Some Results for Nonlinear (𝑛+1)-Term Fractional Integrodifferential Inclusions with Multipoint Boundary Conditions

Abstract

We are concerned with the nonlinear (𝑛+1)-term fractional integrodifferential inclusions 𝐿(𝐷)𝑢∈𝐹(𝑡,𝑢,𝑇𝑢,𝑆𝑢), 𝑎.𝑒.𝑡∈[0,1], where 𝐿(𝐷)=𝐷^𝛼−𝑏_𝑛𝐷^𝛽𝑛−𝑏_𝑛−1𝐷^𝛽𝑛−1−⋯−𝑏₁𝐷^𝛽1, 0<𝛽₁<𝛽₂<⋯<𝛽_𝑛−1<𝛽_𝑛<1<𝛼<2, 𝑏₁,𝑏₂,…,𝑏_𝑛 are constant coefficients, and (𝑇𝑢)(𝑡)=∫¹⁰𝑘(𝑡,𝑠)𝑢(𝑠)𝑑𝑠, (𝑆𝑢)(𝑡)=∫^𝑡0𝑙(𝑡,𝑠)𝑢(𝑠)𝑑𝑠, subject to the nonlocal conditions 𝑢(0)=0, 𝑢(1)=∑^𝑚𝑖=1𝛾_𝑖𝑢(𝜂_𝑖). The existence results are obtained by using two fixed-point theorems due to Bohnenblust-Karlin and Covitz-Nadler, respectively. Our results partly generalize and improve the known ones.

1. Introduction

Fractional differential equations (FDEs) have received increasing interest for the last three decades. It is benefited by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in the modeling of many phenomena in various fields of science, engineering economics, and other fields, see for instance [1–12] and references therein.

Babakhani and Gejji [7] considered the existence of positive solutions for the nonlinear fractional differential equations𝐿(𝐷)𝑢=𝑓(𝑡,𝑢),𝑢(0)=0,0<𝑡<1,(1.1) where 𝐿(𝐷)=𝐷^𝑠𝑛−𝑎_𝑛−1𝐷^𝑠𝑛−1−⋯−𝑎₁𝐷^𝑠1, 0<𝑠₁<𝑠₂<⋯<𝑠_𝑛<1, 𝑎_𝑗>0, 𝑗=1,2,…,𝑛−1, and 𝐷^𝑠𝑗 is the standard Riemann-Liouville fractional derivative. Some existence results of positive solutions are obtained by using some fixed-point theorems on a cone.

Stojanović [11] considered the existenceuniqueness of solutions for a nonlinear 𝑛-term fractional differential equation𝑏₀𝐷^𝛽0𝑢(𝑡)+^𝑚−1_𝑖=1𝑏_𝑖𝐷^𝛽𝑖𝑢(𝑡)+^𝑛−1_𝑖=𝑚𝑏_𝑖𝐷^𝛼𝑖𝑢(𝑡)+𝑏_𝑛𝐷^𝛼𝑛𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡∈(0,1),(1.2) where 0<𝛽₁<𝛽₂<⋯<𝛽_𝑚−1<1<𝛼_𝑚<𝛼_𝑚+1<⋯<𝛼_𝑛<2, with initial data 𝑢(0)=𝑓(0), 𝑢_𝑡(0)=𝑔(0).

On the other hand, realistic problems arising from economics, optimal control, and so on can be modeled as differential inclusions. Recently, El-Sayed and Ibrahim [13] initiated the study of fractional differential equations inclusions. Differential inclusions have been widely investigated by many authors, see [14–30] and references therein.

Very recently, in the survey paper [16], Agarwal et al. establish sufficient conditions for the existence and uniqueness of solutions for various classes of initial and boundary value problems for fractional differential equations and inclusions involving the Caputo fractional derivative.

Ouahab [26] studied the following boundary value problem of fractional differential inclusions:−𝐷^𝛼𝑦(𝑡)∈𝐹(𝑡,𝑦(𝑡𝑦))(0,)=a.e𝑦.(1𝑡)∈=𝐽0=,[0,1],1<𝛼≤2,(1.3) where 𝐷^𝛼 is the standard Riemann-Liouville fractional derivative, 𝐹∶𝐽×ℝ→𝒫(ℝ) is a multivalued map with compact values.

Chang and Nieto [27] studied boundary value problem of fractional differential inclusions^𝐶0𝐷^𝛿𝑦(𝑡)𝑦∈(0𝐹)(=𝑡,𝑦𝛼,(𝑡)),𝑦(𝑡1∈)=𝐽𝛽∶=,[𝛼0,,1𝛽]≠,0𝛿,∈(1,2),(1.4) where ^𝐶0𝐷^𝛿𝑦(𝑡) is the Caputo's derivative, 𝐹∶𝐽×ℝ→𝒫(ℝ).

However, to the best of our knowledge, the existence of solutions for fractional integro-differential inclusions with multipoint boundary conditions has not been paid much attention. Our goal is to fill this gap in literature.

In the present work, we consider more general fractional integro-differential inclusions𝐿(𝐷)𝑢∈𝐹(𝑡,𝑢,𝑇𝑢,𝑆𝑢),a.e.𝑡∈[0,1](1.5) with multipoint boundary conditions𝑢(0)=0,𝑢(1)=^𝑚_𝑖=1𝛾_𝑖𝑢𝜂_𝑖,(1.6) where 𝐿(𝐷)=𝐷^𝛼−𝑏_𝑛𝐷^𝛽𝑛−𝑏_𝑛−1𝐷^𝛽𝑛−1−⋯−𝑏₁𝐷^𝛽1, 0<𝛽₁<𝛽₂<⋯<𝛽_𝑛−1<𝛽_𝑛<1<𝛼<2, 𝐷^𝛼,𝐷^𝛽𝑗, 𝑗=1,2,…,𝑛, is the standard Riemann-Liouville fractional derivative, 0<𝜂₁<𝜂₂⋯<𝜂_𝑚<1, 0≤∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1<1, (𝑇𝑢)(𝑡)=∫¹⁰𝑘(𝑡,𝑠)𝑢(𝑠)d𝑠, (𝑆𝑢)(𝑡)=∫^𝑡0𝑙(𝑡,𝑠)𝑢(𝑠)d𝑠, 𝑘,𝑙 are two continuous functions on [0,1]×[0,1], and 𝐹∶[0,1]×ℝ×ℝ×ℝ→𝒫(ℝ) is a given multivalued function (𝒫(ℝ) is the family of all nonempty subsets of ℝ). We shall consider both the cases of convex and nonconvex valued right hand side and establish some sufficient conditions which admit that the integro-differential inclusions problem has at least one solution. These results obtained by applying two fixed point theorems due to Bohnenblust-Karlin and Covitz-Nadler are complement of previously known results.

