D, 37C, 65P

1.

Introduction

During the past three decades fractional calculus had many applications in the area of physics and engineering. The applications of fractional calculus used in many fields such as electrical networks, control theory of dynamical systems, probability and statistics, electrochemistry of corrosion, chemical physics, optics and signal processing can be successfully modelled by linear or nonlinear fractional differential equations (FDEs). Finding approximate or exact solutions of FDEs is an important task. Except for a limited number of these equations, we have difficulty in finding their analytical solutions. Therefore, there have been attempts to develop new methods for obtaining numerical solutions which reasonably approximate the exact solutions. The operational matrix method using orthogonal functions has been considered important to deal with various problems. The main feature of this technique is that it reduces these equations into a system of algebraic equations and thus greatly simplifies the problem. In this method, a truncated orthogonal series is used for numerical integration of differential equations, with the goal of obtaining efficient computational solutions. A lot of work has been done using shifted Legendre polynomials in the literature. In this chapter, we have mainly focused on the application of Legendre polynomials to create operational matrices for solving a class of fractional-order differential equations and fractional integro-differential equations.

Operational Matrices for Solving Fractional Order Integral ...

151

The chapter is organized as follows: In Section 2., we have introduced some necessary definitions and mathematical preliminaries of the fractional calculus theory. In Section 3. definition and properties of Legendre polynomials are discussed. In Section 4. the Legendre polynomials are used to approximate the function. Section 5. operational matrices of the fractional derivative and integration are constructed. In Section 6. the proposed method is applied to several examples. Section 7. consists of existence, uniqueness, stability and convergence of the method. Also a conclusion is given in Section 7.3..

2.

Fractional Calculus

In this section, several common definitions and properties of fractional integrals and derivatives will be introduced.

2.1.

Riemann-Liouville Fractional Integral and Derivative

The Riemann-Liouville fractional order integral and derivative are presented in the following definitions [1–3]. Definition 1. Suppose a is a positive real number and the function f (t) is defined on the interval [a, b]. The α-th order fractional integral of the function f (t) is defined as RL α It f (t) =

1 Γ(α)

Z t a

(t − τ)α−1 f (τ)dτ,

where t ∈ [a, b] and Γ(x) is the Gamma function defined by Γ(x) =

Z ∞ 0

e−t t x−1 dt,

Re(x) > 0.

Some of the important properties of the Riemann-Liouville fractional integral are given as: 1.

β α RL α RL β It It f (t) = RLIt RLIt f (t),

2.

α+β RL α RL β It It f (t) = RLIt f (t),

3.

RL α It (λ f (t) + µg(t)) =

4.

RLI αC t

=

C t α, Γ(α+1)

α

α

λ RLI t f (t) + µ RLIt g(t),

C is a constant,

152

H. Tajadodi, R. M. Ganji, S. M. Narsale et al.

5.

RLI α (t − a) p t

=

Γ(p+1) (t − a)α+p , Γ(α+p+1)

Re(p) > −1.

Definition 2. Suppose a is a positive real number, n−1 ≤ α < n with n a positive integer and the function f (t) is defined on the interval [a, b]. The α-th order Riemann-Liouville (RL) fractional derivative of function f (t) is defined as Z  1 dn t f (τ) d n RL n−α RL α Dt f (t) = dτ = I f (t) . t Γ(n − α) dt n a (t − τ)α−n+1 dt n In particular if n = 1, then the Riemann-Liouville fractional derivative of order α ∈ (0, 1) is defined as: Z  d t f (τ) d RL 1−α 1 RL α dτ = I f (t) . Dt f (t) = t Γ(1 − α) dt a (t − τ)α dt A direct calculation gives RL α Dt (t − a) p

=

Γ(p + 1) (t − a) p−α, Re(p) > −1. Γ(p − α + 1)

It can be proved that 1.

d m RL α Dt f (t) = RLDtm+α f (t), dt m

2.

RL α RL α Dt It

2.2.

α > 0, m > 0, m ∈ Z+ ,

f (t) = f (t).

Caputo Fractional Derivative

Although the Riemann-Liouville definition of fractional derivatives seems to play an important role in the development of fractional calculus, several authors including Caputo (1967, 1969) [1–3] realized that the Riemann-Liouville definition needs revision because the applied problems in viscoelaticity, solid mechanics and in rheology require physically interpretable initial conditions such 00 as f (a), f 0 (a), f (a). Definition 3. Suppose a is a positive real number, n−1 ≤ α < n with n a positive integer and the function f (t) is defined on the interval [a, b]. The α-th order Caputo (C) fractional derivative of function f (t) is defined as C α Dt f (t) =

where t ∈ [a, b].

1 Γ(n − α)

Z t a

f (n) (τ) dτ, (t − τ)α−n+1

Operational Matrices for Solving Fractional Order Integral ... It is clear that C α Dt f (t) = RLItn−α

A direct calculation gives C α Dt (t − a) p

=



153

 f (n) (t) .

Γ(p + 1) (t − a) p−α , Re(p) > n − 1 ≥ 0. Γ(p − α + 1)

It can be proved that 1. CDtα C = 0,

C is a constant,

2. limα→n CDtα f (t) = f (n)(t),
m 3. CDtα dtd m f (t) = CDtm+α f (t),

n − 1 < α < n, m ∈ N,

f (k)(0+) (t − a)k−α, Γ(k − α + 1) k=0

n−1

4. CDtα f (t) = RLDtα f (t) − ∑

n−1

5.

RL α C α It Dt f (t) =

f (t) − ∑

k=0

3.

f (k)(0+) (t − a)k . k!

Legendre Polynomials and Related Results

Polynomials play the most important role in spectral methods, so it is necessary to have a thorough study of their relevant properties. The well-known Legendre polynomials Li (x), i = 0, 1, · · · , are the eigenfunctions of the singular SturmLiouville problem. Consider the Sturm-Liouville problem in the interval (−1, 1) as
0 − pu0 + qu = λwu,

the coefficients p, q and w are three given, real-valued functions such that: p is continuously differentiable, strictly positive in (−1, 1) and continuous at x = ±1; q is continuous, nonnegative and bounded in (−1, 1); the weight function w is continuous, nonnegative and integrable over (−1, 1). Now let us suppose p(x) = 1 − x2 , q(x) = 0 and w(x) = 1, then

0 1 − x2 L0i (x) + i(i + 1)Li (x) = 0.

154

H. Tajadodi, R. M. Ganji, S. M. Narsale et al.

Li (x) is even if i is even and odd if i is odd. If Li (x) is normalized so that Li (1) = 1, then for any i:    [i] 1 2 2i − 2` i−2` ` i Li (x) = i ∑ (−1) x , 2 `=0 ` i   where 2i denotes the integral part of 2i . The Legendre polynomials satisfy the recursion relation [4]: Li+1 (x) =

i 2i + 1 x Li (x) − Li−1 (x), i+1 i+1

i = 1, 2, 3, · · · ,

where L0 (x) = 1 and L1 (x) = x. Relevant properties are 1. |Li (x)| ≤ 1,

−1 ≤ x ≤ 1,

2. Li (±1) = (±1)i , 3. |L0i (x)| ≤ 12 i(i + 1),

−1 ≤ x ≤ 1,

4. L0i (±1) = (±1)i+1 21 i(i + 1), 5.

R1

2 −1 Li (x)dx =

i + 12


−1

.

In order to use these polynomials on the interval t ∈ [0, 1] we define the so-called shifted Legendre polynomials by introducing the change of variable x = 2t − 1. Let the shifted Legendre polynomials Li (2t − 1) be denoted by L∗i (t). Then L∗i (t) can be obtained as follows: L∗i (t) = Li (2t − 1),

i = 0, 1, 2, · · · .

The analytic form of the shifted Legendre polynomial of degree i is defined by L∗i (t) =

i

∑ li,k t k ,

(1)

k=0

where li,k =

(−1)i+k(i + k)! . (i − k)!(k!)2

(2)

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155

The orthogonality property is satisfied for these polynomials with respect to the weight function w(t) = 1, as follows  Z 1  1 , m = n, w(t) L∗m (t) L∗n (t) dt = 2m + 1 0, 0 m 6= n.

4.

Approximation of Functions in Terms of Legendre Polynomials

Any arbitrary function y ∈ L2 (0, 1) can be expanded in terms of the Legendre polynomials, L∗i (t), described in the previous section by the following infinite series [5]: ∞

y(t) = ∑ yi L∗i (t),

(3)

i=0

where yi =

hy(t), L∗i (t)i 1 = ∗ ∗ ∗ hLi (t), Li (t)i hLi (t), L∗i (t)i

Z 1 0

y(τ) L∗i (τ) dτ.

(4)

By taking only the first M + 1 terms in (3), y(t) can be approximated as M

y(t) ' yM (t) = ∑ yi L∗i (t) = Y T ϕ(t), i=0

where ϕ(t) = [L∗0 (t), L∗1 (t), · · · , L∗M (t)]T ,

(5)

and Y = [y0 , y1 , · · · , yM ]T , which Y can be calculated by Y =Q

−1

Z 1

y(τ)ϕ(τ) dτ,

0

with Q=

Z 1 0

ϕ(τ)ϕT (τ)dτ.

(6)

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H. Tajadodi, R. M. Ganji, S. M. Narsale et al. The basis vector ϕ(t) can be written in terms of the Taylor basis functions

as ϕ(t) = A TM (t), where TM (t) = [1,t, · · · ,t M ]T , is the Taylor basis vector and A = [ai, j ] , i, j = 0, 1, · · · , M, with ( li, j , i ≥ j, ai, j = 0, i < j, where li, j is given by (2). Similarly, we can expand a function y(x,t) in L2 ((0, 1) × (0, 1)) in terms of the Legendre polynomials as ∞

∞

y(x,t) = ∑ ∑ yi, j L∗i (x) L∗j (t).

(7)

i=0 j=0

By considering only the first (M + 1) × (M + 1) terms in (7), y(x,t) can be approximated as M M

y(x,t) ' ∑ ∑ yi, j L∗i (x) L∗j (t) = ϕT (x)Y ϕ(t), i=0 j=0

where Y is an (M + 1) × (M + 1) matrix and is given by Y = Q−1 hϕ(x), hy(x,t), ϕ(t)iiQ−1 .

5.

(8)

Operational Matrices Based on Legendre Polynomials

In this section, we obtain the operational matrices of the fractional integral and the product of basis function based on the shifted Legendre Polynomials.

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5.1.

157

The Operational Matrix of Riemann-Liouville Fractional Integral

Let ϕ(t) be the shifted Legendre polynomials vector given by (5), then the Riemann-Liouville fractional integral of ϕ(t) can be approximated as: RL α I t ϕ(t) ' IαRL ϕ(t),

where IαRL is the operational matrix of Riemann-Liouville fractional integral. To evaluate this matrix, first we apply the Riemann-Liouville fractional integral, RLI α , onto L∗ (t), i = 0, 1, · · · , M, as t i !   i i i Γ(k + 1)li,k k+α α RL α ∗ RL α k t . I t Li (t) = I t ∑ li,kt = ∑ li,k RLIt t k = ∑ Γ(k + α + 1) k=0 k=0 k=0 By approximating the function t k+α in terms of the the shifted Legendre polynomials, we have M

t k+α '

∑ dk,sL∗s (t),

(9)

s=0

where dk,s can be calculated by (4). In view of (9) and for i = 0, 1, · · · , M, we get ! ! i M M i Γ(k + 1)li,k Γ(k + 1)li,k dk,s RL α ∗ ∗ I t Li (t) ' ∑ ∑ dk,sLs (t) = ∑ ∑ Γ(k + α + 1) L∗s (t) Γ(k + α + 1) k=0 s=0 s=0 k=0 ! M

=∑

i

s=0

∑ σi,s,k

L∗s (t).

k=0

Therefore, for i = 0, 1, · · · , M, we can write

RL α I t ϕ(t) ' IαRL ϕ(t),

where



σ0,0,0

 1   ∑ σ1,0,k  k=0 α IRL =  ..  .  M  ∑ σM,0,k k=0

σ0,1,0 1

∑ σ1,1,k

k=0

.. .

M

∑ σM,1,k k=0

··· ··· ··· ···

σ0,M,0



 σ ∑ 1,M,k  k=0 , ..  .   M  ∑ σM,M,k 1

k=0

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with

Γ(k + 1)li,k dk,s . Γ(k + α + 1)

σi,s,k =

5.2.

The Operational Matrix of the Product

Suppose that C = [c0 , c1 , · · · , cM ]T is an arbitrary (M +1)×1 matrix, then Cb is an (M + 1) × (M + 1) operational matrix of product based on the shifted Legendre polynomials whenever b ϕ(t) ϕT (t)C ' Cϕ(t).

To evaluate the product of ϕ(t) and ϕT (t), that will give the operational matrix of the product based on the shifted Legendre basis, first, we write i+ j

L∗i (t) L∗j (t) =

∑ ar L∗r (t).

(10)

r=0

The coefficients ar , r = 0, 1, . . ., i + j, are computed in the following manner. Multiplying both sides of equation (10) by L∗m (t), m = 0, 1, · · · , i + j, and integrating the result yields Z 1 0

i+ j

L∗i (τ) L∗j (τ) L∗m(τ) dt =

∑ ar

r=0

Z 1 0

L∗r (τ) L∗m (τ)dτ =

1 am . 2m + 1

(11)

Using (11) and the analytic form of the shifted Legendre polynomial given by (1), we obtain am = (2m + 1)

Z 1 0 i

L∗i (τ) L∗j (τ) L∗m (τ) dτ j

m

= (2m + 1) ∑ ∑ ∑ li,k l j,ι lm,s k=0 ι=0 s=0

Z 1

τk+ι+s dτ

0

= (2m + 1)∆i, j,m , where i

∆i, j,m =

j

m

li,k l j,ι lm,s

∑ ∑ ∑ k+ι+s+1 .

k=0 l=0 s=0

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159

By substituting am into equation (10), we have i+ j

L∗i (t) L∗j (t) =

∑ (2m + 1)∆i, j,m .

(12)

m=0

then Eq. (12) can be written as

where Cb is given by where

6.

Applications

6.1.

b ϕ(t) ϕT (t)C ' Cϕ(t), Cb = [b ci, j ],

i, j = 0, 1, · · · , M, M

cbi, j = (2 j + 1)

∑ ∆i, j,m cm .

m=0

Solving Fractional Differential Equation

Consider the inhomogeneous Bagley-Torvik equation as a multi-order fractional differential equations (FDEs) [6–8] 3

y00 (t) + C Dt2 y(t) + y(t) = 1 + t, y(0) = 1,

0 < t < 1,

y0 (t) = 1.

For solving Bagley-Torvik equation using the operational matrix, first we apply the integral operator, I 2 , twice on both sides of this equation. It gives the following relation ! 3 1 1 2 2 t t t2 t3 2

y(t) − (1 + t) + RLIt2 y(t) − + + I y(t) = + . 2 6 Γ 23 Γ 52 This implies that 1

y(t) + RLIt2 y(t) + I 2 y(t) =

1

3

t2 t3 t2 t2
+
. + +1 +t + 3 2 6 Γ 2 Γ 25

(13)

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Now, we approximate y(t) and the right side of the above equation by the Legendre basis as follows: M

y(t) ' ∑ yi L∗i (t) = Y T ϕ(t),

(14)

i=0

where Y T = [y0 , y1 , · · · , yM ],

ϕ(t) = [L∗0 (t), L∗1(t), · · · , L∗M (t)]T ,

and 1

3

t2 t2 t2 t3
' F T ϕ(t),
+ + +1 +t + 3 2 6 Γ 2 Γ 52

(15)

where F can be calculated by (6) as follows: F T ' Q−1

Z 1 0

1

3

τ2 τ3 τ2 τ2

+ + +1+τ+ 3 2 6 Γ 2 Γ 25

!

ϕ(τ) dτ.

1

In view of (13), we need to approximate I 2 y(t) and I 2 y(t). To do that, we use the operational matrix of the integral. Applying this matrix on (14) yields:   1 1 1 1
RLI 2 y(t) ' RLI 2 Y T ϕ(t) = Y T RLI 2 ϕ(t) = Y T I 2 ϕ(t), t t t RL {z } | 1

2 ϕ(t) IRL







I 2 y(t) ' I 2 Y T ϕ(t) = Y T I 2 ϕ(t) = Y T I2RL ϕ(t). | {z } I2RL ϕ(t)

Substituting (14), (15) and (16) into (13), yields 1

2 Y T ϕ(t) +Y T IRL ϕ(t) +Y T I2RL ϕ(t) = F T ϕ(t).

Consequently, we can write 1

2 +Y T I2RL = F T . Y T +Y T IRL

(16)

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161

Finally, T

Y =F

T



1 2

I + IRL + I2RL

−1

,

(17)

where I is an (M + 1) × (M + 1) identity matrix. By solving (17), we can calculate Y . Then, the approximation y(t) can be computed using (14). Now, let us suppose in (14), M = 2, then we calculate the following results T Y = [y 0 , y1 , y2 ] ,  1 0 0 Q = 0 31 0  , 0 0 1  4 5 √ 3 π − √ IRL =  154 π − 1054√ π 1 2

F=



4 √ 5 π 4 √ 7 π 4 − 15√ π

− 214√ π 4 √ 9 π 100 √ 231 π

ϕ(t) = [1, 1 − 6t + 6t 2 ]T ,  −1 + 2t,  1 0 0 Q−1 = 0 3 0 , 0 0 5  1  1 1



 ,

41 28 33 52 1 4 + √ , + √ , − √ 24 15 π 40 35 π 8 63 π

6

T

1 I2RL = − 12 1 60

4

1 − 10 0

12

(18)

0 , 1 − 42

,

then, according to (18), Y is calculated as   3 1 T Y = , ,0 , 2 2 when put into (14) gives the exact solution (y(t) = 1 + t).

6.2.

Solving Fractional Integro-Differential Equation

Consider the following fractional integro-differential equation (FIDE) :  √ R R 8t 3 4 t 17t 3t 2 5t 3 7t 4 C 12 Dt y(t) = 01 t τ y(τ) dτ + 0t (t + τ) y(τ) dτ + √ − + √2 − − − , 12 2 3 12 π 3 π  y(0) = 1,

(19)

For solving this equation using the operational matrix, first we approximate 1 2

CD t

y(t) as 1

M

Dt2 y(t) ' ∑ yi L∗i (t) = Y T ϕ(t).

C

(20)

i=0

1

By taking the Riemann-Liouville integral, RLIt2 , of (20) and using the initial condition, we have 1

2 y(t) ' Y T IRL ϕ(t) + 1.

162

H. Tajadodi, R. M. Ganji, S. M. Narsale et al. Approximating 1 ' Y0T ϕ(t), where Y0 can be calculated by (6), yields y(t) ' γ ϕ(t),

(21)

1 2

where γ = Y T IRL +Y0T . Now, approximating t τ and t + τ as ( t τ ' ϕT (t) K1 ϕ(τ), t + τ ' ϕT (t) K2 ϕ(τ),

(22)

where K1 and K2 can be computed by (8), and using (21), the integral parts are approximated by Z 1 0

t τ y(τ) dτ '

Z 1 0

ϕT (t) K1 ϕ(τ) ϕT (τ) γT dτ  

= ϕT (t) K1 T

Z 1

ϕ(τ) ϕT (τ) dτγT | {z } 0

Q

T

= ϕ (t) K1Qγ , Z t 0

(t + τ) y(τ) dτ '

Z t 0

(23)

ϕT (t) K2 ϕ(τ) ϕT (τ) γT dτ

= ϕ (t) K2

Z t

= ϕT (t) K2

Z t

T

0

0

= ϕT (t) K2γbT

ϕ(τ) ϕT (τ) γT dτ | {z } γbT ϕ(τ)

γbT ϕ(τ) dτ

Z t

ϕ(τ) dτ | 0 {z } I1RL ϕ(t)

= ϕT (t) K2γbT I1RL ϕ(t). (24) √ 8t 3 3t 2 5t 3 4 t 17t Now, it is enough to approximate the term √ − + √2 − − − 12 3 π 2 3 π 7t 4 . Therefore, 12

√ 8t 3 4 t 17t 3t 2 5t 3 7t 4 √ − + √2 − − − = F T ϕ(t). 12 3 π 2 3 12 π

(25)

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Substituting (20) and (23)-(25) in equation (19), we have Y T ϕ(t) = ϕT (t) K1QγT + ϕT (t) K2γbT I1RL ϕ(t) + F T ϕ(t).

