Existence Results for Fractional Differential Inclusions with Multivalued Term Depending on Lower-Order Derivative

Abstract

This paper is concerned with a class of fractional differential inclusions whose multivalued term depends on lower-order fractional derivative with fractional (non)separated boundary conditions. The cases of convex-valued and non-convex-valued right-hand sides are considered. Some existence results are obtained by using standard fixed point theorems. A possible generalization for the inclusion problem with integral boundary conditions is also discussed. Examples are given to illustrate the results.

1. Introduction

Recently, the subject of fractional differential equations has emerged as an important area of investigation. Indeed, we can find numerous applications of fractional-order derivatives in the mathematical modeling of physical and biological phenomena in various fields of science and engineering [1–3]. A variety of results on initial and boundary value problems of fractional differential equations and inclusions can easily be found in the literature on this topic. For some recent results, we can refer to, for instance, [4–20] (equations) [21–27] (inclusions) and the references therein.

Ahmad and Ntouyas [22] considered a boundary value problem of fractional differential inclusions with fractional separated boundary conditions given by ^𝑐𝐷^𝑞𝑥(𝑡)((𝑡[]𝛼∈𝐹𝑡,𝑥)),𝑡∈0,1,1<𝑞≤2,₁𝑥(0)+𝛽₁^𝑐𝐷^𝑝𝑥(0)=𝛾₁,𝛼₂𝑥(1)+𝛽₂^𝑐𝐷^𝑝𝑥(1)=𝛾₂,0<𝑝<1, (1.1) where ^𝑐𝐷^𝑞 denotes the Caputo fractional derivative of order 𝑞, 𝐹∶[0,1]×ℝ→2^ℝ is a multivalued map, and 𝛼_𝑖, 𝛽_𝑖, 𝛾_𝑖 (𝑖=1,2) are real constants, with 𝛼₁≠0.

In Cernea [24], the following multipoint boundary value problem for a fractional-order differential inclusion was studied 𝐷^𝛼𝑥(𝑡)∈𝐹𝑡,𝑥(𝑡),𝑥^′(𝑡)a.e[]𝑥(0).𝑡∈0,1,2<𝛼≤3,=𝑥^′(0)=0,𝑥(1)−^𝑚_𝑖=1𝑎_𝑖𝑥𝜉_𝑖=𝜆, (1.2) where 𝐷^𝛼 is the standard Riemann-Liouville fractional derivative, 𝑚≥1, 0<𝜉₁<𝜉₂<⋯<𝜉_𝑚<1, ∑^𝑚_𝑖=1𝑎_𝑖𝜉_𝑖^𝛼−1<1, 𝜆>0, 𝑎_𝑖>0, 𝑖=1,2,…,𝑚, and 𝐹∶[0,1]×ℝ×ℝ→2^ℝ is a multivalued map.

In Khan et al. [11], the authors studied the existence and uniqueness results of nonlinear fractional differential equation of the type ^𝑐𝐷^𝑞𝑥(𝑡)=𝑓𝑡,𝑥(𝑡),^𝑐𝐷^𝜎𝑥(𝑡[],(0)),𝑡∈0,𝑇𝛼𝑥−𝛽𝑥^′(0)=^𝑇₀𝑔(𝑠,𝑥)𝑑𝑠,𝛾𝑥(𝑇)+𝛿𝑥^′(𝑇)=^𝑇₀ℎ(𝑠,𝑥)𝑑𝑠, (1.3) where 0<𝜎<1, 1<𝑞<2, 𝛼,𝛿>0, 𝛽,𝛾≥0 (or 𝛼,𝛿≥0, 𝛽,𝛾>0) and ^𝑐𝐷^𝑞, ^𝑐𝐷^𝜎 are the Caputo fractional derivatives. The results in [11, 22, 24] are obtained by using appropriate standard fixed point theorems.

Motivated by the papers cited above, in this paper, we consider the existence results for a new class of fractional differential inclusions of the form ^𝑐𝐷^𝛼𝑥(𝑡)∈𝐹𝑡,𝑥(𝑡),^𝑐𝐷^𝛽𝑥(𝑡),a.e.𝑡∈[0,𝑇], (1.4) where ^𝑐𝐷^𝛼 denotes the Caputo fractional derivative of order 𝛼, 𝐹∶[0,1]×ℝ×ℝ→2^ℝ is a multivalued map, 1<𝛼≤2, 0<𝛽≤1, and 𝑇>0. We study (1.4) subject to two families of boundary conditions:(1) separated boundary conditions 𝑎₁𝑥(0)+𝑏₁^𝑐𝐷^𝛾𝑥(0)=𝑐₁,𝑎₂𝑥(𝑇)+𝑏₂^𝑐𝐷^𝛾𝑥(𝑇)=𝑐₂, (1.5)(2) Nonseparated boundary conditions 𝑎₁𝑥(0)+𝑏₁𝑥(𝑇)=𝑐₁,𝑎₂^𝑐𝐷^𝛾𝑥(0)+𝑏₂^𝑐𝐷^𝛾𝑥(𝑇)=𝑐₂, (1.6) where 𝑎_𝑖, 𝑏_𝑖, 𝑐_𝑖, 𝑖=1,2 are real constants and 0<𝛾<1.

The results of this paper can easily to be generalized to the boundary value problems of fractional differential inclusions (1.4) with the following integral boundary conditions: 𝑎₁𝑥(0)+𝑏₁^𝑐𝐷^𝛾𝑥(0)=𝑐₁^𝑇₀𝑔((𝑠𝑎𝑠,𝑥))𝑑𝑠,₂𝑥(𝑇)+𝑏₂^𝑐𝐷^𝛾𝑥(𝑇)=𝑐₂^𝑇₀ℎ(𝑠,𝑥(𝑠))𝑑𝑠,𝑎₁𝑥(0)+𝑏₁𝑥(𝑇)=𝑐₁^𝑇₀𝑔((𝑠𝑎𝑠,𝑥))𝑑𝑠,_2𝑐𝐷^𝛾𝑥(0)+𝑏₂^𝑐𝐷^𝛾𝑥(𝑇)=𝑐₂^𝑇₀ℎ(𝑠,𝑥(𝑠))𝑑𝑠, (1.7)(1.8) where 𝑔,ℎ∶[0,𝑇]×ℝ→ℝ are given functions.

We remark that when the third variable of the multifunction 𝐹 in (1.4) vanishes, the problem (1.4), (1.5) reduces to the case considered in [22]. When 𝑎₁=𝑏₁=1, 𝑎₂=𝑏₂=1, and 𝑐₁=𝑐₂=0, the problem (1.4), (1.6) reduces to an antiperiodic fractional boundary value problem (the case of 𝐹=𝑓 a given continuous function was studied in [4, 15]). Our results generalize some results from the literature cited above and constitute a contribution to this emerging field of research.

The rest of the paper is organized as follows: in Section 2 we present the notations and definitions and give some preliminary results that we need in the sequel, Section 3 is dedicated to the existence results of the fractional differential inclusion (1.4) with boundary conditions (1.5) and (1.6), in Section 4 we indicate a possible generalization for the inclusion problem (1.4) with integral boundary conditions (1.7) and (1.8), and two illustrative examples are given in Section 5.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper.

Let (𝑋,‖⋅‖) be a normed space. We use the notations 𝑃(𝑋)={𝑌⊆𝑋∶𝑌≠∅}, 𝑃_cl(𝑋)={𝑌∈𝑃(𝑋)∶𝑌closed}, 𝑃_𝑏(𝑋)={𝑌∈𝑃(𝑋)∶𝑌bounded}, 𝑃_cp(𝑋)={𝑌∈𝑃(𝑋)∶𝑌compact}, 𝑃_cp,𝑐(𝑋)={𝑌∈𝑃(𝑋)∶𝑌compact, convex}, and so on.

Let 𝐴,𝐵∈𝑃_cl(𝑋); the Pompeiu-Hausdorff distance of 𝐴, 𝐵 is defined as ℎ(𝐴,𝐵)=maxsup_𝑎∈𝐴𝑑(𝑎,𝐵),sup_𝑏∈𝐵𝑑(𝑏,𝐴). (2.1)

A multivalued map 𝐹∶𝑋→𝑃(𝑋) is convex (closed) valued if 𝐹(𝑥) is convex (closed) for all 𝑥∈𝑋. 𝐹 is said to be completely continuous if 𝐹(𝐵) is relatively compact for every 𝐵∈𝑃_𝑏(𝑋). 𝐹 is called upper semicontinuous on 𝑋, if for every 𝑥₀∈𝑋, the set 𝐹(𝑥₀) is a nonempty closed subset of 𝑋, and for every open set 𝑂 of 𝑋 containing 𝐹(𝑥₀), there exists an open neighborhood 𝑈₀ of 𝑥₀ such that 𝐹(𝑈₀)⊆𝑂. Equivalently, 𝐹 is upper semicontinuous if the set {𝑥∈𝑋∶𝐹(𝑥)⊆𝑂} is open for any open set 𝑂 of 𝑋. 𝐹 is called lower semicontinuous if the set {𝑥∈𝑋∶𝐹(𝑥)∩𝑂≠∅} is open for each open set 𝑂 in 𝑋. If a multivalued map 𝐹 is completely continuous with nonempty compact values, then 𝐹 is upper semicontinuous if and only if 𝐹 has a closed graph, that is, if 𝑥_𝑛→𝑥_∗ and 𝑦_𝑛→𝑦_∗, then 𝑦_𝑛∈𝐹(𝑥_𝑛) implies 𝑦_∗∈𝐹(𝑥_∗) [28].

