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Research article Special Issues

A reliable analytic technique and physical interpretation for the two-dimensional nonlinear Schrödinger equations

  • Received: 26 April 2024 Revised: 30 June 2024 Accepted: 08 July 2024 Published: 19 August 2024
  • MSC : 35C08, 35Q40, 35Q55

  • Nonlinear Schrödinger equations are a key paradigm in nonlinear research, attracting both mathematical and physical attention. This work was primarily concerned with the usage of a reliable analytic technique in order to solve two models of (2+1)-dimensional nonlinear Schrödinger equations. By applying a comprehensible wave transformation, every nonlinear model was simplified to an ordinary differential equation. A number of critical solutions were observed that correlated to various parameters. The provided approach has various advantages, including reducing difficult computations and succinctly presenting key results. Some 2D and 3D graphical representations regarding presented solitons were considered for the appropriate values of the parameters. We also showed the effect of the physical parameters on the dynamical behavior of the presented solutions. Finally, the proposed approach may be expanded to tackle increasingly complicated problems in applied science.

    Citation: Mahmoud A. E. Abdelrahman, H. S. Alayachi. A reliable analytic technique and physical interpretation for the two-dimensional nonlinear Schrödinger equations[J]. AIMS Mathematics, 2024, 9(9): 24359-24371. doi: 10.3934/math.20241185

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  • Nonlinear Schrödinger equations are a key paradigm in nonlinear research, attracting both mathematical and physical attention. This work was primarily concerned with the usage of a reliable analytic technique in order to solve two models of (2+1)-dimensional nonlinear Schrödinger equations. By applying a comprehensible wave transformation, every nonlinear model was simplified to an ordinary differential equation. A number of critical solutions were observed that correlated to various parameters. The provided approach has various advantages, including reducing difficult computations and succinctly presenting key results. Some 2D and 3D graphical representations regarding presented solitons were considered for the appropriate values of the parameters. We also showed the effect of the physical parameters on the dynamical behavior of the presented solutions. Finally, the proposed approach may be expanded to tackle increasingly complicated problems in applied science.



    Fractional calculus has emerged as a powerful tool to study complex phenomena in numerous scientific and engineering disciplines such as biology, physics, chemistry, economics, signal and image processing, control theory and so on. Fractional differential equations describe many real world process related to memory and hereditary properties of various materials more accurately as compared to classical order differential equations. For examples and applications see the monographs as [1,2,3,4,5,7,6,8].

    In the literature, many authors focused on Riemann-Liouville and Caputo type derivatives in investigating fractional differential equations. A generalization of derivatives of both Riemann-Liouville and Caputo was given by R. Hilfer in [9], the known as the Hilfer fractional derivative of order α and a type β[0,1], which interpolates between the Riemann-Liouville and Caputo derivative, since it is reduced to the Riemann-Liouville and Caputo fractional derivatives when β=0 and β=1, respectively. Some properties and applications of the Hilfer fractional derivative are given in [10,11] and references cited therein.

    Initial value problems involving Hilfer fractional derivatives were studied by several authors, see for example [12,13,14,15] and references therein. In [16] the authors initiated the study of nonlocal boundary value problems for Hilfer fractional derivative, by studying boundary value problem of Hilfer-type fractional differential equations with nonlocal integral boundary conditions

    HDα,βx(t)=f(t,x(t)),t[a,b],1<α<2,0β1, (1.1)
    x(a)=0,x(b)=mi=1δiIφix(ξi),φi>0,δiR,ξi[a,b], (1.2)

    where HDα,β is the Hilfer fractional derivative of order α, 1<α<2 and parameter β, 0β1, Iφi is the Riemann-Liouville fractional integral of order φi>0, ξi[a,b], a0 and δiR. Several existence and uniqueness results were proved by using a variety of fixed point theorems.

    In [17] the existence and uniqueness of solutions were studied, for a new class of system of Hilfer-Hadamard sequential fractional differential equations

    {(HDα1,β11++k1HDα11,β11+)u(t)=f(t,u(t),v(t)),  1<α12,  t[1,e],(HDα2,β21++k2HDα21,β21+)v(t)=g(t,u(t),v(t)),  1<α22,  t[1,e], (1.3)

    with two point boundary conditions

    {u(1)=0,  u(e)=A1,v(1)=0,  v(e)=A2, (1.4)

    where HDαi,βi is the Hilfer-Hadamard fractional derivative of order αi(1,2] and type βi[0,1] for i{1,2}, k1,k2,A1,A2R+ and f, g:[1,e]×R2R are given continuous functions.

    The fractional derivative with another function, in the Hilfer sense, called ψ-Hilfer fractional derivative, has been introduced in [18]. For some recent results on existence and uniqueness of initial value problems and results on Ulam-Hyers-Rassias stability see [19,20,21,22,23,24,25,26,27,28,29] and references therein. Recently, in [30] the authors extended the results in [16] to ψ-Hilfer nonlocal implicit fractional boundary value problems.

    Recently in [31] the existence and uniqueness of solutions were studied, for a new class of boundary value problems of sequential ψ-Hilfer-type fractional differential equations with multi-point boundary conditions of the form

    (HDα,β;ψ+kHDα1,β;ψ)x(t)=f(t,x(t)),t[a,b], (1.5)
    x(a)=0,x(b)=mi=1λix(θi), (1.6)

    where HDα,β;ψ is the ψ-Hilfer fractional derivative of order α, 1<α<2 and parameter β, 0β1, f:[a,b]×RR is a continuous function, a<b, k,λiR,i=1,2,,m and a<θ1<θ2<<θm<b.

    In this paper, motivated by the research going on in this direction, we study a new class of boundary value problems of ψ-Hilfer fractional integro-differential equations with mixed nonlocal boundary conditions of the form

    {HDα,ρ;ψ0+x(t)=f(t,x(t),Iϕ;ψ0+x(t)),t(0,T],x(0)=0,mi=1δix(ηi)+nj=1ωjIβj;ψ0+x(θj)+rk=1λkHDμk,ρ;ψ0+x(ξk)=κ, (1.7)

    where HDu,ρ;ψ0+ is ψ-Hilfer fractional derivatives of order u={α,μk} with 1<μk<α2, 0ρ1, Iv;ψ0+ is ψ-Riemann-Liouville fractional integral of order v={ϕ,βj}, ϕ,βj>0 for j=1,2,,n, κ,δi,ωj,λkR are given constants, the points ηi,θj,ξkJ, i=1,2,,m, j=1,2,,n, k=1,2,,r and f:J×R2R is a given continuous function, and J:=[0,T], T>0. It is imperative to note that the problems addressed in this paper provide more insight into the study of ψ-Hilfer fractional differential equations involving mixed nonlocal boundary conditions. Our results are not only interesting from theoretical point of view, but also helpful in studying the applied problems containing the systems like the ones considered in this paper. Our nonlocal boundary conditions are also useful, since they are the most general mixed type. We emphasize that the mixed nonlocal boundary conditions include multi-point, fractional derivative multi-order and fractional integral multi-order boundary conditions.

    This paper is organized as follows: In Section 2, we present some necessary definitions and preliminaries results that will be used to prove our main results. The existence and uniqueness of the solutions for the problem (1.7) are established in Section 3. Our methodology for obtaining the desired results is standard, but its application in the framework of the present problem is new. In Section 4, we discuss the Ulam's stability of the solutions of the problem (1.7) in the frame of Ulam-Hyers (UH) stability, generalized Ulam-Hyers (UH) stability, Ulam-Hyers-Rassias (UHR) stability and generalized Ulam-Hyers-Rassias (UHR) stability is investigated. Finally, examples are given in Section 5 to illustrate the theoretical results.

    In this section, we introduce some notation, spaces, definitions and fundamental lemmas which are useful throughout this paper.

    Let C=C(J,R) denote the Banach space of all continuous functions from J into R with the norm defined by

    f=suptJ{|f(t)|}.

    On the order hand, we have n-times absolutely continuous functions given by

    ACn(J,R)={f:JR;f(n1)AC(J,R)}.

    Definition 2.1. [2] Let (a,b), (a<b), be a finite or infinite interval of the half-axis R+ and αR+. Also let ψ(x) be an increasing and positive monotone function on (a,b], having a continuous derivative ψ(x) on (a,b). The ψ-Riemann-Liouville fractional integral of a function f with respect to another function ψ on [a,b] is defined by

    Iα;ψa+f(t)=1Γ(α)taψ(s)(ψ(t)ψ(s))α1f(s)ds,t>a>0, (2.1)

    where Γ() is represent the Gamma function.

    Definition 2.2. [2] Let ψ(t)0 and α>0, nN. The Riemann–Liouville derivatives of a function f with respect to another function ψ of order α correspondent to the Riemann–Liouville, is defined by

    Dα;ψa+f(t)=(1ψ(t)ddt)nInα;ψa+f(t)=1Γ(nα)(1ψ(t)ddt)ntaψ(s)(ψ(t)ψ(s))nα1f(s)ds, (2.2)

    where n=[α]+1, [α] is represent the integer part of the real number α.

    Definition 2.3. [18] Let n1<α<n with nN, [a,b] is the interval such that a<b and f,ψCn([a,b],R) two functions such that ψ is increasing and ψ(t)0, for all t[a,b]. The ψ-Hilfer fractional derivative of a function f of order α and type 0ρ1, is defined by

    HDα,ρ;ψa+f(t)=Iρ(nα);ψa+(1ψ(t)ddt)nI(1ρ)(nα);ψa+f(t)=Iγα;ψa+Dγ;ψa+f(t), (2.3)

    where n=[α]+1, [α] represents the integer part of the real number α with γ=α+ρ(nα).

    Lemma 2.4. [2] Let α,β>0. Then we have the following semigroup property given by,

    Iα;ψa+Iβ;ψa+f(t)=Iα+β;ψa+f(t),t>a. (2.4)

    Next, we present the ψ-fractional integral and derivatives of a power function.

    Proposition 2.5. [2,18] Let α0, υ>0 and t>a. Then, ψ-fractional integral and derivative of a power function are given by

    (i) Iα;ψa+(ψ(s)ψ(a))υ1(t)=Γ(υ)Γ(υ+α)(ψ(t)ψ(a))υ+α1.

    (ii) Dα,ρ;ψa+(ψ(s)ψ(a))υ1(t)=Γ(υ)Γ(υα)(ψ(t)ψ(a))υα1.

    (iii) HDα,ρ;ψa+(ψ(s)ψ(a))υ1(t)=Γ(υ)Γ(υα)(ψ(t)ψ(a))υα1,υ>γ=α+ρ(2α).