The organization of this paper is as follows. In Section 2, we present some necessary definitions and preliminaries which are used throughout this paper. Main results and their proofs are given in Section 3.

2. Preliminaries and Several Lemmas

In this section, we recall some basic definitions and notations and give several lemmas which are useful in our discussion.

Definition 2.1. The Riemann-Liouville fractional integral of order 𝛼>0 of a function 𝑓∶ℝ⁺→ℝ is given by 𝐼^𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡0(𝑡−𝑠)^𝛼−1𝑓(𝑠)d𝑡,(2.1) provided the right side is pointwise defined on ℝ⁺.

Definition 2.2. The Riemann-Liouville fractional derivative of function 𝑓∶ℝ⁺→ℝ is given by 𝐷^𝛼𝑓(𝑡)=𝐷^𝑛𝐼^𝑛−𝛼𝑓(𝑡),(2.2) where 𝑛=[𝛼]+1, 𝐷^𝑛=d^𝑛/d𝑡^𝑛, provided the right side is pointwise defined on ℝ⁺.

Lemma 2.3. Let 𝛼>0, then 𝐼^𝛼𝐷^𝛼𝑥(𝑡)=𝑥(𝑡)+𝑐₁𝑡^𝛼−1+𝑐₂𝑡^𝛼−2+⋯+𝑐_𝑛𝑡^𝛼−𝑛,(2.3) for some 𝑐_𝑖∈ℝ, 𝑖=0,1,2,…,𝑛−1, 𝑛=−[−𝛼].

Remark 2.4. If the fractional derivative 𝐷^𝛼𝑥(𝑡) is integrable, then 𝐼^𝛼𝐷^𝛽𝑥(𝑡)=𝐼^𝛼−𝛽𝑥(𝑡)−𝐼^1−𝛽𝑥(𝑡)_𝑡=0𝑡^𝛼−1Γ(𝛼),0<𝛽≤𝛼<1.(2.4) If 𝑥 is continuous on [0,1], then [𝐼^1−𝛽𝑥(𝑡)]_𝑡=0=0, and (2.4) reduces to 𝐼^𝛼𝐷^𝛽𝑥(𝑡)=𝐼^𝛼−𝛽𝑥(𝑡),0<𝛽≤𝛼<1.(2.5)

The reader is referred to [4, 8, 9] for more details on fractional integrals and fractional derivatives.

Lemma 2.5. Let 𝑔∈𝐿([0,1],ℝ), 0<𝛽₁<𝛽₂<⋯<𝛽_𝑛−1<𝛽_𝑛<1<𝛼<2, and 0<∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1<1. Then 𝑢(𝑡) is a solution of the BVP 𝐷^𝛼−𝑏_𝑛𝐷^𝛽𝑛−𝑏_𝑛−1𝐷^𝛽𝑛−1−⋯−𝑏₁𝐷^𝛽1𝑢(𝑡)=𝑔(𝑡),a.e.𝑡∈[0,1],(2.6)𝑢(0)=0,𝑢(1)=^𝑚_𝑖=1𝛾_𝑖𝑢𝜂_𝑖.(2.7) if and only if 𝑢(𝑡) satisfies the integral equation 𝑢(𝑡)=^𝑚_𝑛𝑖=1_𝑗=1𝛾_𝑖𝑏_𝑗1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑗^𝜂𝑖0−_𝑛𝜂_𝑖−𝑠^{𝛼−𝛽𝑗−1}𝑡^𝛼−1𝑢(𝑠)d𝑠_𝑖=1𝑏_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑖¹⁰+_𝑛(1−𝑠)^{𝛼−𝛽𝑖−1}𝑡^𝛼−1𝑢(𝑠)d𝑠_𝑖=1𝑏_𝑖Γ𝛼−𝛽_𝑖^𝑡0(𝑡−𝑠)^{𝛼−𝛽𝑖−1}𝑢(𝑠)d𝑠+1Γ(𝛼)^𝑡0+_𝑚(𝑡−𝑠)^𝛼−1𝑔(𝑠)d𝑠_𝑖=1𝛾_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)^𝜂𝑖0−1𝜂_𝑖−𝑠^𝛼−1𝑡^𝛼−1𝑔(𝑠)d𝑠1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)¹⁰(1−𝑠)^𝛼−1𝑡^𝛼−1𝑔(𝑠)d𝑠.(2.8)

Proof. In view of Lemma 2.3 and Remark 2.4, (2.6) is equivalent to the integral equation 𝑢(𝑡)=𝑐₁𝑡^𝛼−1+𝑐₂𝑡^𝛼−2+^𝑛_𝑖=1𝑏_𝑖𝐼^{𝛼−𝛽𝑖}𝑢(𝑡)+𝐼^𝛼𝑔(𝑡),(2.9) for some 𝑐₁,𝑐₂∈ℝ, that is, 𝑢(𝑡)=𝑐₁𝑡^𝛼−1+𝑐₂𝑡^𝛼−2+^𝑛_𝑖=1𝑏_𝑖Γ𝛼−𝛽_𝑖^𝑡0(𝑡−𝑠)^{𝛼−𝛽𝑖−1}𝑢(𝑠)d𝑠+1Γ(𝛼)^𝑡0(𝑡−𝑠)^𝛼−1𝑔(𝑠)d𝑠.(2.10) The boundary condition 𝑢(0)=0 implies 𝑐₂=0. Thus, 𝑢(𝑡)=𝑐₁𝑡^𝛼−1+^𝑛_𝑖=1𝑏_𝑖Γ𝛼−𝛽_𝑖^𝑡0(𝑡−𝑠)^{𝛼−𝛽𝑖−1}𝑢(𝑠)d𝑠+1Γ(𝛼)^𝑡0(𝑡−𝑠)^𝛼−1𝑔(𝑠)d𝑠.(2.11) In view of the boundary condition 𝑢(1)=∑^𝑚𝑖=1𝛾_𝑖𝑢(𝜂_𝑖), we conclude that 𝑐₁=^𝑚_𝑛𝑖=1_𝑗=1𝛾_𝑖𝑏_𝑗1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑗^𝜂𝑖0−_𝑛𝜂_𝑖−𝑠^{𝛼−𝛽𝑗−1}𝑢(𝑠)d𝑠_𝑖=1𝑏_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑖¹⁰+_𝑚(1−𝑠)^{𝛼−𝛽𝑖−1}𝑢(𝑠)d𝑠_𝑖=1𝛾_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)^𝜂𝑖0−1𝜂_𝑖−𝑠^𝛼−1𝑔(𝑠)d𝑠1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)¹⁰(1−𝑠)^𝛼−1𝑔(𝑠)d𝑠.(2.12) Therefore, the solution 𝑢(𝑡) of (2.6) and (2.7) satisfies (2.8).
Conversely, if 𝑢(𝑡) is a solution of (2.8), it is easy to verify that 𝑢(𝑡) satisfies (2.6) and (2.7). The proof is complete.

