Existence, Uniqueness and HU-Stability Results for Nonlinear Fuzzy Fractional Volterra-Fredholm Integro-Differential Equations

Ahmed A. Hamoud; Nedal M. Mohammed; Homan Emadifar; Foroud Parvaneh; Faraidum K. Hamasalh; Soubhagya Kumar Sahoo; Masoumeh Khademi

doi:10.5391/IJFIS.2022.22.4.391

Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(4): 391-400

Published online December 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.4.391

Existence, Uniqueness and HU-Stability Results for Nonlinear Fuzzy Fractional Volterra-Fredholm Integro-Differential Equations

Ahmed A. Hamoud¹, Nedal M. Mohammed², Homan Emadifar³;⁵, Foroud Parvaneh⁴, Faraidum K. Hamasalh⁵, Soubhagya Kumar Sahoo⁶, and Masoumeh Khademi³

¹Department of Mathematics, Taiz University, Taiz P.O. Box 6803, Yemen
²Department of Mathematics & Computer Science, Taiz University, Taiz P.O. Box 6803, Yemen
³Department of Mathematics, Islamic Azad University, Hamedan Branch, Hamedan, Iran
⁴Department of Mathematics, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran
⁵Department of Mathematics, College of Education, University of Sulaimani, Kurdistan Region, Iraq
⁶Department of Mathematics, Institute of Technical Education and Research, Siksha O Anusandhan University, Bhubaneswar 751031, Odisha, India

Correspondence to :
Homan Emadifar (homan_emadi@yahoo.com)

Received: February 16, 2022; Revised: November 12, 2022; Accepted: December 12, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction
2. Preliminaries
3. Existence and Uniqueness Results
4. Ulam-Hyers Stabilities
5. Examples
6. Conclusions
Conflict of Interest
References
Biographies

In this paper, we established some new results concerning the existence and uniqueness of the solutions of nonlinear Atangana-Baleanu-Caputo fractional (ABC-fractional) Fuzzy Volterra-Fredholm integro-differential equations. These new findings are obtained using the theorems of a fixed point. In addition, we investigate Hyers-Ulam stability for this fractional system. Our work extends and improves the results in the literature. Finally, some examples demonstrate the validity of the obtained theoretical results.

Keywords: Fuzzy Volterra-Fredholm equation, ABC-fractional derivative, Fixed point techniques, Hyers-Ulam stability

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
3. Existence and Uniqueness Results
4. Ulam-Hyers Stabilities
5. Examples
6. Conclusions
Conflict of Interest
References
Biographies

The history of fractional calculus goes back to more than 300 years ago. The idea of the fractional derivative returns back to L’Hospital who raised the question about derivative 1/2. To describe complex problems, the concept of a fractional-order derivative and a partial differential equation are used. One of the difficulties encountered in solving such equations is to predict the future behavior of the physical problem. Using fractional derivative operators to cope with this situation helps researchers [1–8]. Many fractional derivative definitions are used in the literature. The Caputo version [1] of the captive derivative is mostly used to model real world problems because it allows the use of initial conditions. The problem encountered in this version, however, is singularity due to the function used to induce the local derivative. Atangana-Baleanu fractional derivative [3] in the sense of Caputo is also quite assertive in this regard. Because the kernel used in this definition is both non-local and non-singular. This allows us to get rid of the singularity problem in the Caputo fractional derivative.

Recently, application of integro-differential equations of fractional order in mathematical modelling of physics and engineering subjects has increased progressively and they are enormous [2, 9–14].

The interval-approach fuzzy systems employ an infinite valued parameter of [0.1] as trust. This parameter increases the complicity while playing the leading role in producing the fuzzy system solution. The Atangana-Baleanu derivative has a memory to utilize all the information given in solving the models in the fractional case of integro-differential equations. This is therefore a significant aspect and benefit of this derivative in contrast with other known derivatives to minimize the complexity of numerical results. Although the instruments for the fractional integro-differential calculus were accessible for modeling various physical processes, the study has begun in the past three decades or so of the theory of fractional integro-differential equations [15, 16].

In fact the derivative Atangana-Baleanu is uniquely based on the Mittag-Leffler function that is more adapted to describe nature than power functions [3, 17–20]. A various classes of fractional differential equations have been taken into consideration by some authors [21–31].

Allahviranloo and Ghanbari [15] the following problem was investigated of A-B-FFDE for the existence of

DABC0ϑX(ν)=g(ν,X(ν)),0<ν<T<∞,X(0)=X0.

In the context of the A-B fractional operator, Arqub and Maayah [24] used the numeric approach based on the reproductive kernel Hilbert space method, by using certain tables and visual images for Fredholm type numeric solutions:

DABC0ϑX(ν)=Δ(ν,X(ν),∫01K(ν,ρ,X(ρ))dρ),X(0)=α,ν∈[0,1].

Motivated by the recent interest shown by many researchers in the theoretical treatment of ABC-fractional fuzzy differential equations, we consider and investigate a new problem consisting of a nonlinear ABC-fractional fuzzy Volterra-Fredholm integro-differential equation and initial conditions. In precise terms, we explore the existence, uniqueness and U-H stability criteria for the solutions of a nonlinear ABC-fractional fuzzy Volterra- Fredholm integro-differential equation of the form

(1)DABCιϑX(ν)=Δ(ν,X(ν))⊕∫ινΞ1(ν,ρ,X(ρ))dρ ⊕∫ιℓΞ2(ν,ρ,X(ρ))dρ,

(2)X(ι)=0, ν∈J:=[ι,ℓ],

where DABCιϑ is ABC-fractional operator with 0 < ϑ ≤ 1, Δ : J × ℝ → ℝ, Ξ₁, Ξ₂ : J × J × ℝ → ℝ, are continuous functions in the arguments with .

We organize the chapter as follows: In Section 2, we indicate some basic fractional calculus ideas, fixed point theorems, and also show an auxiliary lemma which is utilised in the this document. Section 3, provide the proofs of the existence and uniqueness of solution to the problem (1)–(2). The generalized Ulam stability is proved in Section 4. In Section 5, some examples are presented to illustrate the theory. Finally, a short conclusion is given in Section 6.

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
3. Existence and Uniqueness Results
4. Ulam-Hyers Stabilities
5. Examples
6. Conclusions
Conflict of Interest
References
Biographies

We mention in this part the notes, definitions, lemmas, fixed-point theorems and preparatory facts required to produce our major findings. For more details, see [15, 16, 19, 20, 22, 23, 29, 30, 32, 33].

The following concepts are needed in this paper:

(1) G^* is indicated in the set of FV measurable L^E(a, b) at [a, b].
(2) C^E(a, b) is indicated in the set of continuous on [a, b].
(3) All the fuzzy numbers are marked with ℝFc.
(4) ∑([0,1],ℝFC) (ξ)={ζ(ν)∈C([0,1],ℝFc):ζ≤ξ}.

The distance of Hausdorff is D:ℝFc×ℝFc→ℝ∪{0} with D(ζ1,ζ2)=supt∈[0,1]max{∣ζ1l(ν)-ζ2l(ν)∣,∣ζ1τ(ν)-ζ2τ(ν)∣}. A whole metric space (ℝ_F, ) is fulfilling:

(1) D(ζ1⊕ς,ζ2⊕ς)=D(ζ1,ζ2),∀ζ1,ζ2,ς∈ℝFc.
(2) D(ζ1⊕ς,0)=D(ζ1,0)+D(ζ1,0),∀ζ1,ζ2,ς∈ℝFc.
(3) D(ζ1⊕ζ2,ζ1⊕ς)=D(ζ2,ς),∀ζ1,ζ2,ς∈ℝFc.
(4) D(ζ1⊕ζ2,u⊕ς)≤D(ζ1,u)+D(ζ2,ς),∀ζ1,ζ2,ς∈ℝFc.
(5) D(ζ1⊖ζ2,u⊖ς)≤D(ζ1,u)+D(ζ2,ς),∀ζ1,ζ2,ς∈ℝFc.
(6) D(η⊙ζ1,η⊙ζ2)≤∣η∣D(ζ1,ζ2),∀ζ1,ζ2∈ℝF,η∈ℝFc.

Definition 2.1. [3]

ABC-fractional derivative of ζ ∈ H^*(u, v), v > u, ϖ ∈ [0, 1] is given by

(3)Du ABCτϖζ(τ)=∑(ϖ)1-ϖ∫uτζ′(ν)Eϖ[-ϖ(τ-ν)ϖ1-ϖ]dν,

where ∑(ϖ) > 0 is a normalizing function with ∑(0) = ∑(1) = 1, and E_ϖ is the well-Known Mittag-Leffler function of one variable.

Definition 2.2 [32]

ABR-fractional derivative of ζ ∈ H^*(u, v), v > u, ϖ ∈ [0, 1] is given by

(4)Du ABRτϖζ(τ)=∑(ϖ)1-ϖddτ∫uτζ(ν)Eϖ[-ϖ(τ-ν)ϖ1-ϖ]dν.

Definition 2.3 [3]

A–B integral of ζ ∈ H^*(u, v), v > u, ϖ ∈ (0, 1) is given by

(5)Iu ABRτϖζ(τ)= 1-ϖ∑(ϖ)ζ(τ) +ϖ∑(ϖ)Γ(ϖ)∫uτζ(ν) (τ-ν)ϖ-1dν.