    Lemma 2.6. Let m1<α<m, n1<β<n, n,mN, nm, 0ρ1 and αβ+ρ(nβ). If hCn(J,R), then

    HDβ,ρ;ψa+Iα;ψa+h(t)=Iαβ;ψa+h(t). (2.5)

    Proof. Let λ=β+ρ(nβ) with n1<λ<n, we get

    HDβ,ρ;ψa+(Iα;ψa+h(t))=Iλβ;ψa+Dλ;ψa+(Iα;ψa+h(t))=Iλβ;ψa+(1ψ(t)ddt)nInλ;ψa+(Iα;ψa+h(t))=Iλβ;ψa+(1ψ(t)ddt)nInλ+α;ψa+h(t).

    By using Definition 2.1, we obtain

    (1ψ(t)ddt)Inλ+α;ψa+h(t)=1ψ(t)ddt(1Γ(nλ+α)taψ(τ)(ψ(t)ψ(τ))n+αλ1h(τ)dτ)=1Γ(nλ+α)1ψ(t)(ta(n+αλ1)ψ(τ)ψ(t)(ψ(t)ψ(τ))n+αλ2h(τ)dτ)=1Γ(nλ+α1)taψ(τ)(ψ(t)ψ(τ))n+αλ2h(τ)dτ=Inλ+α1;ψa+h(t),

    and

    (1ψ(t)ddt)2Inλ+α;ψa+h(t)=1ψ(t)ddt(1Γ(nλ+α1)taψ(τ)(ψ(t)ψ(τ))n+αλ2h(τ)dτ)=1Γ(nλ+α1)1ψ(t)(ta(n+αλ2)ψ(τ)ψ(t)(ψ(t)ψ(τ))n+αλ3h(τ)dτ)=1Γ(nλ+α2)taψ(τ)(ψ(t)ψ(τ))n+αλ3h(τ)dτ=Inλ+α2;ψa+h(t).

    Repeat the above process, we have

    (1ψ(t)ddt)nInλ+α;ψa+h(t)=1ψ(t)ddt(1Γ(αλ)taψ(τ)(ψ(t)ψ(τ))αλ1h(τ)dτ)=1Γ(αλ+1)1ψ(t)(ta(αλ)ψ(τ)ψ(t)(ψ(t)ψ(τ))αλ1h(τ)dτ)=1Γ(λ+α)taψ(τ)(ψ(t)ψ(τ))αλ1h(τ)dτ=Iαλ;ψa+h(t),

    which implies that

    HDβ,ρ;ψa+(Iα;ψa+h(t))=Iλβ;ψa+Iαλ;ψa+h(t)=Iαβ;ψa+h(t).

    This completes the proof.

    Lemma 2.7. [18] If fCn(J,R), n1<α<n, 0ρ1 and γ=α+ρ(nα) then

    Iα;ψa+HDα,ρ;ψa+f(t)=f(t)nk=1(ψ(t)ψ(a))γkΓ(γk+1)f[nk]ψI(1ρ)(nα);ψa+f(a), (2.6)

    for all tJ, where f[n]ψf(t):=(1ψ(t)ddt)nf(t).

    Fixed point theorems play a major role in establishing the existence theory for the problem (1.7). We collect here some well-known fixed point theorems used in this paper.

    Lemma 2.8. (Banach contraction principle [32]). Let D be a non-empty closed subset of a Banach space E. Then any contraction mapping T from D into itself has a unique fixed point.

    Lemma 2.9. (Krasnosel'skii's fixed point theorem [33]). Let M be a closed, bounded, convex, and nonempty subset of a Banach space. Let A,B be the operators such that (i) Ax+ByM whenever x, yM; (ii) A is compact and continuous; (iii) B is contraction mapping. Then there exists zM such that z=Az+bz.

    Lemma 2.10. (Leray-Schauder nonlinear alternative [32]). Let E be a Banach space, C a closed, convex subset of E,U an open subset of C and 0U. Suppose that D:¯UC is a continuous, compact (that is, D(¯U) is a relatively compact subset of C) map. Then either

    (i) D has a fixed point in ¯U, or

    (ii) there is a xU (the boundary of U in C) and ν(0,1) with x=νD(x).

    In order to transform the problem (1.7) into a fixed point problem, we must convert it into an equivalent Voltera integral equation. We provide the following auxiliary lemma, which is important in our main results and concern a linear variant of the boundary value problem (1.7).

    Lemma 2.11. Let 1<μk<α2, 0ρ1, γ=α+ρ(2α), k=1,2,,r and Ω0. Suppose that hC. Then xC2 is a solution of the problem

    {HDα,ρ;ψ0+x(t)=h(t),t(0,T],x(0)=0,mi=1δix(ηi)+nj=1ωjIβj;ψ0+x(θj)+rk=1λkHDμk,ρ;ψ0+x(ξk)=κ, (2.7)

    if and only if x satisfies the integral equation

    x(t)=Iα;ψ0+h(t)+(ψ(t)ψ(0))γ1ΩΓ(γ)[κ(mi=1δiIα;ψ0+h(ηi)+nj=1ωjIα+βj;ψ0+h(s)(θj)+rk=1λkIαμk;ψ0+h(s)(ξk))], (2.8)

    where

    Ω=mi=1δi(ψ(ηi)ψ(0))γ1Γ(γ)+nj=1ωj(ψ(θj)ψ(0))γ+βj1Γ(γ+βj)+rk=1λk(ψ(ξk)ψ(0))γμk1Γ(γμk). (2.9)

    Proof. Let xC be a solution of the problem (1.7). By using Lemma 2.7, we have

    x(t)=Iα;ψ0+h(t)+(ψ(t)ψ(0))γ1Γ(γ)c1+(ψ(t)ψ(0))γ2Γ(γ1)c2, (2.10)

    where c1,c2R are arbitrary constants.

    For t=0, we get c2=0, and thus

    x(t)=Iα;ψ0+h(t)+(ψ(t)ψ(0))γ1Γ(γ)c1. (2.11)

    Taking the operators HDμk,ρ;ψ0+ and Iβj;ψ0+ into (2.10), we obtain

    HDμk,ρ;ψ0+x(t)=Iαμk;ψ0+h(t)+(ψ(t)ψ(0))γμk1Γ(γμk)c1,Iβj;ψ0+x(t)=Iα+βj;ψ0+h(t)+(ψ(t)ψ(0))γ+βj1Γ(γ+βj)c1.

    Applying the second boundary condition in (1.7), we have

    c1[mi=1δi(ψ(ηi)ψ(0))γ1Γ(γ)+nj=1ωj(ψ(θj)ψ(0))γ+βj1Γ(γ+βj)+rk=1λk(ψ(ξk)ψ(0))γμk1Γ(γμk)]+mi=1δiIα;ψ0+h(ηi)+nj=1ωjIα+βj;ψ0+h(θj)+rk=1λkIαμk;ψ0+h(ξk)=κ,

    from which we get

    c1=1Ω[κ(mi=1δiIα;ψ0+h(ηi)+nj=1ωjIα+βj;ψ0+h(θj)+rk=1λkIαμk;ψ0+h(ξk))],

    where Ω is defined by (2.9). Substituting the value of c1 in (2.11), we obtain (2.8).

    Conversely, it is easily to shown, by a direct calculation, that the solution x given by (2.8) satisfies the problem (2.7). The Lemma 2.11 is proved.

    In this section, we present existence and uniqueness results to the considered problem (1.7).

    For the sake of convenience, we use the following notations:

    A(χ,ε)=(ψ(χ)ψ(0))εΓ(ε+1), (3.1)
    Λ0=1+A(T,ϕ), (3.2)
    Λ1=A(T,α)+A(T,γ1)|Ω|(mi=1|δi|A(ηi,α)+nj=1|ωj|A(θj,α+βj)+rk=1|λk|A(ξk,αμk)). (3.3)

    In view of Lemma 2.11, an operator Q:CC is defined by

    (Qx)(t)=Iα;ψ0+Fx(s)(t)+A(t,γ1)Ω[κ(mi=1δiIα;ψ0+Fx(s)(ηi)+nj=1ωjIα+βj;ψ0+Fx(s)(θj)+rk=1λkIαμk;ψ0+Fx(s)(ξk))], (3.4)

    where

    Fx(t)=f(t,x(t),Iϕ;ψ0+x(t)),tJ.

    Throughout this paper, the expression Iq,ρ0+Fx(s)(c) means that

    Iu;ψ0+Fx(s)(c)=1Γ(u)c0ψ(s)(ψ(c)ψ(s))u1Fx(s)ds,

    where u={ϕ,βj} and c={t,σ,θj}, j=1,2,,n.

    It should be noticed that the problem (1.7) has solutions if and only if the operator Q has fixed points.

    In the first result, we establish the existence and uniqueness of solutions for the problem (1.7), by applying Banach's fixed point theorem.

    Theorem 3.1. Assume that f:J×R2R is a continuous function such that:

    (H1) there exist a constant L1>0 such that

    |f(t,u1,v1)f(t,u2,v2)|L1(|u1u2|+|v1v2|)

    for any ui, viR, i=1,2 and tJ.

    If

    Λ0Λ1L1<1, (3.5)

    where Λ0 and Λ1 are given by (3.2) and (3.3) respectively, then the problem (1.7) has a unique solution on J.

    Proof. Firstly, we transform the problem (1.7) into a fixed point problem, x=Qx, where the operator Q is defined as in (3.4). Applying the Banach contraction mapping principle, we shall show that the operator Q has a unique fixed point, which is the unique solution of the problem (1.7)

    Let suptJ|f(t,0,0)|:=M1<. Next, we set Br1:={xC:xr1} with

    r1Λ1M1+(|κ|A(T,γ1))/|Ω|1Λ0Λ1L1, (3.6)

    where Ω, A(T,γ1), Λ0, Λ1 are given by (2.9), (3.1)–(3.3), respectively. Observe that Br1 is a bounded, closed, and convex subset of C. The proof is divided into two steps:

    Step I. We show that QBr1Br1.

    For any xBr1, we have

    |(Qx)(t)|Iα;ψ0+|Fx(s)|(T)+A(T,γ1)|Ω|(|κ|+mi=1|δi|Iα;ψ0+|Fx(s)|(ηi)+nj=1|ωj|Iα+βj;ψ0+|Fx(s)|(θj)+rk=1|λk|Iαμk;ψ0+|Fx(s)|(ξk)).

    We note that

    Iϕ;ψ0+|x(τ)|(s)=1Γ(ϕ)s0ψ(τ)(ψ(s)ψ(τ))ϕ1|x(τ)|dτA(s,ϕ)x.

    It follows from conditions (H1) that

    |Fx(t)||f(t,x(t),Iϕ;ψ0+x(s)(t))f(t,0,0)|+|f(t,0,0)|L1(|x(t)|+Iϕ;ψ0+|x(s)|(t))+M1,L1(1+(ψ(T)ψ(0))ϕΓ(ϕ+1))x+M1=L1[1+A(T,ϕ)]x+M1=L1Λ0x+M1.

