# Fixed Point Theory and the Liouville–Caputo Integro-Differential FBVP with Multiple Nonlinear Terms

Authors: Shahram Rezapour; Ali Boulfoul; Brahim Tellab; Mohammad Esmael Samei; Sina Etemad; Reny George
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This work is reserved for the study of a special category of boundary value problems (BVPs) consisting of Liouville–Caputo integro-differential equations with multiple nonlinear terms. This fractional model and its boundary value conditions (BVCs) involve different simple BVPs, in which the second BVC as a linear combination of two Caputo derivatives of the unknown function equals a nonzero constant. The Banach principle gives a unique solution for this Liouville–Caputo BVP. Further, the Krasnoselskii and Leray–Schauder criteria give the existence property regarding solutions of the mentioned problem. For each theorem, we provide an example based on the required hypotheses and derive numerical data in the framework of tables and figures to show the consistency of results from different points of view.


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## Body

## 1. Introduction

In recent years, fractional differential equations have attracted the attention of many authors because of the numerous applications in various branches of science and engineering, in particular, fluid mechanics, image and signal processing, electromagnetic theory, potential theory, fractals theory, biology, control theory, viscoelasticity, and so on [1–3]. From the mathematical point of view, a number of researchers working on fractional calculus conduct their research in the field of applications of different fractional operators and various structures of BVPs in modeling abstract and real-world phenomena, but the discussion related to the fractional derivatives is an old problem and continues to receive many kinds of feedback. The physical aspect of the fractional derivative is now proved in many investigations. As we know, fractional-order derivatives have many advantages in comparison to the first-order derivatives. For example, one of the most simple examples in which the fractional derivative has a significant impact can be observed in diffusion processes. It is established that the subdiffusion is obtained when the order of the fractional derivative belongs to the interval (0, 1). Another impact of fractional derivatives can be observed in stability analysis. There are many differential equations that are not stable with the first-order derivative but are stable when we replace the first-order derivative by the fractional-order derivatives. By considering these cases, we can understand the importance of fractional operators, and the Liouville–Caputo derivative is one of the most important examples in this field. For better and more accurate simulations and better numerical results, we use the Liouville–Caputo derivative in this paper.Along with these important abilities, fixed point theory is regarded as one of the most important tools to derive existence criterion of solutions. To better understand the subject, some research in this field can be enumerated. In [4], Ahmad and Agarwal turned to the existence of solution for several new structures of fractional BVPs via slit-strip BVCs. Then, Ahmad and Ntouyas [5] and Alsaedi et al. [6] investigated similar results regarding solutions of a sequential BVPs consisting of nonlocal integro-differential inclusions of the Caputo type. In [7], Boucenna et al. defined a nonlinear ℙ on the Sobolev space and utilized the special operators for proving theorems with the help of some tools in functional analysis. Similarly, Azzaoui et al. [8] defined a Sobolev space again and derived the existence criterion for positive solutions on such a space. Bai and Sun [9] not only established the aforesaid existence criterion regarding positive solutions but also derived their multiplicity to a singular BVP. In [10], Islam et al. proved some results about the existence of a solution of an infinite system of integral equations by using a new family of contractions entitled the generalized α-admissible Hardy–Rogers contractions in cone b2-metric spaces over Banach algebras. In [11], Shoaib et al. studied other existence results via f-contractions of Nadler type in 2020. After that, recently, Ali et al. [12] considered a nonlinear fractional differential equation equipped with the integral type boundary conditions and proved the existence results with the help of topological degree theory.Recently, Boulfoul et al. [13] considered a weighted space of the Banach type by defining a nonlinear integro-differential BVP on an unbounded domain and checked two properties of existence and uniqueness under fixed point techniques. In the sequel of this path, a new type of generalized fractional operator in the Hilfer settings was utilized by Shatanawi et al. to prove the main existence conditions for a nonlocal multipoint BVP [14]. Zada et al. [15] continued similar area of existence theory by studying an impulsive integro-differential BVP in the sense of Riemann–Liouville and reviewed the property of the stability. In 2021, the authors in [16, 17] used two numerical algorithms for approximating solutions of two similar multiorder multiterm BVPs, with RL operators and the generalized RL-ψ-operators. By developing studies in this regard, new classes of BVPs were designed in the context of p-Laplacian operators. Khan et al. introduced an advanced singular fractional in the framework of the Atangana–Baleanu derivation operators along with p-Laplacian structure [18]; then, in another work, Hasib Khan et al. [19] extended the above system in the form of a p-Laplacian hybrid BVP.Thereafter, some researchers expanded their existence results on real systems and models. For example, Rizwan et al. [20] designed a switched system of coupled impulsive implicit model of Langevin equation, and both stability and existence theories can be found in their paper for such a fractional physical structure. Etemad et al. [21] continued their study by considering an inclusion BVP of the Caputo–Hadamard type and accomplished some results by terms of a new notion called end-points along with approximate property for these points. In the same year, Samei et al. [22] reformulated similar Caputo–Hadamard inclusion BVP of the hybrid type and turned to deriving existence criteria. The theory of topological degree is another tool for obtaining some results regarding solutions of a multiterm delay BVP which Sher et al. implemented it in their newly published article [23]. Abdeljawad et al. [24] modeled a new fractional BVP and proved the relevant existence theorems on the extended b-metric space. Also, Boutiara et al. [25] applied the Caputo type and Erdélyi–Kober type operators for modeling a nonlocal fractional BVP and deriving existence aspects of solutions.Along with above works, some researchers generalized existence theorems by terms of the existing notions in quantum fractional calculus. For instance, Etemad et al. [26] investigated a 3-point quantum inclusion BVP in the context of α-ψ-contractions. Sitthiwirattham [27] studied q-integro-difference BVP containing different values of q and orders, and Sitho et al. [28] designed a noninstantaneous impulsive q-integro-difference BVP with quantum Hahn operators. Samei et al. also introduced a singular quantum BVP for the first time [29]. Even, some applications of fixed point can be followed in the papers regarding mathematical biological models (see [30–32]).In [33], Ntouyas and Tariboon discussed the multiorder BVP with a linear combination of fractional integrals in the BVCs:(1)rD0σ´1zs+1−r´D0σ´2zs=us,zs,0<s<τ∗,z0=0,r´DI0ν´1zτ∗+1−r´DI0ν´2zτ∗=a0,where D0η´ stands for the Riemann–Liouville η´th derivative with η´∈σ´1,σ´2 provided that 1<σ´1,σ´2<2 and I0η´ is the Riemann–Liouville η´th integral with η´∈ν´1,ν´2, a0∈ℝ, 0<r´≤1, and 0≤r´≤1. Green’s function for this corresponding problem has been investigated and some existence results have been obtained using fixed point theorems. Xu et al. [34] turned to investigating the existence property and Hyers–Ulam stability to fractional multiple order BVP:(2)r´D0σ´zs+D0ν´zs=us,zs,0<s<τ∗,z0=0,r´D0σ´1zτ∗+I0σ´2zs0=a0,where D0σ´ and D0ν´ are Riemann–Liouville fractional derivatives, with 1<σ´≤2 and 1≤ν´<σ´, 0<r´≤1, 0≤r´≤1, 0≤σ´1≤σ´−ν´, σ´2≥0, a0∈ℝ, and 0<s0<τ∗.Inspired by the works cited above and to continue the study of existence theory in the context of fractionalBVPs, we focus on surveying some results regarding solutions of the following Liouville–Caputo integro-differential BVP:(3)r´DC0+σ´1+1−r´I0+σ´2zs=us,zs+DC0+σ´3u^s,zs,0≤s≤τ∗,z0=0,r´DC0+v´1zτ∗+1−r´DC0+v´1zτ∗=a0,so that DC0+η´1 is the Caputo η´th derivative with η´∈σ´1,σ´3,ν´1,ν´2, a0∈ℝ and I0+σ´2 stands for the Riemann–Liouville fractional σ´2th integral such that s1<σ´1,σ´3≤2,σ´1>σ´3, 0<σ´2≤1, 0<r´≤1, 0≤r´≤1, 0<ν´1,ν´2<σ´1−σ´3, and u,u^∈CO×ℝ,ℝ are two given functions, where O≕0,τ∗.In fact, the existing ideas of two papers published in [33, 34] motivate us to design a combined model of Liouville–Caputo integro-differential BVP. Also, by assuming special values for coefficients, our problem is reduced to some simpler forms of the fractional boundary value problems. In other words, if r´=1 and τ∗=1, then Liouville–Caputo BVP (3) is reduced to the following one:(4)Dℂ0+σ´1zs=us,zs+Dℂ0+σ´3u^s,zs,s∈O,z0=0,r´Dℂ0+ν´1u1+1−r´CD0+ν´2u1=a0,and if r´=0 and τ∗=1, then Liouville–Caputo BVP (3) becomes(5)r´DC0+σ´1+1−r´I0+σ´σ´2zs=us,zs+DC0+σ´3u^s,zs,s∈O,z0=0,DC0+ν´2u1=a0.Precisely, and in comparison to some similar works, we have studied a more general problem in which we have illustrated our theoretical results by numerical examples and plots. These items make the novelty of our work because it is important for us that we can analyze the mentioned system analytically, numerically, and graphically. We also consider two different nonlinear terms in the right-hand side of the problem to cover a vast range of nonlinear functions arising in particular real fractional nonlinear mathematical models.This paper is organized as follows. In Section2, we recollect several assembled definitions of fractional calculus, useful lemmas, and some theorems about the fixed point that we need subsequently. Section 3 is divided into three parts. First, we utilize Banach’s criterion of contraction mapping to establish our result regarding unique solution. In the next subsection, we give the proof of the first fundamental existence theorem of this paper by utilizing a fixed point criterion due to Krasnoselskii. Also, in the third subsection, we verify another result regarding existence theory with the aid of Leray–Schauder theorem. Along with these, appropriate applications in the framework of illustrative examples are provided, in which numerical simulation and the corresponding data are given in each part graphically. Finally, in Section 4, we point out the conclusions of our article.


