# Study of a Class of Generalized Multiterm Fractional Differential Equations with Generalized Fractional Integral Boundary Conditions

Authors: Wafa Shammakh; Hadeel Z. Alzumi; Zahra Albarqi
Journal: Discrete Dynamics in Nature and Society
 (2020)

---

## Abstract

The aim of this work is to study the new generalized fractional differential equations involving generalized multiterms and equipped with multipoint boundary conditions. The nonlinear term is taken from Orlicz space. The existence and uniqueness results, with the help of contemporary tools of fixed point theory, are investigated. The Ulam stability of our problem is also presented. The obtained results are well illustrated by examples.


---

## Body

## 1. Introduction

Fractional differential equations (FDEs) have a considerable interest both in mathematics and in applications. They were used in modeling in various disciplines of engineering, physics, chemical technology, population dynamics, biotechnology, economics, etc. (see, for example, [1–10], and the references cited therein). Recently, some authors explored FDEs with various types of conditions as multipoint, nonlocal, antiperiodic, and integral boundary conditions [11–15].Furthermore, there has been a significant development in fractional derivatives and integrals due to the necessity of a better model for real phenomena. In [16], Katugampola suggested a new fractional integral that combines the Riemann-Liouville and Hadamard integrals into a single integral. In 2015, Caputo and Fabrizio [17] defined the so-called Caputo–Fabrizio fractional derivatives by imposing a nonsingular exponential kernel. Later, the exponential kernel was replaced by the Mittag–Leffler function [18, 19]. Another definition of generalized proportional fractional derivatives-generated Caputo and Riemann–Liouville involving exponential functions in their kernels was introduced by Jarad et al. [20]. Another new type of new fractional derivative can be found in [21]. An application on such derivatives can be seen in [22]. More recently, generalized fractional derivatives that contain kernels depending on the function on the space of absolutely continuous functions were investigated [23]. Following the above points, many authors established various generalized fractional boundary value problems (GFBVPs) (see, for example, [24–27]). It is worth to mention that some authors worked on the existence of solutions and studied the stability analysis for nonlinear singular fractional differential equations. For more details, we refer to [28–30].On the other hand, Birnbaum and Orlicz [31] proposed a more general setting of function spaces in 1931, which were called Orlicz spaces. These spaces are the generalization of the classical Lebesgue spaces Lp1<p<∞, where the kernel is given by a convex function instead of xp. Some results that deal with differential equations in the framework of Orlicz spaces can be found in [32, 33].In this paper, we investigate the boundary value problem as follows:(1)DC0+α,ρut=ft,ut,DC0+α−1,ρut,DC0+α−2,ρut,DC0+α−3,ρut,t∈0,1,ρ>0,u0=δu0=δ2u0=0,∫01usds=∑i=1mδiIβ,ρuξi,0<ξi<1,δ=t1−ρddt,where DC0+α,ρ is the generalized fractional derivative of Caputo type of order α, 3<α≤4,I0β,ρ denotes the Katugampola-type fractional integral of order 0<β<1, f is defined on an Orlicz space LF0,1,andδi∈ℝ,i=1,…,m.The rest of the paper is organized as follows. InSection 2, we will briefly recall some preliminary materials related to our problem. In Section 3, we discuss the existence and uniqueness results of GFBVP using some fixed point theorems and support the obtained results by using examples to well illustrate. The Ulam stability of our problem is given in Section 4.