    Then we have

    |(Qx)(t)|(L1Λ0x+M1)(ψ(T)ψ(0))αΓ(α+1)+A(T,γ1)|Ω|[|κ|+(L1Λ0x+M1)(mi=1|δi|(ψ(ηi)ψ(0))αΓ(α+1)+nj=1|ωj|(ψ(θj)ψ(0))α+βjΓ(α+βj+1)+rk=1|λk|(ψ(ξk)ψ(0))αμkΓ(αμk+1))]=L1Λ0[A(T,α)+A(T,γ1)|Ω|(mi=1|δi|A(ηi,α)+nj=1|ωj|A(θj,α+βj)+rk=1|λk|A(ξk,αμk))]x+[A(T,α)+A(T,γ1)|Ω|(mi=1|δi|A(ηi,α)+nj=1|ωj|A(θj,α+βj)+rk=1|λk|A(ξk,αμk))]M1+|κ|A(T,γ1)|Ω|Λ0Λ1L1r1+Λ1M1+|κ|A(T,γ1)|Ω|r1,

    which implies that QBr1Br1.

    Step II. We show that Q:CC is a contraction.

    For any x, yC and for each tJ, we have

    |(Qx)(t)(Qy)(t)|Iα;ψ0+|Fx(s)Fy(s)|(T)+A(T,γ1)|Ω|(mi=1|δi|Iα;ψ0+|Fx(s)Fy(s)|(ηi)+nj=1|ωj|Iα+βj;ψ0+|Fx(s)Fy(s)|(θj)+rk=1|λk|Iαμk;ψ0+|Fx(s)Fy(s)|(ξk)){(ψ(T)ψ(0))αΓ(α+1)+A(T,γ1)|Ω|(mi=1|δi|(ψ(ηi)ψ(0))αΓ(α+1)+nj=1|ωj|(ψ(θj)ψ(0))α+βjΓ(α+βj+1)+rk=1|λk|(ψ(ξk)ψ(0))αμkΓ(αμk+1))}L1Λ0xy={A(T,α)+A(T,γ1)|Ω|(mi=1|δi|A(ηi,α)+nj=1|ωj|A(θj,α+βj)+rk=1|λk|A(ξk,αμk))}L1Λ0xy=Λ0Λ1L1xy,

    which implies that QxQyΛ0Λ1L1xy. As Λ0Λ1L1<1, hence, the operator Q is a contraction. Therefore, by the Banach contraction mapping principle (Lemma 2.8) the operator Q has a fixed point, and hence the problem (1.7) has a unique solution on J. The proof is completed.

    Next, we present an existence theorem by using Krasnosel'skii's fixed point theorem.

    Theorem 3.2. Assume that f:J×R2R is a continuous function satisfying (H1). In addition, we assume that:

    (H2) |f(t,u,v)|σ(t), (t,u,v)J×R2, and σC(J,R+).

    If

    L1Λ0[Λ1A(T,α)]<1, (3.7)

    where Λ0, Λ1, A(T,α) are defined by (3.2), (3.3) and (3.1), respectively, then the problem (1.7) has at least one solution on J.

    Proof. Let suptJ|σ(t)|=σ and Br2:={xC:xr2}, where

    r2σΛ1+|κ|A(T,γ1)|Ω|.

    We define the operators Q1 and Q2 on Br2 by

    (Q1x)(t)=Iα;ψ0+Fx(s)(t),tJ,(Q2x)(t)=A(t,γ1)Ω[κ(mi=1δiIα;ψ0+Fx(s)(ηi)+nj=1ωjIα+βj;ψ0+Fx(s)(θj)+rk=1λkIαμk;ψ0+Fx(s)(ξk))],tJ.

    Note that Q=Q1+Q2. For any x,yBr2, we have

    |(Q1x)(t)+(Q2y)(t)|suptJ{Iα;ψ0+|Fx(s)|(t)+A(t,γ1)|Ω|(|κ|+mi=1|δi|Iα;ψ0+|Fy(s)|(ηi)+nj=1|ωj|Iα+βj;ψ0+|Fy(s)|(θj)+rk=1|λk|Iαμk;ψ0+|Fy(s)|(ξk))}σ{A(T,α)+A(T,γ1)|Ω|(mi=1|δi|A(ηi,α)+nj=1|ωj|A(θj,α+βj)+rk=1|λk|A(ξk,αμk))}+|κ|A(T,γ1)|Ω|σΛ1+|κ|A(T,γ1)|Ω|r2.

    This implies that Q1x+Q2xBr2, which satisfies the assumption (i) of Lemma 2.9.

    We show that the assumption (ii) of Lemma 2.9 is satisfied.

    Let xn be a sequence such that xnx in C. Then for each tJ, we have

    |(Q1xn)(t)(Q1x)(t)|Iα;ψ0+|Fxn(s)Fx(s)|(T)A(T,α)FxnFx.

    Since f is continuous, this implies that the operator Fx is also continuous. Hence, we obtain

    FxnFx0asn.

    Thus, this shows that the operator Q1x is continuous. Also, the set Q1Br2 is uniformly bounded on Br2 as

    Q1xA(T,α)σ.

    Next, we prove the compactness of the operator Q1. Let sup(t,u,v)J×B2r2|f(t,u,v)|=ˆf<, then for each t1,t2J with 0t1<t2T, we obtain

    |(Q1x)(t2)(Q1x)(t1)|=1Γ(α)|t10ψ(s)[(ψ(t2)ψ(s))α1(ψ(t1)ψ(s))α1]Fx(s)ds+t2t1ψ(s)(ψ(t2)ψ(s))α1Fx(s)ds|ˆfΓ(α+1)[2(ψ(t2)ψ(t1))α+|(ψ(t2)ψ(0))α(ψ(t1)ψ(0))α|].

    Obviously, the right hand side in the above inequality is independent of x and tends to zero as t2t1. Therefore, the operator Q1 is equicontinuous. So Q1 is relatively compact on Br2. Then, by the Arzelá-Ascoli theorem, Q1 is compact on Br2.

    Moreover, it is easy to prove, using condition (3.7), that the operator Q2 is a contraction and thus the assumption (iii) of Lemma 2.9 holds. Thus all the assumptions of Lemma 2.9 are satisfied. So the conclusion of Lemma 2.9 implies that the problem (1.7) has at least one solution on J. The proof is completed.

    The Leray-Schauder's nonlinear alternative [32] is used to prove our last existence result.

    Theorem 3.3. Assume that:

    (H3) there exist a function qC(J,R+) and a continuous nondecreasing function Φ:[0,)[0,) which is subhomogeneous (that is, Φ(μx)μΦ(x), for all μ1 and xC), such that

    |f(t,u,v)|q(t)Φ(|u|+|v|)for each(t,u,v)J×R2;

    (H4) there exist a constant M2>0 such that

    M2Λ0Λ1Φ(M2)q+(|κ|A(T,γ1))/|Ω|>1,

    with Ω, A(T,α) Λ0 and Λ1 by (2.9), (3.1), (3.2) and (3.3).

    Then, the problem (1.7) has at least one solution on J.

    Proof. Let the operator Q be defined by (3.4). Firstly, we show that Q maps bounded sets (balls) into bounded set in C. For a constant r3>0, let Br3={xC:xr3} be a bounded ball in C. Then, for tJ, we obtain

    |(Qx)(t)|suptJ{Iα;ψ0+|Fx(s)|(t)+A(t,γ1)|Ω|(|κ|+mi=1|δi|Iα;ψ0+|Fx(s)|(ηi)+nj=1|ωj|Iα+βj;ψ0+|Fx(s)|(θj)+rk=1|λk|Iαμk;ψ0+|Fx(s)|(ξk))}qΦ{(1+(ψ(T)ψ(0))ϕΓ(ϕ+1))x}{Iα;ψ0+(1)(T)+A(T,γ1)|Ω|×(mi=1|δi|Iα;ψ0+(1)(ηi)+nj=1|ωj|Iα+βj;ψ0+(1)(θj)+rk=1|λk|Iαμk;ψ0+(1)(ξk))}+|κ|(ψ(T)ψ(0))γ1|Ω|Γ(γ)=qΦ(Λ0x){A(T,α)+A(T,γ1)|Ω|(mi=1|δi|A(ηi,α)+nj=1|ωj|A(θj,α+βj)+rk=1|λk|A(ξk,αμk))}+|κ|A(T,γ1)|Ω|Λ0Λ1Φ(x)q+|κ|A(T,γ1)|Ω|.

    Consequently

    \begin{equation*} \|Qx|| \le \Lambda_0\Lambda_{1}\Phi(r_3)\Vert q \Vert + \frac{\vert \kappa \vert A(T, \gamma-1)}{|\Omega|}. \end{equation*}

    Next, we show that the operator \mathcal{Q} maps bounded sets into equicontinuous sets of \mathcal{C} . Let t_1, t_2 \in J with t_1 < t_2 and x \in B_{r_3} . Then we get

    \begin{eqnarray} \vert(\mathcal{Q}x)(t_2) - (\mathcal{Q}x)(t_1)\vert &\leq& \frac{1}{\Gamma (\alpha )}\Bigg|\int_{0}^{t_1}{\psi }^{\prime}(s)\left[\left(\psi(t_2) - \psi(s)\right)^{\alpha -1} - \left(\psi(t_1) - \psi(s)\right)^{\alpha -1}\right]F_{x}(s)ds\\ && + \int_{t_1}^{t_2}{\psi }^{\prime}(s)\left(\psi(t_2) - \psi(s)\right)^{\alpha -1}F_{x}(s)ds\Bigg|\\ && + \frac{\left( \psi (t_2)-\psi (0) \right)^{\gamma-1} - \left( \psi (t_1)-\psi (0) \right)^{\gamma-1}}{\vert\Omega\vert\Gamma(\gamma)} \Bigg(\vert \kappa \vert + \sum\limits_{i = 1}^{m}\vert\delta_{i}\vert\mathcal{I}_{0^+}^{\alpha ;\psi }\vert F_{x}(s) \vert(\eta_{i})\\ && + \sum\limits_{j = 1}^{n}\vert\omega_{j}\vert\mathcal{I}_{0^+}^{\alpha+\beta_{j};\psi }\vert F_{x}(s)\vert(\theta_{j}) + \sum\limits_{k = 1}^{r}\vert\lambda_{k}\vert\mathcal{I}_{0^+}^{\alpha -\mu_{k};\psi }\vert F_{x}(s)\vert(\xi_{k}) \Bigg)\\ &\leq& \frac{\Lambda_0\Phi(r_3)\Vert q \Vert}{\Gamma (\alpha+1)} \Big[2\left(\psi(t_2) - \psi(t_1)\right)^{\alpha} + \left|\left(\psi(t_2) - \psi(0)\right)^{\alpha} - \left(\psi(t_1) - \psi(0)\right)^{\alpha}\right|\Big]\\ && + \frac{|\kappa| + \Lambda_0\Lambda_{1}\Phi(r_3)\Vert q \Vert}{\vert\Omega\vert\Gamma(\gamma)} \left|\left( \psi (t_2)-\psi (0) \right)^{\gamma-1} - \left( \psi (t_1)-\psi (0) \right)^{\gamma-1}\right|. \end{eqnarray} (3.8)

    As t_2 - t_1 \to 0 , the right hand side of (3.8) tends to zero independently of x\in B_{r_3} . Hence, by the Arzelá-Ascoli theorem, the operator \mathcal{Q} is completely continuous.