## 2. Preliminaries

Before establishing our main results, we need to present some useful definitions and properties which help us to prove the essential lemmas and theorems.Definition 1.
(see [3]). Let σ´>0 and z:0,+∞⟶ℝ be continuous. The integral(6)I0+σ´zs=1Γσ´∫0ss−ξσ´−1zξdξis named as the fractional integral in the Riemann–Liouville (FRL-integral) framework of order σ´ provided this integral possesses a finite value.Definition 2.
(see [3]). Let σ´>0, κ=σ´+1, and z:0,+∞⟶ℝ belong to ACκ0,∞,ℝ. Then, the integral(7)DC0+σ´zs=1Γκ−σ´∫0ss−ξκ−σ´−1zκξdξis named as the fractional derivative in the Caputo framework of order σ´ provided its value is finite.Remark 1.
We have the following:(E1) For0≤ν´<σ´, the equality Dℂ0+ν´I0+σ´zs=I0+σ´−ν´zs holds.(E2) Forσ´>−1 such that σ´≠ν´−j,j=1,2,…,n, we have for s≥0,(8)Dℂ0+σ´sν´=Γ1+ν´Γν´−σ´+1sν´−σ´andCD0+σ´sν´−j=0,j=1,2,…,n.Proposition 1 (see [35]).
Suppose thatz is contained in the space ℒO∩CO and κ=σ´+1, where O=0,1. Then,(9)I0+σ´CDℂ0+σ´zs=zs+d1+d2s+d3s2+⋯+dκ−1sκ,with d1,d2,…,dk−1∈ℝ.Lemma 1 (see [36]).
LetV be a nonempty, closed, and convex subset of a Banach space Ξ. Let Υ1,Υ2 be such that HiΥ1v+Υ2w∈V whenever z,w∈V, HiiΥ1 is compact and continuous, and HiiiΥ2 is contraction. Then, ∃z∈V with z=Υ1z+Υ2z.Lemma 2 (see [37]).
LetΞ be a Banach space, V⊂Ξ be closed and convex in Ξ, E⊂V be open, 0∈E, and let Θ:E¯⟶V be completely continuous. Then, either Hi a fixed point is found for Θ in E or Hii∃z∈∂Ε and ℓ´∈O with z=ℓ´Θz, where O=0,1.This key lemma will be useful for our study.Lemma 3.
Let1<σ´1,σ´3≤2,σ´1>σ´3, 0<σ´2≤1, 0<r´≤1, 0≤r´≤1, and 0<ν´1,ν´2<σ´1−σ´3. Then, the integral equation(10)zs=r´−1r´Γσ´1+σ´2∫0ss−ξσ´1+σ´2−1zξd+1r´Γσ´1∫0ss−ξσ´1−1Uξdξ+1r´Γσ´1−σ´3∫0ss−ξσ´1−σ´3−1U^ξdξ+Πsa0−r´r´−1r´Γσ´1+σ´2−ν´1∫0τ∗τ∗−ξσ´1+σ´2−ν´1−1zξdξ−r´r´Γσ´1−ν´1∫0τ∗τ∗−ξσ´1−ν´1−1Uξdξ−r´r´Γσ´1−a0∫0τ∗τ∗−ξσ´1−σ´3−1U^ξdξ−1−r´r´−1r´Γσ´1+σ´2−ν´2∫0τ∗τ∗−ξσ´1+σ´2−ν´2−1zξdξ−1−r´r´Γσ´1−ν´2∫0τ∗τ∗−ξσ´1−ν´2−1Uξdξ−1−r´r´Γσ´1−σ´3∫0τ∗τ∗−ξσ´1−σ´3−1U^ξdξ,with(11)Π=r´Γ2−ν´1+1−r´Γ2−ν´2−1,is the solution of the linear fractional BVP:(12)r´DC0+σ´1+1−r´I0+σ´2zs=Us+CD0+σ´3U^s,0≤s≤τ∗,z0=0,r´DC0+ν´1zτ∗+1−r´DC0+ν´2zτ∗=a0.Proof.
In view of the first equation of (12), we can write(13)DC0+σ´1zs=r´−1r´I0+σ´2zs+1r´Us+1r´DC0+σ´3U^s.
Taking theσ´1th FRL-integral on (13), we find(14)zs=r´−1r´I0+σ´1+σ´2zs+1r´I0+σ´1Us+1r´I0+σ´1−σ´3U^s+d1+d2s=r´−1r´Γσ´1+σ´2∫0ss−ξσ´1+σ´2−1zξdξ+1r´Γσ´1∫0ss−ξσ´1−1Uξdξ+1r´Γσ´1−σ´3∫0ss−ξσ´1−σ´3−1U^ξdξ+d1+d2s,where d1,d2∈ℝ. The first boundary condition of (12) gives us d1=0; then,(15)zs=r´−1r´Γσ´1+σ´2∫0ss−ξσ´1+σ´2−1zξdξ+1r´Γσ´1∫0ss−ξσ´1−1Uξdξ+1r´Γσ´1−σ´3∫0ss−ξσ´1−σ´3−1U^ξdξ+d2s,and by applying the ηth-Caputo derivative η∈ν´1,ν´2 with 0<η<σ´1−σ´3 to (15), we obtain(16)DC0+ηzs=r´−1r´Γσ´1+σ´2−η∫0ss−ξσ´1+σ´2−η´−1zsdξ+1r´Γσ´1−η∫0ss−ξσ´1−η´−1Uξdξ+1r´Γσ´1−σ´3−η∫0ss−ξσ´1−σ´3−η´−1U^ξdξ+d2Γ2−ηs1−η´,because DC0+ηs=Γ2/Γ2−ηs1−η´. Taking η´=ν´1 and η´=ν´2 in expression (16) and applying the second boundary condition of (12), we get(17)a0=r´r´−1r´Γσ´1+σ´2−ν´1∫011−ξσ´1+σ´2−ν´1−1zsdξ+r´r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1Uξdξ+r´r´Γσ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1U^ξdξ+d2r´Γ2−ν´1+1−r´r´−1r´Γσ´1+σ´2−ν´2∫011−ξσ´1+σ´2−ν´2−1zξdξ+1−r´r´Γσ´1−ν´2∫011−ξσ´1−ν´2−1Uξdξ+1−r´r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1U^ξdξ+d21−r´Γ2−ν´2.
Therefore,(18)d2=Πa0−r´r´−1r´Γσ´1+σ´2−ν´1∫011−ξσ´1+σ´2−ν´1−1zξdξ−r´r´Γβ1−ν´1∫011−ξσ´1−ν´1−1Uξdξ−r´r´Γσ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1U^ξdξ−1−r´r´−1r´Γσ´1+σ´2−ν´2∫011−ξσ´1+σ´2−ν´2−1zξdξ−1−r´r´Γσ´1−ν´2∫011−ξσ´1−ν´2−1Uξdξ−1−r´r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1U^ξdξ.
By substituting the value ofd2 in equation (15), we obtain integral equation (10). This ends the proof.


## 3. Basic Theorems with Illustrative Examples

LetO=0,1 throughout the paper. Consider the Banach space CO,ℝ of all continuous functions with the norm of uniform convergence(19)z=sups∈Ozs.In accordance with Lemma3, it is obvious that we can transform our BVP (3) to the following fixed point problem z=Pz, where P is an operator P:CO,ℝ⟶CO,ℝ defined as(20)Pzs=r´−1r´Γσ´1+σ´2∫0ss−ξσ´1+σ´2−1zξdξ=r´−1r´Γσ´1+σ´2∫0ss−ξσ´1+σ´2−1zξdξ+1r´Γσ´1−σ´3∫0ss−ξσ´1−σ´3−1u^ξ,zξdξ+Πsa0−r´r´−1r´Γσ´1+σ´2−ν´1∫011−ξσ´1+σ´2−ν´1−1zξdξ−r´r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1uξ,zξdξ−r´r´Γσ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1u^ξ,zξdξ−1−r´r´−1r´Γσ´1+σ´2−ν´2∫011−ξσ´1+σ´2−ν´2−1zξdξ−1−r´r´Γσ´1−ν´2∫011−ξσ´1−ν´2−1uξ,zξdξ−1−r´r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1u^ξ,zξdξ.Therefore,BVP (3) admits a solution equivalent to saying that P has a fixed point.