A multivalued map 𝐹∶[0,𝑇]→𝑃_cl(𝑋) is said to be measurable if, for every 𝑥∈𝑋, the function 𝑡→𝑑(𝑥,𝐹(𝑡))=inf{𝑑(𝑥,𝑦)∶𝑦∈𝐹(𝑡)} is a measurable function.

Definition 2.1. A multivalued map 𝐹∶𝑋→𝑃_cl(𝑋) is called(1)𝛾-Lipschitz if there exists 𝛾>0 such that ℎ𝐹(𝑥),𝐹𝑦≤𝛾𝑑𝑥,𝑦,foreach𝑥,𝑦∈𝑋 (2.2)(2) a contraction if it is 𝛾-Lipschitz with 𝛾<1.

Definition 2.2. A multivalued map 𝐹∶[0,𝑇]×ℝ×ℝ→𝑃(ℝ) is said to be Carathéodory if(1)𝑡→𝐹(𝑡,𝑥,𝑦) is measurable for each 𝑥,𝑦∈ℝ;(2)(𝑥,𝑦)→𝐹(𝑡,𝑥,𝑦) is upper semicontinuous for a.e. 𝑡∈[0,𝑇]. Further, a Carathéodory function 𝐹 is said to be 𝐿¹- Carathéodory if(3) for each 𝑙>0, there exists 𝜑_𝑙∈𝐿¹([0,𝑇],ℝ⁺) such that 𝐹𝑡,𝑥,𝑦=sup|𝑣|∶𝑣∈𝐹𝑡,𝑥,𝑦≤𝜑_𝑙(𝑡) (2.3) for all |𝑥|≤𝑙, |𝑦|≤𝑙 and a.e. 𝑡∈[0,𝑇].

Lemma 2.3 (see [29]). Let 𝑋 be a Banach space. Let 𝐺∶[0,𝑇]×𝑋→𝑃_cp,c(𝑋) be an 𝐿¹- Carathéodory multivalued map and Γ a linear continuous map from 𝐿¹([0,𝑇],𝑋) to 𝐶([0,𝑇],𝑋), then the operator Γ∘𝑆_𝐺∶𝐶([0,𝑇],𝑋)→𝑃_{𝑐𝑝,𝑐}(𝐶([0,𝑇],𝑋)),𝑦↦Γ∘𝑆_𝐺𝑦=Γ𝑆_𝐺,𝑦 (2.4) is a closed graph operator in 𝐶([0,𝑇],𝑋)×𝐶([0,𝑇],𝑋).

Here 𝑆_𝐺,𝑦={𝑣∈𝐿¹([0,𝑇],𝑋)∶𝑣(𝑡)∈𝐺(𝑡,𝑦(𝑡)) for a.e. 𝑡∈[0,𝑇]}.

Definition 2.4 (see [30]). The Riemann-Liouville fractional integral of order 𝑞 for a function 𝑓 is defined as 𝐼^𝑞𝑓(𝑡)=1Γ𝑞^𝑡₀𝑓(𝑠)(𝑡−𝑠)^1−𝑞𝑑𝑠,𝑞>0, (2.5) provided the integral exists.

Definition 2.5 (see [30]). For at least 𝑛-times differentiable function 𝑓, the Caputo derivative of order 𝑞 is defined as ^𝑐𝐷^𝑞𝑓(𝑡)=1Γ𝑛−𝑞^𝑡₀(𝑡−𝑠)^{𝑛−𝑞−1}𝑓^(𝑛)(𝑠)𝑑𝑠,𝑛−1<𝑞<𝑛,𝑛=𝑞+1, (2.6) where [𝑞] denotes the integer part of the real number 𝑞.

Lemma 2.6 (see [20]). Let 𝛼>0; then the differential equation ^𝑐𝐷^𝛼ℎ(𝑡)=0 (2.7) has solutions ℎ(𝑡)=𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+⋯+𝑐_𝑛−1𝑡^𝑛−1 and 𝐼^𝛼𝑐𝐷^𝛼ℎ(𝑡)=ℎ(𝑡)+𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+⋯+𝑐_𝑛−1𝑡^𝑛−1; (2.8) here 𝑐_𝑖∈ℝ, 𝑖=0,1,2,…,𝑛−1, 𝑛=[𝛼]+1.

The following lemma obtained in [6] is useful in the rest of the paper.

Lemma 2.7 (see [6]). For a given 𝑦∈𝐶([0,𝑇],ℝ), the unique solution of the fractional separated boundary value problem ^𝑐𝐷^𝛼𝑥(𝑡)(𝑡)[]𝑎=𝑦,𝑡∈0,𝑇,1<𝛼≤2,₁𝑥(0)+𝑏₁^𝑐𝐷^𝛾𝑥(0)=𝑐₁,𝑎₂𝑥(𝑇)+𝑏₂^𝑐𝐷^𝛾𝑥(𝑇)=𝑐₂,0<𝛾<1, (2.9) is given by 𝑥(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)−𝑡𝑑𝑠𝑣₁𝑎₂^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑏₂^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)𝛼−𝛾𝑑𝑠+𝑣₂𝑐𝑡+¹𝑎₁, (2.10) where 𝑣₁=𝑎₂𝑇Γ2−𝛾+𝑏₂𝑇^1−𝛾Γ2−𝛾,𝑣₂=𝑎₁𝑐₂−𝑎₂𝑐₁𝑎₁𝑣₁. (2.11)

We notice that the solution (2.10) of the problem (2.9) does not depend on the parameter 𝑏₁, that is to say, the parameter 𝑏₁ is of arbitrary nature for this problem. And by (2.10), we should assume that 𝑎₁≠0 and 𝑎₂𝑇^𝛾Γ(2−𝛾)≠−𝑏₂.

Lemma 2.8. For any 𝑦∈𝐶([0,𝑇],ℝ), the unique solution of the fractional nonseparated boundary value problem ^𝑐𝐷^𝛼𝑥(𝑡)(𝑡)[]𝑎=𝑦,𝑡∈0,𝑇,1<𝛼≤2,₁𝑥(0)+𝑏₁𝑥(𝑇)=𝑐₁,𝑎₂^𝑐𝐷^𝛾𝑥(0)+𝑏₂^𝑐𝐷^𝛾𝑥(𝑇)=𝑐₂,0<𝛾<1, (2.12) is given by 𝑥(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠−𝑡Γ2−𝛾𝑇^1−𝛾𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)𝑐𝛼−𝛾𝑑𝑠+𝑡Γ2−𝛾₂𝑇^1−𝛾𝑏₂−𝑏₁𝑎₁+𝑏₁^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠−𝑇^𝛾Γ2−𝛾^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)−1𝛼−𝛾𝑑𝑠𝑎₁+𝑏₁𝑏₁𝑐₂𝑇^𝛾Γ2−𝛾𝑏₂−𝑐₁. (2.13)

Proof. For 1<𝛼≤2, by Lemma 2.6, we know that the general solution of the equation ^𝑐𝐷^𝛼𝑥(𝑡)=𝑦(𝑡) can be written as 𝑥(𝑡)=𝐼^𝛼𝑦(𝑡)−𝑘₁−𝑘₂𝑡=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠−𝑘₁−𝑘₂𝑡, (2.14) where 𝑘₁, 𝑘₂∈ℝ are arbitrary constants. Since ^𝑐𝐷^𝛾𝑘=0 (𝑘 is a constant), ^𝑐𝐷^𝛾𝑡=𝑡^1−𝛾/Γ(2−𝛾), ^𝑐𝐷^𝛾𝐼^𝛼𝑦(𝑡)=𝐼^𝛼−𝛾𝑦(𝑡) (see [30]), from (2.14), we have ^𝑐𝐷^𝛾𝑥(𝑡)=𝐼^𝛼−𝛾𝑦(𝑡)−𝑘₂𝑡^1−𝛾Γ=2−𝛾^𝑡₀(𝑡−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)𝑘𝛼−𝛾𝑑𝑠−²𝑡^1−𝛾Γ.2−𝛾 (2.15) Using the boundary conditions, we obtain 𝑎₁−𝑘₁+𝑏₁^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠−𝑘₁−𝑘₂𝑇=𝑐₁,𝑎₂×0+𝑏₂^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)𝑘𝛼−𝛾𝑑𝑠−²𝑇^1−𝛾Γ2−𝛾=𝑐₂. (2.16) Therefore, we have 𝑘₁=1𝑎₁+𝑏₁𝑏₁𝑐₂𝑇^𝛾Γ2−𝛾𝑏₂−𝑐₁+𝑏₁𝑎₁+𝑏₁^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠−𝑇^𝛾Γ2−𝛾^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠),𝑘𝛼−𝛾𝑑𝑠₂=Γ2−𝛾𝑇^1−𝛾𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)𝑐𝛼−𝛾𝑑𝑠−²𝑏₂. (2.17) Substituting the values of 𝑘₁, 𝑘₂ in (2.14), we obtain (2.13). This completes the proof.

