### Abstract

This article studies a new class of nonlocal boundary value problems of nonlinear
differential equations and inclusions of fractional order with strip conditions. We
extend the idea of four-point nonlocal boundary conditions

**MSC 2000**: 26A33; 34A12; 34A40.

##### Keywords:

fractional differential equations; fractional differential inclusions; nonlocal boundary conditions; fixed point theorems; Leray-Schauder degree### 1 Introduction

The subject of fractional calculus has recently evolved as an interesting and popular field of research. A variety of results on initial and boundary value problems of fractional order can easily be found in the recent literature on the topic. These results involve the theoretical development as well as the methods of solution for the fractional-order problems. It is mainly due to the extensive application of fractional calculus in the mathematical modeling of physical, engineering, and biological phenomena. For some recent results on the topic, see [1-19] and the references therein.

In this article, we discuss the existence and uniqueness of solutions for a boundary
value problem of nonlinear fractional differential equations and inclusions of order
*q *∈ (1, 2] with nonlocal strip conditions. As a first problem, we consider the following
boundary value problem of fractional differential equations

where * ^{c}D^{q }*denotes the Caputo fractional derivative of order

*q*,

*η*are appropriately chosen real numbers.

The boundary conditions in the problem (1.1) can be regarded as six-point nonlocal
boundary conditions, which reduces to the typical integral boundary conditions in
the limit *α*, *γ *→ 0, *β*, *δ *→ 1. Integral boundary conditions have various applications in applied fields such
as blood flow problems, chemical engineering, thermoelasticity, underground water
flow, population dynamics, etc. For a detailed description of the integral boundary
conditions, we refer the reader to the articles [20,21] and references therein. Regarding the application of the strip conditions of fixed
size, we know that such conditions appear in the mathematical modeling of real world
problems, for example, see [22,23].

As a second problem, we study a two-strip boundary value problem of fractional differential inclusions given by

where

We establish existence results for the problem (1.2), when the right-hand side is convex as well as non-convex valued. The first result relies on the nonlinear alternative of Leray-Schauder type. In the second result, we shall combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values, while in the third result, we shall use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler.

The methods used are standard, however their exposition in the framework of problems (1.1) and (1.2) is new.

### 2 Linear problem

Let us recall some basic definitions of fractional calculus [24-26].

**Definition 2.1 ***For at least n-times continuously differentiable function *
*the Caputo derivative of fractional order q is defined as*

*where *[*q*] *denotes the integer part of the real number q*.

**Definition 2.2 ***The Riemann-Liouville fractional integral of order q is defined as*

*provided the integral exists*.

By a solution of (1.1), we mean a continuous function *x*(*t*) which satisfies the equation * ^{c}D^{q}x*(

*t*) =

*f*(

*t*,

*x*(

*t*)), 0

*< t <*1, together with the boundary conditions of (1.1).

To define a fixed point problem associated with (1.1), we need the following lemma, which deals with the linear variant of problem (1.1).

**Lemma 2.3 ***For a given *
*the solution of the fractional differential equation*

*subject to the boundary conditions in (1.1) is given by*

*where*

**Proof**. It is well known that the solution of (2.1) can be written as [24]

where

Solving these equations simultaneously, we find that

Substituting the values of *c*_{0 }and *c*_{1 }in (2.3), we obtain the solution (2.2). □

### 3 Existence results for single-valued case

Let

In view of Lemma 2.3, we define an operator

Observe that the problem (1.1) has solutions if and only if the operator equation
**F***x *= *x *has fixed points.

For the forthcoming analysis, we need the following assumptions:

(**A _{1}**)

*|f*(

*t*,

*x*) -

*f*(

*t*,

*y*)| ≤

*L|x*-

*y|*, ∀

*t*∈ [0, 1],

*L >*0,

*x*,

*y*∈ ℝ;

(**A _{2}**) |

*f*(

*t*,

*x*)|

*≤ μ*(

*t*), ∀(

*t*,

*x*) ∈ [0, 1] × ℝ, and

*μ*∈

*C*([0, 1], ℝ

^{+}).