Definition 2.4 [16]

A fuzzy number ζ(τ ) is a map ζ : ℝ → [0, 1], satisfy:

(1) ζ is upper semi-continuous.
(2) ζ(ρq +(1−ρ)s) ≥ min{ζ(q), ζ(s)} ∀q, s ∈ ℝ, ρ ∈ [0, 1], i.e., ζ is convex.
(3) ζ is normal i.e., ∃ ν₀ ∈ ℝ; ζ(ν₀) = 1.
(4) The closure of Supp (ζ) = {q ∈ ℝ: ζ(q) > 0} is compact.

Definition 2.5 [16]

The arithmetical operations in parametric form are specified as

(ζ1⊕ζ2) [c]=[ζ1l(c)+ζ2l(c),ζ1u(c)+ζ2u(c)],(ξ-ζ1) [c]={[ξζ1l(c),ξζ2u(c)],ξ≥0,[ξζ1u(c),ξζ2l(c)],ξ>0,

where c ∈ [0, 1].

We refer to a closed, bounded range, [ζ1l(c),ζ1u(c)] by the c-Level set with a fuzzy number, where ζ1l(c), is a left-hand ζ1u(c) is a right-hand end with [ζ₁]_c.

Definition 2.6 [15]

A ζ₁ fuzzy parameter is a (ζ1l,ζ1u) pair, where ζ1l(c),ζ1u(c), for c ∈ [0, 1], fulfill the following:

(1) ζ1l is left nondecreasing, bounded and continuous function on (0, 1].
(2) ζ1u is right nondecreasing, bounded and nondecreasing function on (0, 1].
(3) ζ1u(c)≥ζ1l(c), for c ∈ [0, 1].

Definition 2.7 [15]

The fused and fractional δ > 0 integrated Riemann-Liouville of a function G∈LFE(I)∩CFE(I) is shown by

IδG(ν)=1Γ(δ)∫0ν(ν-ρ)δ-1G(ρ)dρ,ν>0,

the parametric form is shown

IδG(ν)=(1Γ(δ)∫0ν(ν-ρ)δ-1Gl(ρ;η)dρ,1Γ(δ)∫0ν(ν-ρ)δ-1Gu(ρ;η)dρ),η∈[0,1].

Definition 2.8. [14]

Parametric form of order δ ∈ (n−1, n] of fractional Caputo derivative is given by

1) [Dρi,δζ(ν)]η=[Di,δζl(ν;η),Di,δζu(ν;η)],
2) [Dνii,δζ(ν)]η=[Di,δζu(ν;η),Di,δζl(ν;η)],

where

Di,δG(ν)=1Γ(n-δ)∫0νGi-(k)(ρ) (ν-ρ)n-δ-1dρ,Dii,δG(ν)=1Γ(n-δ)∫0νGii-(k)(ρ) (ν-ρ)n-δ-1dρ,

for n = [δ] + 1.

Lemma 2.9

Let 0 < ϑ ≤ 1, assume that Δ, Ξ₁ and Ξ₂ are continuous functions. If then is solution of satisfies

(6)X(ν)= 1-ϑ∑(ϑ)⊙[Δ(ν,X(ν))⊕∫ινΞ1(ν,ρ,X(ρ))dρ ⊕∫ιℓΞ2(ν,ρ,X(ρ))dρ] ⊕ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1[Δ(ρ,X(ρ)) ⊕∫ιρΞ1(ρ,η,X(η))dη ⊕∫ιℓΞ2(ρ,η,X(η))dη] dρ.

3. Existence and Uniqueness Results

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Abstract
1. Introduction
2. Preliminaries
3. Existence and Uniqueness Results
4. Ulam-Hyers Stabilities
5. Examples
6. Conclusions
Conflict of Interest
References
Biographies

We will establish that the system (1)–(2) has a unique solution in C(J,ℝ). For this end, we transform the system (1)–(2) into fixed point as , where

℧:C(J,ℝ)→C(J,ℝ),

is determined by

(7)℧(ν,X(ν))= 1-ϑ∑(ϑ)⊙[Δ(ν,X(ν))⊕∫ινΞ1(ν,ρ,X(ρ))dρ ⊕∫ιℓΞ2(ν,ρ,X(ρ))dρ]⊕ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1 ×[Δ(ρ,X(ρ))⊕∫ιρΞ1(ρ,η,X(η))dη ⊕∫ιℓΞ2(ρ,η,X(η))dη] dρ.

Theorem 3.1

Let Δ and Ξ₁, Ξ₂ are continuous functions. Suppose that

(H1) There exist constants L_g, L_k, and L_h > 0 such that

D(Δ(ν,X(ν)),Δ(ν,N(ν)))≤LgD(X,N),D(Ξ1(ν,ρ,X(ν)),Ξ1(ν,ρ,N(ν)))≤LkD(X,N),D(Ξ2(ν,ρ,X(ν)),Ξ2(ν,ρ,N(ν)))≤LhD(X,N),

and

(8)(1-ϑ∑(ϑ)+ϑ(ℓϑ-ιϑ)∑(ϑ)Γ(ϑ+1)) [Lg+(Lk+Lh) (ℓ-ι)]<1.

Then operator satisfies the Lipschitz condition.

Proof

To prove that Lipschitz condition fulfilled by . We have to do this

(9)D(℧(ν,X(ν)),℧(ν,N(ν)))=D(1-ϑ∑(ϑ)⊙[Δ(ν,X(ν))⊕∫ινΞ1(ν,ρ,X(ρ))dρ ⊕∫ιℓΞ2(ν,ρ,X(ρ))dρ]⊕ϑ∑(ϑ)Γ(ϑ) ∫ιν(ν-ρ)ϑ-1[Δ(ρ,X(ρ))⊕∫ιρΞ1(ρ,η,X(η))dη ⊕∫ιℓΞ2(ρ,η,X(η))dη] dρ,1-ϑ∑(ϑ)⊙[Δ(ν,N(ν)) ⊕∫ινΞ1(ν,ρ,N(ρ))dρ⊕∫ιℓΞ2(ν,ρ,N(ρ))dρ] ⊕ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1[Δ(ρ,N(ρ)) ⊕∫ιρΞ1(ρ,η,N(η))dη⊕∫ιℓΞ2(ρ,η,N(η))dη] dp)≤1-ϑ∑(ϑ)[D(Δ(ν,X(ν)),Δ(ν,N(ν))) +∫ινD(Ξ1(ν,ρ,X(ρ)),Ξ1(ν,ρ,N(ρ)))dρ +∫ιℓD(Ξ2(ν,ρ,X(ρ)),Ξ2(ν,ρ,N(ρ)))dρ +ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1 ×[D(Δ(ρ,X(ρ)),Δ(ρ,N(ρ))) +∫ιρD(Ξ1(ρ,η,X(η)),Ξ1(ρ,η,N(η)))dη +∫ιℓD(Ξ2(ρ,η,X(η)),Ξ2(ρ,η,N(η)))dη] dρ≤1-ϑ∑(ϑ)[LgD(X,N)+∫ινLkD(X,N)dρ +∫ιℓLhD(X,N)dρ] +ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1[LgD(X,N) +∫ιρLkD(X,N)dη+∫ιℓLhD(X,N)dη] dρ≤1-ϑ∑(ϑ)[Lg+Lk(ℓ-ι)+Lh(ℓ-ι)]D(X,N) +ϑ(ℓϑ-ιϑ)∑(ϑ)Γ(ϑ+1) ×[Lg+Lk(ℓ-ι)+Lh(ℓ-ι)]D(X,N)= (1-ϑ∑(ϑ)+ϑ(ℓℓ-ιϑ)∑(ϑ)Γ(ϑ+1)) ×[Lg+Lk(ℓ-ι)+Lh(ℓ-ι)]D(X,N).

The operator ʊ fulfils the requirement of Lipschitz using condition (8). Hence, proof is verified.

Now, we consider the recurrence formula:

(10)Xn(ν)= 1-ϑ∑(ϑ)⊙[Δ(ν,Xn(ν))⊕∫ινΞ1(ν,ρ,Xn(ρ))dρ ⊕∫ιℓΞ2(ν,ρ,Xn(ρ))dρ]⊕ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1 ×[Δ(ρ,Xn(ρ))⊕∫ιρΞ1(ρ,η,Xn(η))dη ⊕∫ιℓΞ2(ρ,η,Xn(η))dη] dρ.

Theorem 3.2

Assume that (H1) is satisfied, and if

Ω: =(1-ϑ∑(ϑ)+ϑ(ℓϑ-ιϑ)∑(ϑ)Γ(ϑ+1)) [Lg+(Lk+Lh) (ℓ-ι)] <1.

Then the problem (1)–(2) has at least a solution on J.