## 2. Preliminaries

For convenience of the reader, we present some basic definitions about generalized fractional calculus theory, which can be found in [16, 34, 35]. Also, we introduce some necessary concepts for Orlicz spaces which are used throughout this paper. For more details about Orlicz spaces, one can see [36, 37].Definition 1.
LetQ:0,∞⟶0,∞ be a right continuous, monotone, increasing function with the following:(i)
Q0=0(ii)
limt⟶∞Qt=∞(iii)
Qt>0 whenever t>0
Then, the function defined by(2)Fx=∫0xQtdt,x≥0,is called N−function. Alternatively, the function F is an N−function iff F is continuous, even, and convex with the following:(i)
limx⟶0Fx/x=0(ii)
limx⟶0Fx/x=∞(iii)
Fx>0 if x>0Definition 2.
For anN-function, we define(3)F∗x=∫0xQ−1tdt,x≥0,where Q−1 is the right inverse of the right derivative of F and is called the complementary of F and it satisfies the following condition:(4)F∗x=suptx−Ft:t≥0,∀x≥0.Remark 1.
Note that the functionF∗ is also N-function and the complementary pairs F and F∗ satisfy the following Young inequality:(5)xt≤Fx+F∗t,∀x,t≥0.Definition 3.
A functionF:0,∞⟶0,∞ is called a Young function if it is convex and satisfies the conditions F0=limx⟶0+Fx=0 and limx⟶∞Fx=∞.Definition 4.
(Orlicz space). For anN-function F, the Orlicz space LF0,1 is the space of measurable functions u:0,1⟶ℝ such that ∫01Fuxdx<∞. This space endowed with the Luxemburg norm, i.e.,(6)uF=infλ>0:∫01Fuxλdx≤1,and the pair LF0,1,uF is a Banach space.Remark 2.
For an Orlicz space, the Hölder inequality holds, that is,(7)∫01uvdx≤2uFvF∗,where u∈LF0,1 and v∈LF∗0,1.
For generalized fractional calculus, Katugampola [16, 34, 38] introduced the following definitions.Definition 5.
(Katugampola generalized fractional integral). The generalized fractional integral of orderα∈ℂ with ℜα>0 and ρ>0 for −∞<a<t<∞ is defined by(8)Ia+α,ρut=ρ1−αΓα∫atsρ−1tρ−sρ1−αusds,where u belongs to the space Xcpa,b, which denotes the space of all complex-valued Lebesgue measurable functions f on a,b for which fXcp<∞, where the norm is defined by(9)fXcp=∫abtcftpdtt1/p<∞,1≤p<∞,c∈ℝ.
Note that integral (8) is called the left-sided generalized fractional integral.Definition 6.
(Katugampola generalized Caputo fractional derivative). The left generalized fractional derivative of orderα∈ℂ with ℜα>0 and ρ>0 of u∈ACδna,b, where −∞<a<t<∞, is defined by(10)DCa+α,ρut=1Γn−α∫attρ−sρρn−α−1δnusdss1−ρ=Ia+n−α,ρδnus,where n=α+1 and ACδna,b denotes the space of all absolutely continuous real valued functions on a,b, where(11)ACδna,b=f:a,b⟶ℂandδn−1f∈ACa,b,δ=t1−ρddt,(12)Cδ,ϵna,b=f:a,b⟶ℂandδn−1f∈Ca,b,δnf∈Cϵ,ρa,b,δ=t1−ρddt,for 0≤ϵ≤1, endowed with the norms fCδn=∑k=0nδkfC and fCδ,ϵn=∑k=0n−1δkfC+δnfCϵ,ρ, respectively.
Next, we recall some basic properties of generalized fractional integral and derivative [39].Lemma 1.
Letℜα≥0,n=ℜα+1, and f∈ACδna,b, with 0<a<b<∞. For k,m∈ℕ,(13)Iα,ρakDaCα,ρmft=DaCα,ρmfτΓkα+1tρ−aρρkα,τ∈a,t.Theorem 1.
Letℜα≥0,n=ℜα+1, and f∈ACδna,b, where 0<a<b<∞. Then,(i)
Ifα∉ℕ0,DaCα,ρfx=aIn−α,ρδnfx(ii)
Ifα∈ℕ,DaCα,ρfx=δnfx(iii)
DaC0,ρf=fTheorem 2.
Letfx∈Cδna,b,0<a<b<∞, and α,β∈ℂ such that ℜα≥0andℜβ≥0. Then,(14)DaCα,ρIaβ,ρfx=Iaβ−α,ρfx.
Our main results are based on utilizing the following fixed point theorems.Theorem 3. (Schaefer’s fixed point theorem [40]).
LetX be a Banach space. Assume that ℒ:X⟶X is a completely continuous operator and the set T=u∈X:u=λℒu,0<λ<1 is bounded. Then, ℒ has a fixed point in X.Theorem 4. (Krasnoselskii’s fixed point theorem [40]).
LetN be a closed, convex, bounded, and nonempty subset of a Banach space X. Let T1andT2 be operators such that(i)
T1u1+T2u2 belong to N whenever u1,u2∈N(ii)
T1 is a compact and continuous and T2 is a contraction mapping
Then, there existsu0∈N such that u0=T1u0+T2u0.
For computational convenience, we introduce the following notations:(15)ψ=1ρ33ρ+1−∑i=1mδiξiρβ+3Γ4ρβ+3Γβ+4.Lemma 2.
Leth∈LF0,1 and ψ be given by (15). Then, the solution of the GFBVP(16)DC0+α,ρut=ht,t∈0,1,ρ>0,u0=δu0=δ2u0=0,∫01usds=∑i=1mδiIβ,ρuξi,ξi∈0,1,δ=t1−ρddt,is given by(17)ut=Iα,ρht+t3ρρ3ψ∑i=1mδiIα+β,ρhξi−∫01Iα,ρhsds.Proof.
By using Lemma1, we obtain(18)ut=Iα,ρht+C0+C1tρρ+C2t2ρρ2+C3t3ρρ3,where Ci∈ℝ,i=0,1,2,3 are the arbitrary constants. Using the boundary conditions u0=δu0=δ2u0=0, we get C0=C1=C2=0. Then, from ∫01usds=∑i=1mδiIβ,ρuξi, it follows that(19)C3=1ψ∑i=1mδiρ1−α+βΓα+β∫0ξiξiρ−sρα+β−1sρ−1hsds−ρ1−αΓα∫01∫0ssρ−τρατρ−1hτdτds.
Substituting the values ofC0,C1,C2, and C3 in (18), we get (17).


