# Onψ-Caputo Partial Hyperbolic Differential Equations with a Finite Delay

Authors: Mohammed S. Abdo; Mohammed Zakarya; Emad E. Mahmoud; Saeed M. Ali; Abdel-Haleem Abdel-Aty
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this work, we are concerned with some qualitative analyses of fractional-order partial hyperbolic functional differential equations under theψ-Caputo type. To be precise, we investigate the existence and uniqueness results based on the nonlinear alternative of the Leray-Schauder type and Banach contraction mapping. Moreover, we present two similar results to nonlocal problems. Then, the guarantee of the existence of solutions is shown by Ulam-Hyer’s stability. Two examples will be given to illustrate the abstract results. Eventually, some known results in the literature are extended.


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## Body

## 1. Introduction

Since fractional calculus (FC) has a decent global correlation execution to reflect the historical reliance process of the improvement of system functions and can likewise describe the characteristics of the dynamic system itself, it turned into a strong mathematical gadget to describe a few complex developments, unpredictable phenomena, memory highlights, and other aspects. FC theory was vastly utilized by mathematicians as well as scientific experts, engineers, financial analysts, scholars, and physicists (see [1–4]). Riemann, in 1876, suggested the definition of the Riemann-Liouville (RL) fractional derivative (FD). Caputo originally proposed one more definition of FD through a changed RL fractional integral (FI) toward the start of the twentieth century, to be specific, a Caputo FD. One issue in this field is the major and extraordinary number of possible various definitions of FD and FI; settling on the choice of the best operator for every specific framework is a significant issue. One method for conquering this issue is to consider overall definitions, of which the classical ones can be viewed as specific cases [5, 6].In this regard, Almeida [7] and Sousa and de Oliveira [8] recently introduced ψ-Caputo FD and ψ-Hilfer FD of one variable, respectively, from which it is feasible to obtain a wide class of FDs already well established. Sousa and de Oliveira [9] have very recently expanded ψ-Hilfer FD with two variables. Therefore, one of the aims of this work is to introduce some qualitative analyses of solutions based on ψ-Caputo FD with two variables.Then again, functional differential equations (FDEs) and fractional FDEs with finite delay show up frequently in applications as models of equations, and consequently, the investigation of these kinds of equations has gotten incredible consideration somewhat recently; see, for instance, [10–14] and the references in those. The literature connected with the existence of solutions of fractional partial FDEs with a finite delay was processed very slowly; see, for instance, [15–19].The background and survey in the literature relative to classical fractional partial hyperbolic FDEs can be found in the monograph of Abbas et al. [19]. Sousa and de Oliveira [9] discussed the stability of fractional partial hyperbolic DEs without delay under the ψ-Hilfer operator. Baitiche et al. [20] established the existence result of coupled systems of fractional partial hyperbolic DEs without delay.This work is concerned with the existence, uniqueness, and Ulam-Hyer (HU) stability of the solution to theψ-Caputo-type fractional partial hyperbolic FDE with finite delay:
(1)CD0+r;ψzϰ,τ=Fϰ,τ,zϰ,τ,ϰ,τ∈J1≔0,c×0,d,zϰ,τ=φϰ,τ,ϰ,τ∈J2≔−κ1,c×−κ2,d\0,c×0,d,zϰ,0=ϕ1ϰ,z0,τ=ϕ2τ,ϰ∈0,c,τ∈0,d,and the ψ-Caputo-type fractional nonlocal partial hyperbolic FDE with finite delay:
(2)CDℓ+r;ψzϰ,τ=Fϰ,τ,zϰ,τ,ϰ,τ∈J1≔0,c×0,d,zϰ,τ=φϰ,τ,ϰ,τ∈J2≔−κ1,c×−κ2,d\0,c×0,d,zϰ,0+h1z=ϕ1τ,z0,τ+h1z=ϕ2τ,ϰ∈0,c,τ∈0,d,where c,d,κ1,κ2>0,r=μ,ν∈0,1×0,1,DCℓ+r;ψ is the ψ-Caputo FD of order r with respect to another function ψ, which is increasing, and ∂ψ/∂ù,∂ψ/∂τ≠0, for ϰ̀,τ∈J1,ℓ=0,0,φ·,·∈C≔C−κ1,0×−κ2,0,ℝ,F:J1×C⟶ℝ,ϕ1:0,c⟶ℝ,ϕ2:0,d⟶ℝ are absolutely continuous with ϕ1ϰ=φϰ,0,ϕ2τ=φ0,τ,∀ϰ∈0,c,∀τ∈0,d, and h1,h2:CJ1,ℝ⟶ℝ are continuous.This paper is concerned with the qualitative analyses of fractional partial hyperbolic FDEs, which are very new, and the implementation of theψ-fractional operator makes it more general and novel, unlike the classical fractional operators. To be precise, we are interested in investigating the existence, uniqueness, and Ulam-Hyer’s stability results for our problems (1)–(2). These results initiate the investigation of ψ-Caputo fractional partial hyperbolic FDEs with a finite delay, which mainly includes a more general fractional operator based on another function ψ. To be certain, in the analysis of our results, we essentially use fixed point theorems (FPTs) of the Leray-Schauder type and Banach type. Our outcomes can be interpreted as extensions of preceding results that Abbas et al. [19] and Sousa and de Oliveira [9] obtained for classical FHDEs, which can be considered a contribution to the literature.The rest of the work has been organized as follows. Section2 is devoted to some essential connotations of ψ-fractional calculus with auxiliary lemmas to problems at hand. The existence, uniqueness, and UH stability results based on fixed point techniques are provided in Section 3. Suitable examples are given in Section 4. In Section 5, we present the conclusions.