Proof

We define the sequence . Then, by using (10),

(11)D(Xn(ν),X(ν))= D(1-ϑ∑(ϑ)⊙[Δ(ν,Xn(ν))⊕∫ινΞ1(ν,ρ,Xn(ρ))dρ ⊕∫ιℓΞ2(ν,ρ,Xn(ρ))dρ]⊕ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1 ×[Δ(ρ,Xn(ρ))⊕∫ιρΞ1(ρ,η,Xn(η))dη ⊕∫ιℓΞ2(ρ,η,Xn(η))dη] dρ,1-ϑ∑(ϑ)⊙[Δ(ν,X(ν)) ⊕∫ινΞ1(ν,ρ,X(ρ))dρ⊕∫ιℓΞ2(ν,ρ,X(ρ))dρ] ⊕ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1[Δ(ρ,X(ρ)) ⊕∫ιρΞ1(ρ,η,X(η))dη⊕∫ιℓΞ2(ρ,η,X(η))dη] dp)≤ 1-ϑ∑(ϑ)[D(Δ(ν,Xn(ν)),Δ(ν,X(ν))) +∫ινD(Ξ1(ν,ρ,Xn(ρ)),Ξ1(ν,ρ,X(ρ)))dρ +∫ιℓD(H(ν,ρ,Xn(ρ)),Ξ2(ν,ρ,X(ρ)))dρ] +ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1 ×[D(Δ(ρ,Xn(ρ)),Δ(ρ,X(ρ))) +∫ιρD(Ξ1(ρ,η,Xn(ρ)),Ξ1(ρ,η,X(η)))dη +∫ιℓD(H(ρ,η,Xn(η)),Ξ2(ρ,η,X(η)))dη] dρ≤ 1-ϑ∑(ϑ)[LgD(Xn,X)+∫ινLkD(Xn,X)dρ +∫ιℓLhD(Xn,X)dρ]+ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1 ×[LgD(Xn,X)+∫ιρLkD(Xn,X)dη +∫ιℓLhD(Xn,X)dη] dρ≤1-ϑ∑(ϑ)[Lg+(ℓ-ι)Lk+(ℓ-ι)Lh]D(Xn,X) +ϑ(ℓϑ-ιϑ)∑(ϑ)Γ(ϑ+1)[Lg+(ℓ-ι)Lk+(ℓ-ι)Lh]D(Xn,X)= (1-ϑ∑(ϑ)+ϑ(ℓℓ-ιϑ)∑(ϑ)Γ(ϑ+1)) ×[Lg+(Lk+Lh) (ℓ-ι)]D(Xn,X)= ΩDn(Xn,X).

Since Ω < 1. Thus, the sequence , as n → ∞. Hence, proof of the existence is verified.

Theorem 3.3

Assume that Δ, Ξ₁, Ξ₂ are continuous functions satisfy the assumption (H1). Then the IVP (1)–(2) has a unique solutions on J if Ω < 1.

Proof

For the uniqueness of solution of the IVP (1)–(2), we assume that there exists another solution say , satisfying

(12)N(ν)= 1-ϑ∑(ϑ)⊙[Δ(ν,N(ν))⊕∫ινΞ1(ν,ρ,N(ρ))dρ ⊕∫ιℓΞ2(ν,ρ,N(ρ))dρ]⊕ϑ∑(ϑ)Γ(ϑ) ×∫ιν(ν-ρ)ϑ-1[Δ(ρ,N(ρ)) ⊕∫ιρΞ1(ρ,η,N(η))dη⊕∫ιℓΞ2(ρ,η,N(η))dη] dρ.

Now, consider

(13)D(X(ν),N(ν))= D(1-ϑ∑(ϑ)⊙[Δ(ν,X(ν))⊕∫ινΞ1(ν,ρ,X(ρ))dρ ⊕∫ιℓΞ2(ν,ρ,X(ρ))dρ]⊕ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1 ×[Δ(ρ,X(ρ))⊕∫ιρΞ1(ρ,η,X(η))dη ⊕∫ιℓΞ2(ρ,η,X(η))dη] dρ, 1-ϑ∑(ϑ)⊙[Δ(ν,N(ν))⊕∫ινΞ1(ν,ρ,N(ρ))dρ ⊕∫ιℓΞ2(ν,ρ,N(ρ))dρ]⊕ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1 ×[Δ(ρ,N(ρ))⊕∫ιρΞ1(ρ,η,N(η))dη ⊕∫ιℓΞ2(ρ,η,N(η))dη] dp)≤ 1-ϑ∑(ϑ)[D(Δ(ν,X(ν)),Δ(ν,N(ν))) +∫ινD(Ξ1(ν,ρ,X(ρ)),Ξ1(ν,ρ,N(ρ)))dρ +∫ιℓD(Ξ2(ν,ρ,X(ρ)),Ξ2(ν,ρ,N(ρ)))dρ +ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1 ×[D(Δ(ρ,X(ρ)),Δ(ρ,N(ρ))) +∫ιρD(Ξ1(ρ,η,X(η)),Ξ1(ρ,η,N(η)))dη +∫ιℓD(Ξ2(ρ,η,X(η)),Ξ2(ρ,η,N(η)))dη]dρ≤1-ϑ∑(ϑ)[LgD(X,N)+∫ινLkD(X,N)dρ +∫ιℓLhD(X,N)dρ] +ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1[LgD(X,N) +∫ιρLkD(X,N)dη+∫ιℓLhD(X,N)dη]dρ≤ 1-ϑ∑(ϑ)[Lg+Lk(ℓ-ι)+Lh(ℓ-ι)]D(X,N) +ϑ(ℓϑ-ιϑ)∑(ϑ)Γ(ϑ+1)[Lg+Lk(ℓ-ι)+Lh(ℓ-ι)]D(X,N).

This also means that

(14)1-ϑ∑(ϑ)[Lg+Lk(ℓ-ι)+Lh(ℓ-ι)]D(X,N)+ϑ(ℓϑ-ιϑ)∑(ϑ)Γ(ϑ+1)[Lg+Lk(ℓ-ι)+Lh(ℓ-ι)]×D(X,N)≥0.

Thus , which implies that . In the final analysis, the IVP (1)–(2) has a unique solution.

4. Ulam-Hyers Stabilities

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Abstract
1. Introduction
2. Preliminaries
3. Existence and Uniqueness Results
4. Ulam-Hyers Stabilities
5. Examples
6. Conclusions
Conflict of Interest
References
Biographies

We will discuss the Ulam-Hyers stability for the IVP (1)–(2) by using the integration.

Definition 4.1

For each β > 0 and for each solution of the IVP (1)–(2)

(15)X1(ν)=1-ϑ∑(ϑ)⊙[Δ(ν,X1(ν))⊕∫ινΞ1(ν,ρ,X1(ρ))dρ ⊕∫ιℓΞ2(ν,ρ,X1(ρ))dρ]⊕ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1 [Δ(ρ,X1(ρ))⊕∫ιρΞ1(ρ,η,X1(η))dη ⊕∫ιℓΞ2(ρ,η,X1(η))dη]dρ

is said to be Ulam-Hyers stable if there exist constant λ^* is a positive real number depending on β such that

(16)D(X(ν),X1(ν))≤λ*β.

Theorem 4.2

Under assumption (H1) in Theorem 3.1, with Ω < 1. The IVP (1)–(2) is Ulam-Hyers stable.

Proof

Let be a solution of the IVP (1)–(2) satisfying (15) in the sense of Theorem 3.1.

(17)D(X(ν),X1(ν))= D(1-ϑ∑(ϑ)⊙[Δ(ν,X(ν))⊕∫ινΞ1(ν,ρ,X(ρ))dρ ⊕∫ιℓΞ2(ν,ρ,X(ρ))dρ]⊕ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1 ×[Δ(ρ,X(ρ))⊕∫ιρΞ1(ρ,η,X(η))dη ⊕∫ιℓΞ2(ρ,η,X(η))dη]dρ,1-ϑ∑(ϑ)⊙[Δ(ν,X1(ν)) ⊕∫ινΞ1(ν,ρ,X1(ρ))dρ⊕∫ιℓΞ2(ν,ρ,X1(ρ))dρ] ⊕ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1[Δ(ρ,X1(ρ)) ⊕∫ιρΞ1(ρ,η,X1(η))dη⊕∫ιℓΞ2(ρ,η,X1(η))dη]dp)≤ 1-ϑ∑(ϑ)[D(Δ(ν,X(ν)),Δ(ν,X1(ν))) +∫ινD(Ξ1(ν,ρ,X(ρ)),Ξ1(ν,ρ,X1(ρ)))dρ +∫ιℓD(Ξ2(ν,ρ,X(ρ)),Ξ2(ν,ρ,X1(ρ)))dρ] +ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1[D(Δ(ρ,X(ρ)),Δ(ρ,X1(ρ))) +∫ιρD(Ξ1(ρ,η,X(η)),Ξ1(ρ,η,X1(η)))dη +∫ιℓD(Ξ2(ρ,η,X(η)),Ξ2(ρ,η,X1(η)))dη]dρ≤1-ϑ∑(ϑ)[LgD(X,X1)+∫ινLkD(X,X1)dρ +∫ιℓLhD(X,X1)dρ]+ϑ∑(ϑ)Γ(ϑ)∫ιν(ν-ρ)ϑ-1 ×[LgD(X,X1)+∫ιρLkD(X,X1)dη +∫ιℓLhD(X,X1)dη]dρ≤1-ϑ∑(ϑ)[Lg+Lk(ℓ-ι)+Lh(ℓ-ι)]D(X,X1) +ϑ(ℓϑ-ιϑ)∑(ϑ)Γ(ϑ+1)[Lg+Lk(ℓ-ι)+Lh(ℓ-ι)]× D(X,X1).

This further implies that

D(X(ν),X1(ν))≤λ*β,

for λ*=1-ϑ∑(ϑ)[Lg+Lk(ℓ-ι)+Lh(ℓ-ι)]+ϑ(ℓϑ-ιϑ)∑(ϑ)Γ(ϑ+1)[Lg+Lk(ℓ-ι)+Lh(ℓ-ι)], , we obtain the Ulam-Hyers stability condition.

5. Examples

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Abstract
1. Introduction
2. Preliminaries
3. Existence and Uniqueness Results
4. Ulam-Hyers Stabilities
5. Examples
6. Conclusions
Conflict of Interest
References
Biographies

Example 1

Consider the fractional Volterra-Fredholm integro-differential equation

(18)D0.5X(ν)= X(ν)6eν⊕∫0ν(X(ρ)(6+e-ρ+ν)-12)dρ ⊕∫01(X(ρ)(7+e-ρ+ν)-12)dρ,

with condition

(19)X(0)=0.