## 3. Main Results

In this section, we prove the existence and uniqueness of solutions of GFBVP (1). We assume that f belongs to an Orlicz space LF0,1. For 3<α≤4, let X=u:u,DC0+α−1,ρu,DC0+α−2,ρu,DC0+α−3,ρu∈C0,1,ℝ denote the Banach space of all continuous functions defined on 0,1 into ℝ endowed with the norm(20)u=suput+DC0+α−1,ρut+DC0+α−2,ρut+DC0+α−3,ρut,t∈0,1.Relative to GFBVP (1), in view of Lemma 2, we define an operator ℒ:X⟶X as(21)ℒut=ρ1−αΓα∫0ttρ−sρα−1sρ−1fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds+t3ρρ3ψ∑i=1mδiρ1−α+βΓα+β∫0ξiξiρ−sρα+β−1sρ−1fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds−∫01Iα,ρfs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds,t∈0,1.Notice that GFBVP (1) has solutions if and only if the operator ℒ has fixed points.The following result plays a major role in our analysis.Lemma 3.
LetF be a Young function which has a Young complement F∗ satisfying(22)∫0tF∗sα−1ds<∞,∫0tF∗sα+β−1ds<∞,t>0,for α∈3,4 and β∈0,1. Then, the operator ℒ exists and is well defined.Proof.
Letu∈X. Define a function as follows:(23)ϱ1s=tρ−sρα−1sρ−1,ifs∈0,t,t>0,0,otherwise.
At the beginning, we will show thatϱ1∈LF∗0,1. By using appropriate substitution and properties of the Young functions, one obtains(24)∫01F∗ϱ1sκds=∫0tF∗tρ−sρα−1κsρ−1ds=1ρ1κ1/1−α∫0κ1/1−αtρF∗sα−1ds,by the assumption of the theorem, and we get ϱ1∈LF∗0,1. Similarly, setting(25)ϱ2s=tρ−sρα+β−1sρ−1,ifs∈0,t,t>0,0,otherwise,one can get ϱ2∈LF∗0,1. Next, we show that ℒ is well defined, i.e., ℒut∈C0,1,ℝ. Let 0≤τ<t≤1. Thus, we have(26)ℒut−ℒuτ≤ρ1−αΓα∫0τtρ−sρα−1−τρ−sρα−1sρ−1fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds+∫τttρ−sρα−1sρ−1fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds+t3ρ−τ3ρρ3|ψ∑i=1mδiρ1−α+βΓα+β∫0ξiξiρ−sρα+β−1sρ−1fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds−ρ1−αΓα∫01∫0ssρ−τρα−1τρ−1fτ,uτ,DC0+α−1,ρuτ,DC0+α−2,ρuτ,DC0+α−3,ρuτdτds≤ρ1−αΓα∫01G1s+G2sfs,us,DC0+α−1,ρusDC0+α−2,ρus,DC0+α−3,ρusds+t3ρ−τ3ρρ3ψ∑i=1mδiρ1−α+βΓα+β∫0ξiξiρ−sρα+β−1sρ−1fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds−ρ1−αΓα∫01∫0ssρ−τρα−1τρ−1fτ,uτ,DC0+α−1,ρuτ,DC0+α−2,ρuτDC0+α−3,ρuτdτds,where(27)G1s=tρ−sρα−1−τρ−sρα−1sρ−1,ifs∈0,τ,0,otherwise,(28)G2s=tρ−sρα−1sρ−1,ifs∈τ,t,0,otherwise.
The functionsGi,i=1,2, belong to LF∗0,1 with GiF∗≤gtρ−τρ,i=1,2, where g:ℝ+⟶ℝ+ is a continuous and increasing function with g0=0. Using the Hölder inequality, we have(29)ℒut−ℒuτ≤fF2ρα−1ΓαG1F∗+G2F∗+t3ρ−τ3ρρ3ψ∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1.
Then,0<tρ−τρ<ϵ and by the continuity of g, we see that ℒu is continuous, which completes the proof.
Our first existence result relies on Schaefer’s fixed point theorem.Theorem 5.
Assume that there existsℳ∈C0,1,ℝ such that(30)H1ft,ut,DC0+α−1,ρut,DC0+α−2,ρut,DC0+α−3,ρut≤ℳt,fort∈0,1,withℳ=maxt∈0,1ℳt. Then, the GFBVP (1) has at least one solution on 0,1.Proof.