### 3.1. Banach Principle and Unique Solution

First, we apply Banach’s principle of contraction mapping to prove our result of existence and uniqueness. To have computations with more convenience and clarity, we use these notations:(21)η1=1−r´r´Γσ´1+σ´2+1+r´1−r´Πr´Γσ´1+σ´2−ν´1+1+1−r´1−r´Πr´Γσ´1+σ´2−ν´2+1,(22)η2=1r´Γσ´1+1+r´Πr´Γσ´1−ν´1+1+1−r´Πr´Γσ´1−ν´2+1,(23)η3=1r´Γσ´1−σ´3+1+r´Πr´Γσ´1−σ´3−ν´1+1+1−r´Πr´Γσ´1−σ´3−ν´2+1.Theorem 1.
Assume thatu,u^:O×ℝ⟶ℝ are two continuous functions subject to the following two conditions:S1: us,z−us,z⋆≤Θ1z−z⋆,S2: u^s,z−u^s,z⋆≤Θ2z−z⋆,for s∈O, z,z⋆∈ℝ, where Θ1,Θ2 are two real positive constants. If(24)η1+Θ1η2+Θ2η3<1,then the supposed BVP (3) admits a unique solution on O.Proof.
By fixingu⋆=sups∈Ous,0 and u^⋆=sups∈Ou^s,0 with the choice Δ¯1>0, so that(25)Δ¯1≥u2⋆+u^⋆η3+Πa01−η1−Θ1η2−Θ2η3,where Π is the positive constant expressed by (11), at first, we show that the image of the ball BΔ¯1 by P is included in BΔ¯1, where(26)BΔ¯1=z∈CO,ℝ:z≤Δ¯1.
So, for eachz∈BΔ¯1, we have(27)Pzs≤1−r´r´Γσ´1+σ´2∫011−ξσ´1+σ´2−1zξdξ+1r´Γσ´1∫011−ξσ´1−1uξ,zξ−uξ,0+uξ,0dξ+1r´Γσ´1−σ´3∫011−ξσ´1−σ´3−1u^ξ,zξ−uξ,0+u^ξ,0dξ+Πa0+r´1−r´r´Γσ´1+σ´2−ν´1∫011−ξσ´1+σ´2−ν´1−1zξdξ+r´r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1uξ,zξ−uξ,0+uξ,0dξ+r´r´Γσ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1u^ξ,zξ−u^ξ,0+u^ξ,0dξ+1−r´1−r´r´Γσ´1+σ´2−ν´2∫011−ξσ´1+σ´2−ν´2−1zξdξ+1−r´r´Γσ´1−ν´2∫011−ξσ´1−ν´2−1uξ,zξ−uξ,0+uξ,0dξ+1−r´r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1u^ξ,zξ−u^ξ,0+|u^ξ,0dξ≤z1−r´r´Γσ´1+σ´2+1+r´1−r´Πr´Γσ´1+σ´2−ν´1+1+1−r´1−r´Πr´Γσ´1+σ´2−ν´2+1+Θ1z+z⋆1r´Γσ´1+1+r´Πr´Γσ´1−ν´1+1+1−r´Πr´Γσ´1−ν´2+1+Θ2z+z⋆1r´Γσ´1−σ´3+1+r´Πr´Γσ´1−σ´3−ν´1+1+1−r´Πr´Γσ´1−σ´3−ν´2+1+Πa0≤Δ¯1η1+Θ1η2+Θ2η3+u⋆η2+u^⋆η3+Πa0≤Δ¯1.
This implies thatPz≤Δ¯1. Thus PBΔ¯1⊂BΔ¯1. Next, for all z,z⋆∈CO,ℝ and each s∈O, we can write(28a)Pzs−Pz⋆s≤1−r´r´Γσ´1+σ´2∫011−ξσ´1+σ´2−1zξ−z⋆ξdξ+1r´Γσ´1∫011−ξσ´1−1uξ,zξ−uξ,z⋆ξdξ+1r´Γσ´1−σ´3∫011−ξσ´1−σ´3−1u^ξ,zξ−u^ξ,z⋆ξdξ+Π|a0|+r´1−r´r´Γσ´1+σ´2−ν´1∫011−ξσ´1+σ´2−ν´1−1zξ−z⋆ξdξ+r´r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1uξ,zξ−uξ,z⋆ξdξ+r´r´Γσ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1u^ξ,zξ−u^ξ,z⋆ξdξu+1−r´1−r´r´Γσ´1+σ´2−ν´2∫011−ξσ´1+σ´2−ν´2−1zξ−z⋆ξdξ+1−r´r´Γσ´1−ν´2∫011−ξσ´1−ν´2−1uξ,zξ−uξ,z⋆ξdξ+1−r´r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1u^ξ,zξ−u^ξ,z⋆ξdξ≤z−z⋆1−r´r´Γσ´1+σ´2+1+r´1−r´Πr´Γσ´1+σ´2−ν´1+1+1−r´1−r´Πr´Γσ´1+σ´2−ν´2+1(28b)+Θ1z−z⋆1r´Γσ´1+1+r´Πr´Γσ´1−ν´1+1+1−r´Πr´Γσ´1−ν´2+1+Θ2z−z⋆1r´Γσ´1−σ´3+1+r´Πr´Γσ´1−σ´3−ν´1+1+1−r´Πr´Γσ´1−σ´3−ν´2+1=η1+Θ1η2+Θ2η3z−z⋆,which means that Pz−Pz⋆≤η1+Θ1η2+Θ2η3z−z⋆. Therefore, from (24), it follows that P is a contraction. Consequently, the Banach principle of contraction mapping ensures that P has a fixed point which represents the unique solution of our BVP (3). This ends the argument.Example 1.
Consider the Liouville–Caputo fractionalBVP(29)45DC0+5/3+15I0+1/4zs=2zs5+s21+expszs+DC0+4/3exp−szsπ2+zs,z0=0,14DC0+1/6z1+34DC0+1/2z1=5.
In the present example, we haveσ´1=5/3∈1,2, σ´2=1/4∈0,1, σ´3=4/3∈1,2, r´=4/5∈0,1, r´=1/4∈0,1, ν´1=1/6∈0,1, ν´2=1/12∈0,1, a0=5, τ∗=1, and(30)us,z=2z5+s21+expsz,u^s,z=exp−szπ2+z.
Then,(31)us,z−us,z⋆=2z5+s21+expsz−2z⋆5+s21+expsz⋆=25+s2z1+expsz−z⋆1+expsz⋆≤225z−z⋆,u^s,z−u^s,z⋆=exp−szπ2+z−expsz⋆π2+z⋆≤1π2z−z⋆,i.e., Θ1=2/25≈0.0800 and Θ2=1/π2≈0.1013. A simple computation gives us(32)Π=r´Γ2−ν´1+1−r´Γ2−ν´2−1=1/4Γ2−1/6+1−4/5Γ2−4/12−1,and(33)η1=1−r´r´Γσ´1+σ´2+1+r´1−r´Πr´Γσ´1+σ´2−ν´1+1+1−r´1−r´Πr´Γσ´1+σ´2−ν´2+1=1−4/54/5Γ5/3+1/4+1+1/41−4/5Π4/5Γ5/3+1/4−1/6+1+1−1/41−4/5Π4/5Γ5/3+1/4−1/12+1,η2=1r´Γσ´1+1+r´Πr´Γσ´1−ν´1+1+1−r´Πr´Γσ´1−ν´2+1=14/5Γ5/3+1+1/4Π4/5Γ5/3−1/6+1+1−1/4Π4/5Γ5/3−1/12+1,η3=1r´Γσ´1−σ´3+1+r´Πr´Γσ´1−σ´3−ν´1+1+1−r´Πr´Γσ´1−σ´3−ν´2+1=14/5Γ5/3−4/3+1+1/4Π4/5Γ5/3−4/3−1/6+1+1−1/4Π4/5Γ5/3−4/3−1/12+1.
So,Π≈0.9607, η1≈0.2766, η1≈1.6945, and η3≈2.7171 which lead to(34)η1+Θ1η2+Θ2η3≈0.6875<1.
Table1 shows these results. These values are plotted in Figure 1. By using the result of Theorem 1, we conclude that our BVP (29) admits only one solution on 0,1.Table 1
Numerical values ofη1, η2, η3, and Π, for r´∈0,1, in Example 1.
r´∈0,1r´Πη1η2η3η1+Θ1η2+Θ2η30.050.960 721.020 927.112 643.473 427.594 60.100.960 79.957 213.556 321.736 713.244 10.150.960 76.269 49.037 514.491 18.460 60.200.960 74.425 46.778 110.868 46.068 90.250.960 73.319 15.422 58.694 74.633 80.300.960 72.581 54.518 87.245 63.677 10.350.960 72.054 73.873 26.210 52.993 80.400.960 71.659 53.389 15.434 22.481 30.450.960 71.352 23.012 54.830 42.082 60.500.960 71.106 42.711 34.347 31.763 70.550.960 70.905 22.464 83.952 11.502 80.600.960 70.737 62.259 43.622 81.285 40.650.960 70.595 72.085 63.344 11.101 40.700.960 70.474 21.936 63.105 20.943 70.750.960 70.368 81.807 52.898 20.807 00.800.960 70.276 61.694 52.717 10.687 50.850.960 70.195 21.594 92.557 30.581 90.900.960 70.122 91.506 32.415 20.488 10.950.960 70.058 21.427 02.288 10.404 2Figure 1
Graphical representation ofη1+Θ1η2+Θ2η3 in Example 1. (a) r∈0,1, r´=1/4. (b) r=4/5r´∈0,1.
(a)(b)


