A fixed point approach to the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation

Abstract In this paper, we have presented and studied two types of the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation. We prove that the fractional order delay integral equation is Mittag-Leffler-Hyers-Ulam stable on a compact interval with respect to the Chebyshev and Bielecki norms by two notions.


Introduction
Fractional differential and integral equations can serve as excellent tools for description of mathematical modelling of systems and processes in the fields of economics, physics, chemistry, aerodynamics, and polymerrheology. It also serves as an excellent tool for description of hereditary properties of various materials and processes. For more details on fractional calculus theory, one can see the monographs of Kilbas et al. [1], Miller and Ross [2] and Podlubny [3]. The stability of functional equations was originally raised by Ulam in 1940 in a talk given at Wisconsin University. The stability problem posed by Ulam was the following: Under what conditions there exists an additive mapping near an approximately additive mapping? (for more details see [4]).The first answer to the question of Ulam was given by Hyers [5] in 1941 in the case of Banach spaces: Let X 1 ; X 2 be two Banach spaces and " > 0: Then for every mapping f W X 1 ! X 2 satisfying kf .x C y/ f .x/ f .y/k Ä " for all x; y 2 X 1 , there exists a unique additive mapping g W X 1 ! X 2 with the property kf .x/ g.x/k Ä "; 8x 2 X 1 : This type of stability is called Hyers-Ulam stability. In 1978, Th. M. Rassias [6] provided a remarkable generalization of the Hyers-Ulam stability by considering variables on the right-hand side of the inequalities. The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the initial equation (see [7][8][9][10]). Recently some authors ( [11][12][13][14][15][16][17][18]) extended the Ulam stability problem from an integer-order differential equation to a fractional-order differential equation. For more results on Ulam type stability of fractional differential equations see [19][20][21][22][23].
In this paper we present both Mittag-Leffler-Hyers-Ulam stability and Mittag-Leffler-Hyers-Ulam-Rassias stability for the following fractional Volterra type integral equations with delay of the form x / q 1 f .x; ; y. /; y.˛. ///d ; where q 2 .0; 1/, I q c C is the fractional integral of the order q, .:/ is the Gamma function, a; b and c are fixed real numbers such that 1 < a Ä x Ä b < C1; and c 2 .a; b/. Also f W OEa; b OEa; b R R ! R is a continuous function and˛W OEa; b ! OEa; b is a continuous delay function which fulfils˛.x/ Ä x; for all x 2 OEa; b:

Preliminaries
In this section, we introduce notations, definitions and preliminaries which are used throughout this paper.
Definition 2.1. Given an interval OEa; b of R, the fractional order integral of a function h 2 L 1 .OEa; b; R/ of order 2 R C is defined by where .:/ is the Gamma function.
In the sequel, we will use a Banach 0 s fixed point theorem in a framework of a generalized complete metric space. For a nonempty set X , we introduce the definition of the generalized metric on X. The above concept differs from the usual concept of a complete metric space by the fact that not every two points in X have necessarily a finite distance. One might call such a space a generalized complete metric space. We now introduce one of the fundamental results of the Banach 0 s fixed point theorem in a generalized complete metric space. Theorem 2.5. Let .X; d / be a generalized complete metric space. Assume that ƒ W X ! X is a strictly contractive operator with the Lipschitz constant L < 1: If there exists a non negative integer k such that d.ƒ kC1 x; ƒ k x/ < 1 for some x 2 X; then, the following properties are true: (a) The sequence ƒ n x convergences to a fixed point x of ƒI (b) x is the unique fixed point of ƒ in X D fy 2 X jd.ƒ k x; y/ < 1gI (c) If y 2 X ; then d.y; x / Ä 1 1 L d.ƒy; y/: Theorem 2.6 ([24, Thorem 1]). Suppose that Q a is a nonnegative function locally integrable on OE0; 1/ and Q g.t / is a nonnegative, nondecreasing continuous function defined on Q g.t / Ä M; t 2 OE0; 1/, and suppose u.t / is nonnegative and locally integrable on OE0; .t s/ q 1 u.s/ds; t 2 OE0; 1/:   (1) is Mittag-Leffler-Ulam-Hyers stable of first type, with respect to E q , if there exists a real number c > 0 such that for each " > 0 and for each solution y of the inequality there exists a unique solution y 0 of equation (1) satisfying the following inequality: for any x; 2 OEa; b and y 1 ; y 2 2 R and equation (2). Then, the equation (1) is Mittag-Leffler-Hyers-Ulam stable of the first type.
Proof. Let us consider the space of continuous functions Similar to the well-known Theorem 3.1 of [25], endowed with the generalized metric defined by it is known that .X; d / is a complete generalized metric space. Define an operator ƒ W X ! X by for all g 2 X and x 2 OEa; b. Since g is a continuous function, it follows that ƒg is also continuous and this ensures that ƒ is a well-defined operator. For any g; h 2 X; let K gh 2 OE0; 1 such that for any x 2 OEa; b: From the definition of ƒ, (3) and (6) we have for all x 2 OEa; b; that is, d.ƒg; ƒh/ Ä LK gh "E q .x q /: Hence, we can conclude that d.ƒg; ƒh/ Ä Ld.g; h/ for any g; h 2 X; and since 0 < L < 1; the strictly continuous property is verified. Let us take g 0 2 X: From the continuous property of g 0 and ƒg 0 ; it follows that there exists a constant 0 < K 1 < 1 such that j.ƒg 0 /.x/ g 0 .x/j D j for all x 2 OEa; b; since f and g 0 are bounded on OEa; b and min x2OEa;b E q .x q / > 0: Thus, (4) implies that d.ƒg 0 ; g 0 / < 1: Therefore, according to Theorem 2.5 (a), there exists a continuous function y 0 W OEa; b ! R such that ƒ n g 0 ! y 0 in .X; d / as n ! 1 and ƒy 0 D y 0 I that is, y 0 satisfies the equation (1) for every x 2 OEa; b: We will now prove that fg 2 Xjd.g 0 ; g/ < 1g D X: For any g 2 X; since g and g 0 are bounded on OEa; b and min x2OEa;b E q .x q / > 0; there exists a constant 0 < C g < 1 such that for any x 2 OEa; b: Hence, we have d.g 0 ; g/ < 1 for all g 2 X I that is, fg 2 X j d.g 0 ; g/ < 1g D X: Hence, in view of Theorem 2.5 (b), we conclude that y 0 is the unique continuous function which satisfies the equation (1). Now we have d.y; ƒy/ Ä "E q .x q /. Finally, Theorem 2.5 (c) together with the above inequality imply that d.y; y 0 / Ä 1 1 L d.ƒy; y/ Ä 1 1 L "E q .x q /: This means that the equation (1) is Mittag-Leffler-Hyers-Ulam stable. for any x; 2 OEa; b and y 1 ; y 2 2 R and equation (2). Also suppose that 0 < 2LE q .b/ < 1. Then, the initial integral equation (1) is Mittag-Leffler-Hyers-Ulam stable of the first type via the Chebyshev norm.
Proof. Just like the discussion in Theorem 3.2, we only prove that ƒ defined in (5)  .q/ kg hk c for all x 2 OEa; bI that is, d.ƒg; ƒh/ Ä 2Lkg hk c E q .b/: Hence, we can conclude that d.ƒg; ƒh/ Ä 2LE q .b/d.g; h/ for any g; h 2 X . By letting 0 < 2LE q .b/ < 1, the strictly continuous property is verified. Now by proceeding a proof similar to the proof of Theorem 3.2, we have which means that the equation (1) is Mittag-Leffler-Hyers-Ulam stable of the first type via the Chebyshev norm.
In the following Theorem we have used the Bielecki norm to derive the similar Theorem 3.2 for the fundamental equation (1) via the Bielecki norm. for any x; 2 OEa; b and y 1 ; y 2 2 R and equation (2). Also suppose that 0 < 2L .q/ b q e Âb p 2Â.2q 1/ < 1. Then, equation (1) is Mittag-Leffler-Hyers-Ulam stable of the first type via the Bielecki norm.
Proof. Just like the discussion in Theorem 3.4, we only prove that ƒ defined in (5) is a contraction mapping on X with respect to the Bielecki norm: which means that equation (1) is Mittag-Leffler-Hyers-Ulam stable of the first type via the Bielecki norm.