## 2. Preliminary Results

In this section, we give some notations and essential definitions of fractional partial integrals and derivatives (FPIs and FPDs) and some function spaces to simplify the forthcoming analysis. LetJ1=0,c×0,d,J2≔−κ1,c×−κ2,d\0,c×0,d, where c,d,κ1,κ2>0,ℓ=0,0, and r=μ,ν∈0,1×0,1. Denote C≔C−κ1,0×−κ2,0,ℝ the space of continuous functions on −κ1,0×−κ2,0. Note that C is the Banach space with the norm
(3)zC=supϰ,τ∈−κ1,0×−κ2,0zϰ,τ,and let CJ1,ℝ be the Banach space with the norm
(4)z∞=supϰ,τ∈0,c×0,dzϰ,τ.The spaceL1J1,ℝ is endowed with the norm
(5)zL1=∫0c∫0dzϰ,τdϰdτ.For anyzϰ,τ:−κ1,c×−κ2,d⟶ℝ, where ϰ,τ∈J1, we have
(6)zϰ,τθ,θ=zϰ+θ,τ+θ,forθ,θ∈−κ1,0×−κ2,0.Define the spaceC−κ1,c×−κ2,d,ℝ as
(7)Cc,d=z:−κ1,c×−κ2,d⟶ℝ:zJ2=φ∈C,zJ1∈CJ1,ℝ,where zJ1is the restriction of z to J1, which is a Banach space with the norm
(8)zCc,d=supϰ,τ∈−κ1,c×−κ2,dzϰ,τ.In the forthcoming analysis, let us considerψ· to be an increasing and positive monotone function on J1 with ψϰ̀·,ψτ·≠0 on J1, where ψϰ=∂ψ/∂ϰ and ψτ=∂ψ/∂τ. On the whole paper, keep in mind ψθ−1y,m≔ψy−ψmθ−1.Definition 1 (see [9]).
Letℓ=0,0,r=μ,ν, where μ,ν>0. Then, the ψ-RL FPI of a function of two variables zϰ,τ∈L1J1,ℝ of order r is given by
(9)Iℓ+r;ψzϰ,τ=1ΓμΓν∫0ϰ∫0τψϰθψτζψμ−1ϰ,θψν−1τ,ζzθ,ζdθdζ.
Also, we have(10)I0+μzϰ,τ=1Γμ∫0ϰψϰθψμ−1ϰ,θzθ,τdθ,I0+νzϰ,τ=1Γν∫0τψτθψν−1τ,θzϰ,θdθ.Definition 2 (see [9]).
Letℓ=0,0, and r=μ,ν, where 0<μ,ν≤1. Then, the ψ-RL FPD of a function zϰ,τ∈L1J1,ℝ of order r is defined by
(11)Dℓ+r;ψzϰ,τ=1ψϰψτ∂2∂ϰ∂τIℓ+1−r;ψzϰ,τ.Definition 3 (see [9]).