### 3.2. Existence Result Based on Krasnoselskii’s Criterion

Our existence analysis in this part is a consequence of Krasnoselskii’s criterion (Lemma1). For this fact, we introduce two operators P1 and P2 defined on the ball(35)BΔ¯2=z∈CO,ℝ:zΔ¯2,such that, for all s∈O,(36)P1zs=r´−1r´Γσ´1+σ´2∫0ss−ξσ´1+σ´2−1zξdξ−Πsr´r´−1r´Γσ´1+σ´2−ν´1∫011−ξσ´1+σ´2−ν´1−1zξdξ−Πs1−r´r´−1r´Γσ´1+σ´2−ν´2∫011−ξσ´1+σ´2−ν´2−1zξdξ,and(37)P2zs=1r´Γσ´1∫0ss−ξσ´1−1uξ,zξdξ+1r´Γσ´1−σ´3∫0ss−ξσ´1−σ´3−1u^ξ,zξdξ+Πsa0−r´r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1uξ,zξdξ−r´r´Γσ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1u^ξ,zξdξ−1−r´r´Γβ1−ν´2∫011−ξσ´1−ν´2−1uξ,zξdξ−1−r´r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1u^ξ,zξdξ.Theorem 2.
Consider the continuous functionsu,u^:O×ℝ⟶ℝ which, respectively, satisfy the conditions S1 and S22 of Theorem 1. Furthermore, suppose thatS3: us,z≤ϱ1s,S4: u^s,z≤ϱ2s,for s,z∈O×ℝ, and ϱj∈CO,ℝ+, j=1,2. If η1<1 which is defined in equation (21), then the supposed BVP (3) admits at least one solution defined on O.Proof.
Putϱj=sups∈Oϱjs, j=1,2. We choose Δ¯2 so that(38)Δ¯2≥ϱ1η2+ϱ2η3+Πa01−η1.
In the first place, we prove thatP1z+P2z⋆∈BΔ¯2. So, for all z,z⋆∈BΔ¯2, we have(39)P1zs+P2z⋆s≤1−r´r´Γσ´1+σ´2∫011−ξσ´1+σ´2−1zξdξ+1r´Γσ´1∫011−ξσ´1−1uξ,z⋆ξdξ+Πr´1−r´r´Γσ´1+σ´2−ν´1∫011−ξσ´1+σ´2−ν´1−1zξdξ+Π1−r´1−r´r´Γσ´1+σ´2−ν´2∫011−ξσ´1+σ´2−ν´2−1zξdξ+1r´Γσ´1−σ´3∫011−ξσ´1−σ´3−1u^ξ,z⋆ξdξ+Πa0+r´r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1uξ,z⋆ξdξ+r´r´Γσ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1u^ξ,z⋆ξdξ+1−r´r´Γσ´1−ν´2∫011−ξσ´1−ν´2−1uξ,z⋆ξdξ+1−r´r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1u^ξ,z⋆ξdξ≤z1−r´r´Γσ´1+σ´2+1+r´1−r´Πr´Γσ´1+σ´2−ν´1+1+1−r´1−r´Πr´Γσ´1+σ´2−ν´2+1+ϱ11r´Γσ´1+1+r´Πr´Γσ´1−ν´1+1+1−r´Πr´Γσ´1−ν´2+1+ϱ21r´Γσ´1−σ´3+1+r´Πr´Γσ´1−σ´3−ν´1+1+1−r´Πr´Γσ´1−σ´3−ν´2+1+Πa0≤Δ¯2η1+ϱ1η2+ϱ2η3+Πa0≤Δ¯2.
Thus,‖P1z+P2z⋆‖≤Δ¯2, which means that P1z+P2z⋆∈BΔ¯2. Now, we establish that P1 is a contraction. For z,z⋆∈BΔ¯2, we can write(40)P1zs−P1z⋆s≤1−r´r´Γσ´1+σ´2∫011−ξσ´1+σ´2−1zξ−z⋆ξdξ+Πr´1−r´r´Γσ´1+σ´2−ν´1∫011−ξσ´1+σ´2−ν´1−1zξ−z⋆ξdξ+Π1−r´1−r´r´Γσ´1+σ´2−ν´2∫011−ξσ´1+σ´2−ν´2−1zξ−z⋆ξdξ≤z−z⋆1−r´r´Γσ´1+σ´2+1+r´1−r´Πr´Γσ´1+σ´2−ν´1+1+1−r´1−r´Πr´Γσ´1+σ´2−ν´2+1=η1z−z⋆.
Then,(41)P1z−P1z⋆≤η1z−z⋆.
From the conditionη1<1, it follows that P1 is a contraction mapping. On the other hand, we know that the continuity of P2 occurs immediately from that of the functions u and u^. Also, it is simple to establish that for z∈BΔ¯2,(42)P2z≤ϱ1η2+ϱ2η3.
In other words,P2 is uniformly bounded on BΔ¯2. In this moment, we need to show that P2 is equicontinuous. Let(43)u⋆=sups,z∈O×ℝus,z,andu^⋆=sups,z∈O×ℝu^s,z.
This allows us to write, for anys1,s2∈O×O, where s1<s2, and for all z∈BΔ¯2:(44a)P2us2−P2us1=1r´Γσ´1∫0s2s2−ξσ´1−1uξ,zξdξ−∫0s1s1−ξσ´1−1uξ,zξdξ+1r´Γσ´1−σ´3∫0s2s2−ξσ´1−σ´3−1u^ξ,zξdξ−∫0s1s1−ξσ´1−σ´3−1u^ξ,zξdξ+Πs2−s1a0−r´r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1uξ,zξdξ−r´r´Γσ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1u^ξ,zξdξ−1−r´r´Γσ´1−ν´2∫011−ξσ´1−ν´2−1uξ,zξdξ−1−r´r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1u^ξ,zξdξ≤u⋆r´Γσ´1∫0s1s2−ξσ´1−1−s1−ξσ´1−1dξ+∫s1s2s2−ξσ´1−1dξ+u^⋆r´Γσ´1−σ´3∫0s1s2−ξσ´1−σ´3−1−s1−ξσ´1−σ´3−1dξ+∫s1s2s2−ξσ´1−σ´3−1dξ+Πs2−s1a0+r´ν⋆r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1dξ+r´u^⋆r´Γσ´σ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1dξ(44b)+1−r´u⋆r´Γσ´1−ν´2∫011−ξσ´1−ν´2−1dξ+1−r´u^⋆r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1dξ≤u⋆r´Γσ´1s2σ´1−s1σ´1+u^⋆r´Γσ´1−σ´3s2σ´1−σ´3−s1σ´1−σ´3+Πs2−s1a0+r´u⋆r´Γσ´1−ν´1+1+r´u^⋆r´Γσ´1−σ´3−ν´1+1+1−r´u⋆r´Γσ´1−ν´2+1+1−r´u^⋆r´Γσ´1−σ´3−ν´2+1.
The right-hand side of the above inequality is not dependent on and converges to 0, ass2−s1⟶0. This means that P2 is equicontinuous and admits the relative compactness on BΔ¯2. Thus, Arzelà–Ascoli theorem ensures that P2 is compact on BΔ¯2. Consequently, our BVP (3) possesses at least one solution on O. The proof is completed.Example 2.
Consider the following Liouville–CaputoBVP:(45)223DC0+15/8+123I0+2/7zs=sin2sπ+s2zszs+1+DC0+3/213exps+1zsπ2+expszs,z0=0,25DC0+1/8z1+35DC0+1/4z1=227.
Now, we haveσ´1=15/8∈1,2., σ´2=2/7∈0,1, σ´3=3/2∈0,1, r´=22/23∈0,1, r´=2/5∈0,1, ν´1=1/8∈0,1, ν´2=1/4∈0,1, a0=22/7∈ℝν´1, τ∗=1, and(46)us,z=sin2sπ+s2zz+1,u^s,z=13exps+1zπ2+expsz.
Hence,(47)us,z−us,z⋆=sin2sπ+s2zz+1−sin2sπ+s2z⋆z⋆+1=sin2sπ+s2zz+1−z⋆z⋆+1≤1π2z−z⋆,u^s,z−u^s,z⋆=13exps+1zπ2+expsz−13exps+1z⋆π2+expsz⋆=13exps+1zπ2+expsz−z⋆π2+expsz⋆≤14z−z⋆,i.e., Θ1=2/π2, Θ2=1/4, and accordingly,(48)Π=r´Γ2−ν´1+1−r´Γ2−ν´2−1=2/5Γ2−1/8+1−2/5Γ2−1/4−1,and
By some computations, we getΠ≈0.9325, η1≈0.0415<1, η2≈1.2287, η3≈2.2275, and(49)η1+Θ1η2+Θ2η3≈0.8473<1.
Also, we get(50)us,z=sin2sπ+s2zz+1=sin2sπ+s2zz+1≤sin2sπ+s2≕ϱ1s,u^s,z=13exps+1zπ2+expsz=13exps+1zπ2+expsz≤13exps+1≕ϱ2s.
Table2 shows these results. These numerical data are plotted in Figure 2. Then, Theorem 2 states that the Liouville–Caputo BVP (45) admits at least one solution on O.Table 2
Numerical values ofη1, η2, η3, and Π, for r∈0,1, in Example 2.
r´∈0,1r´Πη1η2η3η1+Θ1η2+Θ2η30.040.932 520.078 127.032 449.004 137.807 10.090.932 59.582 813.516 224.502 018.447 20.130.932 56.084 39.010 816.334 711.993 90.170.932 54.335 16.758 112.251 08.767 30.220.932 53.285 55.406 59.800 86.831 30.260.932 52.585 84.505 48.167 35.540 60.300.932 52.086 03.861 87.000 64.618 70.350.932 51.711 23.379 16.125 53.927 30.390.932 51.419 73.003 65.444 93.389 50.430.932 51.186 42.703 24.900 42.959 30.480.932 50.995 62.457 54.454 92.607 30.520.932 50.836 62.252 74.083 72.314 00.570.932 50.702 02.079 43.769 52.065 80.610.932 50.586 71.930 93.500 31.853 10.650.932 50.486 71.802 23.266 91.668 70.700.932 50.399 31.689 53.062 81.507 30.740.932 50.322 11.590 12.882 61.365 00.780.932 50.253 51.501 82.722 41.238 50.830.932 50.192 11.422 82.579 21.125 20.870.932 50.136 91.351 62.450 21.023 30.910.932 50.086 91.287 32.333 50.931 222/23≃0.960.932 50.041 51.228 72.227 50.847 31.000.932 50.000 01.175 32.130 60.770 8(51)η1=1−r´r´Γσ´1+σ´2+1+r´1−r´Πr´Γσ´1+σ´2−ν´1+1+1−r´1−r´Πr´Γσ´1+σ´2−ν´2+1η2=1r´Γσ´1+1+r´Πr´Γσ´1−ν´1+1+1−r´Πr´Γσ´1−ν´2+1=1−22/2322/23Γ15/8+2/7+1+2/51−22/23Π22/23Γ15/8+2/7−1/4+1+1−2/51−22/23Π22/23Γ15/8+2/7−1/4+1,=122/23Γ15/8+1+2/5Π22/23Γ15/8−1/8+1+1−2/5Π22/23Γ15/8−1/4+1,η3=1r´Γσ´1−σ´3+1+r´Πr´Γσ´1−σ´3−ν´1+1+1−r´Πr´Γσ´1−σ´3−ν´2+1=122/23Γ15/8−3/2+1+2/5Π22/23Γ15/8−3/2−1/8+1+1−2/5Π22/23Γ15/8−3/2−1/4+1.Figure 2
Graphical representation ofη1+Θ1η2+Θ2η3 and η1<1 in Example 2. (a) r∈0,1,r´=2/5. (b) η1vsr=22/23,r´=2/5.
(a)(b)