Then we have

|Ξ1(ν,ρ,w(ρ))-Ξ1(ν,ρ,Λ(ρ))|≤17∣w-Λ∣,|Ξ2(ν,ρ,w(ρ))-Ξ2(ν,ρ,Λ(ρ))|≤18∣w-Λ∣.

From equation (18) and (19), we find

ϑ=0.5,ι=0,ℓ=1,

then we calculate L_g = 0.16666, L_k = 0.142857, L_h = 0.125, then we have

New, we have

(1-ϑ∑(ϑ)+ϑ(ℓϑ-ιϑ)∑(ϑ)Γ(ϑ+1)) [Lg+(Lk+Lh) (ℓ-ι)]≅0.46.

Since all the hypotheses of Theorems 3.2 and 3.3 are satisfied, then there exists a solution and the solution is unique of the given problem.

Also, the hypotheses of Theorem 4.2 are satisfied, then the given problem is U-H stable.

Example 2

Consider the fractional Volterra-Fredholm integro-differential equation (1)–(2) with

ϑ=0.5,ι=0,ℓ=1,Lg=0.2,Lk=0.3,Lh=0.2,X(0)=0.

New, we have

(1-ϑ∑(ϑ)+ϑ(ℓϑ-ιϑ)∑(ϑ)Γ(ϑ+1)) [Lg+(Lk+Lh) (ℓ-ι)]≅0.7<1.

Since all the hypotheses of Theorems 3.2 and 3.3 are satisfied, then there exists a solution and the solution is unique of the given problem.

Also, the hypotheses of Theorem 4.2 are satisfied, then the given problem is U-H stable.

6. Conclusions

Go to

Abstract
1. Introduction
2. Preliminaries
3. Existence and Uniqueness Results
4. Ulam-Hyers Stabilities
5. Examples
6. Conclusions
Conflict of Interest
References
Biographies

In this current paper, we focused on establishing the existence and uniqueness results for a class of nonlinear fuzzy Volterra-Fredholm integro-differential equations involving the Atangana-Baleanu-Caputo fractional derivative in Banach Space. Upon making some appropriate assumptions, by utilizing the ideas and techniques of solution operators, the theory of fractional calculus, fixed point theorems, sufficient conditions for controllability results are obtained. In addition, we investigated, Ulam-Hyers stability for this fractional system. We emphasize that the results established in this paper are indeed new and enrich the existing literature on ABC-fractional fuzzy Volterra-Fredholm equations. In the future study, we plan to discuss the inclusion variant of the problem at hand. In view of the occurrence of systems of fractional differential equations in several natural and scientific disciplines, we will also work on a coupled system of ABC-fractional Volterra-Fredholm integro-differential equations equipped with coupled nonlocal ABC integral and multi-point boundary conditions.

Conflict of Interest

Go to

Abstract
1. Introduction
2. Preliminaries
3. Existence and Uniqueness Results
4. Ulam-Hyers Stabilities
5. Examples
6. Conclusions
Conflict of Interest
References
Biographies

No potential conflict of interest relevant to this article was reported.

References

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Abstract
1. Introduction
2. Preliminaries
3. Existence and Uniqueness Results
4. Ulam-Hyers Stabilities
5. Examples
6. Conclusions
Conflict of Interest
References
Biographies

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Lakshmikantham, V (1995). Theory of Integro-Differential Equations: CRC press
Atangana, A, and Baleanu, D (2016). New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model. Thermal Science. 20, 763-769. https://doi.org/10.2298/TSCI160111018A
Achar, SJ, Baishya, C, and Kaabar, MKA (2021). Dynamics of the worm transmission in wireless sensor network in the framework of fractional derivatives. Mathematical Methods in the Applied Sciences. 45. https://doi.org/10.1002/mma.8039
Rashid, S, Kaabar, MKA, Althobaiti, A, and Alqurashi, MS (2022). Constructing analytical estimates of the fuzzy fractional-order Boussinesq model and their application in oceanography. Journal of Ocean Engineering and Science. 2022, 1-20. https://doi.org/10.1016/j.joes.2022.01.003
Abu-Shady, M, and Kaabar, MKA (2021). A generalized definition of the fractional derivative with applications. Mathematical Problems in Engineering. 2021. Article ID 9444803
Kaabar, MKA, Martínez, F, Gómez-Aguilar, JF, Ghanbari, B, Kaplan, M, and Günerhan, H (2021). New approximate analytical solutions for the nonlinear fractional Schrödinger equation with second-order spatiotemporal dispersion via double Laplace transform method. Mathematical Methods in the Applied Sciences. 44, 11138-11156. https://doi.org/10.1002/mma.7476
Samei, ME, Ghaffari, R, Yao, S, Kaabar, WMKA, Martinez, F, and Inc, M (2021). Existence of solutions for a singular fractional q-differential equations under Riemann-Liouville integral boundary condition. Symmetry. 13, 1235. https://doi.org/10.3390/sym13071235
Tien, DN (2013). Fractional stochastic differential equations with applications to finance. Journal of Mathematical Analysis and Applications. 397, 334-348. https://doi.org/10.1016/j.jmaa.2012.07.062
Benson, DA, Wheatcraft, SW, and Meerschaert, MM (2000). Application of a fractional Advection-dispersion equation. Water Resources Research. 36, 1403-1412. https://doi.org/10.1029/2000WR900031
Hilfer, R (2000). Applications of Fractional Calculus in Physics: World Scientific
Zeinali, M, and Shahmorad, S (2018). An equivalence lemma for a class of fuzzy implicit integro-differential equations. Journal of Computational and Applied Mathematics. 327, 388-399. https://doi.org/10.1016/j.cam.2017.06.001
Ravichandran, C, Logeswari, K, and Jarad, F (2019). New results on existence in the framework of Atangana-Baleanu derivative for fractional integro-differential equations. Chaos, Solitons & Fractals. 125, 194-200. https://doi.org/10.1016/j.chaos.2019.05.014
Kilbas, AA, Srivastava, HM, and Trujillo, JJ (2006). Theory and Applications of Fractional Differential Equations. Elsevier, North-Holland Math. Studies
Allahviranloo, T, and Ghanbari, B (2020). On the fuzzy fractional differential equation with interval Atangana- Baleanu fractional derivative approach. Chaos, Solitons & Fractals. 130, 109397. 109397. https://doi.org/10.1016/j.chaos.2019.109397
Van Hoa, N (2015). Fuzzy fractional functional differential equations under Caputo gh-differentiability. Communications in Nonlinear Science and Numerical Simulation. 22, 1134-1157. https://doi.org/10.1016/j.cnsns.2014.08.006
Khan, I (2019). New idea of Atangana and Baleanu fractional derivatives to human blood flow in nanofluids. Chaos: An Interdisciplinary Journal of Nonlinear Science. 29, 013121. https://doi.org/10.1063/1.5078738
Aliyu, AI, Inc, M, Yusuf, A, and Baleanu, D (2018). A fractional model of vertical transmission and cure of vector-borne diseases pertaining to the Atangana-Baleanu fractional derivatives. Chaos, Solitons & Fractals. 116, 268-277. https://doi.org/10.1016/j.chaos.2018.09.043
Ahmad, B, and Sivasundaram, S (2008). Some existence results for fractional integro-differential equations with nonlinear conditions. Communications in Applied Analysis. 12, 107-112. http://www.acadsol.eu/en/articles/12/2/1.pdf
Ahmad, B, Alruwaily, Y, Alsaedi, A, and Ntouyas, SK (2019). Existence and stability results for a fractional order differential equation with non-conjugateriemann-stieltjes integro-multipoint boundary conditions. Mathematics. 7, 249. https://doi.org/10.3390/math7030249
Albzeirat, AK, Ahmad, MZ, Momani, S, and Ahmad, I (2018). New implementation of reproducing kernel Hilbert space method for solving a fuzzy integro-differential equation of integer and fractional orders. Journal of King Saud University-Science. 30, 352-358. https://doi.org/10.1016/j.jksus.2017.04.002
Alsaedi, A, Nieto, JJ, and Venktesh, V (2015). Fractional electrical circuits. Advances in Mechanical Engineering. 7, 1-7. https://doi.org/10.1177/1687814015618127
Bahuguna, D, and Dabas, J (Array). Existence and uniqueness of a solution to a partial integro-differential equation by the method of lines. Electronic Journal of Qualitative Theory of Differential Equations, 1-12. http://real.mtak.hu/id/eprint/22933
Arqub, OA, and Maayah, B (2018). Numerical solutions of integrodifferential equations of Fredholm operator type in the sense of the Atangana-Baleanu fractional operator. Chaos, Solitons & Fractals. 117, 117-124. https://doi.org/10.1016/j.chaos.2018.10.007
Balachandran, K, and Trujillo, JJ (2010). The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications. 72, 4587-4593. https://doi.org/10.1016/j.na.2010.02.035
Delouei, AA, Emamian, A, Karimnejad, S, Sajjadi, H, and Jing, D (2020). Two-dimensional analytical solution for temperature distribution in fg hollow spheres: General thermal boundary conditions. International Communications in Heat and Mass Transfer. 113, 104531. https://doi.org/10.1016/j.icheatmasstransfer.2020.104531
Hussain, K, Hamoud, A, and Mohammed, N (2019). Some new uniqueness results for fractional integrodifferential equations. Nonlinear Functional Analysis and Applications. 24, 827-836. https://doi.org/10.22771/nfaa.2019.24.04.13
Alikhani, R, Bahrami, F, and Jabbari, A (2012). Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations. Nonlinear Analysis: Theory, Methods & Applications. 75, 1810-1821. https://doi.org/10.1016/j.na.2011.09.021
Hamoud, AA, and Ghadle, KP (2018). Existence and uniqueness of solutions for fractional mixed Volterra-Fredholm integro-differential equations. Indian Journal of Mathematics. 60, 375-395.
Hamoud, A, and Ghadle, K (2018). Usage of the Homotopy analysis method for solving fractional Volterra-Fredholm integro-differential equation of the second kind. Tamkang Journal of Mathematics. 49, 301-315. https://doi.org/10.5556/j.tkjm.49.2018.2718
Hamoud, AA, Sharif, AA, and Ghadle, KP (2021). Existence, uniqueness and stability results of fractional Volterra-Fredholm integro differential equations of ψ-Hilfer type. Discontinuity, Nonlinearity, and Complexity. 10, 535-545. https://doi.org/10.5890/DNC.2021.09.013
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Biographies