(26)

For solving (26), there are many methods, such as, Collocation Method, Standard Least Squares Method or other methods. By computing Y and substituting into (21), the approximation y(t) can be computed. Now, let us suppose in (20), M = 2, then we calculate the following results Y = [y0 , y1 , y2 ]T ,   1 0 0 Q = 0 31 0  , 0 0 51 1 1  4 4 0 K1 =  14 14 0 , 0 0 0   1 1 0 2 2 1 I1RL = − 16 0 6 , 1 0 − 10 0

ϕ(t) = [1, −1 + 2t, 1 − 6t + 6t 2 ]T ,   2 √0 − 4y√1 − 4y√ 1 + 34y π 15 π 105 π  4y  4y 4y γ =  5√0π + 7√1π − 15√2π  , √1 + 100y √2 − 4y√0 + 94y π 231 π  21 1π  1 2 0 K2 =  21 0 0 , 0 40 0  4 √ √ − 214√π 3 π 5 π 1   4 4 2 √ √ IRL =  − 154√π 7 π 9 π , 100 √ − 1054√π − 154√π 231 π

h    iT 56 104 8√ 293 1 √ √ F = − 209 + 105 − 3 − 360 5 − , 120√+ 15 π 6 π 315 π  105 π+140y −28y −4y 4(21y0 +15y1−7y2)  γbT =  

0 1 √ 105 π 4(21y0+15y −7y 1 2) √ 315 π 4(−33y0+77y √1+75y2) 3465 π

2

√ 105 π √ 3465 π+4356y0−308y 1+468y2 √ 3465 π 8(21y0 +15y √ 1−7y2) 525 π

 4(−33y0+77y √ 1+75y2) 693 π  8(21y0+15y , √ 1−7y2 )  315 π √ 24255 π+31020y√ 0−3388y1+207 24255 π

then, according to (26) and using the above results, the residual function is written as R (t, y0, y1 , y2 ) = √ √ 1 √ −44000 + 9240 π − 464640t − 42735 πt + 52800t 2 − 884y2 69300 π
√ √ − 5280t 2y2 + 242550 πt 2 + 132 −9 + 525 π − 272t − 840t 2 − 420t 3 y0
√ + 220 −13 + 112t + 216t 2 − 180t 3 + 315 π(−1 + 2t) y1 + 18480t 3y2 √
√ √ + 69300 πy2 + 3008ty2 − 415800 πty2 + 415800 πt 2 y2 .

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` Finally, substituting the collocation points t` = M+2 , ` = 1, 2, · · · , M +1, into the residual function and solving the obtained system, the unknown parameters are computed as

y0 = 2.11973, y1 = 1.62850, y2 = −0.06854. Therefore, putting these parameters into (21), the approximate solution is calculated as y2 (t) ' 1.02277 + 2.00127t + 0.98287t 2. With a similar process, and setting M = 3, 4 and 5, the approximate solutions are calculated as y3 (t) ' 1.00358 + 1.92942t + 1.19507t 2 − 0.141377t 3,

y4 (t) ' 1.00303 + 1.93623t + 1.29922t 2 − 0.481707t 3 + 0.248103t 4,

y5 (t) ' 1.00201 + 1.93709t + 1.45666t 2 − 1.25618t 3 + 1.45187t 4 − 0.594764t 5.

7.

Existence, Uniqueness, Stability and Convergence Results

Let us consider a general form of the fractional integro-differential equation as follows: C α Dt y(t) =

λ1

Z 1 0

K1 (t, τ) Ψ1 (τ, y(τ)) dτ + λ2

Z t 0

K2 (t, τ) Ψ2 (τ, y(τ)) dτ + f (t, y(t)), (27)

with initial condition y(0) = y0 ,

(28)

where 0 < α ≤ 1, the parameters λ1 and λ2 are real constants, K1 , K2 , Ψ1 , Ψ2 and f are given known functions. In this section, we prove the existence and uniqueness of a solution to problem (27)–(28). Furthermore, Ulam-Hyers stability of this problem is discussed. Finally, the convergence of the approximate solution is considered.

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7.1.

165

Existence and Uniqueness

In the following theorem, we suggest some conditions to have a unique solution for problem (27)–(28). Theorem 1. Suppose that CDtα y ∈ C[0, 1] and K1 , K2 ∈ C([0, 1] × [0, 1]). Let Ψ1 , Ψ2 , f ∈ C ([0, 1] × R) satisfy the Lipschitz conditions |Ψ1 (t, y1 ) − Ψ1 (t, y2 )| ≤ L1 |y1 − y2 |,

|Ψ2 (t, y1 ) − Ψ2 (t, y2 )| ≤ L2 |y1 − y2 |,

| f (t, y1) − f (t, y2 )| ≤ L3 |y1 − y2 |. Define I1 y(t) =

Z 1

K1 (t, τ) Ψ1 (τ, y(τ)) dτ,

I2 y(t) =

Z t

K2 (t, τ) Ψ2 (τ, y(τ)) dτ,

0

0

M1 = max |K1 (t, τ)|, 0≤t,τ≤1

M2 = max |K2 (t, τ)|. 0≤t,τ≤1

5 (|λ1 | L1 M1 + |λ2 | L2 M2 + L3 ) < 1, then, 4 problem (27)–(28) has a unique solution y ∈ C[0, 1]. If λ1 I1 y(0) + f (0, y(0)) = 0 and

Proof. First, we prove that y(t) is a solution of problem (27)–(28) if and only if it is a solution of the following integral equation α

y(t) = y0 + RLI t (λ1 I1 y(t) + λ2 I2 y(t) + f (t, y(t))).

(29)

To this aim, let y(t) be a solution of problem (27)–(28). By applying the Riemann-Liouville integral operator of order α to both sides of equation (27), we get α y(t) − y(0) = RLIt (λ1 I1 y(t) + λ2 I2 y(t) + f (t, y(t))). Employing y(0) = y0 in the above equation leads to (29). On the other hand, suppose that y is a solution of (29), then, by applying the Caputo derivative operator of order α to both sides of (29), we obtain equation (27). Furthermore, by setting t = 0 in (29) and using the condition λ1 I1 y(0) + f (0, y(0)) = 0, we get the initial condition y(0) = y0 . Therefore, y is a solution of problem

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(27)–(28). Now, we prove that (29) has a unique solution. To do this, we define the operator A : C[0, 1] → C[0, 1] by α

Ay(t) = y0 + RLIt (λ1 I1 y(t) + λ2 I2 y(t) + f (t, y(t))). Then, finding a solution of problem (27)–(28) in C[0, 1] is equivalent to finding a unique fixed point of the operator A. For y, z ∈ C[0, 1], we have

α α |Ay(t) − Az(t)| = RLIt (λ1 I1 y(t) + λ2 I2 y(t) + f (t,y(t)))− RLIt (λ1 I1 z(t) + λ2 I2 z(t) + f (t,z(t)))   α ≤ |λ1 I1 (y − z)(t) + λ2 I2 (y − z) (t) + f (t,y(t)) − f (t,z(t))| RLIt 1 . (30)

Also, using the Lipschitz condition for the operators I1 and I2 and taking into account the maximum of K1 and K2 , we have |I1 (y − z) (t)| ≤ L1 M1 |y − z| ,

(31)

|I2 (y − z) (t)| ≤ L2 M2 |y − z| .

By utilizing (31) and the Lipschitz condition for the function f , (30) yields |Ay(t) − Az(t)| ≤ (|λ1 | L1 M1 + |λ2 |L2 M2 + L3 ) Since 0 < α ≤ 1, we get

tα |y − z| . Γ(α + 1)

tα 5 ≤ for t ∈ [0, 1] [9]. Therefore, we Γ(α + 1) 4

obtain |Ay(t) − Az(t)| ≤

5 (|λ1 |L1 M1 + |λ2 | L2 M2 + L3 ) |y(t) − z(t)| < |y(t) − z(t)|, 4

which yields kAy − Azk < ky − zk, where k · k is the max-norm in the space C[0, 1]. The above inequality implies that the operator A is a contraction. Consequently, it has a unique fixed point by the Banach fixed-point theorem.

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7.2.

167

Ulam-Hyers Stability

Theorem 2. [10] Suppose that α > 0, h1 (t) is nonnegative, nondecreasing and locally integrable on a < t < b < ∞, h2 (t) ≤ K and y(t) is nonnegative and locally integrable on [a, b) with   α y(t) ≤ h1 (t) + h2 (t) RLIt y(t) , then,

y(t) ≤ h1 (t)Eα (h2 (t)(t − a)α ) ,

(32) ∞

where Eα is the Mittag-Leffler function defined by Eα (t) =

tk

∑ Γ(αk + 1) .

k=0

Remark 1. [11] If in Theorem 2, h1 (t) is not necessarily nondecreasing then the inequality (32) takes the most general form ! Z t ∞ (h2 (t))k y(t) ≤ h1 (t) + ∑ Γ(αk) (t − τ)αk−1 dτ, a ≤ t < b. a k=1 Definition 4. Equation (27) is said to be Ulam-Hyers stable if for each ε > 0 and ∀z(t) satisfying the inequality C α Dt z(t) − (λ1 I1 z(t) + λ2 I2 z(t) + f (t, z(t))) < ε, (33)

there exists a solution y(t) of equation (27) satisfying |z(t) − y(t)| < dε,

d ∈ R.

Theorem 3. According to Theorem 1, equation (27) is Ulam-Hyers stable. Proof. If z(t) satisfies (33), there exists a function g(t) satisfying |g(t)| < ε such that C α Dt z(t) − (λ1 I1 z(t) + λ2 I2 z(t) + f (t, z(t))) = g(t), which is equivalent to α

α

z(t) − z(0) − RLI t (λ1 I1 z(t) + λ2 I2 z(t) + f (t, z(t))) = RLI t g(t). Therefore, we have α

α

|z(t) − z(0) − RLIt (λ1 I1 z(t) + λ2 I2 z(t) + f (t, z(t)))| = |RLIt g(t)| ≤ |g(t)|



RL α It 1



5 ≤ ε. 4

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Now, let y(t) be the solution of equation (27) satisfying y(0) = z(0) = y0 . Then, we have α y(t) = z(0) + RLIt (λ1 I1 y(t) + λ2 I2 y(t) + f (t, y(t))). Note that the existence and uniqueness of y(t) is assured by Theorem 1. We have |z(t) − y(t)| α = z(t) − z(0) − RLIt (λ1 I1 y(t) + λ2 I2 y(t) + f (t, y(t))) α = z(t) − z(0) − RLIt (λ1 I1 z(t) + λ2 I2 z(t) + f (t, z(t)))

α α +RLIt (λ1 I1 z(t) + λ2 I2 z(t) + f (t, z(t)))− RLIt (λ1 I1 y(t) + λ2 I2 y(t) + f (t, y(t)))   5 α ≤ ε + (|λ1 |L1 M1 + |λ2 |L2 M2 + L3 ) RLIt |z(t) − y(t)| . 4 Now, using the Gronwall’s inequality with h1 (t) =

|λ2 |L2 M2 + L3 , we get

5 ε and h2 (t) = |λ1 |L1 M1 + 4

5 |z(t) − y(t)| ≤ εEα ((|λ1 |L1 M1 + |λ2 |L2 M2 + L3 )t α ) 4 5 ≤ εEα (|λ1 |L1 M1 + |λ2 |L2 M2 + L3 ) , 4 which gives |z(t) − y(t)| ≤ dε, with 5 d = Eα (|λ1 |L1 M1 + |λ2|L2 M2 + L3 ). 4 Hence, equation (27) is Ulam-Hyers stable.

7.3.

Convergence Analysis

Here, we consider the convergence of the approximate solution obtained by the proposed scheme in the Section 6. to the analytical solution of problem (27)– (28). To do this, we suppose that (a, b) is a bounded interval in the real line (R) and H m (a, b) (m is an integer number greater than or equal to zero) is the following vector space n o H m (a, b) = y : (a, b) −→ R : for 0 ≤ j ≤ m, y( j) (t) ∈ L2 (a, b) ,

Operational Matrices for Solving Fractional Order Integral ...

169

where y( j) denotes the j-th order derivative of the function y. The Sobolev space H m (a, b) is a Hilbert space equipped with the following norm

∑ ky( j) k2L (a,b)

kykH m (a,b) =

2

j=0

!1

2

m

.

To continue the error discussion, the following lemma from [4] is recalled. lemma 1 (See [4]). Suppose that y ∈ H m (−1, 1) with m ≥ 0. Let PM (y) = ∑M i=0 yi Li (t) be the truncated Legendre series of y. Then, ky − PM (y)kL2 (−1,1) ≤ CM −m |y|H m;M (−1,1), where m

|y|H m;M (−1,1) =

∑ j=min{m,M+1}

!1

2

ky( j) k2L2 (−1,1)

,

and C is a positive constant independent of function y and integer M. lemma 2. Let y : (0, 1) −→ R be a function in H m (0, 1). Suppose that function y¯ : (−1, 1) −→ R is defined by y(t) ¯ = y( 12 (t + 1)) for all t ∈ (−1, 1). Then, for 0≤ j≤m 1 ky¯( j) kL2 (−1,1) = 2 2 − j ky( j) kL2 (0,1). Proof. Using the change of variable τ = 21 (t + 1), we have ky¯( j) k2L2 (−1,1) =

Z 1

|y¯( j) (t)|2dt   2 Z 1 ( j) 1 = y (t + 1) dt 2 −1 Z 1 2 = 2−2 j y( j) (τ) 2dτ −1

0

= 21−2 j ky( j) k2L2 (0,1),

Then, taking the square root from its both sides of this equation, gives the desired result.

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H. Tajadodi, R. M. Ganji, S. M. Narsale et al. M

Theorem 4. Suppose m ≥ 0 and y ∈ H m (0, 1). Let yM (t) = ∑ yi L∗i (t) be the i=0

approximate solution obtained by the operational matrix method in the Section 6.. Then, ky − yM kL2 (0,1) ≤ CM −m |y|H m;M;0 (0,1), where

!1  2k 2 1 (k) 2 ky kL2 (0,1) . ∑ 2 k=min{m,M+1} m

|y|H m;M;0 (0,1) =

Proof. With the help Lemma 2, we obtain 1 ky − yM k2L2 (0,1) = ky − PM (y)k2L2 (−1,1) 2 m 1 0 −2m ≤ CM ∑ ky¯(k)k2L2 (−1,1) 2 k=min{m,M+1}  2k m 1 0 −2m =C M ky(k)k2L2 (0,1). ∑ 2 k=min{m,M+1} By definition ! 12  2k 1 ky(k)k2L2 (0,1) , ∑ 2 k=min{m,M+1} m

|y|H m;M;0 (0,1) = which completes the proof.

Theorem 5. Let yM (t) be the approximate solution of problem (27)–(28) obtained by the operational matrix method in Section 6., y(t) be its analytical solution and RM (t) be the residual error for the approximate solution. Also, the conditions of Theorem 1 are satisfied. Then, if y(t) satisfies the conditions of Theorem 4, RM (t) tends to zero when M → ∞. α

Proof. By applying the Riemann-Liouville integral, RLIt , to both sides of equation (27), we can rewrite equation (27) as follows: α

y(t) = y0 + RLI t (λ1 I1 y(t) + λ2 I2 y(t) + f (t, y(t))).

Operational Matrices for Solving Fractional Order Integral ... where I1 y(t) =

Z 1

K1 (t, τ) Ψ1 (τ, y(τ)) dτ,

I2 y(t) =

Z t

K2 (t, τ) Ψ2 (τ, y(τ)) dτ.

0

0

So, yM (t) satisfies the following equation α

yM (t) = y0 + RLI t (λ1 I1 yM (t) + λ2 I2 yM (t) + f (t, yM (t))) + RM (t), where RM (t) is the residual function given by RM (t) = yM (t) − y(t)+ α

+RLIt (λ1 I1 (y(t) − yM (t)) + λ2 I2 (y(t) − yM (t)) + f (t, y(t)) − f (t, yM(t))) .

Then, kRM (t)k =

α

yM (t) − y(t) + RLIt (λ1 I1 (y(t) − yM (t)) + λ2 I2 (y(t) − yM (t)) + f (t, y(t)) − f (t, yM (t))) .

According to Theorem 1, we have   5 kRM (t)k ≤ 1 + (|λ1 | K1 L1 + |λ2 |K2 L2 + L3 ) kyM (t) − y(t)k. 4 Now applying Theorem 4, yields   5 −m kRM (t)k ≤ CM 1 + (|λ1 |K1 L1 + |λ2 | K2 L2 + L3 ) |y|H m;M;0 (0,1). 4 Therefore, it is clear that RM (t) tends to zero when M → ∞.

171

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H. Tajadodi, R. M. Ganji, S. M. Narsale et al.

Conclusion A numerical method using operational matrices based on the shifted Legendre polynomials has been used to obtain numerical solutions of fractional-order differential equations, as well as fractional integro-differential equations. The main problems are converted into a system of algebraic equations by using operational matrices. After solving the algebraic equations, the values are substituted in the approximate function to obtain a solution of the governing equation. The operational matrices could be obtained with other basic functions. Finally, stability and convergence of the given method are discussed.

References [1] K. S. Miller, B. Ross (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York. [2] I. Podlubny (1999): Fractional Differential Equations. Academic Press, San Diego. [3] S. G. Samko, A. A. Kilbas and O. I. Marichev (1993): Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon. [4] C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang (2006): Spectral Methods. Scientific Computation, Springer-Verlag, Berlin. [5] E. Kreyszig (1989): Introductory Functional Analysis with Applications. John Wiley and Sons. [6] R. L. Bagley, P. J. Torvik (1983): Fractional calculus: a different approach to the analysis of viscoelastically damped structures, AIAA J., 21(5), 741–748. [7] H. Jafari (2016): Numerical Solution of Time-Fractional Klein–Gordon Equation by Using the Decomposition Methods. ASME. J. Comput. Nonlinear Dynam., 11(4). [8] A. Saadatmandi, M. Dehghan (2010): A new operational matrix for solving fractional-order differential equations. Computers and Mathematics with Applications, 59, 1326–1336.

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[9] R. M. Ganji, H. Jafari and S. Nemati, A new approach for solving integrodifferential equations of variable order, Journal of Computational and Applied Mathematics, 379 (2020), 112940. [10] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications, 328(2) (2007), 1075–1081. [11] Y. Adjabi, F. Jarad and T. Abdeljawadc, On generalized fractional operators and a Gronwall type inequality with applications, Filomat, 31(17) (2017), 5457–5473.

In: Understanding Integro-Differential Equations ISBN: 979-8-89113-040-1 c 2023 Nova Science Publishers, Inc. Editors: J. Vasundhara Devi et al.

