# A Study on the Solutions of a Multiterm FBVP of Variable Order

Authors: Zoubida Bouazza; Sina Etemad; Mohammed Said Souid; Shahram Rezapour; Francisco Martínez; Mohammed K. A. Kaabar
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In the present research study, for a given multiterm boundary value problem (BVP) involving the nonlinear fractional differential equation (NnLFDEq) of variable order, the uniqueness-existence properties are analyzed. To arrive at such an aim, we first investigate some specifications of this kind of variable order operator and then derive required criteria confirming the existence of solution. All results in this study are established with the help of two fixed-point theorems and examined by a practical example.


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## Body

## 1. Introduction

The first historical resource to the invention of fractional derivative (FrDr) was proposed in1695, when L’Hopital proposed the question about the meaning of dσy/dxσ if σ=1/2. The proposed FrDr’s definitions are classified into two categories: global nature and local nature. On the one hand, under the global one, the FrDr is defined as integral, Fourier, or Mellin transformations; hence, its nonlocal characteristic is given with a memory. On the other hand, under the local one, FrDr is relied on a local definition through certain incremental ratios. As a result of this classical formulation, fractional calculus (FrCa) has appeared since the time of well-known mathematicians such as Lacroix, Fourier, Liouville, Euler, Laplace, and Abel until the creation of the first modern fractional definitions of Caputo and Riemann-Liouville.The FrCa theory is a representation of a powerful tool of mathematical analysis for investigating the integrals and derivatives of arbitrary order, which constitutes the unifying and generalizing element of the integer-order differentiation andn-fold integration [1, 2]. Studying fractional integrals and derivatives was only devoted to the theoretical mathematical context. However, their applications have been recently seen in multidisciplinary sciences such as theoretical physics, entropy theory, fluid mechanics, biology, and image processing [3–13].Furthermore, studying both of the theoretical and practical aspects of fractional differential equations (FDEqs) has become a focus of international academic research [14–18]. A recent improvement on this investigation is the consideration of the notion of variable order operators. In this sense, various definitions of fractional operators involving the variable order have been introduced. This type of operators which are dependent on their power-law kernel can describe some hereditary specifications of numerous processes and phenomena [19, 20]. In general, it is often difficult to find the analytical solution of FDEqs of variable order; therefore, numerical methods for the approximation of FDEqs of variable order are widespread. Regarding the study of solutions’ existence to the problems of variable order, we refer to [21–24]. On the contrary, a consistent approach with the first-order precision for the solution of FDEqs of variable order is applied by Coimbra in [25]. Lin et al. [26] discussed the convergence and stability of an explicit approximation related to the diffusion equation of variable order with a nonlinear source term. In [27], Zhuang et al. introduced the implicit and explicit Euler approximations for the nonlinear diffusion-advection equation of variable order.While several research studies have been performed on investigating the solutions’ existence of the fractional constant-order problems, the solutions’ existence of the variable-order problems are rarely discussed in literature; we refer to [28–34]. Therefore, investigating this interesting special research topic makes all our results novel and worthy.Inspired by all previous studies, this work investigates the solutions’ existence to the proposed multiterm BVP for NnLFDEqs of variable order in the format(1)D0+Ωτxτ+ψτ,xτ,I0+Ωτxτ=0,τ∈J≔0,K,x0=0,xK=0,where 0<K<+∞, 1<Ωτ≤2, ψ:J×ℝ2→ℝ is continuous, and D0+Ωτ and I0+Ωτ are the Riemann-Liouville fractional derivative (RLFrD) and integral (RLFrIn) of variable-order Ωτ, respectively. We here try to express some differences of variable-order operators than standard ones and then implement our techniques to confirm the existence property related to the above BVP.This work is divided into the following sections: some essential notions are presented in Section2 that will be used later. Two important results are as follows: one is relied on Schauder fixed-point theorem (Schauder-FixPThm), and the other one is relied on the Banach contraction principle (Banach-CoPrp), which are provided in Section 3. In Section 4, a numerical example is provided to validate and apply our theoretical results.