### 3.3. Existence Result by Using Nonlinear Alternative of Leray–Schauder

Another result of existence criterion is realized by implementing the hypotheses in Lemma2. The desired criterion is proved below by the next theorem.Theorem 3.
Assume thatu,u^:O×ℝ⟶ℝ are two continuous functions which satisfy the following assumption.
S5: there are two continuous nondecreasing functions ℘1,℘2:0,+∞⟶0,+∞ and two functions ℏ1,ℏ2∈CO,ℝ provided that(52)us,z≤ℏ1s℘1z,u^s,z≤ℏ2s℘2z,for all s,z∈O×ℝ; moreover, the following assumption holds.
S6: there exists a positive real constant Δ¯3 so that(53)ℏ1℘1Δ¯3η2+ℏ2℘2Δ¯3η3+Πa0Δ¯31−η1>1,η1<1.Then, the Liouville–CaputoBVP (3) has at least one solution on O, where Π,η1,η2,η3 stand for the same constants introduced, respectively, by expressions (11), (21), (22), and (23).Proof.
Consider again the operatorP expressed as (20). First, we will prove that P maps bounded sets into bounded sets in CO,ℝ. Let(54)Bδ=z∈CO,ℝ:z≤δbe a bounded set in CO,ℝ, where δ is a real positive number δ>0. For each s∈O, we have(55)Pzs≤1−r´r´Γσ´1+σ´2∫011−ξσ´1+σ´2−1zξdξ+1r´Γσ´1∫011−ξσ´1−1uξ,zξdξ+1r´Γσ´1−σ´3∫011−ξσ´1−σ´3−1u^ξ,zξdξ+Π|a0|+r´1−r´r´Γσ´1+σ´2−ν´1∫011−ξσ´1+σ´2−ν´1−1zξdξ+r´r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1uξ,zξdξ+r´r´Γσ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1u^ξ,zξdξ+1−r´1−r´r´Γσ´1+σ´2−ν´2∫011−ξσ´1+σ´2−ν´2−1zξdξ+1−r´r´Γσ´1−ν´2∫011−ξσ´1−ν´2−1uξ,zξdξ+1−r´r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1uξ,zξdξ≤zη1+ℏ1℘1zη2+ℏ2℘2zη3+Πa0≤δη1+ℏ1℘1δη2+ℏ2℘2δη3+Πa0,and consequently,(56)Pz≤δη1+ℏ1℘1δη2+ℏ2℘2δη3+Πa0.
The next property is that we prove that the operatorP maps the bounded sets to the equicontinuous sets. Let(57)u⋆=sups,z∈O×Bδus,z,andu^⋆=sups,z∈O×Bδu^s,z.
So, fors1,s2∈O with s1<s2 and z∈Bδ, we have(58a)Pzs2−Pzs1≤δ1−r´r´Γσ´1+σ´2∫0s1s2−ξσ´1+σ´2−1−s1−ξσ´1+σ´2−1dξ+∫s1s2s2−ξσ´1+σ´2−1dξ+u⋆r´Γσ´1∫0s1s2−ξσ´1−1−s1−ξσ´1−1dξ+∫s1s2s2−ξσ´1−1dξ+u^⋆r´Γσ´1−σ´3∫0s1s2−ξσ´1−σ´3−1−t1−ξσ´1−σ´3−1dξ+∫s1s2s2−ξσ´1−σ´3−1dξ+Πs2−s1a0+r´1−r´δr´Γσ´1+σ´2−ν´1∫011−ξσ´1+σ´2−ν´1−1dξ+r´u⋆r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1dξ+r´u^⋆r´Γσ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1dξ+1−r´1−r´δr´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1dξ(58b)+1−r´u⋆r´Γσ´1−ν´2∫011−ξσ´1−ν´2−1dξ+1−r´u^⋆r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1dξ≤δ1−r´s2σ´1+σ´2−s1σ´1+σ´2r´Γσ´1+σ´2+1+Πr´1−r´s2−s1r´Γσ´1+σ´2−ν´1+1+Π1−r´1−r´s2−s1r´Γσ´1−σ´3−ν´2+1+u⋆s2σ´1−s1σ´1r´Γσ´1+1+Πr´s2−s1r´Γσ´1−ν´1+1+Π1−r´s2−s1r´Γσ´1−ν´2+1+u^⋆s2σ´1−σ´3−s1σ´1−σ´3r´Γσ´1−σ´3+1+Πr´s2−s1r´Γσ´1−σ´3−ν´1+1+Π1−r´s2−s1r´Γσ´1−σ´3−ν´2+1+Πa0s2−s1.
Note that the second side of the last inequality is not dependent onz and goes to 0 when s2−s1⟶0. Hence, by Arzelà–Ascoli theorem, it is figured out that P is completely continuous. At this moment, we have established the boundedness of the set of solutions for the operator equation z=ℓ´Pz, where ℓ´∈O. Let now z be a solution of BVP (3). With the same arguments used in (56), one can find(59)z≤zη1+ℏ1℘1zη2+ℏ2℘2zη3+Πa0,and we can also write it as(60)z1−η1ℏ1℘1zη2+ℏ2℘2zη3+Πa0≤1.
From assumptionS6, there exists a constant Δ¯3>0 so that Δ¯3≠z. Consider the set(61)Ε=z∈CO,ℝ:z<Δ¯3.
It was proved thatP:E¯⟶CO,ℝ is a continuous and completely continuous operator. The selection of the set E allows us to confirm that there is no z∈∂Ε which satisfies z=ℓ´Pz, for ℓ´∈O. Hence, the requirements of Lemma 2 ensure that E involves a fixed point z∗∈Ε¯ which stands for the solution to our Liouville–Caputo BVP (3), and the proof is completed.Example 3.
Consider the Liouville–CaputoBVP(62)1719DC0+19/2+219I0+5/12zs=2101s+1zs2zs+1+1+DC0+17/121103exps+1zs3z2s+1+5z2sz2s+3,u0=0,511DC0+1/8z1+611DC0+1/7z1=47.
Now, we haveσ´1=19/12∈1,2, σ´2=5/12∈0,1, σ´3=17/12∈1,2, r´=17/19∈0,1, r´=5/1∈0,1, τ∗=1, ν´1=1/8∈0,1, ν´2=1/7∈0,1, a0=4/7, and(63)us,z=2101s+1zs2zs+1+1,u^s,z=1103exps+1zs3z2s+1+5z2sz2s+3,and(64)us,z≤2101s+1z+1,z^s,z≤1103exps+1z+5.
Then,(65)ℏ1s=2101s+1,℘1z=z+1,ℏ2s=1103exps+1,℘2z=z+5.
On the other hand(66)Π=r´Γ2−ν´1+1−r´Γ2−ν´2−1=5/11Γ2−1/8+1−5/11Γ2−1/7−1,and(67)η1=1−r´r´Γσ´1+σ´2+1+r´1−r´Πr´Γσ´1+σ´2−ν´1+1+1−r´1−r´Πr´Γσ´1+σ´2−ν´2+1=1−17/1917/19Γ19/12+5/12+1+5/111−17/19Π17/19Γ19/12+5/12−1/8+1+1−5111−17/19Π/17/19Γ19/12+5/12−1/7+1,η2=1r´Γσ´1+1+r´Πr´Γσ´1−ν´1+1+1−r´Πr´Γσ´1−ν´2+1=117/19Γ19/12+1+5/11Π17/19Γ19/12−1/8+1+1−5/11Π17/19Γ19/12−1/7+1,η3=1r´Γσ´1−σ´3+1+r´Πr´Γσ´1−σ´3−ν´1+1+1−r´Πr´Γσ´1−σ´3−ν´2+1=117/19Γ19/12−17/12+1+5/11Π/17/19Γ19/12−17/12−18+1+1−5/11Π/17/19Γ19/12−17/12−17+1.
A simple computation leads toΠ≈0.9504, η1≈0.1219, η2≈1.6195, and η3≈2.2857. By solving the inequality (Table 3)(68)A=ℏ1℘1Δ¯3η2+ℏ2℘2Δ¯3η3+Πa0Δ¯31−η1=2×1.6195/101Δ¯3+1+2.2857/40Δ¯3+5+0.9504×4/71−0.1219Δ¯3>1,we get A>1.1000>1. Table 4 shows these data. These numerical values are plotted in Figure 3.
Then, the assumption6 holds for any Δ¯3>0.8456. Consequently, from Theorem 3, we conclude that for Liouville–Caputo BVP (62), at least one solution is found on O.Table 3
Numerical values ofη1, η2, η3, and Π, for and r´´=5/11, in Example 3.
r´Πη1η2η30.050.950 418.650 527.530 838.856 90.110.950 48.807 213.765 419.428 40.160.950 45.526 19.176 912.952 30.210.950 43.885 56.882 79.714 20.260.950 42.901 25.506 27.771 40.320.950 42.245 04.588 56.476 10.370.950 41.776 23.933 05.551 00.420.950 41.424 73.441 34.857 10.470.950 41.151 33.059 04.317 40.530.950 40.932 52.753 13.885 70.580.950 40.753 62.502 83.532 40.630.950 40.604 42.294 23.238 10.680.950 40.478 22.117 82.989 00.740.950 40.370 01.966 52.775 50.790.950 40.276 31.835 42.590 50.840.950 40.194 31.720 72.428 617/19≃0.890.950 40.121 91.619 52.285 70.950.950 40.057 61.529 52.158 71.000.950 40.000 01.449 02.045 1Table 4
Numerical results ofΔ¯3 and A based on Table 3 for η1, η2, η3, and Π in Example 3.
Π=0.9504, η1=0.1219, η2=1.6195, η3=2.2857Δ¯3A>10.520.535 70.560.572 00.600.607 60.640.642 60.680.677 00.720.710 90.760.744 20.800.776 90.840.809 10.880.840 80.920.872 00.960.902 71.000.932 91.040.962 61.080.991 91.121.0207>11.161.049 11.201.077 01.241.104 61.281.131 71.321.158 4Figure 3
Graphical representation ofA>1 for s∈0,1 vs.Δ¯3


