problem in the frame of the generalized Caputo fractional derivatives

R E S E A R C H Open Access

A generalized neutral-type inclusion

problem in the frame of the generalized Caputo fractional derivatives

Adel Lachouri¹, Mohammed S. Abdo², Abdelouaheb Ardjouni³, Sina Etemad⁴and Shahram Rezapour^4,5*

*Correspondence:

sh.rezapour@azaruniv.ac.ir;

sh.rezapour@mail.cmuh.org.tw;

rezapourshahram@yahoo.ca

4Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran

5Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Full list of author information is available at the end of the article

Abstract

In this paper, we study the existence of solutions for a generalized sequential Caputo-type fractional neutral differential inclusion with generalized integral conditions. The used fractional operator has the generalized kernel in the format of (ϑ(t) –ϑ(s)) along with differential operator_ϑ¹_{(t) dt}^d. We obtain existence results for two cases of convex-valued and nonconvex-valued multifunctions in two separated sections. We derive our findings by means of the fixed point principles in the context of the set-valued analysis. We give two suitable examples to validate the theoretical results.

MSC: Primary 34A08; secondary 34A12; 34B15

Keywords: ϑ-Caputo fractional derivatives; Fixed point; Fractional differential inclusion; Generalized Caputo derivative

1 Introduction

Fractional differential inclusions as a generalization of fractional differential equations are established to be of considerable interest and value in optimizations and stochastic pro- cesses [1]. Fractional differential inclusions additionally help us study dynamical systems in which speeds are not remarkably specific by the condition of the system, regardless of relying upon it. In recent periods the theory of fractional differential equations has gained a lot of interest in all areas of mathematics; see [2–4]. Also, fractional differential equa- tions and fractional differential inclusions appear naturally in a variety of scientific fields and have a wide range of applications; see [5–8]. Almeida [9] introduced a new opera- tor called theψ-Caputo fractional derivative combining a fractional operator with other different types of fractional derivatives and thus opened a new window to modern and complicated applications.

Throughout the years, many researchers have been interested in discussing the existence of solutions for fractional differential equations and fractional differential inclusions in- volving various types of fractional derivatives; see [10–34].

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In 2018, Asawasamrit et al. [35] studied the following class of fractional differential equa- tions involving the Hilfer fractional derivatives:

⎧⎪

⎨

⎪⎩

HD^b1,b2ϕ(t) =h(t,ϕ(t)), t∈[a,b], ϕ(a) = 0, ϕ(b) =

m i=1

θ_iI^λa+ⁱϕ(σi), σ_i∈[a,b], (1)

whereb1∈(1, 2),b2∈[0, 1],θ_i∈R,λ_i> 0,^HD^b1,b2is theb1-Hilfer fractional derivative of typeb₂,I_a+^λi is theλ_i-Riemann–Liouville fractional integral, andh∈C([a,b]×R,R).

Mali and Kucche [36] discussed the existence and stability ofψ-Hilfer-type implicit BVP for given fractional differential equations (1), and then Wongcharoen et al. [37] studied the set-valued case of (1) in the same year. Adjimi et al. [38] used fixed point theorems to prove the uniqueness and existence of possible solutions to the generalized Caputo-type problem

⎧⎨

⎩

CD^b₀₊^1,ϑ(^CD^b₀₊^2,ϑϕ(t) –K(t,ϕ(t))) =ψ(t,ϕ(t)),

ϕ(a) = 0, I₀₊^b3,ϑϕ(T) = 0, a∈(0,T), (2) wheret∈[0,T),^CD^θ,ϑ₀₊ is theϑ-Caputo fractional derivative of orderθ∈ {b1,b2∈(0, 1]}.

I₀₊^b3,ϑis theϑ-Riemann–Liouville fractional integral of orderb3> 0, andK,ψ∈C([0,T]× R,R).