Go to

Abstract
1. Introduction
2. Preliminaries
3. Existence and Uniqueness Results
4. Ulam-Hyers Stabilities
5. Examples
6. Conclusions
Conflict of Interest
References
Biographies

Ahmed A. Hamoud possesses his Ph.D. degree in Applied Mathematics from BAMU University, Aurangabad, India. He is currently an assistant professor of Mathematics at Taiz University, Taiz, Yemen. His research interests lie in mathematical modelling, fractional integro differential equations, integral equations and numerical analysis. His other interests include the applications of the fuzzy fractional integro differential equations.

E-mail: ahmed.hamoud@taiz.edu.ye

https://orcid.org/0000-0002-8877-7337

Nedal M. Mohammed received her Ph.D. degree in Computer Science & IT from BAMUUniversity, Aurangabad, India. She is currently Head and Assistant Professor, Dept. of Computer Science & IT, Taiz University, Taiz, Yemen. Her research interests lie in Computer Networks Security, Practical Secure Computation Outsourcing, Secure Outsourced Computation of the Optimization Problems. Her other interests include the applications of the fuzzy Fractional Integro-Differential Equations.

E-mail: dr.nedal.mohammed@taiz.edu.ye

https://orcid.org/0000-0002-9997-7297

Homan Emadifar is a Professor of Mathematics (Numerical Analysis). He is live in Sulaimani City- Kurdistan of Iraq. Department of Mathematics, College of Education, University of Sulaimani/Sulaimani, Iraq. He was awarded a PhD in Mathematics (Numerical Analysis) from Sulaimani University, Department of Mathematics, College of Science. His research interests include numerical solution of ODE, approximation by spline functions, Fluid Mechanics and Fractional Derivatives.

E-mail: homan emadi@yahoo.com

https://orcid.org/0000-0002-8034-1475

Foroud Parvane has a doctorate in mathematics and is a member of the academic staff of Azad University, Kermanshah branch, and has many articles.

Faraidum K. Hamasalh has a doctorate in mathematics and is a member of the academic staff of Azad University, Kermanshah branch, and has many articles.

Soubhagya Kumar Sahoo is currently working in the Department of Mathematics, SOA University, Odisha, India. His research interests are integral inequalities and their applications, fractional calculus, quantum calculus, etc. He has published several research articles in well-reputed international journals. He has reviewed articles for several International journals and also edited few special issues in different journals.

https://orcid.org/0000-0003-4524-1951

Masoumeh Khademi is a researcher in mathematics and has several ISI articles. Her main focus is on equations with fractional derivatives.

https://orcid.org/0000-0002-5276-4075

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(4): 391-400

Published online December 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.4.391

Existence, Uniqueness and HU-Stability Results for Nonlinear Fuzzy Fractional Volterra-Fredholm Integro-Differential Equations

Ahmed A. Hamoud¹, Nedal M. Mohammed², Homan Emadifar³;⁵, Foroud Parvaneh⁴, Faraidum K. Hamasalh⁵, Soubhagya Kumar Sahoo⁶, and Masoumeh Khademi³

Correspondence to:Homan Emadifar (homan_emadi@yahoo.com)

Received: February 16, 2022; Revised: November 12, 2022; Accepted: December 12, 2022

Abstract

Keywords: Fuzzy Volterra-Fredholm equation, ABC-fractional derivative, Fixed point techniques, Hyers-Ulam stability

1. Introduction

Allahviranloo and Ghanbari [15] the following problem was investigated of A-B-FFDE for the existence of

\begin{array}{l} {{}^{A B C}D}_{0}^{ϑ} X (ν) = g (ν, X (ν)), 0 < ν < T < \infty, \\ X (0) = X_{0} . \end{array}

\begin{array}{l} {{}^{A B C}D}_{0}^{ϑ} X (ν) = Δ (ν, X (ν), \int_{0}^{1} K (ν, ρ, X (ρ)) d ρ), \\ X (0) = α, ν \in [0, 1] . \end{array}

(1)

\begin{array}{l} {{}^{A B C}D}_{ι}^{ϑ} X (ν) = & Δ (ν, X (ν)) \oplus \int_{ι}^{ν} Ξ_{1} (ν, ρ, X (ρ)) d ρ \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ν, ρ, X (ρ)) d ρ, \end{array}

(2)

X (ι) = 0, ν \in J : = [ι, ℓ],

where ${{}^{A B C}D}_{ι}^{ϑ}$ is ABC-fractional operator with 0 < ϑ ≤ 1, Δ : J × ℝ → ℝ, Ξ₁, Ξ₂ : J × J × ℝ → ℝ, are continuous functions in the arguments with .

2. Preliminaries

The following concepts are needed in this paper:

(1) G^* is indicated in the set of FV measurable L^E(a, b) at [a, b].
(2) C^E(a, b) is indicated in the set of continuous on [a, b].
(3) All the fuzzy numbers are marked with $ℝ_{F}^{c}$ .
(4) $\sum ([0, 1], ℝ_{F}^{C}) (ξ) = {ζ (ν) \in C ([0, 1], ℝ_{F}^{c}) : ζ \leq ξ}$ .

The distance of Hausdorff is $D : ℝ_{F}^{c} \times ℝ_{F}^{c} \to ℝ \cup {0}$ with $D (ζ_{1}, ζ_{2}) = {sup}_{t \in [0, 1]} max {∣ ζ_{1}^{l} (ν) - ζ_{2}^{l} (ν) ∣, ∣ ζ_{1}^{τ} (ν) - ζ_{2}^{τ} (ν) ∣}$ . A whole metric space (ℝ_F, ) is fulfilling:

(1) $D (ζ_{1} \oplus ς, ζ_{2} \oplus ς) = D (ζ_{1}, ζ_{2}), \forall ζ_{1}, ζ_{2}, ς \in ℝ_{F}^{c}$ .
(2) $D (ζ_{1} \oplus ς, 0) = D (ζ_{1}, 0) + D (ζ_{1}, 0), \forall ζ_{1}, ζ_{2}, ς \in ℝ_{F}^{c}$ .
(3) $D (ζ_{1} \oplus ζ_{2}, ζ_{1} \oplus ς) = D (ζ_{2}, ς), \forall ζ_{1}, ζ_{2}, ς \in ℝ_{F}^{c}$ .
(4) $D (ζ_{1} \oplus ζ_{2}, u \oplus ς) \leq D (ζ_{1}, u) + D (ζ_{2}, ς), \forall ζ_{1}, ζ_{2}, ς \in ℝ_{F}^{c}$ .
(5) $D (ζ_{1} ⊖ ζ_{2}, u ⊖ ς) \leq D (ζ_{1}, u) + D (ζ_{2}, ς), \forall ζ_{1}, ζ_{2}, ς \in ℝ_{F}^{c}$ .
(6) $D (η ⊙ ζ_{1}, η ⊙ ζ_{2}) \leq ∣ η ∣ D (ζ_{1}, ζ_{2}), \forall ζ_{1}, ζ_{2} \in ℝ_{F}, η \in ℝ_{F}^{c}$ .

Definition 2.1. [3]

ABC-fractional derivative of ζ ∈ H^*(u, v), v > u, ϖ ∈ [0, 1] is given by

(3)

{{}_{u}^{A B C}D}_{τ}^{ϖ} ζ (τ) = \frac{\sum (ϖ)}{1 - ϖ} \int_{u}^{τ} ζ^{'} (ν) E_{ϖ} [\frac{- ϖ {(τ - ν)}^{ϖ}}{1 - ϖ}] d ν,

where ∑(ϖ) > 0 is a normalizing function with ∑(0) = ∑(1) = 1, and E_ϖ is the well-Known Mittag-Leffler function of one variable.

Definition 2.2 [32]

ABR-fractional derivative of ζ ∈ H^*(u, v), v > u, ϖ ∈ [0, 1] is given by

(4)

{{}_{u}^{A B R}D}_{τ}^{ϖ} ζ (τ) = \frac{\sum (ϖ)}{1 - ϖ} \frac{d}{d τ} \int_{u}^{τ} ζ (ν) E_{ϖ} [\frac{- ϖ {(τ - ν)}^{ϖ}}{1 - ϖ}] d ν .

