Table of Contents
ISRN Mathematical Analysis
VolumeΒ 2011, Article IDΒ 436045, 20 pages
http://dx.doi.org/10.5402/2011/436045
Research Article

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

Department of Applied Mathematics, Donghua University, Shanghai 201620, China

Received 18 April 2011; Accepted 24 May 2011

Academic Editors: B.Β Djafari-Rouhani, G. L.Β Karakostas, and W.Β Kryszewski

Copyright Β© 2011 Huacheng Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We are concerned with the nonlinear (𝑛+1)-term fractional integrodifferential inclusions 𝐿(𝐷)π‘’βˆˆπΉ(𝑑,𝑒,𝑇𝑒,𝑆𝑒), π‘Ž.𝑒.π‘‘βˆˆ[0,1], where 𝐿(𝐷)=π·π›Όβˆ’π‘π‘›π·π›½π‘›βˆ’π‘π‘›βˆ’1π·π›½π‘›βˆ’1βˆ’β‹―βˆ’π‘1𝐷𝛽1, 0<𝛽1<𝛽2<β‹―<π›½π‘›βˆ’1<𝛽𝑛<1<𝛼<2, 𝑏1,𝑏2,…,𝑏𝑛 are constant coefficients, and ∫(𝑇𝑒)(𝑑)=10π‘˜(𝑑,𝑠)𝑒(𝑠)𝑑𝑠, ∫(𝑆𝑒)(𝑑)=𝑑0𝑙(𝑑,𝑠)𝑒(𝑠)𝑑𝑠, subject to the nonlocal conditions 𝑒(0)=0, βˆ‘π‘’(1)=π‘šπ‘–=1𝛾𝑖𝑒(πœ‚π‘–). 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)𝑒=𝑓(𝑑,𝑒),𝑒(0)=0,0<𝑑<1,(1.1) where 𝐿(𝐷)=π·π‘ π‘›βˆ’π‘Žπ‘›βˆ’1π·π‘ π‘›βˆ’1βˆ’β‹―βˆ’π‘Ž1𝐷𝑠1, 0<𝑠1<𝑠2<β‹―<𝑠𝑛<1, π‘Žπ‘—>0, 𝑗=1,2,…,π‘›βˆ’1, and 𝐷𝑠𝑗 is the standard Riemann-Liouville fractional derivative. Some existence results of positive solutions are obtained by using some fixed-point theorems on a cone.

StojanoviΔ‡ [11] considered the existenceuniqueness of solutions for a nonlinear 𝑛-term fractional differential equation𝑏0𝐷𝛽0𝑒(𝑑)+π‘šβˆ’1𝑖=1𝑏𝑖𝐷𝛽𝑖𝑒(𝑑)+π‘›βˆ’1𝑖=π‘šπ‘π‘–π·π›Όπ‘–π‘’(𝑑)+𝑏𝑛𝐷𝛼𝑛𝑒(𝑑)=𝑓(𝑑,𝑒(𝑑)),π‘‘βˆˆ(0,1),(1.2) where 0<𝛽1<𝛽2<β‹―<π›½π‘šβˆ’1<1<π›Όπ‘š<π›Όπ‘š+1<β‹―<𝛼𝑛<2, with initial data 𝑒(0)=𝑓(0), 𝑒𝑑(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:βˆ’π·π›Όπ‘¦(𝑑)∈𝐹(𝑑,𝑦(𝑑)),a.e[].π‘‘βˆˆπ½=0,1,1<𝛼≀2,𝑦(0)=𝑦(1)=0,(1.3) where 𝐷𝛼 is the standard Riemann-Liouville fractional derivative, πΉβˆΆπ½Γ—β„β†’π’«(ℝ) is a multivalued map with compact values.

Chang and Nieto [27] studied boundary value problem of fractional differential inclusions𝐢0𝐷𝛿[]𝑦(𝑑)∈𝐹(𝑑,𝑦(𝑑)),π‘‘βˆˆπ½βˆΆ=0,1,π›Ώβˆˆ(1,2),𝑦(0)=𝛼,𝑦(1)=𝛽,𝛼,𝛽≠0,(1.4) where 𝐢0𝐷𝛿𝑦(𝑑) is the Caputo's derivative, πΉβˆΆπ½Γ—β„β†’π’«(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)π‘’βˆˆπΉ(𝑑,𝑒,𝑇𝑒,𝑆𝑒),a.e[].π‘‘βˆˆ0,1(1.5) with multipoint boundary conditions𝑒(0)=0,𝑒(1)=π‘šξ“π‘–=1π›Ύπ‘–π‘’ξ€·πœ‚π‘–ξ€Έ,(1.6) where 𝐿(𝐷)=π·π›Όβˆ’π‘π‘›π·π›½π‘›βˆ’π‘π‘›βˆ’1π·π›½π‘›βˆ’1βˆ’β‹―βˆ’π‘1𝐷𝛽1, 0<𝛽1<𝛽2<β‹―<π›½π‘›βˆ’1<𝛽𝑛<1<𝛼<2, 𝐷𝛼,𝐷𝛽𝑗, 𝑗=1,2,…,𝑛, is the standard Riemann-Liouville fractional derivative, 0<πœ‚1<πœ‚2β‹―<πœ‚π‘š<1, βˆ‘0β‰€π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1<1, ∫(𝑇𝑒)(𝑑)=10π‘˜(𝑑,𝑠)𝑒(𝑠)d𝑠, ∫(𝑆𝑒)(𝑑)=𝑑0𝑙(𝑑,𝑠)𝑒(𝑠)d𝑠, π‘˜,𝑙 are two continuous functions on [0,1]Γ—[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 π‘“βˆΆβ„+→ℝ is given by 𝐼𝛼1𝑓(𝑑)=ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑓(𝑠)d𝑑,(2.1) provided the right side is pointwise defined on ℝ+.

Definition 2.2. The Riemann-Liouville fractional derivative of function π‘“βˆΆβ„+→ℝ is given by 𝐷𝛼𝑓(𝑑)=π·π‘›πΌπ‘›βˆ’π›Όπ‘“(𝑑),(2.2) where 𝑛=[𝛼]+1, 𝐷𝑛=d𝑛/d𝑑𝑛, provided the right side is pointwise defined on ℝ+.

Lemma 2.3. Let 𝛼>0, then 𝐼𝛼𝐷𝛼π‘₯(𝑑)=π‘₯(𝑑)+𝑐1π‘‘π›Όβˆ’1+𝑐2π‘‘π›Όβˆ’2+β‹―+π‘π‘›π‘‘π›Όβˆ’π‘›,(2.3) for some π‘π‘–βˆˆβ„, 𝑖=0,1,2,…,π‘›βˆ’1, 𝑛=βˆ’[βˆ’π›Ό].