Now, we recall some facts from multivalued analysis.

Let (𝑋,𝑑) be a metric space, and 𝒫(𝑋)={𝑌⊂𝑋∶𝑌≠∅}, 𝒫_𝑏𝑑(𝑋)={𝑌∈𝒫(𝑋)∶𝑌bounded}, 𝒫_𝑐𝑙(𝑋)={𝑌∈𝒫(𝑋)∶𝑌closed}, 𝒫_𝑐𝑣(𝑋)={𝑌∈𝒫(𝑋)∶𝑌convex}, 𝒫_𝑐𝑝(𝑋)={𝑌∈𝒫(𝑋)∶𝑌compact}, 𝒫_{𝑐𝑣,𝑐𝑝}(𝑋)=𝒫_𝑐𝑣(𝑋)∩𝒫_𝑐𝑝(𝑋), and so forth.

Consider 𝐻_𝑑∶𝒫(𝑋)×𝒫(𝑋)→ℝ₊∪{∞}, given by𝐻_𝑑(𝐴,𝐵)=maxsup_𝑥∈𝐴𝑑(𝑥,𝐵),sup_𝑦∈𝐵𝑑(𝐴,𝑦),(2.13) where 𝑑(𝑥,𝐵)=inf_𝑦∈𝐵𝑑(𝑥,𝑦), 𝑑(𝐴,𝑦)=inf_𝑥∈𝐴𝑑(𝑥,𝑦). Then (𝒫_{𝑏𝑑,𝑐𝑙}(𝑋),𝐻_𝑑) is a metric space and (𝒫_𝑐𝑙(𝑋),𝐻_𝑑) is a generalized metric space (see [31]).

A multivalued map 𝑁∶[0,1]→𝒫_𝑐𝑙(𝑋) is said to be measurable if for each 𝑢∈𝑋, the function 𝑌∶[0,1]→ℝ, defined by𝑌(𝑡)=𝑑(𝑢,𝑁(𝑡))=inf{𝑑(𝑢,𝑧)∶𝑧∈𝑁(𝑡)},(2.14) is measurable.

A multivalued map 𝐹∶𝑋→𝒫(𝑋) is convex (closed) valued if 𝐹(𝑢) is convex (closed) for all 𝑢∈𝑋. 𝐹 is bounded on bounded sets if 𝐹(𝐵)=∪_𝑢∈𝐵𝐹(𝑢) is bounded in 𝑋 for all 𝐵∈𝒫(𝑋). That is, sup_𝑢∈𝐵{sup{|𝑦|∶𝑦∈𝐹(𝑢)}}<∞. 𝐹 is called upper semicontinuous (u.s.c for short) on 𝑋 if for each 𝑢₀∈𝑋, the set 𝐹(𝑢₀) is nonempty closed subset of 𝑋, and if for each open set 𝒰 of 𝑋 containing 𝐹(𝑢₀), there exists an open neighborhood 𝒱 of 𝑢₀ such that 𝐹(𝒱)⊂𝒰. 𝐺 is said to be completely continuous if 𝐹(𝐵) is relatively compact for every 𝐵∈𝒫_𝑏𝑑(𝑋). If the multivalued map 𝐹 is completely continuous with nonempty compact valued, then 𝐺 is u.s.c. if and only if 𝐹 has closed graph, that is, 𝑥_𝑛→𝑥_∗, 𝑦_𝑛→𝑦_∗, 𝑦_𝑛∈𝐺(𝑥_∗) imply 𝑦_∗∈𝐺(𝑥_∗).

More details on multivalued maps can be found in the books of Deimling [3], Górniewicz [32], Hu and Papageorgiou [33], and Tolstonogov [34].

Definition 2.6. The multivalued map 𝐹∶[0,1]×ℝ×ℝ×ℝ→𝒫(ℝ) is 𝐿¹-Carathéodory if (i)𝑡↦𝐹(𝑡,𝑢,𝑣,𝑤) is measurable for each (𝑢,𝑣,𝑤)∈ℝ×ℝ×ℝ; (ii)(𝑢,𝑣,𝑤)↦𝐹(𝑡,𝑢,𝑣,𝑤) is upper semicontinuous for almost all 𝑡∈[0,1]; (iii)for each 𝑟>0, there exists 𝜑_𝑟∈𝐿¹([0,1],ℝ₊) such that for all |𝑢|,|𝑣|,|𝑤|≤𝑟 and for almost all 𝑡∈[0,1],‖𝐹(𝑡,𝑢,𝑣,𝑤)‖_𝒫=sup||𝑓||∶𝑓∈𝐹(𝑡,𝑢,𝑣,𝑤)≤𝜑_𝑟(𝑡).(2.15)

For any 𝑢∈𝐶([0,1],ℝ), we define the set𝑆_𝐹,𝑢=𝑓∈𝐿¹([0,1],ℝ)∶𝑓(𝑡)∈𝐹(𝑡,𝑢(𝑡),(𝑇𝑢)(𝑡),(𝑆𝑢)(𝑡))fora.e.𝑡∈[0,1],(2.16) which is known as the set of selection functions.

Lemma 2.7 (see [35]). Let 𝑋 be a Banach space. Let 𝐹∶[0,1]×𝑋→𝒫_{𝑐𝑝,𝑐𝑣}(𝑋) be an 𝐿¹-Carathéodory multivalued map with 𝑆_𝐹,𝑢≠∅, and let Γ be a linear continuous mapping from 𝐿¹([0,1],𝑋) into 𝐶([0,1],𝑋). Then the operator Γ∘𝑆_𝐹∶𝑢𝐶⟼([0,1]Γ,∘𝑋𝑆)_𝐹⟶𝒫_{𝑐𝑝,𝑐𝑣}(𝐶([0,1],𝑋)),(𝑦)∶=Γ𝑆_𝐹,𝑢(2.17) is a closed graph operator in 𝐶([0,1],𝑋)×𝐶([0,1],𝑋).

The following Bohnenblust-Karlin fixed-point lemma and the fixed-point theorem for contractive multivalued operators given by Covitz and Nadler are of great importance in the proofs of our main results. The proofs of these results can be found in Bohnenblust and Karlin [30] and in Covitz and Nadler [36].

Lemma 2.8 (see [30]). Let 𝑋 be a Banach space, 𝐷 be a nonempty subset of 𝑋, which is bounded, closed, and convex. Suppose 𝐺∶𝐷→𝒫(𝑋) is u.s.c. with closed, convex values, and such that 𝐺(𝐷)⊂𝐷 and 𝐺(𝐷) is relatively compact. Then 𝐺 has a fixed point.