Chapter 6

O N A S ECOND O RDER N ONLINEAR I NTEGRO -D IFFERENTIAL E QUATION OF F REDHOLM T YPE IN A C OMPLEX P LANE Lemita Samir1 ,∗ Selim Raja2 ,† and Belahbib Zeineb2,‡ 1 Ecole normale sup´ erieure de Ouargla, Algeria; Laboratoire de Math´ematiques Appliqu´ees et de Mod´elisation, Universit´e 8 Mai 1945 Guelma, Guelma, Algeria 2 Ecole normale sup´ erieure de Ouargla, Algeria

Abstract In this paper, we deal with the study of a nonlinear Fredholm integrodifferential equation of second order in the complex plane. At first, under appropriate conditions, the existence and uniqueness of the solution of the proposed equation have been demonstrated by using the Banach fixed point theorem. After that, we have resorted to construct a numerical process based on the combination between Nystr¨om and Picard iterative methods in order to obtain an approximate solution of our equation. Our study has been supported by illustrative examples to confirm the efficiency of the used numerical process. ∗ Corresponding Author’s

Emails: [email protected]; [email protected]. Email: [email protected]. ‡ Corresponding Author’s Email: [email protected]. † Corresponding Author’s

176

Lemita Samir, Selim Raja and Belahbib Zeineb

Keywords: nonlinear Fredholm equation, integro-differential equation, fixed point theorem, Nystr¨om method, Picard iterative method, complex plane MSC (2010): 45B05, 47G20, 30D05, 37C25

1.

Introduction

Integral and integro-differential equations in complex analysis are considered to be an important branch of mathematics. They are used for modeling fundamental problems and have applications in diverse scientific fields, such as, electrical engineering, heat and mass transfer, elasticity, oscillation, fluid dynamics, control, etc [1, 2]. For this reason, various articles have discussed linear, nonlinear, integral and integro-differential equations in the complex plane [3, 4, 5, 6], and have chosen appropriate techniques to obtain solutions. In our work, we propose a different form of Fredholm integral equation. This equation is a second order nonlinear integro-differential in the complex plane, where all derivatives (first and second) of the unknown function appearing inside of the nonlinear integral operator. Consider the following proposed equation: Z b

ψ (s) = g(s) +

k(s, r, ψ (r), ψ 0(r), ψ 00(r)) dr,

∀s ∈ [a, b],

(1)

a

where, g(s) ∈ C2 ([a, b], C) and k ∈ C2 ([a, b]2 × C3 , C) are some given complex-valued continuously differentiable functions, and ψ (s) is the unknown function in C2 ([a, b], C). Our work is concerned, first, with the treatment of existence and uniqueness of solutions. Then, we present a numerical process to approach this solution. Furthermore, we prove the convergence of the suggested numerical process and give some illustrative examples.

2.

Preliminaries

We introduce some notions and concepts, which we will use in the next section. First, we differentiate twice both sides of the proposed equation (1) to get more information about the unknown function ψ (s) and obtain the following system

On a Second Order Nonlinear Integro-Differential Equation ...

177

in the complex plane:

ψ (s) = g(s) +

Z b

k(s, r, ψ (r), ψ 0(r), ψ 00(r)) dr,

(2)

a

ψ 0 (s) = g0 (s) +

Z b ∂k

(s, r, ψ (r), ψ 0(r), ψ 00(r)) dr, ∂s Z b 2 ∂ k (s, r, ψ (r), ψ 0(r), ψ 00(r)) dr. ψ 00 (s) = g00 (s) + 2 a ∂s

(3)

a

(4)

For a complex variable z = x + iy, any complex-valued function f (z) can be represented as:

f (z) = f (x + iy) = f 1 (x, y) + i f 2 (x, y), (real and imaginary parts).

This allows us to give the following representations:

ψ (s) = x(s) + iy(s), g(s) = g1 (s) + ig2(s), ψ 0 (s) = x0 (s) + iy0 (s), g0 (s) = g01 (s) + ig02 (s), ψ 00 (x) = x00 (s) + iy00 (s), g00 (s) = g001 (s) + ig002 (s),

where, x(s), y(s), g1(s) and g2 (s) ∈ C2 ([a, b], R). Also, we obtain: k(s, r, ψ , ψ 0 , ψ 00 ) = ∂k (s, r, ψ , ψ 0 , ψ 00 ) = ∂s ∂ 2k (s, r, ψ , ψ 0 , ψ 00 ) = ∂ s2

φ1 (s, r, x, y, x0, y0 , x00 , y00 ) + iφ2 (s, r, x, y, x0, y0 , x00 , y00 ), ∂ φ1 ∂ φ2 (s, r, x, y, x0, y0 , x00 , y00 ) + i (s, r, x, y, x0, y0 , x00 , y00 ), ∂s ∂s ∂ 2 φ1 ∂ 2 φ2 0 0 00 00 (s, r, x, y, x , y , x , y ) + i (s, r, x, y, x0, y0 , x00 , y00 ), ∂ s2 ∂ s2

178

Lemita Samir, Selim Raja and Belahbib Zeineb

where, φ1 , φ2 ∈ C2 ([a, b]2 × R6 , R). As well as, system of equations (2)-(4), can be transformed as follows: x(s) = g1 (s) + y(s) = g2 (s) + x0 (s) = g01 (s) +

Z b a

Z b a

φ1 (s, r, x(r), y(r), x0(r), y0(r), x00 (r), y00(r)) dr,

(5)

φ2 (s, r, x(r), y(r), x0(r), y0(r), x00 (r), y00(r)) dr,

(6)

Z b ∂ φ1

(s, r, x(r), y(r), x0(r), y0(r), x00 (r), y00(r)) dr, (7) ∂s Z b ∂ φ2 (s, r, x(r), y(r), x0(r), y0(r), x00 (r), y00(r)) dr, (8) y0 (s) = g02 (s) + a ∂s Z b 2 ∂ φ1 00 00 x (s) = g1 (s) + (s, r, x(r), y(r),x0(r), y0(r), x00(r), y00(r)) dr, (9) 2 ∂ s a Z b 2 ∂ φ2 y00 (s) = g002 (s) + (s, r, x(r), y(r),x0(r), y0(r), x00(r), y00(r)) dr. (10) 2 a ∂s a

However, to rewrite the previous system in a simple formula, we consider the Banach product space P = ∏6i=1 C([a, b], R) equipped with the following norm: 6

∀ V = (v1 , v2 , v3 , v4 , v5 , v6 ) ∈ P, kV kP = ∑ kvi k∞. i=1

Then, our system of equations (5)-(10) can be reformulated as: ∀s ∈ [a, b], ∀ V = (x(s), y(s), x0(s), y0(s), x00(s), y00(s)) ∈ P, V = TG (V ), where, the functional TG is defined from P into itself by: TG : P −→ P
V 7−→ TG (V) = TG1 (V ), TG2 (V ), TG0 1 (V ), TG0 2 (V ), TG001 (V ), TG002 (V ) ,

where,

TG1 (V ) = g1 (s) + TG2 (V ) = g2 (s) +

Z b a

Z b a

φ1 (s, r, x(r), y(r),x0(r), y0(r), x00(r), y00(r)) dr, φ2 (s, r, x(r), y(r),x0(r), y0(r), x00(r), y00(r)) dr,

TG0 1 (V ), TG0 2 (V ), TG001 (V ), TG002 (V )

and respectively.

are the derivatives of TG1 (V ) and TG2 (V ),

On a Second Order Nonlinear Integro-Differential Equation ...

3.

179

Existence and Uniqueness

The current section shows us how to take the conditions that are ensuring the existence and uniqueness of solutions of system (5)-(10). At first, we give the following necessary definitions and hypothesis: Definition 1. (6-Lipschitzian function) We say that the function K(s, r, x1, x2 , x3 , x4 , x5 , x6 ) is 6 − Lipschitzian with constants l1 , l2 , l3 , l4 , l5 , l6 ∈ R+ , if it satisfies the following condition: ∀s, r ∈ [a, b], ∀xi, yi ∈ R, i = 1 : 6, then: 6

|K(s, r, x1, x2 , x3 , x4 , x5 , x6 ) − K(s, r, y1 , y2 , y3 , y4 , y5 , y6 )| ≤ ∑ li |xi − yi |. i=1

For short, we write: K is 6 − Lipsch. Hypothesis H : 2 2 We suppose that kernels: φ1 , φ2 , ∂∂φs1 , ∂∂φs1 , ∂∂ sφ21 and ∂∂ sφ22 are 6 − Lipsch with constants l2i+1 , l2i+2, m2i+1 , m2i+2 ,t2i+1,t2i+2 for i = 0 : 5, respectively. Moreover, let θ be a positive parameter that verifies: 2

max max

6

8

10

12

∑ li , ∑ li , ∑ li , ∑ li , ∑ li , ∑ li

i=1 2

4

4

i=3

6

i=7

i=5 8

i=9

10

i=11

12

!

!

1 (b − a) ≤ θ < , 3

1 (b − a) ≤ θ < , 3 i=1 i=3 i=7 i=9 i=11 i=5 ! 2 4 6 8 10 12 1 max ∑ ti , ∑ ti , ∑ ti , ∑ ti , ∑ ti, ∑ ti (b − a) ≤ θ < . 3 i=1 i=3 i=5 i=7 i=9 i=11

∑ mi , ∑ mi , ∑ mi , ∑ mi , ∑ mi , ∑ mi

Theorem 3..1. Let hypothesis (H ) be satisfied. Then, the functional TG is a contraction from P into itself. As well as, the system V = TG (V ) has a one solution V ∈ P.

180

Lemita Samir, Selim Raja and Belahbib Zeineb

Proof. For V1 = (x1 , y1 , x01 , y01 , x001 , y001 ),V2 = (x2 , y2 , x02 , y02 , x002 , y002 ) ∈ P, we have: 2

∑

|

TGi (V1 ) − TGi (V2 )|

≤

Z b

i=1

a

Z b

+

a

Z b

≤

a

Z b

+

a

≤

|φ1 (s,r,x1 ,y1 ,x01 ,y01 ,x001 ,y001 ) − φ1 (s,r,x2,y2 ,x02 ,y02 ,x002 ,y002 )|dr |φ2 (s,r,x1 ,y1 ,x01 ,y01 ,x001 ,y001 ) − φ2 (s,r,x2,y2 ,x02 ,y02 ,x002 ,y002 )|dr, (l1 |x1 − x2 | + l3 |y1 − y2 | + l5 |x01 − x02 | + l7 |y01 − y02 | + l9 |x001 − x002 | + l11 |y001 − y002 |) dr (l2 |x1 − x2 | + l4 |y1 − y2 | + l6 |x01 − x02 | + l8 |y01 − y02 | + l10 |x001 − x002 | + l12 |y001 − y002 |)dr, !

+

(b − a)

≤

max

2

4

6

i=1

i=3

i=5

8

10

∑

∑

∑ (li )kx1 − x2k∞ + ∑ (li )ky1 − y2k∞ + ∑ (li )kx01 − x02k∞

(b − a)

12 (li )ky001 − y002 k∞ (li )ky01 − y02 k∞ + (li )kx001 − x002 k∞ + i=9 i=11 i=7 2

4

6

8

∑

10

12

∑ li , ∑ li , ∑ li , ∑ li , ∑ li , i=1

i=3

i=5

i=7

i=9

∑ i=11

!

!

,

li (b − a)kV1 −V2 kP ,

θ kV1 −V2 kP .

≤

Similarly, we get: 2

∑

|

TG0 i (V1 ) − TG0 i (V2 )|

i=1 2

≤ max

4

6

8

10

12

∑ mi , ∑ mi , ∑ mi , ∑ mi , ∑ mi , ∑ mi

i=1

i=3

i=5

i=7

i=9

8

10

i=11

!

(b − a)kV1 −V2 kP ,

≤ θ kV1 −V2 kP . Also, 2

∑

|

TG00i (V1 ) − TG00i (V2 )|

i=1 2

≤ max

4

6

12

∑ t i , ∑ t i , ∑ t i , ∑ mi , ∑ t i , ∑ t i

i=1

i=3

≤ θ kV1 −V2 kP .

i=5

i=7

i=9

i=11

!

(b − a)kV1 −V2 kP ,

On a Second Order Nonlinear Integro-Differential Equation ...

181

Thus, kTG (V1) − TG (V2)kP ≤ 3θ kV1 −V2 kP .

(11)

Since 0 ≤ θ < 31 and by applying Banach fixed point theorem, the functional TG has a one fixed point V ∈ P, which represents the solution of the system (5)-(10). In practice, the solution of the system V = TG (V ) cannot be found exactly. For this reason, we must construct it according the Picard successive iterative method, that we will see in the next theorem. Theorem 3..2. Consider the sequences Vn = (xn , yn , x0n , y0n , x00n , y00n ) ∈ P defined by: for V0 ∈ P: Vn = TG (Vn−1),

n ≥ 1.

Then, Vn converges to V : Vn −→ V,

as n −→ ∞

in the space

P.

Further, we have the following estimate: kV −Vn kP ≤

(3θ )n kV1 −V0 kP . 1 − 3θ

Proof. First, the induction on n ∈ N for inequality (11) gives us: kTGn (V1) − TGn (V2 )kP ≤ (3θ )n kV1 −V2 kP .

(12)

Then, according to the definition of Vn and inequality (12), we have that: for V0 ∈ P: Vn = TGn (V0 ) −→ V,

as n −→ ∞

in the space

P.

However, we have: kV −Vn kP = kV − TGn (V0)kP ≤

∞

∑ (3θ )kkTG (V0) −V0 kP ,

k=n

≤

(3θ )n kV1 −V0 kP . 1 − 3θ

(13)

182

4.

Lemita Samir, Selim Raja and Belahbib Zeineb

Numerical Study

First, as we have seen above in Theorem 3..2, we have constructed a sequence Vn ∈ P, to approach solutions of the system (5)-(10) using Picard iterative method. However, we can see that it is very hard to calculate exactly each integral term in the previous iterated sequences. Therefore, we must apply the well-known Nystr¨om method [7], as a second approximate part, in order to approach all integral terms of the iterated sequences Vn . We describe that process in this section. Let N ∈ N∗ , consider the following equidistance partition δN :   b−a δN = si = a + ih, h = , i = 0, . . ., N . N According to the partition δN , we construct our numerical process based on the conjunction between Nystr¨om and Picard iterative methods. This numerical procedure permits us to give approximate solutions Ven = (e xn , yen , xe0n , ye0n , xe00n , ye00n ) ∈ P of the system (5)-(10), which are given by the following formulas: For all i = 0, 1, . . ., N, for n ≥ 1: N

xen (si) = g1 (si) + ∑ w j φ1 (si ,r j , xen−1 (r j ), yen−1(r j ), xe0n−1 (r j ), ye0n−1(r j ), xe00n−1(r j ), ye00n−1(r j )), j=0 N

yen (si) = g2 (si) + ∑ w j φ2 (si ,r j , xen−1 (r j ), yen−1(r j ), xe0n−1 (r j ), ye0n−1(r j ), xe00n−1(r j ), ye00n−1(r j )), j=0

N

xe0n (si) = g01 (si) + ∑ w j j=0 N

ye0n (si) = g02 (si) + ∑ w j j=0 N

xe00n (si) = g001 (si) + ∑ w j j=0 N

ye00n (si) = g002 (si) + ∑ w j j=0

∂ φ1 (si,r j , xen−1 (r j ), yen−1 (r j ), xe0n−1(r j ), ye0n−1(r j ), xe00n−1(r j ), ye00n−1(r j )), ∂s ∂ φ2 (si,r j , xen−1 (r j ), yen−1 (r j ), xe0n−1(r j ), ye0n−1(r j ), xe00n−1(r j ), ye00n−1(r j )), ∂s

∂ 2 φ1 (si,r j , xen−1 (r j ), yen−1 (r j ), xe0n−1(r j ), ye0n−1 (r j ), xe00n−1(r j ), ye00n−1 (r j )), ∂ s2 ∂ 2 φ2 (si,r j , xen−1 (r j ), yen−1(r j ), xe0n−1(r j ), ye0n−1 (r j ), xe00n−1(r j ), ye00n−1 (r j )), ∂ s2

 N where, w j j=0 are real weights of the used Nystr¨om method, such that: supN≥2 ∑Nj=0 w j < ∞.

The next theorem, confirms the convergence of the proposed numerical process.

On a Second Order Nonlinear Integro-Differential Equation ...

183

Theorem 4..1. Let hypothesis (H ) be satisfied. Then, the approximate soluen converge to V : tions V Ven −→ V,

as n −→ ∞ and N −→ ∞ in the space P.

Further, we have the following estimate: n (3θ )n kV1 −V0 kP + ∑ (3θ )n−p ε p , kV − Ven kP ≤ 1 − 3θ p=1

where, lim ε p = 0, p = 1, ..., n. N→∞

Proof. . By applying the triangle inequality twice and from (13), we obtain:

≤

kV − Ven kP ≤ kV −Vn kP + kVn − Ven kP , en−1)kP + kTG (V en−1) − V en kP . + kTG (Vn−1) − TG (V

(3θ )n 1−3θ kV1 −V0 kP

In the same way of inequality (11), we can write:

en−1)kP ≤ (3θ )kVn−1 − Ven−1 kP . kTG (Vn−1) − TG (V

(14)

en−1) − Ven kP ≤ εn , kTG(V

(15)

On the other hand, from [7], we have the following estimate:

where, εn represents the convergence’s order of the used Nystr¨om method. In addition, inequalities (14) and (15) give us: en kP ≤ (3θ )kVn−1 − Ven−1 kP + εn . kVn − V

Repeating the last inequality n-times, we get the requested estimate: kV − Ven kP ≤

n (3θ )n kV1 −V0 kP + ∑ (3θ )n−p ε p . 1 − 3θ p=1

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Lemita Samir, Selim Raja and Belahbib Zeineb

5.

Extended Study Overview

Consider the general form of the equation proposed above: ψ (s) = g(s) +

Z b

k(s, r, ψ (r), ψ 0 (r), . . ., ψ (m) (r)) dr,

∀s ∈ [a, b],

for m > 2 ∈ N,

a

where, ψ (s), g(s) ∈ Cm ([a, b], C) and k ∈ Cm ([a, b]2 × Cm+1 , C). In the same way that we have shown before we can get the following system: x(s) = g1 (s) + y(s) = g2 (s) + ...

Z b a

Z b a

... (m)

(m) g1 (s) +

φ1 (s, r, x(r), y(r), . .., x(m) (r), y(m)(r)) dr, φ2 (s, r, x(r), y(r), . .., x(m) (r), y(m)(r)) dr,

Z b m ∂ φ1

(s, r, x(r), y(r),. . ., x(m) (r), y(m)(r)) dr, m ∂ s a Z b m ∂ φ2 (m) y(m) (s) = g2 (s) + (s, r, x(r), y(r),. . ., x(m) (r), y(m)(r)) dr. m a ∂s

x

(s) =

However, the extended vision of Definition 1 and hypothesis (H ), which confirm the existence and uniqueness of solutions of this system will be taken as follows: m

Suppose that kernels: φ1 , φ2 , ∂∂φs1 , ∂∂φs2 , ..., ∂∂ sφm1 , with constants: j j l2i+1 , l2i+2

∂ m φ2 ∂ sm

are (2m + 2) − Lipsch

for i = 0 : 2m + 1, j = 1 : m + 1.

Moreover, choose the positive parameter θ as follows: for j = 1 : m + 1 as follows: ! 2 4 4m+2 4m+4 1 j j j j max ∑ li , ∑ li , . . ., ∑ li , ∑ li (b − a) ≤ θ < . m+1 i=1 i=3 i=4m+1 i=4m+3 The numerical process, based on Nystr¨om and Picard iterative methods, (m) (m) gives us an approximate solution Ven = (e xn , yen , xe0n , ye0n , ..., xen , yen ) of the gen-

On a Second Order Nonlinear Integro-Differential Equation ...