The proof proceeds in few steps as follows:Step 1: we will prove the operatorℒ maps bounded sets into bounded sets in C0,1,ℝ. Let E⊂X be a bounded set. Then, for all u∈E, by using the assumption H1, for t∈0,1, we are able to obtain(31)ℒut≤ρ1−αΓα∫0ttρ−sρα−1sρ−1fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds+t3ρρ3ψ∑i=1mδiρ1−α+βΓα+β∫0ξiξiρ−sρα+β−1sρ−1⋅fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds−∫01Iα,ρfs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds≤ρ1−αΓα∫0ttρ−sρα−1sρ−1ℳsds+1ρ3ψ∑i=1mδiρ1−α+βΓα+β∫0ξiξiρ−sρα+β−1sρ−1ℳsds−ρ1−αΓα∫01∫0ssρ−τρα−1τρ−1ℳτdτds≤ρα−1ℳΓα∫0ttρ−sρα−1sρ−1ds+1ρ3ψ∑i=1mδiα+β∫0ξiξiρ−sρα+β−1sρ−1ds−ℳραΓα+1ρα+1.Thus, we obtain(32)ℒu≤ℳ1ραΓα+1+1ρ3ψ∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1=μ1.Now, with some efforts of computation, we have(33)DC0+α−1,ρℒut=I1,ρft+DC0+α−1,ρt3ρρ3ψ∑i=1mδiρ1−α+βΓα+β∫0ξiξiρ−sρα+β−1sρ−1fs,us,DC0+α−1,ρus,DC0+α−2,ρus,CD0+α−3,ρusds−∫01Iα,ρfs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds=∫0tsρ−1fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds+6tρ4−αρ4−αψΓ5−α∑i=1mδiρ1−α+βΓα+β∫0ξiξiρ−sρα+β−1sρ−1fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds−∫01Iα,ρfs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds≤∫0tsρ−1fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds+6tρ4−αρ4−αψΓ5−α∑i=1mδiρ1−α+βΓα+β∫0ξiξiρ−sρα+β−1sρ−1fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds−ρ1−αΓα∫01∫0ssρ−τρα−1τρ−1fτ,uτ,DC0+α−1,ρuτ,DC0+α−2,ρuτ,DC0+α−3,ρuττρ−1dτds.Therefore, we have(34)DC0+α−1,ρℒu≤ℳ∫0tsρ−1ds+6tρ4−αψρ4−αΓ5−α∑i=1mδiℳρ1−α+βΓα+β∫0ξiξiρ−sρα+β−1sρ−1ds−ℳραΓα+1ρα+1≤ℳtρρ+6tρ4−αψρ4−αΓ5−α∑i=1mδiℳξiρα+βρα+βΓα+β+1−ℳραΓα+1ρα+1≤ℳ1ρ+6ψρ4−αΓ5−α∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1=μ2.In a similar way, we arrive at(35)DC0+α−2ℒu≤ℳ1ρ2Γ3+6ψρ5−αΓ6−α∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1=μ3.(36)DC0+α−3ℒu≤ℳ1ρ3Γ4+6ψ|ρ6−αΓ7−α∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1=μ4.Step 2: from (32)–(36), we can immediately get the operator ℒ:X⟶X to map bounded sets into bounded sets. Let 0<t1<t2<1, and for all u∈E, we get(37)ℒut2−ℒut1=ρ1−αΓα∫0t2t2ρ−sρα−1sρ−1.fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds+ρ1−αΓα∫0t1t1ρ−sρα−1sρ−1.fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds+t23ρ−t13ρψρ3∑i=1mδiρ1−α+βΓα+β∫0ξiξiρ−sρα+β−1sρ−1.fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds−∫01Iα,ρfs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds≤ρ1−αΓα∫0t1t2ρ−sρα−1−t1ρ−sρα−1sρ−1.fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds+∫t1t2t2ρ−sρα−1.fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds+t23ρ−t13ρψ|ρ3∑i=1mδiρ1−α+βΓα+β∫0ξiξiρ−sρα+β−1sρ−1.fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds−∫01Iα,ρfs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds.
Thus, we have(38)ℒut2−ℒut1≤ℳ1ραΓα+12t2ρ−t1ρα−t2αρ−t1αρ+t23ρ−t13ρψρ3∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1.
Similarly, it can be easily shown that(39)DC0+α−1,ρℒut2−DC0+α−1,ρℒut1≤ℳt2ρ−t1ρρ+6t2ρ4−α−t1ρ4−αψρ4−αΓ5−α.∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1.(40)DC0+α−2,ρℒut2−DC0+α−2,ρℒut1≤ℳt22ρ−t12ρρ2Γ3+6t2ρ5−α−t1ρ5−αψρ5−αΓ6−α.∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1.(41)DC0+α−3,ρℒut2−DC0+α−3,ρℒut1≤ℳt23ρ−t13ρρ3Γ4+6t2ρ6−α−t1ρ6−αψρ6−αΓ7−α.∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1.
