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

Nasrin Eghbali 1 , Vida Kalvandi 1 , and John M. Rassias 2
• 1 Department of Mathematics, Faculty of Mathematical Sciences, University of Mohaghegh Ardabili, 56199-11367, Ardabil, Iran (Islamic Republic of), eghbali@uma.ac.ir
• 2 Pedagogical Department E.E., Section of Mathematics and Informatics, National and Capodistrian University of Athens, 4 Agamemnonos Str., Aghia Paraskevi, Athens 15342, Greece, Ioannis.Rassias@primedu.uoa.gr
Nasrin Eghbali
, Vida Kalvandi
and John M. Rassias

## 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.

## 1. 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 X1, X2 be two Banach spaces and ε > 0. Then for every mapping f: X1X2 satisfying ||f(x + y) − f(x) − f(y)|| ≤ ε for all x, yX1, there exists a unique additive mapping g: X1X2 with the property ||f(x) − g(x)|| ≤ ε. ∀xX1.

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 [7–10]). Recently some authors ([1118]) 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 [1923].

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
$y(x)=Ic+qf(x,x,y(x),y(α(x)))=1Γ(q)∫cx(x−τ)q−1f(x,τ,y(τ),y(α(τ)))dτ,$

where $q∈(0, 1), Ic+q$ is the fractional integral of the order q, Γ(.) is the Gamma function, a, b and c are fixed real numbers such that −∞ < axb < +∞, and c ∈ (a, b). Also f: [a, b] × [a, b] × ℝ × ℝ → ℝ is a continuous function and α: [a, b] → [a, b] is a continuous delay function which fulfils α(x) ≤ x, for all x ∈ [a, b].

## 2 Preliminaries

In this section, we introduce notations, definitions and preliminaries which are used throughout this paper.

Given an interval [a, b] of ℝ, the fractional order integral of a functionhL1 ([a, b], ℝ) of orderγ ∈ ℝ+is defined by
$Ia+γh(t)=1Γ(γ)∫at(t−s)γ−1h(s)ds,$

where Γ(.) is the Gamma function.

In the sequel, we will use a Banach′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.

For a functionhgiven on the interval [a, b], theαth Riemann-Liouville fractional order derivative of h, is defined by
$(Da+αh)(t)=1Γ(n−α)(ddt)n∫at(t−s)(n−α−1)h(s)ds,$

where n = [α] + 1 and [α] denotes the integer part ofα.

For a function h given on the interval [a, b], the Caputo fractional order derivative of h, is defined by
$(cDa+αh(t)=1Γ(n−α)∫at(t−s)n−α−1h(n)(s)ds,$

where n = [α] + 1.

A function d: X × X → [0, +∞] is called a generalized metric on X if and only if it satisfies the following three properties:

1. 1)d(x, y) = 0 if and only if x = y;
2. 2)d(x, y) = d(y, x) for allx, yX;
3. 3)d(x, z) ≤ d(x, y + d(y, z) for all x, y, zX.

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's fixed point theorem in a generalized complete metric space.

Let (X, d) be a generalized complete metric space. Assume that ∧ : XX is a strictly contractive operator with the Lipschitz constant L < 1. If there exists a non negative integer k such thatd(∧k + 1x, ∧kx) < ∞ for somexX, then, the following properties are true:

1. (a)The sequencenxconvergences to a fixed pointx* of ∧;
2. (b)x* is the unique fixed point ofin X* = {yX|d(∧kx, y) < ∞};
3. (c)If yX*, then d$d(y, x*)≤11−Ld(Λy, y)$.