Remark 2.4. If the fractional derivative 𝐷𝛼π‘₯(𝑑) is integrable, then 𝐼𝛼𝐷𝛽π‘₯(𝑑)=πΌπ›Όβˆ’π›½ξ€ΊπΌπ‘₯(𝑑)βˆ’1βˆ’π›½ξ€»π‘₯(𝑑)𝑑=0π‘‘π›Όβˆ’1Ξ“(𝛼),0<𝛽≀𝛼<1.(2.4) If π‘₯ is continuous on [0,1], then [𝐼1βˆ’π›½π‘₯(𝑑)]𝑑=0=0, and (2.4) reduces to 𝐼𝛼𝐷𝛽π‘₯(𝑑)=πΌπ›Όβˆ’π›½π‘₯(𝑑),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 π‘”βˆˆπΏ([0,1],ℝ), 0<𝛽1<𝛽2<β‹―<π›½π‘›βˆ’1<𝛽𝑛<1<𝛼<2, and βˆ‘0<π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1<1. Then 𝑒(𝑑) is a solution of the BVP ξ€·π·π›Όβˆ’π‘π‘›π·π›½π‘›βˆ’π‘π‘›βˆ’1π·π›½π‘›βˆ’1βˆ’β‹―βˆ’π‘1𝐷𝛽1𝑒(𝑑)=𝑔(𝑑),a.e[],.π‘‘βˆˆ0,1(2.6)𝑒(0)=0,𝑒(1)=π‘šξ“π‘–=1π›Ύπ‘–π‘’ξ€·πœ‚π‘–ξ€Έ.(2.7) if and only if 𝑒(𝑑) satisfies the integral equation 𝑒(𝑑)=π‘šξ“π‘›π‘–=1𝑗=1π›Ύπ‘–π‘π‘—ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘—ξ€Έξ€œπœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’π›½π‘—βˆ’1π‘‘π›Όβˆ’1𝑒(𝑠)dπ‘ βˆ’π‘›ξ“π‘–=1π‘π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œ10(1βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1π‘‘π›Όβˆ’1𝑒(𝑠)d𝑠+𝑛𝑖=1π‘π‘–Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1𝑒(𝑠)d1𝑠+ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑔(𝑠)d𝑠+π‘šξ“π‘–=1π›Ύπ‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1π‘‘π›Όβˆ’1𝑔(𝑠)dπ‘ βˆ’1ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€œ(𝛼)10(1βˆ’π‘ )π›Όβˆ’1π‘‘π›Όβˆ’1𝑔(𝑠)d𝑠.(2.8)

Proof. In view of Lemma 2.3 and Remark 2.4, (2.6) is equivalent to the integral equation 𝑒(𝑑)=𝑐1π‘‘π›Όβˆ’1+𝑐2π‘‘π›Όβˆ’2+𝑛𝑖=1π‘π‘–πΌπ›Όβˆ’π›½π‘–π‘’(𝑑)+𝐼𝛼𝑔(𝑑),(2.9) for some 𝑐1,𝑐2βˆˆβ„, that is, 𝑒(𝑑)=𝑐1π‘‘π›Όβˆ’1+𝑐2π‘‘π›Όβˆ’2+𝑛𝑖=1π‘π‘–Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1𝑒(𝑠)d1𝑠+ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑔(𝑠)d𝑠.(2.10) The boundary condition 𝑒(0)=0 implies 𝑐2=0. Thus, 𝑒(𝑑)=𝑐1π‘‘π›Όβˆ’1+𝑛𝑖=1π‘π‘–Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1𝑒(𝑠)d1𝑠+ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑔(𝑠)d𝑠.(2.11) In view of the boundary condition βˆ‘π‘’(1)=π‘šπ‘–=1𝛾𝑖𝑒(πœ‚π‘–), we conclude that 𝑐1=π‘šξ“π‘›π‘–=1𝑗=1π›Ύπ‘–π‘π‘—ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘—ξ€Έξ€œπœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’π›½π‘—βˆ’1𝑒(𝑠)dπ‘ βˆ’π‘›ξ“π‘–=1π‘π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œ10(1βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1𝑒(𝑠)d𝑠+π‘šξ“π‘–=1π›Ύπ‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€œ(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1𝑔(𝑠)dπ‘ βˆ’1ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)10(1βˆ’π‘ )π›Όβˆ’1𝑔(𝑠)d𝑠.(2.12) Therefore, the solution 𝑒(𝑑) of (2.6) and (2.7) satisfies (2.8).
Conversely, if 𝑒(𝑑) is a solution of (2.8), it is easy to verify that 𝑒(𝑑) 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)={π‘ŒβŠ‚π‘‹βˆΆπ‘Œβ‰ βˆ…}, 𝒫𝑏𝑑(𝑋)={π‘Œβˆˆπ’«(𝑋)βˆΆπ‘Œbounded}, 𝒫𝑐𝑙(𝑋)={π‘Œβˆˆπ’«(𝑋)βˆΆπ‘Œclosed}, 𝒫𝑐𝑣(𝑋)={π‘Œβˆˆπ’«(𝑋)βˆΆπ‘Œconvex}, 𝒫𝑐𝑝(𝑋)={π‘Œβˆˆπ’«(𝑋)βˆΆπ‘Œcompact}, 𝒫𝑐𝑣,𝑐𝑝(𝑋)=𝒫𝑐𝑣(𝑋)βˆ©π’«π‘π‘(𝑋), and so forth.

Consider π»π‘‘βˆΆπ’«(𝑋)×𝒫(𝑋)→ℝ+βˆͺ{∞}, given by𝐻𝑑(𝐴,𝐡)=maxsupπ‘₯βˆˆπ΄π‘‘(π‘₯,𝐡),supπ‘¦βˆˆπ΅ξƒ°,𝑑(𝐴,𝑦)(2.13) where 𝑑(π‘₯,𝐡)=infπ‘¦βˆˆπ΅π‘‘(π‘₯,𝑦), 𝑑(𝐴,𝑦)=infπ‘₯βˆˆπ΄π‘‘(π‘₯,𝑦). Then (𝒫𝑏𝑑,𝑐𝑙(𝑋),𝐻𝑑) is a metric space and (𝒫𝑐𝑙(𝑋),𝐻𝑑) is a generalized metric space (see [31]).

A multivalued map π‘βˆΆ[0,1]→𝒫𝑐𝑙(𝑋) is said to be measurable if for each π‘’βˆˆπ‘‹, the function π‘ŒβˆΆ[0,1]→ℝ, defined byπ‘Œ(𝑑)=𝑑(𝑒,𝑁(𝑑))=inf{𝑑(𝑒,𝑧)βˆΆπ‘§βˆˆπ‘(𝑑)},(2.14) is measurable.

A multivalued map πΉβˆΆπ‘‹β†’π’«(𝑋) is convex (closed) valued if 𝐹(𝑒) is convex (closed) for all π‘’βˆˆπ‘‹. 𝐹 is bounded on bounded sets if 𝐹(𝐡)=βˆͺπ‘’βˆˆπ΅πΉ(𝑒) is bounded in 𝑋 for all π΅βˆˆπ’«(𝑋). That is, supπ‘’βˆˆπ΅{sup{|𝑦|βˆΆπ‘¦βˆˆπΉ(𝑒)}}<∞. 𝐹 is called upper semicontinuous (u.s.c for short) on 𝑋 if for each 𝑒0βˆˆπ‘‹, the set 𝐹(𝑒0) is nonempty closed subset of 𝑋, and if for each open set 𝒰 of 𝑋 containing 𝐹(𝑒0), there exists an open neighborhood 𝒱 of 𝑒0 such that 𝐹(𝒱)βŠ‚π’°. 𝐺 is said to be completely continuous if 𝐹(𝐡) is relatively compact for every π΅βˆˆπ’«π‘π‘‘(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, π‘₯𝑛→π‘₯βˆ—, π‘¦π‘›β†’π‘¦βˆ—, π‘¦π‘›βˆˆπΊ(π‘₯βˆ—) imply π‘¦βˆ—βˆˆπΊ(π‘₯βˆ—).

