Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year


IMPACT FACTOR 2016 (Open Mathematics): 0.682
IMPACT FACTOR 2016 (Central European Journal of Mathematics): 0.489

CiteScore 2016: 0.62

SCImago Journal Rank (SJR) 2016: 0.454
Source Normalized Impact per Paper (SNIP) 2016: 0.850

Mathematical Citation Quotient (MCQ) 2016: 0.23

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 14, Issue 1 (Jan 2016)

Issues

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

Nasrin Eghbali
  • Corresponding author
  • Department of Mathematics, Faculty of Mathematical Sciences, University of Mohaghegh Ardabili, 56199-11367, Ardabil, Iran (Islamic Republic of),
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Vida Kalvandi
  • Department of Mathematics, Faculty of Mathematical Sciences, University of Mohaghegh Ardabili, 56199-11367, Ardabil, Iran (Islamic Republic of),
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ John M. Rassias
  • Pedagogical Department E.E., Section of Mathematics and Informatics, National and Capodistrian University of Athens, 4 Agamemnonos Str., Aghia Paraskevi, Athens 15342, Greece,
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-04-23 | DOI: https://doi.org/10.1515/math-2016-0019

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.

Keywords: Fractional order delay integral equation; Mittag-Leffler-Hyers-Ulam stability; Chebyshev norm; Bielecki norm

MSC 2010: 26A33; 34D10; 45N05

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(xτ)q1f(x,τ,y(τ),y(α(τ)))dτ,(1)

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 function hL1 ([a, b], ℝ) of order γ ∈ ℝ+ is defined by Ia+γh(t)=1Γ(γ)at(ts)γ1h(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 function h given on the interval [a, b], the αth Riemann-Liouville fractional order derivative of h, is defined by (Da+αh)(t)=1Γ(nα)(ddt)nat(ts)(nα1)h(s)ds,

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

For a function h given on the interval [a, b], the Caputo fractional order derivative of h, is defined by (cDa+αh(t)=1Γ(nα)at(ts)nα1h(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)

    d(x, y) = 0 if and only if x = y;

  • 2)

    d(x, y) = d(y, x) for all x, yX;

  • 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 that d(∧k + 1x, ∧kx) < ∞ for some xX, then, the following properties are true:

  • (a)

    The sequencenx convergences to a fixed point x* of ∧;

  • (b)

    x* is the unique fixed point ofin X* = {yX|d(∧kx, y) < ∞};

  • (c)

    If yX*, then d d(y, x*)11Ld(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(ts)q1u(s)ds, t[0, ).

Then u(t)a˜(t)+0t[n=1(g˜(t)γ(q))nΓ(nq)(ts)nq1a˜(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) ≤ ã(t)Eq[(t) Γ(q)tq], where Eq is the Mittag-Leffler function defined by Eq[z]=k=0zkΓ(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(ts)α1f(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(xτ)q1f(x, τ, y(τ), y(α(τ)))dτ|εEq(τq), (2)

there exists a unique solution y0 of 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|y1y2|, (3)

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]|gis  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]}, (4)

it is known that (X, d) is a complete generalized metric space. Define an operator ∧ : XX by (Λg)(x)1Γ(q)0x(xτ)q1g(x, τ, g(τ), g(α(τ)))dτ, (5)

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)(6)

for any x ∈ [a, b]. From the definition of ∧, (3) and (6) we have |(Λg)(x)(Λh)(x)|=1Γ(q)|0x(xτ)q1(f(x, τ, g(τ), g(α(τ))f(x, τ, h(τ), h(α(τ)))dτ|1Γ(q)0x(xτ)q1|g(τ)h(τ)|dτLKghεΓ(q)0x(xτ)q1Eq(τq)dτLKghεΓ(q)0x(xτ)q1k=0τkqΓ(kq+1)dτ=LKghεΓ(q)k=01Γ(kq+1)0x(xτ)q1τkqdτ=LKghεΓ(q)k=01Γ(kq+1)0x(xxt)q1(xt)kqxdt=LKghεΓ(q)k=0x(k+1)qΓ(kq+1)0x(1t)q1tkqdt=LKghεΓ(q)k=0x(k+1)qΓ(kq+1)Γ(q)Γ(kq+1)Γ(q+kq+1))=LKghεk=0x(k+1)qΓ((k+1)q+1)LKghεk=0xnqΓ(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(xτ)q1f(x, τ, g0(τ), g0(α(τ)))dτg0(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, {gX|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)11Ld(Λy, y)11LεEq(xq).

