Existence Results for Langevin Fractional Differential Inclusions Involving Two Fractional Orders with Four-Point Multiterm Fractional Integral Boundary Conditions

Abstract

We discuss the existence of solutions for Langevin fractional differential inclusions involving two fractional orders with four-point multiterm fractional integral boundary conditions. Our study relies on standard fixed point theorems for multivalued maps and covers the cases when the right-hand side of the inclusion has convex as well as nonconvex values. Illustrative examples are also presented.

1. Introduction

We consider a boundary value problem of nonlinear Langevin fractional differential inclusions involving two fractional orders with four-point multiterm fractional integral boundary conditions given by ^𝑐𝐷^𝑝𝑐𝐷^𝑞𝑥(𝑡)((𝑡𝑥(0)=+𝜆∈𝐹𝑡,𝑥)),0<𝑡<1,_𝑛𝑖=1𝛽_𝑖𝐼^𝜇𝑖𝑥𝜁),𝑥(1)=_𝑛𝑖=1(𝛼_𝑖𝐼^𝜈𝑖𝑥𝜂,0<𝜁<𝜂<1, (1) where 0<𝑝,𝑞≤1, ^𝑐𝐷^𝑞 denotes the Caputo fractional derivative of order 𝑞, 𝜆 is a real number, 𝐼^𝜅 is the Riemann-Liouville fractional integral of order 𝜅>0  (𝜅=𝜈_𝑖,𝜇_𝑖;𝑖=1,2,…,𝑛,𝑛∈ℕ={1,2,…}), and 𝛼_𝑖,𝛽_𝑖∈ℝ are suitably chosen constants.

In recent years, the boundary value problems of fractional order differential equations have emerged as an important area of research, since these problems have applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, biophysics, aerodynamics, viscoelasticity and damping, electrodynamics of complex medium, wave propagation, and blood flow phenomena [1–5]. Many researchers have studied the existence theory for nonlinear fractional differential equations with a variety of boundary conditions; for instance, see the papers [6–17] and the references therein.

The Langevin equation (first formulated by Langevin in 1908) is found to be an effective tool to describe the evolution of physical phenomena in fluctuating environments [18]. For some new developments on the fractional Langevin equation, see, for example, [19–26].

The main objective of this paper is to develop the existence theory for a class of problems of the type (1), when the right-hand side is convex as well as nonconvex valued. We establish three existence results: the first result is obtained by means of the nonlinear alternative of Leray-Schauder type; the second one relies on the nonlinear alternative of Leray-Schauder type for single-valued maps together with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values; and a fixed point theorem due to Covitz and Nadler for contraction multivalued maps is applied to get the third result. The methods used are well known; however their exposition in the framework of problem (1) is new.

The paper is organized as follows: in Section 2, we recall some preliminary facts that we need in the sequel, and Section 3 contains our main results.

2. Preliminaries

2.1. Fractional Calculus

Let us recall some basic definitions of fractional calculus [1–3].

Definition 1. For at least 𝑛-times differentiable function 𝑔∶[0,∞)→ℝ, the Caputo derivative of fractional order 𝑞 is defined as ^𝑐𝐷^𝑞𝑔(𝑡)=1Γ𝑛−𝑞^𝑡₀(𝑡−𝑠)^{𝑛−𝑞−1}𝑔^(𝑛)(𝑠)𝑞𝑑𝑠,𝑛−1<𝑞<𝑛,𝑛=+1, (2) where [𝑞] denotes the integer part of the real number 𝑞.

Definition 2. The Riemann-Liouville fractional integral of order 𝑞 is defined as 𝐼^𝑞𝑔(𝑡)=1Γ𝑞^𝑡₀𝑔(𝑠)(𝑡−𝑠)^1−𝑞𝑑𝑠,𝑞>0, (3) provided the integral exists.

Lemma 3. For 𝑞>0, the general solution of the fractional differential equation 𝐷^𝑞𝑥(𝑡)=0 is given by 𝑥(𝑡)=𝑐₀+𝑐₁𝑡+⋯+𝑐_𝑛−1𝑡^𝑛−1, (4) where 𝑐_𝑖∈ℝ,𝑖=1,2,…,𝑛−1  (𝑛=[𝑞]+1).

In view of Lemma 3, it follows that 𝐼^𝑞𝐷^𝑞𝑥(𝑡)=𝑥(𝑡)+𝑐₀+𝑐₁𝑡+⋯+𝑐_𝑛−1𝑡^𝑛−1, (5) for some 𝑐_𝑖∈ℝ,𝑖=1,2,…,𝑛−1  (𝑛=[𝑞]+1).

In the following, 𝐴𝐶¹([0,𝑇],ℝ) will denote the space of functions 𝑥∶[0,𝑇]→ℝ that are absolutely continuous and whose first derivative is absolutely continuous.

Definition 4. A function 𝑥∈𝐴𝐶¹([0,𝑇],ℝ) is called a solution of problem (1) if there exists a function 𝑣∈𝐿¹([0,𝑇],ℝ) with 𝑣(𝑡)∈𝐹(𝑡,𝑥(𝑡)), a.e. [0,𝑇] such that ^𝑐𝐷^𝑝(^𝑐𝐷^𝑞+𝜆)𝑥(𝑡)=𝑣(𝑡), a.e. [0,𝑇], and 𝑥(0)=∑^𝑛_𝑖=1𝛽_𝑖(𝐼^𝜇𝑖𝑥)(𝜁),𝑥(1)=∑^𝑛_𝑖=1𝛼_𝑖(𝐼^𝜈𝑖𝑥)(𝜂).

Lemma 5. Let ℎ∈𝐶[0,1]. Then the boundary value problem ^𝑐𝐷^𝑝𝑐𝐷^𝑞𝑥(𝑡)(𝑡)𝑥(0)=+𝜆=ℎ,0<𝑡<1,_𝑛𝑖=1𝛽_𝑖𝐼^𝜇𝑖𝑥𝜁),𝑥(1)=_𝑛𝑖=1(𝛼_𝑖𝐼^𝜈𝑖𝑥𝜂,0<𝜁<𝜂<1, (6) has a unique solution𝑥(𝑡)=^𝑡₀(𝑡−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝ℎ(𝑟)(𝑠−1𝑑𝑟−𝜆𝑥)𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ×𝑞+1₁₀(1−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝ℎ(𝑟)(𝑠−𝑑𝑟−𝜆𝑥)𝑑𝑠_𝑛𝑖=1𝛼_𝑖^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝ℎ(𝑟)(𝑠−1𝑑𝑟−𝜆𝑥)𝑑𝑠𝑑𝑢Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ×𝑞+1_𝑛𝑖=1𝛽_𝑖^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝ℎ(𝑟)𝑑𝑟−𝜆𝑥(𝑠)𝑑𝑠𝑑𝑢, (7)where Δ=Δ₁Δ₂−Δ₃Δ₄Δ≠0,₁=^𝑛_𝑖=1𝛽_𝑖𝜁^{𝜇𝑖+𝑞}Γ𝜇_𝑖,Δ+𝑞+1₂=1−^𝑛_𝑖=1𝛼_𝑖𝜂^𝜈𝑖Γ𝜈_𝑖,Δ+1₃=1−^𝑛_𝑖=1𝛽_𝑖𝜁^𝜇𝑖Γ𝜇_𝑖,Δ+1₄=^𝑛_𝑖=1𝛼_𝑖𝜂^{𝜈𝑖+𝑞}Γ𝜈_𝑖−1+𝑞+1Γ.𝑞+1 (8)