([24, Thorem 1]). Suppose that ã is a nonnegative function locally integrable on [0, ∞) and g̃(t) is a nonnegative, nondecreasing continuous function defined on g̃(t) ≤ M, t ∈ [0, ∞), and suppose u(t) is nonnegative and locally integrable on [0, ∞)with
$a˜(t)+g˜(t)∫0t(t−s)q−1u(s)ds, t∈[0, ∞).$
Then
$u(t)≤a˜(t)+∫0t[∑n=1∞(g˜(t)γ(q))nΓ(nq)(t−s)nq−1a˜(s)]ds, t∈[0, ∞).$

([24]). Under the hypothesis of Theorem 2.6, let ã(t) be a nondecreasing function on [0, ∞). Then we have u(t) ≤ (t)Eq[(t) Γ(q)tq], where Eq is the Mittag-Leffler function defined by$Eq[z]=∑k=0∞zkΓ(kq+1)$, z ∈ 𝕔.

([16]). Let yC(I, ℝ) be a solution of the following inequality
$|CDtαy(t)−f(t, y(t), y(g(t)))|≤ε.$
Then y is a solution of the following integral inequality:
$|y(t)−y(0)−1Γ(α)∫0t(t−s)α−1f(s, y(s), y(g(s)))ds|≤ε.$

## 3 Mittag-Leffler-Hyers-Ulam stability of the first type

Equation (1) is Mittag-Leffler-Ulam-Hyers stable of first type, with respect to Eq, if there exists a real number c > 0 such that for each ε > 0 and for each solution y of the inequality
$|y(x)−1Γ(q)∫cx(x−τ)q−1f(x, τ, y(τ), y(α(τ)))dτ|≤εEq(τq),$
there exists a unique solution y0of equation (1) satisfying the following inequality:
$|y(x)−y0(x)|≤cεEq(xq).$
Suppose that α: [a, b] → [a, b] is a continuous function such that α(x) ≤ x, for all x ∈ [a, b] and f: [a, b] × [a, b] × ℝ × ℝ → ℝ is a continuous function which satisfies the following Lipschitz condition
$|f(x, τ, y1(τ), y1(α(τ))−f(x, τ, y2(τ), y2(α(τ))|≤L|y1−y2|,$

for any x, τ ∈ [a, b] and y1, y2 ∈ ℝ and equation (2). Then, the equation (1) is Mittag-Leffler-Hyers-Ulam stable of the first type.

Let us consider the space of continuous functions
$X={g:[a, b]→ℝ| g is continuous}.$
Similar to the well-known Theorem 3.1 of [25], endowed with the generalized metric defined by
$d(g, h)=inf{K∈[0, ∞]||g(x)−h(x)|≤KεEq(xq)∀x∈[a, b]},$
it is known that (X, d) is a complete generalized metric space. Define an operator ∧ : XX by
$(Λg)(x)1Γ(q)∫0x(x−τ)q−1g(x, τ, g(τ), g(α(τ)))dτ,$
for all gX and x ∈ [a, b]. Since g is a continuous function, it follows that ∧g is also continuous and this ensures that ∧ is a well-defined operator. For any g, hX, let Kgh ∈ [0, ∞] such that
$|g(x)−h(x)|≤KghεEq(xq)$
for any x ∈ [a, b]. From the definition of ∧, (3) and (6) we have
$|(Λg)(x)−(Λh)(x)|=1Γ(q)|∫0x(x−τ)q−1(f(x, τ, g(τ), g(α(τ))−f(x, τ, h(τ), h(α(τ)))dτ|≤1Γ(q)∫0x(x−τ)q−1|g(τ)−h(τ)|dτ≤LKghεΓ(q)∫0x(x−τ)q−1Eq(τq)dτ≤LKghεΓ(q)∫0x(x−τ)q−1∑k=0∞τkqΓ(kq+1)dτ=LKghεΓ(q)∑k=0∞1Γ(kq+1)∫0x(x−τ)q−1τkqdτ=LKghεΓ(q)∑k=0∞1Γ(kq+1)∫0x(x−xt)q−1(xt)kqxdt=LKghεΓ(q)∑k=0∞x(k+1)qΓ(kq+1)∫0x(1−t)q−1tkqdt=LKghεΓ(q)∑k=0∞x(k+1)qΓ(kq+1)Γ(q)Γ(kq+1)Γ(q+kq+1))=LKghε∑k=0∞x(k+1)qΓ((k+1)q+1)≤LKghε∑k=0∞xnqΓ(nq+1)=LKghεEq(xq)$

for all x ∈ [a, b]; that is, d(∧g, ∧h) ≤ LKghεEq(xq). Hence, we can conclude that d(∧g, ∧h) ≤ Ld(g, h) for any g, hX, and since 0 < L < 1, the strictly continuous property is verified.