## 3.1. Banach Principle and Unique Solution

First, we apply Banach’s principle of contraction mapping to prove our result of existence and uniqueness. To have computations with more convenience and clarity, we use these notations:(21)η1=1−r´r´Γσ´1+σ´2+1+r´1−r´Πr´Γσ´1+σ´2−ν´1+1+1−r´1−r´Πr´Γσ´1+σ´2−ν´2+1,(22)η2=1r´Γσ´1+1+r´Πr´Γσ´1−ν´1+1+1−r´Πr´Γσ´1−ν´2+1,(23)η3=1r´Γσ´1−σ´3+1+r´Πr´Γσ´1−σ´3−ν´1+1+1−r´Πr´Γσ´1−σ´3−ν´2+1.Theorem 1.
Assume thatu,u^:O×ℝ⟶ℝ are two continuous functions subject to the following two conditions:S1: us,z−us,z⋆≤Θ1z−z⋆,S2: u^s,z−u^s,z⋆≤Θ2z−z⋆,for s∈O, z,z⋆∈ℝ, where Θ1,Θ2 are two real positive constants. If(24)η1+Θ1η2+Θ2η3<1,then the supposed BVP (3) admits a unique solution on O.Proof.
By fixingu⋆=sups∈Ous,0 and u^⋆=sups∈Ou^s,0 with the choice Δ¯1>0, so that(25)Δ¯1≥u2⋆+u^⋆η3+Πa01−η1−Θ1η2−Θ2η3,where Π is the positive constant expressed by (11), at first, we show that the image of the ball BΔ¯1 by P is included in BΔ¯1, where(26)BΔ¯1=z∈CO,ℝ:z≤Δ¯1.
So, for eachz∈BΔ¯1, we have(27)Pzs≤1−r´r´Γσ´1+σ´2∫011−ξσ´1+σ´2−1zξdξ+1r´Γσ´1∫011−ξσ´1−1uξ,zξ−uξ,0+uξ,0dξ+1r´Γσ´1−σ´3∫011−ξσ´1−σ´3−1u^ξ,zξ−uξ,0+u^ξ,0dξ+Πa0+r´1−r´r´Γσ´1+σ´2−ν´1∫011−ξσ´1+σ´2−ν´1−1zξdξ+r´r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1uξ,zξ−uξ,0+uξ,0dξ+r´r´Γσ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1u^ξ,zξ−u^ξ,0+u^ξ,0dξ+1−r´1−r´r´Γσ´1+σ´2−ν´2∫011−ξσ´1+σ´2−ν´2−1zξdξ+1−r´r´Γσ´1−ν´2∫011−ξσ´1−ν´2−1uξ,zξ−uξ,0+uξ,0dξ+1−r´r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1u^ξ,zξ−u^ξ,0+|u^ξ,0dξ≤z1−r´r´Γσ´1+σ´2+1+r´1−r´Πr´Γσ´1+σ´2−ν´1+1+1−r´1−r´Πr´Γσ´1+σ´2−ν´2+1+Θ1z+z⋆1r´Γσ´1+1+r´Πr´Γσ´1−ν´1+1+1−r´Πr´Γσ´1−ν´2+1+Θ2z+z⋆1r´Γσ´1−σ´3+1+r´Πr´Γσ´1−σ´3−ν´1+1+1−r´Πr´Γσ´1−σ´3−ν´2+1+Πa0≤Δ¯1η1+Θ1η2+Θ2η3+u⋆η2+u^⋆η3+Πa0≤Δ¯1.
This implies thatPz≤Δ¯1. Thus PBΔ¯1⊂BΔ¯1. Next, for all z,z⋆∈CO,ℝ and each s∈O, we can write(28a)Pzs−Pz⋆s≤1−r´r´Γσ´1+σ´2∫011−ξσ´1+σ´2−1zξ−z⋆ξdξ+1r´Γσ´1∫011−ξσ´1−1uξ,zξ−uξ,z⋆ξdξ+1r´Γσ´1−σ´3∫011−ξσ´1−σ´3−1u^ξ,zξ−u^ξ,z⋆ξdξ+Π|a0|+r´1−r´r´Γσ´1+σ´2−ν´1∫011−ξσ´1+σ´2−ν´1−1zξ−z⋆ξdξ+r´r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1uξ,zξ−uξ,z⋆ξdξ+r´r´Γσ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1u^ξ,zξ−u^ξ,z⋆ξdξu+1−r´1−r´r´Γσ´1+σ´2−ν´2∫011−ξσ´1+σ´2−ν´2−1zξ−z⋆ξdξ+1−r´r´Γσ´1−ν´2∫011−ξσ´1−ν´2−1uξ,zξ−uξ,z⋆ξdξ+1−r´r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1u^ξ,zξ−u^ξ,z⋆ξdξ≤z−z⋆1−r´r´Γσ´1+σ´2+1+r´1−r´Πr´Γσ´1+σ´2−ν´1+1+1−r´1−r´Πr´Γσ´1+σ´2−ν´2+1(28b)+Θ1z−z⋆1r´Γσ´1+1+r´Πr´Γσ´1−ν´1+1+1−r´Πr´Γσ´1−ν´2+1+Θ2z−z⋆1r´Γσ´1−σ´3+1+r´Πr´Γσ´1−σ´3−ν´1+1+1−r´Πr´Γσ´1−σ´3−ν´2+1=η1+Θ1η2+Θ2η3z−z⋆,which means that Pz−Pz⋆≤η1+Θ1η2+Θ2η3z−z⋆. Therefore, from (24), it follows that P is a contraction. Consequently, the Banach principle of contraction mapping ensures that P has a fixed point which represents the unique solution of our BVP (3). This ends the argument.Example 1.
Consider the Liouville–Caputo fractionalBVP(29)45DC0+5/3+15I0+1/4zs=2zs5+s21+expszs+DC0+4/3exp−szsπ2+zs,z0=0,14DC0+1/6z1+34DC0+1/2z1=5.
In the present example, we haveσ´1=5/3∈1,2, σ´2=1/4∈0,1, σ´3=4/3∈1,2, r´=4/5∈0,1, r´=1/4∈0,1, ν´1=1/6∈0,1, ν´2=1/12∈0,1, a0=5, τ∗=1, and(30)us,z=2z5+s21+expsz,u^s,z=exp−szπ2+z.
Then,(31)us,z−us,z⋆=2z5+s21+expsz−2z⋆5+s21+expsz⋆=25+s2z1+expsz−z⋆1+expsz⋆≤225z−z⋆,u^s,z−u^s,z⋆=exp−szπ2+z−expsz⋆π2+z⋆≤1π2z−z⋆,i.e., Θ1=2/25≈0.0800 and Θ2=1/π2≈0.1013. A simple computation gives us(32)Π=r´Γ2−ν´1+1−r´Γ2−ν´2−1=1/4Γ2−1/6+1−4/5Γ2−4/12−1,and(33)η1=1−r´r´Γσ´1+σ´2+1+r´1−r´Πr´Γσ´1+σ´2−ν´1+1+1−r´1−r´Πr´Γσ´1+σ´2−ν´2+1=1−4/54/5Γ5/3+1/4+1+1/41−4/5Π4/5Γ5/3+1/4−1/6+1+1−1/41−4/5Π4/5Γ5/3+1/4−1/12+1,η2=1r´Γσ´1+1+r´Πr´Γσ´1−ν´1+1+1−r´Πr´Γσ´1−ν´2+1=14/5Γ5/3+1+1/4Π4/5Γ5/3−1/6+1+1−1/4Π4/5Γ5/3−1/12+1,η3=1r´Γσ´1−σ´3+1+r´Πr´Γσ´1−σ´3−ν´1+1+1−r´Πr´Γσ´1−σ´3−ν´2+1=14/5Γ5/3−4/3+1+1/4Π4/5Γ5/3−4/3−1/6+1+1−1/4Π4/5Γ5/3−4/3−1/12+1.
So,Π≈0.9607, η1≈0.2766, η1≈1.6945, and η3≈2.7171 which lead to(34)η1+Θ1η2+Θ2η3≈0.6875<1.
Table1 shows these results. These values are plotted in Figure 1. By using the result of Theorem 1, we conclude that our BVP (29) admits only one solution on 0,1.Table 1
Numerical values ofη1, η2, η3, and Π, for r´∈0,1, in Example 1.
r´∈0,1r´Πη1η2η3η1+Θ1η2+Θ2η30.050.960 721.020 927.112 643.473 427.594 60.100.960 79.957 213.556 321.736 713.244 10.150.960 76.269 49.037 514.491 18.460 60.200.960 74.425 46.778 110.868 46.068 90.250.960 73.319 15.422 58.694 74.633 80.300.960 72.581 54.518 87.245 63.677 10.350.960 72.054 73.873 26.210 52.993 80.400.960 71.659 53.389 15.434 22.481 30.450.960 71.352 23.012 54.830 42.082 60.500.960 71.106 42.711 34.347 31.763 70.550.960 70.905 22.464 83.952 11.502 80.600.960 70.737 62.259 43.622 81.285 40.650.960 70.595 72.085 63.344 11.101 40.700.960 70.474 21.936 63.105 20.943 70.750.960 70.368 81.807 52.898 20.807 00.800.960 70.276 61.694 52.717 10.687 50.850.960 70.195 21.594 92.557 30.581 90.900.960 70.122 91.506 32.415 20.488 10.950.960 70.058 21.427 02.288 10.404 2Figure 1
Graphical representation ofη1+Θ1η2+Θ2η3 in Example 1. (a) r∈0,1, r´=1/4. (b) r=4/5r´∈0,1.
(a)(b)