Lemma 2.9 (see [36]). Let (𝑋,𝑑) be a complete metric space. If 𝑁 is a contraction operator, then 𝐹𝑖𝑥𝑁≠∅.
For convenience, let us list some conditions. (H1)∑^𝑚𝑖=1∑^𝑛𝑗=1(𝛾_𝑖𝑏_𝑗𝜂^{𝛼−𝛽𝑗}_𝑖/(1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1)Γ(𝛼−𝛽_𝑗+1))+∑^𝑛𝑖=1(𝑏_𝑖/(1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1)Γ(𝛼−𝛽_𝑖+1))+∑^𝑛𝑖=1(𝑏_𝑖/Γ(𝛼−𝛽_𝑖+1))<1;(H2)𝐹∶[0,1]×ℝ×ℝ×ℝ→𝒫_{𝑐𝑝,𝑐𝑣}(ℝ) is Carethéodory multivalued map.

Lemma 2.10. Assume that hypothesis (H1) is satisfied. For any 𝑔∈𝐶([0,1],ℝ), the integral equation 𝑢(𝑡)=^𝑚_𝑛𝑖=1_𝑗=1𝛾_𝑖𝑏_𝑗1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑗^𝜂𝑖0−_𝑛𝜂_𝑖−𝑠^{𝛼−𝛽𝑗−1}𝑡^𝛼−1𝑢(𝑠)d𝑠_𝑖=1𝑏_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑖¹⁰+_𝑛(1−𝑠)^{𝛼−𝛽𝑖−1}𝑡^𝛼−1𝑢(𝑠)d𝑠_𝑖=1𝑏_𝑖Γ𝛼−𝛽_𝑖^𝑡0(𝑡−𝑠)^{𝛼−𝛽𝑖−1}𝑢(𝑠)d𝑠+𝑔(𝑡)(2.18) has a unique solution in 𝐶([0,1],ℝ).

Proof. We define the operator 𝐴 as follows: (𝐴𝑢)(𝑡)∶=^𝑚_𝑛𝑖=1_𝑗=1𝛾_𝑖𝑏_𝑗1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑗^𝜂𝑖0−_𝑛𝜂_𝑖−𝑠^{𝛼−𝛽𝑗−1}𝑡^𝛼−1𝑢(𝑠)d𝑠_𝑖=1𝑏_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑖¹⁰+_𝑛(1−𝑠)^{𝛼−𝛽𝑖−1}𝑡^𝛼−1𝑢(𝑠)d𝑠_𝑖=1𝑏_𝑖Γ𝛼−𝛽_𝑖^𝑡0(𝑡−𝑠)^{𝛼−𝛽𝑖−1}𝑢(𝑠)d𝑠+𝑔(𝑡)(2.19) Obviously, 𝐴 is a map from 𝐶([0,1],ℝ) into 𝐶([0,1],ℝ). Also, we have ||𝐴𝑢₁(𝑡)−𝐴𝑢₂(𝑡)||≤^𝑚_𝑛𝑖=1_𝑗=1𝛾_𝑖𝑏_𝑗1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑗^𝜂𝑖0+_𝑛𝜂_𝑖−𝑠^{𝛼−𝛽𝑗−1}𝑡^𝛼−1||𝑢₁(𝑠)−𝑢₂(𝑠)||d𝑠_𝑖=1𝑏_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑖¹⁰+_𝑛(1−𝑠)^{𝛼−𝛽𝑖−1}𝑡^𝛼−1||𝑢₁(𝑠)−𝑢₂(𝑠)||d𝑠_𝑖=1𝑏_𝑖Γ𝛼−𝛽_𝑖^𝑡0≤⎛⎜⎜⎝_𝑚(𝑡−𝑠)^{𝛼−𝛽𝑖−1}||𝑢₁(𝑠)−𝑢₂(𝑠)||d𝑠_𝑛𝑖=1_𝑗=1𝛾_𝑖𝑏_𝑗𝜂^{𝛼−𝛽𝑗}_𝑖+1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑗+1_𝑛_𝑖=1𝑏_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑖+1+^𝑛_𝑖=1𝑏_𝑖Γ𝛼−𝛽_𝑖+1‖‖𝑢¹−𝑢₂‖‖.(2.20) Therefore, ‖𝐴𝑢₁−𝐴𝑢₂‖≤𝜆‖𝑢₁−𝑢₂‖, where 𝜆=∑^𝑚𝑖=1∑^𝑛𝑗=1(𝛾_𝑖𝑏_𝑗𝜂^{𝛼−𝛽𝑗}_𝑖/(1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1)Γ(𝛼−𝛽_𝑗+1))+∑^𝑛𝑖=1(𝑏_𝑖/(1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1)Γ(𝛼−𝛽_𝑖+1))+∑^𝑛𝑖=1(𝑏_𝑖/Γ(𝛼−𝛽_𝑖+1)). By (H1) and Banach contraction principle, the conclusion of lemma is true.

Definition 2.11. A function 𝑢∈𝐶[0,1] is said to be a solution of the BVP (1.5) and (1.6), if there exists a function 𝑓∈𝐿¹([0,1],ℝ) such that 𝑓(𝑡)∈𝐹(𝑡,𝑢(𝑡),(𝑇𝑢)(𝑡),(𝑆𝑢)(𝑡)) a.e. on [0,1] and 𝑢(𝑡)=^𝑚_𝑛𝑖=1_𝑗=1𝛾_𝑖𝑏_𝑗1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑗^𝜂𝑖0−_𝑛𝜂_𝑖−𝑠^{𝛼−𝛽𝑗−1}𝑡^𝛼−1𝑢(𝑠)d𝑠_𝑖=1𝑏_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑖¹⁰+_𝑛(1−𝑠)^{𝛼−𝛽𝑖−1}𝑡^𝛼−1𝑢(𝑠)d𝑠_𝑖=1𝑏_𝑖Γ𝛼−𝛽_𝑖^𝑡0(𝑡−𝑠)^{𝛼−𝛽𝑖−1}𝑢(𝑠)d𝑠+1Γ(𝛼)^𝑡0+_𝑚(𝑡−𝑠)^𝛼−1𝑓(𝑠)d𝑠_𝑖=1𝛾_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)^𝜂𝑖0−1𝜂_𝑖−𝑠^𝛼−1𝑡^𝛼−1𝑓(𝑠)d𝑠1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)¹⁰(1−𝑠)^𝛼−1𝑡^𝛼−1𝑓(𝑠)d𝑠.(2.21)

3. Main Results

In this section, we present our main results and prove them. Firstly, under convexity condition on the multivalued right-hand side, we are to establish the existence theorem of solutions for fractional differential inclusions (1.5) and (1.6), by employing the Bohnenblust-Karlin fixed-point theorem. Then, under nonconvexity condition on the multivalued right-hand side, the existence theorem of solutions are gotten, by employing the Covitz-Nadler fixed-point theorem.