Motivated by the aforementioned works and inspired by [9], we prove the existence of solutions to the following nonlinear neutral fractional differential inclusion involvingϑ- Caputo fractional derivative withϑ-Riemann–Liouville fractional integral boundary con- ditions:

⎧⎨

⎩

CD^b₀₊^1,ϑ(^CD^b₀₊^2;ϑϕ(t) –K(t,ϕ(t)))∈H(t,ϕ(t)),

ϕ(a) = 0, I₀₊^b3;ϑϕ(T) = 0, a∈(0,T), (3) wheret∈[0,T), andH: [0,T]×R→P(R) is a set-valued map from [0,T]×Rto the collectionP(R)⊂R.

We obtain the desired results for the suggestedϑ-Caputo inclusion FBVP (3) involving convex and nonconvex set-valued maps using some well-known fixed point theorems.

We also construct two examples to validate our results. Reported findings are new in the frame of the generalized sequential Caputo fractional derivatives implemented on a novel neutral-type generalized fractional differential inclusion.

Observe that our problem (3) involves a general structure and is reduced to an Erdelyi–

Kober-type (and Hadamard-type) inclusion problem when we takeϑ(t) =t^η(andϑ(t) = log(t), respectively). Moreover, problem (3) is more general than problem (2).

This paper is organized as follows. Some fundamentals ideas of fractional calculus and theory of multifunctions are presented in Sect.2. The main results on the existence of solu- tions to theϑ-Caputo inclusion problem (3) using some fixed point theorems are obtained in Sect.3. Two examples are provided in Sect.4. In the final section, we give conclusive remarks.

2 Preliminary notions 2.1 Fractional calculus

In this section, we present some basic concepts on fractional calculus and necessary lem- mas.

LetJ_T= [0,T]. ByC=C(JT,R) we denote the Banach space of all continuous functions z:J_T→Rwith the norm

z=supz(t):t∈J_T ,

and byL¹(J_T,R) we denote the Banach space of Lebesgue-integrable functionsz:J_T→R with the norm

zL¹=

J_T

z(t)dt.

Letz:J_T→Rbe integrable, and letϑ∈Cⁿ(J_T,R) be increasing such thatϑ(t)= 0 for all t∈J_T.

Definition 2.1([39]) Theb1-ϑ-Riemann–Liouville integral of a functionzis given by I₀₊^b1;ϑz(t) = 1

(b1) t

0

ϑ(ς)

ϑ(t) –ϑ(ς)b1–1

z(ς)dς, b₁> 0.

Definition 2.2([39]) Theb1-ϑ-Riemann–Liouville fractional derivative of a functionzis defined by

D^b₀₊^1;ϑz(t) = 1

ϑ(t) d dt

n

I₀₊^(n–b1);ϑz(t), wheren= [b₁] + 1.

Definition 2.3([9,39,40]) Theϑ-Caputo fractional derivative of a functionz∈ACⁿ(JT, R) of orderb1is defined by

CD^b₀₊^1;ϑz(t) =I₀₊^(n–b1);ϑz^[n](t), wherez^[n](t) = (_ϑ¹(t)

d

dt)ⁿz(t), andn= [b1] + 1,n∈N.

Lemma 2.4([39,40]) Let b₁,b₂,μ> 0.Then 1)I₀₊^b1;ϑ(ϑ(ς) –ϑ(0))^b2–1(t) = _(b ^(b2)

1+b2)(ϑ(t) –ϑ(0))^b1+b2–1, 2)^CD^b₀₊^1;ϑ(ϑ(ς) –ϑ(0))^b2–1(t) = _(b ^(b2)

2–b1)(ϑ(t) –ϑ(0))^b2–b1–1. Lemma 2.5([39]) If z∈ACⁿ(JT,R)and b1∈(n– 1,n),then

I₀₊^b1;ϑCD^b₀₊^1;ϑz(t) =z(t) – n–1

k=0

z^[n](0⁺) k!

ϑ(t) –ϑ(0)k

.

In particular,for b1∈(0, 1),we have I₀₊^b1;ϑCD^b₀₊^1;ϑz(t) =z(t) –z(0).

