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<2,0≤β≤1, | (1.1) |
x(a)=0,x(b)=m∑i=1δiIφix(ξi),φi>0,δi∈R,ξi∈[a,b], | (1.2) |
where HDα,β is the Hilfer fractional derivative of order α, 1<α<2 and parameter β, 0≤β≤1, Iφi is the Riemann-Liouville fractional integral of order φi>0, ξi∈[a,b], a≥0 and δi∈R. 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++k1HDα1−1,β11+)u(t)=f(t,u(t),v(t)), 1<α1≤2, t∈[1,e],(HDα2,β21++k2HDα2−1,β21+)v(t)=g(t,u(t),v(t)), 1<α2≤2, 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,A2∈R+ and f, g:[1,e]×R2→R 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+kHDα−1,β;ψ)x(t)=f(t,x(t)),t∈[a,b], | (1.5) |
x(a)=0,x(b)=m∑i=1λix(θi), | (1.6) |
where HDα,β;ψ is the ψ-Hilfer fractional derivative of order α, 1<α<2 and parameter β, 0≤β≤1, f:[a,b]×R→R is a continuous function, a<b, k,λi∈R,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,m∑i=1δix(ηi)+n∑j=1ωjIβj;ψ0+x(θj)+r∑k=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<α≤2, 0≤ρ≤1, Iv;ψ0+ is ψ-Riemann-Liouville fractional integral of order v={ϕ,βj}, ϕ,βj>0 for j=1,2,…,n, κ,δi,ωj,λk∈R are given constants, the points ηi,θj,ξk∈J, i=1,2,…,m, j=1,2,…,n, k=1,2,…,r and f:J×R2→R 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‖=supt∈J{|f(t)|}. |
On the order hand, we have n-times absolutely continuous functions given by
ACn(J,R)={f:J→R;f(n−1)∈AC(J,R)}. |
Definition 2.1. [2] Let (a,b), (−∞≤a<b≤∞), be a finite or infinite interval of the half-axis R+ and α∈R+. 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))α−1f(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, n∈N. 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)nIn−α;ψa+f(t)=1Γ(n−α)(1ψ′(t)ddt)n∫taψ′(s)(ψ(t)−ψ(s))n−α−1f(s)ds, | (2.2) |
where n=[α]+1, [α] is represent the integer part of the real number α.
Definition 2.3. [18] Let n−1<α<n with n∈N, [a,b] is the interval such that −∞≤a<b≤∞ and f,ψ∈Cn([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 0≤ρ≤1, is defined by
HDα,ρ;ψa+f(t)=Iρ(n−α);ψa+(1ψ′(t)ddt)nI(1−ρ)(n−α);ψa+f(t)=Iγ−α;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(n−α).
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 α≥0, 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))υ−1(t)=Γ(υ)Γ(υ+α)(ψ(t)−ψ(a))υ+α−1.
(ii) Dα,ρ;ψa+(ψ(s)−ψ(a))υ−1(t)=Γ(υ)Γ(υ−α)(ψ(t)−ψ(a))υ−α−1.
(iii) HDα,ρ;ψa+(ψ(s)−ψ(a))υ−1(t)=Γ(υ)Γ(υ−α)(ψ(t)−ψ(a))υ−α−1,υ>γ=α+ρ(2−α).
Lemma 2.6. Let m−1<α<m, n−1<β<n, n,m∈N, n≤m, 0≤ρ≤1 and α≥β+ρ(n−β). If h∈Cn(J,R), then
HDβ,ρ;ψa+Iα;ψa+h(t)=Iα−β;ψa+h(t). | (2.5) |
Proof. Let λ=β+ρ(n−β) with n−1<λ<n, we get
HDβ,ρ;ψa+(Iα;ψa+h(t))=Iλ−β;ψa+Dλ;ψa+(Iα;ψa+h(t))=Iλ−β;ψa+(1ψ′(t)ddt)nIn−λ;ψa+(Iα;ψa+h(t))=Iλ−β;ψa+(1ψ′(t)ddt)nIn−λ+α;ψa+h(t). |
By using Definition 2.1, we obtain
(1ψ′(t)ddt)In−λ+α;ψa+h(t)=1ψ′(t)ddt(1Γ(n−λ+α)∫taψ′(τ)(ψ(t)−ψ(τ))n+α−λ−1h(τ)dτ)=1Γ(n−λ+α)1ψ′(t)(∫ta(n+α−λ−1)ψ′(τ)ψ′(t)(ψ(t)−ψ(τ))n+α−λ−2h(τ)dτ)=1Γ(n−λ+α−1)∫taψ′(τ)(ψ(t)−ψ(τ))n+α−λ−2h(τ)dτ=In−λ+α−1;ψa+h(t), |