The right side of (38)–(41) goes to zero as t2−t1⟶0. In view of steps I and II, it follows that by Arzela–Ascoli theorem, the sets ℒu:u∈E,DC0+α−1,ρℒu:u∈E,DC0+α−2,ρℒuℒu:u∈E, and DC0+α−3,ρℒu:u∈E are relatively compact in C0,1. Therefore, ℒE is a relatively compact set in X. We consider the set T=u∈X;u=λℒu,0<λ<1. Then, T is bounded. Indeed, let u∈T. So, u=λℒu,0<λ<1, for any t∈0,1, and it follows that(42)u≤ℳραΓα+1+1ρ3ψ∑i=1mδiℳξiρα+βρα+βΓα+β+1−ℳραΓα+1ρα+1.
Thus, all the hypotheses of Theorem3 are satisfied. Therefore, we can conclude that the operator ℒ has at least one fixed point. Hence, the GFBVP (1) has at least one solution on [0, 1].
For computation convenience, we set(43)V=∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1,(44)K=6ρ4−αψΓ5−α,(45)σ=1ρ.
Now, we make use of Theorem4 to prove the existence of solutions of GFBVP (1).Theorem 6.
Letf:0,1×ℝ4⟶ℝ be a continuous function such that the following assumptions hold:(46)H2ft,u,p,q,w−ft,v,p˜,q˜,w˜≤Lu−v+p−p˜+q−q˜+w−w˜,for allt∈0,1,u,v,p,q,w,p˜,q˜,w˜∈RandL>0,H3ft,ut,CD0+α−1,ρut,CD0+α−2,ρut,CD0+α−3,ρut≤ϕt,fort∈0,1andϕ∈C0,1,ℝwithϕ=maxt∈0,1ϕt.
Then, GFBVP (1) has at least one solution on 0,1, provided that(47)LKV<1.Proof.
We defineBr=u∈X:u≤r, where r⩾ϕσ+KV. We split the operator ℒ defined by (21) on Br as ℒ=ℒ1+ℒ2, where ℒ1 and ℒ2 are given by(48)ℒ1ut=ρ1−αΓα∫0ttρ−sρα−1sρ−1fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds.ℒ2ut=t3ρρ3ψ∑i=1mδiρ1−α+βΓα+β∫0ξiξiρ−sρα+β−1sρ−1⋅fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds−∫01Iα,ρfs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds.
Foru,v∈Br, we find that(49)ℒ1u+ℒ2u≤maxt∈0,1ρ1−αΓα∫0ttρ−sρα−1sρ−1fs,us,DC0+α−1,ρ,us,DC0+α−2,ρus,DC0+α−3,ρusds+t3ρρ3ψ∑i=1mδiρ1−α+βΓα+β∫0ξiξiρ−sρα+β−1sρ−1fs,us,DC0+α−1,ρ,us,DC0+α−2,ρus,DC0+α−3,ρusds−∫01Iα,ρfs,us,DC0+α−1,ρ,us,DC0+α−2,ρus,DC0+α−3,ρusds≤ϕ1ραΓα+1+1ρ3ψ∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1≤ϕσ+KV≤r.
Furthermore,(50)DC0+α−1,ρℒ1u+DC0+α−1,ρℒ2u≤ϕ1ρ+6ψρ4−αΓ5−α∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1≤ϕσ+KV≤r,DC0+α−2,ρℒ1u+DC0+α−2,ρℒ2u≤ϕ1ρ2Γ3+6ψρ5−αΓ6−α∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1≤ϕσ+KV≤r,DC0+α−3,ρℒ1u+DC0+α−3,ρT3u≤ϕ1ρ3Γ4+6ψρ6−αΓ7−α∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1≤ϕσ+KV≤r.
Consequently, we obtain(51)ℒ1u+ℒ2uX≤ϕσ+KV≤r,which shows that ℒ1u+ℒ2u∈Br. In what follows, we prove that ℒ2 is a contraction. Let u,v∈Br and for all t∈0,1, we get(52)ℒ2u−ℒ2v=supt∈0,1ℒ2u−ℒ2v=supt∈0,1t3ρρ3ψ∑i=1mδiξiρα+βρα+β∫0ξiξip−spα+β+1sp−1⋅fs,us,DC0+α−1,ρus,DC0+α−2,ρ,DC0+α−3,ρusds−∫01Iα,ρfs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds−t3ρρ3ψ∑i=1mδiξiρα+βρα+β∫0ξiξip−spα+β+1sp−1⋅fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds+∫01Iα,ρfs,vs,DC0+α−1,ρvs,DC0+α−2,ρvs,DC0+α−3,ρvsds=supt∈0,1t3ρρ3ψ∑i=1mδiξiρα+βρα+β∫0ξiξip−spα+β+1sp−1⋅fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρus−fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds−∫01Iα,ρfs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρus−fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds≤supt∈0,1Lut−vt+DC0+α−1,ρut−DC0+α−1,ρvt+DC0+α−2,ρut−DC0+α−2,ρvt+DC0+α−3,ρut−DC0+α−3,ρvt⋅t3ρρ3ψ∑i=1mδiξiρα+βρα+βΓα+β+1∫0ξiξip−spα+β+1sp−1ds−1ραΓα+1ρα+1≤L1ρ3ψ∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1u−v≤LKVu−v.