From the proof of the above lemma, we notice that the solution (2.13) of the problem (2.12) does not depend on the parameter 𝑎₂, that is to say, the parameter 𝑎₂ is of arbitrary nature for this problem. In this situation, we need to assume that 𝑎₁+𝑏₁≠0 and 𝑏₂≠0.

Let us define what we mean by a solution of the problem (1.4), (1.5) and the problem (1.4), (1.6).

Definition 2.9. A function 𝑥∈𝐴𝐶¹([0,𝑇],ℝ) is a solution of the problem (1.4), (1.5) if it satisfies the boundary conditions (1.5) and there exists a function 𝑓∈𝐿¹([0,𝑇],ℝ) such that 𝑓(𝑡)∈𝐹(𝑡,𝑥(𝑡),^𝑐𝐷^𝛽𝑥(𝑡)) a.e. on 𝑡∈[0,𝑇] and 𝑥(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠)−𝑡𝑑𝑠𝑣₁𝑎₂^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠)𝑑𝑠+𝑏₂^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑓(𝑠)𝛼−𝛾𝑑𝑠+𝑣₂𝑐𝑡+¹𝑎₁. (2.18)

Definition 2.10. A function 𝑥∈𝐴𝐶¹([0,𝑇],ℝ) is a solution of the problem (1.4), (1.6) if it satisfies the boundary conditions (1.6) and there exists a function 𝑓∈𝐿¹([0,𝑇],ℝ) such that 𝑓(𝑡)∈𝐹(𝑡,𝑥(𝑡),^𝑐𝐷^𝛽𝑥(𝑡)) a.e. on 𝑡∈[0,𝑇] and 𝑥(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠)𝑑𝑠−𝑡Γ2−𝛾𝑇^1−𝛾𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑓(𝑠)𝑐𝛼−𝛾𝑑𝑠+𝑡Γ2−𝛾₂𝑇^1−𝛾𝑏₂−𝑏₁𝑎₁+𝑏₁^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠)𝑑𝑠−𝑇^𝛾Γ2−𝛾^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑓(𝑠)−1𝛼−𝛾𝑑𝑠𝑎₁+𝑏₁𝑏₁𝑐₂𝑇^𝛾Γ2−𝛾𝑏₂−𝑐₁. (2.19)

Let 𝐶([0,𝑇],ℝ) be the space of all continuous functions defined on [0,𝑇]. Define the space 𝒳={𝑥∶𝑥 and ^𝑐𝐷^𝛽𝑥∈𝐶([0,𝑇],ℝ)} (0<𝛽≤1) endowed with the norm ‖𝑥‖=max_{𝑡∈[0,𝑇]}|𝑥(𝑡)|+max_{𝑡∈[0,𝑇]}|^𝑐𝐷^𝛽𝑥(𝑡)|. We know that (𝒳,‖⋅‖) is a Banach space (see [14]).

We end this section with two fixed point theorems, which will be used in the sequel.

Theorem 2.11 (nonlinear alternative of Leray-Schauder type [31]). Let 𝑋 be a Banach space, 𝐶 a closed convex subset of 𝑋, 𝑈 an open subset of 𝐶 with 0∈𝑈. Suppose that 𝐹∶𝑈→𝑃_{𝑐𝑝,𝑐}(𝐶) is an upper semicontinuous compact map. Then either (1)  𝐹 has a fixed point in 𝑈, or (2) there is a 𝑥∈𝜕𝑈 and 𝜆∈(0,1) such that 𝑥∈𝜆𝐹(𝑥).

Theorem 2.12 (Covitz and Nadler Jr. [32]). Let (𝑋,𝑑) be a complete metric space. If 𝐹∶𝑋→𝑃_𝑐𝑙(𝑋) is a contraction, then 𝐹 has a fixed point.

3. Existence Results

In this section, we will give some existence results for the problems (1.4), (1.5) and (1.4), (1.6).

For each 𝑥∈𝒳, define the set of selections of 𝐹 by 𝑆_𝐹,𝑥=𝑣∈𝐿¹([0,𝑇],ℝ)∶𝑣(𝑡)∈𝐹𝑡,𝑥(𝑡),^𝑐𝐷^𝛽𝑥(𝑡)fora.e.𝑡∈[0,𝑇]. (3.1) In view of Lemmas 2.7 and 2.8, we define operators 𝑁,𝑀∶𝒳→𝑃(𝒳) as 𝑁(𝑥)=ℎ∈𝒳∶ℎ(𝑡)=(𝑆𝑢)(𝑡),𝑢∈𝑆_𝐹,𝑥,𝑀(𝑥)=ℎ∈𝒳∶ℎ(𝑡)=(𝐾𝑢)(𝑡),𝑢∈𝑆_𝐹,𝑥 (3.2)(3.3) with (𝑆𝑢)(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑢(𝑠)−𝑡𝑑𝑠𝑣₁𝑎₂^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑢(𝑠)𝑑𝑠+𝑏₂^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑢(𝑠)𝛼−𝛾𝑑𝑠+𝑣₂𝑐𝑡+¹𝑎₁,(𝐾𝑢)(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑢(𝑠)𝑑𝑠−𝑡Γ2−𝛾𝑇^1−𝛾𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑢(𝑠)𝑐𝛼−𝛾𝑑𝑠+𝑡Γ2−𝛾₂𝑇^1−𝛾𝑏₂−𝑏₁𝑎₁+𝑏₁^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑢(𝑠)𝑑𝑠−𝑇^𝛾Γ2−𝛾^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑢(𝑠)−1𝛼−𝛾𝑑𝑠𝑎₁+𝑏₁𝑏₁𝑐₂𝑇^𝛾Γ2−𝛾𝑏₂−𝑐₁. (3.4) It is clear that if 𝑥∈𝒳 is a fixed point of the operator 𝑁 (the operator 𝑀), then 𝑥 is a solution of the problem (1.4), (1.5) (the problem (1.4), (1.6)).

Now we are in a position to present our main results. The methods used to prove the existence results are standard; however, their exposition in the framework of problems (1.4), (1.5) and (1.4), (1.6) is new.

3.1. Convex Case

We consider first the case when 𝐹 is convex valued.(H1):(1)  𝐹∶[0,𝑇]×ℝ×ℝ→𝑃_cp,𝑐(ℝ) is a Carathéodory multivalued map; (2) there exist 𝑚∈𝐿^∞([0,𝑇],ℝ⁺) and 𝜑,𝜓∶[0,∞)→(0,∞) continuous, nondecreasing such that 𝐹𝑡,𝑥,𝑦=sup|𝑣|∶𝑣∈𝐹𝑡,𝑥,𝑦≤𝑚(𝑡)𝜑(|𝑥|)+𝜓𝑦 (3.5) for 𝑥,𝑦∈ℝ and a.e. 𝑡∈[0,𝑇].

Theorem 3.1. Assume that (H1) is satisfied and there exists 𝐿>0 such that 𝐿𝑃+𝜑(𝐿)+𝜓(𝐿)‖𝑚‖_𝐿^∞𝑄>1, (3.6) where 𝑃=𝑣₂𝑐𝑇+¹𝑎₁+𝑣₂𝑇^1−𝛽Γ,𝑇2−𝛽𝑄=^𝛼−𝛽Γ+𝑇𝛼−𝛽+1Γ(^𝛼)+𝑇𝛼+1𝑣₁𝑇1+^−𝛽Γ𝑎2−𝛽²𝑇^𝛼Γ()+𝑏𝛼+1²𝑇^𝛼−𝛾Γ.𝛼−𝛾+1 (3.7) Then the problem (1.4), (1.5) has at least one solution on [0,𝑇].