## 2. Essential Preliminaries

Some essential mathematical notations that will be used later are provided in this section. By means ofCJ,ℝ, let us represent the Banach space (Banach-Sp) of continuous mappings from J into ℝ via(2)∥x∥=supxτ:τ∈J.Definition 1 (see [22, 35]).
Let−∞<q<p<+∞, and Ωτ:q,p→0,+∞. The left Ωτth-RLFrIn of a function ξτ is expressed by(3)Iq+Ωτξτ=∫qττ−μΩμ−1ΓΩμξμdμ,τ>q,where the gamma function is denoted by Γ·.Definition 2 (see [22, 35]).
Let−∞<q<p<+∞,m∈ℕ and Ωτ:q,p→m−1,m. The left Ωτth-RLFrD of a function ξτ is expressed by(4)Dq+Ωτξτ=ddτmIq+m−Ωτξτ=ddτm∫qττ−μm−Ωμ−1Γm−Ωμξμdμ,τ>q.Obviously, if the orderΩτ is a constant function Ω, then the Riemann-Liouville variable-order fractional derivative (RLVoFrD) (4) and integral (RLVoIn) (3) are the usual RLFrD and RLFrIn, respectively; see [22, 35].Let us now discuss some essential properties.Lemma 3 (see [2]).
Letγ>0. Then, the differential equation(5)Dqγξ=0has a unique solution:(6)ξτ=λ1τ−qγ−1+λ2τ−qγ−2+⋯+λmτ−qγ−m,
λj∈ℝ,j=1,2,⋯,m, here m−1<γ≤m.Lemma 4 (see [2]).
Lettingq>0, ξ∈Lq,p, Dqγξ∈Lq,p, then(7)Iq+γDq+γξτ=ξτ+λ1τ−qγ−1+λ2τ−qγ−2+⋯+λmτ−qγ−m,
λj∈ℝ,j=1,2,⋯,m, here m−1<γ≤m.Lemma 5 (see [2]).
Lettingγ>0, then we obtain(8)Dq+γIq+γξτ=ξτ.Lemma 6 (see [2]).
Lettingγ,δ>0, then we get(9)Iq+γIq+δξτ=Iq+δIq+γξτ=Iq+γ+δξτ.Remark 7 (see [30, 32]).
Generally, for functionsΩτ and ντ, the semigroup property does not hold, i.e.,(10)Iq+ΩτIq+ντξτ≠Iq+Ωτ+ντξτ.Example 1.
Let(11)Ωτ=2,τ∈0,1,1,τ∈1,3,ντ=1,τ∈0,1,2,τ∈1,3,and ξτ=τ and τ∈0,3. Then, we compute(12)I0+ΩτI0+ντξτ=∫01τ−μΩμ−1ΓΩμ∫0μμ−ωνω−1Γνωξωdωdμ+∫1ττ−μΩμ−1ΓΩμ∫0μμ−ωνω−1Γνωξωdωdμ=∫01τ−μ1Γ2∫0μμ−ω0Γ1ωdωdμ+∫1ττ−μ0Γ1∫01μ−ω0Γ1ωdω+∫1μμ−ω1Γ2ωdωdμ=∫01τ−μμ22Γ2dμ+∫1τμ36−μ2+56dμ,I0+Ωτ+ντξτ=∫0ττ−μΩμ+νμ−1ΓΩμ+νμξμdμ.
We see that(13)I0+ΩτIq+ντξττ=2=∫012−μμ22Γ2dμ+∫12μ36−μ2+56dμ=524+1724=2224,I0+Ωτ+ντξττ=2=∫012−μ2+1−1Γ2+1μdμ+∫122−μ1+2−1Γ1+2μdμ=1124+524=1624.
Therefore, we obtain(14)I0+ΩτI0+ντξττ=2≠I0+Ωτ+ντξττ=2.Lemma 8 (see [24]).
Let0≤ζ<1 and Ω:J→1,2 be continuous. Then, for y∈CζJ,ℝ=yτ∈CJ,ℝ,τζyτ∈CJ,ℝ,τ∈minτ∈J∣Ωτ∣, the variable order fractional integral I0+Ωτyτ exists for any points on J.Lemma 9 (see [24]).
LetΩ:J→1,2 be a continuous function, then I0+Ωτyτ∈CJ,ℝ for y∈CJ,ℝ.Definition 10 (see [34]).
We say that the setI⊂ℝ is a generalized interval if it is either an interval (one of the formats c1,c2,c1,c2,c1,c2, or c1,c2 or a single-point c1, or ∅.Definition 11 (see [34]).
By assumingI as a generalized interval, a finite set ℙ consisting of generalized intervals belonging to I is named as a partition of I if each x∈I is contained in exactly one of the generalized intervals J in P.Definition 12 (see [34]).
Assume thatI is a generalized interval and suppose that ρ:I→ℝ is a function, and assume that ℙ is a partition of I. Then, ρ is named as a piecewise constant w.r.t. ℙ if, for each J∈ℙ, ρ is constant on J.Theorem 13 (see [36] (Schauder-FixPThm)).
Assume thatE is a Banach-Sp, and Q as its subset via convexity, closedness, and boundedness and F:Q→Q via the compactness and continuity. Then, it is found at least a fixed point in Q for F.