With the same process, one has(53)DC0+α−1,ρℒ2u−CDC0+α−1,ρℒ2v≤LKVu−v,DC0+α−2,ρℒ2u−DC0+α−2,ρℒ2v≤LKVu−v,DC0+α−3,ρℒ2u−DC0+α−3,ρℒ2v≤LKVu−v.
Hence, we get(54)ℒ2u−ℒ2vX≤LKVu−vX.
It remains to show thatℒ1 is continuous and compact. The continuity of the function f implies that the operator ℒ1 is continuous. To achieve the compactness of the operator ℒ1, we first prove that ℒ1 is uniformly bounded on Br as follows:(55)ℒ1u=ρ1−αΓαsupt∈0,1∫0ttρ−sρα−1sρ−1⋅fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds≤ϕραΓα+1≤ϕσ,DC0+α−1,ρℒ1u=supt∈0,1ϕtρρ≤ϕρ≤ϕσ,DC0+α−2,ρℒ1u=supt∈0,1ϕt2ρρ2Γ3≤ϕρ2Γ3≤ϕσ,DC0+α−3,ρℒ1u=supt∈0,1ϕt3ρρ3Γ4≤ϕρ3Γ4≤ϕσ.
So,(56)ℒ1uX≤ϕσ.
Now, letsupt∈0,1×Br4ft,u,p,q,w=η. Consequently, for t1,t2∈0,1,t1<t2, we have(57)ℒ1ut2−ℒ1ut1≤ηρ2t2ρ−t1ρα−t2αρ−t1αρ.
Furthermore,(58)DC0+α−1,ρℒ1ut2−DC0+α−1,ρℒ1ut1≤ηρt2ρ−t1ρ,DC0+α−2,ρℒ1ut2−DC0+α−2,ρℒ1ut1≤ηρt22ρ−t12ρ,DC0+α−2,ρℒ1ut2−DC0+α−2,ρℒ1ut1≤ηρt23ρ−t13ρ,
Therefore, ast2−t1⟶0, the right-hand sides of the above inequalities tend to zero independently of u∈Br. Thus, ℒ1 is equicontinuous and so it is relatively compact on Br. According to the Arzela–Ascoli theorem, the operator ℒ1 is compact. By using Theorem 4, there exists at least one solution of GFBVP (1) on 0,1.
In the next result, we prove the existence of solutions for GFBVP (1) by applying Banach’s fixed point theorem.Theorem 7.
Assume that the conditionH2 holds. Then, GFBVP (1) has a unique solution on 0,1, provided that(59)LA<1,where A=σ+KV and V,K, and σ are given by (43)–(45).Proof.
Consider the operatorℒ:X⟶X defined by(60)ℒut=ρ1−αΓα∫0ttρ−sρα−1sρ−1fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds+t3ρρ3ψ∑i=1mδiρ1−α+βΓα+β∫0ξiξiρ−sρα+β−1sρ−1⋅fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds−∫01Iα,ρs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds.
Set(61)supt∈0,1ft,0,0,0,0=f0,r≥f0A1−LA.
We shall show thatℒBr⊂Br, where Br=u∈C0,1,R:u≤r. For u∈Br, we have(62)ℒut≤Iα,ρfs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρus+t3ρρ3ψ∑i=1mδiIα+β,ρfs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρus−∫01Iα,ρfs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds≤Iα,ρfs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρus−fs,0,0,0,0+fs,0,0,0,0+t3ρρ3ψ∑i=1mδiIα+β,ρ⋅fs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρus−fs,0,0,0,0+fs,0,0,0,0−∫01Iα,ρfs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρus−fs,0,0,0,0+fs,0,0,0,0ds.
Therefore,(63)ℒut≤supt∈0,1Lut+DC0+α−1,ρut+DC0+α−2,ρut+DC0+α−3,ρut+f0⋅tαρραΓα+1+t3ρρ3ψ∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1≤Lu+f01ραΓα+1+1ρ3ψ∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1≤Lu+f0A≤Lr+f0A≤r.
Similarly,(64)DC0+α−1,ρℒut≤Lu+f0A≤Lr+f0A≤r,DC0+α−2,ρℒut≤Lu+f0A≤Lr+f0A≤r,DC0+α−3,ρℒut≤Lu+f0A≤Lr+f0A≤r,