More details on multivalued maps can be found in the books of Deimling [3], GΓ³rniewicz [32], Hu and Papageorgiou [33], and Tolstonogov [34].

Definition 2.6. The multivalued map 𝐹∢[0,1]×ℝ×ℝ×ℝ→𝒫(ℝ) is 𝐿1-CarathΓ©odory if (i)𝑑↦𝐹(𝑑,𝑒,𝑣,𝑀) is measurable for each (𝑒,𝑣,𝑀)βˆˆβ„Γ—β„Γ—β„; (ii)(𝑒,𝑣,𝑀)↦𝐹(𝑑,𝑒,𝑣,𝑀) is upper semicontinuous for almost all π‘‘βˆˆ[0,1]; (iii)for each π‘Ÿ>0, there exists πœ‘π‘ŸβˆˆπΏ1([0,1],ℝ+) such that for all |𝑒|,|𝑣|,|𝑀|β‰€π‘Ÿ and for almost all π‘‘βˆˆ[0,1],‖‖𝐹(𝑑,𝑒,𝑣,𝑀)𝒫||𝑓||ξ€Ύ=supβˆΆπ‘“βˆˆπΉ(𝑑,𝑒,𝑣,𝑀)β‰€πœ‘π‘Ÿ(𝑑).(2.15)

For any π‘’βˆˆπΆ([0,1],ℝ), we define the set𝑆𝐹,𝑒=ξ€½π‘“βˆˆπΏ1([]0,1,ℝ)βˆΆπ‘“(𝑑)∈𝐹(𝑑,𝑒(𝑑),(𝑇𝑒)(𝑑),(𝑆𝑒)(𝑑))fora.e[]ξ€Ύ,.π‘‘βˆˆ0,1(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→𝒫𝑐𝑝,𝑐𝑣(𝑋) be an 𝐿1-CarathΓ©odory multivalued map with 𝑆𝐹,π‘’β‰ βˆ…, and let Ξ“ be a linear continuous mapping from 𝐿1([0,1],𝑋) into 𝐢([0,1],𝑋). Then the operator Ξ“βˆ˜π‘†πΉ[]∢𝐢(0,1,𝑋)βŸΆπ’«π‘π‘,𝑐𝑣[]ξ€·(𝐢(0,1,𝑋)),π‘’βŸΌΞ“βˆ˜π‘†πΉξ€Έ(𝑆𝑦)∢=Γ𝐹,𝑒(2.17) is a closed graph operator in 𝐢([0,1],𝑋)×𝐢([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 πΊβˆΆπ·β†’π’«(𝑋) is u.s.c. with closed, convex values, and such that 𝐺(𝐷)βŠ‚π· 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 𝐹𝑖π‘₯π‘β‰ βˆ….
For convenience, let us list some conditions. (H1)βˆ‘π‘šπ‘–=1βˆ‘π‘›π‘—=1(π›Ύπ‘–π‘π‘—πœ‚π›Όβˆ’π›½π‘—π‘–βˆ‘/(1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1)Ξ“(π›Όβˆ’π›½π‘—+1))+βˆ‘π‘›π‘–=1(π‘π‘–βˆ‘/(1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1)Ξ“(π›Όβˆ’π›½π‘–βˆ‘+1))+𝑛𝑖=1(𝑏𝑖/Ξ“(π›Όβˆ’π›½π‘–+1))<1;(H2)𝐹∢[0,1]×ℝ×ℝ×ℝ→𝒫𝑐𝑝,𝑐𝑣(ℝ) is CarethΓ©odory multivalued map.

Lemma 2.10. Assume that hypothesis (H1) is satisfied. For any π‘”βˆˆπΆ([0,1],ℝ), the integral equation 𝑒(𝑑)=π‘šξ“π‘›π‘–=1𝑗=1π›Ύπ‘–π‘π‘—ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘—ξ€Έξ€œπœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’π›½π‘—βˆ’1π‘‘π›Όβˆ’1𝑒(𝑠)dπ‘ βˆ’π‘›ξ“π‘–=1π‘π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œ10(1βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1π‘‘π›Όβˆ’1𝑒(𝑠)d𝑠+𝑛𝑖=1π‘π‘–Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1𝑒(𝑠)d𝑠+𝑔(𝑑)(2.18) has a unique solution in 𝐢([0,1],ℝ).

Proof. We define the operator 𝐴 as follows: (𝐴𝑒)(𝑑)∢=π‘šξ“π‘›π‘–=1𝑗=1π›Ύπ‘–π‘π‘—ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘—ξ€Έξ€œπœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’π›½π‘—βˆ’1π‘‘π›Όβˆ’1𝑒(𝑠)dπ‘ βˆ’π‘›ξ“π‘–=1π‘π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œ10(1βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1π‘‘π›Όβˆ’1𝑒(𝑠)d𝑠+𝑛𝑖=1π‘π‘–Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1𝑒(𝑠)d𝑠+𝑔(𝑑)(2.19) Obviously, 𝐴 is a map from 𝐢([0,1],ℝ) into 𝐢([0,1],ℝ). Also, we have ||𝐴𝑒1ξ€Έξ€·(𝑑)βˆ’π΄π‘’2ξ€Έ||≀(𝑑)π‘šξ“π‘›π‘–=1𝑗=1π›Ύπ‘–π‘π‘—ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘—ξ€Έξ€œπœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’π›½π‘—βˆ’1π‘‘π›Όβˆ’1||𝑒1(𝑠)βˆ’π‘’2||(𝑠)d𝑠+𝑛𝑖=1π‘π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œ10(1βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1π‘‘π›Όβˆ’1||𝑒1(𝑠)βˆ’π‘’2||(𝑠)d𝑠+𝑛𝑖=1π‘π‘–Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1||𝑒1(𝑠)βˆ’π‘’2||(𝑠)dπ‘ β‰€βŽ›βŽœβŽœβŽπ‘šξ“π‘›π‘–=1𝑗=1π›Ύπ‘–π‘π‘—πœ‚π›Όβˆ’π›½π‘—π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘—ξ€Έ++1𝑛𝑖=1π‘π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘–ξ€Έ++1𝑛𝑖=1π‘π‘–Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξƒͺ‖‖𝑒+11βˆ’π‘’2β€–β€–.(2.20) Therefore, ‖𝐴𝑒1βˆ’π΄π‘’2β€–β‰€πœ†β€–π‘’1βˆ’π‘’2β€–, where βˆ‘πœ†=π‘šπ‘–=1βˆ‘π‘›π‘—=1(π›Ύπ‘–π‘π‘—πœ‚π›Όβˆ’π›½π‘—π‘–βˆ‘/(1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1)Ξ“(π›Όβˆ’π›½π‘—+1))+βˆ‘π‘›π‘–=1(π‘π‘–βˆ‘/(1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1)Ξ“(π›Όβˆ’π›½π‘–βˆ‘+1))+𝑛𝑖=1(𝑏𝑖/Ξ“(π›Όβˆ’π›½π‘–+1)). By (H1) and Banach contraction principle, the conclusion of lemma is true.