Regarding problem (3), we indicate the following essential lemma, which was proven in [38].

Lemma 2.6([38]) Let

φ_T=ϑ(T) –ϑ(0), φa=ϑ(a) –ϑ(0), = φ_T^b2φ^ba³

(b2+ 1) (b3+ 1)– φ_T^b2+b3

(b2+b3+ 1)= 0, (4) andK,q∈C.Then the solution of linear-type problem

⎧⎨

⎩

CD^b₀₊^1,ϑ(^CD^b₀₊^2;ϑϕ(t) –K(t)) =q(t),t∈[0,T),

ϕ(a) = 0, I₀₊^b3;ϑϕ(T) = 0, a∈(0,T), (5) is given by

ϕ(t) =I₀₊^b2;ϑK(t) +I₀₊^b1+b2;ϑq(t) +(ϑ(t) –ϑ(0))^b2

(b2+ 1)

I₀₊^b2+b3;ϑK(T) +I₀₊^b1+b2+b3;ϑq(T) – φ_T^b3

(b3+ 1)

I₀₊^b2;ϑK(a) +I₀₊^b1+b2;ϑq(a)

+ 1

φ_T^b2+b3 (b2+b3+ 1)

I₀₊^b2;ϑK(a) +I₀₊^b1+b2;ϑq(a)

– φa^b2

(b2+ 1)

I₀₊^b2+b3;ϑK(T) +I₀₊^b1+b2+b3;ϑq(T)

. (6)

2.2 Multifunction theory

We present some concepts regarding the multifunctions (set-valued maps) [41]. For this aim, consider the Banach space (C, · ) andS:C→P(C) as a multifunction that:

(I) is closed-(convex-)valued ifS(ϕ) is a closed (convex) set for eachϕ∈C;

(II) is bounded ifS(B) =∪ϕ∈BS(ϕ) is bounded with respect toϕfor any bounded set B⊂C, that is,

sup

ϕ∈B

sup|f|:f ∈S(ϕ)

<∞;

(III) is measurable whenever for eachη∈R, the function t→d

η,S(t)

=inf

|η–λ|:λ∈S(t) is measurable.

For other notions such as the complete continuity or upper semicontinuity (u.s.c.), see [41]. Furthermore, the set of selections ofHis given by

RH,η=

ω∈L¹(JT,R)|ω(t)∈H(t,η) ∀(a.e. )t∈J_T .

Next, we define Pδ(C) =

W∈P(C) :W=∅and has propertyδ ,

whereP_cl,P_c,P_b, andP_cpdenote the classes of all closed, convex, bounded, and compact sets inC.

Definition 2.7([42]) A multifunctionH:J_T×R→P(R) is Carathéodory ift→H(t,ϕ) is measurable for eachϕ∈R, andϕ→H(t,ϕ) is u.s.c. for almost allt∈J_T.

Furthermore,His calledL¹-Carathéodory if for eachl> 0, there isk^∗∈L¹(JT,R⁺) such that

H(t,ϕ)=sup

|ω|:ω∈H(ω,ϕ) ≤k^∗(t)

for everyk^∗ ≤land for almost allt∈J_T.

The forthcoming lemmas are required to attain the desired outcomes in the current research study.

Lemma 2.8([42]) LetCand S be two Banach spaces,and let Gb(S) =

(ϕ,φ)∈C×S,φ∈S(ϕ)

be the graph ofS.IfS:C→P_cl(S)is u.s.c.ThenGb(S)is closed inC×S.Moreover,ifS is completely continuous and has a closed graph,thenSis u.s.c.

Lemma 2.9([43]) LetCbe a separable Banach space,letH:J_T×C→P_cp,c(C)be L¹- Carathéodory,and letZ:L¹(JT,C)→C(JT,C)be linear and continuous.Then

Z◦RH:C(JT,C)→P_cp,c

C(JT,C)

, ϕ→(Z◦RH)(ϕ) =Z(RH,ϕ), is a map with closed graph in C(JT,C)×C(JT,C).