    The result will follow from the Leray-Schauder's nonlinear alternative once we have proved the boundedness of the set of all solutions to the equations x = \varrho\mathcal{Q}x for \varrho \in (0, 1) .

    Let x be a solution. Then, for t\in J , and following calculations similar to the first step, we obtain

    \begin{equation*} \vert x(t) \vert = \vert \varrho (\mathcal{Q}x)(t) \vert \leq \Lambda_0\Lambda_{1}\Phi(\Vert x \Vert)\Vert q \Vert +\frac{\vert \kappa \vert A(T, \gamma-1)}{|\Omega|}, \end{equation*}

    which leads to

    \begin{equation*} \frac{\Vert x \Vert}{\Lambda_0\Lambda_{1}\Phi( \Vert x \Vert)\Vert q \Vert +(\vert \kappa \vert A(T, \gamma-1))/|\Omega|}\leq 1. \end{equation*}

    In view of (H_4) , there exists a constant M_2 > 0 such that \Vert x \Vert \neq M_2 . Let us set

    \begin{equation*} K : = \{x \in \mathcal{C} : \Vert x \Vert < M_2\}. \end{equation*}

    We see that the operator \mathcal{Q} : \overline{K} \to \mathcal{C} is continuous and completely continuous. From the choice of \overline{K} , there is no x \in \partial K such that x = \varrho\mathcal{Q}x for some \varrho \in (0, 1) . Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 2.10), we deduce that the operator \mathcal{Q} has a fixed point x \in \overline{K} which is a solution of the problem (1.7). The proof is completed.

    In this section, we are developing some results on the different types of Ulam's stability such as Ulam-Hyers ( \mathbb{UH} ), generalized Ulam-Hyers ( \mathbb{UH} ), Ulam-Hyers-Rassias ( \mathbb{UHR} ) and generalized Ulam-Hyers-Rassias ( \mathbb{UHR} ) stability for the proposed problem (1.7).

    We start with needed definitions. Let \epsilon > 0 be a positive real number and \Theta : J \to \mathbb{R}^+ be a continuous function. We consider the following inequalities:

    \begin{eqnarray} \left\vert {}^{H}\mathfrak{D}_{0^+}^{\alpha , \rho;\psi }z(t) - f(t, z(t), \mathcal{I}_{{0}^{+}}^{\phi ;\psi }z(t))\right\vert &\leq& \epsilon, \end{eqnarray} (4.1)
    \begin{eqnarray} \left\vert {}^{H}\mathfrak{D}_{0^+}^{\alpha , \rho;\psi }z(t) - f(t, z(t), \mathcal{I}_{{0}^{+}}^{\phi ;\psi }z(t))\right\vert &\leq& \epsilon \Theta(t), \end{eqnarray} (4.2)
    \begin{eqnarray} \left\vert {}^{H}\mathfrak{D}_{0^+}^{\alpha , \rho;\psi }z(t) - f(t, z(t), \mathcal{I}_{{0}^{+}}^{\phi ;\psi }z(t))\right\vert &\leq& \Theta(t). \end{eqnarray} (4.3)

    Definition 4.1. [34] The problem (1.7) is said to be \mathbb{UH} stable if there exists a real number M_{f} > 0 such that for each \epsilon > 0 and for each solution z\in \mathcal{C} of the inequality (4.1), there exists a solution x\in \mathcal{C} of the problem (1.7) with

    \begin{equation} \left\vert z(t)-x(t)\right\vert \leq M_{f}\epsilon , \quad t\in J. \end{equation} (4.4)

    Definition 4.2. [34] The problem (1.7) is said to be generalized \mathbb{UH} stable if there exists a function \Theta \in \mathcal{C}(\mathbb{R}^{+}, \mathbb{R}^{+}) with \Theta(0) = 0 such that, for each solution z\in \mathcal{C} of inequality (4.2), there exists a solution x\in \mathcal{C} of the problem (1.7) with

    \begin{equation} \left\vert z(t)-x(t)\right\vert \leq \Theta(\epsilon), \quad t\in J. \end{equation} (4.5)

    Definition 4.3. [34] The problem (1.7) is said to be \mathbb{UHR} stable with respect to \Theta \in \mathcal{C}(J, \mathbb{R}^{+}) if there exists a real number M_{f, \Theta} > 0 such that for each \epsilon > 0 and for each solution z\in \mathcal{C} of the inequality (4.2) there exists a solution x\in \mathcal{C} of the problem (1.7) with

    \begin{equation} \left\vert z(t)-x(t)\right\vert \leq M_{f, \Theta}\epsilon \Theta(t), \quad t\in J. \end{equation} (4.6)

    Definition 4.4. [34] The problem (1.7) is said to be generalized \mathbb{UHR} stable with respect to \Theta \in \mathcal{C} (J, \mathbb{R}^{+}) if there exists a real number M_{f, \Theta} > 0 such that for each solution z\in \mathcal{C} of the inequality (4.3), there exists a solution x\in \mathcal{C} of the problem (1.7) with

    \begin{equation} \left\vert z(t)-x(t)\right\vert \leq M_{f, \Theta}\Theta(t), \quad t\in J. \end{equation} (4.7)

    Remark 4.5. It is clear that (i) Definition 4.1 \Rightarrow Definition 4.2 ; (ii) Definition 4.3 \Rightarrow Definition 4.4 ; (iii) Definition 4.3 for \Theta(t) = 1 \Rightarrow Definition 4.1 .

    Remark 4.6. A function z\in \mathcal{C}(J, \mathbb{R}) is a solution of the inequality (4.1) if and only if there exists a function w\in \mathcal{C}(J, \mathbb{R}) (which depends on z ) such that:

    (i) |w(t)|\leq \epsilon , \forall t \in J .

    (ii) ^{H}\mathfrak{D}_{0^+}^{\alpha, \rho; \psi }z(t) = F_{z}(t) + w(t) , t\in J .

    Remark 4.7. A function z\in \mathcal{C} is a solution of the inequality (4.2) if and only if there exists a function v\in \mathcal{C} (which depends on z ) such that:

    (i) |v(t)|\leq \epsilon \Theta(t) , \forall t \in J .

    (ii) ^{H}\mathfrak{D}_{0^+}^{\alpha, \rho; \psi }z(t) = F_{z}(t) + v(t) , t\in J .

    Firstly, we present an important lemma that will be used in the proofs of \mathbb{UH} stability and \mathbb{GUH} stability.

    Lemma 4.8. Let \alpha \in (1, 2] , \rho \in[0, 1). If z \in \mathcal{C} is a solution of the inequality (4.1), then z is a solution of the following inequality

    \begin{equation} \left| z(t) - \mathcal{R}_{z} - \mathcal{I}_{0^+}^{\alpha;\psi }F_{z}(s)(t) \right| \leq \Lambda_{1}\epsilon, \end{equation} (4.8)

    where

    \begin{equation*} \mathcal{R}_{z} = \frac{A(t, \gamma-1)}{\Omega} \left[\kappa - \sum\limits_{i = 1}^{m}\delta_{i} \mathcal{I}_{0^+}^{\alpha ;\psi }F_{z}(s)(\eta_i) - \sum\limits_{j = 1}^{n}\omega_{j} \mathcal{I}_{0^+}^{\alpha +\beta_j;\psi }F_{z}(s)(\theta_j) - \sum\limits_{k = 1}^{r}\lambda_{k}\mathcal{I}_{0^+}^{\alpha -{{\mu }_{k}};\psi }F_{z}(s)(\xi_k) \right], \end{equation*}

    and \Lambda_{1} is given by (3.3).

    Proof. Let z be a solution of the inequality (4.1). So, in view of Remark 4.6 (ii) and Lemma 2.11 , we have

    \begin{equation} \left\{ \begin{array}{c} {}^{H}\mathfrak{D}_{0^+}^{\alpha , \rho;\psi }z(s)(t) = F_{z}(t) + w(t), \quad t\in (0, T], \\ z(0) = 0, \qquad \sum\limits_{i = 1}^{m}\delta_{i} z(\eta_{i}) + \sum\limits_{j = 1}^{n}\omega_{j}\mathcal{I}_{0^+}^{\beta_j; \psi }z(s)(\theta_j) + \sum\limits_{k = 1}^{r}{\lambda}_{k}{}^{H}\mathfrak{D}_{0^+}^{\mu_{k}, \rho; \psi}z(s)(\xi_{k}) = \kappa. \end{array} \right. \end{equation} (4.9)

    Thus, the solution of (4.9) will be in the following term

    \begin{eqnarray*} z(t) & = & \mathcal{I}_{0^+}^{\alpha ;\psi }F_{z}(s)(t) + \frac{A(t, \gamma-1)}{\Omega}\left(\kappa - \sum\limits_{i = 1}^{m}\delta_{i} \mathcal{I}_{0^+}^{\alpha ;\psi }F_{z}(\eta_i) - \sum\limits_{j = 1}^{n}\omega_{j} \mathcal{I}_{0^+}^{\alpha +\beta_j;\psi }F_{z}(\theta_j) - \sum\limits_{k = 1}^{r}\lambda_{k}\mathcal{I}_{0^+}^{\alpha -{{\mu }_{k}};\psi }F_{z}(\xi_k) \right)\nonumber\\ && + \mathcal{I}_{0^+}^{\alpha ;\psi }w(s)(t) - \frac{A(t, \gamma-1)}{\Omega} \left( \sum\limits_{i = 1}^{m}\delta_{i} \mathcal{I}_{0^+}^{\alpha ;\psi }w(s)(\eta_i) + \sum\limits_{j = 1}^{n}\omega_{j} \mathcal{I}_{0^+}^{\alpha +\beta_j;\psi }w(s)(\theta_j) + \sum\limits_{k = 1}^{r}\lambda_{k}\mathcal{I}_{0^+}^{\alpha -{{\mu }_{k}};\psi }w(s)(\xi_k)\right). \end{eqnarray*}

    Then, by using Remark 4.6 (i), it follows that

    \begin{eqnarray*} \left\vert z(t) - \mathcal{R}_{z} - \mathcal{I}_{0^+}^{\alpha ;\psi }F_{z}(s)(t)\right\vert & = & \Bigg\vert \mathcal{I}_{0^+}^{\alpha ;\psi }w(s)(t) - \frac{A(t, \gamma-1)}{\Omega} \Bigg( \sum\limits_{i = 1}^{m}\delta_{i} \mathcal{I}_{0^+}^{\alpha ;\psi }w(s)(\eta_i) + \sum\limits_{j = 1}^{n}\omega_{j} \mathcal{I}_{0^+}^{\alpha +\beta_j;\psi }w(s)(\theta_j)\\ && + \sum\limits_{k = 1}^{r}\lambda_{k}\mathcal{I}_{0^+}^{\alpha -{{\mu }_{k}};\psi }w(s)(\xi_k)\Bigg) \Bigg\vert\\ &\leq& \Bigg[A(T, \alpha) + \frac{A(T, \gamma-1)}{|\Omega|}\Bigg(\sum\limits_{i = 1}^{m}|\delta_i|A(\eta_i, \alpha) +\sum\limits_{j = 1}^{n}|\omega_j|A(\theta_j, \alpha+\beta_j)\\ && + \sum\limits_{k = 1}^{r}|\lambda_k|A(\xi_k, \alpha-\mu_k)\Bigg)\Bigg] \epsilon\\ & = & \Lambda_{1}\epsilon, \end{eqnarray*}

    from which inequality (4.8) is obtained. The proof is completed.