Theorem 3.1. Assume that hypotheses (H1) and (H2) are satisfied. Then BVP (1.5) and (1.6) has at least one solution in 𝐶([0,1],ℝ), provided that (H3)𝜏/Γ(𝛼)+∑^𝑚𝑖=1(𝛾_𝑖𝜇_𝑖/(1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1)Γ(𝛼))+(𝜔/(1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1)Γ(𝛼))<1−𝜆,
where 𝜏=lim_𝑟→∞inf¹⁰max𝜑_𝑟(𝑠),𝜑_̃𝑘𝑟(𝑠),𝜑_̃𝑙𝑟(𝑠)𝑟𝜇d𝑠,_𝑖=lim_𝑟→∞inf^𝜂𝑖₀𝜂_𝑖−𝑠^𝛼−1max𝜑_𝑟(𝑠),𝜑_̃𝑘𝑟(𝑠),𝜑_̃𝑙𝑟(𝑠)𝑟𝜔=lim_𝑟→∞d𝑠,inf¹⁰(1−𝑠)^𝛼−1max𝜑_𝑟(𝑠),𝜑_̃𝑘𝑟(𝑠),𝜑_̃𝑙𝑟(𝑠)𝑟𝑘̃=sup_[]𝑡∈0,1d𝑠,¹⁰𝑘(𝑡,𝑠)d𝑠,𝑙̃=sup_[]𝑡∈0,1^𝑡0𝑙(𝑡,𝑠)d𝑠,(3.1) and 𝜆 is defined in the proof of Lemma 2.10.