Letℓ=0,0,r=μ,ν, where 0<μ,ν≤1, and ψ∈C1J1,ℝ. Then, the ψ-Caputo FPD of a function zϰ,τ∈C1J1,ℝ of order r is defined by
(12)DCℓ+r;ψzϰ,τ=Iℓ+1−r;ψ1ψϰψτ∂2∂ϰ∂τzϰ,τ.Lemma 4 (see [9]).
Letr=μ,ν∈0,∞×0,∞, and ξ1,ξ2>−1. Then,
(13)Iℓ+r;ψψξ1−1ϰ,0ψξ2−1τ,0=Γξ1Γμ+ξ1Γξ2Γν+ξ2ψμ+ξ1−1ϰ,0ψν+ξ2−1τ,0.Lemma 5 (see [9]).
Letr=μ,ν∈0,1×0,1, and ξ1,ξ2>−1. Then,
(14)Dℓ+r;ψψξ1−1ϰ,0ψξ2−1τ,0=Γξ1Γμ−ξ1Γξ2Γν−ξ2ψμ−ξ1−1ϰ,0ψν−ξ2−1τ,0.Lemma 6 (see [7]).
Let0<r<1, and h:c,d⟶ℝ is continuous. Then,
(15)CDc+r,ψIc+r,ψhϰ=hϰ,Ic+r,ψCDc+r,ψhϰ=hϰ−hc.Lemma 7.
The following problem(16)CD0+r;ψzϰ,τ=fϰ,τ,ϰ,τ∈0,c×0,d,zϰ,0=ϕ1ϰ,z0,τ=ϕ2τ,ϰ∈0,c,τ∈0,d,
withϕ10=ϕ20 which has a solution zϰ,τ∈C0,c×0,d,ℝ if and only if zϰ,τ satisfies
(17)zϰ,τ=ηϰ,τ+I0+r;ψfϰ,τ,ϰ,τ∈0,c×0,d,where ηϰ,τ=ϕ1ϰ+ϕ2τ−ϕ10.Proof.
The proof is primitive and similar to the proof of Lemma 3.2 given in [21], so it can be omitted.Here, we only refer to source [22] of the results of Leray-Schauder and Banach FPT.


## 3. Main Results

Let us begin by describing what we mean by a solution to problem (1).Definition 8.
A functionz is a solution of (1), if z∈Cc,d and D2zϰτϰ,τ exists and is integrable.Theorem 9.
Let the following assumptions hold:
(A1)F:J1×C⟶ℝ is continuous.
(A2) There existsLF>0 such that
(18)Fϰ,τ,z−Fϰ,τ,υ≤LFz−υC,ϰ,τ∈J1,z,υ∈C.
If(19)σ≔ψμc,0ψνd,0Γμ+1Γν+1LF<1,then there exists a unique solution for the ψ-Caputo problem (1) on −κ1,c×−κ2,d.Proof.
Consider the operatorK:Cc,d⟶Cc,d defined by Kzϰ,τ=zϰ,τ, i.e.,
(20)Kzϰ,τ=φϰ,τ,ϰ,τ∈J2,ηϰ,τ+1ΓμΓν∫0ϰ∫0τψùθψμ−1ϰ,θψτζψν−1τ,ζ×Fθ,ζ,zθ,ζdζdθ,ϰ,τ∈J1,where ηϰ,τ≔ϕ1ϰ+ϕ2τ−ϕ10.
Letz,ω∈Cc,d, and ϰ,τ∈−κ1,c×−κ2,d. Then,
(21)Kzϰ,τ−Kωϰ,τ≤1ΓμΓν∫0ϰ∫0τψϰθψμ−1ϰ,θψτζψν−1τ,ζ×Fθ,ζ,zθ,ζ−Fθ,ζ,ωθ,ζdζdθ≤LFΓμΓν∫0ϰ∫0τψϰθψμ−1ϰ,θψτζψν−1τ,ζ×zθ,ζ−ωθ,ζCdζdθ≤LFΓμΓνz−ωCc,d∫0ϰψùθψμ−1ϰ,θ∫0τψτζψν−1τ,ζdζdθ≤LFΓμΓνz−ωCc,dψνd,0ν∫0ϰψϰθψμ−1ϰ,θdθ≤LFψμc,0ψνd,0Γμ+1Γν+1z−ωCc,d.which implies
(22)Kz−KωCc,d≤σz−ωCc,d.
Sinceσ<1, the operator K is a contraction. This means that K has a unique fixed point by Banach’s FPT.Theorem 10.