185

eralized previous system by: For all i = 0, 1, . . ., N, for n ≥ 1: N

xn (si ) e

yn (si ) e

=

(m)

(m)

yen (si )

(m)

(m)

(m)

j=0 N

=

g2 (s) + ∑ w j φ2 (si , r j , e xn−1 (r j ), e yn−1 (r j ), ..., xen−1(r j ), e yn−1 (r j )), j=0

... ...

xen (si )

(m)

xn−1 (r j ), e yn−1 (r j ), ..., xen−1(r j ), e yn−1 (r j )), g1 (s) + ∑ w j φ1 (si , r j , e

=

(m)

N

g1 (s) + ∑ w j j=0

=

(m)

N

g2 (s) + ∑ w j j=0

∂ m φ1 (m) (m) (si , r j , e xn−1 (r j ), e yn−1 (r j ), ..., e xn−1(r j ), e yn−1 (r j )), m ∂s ∂ m φ2 (m) (m) (si , r j , e xn−1 (r j ), e yn−1 (r j ), ..., e xn−1(r j ), e yn−1 (r j )). ∂ sm

Therefore, the approximate solution Ven converges to the exact solution V , in the product Banach space P = ∏2m+2 i=1 C([a, b], R), with the following estimated error: n ((m + 1)θ )n kV − Ven kP ≤ kV1 −V0 kP + ∑ ((m + 1)θ )n−pε p , 1 − (m + 1)θ p=1

where, lim ε p = 0, p = 1, ..., n. N→∞

6.

Simulation Results

In this section, we apply our numerical process to approach the solutions of two illustrative examples. These examples illustrate the effectiveness of the proposed process. Example 1. Consider the following integro-differential equation: ψ (s) = sin3 (2π s) + i cos3 (2π s) −

t2 + 3

Z 1 0

α (s, r) dr, 1 + (ψ (r) + ψ 0 (r) + ψ 00 (r))2

s ∈ [0, 1],

where, α (s, r) = s2 r2 (1 + [(1 − 12π 2)(sin3 (2π r) + i cos3 (2π r)) + 3π sin(4π r)[sin(2π r) + 4π cos(2π r) − i(cos(2π r) + 4π sin(2π r))]]2 ).

The exact solution of this example is: ψ (s) = sin3 (2π s) + i cos3 (2π s), s ∈ [0, 1].

186

Lemita Samir, Selim Raja and Belahbib Zeineb Table 1. Computed error E for examples (1) and (2), respectively N

30

50

100

250

500

E : for example (1)

9.30E-4

3.35E-4

8.37E-5

1.34E-5

3.35E-6

E : for example (2)

2.28E-5

2.96E-6

1.85E-7

4.80E-9

3.54E-10

Example 2. Consider the second equation: ψ (s) = cos(2π s) + i sin(4π s) −

t3 + 8

Z 1

β (s, r)

0 00 000 0 2 + eψ (r)+ψ (r)+ψ (r)+ψ (r)

dr,

s ∈ [0, 1],

where, β (s, r) = s3 r7 (2 + e[(1−4π

2 )(cos(2π r)+(8π 3 −2π )sin(2π r))+i[(1−16π 2 )sin(4π r)+(4π −64π 3 )cos(4π r)]]

)

with exact solution: ψ (s) = cos(2π s) + i sin(4π s), s ∈ [0, 1]. Using the numerical process described above, we give an approximate solution for examples (1) and (2), respectively. We mention that we have taken the e0 = 0 as a starting point in the iterative scheme and we have chosen zero vector V en−1 − V en kP < 10−9 . We also calculate the the following stopped condition: kV absolute error E, between the approximate and exact solutions by the formula: E = kV − Ven kP . According to different values of steps N, Table (1) shows us the absolute error E for examples (1) and (2), which is close to 0. Therefore, the accuracy of the error increases with N. On the other hand, Figures (1)-(7) display a graphical comparison between the exact and approximate solution and their derivatives, which appear to be almost identical. So, from these simulation results, the performance and efficacy of the presented process, have been proved.

Conclusion In this work, we have suggested a numerical process which is based on the following two procedures: Picard iterative scheme coupled with Nystr¨om method. We have applied this numerical process to obtain approximate solutions of a type of Fredholm equation with a second order integro-differential in the complex plane. However, the illustrative examples give us very interesting results,

On a Second Order Nonlinear Integro-Differential Equation ...

en with N = Figure 1. ψ versus ψ 50, for Example (1).

f0 with Figure 2. ψ 0 versus ψ n N = 50, for Example (1).

f00 with Figure 3. ψ 00 versus ψ n N = 50, for Example (1).

187

188

Lemita Samir, Selim Raja and Belahbib Zeineb

.

fn with N = Figure 4. ψ versus ψ 50, for Example (2).

f00 with Figure 6. ψ 00 versus ψ n N = 50, for Example (2).

f0 with Figure 5. ψ 0 versus ψ n N = 50, for Example (2).

f000 with Figure 7. ψ 000 versus ψ n N = 50, for Example (2).

On a Second Order Nonlinear Integro-Differential Equation ...

189

which affirm the effectiveness of this numerical scheme. In future, we seek to develop this numerical process that will be suitable for equations with general kernels, for example: Integral equations with weakly singular kernels described in [8].

References [1] Kwok, Y.K. (2010). Applied Complex Variables for Scientists and Engineers. Cambridge University Press, Cambridge. [2] Kythe, P.K. and Puri, P. (1992). Computational methods for linear integral equations. University of New Orleans, New Orleans. [3] Erfanian, M. and Akrami, A. (2017). RH wavelet bases to approximate solution of nonlinear Fredholm-Hammerstein integral equations in complex plane. Mathematical Modelling of Systems, 4(1), 1-8. [4] Erfanian, M. (2018). The approximate solution of nonlinear mixed Volterra-Fredholm-Hammerstein integral equations with RH wavelet bases in a complex plane. Mathematical Methods in the Applied Sciences, 41(18), 8942-8952. [5] Erfanian, M. and Zeidabadi, H. (2019). Solving of nonlinear Fredholm integro-differential equation in a complex plane with rationalized Haar wavelet bases. Asian-European Journal of Mathematics, 12(04), 1950055. [6] Erfanian, M., Zeidabadi, H. and Parsamanesh, M. (2020). Using of PQWs for solving NFID in the complex plane. Advances in Difference Equations, 2020(1), 1-13. [7] Atkinson, K. and Han, H. (2001). Theoretical numerical analysis: a functional analysis framework. Springer, New York. [8] Touati, S., Lemita, S., Ghiat, M. and Aissaoui, M.Z. (2019). Solving a nonlinear Volterra-Fredholm integro-differential equation with weakly singular kernels. Fasciculi Mathematici, 62, 1-14.

In: Understanding Integro-Differential Equations ISBN: 979-8-89113-040-1 c 2023 Nova Science Publishers, Inc.

Editors: J. Vasundhara Devi et al.

Chapter 7

L INEAR AND N ONLINEAR PARTIAL I NTEGRO -D IFFERENTIAL E QUATIONS A RISING FROM F INANCE Jos´e Cruz1 Maria Grossinho1 ˇ coviˇc2,∗ Daniel Sevˇ and Cyril Udeani2 1

2

ISEG, University of Lisbon, Portugal Comenius University in Bratislava, Slovakia

Abstract The purpose of this review chapter is to present our recent results on nonlinear and nonlocal mathematical models arising from modern financial mathematics. It is based on our four papers written ˇevˇcoviˇc, and C. Udeani [1], [2], jointly by J. Cruz, M. Grossinho, D. S [3], [4], as well as parts of the PhD thesis by J. Cruz [5]. We investigated linear and nonlinear partial integro-differential equations (PIDEs) arising from option pricing and portfolio selection problems and studied the systematic relationships between the PIDEs with option pricing theory and Black–Scholes models. First, we relax the liquid and complete market assumptions and extend the models ∗

Corresponding Author’s Email: [email protected].

192

ˇevˇcoviˇc et al. Jos´ e Cruz, Maria Grossinho, Daniel S that study the market illiquidity to the case where the underlying asset price follows a L´ evy stochastic process with jumps. Then, we establish the corresponding PIDE for option pricing under suitable assumptions. The qualitative properties of solutions to nonlocal linear and nonlinear PIDE are presented using the theory of abstract semilinear parabolic equation in the scale of Bessel potential spaces. The existence and uniqueness of solutions to the PIDE for a general class of the so-called admissible L´ evy measures satisfying suitable growth conditions at infinity and origin are also established in the multidimensional space. Additionally, the qualitative properties of solutions to the generalized PIDE are investigated by considering a general shift function arising from nonlinear option pricing models, which takes into account a large trader stock-trading strategy with the underlying asset price following the L´ evy process. For the portfolio management problem, we present the existence and uniqueness results of the fully nonlinear Hamilton-Jacobi-Bellman equation arising from the stochastic dynamic optimization problem in Sobolev spaces using the theory of monotone operator technique, which can also be viewed as PIDE in some sense. Furthermore, a stable, convergent, and consistent numerical scheme that can give an approximate solution to such PIDE is presented, and various numerical experiments are conducted to illustrate the influence of a large trader and the intensity of jumps on the option price.

Keywords: L´ evy measure, option pricing, partial integro-differential equation, Hamilton-Jacobi-Bellman equation, maximal monotone operator, dynamic stochastic portfolio optimization

1. Introduction This review chapter contains our recent advances in research focused on nonlinear and nonlocal mathematical models arising from modern financial mathematics. The main parts of this chapter are based on our ˇevˇcoviˇc, and four chapters jointly written by J. Cruz, M. Grossinho, D. S C. Udeani [1], [2], [3], [4], as well as parts of the PhD thesis by J. Cruz [5]. The classical Black–Scholes model has been widely used in financial industry because of its simplicity and the existence of analytical formula

Linear and Nonlinear Partial Integro-Differential Equations ... 193 for pricing derivative securities. This model relies on restrictive assumptions, such as completeness, frictionlessness of the market, and the assumption that the underlying asset price follows a geometric Brownian motion. However, the assumption that an investor can trade a large amount of assets without affecting the underlying asset price is generally not satisfied, especially in illiquid markets. It is also known that the fully nonlinear Hamilton–Jacobi–Bellman (HJB) equation plays an essential role in finance. For instance, it gives the necessary and sufficient condition for a control with respect to the value function. Therefore, this chapter investigates linear and nonlinear partial integro-differential equations (PIDEs) arising from the option pricing and portfolio selection problem. We investigate the systematic relationships of the PIDEs with option pricing theory and Black–Scholes models. First, we relax the liquid and complete market assumptions and extend the models that study market illiquidity to the case where the underlying asset price follows a L´ evy stochastic process with jumps. Then, we establish the corresponding PIDE for option pricing under suitable assumptions. The qualitative properties of solutions to nonlocal linear and nonlinear PIDEs are presented using the theory of abstract semilinear parabolic equations in the scale of Bessel potential spaces. The existence and uniqueness of solutions to the PIDE for a general class of the so-called admissible L´ evy measures satisfying suitable growth conditions at infinity and origin are also established in the multidimensional space. Additionally, the qualitative properties of solutions to the generalized PIDE are investigated by considering a general shift function arising from nonlinear option pricing models, which takes into account a large trader stock-trading strategy with the underlying asset price following the L´ evy process. For the portfolio management problem, we present the existence and uniqueness results to the fully nonlinear HJB equation arising from stochastic dynamic optimization problem in Sobolev spaces using the theory of monotone operator technique, which can also be viewed as PIDE in some sense. Furthermore, a stable, convergent and consistent numerical scheme is presented that can efficiently approximate the solution of this PIDE, and various numerical experiments are conducted to illustrate the influence of a large trader and the intensity of jumps on the option price. The Black–Scholes model and the HJB equation have been widely used in financial markets. However, evidence from the stock market ob-

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servation shows that the Black–Scholes model is not the most realistic one because it depends on some restrictive assumptions, such as the liquidity, completeness, and frictionless of the market. Additionally, the linear Black–Scholes equation provides a solution that corresponds to a perfectly replicated portfolio, which is not a desirable property. For this reason, several attempts have been made to generalize and relax some of these assumptions. Some authors relaxed these assumptions by (i) considering the presence of transaction costs (see Kwok [6] and Avellaneda and Paras [7]), (ii) feedback and illiquid market effects due to large traders choosing given stock-trading strategies (Sch¨ onbucher and Willmott [8], Frey and Patie [9], Frey and Stremme [10]), and (iii) the ˇevˇcoviˇc [11]). In risk from the unprotected portfolio (Jandaˇcka and S these generalizations, the constant volatility was replaced by a nonlinear function based on the second derivative of the option price. Frey and Stremme derived a nonlinear Black–Scholes model that plays an essential role in the class of the generalized Black–Scholes equation with such a nonlinear diffusion function [9, 12, 11]). In this model, the asset dynamics considers the presence of feedback effects due to a large trader choosing his/her stock-trading strategy [8]. Another important direction in generalizing the original Black–Scholes equation arises from the fact that the sample paths of a Brownian motion are continuous; however, the realized stock price of a typical company exhibits random jumps over the intraday scale, making the price trajectories discontinuous. In the classical Black–Scholes model, the logarithm of the price process has a normal distribution. However, the empirical distribution of stock returns exhibits fat tails. Meanwhile, when calibrating the theoretical prices to the market prices, the implied volatility is not constant as a function of strike price nor as a function of time to maturity, contradicting the prediction of the Black–Scholes model. However, the models with jumps and diffusion can solve the problems inherent to the Black–Scholes model. Jump models also play an essential role in the option market. In the Black–Scholes model, the market is complete, implying that every payoff can exactly be replicated; meanwhile, there is no perfect hedge in jump models, making the way of options not redundant. Market illiquidity has been widely studied in the literature [13, 14, 15, 16, 17]. The first major contribution was made by Robert Jarrow, in 1994, who studied the market manipulation strategies that may arise in

Linear and Nonlinear Partial Integro-Differential Equations ... 195 illiquid markets. The author also studied option pricing theory in discrete time when there is a large trader. The pricing-arbitrage condition was used to ensure that no market manipulation strategy is used by the large trader and large trader’s optimality conditions; thus, replacing the usual free-arbitrage argument. Then, Frey (1998) extended Jarrow’s analysis to the continuous time case and established the existence and uniqueness of solution of a nonlinear partial differential equation (PDE) satisfied by the large trader’s hedging strategy. Additionally, Platten and Schweizer (1998) proposed an explanation for the smile and skewness of the implied volatilities and showed that hedging strategies followed by large traders can lead to option price bias. Sircar and Papanicolaou (1998) also proposed a model where the derivative security price is characterized by a nonlinear PDE that becomes the Black–Scholes equation when there is no feedback. When the program traders are a small fraction of the economy, numerical and analytical methods can be used to analyze the nonlinear PDE through perturbation. This equation is derived using an argument similar to the one used in deriving the classical Black–Scholes equation. Consequently, they obtained that this model also predicts increased implied volatilities as in Platten and Schweizer. Furthermore, Schonbucher and Willmott (2000) analyzed the feedback effects from the presence of hedging strategies. They also derived a nonlinear PDE for an option replication strategy and studied these effects for a put option. The effects are more pronounced in markets with low liquidity, which can induce discontinuities in the price process. However, none of these studies that investigated jump models [18, 19, 20, 21, 22, 23, 24] considered the market illiquidity. Meanwhile, investors and risk managers have realized that financial models based on the assumption that an investor can trade large amounts of an asset without affecting its price are no longer true in markets that are not liquid. Therefore, in this chapter, we relax the liquid and complete market hypothesis and extend the models that study market illiquidity to the case where the underlying asset price follows a L´ evy stochastic process with jumps to obtain a model for pricing European and American call and put options on an underlying asset characterized by a L´ evy measure. In this way, it is assumed that trading strategies affect the stock price and the possibility to account for sudden jumps that might occur when the market is under stress .

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Recently, the relationships between more general nonlocal operators and jump processes have been widely investigated. For instance, there is an actual connection between the solution to PIDEs and properties of the corresponding Markov jump process (cf. Abels and Kassmann [25]; Florescu and Mariani [26]). In recent decades, the role of PIDEs has been investigated in various fields, such as pure mathematics, biological sciences, and economics [27, 28, 29]. PIDE problems arising from financial mathematics, especially option pricing models, have been of great interest to many researchers. In most cases, standard methods for solving these problems lead to the study of parabolic equations. Mikuleviˇcius and Pragaraustas [30] investigated solutions of the Cauchy problem to the parabolic PIDE with variable coefficients in Sobolev spaces. They used their results to obtain solutions of the corresponding martingale problem. Crandal et al. [31] employed the notion of a viscosity solution to investigate the qualitative results. Soner et al. [32] and Barles et al. [33] extended and generalized their results for the first and second order operators, respectively. Florescu and Mariani [26] employed the Schaefer fixed point argument to establish existence of a weak solution of the generalized PIDE. Amster et al. [34] used the notion of upper and lower solutions to obtain the solution of such PIDEs. They proved the existence of solutions in a general domain for multiple assets and the regime switching jump-diffusion model. Cont et al. [35] investigated the actual connection between option pricing in exponential L´ evy models and the corresponding PIDEs for European options and those with single or double barriers. They discussed and established the conditions for which the prices of the option are the classical solution of the corresponding PIDE. In this chapter, we obtain a certain PIDE for option pricing in an illiquid market by assuming a certain dynamics for the stock price. The existence of a solution and the localization results of the associated PIDE are also established. We investigated and established the qualitative properties of solutions to the nonlocal linear and nonlinear PIDE in the scale of Bessel potential spaces using the theory of abstract semilinear parabolic equation. Furthermore, we present the existence and uniqueness results for nonlinear parabolic equations using the monotone operator technique, Fourier transform, and Banach fixed point argument. We considered the fully nonlinear HJB equation arising from the portfolio selection problem, where the goal of an investor is to optimize the conditional expected

Linear and Nonlinear Partial Integro-Differential Equations ... 197 value of the terminal utility of the portfolio. Such a nonlinear parabolic equation is presented in an abstract setting, which can also be viewed as a nonlinear PIDE. Many previous studies have developed numerical methods for PIDEs, such as finite difference and finite element methods. However, the equation corresponding to the case of illiquid markets is more difficult. Therefore, this chapter also presents a stable, convergent, and consistent numerical scheme that can give an approximate solution of such PIDEs. Various experiments are presented to illustrate the influence of a large trader and the intensity of jumps on the option price.