Proof. As argued in [23], the solution of ^𝑐𝐷^𝑝(^𝑐𝐷^𝑞+𝜆)𝑥(𝑡)=ℎ(𝑡) can be written as 𝑥(𝑡)=^𝑡₀(𝑡−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝ℎ(𝑟)(𝑠−𝑐𝑑𝑟−𝜆𝑥)𝑑𝑠₀Γ𝑡𝑞+1^𝑞−𝑐₁. (9) Using the given conditions in (9) together with (8), we find that Δ₃𝑐₁−Δ₁𝑐₀=−𝒥₁,Δ₂𝑐₁−Δ₄𝑐₀=𝒥₂, (10) where𝒥₁=^𝑛_𝑖=1𝛽_𝑖^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝ℎ(𝑟)(𝑠𝒥𝑑𝑟−𝜆𝑥)𝑑𝑠𝑑𝑢,₂=¹₀(1−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝ℎ(𝑟)(𝑠)−𝑑𝑟−𝜆𝑥𝑑𝑠_𝑛𝑖=1𝛼_𝑖^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝ℎ(𝑟)𝑑𝑟−𝜆𝑥(𝑠)𝑑𝑠𝑑𝑢. (11)Solving (10) for 𝑐₀ and 𝑐₁, we find that 𝑐₀=1ΔΔ₂𝒥₁+Δ₃𝒥₂,𝑐₁=1ΔΔ₄𝒥₁+Δ₁𝒥₂. (12)
Substituting these values in (9), we find the desired solution.

In order to simplify the computations in the main results, we present a technical lemma, concerning the bounds of the operators 𝒥₁ and 𝒥₂ defined in the proof of the above lemma.

Lemma 6. One has 𝒥₁≤^𝑛_𝑖=1𝛽_𝑖𝜁^{𝜇𝑖+𝑝+𝑞}‖ℎ‖Γ𝜇_𝑖++𝑝+𝑞+1_𝑛𝑖=1𝛽_𝑖𝜁^{𝜇𝑖+𝑞}|𝜆|‖𝑥‖Γ𝜇_𝑖,𝒥+𝑞+1₂≤‖ℎ‖Γ+|𝜆𝑥‖𝑝+𝑞+1|‖Γ+𝑞+1_𝑛𝑖=1𝛼_𝑖𝜂^{𝜈𝑖+𝑝+𝑞}‖ℎ‖Γ𝜈_𝑖++𝑝+𝑞+1_𝑛𝑖=1𝛼_𝑖𝜂^{𝜈𝑖+𝑞}|𝜆|‖𝑥‖Γ𝜈_𝑖.+𝑞+1 (13)

Proof. By using the following property of beta function 𝐵(𝑏+1,𝑎)=¹₀(1−𝑠)^𝑎−1𝑠^𝑏=Γ(𝑎)Γ()𝑑𝑠𝑏+1Γ(),𝑎+𝑏+1 (14) we have𝒥₁≤^𝑛_𝑖=1𝛽_𝑖×^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝|ℎ(𝑟|𝜆𝑥(𝑠≤)|𝑑𝑟+||)|𝑑𝑠𝑑𝑢_𝑛𝑖=1𝛽_𝑖^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×𝑠^𝑝Γ‖ℎ‖+|𝜆𝑥≤𝑝+1|‖‖𝑑𝑠𝑑𝑢_𝑛𝑖=1𝛽_𝑖^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1𝑠^𝑝‖ℎ‖Γ𝑞Γ+𝑝+1𝑑𝑠_𝑢0(𝑢−𝑠)^𝑞−1Γ𝑞|𝜆𝑥‖≤|‖𝑑𝑠𝑑𝑢_𝑛𝑖=1𝛽_𝑖^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖𝑢^𝑝+𝑞‖ℎ‖Γ𝑞Γ+𝑝+1×𝐵𝑝+1,𝑞𝑑𝑢_𝑛𝑖=1𝛽_𝑖^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖𝑢^𝑞Γ|𝜆𝑥‖≤𝑞+1|‖𝑑𝑢_𝑛𝑖=1𝛽_𝑖𝜁^{𝜇𝑖+𝑝+𝑞}‖ℎ‖Γ𝜇_𝑖Γ𝑞Γ𝐵𝑝+1×𝐵𝑝+1,𝑞𝑝+𝑞+1,𝜇_𝑖+^𝑛_𝑖=1𝛽_𝑖𝜁^{𝜇𝑖+𝑞}Γ𝜇_𝑖Γ𝐵𝑞+1𝑞+1,𝜇_𝑖|𝜆𝑥‖=|‖_𝑛𝑖=1𝛽_𝑖𝜁^{𝜇𝑖+𝑝+𝑞}‖ℎ‖Γ𝜇_𝑖++𝑝+𝑞+1_𝑛𝑖=1𝛽_𝑖𝜁^{𝜇𝑖+𝑞}|𝜆|‖𝑥‖Γ𝜇_𝑖.+𝑞+1 (15)By a similar way, we have 𝒥₂≤|𝑣(𝑢)|Γ+|𝜆𝑥(𝑢𝑝+𝑞+1||)|Γ+𝑞+1_𝑛𝑖=1𝛼_𝑖𝜂^{𝜈𝑖+𝑝+𝑞}|𝑣(𝑢)|Γ𝜈_𝑖++𝑝+𝑞+1_𝑛𝑖=1𝛼_𝑖𝜂^{𝜈𝑖+𝑞}|𝜆||𝑥(𝑢)|Γ𝜈_𝑖,+𝑞+1 (16) which completes the proof.

In the following, for convenience, we put 𝐴₁=Δ₃+Δ₁Γ𝑞+1|Δ|Γ,𝐴𝑞+1₂=Δ₂+Δ₄Γ𝑞+1|Δ|Γ,𝑞+1Ω=1+𝐴₁Γ𝑝+𝑞+1+𝐴₁^𝑛_𝑖=1𝛼_𝑖𝜂^{𝜈𝑖+𝑝+𝑞}Γ𝜈_𝑖+𝑝+𝑞+1+𝐴₂^𝑛_𝑖=1𝛽_𝑖𝜁^{𝜇𝑖+𝑝+𝑞}Γ𝜇_𝑖,|𝜆+𝑝+𝑞+1Ψ=|1+𝐴₁Γ+|𝜆|𝐴𝑞+1_1𝑛𝑖=1𝛼_𝑖𝜂^{𝜈𝑖+𝑞}Γ𝜈_𝑖+𝑞+1+𝐴₂^𝑛_𝑖=1𝛽_𝑖𝜁^{𝜇𝑖+𝑞}Γ𝜇_𝑖.+𝑞+1 (17)

2.2. Background Material for Multivalued Analysis

Now we recall some basic definitions on multivalued maps [27–29].

Let 𝐶([0,1]) denote a Banach space of continuous functions from [0,1] into ℝ with the norm ‖𝑥‖=sup_𝑡∈[0,1]|𝑥(𝑡)|. Let 𝐿¹([0,1],ℝ) be the Banach space of measurable functions 𝑥∶[0,1]→ℝ which are Lebesgue integrable and normed by ‖𝑥‖_𝐿¹=∫¹₀|𝑥(𝑡)|𝑑𝑡.