Let (A1) and the following assumption hold:
(A3) There existp,q∈CJ1,ℝ such that
(23)Fϰ,τ,z≤pϰ,τ+qϰ,τzC,ϰ,τ∈J1,z∈C.
Ifρ≔q∞ψμc,0ψνd,0/Γμ+1Γν+1<1, then there exists at least one solution for the ψ-Caputo problem (1) on −κ1,c×−κ2,d.Proof.
Consider the operatorK:Cc,d⟶Cc,d defined by (20); then, we show thatKis completely continuous.
Step 1:K is continuous. Let zn be a sequence such that zn⟶z in Cc,d. Then,
(24)Kznϰ,τ−Kzϰ,τ≤1ΓμΓν∫0ϰ∫0τψϰθψμ−1ϰ,θψτζψν−1τ,ζ×Fθ,ζ,znθ,ζ−Fθ,ζ,zθ,ζdζdθ≤1ΓμΓν∫0c∫0dψϰθψμ−1ϰ,θψτζψν−1τ,ζ×supθ,ζ∈J1Fθ,ζ,znθ,ζ−Fθ,ζ,zθ,ζdζdθ≤ψμc,0ψνd,0Γμ+1Γν+1F.,.,zn.,.−F.,.,z.,.∞.
SinceF is continuous, Kzn−KzCc,d⟶0, as n⟶∞.
Step 2:KBξ is bounded in Cc,d, whereBξ=z∈Cc,d:zCc,d≤ξ, for any ξ>0.
Set(25)ξ>maxφC,λ1−ρ,where
(26)λ≔η∞+p∞ψμc,0ψνd,0Γμ+1Γν+1.
Forϰ,τ∈J2, we get
(27)Kzϰ,τ≤supϰ,τ∈J2φϰ,τ=φC.
Letϰ,τ∈J1, and z∈Bξ. Then,
(28)Kzϰ,τ≤ηϰ,τ+1ΓμΓν∫0ϰ∫0τψϰθψμ−1ϰ,θψτζψν−1τ,ζ×Fθ,ζ,zθ,ζdζdθ≤ηϰ,τ+1ΓμΓν∫0ϰ∫0τψϰθψμ−1ϰ,θψτζψν−1τ,ζpθ,ζdζdθ+1ΓμΓν∫0ϰ∫0τψϰθψμ−1ϰ,θψτζψν−1τ,ζqθ,ζzθ,ζCdζdθ≤η∞+p∞ΓμΓν∫0ϰ∫0τψϰθψμ−1ϰ,θψτζψν−1τ,ζdζdθ+q∞zCc,dΓμΓν∫0ϰ∫0τψϰθψμ−1ϰ,θψτζψν−1τ,ζdζdθ≤η∞+p∞+q∞ξψμc,0ψνd,0Γμ+1Γν+1=λ+ρξ.
Due to (25), (27), and (28), KzCc,d≤ξ, or Kz∈Bξ, which implies that KBξ is bounded in Cc,d.
Step 3:KBξ is equicontinuous in Cc,d. Let z∈Bξ, and ϰ1,τ1,ϰ2,τ2∈−κ1,c×−κ2,d with ϰ1<ϰ2,τ1<τ2. If ϰ1,τ1,ϰ2,τ2∈J1 and z∈Bξ. Then,
(29)Kzϰ2,τ2−Kzϰ1,τ1≤ηϰ2,τ2−ηϰ1,τ1+1ΓμΓν∫0ϰ1∫0τ1ψϰθψτζψμ−1ϰ2,θψν−1τ2,ζ−ψμ−1ϰ1,θψν−1τ1,ζ×Fθ,ζ,zθ,ζdζdθ+1ΓμΓν∫ϰ1ϰ2∫τ1τ2ψϰθψτζψμ−1ϰ2,θψν−1τ2,ζFθ,ζ,zθ,ζdζdθ+1ΓμΓν∫0ϰ1∫τ1τ2ψϰθψτζψμ−1ϰ2,θψν−1τ2,ζFθ,ζ,zθ,ζdζdθ+1ΓμΓν∫ϰ1ϰ2∫0τ1ψϰθψτζψμ−1ϰ2,θψν−1τ2,ζFθ,ζ,zθ,ζdζdθ≤ηϰ2,τ2−ηϰ1,τ1∞+p∞+q∞ξΓμΓν∫0ϰ1∫0τ1ψùθψτζψμ−1ϰ2,θψν−1τ2,ζ−ψμ−1ϰ1,θψν−1τ1,ζdζdθ+p∞+q∞ξΓμΓν∫ϰ1ϰ2∫τ1τ2ψϰθψτζψμ−1ϰ2,θψν−1τ2,ζdζdθ+p∞+q∞ξΓμΓν∫0ϰ1∫τ1τ2ψϰθψτζψμ−1ϰ2,θψν−1τ2,ζdζdθ+p∞+q∞ξΓμΓν∫ϰ1ϰ2∫0τ1ψϰθψτζψμ−1ϰ2,θψν−1τ2,ζdζdθ≤ηϰ2,τ2−ηϰ1,τ1∞+p∞+q∞ξΓμ+1Γν+12ψντ2,0ψμϰ2,ϰ1+2ψμϰ2,0ψντ2,τ1+ψντ1,0ψμϰ1,0−ψντ2,0ψμϰ2,0−2ψντ2,τ1ψμϰ2,ϰ1.