Let us take g0X. From the continuous property of g0 and ∧g0, it follows that there exists a constant 0 < K1 < ∞ such that
$|(Λg0)(x)−g0(x)|=|1Γ(q)∫0x(x−τ)q−1f(x, τ, g0(τ), g0(α(τ)))dτ−g0(x)|≤K1Eq(xq),$

for all x ∈ [a, b], since f and g0 are bounded on [a, b] and minx∈[a, b]Eq(xq) > 0. Thus, (4) implies that d(∧g0, g0) < ∞.

Therefore, according to Theorem 2.5 (a), there exists a continuous function y0 : [a, b] → ℝ such that ∧ng0y0 in (X, d) as n → ∞ and ∧y0 = y0; that is, y0 satisfies the equation (1) for every x ∈ [a, b].

We will now prove that {gX|d(g0, g) < ∞} = X. For any gX, since g and g0 are bounded on [a, b] and minx∈[a, b]Eq(xq) > 0, there exists a constant 0 < Cg < ∞ such that
$|g0(x)−g(x)|≤CgEq(xq),$
for any x ∈ [a, b]. Hence, we have d(g0, g) < ∞ for all gX; that is,
${g∈X|d(g0, g)<∞}=X.$
Hence, in view of Theorem 2.5 (b), we conclude that y0 is the unique continuous function which satisfies the equation (1). Now we have d(y, ∧ y) ≤ εEq(xq). Finally, Theorem 2.5 (c) together with the above inequality imply that
$d(y, y0)≤11−Ld(Λy, y)≤11−LεEq(xq).$

This means that the equation (1) is Mittag-Leffler-Hyers-Ulam stable. □

Consider the following fractional order system
$CDt12x(t)=15x2(t−1)1+x2(t−1)+15sin(2x(t)), t∈[0, 1]$
and set x(0) = 0. The following inequality holds:
$|CDt12y(t)−f(t, y(t), y(t−1))| ≤εE12(t12).$
By Remark 15, x(0) = 0, $L=25$and above inequality, all assumptions in Theorem 3.2 are satisfied. So our fractional integral is Mittag-Leffler-Hyers-Ulam stable of the first type and
$|y(t)−x(t)|≤CεE12(t12).$

Next, we use the Chebyshev norm ||.||c to derive the above similar result for the equation (1).

Suppose that α: [a, b] → [a, b] is a continuous function such that α(x) ≤ x, for all x ∈ [a, b] and f : [a, b] × [a, b] × ℝ × ℝ → ℝ is a continuous function which additionally satisfies the following Lipschitz condition
$|f(x, τ, y1(τ), y1(α(τ))−f(x, τ, y2(τ), y2(α(τ))|≤L|y1−y2|$

for any x, τ ∈ [a, b] and y1, y2 ∈ ℝ and equation (2). Also suppose that 0 < 2LEq(b) < 1. Then, the initial integral equation (1) is Mittag-Leffler-Hyers-Ulam stable of the first type via the Chebyshev norm.