For a normed space (𝑋,‖⋅‖), let 𝒫_cl(𝑋)={𝑌∈𝒫(𝑋)∶𝑌isclosed},𝒫_𝑏(𝑋)={𝑌∈𝒫(𝑋)∶𝑌isbounded},𝒫_𝑐𝑝(𝑋)=𝑌∈𝒫(𝑋)∶𝑌iscompact,𝒫_{𝑐𝑝,𝑐}(𝑋)=𝑌∈𝒫(𝑋)∶𝑌iscompactandconvex. (18) A multivalued map 𝐺∶𝑋→𝒫(𝑋)(i) is convex (closed) valued if 𝐺(𝑥) is convex (closed) for all 𝑥∈𝑋;(ii) is bounded on bounded sets if 𝐺(𝔹)=∪_𝑥∈𝔹𝐺(𝑥) is bounded in 𝑋 for all 𝔹∈𝒫_𝑏(𝑋) (i.e., sup_𝑥∈𝔹{sup{|𝑦|∶𝑦∈𝐺(𝑥)}}<∞);(iii) is called upper semicontinuous (u.s.c.) on 𝑋 if, for each 𝑥₀∈𝑋, the set 𝐺(𝑥₀) is a nonempty closed subset of 𝑋 and if, for each open set 𝑁 of 𝑋 containing 𝐺(𝑥₀), there exists an open neighborhood 𝒩₀ of 𝑥₀ such that 𝐺(𝒩₀)⊆𝑁;(iv)𝐺 is lower semicontinuous (l.s.c.) if the set {𝑦∈𝑋∶𝐺(𝑦)∩𝐵≠∅} is open for any open set 𝐵 in 𝐸;(v) is said to be completely continuous if 𝐺(𝔹) is relatively compact for every 𝔹∈𝒫_𝑏(𝑋);(vi) is said to be measurable if, for every 𝑦∈ℝ, the function 𝑡↦𝑑𝑦,𝐺(𝑡)=inf𝑦−𝑧∶𝑧∈𝐺(𝑡) (19)  is measurable;(vii)has a fixed point if there is 𝑥∈𝑋 such that 𝑥∈𝐺(𝑥). The fixed point set of the multivalued operator 𝐺 will be denoted by Fix𝐺.

Definition 7. A multivalued map 𝐹∶[0,1]×ℝ→𝒫(ℝ) is said to be Carathéodory if(i)𝑡↦𝐹(𝑡,𝑥) is measurable for each 𝑥∈ℝ; (ii)𝑥↦𝐹(𝑡,𝑥) is upper semicontinuous for almost all 𝑡∈[0,1];
Further a Carathéodory function 𝐹 is called 𝐿¹-Carathéodory if (iii) for each 𝛼>0, there exists 𝜑_𝛼∈𝐿¹([0,1],ℝ⁺) such that ‖𝐹(𝑡,𝑥)‖=sup{|𝑣|∶𝑣∈𝐹(𝑡,𝑥)}≤𝜑_𝛼(𝑡) (20) for all ‖𝑥‖_∞≤𝛼 and for a. e. 𝑡∈[0,1].

For each 𝑥∈𝐶([0,1],ℝ), define the set of selections of 𝐹 by 𝑆_𝐹,𝑥∶=𝑣∈𝐿¹([0,1],ℝ)∶𝑣(𝑡)∈𝐹(𝑡,𝑥(𝑡))fora.e.t∈[0,1]. (21)

We define the graph of 𝐺 to be the set 𝐺𝑟(𝐺)={(𝑥,𝑦)∈𝑋×𝑌,𝑦∈𝐺(𝑥)} and recall two useful results regarding closed graphs and upper semicontinuity.

Lemma 8 (see [27, Proposition 1.2]). If 𝐺∶𝑋→𝒫_cl(𝑌) is u.s.c., then 𝐺𝑟(𝐺) is a closed subset of 𝑋×𝑌; that is, for every sequence {𝑥_𝑛}_𝑛∈ℕ⊂𝑋 and {𝑦_𝑛}_𝑛∈ℕ⊂𝑌, if, when 𝑛→∞, 𝑥_𝑛→𝑥_∗, 𝑦_𝑛→𝑦_∗, and 𝑦_𝑛∈𝐺(𝑥_𝑛), then 𝑦_∗∈𝐺(𝑥_∗). Conversely, if 𝐺 is completely continuous and has a closed graph, then it is upper semicontinuous.

Lemma 9 (see [30]). Let 𝑋 be a Banach space. Let 𝐹∶[0,𝑇]×ℝ→𝒫_{𝑐𝑝,𝑐}(𝑋) be an 𝐿¹-Carathéodory multivalued map, and let Θ be a linear continuous mapping from 𝐿¹([0,1],𝑋) to 𝐶([0,1],𝑋). Then the operator Θ∘𝑆_F∶𝐶([0,1],𝑋)→𝒫_{𝑐𝑝,𝑐}(𝐶([0,1],𝑋)),𝑥↦Θ∘𝑆_𝐹(𝑥)=Θ𝑆_{𝐹,𝑥,𝑦} (22) is a closed graph operator in 𝐶([0,1],𝑋)×𝐶([0,1],𝑋).

We recall the well-known nonlinear alternative of Leray-Schauder for multivalued maps.

Lemma 10 (nonlinear alternative for Kakutani maps [31]). Let 𝐸 be a Banach space, 𝐶 a closed convex subset of 𝐸,𝑈 an open subset of 𝐶, and 0∈𝑈. Suppose that 𝐹∶𝑈→𝒫_{𝑐,𝑐𝑣}(𝐶) is an upper semicontinuous compact map; here 𝒫_{𝑐,𝑐𝑣}(𝐶) denotes the family of nonempty, compact convex subsets of 𝐶. Then either(i)𝐹 has a fixed point in 𝑈, or(ii)there is a 𝑢∈𝜕𝑈 and 𝜆∈(0,1) with 𝑢∈𝜆𝐹(𝑢).

Definition 11. Let 𝐴 be a subset of [0,1]×ℝ. 𝐴 is ℒ⊗ℬ measurable if 𝐴 belongs to the 𝜎-algebra generated by all sets of the form 𝒥×𝒟, where 𝒥 is Lebesgue measurable in [0,1] and 𝒟 is Borel measurable in ℝ.

Definition 12. A subset 𝒜 of 𝐿¹([0,1],ℝ) is decomposable if, for all 𝑢,𝑣∈𝒜, and measurable 𝒥⊂[0,1]=𝐽, the function 𝑢𝜒_𝒥+𝑣𝜒_𝐽−𝒥∈𝒜, where 𝜒_𝒥 stands for the characteristic function of 𝒥.

Lemma 13 (see [32]). Let 𝑌 be a separable metric space, and let 𝑁∶𝑌→𝒫(𝐿¹([0,1],ℝ)) be a lower semicontinuous (l.s.c.) multivalued operator with nonempty closed and decomposable values. Then 𝑁 has a continuous selection; that is, there exists a continuous function (single-valued) 𝑔∶𝑌→𝐿¹([0,1],ℝ) such that 𝑔(𝑥)∈𝑁(𝑥) for every 𝑥∈𝑌.

Let (𝑋,𝑑) be a metric space induced from the normed space (𝑋;‖⋅‖). Consider 𝐻_𝑑∶𝒫(𝑋)×𝒫(𝑋)→ℝ∪{∞} given by 𝐻_𝑑(𝐴,𝐵)=maxsup_𝑎∈𝐴𝑑(𝑎,𝐵),sup_𝑏∈𝐵𝑑(𝐴,𝑏), (23) where 𝑑(𝐴,𝑏)=inf_𝑎∈𝐴𝑑(𝑎;𝑏) and 𝑑(𝑎,𝐵)=inf_𝑏∈𝐵𝑑(𝑎;𝑏). Then (𝒫_𝑏,cl(𝑋),𝐻_𝑑) is a metric space (see [33]).