If−κ1≤ϰ1≤ϰ2≤0, and −κ2≤τ1≤τ2≤0, then
(30)Kzϰ2,τ2−Kzϰ1,τ1≤φϰ2,τ2−φϰ1,τ1.
If−κ1≤ϰ1<0<ϰ2≤c, and −κ2≤τ1<0<τ2≤d, then
(31)Kzϰ2,τ2−Kzϰ1,τ1≤Kzϰ2,τ2−Kz0,0+Kz0,0−Kzϰ1,τ1≤ηϰ2,τ2−η0,0∞+p∞+q∞ξΓμ+1Γν+12ψντ2,0ψμϰ2,0+2ψμϰ2,0ψντ2,0+ψν0,0ψμ0,0−ψντ2,0ψμϰ2,0−2ψντ2,0ψμϰ2,0+φ0,0−φϰ1,τ1.
In all previous cases, asϰ1⟶ϰ2, τ1⟶τ2, and the uniform continuity of η on J1 and φ on J2 implies that for any ε>0, there exists δ>0, independent of ϰ1,ϰ2, τ1,τ2 and z, such that Kzϰ2,τ2−Kzϰ1,τ1≤ε whenever ψϰ2−ψϰ1≤δ/2 and ψτ2−ψτ1≤δ/2. Therefore, KBξ is equicontinuous. It follows from the Arzela–Ascoli theorem that K is compact.
Step 4:KBξ a priori bounds. ∃ an open set Ω⊆Cc,d with z≠ℵKz, for ℵ∈0,1, and z∈∂Ω. Let ϰ,τ∈−κ1,c×−κ2,d and z∈Cc,d with z≠ℵKz, for some ℵ∈0,1. Then,
(32)zϰ,τ≤ℵηϰ,τ+ℵΓμΓν∫0ϰ∫0τψϰθψμ−1ϰ,θψτζψν−1τ,ζFθ,ζ,zθ,ζdζdθ≤ηϰ,τ+1ΓμΓν∫0ϰ∫0τψϰθψμ−1ϰ,θψτζψν−1τ,ζpθ,ζ+qθ,ζzθ,ζCdζdθ≤η∞+p∞ψμc,0ψνd,0Γμ+1Γν+1+q∞ΓμΓν∫0ϰ∫0τψùθψμ−1ϰ,θψτζψν−1τ,ζzCc,ddζdθ.
Ifϰ,τ∈J1, then (32) becomes
(33)zϰ,τ∞≤η∞+p∞ψμc,0ψνd,0Γμ+1Γν+1+q∞ΓμΓν∫0ϰ∫0τψϰθψμ−1ϰ,θψτζψν−1τ,ζzθ,ζ∞dζdθ≤η∞+p∞+q∞zϰ,τ∞Γμ+1Γν+1ψμc,0ψνd,0,which implies
(34)zϰ,τ∞≤λ1−ρ≔M.
Forϰ,τ∈J2,zϰ,τ∞=φC.
Consequently,(35)z∞=maxM,φC≔ξ∗.