2. Background and Motivation Based on the classical theory developed by Black, Scholes, and Merton, the price V (t, S) of an option in a stylized market at time t ∈ [0, T ] and the underlying asset price S can be calculated as a solution to the following linear Black–Scholes parabolic equation: ∂V 1 ∂2V ∂V (t, S)+ σ 2 S 2 2 (t, S)+rS (t, S)−rV (t, S) = 0, t ∈ [0, T ), S > 0. ∂t 2 ∂S ∂S (1) Here, σ > 0 is the historical volatility of the underlying asset driven by the geometric Brownian motion, and r > 0 is the risk-free interest rate of zero-coupon bond. The solution of the above equation is subject to the terminal payoff condition V (T, S) = Φ(S) at maturity t = T . Meanwhile, evidence from stock market observations indicates that the model is not the most realistic one because it assumes that the market is liquid, complete, frictionless and without transaction costs. It is also known that the linear Black–Scholes equation provides a solution corresponding to a perfectly replicated portfolio, which need not be a desirable property. To solve these problems, several attempts have been made to generalized the linear Black–Scholes equation (1) by replacing the constant volatility σ with a nonlinear function σ ˜ (S∂S2 V ) depending on the second derivative ∂S2 V of the option price. In this regard, Frey and Stremme derived a nonlinear Black–Scholes model, which plays an essential role in the class of generalized Black–Scholes equation with such a nonlinear diffusion function [9, 12]). They considered the case in which asset dynamics takes into account the presence of feedback effects due to a large trader choos-

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ing his/her stock-trading strategy (see also [8]). The diffusion coefficient is non-constant, and it is given by
−2 σ ˜ (S∂S2 V )2 = σ 2 1 − %S∂S2 V , (2)

where σ and % > 0 are constants. Furthermore, several researchers have attempted to generalize the original Black–Scholes equation, which arises from the fact that the sample paths of a Brownian motion are continuous. However, the realized stock price of a typical company exhibits random jumps over the intraday scale, making the price trajectories discontinuous. The underlying asset price process is usually assumed to follow a geometric Brownian motion in the classical Black–Scholes model. However, the empirical distribution of stock returns exhibits fat tails. The models with jumps and diffusion can solve the problems inherent to the linear Black–Scholes model and play an essential role in options pricing. It is well known that the market is complete in the Black–Scholes model, illustrating that each payoff can be perfectly replicated; however, there is no perfect hedge in jumpdiffusion models, making the options not redundant. It turns out that the option price can be computed from the solution V (t, S) to the following PIDE Black–Scholes equation [1]: 1 ∂ 2V ∂V ∂V (t, S) + σ 2 S 2 (t, S) + rS (t, S) − rV (t, S) ∂tZ 2 ∂S 2 ∂S ∂V + V (t, S + H(z, S)) − V (t, S) − H(z, S) (t, S)ν(dz) = 0,(3) ∂S R where H(z, S) = S(ez −1), and ν is the so-called L´ evy measure characterizing the underlying asset process with random jumps in time and space. It is worth noting that (3) reduces to the classical linear Black–Scholes equation (1) if ν = 0. In this chapter, we consider both directions of generalization of the Black–Scholes equation. First, we relax the assumption of a liquid market following the Frey–Stremme model by assuming that the underlying asset price follows a L´ evy stochastic process with jumps and establish the corresponding PIDEs. Then, we present the existence and uniqueness results to the linear and nonlinear nonlocal PIDE in the framework of Bessel potential spaces for the multidimensional case. A more generalized nonlinear nonlocal PIDE is also presented by considering a shift

Linear and Nonlinear Partial Integro-Differential Equations ... 199 function ξ = ξ(τ, x, z) depending on variables x, z ∈ Rn . In addition, we derive, analyze and perform numerical computations of the model. We also show that the corresponding nonlinear PIDE has the following form: ∂2V ∂V 1 σ2 ∂V + S 2 2 + rS − rV 2 ∂t 2 (1 − %S∂S φ) ∂S ∂S Z ∂V + V (t, S + H(t, z, S)) − V (t, S) − H(t, z, S) ν(dz) = 0. (4) ∂S R It is worth noting that the function H(t, z, S) may depend on the large trader strategy function φ = φ(t, S) and the delta ∂S V of the price V if % > 0. We consider a stylized economy with two traded assets: a riskless asset (a bond with a price Bt taken as numeraire) and a risky asset (stock with a price St ). Here, we assume that the bond market is perfectly elastic, since it is more liquid than stocks, and consider two types of traders: reference and program traders. Program traders are also known as portfolio insurers because they use dynamic hedging strategies to hedge the portfolio against jumps in stock prices. They are classified as single traders or a group of traders acting together. It is assumed that their trades influence the stock price equilibrium. On the contrary, reference traders can be considered as representative traders of many small ˜ Yt, St) agents. We assume that they act as price takers. Generally, D(t, represents the demand function of the reference trader that depends on the income process Yt or some other fundamental state variable that influences the demand of the reference trader. The aggregate demand of program traders is denoted by ϕ(t, St) = ξφ(t, St), where ξ is the number of identical written securities that program traders are trying to hedge, and φ(t, St) is the demand per unit of the security being hedged. For simplicity, we assume that ξ is the same for every program trader. See [17] for a more general case where different securities are considered. Suppose that the supply of a stock with the price S˜0 is constant, and let ˜ be the quantity demanded by a reference trader per D(t, Y, S) = D(t,Y,S) S˜0 unit supply. Then, the total demand relative to the supply at time t is given by G(t, Y, S) = D(t, Y, S) + ρφ(t, S), where ρ = Sξ˜ , and ρφ(t, S) 0 is the proportion of the total supply of the stock traded by the program traders. Therefore, to obtain the market equilibrium, the variables Y and

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S should satisfy G(t, Y, S) = 1. Assume that the function G is monotone with respect to the variables Y and S, and it is sufficiently smooth. Then, we can solve the implicit equation G(t, Yt, St) = 1 to obtain St = ψ(t, Yt), where ψ is a sufficiently smooth function. Using the approach in [17], we assume that the stochastic process Yt has the following dynamics: dYt = µ(t, Yt )dt + η(t, Yt)dWt. Then, using Itˆ o’s lemma for the process St = ψ(t, Yt), we have   η2 2 dSt = ∂t ψ + µ∂y ψ + ∂y ψ dt+η∂y ψdWt ≡ b(t, St)Stdt+v(t, St )StdWt . 2 (5) It means that St follows a geometric Brownian motion with a nonconstant volatility function v(t, S) = η(t, Y )∂Y ψ(t, Y )/ψ(t, Y ), where Y = ψ −1 (t, S). Thus, we follow the argument used in the derivation of the original Black–Scholes equation to obtain a generalization of the Black–Scholes PDE with a nonconstant volatility function σ = v(t, S). We employ the Frey–Stremme’s approach (cf. [9, 12]) to prescribe a dynamics for the underlying stock price instead of deriving it using the market equilibrium and dynamics for the income process Yt as is done in [17]. In this way, Frey and Stremme derived the same stock price dynamics as in [17] corresponding to a situation where the demand function is of logγ arithmic type, D(Y, S) = ln( YS ), where γ = ησ0 , and the income process Yt follows a geometric Brownian motion, i.e., 1 1 , ∂S D(Y, S) = − , dYt = µ0 Yt dt + η0 Yt dWt ,(6) Y S γ Y1 ∂Y ψ(t, Y ) η0 Y σ v(t, S) = η(t, Y ) =− = . ∂φ 1 ψ(t, Y ) S − +ρ 1 − ρS ∂φ

∂Y D(Y, S) = γ

S

∂S

∂S

Assuming the delta hedging strategy with φ(t, S) = ∂S V (t, S) and substituting the volatility function v(t, S) in (5), we obtain the generalized Black–Scholes equation with the nonlinear diffusion function of the form (2). In this chapter, we first generalized the Frey–Stremme model by considering an underlying asset following a L´ evy process with jumps. After that, we establish the corresponding PIDE for option pricing. Furthermore, we investigate the existence and uniqueness of solutions to such PIDE in multidimensional spaces.

Linear and Nonlinear Partial Integro-Differential Equations ... 201

3. Preliminaries and Definitions This section presents some basic definitions and properties of L´ evy measures and notion of admissible activity L´ evy measures. Here,| · | and k · k represent the Euclidean norm in Rn and the norm in an infinite dimensional function space (e.g., Lp (Rn ), X γ ). In what follows, a · b stands for √ the usual Euclidean product in Rn with the norm |z| = z · z. Definition 1. [3] A L´evy process on Rn is a stochastic (right continuous) process X = {Xt , t ≥ 0} having the left limit with independent stationary increments. It is uniquely characterized by its L´evy exponent φ: Ex (eiy·Xt ) = e−tφ(y) , y ∈ Rn . The subscript x in the expectation operator Ex indicates that the process Xt starts with a given value x at the origin t = 0. The L’evy exponent φ has the following unique decomposition: Z n X
φ(y) = ib · y + aij yi yj + 1 − eiy·z + iy · z1|z|≤1 ν(dz), Rn

i,j=1

where b ∈ Rn is a constant vector; (aij ) is a constant matrix, which is positive semidefinite; ν(dz) is a nonnegative measure in Rn \ {0} such that R 2 Rn min(1, |z| )ν(dz) < ∞ (cf. [36]).

3.1. Exponential L´ evy Models

Let Xt , t ≥ 0, be a stochastic process. The Poisson random measure ν(A) of a Borel set A ∈ B(R) is defined by ν(A) = E [JX ([0, 1] × A)], where JX ([0, t] × A) = # {s ∈ [0, t] : ∆Xs ∈ A}. This measure gives the mean number per unit of time of jumps whose amplitude belongs to the set A. It is worth noting that the L´ evy–Itˆ o decomposition provides a representation of Xt interpreted as a combination of a Brownian motion with a drift ω and an infinite sum of independent compensated Poisson processes with variable jump sizes (see [2]), i.e., Z Z dXt = ωdt + σdWt + xJX (dt, dx) + xJeX (dt, dx) , |x|≥1

|x| 0 if µ = 0), where C0 > 0 is a positive constant. Remark 2. It is worth noting that the additional conditions R R 2 z min(|z| , 1)ν(dz) < ∞ and R |z|>1 e ν(dz) < ∞ are satisfied provided that ν is an admissible L´evy measure with shape parameters α < 3, and either µ > 0, D± ∈ R, or µ = 0 and D − + 1 < 0 < D+ . For the Merton model, we have α = 0, D± = 0 and µ = 1/(2δ 2 ) > 0. Meanwhile, for the Kou model, we have α = µ = 0, D+ = λ− , D− = −λ+ . For the variance Gamma process, we have α = 1, µ = 0, D± = A ± B.

4. Multidimensional Linear and Nonlinear PIDE This section focuses on qualitative properties of solutions to the linear and nonlinear nonlocal parabolic PIDE of the form: ∂u ∂τ

=

σ2 ∆u 2

+

R

Rn

[u(τ, x + z) − u(τ, x) − z · ∇x u(τ, x)] ν(dz) + g(τ, x, u, ∇x u)(21) u(0, x) = u0 (x),

x ∈ Rn , τ ∈ (0, T ),

where g is a given sufficiently smooth function; ν is a positive measure on Rn such that its Radon derivative is a nonnegative Lebesgue measurable function h in Rn , i.e., ν(dz) = h(z)dz. Additionally, we will analyze the solution of the following generalization of the above PIDE, in which the shift function may depend on the variables τ > 0, x, z ∈ R:

ˇevˇcoviˇc et al. Jos´ e Cruz, Maria Grossinho, Daniel S

208

∂u σ2 = ∆u + ∂τ 2

Z

[u(τ, x + ξ) − u(τ, x) − ξ · ∇x u(τ, x)] ν(dz) + g(τ, x, u, ∇x u), (22)

Rn

where ξ = ξ(τ, x, z) is the shift function. An application of such a general shift function ξ can be found in nonlinear option pricing models considering a large trader stock-trading strategy with the underlying asset ˇ coviˇc [1]). If price dynamic following the L´ evy process (cf., Cruz and Sevˇ ξ(x, z) ≡ z, then (22) reduces to equation (21). For example, nonlinearity g often arises from applications occurring in pricing XVA derivatives (cf., Arregui et al. [28, 38]) or applications of the penalty method for ˇ coviˇc [2]). American option pricing under a PIDE model (cf., Cruz and Sevˇ

4.1. Existence and Uniqueness Results of PIDE In this section, we present the existence and uniqueness results for the general equation (22) for a class of L´ evy measures using the theory of abstract semilinear parabolic equation in the scale of Bessel potential spaces. First, we rewrite the PIDE (22) in high-dimensional space as follows: ∂u ∂τ

+ Au = f (u) + g(τ, x, u, ∇x u), u(0, x) = u0 (x), x ∈ Rn , τ ∈ (0, T ),

(23)

where A = −(σ 2 /2)∆. The linear nonlocal operator f is defined by Z f (u)(·) = [u(· + ξ) − u(·) − ξ · ∇x u(·) ] ν(dz), (24) Rn

where ξ = ξ(τ, x, z) is a given shift function. The function g is assumed to be H¨ older and Lipschitz continuous in τ and other variables, respectively. Then, we employ the theory of abstract semilinear parabolic equations presented by Henry [39] to establish the existence, continuation, and uniqueness of a solution. A solution to the PIDE (23) is constructed in the p scale of the Bessel potential spaces L2γ (Rn ), γ ≥ 0 in high-dimensional space, n ≥ 1. These spaces can be viewed as a natural extension of the classical Sobolev spaces W k,p (Rn ) for non-integer values of order k. It is worth noting that the nested scale of Bessel potential spaces allows for

Linear and Nonlinear Partial Integro-Differential Equations ... 209 a finer formulation of existence and uniqueness results than the classical Sobolev spaces. Definition 3. [39, Definition 1] An analytic semigroup is a family of bounded linear operators {S(t), t ≥ 0} in a Banach space X satisfying the following conditions: i) S(0) = I, S(t)S(s) = S(s)S(t) = S(t + s), for all t, s ≥ 0; ii) S(t)u → u when t → 0+ for all u ∈ X; iii) t → S(t)u is a real analytic function on 0 < t < ∞ for each u ∈ X. The associated infinitesimal generator A is defined as follows: Au = limt→0+ 1t (S(t)u − u) and its domain D(A) ⊆ X consists of those elements u ∈ X for which the limit exists in the space X. Definition 4. [39] Let Sa,φ = {λ ∈ C : φ ≤ arg(λ − a) ≤ 2π − φ} be a sector of complex numbers. A closed densely defined linear operator A : D(A) ⊂ X → X is called a sectorial operator if there exists a constant M ≥ 0 such that k(A − λ)−1 k ≤ M/|λ − a| for all λ ∈ Sa,φ ⊂ C \ σ(A). Next, we briefly recall the construction and basic properties of Bessel potential spaces. It is worth noting that if A is a sectorial operator in a Banach space X, then −A is a generator of an ana lytic semigroup e−At , t ≥ 0 acting on X (cf., [39, Chapter I]). For any γ > 0, we can the operator A−γ : X → X as folR ∞introduce 1 γ−1 −Aξ −γ lows: A = Γ(γ) 0 ξ e dξ. Then, the fractional power space γ γ X = D(A ) is the domain of the operator Aγ = (A−γ )−1 , i.e., X γ = {u ∈ X : ∃ϕ ∈ X, u = A−γ ϕ}. The norm is defined as follows: kukX γ = kAγ ukX = kϕkX . Furthermore, we have continuous embedding: D(A) ≡ X 1 ,→ X γ1 ,→ X γ2 ,→ X 0 ≡ X, for 0 ≤ R γ2 ≤ γ1 ≤ 1. Recall the convolution operator (G ∗ ϕ)(x) = Rn G(x − y)ϕ(y)dy. According to [39, Section 1.6], A = −(σ 2 /2)∆ is a sectorial operator in the Lebesgue space X = Lp (Rn ) for any p ≥ 1, n ≥ 1, and D(A) ⊂ W 2,p (Rn ). It follows from [40, Chapter 5] that the space X γ , γ > 0, can p be identified with the Bessel potential space L2γ (Rn ), where p L2γ (Rn ) := {u ∈ X : ∃ϕ ∈ X, u = G2γ ∗ ϕ}.

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Here, G2γ is the Bessel potential function, Z ∞ 2 1 G2γ (x) = y −1+γ−n/2 e−(y+|x| /(4y))dy. n/2 (4π) Γ(γ) 0 The norm of u = G2γ ∗ ϕ is given by kukX γ = kϕkLp . The space X γ is continuously embedded in the fractional Sobolev–Slobodeckii space W 2γ,p (Rn ) (cf., [39, Section 1.6]). In what follows, we denote C0 > 0 as a generic constant which is independent of the solution u; however, it may depend on the model parameters, e.g., n ≥ 1, p ≥ 1, γ ∈ [0, 1). Proposition 1. [3, Proposition 1] Let us define the mapping Q(u, ξ) as follows: Q(u, ξ) = u(x + ξ(x)) − ξ(x) · ∇x u(x), x ∈ Rn .

Then, there exists a constant Cˆ > 0 such that, for any vector valued functions ξ1 , ξ2 ∈ (L∞ (Rn ))n , and u such that ∇x u ∈ (X γ−1/2)n , 1/2 ≤ γ < 1, the following estimate holds: ˆ 1 −ξ2 k2γ−1 (kξ1 k∞ +kξ2 k∞ )k∇x uk γ−1/2 . kQ(u, ξ1)−Q(u, ξ2)kLp(Rn ) ≤ Ckξ ∞ X Proof. Let u ∈ X be such that ∇x u ∈ (X γ−1/2)n , i.e., ∂xi u ∈ X γ−1/2 for each i = 1, · · · , n. Then, ∇x u = A−(2γ−1)/2 ϕ = G2γ−1 ∗ ϕ for some ϕ ∈ (Lp(Rn ))n , and k∇x ukX γ−1/2 = kA(2γ−1)/2∇x ukX = kϕkLp . Here, ϕ = (ϕ1 , · · · , ϕn) and ∂xi u = G2γ−1 ∗ ϕi . Let x, ξ ∈ Rn . Then, ∇x u(x + ξ) = G2γ−1 (x + ξ − ·) ∗ ϕ(·),

∇x u(x) = G2γ−1 (x − ·) ∗ ϕ(·).

Recall that the following inequality holds for convolution operator: kψ ∗ ϕkLp (Rn ) ≤ kψkLq (Rn ) kϕkLr (Rn ) , where p, q, r ≥ 1 and 1/p + 1 = 1/q + 1/r (see [39, Section 1.6]). In particular, for q = 1, we have kψ ∗ ϕkLp ≤ kψkL1 kϕkLp . The following estimate holds for the modulus of continuity of the Bessel potential function Gα , α ∈ (0, 1): kGα(· + h) − Gα (·)kL1 ≤ C0 |h|α,

Linear and Nonlinear Partial Integro-Differential Equations ... 211 for any h ∈ Rn (cf., [40, Chapter 5.4, Proposition 7]). Let ξ1 , ξ2 be bounded vector valued functions, i.e., ξ1 , ξ2 ∈ (L∞ (Rn ))n . Then, for any x ∈ Rn and θ ∈ [0, 1], we have u(x + ξ1 (x)) − u(x + ξ2 (x)) − (ξ1 (x) − ξ2 (x)) · ∇x u(x)

= u(x + ξ1 (x)) − ∇x u(x) − ξ1 (x) · ∇x u(x)

−[u(x + ξ2 (x)) − ∇x u(x) − ξ2 (x) · ∇x u(x)] Z 1 = (ξ1 (x) − ξ2 (x)) ∇xu(x + θξ1 (x)) − ∇x u(x)dθ 0 Z 1 + ∇x u(x + θξ1 (x)) − ∇x u(x + θξ2 (x))dθ . 0

Now,

kQ(u, ξ1 ) − Q(u, ξ2 )kpLp (Rn ) |u(x + ξ1 (x)) − u(x + ξ2 (x)) − (ξ1 (x) − ξ2 (x)) · ∇x u(x)|p dx ˛p R ˛˛ R1 ˛ ≤ Rn ˛(ξ1 (x) − ξ2 (x)) 0 ∇x u(x + θξ1 (x)) − ∇x u(x)dθ˛ dx ˛p R ˛˛ R1 ˛ + Rn ˛ξ2 (x) 0 ∇x u(x + θξ1 (x)) − ∇x u(x + θξ2 (x))dθ˛ dx R1R ≤ kξ1 − ξ2 kp∞ 0 Rn |∇x u(x + θξ1 (x)) − ∇x u(x)|p dxdθ R1R +kξ2 kp∞ 0 Rn |∇x u(x + θξ1 (x)) − ∇x u(x + θξ2 (x)|pdxdθ R1 ≤ kξ1 − ξ2 kp∞ 0 k (G2γ−1 (· + θξ1 ) − G2γ−1 (·)) ∗ ϕkpLp dθ R1 +kξ2 kp∞ 0 k (G2γ−1 (· + θξ1 ) − G2γ−1 (· + θξ2 )) ∗ ϕkpLp dθ R1 ≤ kξ1 − ξ2 kp∞ 0 kG2γ−1 (· + θξ1 ) − G2γ−1 (·)kpL1 dθkϕkpLp R1 +kξ2 kp∞ 0 kG2γ−1 (· + θξ1 ) − G2γ−1 (· + θξ2 )kpL1 dθkϕkpLp ” “ (2γ−1)p (2γ−1)p C0p k∇x ukpX γ−1/2 ≤ kξ1 − ξ2 kp∞ kξ1 k∞ + kξ2 kp∞ kξ1 − ξ2 k∞ “ ” (2−2γ)p (2γ−1)p (2γ−1)p kξ1 kp∞ + kξ2 k∞ kξ1 k∞ + kξ2 kp∞ C0p k∇x ukpX γ−1/2 . ≤ kξ1 − ξ2 k∞ =

R

Rn

α

β

By Young’s inequality, we have ab ≤ aα + bβ for any a, b ≥ 0, and α, β > 1 with 1/α + 1/β = 1 (cf., [41]). Set α = 1/(2 − 2γ), β = 1/(2γ − 1). Then, 1/α + 1/β = 1, and we obtain (2−2γ)p (2γ−1)p p p p p kξ2 k∞ kξ1 k∞ ≤ (2−2γ)kξ2k∞ +(2γ−1)kξ1k∞ ≤ 2kξ2 k∞ +kξ1 k∞ . Therefore, kQ(u, ξ1 ) − Q(u, ξ2 )kpLp (Rn )

≤

2kξ1 − ξ2 k(2γ−1)p (kξ1 kp∞ + kξ2 kp∞ ) C0p k∇x ukpX γ−1/2 ∞

≤

2C0p kξ1 − ξ2 k(2γ−1)p (kξ1 k∞ + kξ2 k∞ )p k∇x ukpX γ−1/2 . ∞

212

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Therefore, the pointwise estimate holds with the constant Cˆ = C0 > 0.