Definition 14. A multivalued operator 𝑁∶𝑋→𝒫_cl(𝑋) is called(a)𝛾-Lipschitz if and only if there exists 𝛾>0 such that 𝐻_𝑑𝑁(𝑥),𝑁𝑦≤𝛾𝑑𝑥,𝑦foreach𝑥,𝑦∈𝑋; (24)(b) a contraction if and only if it is 𝛾-Lipschitz with 𝛾<1.

Lemma 15 (see [34]). Let (𝑋,𝑑) be a complete metric space. If 𝑁∶𝑋→𝒫_cl(𝑋) is a contraction, then Fix𝑁≠∅.

3. Main Results

3.1. The Carathéodory Case

In this section, we are concerned with the existence of solutions for the problem (1) when the right-hand side has convex as well as nonconvex values. Initially, we assume that 𝐹 is a compact and convex valued multivalued map.

Theorem 16. Suppose that(𝐻₁) the map 𝐹∶[0,1]×ℝ→𝒫(ℝ) is Carathéodory and has nonempty compact and convex values; (𝐻₂) there exist a continuous nondecreasing function 𝜓∶[0,∞)→(0,∞) and function 𝑝∈𝐿¹([0,1],ℝ⁺) such that ‖𝐹(𝑡,𝑥)‖_𝒫∶=sup{|𝑣|∶𝑣∈𝐹(𝑡,𝑥)}≤𝑝(𝑡)𝜓(‖𝑥‖) (25)  for each (𝑡,𝑢)∈[0,1]×ℝ;(𝐻₃) there exists a number 𝑀>0 such that 𝑀(1−Ψ)𝜓(𝑀)𝑝_𝐿1Ω>1, (26)with Ψ<1, where Ω,Ψ are defined in (17).
Then BVP (1) has at least one solution.