Definition 2.11. A function π‘’βˆˆπΆ[0,1] is said to be a solution of the BVP (1.5) and (1.6), if there exists a function π‘“βˆˆπΏ1([0,1],ℝ) such that 𝑓(𝑑)∈𝐹(𝑑,𝑒(𝑑),(𝑇𝑒)(𝑑),(𝑆𝑒)(𝑑)) a.e. on [0,1] and 𝑒(𝑑)=π‘šξ“π‘›π‘–=1𝑗=1π›Ύπ‘–π‘π‘—ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘—ξ€Έξ€œπœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’π›½π‘—βˆ’1π‘‘π›Όβˆ’1𝑒(𝑠)dπ‘ βˆ’π‘›ξ“π‘–=1π‘π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œ10(1βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1π‘‘π›Όβˆ’1𝑒(𝑠)d𝑠+𝑛𝑖=1π‘π‘–Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1𝑒(𝑠)d1𝑠+ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑓(𝑠)d𝑠+π‘šξ“π‘–=1π›Ύπ‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1π‘‘π›Όβˆ’1𝑓(𝑠)dπ‘ βˆ’1ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€œ(𝛼)10(1βˆ’π‘ )π›Όβˆ’1π‘‘π›Όβˆ’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 𝐢([0,1],ℝ), provided that (H3)βˆ‘πœ/Ξ“(𝛼)+π‘šπ‘–=1(π›Ύπ‘–πœ‡π‘–βˆ‘/(1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1βˆ‘)Ξ“(𝛼))+(πœ”/(1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1)Ξ“(𝛼))<1βˆ’πœ†,
where 𝜏=limπ‘Ÿβ†’βˆžξ€œinf10ξ€½πœ‘maxπ‘Ÿ(𝑠),πœ‘Μƒπ‘˜π‘Ÿ(𝑠),πœ‘Μƒπ‘™π‘Ÿξ€Ύ(𝑠)π‘Ÿdπœ‡π‘ ,𝑖=limπ‘Ÿβ†’βˆžξ€œinfπœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1ξ€½πœ‘maxπ‘Ÿ(𝑠),πœ‘Μƒπ‘˜π‘Ÿ(𝑠),πœ‘Μƒπ‘™π‘Ÿξ€Ύ(𝑠)π‘Ÿd𝑠,πœ”=limπ‘Ÿβ†’βˆžξ€œinf10(1βˆ’π‘ )π›Όβˆ’1ξ€½πœ‘maxπ‘Ÿ(𝑠),πœ‘Μƒπ‘˜π‘Ÿ(𝑠),πœ‘Μƒπ‘™π‘Ÿξ€Ύ(𝑠)π‘Ÿd̃𝑠,π‘˜=sup[]π‘‘βˆˆ0,1ξ€œ10π‘˜(𝑑,𝑠)d̃𝑠,𝑙=sup[]π‘‘βˆˆ0,1ξ€œπ‘‘0𝑙(𝑑,𝑠)d𝑠,(3.1) and πœ† is defined in the proof of Lemma 2.10.