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

Consider the following fractional order system CDt12x(t)=15x2(t1)1+x2(t1)+15sin(2x(t)), t[0, 1]

and set x(0) = 0. The following inequality holds: |CDt12y(t)f(t, y(t), y(t1))|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|y1y2|

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(xτ)q1f(x, τ, g(τ), g(α(τ))f(x, τ, h(τ), h(α(τ)))dτ|1Γ(q)0x(xτ)q1(maxx[a, b]|g(τ)h(τ)|+maxx[a, b]|g(α(τ))h(α(τ))|)dτ2LΓ(q)ghc0x(xτ)q1dτ2LbqΓ(q+1)ghc2LghcEq(b)

for all x ∈ [a, b]; that is, d(∧g, ∧h) ≤ 2L||gh||c Eq(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)112LEq(b)d(Λy, y)112LEq(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 xB:=maxtJ|x(t)|eθt, θ>0, J+

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|y1y2|

for any x, τ ∈ [a, b] and y1, y2 ∈ ℝ and equation (2). Also suppose that 0<2LΓ(q)bqeθb2θ(2q1)<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(xτ)q1(f(x, τ, g(τ), g(α(τ))f(x, τ, h(τ), h(α(τ)))dτ|1Γ(q)0x(xτ)q1eθτ(maxx[a, b]|g(τ)h(τ)|eθτ+maxx[a, b]|g(α(τ))h(α(τ))|eθτ)dτ1Γ(q)ghB0x(xτ)q1eθτ2LΓ(q)ghB[(0x(xτ)2(q1)dτ)12(0x(e2θτdτ)12]2LΓ(q)ghBbqeθb2θ(2q1)

for all x ∈ [a, b]; that is, d(Λg, Λh)2LΓ(q)bqeθb2θ(2q1)ghB. Hence, we can conclude that d(Λg, Λh)2LΓ(q)bqeθb2θ(2q1)d(g, h) for any g, hX. By letting 0<2LΓ(q)bqeθb2θ(2q1)<1, the strictly continuous property is verified. Now by a similar process with Theorem 3.4, we have d(y, y0)112LΓ(q)bqeθb2θ(2q1)d(Λy, y)112LΓ(q)bqeθb2θ(2q1)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(xτ)q1f(x, τ, y(τ), y(α(τ)))dτ|φ(x)εEq(τq), (7)

there exists a unique solution y0 of equation (1) satisfying |y(x)y0(x)|Cφ(x)εEq(xq),

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

Set M=1Γ(q)(1pqp)1p(ba)qp 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|y1y2|(8)

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)pKφ(x), (9)

then there exists a unique continuous function y0: [a, b] → ℝ such that y0 satisfies equation (1) and |y(x)y0(x)|Cφ(x)εEq(xq), (10)

for all x ∈ [a, b].

Let us consider the space of continuous functions X={g:[a, b]|gis  continuous}.(11)

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(xτ)q1g(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)(12)

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(xτ)q1(f(x, τ, g(τ), g(α(τ))f(x, τ, h(τ), h(α(τ)))dτ|1Γ(q)0x(xτ)q1|g(τ)h(τ)|dτ1Γ(q)Kgh0x(xτ)q1φ(τ)dτLΓ(q)Kgh(0x(xτ)q11pdτ)1p(0x(φ(τ))1pdτ)pLKghKMφ(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(xτ)q1f(x, τ, g0(τ), g0(α(τ)))dτg0(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 ∧n g0y0 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, {gX|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).(13)

Finally, Theorem 2.5 (c) together with (13) imply that d(y, y0)11KMLd(Λy, y)11KMLφ(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(xτ)q1f(x, τ, y(τ), y(α(τ)))dτ|ε, (14)

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|w1w2|,

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(xτ)q1(f(x, τ, y(τ), y(α(τ))dτ|+|1Γ(q)0x(xτ)q1f(x, τ, y(τ), y(α(τ))dτ1Γ(q)0x(xτ)q1f(x, τ, x(τ), x(α(τ))dτ|+|1Γ(q)0x(xτ)q1f(x, τ, x(τ), x(α(τ))dτx(t)|ε+|1Γ(q)0x(xτ)q1(f(x, τ, y(τ), y(α(τ))f(x, τ, x(τ), x(α(τ))dτ|ε+|1Γ(q)0x(xτ)q1|f(x, τ, y(τ), y(α(τ))f(x, τ, x(τ), x(α(τ))|dτε+|LΓ(q)0x(xτ)q1|yx|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(xτ)q1f(x, τ, y(τ), y(α(τ)))dτ|εφ(x), (15)

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|w1w2|,

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.

References

  • [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. Google Scholar

  • [2]

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

  • [3]

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

  • [4]

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

  • [5]

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

  • [6]

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

  • [7]

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

  • [8]

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

  • [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.Google Scholar

  • [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.Google Scholar

  • [11]

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

  • [12]

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

  • [13]

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

  • [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. 20Google Scholar

  • [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.Google Scholar

  • [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. Google Scholar

  • [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.Google Scholar

  • [18]

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

  • [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. Google Scholar

  • [20]

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

  • [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.Google Scholar

  • [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.Google Scholar

  • [23]

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

  • [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. Google Scholar

  • [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. Google Scholar

About the article

Received: 2015-08-01

Accepted: 2015-11-03

Published Online: 2016-04-23

Published in Print: 2016-01-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0019.

Export Citation

© 2016 Eghbali et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in