Definition 2.3 [3]

A–B integral of ζ ∈ H^*(u, v), v > u, ϖ ∈ (0, 1) is given by

(5)

\begin{array}{l} {{}_{u}^{A B R}I}_{τ}^{ϖ} ζ (τ) = & \frac{1 - ϖ}{\sum (ϖ)} ζ (τ) \\ + \frac{ϖ}{\sum (ϖ) Γ (ϖ)} \int_{u}^{τ} ζ (ν) {(τ - ν)}^{ϖ - 1} d ν . \end{array}

Definition 2.4 [16]

A fuzzy number ζ(τ ) is a map ζ : ℝ → [0, 1], satisfy:

(1) ζ is upper semi-continuous.
(2) ζ(ρq +(1−ρ)s) ≥ min{ζ(q), ζ(s)} ∀q, s ∈ ℝ, ρ ∈ [0, 1], i.e., ζ is convex.
(3) ζ is normal i.e., ∃ ν₀ ∈ ℝ; ζ(ν₀) = 1.
(4) The closure of Supp (ζ) = {q ∈ ℝ: ζ(q) > 0} is compact.

Definition 2.5 [16]

The arithmetical operations in parametric form are specified as

\begin{array}{l} (ζ_{1} \oplus ζ_{2}) [c] = [ζ_{1}^{l} (c) + ζ_{2}^{l} (c), ζ_{1}^{u} (c) + ζ_{2}^{u} (c)], (ξ - ζ_{1}) [c] \\ = {\begin{array}{l} [ξ ζ_{1}^{l} (c), ξ ζ_{2}^{u} (c)], & ξ \geq 0, \\ [ξ ζ_{1}^{u} (c), ξ ζ_{2}^{l} (c)], & ξ > 0, \end{array} \end{array}

where c ∈ [0, 1].

We refer to a closed, bounded range, [ $ζ_{1}^{l} (c), ζ_{1}^{u} (c)$ ] by the c-Level set with a fuzzy number, where $ζ_{1}^{l} (c)$ , is a left-hand $ζ_{1}^{u} (c)$ is a right-hand end with [ζ₁]_c.

Definition 2.6 [15]

A ζ₁ fuzzy parameter is a ( $ζ_{1}^{l}, ζ_{1}^{u}$ ) pair, where $ζ_{1}^{l} (c), ζ_{1}^{u} (c)$ , for c ∈ [0, 1], fulfill the following:

(1) $ζ_{1}^{l}$ is left nondecreasing, bounded and continuous function on (0, 1].
(2) $ζ_{1}^{u}$ is right nondecreasing, bounded and nondecreasing function on (0, 1].
(3) $ζ_{1}^{u} (c) \geq ζ_{1}^{l} (c)$ , for c ∈ [0, 1].

Definition 2.7 [15]

The fused and fractional δ > 0 integrated Riemann-Liouville of a function $G \in L_{F}^{E} (I) \cap C_{F}^{E} (I)$ is shown by

I^{δ} G (ν) = \frac{1}{Γ (δ)} \int_{0}^{ν} {(ν - ρ)}^{δ - 1} G (ρ) d ρ, ν > 0,

the parametric form is shown

I^{δ} G (ν) = (\frac{1}{Γ (δ)} \int_{0}^{ν} {(ν - ρ)}^{δ - 1} G_{l} (ρ; η) d ρ, \frac{1}{Γ (δ)} \int_{0}^{ν} {(ν - ρ)}^{δ - 1} G_{u} (ρ; η) d ρ), η \in [0, 1] .

Definition 2.8. [14]

Parametric form of order δ ∈ (n−1, n] of fractional Caputo derivative is given by

1) ${[D_{ρ}^{i, δ} ζ (ν)]}_{η} = [D^{i, δ} ζ_{l} (ν; η), D^{i, δ} ζ_{u} (ν; η)],$
2) ${[D_{ν}^{i i, δ} ζ (ν)]}_{η} = [D^{i, δ} ζ_{u} (ν; η), D^{i, δ} ζ_{l} (ν; η)],$

where

\begin{array}{l} D^{i, δ} G (ν) = \frac{1}{Γ (n - δ)} \int_{0}^{ν} {{}_{i -}G}^{(k)} (ρ) {(ν - ρ)}^{n - δ - 1} d ρ, \\ D^{i i, δ} G (ν) = \frac{1}{Γ (n - δ)} \int_{0}^{ν} {{}_{i i -}G}^{(k)} (ρ) {(ν - ρ)}^{n - δ - 1} d ρ, \end{array}

for n = [δ] + 1.

Lemma 2.9

Let 0 < ϑ ≤ 1, assume that Δ, Ξ₁ and Ξ₂ are continuous functions. If then is solution of satisfies

(6)

\begin{array}{l} X (ν) = & \frac{1 - ϑ}{\sum (ϑ)} ⊙ [Δ (ν, X (ν)) \oplus \int_{ι}^{ν} Ξ_{1} (ν, ρ, X (ρ)) d ρ \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ν, ρ, X (ρ)) d ρ] \\ \oplus \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} [Δ (ρ, X (ρ)) \\ \oplus \int_{ι}^{ρ} Ξ_{1} (ρ, η, X (η)) d η \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ρ, η, X (η)) d η] d ρ . \end{array}

3. Existence and Uniqueness Results

We will establish that the system (1)–(2) has a unique solution in C(J,ℝ). For this end, we transform the system (1)–(2) into fixed point as , where

℧ : C (J, ℝ) \to C (J, ℝ),

is determined by

(7)

\begin{array}{l} ℧ (ν, X (ν)) = & \frac{1 - ϑ}{\sum (ϑ)} ⊙ [Δ (ν, X (ν)) \oplus \int_{ι}^{ν} Ξ_{1} (ν, ρ, X (ρ)) d ρ \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ν, ρ, X (ρ)) d ρ] \oplus \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} \\ \times [Δ (ρ, X (ρ)) \oplus \int_{ι}^{ρ} Ξ_{1} (ρ, η, X (η)) d η \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ρ, η, X (η)) d η] d ρ . \end{array}

Theorem 3.1

Let Δ and Ξ₁, Ξ₂ are continuous functions. Suppose that

(H1) There exist constants L_g, L_k, and L_h > 0 such that

\begin{array}{l} D (Δ (ν, X (ν)), Δ (ν, N (ν))) \leq L_{g} D (X, N), \\ D (Ξ_{1} (ν, ρ, X (ν)), Ξ_{1} (ν, ρ, N (ν))) \leq L_{k} D (X, N), \\ D (Ξ_{2} (ν, ρ, X (ν)), Ξ_{2} (ν, ρ, N (ν))) \leq L_{h} D (X, N), \end{array}

and

(8)

(\frac{1 - ϑ}{\sum (ϑ)} + \frac{ϑ (ℓ^{ϑ} - ι^{ϑ})}{\sum (ϑ) Γ (ϑ + 1)}) [L_{g} + (L_{k} + L_{h}) (ℓ - ι)] < 1.

Then operator satisfies the Lipschitz condition.

Proof

To prove that Lipschitz condition fulfilled by . We have to do this

(9)

\begin{array}{l} D (℧ (ν, X (ν)), ℧ (ν, N (ν))) = & D (\frac{1 - ϑ}{\sum (ϑ)} ⊙ [Δ (ν, X (ν)) \oplus \int_{ι}^{ν} Ξ_{1} (ν, ρ, X (ρ)) d ρ \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ν, ρ, X (ρ)) d ρ] \oplus \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \\ \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} [Δ (ρ, X (ρ)) \oplus \int_{ι}^{ρ} Ξ_{1} (ρ, η, X (η)) d η \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ρ, η, X (η)) d η] d ρ, \frac{1 - ϑ}{\sum (ϑ)} ⊙ [Δ (ν, N (ν)) \\ \oplus \int_{ι}^{ν} Ξ_{1} (ν, ρ, N (ρ)) d ρ \oplus \int_{ι}^{ℓ} Ξ_{2} (ν, ρ, N (ρ)) d ρ] \\ \oplus \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} [Δ (ρ, N (ρ)) \\ \oplus \int_{ι}^{ρ} Ξ_{1} (ρ, η, N (η)) d η \oplus \int_{ι}^{ℓ} Ξ_{2} (ρ, η, N (η)) d η] d p) \\ \leq & \frac{1 - ϑ}{\sum (ϑ)} [D (Δ (ν, X (ν)), Δ (ν, N (ν))) \\ + \int_{ι}^{ν} D (Ξ_{1} (ν, ρ, X (ρ)), Ξ_{1} (ν, ρ, N (ρ))) d ρ \\ + \int_{ι}^{ℓ} D (Ξ_{2} (ν, ρ, X (ρ)), Ξ_{2} (ν, ρ, N (ρ))) d ρ \\ + \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} \\ \times [D (Δ (ρ, X (ρ)), Δ (ρ, N (ρ))) \\ + \int_{ι}^{ρ} D (Ξ_{1} (ρ, η, X (η)), Ξ_{1} (ρ, η, N (η))) d η \\ + \int_{ι}^{ℓ} D (Ξ_{2} (ρ, η, X (η)), Ξ_{2} (ρ, η, N (η))) d η] d ρ \\ \leq & \frac{1 - ϑ}{\sum (ϑ)} [L_{g} D (X, N) + \int_{ι}^{ν} L_{k} D (X, N) d ρ \\ + \int_{ι}^{ℓ} L_{h} D (X, N) d ρ] \\ + \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} [L_{g} D (X, N) \\ + \int_{ι}^{ρ} L_{k} D (X, N) d η + \int_{ι}^{ℓ} L_{h} D (X, N) d η] d ρ \\ \leq & \frac{1 - ϑ}{\sum (ϑ)} [L_{g} + L_{k} (ℓ - ι) + L_{h} (ℓ - ι)] D (X, N) \\ + \frac{ϑ (ℓ^{ϑ} - ι^{ϑ})}{\sum (ϑ) Γ (ϑ + 1)} \\ \times [L_{g} + L_{k} (ℓ - ι) + L_{h} (ℓ - ι)] D (X, N) \\ = & (\frac{1 - ϑ}{\sum (ϑ)} + \frac{ϑ (ℓ^{ℓ} - ι^{ϑ})}{\sum (ϑ) Γ (ϑ + 1)}) \\ \times [L_{g} + L_{k} (ℓ - ι) + L_{h} (ℓ - ι)] D (X, N) . \end{array}

The operator ʊ fulfils the requirement of Lipschitz using condition (8). Hence, proof is verified.