## 3. Existence of Solutions

All our original main results in this work are discussed in this section. Some assumptions are presented as follows:(ASP1) Letm∈ℕ be an integer, P=J1≔0,K1,J2≔K1,K2,J3≔K2,K3,⋯Jm≔Km−1,K be a partition of the interval J, and let Ωτ:J→1,2 be a piecewise constant function w.r.t. P as follows:(15)Ωτ=∑j=1mΩjIjτ=Ω1,ifτ∈J1,Ω2,ifτ∈J2,..Ωm,ifτ∈Jm,where 1<Ωj≤2 and Ij is an indicator of intervals Jj≔Kj−1,Kj,j=1,2,⋯,m, such that,(16)Ijτ=1,forτ∈Jj,0,forelsewhere.(ASP2) Letτζψ:J×ℝ×ℝ→ℝ be a continuous function 0≤ζ<1. ∃ constants ci>0,i=1,2,3 and 0<σ<1,0<η<1, such that(17)τζψτ,y,z≤c1+c2yσ+c3zη,for any y,z∈ℝ and τ∈J.(ASP3)∃ constants W,L>0,0≤ζ<1 such that(18)τζψτ,s1,z1−ψτ,s2,z2≤Ws1−s2+Lz1−z2,for any s1,s2,z1,z2∈ℝ and τ∈J.ByEj=CJj,ℝ, let us represent the Banach-Sp of continuous mappings from Jj into ℝ via(19)xEj=supτ∈Jjxτ,where j∈1,2,⋯,m.To get our original results, let us first perform an essential analysis to our proposed BVP (1). By (4), the equation of the BVP (1) can be written as(20)d2dτ2∫0ττ−μ1−ΩμΓ2−Ωμxμdμ+ψτ,xτ,I0+Ωτxτ=0,τ∈J.According toASP1, equation (20) on the interval Jj,j=1,2,⋯,m can be written as(21)d2dτ2∫0K1τ−μ1−Ω1Γ2−Ω1xμdμ+⋯+∫Kj−1ττ−μ1−ΩjΓ2−Ωjxμdμ+ψτ,xτ,I0+Ωjxτ=0,for τ∈Jj. Let us now define the solution to the BVP (1), which is essential in this research study.Definition 14.
A BVP (1) has a solution, if ∃ functions xj,j=1,2,⋯,m, so that xj∈C0,Kj,ℝ that satisfy equation (21) and xj0=0=xjKj.From our previous analysis above, (1) can be expressed as equation (20), which can be written on the intervals Jj,j∈1,2,⋯,m as (21). For 0≤τ≤Kj−1, by taking xτ≡0, then (21) is written as follows:(22)DKi−1+uixτ+ψτ,xτ,IKi−1+uixτ=0,τ∈Ji.Let us consider the following BVP:(23)DKj−1+Ωjxτ+ψτ,xτ,IKj−1+Ωjxτ=0,τ∈Jj,xKj−1=0,xKj=0.For the solutions’ existence of problem (23), an auxiliary lemma is needed as follows:Lemma 15.