Theorem 2.10(Nonlinear alternative for contractive maps [42]) LetCbe a Banach space, and letDbe a bounded neighborhood of0∈C.Let₁:C→P_cp,c(C)and₂:D→P_cp,c(C) be two set-valued operators satisfying:

(i)₁is a contraction,and (ii)₂is u.s.c.and compact.

IfS˜=₁+₂,then either (a)S˜has a fixed-point inD,or

(b)there existϕ∈∂Dandμ∈(0, 1)such thatϕ∈μS˜(ϕ).

Theorem 2.11 (Nadler–Covitz fixed point theorem [44]) Let C be a complete metric space.IfH:C→P_cl(C)is a contraction,thenHhas a fixed point.

3 Existence results for set-valued problems

In this section, we establish the main existence theorems.

Definition 3.1 The functionϕ∈C¹(JT,R) is a solution of (3) if there isω∈L¹(JT,R) such thatω(t)∈H(t,ϕ) for everyt∈J_Tsatisfying the generalized integral boundary conditions

ϕ(a) = 0, I₀₊^b3;ϑϕ(T) = 0, a∈(0,T), and

ϕ(t) =I₀₊^b2;ϑK t,ϕ(t)

+I₀₊^b1+b2;ϑω(t) +(ϑ(t) –ϑ(0))^b2 (b2+ 1)

×

I₀₊^b2+b3;ϑK

T,ϕ(T)

+I₀₊^b1+b2+b3;ϑω(T) – φ_T^b3

(b₃+ 1)

I₀₊^b2;ϑK a,ϕ(a)

+I₀₊^b1+b2;ϑω(a) + 1

φ_T^b2+b3 (b2+b3+ 1)

I₀₊^b2;ϑK a,ϕ(a)

+I₀₊^b1+b2;ϑω(a) – φa^b2

(b2+ 1)

I₀₊^b2+b3;ϑK

T,ϕ(T)

+I^b₀₊^1+b2+b3;ϑω(T) . 3.1 Case 1: convex-valued multifunctions

The first theorem deals with convex-valued multifunctionHusing the nonlinear alterna- tive for contractive maps (Theorem2.10). For convenience, we put

ζ₁= φ_T^b1+b2 (b1+b2+ 1) + φ_T^b2

|| (b2+ 1)

φ_T^b1+b2+b3

(b1+b2+b3+ 1)+ φ_T^b1+b2+b3 (b3+ 1) (b1+b2+ 1)

+ 1

||

φ^b_T^1+2b2+b3

(b₂+b₃+ 1) (b₁+b₂+ 1)+ φ_T^b1+2b2+b3

(b₂+ 1) (b₁+b₂+b₃+ 1)

, ζ₂= φ^b_T²

(b₂+ 1)+ φ_T^b2

|| (b₂+ 1)

φ_T^b2+b3

(b₂+b₃+ 1)+ φ^b_T^2+b3 (b₃+ 1) (b₂+ 1)

+ 1

||

φ_T^2b2+b3

(b₂+b₃+ 1) (b₂+ 1)+ φ_T^2b2+b3 (b₂+ 1) (b₂+b₃+ 1)

.

(7)

Theorem 3.2 Suppose that:

(Hyp1)The set-valued mapH:J_T×R→P_cp,c(R)is L¹-Carathéodory;

(Hyp2)There existR1∈C(JT,R⁺)and a nondecreasing functionR2∈C((0, +∞), (0, +∞)) such that

H(t,ϕ)

P=sup

|η|:η∈H(t,ϕ) ≤R1(t)R2

ϕ

, ∀(t,ϕ)∈J_T×R;

(Hyp3)There is a constant n_K<ζ₂^–1such that K(t,ϕ) –K(t,ϕ)≤n_K|ϕ–ϕ|;

(Hyp4)There isψ_K∈C(JT,R⁺)such that K(t,ϕ)≤ψ_K(t), ∀(t,ϕ)∈J_T×R;

(Hyp5)There isL> 0such that L

ζ₁R₁R₂(L) +ζ₂ψ_K> 1. (8)

Then(3)has a solution onJ_T.