    Now, we prove \mathbb{UH} stability and generalized \mathbb{UH} stability results for the problem (1.7).

    Theorem 4.9. Assume that the function f:J\times\mathbb{R}^2 \to \mathbb{R} is continuous and (H_1) holds with \Lambda_{0} A(T, \alpha) L_{1} < 1 . Then the problem (1.7) is \mathbb{UH} stable on J and consequently generalized \mathbb{UH} stable.

    Proof. Let \epsilon > 0 and z\in \mathcal{C} be any solution of the inequality (4.1). Let x \in \mathcal{C} be the unique solution of the following problem (1.7)

    \begin{equation*} \left\{ \begin{array}{c} {}^{H}\mathfrak{D}_{0^+}^{\alpha , \rho;\psi }x(s)(t) = F_{x}(t), \quad t\in (0, T], \\ x(0) = 0, \qquad \sum\limits_{i = 1}^{m}\delta_{i} x(\eta_{i}) + \sum\limits_{j = 1}^{n}\omega_{j}\mathcal{I}_{0^+}^{\beta_j; \psi }x(s)(\theta_j) + \sum\limits_{k = 1}^{r}{\lambda}_{k}{}^{H}\mathfrak{D}_{0^+}^{\mu_{k}, \rho; \psi}x(s)(\xi_{k}) = \kappa. \end{array} \right. \end{equation*}

    Using Lemma 2.11 , we obtain

    \begin{eqnarray*} x(t) & = & \mathcal{R}_{x} + \mathcal{I}_{0^+}^{\alpha ;\psi }F_{x}(s)(t), \end{eqnarray*}

    where

    \begin{equation} \label{Rx} \mathcal{R}_{x} = \frac{A(t, \gamma-1)}{\Omega}\left(\kappa - \sum\limits_{i = 1}^{m}\delta_{i} \mathcal{I}_{0^+}^{\alpha ;\psi }F_{x}(s)(\eta_i) - \sum\limits_{j = 1}^{n}\omega_{j} \mathcal{I}_{0^+}^{\alpha +\beta_j;\psi }F_{x}(s)(\theta_j)\nonumber - \sum\limits_{k = 1}^{r}\lambda_{k}\mathcal{I}_{0^+}^{\alpha -{{\mu }_{k}};\psi }F_{x}(\xi_k) \right). \end{equation}

    On the other hand, if x(0) = z(0) , x(\eta_{i}) = z(\eta_{i}) , \mathcal{I}_{0^+}^{\beta_j; \psi }x(s)(\theta_j) = \mathcal{I}_{0^+}^{\beta_j; \psi }z(s)(\theta_j) and {}^{H}\mathfrak{D}_{0^+}^{\mu_{k}, \rho; \psi}x(s)(\xi_{k}) = {}^{H}\mathfrak{D}_{0^+}^{\mu_{k}, \rho; \psi}z(s)(\xi_{k}) , then \mathcal{R}_{x} = \mathcal{R}_{z}. Indeed, we have

    \begin{eqnarray*} |\mathcal{R}_{x} - \mathcal{R}_{z}| &\leq& \frac{A(t, \gamma-1)}{|\Omega|} \Bigg( \sum\limits_{i = 1}^{m}|\delta_{i}| \mathcal{I}_{0^+}^{\alpha ;\psi }|F_{x}(s) - F_{z}(s)|(\eta_i)\\ && + \sum\limits_{j = 1}^{n}|\omega_{j}| \mathcal{I}_{0^+}^{\alpha +\beta_j;\psi }|F_{x}(s) - F_{z}(s)|(\theta_j) + \sum\limits_{k = 1}^{r}|\lambda_{k}|\mathcal{I}_{0^+}^{\alpha -{{\mu }_{k}};\psi }|F_{x}(s) - F_{z}(s)|(\xi_k)\Bigg)\\ &\leq&\frac{A(t, \gamma-1)}{|\Omega|} \Bigg( \sum\limits_{i = 1}^{m}|\delta_{i}| \mathcal{I}_{0^+}^{\alpha ;\psi }|x(s) - z(s)|(\eta_i) + \sum\limits_{j = 1}^{n}|\omega_{j}| \mathcal{I}_{0^+}^{\alpha +\beta_j;\psi }|x(s) - z(s)|(\theta_j)\\ && + \sum\limits_{k = 1}^{r}|\lambda_{k}|\mathcal{I}_{0^+}^{\alpha -{{\mu }_{k}};\psi }|x(s) - z(s)|(\xi_k)\Bigg) \Lambda_{0}\Lambda_{1}L_{1}\\ & = & 0. \end{eqnarray*}

    Thus \mathcal{R}_{x} = \mathcal{R}_{z}. Now, by applying the triangle inequality, |u-v| \leq |u| + |v| , and Lemma 4.8 , for any t\in J , we have

    \begin{eqnarray*} |z(t) - x(t)| &\leq& \left| z(t) - \mathcal{R}_{x} - \mathcal{I}_{0^+}^{\alpha ;\psi }F_{x}(s)(t)\right|\\ &\leq& \left| z(t) - \mathcal{R}_{z} - \mathcal{I}_{0^+}^{\alpha;\psi }F_{z}(s)(t) \right| + \mathcal{I}_{0^+}^{\alpha;\psi }|F_{z}(s) - F_{x}(s)|(t) + \left|\mathcal{R}_{z} - \mathcal{R}_{x} \right|\\ &\leq& \Lambda_{1}\epsilon + \Lambda_{0} A(T, \alpha) L_{1}|z(t) - x(t)|. \end{eqnarray*}

    This implies that

    \begin{equation*} |z(t) - x(t)| \leq \frac{\Lambda_{1}}{1 - \Lambda_0 A(T, \alpha) L_1 }\; \epsilon. \end{equation*}

    By setting

    \begin{equation*} M_{f} = \frac{\Lambda_{1}}{1 - \Lambda_0 A(T, \alpha) L_1 }, \end{equation*}

    we obtain

    \begin{equation*} |z(t) - x(t)| \leq M_{f}\; \epsilon. \end{equation*}

    Hence, the problem (1.7) is \mathbb{UH} stable. Further, if we set \Theta(\epsilon) = M_{f}\epsilon and \Theta(0) = 0 we have

    \begin{equation*} |z(t) - x(t)| \leq \Theta(\epsilon), \end{equation*}

    which implies that the solution of the problem (1.7) is generalized \mathbb{UH} stable. The proof is completed.

    For the proof of our next lemma, we assume the following assumption:

    (H_3) There exists an increasing function \Theta\in \mathcal{C}(J, \mathbb{R}^+) and there exists n_{\Theta} > 0 , such that, for any t\in J , the following integral inequality

    \begin{equation} \mathcal{I}_{{0}^{+}}^{\alpha ;\psi }\Theta(t) \leq n_{\Theta} \Theta(t). \end{equation} (4.10)

    Next, we present an important lemma that will be used in the proofs of \mathbb{UHR} and generalized \mathbb{UHR} stability results.

    Lemma 4.10. Let \alpha \in (1, 2] , \rho\in[0, 1]. If z \in \mathcal{C} is a solution of the inequality (4.2), then z is a solution of the following inequality

    \begin{equation} \left\vert z(t) - \mathcal{R}_{z} - \mathcal{I}_{0^+}^{\alpha;\psi }F_{z}(s)(t)\right\vert \leq \Lambda_{2}\epsilon n_{\Theta} \Theta(t), \end{equation} (4.11)

    where

    \begin{equation} \Lambda_{2} = 1 + \frac{A(T, \gamma-1)}{|\Omega|}\left(\sum\limits_{i = 1}^{m}|\delta_i| +\sum\limits_{j = 1}^{n}|\omega_j| + \sum\limits_{k = 1}^{r}|\lambda_k|\right). \end{equation} (4.12)

    Proof. Let z be a solution of the inequality (4.2). So, in view of Remark 4.7 (ii) and Lemma 2.11 , the solution of (4.9) can be written by

    \begin{eqnarray*} z(t) & = & \mathcal{I}_{0^+}^{\alpha ;\psi }F_{z}(s)(t) + \frac{A(t, \gamma-1)}{\Omega}\Bigg(\kappa - \sum\limits_{i = 1}^{m}\delta_{i} \mathcal{I}_{0^+}^{\alpha ;\psi }F_{z}(s)(\eta_i) - \sum\limits_{j = 1}^{n}\omega_{j} \mathcal{I}_{0^+}^{\alpha +\beta_j;\psi }F_{z}(s)(\theta_j)\nonumber\\ && - \sum\limits_{k = 1}^{r}\lambda_{k}\mathcal{I}_{0^+}^{\alpha -{{\mu }_{k}};\psi }F_{z}(s)(\xi_k)\Bigg) + \mathcal{I}_{0^+}^{\alpha ;\psi }v(s)(t) - \frac{A(t, \gamma-1)}{\Omega} \Bigg( \sum\limits_{i = 1}^{m}\delta_{i} \mathcal{I}_{0^+}^{\alpha ;\psi }v(s)(\eta_i)\nonumber\\ && + \sum\limits_{j = 1}^{n}\omega_{j} \mathcal{I}_{0^+}^{\alpha +\beta_j;\psi }v(s)(\theta_j) + \sum\limits_{k = 1}^{r}\lambda_{k}\mathcal{I}_{0^+}^{\alpha -{{\mu }_{k}};\psi }v(s)(\xi_k)\Bigg). \end{eqnarray*}