Proof. Consider the operator 𝑁∶𝒳→𝑃(𝒳) defined by (3.2). From (H1), we have for each 𝑥∈𝒳, the set 𝑆_𝐹,𝑥 is nonempty [29]. For 𝑥∈𝒳, let 𝑢∈𝑆_𝐹,𝑥 and ℎ=𝑆𝑢, that is, ℎ∈𝑁(𝑥), we have ^𝑐𝐷^𝛽ℎ(𝑡)=^𝑡₀(𝑡−𝑠)^{𝛼−𝛽−1}Γ𝑢(𝑠)𝛼−𝛽𝑑𝑠−𝑘𝑡^1−𝛽Γ,2−𝛽 (3.8) where 𝑘 is a constant given by 1𝑘=𝑣₁𝑎₂^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑢(𝑠)𝑑𝑠+𝑏₂^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑢(𝑠)𝛼−𝛾𝑑𝑠−𝑣₂. (3.9) Hence we know that the operator 𝑁∶𝒳→𝑃(𝒳) is well defined.
We put 𝑆𝑢=𝑆₁𝑢+𝑆₂𝑢 where 𝑆₁𝑢(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑢(𝑠)𝑑𝑠,𝑆₂𝑢(𝑡)=−𝑘_𝑢𝑐𝑡+¹𝑎₁. (3.10) Here 𝑘_𝑢 means that the constant 𝑘 defined by (3.9) is related to 𝑢.
We will show that 𝑁 satisfies the requirements of the nonlinear alternative of Leray-Schauder type. The proof will be given in five steps.
Step 1 (𝑁(𝑥) is convex valued). Since 𝐹 is convex valued, we know that 𝑆_𝐹,𝑥 is convex and therefore it is obvious that 𝑁(𝑥) is convex for each 𝑥∈𝒳.
Step 2 (𝑁 maps bounded sets into bounded sets in 𝒳). Let 𝐵_𝑟 be a bounded subset of 𝒳 such that for any 𝑥∈𝐵_𝑟, ‖𝑥‖≤𝑟, 𝑟>0. We prove that there exists a constant 𝑙>0 such that for each 𝑥∈𝐵_𝑟, one has ‖ℎ‖≤𝑙 for each ℎ∈𝑁(𝑥). Let 𝑥∈𝐵_𝑟 and ℎ∈𝑁(𝑥), then there exists 𝑢∈𝑆_𝐹,𝑥 such that ℎ(𝑡)=(𝑆𝑢)(𝑡)for𝑡∈[0,𝑇]. (3.11) By simple calculations, we have 𝑆₁𝑢(𝑡)≤^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)|𝑢(𝑠)|𝑑𝑠≤𝜑(𝑟)+𝜓(𝑟)‖𝑚‖_𝐿^∞𝑇^𝛼Γ(),𝑆𝛼+1₂𝑢(𝑡≤𝜑(𝑟)(𝑟)‖𝑚‖)+𝜓_𝐿^∞𝑇𝑣₁𝑎₂𝑇^𝛼Γ()+𝑏𝛼+1²𝑇^𝛼−𝛾Γ+𝑣𝛼−𝛾+1²𝑐𝑇+¹𝑎_1𝑐,𝐷^𝛽ℎ(𝑡)≤^𝑡₀(𝑡−𝑠)^{𝛼−𝛽−1}Γ|𝑢(𝑠𝑘𝛼−𝛽)|𝑑𝑠+^𝑢𝑇^1−𝛽Γ≤𝜑(𝑟)(𝑟)‖𝑚‖2−𝛽+𝜓_𝐿^∞𝑇^𝛼−𝛽Γ+𝑣𝛼−𝛽+1²𝑇^1−𝛽Γ+𝜑(𝑟)(𝑟)𝑚‖2−𝛽+𝜓‖_𝐿^∞𝑇^1−𝛽𝑣₁Γ𝑎2−𝛽²𝑇^𝛼Γ()+𝑏𝛼+1²𝑇^𝛼−𝛾Γ.𝛼−𝛾+1 (3.12) Hence we obtain ‖ℎ‖≤𝑣₂𝑐𝑇+¹𝑎₁+𝑣₂𝑇^1−𝛽Γ+𝜑(𝑟)(𝑟)‖𝑚‖2−𝛽+𝜓_𝐿^∞×𝑇^𝛼−𝛽Γ+𝑇𝛼−𝛽+1Γ(^𝛼)+𝑇𝛼+1𝑣₁+𝑇^1−𝛽𝑣₁Γ𝑎2−𝛽²𝑇^𝛼Γ()+𝑏𝛼+1²𝑇^𝛼−𝛾Γ𝜑(𝑟)(𝑟)‖𝑚‖𝛼−𝛾+1≤𝑃++𝜓_𝐿^∞𝑄=𝑙(aconstant). (3.13)
Step 3 (𝑁 maps bounded sets into equicontinuous sets in 𝒳). Let 𝐵_𝑟 be a bounded set of 𝒳 as in Step 2. Let 0≤𝑡₁<𝑡₂≤𝑇 and 𝑥∈𝐵_𝑟. For each ℎ∈𝑁(𝑥), then there is 𝑢∈𝑆_𝐹,𝑥 such that ℎ(𝑡)=(𝑆𝑢)(𝑡). Since 𝑆₁𝑢𝑡₂−𝑆₁𝑢𝑡₁=^𝑡2_𝑡1𝑡₂−𝑠^𝛼−1Γ(𝛼)𝑢(𝑠)𝑑𝑠+^𝑡1₀𝑡₂−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−1Γ(𝛼)𝑢(𝑠)≤𝜑(𝑟)(𝑟)‖𝑚‖𝑑𝑠+𝜓_𝐿^∞Γ()𝑡𝛼+1²−𝑡₁^𝛼+𝑡^𝛼₂−𝑡₂−𝑡₁^𝛼−𝑡^𝛼₁≤𝜑(𝑟)+𝜓(𝑟)‖𝑚‖_𝐿^∞𝑡^𝛼₂−𝑡^𝛼₁Γ(),𝑆𝛼+1₂𝑢𝑡₂−𝑆₂𝑢𝑡₁=−𝑘_𝑢𝑡₂+𝑐₁𝑎₁+𝑘_𝑢𝑡₁−𝑐₁𝑎₁≤𝜑(𝑟)+𝜓(𝑟)‖𝑚‖_𝐿^∞𝑣₁𝑎₂𝑇^𝛼Γ()+𝑏𝛼+1²𝑇^𝛼−𝛾Γ+𝑣𝛼−𝛾+1²𝑡₂−𝑡₁,_𝑐𝐷^𝛽ℎ𝑡₂−^𝑐𝐷^𝛽ℎ𝑡₁=𝐼^𝛼−𝛽𝑢𝑡₂−𝑘_𝑢𝑡₂^1−𝛽Γ2−𝛽−𝐼^𝛼−𝛽𝑢𝑡₁+𝑘_𝑢𝑡₁^1−𝛽Γ≤𝜑(𝑟)(𝑟)‖𝑚‖2−𝛽+𝜓_𝐿^∞𝑣₁Γ𝑎2−𝛽²𝑇^𝛼Γ()+𝑏𝛼+1²𝑇^𝛼−𝛾Γ+𝑣𝛼−𝛾+1₂Γ𝑡2−𝛽^21−𝛽−𝑡₁^1−𝛽+𝜑(𝑟)+𝜓(𝑟)‖𝑚‖_𝐿^∞𝑡₂^𝛼−𝛽−𝑡₁^𝛼−𝛽Γ,𝛼−𝛽+1 (3.14) we obtain that (since 𝛼>1, 𝛼−𝛽>0 and 1−𝛽≥0) ℎ𝑡₂−ℎ𝑡₁→0,^𝑐𝐷^𝛽ℎ𝑡₂−^𝑐𝐷^𝛽ℎ𝑡₁→0as𝑡₂→𝑡₁ (3.15) and the limits are independent of 𝑥∈𝐵_𝑟 and ℎ∈𝑁(𝑥).
Step 4 (𝑁 has a closed graph). Let 𝑥_𝑛→𝑥_∗, ℎ_𝑛∈𝑁(𝑥_𝑛), and ℎ_𝑛→ℎ_∗; we need to show ℎ_∗∈𝑁(𝑥_∗). Now ℎ_𝑛∈𝑁(𝑥_𝑛) implies that there exists 𝑢_𝑛∈𝑆_{𝐹,𝑥𝑛} such that ℎ_𝑛(𝑡)=(𝑆𝑢_𝑛)(𝑡) for 𝑡∈[0,𝑇]. Let us consider the continuous linear operator Γ∶𝐿¹([0,𝑇],ℝ)→𝒳 given by (Γ𝑢)(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑢(𝑠)𝑡𝑑𝑠−𝑣₁𝑎₂^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑢(𝑠)𝑑𝑠+𝑏₂^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑢(𝑠)𝛼−𝛾𝑑𝑠 (3.16) and denote 𝑤(𝑡)=𝑣₂𝑡+𝑐₁/𝑎₁. Then ℎ_𝑛(𝑡)−𝑤(𝑡)=(Γ𝑢_𝑛)(𝑡) and ℎ_𝑛−ℎ_∗=max_{[]𝑡∈0,𝑇}ℎ_𝑛(𝑡)−𝑤(𝑡)−ℎ_∗(𝑡)−𝑤(𝑡)+max_{[]𝑡∈0,𝑇}^𝑐𝐷^𝛽ℎ_𝑛−𝑤(𝑡)−^𝑐𝐷^𝛽ℎ_∗−𝑤(𝑡)→0as𝑛→∞. (3.17) We apply Lemma 2.3 to find that Γ∘𝑆_𝐹 has closed graph and from the definition of Γ we get ℎ_𝑛−𝑤∈Γ∘𝑆_𝐹(𝑥_𝑛). Since 𝑥_𝑛→𝑥_∗, ℎ_𝑛−𝑤→ℎ_∗−𝑤, it follows the existence of 𝑢_∗∈𝑆_𝐹,𝑥∗ such that ℎ_∗−𝑤=Γ(𝑢_∗). This means that ℎ_∗∈𝑁(𝑥_∗).
Step 5 (a priori bounds on solutions). Let 𝑥∈𝜆𝑁(𝑥) for some 𝜆∈(0,1). Then there exists 𝑢∈𝑆_𝐹,𝑥 such that 𝑥(𝑡)=𝜆(𝑆𝑢)(𝑡) for 𝑡∈[0,𝑇]. With the same arguments as in Step 2 of our proof, for each 𝑡∈[0,𝑇], we obtain |𝑥(𝑡)|+^𝑐𝐷^𝛽𝑥(𝑡)≤𝑃+𝜑(‖𝑥‖)+𝜓(‖𝑥‖)‖𝑚‖_𝐿^∞𝑄. (3.18) Thus ‖𝑥‖≤𝑃+𝜑(‖𝑥‖)+𝜓(‖𝑥‖)‖𝑚‖_𝐿^∞𝑄. (3.19) Now we set 𝑈={𝑥∈𝒳∶‖𝑥‖<𝐿}. (3.20) Clearly, 𝑈 is an open subset of 𝒳 and 0∈𝑈. As a consequence of Steps 1–4, together with the Arzela-Ascoli theorem, we can conclude that 𝑁∶𝑈→𝑃_{𝑐𝑝,𝑐}(𝒳) is upper semicontinuous and completely continuous. From the choice of the 𝑈, there is no 𝑥∈𝜕𝑈 such that 𝑥∈𝜆𝑁(𝑥) for some 𝜆∈(0,1). Therefore, by the nonlinear alternative of Leary-Schauder type (Theorem 2.11), we deduce that 𝑁 has a fixed point 𝑥∈𝑈, which is a solution of the problem (1.4), (1.5). This completes the proof.