Now, we consider the recurrence formula:

(10)

\begin{array}{l} X_{n} (ν) = & \frac{1 - ϑ}{\sum (ϑ)} ⊙ [Δ (ν, X_{n} (ν)) \oplus \int_{ι}^{ν} Ξ_{1} (ν, ρ, X_{n} (ρ)) d ρ \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ν, ρ, X_{n} (ρ)) d ρ] \oplus \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} \\ \times [Δ (ρ, X_{n} (ρ)) \oplus \int_{ι}^{ρ} Ξ_{1} (ρ, η, X_{n} (η)) d η \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ρ, η, X_{n} (η)) d η] d ρ . \end{array}

Theorem 3.2

Assume that (H1) is satisfied, and if

\begin{array}{l} Ω : & = (\frac{1 - ϑ}{\sum (ϑ)} + \frac{ϑ (ℓ^{ϑ} - ι^{ϑ})}{\sum (ϑ) Γ (ϑ + 1)}) [L_{g} + (L_{k} + L_{h}) (ℓ - ι)] \\ < 1. \end{array}

Then the problem (1)–(2) has at least a solution on J.

Proof

We define the sequence . Then, by using (10),

(11)

\begin{array}{l} D (X_{n} (ν), X (ν)) = & D (\frac{1 - ϑ}{\sum (ϑ)} ⊙ [Δ (ν, X_{n} (ν)) \oplus \int_{ι}^{ν} Ξ_{1} (ν, ρ, X_{n} (ρ)) d ρ \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ν, ρ, X_{n} (ρ)) d ρ] \oplus \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} \\ \times [Δ (ρ, X_{n} (ρ)) \oplus \int_{ι}^{ρ} Ξ_{1} (ρ, η, X_{n} (η)) d η \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ρ, η, X_{n} (η)) d η] d ρ, \frac{1 - ϑ}{\sum (ϑ)} ⊙ [Δ (ν, X (ν)) \\ \oplus \int_{ι}^{ν} Ξ_{1} (ν, ρ, X (ρ)) d ρ \oplus \int_{ι}^{ℓ} Ξ_{2} (ν, ρ, X (ρ)) d ρ] \\ \oplus \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} [Δ (ρ, X (ρ)) \\ \oplus \int_{ι}^{ρ} Ξ_{1} (ρ, η, X (η)) d η \oplus \int_{ι}^{ℓ} Ξ_{2} (ρ, η, X (η)) d η] d p) \\ \leq & \frac{1 - ϑ}{\sum (ϑ)} [D (Δ (ν, X_{n} (ν)), Δ (ν, X (ν))) \\ + \int_{ι}^{ν} D (Ξ_{1} (ν, ρ, X_{n} (ρ)), Ξ_{1} (ν, ρ, X (ρ))) d ρ \\ + \int_{ι}^{ℓ} D (H (ν, ρ, X_{n} (ρ)), Ξ_{2} (ν, ρ, X (ρ))) d ρ] \\ + \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} \\ \times [D (Δ (ρ, X_{n} (ρ)), Δ (ρ, X (ρ))) \\ + \int_{ι}^{ρ} D (Ξ_{1} (ρ, η, X_{n} (ρ)), Ξ_{1} (ρ, η, X (η))) d η \\ + \int_{ι}^{ℓ} D (H (ρ, η, X_{n} (η)), Ξ_{2} (ρ, η, X (η))) d η] d ρ \\ \leq & \frac{1 - ϑ}{\sum (ϑ)} [L_{g} D (X_{n}, X) + \int_{ι}^{ν} L_{k} D (X_{n}, X) d ρ \\ + \int_{ι}^{ℓ} L_{h} D (X_{n}, X) d ρ] + \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} \\ \times [L_{g} D (X_{n}, X) + \int_{ι}^{ρ} L_{k} D (X_{n}, X) d η \\ + \int_{ι}^{ℓ} L_{h} D (X_{n}, X) d η] d ρ \\ \leq & \frac{1 - ϑ}{\sum (ϑ)} [L_{g} + (ℓ - ι) L_{k} + (ℓ - ι) L_{h}] D (X_{n}, X) \\ + \frac{ϑ (ℓ^{ϑ} - ι^{ϑ})}{\sum (ϑ) Γ (ϑ + 1)} [L_{g} + (ℓ - ι) L_{k} + (ℓ - ι) L_{h}] D (X_{n}, X) \\ = & (\frac{1 - ϑ}{\sum (ϑ)} + \frac{ϑ (ℓ^{ℓ} - ι^{ϑ})}{\sum (ϑ) Γ (ϑ + 1)}) \\ \times [L_{g} + (L_{k} + L_{h}) (ℓ - ι)] D (X_{n}, X) \\ = & Ω {}^{n}D (X_{n}, X) . \end{array}

Since Ω < 1. Thus, the sequence , as n → ∞. Hence, proof of the existence is verified.

Theorem 3.3

Assume that Δ, Ξ₁, Ξ₂ are continuous functions satisfy the assumption (H1). Then the IVP (1)–(2) has a unique solutions on J if Ω < 1.

Proof

For the uniqueness of solution of the IVP (1)–(2), we assume that there exists another solution say , satisfying

(12)

\begin{array}{l} N (ν) = & \frac{1 - ϑ}{\sum (ϑ)} ⊙ [Δ (ν, N (ν)) \oplus \int_{ι}^{ν} Ξ_{1} (ν, ρ, N (ρ)) d ρ \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ν, ρ, N (ρ)) d ρ] \oplus \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \\ \times \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} [Δ (ρ, N (ρ)) \\ \oplus \int_{ι}^{ρ} Ξ_{1} (ρ, η, N (η)) d η \oplus \int_{ι}^{ℓ} Ξ_{2} (ρ, η, N (η)) d η] d ρ . \end{array}

Now, consider

(13)

\begin{array}{l} D (X (ν), N (ν)) = & D (\frac{1 - ϑ}{\sum (ϑ)} ⊙ [Δ (ν, X (ν)) \oplus \int_{ι}^{ν} Ξ_{1} (ν, ρ, X (ρ)) d ρ \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ν, ρ, X (ρ)) d ρ] \oplus \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} \\ \times [Δ (ρ, X (ρ)) \oplus \int_{ι}^{ρ} Ξ_{1} (ρ, η, X (η)) d η \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ρ, η, X (η)) d η] d ρ, \\ \frac{1 - ϑ}{\sum (ϑ)} ⊙ [Δ (ν, N (ν)) \oplus \int_{ι}^{ν} Ξ_{1} (ν, ρ, N (ρ)) d ρ \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ν, ρ, N (ρ)) d ρ] \oplus \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} \\ \times [Δ (ρ, N (ρ)) \oplus \int_{ι}^{ρ} Ξ_{1} (ρ, η, N (η)) d η \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ρ, η, N (η)) d η] d p) \\ \leq & \frac{1 - ϑ}{\sum (ϑ)} [D (Δ (ν, X (ν)), Δ (ν, N (ν))) \\ + \int_{ι}^{ν} D (Ξ_{1} (ν, ρ, X (ρ)), Ξ_{1} (ν, ρ, N (ρ))) d ρ \\ + \int_{ι}^{ℓ} D (Ξ_{2} (ν, ρ, X (ρ)), Ξ_{2} (ν, ρ, N (ρ))) d ρ \\ + \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} \\ \times [D (Δ (ρ, X (ρ)), Δ (ρ, N (ρ))) \\ + \int_{ι}^{ρ} D (Ξ_{1} (ρ, η, X (η)), Ξ_{1} (ρ, η, N (η))) d η \\ + \int_{ι}^{ℓ} D (Ξ_{2} (ρ, η, X (η)), Ξ_{2} (ρ, η, N (η))) d η] d ρ \\ \leq & \frac{1 - ϑ}{\sum (ϑ)} [L_{g} D (X, N) + \int_{ι}^{ν} L_{k} D (X, N) d ρ \\ + \int_{ι}^{ℓ} L_{h} D (X, N) d ρ] \\ + \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} [L_{g} D (X, N) \\ + \int_{ι}^{ρ} L_{k} D (X, N) d η + \int_{ι}^{ℓ} L_{h} D (X, N) d η] d ρ \\ \leq & \frac{1 - ϑ}{\sum (ϑ)} [L_{g} + L_{k} (ℓ - ι) + L_{h} (ℓ - ι)] D (X, N) \\ + \frac{ϑ (ℓ^{ϑ} - ι^{ϑ})}{\sum (ϑ) Γ (ϑ + 1)} [L_{g} + L_{k} (ℓ - ι) + L_{h} (ℓ - ι)] D (X, N) . \end{array}

This also means that

(14)

\begin{array}{l} \frac{1 - ϑ}{\sum (ϑ)} [L_{g} + L_{k} (ℓ - ι) + L_{h} (ℓ - ι)] D (X, N) \\ + \frac{ϑ (ℓ^{ϑ} - ι^{ϑ})}{\sum (ϑ) Γ (ϑ + 1)} [L_{g} + L_{k} (ℓ - ι) + L_{h} (ℓ - ι)] \\ \times D (X, N) \geq 0. \end{array}

Thus , which implies that . In the final analysis, the IVP (1)–(2) has a unique solution.