The functionx∈Ej is a solution of problem (23) iff x satisfies the integral equation as follows:(24)xτ=∫Kj−1KjGjτ,μψμ,xμ,IKj−1+Ωjxμdμ,where Gjτ,μ is Green’s function defined by(25)Gjτ,μ=1ΓΩjKj−Kj−11−Ωjτ−Kj−1Ωj−1Kj−μΩj−1−τ−μΩj−1,Kj−1≤μ≤τ≤Kj,1ΓΩjKj−Kj−11−Ωjτ−Kj−1Ωj−1Kj−μΩj−1,Kj−1≤τ≤μ≤Kj,where j=1,2,⋯,m.Proof.
Letx∈Ej be a solution of the BVP (23). Now, let us apply the operator IKj−1+Ωj to both sides of the equation of the supposed BVP (23). By Lemma 4, we obtain(26)xτ=λ1τ−Kj−1Ωj−1+λ2τ−Kj−1Ωj−2−1ΓΩj∫Kj−1ττ−μΩj−1ψμ,xμ,IKj−1+Ωjxμdμ,τ∈Jj.
ByxKj−1=0 and the function ψ, we obtain λ2=0. Let xτ satisfies xKj=0. Thus, we get λ1=Kj−Kj−11−ΩjIKj−1+ΩjψKj,xKj,IKj−1+ΩjxKj. Then, we have(27)xτ=Kj−Kj−11−Ωjτ−Kj−1Ωj−1IKj−1+ΩjψKj,xKj,IKj−1+ΩjxKj−IKj−1+Ωjψτ,xτ,IKj−1+Ωjxτ,τ∈Jj,by the continuity of Green’s function which implies that(28)xτ=∫Kj−1KjGjτ,μψμ,xμ,IKj−1+Ωjxμdμ.
Conversely, letx∈Ej be a solution of integral equation (24); then, by the continuity of function τζψ and Lemma 5, we can easily get that x is the solution of BVP (23).The following proposition will be needed:Proposition 16 (see [34]).
Let0≤ζ<1 and assume that τζψ:J×ℝ×ℝ→ℝ is continuous, Ωτ:J→1,2 satisfies ASP1. Then, Green’s function of BVP (23) satisfies the following properties:(I)
Gjτ,μ≥0,∀Kj−1≤τ,μ≤Kj(II)
maxτ∈JjGjτ,μ=Gjμ,μ,μ∈Jj(III)
Gjμ,μ has one unique maximum given by(29)maxμ∈JjGjμ,μ=1ΓΩjKj−Kj−14Ωj−1,where j=1,2,⋯,m.
The first existence result is relied on Theorem13.Theorem 17.
Suppose that (ASP1)–(ASP3) hold; then, it is found a solution to the BVP (1) on J.Proof.
Problem (23) can be transformed into a fixed-point problem. Let us construct the following operator:(30)M:Ej→Ej,formulated by(31)Mxτ=∫Kj−1KjGjτ,μψμ,xμ,IKj−1+Ωjxμdμ,τ∈Jj.
It follows from the properties of fractional integrals and from the continuity of functionτζψ that the operator M:Ej→Ej defined in (31) is well-defined. We consider the set(32)BRj=x∈Ej,xEj≤Rj,where(33)Rj=max3c1ΓΩjKj-Kj-14Ωj-1Kj1-ζ-Kj-11-ζ1-ζ,3c2ΓΩjKj-Kj-14Ωj-1Kj1-ζ-Kj-11-ζ1-ζ1/1-σ,3c3ΓΩjKj-Kj-14Ωj-1⋅Kj1-ζ-Kj-11-ζ1-ζKj-Kj-1ΩjΓΩj+1η1/1-η.