Proof First, to switch the neutral-type fractional differential inclusion (3) into a fixed- point problem, we defineS˜:C→P(C) as

S(ϕ) =˜

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩ φ∈C:

φ(t) =

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

I₀₊^b2;ϑK(t,ϕ(t)) +I₀₊^b1+b2;ϑω(t) +(ϑ(t)–ϑ(0))^b2

(b2+1) [I₀₊^b2+b3;ϑK(T,ϕ(T)) +I₀₊^b1+b2+b3;ϑω(T) – ^φ

b3 T

(b3+1)(I₀₊^b2;ϑK(a,ϕ(a)) +I₀₊^b1+b2;ϑω(a))]

+¹[ ^φ

b2+b3 T

(b2+b3+1)(I₀₊^b2;ϑK(a,ϕ(a)) +I₀₊^b1+b2;ϑω(a)) – ^φ

b2 a

(b2+1)(I₀₊^b2+b3;ϑK(T,ϕ(T)) +I₀₊^b1+b2+b3;ϑω(T))].

⎫⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎭

(9)

forω∈RH,ϕ. Consider two operators₁:C→Cand₂:C→P(C) defined as

₁ϕ(t) =I₀₊^b2;ϑK t,ϕ(t)

+(ϑ(t) –ϑ(0))^b2 (b2+ 1)

×

I₀₊^b2+b3;ϑK

T,ϕ(T)

– φ_T^b3

(b₃+ 1)I₀₊^b2;ϑK

a,ϕ(a) + 1

φ^b_T^2+b3

(b₂+b₃+ 1)I₀₊^b2;ϑK a,ϕ(a)

– φ^ba²

(b₂+ 1)I₀₊^b2+b3;ϑK

T,ϕ(T) , and

₂(ϕ) =

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩ φ∈C:

φ(t) =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

I₀₊^b1+b2;ϑω(t) +(ϑ(t)–ϑ(0))^b2

(b2+1) [I^b₀₊^1+b2+b3;ϑω(T) – ^φ

b3 T

(b3+1)I₀₊^b1+b2;ϑω(a)]

+¹[ ^φ

b2+b3 T

(b2+b3+1)I₀₊^b1+b2;ϑω(a) – ^φ

b2 a

(b2+1)I₀₊^b1+b2+b3;ϑω(T)]

⎫⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎪⎭ .

Obviously,S˜=₁+₂. In what follows, we will show that the operators satisfy the hy- potheses of the nonlinear alternative for contractive maps (Theorem2.10). First, we define the bounded set

Bc=

ϕ∈C:ϕ ≤c , c> 0, (10)

and show that₁ and₂define the set-valued operators₁,₂:Bc→P_cp,c(C). To do this, we show that₁and₂are compact and convex-valued. We consider two steps.

Step 1.₂is bounded on bounded sets ofC.

LetBcbe bounded inC. Forφ∈₂(ϕ) andϕ∈Bc, there existsω∈RH,ϕsuch that φ(t) =I₀₊^b1+b2;ϑω(t) +(ϑ(t) –ϑ(0))^b2

(b₂+ 1)

I₀₊^b1+b2+b3;ϑω(T) – φ_T^b3

(b₃+ 1)I₀₊^b1+b2;ϑω(a)

+ 1

φ_T^b2+b3

(b₂+b₃+ 1)I₀₊^b1+b2;ϑω(a) – φa^b2

(b₂+ 1)I₀₊^b1+b2+b3;ϑω(T)

.