Theorem 3.2. Assume that (H1) is satisfied and there exists 𝐿₁>0 such that 𝐿₁𝑃₁+𝜑𝐿₁+𝜓𝐿₁‖𝑚‖_𝐿^∞𝑄₁>1, (3.21) where 𝑃₁=𝑇^𝛾Γ2−𝛾𝑐₂𝑏₂𝑇1+^−𝛽Γ+𝑏2−𝛽¹𝑐₂𝑇^𝛾Γ2−𝛾𝑎₁+𝑏₁𝑏₂−𝑐₁𝑎₁+𝑏₁,𝑄₁=𝑇^𝛼𝑏1+¹𝑎₁+𝑏₁1Γ()+Γ𝛼+12−𝛾Γ𝛼−𝛾+1+𝑇^𝛼−𝛽1Γ+Γ𝛼−𝛽+12−𝛾ΓΓ.2−𝛽𝛼−𝛾+1 (3.22) Then the problem (1.4), (1.6) has at least one solution on [0,𝑇].

Proof. To obtain the result, the main aim is to study the properties of the operator 𝑀 defined in (3.3). The proof of them is similar to those of Theorem 3.1, so we omit the details. Here we just give some estimations, which are needed in the following theorems. Let 𝑥∈𝒳 and ℎ∈𝑀(𝑥); then there exists 𝑢∈𝑆_𝐹,𝑥 such that ℎ(𝑡)=(𝐾𝑢)(𝑡),for𝑡∈[0,𝑇]. (3.23) We put 𝐾𝑢=𝐾₁𝑢+𝐾₂𝑢 and 𝐾₁𝑢(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑢(𝑠)𝑑𝑠,𝐾₂𝑢(𝑡)=−𝑘^𝑢₂𝑡−𝑘^𝑢₁; (3.24) here 𝑘^𝑢₁ and 𝑘^𝑢₂ are constants given by 𝑘^𝑢₁=𝑏₁𝑐₂𝑇^𝛾Γ2−𝛾𝑎₁+𝑏₁𝑏₂−𝑐₁𝑎₁+𝑏₁+𝑏₁𝑎₁+𝑏₁^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑢(𝑠)𝑑𝑠−𝑇^𝛾Γ2−𝛾^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑢(𝑠),𝑘𝛼−𝛾𝑑𝑠_𝑢2=Γ2−𝛾𝑇^1−𝛾𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑢(𝑠)𝑐𝛼−𝛾𝑑𝑠−²𝑏₂. (3.25) By simple calculations, we have 𝐾₁𝑢(𝑡)≤^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)|𝑢(𝑠)|𝑑𝑠≤𝜑(‖𝑥‖)+𝜓(‖𝑥‖)‖𝑚‖_𝐿^∞𝑇^𝛼Γ(),𝐾𝛼+1₂𝑢(𝑡)≤𝑇𝑘^𝑢₂+𝑘^𝑢₁,𝑇𝑘^𝑢₂≤𝑇^𝛾Γ2−𝛾𝜑(‖𝑥‖)+𝜓(‖𝑥‖)‖𝑚‖_𝐿^∞𝑇^𝛼−𝛾Γ+𝑐𝛼−𝛾+1𝑏²₂,𝑘^𝑢₁≤𝑏₁𝑎₁+𝑏₁𝜑(‖𝑥‖)+𝜓(‖𝑥‖)‖𝑚‖_𝐿^∞𝑇^𝛼Γ()+𝜑𝑥𝑥‖𝑚‖𝛼+1(‖‖)+𝜓(‖‖)_𝐿^∞Γ2−𝛾𝑇^𝛼Γ+𝑏𝛼−𝛾+1₁𝑐₂𝑇^𝛾Γ2−𝛾𝑎₁+𝑏₁𝑏₂−𝑐₁𝑎₁+𝑏_1𝑐,𝐷^𝛽ℎ(𝑡)≤^𝑡₀(𝑡−𝑠)^{𝛼−𝛽−1}Γ|𝑢(𝑠𝑘𝛼−𝛽)|𝑑𝑠+^𝑢2𝑇^1−𝛽Γ≤𝜑𝑥𝑥‖𝑚‖2−𝛽(‖‖)+𝜓(‖‖)_𝐿^∞𝑇^𝛼−𝛽Γ+Γ𝑇𝛼−𝛽+12−𝛾^𝛾−𝛽Γ×𝜑𝑥𝑥‖𝑚‖2−𝛽(‖‖)+𝜓(‖‖)_𝐿^∞𝑇^𝛼−𝛾Γ+𝑐𝛼−𝛾+1𝑏²₂. (3.26) Hence we obtain ‖ℎ‖≤𝑇^𝛾Γ2−𝛾𝑐₂𝑏₂𝑇1+^−𝛽Γ+𝑏2−𝛽¹𝑐₂𝑇^𝛾Γ2−𝛾𝑎₁+𝑏₁𝑏₂−𝑐₁𝑎₁+𝑏₁+𝜑(‖𝑥‖)+𝜓(‖𝑥‖)‖𝑚‖_𝐿^∞𝑇^𝛼𝑏1+¹𝑎₁+𝑏₁1Γ()+Γ𝛼+12−𝛾Γ+𝜑𝑥𝑥𝑚‖𝛼−𝛾+1(‖‖)+𝜓(‖‖)‖_𝐿^∞𝑇^𝛼−𝛽1Γ+Γ𝛼−𝛽+12−𝛾ΓΓ.2−𝛽𝛼−𝛾+1 (3.27) This is the end of the proof.

3.2. Nonconvex Case

Now we study the case when 𝐹 is not necessarily convex valued.

A subset 𝐴 of 𝐿¹([0,𝑇],ℝ) is decomposable if for all 𝑢,𝑣∈𝐴 and 𝐽⊆[0,𝑇] Lebesgue measurable, then 𝑢𝜒_𝐽+𝑣𝜒_{[0,𝑇]−𝐽}∈𝐴, where 𝜒 stands for the characteristic function.(H2):𝐹∶[0,𝑇]×ℝ×ℝ→𝑃_cp(ℝ) is a multivalued map such that (1)  (𝑡,𝑥,𝑦)→𝐹(𝑡,𝑥,𝑦) is Σ⊗ℬ_ℝ⊗ℬ_ℝ measurable; (2)  (𝑥,𝑦)→𝐹(𝑡,𝑥,𝑦) is lower semicontinuous for a.e. 𝑡∈[0,𝑇].

Theorem 3.3. Let (H1)(1.5), (H2), and relation (3.6) hold; then the problem (1.4), (1.5) has at least one solution on [0,𝑇].