Proof. To transform the problem into a fixed-point problem, we consider the multivalued operator, 𝑁∶𝐶([0,1],ℝ)→𝒫(𝐶([0,1],ℝ)), where for any 𝑢∈𝐶([0,1],ℝ),𝑁(𝑢) is defined by ℎ∈𝐶([0,1],ℝ)∶ℎ(𝑡)=^𝑚_𝑛𝑖=1_𝑗=1𝛾_𝑖𝑏_𝑗1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑗^𝜂𝑖0−_𝑛𝜂_𝑖−𝑠^{𝛼−𝛽𝑗−1}𝑡^𝛼−1ℎ(𝑠)d𝑠_𝑖=1𝑏_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑖¹⁰+_𝑛(1−𝑠)^{𝛼−𝛽𝑖−1}𝑡^𝛼−1ℎ(𝑠)d𝑠_𝑖=1𝑏_𝑖Γ𝛼−𝛽_𝑖^𝑡0(𝑡−𝑠)^{𝛼−𝛽𝑖−1}ℎ(𝑠)d𝑠+1Γ(𝛼)^𝑡0+_𝑚(𝑡−𝑠)^𝛼−1𝑓(𝑠)d𝑠_𝑖=1𝛾_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)^𝜂𝑖0−1𝜂_𝑖−𝑠^𝛼−1𝑡^𝛼−1𝑓(𝑠)d𝑠1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)¹⁰(1−𝑠)^𝛼−1𝑡^𝛼−1𝑓(𝑠)d𝑠,𝑓∈𝑆_𝐹,𝑢.(3.2)
For any 𝑓∈𝐿¹([0,1],ℝ), we have ∫^𝑡0(𝑡−𝑠)^𝛼−1𝑓(𝑠)d𝑠, ∫^𝜂𝑖₀(𝜂_𝑖−𝑠)^𝛼−1𝑡^𝛼−1𝑓(𝑠)d𝑠, ∫¹⁰(1−𝑠)^𝛼−1𝑡^𝛼−1𝑓(𝑠)d𝑠∈𝐶([0,1],ℝ). In view of Lemma 2.10, 𝑁 is well defined. Moreover, it follows from the convexity of 𝑆_𝐹,𝑢 (because 𝐹 has convex values) that 𝑁(𝑢) is convex for each 𝑢∈𝐶([0,1],ℝ). Clearly, the fixed points of 𝑁 are solutions of (1.5) and (1.6).
We shall show that 𝑁 has a fixed point in three steps.
Step 1. we claim that there exists a 𝑟^∗>0, such that 𝑁(𝐵_𝑟∗)⊂𝐵_𝑟∗, where 𝐵_𝑟∗={𝑢∈𝐶([0,1],ℝ)∶‖𝑢‖≤𝑟^∗}.
In fact, if it is not true, then for any 𝑟>0, there exists a function 𝑢_𝑟∈𝐵_𝑟, ℎ_𝑟∈𝑁(𝑢_𝑟) but ℎ_𝑟∉𝐵_𝑟, that is ‖ℎ_𝑟‖>𝑟 and for some 𝑓_𝑟∈𝑆_𝐹,𝑢, ℎ_𝑟(𝑡)=^𝑚_𝑛𝑖=1_𝑗=1𝛾_𝑖𝑏_𝑗1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑗^𝜂𝑖0−_𝑛𝜂_𝑖−𝑠^{𝛼−𝛽𝑗−1}𝑡^𝛼−1ℎ_𝑟(𝑠)d𝑠_𝑖=1𝑏_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑖¹⁰+_𝑛(1−𝑠)^{𝛼−𝛽𝑖−1}𝑡^𝛼−1ℎ_𝑟(𝑠)d𝑠_𝑖=1𝑏_𝑖Γ𝛼−𝛽_𝑖^𝑡0(𝑡−𝑠)^{𝛼−𝛽𝑖−1}ℎ_𝑟(s)d𝑠+1Γ(𝛼)^𝑡0+_𝑚(𝑡−𝑠)^𝛼−1𝑓_𝑟(𝑠)d𝑠_𝑖=1𝛾_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)^𝜂𝑖0−1𝜂_𝑖−𝑠^𝛼−1𝑡^𝛼−1𝑓_𝑟(𝑠)d𝑠1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)¹⁰(1−𝑠)^𝛼−1𝑡^𝛼−1𝑓_𝑟(𝑠)d𝑠.(3.3)
On the other hand, from (H2), we obtain ||ℎ_𝑟(𝑡)||≤𝜆‖‖ℎ_𝑟‖‖+1Γ(𝛼)^𝑡0+1(𝑡−𝑠)^𝛼−1𝑓_𝑟(𝑠)d𝑠1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)¹⁰+_𝑚(1−𝑠)^𝛼−1𝑡^𝛼−1𝑓_𝑟(𝑠)d𝑠_𝑖=1𝛾_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)^𝜂𝑖0≤𝜆‖‖ℎ_𝑟𝜂_𝑖−𝑠^𝛼−1𝑡^𝛼−1𝑓_𝑟(𝑠)d𝑠‖‖+1Γ(𝛼)¹⁰+_𝑚(|𝑡−𝑠|)^𝛼−1𝜑_max{𝑟,𝑘̃_𝑟,𝑙̃_𝑟}(𝑠)d𝑠_𝑖=1𝛾_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)^𝜂𝑖0+1𝜂_𝑖−𝑠^𝛼−1𝑡^𝛼−1𝜑_max{𝑟,𝑘̃_𝑟,𝑙̃_𝑟}(𝑠)d𝑠1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)¹⁰(1−𝑠)^𝛼−1𝑡^𝛼−1𝜑_max{𝑟,𝑘̃_𝑟,𝑙̃_𝑟}(𝑠)d𝑠.(3.4) Thus, 𝑟<‖‖ℎ_𝑟‖‖≤1(1−𝜆)Γ(𝛼)¹⁰+_𝑚(|𝑡−𝑠|)^𝛼−1𝜑_max{𝑟,𝑘̃_𝑟,𝑙̃_𝑟}(𝑠)d𝑠_𝑖=1𝛾_𝑖(1−𝜆)1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)^𝜂𝑖0+1𝜂_𝑖−𝑠^𝛼−1𝜑_max{𝑟,𝑘̃_𝑟,𝑙̃_𝑟}(𝑠)d𝑠(1−𝜆)1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)¹⁰(1−𝑠)^𝛼−1𝜑_max{𝑟,𝑘̃_𝑟,𝑙̃_𝑟}(𝑠)d𝑠.(3.5)
Dividing both sides by 𝑟 and taking the lower limit as 𝑟→+∞, we get 1≤𝜏(1−𝜆)Γ(𝛼)+^𝑚_𝑖=1𝛾_𝑖𝜇_𝑖(1−𝜆)1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)+(1−𝜆)1−∑𝜔^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼).(3.6)
This is a contraction to (H3). Hence there exists a 𝑟^∗>0 such that 𝑁(𝐵_𝑟∗)⊂𝐵_𝑟∗.Step 2. 𝑁(𝐵_𝑟∗) is equicontinuous.
Let 𝑡₁,𝑡₂∈[0,1], 𝑡₁<𝑡₂, 𝑢∈𝐵_𝑟∗, and ℎ∈𝑁(𝑢). Then there exists 𝑓∈𝑆_𝐹,𝑢 such that for each 𝑡∈[0,1], we have ℎ(𝑡)=^𝑚_𝑛𝑖=1_𝑗=1𝛾_𝑖𝑏_𝑗1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑗^𝜂𝑖0−_𝑛𝜂_𝑖−𝑠^{𝛼−𝛽𝑗−1}𝑡^𝛼−1ℎ(𝑠)d𝑠_𝑖=1𝑏_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑖¹⁰+_𝑛(1−𝑠)^{𝛼−𝛽𝑖−1}𝑡^𝛼−1ℎ(𝑠)d𝑠_𝑖=1𝑏_𝑖Γ𝛼−𝛽_𝑖^𝑡0(𝑡−𝑠)^{𝛼−𝛽𝑖−1}ℎ(𝑠)d𝑠+1Γ(𝛼)^𝑡0+_𝑚(𝑡−𝑠)^𝛼−1𝑓(𝑠)d𝑠_𝑖=1𝛾_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)^𝜂𝑖0−1𝜂_𝑖−𝑠^𝛼−1𝑡^𝛼−1𝑓(𝑠)d𝑠1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)¹⁰(1−𝑠)^𝛼−1𝑡^𝛼−1𝑓(𝑠)d𝑠.(3.7) Therefore, ||ℎ𝑡₁−ℎ𝑡₂||≤𝑎𝑡^2𝛼−1−𝑡^1𝛼−1+1Γ(𝛼)^𝑡2_𝑡1+1𝜑_max{𝑟∗,𝑘̃_𝑟∗,𝑙̃_𝑟∗}(𝑠)d𝑠Γ(𝛼)^𝑡1₀𝑡²+_𝑛−𝑠^𝛼−1−𝑡₁−𝑠^𝛼−1𝜑_max{𝑟^∗_,𝑘̃_𝑟∗,𝑙̃_𝑟∗}(𝑠)d𝑠_𝑖=1𝑏_𝑖𝑟^∗Γ𝛼−𝛽_𝑖+1𝑡^{𝛼−𝛽𝑖2}−𝑡^{𝛼−𝛽𝑖}₁+2𝑡₂−𝑡₁^{𝛼−𝛽𝑖},(3.8) where 𝑎=^𝑚_𝑛𝑖=1_𝑗=1𝛾_𝑖𝑏_𝑗𝜂^{𝛼−𝛽𝑗}_𝑖𝑟^∗1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑗+1+^𝑛_𝑖=1𝑏_𝑖𝑟^∗+1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑖+1_𝑚_𝑖=1𝛾_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)^𝜂𝑖0+1𝜂_𝑖−𝑠^𝛼−1𝜑_max{𝑟∗,𝑘̃_𝑟∗,𝑙̃_𝑟∗}(𝑠)d𝑠1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)¹⁰(1−𝑠)^𝛼−1𝜑_max{𝑟∗,𝑘̃_𝑟∗,𝑙̃_𝑟∗}(𝑠)d𝑠.(3.9)
The right-hand side of the above inequality tends to zeros independently of ℎ∈𝐵_𝑟∗, as 𝑡₂−𝑡₁→0. This shows that 𝑁(𝐵_𝑟∗) is equicontinuous. Step 3. 𝑁 has a closed graph.
Let 𝑢_𝑛→𝑢_∗, ℎ_𝑛∈𝑁(𝑢_𝑛) and ℎ_𝑛→ℎ_∗. We will prove that ℎ_∗∈𝑁(𝑢_∗). Now ℎ_𝑛∈𝑁(𝑢_𝑛) implies that there exists 𝑓_𝑛∈𝑆_{𝐹,𝑢𝑛} such that for each 𝑡∈[0,1], ℎ_𝑛(𝑡)=^𝑚_𝑛𝑖=1_𝑗=1𝛾_𝑖𝑏_𝑗1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑗^𝜂𝑖0−_𝑛𝜂_𝑖−𝑠^{𝛼−𝛽𝑗−1}𝑡^𝛼−1ℎ_𝑛(𝑠)d𝑠_𝑖=1𝑏_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑖¹⁰+_𝑛(1−𝑠)^{𝛼−𝛽𝑖−1}𝑡^𝛼−1ℎ_𝑛(𝑠)d𝑠_𝑖=1𝑏_𝑖Γ𝛼−𝛽_𝑖^𝑡0(𝑡−𝑠)^{𝛼−𝛽𝑖−1}ℎ_𝑛(𝑠)d𝑠+1Γ(𝛼)^𝑡0+_𝑚(𝑡−𝑠)^𝛼−1𝑓_𝑛(𝑠)d𝑠_𝑖=1𝛾_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)^𝜂𝑖0−1𝜂_𝑖−𝑠^𝛼−1𝑡^𝛼−1𝑓_𝑛(𝑠)d𝑠1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)¹⁰(1−𝑠)^𝛼−1𝑡^𝛼−1𝑓_𝑛(𝑠)d𝑠.(3.10)
We need to show that there exists 𝑓_∗∈𝑆_𝐹,𝑢 such that for each 𝑡∈[0,1], ℎ_∗(𝑡)=^𝑚_𝑛𝑖=1_𝑗=1𝛾_𝑖𝑏_𝑗1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑗^𝜂𝑖0−_𝑛𝜂_𝑖−𝑠^{𝛼−𝛽𝑗−1}𝑡^𝛼−1ℎ_∗(𝑠)d𝑠_𝑖=1𝑏_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑖¹⁰+_𝑛(1−𝑠)^{𝛼−𝛽𝑖−1}𝑡^𝛼−1ℎ_∗(𝑠)d𝑠_𝑖=1𝑏_𝑖Γ𝛼−𝛽_𝑖^𝑡0(𝑡−𝑠)^{𝛼−𝛽𝑖−1}ℎ_∗(𝑠)d𝑠+1Γ(𝛼)^𝑡0+_𝑚(𝑡−𝑠)^𝛼−1𝑓_∗(𝑠)d𝑠_𝑖=1𝛾_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)^𝜂𝑖0−1𝜂_𝑖−𝑠^𝛼−1𝑡^𝛼−1𝑓_∗(𝑠)d𝑠1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)¹⁰(1−𝑠)^𝛼−1𝑡^𝛼−1𝑓_∗(𝑠)d𝑠.(3.11) In (2.18), taking 𝑔(𝑡)=1Γ(𝛼)^𝑡0(𝑡−𝑠)^𝛼−1𝑓(𝑠)d𝑠+^𝑚_𝑖=1𝛾_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)^𝜂𝑖0−1𝜂_𝑖−𝑠^𝛼−1𝑡^𝛼−1𝑓(𝑠)d𝑠1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)¹⁰(1−𝑠)^𝛼−1𝑡^𝛼−1𝑓(𝑠)d𝑠.(3.12) In view of Lemma 2.10, we know that for any 𝑓∈𝐿([0,1],ℝ), the integral equation (2.18) has a unique solution, which we denote by ℎ_𝑓. Because of this, we can define the operator Φ∶𝐿([0,1]𝑓,ℝ⟼)⟶ℎ_𝑓𝐶([0,1],ℝ),.(3.13) It is easy to verify that the operator Φ is linear. On the other hand, we can get ‖‖ℎ_𝑓‖‖≤1(1−𝜆)Γ(𝛼)¹⁰+(1|𝑡−𝑠|)^𝛼−1||𝑓(𝑠)||d𝑠(1−𝜆)1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)¹⁰+_𝑚(1−𝑠)^𝛼−1||𝑓(𝑠)||d𝑠_𝑖=1𝛾_𝑖(1−𝜆)1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)^𝜂𝑖0≤𝜇₁₀𝜂_𝑖−𝑠^𝛼−1||𝑓(𝑠)||d𝑠||𝑓(𝑠)||d𝑠,(3.14) where 𝜇=1(1−𝜆)Γ(𝛼)+^𝑚_𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1(1−𝜆)1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)+(1−𝜆)1−∑1^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼),(3.15) that is, ‖Φ(𝑓)‖≤𝜇‖𝑓‖_𝐿¹, which implies Φ is continuous. From Lemma 2.7, it follows that Φ∘𝑆_𝐹,𝑢 is a closed graph operator. Moreover, we know ℎ_𝑛∈Φ(𝑆_{𝐹,𝑢𝑛}). Since 𝑢_𝑛→𝑢_∗, it follows from Lemma 2.7 that (3.11) holds for some 𝑓_∗∈𝑆_𝐹,𝑢∗.
Therefore, 𝑁 is a compact multivalued map, u.s.c. with convex closed values. As a consequence of Lemma 2.8, we immediately conclude that 𝑁 has a fixed-point 𝑢 which is a solution of the problem (1.5) and (1.6). The proof is complete.