Just like the discussion in Theorem 3.2, we only prove that ∧ defined in (5) is a contraction mapping on X with respect to the Chebyshev norm:
$|(Λg)(x)−(Λh)(x)|=1Γ(q)|∫0x(x−τ)q−1f(x, τ, g(τ), g(α(τ))−f(x, τ, h(τ), h(α(τ)))dτ|≤1Γ(q)∫0x(x−τ)q−1(maxx∈[a, b]|g(τ)−h(τ)|+maxx∈[a, b]|g(α(τ))−h(α(τ))|)dτ≤2LΓ(q)g−hc∫0x(x−τ)q−1dτ≤2LbqΓ(q+1)g−hc≤2Lg−hcEq(b)$
for all x ∈ [a, b]; that is, d(∧g, ∧h) ≤ 2L||gh||cEq(b). Hence, we can conclude that d(∧g, ∧h) ≤ 2LEq(b)d(g, h) for any g, hX. By letting 0 < 2LEq(b) < 1, the strictly continuous property is verified. Now by proceeding a proof similar to the proof of Theorem 3.2, we have
$d(y, y0)≤11−2LEq(b)d(Λy, y)≤11−2LEq(b)εEq(xq)≤CεEq(xq),$

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
$‖x‖B:=maxt∈J|x(t)|e−θt, θ>0, J⊂ℝ+$

to derive the similar Theorem 3.2 for the fundamental equation (1) via the Bielecki norm.

Suppose that α: [a, b] → [a, b] is a continuous function such that α(x) ≤ x, for all x ∈ [a, b] and f: [a, b] × [a, b] × ℝ × ℝ → ℝ is a continuous function which additionally satisfies the Lipschitz condition
$|f(x, τ, y1(τ), y1(α(τ))−f(x, τ, y2(τ), y2(α(τ))|≤L|y1−y2|$

for any x, τ ∈ [a, b] and y1, y2 ∈ ℝ and equation (2). Also suppose that$0<2LΓ(q)bqeθb2θ(2q−1)<1$. Then, equation (1) is Mittag-Leffler-Hyers-Ulam stable of the first type via the Bielecki norm.

Just like the discussion in Theorem 3.4, we only prove that ∧ defined in (5) is a contraction mapping on X with respect to the Bielecki norm:
$|(Λg)(x)−(Λh)(x)|=1Γ(q)|∫0x(x−τ)q−1(f(x, τ, g(τ), g(α(τ))−f(x, τ, h(τ), h(α(τ)))dτ|≤1Γ(q)∫0x(x−τ)q−1eθτ(maxx∈[a, b]|g(τ)−h(τ)|e−θτ+maxx∈[a, b]|g(α(τ))−h(α(τ))|e−θτ)dτ≤1Γ(q)g−hB∫0x(x−τ)q−1eθτ≤2LΓ(q)g−hB[(∫0x(x−τ)2(q−1)dτ)12(∫0x(e2θτdτ)12]≤2LΓ(q)g−hBbqeθb2θ(2q−1)$
for all x ∈ [a, b]; that is, $d(Λg, Λh)≤2LΓ(q)bqeθb2θ(2q−1)‖g−h‖B$. Hence, we can conclude that $d(Λg, Λh)≤2LΓ(q)bqeθb2θ(2q−1)d(g, h)$ for any g, hX. By letting $0<2LΓ(q)bqeθb2θ(2q−1)<1$, the strictly continuous property is verified. Now by a similar process with Theorem 3.4, we have
$d(y, y0)≤11−2LΓ(q)bqeθb2θ(2q−1)d(Λy, y)≤11−2LΓ(q)bqeθb2θ(2q−1)εEq(xq)≤CεEq(xq),$

which means that equation (1) is Mittag-Leffler-Hyers-Ulam stable of the first type via the Bielecki norm. □

## 4 Mittag-Leffler-Hyers-Ulam-Rassias stability of the first type

Equation (1) is Mittag-Leffler-Hyers-Ulam-Rassias stable of the first type, with respect to Eq, if there exists a real number C > 0 such that for each ε > 0 and for each solution y of the following inequality
$|y(x)−1Γ(q)∫cx(x−τ)q−1f(x, τ, y(τ), y(α(τ)))dτ|≤φ(x)εEq(τq),$
there exists a unique solution y0of equation (1) satisfying
$|y(x)−y0(x)|≤Cφ(x)εEq(xq),$

where φ: X → [0, ∞) is a continuous function.