Set(36)Ω=z∈Cc,d:z∞<ξ∗+1.Through our choiceΩ, nothing z∈∂Ω such that z=ℵKz,0<ℵ<1.As conclusion, the Leray-Schauder FPT shows thatK has a fixed point z∈Ω⊂Cc,d such that z=Kz which is a solution to problem.We now provide two results on the nonlocal problem (2), and their proofs are quite similar to the preceding results. In addition, the results in Theorems 9 and 10 can be presented by
(37)Kzϰ,τ=φϰ,τ,ϰ,τ∈J2,ηϰ,τ+h1z+h2z+1ΓμΓν×∫0ϰ∫0τψϰθψμ−1ϰ,θψτζψν−1τ,ζFθ,ζ,zθ,ζdζdθϰ,τ∈J1.Theorem 11.
Let (A1) and (A2) be satisfied. If there existLh1,Lh2>0 such that
(38)h1z−h1υ≤Lh1z−υ∞,forz,υ∈CJ1,ℝ,h2z−h2υ≤Lh2z−υ∞,forz,υ∈CJ1,ℝ,with Λ≔Lh1+Lh2+σ<1, where σ is defined by (19); then, there exists a unique solution for the ψ-Caputo problem (2) on −κ1,c×−κ2,d.Theorem 12.
Let (A1) and (A3) be satisfied. If there existdh1,dh2>0 such that
(39)h1z≤dh11+z∞,forz∈CJ1,ℝn,h2z≤dh21+z∞,forz∈CJ1,ℝn,with ρ<1, where ρ is defined by (A3); then, there exists at least one solution for the ψ-Caputo problem (2) on −κ1,c×−κ2,d.Now, we provide the UH and GUH stability of theψ-problem (2).Definition 13. (see (2)).
Problem ((2)) is UH stable if there exists a χφ>0 such that ∀ε>0 and each solution ωϰ,τ∈Cc,d of the inequality
(40)CDℓ+r;ψωϰ,τ−Fϰ,τ,ωϰ,τ≤ε,ϰ,τ∈J1,ωϰ,τ−φϰ,τ≤ε,ϰ,τ∈J2,there exists a solution zϰ,τ∈Cc,d of (2) satisfies
(41)ωϰ,τ−zϰ,τCc,d≤χφε.Remark 14.
ωϰ,τ∈Cc,d satisfies (40) iff there exists ςϰ,τ∈Cc,d with
(i)
ςϰ,τ≤ε,ϰ∈J1(ii)
for allϰ∈J1(42)CDℓ+r;ψωϰ,τ=Fϰ,τ,ωϰ,τ+ςϰ,τ.Lemma 15.
Letr=μ,ν∈0,1×0,1, and ωϰ,τ∈Cc,d is a solution of (40). Then, ωϰ,τ satisfies
(43)ωϰ,τ−ω0ϰ,τ−1ΓμΓν∫0ϰψϰθψμ−1ϰ,θ∫0τψτζψν−1τ,ζFθ,ζ,ωθ,ζdζdθ≤εψνd,0Γν+1ψμc,0Γμ+1,for ϰ,τ∈J1, where ω0ϰ,τ=ηϰ,τ+h1ω+h2ω. Moreover, ωϰ,τ−φϰ,τ=0, for ϰ,τ∈J2.Proof.
Letωϰ,τ is a solution of (40). It follows from (ii) of Remark 14that
(44)CDℓ+r;ψωϰ,τ=Fϰ,τ,ωϰ,τ+ςϰ,τ,ϰ,τ∈J1,ωϰ,τ=φϰ,τ,ϰ,τ∈J2,ωϰ,0+h1ω=ϕ1ϰ,ω0,τ+h2ω=ϕ2τ,ϰ,τ∈J1.
Then, the solution of problem (44) is
(45)ωϰ,τ=φϰ,τ,ϰ,τ∈J2,ηϰ,τ+h1ω+h2ω+1ΓμΓν×∫0ϰ∫0τψϰθψμ−1ϰ,θψτζψν−1τ,ζFθ,ζ,ωθ,ζ+ςθ,ζdζdθ,ϰ,τ∈J1.
Once more by (i) of Remark14, we get
(46)ωϰ,τ−ω0ϰ,τ−1ΓμΓν∫0ϰψϰθψμ−1ϰ,θ∫0τψτζψν−1τ,ζFθ,ζ,ωθ,ζdζdθ≤1ΓμΓν∫0ϰ∫0τψϰθψμ−1ϰ,θψτζψν−1τ,ζςθ,ζdζdθ≤εΓμΓν∫0ϰψϰθψμ−1ϰ,θ∫0τψτζψν−1τ,ζdζdθ=εψντ,0Γν+11Γμ∫0ϰψϰθψμ−1ϰ,θdθ=εψντ,0Γν+1ψμϰ,0Γμ+1≤εψνd,0Γν+1ψμc,0Γμ+1,for ϰ,τ∈J2. For ϰ,τ∈J2, we obtain ωϰ,τ−φϰ,τ=φϰ,τ−φϰ,τ=0.Theorem 16.