By taking the norm fromt∈0,1, it yields ℒu≤r,DC0+α−1,ρℒu≤r,DC0+α−2,ρℒu≤r, and DC0+α−3,ρℒu≤r. Consequently, we have(65)ℒuX≤Lr+f0A≤r,which shows that ℒ maps Br into itself. In order to show that the operator ℒ is a contraction, let u,v∈C0,1,ℝ. Then, for each t∈0,1, we obtain(66)ℒu−ℒv=supt∈0,1ℒut−ℒvt=supt∈0,1Iα,ρfs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρus+t3ρρ3ψ∑i=1mδiIα+β,ρfs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρus−∫01Iα,ρfs,us,DC0+α−1,ρus,DC0+α−2,ρus,DC0+α−3,ρusds−Iα,ρfs,vs,DC0+α−1,ρvs,DC0+α−2,ρvs,DC0+α−3,ρvs−t3ρρ3ψ∑i=1mδiIα+β,ρfs,vs,DC0+α−1,ρvs,DC0+α−2,ρvs,DC0+α−3,ρvs+∫01Iα,ρfs,vs,DC0+α−1,ρvs,DC0+α−2,ρvs,DC0+α−3,ρvsds≤supt∈0,1Lut−vt+DC0+α−1,ρut−DC0+α−1,ρvt+DC0+α−2,ρut−DC0+α−2,ρvt+DC0+α−3,ρut−DC0+α−3,ρvt1ραΓα+1+1ρ3ψ∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1≤Lu−v1ραΓα+1+1ρ3ψ∑i=1mδiξiρα+βρα+βΓα+β+1−1ραΓα+1ρα+1≤LAu−v.
Similarly,(67)DC0+α−1,ρℒu−DC0+α−1,ρℒv≤LAu−v,DC0+α−2,ρℒu−DC0+α−2,ρℒv≤LAu−v,DC0+α−3,ρℒu−DC0+α−3,ρℒv≤LAu−v.
Consequently,(68)ℒu−ℒvX≤LAu−vX.
Thus, in view of condition (59), it follows that the operator ℒ is a contraction. Hence, the operator ℒ has a unique fixed point which corresponds to a unique solution of GFBVP (1).
We conclude this section with some examples showing the applicability of our main results.Example 1.
Let us consider the following boundary value problem:(69)DC0+7/2,1/3ut=ft,ut,DC0+α−1,ρut,DC0+α−2,ρut,DC0+α−3,ρut,t∈0,1,u0=δu0=δ2u0=0,∫01usds=∑i=12δiI1/2,1/3uξi,where α=7/2,β=1/2,ρ=1/3,δ1=2/3,δ2=7/6,ξ1=1/3,ξ2=5/6, and(70)ft,ut,DC0+5/2,1/3ut,DC0+3/2,1/3ut,DC0+1/2,1/3ut=19etsint+utt+32+DC0+5/2,1/3utt2+20+DC0+3/2,1/3ut510t+64+DC0+1/2,1/3ut11t+1.
To check that the conditionH2 holds, let u,v∈C0,1,ℝ and t∈0,1; then, we get(71)ft,ut,DC0+5/2,1/3ut,DC0+3/2,1/3ut,DC0+1/2,1/3ut−ft,vt,DC0+5/2,1/3vt,DC0+3/2,1/3vt,DC0+1/2,1/3vt≤19ut−vt+120DC0+5/2,1/3ut−DC0+5/2,1/3vt+140DC0+3/2,1/3ut−DC0+3/2,1/3vt+111DC0+1/2,1/3ut−DC0+1/2,1/3vt≤19u−vX.
Hence, the conditionH2 holds with L=1/9 and it follows that LA=0.497230703<1, where A=σ+KV and V,K and σ are given by (43)–(45). Then, the hypotheses of Theorem 7 are satisfied and problem (69) has a unique solution on 0,1.Example 2.
We consider the following boundary value problem:(72)DC0+7/2,1/3ut=ft,ut,DC0+α−1,ρut,DC0+α−2,ρut,DC0+α−3,ρut,t∈0,1,u0=δu0=δ2u0=0,∫01usds=∑i=12δiI1/2,1/3uξi.where α=7/2,β=1/2,ρ=1/3,δ1=2/3,δ2=7/6,ξ1=1/3,andξ2=5/6. Choose(73)ft,ut,DC0+5/2,1/3ut,DC0+3/2,1/3ut,DC0+1/2,1/3ut=130et+tan−1utcost+145+t8t3+30DC0+5/2,1/3utDC0+5/2,1/3ut+1+36t+10DC0+3/2,1/3utDC0+3/2,1/3ut+1+1sint+169DC0+1/2,1/3utDC0+1/2,1/3ut+1.
SetFu=uα/α, α>1. In particular, if α=2, then F is an N-function and satisfies(74)∫01Ffusds<∞,which shows that f belongs to an Orlicz space LF0,1. Furthermore,(75)ft,ut,DC0+5/2,1/3ut,DC0+3/2,1/3ut,DC0+1/2,1/3ut≤ℳt,with ℳt=et/30+π/2cost+145+t/8t3+30+3/6t+10+1/sint+169 and ℳ=1.225. Thus, the condition H1 holds and by the conclusion of Theorem 5, problem (72) has at least one solution on 0,1.
Moreover,(76)ft,ut,DC0+5/2,1/3ut,DC0+3/2,1/3ut,DC0+1/2,1/3ut−ft,vt,DC0+5/2,1/3vt,DC0+3/2,1/3vt,DC0+1/2,1/3vt≤112ut−vt+130DC0+5/2,1/3ut−DC0+5/2,1/3vt+120DC0+3/2,1/3ut−DC0+3/2,1/3vt+113DC0+1/2,1/3ut−DC0+1/2,1/3vt≤112u−vX.which proves the validity of condition H2 with LA=0.372923027<1, where A=σ+KV and V,K, and σ are given by (43)–(45). Hence, Theorem 7 can be applicable and so problem (72) has a unique solution on 0,1.