    Then, by using Remark 4.7 (i) with (H_3) , we have the following estimation

    \begin{eqnarray*} \left\vert z(t) - \mathcal{R}_{z} - \mathcal{I}_{0^+}^{\alpha ;\psi }F_{z}(s)(t)\right\vert & = & \Bigg\vert \mathcal{I}_{0^+}^{\alpha ;\psi }v(s)(t) - \frac{A(t, \gamma-1)}{\Omega} \Bigg( \sum\limits_{i = 1}^{m}\delta_{i} \mathcal{I}_{0^+}^{\alpha ;\psi }v(s)(\eta_i) + \sum\limits_{j = 1}^{n}\omega_{j} \mathcal{I}_{0^+}^{\alpha +\beta_j;\psi }v(s)(\theta_j)\\ && + \sum\limits_{k = 1}^{r}\lambda_{k}\mathcal{I}_{0^+}^{\alpha -{{\mu }_{k}};\psi }v(s)(\xi_k)\Bigg) \Bigg\vert\\ &\leq& \left[1 + \frac{A(T, \gamma-1)}{|\Omega|}\left(\sum\limits_{i = 1}^{m}|\delta_i| +\sum\limits_{j = 1}^{n}|\omega_j| + \sum\limits_{k = 1}^{r}|\lambda_k|\right)\right] \epsilon n_{\Theta} \Theta(t)\\ & = & \Lambda_{2}\epsilon n_{\Theta} \Theta(t), \end{eqnarray*}

    from which inequality (4.11) is obtained. The proof is completed.

    Finally, we present \mathbb{UHR} and generalized \mathbb{UHR} stability results for the problem (1.7).

    Theorem 4.11. Assume that the function f:J\times\mathbb{R}^2 \to \mathbb{R} is continuous and (H_1) holds. Then the problem (1.7) is \mathbb{UHR} stable on J and consequently generalized \mathbb{UHR} stable.

    Proof. Let \epsilon > 0 and z\in \mathcal{C} be the solution of the inequality (4.3). Let x \in \mathcal{C} be the unique solution of the problem (1.7). By using Lemma 2.11 , we obtain

    \begin{equation*} x(t) = \mathcal{R}_{x} + \mathcal{I}_{0^+}^{\alpha ;\psi }F_{x}(s)(t), \end{equation*}

    where

    {{\mathcal{R}}_{x}} = \frac{A(t, \gamma -1)}{\Omega }\left( \kappa -\sum\limits_{i = 1}^{m}{{{\delta }_{i}}}\mathcal{I}_{{{0}^{+}}}^{\alpha ;\psi }{{F}_{x}}(s)({{\eta }_{i}})-\sum\limits_{j = 1}^{n}{{{\omega }_{j}}}\mathcal{I}_{{{0}^{+}}}^{\alpha +{{\beta }_{j}};\psi }{{F}_{x}}(s)({{\theta }_{j}})-\sum\limits_{k = 1}^{r}{{{\lambda }_{k}}}\mathcal{I}_{{{0}^{+}}}^{\alpha -{{\mu }_{k}};\psi }{{F}_{x}}(s)({{\xi }_{k}}) \right).

    On the other hand, if x(0) = z(0) , x(\eta_{i}) = z(\eta_{i}) , \mathcal{I}_{0^+}^{\beta_j; \psi }x(s)(\theta_j) = \mathcal{I}_{0^+}^{\beta_j; \psi }z(s)(\theta_j) and {}^{H}\mathfrak{D}_{0^+}^{\mu_{k}, \rho; \psi}x(s)(\xi_{k}) = {}^{H}\mathfrak{D}_{0^+}^{\mu_{k}, \rho; \psi}z(s)(\xi_{k}) , then it is easy to see that \mathcal{R}_{x} = \mathcal{R}_{z}.

    Now, by appying |u-v| \leq |u| + |v| and Lemma 4.10 , for any t\in J , we have

    \begin{eqnarray*} |z(t) - x(t)| &\leq& \left| z(t) - \mathcal{R}_{x} - \mathcal{I}_{0^+}^{\alpha ;\psi }F_{x}(s)(t)\right|\\ &\leq& \left| z(t) - \mathcal{R}_{z} - \mathcal{I}_{0^+}^{\alpha;\psi }F_{z}(s)(t) \right| + \mathcal{I}_{0^+}^{\alpha;\psi }|F_{z}(s) - F_{x}(s)|(t) + \left|\mathcal{R}_{z} - \mathcal{R}_{x} \right|\\ &\leq& \Lambda_{2}\epsilon n_{\Theta} \Theta(t) + \Lambda_{0} A(T, \alpha) L_{1}|z(t) - x(t)| \end{eqnarray*}

    This implies that

    \begin{equation*} |z(t) - x(t)| \leq \frac{\Lambda_{2}n_{\Theta}}{1 - \Lambda_0 A(T, \alpha) L_1 }\; \epsilon\Theta(t). \end{equation*}

    By setting

    \begin{equation*} M_{f, \Theta} = \frac{\Lambda_{2}n_{\Theta} }{1 - \Lambda_0 A(T, \alpha) L_1 }, \end{equation*}

    we obtain

    \begin{equation*} |z(t) - x(t)| \leq M_{f, \Theta}\; \epsilon \Theta(t). \end{equation*}

    Therefore, the problem (1.7) is \mathbb{UHR} stable. Further, in the same fashion, it is easy to check that the solution of the problem (1.7) is generalized \mathbb{UHR} stable. This completes the proof.

    This section presents some examples which illustrate the validity and applicability of our main results.

    Example 5.1. Consider the following mixed nonlocal boundary problem of the form:

    \begin{equation} \left\{ \begin{array}{c} {}^{H}\mathfrak{D}_{0^+}^{\frac{8}{5}, \frac{1}{4}; e^{\frac{t}{2}}}x(t) = f(t, x(t), \mathcal{I}_{0^+}^{\frac{1}{3}; e^{\frac{t}{2}}}x(t)), \quad t\in (0, 1], \\[0.4cm] x(0) = 0, \, \, \, \, \sum\limits_{i = 1}^{3}{\left(\frac{-i}{i+5}\right)^{i+1}x\left({\frac{i}{3}}\right)} + \sum\limits_{j = 1}^{2}\left(\frac{j+1}{j+2}\right)\mathcal{I}_{{{0}^{+}}}^{{\frac{j}{3}};e^{\frac{t}{2}}}x\left({\frac{j}{2}}\right) + \sum\limits_{k = 1}^{4}{{\left(\frac{-k}{k+2}\right)^{k}}{}^{H}\mathfrak{D}_{{{0}^{+}}}^{{\frac{k+8}{8}}, {{\frac{1}{4}}};e^{\frac{t}{2}} }x\left({\frac{k}{4}}\right)} = \frac{1}{2}. \end{array} \right. \end{equation} (5.1)

    Here \alpha = 8/5 , \rho = 1/4 , \phi = 1/3 , T = 1 , \kappa = 1/2 , m = 3 , n = 2 , r = 4 , \delta_{i} = ((-i)/(i+5))^{(i+1)} , \omega_{j} = (j+1)/(j+2) , \lambda_{k} = ((-k)/(k+2))^k , \eta_i = i/3 , \theta_j = j/2 , \xi_k = k/4 , \beta_j = j/3 , \mu_k = (k+8)/8 for i = 1, 2, 3 , j = 1, 2 and k = 1, 2, 3, 4 . From the given all data, we obtain that \Omega \approx 0.5377547471 \neq 0 , \Lambda_{0} \approx 1.96941831 , \Lambda_{1} \approx 2.131548185 and \Lambda_{2} \approx 6.661728461 .

    (I) Consider the function

    \begin{equation} f(t, x(t), \mathcal{I}_{0^+}^{\frac{1}{3}; e^{\frac{t}{2}} }x(t)) : = \frac{t^2+1}{(3-\sin^2\pi t)^2}\cdot\frac{\vert x(t) \vert}{2+\vert x(t) \vert} + (2t-1)\cdot\frac{\vert \mathcal{I}_{{{0}^{+}}}^{\frac{1}{3}; e^{\frac{t}{2}}}x(t) \vert}{9+\vert \mathcal{I}_{{{0}^{+}}}^{\frac{1}{3}; e^{\frac{t}{2}}}x(t) \vert}. \end{equation} (5.2)

    For x_1 , x_2 , y_1 , y_2\in \mathbb{R} and t\in [0, 1] , we have

    \begin{equation*} \vert f(t, x_1, y_1)-f(t, x_2, y_2) \vert \leq \frac{1}{9}\left(\left\vert x_1-x_2\right\vert+\vert y_1-y_2 \vert\right). \end{equation*}

    The assumptions (H_1) is satisfied with L_{1} = 1/9 . Hence

    \begin{equation*} \Lambda_{0}\Lambda_{1} L_{1} \approx 0.4664344471 < 1. \end{equation*}

    Since, all the assumptions of Theorem 3.1 are satisfied, then the problem (5.1) has a unique solution on [0, 1] . Further, we can also compute that

    \begin{equation*} M_{f} = \frac{\Lambda_{1}}{1 - \Lambda_0 A(T, \alpha) L_1 } \approx 2.30834181 > 1. \end{equation*}

    Therefore, by Theorem 4.9, the problem (5.1) is both \mathbb{UH} and generalized \mathbb{UH} stable on [0.1] . In addition, by setting \Theta(t) = \psi(t) - \psi(0) with Proposition 2.5 (i), it is easy to calculate that

    \begin{equation*} \mathcal{I}_{{0}^{+}}^{\alpha ;\psi }\Theta(t) = \frac{1}{\Gamma(\frac{7}{2})}(\psi(t) - \psi(0))^{\frac{5}{2}} \Theta(t) \leq \frac{4(e^{0.5} - 1)^{\frac{5}{2}} }{15\sqrt{\pi}}\Theta(t). \end{equation*}

    Thus, the inequality (4.10) is satisfied with n_{\Theta} = \frac{4(e^{0.5} - 1)^{\frac{5}{2}} }{15\sqrt{\pi}} > 0 . It follows that

    \begin{equation*} M_{f, \Theta} = \frac{\Lambda_{2}n_{\Theta} }{1 - \Lambda_0 A(T, \alpha) L_1 } \approx 0.3679010534 > 0. \end{equation*}

    Hence, by Theorem 4.11 , the problem (5.1), with f given by (5.2), is both \mathbb{UHR} and also generalized \mathbb{UHR} stable on [0.1] .