## 3.2. Existence Result Based on Krasnoselskii’s Criterion

Our existence analysis in this part is a consequence of Krasnoselskii’s criterion (Lemma1). For this fact, we introduce two operators P1 and P2 defined on the ball(35)BΔ¯2=z∈CO,ℝ:zΔ¯2,such that, for all s∈O,(36)P1zs=r´−1r´Γσ´1+σ´2∫0ss−ξσ´1+σ´2−1zξdξ−Πsr´r´−1r´Γσ´1+σ´2−ν´1∫011−ξσ´1+σ´2−ν´1−1zξdξ−Πs1−r´r´−1r´Γσ´1+σ´2−ν´2∫011−ξσ´1+σ´2−ν´2−1zξdξ,and(37)P2zs=1r´Γσ´1∫0ss−ξσ´1−1uξ,zξdξ+1r´Γσ´1−σ´3∫0ss−ξσ´1−σ´3−1u^ξ,zξdξ+Πsa0−r´r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1uξ,zξdξ−r´r´Γσ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1u^ξ,zξdξ−1−r´r´Γβ1−ν´2∫011−ξσ´1−ν´2−1uξ,zξdξ−1−r´r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1u^ξ,zξdξ.Theorem 2.
Consider the continuous functionsu,u^:O×ℝ⟶ℝ which, respectively, satisfy the conditions S1 and S22 of Theorem 1. Furthermore, suppose thatS3: us,z≤ϱ1s,S4: u^s,z≤ϱ2s,for s,z∈O×ℝ, and ϱj∈CO,ℝ+, j=1,2. If η1<1 which is defined in equation (21), then the supposed BVP (3) admits at least one solution defined on O.Proof.
Putϱj=sups∈Oϱjs, j=1,2. We choose Δ¯2 so that(38)Δ¯2≥ϱ1η2+ϱ2η3+Πa01−η1.
In the first place, we prove thatP1z+P2z⋆∈BΔ¯2. So, for all z,z⋆∈BΔ¯2, we have(39)P1zs+P2z⋆s≤1−r´r´Γσ´1+σ´2∫011−ξσ´1+σ´2−1zξdξ+1r´Γσ´1∫011−ξσ´1−1uξ,z⋆ξdξ+Πr´1−r´r´Γσ´1+σ´2−ν´1∫011−ξσ´1+σ´2−ν´1−1zξdξ+Π1−r´1−r´r´Γσ´1+σ´2−ν´2∫011−ξσ´1+σ´2−ν´2−1zξdξ+1r´Γσ´1−σ´3∫011−ξσ´1−σ´3−1u^ξ,z⋆ξdξ+Πa0+r´r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1uξ,z⋆ξdξ+r´r´Γσ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1u^ξ,z⋆ξdξ+1−r´r´Γσ´1−ν´2∫011−ξσ´1−ν´2−1uξ,z⋆ξdξ+1−r´r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1u^ξ,z⋆ξdξ≤z1−r´r´Γσ´1+σ´2+1+r´1−r´Πr´Γσ´1+σ´2−ν´1+1+1−r´1−r´Πr´Γσ´1+σ´2−ν´2+1+ϱ11r´Γσ´1+1+r´Πr´Γσ´1−ν´1+1+1−r´Πr´Γσ´1−ν´2+1+ϱ21r´Γσ´1−σ´3+1+r´Πr´Γσ´1−σ´3−ν´1+1+1−r´Πr´Γσ´1−σ´3−ν´2+1+Πa0≤Δ¯2η1+ϱ1η2+ϱ2η3+Πa0≤Δ¯2.
Thus,‖P1z+P2z⋆‖≤Δ¯2, which means that P1z+P2z⋆∈BΔ¯2. Now, we establish that P1 is a contraction. For z,z⋆∈BΔ¯2, we can write(40)P1zs−P1z⋆s≤1−r´r´Γσ´1+σ´2∫011−ξσ´1+σ´2−1zξ−z⋆ξdξ+Πr´1−r´r´Γσ´1+σ´2−ν´1∫011−ξσ´1+σ´2−ν´1−1zξ−z⋆ξdξ+Π1−r´1−r´r´Γσ´1+σ´2−ν´2∫011−ξσ´1+σ´2−ν´2−1zξ−z⋆ξdξ≤z−z⋆1−r´r´Γσ´1+σ´2+1+r´1−r´Πr´Γσ´1+σ´2−ν´1+1+1−r´1−r´Πr´Γσ´1+σ´2−ν´2+1=η1z−z⋆.
Then,(41)P1z−P1z⋆≤η1z−z⋆.
From the conditionη1<1, it follows that P1 is a contraction mapping. On the other hand, we know that the continuity of P2 occurs immediately from that of the functions u and u^. Also, it is simple to establish that for z∈BΔ¯2,(42)P2z≤ϱ1η2+ϱ2η3.
In other words,P2 is uniformly bounded on BΔ¯2. In this moment, we need to show that P2 is equicontinuous. Let(43)u⋆=sups,z∈O×ℝus,z,andu^⋆=sups,z∈O×ℝu^s,z.
This allows us to write, for anys1,s2∈O×O, where s1<s2, and for all z∈BΔ¯2:(44a)P2us2−P2us1=1r´Γσ´1∫0s2s2−ξσ´1−1uξ,zξdξ−∫0s1s1−ξσ´1−1uξ,zξdξ+1r´Γσ´1−σ´3∫0s2s2−ξσ´1−σ´3−1u^ξ,zξdξ−∫0s1s1−ξσ´1−σ´3−1u^ξ,zξdξ+Πs2−s1a0−r´r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1uξ,zξdξ−r´r´Γσ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1u^ξ,zξdξ−1−r´r´Γσ´1−ν´2∫011−ξσ´1−ν´2−1uξ,zξdξ−1−r´r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1u^ξ,zξdξ≤u⋆r´Γσ´1∫0s1s2−ξσ´1−1−s1−ξσ´1−1dξ+∫s1s2s2−ξσ´1−1dξ+u^⋆r´Γσ´1−σ´3∫0s1s2−ξσ´1−σ´3−1−s1−ξσ´1−σ´3−1dξ+∫s1s2s2−ξσ´1−σ´3−1dξ+Πs2−s1a0+r´ν⋆r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1dξ+r´u^⋆r´Γσ´σ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1dξ(44b)+1−r´u⋆r´Γσ´1−ν´2∫011−ξσ´1−ν´2−1dξ+1−r´u^⋆r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1dξ≤u⋆r´Γσ´1s2σ´1−s1σ´1+u^⋆r´Γσ´1−σ´3s2σ´1−σ´3−s1σ´1−σ´3+Πs2−s1a0+r´u⋆r´Γσ´1−ν´1+1+r´u^⋆r´Γσ´1−σ´3−ν´1+1+1−r´u⋆r´Γσ´1−ν´2+1+1−r´u^⋆r´Γσ´1−σ´3−ν´2+1.
The right-hand side of the above inequality is not dependent on and converges to 0, ass2−s1⟶0. This means that P2 is equicontinuous and admits the relative compactness on BΔ¯2. Thus, Arzelà–Ascoli theorem ensures that P2 is compact on BΔ¯2. Consequently, our BVP (3) possesses at least one solution on O. The proof is completed.Example 2.
Consider the following Liouville–CaputoBVP:(45)223DC0+15/8+123I0+2/7zs=sin2sπ+s2zszs+1+DC0+3/213exps+1zsπ2+expszs,z0=0,25DC0+1/8z1+35DC0+1/4z1=227.
Now, we haveσ´1=15/8∈1,2., σ´2=2/7∈0,1, σ´3=3/2∈0,1, r´=22/23∈0,1, r´=2/5∈0,1, ν´1=1/8∈0,1, ν´2=1/4∈0,1, a0=22/7∈ℝν´1, τ∗=1, and(46)us,z=sin2sπ+s2zz+1,u^s,z=13exps+1zπ2+expsz.
Hence,(47)us,z−us,z⋆=sin2sπ+s2zz+1−sin2sπ+s2z⋆z⋆+1=sin2sπ+s2zz+1−z⋆z⋆+1≤1π2z−z⋆,u^s,z−u^s,z⋆=13exps+1zπ2+expsz−13exps+1z⋆π2+expsz⋆=13exps+1zπ2+expsz−z⋆π2+expsz⋆≤14z−z⋆,i.e., Θ1=2/π2, Θ2=1/4, and accordingly,(48)Π=r´Γ2−ν´1+1−r´Γ2−ν´2−1=2/5Γ2−1/8+1−2/5Γ2−1/4−1,and
By some computations, we getΠ≈0.9325, η1≈0.0415<1, η2≈1.2287, η3≈2.2275, and(49)η1+Θ1η2+Θ2η3≈0.8473<1.
Also, we get(50)us,z=sin2sπ+s2zz+1=sin2sπ+s2zz+1≤sin2sπ+s2≕ϱ1s,u^s,z=13exps+1zπ2+expsz=13exps+1zπ2+expsz≤13exps+1≕ϱ2s.
Table2 shows these results. These numerical data are plotted in Figure 2. Then, Theorem 2 states that the Liouville–Caputo BVP (45) admits at least one solution on O.Table 2
Numerical values ofη1, η2, η3, and Π, for r∈0,1, in Example 2.
r´∈0,1r´Πη1η2η3η1+Θ1η2+Θ2η30.040.932 520.078 127.032 449.004 137.807 10.090.932 59.582 813.516 224.502 018.447 20.130.932 56.084 39.010 816.334 711.993 90.170.932 54.335 16.758 112.251 08.767 30.220.932 53.285 55.406 59.800 86.831 30.260.932 52.585 84.505 48.167 35.540 60.300.932 52.086 03.861 87.000 64.618 70.350.932 51.711 23.379 16.125 53.927 30.390.932 51.419 73.003 65.444 93.389 50.430.932 51.186 42.703 24.900 42.959 30.480.932 50.995 62.457 54.454 92.607 30.520.932 50.836 62.252 74.083 72.314 00.570.932 50.702 02.079 43.769 52.065 80.610.932 50.586 71.930 93.500 31.853 10.650.932 50.486 71.802 23.266 91.668 70.700.932 50.399 31.689 53.062 81.507 30.740.932 50.322 11.590 12.882 61.365 00.780.932 50.253 51.501 82.722 41.238 50.830.932 50.192 11.422 82.579 21.125 20.870.932 50.136 91.351 62.450 21.023 30.910.932 50.086 91.287 32.333 50.931 222/23≃0.960.932 50.041 51.228 72.227 50.847 31.000.932 50.000 01.175 32.130 60.770 8(51)η1=1−r´r´Γσ´1+σ´2+1+r´1−r´Πr´Γσ´1+σ´2−ν´1+1+1−r´1−r´Πr´Γσ´1+σ´2−ν´2+1η2=1r´Γσ´1+1+r´Πr´Γσ´1−ν´1+1+1−r´Πr´Γσ´1−ν´2+1=1−22/2322/23Γ15/8+2/7+1+2/51−22/23Π22/23Γ15/8+2/7−1/4+1+1−2/51−22/23Π22/23Γ15/8+2/7−1/4+1,=122/23Γ15/8+1+2/5Π22/23Γ15/8−1/8+1+1−2/5Π22/23Γ15/8−1/4+1,η3=1r´Γσ´1−σ´3+1+r´Πr´Γσ´1−σ´3−ν´1+1+1−r´Πr´Γσ´1−σ´3−ν´2+1=122/23Γ15/8−3/2+1+2/5Π22/23Γ15/8−3/2−1/8+1+1−2/5Π22/23Γ15/8−3/2−1/4+1.Figure 2
Graphical representation ofη1+Θ1η2+Θ2η3 and η1<1 in Example 2. (a) r∈0,1,r´=2/5. (b) η1vsr=22/23,r´=2/5.
(a)(b)