Proof. To transform the problem into a fixed-point problem, we consider the multivalued operator, π‘βˆΆπΆ([0,1],ℝ)→𝒫(𝐢([0,1],ℝ)), where for any π‘’βˆˆπΆ([0,1],ℝ),𝑁(𝑒) is defined by ξƒ―[]β„ŽβˆˆπΆ(0,1,ℝ)βˆΆβ„Ž(𝑑)=π‘šξ“π‘›π‘–=1𝑗=1π›Ύπ‘–π‘π‘—ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘—ξ€Έξ€œπœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’π›½π‘—βˆ’1π‘‘π›Όβˆ’1β„Ž(𝑠)dπ‘ βˆ’π‘›ξ“π‘–=1π‘π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œ10(1βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1π‘‘π›Όβˆ’1β„Ž(𝑠)d𝑠+𝑛𝑖=1π‘π‘–Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1β„Ž(𝑠)d1𝑠+ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑓(𝑠)d𝑠+π‘šξ“π‘–=1π›Ύπ‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1π‘‘π›Όβˆ’1𝑓(𝑠)dπ‘ βˆ’1ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)10(1βˆ’π‘ )π›Όβˆ’1π‘‘π›Όβˆ’1𝑓(𝑠)d𝑠,π‘“βˆˆπ‘†πΉ,𝑒.(3.2)
For any π‘“βˆˆπΏ1([0,1],ℝ), we have βˆ«π‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑓(𝑠)d𝑠, βˆ«πœ‚π‘–0(πœ‚π‘–βˆ’π‘ )π›Όβˆ’1π‘‘π›Όβˆ’1𝑓(𝑠)d𝑠, ∫10(1βˆ’π‘ )π›Όβˆ’1π‘‘π›Όβˆ’1𝑓(𝑠)dπ‘ βˆˆπΆ([0,1],ℝ). In view of Lemma 2.10, 𝑁 is well defined. Moreover, it follows from the convexity of 𝑆𝐹,𝑒 (because 𝐹 has convex values) that 𝑁(𝑒) is convex for each π‘’βˆˆπΆ([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 π‘Ÿβˆ—>0, such that 𝑁(π΅π‘Ÿβˆ—)βŠ‚π΅π‘Ÿβˆ—, where π΅π‘Ÿβˆ—={π‘’βˆˆπΆ([0,1],ℝ)βˆΆβ€–π‘’β€–β‰€π‘Ÿβˆ—}.
In fact, if it is not true, then for any π‘Ÿ>0, there exists a function π‘’π‘Ÿβˆˆπ΅π‘Ÿ, β„Žπ‘Ÿβˆˆπ‘(π‘’π‘Ÿ) but β„Žπ‘Ÿβˆ‰π΅π‘Ÿ, that is β€–β„Žπ‘Ÿβ€–>π‘Ÿ and for some π‘“π‘Ÿβˆˆπ‘†πΉ,𝑒, β„Žπ‘Ÿ(𝑑)=π‘šξ“π‘›π‘–=1𝑗=1π›Ύπ‘–π‘π‘—ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘—ξ€Έξ€œπœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’π›½π‘—βˆ’1π‘‘π›Όβˆ’1β„Žπ‘Ÿ(𝑠)dπ‘ βˆ’π‘›ξ“π‘–=1π‘π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œ10(1βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1π‘‘π›Όβˆ’1β„Žπ‘Ÿ(𝑠)d𝑠+𝑛𝑖=1π‘π‘–Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1β„Žπ‘Ÿ(s)d1𝑠+ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1π‘“π‘Ÿ(𝑠)d𝑠+π‘šξ“π‘–=1π›Ύπ‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1π‘‘π›Όβˆ’1π‘“π‘Ÿ(𝑠)dπ‘ βˆ’1ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€œ(𝛼)10(1βˆ’π‘ )π›Όβˆ’1π‘‘π›Όβˆ’1π‘“π‘Ÿ(𝑠)d𝑠.(3.3)
On the other hand, from (H2), we obtain ||β„Žπ‘Ÿ(||β€–β€–β„Žπ‘‘)β‰€πœ†π‘Ÿβ€–β€–+1ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1π‘“π‘Ÿ(𝑠)d𝑠+1ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)10(1βˆ’π‘ )π›Όβˆ’1π‘‘π›Όβˆ’1π‘“π‘Ÿ(𝑠)d𝑠+π‘šξ“π‘–=1π›Ύπ‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1π‘‘π›Όβˆ’1π‘“π‘Ÿ(𝑠)dπ‘ β€–β€–β„Žβ‰€πœ†π‘Ÿβ€–β€–+1ξ€œΞ“(𝛼)10(|π‘‘βˆ’π‘ |)π›Όβˆ’1πœ‘max{π‘Ÿ,Μƒπ‘˜π‘Ÿ,Μƒπ‘™π‘Ÿ}(𝑠)d𝑠+π‘šξ“π‘–=1π›Ύπ‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1π‘‘π›Όβˆ’1πœ‘max{π‘Ÿ,Μƒπ‘˜π‘Ÿ,Μƒπ‘™π‘Ÿ}(𝑠)d𝑠+1ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)10(1βˆ’π‘ )π›Όβˆ’1π‘‘π›Όβˆ’1πœ‘max{π‘Ÿ,Μƒπ‘˜π‘Ÿ,Μƒπ‘™π‘Ÿ}(𝑠)d𝑠.(3.4) Thus, β€–β€–β„Žπ‘Ÿ<π‘Ÿβ€–β€–β‰€1ξ€œ(1βˆ’πœ†)Ξ“(𝛼)10(|π‘‘βˆ’π‘ |)π›Όβˆ’1πœ‘max{π‘Ÿ,Μƒπ‘˜π‘Ÿ,Μƒπ‘™π‘Ÿ}(𝑠)d𝑠+π‘šξ“π‘–=1π›Ύπ‘–ξ€·βˆ‘(1βˆ’πœ†)1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1πœ‘max{π‘Ÿ,Μƒπ‘˜π‘Ÿ,Μƒπ‘™π‘Ÿ}(𝑠)d𝑠+1ξ€·βˆ‘(1βˆ’πœ†)1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)10(1βˆ’π‘ )π›Όβˆ’1πœ‘max{π‘Ÿ,Μƒπ‘˜π‘Ÿ,Μƒπ‘™π‘Ÿ}(𝑠)d𝑠.(3.5)
Dividing both sides by π‘Ÿ and taking the lower limit as π‘Ÿβ†’+∞, we get 𝜏1≀+(1βˆ’πœ†)Ξ“(𝛼)π‘šξ“π‘–=1π›Ύπ‘–πœ‡π‘–ξ€·βˆ‘(1βˆ’πœ†)1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έ+πœ”Ξ“(𝛼)ξ€·βˆ‘(1βˆ’πœ†)1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έ.Ξ“(𝛼)(3.6)
This is a contraction to (H3). Hence there exists a π‘Ÿβˆ—>0 such that 𝑁(π΅π‘Ÿβˆ—)βŠ‚π΅π‘Ÿβˆ—.
Step 2. 𝑁(π΅π‘Ÿβˆ—) is equicontinuous.
Let 𝑑1,𝑑2∈[0,1], 𝑑1<𝑑2, π‘’βˆˆπ΅π‘Ÿβˆ—, and β„Žβˆˆπ‘(𝑒). Then there exists π‘“βˆˆπ‘†πΉ,𝑒 such that for each π‘‘βˆˆ[0,1], we have β„Ž(𝑑)=π‘šξ“π‘›π‘–=1𝑗=1π›Ύπ‘–π‘π‘—ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘—ξ€Έξ€œπœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’π›½π‘—βˆ’1π‘‘π›Όβˆ’1β„Ž(𝑠)dπ‘ βˆ’π‘›ξ“π‘–=1π‘π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œ10(1βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1π‘‘π›Όβˆ’1β„Ž(𝑠)d𝑠+𝑛𝑖=1π‘π‘–Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1β„Ž(𝑠)d1𝑠+ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑓(𝑠)d𝑠+π‘šξ“π‘–=1π›Ύπ‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1π‘‘π›Όβˆ’1𝑓(𝑠)dπ‘ βˆ’1ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€œ(𝛼)10(1βˆ’π‘ )π›Όβˆ’1π‘‘π›Όβˆ’1𝑓(𝑠)d𝑠.