Clearly,BRj≠∅ is convex, bounded, and closed. Now, we prove in the following three steps that M satisfies the hypotheses of Theorem 13.
Step I.MBRj⊆BRj
Forx∈BRj, by Proposition 16 and (ASP2), we get(34)Mxτ=∫Kj−1KjGjτ,μψμ,xμ,IKj−1+Ωjxμdμ≤∫Kj−1KjGjτ,μψμ,xμ,IKj−1+Ωjxμdμ≤∫Kj−1KjGjτ,μμ−ζc1+c2xμσ+c3IKj−1+Ωjxμηdμ≤1ΓΩjKj−Kj−14Ωj−1∫Kj−1Kjμ−ζc1+c2xμσ+c3Kj−Kj−1ΩjΓΩj+1ηxEjηdμ≤1ΓΩjKj−Kj−14Ωj−1Kj1−ζ−Kj−11−ζ1−ζc1+c2Rjσ+c3Kj−Kj−1ΩjΓΩj+1ηRjη≤Rj3+Rj3+Rj3=Rj,
which means thatMBRj⊆BRj.
Step II.M is continuous.
Assume thatxm is a sequence via xm→x in Ej. We verify that(35)Mxm−MxEj→0,m→∞.
Indeed, forτ∈Jj, by Proposition 16 and (ASP3), we obtain(36)Mxmτ−Mxτ≤∫Kj−1KjGjτ,μψμ,xmμ,IKj−1+Ωjxmμ−ψμ,xμ,IKj−1+Ωjxμdμ≤1ΓΩjKj−Kj−14Ωj−1∫Kj−1Kjψμ,xmμ,IKj−1+Ωjxmμ−ψμ,xμ,IKj−1+Ωjxμdμ≤1ΓΩjKj−Kj−14Ωj−1∫Kj−1Kjμ−ζWxmμ−xμ+LIKj−1+Ωjxmμ−xμdμ≤1ΓΩjKj−Kj−14Ωj−1W∥xm−x∥Ej∫Kj−1Kjμ−ζdμ+L∥IKj−1+Ωjxm−x∥Ej∫Kj−1Kjμ−ζdμ≤Kj1−ζ−Kj−11−ζKj−Kj−1Ωj−14Ωj−11−ζΓΩjW+LKj−Kj−1ΩjΓΩj+1∥xm−x∥Ej.
So(37)Mxm−MxEj→0asm→∞.
Consequently,M is a continuous operator on Ej.
Step III.M is compact.
Now, we will prove thatMBRj is relatively compact, meaning that M is compact. Clearly, MBRj is uniformly bounded because by Step II, we have(38)MBRj=Mx:x∈BRj⊂BRj.
Thus, for eachx∈BRj, we have MxEj≤Rj which means that MBRj is uniformly bounded. It remains to prove that MBRj is equicontinuous.
Forτ1,τ2∈Jj,τ1<τ2 and x∈BRj and ψå=ψμ,0,0, we have(39)Mxτ2−Mxτ1=∫Kj−1KjGjτ2,μψμ,xμ,IKj−1+Ωjxμdμ−∫Kj−1KjGjτ1,μψμ,xμ,IKj−1+Ωjxμdμ≤∫Kj−1KjGjτ2,μ−Gjτ1,μψμ,xμ,IKj−1+Ωjxμdμ≤∫Kj−1KjGjτ2,μ−Gjτ1,μψμ,xμ,IKj−1+Ωjxμ−ψμ,0,0+ψμ,0,0dμ≤∫Kj−1KjGjτ2,μ−Gjτ1,μψμ,xμ,IKj−1+Ωjxμ−ψμ,0,0dμ+∫Kj−1KjGjτ2,μ−Gjτ1,μψμ,0,0dμ≤∫Kj−1KjGjτ2,μ−Gjτ1,μμ−ζWxμ+LIKj−1+Ωjxμdμ+ψ⋆∫Kj−1KjGjτ2,μ−Gjτ1,μdμ≤∫Kj−1KjGjτ2,μ−Gjτ1,μμ−ζWxμ+LKj−Kj−1ΩjΓΩj+1xμdμ+ψ⋆∫Kj−1KjGjτ2,μ−Gjτ1,μdμ≤Kj−1−ζW+LKj−Kj−1ΩjΓΩj+1Rj∫Kj−1KjGjτ2,μ−Gjτ1,μdμ+ψ⋆∫Kj−1KjGjτ2,μ−Gjτ1,μdμ,by the continuity of Green’s function Gj. Hence,(40)Mxτ2−Mxτ1→0,as ∣τ2−τ1∣→0, free of x∈BRj. It implies that MBRj is equicontinuous. From Steps I to III and the Arzela-Ascoli theorem, it can be concluded that M is completely continuous.