and
(1ψ′(t)ddt)2In−λ+α;ψa+h(t)=1ψ′(t)ddt(1Γ(n−λ+α−1)∫taψ′(τ)(ψ(t)−ψ(τ))n+α−λ−2h(τ)dτ)=1Γ(n−λ+α−1)1ψ′(t)(∫ta(n+α−λ−2)ψ′(τ)ψ′(t)(ψ(t)−ψ(τ))n+α−λ−3h(τ)dτ)=1Γ(n−λ+α−2)∫taψ′(τ)(ψ(t)−ψ(τ))n+α−λ−3h(τ)dτ=In−λ+α−2;ψa+h(t). |
Repeat the above process, we have
(1ψ′(t)ddt)nIn−λ+α;ψa+h(t)=1ψ′(t)ddt(1Γ(α−λ)∫taψ′(τ)(ψ(t)−ψ(τ))α−λ−1h(τ)dτ)=1Γ(α−λ+1)1ψ′(t)(∫ta(α−λ)ψ′(τ)ψ′(t)(ψ(t)−ψ(τ))α−λ−1h(τ)dτ)=1Γ(λ+α)∫taψ′(τ)(ψ(t)−ψ(τ))α−λ−1h(τ)dτ=Iα−λ;ψa+h(t), |
which implies that
HDβ,ρ;ψa+(Iα;ψa+h(t))=Iλ−β;ψa+Iα−λ;ψa+h(t)=Iα−β;ψa+h(t). |
This completes the proof.
Lemma 2.7. [18] If f∈Cn(J,R), n−1<α<n, 0≤ρ≤1 and γ=α+ρ(n−α) then
Iα;ψa+HDα,ρ;ψa+f(t)=f(t)−n∑k=1(ψ(t)−ψ(a))γ−kΓ(γ−k+1)f[n−k]ψI(1−ρ)(n−α);ψa+f(a), | (2.6) |
for all t∈J, 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'ski⌣i'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+By∈M whenever x, y∈M; (ii) A is compact and continuous; (iii) B is contraction mapping. Then there exists z∈M 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 0∈U. Suppose that D:¯U→C 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 x∈∂U (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<α≤2, 0≤ρ≤1, γ=α+ρ(2−α), k=1,2,…,r and Ω≠0. Suppose that h∈C. Then x∈C2 is a solution of the problem
{HDα,ρ;ψ0+x(t)=h(t),t∈(0,T],x(0)=0,m∑i=1δix(ηi)+n∑j=1ωjIβj;ψ0+x(θj)+r∑k=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))γ−1ΩΓ(γ)[κ−(m∑i=1δiIα;ψ0+h(ηi)+n∑j=1ωjIα+βj;ψ0+h(s)(θj)+r∑k=1λkIα−μk;ψ0+h(s)(ξk))], | (2.8) |
where
Ω=m∑i=1δi(ψ(ηi)−ψ(0))γ−1Γ(γ)+n∑j=1ωj(ψ(θj)−ψ(0))γ+βj−1Γ(γ+βj)+r∑k=1λk(ψ(ξk)−ψ(0))γ−μk−1Γ(γ−μk). | (2.9) |
Proof. Let x∈C 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))γ−1Γ(γ)c1+(ψ(t)−ψ(0))γ−2Γ(γ−1)c2, | (2.10) |
where c1,c2∈R are arbitrary constants.
For t=0, we get c2=0, and thus
x(t)=Iα;ψ0+h(t)+(ψ(t)−ψ(0))γ−1Γ(γ)c1. | (2.11) |
Taking the operators HDμk,ρ;ψ0+ and Iβj;ψ0+ into (2.10), we obtain
HDμk,ρ;ψ0+x(t)=Iα−μk;ψ0+h(t)+(ψ(t)−ψ(0))γ−μk−1Γ(γ−μk)c1,Iβj;ψ0+x(t)=Iα+βj;ψ0+h(t)+(ψ(t)−ψ(0))γ+βj−1Γ(γ+βj)c1. |
Applying the second boundary condition in (1.7), we have
c1[m∑i=1δi(ψ(ηi)−ψ(0))γ−1Γ(γ)+n∑j=1ωj(ψ(θj)−ψ(0))γ+βj−1Γ(γ+βj)+r∑k=1λk(ψ(ξk)−ψ(0))γ−μk−1Γ(γ−μk)]+m∑i=1δiIα;ψ0+h(ηi)+n∑j=1ωjIα+βj;ψ0+h(θj)+r∑k=1λkIα−μk;ψ0+h(ξk)=κ, |
from which we get
c1=1Ω[κ−(m∑i=1δiIα;ψ0+h(ηi)+n∑j=1ωjIα+βj;ψ0+h(θj)+r∑k=1λkIα−μk;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,γ−1)|Ω|(m∑i=1|δi|A(ηi,α)+n∑j=1|ωj|A(θj,α+βj)+r∑k=1|λk|A(ξk,α−μk)). | (3.3) |
In view of Lemma 2.11, an operator Q:C→C is defined by
(Qx)(t)=Iα;ψ0+Fx(s)(t)+A(t,γ−1)Ω[κ−(m∑i=1δiIα;ψ0+Fx(s)(ηi)+n∑j=1ωjIα+βj;ψ0+Fx(s)(θj)+r∑k=1λkIα−μk;ψ0+Fx(s)(ξk))], | (3.4) |
where
Fx(t)=f(t,x(t),Iϕ;ψ0+x(t)),t∈J. |
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))u−1Fx(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×R2→R is a continuous function such that:
(H1) there exist a constant L1>0 such that
|f(t,u1,v1)−f(t,u2,v2)|≤L1(|u1−u2|+|v1−v2|) |
for any ui, vi∈R, i=1,2 and t∈J.
