Nonlinear boundary value problems for fractional differential inclusions with Caputo-Hadamard derivatives on the half line

1.   Introduction

This paper deals with the existence of solutions to boundary value problems (BVP for short) for fractional differential inclusions. In particular, we consider the boundary value inclusion on an infinite interval

where $^{H}_{C} D^{r}$ is the Caputo-Hadamard fractional derivative, ${\mathcal P}({{\mathbb R}})$ is the family of all nonempty subsets of ${{\mathbb R}}$, $F:J \times {{\mathbb R}} \to {\mathcal P}({{\mathbb R}})$ is a multivalued map, and $y_{1} \in {{\mathbb R}}$.

Fractional order differential equations have proven to be effective models of various phenomena in engineering and the sciences such as viscoelasticity, electrochemistry, control theory, flows through porous media, electromagnetism, and others. Recently, they have been applied to problems in biological modeling and social interactions ^[14,15]. The monographs of Abbas et al. ^[1,2,3], Hilfer ^[21], Kilbas et al. ^[23], Podlubny ^[26], Momani et al. ^[25] contain the mathematical background needed to understand the value of this modeling tool. For results on fractional order derivatives in general and Hadamard fractional derivatives in particular, we refer the reader to ^{[5,6,7,10,17,18,20,28]}.

The Caputo left-sided fractional derivative of order $\alpha$ is defined by

where $\alpha > 0$ and $n = [\alpha]+1$. This derivative is very useful in many applied problems because it satisfies its initial data which contains $y(0)$, $y^{\prime}(0)$, etc., as well as the same data for boundary conditions.

The fractional derivative as presented by Hadamard in 1892 ^[19] differs from the well-known Caputo derivative in two significant ways. First, its kernel involves a logarithmic function with an arbitrary exponent, and secondly, the Hadamard derivative of a constant is not 0. The Caputo-Hadamard fractional derivative was introduced by Jarad et al. ^[22] is a modification of the Hadamard fractional derivative that maintains the property that the derivative of a constant is 0. In recent years there have been a number of papers examining problems involving the Caputo-Hadamard derivative, and as examples, we refer the reader to Adjabi et al. ^[4] and Shammack ^[27].

Here we present two results guaranteeing the existence of solutions to the problem (1.1)-(1.2); one is for the case where the right hand side is convex valued, and the other is for the nonconvex case. The nonlinear alternative of Leray-Schauder type is used in the proof for the convex case, and the Covitz-Nadler fixed point theorem for multivalued contraction maps is used in the nonconvex case. We should mention that each of approaches are then combined with the diagonalization method to obtain the results. It should be pointed out that this paper actually initiates the application of the diagonalization method to such classes of problems. The theorems in the present paper extend current results in the literature to the multivalued case.

2.   Preliminaries

We begin by presenting some definitions and preliminary facts that are needed in the proofs of our results. We take $C(J, {{\mathbb R}})$ to be the space of all continuous functions from $J$ into ${{\mathbb R}}$ with the norm

and let $L^{1}(J, {{\mathbb R}})$ be the Banach space of Lebesgue integrable functions $y:J\longrightarrow {{\mathbb R}}$ with the norm

Also, we let $AC(J, {{\mathbb R}})$ denote the space of functions $y:J \to {{\mathbb R}}$ that are absolutely continuous.

Let $\delta = t \frac{d}{dt}$, $\delta^n = \delta(\delta^{n-1)}$, set

and let $AC^{1}(J, {{\mathbb R}})$ be the space of absolutely continuous functions $y:J \to {{\mathbb R}}$ with an absolutely continuous first derivative.

For any Banach space $(X, \|\cdot\|)$, we set:

● $P_{cl}(X) = \{Y\in {\mathcal P}(X): Y\, \hbox {is closed}\},$

● $P_{b}(X) = \{Y\in {\mathcal P}(X): Y\, \hbox {is bounded}\},$

● $P_{cp}(X) = \{Y\in {\mathcal P}(X): Y\, \hbox{is compact}\},$

● $P_{cp, c}(X) = \{Y\in {\mathcal P}(X): Y\, \hbox{is compact and convex}\}.$

We say that a multivalued map $G:X \to {\mathcal P}(X)$ is convex (closed) valued if $G(X)$ is convex (closed) for all $x \in X$. A map $G$ is bounded on bounded sets if $G(B) = \cup_{x \in B} G(x)$ is bounded in $X$ for all $B \in P_{b}(X)$ (i.e., $\sup_{x \in B}\{\sup\{|y|: y \in G(x)\} < \infty)$. The mapping $G$ is upper semi-continuous (u.s.c) on $X$ if for each $x_{0} \in X$, the set $G(x_{0})$ is a nonempty closed subset of $X$, and for each open set $N$ of $X$ containing $G(x_{0})$, there exists an open neighborhood $N_{0}$ of $x_{0}$ such that $G(N_{0}) \subset N$. The mapping $G$ is completely continuous if $G(B)$ is relatively compact for each $B \in P_{b}(X)$.