For convenience, let us set

where

**Theorem 3.1 ***Assume that *
*is a jointly continuous function and satisfies the assumption *(*A*_{1}) *with L < *1/Λ, *where *Λ *is given by (3.2). Then the boundary value problem (1.1) has a unique solution*.

**Proof**. Setting
**F***B _{r }*⊂

*B*, where

_{r}*x*∈

*B*, we have

_{r}

Now, for

where Λ is given by (3.2). Observe that Λ depends only on the parameters involved
in the problem. As *L < *1*/*Λ, therefore **F **is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping
principle (Banach fixed point theorem). □

Now, we prove the existence of solutions of (1.1) by applying Krasnoselskii's fixed point theorem [27].

**Theorem 3.2 ***(Krasnoselskii's fixed point theorem). Let M be a closed, bounded, convex, and nonempty
subset of a Banach space X*. *Let A*, *B be the operators such that (i) Ax *+ *By *∈ *M whenever x*, *y *∈ *M; (ii) A is compact and continuous; (iii) B is a contraction mapping. Then there
exists z *∈ *M such that z *= *Az *+ *Bz*.

**Theorem 3.3 ***Let *
*be a jointly continuous function satisfying the assumptions *(*A*_{1}) *and *(*A*_{2}) *with*

*Then the boundary value problem (1.1) has at least one solution on *[0, 1].

**Proof**. Letting

and consider

For

Thus,
*A*_{1}) together with (3.3) that is a contraction mapping. Continuity of *f *implies that the operator is continuous. Also, is uniformly bounded on

Now we prove the compactness of the operator .

In view of (*A*_{1}), we define

which is independent of *x*. Thus, is equicontinuous. Hence, by the Arzelá-Ascoli Theorem, is compact on

Our next existence result is based on Leray-Schauder degree theory.

**Theorem 3.4 ***Let *
*Assume that there exist constants *
*where *Λ *is given by (3.2) and M > *0 *such that |f*(*t*, *x*)*| *≤*κ|x|*+*M for all t *∈ [0, 1], *x *∈ *C*[0, 1]. *Then the boundary value problem (1.1) has at least one solution*.

**Proof**. Consider the fixed point problem

where **F **is defined by (3.1). In view of the fixed point problem (3.4), we just need to prove
the existence of at least one solution *x *∈ *C*[0, 1] satisfying (3.4). Define a suitable ball *B _{R }*⊂

*C*[0, 1] with radius

*R >*0 as

where *R *will be fixed later. Then, it is sufficient to show that

Let us set

Then, by the Arzelá-Ascoli Theorem, *h _{λ }*(

*x*) =

*x*-

*H*(λ,

*x*) =

*x*- λ

**F**

*x*is completely continuous. If (3.5) is true, then the following Leray-Schauder degrees are well defined and by the homotopy invariance of topological degree, it follows that

where *I *denotes the unit operator. By the nonzero property of Leray-Schauder degree, *h*_{1}(*t*) = *x - λ***F***x *= 0 for at least one *x *∈ *B _{R}*. In order to prove (3.5), we assume that

*x*=

*λ*

**F**

*x*,

*λ*∈ [0, 1]. Then for

*x*∈

*∂B*and

_{R }*t*∈ [0, 1] we have

which, on taking norm
*x*‖, yields

Letting

**Example 3.5 ***Consider the following strip fractional boundary value problem*

Here, *q *= 3/2, *σ *= 1, *η *= 1, *α *= 1/3, *β *= 1/2, *γ *= 2/3, *δ *= 3/4 and
*A*_{1}) is satisfied with
_{1 }= 65/72, Δ_{2 }= 535/288, Δ = 4945/5184, and

Clearly, *L*Λ = 0.282191 *< *1. Thus, by the conclusion of Theorem 3.1, the boundary value problem (3.6) has a
unique solution on [0, 1].