Proof. Let us introduce the operator 𝑁∶𝐶([0,1],ℝ)→𝒫(𝐶([0,1],ℝ)) as𝑁(𝑥)=])∶ℎ(𝑡)=∫ℎ∈𝐶([0,1,ℝ_𝑡0(𝑡−𝑠)^𝑞−1Γ𝑞∫^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)(𝑠)−1𝑑𝑟−𝜆𝑥𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ×∫𝑞+1₁₀(1−𝑠)^𝑞−1Γ𝑞×∫^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)(𝑠)−𝑑𝑟−𝜆𝑥𝑑𝑠_𝑛𝑖∑₌₁𝛼_𝑖∫^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×∫^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×∫^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)(𝑠−1𝑑𝑟−𝜆𝑥)𝑑𝑠𝑑𝑢Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ𝑞+1^𝑛_𝑖∑₌₁𝛽_𝑖∫^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×∫^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞∫^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)𝑑𝑟−𝜆𝑥(𝑠)𝑑𝑠𝑑𝑢 (27)for 𝑣∈𝑆_𝐹,𝑥. We will show that the operator 𝑁 satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that 𝑁(𝑥)  is convex for each 𝑥∈𝐶([0,1],ℝ). For that, let ℎ₁,ℎ₂∈𝑁(𝑥). Then there exist 𝑣₁,𝑣₂∈𝑆_𝐹,𝑥 such that, for each 𝑡∈[0,1], we haveℎ_𝑖(𝑡)=^𝑡₀(𝑡−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_𝑖(𝑟)(𝑠)−1𝑑𝑟−𝜆𝑥𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ×𝑞+1₁₀(1−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_𝑖(𝑟)(𝑠)−𝑑𝑟−𝜆𝑥𝑑𝑠_𝑛𝑖=1𝛼_𝑖^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_𝑖(𝑟)(𝑠−1𝑑𝑟−𝜆𝑥)𝑑𝑠𝑑𝑢Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ×𝑞+1_𝑛𝑖=1𝛽_𝑖×^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_𝑖(𝑟)𝑑𝑟−𝜆𝑥(s)𝑑𝑠𝑑𝑢, (28)for 𝑖=1,2. Let 0≤𝜔≤1. Then, for each 𝑡∈[0,1] and putting 𝑧(𝑟)=𝜔𝑣₁(𝑟)+(1−𝜔)𝑣₂(𝑟), we have 𝜔ℎ₁+(1−𝜔)ℎ₂𝑡)=_𝑡0((𝑡−𝑠)^𝑞−1Γ𝑞^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑧(𝑟)(𝑠)−1𝑑𝑟−𝜆𝑥𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ×𝑞+1₁₀(1−𝑠)^𝑞−1Γ𝑞^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑧(𝑟)(𝑠)−𝑑𝑟−𝜆𝑥𝑑𝑠_𝑛𝑖=1𝛼_𝑖^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑧(𝑟)(𝑠−1𝑑𝑟−𝜆𝑥)𝑑𝑠𝑑𝑢Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ×𝑞+1_𝑛𝑖=1𝛽_𝑖^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑧(𝑟)𝑑𝑟−𝜆𝑥(𝑠)𝑑𝑠𝑑𝑢. (29)
Since 𝑆_𝐹,𝑥 is convex (𝐹 has convex values), therefore it follows that 𝜔ℎ₁+(1−𝜔)ℎ₂∈𝑁(𝑥).
Next, we show that 𝑁(𝑥)  maps bounded sets into bounded sets in 𝐶([0,1],ℝ). For a positive number 𝑟, let 𝐵_𝜌={𝑥∈𝐶([0,1],ℝ)∶‖𝑥‖≤𝜌} be a bounded set in 𝐶([0,1],ℝ). Then, for each ℎ∈𝑁(𝑥),𝑥∈𝐵_𝜌, there exists 𝑣∈𝑆_𝐹,𝑥 such that ℎ(𝑡)=^𝑡₀(𝑡−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)(𝑠)−1𝑑𝑟−𝜆𝑥𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ×𝑞+1₁₀(1−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)(𝑠)−𝑑𝑟−𝜆𝑥𝑑𝑠_𝑛𝑖=1𝛼_𝑖^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)(𝑠−1𝑑𝑟−𝜆𝑥)𝑑𝑠𝑑𝑢Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ×𝑞+1_𝑛𝑖=1𝛽_𝑖^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)(𝑠=𝑑𝑟−𝜆𝑥)𝑑𝑠𝑑𝑢_𝑡0(𝑡−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(r)(𝑠−1𝑑𝑟−𝜆𝑥)𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ𝒥𝑞+1²−1Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ𝒥𝑞+1¹,|ℎ(𝑡≤)|_𝑡0(𝑡−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝|𝑣(𝑟)|𝑑𝑟+|𝜆||𝑥(𝑠)|𝑑𝑠+𝐴₁𝒥₂+𝐴₂𝒥₁≤𝜓(‖𝑥‖)𝑝_𝐿1Γ+|𝜆𝑥‖𝑝+𝑞+1|‖Γ𝑞+1+𝐴₁𝜓(‖𝑥‖)𝑝_𝐿1Γ+|𝜆𝑥‖𝑝+𝑞+1|‖Γ+𝑞+1_𝑛𝑖=1𝛼_𝑖𝜂^{𝜈𝑖+𝑝+𝑞}𝜓(‖𝑥‖)𝑝_𝐿1Γ𝜈_𝑖+𝜂+𝑝+𝑞+1_{𝜈𝑖+𝑞}|𝜆|‖𝑥‖Γ𝜈_𝑖+𝑞+1+𝐴₂^𝑛_𝑖=1𝛽_𝑖𝜁^{𝜇𝑖+𝑝+𝑞}𝜓(‖𝑥‖)𝑝_𝐿1Γ𝜇_𝑖+𝜁+𝑝+𝑞+1_{𝜇𝑖+𝑞}|𝜆|‖𝑥‖Γ𝜇_𝑖𝑥𝑝+𝑞+1=𝜓(‖‖)_𝐿1×1+𝐴₁Γ𝑝+𝑞+1+𝐴₁^𝑛_𝑖=1𝛼_𝑖𝜂^{𝜈𝑖+𝑝+𝑞}Γ𝜈_𝑖+𝑝+𝑞+1+𝐴₂^𝑛_𝑖=1𝛽_𝑖𝜁^{𝜇𝑖+𝑝+𝑞}Γ𝜇_𝑖+‖𝑥|𝜆|+𝑝+𝑞+1‖1+𝐴₁Γ+|𝜆|𝐴𝑞+1_1𝑛𝑖=1𝛼_𝑖𝜂^{𝜈𝑖+𝑞}Γ𝜈_𝑖+𝑞+1+𝐴₂^𝑛_𝑖=1𝛽_𝑖𝜁^{𝜇𝑖+𝑞}Γ𝜇_𝑖𝑥𝑝+𝑞+1=Ω𝜓(‖‖)_𝐿1+Ψ‖𝑥‖. (30) Then ‖ℎ‖≤Ω𝜓𝜌𝑝_𝐿1+Ψ𝜌. (31)
Now we show that 𝑁maps bounded sets into equicontinuous sets of 𝐶([0,1],ℝ). Let 𝑡^′,𝑡^′′∈[0,1] with 𝑡^′<𝑡^′′ and 𝑥∈𝐵_𝜌, where 𝐵_𝜌, as above, is a bounded set of 𝐶([0,1],ℝ). For each ℎ∈𝑁(𝑥), we obtain ℎ𝑡^′′−ℎ𝑡^′≤^𝑡′′₀𝑡^′′−𝑠^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝|𝑣(𝑟)|𝑑𝑟+|𝜆||𝑥(𝑠)|𝑑𝑠−^𝑡′₀𝑡^′−𝑠^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝|𝑣(𝑟|𝜆𝑥(𝑠+Δ)|𝑑𝑟+||)|𝑑𝑠₃|Δ|Γ𝑡𝑞+1^′′𝑞−𝑡^′𝑞𝒥₂+Δ₂|Δ|Γ𝑡𝑞+1^′′𝑞−𝑡^′𝑞𝒥₁≤^𝑡′₀𝑡^′′−𝑠^𝑞−1−𝑡^′−𝑠^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝|𝑣(𝑟|𝜆𝑥(𝑠+)|𝑑𝑟+||)|𝑑𝑠_{𝑡′′𝑡′}𝑡^′′−𝑠^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝|𝑣(𝑟|𝜆𝑥(𝑠+Δ)|𝑑𝑟+||)|𝑑𝑠₃|Δ|Γ𝑡𝑞+1^′′𝑞−𝑡^′𝑞𝒥₂+Δ₂|Δ|Γ𝑡𝑞+1^′′𝑞−𝑡^′𝑞𝒥₁≤^𝑡′₀𝑡^′′−𝑠^𝑞−1−𝑡^′−𝑠^𝑞−1Γ𝑞×𝑝_𝐿1𝜓𝜌^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝|𝜆|𝜌+𝑑𝑟+𝑑𝑠_{𝑡′′𝑡′}𝑡^′′−𝑠^𝑞−1Γ𝑞×𝑝_𝐿1𝜓𝜌^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝|𝜆|𝜌+Δ𝑑𝑟+𝑑𝑠₃|Δ|Γ𝑡𝑞+1^′′𝑞−𝑡^′𝑞𝒥₂+Δ₂|Δ|Γ𝑡𝑞+1^′′𝑞−𝑡^′𝑞𝒥₁. (32)
Obviously the right-hand side of the above inequality tends to zero independently of 𝑥∈𝐵_𝜌 as 𝑡^′′−𝑡^′→0. As 𝑁 satisfies the above three assumptions, therefore it follows by Ascoli-Arzelá theorem that 𝑁∶𝐶([0,1],ℝ)→𝒫(𝐶([0,1],ℝ)) is completely continuous.
In our next step, we show that 𝑁 has a closed graph. Let 𝑥_𝑛→𝑥_∗,ℎ_𝑛∈𝑁(𝑥_𝑛), and ℎ_𝑛→ℎ_∗. Then we need to show that ℎ_∗∈𝑁(𝑥_∗). Associated with ℎ_𝑛∈𝑁(𝑥_𝑛), there exists 𝑣_𝑛∈𝑆_{𝐹,𝑥𝑛} such that, for each 𝑡∈[0,1],ℎ_𝑛(𝑡)=^𝑡₀(𝑡−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_𝑛(𝑟)(𝑠)−1𝑑𝑟−𝜆𝑥𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ×𝑞+1₁₀(1−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_𝑛(𝑟)(𝑠)−𝑑𝑟−𝜆𝑥𝑑𝑠_𝑛i=1𝛼_𝑖^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_𝑛(𝑟)(𝑠−1𝑑𝑟−𝜆𝑥)𝑑𝑠𝑑𝑢Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ×𝑞+1_𝑛𝑖=1𝛽_𝑖^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_𝑛(𝑟)𝑑𝑟−𝜆𝑥(𝑠)𝑑𝑠𝑑𝑢. (33)Thus we have to show that there exists 𝑣_∗∈𝑆_𝐹,𝑥∗ such that, for each 𝑡∈[0,1],ℎ_∗(𝑡)=^𝑡₀(𝑡−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_∗(𝑟)(𝑠)−1𝑑𝑟−𝜆𝑥𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ×𝑞+1₁₀(1−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_∗(𝑟)(𝑠)−𝑑𝑟−𝜆𝑥𝑑𝑠_𝑛𝑖=1𝛼_𝑖^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_∗(𝑟)(𝑠−1𝑑𝑟−𝜆𝑥)𝑑𝑠𝑑𝑢Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ×𝑞+1_𝑛𝑖=1𝛽_𝑖^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_∗(𝑟)𝑑𝑟−𝜆𝑥(𝑠)𝑑𝑠𝑑𝑢. (34)
Let us consider the continuous linear operator Θ∶𝐿¹([0,1],ℝ)→𝐶([0,1],ℝ) so that (𝑣)=𝑣↦Θ_𝑡0(𝑡−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)(𝑠)−1𝑑𝑟−𝜆𝑥𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ×𝑞+1₁₀(1−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)(𝑠)−𝑑𝑟−𝜆𝑥𝑑𝑠_𝑛𝑖=1𝛼_𝑖×^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)(𝑠−1𝑑𝑟−𝜆𝑥)𝑑𝑠𝑑𝑢Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ×𝑞+1_𝑛𝑖=1𝛽_𝑖×^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)𝑑𝑟−𝜆𝑥(𝑠)𝑑𝑠𝑑𝑢. (35) Observe thatℎ_𝑛(𝑡)−ℎ_∗(𝑡=_𝑡0)(𝑡−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_𝑛(𝑟)−𝑣_∗(𝑠)−1𝑑𝑟𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ×𝑞+1₁₀(1−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_𝑛(𝑟)−𝑣_∗(𝑠)−𝑑𝑟𝑑𝑠_𝑛𝑖=1𝛼_𝑖×^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γq×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_𝑛(𝑟)−𝑣_∗(𝑠)−1𝑑𝑟𝑑𝑠𝑑𝑢Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ×𝑞+1_𝑛𝑖=1𝛽_𝑖×^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_𝑛(𝑟)−𝑣_∗(𝑠)𝑑𝑟𝑑𝑠𝑑𝑢, (36)which tends to zero as 𝑛→∞.
Thus, it follows from Lemma 9 that Θ∘𝑆_𝐹 is a closed graph operator. Further, we have ℎ_𝑛(𝑡)∈Θ(𝑆_{𝐹,𝑥𝑛}). Since 𝑥_𝑛→𝑥_∗, it follows thatℎ_∗(𝑡)=^𝑡₀(𝑡−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_∗(𝑟)(𝑠)−1𝑑𝑟−𝜆𝑥𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ×𝑞+1₁₀(1−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_∗(𝑟)(𝑠)−𝑑𝑟−𝜆𝑥𝑑𝑠_𝑛𝑖=1𝛼_𝑖×^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_∗(𝑟)(𝑠−1𝑑𝑟−𝜆𝑥)𝑑𝑠𝑑𝑢Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ×𝑞+1_𝑛𝑖=1𝛽_𝑖×^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_∗(𝑟)𝑑𝑟−𝜆𝑥(𝑠)𝑑𝑠𝑑𝑢, (37)for some 𝑣_∗∈𝑆_𝐹,𝑥∗.
Finally, we discuss a priori bounds on solutions. Let 𝑥 be a solution of (1). Then there exists 𝑣∈𝐿¹([0,1],ℝ) with 𝑣∈𝑆_𝐹,𝑥 such that, for 𝑡∈[0,1], we haveℎ(𝑡)=^𝑡₀(𝑡−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)(𝑠)−1𝑑𝑟−𝜆𝑥𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ×𝑞+1₁₀(1−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)(𝑠)−𝑑𝑟−𝜆𝑥𝑑𝑠_𝑛𝑖=1𝛼_𝑖×^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)(𝑠−1𝑑𝑟−𝜆𝑥)𝑑𝑠𝑑𝑢Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ×𝑞+1_𝑛𝑖=1𝛽_𝑖×^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)𝑑𝑟−𝜆𝑥(𝑠)𝑑𝑠𝑑𝑢. (38)Using the computations proving that 𝑁(𝑥) maps bounded sets into bounded sets and the notations (17), we have ‖𝑥‖≤𝜓(‖𝑥‖)𝑝_𝐿1×1+𝐴₁Γ𝑝+𝑞+1+𝐴₁^𝑛_𝑖=1𝛼_𝑖𝜂^{𝜈𝑖+𝑝+𝑞}Γ𝜈_𝑖+𝑝+𝑞+1+𝐴₂^𝑛_𝑖=1𝛽_𝑖𝜁^{𝜇𝑖+𝑝+𝑞}Γ𝜇_𝑖+‖𝑥|𝜆|+𝑝+𝑞+1‖1+𝐴₁Γ+|𝜆|𝐴𝑞+1_1𝑛𝑖=1𝛼_𝑖𝜂^{𝜈𝑖+𝑞}Γ𝜈_𝑖+𝑞+1+𝐴₂^𝑛_𝑖=1𝛽_𝑖𝜁^{𝜇𝑖+𝑞}Γ𝜇_𝑖𝑥𝑝+𝑞+1=Ω𝜓(‖‖)_𝐿1+Ψ‖𝑥‖. (39)
Consequently ‖𝑥‖(1−Ψ)𝜓(‖𝑥‖)𝑝_𝐿1Ω≤1. (40) In view of (𝐻₃), there exists 𝑀 such that ‖𝑥‖≠𝑀. Let us set 𝑈={𝑥∈𝐶([0,1],ℝ)∶‖𝑥‖<𝑀+1}. (41) Note that the operator 𝑁∶𝑈→𝒫(𝐶([0,1],ℝ)) is upper semicontinuous and completely continuous. From the choice of 𝑈, there is no 𝑥∈𝜕𝑈 such that 𝑥∈𝜇𝑁(𝑥) for some 𝜇∈(0,1). Consequently, by the nonlinear alternative of Leray-Schauder type [31], we deduce that 𝑁 has a fixed point 𝑥∈𝑈 which is a solution of the problem (1). This completes the proof.