Under assumption (Hyp2), for anyt∈J_T, we have φ(t)≤I₀₊^b1+b2;ϑω(t)

+(ϑ(t) –ϑ(0))^b2

|| (b₂+ 1)

I₀₊^b1+b2+b3;ϑω(T)+ φ_T^b3

(b₃+ 1)I₀₊^b1+b2;ϑω(a) + 1

||

φ_T^b2+b3

(b2+b3+ 1)I₀₊^b1+b2;ϑω(a)+ φa^b2

(b2+ 1)I₀₊^b1+b2+b3;ϑω(T)

≤ ˜R₁ ˜R₂(c)(T)

×

φ^b_T^1+b2 (b₁+b₂+ 1) + φ_T^b2

|| (b₂+ 1)

φ^b_T^1+b2+b3

(b₁+b₂+b₃+ 1)+ φ_T^b1+b2+b3 (b₃+ 1) (b₁+b₂+ 1)

+ 1

||

φ_T^b1+2b2+b3

(b2+b3+ 1) (b1+b2+ 1)+ φ_T^b1+2b2+b3

(b2+ 1) (b1+b2+b3+ 1)

.

Thus

φ ≤ζ₁R1R2(c).

Step 2.₂maps bounded sets ofCinto equicontinuous sets.

Letϕ∈Bcandφ∈₂(ϕ). Then there is a functionω∈RH,ϕsuch that φ(t) =I₀₊^b1+b2;ϑω(t) +(ϑ(t) –ϑ(0))^b2

(b2+ 1)

I₀₊^b1+b2+b3;ϑω(T) – φ_T^b3

(b3+ 1)I₀₊^b1+b2;ϑω(a)

+ 1

φ_T^b2+b3

(b2+b3+ 1)I₀₊^b1+b2;ϑω(a) – φa^b2

(b2+ 1)I₀₊^b1+b2+b3;ϑω(T)

, t∈J_T.

Lett₁,t₂∈J_Twitht₁<t₂. Then φ(t2) –φ(t1)

≤ ˜R₁ ˜R₂(c) (b1+b2+ 1)

ϑ(t2) –ϑ(0)b1+b2

–

ϑ(t1) –ϑ(0)b1+b2

×(ϑ(t₂) –ϑ(0))^b2– (ϑ(t₁) –ϑ(0))^b2

|| (b₂+ 1)

I₀₊^b1+b2+b3;ϑω(T) + φ^b_T³

(b₃+ 1)I₀₊^b1+b2;ϑω(a)

.

Ast₁→t₂, we obtain φ(t2) –φ(t1)→0.

Hence₂(B_c) is equicontinuous. From steps 1–2, by the Arzelà–Ascoli theorem,₂ is completely continuous.

Step 3.₂(ϕ) is convex for everyϕ∈C.

Letφ₁,φ₂∈₂(ϕ). Then there existω₁,ω₂∈RH,ϕsuch that for eacht∈J_T, φj(t) =I₀₊^b1+b2;ϑωj(t) +(ϑ(t) –ϑ(0))^b2

(b2+ 1)

I₀₊^b1+b2+b3;ϑωj(T) – φ_T^b3

(b3+ 1)I₀₊^b1+b2;ϑωj(a)

+ 1

φ_T^b2+b3

(b2+b3+ 1)I₀₊^b1+b2;ϑω_j(a) – φa^b2

(b2+ 1)I₀₊^b1+b2+b3;ϑω_j(T)

, j= 1, 2.

Letσ∈[0, 1]. Then, for eacht∈J_T, we write σφ₁(t) + (1 –σ)φ₂(t)

=I₀₊^b1+b2;ϑ

σ ω₁(t) + (1 –σ)ω2(t) +(ϑ(t) –ϑ(0))^b2

(b2+ 1)

I₀₊^b1+b2+b3;ϑ

σ ω₁(T) + (1 –σ)ω2(T) – φ^b_T³

(b3+ 1)I₀₊^b1+b2;ϑ

σ ω₁(a) + (1 –σ)ω2(a) + 1

φ_T^b2+b3

(b₂+b₃+ 1)I₀₊^b1+b2;ϑ

σ ω₁(a) + (1 –σ)ω2(a) – φ^ba²

(b₂+ 1)I₀₊^b1+b2+b3;ϑ

σ ω₁(T) + (1 –σ)ω2(T) .