As a direct corollary of Theorem 3.1, we can immediately obtain the following result when the nonlinearity 𝐹 has sublinear growth in the state variable.

Corollary 3.2. Suppose that (H1) and (H2) are satisfied. Then the problem (1.5) and (1.6) has at least one solution in 𝐶([0,1],ℝ), provided that (H4)there exist functions 𝑝(𝑡),𝑞(𝑡),𝑟(𝑡),𝜃(𝑡)∈𝐿([0,1],ℝ), and 𝜎₁,𝜎₂,𝜎₃∈[0,1) such that‖𝐹(𝑡,𝑢,𝑣,𝑤)‖_𝒫≤𝑝(𝑡)|𝑢|^𝜎1+𝑞(𝑡)|𝑣|^𝜎2+𝑟(𝑡)|𝑢|^𝜎3+𝜃(𝑡)(3.16) for each (𝑡,𝑢,𝑣,𝑤)∈[0,1]×ℝ×ℝ×ℝ.

Proof. We only need to verify that (H4) implies (H3) in Theorem 3.1. Taking 𝜑(𝑡)=𝑝(𝑡)|𝑢|^𝜎1+𝑞(𝑡)|𝑣|^𝜎2+𝑟(𝑡)|𝑢|^𝜎3+𝜃(𝑡), it is easy to prove 𝜏=lim_𝑟→∞inf¹⁰max𝜑_𝑟(𝑠),𝜑_̃𝑘𝑟(𝑠),𝜑_̃𝑙𝑟(𝑠)𝑟𝜇d𝑠=0,_𝑖=lim_𝑟→∞inf^𝜂𝑖₀𝜂_𝑖−𝑠^𝛼−1max𝜑_𝑟(𝑠),𝜑_̃𝑘𝑟(𝑠),𝜑_̃𝑙𝑟(𝑠)𝑟𝜔=lim_𝑟→∞d𝑠=0,inf¹⁰(1−𝑠)^𝛼−1max𝜑_𝑟(𝑠),𝜑_̃𝑘𝑟(𝑠),𝜑_̃𝑙𝑟(𝑠)𝑟d𝑠=0,(3.17) because of 𝜎₁,𝜎₂,𝜎₃∈[0,1). Then (H3) is satisfied. The proof is finished.

In the next part, we are concerned with the BVP (1.5) and (1.6) with nonconvex valued right-hand side. By using Covitz and Nadler fixed-point theorem, we obtain the following result.