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 supt∈J|f(t,0,0)|:=M1<∞. Next, we set Br1:={x∈C:‖x‖≤r1} with
r1≥Λ1M1+(|κ|A(T,γ−1))/|Ω|1−Λ0Λ1L1, | (3.6) |
where Ω, A(T,γ−1), 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 QBr1⊂Br1.
For any x∈Br1, we have
|(Qx)(t)|≤Iα;ψ0+|Fx(s)|(T)+A(T,γ−1)|Ω|(|κ|+m∑i=1|δi|Iα;ψ0+|Fx(s)|(ηi)+n∑j=1|ωj|Iα+βj;ψ0+|Fx(s)|(θj)+r∑k=1|λk|Iα−μk;ψ0+|Fx(s)|(ξk)). |
We note that
Iϕ;ψ0+|x(τ)|(s)=1Γ(ϕ)∫s0ψ′(τ)(ψ(s)−ψ(τ))ϕ−1|x(τ)|dτ≤A(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=L1Λ0‖x‖+M1. |
Then we have
|(Qx)(t)|≤(L1Λ0‖x‖+M1)(ψ(T)−ψ(0))αΓ(α+1)+A(T,γ−1)|Ω|[|κ|+(L1Λ0‖x‖+M1)(m∑i=1|δi|(ψ(ηi)−ψ(0))αΓ(α+1)+n∑j=1|ωj|(ψ(θj)−ψ(0))α+βjΓ(α+βj+1)+r∑k=1|λk|(ψ(ξk)−ψ(0))α−μkΓ(α−μk+1))]=L1Λ0[A(T,α)+A(T,γ−1)|Ω|(m∑i=1|δi|A(ηi,α)+n∑j=1|ωj|A(θj,α+βj)+r∑k=1|λk|A(ξk,α−μk))]‖x‖+[A(T,α)+A(T,γ−1)|Ω|(m∑i=1|δi|A(ηi,α)+n∑j=1|ωj|A(θj,α+βj)+r∑k=1|λk|A(ξk,α−μk))]M1+|κ|A(T,γ−1)|Ω|≤Λ0Λ1L1r1+Λ1M1+|κ|A(T,γ−1)|Ω|≤r1, |
which implies that QBr1⊂Br1.
Step II. We show that Q:C→C is a contraction.
For any x, y∈C and for each t∈J, we have
|(Qx)(t)−(Qy)(t)|≤Iα;ψ0+|Fx(s)−Fy(s)|(T)+A(T,γ−1)|Ω|(m∑i=1|δi|Iα;ψ0+|Fx(s)−Fy(s)|(ηi)+n∑j=1|ωj|Iα+βj;ψ0+|Fx(s)−Fy(s)|(θj)+r∑k=1|λk|Iα−μk;ψ0+|Fx(s)−Fy(s)|(ξk))≤{(ψ(T)−ψ(0))αΓ(α+1)+A(T,γ−1)|Ω|(m∑i=1|δi|(ψ(ηi)−ψ(0))αΓ(α+1)+n∑j=1|ωj|(ψ(θj)−ψ(0))α+βjΓ(α+βj+1)+r∑k=1|λk|(ψ(ξk)−ψ(0))α−μkΓ(α−μk+1))}L1Λ0‖x−y‖={A(T,α)+A(T,γ−1)|Ω|(m∑i=1|δi|A(ηi,α)+n∑j=1|ωj|A(θj,α+βj)+r∑k=1|λk|A(ξk,α−μk))}L1Λ0‖x−y‖=Λ0Λ1L1‖x−y‖, |
which implies that ‖Qx−Qy‖≤Λ0Λ1L1‖x−y‖. 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'ski⌣i's fixed point theorem.