    (II) Consider the function

    \begin{equation} f(t, x(t), \mathcal{I}_{0^+}^{\frac{1}{3}; e^{\frac{t}{2}} }x(t)) : = e^{-t} + \frac{\tan^{-1}\vert x(t) \vert}{4+t} + \frac{2\sin\vert x(t) \vert}{4+t}\cdot\frac{\vert \mathcal{I}_{0^{+}}^{\frac{1}{3}; e^{\frac{t}{2}}} x(t) \vert}{2+\vert \mathcal{I}_{0^{+}}^{\frac{1}{3}; e^{\frac{t}{2}}} x(t) \vert}. \end{equation} (5.3)

    For x_1 , x_2 , y_1 , y_2\in \mathbb{R} and t\in [0, 1] , we have

    \begin{equation*} \vert f(t, x_1, y_1)-f(t, x_2, y_2) \vert \leq \frac{1}{4+t}\left(\left\vert x_1-x_2\right\vert + \vert y_1-y_2 \vert\right) \leq \frac{1}{4}\left(\left\vert x_1-x_2\right\vert + \vert y_1-y_2 \vert\right). \end{equation*}

    This means that the assumption (H_1) is satisfied with L_{1} = 1/4 . We obtain

    \begin{equation*} L_{1}\Lambda_{0}\left(\Lambda_{1} - A(T, \alpha)\right) \approx 0.8771522228 < 1, \end{equation*}

    and

    \begin{equation*} \vert f(t, x, y)\vert \leq e^{-t} + \frac{1}{4+t}\left(\frac{\pi}{2} + 1\right), \end{equation*}

    which satisfy (3.7) and (H_2) , respectively. Using the Theorem 3.2, the problem (5.1), with f given by (5.3), has at least one solution on [0, 1]

    (III) Consider the function

    \begin{equation} f(t, x(t), \mathcal{I}_{0^+}^{\frac{1}{3}; e^{\frac{t}{2}} }x(t)) : = \frac{e^{-t}}{(4+t)^2}\left( \frac{\vert x^{5}(t) \vert}{1 + x^{4}(t)} + \frac{ \mathcal{I}_{0^{+}}^{\frac{1}{3}; e^{\frac{t}{2}}} x^{6}(t)}{1+\vert \mathcal{I}_{0^{+}}^{\frac{1}{3}; e^{\frac{t}{2}}} x^{5}(t)\vert} + 1\right). \end{equation} (5.4)

    Also, the nonlinear function can be expressed as

    \begin{equation*} \vert f(t, x, y)\vert \leq \frac{e^{-t}}{(4+t)^2}\left(\vert x \vert + \vert y \vert + 1\right). \end{equation*}

    By (H_3) , we set q(t) = e^{-t}/(4+t)^2 and \Phi(u) = u + 1 , then \Vert q \Vert = 1/16 and \Phi(|x| + |y|) = |x| + |y| + 1. Thus, we can compute that there exists a constant M_{2} > 1.527092217 satisfying inequality in (H_4) . Therefore, all conditions in Theorem 3.3 are fulfilled. Thus the problem (5.1) with f given by (5.4) has at least one solution on [0, 1] .

    This paper discussed a new class of \psi -Hilfer fractional integro-differential equation supplemented with mixed nonlocal boundary condition which is a combination of multi-point, fractional derivative multi-order and fractional integral multi-order boundary conditions. Existence and uniqueness results are established. The uniqueness result is proved by applying the Banach's fixed point theorem, while the existence results are investigated via Krasnosel'ski{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}'s fixed point theorem and Larey-Schauder nonlinear alternative. Our results are not only new in the given setting but also provide some new special cases by fixing the parameters involved in the problem at hand. For instance, by fixing \omega_{j} = 0, \lambda_k = 0 for all j = 1, 2, \ldots, n, \; k = 1, 2, \ldots, r our results correspond to the ones for boundary value problems for \psi -Hilfer nonlinear fractional integro-differential equations supplemented with multi-point boundary conditions. In case we take \delta_{i} = 0, \lambda_k = 0 for all i = 1, 2, \ldots, m, \; k = 1, 2, \ldots, r we obtain the results for boundary value problems for \psi -Hilfer nonlinear fractional integro-differential equations equipped with multi-term integral boundary conditions. Further, we studied different kinds of Ulam's stability such as \mathbb{UH} , generalized \mathbb{UH} , \mathbb{UHR} and generalized \mathbb{UHR} stability. In the end, we present examples to demonstrate the consistency to the theoretical findings.

    The work accomplished in this paper is new and enrich the literature on boundary value problems for nonlinear \psi -Hilfer fractional differential equations.

    The first author would like to thank King Mongkut's University of Technology North Bangkok and the Center of Excellence in Mathematics (CEM), CHE, Sri Ayutthaya Rd., Bangkok, 10400, Thailand for support this work. The second author would like to thank for funding this work through the Center of Excellence in Mathematics (CEM), CHE, Sri Ayutthaya Rd., Bangkok, 10400, Thailand and Barapha University.

    On behalf of all authors, the corresponding author states that there is no conflict of interest.