Now, from Theorem13, problem (23) possesses at least a solution xj~ in BRj. We let(41)xj=0,τ∈0,Kj−1,x~j,τ∈Jj.
We know thatxj∈C0,Kj,ℝ defined by (41) satisfies the following equation:(42)d2dτ2∫0K1τ−μ1−Ω1Γ2−μ1xjμdμ+⋯+∫Kj−1ττ−μ1−ΩjΓ2−Ωjxjμdμ+ψμ,x~jμ,I0+Ωjx~jμ=0,for τ∈Jj, which means that xj is a solution of (21) with xj0=0,xjKj=x~jKj=0. In consequence, we figure out that the BVP (1) admits at least a solution defined by(43)xτ=x1τ,τ∈J1,x2τ=0,τ∈J1,x~2,τ∈J2...xjτ=0,τ∈0,Kj−1,x~j,τ∈Jj,and the argument is ended.Let us discuss our second result which is relied on the Banach-CoPrp.Theorem 18.
Suppose thatASP1 and ASP3 hold and if(44)Kj1−ζ−Kj−11−ζKj−Kj−1Ωj−14Ωj−11−ζΓΩjW+LKj−Kj−1ΩjΓΩj+1<1,then, it is found a solution uniquely for BVP (1) in Ej.Proof.
Let us use the Banach-CoPrp to show thatM defined in (31) has a unique fixed point. By Proposition 16 and ASP3, and for xτ,yτ∈Ej:(45)Mxτ−Myτ=∫Kj−1KjGjτ,μψμ,xμ,IKj−1+Ωjxμdμ−∫Kj−1KjGjτ,μψμ,yμ,IKj−1+Ωjyμdμ≤∫Kj−1KjGjτ,μψμ,xμ,IKj−1+Ωjxμ−ψμ,yμ,IKj−1+Ωjyμ≤1ΓΩjKj−Kj−14Ωj−1∫Kj−1Kjψμ,xμ,IKj−1+Ωjxμ−ψμ,yμ,IKj−1+Ωjyμ≤1ΓΩjKj−Kj−14Ωj−1∫Kj−1Kjμ−ζWxμ−yμ+LIKj−1+Ωjxμ−yμdμ≤1ΓΩjKj−Kj−14Ωj−1W∥x−y∥Ej∫Kj−1Kjμ−ζdμ+LKj−Kj−1ΩjΓΩj+1∥x−y∥Ej∫Kj−1Kjμ−ζdμ≤Kj1−ζ−Kj−11−ζKj−Kj−1Ωj−14Ωj−11−ζΓΩjW+LKj−Kj−1ΩjΓΩj+1∥x−y∥Ej.
Consequently, by (44), the operator M is a contraction. Thus, by Banach-CoPrp, one can find a fixed-point xj~∈Ej for M uniquely, which is the same unique solution of the BVP (23). We let(46)xj=0,τ∈0,Kj−1,x~j,τ∈Jj.
By assumingC0,Kj,ℝ as the set of all continuous functions from 0,Kj into ℝ, we know that xj∈C0,Kj,ℝ defined by (46) satisfies the following equation:(47)d2dτ2∫0K1τ−μ1−Ω1Γ2−Ω1xjμdμ+⋯+∫Kj−1ττ−μ1−ΩjΓ2−Ωjxjμdμ+ψμ,x~jμ,I0+Ωjx~jμ=0,for τ∈Jj, which means that xj is a unique solution of (21) with xj0=0 and xjKj=x~jKj=0. Then,(48)xτ=x1τ,τ∈J1,x2τ=0,τ∈J1,x~2,τ∈J2,...xjτ=0,τ∈0,Kj−1,x~j,τ∈Jjis one and only one solution to the proposed BVP (1) and the argument is ended.