Proof. From (H1)(1.5), (H2), and [33, Lemma 4.1], the map ℱ∶𝒳→𝑃𝐿¹([0,𝑇],ℝ),𝑥→ℱ(𝑥)=𝑆_𝐹,𝑥 (3.28) is lower semicontinuous and has nonempty closed and decomposable values. Then from a selection theorem due to Bressan and Colombo [34], there exists a continuous function 𝑓∶𝒳→𝐿¹([0,𝑇],ℝ) such that 𝑓(𝑥)∈ℱ(𝑥) for all 𝑥∈𝒳. That is to say, we have 𝑓(𝑥)(𝑡)∈𝐹(𝑡,𝑥(𝑡),^𝑐𝐷^𝛽𝑥(𝑡)) for a.e. 𝑡∈[0,𝑇]. Now consider the problem ^𝑐𝐷^𝛼𝑥(𝑡)=𝑓(𝑥)(𝑡),𝑡∈[0,𝑇] (3.29) with the boundary conditions (1.5). Note that if 𝑥∈𝒳 is a solution of the problem (3.29), then 𝑥 is a solution to the problem (1.4), (1.5).
Problem (3.29) is then reformulated as a fixed point problem for the operator 𝑁₁∶𝒳→𝒳 defined by 𝑁₁(𝑥)(𝑡)=𝑆𝑓(𝑥)(𝑡). (3.30) It can easily be shown that 𝑁₁ is continuous and completely continuous and satisfies all conditions of the Leray-Schauder nonlinear alternative for single-valued maps [31]. The remaining part of the proof is similar to that of Theorem 3.1, so we omit it. This completes the proof.

Theorem 3.4. Let (H1)(1.5), (H2), and relation (3.21) hold, then the problem (1.4), (1.6) has at least one solution on [0,𝑇].

The proof of this theorem is similar to that of Theorem 3.3.(H3):𝐹∶[0,𝑇]×ℝ×ℝ→𝑃_𝑐𝑝(ℝ) is a multivalued map such that (1)  𝐹 is integrably bounded and the map 𝑡→𝐹(𝑡,𝑥,𝑦) is measurable for all 𝑥,𝑦∈ℝ; (2) there exists 𝑚∈𝐿^∞([0,𝑇],ℝ⁺) such that for a.e. 𝑡∈[0,𝑇] and all 𝑥₁, 𝑥₂, 𝑦₁, 𝑦₂∈ℝ, ℎ𝐹𝑡,𝑥₁,𝑦₁,𝐹𝑡,𝑥₂,𝑦₂≤𝑚(𝑡)𝑥₁−𝑥₂+𝑦₁−𝑦₂. (3.31)

Theorem 3.5. Let (H3) hold, if, in addition, ‖𝑚‖_𝐿^∞𝑇^𝛼Γ()+𝑇𝛼+1Γ^𝛼−𝛽+𝑇𝛼−𝛽+1𝑣₁+𝑇^1−𝛽𝑣₁Γ𝑎2−𝛽²𝑇^𝛼Γ()+𝑏𝛼+1²𝑇^𝛼−𝛾Γ𝛼−𝛾+1<1, (3.32) then the problem (1.4), (1.5) has at least one solution on [0,𝑇].

Proof. From (H3), we have that the multivalued map 𝑡→𝐹(𝑡,𝑥(𝑡),^𝑐𝐷^𝛽𝑥(𝑡)) is measurable [28, Proposition 2.7.9] and closed valued for each 𝑥∈𝒳. Hence it has measurable selection [28, Theorem 2.2.1] and the set 𝑆_𝐹,𝑥 is nonempty. Let 𝑁 be defined in (3.2). We will show that, under this situation, 𝑁 satisfies the requirements of Theorem 2.12.
Step 1. For each 𝑥∈𝒳, 𝑁(𝑥)∈𝑃_cl(𝒳). Let ℎ_𝑛∈𝑁(𝑥), 𝑛≥1 such that ℎ_𝑛→ℎ in 𝒳. Then ℎ∈𝒳 and there exists 𝑢_𝑛∈𝑆_𝐹,𝑥, 𝑛≥1 such that ℎ_𝑛(𝑡)=𝑆𝑢_𝑛(𝑡),𝑡∈[0,𝑇]. (3.33) By (H3), the sequence 𝑢_𝑛 is integrable bounded. Since 𝐹 has compact values, we may pass to a subsequence if necessary to get that 𝑢_𝑛 converges to 𝑢 in 𝐿¹([0,𝑇],ℝ). Thus 𝑢∈𝑆_𝐹,𝑥 and for each 𝑡∈[0,𝑇]ℎ_𝑛(𝑡)→ℎ(𝑡)=(𝑆𝑢)(𝑡). (3.34) This means that ℎ∈𝑁(𝑥) and 𝑁(𝑥) is closed.
Step 2. There exists 𝜌<1 such that ℎ𝑁(𝑥),𝑁𝑦≤𝜌𝑥−𝑦,∀𝑥,𝑦∈𝒳. (3.35) Let 𝑥, 𝑦∈𝒳 and ℎ₁∈𝑁(𝑦); then there exists 𝑢₁∈𝑆_𝐹,𝑦 such that ℎ₁(𝑡)=𝑆𝑢₁(𝑡),𝑡∈[0,𝑇]. (3.36) From (H3)(2), we deduce ℎ𝐹𝑡,𝑥(𝑡),^𝑐𝐷^𝛽𝑥(𝑡),𝐹𝑡,𝑦(𝑡),^𝑐𝐷^𝛽𝑦(𝑡)≤𝑚(𝑡)𝑥(𝑡)−𝑦(𝑡)+^𝑐𝐷^𝛽𝑥(𝑡)−^𝑐𝐷^𝛽𝑦(𝑡). (3.37) Hence, for a.e. 𝑡∈[0,𝑇], there exists 𝑣∈𝐹(𝑡,𝑥(𝑡),^𝑐𝐷^𝛽𝑥(𝑡)) such that 𝑢₁(𝑡)−𝑣≤𝑚(𝑡)𝑥(𝑡)−𝑦(𝑡)+^𝑐𝐷^𝛽𝑥(𝑡)−^𝑐𝐷^𝛽𝑦(𝑡). (3.38) Consider the multivalued map 𝑉∶[0,𝑇]→𝑃(ℝ) given by 𝑉(𝑡)=𝑢∈ℝ∶𝑢₁(𝑡)−𝑢≤𝑚(𝑡)𝑥(𝑡)−𝑦(𝑡)+^𝑐𝐷^𝛽𝑥(𝑡)−^𝑐𝐷^𝛽𝑦(𝑡). (3.39) Since 𝑢₁(𝑡), 𝛼(𝑡)=𝑚(𝑡)(|𝑥(𝑡)−𝑦(𝑡)|+|^𝑐𝐷^𝛽𝑥(𝑡)−^𝑐𝐷^𝛽𝑦(𝑡)|) are measurable, [35, Theorem III.41] implies that 𝑉 is measurable. It follows from (H3) that the map 𝑡→𝐹(𝑡,𝑥(𝑡),^𝑐𝐷^𝛽𝑥(𝑡)) is measurable. Hence by (3.38) and [28, Proposition  2.1.43], the multivalued map 𝑡→𝑉(𝑡)∩𝐹(𝑡,𝑥(𝑡),^𝑐𝐷^𝛽𝑥(𝑡)) is measurable and nonempty closed valued. Therefore, we can find 𝑢₂(𝑡)∈𝐹(𝑡,𝑥(𝑡),^𝑐𝐷^𝛽𝑥(𝑡)) such that for a.e. 𝑡∈[0,𝑇], 𝑢₁(𝑡)−𝑢₂(𝑡)≤𝑚(𝑡)𝑥(𝑡)−𝑦(𝑡)+^𝑐𝐷^𝛽𝑥(𝑡)−^𝑐𝐷^𝛽𝑦(𝑡). (3.40) Let ℎ₂(𝑡)=(𝑆𝑢₂)(𝑡), that is, ℎ₂∈𝑁(𝑥). Since 𝑆₁𝑢₁(𝑡)−𝑆₁𝑢₂(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑢₁(𝑠)−𝑢₂(𝑠)≤𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑚(𝑠)𝑥(𝑠)−𝑦(𝑠)+^𝑐𝐷^𝛽𝑥(𝑠)−^𝑐𝐷^𝛽𝑦(𝑠≤‖𝑚‖)𝑑𝑠_𝐿∞𝑇^𝛼Γ(),𝑆𝛼+1𝑥−𝑦₂𝑢₁(𝑡)−𝑆₂𝑢₂(𝑡)=𝑡𝑘_𝑢1−𝑘_𝑢2𝑎≤𝑇²𝑣₁^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑢₁(𝑠)−𝑢₂(𝑠+𝑏)𝑑𝑠₂𝑣₁^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑢𝛼−𝛾¹(𝑠)−𝑢₂(𝑠≤‖𝑚‖)𝑑𝑠_𝐿∞𝑇𝑣₁𝑎₂𝑇^𝛼Γ()+𝑏𝛼+1²𝑇^𝛼−𝛾Γ,𝛼−𝛾+1𝑥−𝑦_𝑐𝐷^𝛽ℎ₁(𝑡)−^𝑐𝐷^𝛽ℎ₂(𝑡)=𝐼^𝛼−𝛽𝑢₁(𝑡)−𝑘_𝑢1𝑡^1−𝛽Γ2−𝛽−𝐼^𝛼−𝛽𝑢₂(𝑡)+𝑘_𝑢2𝑡^1−𝛽Γ≤‖𝑚‖2−𝛽_𝐿∞𝑇^𝛼−𝛽Γ+‖𝑚‖𝛼−𝛽+1𝑥−𝑦_𝐿∞𝑇^1−𝛽𝑣₁Γ𝑎2−𝛽²𝑇^𝛼Γ()+𝑏𝛼+1²𝑇^𝛼−𝛾Γ,𝛼−𝛾+1𝑥−𝑦 (3.41) we obtain ℎ₁−ℎ₂≤‖𝑚‖_𝐿^∞𝑇^𝛼Γ()+𝑇𝛼+1Γ^𝛼−𝛽+𝑇𝛼−𝛽+1𝑣₁+𝑇^1−𝛽𝑣₁Γ𝑎2−𝛽²𝑇^𝛼Γ()+𝑏𝛼+1²𝑇^𝛼−𝛾Γ.𝛼−𝛾+1𝑥−𝑦 (3.42) Denote 𝜌=‖𝑚‖_𝐿^∞𝑇^𝛼Γ()+𝑇𝛼+1Γ^𝛼−𝛽+𝑇𝛼−𝛽+1𝑣₁+𝑇^1−𝛽𝑣₁Γ𝑎2−𝛽²𝑇^𝛼Γ()+𝑏𝛼+1²𝑇^𝛼−𝛾Γ.𝛼−𝛾+1 (3.43) By using an analogous relation obtained by interchanging the roles of 𝑥 and 𝑦, we get ℎ𝑁(𝑥),𝑁𝑦≤𝜌𝑥−𝑦. (3.44) Therefore, from condition (3.32), Theorem 2.12 implies that 𝑁 has a fixed point, which is a solution of the problem (1.4), (1.5). This completes the proof.