(3.7) Therefore, ||β„Žξ€·π‘‘1ξ€Έξ€·π‘‘βˆ’β„Ž2ξ€Έ||ξ€·π‘‘β‰€π‘Ž2π›Όβˆ’1βˆ’π‘‘1π›Όβˆ’1ξ€Έ+1ξ€œΞ“(𝛼)𝑑2𝑑1πœ‘max{π‘Ÿβˆ—,Μƒπ‘˜π‘Ÿβˆ—,Μƒπ‘™π‘Ÿβˆ—}(𝑠)d𝑠+1Ξ“ξ€œ(𝛼)𝑑10𝑑2ξ€Έβˆ’π‘ π›Όβˆ’1βˆ’ξ€·π‘‘1ξ€Έβˆ’π‘ π›Όβˆ’1ξ‚πœ‘max{π‘Ÿβˆ—,Μƒπ‘˜π‘Ÿβˆ—,Μƒπ‘™π‘Ÿβˆ—}(𝑠)d𝑠+𝑛𝑖=1π‘π‘–π‘Ÿβˆ—Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ‚€π‘‘+1π›Όβˆ’π›½π‘–2βˆ’π‘‘π›Όβˆ’π›½π‘–1𝑑+22βˆ’π‘‘1ξ€Έπ›Όβˆ’π›½π‘–ξ‚,(3.8) where π‘Ž=π‘šξ“π‘›π‘–=1𝑗=1π›Ύπ‘–π‘π‘—πœ‚π›Όβˆ’π›½π‘—π‘–π‘Ÿβˆ—ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘—ξ€Έ++1𝑛𝑖=1π‘π‘–π‘Ÿβˆ—ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘–ξ€Έ++1π‘šξ“π‘–=1π›Ύπ‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1πœ‘max{π‘Ÿβˆ—,Μƒπ‘˜π‘Ÿβˆ—,Μƒπ‘™π‘Ÿβˆ—}(𝑠)d𝑠+1ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)10(1βˆ’π‘ )π›Όβˆ’1πœ‘max{π‘Ÿβˆ—,Μƒπ‘˜π‘Ÿβˆ—,Μƒπ‘™π‘Ÿβˆ—}(𝑠)d𝑠.(3.9)
The right-hand side of the above inequality tends to zeros independently of β„Žβˆˆπ΅π‘Ÿβˆ—, as 𝑑2βˆ’π‘‘1β†’0. This shows that 𝑁(π΅π‘Ÿβˆ—) is equicontinuous. Step 3. 𝑁 has a closed graph.
Let π‘’π‘›β†’π‘’βˆ—, β„Žπ‘›βˆˆπ‘(𝑒𝑛) and β„Žπ‘›β†’β„Žβˆ—. We will prove that β„Žβˆ—βˆˆπ‘(π‘’βˆ—). Now β„Žπ‘›βˆˆπ‘(𝑒𝑛) implies that there exists π‘“π‘›βˆˆπ‘†πΉ,𝑒𝑛 such that for each π‘‘βˆˆ[0,1], β„Žπ‘›(𝑑)=π‘šξ“π‘›π‘–=1𝑗=1π›Ύπ‘–π‘π‘—ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘—ξ€Έξ€œπœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’π›½π‘—βˆ’1π‘‘π›Όβˆ’1β„Žπ‘›(𝑠)dπ‘ βˆ’π‘›ξ“π‘–=1π‘π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œ10(1βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1π‘‘π›Όβˆ’1β„Žπ‘›(𝑠)d𝑠+𝑛𝑖=1π‘π‘–Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1β„Žπ‘›(𝑠)d1𝑠+ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑓𝑛(𝑠)d𝑠+π‘šξ“π‘–=1π›Ύπ‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1π‘‘π›Όβˆ’1𝑓𝑛(𝑠)dπ‘ βˆ’1ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€œ(𝛼)10(1βˆ’π‘ )π›Όβˆ’1π‘‘π›Όβˆ’1𝑓𝑛(𝑠)d𝑠.(3.10)
We need to show that there exists π‘“βˆ—βˆˆπ‘†πΉ,𝑒 such that for each π‘‘βˆˆ[0,1], β„Žβˆ—(𝑑)=π‘šξ“π‘›π‘–=1𝑗=1π›Ύπ‘–π‘π‘—ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘—ξ€Έξ€œπœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’π›½π‘—βˆ’1π‘‘π›Όβˆ’1β„Žβˆ—(𝑠)dπ‘ βˆ’π‘›ξ“π‘–=1π‘π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œ10(1βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1π‘‘π›Όβˆ’1β„Žβˆ—(𝑠)d𝑠+𝑛𝑖=1π‘π‘–Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1β„Žβˆ—(𝑠)d1𝑠+ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1π‘“βˆ—(𝑠)d𝑠+π‘šξ“π‘–=1π›Ύπ‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1π‘‘π›Όβˆ’1π‘“βˆ—(𝑠)dπ‘ βˆ’1ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€œ(𝛼)10(1βˆ’π‘ )π›Όβˆ’1π‘‘π›Όβˆ’1π‘“βˆ—(𝑠)d𝑠.(3.11) In (2.18), taking 1𝑔(𝑑)=ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑓(𝑠)d𝑠+π‘šξ“π‘–=1π›Ύπ‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1π‘‘π›Όβˆ’1𝑓(𝑠)dπ‘ βˆ’1ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)10(1βˆ’π‘ )π›Όβˆ’1π‘‘π›Όβˆ’1𝑓(𝑠)d𝑠.(3.12) In view of Lemma 2.10, we know that for any π‘“βˆˆπΏ([0,1],ℝ), the integral equation (2.18) has a unique solution, which we denote by β„Žπ‘“. Because of this, we can define the operator [][]Φ∢𝐿(0,1,ℝ)⟢𝐢(0,1,ℝ),π‘“βŸΌβ„Žπ‘“.(3.13) It is easy to verify that the operator Ξ¦ is linear. On the other hand, we can get β€–β€–β„Žπ‘“β€–β€–β‰€1ξ€œ(1βˆ’πœ†)Ξ“(𝛼)10(|π‘‘βˆ’π‘ |)π›Όβˆ’1||||𝑓(𝑠)d𝑠+1ξ€·βˆ‘(1βˆ’πœ†)1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)10(1βˆ’π‘ )π›Όβˆ’1||𝑓||(𝑠)d𝑠+π‘šξ“π‘–=1π›Ύπ‘–ξ€·βˆ‘(1βˆ’πœ†)1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1||||𝑓(𝑠)dπ‘ ξ€œβ‰€πœ‡10||||𝑓(𝑠)d𝑠,(3.14) where 1πœ‡=+(1βˆ’πœ†)Ξ“(𝛼)π‘šξ“π‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€·βˆ‘(1βˆ’πœ†)1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έ+1Ξ“(𝛼)ξ€·βˆ‘(1βˆ’πœ†)1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έ,Ξ“(𝛼)(3.15) that is, β€–Ξ¦(𝑓)β€–β‰€πœ‡β€–π‘“β€–πΏ1, which implies Ξ¦ is continuous. From Lemma 2.7, it follows that Ξ¦βˆ˜π‘†πΉ,𝑒 is a closed graph operator. Moreover, we know β„Žπ‘›βˆˆΞ¦(𝑆𝐹,𝑒𝑛). Since π‘’π‘›β†’π‘’βˆ—, it follows from Lemma 2.7 that (3.11) holds for some π‘“βˆ—βˆˆπ‘†πΉ,π‘’βˆ—.
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 𝑒 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 𝐢([0,1],ℝ), provided that (H4)there exist functions 𝑝(𝑑),π‘ž(𝑑),π‘Ÿ(𝑑),πœƒ(𝑑)∈𝐿([0,1],ℝ), and 𝜎1,𝜎2,𝜎3∈[0,1) such that‖𝐹(𝑑,𝑒,𝑣,𝑀)‖𝒫≀𝑝(𝑑)|𝑒|𝜎1+π‘ž(𝑑)|𝑣|𝜎2+π‘Ÿ(𝑑)|𝑒|𝜎3+πœƒ(𝑑)(3.16) for each (𝑑,𝑒,𝑣,𝑀)∈[0,1]×ℝ×ℝ×ℝ.