3.2. The Lower Semicontinuous Case

Next, we study the case where 𝐹 is not necessarily convex valued. Our approach here is based on the nonlinear alternative of Leray-Schauder type combined with the selection theorem of Bressan and Colombo for lower semicontinuous maps with decomposable values.

Theorem 17. Assume that (𝐻₂)-(𝐻₃) and the following conditions hold:(𝐻₄)𝐹∶[0,1]×ℝ→𝒫(ℝ) is a nonempty compact-valued multivalued map such that(a)(𝑡,𝑥)↦𝐹(𝑡,𝑥) is ℒ⊗ℬ measurable,(b)𝑥↦𝐹(𝑡,𝑥) is lower semicontinuous for each 𝑡∈[0,1];(𝐻₅) for each 𝜎>0, there exists 𝜑_𝜎∈𝐿¹([0,1],ℝ₊) such that ‖𝐹(𝑡,𝑥)‖=sup𝑦∶𝑦∈𝐹(𝑡,𝑥)≤𝜑_𝜎(𝑡)∀‖𝑥‖≤𝜎andfor𝑎.𝑒.𝑡∈[0,1]. (42)Then the boundary value problem (1) has at least one solution on [0,1].

Proof. It follows from (𝐻₄) and (𝐻₅) that 𝐹 is of l.s.c. type [35]. Then from Lemma 13, there exists a continuous function 𝑓∶𝐶([0,1],ℝ)→𝐿¹([0,1],ℝ) such that 𝑓(𝑥)∈ℱ(𝑥) for all 𝑥∈𝐶([0,1],ℝ).
Consider the problem ^𝑐𝐷^𝑝𝑐𝐷^𝑞𝑥(𝑡)(𝑥(𝑡𝑥(0)=+𝜆=𝐹)),0<𝑡<1,_𝑛𝑖=1𝛽_𝑖𝐼^𝜇𝑖𝑥𝜁),𝑥(1)=_𝑛𝑖=1(𝛼_𝑖𝐼^𝜈𝑖𝑥𝜂,0<𝜁<𝜂<1. (43)
Observe that, if 𝑥∈𝐴𝐶¹([0,1]) is a solution of (43), then 𝑥 is a solution to the problem (1). In order to transform the problem (43) into a fixed point problem, we define the operator 𝑁 as (𝑡)=𝑁𝑥_𝑡0(𝑡−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑓(𝑥(𝑟(𝑠)−1))𝑑𝑟−𝜆𝑥𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ×𝑞+1₁₀(1−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑓(𝑥(𝑟(𝑠)−))𝑑𝑟−𝜆𝑥𝑑𝑠_𝑛𝑖=1𝛼_𝑖×^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑓(𝑥(𝑟(𝑠−1))𝑑𝑟−𝜆𝑥)𝑑𝑠𝑑𝑢Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ×𝑞+1_𝑛𝑖=1𝛽_𝑖×^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑓(𝑥(𝑟))𝑑𝑟−𝜆𝑥(𝑠)𝑑𝑠𝑑𝑢. (44) It can easily be shown that 𝑁 is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 16. So we omit it. This completes the proof.