4. Ulam-Hyers Stabilities

We will discuss the Ulam-Hyers stability for the IVP (1)–(2) by using the integration.

Definition 4.1

For each β > 0 and for each solution of the IVP (1)–(2)

(15)

\begin{array}{l} X_{1} (ν) = & \frac{1 - ϑ}{\sum (ϑ)} ⊙ [Δ (ν, X_{1} (ν)) \oplus \int_{ι}^{ν} Ξ_{1} (ν, ρ, X_{1} (ρ)) d ρ \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ν, ρ, X_{1} (ρ)) d ρ] \oplus \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} \\ [Δ (ρ, X_{1} (ρ)) \oplus \int_{ι}^{ρ} Ξ_{1} (ρ, η, X_{1} (η)) d η \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ρ, η, X_{1} (η)) d η] d ρ \end{array}

is said to be Ulam-Hyers stable if there exist constant λ^* is a positive real number depending on β such that

(16)

D (X (ν), X_{1} (ν)) \leq λ^{*} β .

Theorem 4.2

Under assumption (H1) in Theorem 3.1, with Ω < 1. The IVP (1)–(2) is Ulam-Hyers stable.

Proof

Let be a solution of the IVP (1)–(2) satisfying (15) in the sense of Theorem 3.1.

(17)

\begin{array}{l} D (X (ν), X_{1} (ν)) = & D (\frac{1 - ϑ}{\sum (ϑ)} ⊙ [Δ (ν, X (ν)) \oplus \int_{ι}^{ν} Ξ_{1} (ν, ρ, X (ρ)) d ρ \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ν, ρ, X (ρ)) d ρ] \oplus \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} \\ \times [Δ (ρ, X (ρ)) \oplus \int_{ι}^{ρ} Ξ_{1} (ρ, η, X (η)) d η \\ \oplus \int_{ι}^{ℓ} Ξ_{2} (ρ, η, X (η)) d η] d ρ, \frac{1 - ϑ}{\sum (ϑ)} ⊙ [Δ (ν, X_{1} (ν)) \\ \oplus \int_{ι}^{ν} Ξ_{1} (ν, ρ, X_{1} (ρ)) d ρ \oplus \int_{ι}^{ℓ} Ξ_{2} (ν, ρ, X_{1} (ρ)) d ρ] \\ \oplus \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} [Δ (ρ, X_{1} (ρ)) \\ \oplus \int_{ι}^{ρ} Ξ_{1} (ρ, η, X_{1} (η)) d η \oplus \int_{ι}^{ℓ} Ξ_{2} (ρ, η, X_{1} (η)) d η] d p) \\ \leq & \frac{1 - ϑ}{\sum (ϑ)} [D (Δ (ν, X (ν)), Δ (ν, X_{1} (ν))) \\ + \int_{ι}^{ν} D (Ξ_{1} (ν, ρ, X (ρ)), Ξ_{1} (ν, ρ, X_{1} (ρ))) d ρ \\ + \int_{ι}^{ℓ} D (Ξ_{2} (ν, ρ, X (ρ)), Ξ_{2} (ν, ρ, X_{1} (ρ))) d ρ] \\ + \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} [D (Δ (ρ, X (ρ)), Δ (ρ, X_{1} (ρ))) \\ + \int_{ι}^{ρ} D (Ξ_{1} (ρ, η, X (η)), Ξ_{1} (ρ, η, X_{1} (η))) d η \\ + \int_{ι}^{ℓ} D (Ξ_{2} (ρ, η, X (η)), Ξ_{2} (ρ, η, X_{1} (η))) d η] d ρ \\ \leq & \frac{1 - ϑ}{\sum (ϑ)} [L_{g} D (X, X_{1}) + \int_{ι}^{ν} L_{k} D (X, X_{1}) d ρ \\ + \int_{ι}^{ℓ} L_{h} D (X, X_{1}) d ρ] + \frac{ϑ}{\sum (ϑ) Γ (ϑ)} \int_{ι}^{ν} {(ν - ρ)}^{ϑ - 1} \\ \times [L_{g} D (X, X_{1}) + \int_{ι}^{ρ} L_{k} D (X, X_{1}) d η \\ + \int_{ι}^{ℓ} L_{h} D (X, X_{1}) d η] d ρ \\ \leq & \frac{1 - ϑ}{\sum (ϑ)} [L_{g} + L_{k} (ℓ - ι) + L_{h} (ℓ - ι)] D (X, X_{1}) \\ + \frac{ϑ (ℓ^{ϑ} - ι^{ϑ})}{\sum (ϑ) Γ (ϑ + 1)} [L_{g} + L_{k} (ℓ - ι) + L_{h} (ℓ - ι)] \\ \times & D (X, X_{1}) . \end{array}

This further implies that

D (X (ν), X_{1} (ν)) \leq λ^{*} β,

for $λ^{*} = \frac{1 - ϑ}{\sum (ϑ)} [L_{g} + L_{k} (ℓ - ι) + L_{h} (ℓ - ι)] + \frac{ϑ (ℓ^{ϑ} - ι^{ϑ})}{\sum (ϑ) Γ (ϑ + 1)} [L_{g} + L_{k} (ℓ - ι) + L_{h} (ℓ - ι)]$ , , we obtain the Ulam-Hyers stability condition.

5. Examples

Example 1

Consider the fractional Volterra-Fredholm integro-differential equation

(18)

\begin{array}{l} D^{0.5} X (ν) = & \frac{X (ν)}{6 e^{ν}} \oplus \int_{0}^{ν} (\frac{X (ρ)}{(6 + e^{- ρ} + ν)} - \frac{1}{2}) d ρ \\ \oplus \int_{0}^{1} (\frac{X (ρ)}{(7 + e^{- ρ} + ν)} - \frac{1}{2}) d ρ, \end{array}

with condition

(19)

X (0) = 0.

Then we have

\begin{array}{l} | Ξ_{1} (ν, ρ, w (ρ)) - Ξ_{1} (ν, ρ, Λ (ρ)) | \leq \frac{1}{7} ∣ w - Λ ∣, \\ | Ξ_{2} (ν, ρ, w (ρ)) - Ξ_{2} (ν, ρ, Λ (ρ)) | \leq \frac{1}{8} ∣ w - Λ ∣ . \end{array}

From equation (18) and (19), we find

ϑ = 0.5, ι = 0, ℓ = 1,

then we calculate L_g = 0.16666, L_k = 0.142857, L_h = 0.125, then we have

New, we have

\begin{array}{l} (\frac{1 - ϑ}{\sum (ϑ)} + \frac{ϑ (ℓ^{ϑ} - ι^{ϑ})}{\sum (ϑ) Γ (ϑ + 1)}) [L_{g} + (L_{k} + L_{h}) (ℓ - ι)] \\ ≅ 0.46. \end{array}

Since all the hypotheses of Theorems 3.2 and 3.3 are satisfied, then there exists a solution and the solution is unique of the given problem.

Also, the hypotheses of Theorem 4.2 are satisfied, then the given problem is U-H stable.

Example 2

Consider the fractional Volterra-Fredholm integro-differential equation (1)–(2) with

\begin{array}{l} ϑ = 0.5, ι = 0, ℓ = 1, L_{g} = 0.2, L_{k} = 0.3, L_{h} = 0.2, \\ X (0) = 0. \end{array}

New, we have

\begin{array}{l} (\frac{1 - ϑ}{\sum (ϑ)} + \frac{ϑ (ℓ^{ϑ} - ι^{ϑ})}{\sum (ϑ) Γ (ϑ + 1)}) [L_{g} + (L_{k} + L_{h}) (ℓ - ι)] \\ ≅ 0.7 < 1. \end{array}

Since all the hypotheses of Theorems 3.2 and 3.3 are satisfied, then there exists a solution and the solution is unique of the given problem.

Also, the hypotheses of Theorem 4.2 are satisfied, then the given problem is U-H stable.

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International Journalof Fuzzy Logic andIntelligent Systems

Original Article

Existence, Uniqueness and HU-Stability Results for Nonlinear Fuzzy Fractional Volterra-Fredholm Integro-Differential Equations

Abstract

1. Introduction

2. Preliminaries

Definition 2.1. [3]

Definition 2.2 [32]

Definition 2.3 [3]

Definition 2.4 [16]

Definition 2.5 [16]

Definition 2.6 [15]

Definition 2.7 [15]

Definition 2.8. [14]

Lemma 2.9

3. Existence and Uniqueness Results

Theorem 3.1

Theorem 3.2

Theorem 3.3

4. Ulam-Hyers Stabilities

Definition 4.1

Theorem 4.2

5. Examples

Example 1

Example 2

6. Conclusions

Conflict of Interest

References

Biographies

Article

Original Article

Existence, Uniqueness and HU-Stability Results for Nonlinear Fuzzy Fractional Volterra-Fredholm Integro-Differential Equations

Abstract

1. Introduction

2. Preliminaries

Definition 2.1. [3]

Definition 2.2 [32]

Definition 2.3 [3]

Definition 2.4 [16]

Definition 2.5 [16]

Definition 2.6 [15]

Definition 2.7 [15]

Definition 2.8. [14]

Lemma 2.9

3. Existence and Uniqueness Results

Theorem 3.1

Theorem 3.2

Theorem 3.3

4. Ulam-Hyers Stabilities

Definition 4.1

Theorem 4.2

5. Examples

Example 1

Example 2

6. Conclusions

References

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