SinceHhas convex values,R_H,ϕis convex, and [σ ω1(t) + (1 –σ)ω2(t)]∈R_H,ϕ. Thusσφ1+ (1 –σ)φ₂∈₂(ϕ). In consequence,₂is convex-valued. Additionally,₁is compact and convex-valued.

Step 4.We check that the graph of₂is closed.

Letϕ_n→ϕ_∗,φ_n∈₂(ϕn), andφ_n→φ_∗. We prove thatφ_∗∈₂(ϕ∗). Sinceφ_n∈₂(ϕn), there existsωn∈R_H,ϕnsuch that

φ_n(t) =I₀₊^b1+b2;ϑω_n(t) +(ϑ(t) –ϑ(0))^b2 (b2+ 1)

I₀₊^b1+b2+b3;ϑω_n(T) – φ_T^b3

(b3+ 1)I^b₀₊^1+b2;ϑω_n(a)

+ 1

φ_T^b2+b3

(b₂+b₃+ 1)I^b₀₊^1+b2;ϑω_n(a) – φa^b2

(b₂+ 1)I₀₊^b1+b2+b3;ϑω_n(T)

, t∈J_T. Therefore we have to show that there isω_∗∈R_H,ϕ_∗such that for eacht∈J_T,

φ_∗(t) =I₀₊^b1+b2;ϑω_∗(t) +(ϑ(t) –ϑ(0))^b2 (b2+ 1)

I₀₊^b1+b2+b3;ϑω_∗(T) – φ_T^b3

(b3+ 1)I₀₊^b1+b2;ϑω_∗(a)

+ 1

φ^b_T^2+b3

(b₂+b₃+ 1)I₀₊^b1+b2;ϑω_∗(a) – φ^ba²

(b₂+ 1)I₀₊^b1+b2+b3;ϑω_∗(T)

, t∈J_T.

Define the continuous linear operatorZ:L¹(JT,R)→C(JT,R) by ω→Z(ω)(t)

=I₀₊^b1+b2;ϑω(t) +(ϑ(t) –ϑ(0))^b2

(b2+ 1)

I^b₀₊^1+b2+b3;ϑω(T) – φ_T^b3

(b3+ 1)I^b₀₊^1+b2;ϑω(a)

+ 1

φ_T^b2+b3

(b2+b3+ 1)I₀₊^b1+b2;ϑω(a) – φa^b2

(b2+ 1)I₀₊^b1+b2+b3;ϑω(T)

, t∈J_T.

Note that

φ_n–φ_∗= I₀₊^b1+b2;ϑ

ω_n(t) –ω_∗(t) +(ϑ(t) –ϑ(0))^b2

(b2+ 1)

I₀₊^b1+b2+b3;ϑ

ω_n(T) –ω_∗(T) – φ_T^b3

(b₃+ 1)I₀₊^b1+b2;ϑ

ω_n(a) –ω_∗(a) + 1

φ_T^b2+b3

(b₂+b₃+ 1)I₀₊^b1+b2;ϑ

ω_n(a) –ω_∗(a) – φa^b2

(b₂+ 1)I₀₊^b1+b2+b3;ϑ

ω_n(T) –ω_∗(T)

→0

asn→ ∞. By Lemma2.9,Z◦RH,ϕis a closed graph map. On the other hand, φ_n∈Z(RH,ϕn).

Sinceϕn→ϕ_∗, Lemma2.9gives

φ_∗(t) =I₀₊^b1+b2;ϑω_∗(t) +(ϑ(t) –ϑ(0))^b2 (b2+ 1)

I₀₊^b1+b2+b3;ϑω_∗(T) – φ_T^b3

(b3+ 1)I₀₊^b1+b2;ϑω_∗(a)

+ 1

φ^b_T^2+b3

(b₂+b₃+ 1)I₀₊^b1+b2;ϑω_∗(a) – φ^ba²

(b₂+ 1)I₀₊^b1+b2+b3;ϑω_∗(T)

for someω_∗∈RH,ϕ∗. Thus₂ has a closed graph, In consequence,₂ is compact and u.s.c.