Set$M=1Γ(q)(1−pq−p)1−p(b−a)q−p$with 0 < p < q. Let K and L be positive constants with 0 < KLM < 1. Assume that α: [a, b] → [a, b] is a continuous function such that α(x) ≤ x for all x ∈ [a, b] and f : [a, b] × [a, b] × ℝ × ℝ → ℝ is a continuous function which additionally satisfies the Lipschitz condition:
$|f(x, τ, y1(τ), y1(α(τ))−f(x, τ, y2(τ), y2(α(τ))|≤L|y1−y2|$
for any x, τ ∈ [a, b] and y1, y2 ∈ ℝ. If a continuous function y : [a, b] → ℝ satisfies (7) for all x ∈ [a, b], where φ : [a, b] → (0, ∞) is a$L1p$-integrable function satisfying.
$(∫0x(φ(τ))1pdτ)p≤Kφ(x),$
then there exists a unique continuous function y0: [a, b] → ℝ such that y0satisfies equation (1) and
$|y(x)−y0(x)|≤Cφ(x)εEq(xq),$

for all x ∈ [a, b].

Let us consider the space of continuous functions
$X={g:[a, b]⟶ℝ| g is continuous}.$
Similar to Theorem 3.1 of [25], endowed with the generalized metric defined by
$d(g, h)=inf{K∈[0, ∞]||g(x)−h(x)|≤Kφ(x)∀x∈[a, b]},$
it is known that (X, d) is a complete generalized metric space. Define an operator ∧ : XX by the formula
$(Λg)(x)=1Γ(q)∫0x(x−τ)q−1g(x, τ, g(τ), g(α(τ)))dτ,$
for all gX and x ∈ [a, b]. Since g is a continuous function, it follows that ∧g is also continuous and this ensures that ∧ is a well-defined operator. For any g, hX, let Kgh ∈ [0, ∞] such that inequality
$|g(x)−h(x)|≤Kghφ(x)$
holds for any x ∈ [a, b]. From the definition of ∧ and inequalities (8), (9) and (12), we have
$|(Λg)(x)−(Λh)(x)|=1Γ(q)|∫0x(x−τ)q−1(f(x, τ, g(τ), g(α(τ))−f(x, τ, h(τ), h(α(τ)))dτ|≤1Γ(q)∫0x(x−τ)q−1|g(τ)−h(τ)|dτ≤1Γ(q)Kgh∫0x(x−τ)q−1φ(τ)dτ≤LΓ(q)Kgh(∫0x(x−τ)q−11−pdτ)1−p(∫0x(φ(τ))1pdτ)p≤LKghKMφ(x),$

for all x ∈ [a, b]; that is, d(∧g, ∧h) ≤ KLMKghφ(x). Hence, we conclude that d(∧g, ∧h) ≤ KLMd(g, h) for any g, hX, and since 0 < KLM < 1, the strictly continuous property is verified.

Let us take g0X. There exists a constant 0 < K1 < ∞ such that
$|(Λg0)(x)−g0(x)|=|1Γ(q)∫0x(x−τ)q−1f(x, τ, g0(τ), g0(α(τ)))dτ−g0(x)|≤K1φ(x),$

for all x ∈ [a, b], since f and g0 are bounded on [a, b] and minx∈[a, b]φ(x) > 0. Thus, (11) implies that d(∧g0, g0) < ∞. Therefore, according to Theorem 2.5 (a), there exists a continuous function y0: [a, b] → ℝ such that ∧ng0y0 in (X, d) as n → ∞ and ∧y0 = y0; that is, y0 satisfies equation (1) for every x ∈ [a, b].