1/p

Applying Proposition 1 with ξ1 = ξ and ξ2 = 0, we obtain the following corollary. Corollary 1. [3, Corollary 1] Let u be such that ∇xu ∈ (X γ−1/2)n where 1 > γ ≥ 1/2. Then, for any ξ ∈ Rn , the following pointwise estimate holds: kQ(u, ξ)kLp(Rn ) ≤ C0 |ξ|2γ k∇xukX γ−1/2 .

Next, we consider the case where the nonlocal integral term depends on the variables x and z. It is a generalization of the result [1, Lemma ˇ coviˇc proven for the case where ξ(x, z) ≡ z. 3.4] due to Cruz and Sevˇ Proposition 2. [3, Proposition 2] Suppose that the shift mapping ξ = ξ(x, z) satisfies supx∈Rn |ξ(x, z)| ≤ C0 |z|ω (1 + eD0 |z| ) for some constants C0 > 0, D0 ≥ 0, ω > 0 and any z ∈ Rn . Assume ν is a L´evy measure with the shape parameters α, D, and either µ > 0, D ∈ R, or µ = 0 and D > D0 ≥ 0. Assume 1/2 ≤ γ < 1, and γ > (α − n)/(2ω). Then there exists a constant C0 > 0 such that kf (u)kLp ≤ C0 k∇xukX γ−1/2 , provided that ∇x u ∈ (X γ−1/2)n . If u ∈ X γ then kf (u)kLp ≤ CkukX γ , i.e., f : X γ → X is a bounded linear operator.

Proof. The L´ evy measure ν(dz) is given by ν(dz) = h(z)dz. Let us denote 2 ˜ the auxiliary function ˜h(z) = |z|αh(z). Then, 0 ≤ h(z) ≤ C0 e−D|z|−µ|z| 1 ˜ since h(z) = |z|−αh(z) = h1 (z)h2 (z), where h1 (z) = |z|−β ˜h(z) 2 and 1 h2 (z) = |z|β−α˜h(z) 2 . Applying Proposition 1 with ξ1 = ξ, ξ2 = 0, and using the H¨ older inequality, we obtain ˛p R ˛R kf (u)kpLp = Rn ˛ Rn (u(x + ξ(x, z)) − u(x) − ξ(x, z) · ∇x u(x))h(z)dz ˛ dx R R ≤ Rn Rn |u(x + ξ(x, z)) − u(x) − ξ(x, z) · ∇x u(x)|p h1 (z)pdz `R ´p/q × Rn h2 (z)q dz dx R `R ´ = Rn Rn |u(x + ξ(x, z)) − u(x) − ξ(x, z) · ∇x u(x)|p dx h1 (z)p dz `R ´p/q × Rn h2 (z)q dz R `R ´p/q ˜ p/2 dz ≤ C0p k∇x ukpX γ−1/2 Rn |ξ(x, z)|2γp |z|−βph(z) h (z)q dz Rn 2 R `R ´p/q ≤ C0p k∇x ukpX γ−1/2 Rn |z|(2γω−β)p ˜ h(z)p/2 dz Rn h2 (z)q dz .

Linear and Nonlinear Partial Integro-Differential Equations ... 213 Assuming p, q ≥ 1, 1/p + 1/q = 1 are such that

p > −n, p−1 R R then, the integrals Rn |z|(2γω−β)p˜h(z)p/2 dz and Rn h2 (z)q dz = R (β−α)q ˜ h(z)q/2 dz are finite, provided that the shape parameters satRn |z| isfy either µ > 0, D ∈ R, or µ = 0, D > D0 ≥ 0. As γ > (α − n)/(2ω), there exists β > 1 satisfying (2γω − β)p > −n,

(β − α)q = (β − α)

α − n + n/p < β < 2γω + n/p. Therefore, there exists C0 > 0 such that kf (u)kLp ≤ C0 k∇xukX γ−1/2 . Let C([0, T ], X γ ) be the Banach space consisting of continuous functions from [0, T ] to X γ with the maximum norm. The following proposiˇevˇcoviˇc [2]). tion is due to Henry [39] (see also Cruz and S Proposition 3. [39, Proposition 3.5] Suppose the linear operator −A  −At that is a generator of an analytic semigroup e , t ≥ 0 in a Banach space X. Assume the initial condition U0 belongs to the space X γ where 0 ≤ γ < 1. Suppose that the mappings F : [0, T ] × X γ → X and h : (0, T ] → X RT are H¨ older continuous in the τ variable, 0 kh(τ )kX dτ < ∞, and F is Lipschitz continuous in the U variable. Then, for any T > 0, there exists a unique solution to the abstract semilinear evolution equation: ∂τ U + AU = F (τ, U ) + h(τ ) such that U ∈ C([0, T ], X γ), U (0) = U0 , ∂τ U (τ ) ∈ X, U (τ ) ∈ D(A) for any τ ∈ (0, T ). The Rfunction U is a solution in the mild τ (integral) sense, i.e., U (τ ) = e−Aτ U0 + 0 e−A(τ −s) (F (s, U (s)) + h(s))ds, τ ∈ [0, T ]. Applying Propositions 2 and 3, we can state the following result, ˇevˇcoviˇc and which is a nontrivial generalization of the result shown by S Cruz [2] for n = 1. Theorem 2. [3, Theorem 1] Suppose that the shift mapping ξ = ξ(x, z) satisfies supx∈R |ξ(x, z)| ≤ C0 |z|ω (1 + eD0 |z| ), z ∈ Rn , for some constants C0 > 0, D0 ≥ 0, ω > 0. Assume ν is an admissible activity L´evy measure with the shape parameters α, D, and, either µ > 0, D ∈ R, or µ = 0, D > D0 ≥ 0. Assume 1/2 ≤ γ < 1 and γ > (α − n)/(2ω), n ≥ 1. Suppose that g(τ, x, u, ∇xu) is H¨ older continuous in the τ variable and Lipschitz

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continuous in the remaining variables, respectively. Assume u0 ∈ X γ , and T > 0. Then, there exists a unique mild solution u to PIDE (22) that satisfies u ∈ C([0, T ], X γ ).

4.2. Maximal Monotone Operator Technique for Solving Nonlinear Parabolic Equations This section presents the existence and uniqueness results of a fully nonlinear parabolic equation using the monotone operator technique. We consider the HJB equation arising from portfolio optimization selection, where the goal is to maximize the conditional expected value of the terminal utility of the portfolio. Such a fully nonlinear HJB equation presented in an abstract setting can be viewed as a PIDE in some sense. First, we employ the so-called Riccati transformation method to transform the fully nonlinear HJB equation into a quasilinear parabolic equation, which can be viewed as the porous media type of equation with source term. Then, we showed that the underlying operator is maximally monotone in some Sobolev spaces. Next, we employed the Banach fixed point theorem and Fourier transform technique to obtain the existence and uniqueness of a solution to the general form of the transformed parabolic equation in an abstract setting in high-dimensional spaces. Furthermore, as a crucial requirement for solving the Cauchy problem, we find that the diffusion function to the quasilinear parabolic equation is globally Lipschitz continuous under some assumptions. We consider the Cauchy problem for the nonlinear parabolic PDE of the following form: ∂τ ϕ − ∆α(τ, ϕ) = g0 (τ, ϕ) + ∇ · g1 (τ, ϕ),

ϕ(·, 0) = ϕ0 ,

(25) (26)

where τ ∈ (0, T ), x ∈ Rd , d ≥ 1. The solution ϕ = ϕ(x, τ ) to such a nonlinear parabolic equation is established in some Sobolev spaces in high-dimensional spaces (see [4]). To achieve such results, we assumed that the diffusion function α = α(x, τ, ϕ) is globally Lipschitz continuous and strictly increasing in the ϕ-variable. An example of such a Lipschitz continuous function α(x, τ, ϕ) is the value function of the following para-

Linear and Nonlinear Partial Integro-Differential Equations ... 215 metric optimization problem:   ϕ α(x, τ, ϕ) = min −µ(x, t, θ) + σ(x, t, θ)2 , θ∈4 2

τ ∈ (0, T ), x ∈ Rd , ϕ > ϕmin , (27)

where µ and σ 2 are given C 1 functions, and 4 ⊂ Rn is a compact decision set. The properties of the value function depend on the structure of the decision set 4. It is smooth if 4 is a convex set; meanwhile, it can only be C 0,1 smooth if 4 is not connected. 4.2.1.

Existence and Uniqueness of a Solution to the Cauchy Problem

First, we define our underlying function spaces. Let V ,→ H ,→ V 0 be a Gelfand triple, where Z 2 d d 2 H = L (R ) = {f : R → R, kf kL2 = |f (x)|2dx < ∞} Rd

Ris a Hilbert space endowed with the 0 inner product (f, g) Rd f (x)g(x)dx. The Banach spaces V and V are defined as follows: V = H 1 (Rd),

=

V 0 = H −1 (Rd ),

where the Sobolev spaces H s(Rd ) are defined by means of the Fourier transform Z 1 ˆ e−ix·ξ f (x)dx, ξ = (ξ1 , ξ2 , ..., ξd)T ∈ Rd , f (ξ) = (2π)d/2 Rd H s (Rd ) = {f : Rd → R, (1 + |ξ|2 )s/2 fˆ(ξ) ∈ L2 (Rd )}, s ∈ R R endowed with the norm kf k2H s = Rd (1 + |ξ|2 )s |fˆ(ξ)|2dξ, and |ξ| = (ξ12 + · · · + ξd2 )1/2 . Let the linear operator A : V → V 0 be defined as follows: Aψ = ψ − ∆ψ. It is worth nothing that the operator A is self-adjoint in the Hilbert space H = L2 (Rd) with the following Fourier transform representation: c ˆ Aψ(ξ) = (1 + |ξ|2 )ψ(ξ).

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s ψ(ξ) = (1 + |ξ|2 )s ψ(ξ), d ˆ The fractional power of A is defined by A s ∈ R. In particular, ±1/2 ψ(ξ) = (1 + |ξ|2 )±1/2 ψ(ξ), ˆ A\

and A−1/2 is a self-adjoint operator in the Hilbert space H = L2 (Rd ). Moreover, A−1 = A−1/2 A−1/2 . In the sequel, we denote the duality pairing between spaces V and V 0 by h., .i, i.e., the value of a functional F ∈ V 0 at u ∈ V is denoted by hF, ui. We have the following definitions. Definition 5. [4, 42] An operator (in general, nonlinear) B : V → V 0 is said to be (i) monotone if hB(u) − B(v), u − vi ≥ 0, ∀ u, v ∈ V, (ii) strongly monotone if there exists a constant C > 0 such that hB(u) − B(v), u − vi ≥ Cku − vk2V , ∀ u, v ∈ V, (iii) hemicontinuous if for each u, v ∈ V , the real-valued function t 7→ B(u + tv)(v) is continuous. Theorem 3. [42, 43] Let V be a separable reflexive Banach space, dense, and continuous in a Hilbert space H, which is identified with its dual, so V ,→ H ,→ V 0 . Let p ≥ 2 and set V = Lp((0, T ); V ). Assume a family of operators A(τ, .) : V → V 0 , 0 ≤ τ < T , is given such that (i) for each ϕ ∈ V , the function A(., ϕ) : [0, T ] → V 0 is measurable, (ii) for a.e τ ∈ [0, T ], the operator A(τ, .) : V → V 0 is monotone, hemicontinuous, and bounded by kA(τ, ϕ)k ≤ C(kϕkp−1 + k(τ )), ϕ ∈ 0 V, 0 ≤ τ < T, where k ∈ Lp (0, T ), (iii) and there exists λ > 0 such that hA(τ, ϕ), ϕi ≥ λkϕkp − k(τ ), ϕ ∈ V, 0 ≤ τ < T.

Then, for each fˆ ∈ V 0 and ϕ0 ∈ H, there exists a unique solution ϕ ∈ V of the Cauchy problem ∂τ ϕ(τ ) + A(τ, ϕ(τ )) = fˆ(τ ) in V 0 , ϕ(0) = ϕ0 .

Linear and Nonlinear Partial Integro-Differential Equations ... 217 Consider the spaces V = L2 ((0, T ); V ), H = L2 ((0, T ); H), and V 0 = i.e., p = 2. Thus, we have that these spaces satisfy the Gelfand triple, i.e., V ,→ H ,→ V 0 , where H is a Hilbert space endowed with the norm Z L2 ((0, T ); V 0 ),

kϕk2H =

T

0

kϕ(τ )k2H dτ, ∀ϕ ∈ H.

For a given value ϕmin , we denote D = Rd × (0, T ) × (ϕmin , ∞).

Theorem 4. [4, Theorem 2] Assume that the above settings on H and V hold. Let g0 , g1j : [0, T ] × H → H, j = 1, · · · , n, be globally Lipschitz continuous functions. Suppose α ∈ C 0,1 (D) is such that there exist constants ω, L, L0 > 0 such that 0 < ω ≤ α0ϕ (x, τ, ϕ) ≤ L, |∇x α(x, τ, ϕ)| ≤ p(x, τ ) + L0 |ϕ|, α(x, τ, 0) = h(x, τ ) for a.e. (x, τ, ϕ) ∈ D and p, h ∈ L∞ ((0, T ); H). Then, for any T > 0 and ϕ0 ∈ H, there exists a unique solution ϕ ∈ V of the Cauchy problem ∂τ ϕ + Aα(·, τ, ϕ) = g0 (τ, ϕ) + ∇ · g1 (τ, ϕ),

ϕ(0) = ϕ0 .

(28)

We remark here that the above result and its proof are contained in our recent paper [4, Theorem 2]. Proof: Recall that H = L2 (Rd ) and V = H 1 (Rd ), its dual space being 0 V 0 = H −1 (Rd ). Let the scalar products in V and V be respectively defined by (f, g)V = (A1/2 f, A1/2 g)H = (Af, g)H , (f, g)V 0 = (A−1/2 f, A−1/2 g)H = (A−1 f, g)H .

Let us define the operator A(τ, ·) : V → V 0 by hA(τ, ϕ), ψi = (A−1 Aα(·, τ, ϕ), ψ)H = (α(·, τ, ϕ), ψ)H . Therefore, we conclude that the mapping ϕ 7→ α(·, τ, ϕ) maps V into V under the assumption made on the function α. Indeed, if ϕ ∈ V and η = α(·, τ, ϕ), then η(x) = α(x, τ, ϕ(x)) − α(x, τ, 0) + α(x, τ, 0), and so |η(x)| ≤ (max α0ϕ (x, τ, ϕ))|ϕ(x)| + |h(x, τ )| ≤ L|ϕ(x)| + |h(x, τ )|. ϕ

R

R Thus, Rd |η(x)|2dx ≤ 2 Rd L2 |ϕ(x)|2 + |h(x, τ )|2dx ≤ 2L2 kϕk2H + 2kh(·, τ )k2H . Since ∇η(x) = ∇x α(x, τ, ϕ(x)) + α0ϕ (x, τ, ϕ(x))∇ϕ(x), we have

ˇevˇcoviˇc et al. Jos´ e Cruz, Maria Grossinho, Daniel S

218

Z kηk2V = |η(x)|2 + |∇η(x)|2 dx d R Z Z ≤2 L2 |ϕ(x)|2 + |h(x, τ )|2 dx + 2 |p(x, τ )|2 + L20 |ϕ(x)|2 dx+ Rd Rd Z +2 L2 |∇ϕ(x)|2dx Rd

2

≤2(L kϕk2V + kh(·, τ )k2H + kp(·, τ )k2H + L20 kϕk2H ) < ∞,

because p, h ∈ L∞ ((0, T ); H). Consequently, η ∈ V , as claimed. Next, we show that the operator A is monotone in the space V 0 . According to (77), we have (α(x, τ, ϕ1)−α(x, τ, ϕ2 ))(ϕ1 −ϕ2 ) ≥ ω(ϕ1 −ϕ2 )2 , for any ϕ1 , ϕ2 ≥ ϕmin , x ∈ R, τ ∈ [0, T ]. hA(τ, ϕ1) − A(τ, ϕ2), ϕ1 − ϕ2 i = (α(·, τ, ϕ1) − α(·, τ, ϕ2), ϕ1 − ϕ2 ) Z = (α(x, τ, ϕ1(x)) − α(x, τ, ϕ2(x)))(ϕ1(x) − ϕ2 (x))dx Rd Z ≥ ω|ϕ1 (x) − ϕ2 (x)|2dx = ωkϕ1 − ϕ2 k2H . Rd

This implies that the operator A(τ, ·) is strongly monotone. For a given ϕ˜ ∈ H, we have fˆ ∈ V 0 , where fˆ(τ ) = g0 (τ, ϕ(·, ˜ τ )) + ∇ · g1 (τ, ϕ(·, ˜ τ )), because g0 , g1j : [0, T ] × H → H are globally Lipschitz continuous, H ,→ V 0 , and the operator ∇ maps H into V 0 . The hemicontinuity, boundedness, and coercivity of the operator A follow from the assumption that the function α is globally Lipschitz continuous and strictly increasing. Applying Theorem 3, we deduce the existence of a solution ϕ ∈ V such that ∂τ ϕ + A(τ, ϕ) = fˆ(τ ), ϕ0 ∈ H, (29) where A(τ, ϕ) = Aα(·, τ, ϕ). Next, we multiply (29) by A−1 to obtain ∂τ A−1 ϕ + α(·, τ, ϕ) = f,

(30)

where f = f (τ, ϕ) ˜ = A−1 fˆ(τ ). For τ ∈ [0, T ], we denote f˜(ϕ) ˜ = P −1/2 ˆ −1/2 A f (τ ) = A g0 (τ, ϕ) ˜ + A−1/2 dj=1 ∂xj g1j (τ, ϕ). ˜ The Fourier transform of f˜ is defined by ˜(ϕ)(ξ) f[ ˜ =

d X (−iξj ) 1 \ g0 (τ, ϕ)(ξ) ˜ + g\ (τ, ϕ)(ξ). ˜ 2 1/2 2 )1/2 1j (1 + |ξ| ) (1 + |ξ| j=1

Linear and Nonlinear Partial Integro-Differential Equations ... 219 Let β > 0 be the Lipschitz constant of the mappings g0 , g1j , j = 1, · · · , d. Using Parseval’s identity and Lipschitz continuity of g0 , g1j in H, we obtain, for ϕ˜1 , ϕ ˜2 ∈ H, Z \ \ 2 2 ˜ ˜(ϕ˜1 )(ξ) − f\ ˜(ϕ˜2 )(ξ)|2dξ ˜ ˜ ˜ |f\ kf (ϕ˜1 ) − f (ϕ˜2 )kH = kf (ϕ˜1 ) − f (ϕ˜2 )kH = Rd Z 1 |g \ (τ, ϕ ˜1)(ξ) − g0\ (τ, ϕ ˜2)(ξ)|2 ≤2 2 0 1 + |ξ| d R +

d X j=1

|ξ|2 |g1j\ (τ, ϕ ˜ 1)(ξ) − g1j\ (τ, ϕ ˜ 2)(ξ)|2 dξ 1 + |ξ|2

≤ 2kg0\ (τ, ϕ ˜1) − g0\ (τ, ϕ2)k2H + 2 = 2kg0(τ, ϕ ˜ 1) − ˜2

≤β

g0 (τ, ϕ ˜2)k2H

+2

d X j=1

d X j=1

kϕ˜1 − ϕ˜2 k2H ,

kg1j\ (τ, ϕ ˜1) − g1j\ (τ, ϕ2)k2H kg1j (τ, ϕ ˜1) − g1j (τ, ϕ ˜ 2)k2H

where β˜2 = 2(1 + d)β 2 . Hence, we obtain ˜ ϕ˜1 − ϕ˜2 kH . kf˜(ϕ˜1 ) − f˜(ϕ˜2 )kH ≤ βk

(31)

Suppose ϕ1 , ϕ2 ∈ H are such that ϕ1 = F (ϕ˜1 ) and ϕ2 = F (ϕ˜2 ). Here, the map F : H → H is defined by ϕ = F (ϕ), ˜ where ϕ is a solution to the Cauchy problem ∂τ A−1 ϕ + α(·, τ, ϕ) = f (τ, ϕ), ˜

ϕ(0) = ϕ0 .