If $G$ is a multivalued map that is completely continuous with nonempty compact values, then $G$ is upper semi-continuous if and only if $G$ has a closed graph (that is, if $x_{n} \to x_{*}$, $y_{n} \to y_{*}$, and $y_{n} \in G(x_{n})$, then $y_{*} \in G(x_{*})$). We say that $x \in X$ is a fixed point of $G$ if $x \in G(x)$. The set of fixed points of the multivalued operator $G$ will be denote by $Fix G$. A multivalued map $G:J\to P_{cl}({{\mathbb R}})$ is called measurable if for every $y \in {{\mathbb R}}$,

is a measurable function.

Definition 2.1. A multivalued map $F:J \times {{\mathbb R}} \to {\mathcal P}({{\mathbb R}})$ is said to be Carathéodory if

(1) $t \to F(t, u)$ is measurable for each $u \in {{\mathbb R}}$, and

(2) $u \to F(t, u)$ is upper semicontinuous for almost all $t \in J$.

Let $(X, d)$ be a metric space induced from the normed space $(X, |\cdot|)$. the function $H_{d}:{\mathcal P}(X) \times {\mathcal P}(X) \to {{\mathbb R}}_{+} \cup \{\infty\}$ given by

is known as the Hausdorff-Pompeiu metric.

Definition 2.2. A multivalued operator $N:X \to P_{cl}(X)$ is called:

(1) $\gamma$-Lipschitz if and only if there exists $\gamma > 0$ such that

(2) a contraction if and only if it is $\gamma$-Lipschitz with $\gamma < 1$.

For more details on multivalued maps see the books of Aubin and Cellina ^[8], Aubin and Frankowska ^[9], Castaing and Valadier ^[11], and Deimling ^[13].

Theorem 2.3. ([12,Covitz and Nadler]) Let $(X, d)$ be a complete metric space. If $N:X \to P_{cl}(X)$ is a contraction, then $Fix N \neq\emptyset$.

Lemma 2.4. (^[24]) Let $J$ be a compact real interval, $F$ be a Carathéodory multivalued map, and let $\theta$ be a linear continuous map from $L^{1}(J, E) \mapsto C(J, E)$. Then the operator

is a closed graph operator in $L^{1}(J, E) \times C(J, E)$.

Definition 2.5 (^[23]) The Hadamard fractional integral of order $r$ for a function $h\, : [1, +\infty) \to {{\mathbb R}}$ is defined as

provided that the integral exists.

Definition 2.6 (^[23]) For a function $h$ on the interval $[1, +\infty)$, the Hadamard fractional-order derivative of $h$ of order $r$ is defined by

Here, $[r]$ denotes the integer part of $r$ and $\log(\cdot) = \log_{e}(\cdot)$.

Definition 2.7 (^[22]) For a function $h$ belonging to $AC^{n}_{\delta}([a, b], {{\mathbb R}})$ with $a > 0$, we define the Caputo-type modification of the left-sided Hadamard fractional derivatives to be

where $n = [\alpha]+1$.

Lemma 2.8. (^[22])Let $y \in AC^{n}_{\delta}([a, b], {{\mathbb R}})$ or $C^{n}_{\delta}([a, b], {{\mathbb R}})$. Then

We next recall the nonlinear alternative of Leray-Schauder.

Theorem 2.9. Let $X$ be a Banach space and $C$ a nonempty closed convex subset of $X$. Let $U$ be a nonempty open subset of $C$ with $0 \in U$ and $T : \overline{U} \to P_{cp, c}(C)$ be a upper semicontinuous compact map. Then either

(1) $T$ has fixed points in $\overline{U}$, or

(2) There exist $u \in \partial U$ and $\lambda \in (0, 1)$ with $u \in \lambda T(u)$.

3.   Main results

We begin by defining what we mean by the problem (1.1)-(1.2) having a solution.

Definition 3.1. A function $y \in AC^{2}_{\delta}(J, {{\mathbb R}})$ is said to be a solution of (1.1)-(1.2), if there exists a function $v \in L^{1}(J, {{\mathbb R}})$ with $v(t) \in F(t, y(t))$ for a.e. $t \in J$ such that $^{H}_{C} D^{r}y(t) = v(t)$ and the function $y$ satisfies the boundary condition (1.2).

Lemma 3.2. Let $h:[1, T)\to {{\mathbb R}}$ be continuous functions. A function $y$ is a solution of the fractional integral equation

if and only if $y$ is a solution of the nonlinear fractional problem

Proof. Applying the Hadamard fractional integral of order $r$ to both sides of (3.2) and then using Lemma 2.8, we obtain

Applying (3.3) yields

and

Hence,

Substituting into (3.4), we obtain (3.1).