Mittag-Leffler-Hyers-Ulam-Rassias stability of the first type
Definition 4.1. Equation (1) is Mittag-Leffler-Hyers-Ulam-Rassias stable of the first type, with respect to E q , if there exists a real number C > 0 such that for each " > 0 and for each solution y of the following inequality x / q 1 f .x; ; y. /; y.˛. ///d j Ä '.x/"E q . q /; there exists a unique solution y 0 of equation (1) satisfying where ' W X ! OE0; 1/ is a continuous function.
for any x; 2 OEa; b and y 1 ; y 2 2 R. If a continuous function y W OEa; b ! R satisfies (7) for all then there exists a unique continuous function y 0 W OEa; b ! R such that y 0 satisfies equation (1) and for all x 2 OEa; b: Proof. Let us consider the space of continuous functions X D fg W OEa; b ! R j g is continuousg: (11) Similar to Theorem 3.1 of [25], endowed with the generalized metric defined by it is known that .X; d / is a complete generalized metric space. Define an operator ƒ W X ! X by the formula x / q 1 g.x; ; g. /; g.˛. ///d ; for all g 2 X and x 2 OEa; b. Since g is a continuous function, it follows that ƒg is also continuous and this ensures that ƒ is a well-defined operator. For any g; h 2 X; let K gh 2 OE0; 1 such that inequality holds for any x 2 OEa; b. From the definition of ƒ and inequalities (8), (9) and (12) for all x 2 OEa; bI that is, d.ƒg; ƒh/ Ä KLMK gh '.x/: Hence, we conclude that d.ƒg; ƒh/ Ä KLM d.g; h/ for any g; h 2 X; and since 0 < KLM < 1; the strictly continuous property is verified. Let us take g 0 2 X. There exists a constant 0 < K 1 < 1 such that for all x 2 OEa; b; since f and g 0 are bounded on OEa; b and min x2OEa;b '.x/ > 0: Thus, (11) implies that d.ƒg 0 ; g 0 / < 1: Therefore, according to Theorem 2.5 (a), there exists a continuous function y 0 W OEa; b ! R such that ƒ n g 0 ! y 0 in .X; d / as n ! 1 and ƒy 0 D y 0 I that is, y 0 satisfies equation (1) for every x 2 OEa; b: We will now verify that fg 2 X j d.g 0 ; g/ < 1g D X: For any g 2 X; since g and g 0 are bounded on OEa; b and min x2OEa;b E q .x q / > 0; there exists a constant 0 < C g < 1 such that for any x 2 OEa; b: Hence, we have d.g 0 ; g/ < 1 for all g 2 X I that is, fg 2 X j d.g 0 ; g/ < 1g D X: So, in view of Theorem 2.5 (b), we conclude that y 0 is the unique continuous function such that it satisfies equation (1).

Mittag-Leffler-Hyers-Ulam stability and Mittag-Leffler-Hyers-Ulam-Rassias stability of the second type
Let us consider equation (1) in the case I D OE0; b.
Proof. Let y 2 C.I; B/ satisfy the inequality (14). Let us denote by x 2 C.OE0; b; B/ the unique solution of the (1 Thus, the conclusion of our theorem holds. for all x 2 OEa; b such that the function ' W X ! OE0; 1/ is a non-negative non-decreasing locally integrable function on OE0; 1/ and c 2 R. for all t 2 OE0; b; w 1 ; w 2 2 B and inequality (15). Then the initial equation (1) is Mittag-Leffler-Hyers-Ulam-Rassias stable of the second type.
Proof. By putting "'.x/ instead of " in the proof of Theorem 5.2, the proof is complete.