Proof. We only need to verify that (H4) implies (H3) in Theorem 3.1. Taking πœ‘(𝑑)=𝑝(𝑑)|𝑒|𝜎1+π‘ž(𝑑)|𝑣|𝜎2+π‘Ÿ(𝑑)|𝑒|𝜎3+πœƒ(𝑑), it is easy to prove 𝜏=limπ‘Ÿβ†’βˆžξ€œinf10ξ€½πœ‘maxπ‘Ÿ(𝑠),πœ‘Μƒπ‘˜π‘Ÿ(𝑠),πœ‘Μƒπ‘™π‘Ÿξ€Ύ(𝑠)π‘Ÿdπœ‡π‘ =0,𝑖=limπ‘Ÿβ†’βˆžξ€œinfπœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1ξ€½πœ‘maxπ‘Ÿ(𝑠),πœ‘Μƒπ‘˜π‘Ÿ(𝑠),πœ‘Μƒπ‘™π‘Ÿξ€Ύ(𝑠)π‘Ÿd𝑠=0,πœ”=limπ‘Ÿβ†’βˆžξ€œinf10(1βˆ’π‘ )π›Όβˆ’1ξ€½πœ‘maxπ‘Ÿ(𝑠),πœ‘Μƒπ‘˜π‘Ÿ(𝑠),πœ‘Μƒπ‘™π‘Ÿξ€Ύ(𝑠)π‘Ÿd𝑠=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→𝑃𝑐𝑙(ℝ); 𝑑→𝐹(𝑑,𝑒,𝑣,𝑀) is measurable for each 𝑒,𝑣,π‘€βˆˆβ„;(A2)There exist three functions 𝑝,π‘ž,π‘ŸβˆˆπΏ1([0,1],ℝ) such that for a.e. π‘‘βˆˆ[0,1] and 𝑒1,𝑒2,𝑣1,𝑣2,𝑀1,𝑀2βˆˆβ„,𝐻𝑑𝐹𝑑,𝑒1,𝑣1,𝑀1ξ€Έξ€·,𝐹𝑑,𝑒2,𝑣2,𝑀2||𝑒≀𝑝(𝑑)1βˆ’π‘’2||||𝑣+π‘ž(𝑑)1βˆ’π‘£2||||𝑀+π‘Ÿ(𝑑)1βˆ’π‘€2||,𝑑(0,𝐹(𝑑,0,0,0))β‰€π‘š(𝑑).(3.18) Then the BVP (1.5) and (1.6) has at least one solution in 𝐢([0,1],ℝ) provided that Μƒπ‘Ž0+̃𝑏0+̃𝑐0+π‘š+1𝑖=1Μƒπ‘Žπ‘–+Μƒπ‘˜Μƒπ‘π‘–+Μƒπ‘™Μƒπ‘π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έ<1βˆ’πœ†,(3.19) where Μƒπ‘Ž0=1ξ€œΞ“(𝛼)10𝑝(𝑠)d𝑠,Μƒπ‘Žπ‘–=1ξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1𝑝(𝑠)d̃𝑏𝑠,𝑖=1,2,…,π‘š+1,0=1ξ€œΞ“(𝛼)10π‘ž(𝑠)d̃𝑏𝑠,𝑖=1ξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1π‘ž(𝑠)d𝑠,𝑖=1,2,…,π‘š+1,̃𝑐0=1ξ€œΞ“(𝛼)10𝑝(𝑠)d𝑠,̃𝑐𝑖=1ξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1π‘Ÿ(𝑠)d𝑠,𝑖=1,2,…,π‘š+1,(3.20) β€‰πœ‚π‘š+1=1 and πœ† defined as in Lemma 2.10, ΜƒΜƒπ‘™π‘˜, defined by (3.1) in Theorem 3.1.

