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

Nasrin Eghbali; Vida Kalvandi; John M. Rassias

doi:10.1515/math-2016-0019

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

Nasrin Eghbali , Vida Kalvandi and John M. Rassias

Published/Copyright: April 23, 2016

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$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

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: X₁ → X₂ satisfying ||f(x + y) − f(x) − f(y)|| ≤ ε for all x, y ∈ X₁, there exists a unique additive mapping g: X₁ → X₂ with the property ||f(x) − g(x)|| ≤ ε. ∀x ∈ X₁.

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 ([11–18]) 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 [19–23].

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

(1)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 −∞ < a ≤ x ≤ b < +∞, 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.

Definition 2.1.

Given an interval [a, b] of ℝ, the fractional order integral of a functionh ∈ L¹ ([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.

Definition 2.2.

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.

Definition 2.3.

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.

Definition 2.4.

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

d(x, y) = 0 if and only if x = y;
d(x, y) = d(y, x) for allx, y ∈ X;
d(x, z) ≤ d(x, y + d(y, z) for all x, y, z ∈ 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's fixed point theorem in a generalized complete metric space.

Theorem 2.5.

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

The sequence ∧ⁿxconvergences to a fixed pointx* of ∧;
x* is the unique fixed point of ∧in X* = {y ∈ X|d(∧^kx, y) < ∞};
If y ∈ X*, then dd(y, x*)≤11−Ld(Λy, y).

Theorem 2.6

([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, ∞).

Remark 2.7

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

Remark 2.8

([16]). Let y ∈ C(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

Definition 3.1.

Equation (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 ε > 0 and for each solution y of the inequality

(2)|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).

Theorem 3.2.

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

(3)|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.

Proof

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

(4)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 ∧ : X → X by

(5)(Λg)(x)1Γ(q)∫0x(x−τ)q−1g(x, τ, g(τ), g(α(τ)))dτ,

for all g ∈ X 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, h ∈ X, let K_gh ∈ [0, ∞] such that

(6)|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) ≤ LK_ghεE_q(x^q). Hence, we can conclude that d(∧g, ∧h) ≤ Ld(g, h) for any g, h ∈ X, and since 0 < L < 1, the strictly continuous property is verified.

Let us take g₀ ∈ X. 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 min_{x∈[a, b]}E_q(x^q) > 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 ∧ⁿg₀ → y₀ 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 {g ∈ X|d(g₀, g) < ∞} = X. For any g ∈ X, since g and g₀ are bounded on [a, b] and min_{x∈[a, b]}E_q(x^q) > 0, there exists a constant 0 < C_g < ∞ such that

|g0(x)−g(x)|≤CgEq(xq),

for any x ∈ [a, b]. Hence, we have d(g₀, g) < ∞ for all g ∈ X; 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) ≤ εE_q(x^q). 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. □

Example 3.3.

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=25and 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).

Theorem 3.4.

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 < 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 ∧ defined in (5) is a contraction mapping on X with respect to the Chebyshev norm:

for all x ∈ [a, b]; that is, d(∧g, ∧h) ≤ 2L||g − h||_cE_q(b). Hence, we can conclude that d(∧g, ∧h) ≤ 2LE_q(b)d(g, h) for any g, h ∈ 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

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.

Theorem 3.5.

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 that0<2LΓ(q)bqeθb2θ(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 ∧ defined in (5) is a contraction mapping on X with respect to the Bielecki norm:

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, h ∈ X. 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

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 ε > 0 and for each solution y of the following inequality

(7)|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.

Theorem 4.2.

SetM=1Γ(q)(1−pq−p)1−p(b−a)q−pwith 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:

(8)|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 aL1p-integrable function satisfying.

(9)(∫0x(φ(τ))1pdτ)p≤Kφ(x),

then there exists a unique continuous function y₀: [a, b] → ℝ such that y₀satisfies equation (1) and

(10)|y(x)−y0(x)|≤Cφ(x)εEq(xq),

for all x ∈ [a, b].

Proof

Let us consider the space of continuous functions

(11)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 ∧ : X → X by the formula

(Λg)(x)=1Γ(q)∫0x(x−τ)q−1g(x, τ, g(τ), g(α(τ)))dτ,

(12)|g(x)−h(x)|≤Kghφ(x)

holds for any x ∈ [a, b]. From the definition of ∧ and inequalities (8), (9) and (12), we have

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

Let us take g₀ ∈ X. 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 min_{x∈[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 ∧ⁿg₀ → y₀ in (X, d) as n → ∞ and ∧y₀ = y₀; that is, y₀ satisfies equation (1) for every x ∈ [a, b].

We will now verify that {g ∈ X| d(g₀, g) < ∞} = X. For any g ∈ X, since g and g₀ are bounded on [a, b] and min_x∈[a, b] E_q (x^q) > 0, there exists a constant 0 < C_g < ∞ such that

|g0(x)−g(x)|≤Cgφ(x),

for any x ∈ [a, b]. Hence, we have d(g₀, g) < ∞ for all g ∈ X; 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

(13)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].

Definition 5.1.

Equation (1) is Mittag-Leffler-Hyers-Ulam stable of the second type, with respect to E_q, if for every ε > 0 and solution y ∈ C([a, b], ℝ) of the following equation

(14)|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.

Theorem 5.2.

Let B be a Banach algebra. We suppose that: (i) f ∈ C([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, w₂ ∈ B and equation (14). Then equation (1) is Mittag-Leffler-Hyers-Ulam stable of the second type.

Proof

Let y ∈ C(I, B) satisfy the inequality (14). Let us denote by x ∈ C([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. □

Definition 5.3.

Equation (1) is Mittag-Leffler-Hyers-Ulam-Rassias stable of the second type, with respect to E_q, if for every ε > 0 and solution y ∈ C([a, b], ℝ) of the following inequality

(15)|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.

Theorem 5.4.

Let B be a Banach algebra. We suppose that: (i) f ∈ C([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, w₂ ∈ B and inequality (15). Then the initial equation (1) is Mittag-Leffler-Hyers-Ulam-Rassias stable of the second type.

Proof

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

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Received: 2015-8-1

Accepted: 2015-11-3

Published Online: 2016-4-23

Published in Print: 2016-1-1

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