Theorem 3.3. Assume that the following hypotheses hold: (A1) 𝐹∶[0,1]×ℝ×ℝ×ℝ→𝑃_𝑐𝑙(ℝ); 𝑡→𝐹(𝑡,𝑢,𝑣,𝑤) is measurable for each 𝑢,𝑣,𝑤∈ℝ;(A2)There exist three functions 𝑝,𝑞,𝑟∈𝐿¹([0,1],ℝ) such that for a.e. 𝑡∈[0,1] and 𝑢₁,𝑢₂,𝑣₁,𝑣₂,𝑤₁,𝑤₂∈ℝ,𝐻_𝑑𝐹𝑡,𝑢₁,𝑣₁,𝑤₁,𝐹𝑡,𝑢₂,𝑣𝑑₂(,0𝑤,𝐹₂(𝑡,≤0,𝑝0(,𝑡0))||𝑢)₁≤−𝑚𝑢₂(𝑡||)+.𝑞(𝑡)||𝑣₁−𝑣₂||+𝑟(𝑡)||𝑤₁−𝑤₂||,(3.18) Then the BVP (1.5) and (1.6) has at least one solution in 𝐶([0,1],ℝ) provided that 𝑎₀̃+𝑏̃₀+𝑐̃₀+^𝑚+1_𝑖=1𝑎_𝑖̃+𝑘̃𝑏̃_𝑖+𝑙𝑐̃̃_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1<1−𝜆,(3.19) where 𝑎₀̃=1Γ(𝛼)¹⁰𝑝(𝑠)d𝑠,𝑎_𝑖̃=1Γ(𝛼)^𝜂𝑖₀𝜂_𝑖𝑏̃−𝑠^𝛼−1𝑝(𝑠)d𝑠,𝑖=1,2,…,𝑚+1,₀=1Γ(𝛼)¹⁰𝑞(𝑠)d𝑠,𝑏̃_𝑖=1Γ(𝛼)^𝜂𝑖₀𝜂_𝑖𝑐̃₀−𝑠^𝛼−1𝑞(𝑠)d𝑠,𝑖=1,2,…,𝑚+1,=1Γ(𝛼)¹⁰𝑝(𝑠)d𝑠,𝑐̃_𝑖=1Γ(𝛼)^𝜂𝑖₀𝜂_𝑖−𝑠^𝛼−1𝑟(𝑠)d𝑠,𝑖=1,2,…,𝑚+1,(3.20)  𝜂_𝑚+1=1 and 𝜆 defined as in Lemma 2.10, 𝑘̃,𝑙̃ defined by (3.1) in Theorem 3.1.

Proof. Transform the problem into a fixed-point problem. Let 𝑁∶𝐶([0,1],ℝ)→𝒫(𝐶([0,1],ℝ) be defined as in Theorem 3.1. We shall show that 𝑁 satisfies the assumptions of Lemma 2.9. The proof will be given in two steps.Step 1. 𝑁(𝑢)∈𝒫_𝑐𝑙(𝐶([0,1],ℝ)) for each 𝑢∈𝐶([0,1],ℝ).
Indeed, let (𝑢_𝑛)_𝑛≥0∈𝑁(𝑢) such that 𝑢_𝑛→𝑢̃ in 𝐶([0,1],ℝ). Then there exists 𝑔_𝑛∈𝑆_𝐹,𝑢 such that for each 𝑡∈[0,1], 𝑢_𝑛(𝑡)=^𝑚_𝑛𝑖=1_𝑗=1𝛾_𝑖𝑏_𝑗1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑗^𝜂𝑖0−_𝑛𝜂_𝑖−𝑠^{𝛼−𝛽𝑗−1}𝑡^𝛼−1𝑢_𝑛(𝑠)d𝑠_𝑖=1𝑏_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ𝛼−𝛽_𝑖¹⁰+_𝑛(1−𝑠)^{𝛼−𝛽𝑖−1}𝑡^𝛼−1𝑢_𝑛(𝑠)d𝑠_𝑖=1𝑏_𝑖Γ𝛼−𝛽_𝑖^𝑡0(𝑡−𝑠)^{𝛼−𝛽𝑖−1}𝑢_𝑛(𝑠)d𝑠+1Γ(𝛼)^𝑡0+_𝑚(𝑡−𝑠)^𝛼−1𝑔_𝑛(𝑠)d𝑠_𝑖=1𝛾_𝑖1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)^𝜂𝑖0−1𝜂_𝑖−𝑠^𝛼−1𝑡^𝛼−1𝑔_𝑛(𝑠)d𝑠1−∑^𝑚𝑖=1𝛾_𝑖𝜂^𝑖𝛼−1Γ(𝛼)¹⁰(1−𝑠)^𝛼−1𝑡^𝛼−1𝑔_𝑛(𝑠)d𝑠.(3.21)
For every 𝑤∈𝐹(𝑡,𝑢(𝑡),(𝑇𝑢)(𝑡),(𝑆𝑢)(𝑡)), from (A2), we have |𝑤|≤𝑑(0,𝐹(𝑡,0,0,0))+𝐻_𝑑(𝐹(𝑡,0,0,0),𝐹(𝑡,𝑢(𝑡),(𝑇𝑢)(𝑡),(𝑆𝑢)(𝑡))).(3.22) Then, ||𝑔_𝑛(𝑡)||≤𝑚(𝑡)+𝑝(𝑡)||𝑢(𝑡)||+𝑞(𝑡)||(𝑇𝑢)(𝑡)||+𝑟(𝑡)||(𝑆𝑢)(𝑡)||a.e.𝑡∈[0,1],(3.23) that is, 𝑔_𝑛(𝑡)∈𝐵(0,𝑚(𝑡)+𝑝(𝑡)|𝑢(𝑡)|+𝑞(𝑡)|(𝑇𝑢)(𝑡)|+𝑟(𝑡)|(𝑆𝑢)(𝑡)|), where 𝐵0,=𝑚(𝑤𝑡)+∈𝑝ℝ(𝑡∶)|||𝑢𝑤(|𝑡)≤||+𝑚𝑞(𝑡()𝑡)+||(𝑇𝑝𝑢()𝑡)(𝑡||𝑢)||(+𝑡)||𝑟+(𝑡𝑞)||((𝑡𝑆)𝑢||()𝑇(𝑢𝑡))||(𝑡)||+𝑟(𝑡)||(𝑆𝑢)(𝑡)||∶=Ψ(𝑡).(3.24)
It is clear that Ψ∶[0,1]→𝒫_{𝑐𝑝,𝑐𝑣}(ℝ) is a multivalued integrable bounded map. Since 𝑔_𝑛(⋅)∈Ψ(⋅),𝑛≥1, we may pass to a subsequence if necessary to get that 𝑔_𝑛 converges weakly to 𝑔 in 𝐿^1𝑤([0,1],ℝ). From Mauz's lemma [37], there exists 𝑔∈conv{𝑔_𝑛(𝑡)∶𝑛≥1}, then there exists a subsequence {𝑔_𝑛