We will now verify that {gX| d(g0, g) < ∞} = X. For any gX, since g and g0 are bounded on [a, b] and minx[a, b] Eq (xq) > 0, there exists a constant 0 < Cg < ∞ such that
$|g0(x)−g(x)|≤Cgφ(x),$
for any x ∈ [a, b]. Hence, we have d(g0, g) < ∞ for all gX; that is,
${g∈X | d(g0, g)<∞}=X.$

So, in view of Theorem 2.5 (b), we conclude that y0 is the unique continuous function such that it satisfies equation (1).

On the other hand, from (7) it follows that
$d(y, Λy)≤εEq(xq)φ(x).$
Finally, Theorem 2.5 (c) together with (13) imply that
$d(y, y0)≤11−KMLd(Λy, y)≤11−KMLφ(x)εEq(xq),$

which means that the inequality (10) holds for all x ∈ [a, b].        □

## 5 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 = [0, b].

Equation (1) is Mittag-Leffler-Hyers-Ulam stable of the second type, with respect to Eq, if for every ε > 0 and solution yC([a, b], ℝ) of the following equation
$|y(x)−1Γ(q)∫0x(x−τ)q−1f(x, τ, y(τ), y(α(τ)))dτ|≤ε,$
there exists a solution x ∈ (C([a, b], ℝ) of equation (1) with
$|y(t)−x(t)|≤εEq(cxq),$

for all x ∈ [a, b] and C ∈ ℝ.

Let B be a Banach algebra. We suppose that: (i) fC([0, b] × B, B); (ii) f satisfies the following Lipschitz condition
$|f(t, w1)−f(t, w2)|≤L|w1−w2|,$

for all t ∈ [0, b]; w1, w2B and equation (14). Then equation (1) is Mittag-Leffler-Hyers-Ulam stable of the second type.

Let yC(I, B) satisfy the inequality (14). Let us denote by xC([0, b], B) the unique solution of the (1). We have
$|y(t)−(x)(t)|≤|y(t)−1Γ(q)∫0x(x−τ)q−1(f(x, τ, y(τ), y(α(τ))dτ|+|1Γ(q)∫0x(x−τ)q−1f(x, τ, y(τ), y(α(τ))dτ−1Γ(q)∫0x(x−τ)q−1f(x, τ, x(τ), x(α(τ))dτ|+|1Γ(q)∫0x(x−τ)q−1f(x, τ, x(τ), x(α(τ))dτ−x(t)|≤ε+|1Γ(q)∫0x(x−τ)q−1(f(x, τ, y(τ), y(α(τ))−f(x, τ, x(τ), x(α(τ))dτ|≤ε+|1Γ(q)∫0x(x−τ)q−1|f(x, τ, y(τ), y(α(τ))−f(x, τ, x(τ), x(α(τ))|dτ≤ε+|LΓ(q)∫0x(x−τ)q−1|y−x|dτ.$
Now by Remark 2.7, we have
$u(t)≤εEq(Lxq).$

Thus, the conclusion of our theorem holds. □

Equation (1) is Mittag-Leffler-Hyers-Ulam-Rassias stable of the second type, with respect to Eq, if for every ε > 0 and solution yC([a, b], ℝ) of the following inequality
$|y(x)−1Γ(q)∫0x(x−τ)q−1f(x, τ, y(τ), y(α(τ)))dτ|≤εφ(x),$
there exists a solution x ∈ (C([a, b], ℝ) of equation (1) with
$|y(t)−x(t)|≤λEq(cxq),$

for all x ∈ [a, b] such that the function φ : X → [0, ∞) is a non-negative non-decreasing locally integrable function on [0, ∞)and C ∈ ℝ.

Let B be a Banach algebra. We suppose that: (i) fC([0, b] × B, B) (ii) f satisfies the following Lipschitz condition
$|f(t, w1)−f(t, w2)|≤L|w1−w2|,$

for all t ∈ [0, b], w1, w2B and inequality (15). Then the initial equation (1) is Mittag-Leffler-Hyers-Ulam-Rassias stable of the second type.