3.3. The Lipschitz Case

Now we prove the existence of solutions for the problem (1) with a nonconvex valued right-hand side by applying a fixed point theorem for multivalued map due to Covitz and Nadler [34].

Theorem 18. Assume that the following conditions hold:(𝐻₆)𝐹∶[0,1]×ℝ→𝒫_𝑐𝑝(ℝ) is such that 𝐹(⋅,𝑥)∶[0,1]→𝒫_𝑐𝑝(ℝ) is measurable for each 𝑥∈ℝ;(𝐻₇)𝐻_𝑑(𝐹(𝑡,𝑥),𝐹(𝑡,𝑥))≤𝑚(𝑡)|𝑥−𝑥| for almost all 𝑡∈[0,1] and 𝑥,𝑥∈ℝ with 𝑚∈𝐶([0,1],ℝ⁺) and 𝑑(0,𝐹(𝑡,0))≤𝑚(𝑡) for almost all 𝑡∈[0,1].
Then the boundary value problem (1) has at least one solution on [0,1] if Ω‖𝑚‖_𝐿¹<1, that is, 1+𝐴₁Γ𝑝+𝑞+1+𝐴₁^𝑛_𝑖=1𝛼_𝑖𝜂^{𝜈𝑖+𝑝+𝑞}Γ𝜈_𝑖+𝑝+𝑞+1+𝐴₂^𝑛_𝑖=1𝛽_𝑖𝜁^{𝜇𝑖+𝑝+𝑞}Γ𝜇_𝑖𝑚‖+𝑝+𝑞+1‖_𝐿¹<1. (45)

Proof. Observe that the set 𝑆_𝐹,𝑥 is nonempty for each 𝑥∈𝐶([0,1],ℝ) by the assumption (𝐻₆), so 𝐹 has a measurable selection (see Theorem III.6 [36]). Now we show that the operator 𝑁 satisfies the assumptions of Lemma 15. To show that 𝑁(𝑥)∈𝒫_cl((𝐶[0,1],ℝ)) for each 𝑥∈𝐶([0,1],ℝ), let {𝑢_𝑛}_𝑛≥0∈𝑁(𝑥) be such that 𝑢_𝑛→𝑢(𝑛→∞) in 𝐶([0,1],ℝ). Then 𝑢∈𝐶([0,1],ℝ), and there exists 𝑣_𝑛∈𝑆_𝐹,𝑥 such that, for each 𝑡∈[0,1],𝑢_𝑛(𝑡)=^𝑡₀(𝑡−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_𝑛(𝑟)(𝑠)−1𝑑𝑟−𝜆𝑥𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ×𝑞+1₁₀(1−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_𝑛(𝑟)(𝑠)−𝑑𝑟−𝜆𝑥𝑑𝑠_𝑛𝑖=1𝛼_𝑖×^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_𝑛(𝑟)(𝑠−1𝑑𝑟−𝜆𝑥)𝑑𝑠𝑑𝑢Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ×𝑞+1_𝑛𝑖=1𝛽_𝑖×^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣_𝑛(𝑟)𝑑𝑟−𝜆𝑥(𝑠)𝑑𝑠𝑑𝑢. (46)
As 𝐹 has compact values, we pass onto a subsequence to obtain that 𝑣_𝑛 converges to 𝑣 in 𝐿¹([0,1],ℝ). Thus, 𝑣∈𝑆_𝐹,𝑥 and, for each 𝑡∈[0,1], 𝑢_𝑛(𝑡)(𝑡)=→𝑢_𝑡0(𝑡−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)(𝑠)−1𝑑𝑟−𝜆𝑥𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ×𝑞+1₁₀(1−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)(𝑠)−𝑑𝑟−𝜆𝑥𝑑𝑠_𝑛𝑖=1𝛼_𝑖×^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)(𝑠−1𝑑𝑟−𝜆𝑥)𝑑𝑠𝑑𝑢Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ×𝑞+1_𝑛𝑖=1𝛽_𝑖×^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣(𝑟)𝑑𝑟−𝜆𝑥(𝑠)𝑑𝑠𝑑𝑢. (47)Hence, 𝑢∈𝑁(𝑥).
Next we show that there exists 𝛾<1 such that 𝐻_𝑑(𝑁(𝑥),𝑁(𝑥‖))≤𝛾𝑥−𝑥‖foreach𝑥,𝑥∈𝐶([0,1],ℝ). (48) Let 𝑥,𝑥∈𝐶([0,1],ℝ) and ℎ₁∈𝑁(𝑥). Then there exists 𝑣₁(𝑡)∈𝐹(𝑡,𝑥(𝑡)) such that, for each 𝑡∈[0,1], ℎ₁(𝑡)=^𝑡₀(𝑡−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣₁(𝑟)(𝑠)−1𝑑𝑟−𝜆𝑥𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ×𝑞+1₁₀(1−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣₁(𝑟)(𝑠)−𝑑𝑟−𝜆𝑥𝑑𝑠_𝑛𝑖=1𝛼_𝑖×^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣₁(𝑟)(𝑠−1𝑑𝑟−𝜆𝑥)𝑑𝑠𝑑𝑢Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ×𝑞+1_𝑛𝑖=1𝛽_𝑖×^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣₁(𝑟)𝑑𝑟−𝜆𝑥(𝑠)𝑑𝑠𝑑𝑢. (49)
By (𝐻₇), we have 𝐻_𝑑(𝐹(𝑡,𝑥),𝐹(𝑡,𝑥))≤𝑚(𝑡)|𝑥(𝑡)−𝑥(𝑡)|. (50) So, there exists 𝑤(𝑡)∈𝐹(𝑡,𝑥(𝑡)) such that 𝑣₁(𝑡)−𝑤(𝑡)≤𝑚(𝑡)|𝑥(𝑡)−𝑥(𝑡)|,𝑡∈[0,1]. (51)
Define 𝑈∶[0,1]→𝒫(ℝ) by 𝑈(𝑡)=𝑤∈ℝ∶𝑣₁(𝑡)−𝑤≤𝑚(𝑡)|𝑥(𝑡)−𝑥(𝑡)|. (52) Since the multivalued operator 𝑈(𝑡)∩𝐹(𝑡,𝑥(𝑡)) is measurable (Proposition III.4 [36]), there exists a function 𝑣₂(𝑡) which is a measurable selection for 𝑉. So 𝑣₂(𝑡)∈𝐹(𝑡,𝑥(𝑡)), and for each 𝑡∈[0,1], we have |𝑣₁(𝑡)−𝑣₂(𝑡)|≤𝑚(𝑡)|𝑥(𝑡)−𝑥(𝑡)|.
For each 𝑡∈[0,1], let us define ℎ₂(𝑡)=^𝑡₀(𝑡−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣₂(𝑟)(𝑠)−1𝑑𝑟−𝜆𝑥𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ×𝑞+1₁₀(1−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣₂(𝑟)(𝑠)−𝑑𝑟−𝜆𝑥𝑑𝑠_𝑛𝑖=1𝛼_𝑖×^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣₂(𝑟)(𝑠−1𝑑𝑟−𝜆𝑥)𝑑𝑠𝑑𝑢Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ×𝑞+1_𝑛𝑖=1𝛽_𝑖×^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣₂(𝑟)𝑑𝑟−𝜆𝑥(𝑠)𝑑𝑠𝑑𝑢. (53) Thus, ℎ₁(𝑡)−ℎ₂(𝑡=_𝑡0)(𝑡−𝑠)^q−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣₁(𝑟)−𝑣₂(𝑟−1)𝑑𝑟𝑑𝑠Δ𝑡^𝑞Δ₃+Δ₁Γ𝑞+1Γ×𝑞+1₁₀(1−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣₁(𝑟)−𝑣₂(𝑟−)𝑑𝑟𝑑𝑠_𝑛𝑖=1𝛼_𝑖×^𝜂₀𝜂−𝑢^𝜈𝑖−1Γ𝜈_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣₁(𝑟)−𝑣₂(𝑟−1)𝑑𝑟𝑑𝑠𝑑𝑢Δ𝑡^𝑞Δ₂+Δ₄Γ𝑞+1Γ×𝑞+1_𝑛𝑖=1𝛽_𝑖×^𝜁₀(𝜁−𝑢)^𝜇𝑖−1Γ𝜇_𝑖×^𝑢₀(𝑢−𝑠)^𝑞−1Γ𝑞×^𝑠₀(𝑠−𝑟)^𝑝−1Γ𝑝𝑣₁(𝑟)−𝑣₂(𝑟)𝑑𝑟𝑑𝑠𝑑𝑢. (54)Hence, ℎ₁−ℎ₂≤1+𝐴₁Γ𝑝+𝑞+1+𝐴₁^𝑛_𝑖=1𝛼_𝑖𝜂^{𝜈𝑖+𝑝+𝑞}Γ𝜈_𝑖+𝑝+𝑞+1+𝐴₂^𝑛_𝑖=1𝛽_𝑖𝜁^{𝜇𝑖+𝑝+𝑞}Γ𝜇_𝑖×‖𝑚‖+𝑝+𝑞+1_𝐿1‖𝑥−𝑥‖. (55) Analogously, interchanging the roles of 𝑥 and 𝑥, we obtain 𝐻_𝑑(𝑁(𝑥),𝑁(𝑥))≤𝛾‖𝑥−𝑥‖≤1+𝐴₁Γ𝑝+𝑞+1+𝐴₁^𝑛_𝑖=1𝛼_𝑖𝜂^{𝜈𝑖+𝑝+𝑞}Γ𝜈_𝑖+𝑝+𝑞+1+𝐴₂^𝑛_𝑖=1𝛽_𝑖𝜁^{𝜇𝑖+𝑝+𝑞}Γ𝜇_𝑖×‖𝑚‖+𝑝+𝑞+1_𝐿1‖𝑥−𝑥‖. (56)
Since 𝑁 is a contraction, it follows from Lemma 15 that 𝑁 has a fixed point 𝑥 which is a solution of (1). This completes the proof.