Theorem 3.6. Let (H3) hold, if, in addition, ‖𝑚‖_𝐿^∞𝑇^𝛼𝑏1+¹𝑎₁+𝑏₁1Γ()+Γ𝛼+12−𝛾Γ+‖𝑚‖𝛼−𝛾+1_𝐿∞𝑇^𝛼−𝛽1Γ+Γ𝛼−𝛽+12−𝛾Γ2−𝛽Γ𝛼−𝛾+1<1, (3.45) then the problem (1.4), (1.6) has at least one solution on [0,𝑇].

Using the arguments employed in the proof of Theorem 3.5, we can prove this theorem similarly. Hence the details are omitted here.

4. Integral Boundary Conditions

In this section, the existence results of the problems (1.4), (1.5) and (1.4), (1.6) obtained in the previous section will be extended to the ones of the problems of fractional differential inclusions (1.4) subject to the integral boundary conditions (1.7) and (1.8).

Lemma 4.1. For any 𝑦,𝜉,𝜒∈𝐶([0,𝑇],ℝ), the unique solution of the fractional separated integral boundary value problem, ^𝑐𝐷^𝛼𝑥(𝑡)(𝑡)[]𝑎=𝑦,𝑡∈0,𝑇,1<𝛼≤2,₁𝑥(0)+𝑏₁^𝑐𝐷^𝛾𝑥(0)=𝑐₁^𝑇₀𝜉(𝑠)𝑎𝑑𝑠,₂𝑥(𝑇)+𝑏₂^𝑐𝐷^𝛾𝑥(𝑇)=𝑐₂^𝑇₀𝜒(𝑠)𝑑𝑠,0<𝛾<1, (4.1) is given by 𝑥(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)−𝑡𝑑𝑠𝑣₁𝑎₂^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝑏₂^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)+𝑐𝛼−𝛾𝑑𝑠₂𝑡𝑣₁^𝑇₀𝜒(𝑠)𝑐𝑑𝑠+¹𝑣₁−𝑎₂𝑡𝑎₁𝑣₁^𝑇₀𝜉(𝑠)𝑑𝑠. (4.2)

Lemma 4.2. For any 𝑦, 𝜉, 𝜒∈𝐶([0,𝑇],ℝ), the unique solution of the fractional nonseparated integral boundary value problem, ^𝑐𝐷^𝛼𝑥(𝑡)(𝑡)[]𝑎=𝑦,𝑡∈0,𝑇,1<𝛼≤2,₁𝑥(0)+𝑏₁𝑥(𝑇)=𝑐₁^𝑇₀𝜉(𝑠)𝑎𝑑𝑠,_2𝑐𝐷^𝛾𝑥(0)+𝑏₂^𝑐𝐷^𝛾𝑥(𝑇)=𝑐₂^𝑇₀𝜒(𝑠)𝑑𝑠,0<𝛾<1, (4.3) is given by 𝑥(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠−𝑡Γ2−𝛾𝑇^1−𝛾𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)𝑐𝛼−𝛾𝑑𝑠+𝑡Γ2−𝛾₂𝑇^1−𝛾𝑏₂^𝑇₀𝜒(𝑠)−𝑏𝑑𝑠₁𝑎₁+𝑏₁^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠−𝑇^𝛾Γ2−𝛾^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑦(𝑠)−𝑏𝛼−𝛾𝑑𝑠₁𝑇^𝛾Γ2−𝛾𝑐₂𝑏₂𝑎₁+𝑏₁^𝑇₀𝜒(𝑠)𝑐𝑑𝑠+¹𝑎₁+𝑏₁^𝑇₀𝜉(𝑠)𝑑𝑠. (4.4)

To obtain the existence results of the problems (1.4), (1.7) and (1.4), (1.8), in view of Lemmas 4.1 and 4.2, we define two operators Π,Ω∶𝒳→𝑃(𝒳) as Π(𝑥)=ℎ∈𝒳∶ℎ=𝐻𝑣,𝑣∈𝑆_𝐹,𝑥,Ω(𝑥)=ℎ∈𝒳∶ℎ=𝑍𝑣,𝑣∈𝑆_𝐹,𝑥 (4.5)(4.6) with (𝐻𝑣)(𝑡)=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑣(𝑠)−𝑡𝑑𝑠𝑣₁𝑎₂^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑣(𝑠)𝑑𝑠+𝑏₂^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑣(𝑠)+𝑐𝛼−𝛾𝑑𝑠₂𝑡𝑣₁^𝑇₀ℎ((𝑠𝑐𝑠,𝑥))𝑑𝑠+¹𝑎₁^𝑇₀𝑔((𝑠𝑐𝑠,𝑥))𝑑𝑠−¹𝑎₂𝑡𝑎₁𝑣₁^𝑇₀𝑔((𝑠(𝑡)𝑠,𝑥))𝑑𝑠,𝑍𝑣)(=^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑣(𝑠)𝑑𝑠−𝑡Γ2−𝛾𝑇^1−𝛾𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑣(𝑠)+𝑐𝛼−𝛾𝑑𝑠𝑡Γ2−𝛾₂𝑇^1−𝛾𝑏₂^𝑇₀ℎ((𝑠−𝑏𝑠,𝑥))𝑑𝑠₁𝑎₁+𝑏₁^𝑇₀(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑣(𝑠)𝑑𝑠−𝑇^𝛾Γ2−𝛾^𝑇₀(𝑇−𝑠)^{𝛼−𝛾−1}Γ𝑣(𝑠)−𝑏𝛼−𝛾𝑑𝑠₁𝑇^𝛾Γ2−𝛾𝑐₂𝑏₂𝑎₁+𝑏₁^𝑇₀ℎ((𝑠𝑐𝑠,𝑥))𝑑𝑠+¹𝑎₁+𝑏₁^𝑇₀𝑔(𝑠,𝑥(𝑠))𝑑𝑠. (4.7) Observe that if 𝑥∈𝒳 is a fixed point of the operator Π (the operator Ω), that is, 𝑥∈Π(𝑥) (𝑥∈Ω(𝑥)), then 𝑥 is a solution of the problem (1.4), (1.7) (the problem (1.4), (1.8)).

From the definitions of the operators 𝑁, Π (see (3.2), (4.5)), we know that the difference between them is very apparent, that is, 𝑐₁, 𝑐₂ in (3.2) were replaced by 𝑐₁∫^𝑇₀𝑔(𝑠,𝑥(𝑠))𝑑𝑠 and 𝑐₂∫^𝑇₀ℎ(𝑠,𝑥(𝑠))𝑑𝑠 in (4.5). This fact is also true for the operators 𝑀, Ω (see (3.3), (4.6)).

In the following, we state some existence results for the problems (1.4), (1.7) and (1.4), (1.8). We omit the proofs as these are similar to the ones given in Section 3.(A1): The functions 𝑔, ℎ∶[0,𝑇]×ℝ→ℝ are continuous. There exist functions 𝑚₂, 𝑚₃∈𝐿¹([0,𝑇],ℝ⁺) and 𝜑₂, 𝜑₃∶[0,∞)→(0,∞) continuous, nondecreasing such that 𝑔(𝑡,𝑥)≤𝑚₂(𝑡)𝜑₂(|𝑥|),|ℎ(𝑡,𝑥)|≤𝑚₃(𝑡)𝜑₃(|𝑥|) (4.8) for all 𝑥∈ℝ and a.e. 𝑡∈[0,𝑇].