Step 5.₁is a contraction inC.

Letϕ,ϕ∈C. By the assumption (Hyp3) we get 1ϕ(t) –1ϕ(t)

≤ φ_T^b2

(b2+ 1)+ φ_T^b2

|| (b2+ 1)

φ^b_T^2+b3

(b2+b3+ 1)+ φ_T^b2+b3 (b3+ 1) (b2+ 1)

+ 1

||

φ_T^2b2+b3

(b₂+b₃+ 1) (b₂+ 1)+ φ_T^2b2+b3 (b₂+ 1) (b₂+b₃+ 1)

n_Kϕ–ϕ.

Thus

₁ϕ–₁ϕ ≤n_Kζ₂ϕ–ϕ.

Asn_Kζ₂< 1, we infer that₁is a contraction.

Thus the operators₁ and₂satisfy assumptions of Theorem2.10. So, it yields that either condition (a)S˜has a fixed-point inDor (b) there existϕ∈∂Dandμ∈(0, 1) with ϕ∈μS˜(ϕ). We show that conclusion (b) is not possible. Ifϕ∈μ₁(ϕ) +μ₂(ϕ) forμ∈ (0, 1), then there isω∈RH,ϕsuch that

ϕ(t)=

μI^b₀₊^2;ϑK t,ϕ(t)

+μI₀₊^b1+b2;ϑω(t) +μ(ϑ(t) –ϑ(0))^b2

(b2+ 1)

I₀₊^b2+b3;ϑK

T,ϕ(T)

+I₀₊^b1+b2+b3;ϑω(T) – φ_T^b3

(b3+ 1)

I₀₊^b2;ϑK a,ϕ(a)

+I₀₊^b1+b2;ϑω(a) +μ

φ_T^b2+b3 (b2+b3+ 1)

I₀₊^b2;ϑK a,ϕ(a)

+I₀₊^b1+b2;ϑω(a) – φa^b2

(b2+ 1)

I₀₊^b2+b3;ϑK

T,ϕ(T)

+I₀₊^b1+b2+b3;ϑω(T)

≤ζ₁R1R2(ϕ) +ζ₂ψ_K. Thus

ϕ(t)≤ζ₁R1R2(ϕ) +ζ₂ψ_K, ∀t∈J_T. (11) If condition (b) of Theorem2.10is true, then there areμ∈(0, 1) andϕ∈∂Dwithϕ= μS˜(ϕ). Thenϕis a solution of (3) withϕ=L. Now by (11) we get

L

ζ₁R1R2(L) +ζ₂ψ_K≤1,

contradicting to (8). Thus it follows from Theorem2.10thatS˜has a fixed-point, which is

a solution of (3), and the proof is completed.

3.2 Case 2: nonconvex-valued multifunctions

In this section, we obtain another existence criterion forϑ-Caputo fractional differen- tial inclusion (3) under new assumptions. We will show our desired existence with a nonconvex-valued multifunction by using a theorem of Nadler and Covitz (Theorem 2.11).

Consider (C,d) as a metric space. ConsiderH^d:P(C)×P(C)→R⁺∪ {∞}defined by H^d(B,˜ C) =˜ max

sup

b∈ ˜˜ B

d(b,˜ C),˜ sup

˜ c∈ ˜C

d(B,˜ ˜c)

,

where d(B,˜ ˜c) =inf_{b∈ ˜˜ B}d(b,˜ ˜c) and d(b,˜ C) =˜ inf_{˜c∈ ˜C}d(b,˜ c). Then (P˜ b,cl(C),H^d) is a metric space (see [45]).