Fractional order integral equations of two independent variables
Introduction
Fractional integral equations have recently been applied in various areas of engineering, science, finance, applied mathematics, and bio-engineering and others. However, many researchers remain unaware of this field. There has been a significant development in ordinary and partial fractional differential and integral equations in recent years; see the monographs of Baleanu et al. [4], Kilbas et al. [13], Miller and Ross [15], Lakshmikantham et al. [14], Podlubny [20], Samko et al. [22]. Recently some interesting results on the attractivity of the solutions of some classes of integral equations have been obtained by Abbas et al. [1], [2], Banaś et al. [5], [6], [7], Darwish et al. [8], Dhage [9], [10], [11], Pachpatte [18], [19] and the references therein.
In [17], Mureşan proved some results concerning the existence, uniqueness, data dependence and comparison theorems, by applying some results from Picard and weakly Picard operators’ theory [21], for the following functional integral equation of the formwhere , . In this paper we improve the above results for the following partial integral equation of Riemann–Liouville fractional order of the formwhere are given continuous functions, is the left-sided mixed Riemann–Liouville integral of order r.
Next, we prove some results concerning the existence and the attractivity of solutions for the following partial Riemann–Liouville fractional order integral equation of the formwhere and are given continuous functions.
Our investigations are conducted in Banach spaces with an application of Banach’s contraction principle and Schauder’s fixed point theorem for the existence and uniqueness of solutions of Eq. (2). We use the Schauder fixed point theorem for the existence of solutions of Eq. (3), and we prove that all solutions are globally asymptotically stable. Also, we present some examples illustrating the applicability of the imposed conditions.
Section snippets
Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. By we denote the Banach space of all continuous functions from J into with the normwhere denotes a suitable complete norm on .
Let E be the space of functions , which fulfill the following condition:where is a positive constant. In the space E we define the normAccording
Main Results
Let us start by defining what we mean by a solution of Eq. (2). Definition 3.1 A function is said to be a solution of (2) if u satisfies Eq. (2) on J.
Now, we shall prove the following theorem concerning the existence and uniqueness of a solution of Eq. (2).
Set Theorem 3.2 Assume that the following hypotheses hold The functions and are in E, There exist constants such thatfor each and .
Ifthen Eq. (2)
Examples
As applications and to illustrate our results, we present the following examples. Example 4.1 Consider the following integral equation of fractional orderSet andWe have , then and condition is satisfied. For each and , we haveHence condition
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