By putting εφ(x) instead of ε in the proof of Theorem 5.2, the proof is complete.

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Kilbas A. A., Srivastava H. M. and Trujillo J. J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Stydies, 204, Elsevier Science, B. V., Amsterdam, 2006.

• [2]

Miller K. S., Ross B., An Introduction to the Fractional Calculus and Differential Equations, John wiley, New York, 1993.

• [3]

Podlubny I., Fractional Differential Equations, Academic Press, San Diego, 1999.

• [4]

Ulam S. M., Problems in Modern Mathematics, Chap. VI, Science eds., Wiley, New York, 1960.

• [5]

Hyers D. H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci., U.S.A., 1941, 27, 222–224.

• [6]

Rassias Th. M., On the stability of linear mapping in Banach spaces, Proc. Amer. Math. Soc., 1978, 72, 297–300.

• [7]

Rassias J. M., On approximation of approximately linear mappings by linear mappings, J. Func. Anal., 1982, 46, (1), 126–130.

• [8]

Rassias J. M., Solution of a problem of Ulam, J. Approx. Theory, 1989, 57, (3), 268–273.

• [9]

Rassias J. M., On the stability of the non-linear Euler-Lagrange functional equation in real normed spaces, J. Math. Phys. Sci., 1994, 28, (5), 231–235.

• [10]

Rassias J. M., Mixed type partial differential equations with initial and boundary values in fluid mechanics, Intern. J. Appl. Math. stat., 2008, 13, (J08), 77–107.

• [11]

Ibrahim R. W., Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math., 2012, 23, (5), 9 pp.

• [12]

Ibrahim R. W., Ulam stability for fractional differential equation in complex domain, Abstr. Appl. Anal., 2012, 2012, 1–8.

• [13]

Ibrahim R. W., Ulam-Hyers stability for Cauchy fractional differential equation in the unit disk, Abstr. Appl. Anal., 2012, 2012, 1–10.

• [14]

Wang J. R., Lv L. and Zhou Y., Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 2011, 63, 1–10. 20

• [15]

Wang J. R., Lv L. and Zhou Y., New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simulat., 2012, 17, 2530–2538.

• [16]

Wang J. R., Zhang Y., Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations, Optimization: A Journal of Mathematical Programming and optimization Research, 2014, 63 (8), 1181–1190.

• [17]

Wang J. R., Zhou Y. and Feckan M., Nonlinear impulsive problems for fractional differential equations and Ulam stability, Appl. Math. Comput., 2012, 64, 3389–3405.

• [18]

Wei W., Li Xuezhu. and Li Xia, New stability results for fractional integral equation, Comput. Math. Appl., 2012, 64 (10), 3468– 3476.

• [19]

Peng Sh., Wang J. R., Existence and Ulam-Hyers stability of ODEs involving two Caputo fractional derivatives, Electronic Journal of Qualitative Theory of Differential Equations, 2015, 48-54 (52), 1–16.

• [20]

Wang J., Li X., Ulam-Hyers stability of fractional Langevin equations, Appl. Math. Comput., 2015, 258, 72–83.

• [21]

Wang J., Lin Z., A class of impulsive nonautonomous differential equations and Ulam-Hyers-Rassias stability, Mathematical Methods in the Applied Sciences, 38 (5), (2015), 865–880.

• [22]

Wang J. R., Zhou Y. and Lin Z., On a new class of impulsive fractional differential equations, App. Math. Comput., 2014, 242, 649–657.

• [23]

Wang J., Lin Z., Ulam’s type stability of Hadamard type fractional integral equations, Filomat, 2014, 28 (7), 1323–1331.

• [24]

Ye H., Gao J. and Ding Y., A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 2007, 328, 1075–1081.

• [25]

Jung S. M., A fixed point approach to the stability of differential equations ý = F (x, y), Bull. Malays. Math. Sci. Soc., 2010, 33, 47–56.