Proof. Transform the problem into a fixed-point problem. Let π‘βˆΆπΆ([0,1],ℝ)→𝒫(𝐢([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(𝑒)βˆˆπ’«π‘π‘™(𝐢([0,1],ℝ)) for each π‘’βˆˆπΆ([0,1],ℝ).
Indeed, let (𝑒𝑛)𝑛β‰₯0βˆˆπ‘(𝑒) such that 𝑒𝑛→̃𝑒 in 𝐢([0,1],ℝ). Then there exists π‘”π‘›βˆˆπ‘†πΉ,𝑒 such that for each π‘‘βˆˆ[0,1], 𝑒𝑛(𝑑)=π‘šξ“π‘›π‘–=1𝑗=1π›Ύπ‘–π‘π‘—ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘—ξ€Έξ€œπœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’π›½π‘—βˆ’1π‘‘π›Όβˆ’1𝑒𝑛(𝑠)dπ‘ βˆ’π‘›ξ“π‘–=1π‘π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œ10(1βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1π‘‘π›Όβˆ’1𝑒𝑛(𝑠)d𝑠+𝑛𝑖=1π‘π‘–Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1𝑒𝑛(𝑠)d1𝑠+ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑔𝑛(𝑠)d𝑠+π‘šξ“π‘–=1π›Ύπ‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1π‘‘π›Όβˆ’1𝑔𝑛(𝑠)dπ‘ βˆ’1ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€œ(𝛼)10(1βˆ’π‘ )π›Όβˆ’1π‘‘π›Όβˆ’1𝑔𝑛(𝑠)d𝑠.(3.21)
For every π‘€βˆˆπΉ(𝑑,𝑒(𝑑),(𝑇𝑒)(𝑑),(𝑆𝑒)(𝑑)), from (A2), we have |𝑀|≀𝑑(0,𝐹(𝑑,0,0,0))+𝐻𝑑(𝐹(𝑑,0,0,0),𝐹(𝑑,𝑒(𝑑),(𝑇𝑒)(𝑑),(𝑆𝑒)(𝑑))).(3.22) Then, ||𝑔𝑛||||||||||||||(𝑑)β‰€π‘š(𝑑)+𝑝(𝑑)𝑒(𝑑)+π‘ž(𝑑)(𝑇𝑒)(𝑑)+π‘Ÿ(𝑑)(𝑆𝑒)(𝑑)a.e[],.π‘‘βˆˆ0,1(3.23) that is, 𝑔𝑛(𝑑)∈𝐡(0,π‘š(𝑑)+𝑝(𝑑)|𝑒(𝑑)|+π‘ž(𝑑)|(𝑇𝑒)(𝑑)|+π‘Ÿ(𝑑)|(𝑆𝑒)(𝑑)|), where 𝐡||𝑒||||||||||ξ€Έ=ξ€½||||||(||||(||ξ€Ύ0,π‘š(𝑑)+𝑝(𝑑)(𝑑)+π‘ž(𝑑)(𝑇𝑒)(𝑑)+π‘Ÿ(𝑑)(𝑆𝑒)(𝑑)π‘€βˆˆβ„βˆΆ|𝑀|β‰€π‘š(𝑑)+𝑝(𝑑)𝑒(𝑑)+π‘ž(𝑑)𝑇𝑒)(𝑑)+π‘Ÿ(𝑑)𝑆𝑒)(𝑑)∢=Ξ¨(𝑑).(3.24)
It is clear that Ψ∢[0,1]→𝒫𝑐𝑝,𝑐𝑣(ℝ) is a multivalued integrable bounded map. Since 𝑔𝑛(β‹…)∈Ψ(β‹…),𝑛β‰₯1, we may pass to a subsequence if necessary to get that 𝑔𝑛 converges weakly to 𝑔 in 𝐿1𝑀([0,1],ℝ). From Mauz's lemma [37], there exists π‘”βˆˆconv{𝑔𝑛(𝑑)βˆΆπ‘›β‰₯1}, then there exists a subsequence {π‘”π‘›βˆΆπ‘›β‰₯1} in conv{π‘”π‘›βˆΆπ‘›β‰₯1}, such that 𝑔𝑛 converges strongly to 𝑔 in 𝐿1([0,1],ℝ), which implies π‘”βˆˆπΏ1([0,1],ℝ). Then for each π‘‘βˆˆ[0,1], 𝑒𝑛(𝑑)βŸΆΜƒπ‘’(𝑑)=π‘šξ“π‘›π‘–=1𝑗=1π›Ύπ‘–π‘π‘—ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘—ξ€Έξ€œπœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’π›½π‘—βˆ’1π‘‘π›Όβˆ’1̃𝑒(𝑠)dπ‘ βˆ’π‘›ξ“π‘–=1π‘π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œ10(1βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1π‘‘π›Όβˆ’1̃𝑒(𝑠)d𝑠+𝑛𝑖=1π‘π‘–Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1̃𝑒(𝑠)d1𝑠+ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑔(𝑠)d𝑠+π‘šξ“π‘–=1π›Ύπ‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1π‘‘π›Όβˆ’1𝑔(𝑠)dπ‘ βˆ’1ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€œ(𝛼)10(1βˆ’π‘ )π›Όβˆ’1π‘‘π›Όβˆ’1𝑔(𝑠)d𝑠.(3.25) So, Μƒπ‘’βˆˆπ‘(𝑒)βˆˆπ’«π‘π‘™(𝐢([0,1],ℝ)). Step 2. There exists 𝛾<1, such that 𝐻𝑑𝑁𝑒(𝑒),π‘βˆ—β€–β€–ξ€Έξ€Έβ‰€π›Ύπ‘’βˆ’π‘’βˆ—β€–β€–βˆžforeach𝑒,π‘’βˆ—([]∈𝐢0,1,ℝ).(3.26)
Let 𝑒,π‘’βˆ—βˆˆπΆ([0,1],ℝ) and β„Žβˆˆπ‘(𝑒). Then there exists 𝑓(𝑑)∈𝐹(𝑑,𝑒(t),(𝑇𝑒)(𝑑),(𝑆𝑒)(𝑑)) such that for each π‘‘βˆˆ[0,1]β„Ž(𝑑)=π‘šξ“π‘›π‘–=1𝑗=1π›Ύπ‘–π‘π‘—ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘—ξ€Έξ€œπœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’π›½π‘—βˆ’1π‘‘π›Όβˆ’1β„Ž(𝑠)dπ‘ βˆ’π‘›ξ“π‘–=1π‘π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œ10(1βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1π‘‘π›Όβˆ’1β„Ž(𝑠)d𝑠+𝑛𝑖=1π‘π‘–Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1β„Ž(𝑠)d1𝑠+ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑓(𝑠)d𝑠+π‘šξ“π‘–=1π›Ύπ‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€Έξ€œΞ“(𝛼)πœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’1π‘‘π›Όβˆ’1𝑓(𝑠)dπ‘ βˆ’1ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€œ(𝛼)10(1βˆ’π‘ )π›Όβˆ’1π‘‘π›Όβˆ’1𝑓(𝑠)d𝑠.(3.27) It follows from (A2) that 𝐻𝑑𝐹(𝑑,𝑒(𝑑),(𝑇𝑒)(𝑑),(𝑆𝑒)(𝑑)),𝐹𝑑,π‘’βˆ—ξ€·(𝑑),π‘‡π‘’βˆ—ξ€Έξ€·(𝑑),π‘†π‘’βˆ—ξ€Έ||(𝑑)≀𝑝(𝑑)𝑒(𝑑)βˆ’π‘’βˆ—(||||(𝑑)+π‘ž(𝑑)𝑇𝑒)(𝑑)βˆ’π‘‡π‘’βˆ—ξ€Έ(||||(𝑑)+π‘Ÿ(𝑑)𝑆𝑒)(𝑑)βˆ’π‘†π‘’βˆ—ξ€Έ(||[].𝑑),π‘‘βˆˆ0,1(3.28) Hence, there is π‘”βˆˆπΉ(𝑑,π‘’βˆ—(𝑑),(π‘‡π‘’βˆ—)(𝑑),(π‘†π‘’βˆ—)(𝑑)) such that ||𝑓||||𝑒(𝑑)βˆ’π‘”β‰€π‘(𝑑)(𝑑)βˆ’π‘’βˆ—||||ξ€·(𝑑)+π‘ž(𝑑)(𝑇𝑒)(𝑑)βˆ’π‘‡π‘’βˆ—ξ€Έ||||ξ€·(𝑑)+π‘Ÿ(𝑑)(𝑆𝑒)(𝑑)βˆ’π‘†π‘’βˆ—ξ€Έ||[].(𝑑),π‘‘βˆˆ0,1(3.29)
Consider 𝐺∢[0,1]→𝒫(ℝ), given by =ξ€½||||𝐺(𝑑)π‘”βˆˆβ„βˆΆπ‘“(𝑑)βˆ’π‘”β‰€π‘(t)||𝑒(𝑑)βˆ’π‘’βˆ—||||ξ€·(𝑑)+π‘ž(𝑑)(𝑇𝑒)(𝑑)βˆ’π‘‡π‘’βˆ—ξ€Έ||||ξ€·(𝑑)+π‘Ÿ(𝑑)(𝑆𝑒)(𝑑)βˆ’π‘†π‘’βˆ—ξ€Έ||ξ€Ύ.(𝑑)(3.30)
Since the multivalued operator ξ‚πΊβˆΆπ‘‘β†’πΊ(𝑑)∩𝐹(𝑑,π‘’βˆ—(𝑑),(π‘‡π‘’βˆ—)(𝑑),(π‘†π‘’βˆ—)(𝑑)) is measurable (see in [38, Proposition  III.4]), there exists a function π‘”βˆ—, which is a measurable selection for 𝐺. So, π‘”βˆ—(𝑑)∈𝐹(𝑑,π‘’βˆ—(𝑑),(π‘‡π‘’βˆ—)(𝑑),(π‘†π‘’βˆ—)(𝑑)) and ||𝑓(𝑑)βˆ’π‘”βˆ—||||(𝑑)≀𝑝(𝑑)𝑒(𝑑)βˆ’π‘’βˆ—||||ξ€·(𝑑)+π‘ž(𝑑)(𝑇𝑒)(𝑑)βˆ’π‘‡π‘’βˆ—ξ€Έ||||ξ€·(𝑑)+π‘Ÿ(𝑑)(𝑆𝑒)(𝑑)βˆ’π‘†π‘’βˆ—ξ€Έ||[].(𝑑),π‘‘βˆˆ0,1(3.31)
For each π‘‘βˆˆ[0,1], define β„Žβˆ—(𝑑)=π‘šξ“i𝑛=1𝑗=1π›Ύπ‘–π‘π‘—ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘—ξ€Έξ€œπœ‚π‘–0ξ€·πœ‚π‘–ξ€Έβˆ’π‘ π›Όβˆ’π›½π‘—βˆ’1π‘‘π›Όβˆ’1β„Žβˆ—(𝑠)dπ‘ βˆ’π‘›ξ“π‘–=1π‘π‘–ξ€·βˆ‘1βˆ’π‘šπ‘–=1π›Ύπ‘–πœ‚π‘–π›Όβˆ’1ξ€ΈΞ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œ10(1βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1π‘‘π›Όβˆ’1β„Žβˆ—(𝑠)d𝑠+𝑛𝑖=1π‘π‘–Ξ“ξ€·π›Όβˆ’π›½π‘–ξ€Έξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’π›½π‘–βˆ’1β„Žβˆ—(𝑠)d1𝑠+ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1π‘”βˆ—(π