Theorem 3.2. Assume that f:J×R2→R is a continuous function satisfying (H1). In addition, we assume that:
(H2) |f(t,u,v)|≤σ(t), ∀(t,u,v)∈J×R2, and σ∈C(J,R+).
If
L1Λ0[Λ1−A(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 supt∈J|σ(t)|=‖σ‖ and Br2:={x∈C:‖x‖≤r2}, where
r2≥‖σ‖Λ1+|κ|A(T,γ−1)|Ω|. |
We define the operators Q1 and Q2 on Br2 by
(Q1x)(t)=Iα;ψ0+Fx(s)(t),t∈J,(Q2x)(t)=A(t,γ−1)Ω[κ−(m∑i=1δiIα;ψ0+Fx(s)(ηi)+n∑j=1ωjIα+βj;ψ0+Fx(s)(θj)+r∑k=1λkIα−μk;ψ0+Fx(s)(ξk))],t∈J. |
Note that Q=Q1+Q2. For any x,y∈Br2, we have
|(Q1x)(t)+(Q2y)(t)|≤supt∈J{Iα;ψ0+|Fx(s)|(t)+A(t,γ−1)|Ω|(|κ|+m∑i=1|δi|Iα;ψ0+|Fy(s)|(ηi)+n∑j=1|ωj|Iα+βj;ψ0+|Fy(s)|(θj)+r∑k=1|λk|Iα−μk;ψ0+|Fy(s)|(ξk))}≤‖σ‖{A(T,α)+A(T,γ−1)|Ω|(m∑i=1|δi|A(ηi,α)+n∑j=1|ωj|A(θj,α+βj)+r∑k=1|λk|A(ξk,α−μk))}+|κ|A(T,γ−1)|Ω|≤‖σ‖Λ1+|κ|A(T,γ−1)|Ω|≤r2. |
This implies that Q1x+Q2x∈Br2, 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 xn→x in C. Then for each t∈J, we have
|(Q1xn)(t)−(Q1x)(t)|≤Iα;ψ0+|Fxn(s)−Fx(s)|(T)≤A(T,α)‖Fxn−Fx‖. |
Since f is continuous, this implies that the operator Fx is also continuous. Hence, we obtain
‖Fxn−Fx‖→0asn→∞. |
Thus, this shows that the operator Q1x is continuous. Also, the set Q1Br2 is uniformly bounded on Br2 as
‖Q1x‖≤A(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,t2∈J with 0≤t1<t2≤T, we obtain
|(Q1x)(t2)−(Q1x)(t1)|=1Γ(α)|∫t10ψ′(s)[(ψ(t2)−ψ(s))α−1−(ψ(t1)−ψ(s))α−1]Fx(s)ds+∫t2t1ψ′(s)(ψ(t2)−ψ(s))α−1Fx(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 t2→t1. 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 q∈C(J,R+) and a continuous nondecreasing function Φ:[0,∞)→[0,∞) which is subhomogeneous (that is, Φ(μx)≤μΦ(x), for all μ≥1 and x∈C), 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,γ−1))/|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={x∈C:‖x‖≤r3} be a bounded ball in C. Then, for t∈J, we obtain
|(Qx)(t)|≤supt∈J{Iα;ψ0+|Fx(s)|(t)+A(t,γ−1)|Ω|(|κ|+m∑i=1|δi|Iα;ψ0+|Fx(s)|(ηi)+n∑j=1|ωj|Iα+βj;ψ0+|Fx(s)|(θj)+r∑k=1|λk|Iα−μk;ψ0+|Fx(s)|(ξk))}≤‖q‖Φ{(1+(ψ(T)−ψ(0))ϕΓ(ϕ+1))‖x‖}{Iα;ψ0+(1)(T)+A(T,γ−1)|Ω|×(m∑i=1|δi|Iα;ψ0+(1)(ηi)+n∑j=1|ωj|Iα+βj;ψ0+(1)(θj)+r∑k=1|λk|Iα−μk;ψ0+(1)(ξk))}+|κ|(ψ(T)−ψ(0))γ−1|Ω|Γ(γ)=‖q‖Φ(Λ0‖x‖){A(T,α)+A(T,γ−1)|Ω|(m∑i=1|δi|A(ηi,α)+n∑j=1|ωj|A(θj,α+βj)+r∑k=1|λk|A(ξk,α−μk))}+|κ|A(T,γ−1)|Ω|≤Λ0Λ1Φ(‖x‖)‖q‖+|κ|A(T,γ−1)|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.
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