Under assumptions of Theorem9, the solution of the problem (2) is HU and GHU stable on −κ1,c×−κ2,d.Proof.
Letωϰ,τ∈C be a solution of (40), and zϰ,τ∈Cc,d is a unique solution of the following problem:
(47)CDℓ+r;ψzϰ,τ=Fϰ,τ,zϰ,τ,ϰ,τ∈J1,zϰ,τ=φϰ,τ,ϰ,τ∈J2,zϰ,0+h1z=ωϰ,0+h1ω,z0,τ+h2z=ω0,τ+h2ω,ϰ,τ∈J1.
The previous problem has a solution(48)zϰ,τ=φϰ,τ,ϰ,τ∈J2,z0ϰ,τ+1ΓμΓν∫0ϰψϰθψμ−1ϰ,θ∫0τψτζψν−1τ,ζFθ,ζ,zθ,ζdζdθϰ,τ∈J1,where z0ϰ,τ≔ηϰ,τ+h1z+h2z.
Sincezϰ,0+h1z=ωϰ,0+h1ω and z0,τ+h2z=ω0,τ+h2ω, we have z0ϰ,τ=ω0ϰ,τ. Indeed,
(49)z0ϰ,τ=ηϰ,τ+h1z+h2z=ηϰ,τ−zϰ,0+ωϰ,0+h1ω−z0,τ+ω0,τ+h2ω=ηϰ,τ−ϕ1ϰ+ϕ1ϰ+h1ω−ϕ2τ+ϕ2τ+h2ω=ηϰ,τ+h1ω+h2ω=ω0ϰ,τ.
Hence, (48) becomes
(50)zϰ,τ=φϰ,τ,ϰ,τ∈J2,ω0ϰ,τ+1ΓμΓν∫0ϰψϰθψμ−1ϰ,θ∫0τψτζψν−1τ,ζFθ,ζ,zθ,ζdζdθϰ,τ∈J1.
Note that,ωϰ,τ−zϰ,τ=0, for all ϰ,τ∈J2.
Using Lemma15 and (A2), for ϰ,τ∈J1, we have
(51)ωϰ,τ−zϰ,τ=ωϰ,τ−ω0ϰ,τ−1ΓμΓν∫0ϰψϰθψμ−1ϰ,θ∫0τψτζψν−1τ,ζFθ,ζ,zθ,ζdζdθ≤ωϰ,τ−ω0ϰ,τ−1ΓμΓν∫0ϰψϰθψμ−1ϰ,θ∫0τψτζψν−1τ,ζFθ,ζ,ωθ,ζdζdθ+1ΓμΓν∫0ϰψϰθψμ−1ϰ,θ∫0τψτζψν−1τ,ζFθ,ζ,ωθ,ζ−Fθ,ζ,zθ,ζdζdθ≤εψνd,0Γν+1ψμc,0Γμ+1+LFωθ,ζ−zθ,ζCψνd,0Γν+1ψμc,0Γμ+1≤εσLF+σω−zCc,d,
which implies(52)ω−zCc,d≤σLF1−σε.
Takingχφ≔σ/LF1−σ such that σ<1, then (51) becomes
(53)ω−zCc,d≤χφε.
Hence, problem (2) is UH stable. Moreover, if there exists a nondecreasing function Ψ:0,∞⟶0,∞ such that Ψε=ε, then we have with Ψ0=0,(54)ω−zCc,d≤Ψε,which proves that problem (2) is also GUH stable.


## 4. Examples

In this portion, we provide two examples of partial hyperbolic FDEs having fractional order and satisfying the obtained results. All computational work will be performed through MATLAB.Example 1.