Remark 19. It is important to note that several new interesting special results of the present work can be obtained by fixing the parameters involved in the given problem. Some of these results are listed in the following.(i) Our results correspond to the multivalued extension of the Dirichlet problem considered in [23] for 𝛽_𝑖=0=𝛼_𝑖,𝑖=1,2,…,𝑛.(ii) In case we take 𝛽_𝑖=0,𝑖=1,2,…,𝑛, we obtain the results for Langevin fractional differential inclusions with the three-point integral boundary conditions of the following form: 𝑥(0)=0,𝑥(1)=^𝑛_𝑖=1𝛼_𝑖𝐼^𝜈𝑖𝑥𝜂. (57)(iii) By taking 𝛽_𝑖=0=𝛼_𝑖,𝑖=2,3,…,𝑛, and 𝜈₁=1=𝜇₁, we get the results for Langevin fractional differential inclusions with four-point nonlocal integral boundary conditions of the following type: 𝑥(0)=𝛽₁^𝜁₀𝑥(𝑠)𝑑𝑠,𝑥(1)=𝛼₁^𝜂₀𝑥(𝑠)𝑑𝑠. (58)  Note that we obtain the typical integral boundary condition in the limit 𝜁,𝜂→1⁻.

Example 20. Consider the problem ^𝑐𝐷^2/3𝑐𝐷^4/5+1𝑥(𝑡)((𝑡𝑥(0)=110∈𝐹𝑡,𝑥)),0<𝑡<1,2𝐼^1/2𝑥14−13𝐼^3/4𝑥14+14𝐼^3/2𝑥14−15𝐼^4/3𝑥14,𝑥(1)=12𝐼^1/4𝑥13−23𝐼^3/4𝑥13+34𝐼^5/4𝑥13−45𝐼^7/4𝑥13, (59) where 𝐹∶[0,1]×ℝ→𝒫(ℝ) is a multivalued map given by ()=1𝑥→𝐹𝑡,𝑥4𝑥⁵𝑥⁵+1+38(),1𝑡+141sin𝑥+4(𝑡+1). (60) For 𝑓∈𝐹, we have 𝑓1≤max4𝑥⁵𝑥⁵+1+38(),1𝑡+141sin𝑥+4(≤3𝑡+1)4. (61) Here 𝑝=2/3,  𝑞=4/5,  𝜆=1/10,  𝜂=1/3,  𝑛=4,  𝛼₁=1/2,  𝛼₂=−2/3,  𝛼₃=3/4,  𝛼₄=−4/5,  𝜈₁=1/4,  𝜈₂=3/4,  𝜈₃=5/4,  𝜈₄=7/4,  𝜁=1/4,  𝛽₁=1/2,  𝛽₂=−1/3,  𝛽₃=1/4,  𝛽₄=−1/5,  𝜇₁=1/2,  𝜇₂=3/4,  𝜇₃=3/2,  and 𝜇₄=4/3. Clearly, ‖𝐹(𝑡,𝑥)‖_𝒫∶=sup𝑦∶𝑦∈𝐹(𝑡,𝑥)≤𝑝(𝑡)𝜓(‖𝑥‖),𝑥∈ℝ, (62) with 𝑝(𝑡)=1,𝜓(‖𝑥‖)=3/4. Using the given data, it is found that Δ₁=0.126506,Δ₂Δ=0.804143,₃=0.849082,Δ₄=−0.983656,Δ=0.936934,𝐴₁𝐴=1.108019,₂=1.971367,Ω=1.764724,Ψ=0.47047. (63) Thus, 𝜓(𝑀𝑝𝑀>)_𝐿1Ω1−Ψ≈2.499467. (64) Clearly, all the conditions of Theorem 16 are satisfied. So there exists at least one solution of the problem (59) on [0,1].

Example 21. Consider the fractional inclusion boundary value problem (59) with 𝐹∶[0,1]×ℝ→𝒫(ℝ) given by 𝐹(𝑡,𝑥)=0,sin𝑥(2+𝑡)²+1.12 (65) Then, we have 𝑢|(≤1sup{|∶𝑢∈𝐹𝑡,𝑥)}+112(2+𝑡)²,𝐻_𝑑(𝐹(𝑡,𝑥),𝐹(𝑡,𝑥))≤𝑚(𝑡)|𝑥−𝑥|, (66) where 𝑚(𝑡)=1/(2+𝑡)². With ‖𝑚‖_𝐿¹=1/6 and Ω=1.764724 (from Example 20), it is found that 𝛾=Ω‖𝑚‖_𝐿¹≈0.294121<1. (67) Since all the conditions of Theorem 18 are satisfied, therefore the problem (59) with 𝐹 given by (65) has at least one solution on [0,1].

Acknowledgments

The authors thank the referees for their useful comments. This work was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.