Theorem 4.3. Assume that (H1) and (A1) hold. If there exists a constant 𝐼>0 such that 𝐼𝜑(𝐼)+𝜓(𝐼)‖𝑚‖_𝐿^∞𝑄+𝜑₃(𝐼)𝑚_3𝐿1𝑅+𝜑₂(𝐼)𝑚_2𝐿1𝑊>1, (4.9) here 𝑄 is defined by (3.7) and 𝑐𝑅=²𝑇𝑣₁𝑇1+^−𝛽Γ𝑐2−𝛽,𝑊=¹𝑎₁𝑎1+²𝑇𝑣₁𝑇1+^−𝛽Γ.2−𝛽 (4.10) Then the boundary value problem (1.4), (1.7) has at least one solution on [0,𝑇].

Theorem 4.4. Assume that (H1) and (A1) hold. If there exists a constant 𝐼₁>0 such that 𝐼₁𝜑𝐼₁+𝜓𝐼₁‖𝑚‖_𝐿^∞𝑄₁+𝜑₃𝐼₁𝑚_3𝐿1𝑅₁+𝜑₂𝐼₁𝑚_2𝐿1𝑊₁>1, (4.11) here 𝑄₁ is defined by (3.22) and 𝑅₁=𝑐₂𝑇^𝛾Γ2−𝛾𝑏₂𝑏1+¹𝑎₁+𝑏₁+𝑇^−𝛽Γ2−𝛽,𝑊₁=𝑐₁𝑎₁+𝑏₁. (4.12) Then the boundary value problem (1.4), (1.8) has at least one solution on [0,𝑇].

Theorem 4.5. Assume that (H1)(1.5), (H2), (A1) and condition (4.9) hold. Then the boundary value problem (1.4), (1.7) has at least one solution on [0,𝑇].

Theorem 4.6. Assume that (H1)(1.5), (H2), (A1) and condition (4.11) hold. Then the boundary value problem (1.4), (1.8) has at least one solution on [0,𝑇]. (A2): The functions 𝑔,ℎ∶[0,𝑇]×ℝ→ℝ are continuous and satisfy 𝑔(𝑡,𝑥)−𝑔𝑡,𝑦≤𝑚₂(𝑡)𝑥−𝑦,ℎ(𝑡,𝑥)−ℎ𝑡,𝑦≤𝑚₃(𝑡)𝑥−𝑦 (4.13) for all 𝑥,𝑦∈ℝ and a.e. 𝑡∈[0,𝑇]; here 𝑚₂, 𝑚₃∈𝐿¹([0,𝑇],ℝ⁺).

Theorem 4.7. Assume that (H3) and (A2) hold. If, in addition, ‖𝑚‖_𝐿^∞𝑄+𝑚_3𝐿1𝑅+𝑚_2𝐿1𝑊<1, (4.14) here 𝑄 is defined by (3.7) and 𝑅, 𝑊 are defined by (4.10), then the boundary value problem (1.4), (1.7) has at least one solution on [0,𝑇].

Theorem 4.8. Assume that (H3) and (A2) hold. If, in addition, ‖𝑚‖_𝐿^∞𝑄₁+𝑚_3𝐿1𝑅₁+𝑚_2𝐿1𝑊₁<1, (4.15) here 𝑄₁ is defined by (3.22) and 𝑅₁, 𝑊₁ are defined by (4.12), then the boundary value problem (1.4), (1.8) has at least one solution on [0,𝑇].

5. Examples

In this section, we give two simple examples to show the applicability of our results.

Example 5.1. Consider the following fractional boundary value problem: ^𝑐𝐷^3/2𝑥(𝑡)∈𝐹𝑡,𝑥(𝑡),^𝑐𝐷^3/4𝑥(𝑡)[],𝑥(0)−1,𝑡∈0,12^𝑐𝐷^1/2𝑥(0)(1)+1=2.5,2𝑥3^𝑐𝐷^1/2𝑥(1)1=−3, (5.1) where 𝛼=3/2, 𝛽=3/4, 𝛾=1/2, 𝑎₁=1, 𝑏₁=−1/2, 𝑐₁=2.5, 𝑎₂=2, 𝑏₂=1/3, 𝑐₂=−1/3, 𝑇=1, and 𝐹∶[0,1]×ℝ×ℝ→𝑃(ℝ) is a multivalued map given by 𝐹𝑡,𝑥,𝑦=𝑢∈ℝ∶𝑒^−|𝑥|−𝑦𝑦|𝑥|1++sin𝑡≤𝑢≤5+1+𝑥²+6𝑡³+cos𝑦. (5.2) In the context of this problem, we have 𝐹𝑡,𝑥,𝑦=sup|𝑣|∶𝑣∈𝐹𝑡,𝑥,𝑦≤7+6𝑡³≤13,for𝑡∈[0,1],𝑥,𝑦∈ℝ. (5.3) It is clear that 𝐹 is convex compact valued and is of Carathéodory type. Let 𝑚(𝑡)≡1 and 𝜑(|𝑥|)≡3, 𝜓(|𝑦|)≡10; we get for 𝑡∈[0,1], 𝑥, 𝑦∈ℝ𝐹𝑡,𝑥,𝑦=sup|𝑣|∶𝑣∈𝐹𝑡,𝑥,𝑦≤𝑚(𝑡)𝜑(|𝑥|)+𝜓𝑦. (5.4) As for the condition (3.6), since 𝑃+(𝜑(|𝑥|)+𝜓(|𝑦|))‖𝑚‖_𝐿^∞𝑄=𝑃+13𝑄 (𝑃, 𝑄 defined in (3.6)) is a constant, we can choose 𝐿 large enough so that 𝐿𝑃+𝜑(𝐿)+𝜓(𝐿)‖𝑚‖_𝐿^∞𝑄>1. (5.5) Thus, by the conclusion of Theorem 3.1, the boundary value problem (5.1) has at least one solution on [0,1].

Example 5.2. Consider the following fractional differential inclusion with integral boundary conditions: ^𝑐𝐷^7/4𝑥(𝑡)∈𝐹𝑡,𝑥(𝑡),^𝑐𝐷^1/2𝑥(𝑡)[],(0)+1,𝑡∈0,13𝑥3𝑥(1)=¹₀𝑔((𝑠2𝑠,𝑥))𝑑𝑠,_𝑐𝐷^1/4𝑥(0)+3^𝑐𝐷^1/4𝑥(1)=14¹₀ℎ(𝑠,𝑥(𝑠))𝑑𝑠, (5.6) where 𝛼=7/4, 𝛽=1/2, 𝛾=1/4, 𝑇=1, 𝑎₁=3, 𝑏₁=1/3, 𝑐₁=1, 𝑎₂=2, 𝑏₂=3, 𝑐₂=1/4, 𝐹𝑡,𝑥,𝑦=−𝑙₁(𝑡)−sin𝑥(4+𝑡)²1−2,−1100,𝑦161+𝑦+𝑙₂(𝑡,𝑔()=1)𝑡,𝑥(3+𝑡)²cos𝑥,ℎ(𝑡,𝑥)=𝑥, (5.7) and 𝑙₁, 𝑙₂∈𝐿¹([0,1],ℝ⁺).
From the data given above, we have for 𝑡∈[0,1], 𝑥, 𝑦∈ℝ, 𝑢|1sup|∶𝑢∈𝐹𝑡,𝑥,𝑦≤3+(4+𝑡)²+𝑙₁(𝑡)+𝑙₂(𝑡),ℎ𝐹𝑡,𝑥₁,𝑦₁,𝐹𝑡,𝑥₂,𝑦₂≤1(4+𝑡)²𝑥₁−𝑥₂+1𝑦16¹−𝑦₂,𝑔()≤1𝑡,𝑥−𝑔𝑡,𝑦(3+𝑡)²𝑥−𝑦,ℎ(𝑡,𝑥)−ℎ𝑡,𝑦≤𝑥−𝑦. (5.8) Then let 𝑚₂(𝑡)=1/(3+𝑡)², 𝑚₃(𝑡)=1, and 𝑚(𝑡)=1/16+1/(4+𝑡)²; we have ℎ𝐹𝑡,𝑥₁,𝑦₁,𝐹𝑡,𝑥₂,𝑦₂≤𝑚(𝑡)𝑥₁−𝑥₂+𝑦₁−𝑦₂,‖𝑚‖_𝐿^∞𝑄₁+𝑚_3𝐿1𝑅₁+𝑚_2𝐿1𝑊₁≤181×3.1071+1×0.1707+9×310=0.5924<1. (5.9) Here 𝑄₁ is defined by (3.22) and 𝑅₁, 𝑊₁ are defined by (4.12). Hence all the assumptions of Theorem 4.8 are satisfied, and by the conclusion of it, the boundary value problem (5.6) has at least one solution on [0,1].

Acknowledgment

This project was supported by NNSF of China Grants nos. 11271087 and 61263006.