## 4. Ulam Stability

In this section, we develop the criteria for Ulam stability of GFBVP (1) by means of its equivalent integral equation. For simplicity, we set(77)fs,vs,DC0+α−1,ρvs,DC0+α−2,ρvs,DC0+α−3,ρvs=fs^.Then,(78)vt=ρ1−αΓα∫0ttρ−sρα−1sρ−1fs^ds+t3ρρ3ψ∑i=1mδiρ1−α+βΓα+β∫0ξiξiρ−sρα+β−1sρ−1fs^ds−∫01Iα,ρfs^ds,where v∈X and f:0,1×ℝ4⟶ℝ is a continuous function. Next, we define a continuous nonlinear operator ℱ:X⟶X as(79)ℱvt=DC0+α,ρvt−ft^.For definitions of Ulam–Hyers, generalized Ulam–Hyers and Ulam–Hyers–Rassias stability, and we refer to [41].Definition 7.
GFBVP (1) is said to be Ulam–Hyers stable if there exists a real number b>0 such that, for each ϵ>0 and for each solution v∈X,(80)ℱv≤ϵ,t∈0,1,there exists a solution u∈X of (1) satisfying the inequality(81)u−v≤bϵ1,t∈0,1,where ϵ1 is a positive real number depending on ϵ.Definition 8.
GFBVP (1) is generalized Ulam–Hyers stable if there exists Θ∈Cℝ+,ℝ+ such that, for each solution v∈X of (1), there exists a solution u∈X of (1) with(82)ut−vt≤Θϵ,t∈0,1.Definition 9.
GFBVP (1) is Ulam–Hyers–Rassias stable with respect to Ψ∈C0,1,ℝ if there exists a real number b>0 such that, for each ϵ>0 and for each solution v∈X of (1),(83)ℱvt≤ϵΨt,t∈0,1,we can find a solution u∈X of (1) satisfying the inequality(84)ut−vt≤bϵ1Ψt,t∈0,1,where ϵ1 is a positive real number depending on ϵ.Theorem 8.
Assume that conditions (47) and (59) hold. Then, GFBVP (1) satisfies both Ulam–Hyers and generalized Ulam–Hyers stability criteria.Proof.
We know thatu∈X is a unique solution of (1) (by Theorem 7). Let v∈X be another solution of (1) satisfying ℱv≤ϵ,t∈0,1. Observe that the operators ℱ and ℒ−ℐ are equivalent for every solution v∈X of (1). Therefore, by the fixed point property of the operator ℒ together with ut=ℒut and ℱv≤ϵ,t∈0,1, we have(85)vt−ut=vt−ℒvt+ℒvt−ℒut≤ℒut−ℒvt+ℒvt−vt≤LAu−vX+ϵ,where ϵ>0 and LA<1. Taking the norm for t∈0,1 and solving for u−vX, we obtain(86)u−vX≤ϵ1−LA.
Choosingϵ1=ϵ/1−LA and b=1, the Ulam–Hyers stability condition is satisfied. More generally, defining Θϵ=ϵ/1−LA, the generalized Ulam–Hyers stability condition is also satisfied which completes the proof.Theorem 9.
Assume that conditions (47) and (59) hold and there exists a function z∈C0,1,ℝ satisfying condition ut−vt≤bϵ1Ψt,t∈0,1. Then, GFBVP (1) is Ulam–Hyers–Rassias stable with respect to Ψ.Proof.
Similar to the discussion in the proof of Theorem8, we have(87)u−vX≤ϵ1Ψt,with ϵ1=ϵ/1−LA, and we get the desired result.



---

*Source: 6824672-2020-05-11.xml*