## 3.3. Existence Result by Using Nonlinear Alternative of Leray–Schauder

Another result of existence criterion is realized by implementing the hypotheses in Lemma2. The desired criterion is proved below by the next theorem.Theorem 3.
Assume thatu,u^:O×ℝ⟶ℝ are two continuous functions which satisfy the following assumption.
S5: there are two continuous nondecreasing functions ℘1,℘2:0,+∞⟶0,+∞ and two functions ℏ1,ℏ2∈CO,ℝ provided that(52)us,z≤ℏ1s℘1z,u^s,z≤ℏ2s℘2z,for all s,z∈O×ℝ; moreover, the following assumption holds.
S6: there exists a positive real constant Δ¯3 so that(53)ℏ1℘1Δ¯3η2+ℏ2℘2Δ¯3η3+Πa0Δ¯31−η1>1,η1<1.Then, the Liouville–CaputoBVP (3) has at least one solution on O, where Π,η1,η2,η3 stand for the same constants introduced, respectively, by expressions (11), (21), (22), and (23).Proof.
Consider again the operatorP expressed as (20). First, we will prove that P maps bounded sets into bounded sets in CO,ℝ. Let(54)Bδ=z∈CO,ℝ:z≤δbe a bounded set in CO,ℝ, where δ is a real positive number δ>0. For each s∈O, we have(55)Pzs≤1−r´r´Γσ´1+σ´2∫011−ξσ´1+σ´2−1zξdξ+1r´Γσ´1∫011−ξσ´1−1uξ,zξdξ+1r´Γσ´1−σ´3∫011−ξσ´1−σ´3−1u^ξ,zξdξ+Π|a0|+r´1−r´r´Γσ´1+σ´2−ν´1∫011−ξσ´1+σ´2−ν´1−1zξdξ+r´r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1uξ,zξdξ+r´r´Γσ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1u^ξ,zξdξ+1−r´1−r´r´Γσ´1+σ´2−ν´2∫011−ξσ´1+σ´2−ν´2−1zξdξ+1−r´r´Γσ´1−ν´2∫011−ξσ´1−ν´2−1uξ,zξdξ+1−r´r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1uξ,zξdξ≤zη1+ℏ1℘1zη2+ℏ2℘2zη3+Πa0≤δη1+ℏ1℘1δη2+ℏ2℘2δη3+Πa0,and consequently,(56)Pz≤δη1+ℏ1℘1δη2+ℏ2℘2δη3+Πa0.
The next property is that we prove that the operatorP maps the bounded sets to the equicontinuous sets. Let(57)u⋆=sups,z∈O×Bδus,z,andu^⋆=sups,z∈O×Bδu^s,z.
So, fors1,s2∈O with s1<s2 and z∈Bδ, we have(58a)Pzs2−Pzs1≤δ1−r´r´Γσ´1+σ´2∫0s1s2−ξσ´1+σ´2−1−s1−ξσ´1+σ´2−1dξ+∫s1s2s2−ξσ´1+σ´2−1dξ+u⋆r´Γσ´1∫0s1s2−ξσ´1−1−s1−ξσ´1−1dξ+∫s1s2s2−ξσ´1−1dξ+u^⋆r´Γσ´1−σ´3∫0s1s2−ξσ´1−σ´3−1−t1−ξσ´1−σ´3−1dξ+∫s1s2s2−ξσ´1−σ´3−1dξ+Πs2−s1a0+r´1−r´δr´Γσ´1+σ´2−ν´1∫011−ξσ´1+σ´2−ν´1−1dξ+r´u⋆r´Γσ´1−ν´1∫011−ξσ´1−ν´1−1dξ+r´u^⋆r´Γσ´1−σ´3−ν´1∫011−ξσ´1−σ´3−ν´1−1dξ+1−r´1−r´δr´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1dξ(58b)+1−r´u⋆r´Γσ´1−ν´2∫011−ξσ´1−ν´2−1dξ+1−r´u^⋆r´Γσ´1−σ´3−ν´2∫011−ξσ´1−σ´3−ν´2−1dξ≤δ1−r´s2σ´1+σ´2−s1σ´1+σ´2r´Γσ´1+σ´2+1+Πr´1−r´s2−s1r´Γσ´1+σ´2−ν´1+1+Π1−r´1−r´s2−s1r´Γσ´1−σ´3−ν´2+1+u⋆s2σ´1−s1σ´1r´Γσ´1+1+Πr´s2−s1r´Γσ´1−ν´1+1+Π1−r´s2−s1r´Γσ´1−ν´2+1+u^⋆s2σ´1−σ´3−s1σ´1−σ´3r´Γσ´1−σ´3+1+Πr´s2−s1r´Γσ´1−σ´3−ν´1+1+Π1−r´s2−s1r´Γσ´1−σ´3−ν´2+1+Πa0s2−s1.
Note that the second side of the last inequality is not dependent onz and goes to 0 when s2−s1⟶0. Hence, by Arzelà–Ascoli theorem, it is figured out that P is completely continuous. At this moment, we have established the boundedness of the set of solutions for the operator equation z=ℓ´Pz, where ℓ´∈O. Let now z be a solution of BVP (3). With the same arguments used in (56), one can find(59)z≤zη1+ℏ1℘1zη2+ℏ2℘2zη3+Πa0,and we can also write it as(60)z1−η1ℏ1℘1zη2+ℏ2℘2zη3+Πa0≤1.
From assumptionS6, there exists a constant Δ¯3>0 so that Δ¯3≠z. Consider the set(61)Ε=z∈CO,ℝ:z<Δ¯3.
It was proved thatP:E¯⟶CO,ℝ is a continuous and completely continuous operator. The selection of the set E allows us to confirm that there is no z∈∂Ε which satisfies z=ℓ´Pz, for ℓ´∈O. Hence, the requirements of Lemma 2 ensure that E involves a fixed point z∗∈Ε¯ which stands for the solution to our Liouville–Caputo BVP (3), and the proof is completed.Example 3.
Consider the Liouville–CaputoBVP(62)1719DC0+19/2+219I0+5/12zs=2101s+1zs2zs+1+1+DC0+17/121103exps+1zs3z2s+1+5z2sz2s+3,u0=0,511DC0+1/8z1+611DC0+1/7z1=47.
Now, we haveσ´1=19/12∈1,2, σ´2=5/12∈0,1, σ´3=17/12∈1,2, r´=17/19∈0,1, r´=5/1∈0,1, τ∗=1, ν´1=1/8∈0,1, ν´2=1/7∈0,1, a0=4/7, and(63)us,z=2101s+1zs2zs+1+1,u^s,z=1103exps+1zs3z2s+1+5z2sz2s+3,and(64)us,z≤2101s+1z+1,z^s,z≤1103exps+1z+5.
Then,(65)ℏ1s=2101s+1,℘1z=z+1,ℏ2s=1103exps+1,℘2z=z+5.
On the other hand(66)Π=r´Γ2−ν´1+1−r´Γ2−ν´2−1=5/11Γ2−1/8+1−5/11Γ2−1/7−1,and(67)η1=1−r´r´Γσ´1+σ´2+1+r´1−r´Πr´Γσ´1+σ´2−ν´1+1+1−r´1−r´Πr´Γσ´1+σ´2−ν´2+1=1−17/1917/19Γ19/12+5/12+1+5/111−17/19Π17/19Γ19/12+5/12−1/8+1+1−5111−17/19Π/17/19Γ19/12+5/12−1/7+1,η2=1r´Γσ´1+1+r´Πr´Γσ´1−ν´1+1+1−r´Πr´Γσ´1−ν´2+1=117/19Γ19/12+1+5/11Π17/19Γ19/12−1/8+1+1−5/11Π17/19Γ19/12−1/7+1,η3=1r´Γσ´1−σ´3+1+r´Πr´Γσ´1−σ´3−ν´1+1+1−r´Πr´Γσ´1−σ´3−ν´2+1=117/19Γ19/12−17/12+1+5/11Π/17/19Γ19/12−17/12−18+1+1−5/11Π/17/19Γ19/12−17/12−17+1.
A simple computation leads toΠ≈0.9504, η1≈0.1219, η2≈1.6195, and η3≈2.2857. By solving the inequality (Table 3)(68)A=ℏ1℘1Δ¯3η2+ℏ2℘2Δ¯3η3+Πa0Δ¯31−η1=2×1.6195/101Δ¯3+1+2.2857/40Δ¯3+5+0.9504×4/71−0.1219Δ¯3>1,we get A>1.1000>1. Table 4 shows these data. These numerical values are plotted in Figure 3.
Then, the assumption6 holds for any Δ¯3>0.8456. Consequently, from Theorem 3, we conclude that for Liouville–Caputo BVP (62), at least one solution is found on O.Table 3
Numerical values ofη1, η2, η3, and Π, for and r´´=5/11, in Example 3.
r´Πη1η2η30.050.950 418.650 527.530 838.856 90.110.950 48.807 213.765 419.428 40.160.950 45.526 19.176 912.952 30.210.950 43.885 56.882 79.714 20.260.950 42.901 25.506 27.771 40.320.950 42.245 04.588 56.476 10.370.950 41.776 23.933 05.551 00.420.950 41.424 73.441 34.857 10.470.950 41.151 33.059 04.317 40.530.950 40.932 52.753 13.885 70.580.950 40.753 62.502 83.532 40.630.950 40.604 42.294 23.238 10.680.950 40.478 22.117 82.989 00.740.950 40.370 01.966 52.775 50.790.950 40.276 31.835 42.590 50.840.950 40.194 31.720 72.428 617/19≃0.890.950 40.121 91.619 52.285 70.950.950 40.057 61.529 52.158 71.000.950 40.000 01.449 02.045 1Table 4
Numerical results ofΔ¯3 and A based on Table 3 for η1, η2, η3, and Π in Example 3.
Π=0.9504, η1=0.1219, η2=1.6195, η3=2.2857Δ¯3A>10.520.535 70.560.572 00.600.607 60.640.642 60.680.677 00.720.710 90.760.744 20.800.776 90.840.809 10.880.840 80.920.872 00.960.902 71.000.932 91.040.962 61.080.991 91.121.0207>11.161.049 11.201.077 01.241.104 61.281.131 71.321.158 4Figure 3
Graphical representation ofA>1 for s∈0,1 vs.Δ¯3


## 4. Conclusion

In this paper, we considered a Liouville–CaputoBVP and proved our main results by using three fixed point theorems due to Banach, Krasnoselskii, and Leray–Schauder. Several special cases can be extracted from the mentioned BVP (3). Let us point out them, for example, if r´=1, then the Liouville–Caputo BVP (3) reduces to the following one:(69)DC0+σ´1Czs=us,zs+DC0+σ´3u^s,zs,s∈O,z0=0,r´DC0+ν´1u1+1−r´DC0+ν´2u1=a0.Ifr´=0, then the Liouville–Caputo BVP (3) becomes(70)r´DC0+σ´1+1−r´I0+σ´σ´2zs=us,zs+CD0+σ´3u^s,zs,s∈O,z0=0,DC0+ν´2u1=a0.Consequently, some existence and uniqueness results for this particular case are obtained by exploiting Theorems1–3. For future studies, we aim to combine these BVPs with nonsingular kernels in fractal-fractional operators.



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