**Example 3.6 ***Consider the following boundary value problem*

Here,

Clearly *M *= 1 and

Thus, all the conditions of Theorem 3.4 are satisfied and consequently the problem (3.7) has at least one solution.

### 4 Existence results for multi-valued case

#### 4.1 Preliminaries

Let us recall some basic definitions on multi-valued maps [28,29].

For a normed space (*X*, ‖.‖), let
*G*(*x*) is convex (closed) for all *x *∈ *X*. The map *G *is bounded on bounded sets if
*X *for all
*G *is called upper semi-continuous (u.s.c.) on *X *if for each *x*_{0 }∈ *X*, the set *G*(*x*_{0}) is a nonempty closed subset of *X*, and if for each open set *N *of *X *containing *G*(*x*_{0}), there exists an open neighborhood
*x*_{0 }such that
*. G *is said to be completely continuous if
*G *is completely continuous with nonempty compact values, then *G *is u.s.c. if and only if *G *has a closed graph, i.e., *x _{n }*→

*x*

_{*},

*y*→

_{n }*y*

_{*},

*y*∈

_{n }*G*(

*x*) imply

_{n}*y*

_{* }∈

*G*(

*x*

_{*}).

*G*has a fixed point if there is

*x*∈

*X*such that

*x*∈

*G*(

*x*). The fixed point set of the multivalued operator

*G*will be denoted by Fix

*G*. A multivalued map

is measurable.

Let *C*([0, 1]) denotes a Banach space of continuous functions from [0, 1] into ℝ with the
norm
*L*^{1}([0, 1], ℝ) be the Banach space of measurable functions *x *: [0, 1] → ℝ which are Lebesgue integrable and normed by

**Definition 4.1 ***A multivalued map *
*is said to be Carathéodory if*

(**i**) *t *↦ *F *(*t*, *x*) *is measurable for each x *∈ ℝ*;*

(**ii**) *x *↦ *F *(*t*, *x*) *is upper semicontinuous for almost all t *∈ [0, 1]*;*

*Further a Carathéodory function F is called L*^{1}*-Carathéodory if*

(**iii**) *for each α > *0, *there exists *
*such that*

*for all *‖*x*‖_{∞ }≤ *α and for a. e. t *∈ [0, 1].

For each
*F *by

Let *X *be a nonempty closed subset of a Banach space *E *and
*G *is lower semi-continuous (l.s.c.) if the set {*y *∈ *X *: *G*(*y*) ∩ *B *≠ ∅} is open for any open set *B *in *E*. Let *A *be a subset of [0, 1] × ℝ. *A *is
*A *belongs to the *σ*-algebra generated by all sets of the form
*L*^{1}([0, 1], ℝ) is decomposable if for all

**Definition 4.2 ***Let Y be a separable metric space and let *
*be a multivalued operator. We say N has a property (BC) if N is lower semi-continuous
(l.s.c.) and has nonempty closed and decomposable values*.

Let
*F *as

which is called the Nemytskii operator associated with *F*.

**Definition 4.3 ***Let *
*be a multivalued function with nonempty compact values. We say F is of lower semi-continuous
type (l.s.c. type) if its associated Nemytskii operator **is lower semi-continuous and has nonempty closed and decomposable values*.

Let (*X*, *d*) be a metric space induced from the normed space (*X*; ‖.‖). Consider

where *d*(*A*, *b*) = inf_{a∈A }*d*(*a*; *b*) and *d*(*a*, *B*) = inf_{b∈B }*d*(*a*; *b*). Then (*P*_{b,cl}(*X*), *H _{d}*) is a metric space and (

*P*(

_{cl}*X*),

*H*) is a generalized metric space (see [30]).

_{d}**Definition 4.4 ***A multivalued operator N *: *X *→ *P _{cl}*(

*X*)

*is called:*

(**a**) *γ-Lipschitz if and only if there exists γ > *0 *such that*

(**b**) *a contraction if and only if it is γ-Lipschitz with γ < *1.