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• [1]

Kilbas A. A., Srivastava H. M. and Trujillo J. J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Stydies, 204, Elsevier Science, B. V., Amsterdam, 2006.

• [2]

Miller K. S., Ross B., An Introduction to the Fractional Calculus and Differential Equations, John wiley, New York, 1993.

• [3]

Podlubny I., Fractional Differential Equations, Academic Press, San Diego, 1999.

• [4]

Ulam S. M., Problems in Modern Mathematics, Chap. VI, Science eds., Wiley, New York, 1960.

• [5]

Hyers D. H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci., U.S.A., 1941, 27, 222–224.

• [6]

Rassias Th. M., On the stability of linear mapping in Banach spaces, Proc. Amer. Math. Soc., 1978, 72, 297–300.

• [7]

Rassias J. M., On approximation of approximately linear mappings by linear mappings, J. Func. Anal., 1982, 46, (1), 126–130.

• [8]

Rassias J. M., Solution of a problem of Ulam, J. Approx. Theory, 1989, 57, (3), 268–273.

• [9]

Rassias J. M., On the stability of the non-linear Euler-Lagrange functional equation in real normed spaces, J. Math. Phys. Sci., 1994, 28, (5), 231–235.

• [10]

Rassias J. M., Mixed type partial differential equations with initial and boundary values in fluid mechanics, Intern. J. Appl. Math. stat., 2008, 13, (J08), 77–107.

• [11]

Ibrahim R. W., Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math., 2012, 23, (5), 9 pp.

• [12]

Ibrahim R. W., Ulam stability for fractional differential equation in complex domain, Abstr. Appl. Anal., 2012, 2012, 1–8.

• [13]

Ibrahim R. W., Ulam-Hyers stability for Cauchy fractional differential equation in the unit disk, Abstr. Appl. Anal., 2012, 2012, 1–10.

• [14]

Wang J. R., Lv L. and Zhou Y., Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 2011, 63, 1–10. 20

• [15]

Wang J. R., Lv L. and Zhou Y., New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simulat., 2012, 17, 2530–2538.

• [16]

Wang J. R., Zhang Y., Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations, Optimization: A Journal of Mathematical Programming and optimization Research, 2014, 63 (8), 1181–1190.

• [17]

Wang J. R., Zhou Y. and Feckan M., Nonlinear impulsive problems for fractional differential equations and Ulam stability, Appl. Math. Comput., 2012, 64, 3389–3405.

• [18]

Wei W., Li Xuezhu. and Li Xia, New stability results for fractional integral equation, Comput. Math. Appl., 2012, 64 (10), 3468– 3476.

• [19]

Peng Sh., Wang J. R., Existence and Ulam-Hyers stability of ODEs involving two Caputo fractional derivatives, Electronic Journal of Qualitative Theory of Differential Equations, 2015, 48-54 (52), 1–16.

• [20]

Wang J., Li X., Ulam-Hyers stability of fractional Langevin equations, Appl. Math. Comput., 2015, 258, 72–83.

• [21]

Wang J., Lin Z., A class of impulsive nonautonomous differential equations and Ulam-Hyers-Rassias stability, Mathematical Methods in the Applied Sciences, 38 (5), (2015), 865–880.

• [22]

Wang J. R., Zhou Y. and Lin Z., On a new class of impulsive fractional differential equations, App. Math. Comput., 2014, 242, 649–657.

• [23]

Wang J., Lin Z., Ulam’s type stability of Hadamard type fractional integral equations, Filomat, 2014, 28 (7), 1323–1331.

• [24]

Ye H., Gao J. and Ding Y., A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 2007, 328, 1075–1081.

• [25]

Jung S. M., A fixed point approach to the stability of differential equations ý = F (x, y), Bull. Malays. Math. Sci. Soc., 2010, 33, 47–56.

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