Letting ϕ = ϕ1 − ϕ2 = F (ϕ˜1 ) − F (ϕ˜2 ), we obtain ∂τ A−1 (ϕ1 − ϕ2 ) + α(·, τ, ϕ1) − α(·, τ, ϕ2) = f (ϕ˜1 ) − f (ϕ˜2 ).

(32)

Next, multiplying (32) by ϕ1 − ϕ2 and taking the scalar product in the space H, we obtain (∂τ A−1 (ϕ1 − ϕ2 ), ϕ1 − ϕ2 ) + (α(·, τ, ϕ1) − α(·, τ, ϕ2), ϕ1 − ϕ2 )

= (f (τ, ϕ ˜1) − f (τ, ϕ ˜2), ϕ1 − ϕ2 ). (33)

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Using (31) and the fact that A−1/2 is self-adjoint in H, then (33) gives 1 d kA−1/2 (ϕ1 − ϕ2 )k2H + ωkϕ1 − ϕ2 k2H 2 dτ ≤ hf (τ, ϕ ˜1 ) − f (τ, ϕ ˜2 ), ϕ1 − ϕ2 i = hA1/2 (f (τ, ϕ ˜1 ) − f (τ, ϕ ˜2 )), A−1/2 (ϕ1 − ϕ2 )i ≤ kA1/2 (f (τ, ϕ˜1 ) − f (τ, ϕ˜2 ))kH kϕ1 − ϕ2 kV 0 = kf˜(ϕ ˜1 ) − f˜(ϕ ˜2 )kH kϕ1 − ϕ2 kV 0 ˜ ϕ˜1 − ϕ˜2 kH kϕ1 − ϕ2 kV 0 . ≤ βk

This implies 1 d ˜ ϕ˜1 − ϕ˜2 kH kϕ1 − ϕ2 kV 0 . kϕ1 − ϕ2 k2V 0 + ωkϕ1 − ϕ2 k2H ≤ βk 2 dτ Then, integrating on a small time interval [0, T ] from 0 to t and noting that ϕ1 (0) = ϕ2 (0) = ϕ0 , we obtain Z τ 1 kϕ1 (τ ) − ϕ2 (τ )k2V 0 + ω kϕ1 (s) − ϕ2 (s)k2H ds 2 0 Z τ ≤ β˜ kϕ˜1 (s) − ϕ˜2 (s)kH kϕ1 (s) − ϕ2 (s)kV 0 ds 0 Z T ≤ β˜ max kϕ1 (τ ) − ϕ2 (τ )kV 0 kϕ˜1 (τ ) − ϕ˜2 (τ )kH dτ. τ ∈[0,T ]

0

Taking the maximum over τ ∈ [0, T ] and using the fact that for any a, b ∈ R, ab ≤ 21 a2 + 12 b2 , we obtain Z T 1 ( max kϕ1 (τ ) − ϕ2 (τ )kV 0 )2 + ω kϕ1 (τ ) − ϕ2 (τ )k2H dτ 2 τ ∈[0,T ] 0 Z T ≤ β˜ max kϕ1 (τ ) − ϕ2 (τ )kV 0 kϕ˜1 (τ ) − ϕ˜2 (τ )kH dτ τ ∈[0,T ]

0

Z T 1 β˜2 2 0 ≤ ( max kϕ1 (τ ) − ϕ2 (τ )kV ) + ( kϕ˜1 (τ ) − ϕ˜2 (τ )kH dτ )2 . 2 τ ∈[0,T ] 2 0 RT Using the Cauchy–Schwartz inequality, we obtain ω 0 kϕ1 (τ ) − RT ˜2 R T ˜2 R T ϕ2 (τ )k2H dτ ≤ β2 0 dτ 0 kϕ˜1 (τ ) − ϕ˜2 (τ )k2H dτ = β 2T 0 kϕ˜1 (τ ) − ϕ˜2 (τ )k2H dτ. This implies that kF (ϕ˜1 ) − F (ϕ˜2 )k2H ≤

β˜2 T kϕ˜1 − ϕ˜2 k2H . 2ω

Linear and Nonlinear Partial Integro-Differential Equations ... 221 ˜2

Thus, for sufficiently small value of T such that β2ωT < 1, the operator F is a contraction on the space H. Therefore, by the Banach fixed point theorem, F has a unique fixed point in H. It is worth noting that β˜ and ω are given such that they are independent of T . If T > 0 is arbitrary, then we can apply a simple continuation argument. In other words, if ˜2 the solution exists in (0, T0) interval with β2ωT0 < 1, then starting from the initial condition ϕ0 = ϕ(T0 /2), we can continue the solution ϕ from the interval (0, T0) over the interval (0, T0) ∪ (T0 /2, T0/2 + T0 ) ≡ (0, 3T0/2). Continuing in this manner, we obtain the existence and uniqueness of a solution ϕ ∈ H defined on the time interval (0, T ). Finally, the solution belongs to the space V because the right-hand side, i.e., the function fˆ(τ ) = g0 (τ, ϕ(·, τ )) + ∇ · g1 (τ, ϕ(·, τ )) belongs to V 0 . Applying Theorem 3 we conclude ϕ ∈ V, as claimed. ♦ The following result demonstrates the absolute continuity and a-priori energy estimate property of the solution. Based on the assumption of the previous theorem, we have α(·, 0), g0(·, 0), g1j(·, 0) ∈ H. Here, the space X = L∞ ((0, T ); V 0 ) is endowed with the norm kϕk2X = sup kϕ(τ )k2V 0 , ∀ϕ ∈ X . τ ∈[0,T ]

Again, the following result and its proof are contained in our recent paper [4, Theorem 3]. Theorem 5. [4, Theorem 3] Suppose that the functions α, g0 , g1j satisfy the assumptions of Theorem 4. Then, the unique solution ϕ ∈ V to the Cauchy problem (29) is absolutely continuous, i.e., ϕ ∈ C([0, T ]; H). Moreover, there exists a constant C˜ > 0, such that the unique solution satisfies the following inequality: kϕk2X + kϕk2H ≤ C˜ kϕ0 k2V 0 + kα(·, 0)k2H + kg0 (·, 0)k2H +

d X j=1


kg1j (·, 0)k2H .

(34)

Proof. Since fˆ ∈ V 0 , where fˆ = g0 + ∇ · g1 and A(τ, ϕ) ∈ V 0 , then ∂τ ϕ ∈ V 0 . Therefore, for each ϕ0 ∈ H, we have ϕ ∈ W , where W is the Banach space W = {ϕ, ϕ ∈ V, ∂τ ϕ ∈ V 0 }. According to [43, Proposition

ˇevˇcoviˇc et al. Jos´ e Cruz, Maria Grossinho, Daniel S

222

1.2], we have W ,→ C([0, T ]; H). Hence, the unique solution ϕ to the Cauchy problem (4) belongs to the space C([0, T ]; H), as claimed. Next, we show that the unique solution satisfies a-priori energy estimate (34). Let ϕ be a unique solution to the Cauchy problem (28). Multiply (30) by ϕ and take the scalar product in H to obtain (∂τ A−1 ϕ, ϕ)H + (α(·, τ, ϕ), ϕ)H = (A−1 g0 (τ, ϕ) + A−1 ∇ · g1 (τ, ϕ), ϕ). (35) Using the Lipschitz continuity of g0 , g1 , and strong monotonicity of α, we obtain 1 d 2 2 dτ kϕkV 0

+ ωkϕk2H = (∂τ A−1 ϕ, ϕ) + ωkϕk2H

≤ (∂τ A−1 ϕ, ϕ) + (α(·, ϕ) − α(·, 0), ϕ)

= (A−1 (g0 (·, ϕ) + ∇ · g1 (τ, ϕ)) − α(·, 0), ϕ)

= (A−1 (g0 (·, ϕ) − g0 (·, 0) + ∇ · g1 (·, ϕ) − ∇ · g1 (·, 0)), ϕ) +(A−1 (g0 (·, 0) + ∇ · g1 (·, 0)), ϕ) − (α(·, 0), ϕ)

= (A−1/2 (g0 (·, ϕ) − g0 (·, 0) + ∇ · g1 (·, ϕ) − ∇ · g1 (·, 0)), A−1/2ϕ) +(A−1/2 (g0 (·, 0) + ∇ · g1 (·, 0)), A−1/2ϕ) − (α(·, 0), ϕ) ≤ β(1 + d)kϕkH kϕkV 0 + kA−1/2 (g0 (·, 0) + ∇ · g1 (·, 0))kH kϕkV 0 ≤

ω 2 4 kϕkH

+

+kα(·, 0)kH kϕkH β 2 (1+d)2 kϕk2V 0 + 12 kA−1/2 (g0 (·, 0) + ∇ ω + 12 kϕk2V 0 + ω1 kα(·, 0)k2H + ω4 kϕk2H .

· g1 (·, 0))k2H

Hence, there exist constants C0 , C1 > 0 such that d kϕk2V 0 + ωkϕk2H dτ

≤

d X ` ´ C1 kϕk2V 0 + C0 kg0 (·, 0)k2H + kg1j (·, 0)k2H + kα(·, 0)k2H . j=1

Solving the differential inequality y 0 (τ ) ≤ C1 y(τ ) + r(τ ), where P y(τ ) = kϕ(·, τ )k2V 0 and r(τ ) = C0 kg0 (·, τ, 0)k2H + dj=1 kg1j (·, τ, 0)k2H +
kα(·, τ, 0)k2H , we obtain C1 T

y(τ ) ≤ e

y(0) +

and the proof of the theorem follows.

Z

0

T


r(s)ds ,

Linear and Nonlinear Partial Integro-Differential Equations ... 223

5. Applications to Option Pricing The classical linear Black–Scholes model and its multidimensional generalizations have been widely used in the financial market analysis. It is well known that the price V = V (t, S) of an option on an underlying asset price S at time t ∈ [0, T ] can be obtained as a solution to the linear Black–Scholes parabolic equation of the form (1). Generally, the underlying asset price is assumed to follow the geometric Brownian motion dS/S = µdt + σdW . Here, {Wt , t ≥ 0} is the standard Wiener process. The terminal condition Φ(S) represents the payoff diagram at maturity t = T , Φ(S) = (S − K)+ (call option case) or Φ(S) = (K − S)+ (put option case). For the multidimensional case, where the option price V (t, S1, · · · , Sn) depends on the vector of n underlying stochastic assets S = (S1 , · · · , Sn ) with the volatilities σi and mutual correlations %ij , i, j = 1, · · · , n, the Black–Scholes pricing equation can be expressed as follows: n n n X ∂V 1 XX ∂ 2V ∂V + ρij σi σj Si Sj +r Si − rV = 0, ∂t 2 i=1 j=1 ∂Si ∂Sj ∂S i i=1

V (T, S) = Φ(S). (36)

Equations (1) and (36) can be transformed into equation (21) defined ˇevˇcoviˇc, Stehl´ıkov´ on the whole space Rn (cf., S a, Mikula [44, Chapter 4, Section 5]). According to stock market observations, the models (1) and (36) were derived under some restrictive assumptions, e.g., completeness and frictionless of the financial market, perfect replication of a portfolio and its liquidity, and absence of transaction costs. However, these assumptions are often violated in financial markets. In recent decades, several attempts have been made to investigate the effects of nontrivial transaction costs [7, 6, 45, 46]. For example, Sch¨ onbucher and Willmott [8], Frey and Patie [9], Frey and Stremme [10] investigated the feedback and illiquid market effects due to large traders choosing given stock-trading ˇevˇcoviˇc recently investigated the effects of risk strategies. Jandaˇcka and S arising from an unprotected portfolio. Barles and Soner [47] analyzed option pricing models based on utility maximization. The common feature of these generalizations of the linear Black–Scholes equation (1) is that the constant volatility σ is replaced by a nonlinear function depend-

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ing on the second derivative ∂S2 V of the option price V . Among these generalizations, Frey and Stremme [11] derived a nonlinear Black–Scholes model by assuming that the underlying asset dynamics takes into account the presence of feedback effects due to the influence of a large trader choosing a particular stock trading strategy (see also [8, 9, 12]). ˇevˇcoviˇc [1] generalized the Black–Scholes equaRecently, Cruz and S tion in two important directions. First, they used the ideas of Frey and Stremme [11] to incorporate the effect of a large trader into the model. Second, they relaxed the assumption on liquidity of market by assuming that the underlying asset price follows a L´ evy stochastic process with jumps to obtain the following nonlinear PIDE:

0 = +

1 σ2 S2 ∂V ∂2V ∂V + + rS − rV 2 ∂t 2 (1 − %S∂S φ) ∂S 2 ∂S  Z  ∂V V (t, S + H) − V (t, S) − H ν(dz), ∂S R

(37)

where the shift function H = H(φ, S, z) depends on the large investor stocktrading strategy function φ = φ(t, S). Moreover, this shift function is a solution to the following implicit algebraic equation: H = ρS(φ(t, S + H) − φ(t, S)) + S(ez − 1).

(38)

The large trader strategy function φ may depend on the derivative ∂S V of the option price V , e.g., φ(t, S) = ∂S V (t, S). However, in our application, we assume the trading strategy function φ(t, S) is prescribed and globally H¨ older continuous. Next, we present the analysis of this equation depending the behavior of the parameter ρ = 0. If ρ = 0, then H = S(ez − 1). Thus, the equation (37) can be reduced to a linear PIDE of the form (21) in the one-dimensional space (n = 1). This is S obtained using the standard transformation τ = T − t, x = ln( K ) and setting −rτ V (t, S) = e u(τ, x). However, if ρ > 0, then (37) can be transformed into a nonlinear parabolic PIDE. Indeed, suppose that the transformed large trader stock-trading strategy ψ(τ, x) = φ(t, S). Then, V (t, S) solves equation (37) if and only if the transformed function u(τ, x) is a solution to the following nonlinear parabolic equation:   ∂u σ2 1 ∂2u σ2 1 ∂u ∂τ = 2 (1−ρ∂x ψ)2 ∂ 2 x + r − 2 (1−ρ∂x ψ)2 − δ(τ, x) ∂x R
+ R u(τ, x + ξ) − u(τ, x) − ξ ∂u u(0, x) = Φ(Kex ) (39) ∂x (τ, x) ν(dz),

Linear and Nonlinear Partial Integro-Differential Equations ... 225 τ ∈ [0, T ], x ∈ R. The shift function ξ(τ, x, z) is a solution to the following algebraic equation: eξ = ez + ρ(ψ(τ, x + ξ) − ψ(τ, x)), (40) R R and δ(τ, x) = R (eξ − 1 − ξ)ν(dz) = R (ez − 1 − ξ + ρ(ψ(τ, x + ξ) − ψ(τ, x)))ν(dz). For small values of 0 < ρ  1, we can construct the first order asymptotic expansion ξ(τ, x, z) = ξ0 (τ, x, z) + ρξ1 (τ, x, z). For ρ = 0, we obtain ξ0 (τ, x, z) = z. Hence, ez+ρξ1 = ez + ρ(ψ(τ, x + z + ρξ1 ) − ψ(τ, x)). Taking the first derivative of the above implicit equation with respect to ρ and evaluating it at the origin ρ = 0, we obtain ξ1 = e−z (ψ(τ, x + z) − ψ(τ, x)), i.e., ξ(τ, x, z) = z + ρe−z (ψ(τ, x + z) − ψ(τ, x)). (41) Consequently, we obtain the following lemma. Lemma 6. [3, Lemma 1] Assume that the stock-trading strategy φ = φ(t, S) is a globally ω-H¨ older continuous function, 0 < ω ≤ 1. Then, the transformed function ψ(τ, x) = φ(t, S) is ω-H¨ older continuous, and the first order asymptotic expansion ξ(τ, x, z) of the nonlinear algebraic equation (40) is ω-H¨ older continuous in all variables. Furthermore, there exists a constant C0 > 0 such that supτ,x |ξ(τ, x, z)| ≤ C0 |z|ω (1 + e|z| ) for any z ∈ R.