## 4. Example

An illustrative numerical example is given in this section to apply and validate all our theoretical results.Example 2.
Consider the fractional BVP:(49)D0+Ωτxτ+τ−1/51+∣xτ∣+∣I0+Ωτxτ∣=0,τ∈J≔0,2,x0=0,x2=0.
Let(50)ψτ,s,h=τ−1/51+∣s∣+∣h∣,τ,s,h∈0,2×0,+∞×0,+∞;(51)Ωτ=1.7,τ∈J1≔0,1,1.8,τ∈J2≔1,2.
We see thatΩτ satisfies condition (ASP1). We have(52)τ1/5ψτ,s1,h1−ψτ,s2,h2=τ1/5τ−1/51+∣s1∣+∣h1∣−τ−1/51+∣s2∣+∣h2∣=∣s2∣+∣h2∣−∣s1∣−∣h1∣1+s1+h11+s2+h2≤s2−s1+h2−h1≤s1−s2+h1−h2.
Thus, (ASP3) holds withζ=1/5 and W=L=1. By (5), the equation of problem (49) is divided into two expressions as follows:(53)D0+1.7xτ+τ−1/51+∣xτ∣+∣I0+1.7xτ∣=0,τ∈0,1,D1+1.8xτ+τ−1/51+∣xτ∣+∣I0+1.8xτ∣=0,τ∈1,2.
Forτ∈0,1, the BVP (49) is corresponding to the following BVP:(54)D0+1.7xτ+τ−1/51+∣xτ∣+∣I0+1.7xτ∣=0,τ∈0,1,x0=0,x1=0.
We shall check that condition (44) is satisfied as follows:(55)K11−ζ−K01−ζK1−K0Ω1−14Ω1−11−ζΓΩ1W+LK1−K0Ω1ΓΩ1+1=541.7Γ1.71+1Γ2.7≃0.8587<1.
By Theorem18, it is found a solution x1∈E1 uniquely to the BVP (54).
Forτ∈1,2, problem (49) can be written as follows:(56)D1+1.8xτ+τ−1/51+∣xτ∣+∣I0+1.8xτ∣=0,τ∈1,2,x1=0,x2=0.
We see that(57)K21−ζ−K11−ζK2−K1Ω2−14Ω2−11−ζΓΩ2W+LK2−K1Ω2ΓΩ2+1=524/5−141.8Γ1.81+1Γ2.8≃0.5237<1.
Thus, condition (44) is satisfied. Therefore, by Theorem 18, it is found a solution x~2∈E2 uniquely to the BVP (56). It is known that(58)x2τ=0,τ∈0,1,x~2τ,τ∈1,2.
Now, from Definition14, the BVP (49) has a unique solution defined by(59)xτ=x1τ,τ∈0,1,x2τ=0,τ∈0,1,x~2τ,τ∈1,2.


## 5. Conclusion

Our proposed multiterm BVP has been successfully investigated in this work via two fixed-point theorems: Schauder-FixPThm and Banach-CoPrp to prove the solutions’ existence and uniqueness for NnLFDEqs. A numerical example is given at the end to support and validate the potentiality of all our obtained results. Therefore, all results in this work show a great potential to be applied in various applications of multidisciplinary sciences. Further investigations on this open research problem can be also possible with the help of our original results in this research paper. In other words, one can extend the proposed BVP to other complicated real mathematical fractional models by terms of newly introduced operators with nonsingular kernels in the future.



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*Source: 9939147-2021-05-24.xml*