    [1] H. Zhang, X. Yang, Q. Tang, D. Xu, A robust error analysis of the OSC method for a multi-term fourth-order sub-diffusion equation, Comput. Math. Appl., 109 (2022), 180–190. https://doi.org/10.1016/j.camwa.2022.01.007 doi: 10.1016/j.camwa.2022.01.007
    [2] X. Yang, H. Zhang, J. Tang, The OSC solver for the fourth-order sub-diffusion equation with weakly singular solutions, Comput. Math. Appl., 82 (2021), 1–12. https://doi.org/10.1016/j.camwa.2020.11.015 doi: 10.1016/j.camwa.2020.11.015
    [3] H. Zhang, X. Yang, D. Xu, Unconditional convergence of linearized orthogonal spline collocation algorithm for semilinear subdiffusion equation with nonsmooth solution, Numer. Meth. Part. Differ. Equ., 37 (2021), 1361–1373. https://doi.org/10.1002/num.22583 doi: 10.1002/num.22583
    [4] A. F. Daghistani, A. M. T. Abd El-Bar, A. M. Gemeay, M. A. E. Abdelrahman, S. Z. Hassan, A hyperbolic secant-squared distribution via the nonlinear evolution equation and its application, Mathematics, 11 (2023), 4270. https://doi.org/10.3390/math11204270 doi: 10.3390/math11204270
    [5] M. A. E. Abdelrahman, G. Alshreef, Closed-form solutions to the new coupled Konno–Oono equation and the Kaup-Newell model equation in magnetic field with novel statistic application, Eur. Phys. J. Plus, 136 (2021), 455. https://doi.org/10.1140/epjp/s13360-021-01472-2 doi: 10.1140/epjp/s13360-021-01472-2
    [6] Y. Cheng, A. Chertock, M. Herty, A. Kurganov, T. Wu, A new approach for designing moving-water equilibria preserving schemes for the shallow water equations, J. Sci. Comput., 80 (2019), 538–554. https://doi.org/10.1007/s10915-019-00947-w doi: 10.1007/s10915-019-00947-w
    [7] P. Ripa, Conservation laws for primitive equations models with inhomogeneous layers, Geophys. Astrophys. Fluid Dynam., 70 (1993), 85–111. https://doi.org/10.1080/03091929308203588 doi: 10.1080/03091929308203588
    [8] G. Laibe, D. J. Price, Dusty gas with one fluid, Mon. Not. R. Astron. Soc., 440 (2014), 2136–2146. https://doi.org/10.1093/mnras/stu355 doi: 10.1093/mnras/stu355
    [9] Y. Shi, X. Yang, A time two-grid difference method for nonlinear generalized viscous Burgers' equation, J. Math. Chem., 62 (2024), 1323–1356. https://doi.org/10.1007/s10910-024-01592-x doi: 10.1007/s10910-024-01592-x
    [10] C. Li, H. Zhang, X. Yang, A new nonlinear compact difference scheme for a fourth-order nonlinear Burgers type equation with a weakly singular kernel, J. Appl. Math. Comput., 70 (2024), 2045–2077. https://doi.org/10.1007/s12190-024-02039-x doi: 10.1007/s12190-024-02039-x
    [11] H. Zhang, X. Yang, Y. Liu, Y. Liu, An extrapolated CN-WSGD OSC method for a nonlinear time fractional reaction-diffusion equation, Appl. Numer. Math., 157 (2020), 619–633. https://doi.org/10.1016/j.apnum.2020.07.017 doi: 10.1016/j.apnum.2020.07.017
    [12] H. Zhang, X. Yang, D. Xu, An efficient spline collocation method for a nonlinear fourth-order reaction subdiffusion equation, J. Sci. Comput., 85 (2020), 7. https://doi.org/10.1007/s10915-020-01308-8 doi: 10.1007/s10915-020-01308-8
    [13] X. Yang, H. Zhang, Q. Tang, A spline collocation method for a fractional mobile–immobile equation with variable coefficients, Comput. Appl. Math., 39 (2020), 34. https://doi.org/10.1007/s40314-019-1013-3 doi: 10.1007/s40314-019-1013-3
    [14] H.S. Alayachi, The modulations of higher order solitonic pressure and energy of fluid filled elastic tubes, AIP Adv., 13 (2023), 115214. https://doi.org/10.1063/5.0179155 doi: 10.1063/5.0179155
    [15] X. Yang, W. Qiu, H. Zhang, L. Tang, An efficient alternating direction implicit finite difference scheme for the three-dimensional time-fractional telegraph equation, Comput. Math. Appl., 102 (2021), 233–247. https://doi.org/10.1016/j.camwa.2021.10.021 doi: 10.1016/j.camwa.2021.10.021
    [16] H. Zhang, Y. Liu, X. Yang, An efficient ADI difference scheme for the nonlocal evolution problem in three-dimensional space, J. Appl. Math. Comput., 69 (2023), 651–674. https://doi.org/10.1007/s12190-022-01760-9 doi: 10.1007/s12190-022-01760-9
    [17] X. Yang, W. Qiu, H. Chen, H. Zhang, Second-order BDF ADI Galerkin finite element method for the evolutionary equation with a nonlocal term in three-dimensional space, Appl. Numer. Math., 172 (2022), 497–513. https://doi.org/10.1016/j.apnum.2021.11.004 doi: 10.1016/j.apnum.2021.11.004
    [18] H. Zhang, X. Jiang, F. Wang, X. Yang, The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation, J. Appl. Math. Comput., 70 (2024), 1127–1151. https://doi.org/10.1007/s12190-024-02000-y doi: 10.1007/s12190-024-02000-y
    [19] H. G. Abdelwahed, M. A. E. Abdelrahman, M. Inc, R. Sabry, New soliton applications in earth's magnetotail plasma at critical densities, Front. Phys., 8 (2020), 181. https://doi.org/10.3389/fphy.2020.00181 doi: 10.3389/fphy.2020.00181
    [20] S. Zhang, C. Tian, W. Y. Qian, Bilinearization and new multi-soliton solutions for the (4+1)-dimensional Fokas equation, Pramana-J. Phys., 86 (2016), 1259–1267. https://doi.org/10.1007/s12043-015-1173-7 doi: 10.1007/s12043-015-1173-7
    [21] L. Akinyemi, M. Şenol, U. Akpan, K. Oluwasegun, The optical soliton solutions of generalized coupled nonlinear Schrödinger-Korteweg-de Vries equations, Opt. Quant. Electron., 53 (2021), 394. https://doi.org/10.1007/s11082-021-03030-7 doi: 10.1007/s11082-021-03030-7
    [22] F. Mirzaee, S. Rezaei, N. Samadyar, Numerical solution of two-dimensional stochastic time-fractional sine-Gordon equation on non-rectangular domains using finite difference and meshfree methods, Eng. Anal. Bound. Elem., 127 (2021), 53–63. https://doi.org/10.1016/j.enganabound.2021.03.009 doi: 10.1016/j.enganabound.2021.03.009
    [23] M. A. E. Abdelrahman, H. AlKhidhr, A robust and accurate solver for some nonlinear partial differential equations and tow applications, Phys. Scr., 95 (2020), 065212. https://doi.org/10.1088/1402-4896/ab80e7 doi: 10.1088/1402-4896/ab80e7
    [24] Z. Zhou, H. Zhang, X. Yang, CN ADI fast algorithm on non-uniform meshes for the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamics, Appl. Math. Comput., 474 (2024), 128680. https://doi.org/10.1016/j.amc.2024.128680 doi: 10.1016/j.amc.2024.128680
    [25] X. F. Yang, Z. C. Deng, Y. Wei, A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application, Adv. Differ. Equ., 2015 (2015), 117. https://doi.org/10.1186/s13662-015-0452-4 doi: 10.1186/s13662-015-0452-4
    [26] W. Wang, H. Zhang, Z. Zhou, X. Yang, A fast compact finite difference scheme for the fourth-order diffusion-wave equation, Int. J. Comput. Math., 101 (2024), 170–193. https://doi.org/10.1080/00207160.2024.2323985 doi: 10.1080/00207160.2024.2323985
    [27] B. Q. Li, Y. L. Ma, Interaction properties between rogue wave and breathers to the manakov system arising from stationary self-focusing electromagnetic systems, Chaos Soliton. Fract., 156 (2022), 111832. https://doi.org/10.1016/j.chaos.2022.111832 doi: 10.1016/j.chaos.2022.111832
    [28] X. Jin, J. Jiang, J. Chi, X. Wu, Adaptive finite-time pinned and regulation synchronization of disturbed complex networks, Commun. Nonlinear Sci., 124 (2023), 107319. https://doi.org/10.1016/j.cnsns.2023.107319 doi: 10.1016/j.cnsns.2023.107319
    [29] Z. J. Yang, S. M. Zhang, X. L. Li, Z. G. Pang, H. X. Bu, High-order revivable complex-valued hyperbolic-sine-Gaussian solitons and breathers in nonlinear media with a spatial nonlocality, Nonlinear Dyn., 94 (2018), 2563–2573. https://doi.org/10.1007/s11071-018-4510-9 doi: 10.1007/s11071-018-4510-9
    [30] Z. Sun, J. Li, R. Bian, D. Deng, Z. Yang, Transmission mode transformation of rotating controllable beams induced by the cross phase, Opt. Express, 32 (2024), 9201–9212. https://doi.org/10.1364/OE.520342 doi: 10.1364/OE.520342
    [31] M. A. E. Abdelrahman, N. F. Abdo, On the nonlinear new wave solutions in unstable dispersive environments, Phys. Scr., 95 (2020), 045220. https://doi.org/10.1088/1402-4896/ab62d7 doi: 10.1088/1402-4896/ab62d7
    [32] H. G. Abdelwahed, M. A. E. Abdelrahman, S. Alghanim, N. F. Abdo, Higher-order Kerr nonlinear and dispersion effects on fiber optics, Results Phys., 26 (2021), 104268. https://doi.org/10.1016/j.rinp.2021.104268 doi: 10.1016/j.rinp.2021.104268
    [33] J. L. Lebowitz, H. A. Rose, E. R. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Stat. Phys., 50 (1988), 657–687. https://doi.org/10.1007/BF01026495 doi: 10.1007/BF01026495
    [34] G. D. McDonald, C. C. N. Kuhn, K. S. Hardman, S. Bennetts, P. J. Everitt, P. A. Altin, et al., Bright solitonic matter-wave interferometer, Phys. Rev. Lett., 113 (2014), 013002. https://doi.org/10.1103/PhysRevLett.113.013002 doi: 10.1103/PhysRevLett.113.013002
    [35] Y. L. Ma, N th-order rogue wave solutions for a variable coefficient Schrödinger equation in inhomogeneous optical fibers, Optik, 251 (2022), 168103. https://doi.org/10.1016/j.ijleo.2021.168103 doi: 10.1016/j.ijleo.2021.168103
    [36] B. Q. Li, Y. L. Ma, Interaction properties between rogue wave and breathers to the manakov system arising from stationary self-focusing electromagnetic systems, Chaos Soliton. Fract., 156 (2022), 111832. https://doi.org/10.1016/j.chaos.2022.111832 doi: 10.1016/j.chaos.2022.111832
    [37] O. V. Marchukov, B. A. Malomed, V. A. Yurovsky, M. Olshanii, V. Dunjko, R. G. Hulet, Splitting of nonlinear-Schrödinger-equation breathers by linear and nonlinear localized potentials, Phys. Rev. A, 99 (2019), 063623. https://doi.org/10.1103/PhysRevA.99.063623 doi: 10.1103/PhysRevA.99.063623
    [38] S. Shen, Z. Yang, X. Li, S. Zhang, Periodic propagation of complex-valued hyperbolic-cosine-Gaussian solitons and breathers with complicated light field structure in strongly nonlocal nonlinear media, Commun. Nonlinear Sci., 103 (2021), 106005. https://doi.org/10.1016/j.cnsns.2021.106005 doi: 10.1016/j.cnsns.2021.106005
    [39] Z. Y. Sun, D. Deng, Z. G. Pang, Z. J. Yang, Nonlinear transmission dynamics of mutual transformation between array modes and hollow modes in elliptical sine-Gaussian cross-phase beams, Chaos Soliton. Fract., 178 (2024), 114398. https://doi.org/10.1016/j.chaos.2023.114398 doi: 10.1016/j.chaos.2023.114398
    [40] S. Shen, Z. J. Yang, Z. G. Pang, Y. R. Ge, The complex-valued astigmatic cosine-Gaussian soliton solution of the nonlocal nonlinear Schrödinger equation and its transmission characteristics, Appl. Math. Lett., 125 (2022), 107755. https://doi.org/10.1016/j.aml.2021.107755 doi: 10.1016/j.aml.2021.107755
    [41] L. M. Song, Z. J. Yang, X. L. Li, S. M. Zhang, Coherent superposition propagation of Laguerre-Gaussian and Hermite-Gaussian solitons, Appl. Math. Lett., 102 (2020), 106114. https://doi.org/10.1016/j.aml.2019.106114 doi: 10.1016/j.aml.2019.106114
    [42] M. Najafi, S. Arbabi, Traveling wave solutions for nonlinear Schrödinger equations, Optik, 126 (2015), 3992–3997. https://doi.org/10.1016/j.ijleo.2015.07.165 doi: 10.1016/j.ijleo.2015.07.165
    [43] M. Dehghan, A. Shokri, A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions, Comput. Math. Appl., 54 (2007), 136–146. https://doi.org/10.1016/j.camwa.2007.01.038 doi: 10.1016/j.camwa.2007.01.038
    [44] S. V. Mousavi, S. Miret-Artés, On non-linear Schrödinger equations for open quantum systems, Eur. Phys. J. Plus, 134 (2019), 431. https://doi.org/10.1140/epjp/i2019-12965-6 doi: 10.1140/epjp/i2019-12965-6
    [45] W. Huang, C. Xu, S. T. Chu, S. K. Chaudhuri, The finite-difference vector beam propagation method: Analysis and assessment, J. Lightwave Technol., 10 (1992), 295–305. https://doi.org/10.1109/50.124490 doi: 10.1109/50.124490
    [46] A. I. Aliyu, M. Inc, A. Yusuf, D. Baleanu, Optical solitary waves and conservation laws to the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation, Mod. Phys. Lett. B, 32 (2018), 1850373. https://doi.org/10.1142/S0217984918503736 doi: 10.1142/S0217984918503736
    [47] H. Durur, E. Ilhan, H. Bulut, Novel complex wave solutions of the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation, Fractal Fract., 4 (2020), 41. https://doi.org/10.3390/fractalfract4030041 doi: 10.3390/fractalfract4030041
    [48] D. Baleanu, K. Hosseini, S. Salahshour, K. Sadri, M. Mirzazadeh, C. Park, A. Ahmadian, The (2+1)-dimensional hyperbolic nonlinear Schrödinger equation and its optical solitons, AIMS Mathematics, 6 (2021), 9568–9581. https://doi.org/10.3934/math.2021556 doi: 10.3934/math.2021556
    [49] G. Ai-Lin, L. Ji, Exact solutions of (2+1)-dimensional HNLS equation, Commun. Theor. Phys., 54 (2010), 401. https://doi.org/10.1088/0253-6102/54/3/04 doi: 10.1088/0253-6102/54/3/04
    [50] X. Yang, H. Zhang, The uniform l^{1} long-time behavior of time discretization for time-fractional partial differential equations with nonsmooth data, Appl. Math. Lett., 124 (2022), 107644. https://doi.org/10.1016/j.aml.2021.107644 doi: 10.1016/j.aml.2021.107644
    [51] X. Yang, H. Zhang, Q. Zhang, G. Yuan, Simple positivity-preserving nonlinear finite volume scheme for subdiffusion equations on general non-conforming distorted meshes, Nonlinear Dyn., 108 (2022), 3859–3886. https://doi.org/10.1007/s11071-022-07399-2 doi: 10.1007/s11071-022-07399-2
    [52] X. Yang, Z. Zhang, Analysis of a new NFV scheme preserving DMP for two-dimensional sub-diffusion equation on distorted meshes, J. Sci. Comput., 99 (2024), 80. https://doi.org/10.1007/s10915-024-02511-7 doi: 10.1007/s10915-024-02511-7
    [53] X. Yang, Z. Zhang, On conservative, positivity preserving, nonlinear FV scheme on distorted meshes for the multi-term nonlocal Nagumo-type equations, Appl. Math. Lett., 150 (2024), 108972. https://doi.org/10.1016/j.aml.2023.108972 doi: 10.1016/j.aml.2023.108972
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