The following lemmas will be used in the sequel.

**Lemma 4.5 ***(Nonlinear alternative for Kakutani maps) *[31]. *Let E be a Banach space*, *C is a closed convex subset of E*, *U is an open subset of C and *0 ∈ *U*. *Suppose that *
*is a upper semicontinuous compact map; here *
*denotes the family of nonempty, compact convex subsets of C. Then either*

*(i) F has a fixed point in *
*or*

*(ii) there is a u *∈ *∂U and λ *∈ (0, 1) *with u *∈ *λF*(*u*).

**Lemma 4.6 **[32]*Let X be a Banach space. Let *
*be an L*^{1}*-Carathéodory multivalued map and let *θ *be a linear continuous mapping from L*^{1}([0, 1], *X*) *to C*([0, 1], *X*)*. Then the operator*

*is a closed graph operator in C*([0, 1], *X*) × *C*([0, 1], *X*).

**Lemma 4.7 **[33]*Let Y be a separable metric space and let *
*be a multivalued operator satisfying the property (BC). Then N has a continuous selection*, *that is, there exists a continuous function (single-valued) *
*such that g*(*x*) ∈ *N*(*x*) *for every x *∈ *Y *.

**Lemma 4.8 **[34]*Let *(*X*, *d*) *be a complete metric space. If N *: *X *→ *P _{cl}*(

*X*)

*is a contraction, then FixN ≠*∅.

**Definition 4.9 ***A function x *∈ *C*^{2}([0, 1], ℝ) *is a solution of the problem (1.2) if *
*and there exists a function f *∈ *L*^{1}([0, 1], ℝ) *such that*

*f*(*t*) ∈ *F *(*t*, *x*(*t*)) *a.e. on *[0, 1] *and*

#### 4.2 The Carathéodory case

**Theorem 4.10 ***Assume that:*

(*H*_{1})
*is Carathéodory and has nonempty compact and convex values;*

(*H*_{2}) *there exists a continuous nondecreasing function ψ *: [0, ∞) → (0, ∞) *and a function *
*such that*

(*H*_{3}) *there exists a constant M > *0 *such that*

*Then the boundary value problem (1.2) has at least one solution on *[0, 1].

**Proof**. Define the operator

for *f *∈ *S*_{F,x}. We will show that Ω* _{F }*satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The
proof consists of several steps. As a first step, we show that Ω

*∈*

_{F }is convex for each x*C*([0, 1], ℝ). This step is obvious since

*S*

_{F,x }is convex (

*F*has convex values), and therefore we omit the proof.

In the second step, we show that Ω* _{F }maps bounded sets (balls) into bounded sets in C*([0, 1], ℝ). For a positive number

*ρ*, let

*B*= {

_{ρ }*x*∈

*C*([0, 1], ℝ): ‖

*x*‖ ≤

*ρ*} be a bounded ball in

*C*([0, 1], ℝ). Then, for each

*h*∈ Ω

_{F }(

*x*),

*x*∈

*B*, there exists

_{ρ}*f*∈

*S*

_{F,x }such that

Then for *t*∈[0, 1] we have

Thus,

Now we show that Ω_{F }maps bounded sets into ^{;}*equicontinuous sets of C*([0, 1], ℝ).

Let *t*', *t*'' ∈ [0, 1] with *t*' *< t*'' and *x *∈ *B _{ρ}*. For each

*h*∈ Ω

_{F}(

*x*), we obtain

Obviously the right-hand side of the above inequality tends to zero independently
of *x *∈ *B _{ρ }*as

*t*'' -

*t*' → 0. As Ω

_{F }satisfies the above three assumptions, therefore it follows by the Ascoli-Arzelá theorem that

In our next step, we show that Ω_{F }*has a closed graph*. Let
*h _{n }*∈ Ω

_{F}(

*x*), there exists

_{n}*t*∈ [0, 1],

Thus it suffices to show that there exists
*t *∈ [0, 1],