Consider aψ-Caputo fractional partial hyperbolic FDE
(55)CD0+r;ψzϰ,τ=12eϰ+τ+21+zϰ−1,τ−2,ϰ,τ∈0,1×0,1,zϰ,τ=ϰ+τ2,ϰ,τ∈−1,1×−2,1\0,1×0,1,zϰ,0=ϰ,z0,τ=τ2,ϰ,τ∈0,1,
wherer=μ,ν, μ=1/2,ν=1/3,φϰ,τ=ϰ+τ2,c=d=1,κ1=1,κ2=2,ϕ1ϰ=ϰ,ϕ2τ=τ2. Consider Fϰ,τ,z¯=1/3eϰ+τ+21+z¯ϰ−1,τ−2, for ϰ,τ,z¯∈0,1×0,1×C−1,0×−2,0,ℝ. Let z,υ∈C−1,0×−2,0,ℝ, and ϰ,τ∈0,1×0,1. Then,
(56)Fϰ,τ,zϰ,τ−Fϰ,τ,υϰ,τ≤13e2zϰ−1,τ−2−υϰ−1,τ−2≤13e2z−υC.So, assumptions (A1) and (A2) are satisfied withLF=1/3e2. Moreover, the condition σ=2/35/6e2πΓ1/3<1 with ψϰ=ϰ/3,ψτ=τeτ−1/3 and c=d=1. Hence, Theorem 9 shows that problem (55) has a unique solution defined on −1,1×−2,1.Example 2.
Consider aψ-Caputo fractional partial hyperbolic FDE
(57)CD0+r;ψzϰ,τ=e−ϰ−τ4+eϰ+τ1+zϰ−1,τ−21+zϰ−1,τ−2,ϰ,τ∈0,13×0,13,zϰ,τ=ϰ2+τ,ϰ,τ∈−1,13×−2,13\0,13×0,13,zϰ,0=ϰ2,z0,τ=τ,ϰ,τ∈0,13,
wherer=μ,ν, μ=1/2,ν=1/3,φϰ,τ=ϰ2+τ,c=d=1/3,κ1=1,κ2=2,ϕ1ϰ=ϰ2,ϕ2τ=τ. Consider Fϰ,τ,z¯=e−ϰ−τ/4+eϰ+τ1+z¯/1+z¯, for ϰ,τ,z¯∈0,1/3×0,1/3×C−1,0×−2,0,ℝ. Let z∈C−1,0×−2,0,ℝ, and ϰ,τ∈0,1/3×0,1/3. Then,
(58)Fϰ,τ,zϰ,τ≤e−ϰ−τ4+eϰ+τ1+zϰ,τ1+zϰ,τ≤e−ϰ−τ4+eϰ+τ+e−ϰ−τ4+eϰ+τzϰ,τ≤e−ϰ−τ4+eϰ+τ+e−ϰ−τ4+eϰ+τzC.Thus, (A3) holds withpϰ,τ=qϰ,τ=e−ϰ−τ/4+eϰ+τ, where q∞=1/5. To verify that ρ<1, we select ψϰ=eϰ/3 and ψτ=τ+1, then find that
(59)ψμc,0=ψc−ψ0μ=ec/31/2=e1/9,ψνd,0=ψd−ψ0ν=d+11/3=2231/3,and ρ≈0.314<1. So, all assumptions of Theorem 10 are satisfied. Hence, Theorem 10 shows that problem (57) has a solution defined on −1,1/3×−2,1/3.Remark 3.
Our current outcomes on problems (1) and (2) can be interpreted as extensions of preceding results of Abbas et al. [19], for ψϰ=ϰ.Remark 4.
As special cases, it is possible to obtain other results for similar problems involving various FDs such as Caputo-Katugampola FD (forϰ=ϰρ,ρ>0), Caputo-Hadamard FD (for ψϰ=lnϰ), and other FDs, for different choices of ψ·.


## 5. Conclusion

Somewhat recently, several fractional definitions have been proposed to describe the behaviors of some complex world problems arising in many scientific fields. In this regard, Sousa and de Oliveira [9] introduced the concept of the multivariate partial fractional derivative with respect to another function. As an additional contribution to this topic, existence and uniqueness results have been obtained for two types of Cauchy and nonlocal fractional partial hyperbolic FDEs (1) and (2) involving ψ-Caputo FD with two variables. We have presented several results based on Banach’s and Leray-Schauder’s fixed point theorem. In light of our present results, special cases of similar problems containing several partial fractional operators have been presented according to different choices of the ψ function. Moreover, we have provided the stability results in UH and GUH sense. Lastly, two suitable examples that validate the obtained results were given.It is interesting to approach current problems with infinite delay, and this is what we are thinking of in future research. One can also study the same present problem in terms of the generalized fractional derivative that was recently proposed in [23, 24].



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