5.1. Linearization of PIDE In what follows, we consider a simplified linear approximation of (37) by setting ρ = 0 in the diffusion function, but we keep the shift function H depending on the parameter ρ. Then, the transformed Cauchy problem for the solution u with the first order approximation of the shift function ξ is given as follows: ∂u ∂τ

=

  σ2 ∂ 2 u σ2 ∂u + r − + δ(τ, x) 2 ∂2x 2 ∂x  Z  ∂u + u(τ, x + ξ) − u(τ, x) − ξ (τ, x) ν(dz), ∂x R

(42)

τ ∈ [0, T ], x ∈ R, where ξ(τ, x, z) = z + ρ(ψ(τ, x + z)) − ψ(τ, x)). Note that the call/put option payoff functions Φ(S) = Φ(Kex ) = (S−K)+ = K(ex −1)+ / Φ(S) = Φ(Kex ) = (K −S)+ = K(1 −ex )+ do not belong to the Banach space X γ . According to [2], the procedure on how to overcome this problem and formulate the existence and uniqueness of a solution to the PIDE (42) is based on shifting the solution u by uBS . Here, uBS (τ, x) = erτ V BS (T − τ, Kex )

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is an explicitly given solution to the linear Black–Scholes equation without the PIDE part. In other words, uBS solves the following linear parabolic equation: „ « σ2 ∂ 2 uBS σ2 ∂uBS ∂uBS − − r − = 0, ∂τ 2 ∂x2 2 ∂x

uBS (0, x) = Φ(Kex), τ ∈ (0, T ), x ∈ R. (43)

Recall that uBS (τ, x) = Kex+rτ N (d1 ) − KN (d2 ) (call option case), where Rd √ 2 d1,2 = (x + (r ± σ 2 /2)τ )/(σ τ ) (cf., [6, 44]). Here, N (d) = √12π −∞ e−ξ /2 dξ is the cumulative density function of the normal distribution. The next result and ˇevˇcoviˇc and Udeani [3]. its proof are based on the recent paper by S Theorem 7. [3, Theorem 2] Assume the transformed stock-trading strategy function ψ(τ, x) is globally ω-H¨ older continuous in both variables. Suppose that ν is a L´evy measure with the shape parameters α < 3, D ∈ R, where µ > 0, or p µ = 0 and D > 1. Let X γ be the space of Bessel potentials space L2γ (R), where p+1 1 α−1 < γ < and ≤ γ < 1. Let T > 0. Then, the linear PIDE (42) has a 2ω 2p 2 BS unique mild solution u with the property that the difference U = u − u belongs to the space C([0, T ], X γ ). Proof. We first outline the idea of the proof. The initial condition u(0, ·) 6∈ X γ for two reasons. It is not smooth for x = 0 and grows exponentially for x → ∞ (call option) or x → −∞ (put option). The shift function U = u − uBS satisfies U (0, ·) ≡ 0, and so the initial condition U (0, ·) belongs to X γ . However, the shift function uBS enters the governing PIDE as it includes the term f(uBS (τ, ·)) in the right-hand side. Since uBS (0, x) is not sufficiently smooth for x = 0, the shift term f(uBS (τ, ·)) is singular for τ → 0+ . Following the ideas of [2], for the shift term f(uBS (τ, ·)), we can provide H¨ older estimates, which are sufficient to prove the main result of this theorem (cf., [2, Lemma 4.1]). Furthermore, the exponential growth of the function uBS will be overcome since f˜(ex ) = 0, where f˜(u) = f(u) − δ(τ, ·)∂x u, i.e., Z
f˜(u)(x) = u(x + ξ) − u(x) − (eξ − 1)∂x u(x) ν(dz). R

Next, we present more details of the proof. The function uBS solves the linear PDE (43). Thus, the difference U = u − uBS of the solution u to (42) and uBS satisfies the PIDE with the right-hand side:   ∂U σ2 ∂ 2 U σ2 ∂U ∂uBS BS = + r − − δ(τ, x) + f(U ) + f(u ) − δ(τ, x) ∂τ 2 ∂x2 2 ∂x ∂x σ2 ∂ 2 U = + f(U ) + g(τ, x, ∂xU ) + h(τ, ·), 2 ∂x2

Linear and Nonlinear Partial Integro-Differential Equations ... 227 U (0, x) = 0, x ∈ R, τ ∈ (0, T ). Here g(τ, x, ∂x U ) = (r − σ 2 /2 − δ(τ, x))∂x U , and h(τ, ·) = f˜(uBS (τ, ·)). According to Proposition 2, f : X γ → X is a bounded linear mapping. Consequently, it is Lipschitz continuous, provided that 1/2 ≤ ˜ x ) = 0. Hence, γ < 1 and γ > (α − 1)/(2ω). Clearly, f(e f˜(uBS ) = f˜(uBS − Kerτ+x ),

˜ BS ) = f˜(∂τ (uBS − Kerτ+x )). and ∂τ f(u

Now, it follows from [2, Lemma 4.1] that the following estimate holds true: kh(τ1 , ·) − h(τ2 , ·)kLp = kf˜(uBS (τ1 , ·)) − f˜(uBS (τ2 , ·))kLp ≤ C0 |τ1 − τ2 |−γ+

p+1 2p

,

1 1 kh(τ, ·)kLp = kf˜(uBS )(τ, ·))kLp ≤ C0 |τ −(2γ−1)( 2 − 2p ) ,

for any 0 < τ1 , τ2 , τ ≤ T . The function h : [0, T ] → X ≡ Lp (R) is ((p + 1)/(2p) − γ)-H¨ older continuous because γ < p+1 2p . Moreover, Z

0

T

kh(τ, ·)k

Lp

dτ =

Z

0



T

kf˜(uBS (τ, ·))kLp dτ ≤ C0 

Z

0

T

τ −(2γ−1)( 2 − 2p ) dτ < ∞, 1

1

1 because (2γ − 1) 12 − 2p < 1. Recall that the crucial part of the proof of [2, Lemma 4.1] was based on the estimates:

kf˜(uBS (τ, ·))kLp ≤ C0 kv(τ, ·)kX γ−1/2 , and k∂τ f˜(uBS (τ, ·))kLp ≤ C0 k∂τ v(τ, ·)kX γ−1/2 ,


where v(τ, x) = ∂x uBS (τ, x) − Kerτ+x = Kerτ+x (N (d1 (τ, x)) − 1). This estimate is satisfied because of Proposition 2 under the assumptions made on γ. The proof for the case of a put option is similar. The final estimate on the H¨ older continuity of the mapping h follows from careful estimates of the solution uBS derived in the proof of [2, Lemma 4.1]. Now, the proof follows from Theorem 2 and Proposition 3.

6. Feedback Effects under Jump-Diffusion Asset Price Dynamics Suppose that a large trader uses a stock-holding strategy αt and St is a cadlag process (right continuous with limits to the left). In what follows, we will identify St with St− . We assume St has the following dynamics: Z dSt = µSt dt + σSt dWt + ρSt dαt + St (ex − 1)JX (dt, dx), (44) R

which can seen as a perturbation of the classical jump-diffusion model. For instance, if a large trader does not trade, then αt = 0 or the market liquidity

ˇevˇcoviˇc et al. Jos´ e Cruz, Maria Grossinho, Daniel S

228

parameter ρ is set to zero, then the stock price St follows the classical jumpdiffusion model. We will assume the following structural hypothesis in this chapter: Assumption 1. [1, Assumption 1] Assume the trading strategy αt = φ(t, St ) and ∂φ |. the parameter ρ ≥ 0 satisfy ρL < 1, where L = supS>0 |S ∂S Next, we show an explicit formula for the dynamics of St satisfying (44) under certain regularity assumptions made on the stock-holding function φ(t, S). Proposition 4. Suppose that the stock-holding strategy αt = φ(t, St ) satisfies Assumption 1, where φ ∈ C 1,2([0, T ] × R+ ). If the process St , t ≥ 0, satisfies the implicit stochastic equation (44), then the process St is driven by the following stochastic differential equation (SDE): Z dSt = b(t, St )St dt + v(t, St )St dWt + H(t, x, St )JX (dt, dx), (45) R

where    2 ∂φ 1 2 2∂ φ b(t, S) = µ+ρ + v(t, S) S , ∂φ ∂t 2 ∂S 2 (t, S) 1 − ρS ∂S σ v(t, S) = , ∂φ 1 − ρS ∂S (t, S) H(t, x, S) = S(ex − 1) + ρS [φ(t, S + H(t, x, S)) − φ(t, S)] . 1

(46) (47) (48)

Proof. First, the SDE (45) can be expressed for St as follows: ! Z dSt

=

b(t, St )St +

+

Z

H(t, x, St)ν(dx) dt + v(t, St )St dWt

|x| 0, θ2 = θ3 = 0. For intermediate values of ϕ, two assets are active θ1 > 0, θ2 > 0, θ3 = 0. Furthermore, for larger values of ϕ, all three assets are active, i.e., θ1 > 0, θ2 > 0, θ3 > 0. ˆ The path ϕ 7→ θ(ϕ) has a discontinuity in the first derivative when it leaves a lower-dimensional object (vertex, edge) and enters a higher-dimensional object volume. The advantage of the Riccati transformation of the original HJB is twofold. First, the diffusion function α can be calculated in advance as a result of quadratic optimization problem when the vector µ and the covariance matrix Σ are given or semidefinite programming problem when they belong to a uncertainity set of returns and covariance matrices (cf. [59]). Figure 6 shows the vector of optimal weights θ, as a function of the parameter ϕ, obtained as the optimal solution to the quadratic optimization problem with the covariance matrix from [63] corresponding to the five assets (BASF, Bayer, Degussa–Huls, FMC, Scheringfrom) entering the DAX30 index from 2008. There are more nontrivial weights θi when the parameter ϕ increases. In contrast to the fully nonlinear character of the original HJB equation (66), the transformed equation (75) represents a quasilinear parabolic equa-

Linear and Nonlinear Partial Integro-Differential Equations ... 249 0.1

-0.015

0 -0.02

-0.1

(
)

"(
)

-0.025

-0.03

-0.2 -0.3 -0.4

-0.035

-0.5 -0.04 0

0.2

0.4

0.6

0.8

1

-0.6 0

0.2

0.4

0.6




0.8

1




a)

b)

Figure 5. a) A graph of the value function α(ϕ), and b) the second derivative α00ϕ (ϕ) corresponding to five stocks (BASF, Bayer, Degussa–Huls, FMC Scheringfrom) entering DAX30 index. Source: our computation is based on the method from [4].

1 0.8

( )

0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Figure 6. The optimal vector θ = (θ1 , · · · , θn )T as a function of ϕ for the German DAX30 index. Source: our computation is based on the method from [4, 49].

250

ˇevˇcoviˇc et al. Jos´ e Cruz, Maria Grossinho, Daniel S Solution ϕ(x, τ ) 10 8

ϕ(x, τ)

6 4 2 0 −2 −5

0

5

10

x

Figure 7. A solution ϕ(x, τ ) for the DARA utility function with a0 = 9, a1 = 8, x∗ = 2. Source: our computations based on the numerical method from [4]. tion in divergence form. Thus, efficient numerical schemes can be constructed for this class of equations. In our computational experiments, we employ the finite volume discretization scheme proposed and investigated by Kilianov´ a and ˇevˇcoviˇc [48, 49, 50]). Figure 7 shows the results of a time-dependent sequence S of profiles ϕ(x, τ ) for a constant initial condition ϕ0 ≡ 9. This figure also shows the solution profiles for the initial condition ϕ0 attaining four decreasing values {9, 8, 7, 6}. It represents the DARA utility function. The function ϕ(x, τ ) is increasing in the x variable and decreasing in the τ = T − t variable. Therefore, the optimal vector θ(x, τ ) contains a more diversified portfolio of assets when x increases and the time t → T (see Figure 6). Furthermore, it is reasonable to invest in an asset with the highest expected return when the account value x is low, whereas an investor has to diversify the portfolio when x is large and time t is approaching the terminal maturity T .

Conclusion This review chapter presents the analysis of solutions to nonlinear and nonlocal PDEs arising from the financial market. Specifically, we studied and analyzed linear and nonlinear PIDEs arising from option pricing and portfolio selection problem. For option pricing, we investigated the systematic relationships between the corresponding PIDEs and Black–Scholes models that study market illiquidity when the underlying asset price follows a L´ evy stochastic process with jumps. We employ the theory of abstract semilinear parabolic equations in Bessel potential spaces to establish the qualitative properties of solutions to

Linear and Nonlinear Partial Integro-Differential Equations ... 251 nonlocal linear and nonlinear PIDEs for a general class of the so-called admissible L´ evy measures satisfying suitable growth conditions at infinity and origin in the multidimensional space. We considered a general shift function arising from nonlinear option pricing models, which takes into account a large trader stock-trading strategy with the underlying asset price following the L´ evy process. Then, for the portfolio allocation problem, we presented the qualitative properties results to the fully nonlinear HJB equation arising from the stochastic dynamic optimization problem in Sobolev spaces using the theory of monotone operator technique. Such an HJB equation, presented in abstract setting, arises from portfolio management problem, where the goal of an investor is to maximize the conditional expected terminal utility of a portfolio. We also presented a stable, convergent, and consistent numerical scheme that approximates the solution of such a PIDE. Several numerical simulations were performed to demonstrate the influence of a large trader and the intensity of jumps on the option price.

Acknowledgments Support of the Slovak Research and Development Agency under the project APVV-20-0311 (C.U.) is kindly acknowledged. The research was also supported ˇ.). by the VEGA 1/0611/21 grant (D.S

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Index

A admissible, 192, 193, 201, 207, 213, 251 adomian decomposition method, 125, 126, 127 approximation, 137, 138, 143, 155, 161, 163, 225, 229, 233

B Barbashin, 39, 40, 53 Bessel potential spaces, 192, 193, 198, 208, 209, 250, 251

C calculus, 150, 151, 152, 172 classification, 2, 3, 4, 6, 126, 150 comparison theorems, 10, 32, 46, 52 complex plane, 175, 176, 177, 186, 189 convergence, 24, 26, 27, 29, 64, 150, 151, 164, 168, 172, 176, 182, 183 convolution, 33, 35, 133, 143, 202, 209, 210

D differential, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 18, 19, 21, 23, 25, 27, 29, 30, 31, 32, 33, 36, 40, 41, 45, 46, 52, 53, 54, 55, 56, 57, 67, 74, 106, 109, 113, 115, 116, 117, 118, 119, 120, 121, 122, 125, 126, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137,138, 139, 140, 141, 143, 144, 145, 146, 147, 149, 150, 159, 161, 164, 172, 173, 175, 176, 177, 179, 181, 183, 185, 186, 187, 189, 192, 193, 195, 197,

199, 201, 202, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 222, 223, 225, 227, 228, 229, 231, 233, 235, 237, 239, 240, 241, 243, 244, 245, 247, 249, 251, 252, 253, 254, 255, 256 differential transform method, 126, 127, 136, 139, 146 dynamic stochastic portfolio optimization, 192 dynamic(s), 2, 32, 116, 120, 121, 123, 176, 192, 193, 196, 197, 199, 200, 202, 208, 224, 227, 228, 230, 232, 236, 237, 244, 251, 252, 255, 256

E existence, 1, 2, 3, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 25, 26, 27, 29, 42, 55, 57, 104, 106, 118, 123, 150, 151, 164, 165, 168, 175, 176, 179, 184, 192, 193, 195, 196, 198, 200, 204, 208, 209, 214, 215, 218, 221, 225, 237, 242, 243, 244, 255 existence and uniqueness, 1, 8, 24, 26, 27, 29, 104, 106, 123, 164, 168, 175, 176, 179, 184, 192, 193, 195, 196, 198, 200, 208, 209, 214, 221, 225, 243, 244

F feedback, 194, 195, 197, 223, 224, 227, 252 finance, 193, 203, 204, 251, 252, 253, 254, 255, 256 fixed, 2, 5, 7, 8, 9, 34, 56, 58, 87, 104, 112, 115, 116, 120, 166, 175, 176, 181, 196, 214, 221, 238

258

Index

fixed point method, 56, 116 fixed point theorem, 2, 8, 9, 115, 175, 181, 214 fractional, 56, 115, 117, 118, 119, 122, 123, 149, 150, 151, 152, 153, 155, 156, 157, 159, 161, 163, 164, 165, 167, 169, 171, 172, 173, 209, 210, 216 fractional integro-differential equation(s), 56, 115, 117, 118, 123, 150, 172

M

HJB equations, 237

maximal monotone operator, 192, 251 memory, 11, 35 monotone, 1, 2, 8, 9, 13, 14, 17, 20, 21, 24, 26, 27, 29, 30, 192, 193, 196, 200, 214, 216, 218, 237, 251, 254 monotone iterative technique, 1, 2, 9, 20, 21, 29, 30 monotone operator, 192, 193, 196, 214, 237, 254 multidimensional, 192, 193, 198, 200, 207, 223, 251

I

N

InDE, 2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 33, 35, 36, 37, 39, 40, 43, 44, 45, 49, 51 initial value problem, 1, 51, 108, 109, 125, 126, 130, 146 integral equation(s), 1, 2, 3, 4, 5, 6, 7, 40, 56, 57, 62, 115, 118, 119, 120, 121, 122, 123, 126, 127, 128, 147, 165, 176, 189 integro-differential equation(s), 56, 74, 104, 106, 110, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123

nonlinear fredholm equation, 176

H

L Laplace transform method, 125, 126 Legendre polynomials, 150, 151, 154, 156, 157 linearization, 225 lower and upper solutions, 1, 3, 9, 11, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 30 Lyapunov function, 32, 33, 37, 41, 42, 43, 44, 45, 48, 49, 51, 54, 56, 66, 67, 68, 76, 81, 82, 88, 120 Lyapunov stability, 31, 32, 33 Lyapunov–Krasovskii functional (LKF), 56, 61, 64, 70, 71, 76, 77, 78, 80, 90, 91, 93, 94, 95, 96, 97, 99, 102, 116 Lyapunov-Razumikhin method, 56, 120

O operational, 150, 151, 153, 155, 156, 157, 158, 159, 160, 161, 163, 165, 167, 169, 170, 171, 172, 173 operational matrices, 150, 151, 156, 172 operator(s), 18, 19, 30, 39, 40, 106, 146, 159, 165, 166, 176, 196, 201, 202, 208, 209, 210, 212, 213, 214, 215, 216, 217, 218, 221, 237, 239, 240, 251, 253, 254 option pricing, 191, 192, 193, 195, 196, 200, 208, 223, 231, 250, 251, 253, 254, 255

P parabolic, 192, 193, 196, 197, 207, 208, 214, 223, 224, 226, 236, 237, 239, 240, 242, 250, 253, 254, 256 partial integro-differential equation(s), 191 periodic boundary value problem, 1 perturbation, 33, 36, 53, 118, 195, 227, 239, 240 Picard iterative method, 175, 176, 182, 184 point theorem, 2, 8, 9, 115, 166, 175, 176, 181, 214 polynomials, 142, 150, 151, 153, 154, 155, 156, 157, 158, 172

Index portfolio, 191, 192, 193, 194, 196, 197, 199, 214, 223, 236, 237, 238, 239, 244, 245, 250, 251, 255, 256 portfolio selection problem, 191, 239, 251 practical stability, 32, 33, 51, 52, 54 preliminaries, 151, 176, 201 pricing, 193, 195, 198, 202, 208, 223, 251, 252, 253, 254, 255

Q qualitative, 9, 32, 41, 54, 55, 56, 57, 62, 63, 64, 116, 117, 122, 125, 192, 193, 196, 207, 237, 240, 250, 251 qualitative theory, 9, 56, 116 quasilinearization, 1, 2, 3, 24, 25, 27, 29, 30

R Riccati transformation, 214, 239, 255 risk aversion, 238, 245, 246

259

stability in two measures, 32

T technique(s), 2, 9, 14, 18, 24, 25, 27, 29, 30, 65, 75, 96, 116, 119, 126, 146, 150, 176, 192, 193, 196, 214, 237, 238, 251 transform(s), 33, 38, 125, 126, 128, 132, 133, 142, 143, 144, 146, 196, 214, 215, 236, 252, 253 transformation(s), 45, 83, 89, 127, 130, 132, 133, 135, 137, 138, 146, 214, 224, 237, 238, 239, 245, 255 Tychonoff, 8

U Ulam stability, 56, 106, 112, 113, 114, 115, 116 Ulam-Hyers stability, 56, 108, 115, 123 Ulam-Hyers-Rassias stability, 56, 115, 123 uniqueness, 2, 14, 17, 123, 150, 151, 164, 165, 179, 208, 215, 242

S second kind, 5, 6, 7, 139, 205, 206 stability, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 62, 63, 64, 65, 67, 70, 71, 74, 75, 76, 77, 79, 81, 96, 104, 106, 108, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 150, 151, 164, 167, 172

V variance, 204, 205, 206, 207, 232, 233, 234, 235, 236, 238, 253 Volterra, 1, 2, 4, 5, 6, 7, 8, 29, 31, 36, 41, 49, 52, 53, 54, 55, 56, 57, 65, 67, 93, 94, 104, 106, 109, 110, 112, 115, 116, 117, 118, 119, 120, 121, 122, 123, 126, 128, 139, 141, 143, 144, 146, 147, 189, 253