Conversely, it is clear that if $y$ satisfies equation (3.1), then (3.2) and (3.3) hold.

Remark 3.3. Notice that for $m\in \mathbb{N}$, there exists $J_{m}: = [1, T_{m}] \subset J$ with

such that $T_{m} \to \infty$ as $m \to \infty$.

Definition 3.4. For each $m\in {{\mathbb N}}$ and $y \in AC(J_{m}, {{\mathbb R}})$, define the set of selections of $F$ by

3.1.   The convex case

Our first existence result is for the case where $F$ is convex valued.

Theorem 3.5. Assume that: for each $m \in {\mathbb N}$:

(H1) $F:J_{m} \times {{\mathbb R}} \to {\mathcal P}_{cp, p}({{\mathbb R}})$ is a Carathéodory multi-valued map;

(H2) There exist $p \in C(J, {{\mathbb R}}^{+})$ and a continuous nondecreasing function $\psi:[0, \infty)\to (0, \infty)$ such that

(H3) There exists $C > 0$ such that

(H4) There exists $l \in L^{1}(J_{m}, {{\mathbb R}}^{+})$ with $^{H}I^{r} l(t) < \infty$ such that

and

Then the problem (1.1)-(1.2) has at least one solution on $J$.

Proof. Fix $m \in \mathbb{N}$ and consider the related boundary value problem

First, we shall show that the BVP (3.8)-(3.9) has a solution $y_{m} \in C(J_{m}, {{\mathbb R}})$ with

where $M > 0$ is a constant. To do this, consider the multivalued operator $N : C(J_{m}, {{\mathbb R}}) \to {\mathcal P}(C(J_{m}, {{\mathbb R}}))$ defined by

Clearly, from Lemma 3.2, the fixed points of $N$ are solutions to (3.8)-(3.9). We shall show that $N$ satisfies the hypotheses of the nonlinear Leray-Schauder alternative. We give the proof in steps.

Step 1: $N(y)$ is convex for each $y\in C(J_{m}, E)$. For $h_{1}$, $h_{2} \in N(y)$, there exist $v_{1}$, $v_{2} \in S_{F, y}$ such that

for $t \in J_{m}$ and $i = 1, 2$. Letting $0 \leq d \leq 1$, we see that for each $t \in J_{m}$,

Now $F$ has convex values, so $S_{F, y}$ is convex; hence,

so $N(y)$ is convex.

Step 2: $N$ maps bounded sets into bounded sets in $C(J_{m}, {{\mathbb R}})$. Let $B_{\mu_{*}} = \{ y \in C(J_{m}, {{\mathbb R}})\, : \| y\|_{\infty} \leq \mu_{*} \}$ be a bounded set in $C(J_{m}, {{\mathbb R}})$ and $y \in B_{\mu_{*}}.$ Then for each $h \in N(y)$, there exists $v \in S_{F, y}$ such that

By (H2), we have, for each $t \in J_{m}$,

Thus,

Step 3: $N$ maps bounded sets into equicontinuous sets in $C(J_{m}, {{\mathbb R}})$. Take $t_{1}$, $t_{2} \in J_{m}$ with $t_{1} < t_{2}$, and take $B_{\mu_{*}}$ to be a bounded set in $C(J_{m}, {{\mathbb R}})$ as we did in Step $2$. Let $y \in B_{\mu_{*}}$ and $h \in N(y)$. Then,

As $t_{1} \to t_{2}$, the right hand side of the inequality above approaches zero. Therefore, in view of Steps $1$ to $3$ and the Arzelà-Ascoli theorem, it follows that $N$ is completely continuous.

Step 4: $N$ is upper semicontinuous. We will show this by showing that $N$ has a closed graph. Let $y_{n} \to y_{*}$, $h_{n} \in N(y_{n})$, and $h_{n} \to h_{*}$. We need to prove that $h_{*} \in N(y_{*})$. Now $h_{n} \in N(y_{n})$ implies there exists $v_{n} \in S_{F, y_{n}}$ such that for $t \in J_m$,

We need to show that there is a $v_* \in S_{F, y_*}$ such that, for each $t \in J_m$,

Now $F(t, \cdot)$ is upper semi-continuous, so for every $\epsilon > 0$ there exists $N_{\epsilon} \in {{\mathbb N}}$ such that for every $n > N_{\epsilon}$, we have

Since $F$ has compact values by (H1), there is a subsequence $v_{n_k}$ of $v_{n}$ such that

and

For every $w \in F(t, y_*(t))$, we have

Then,

We can obtain an analogous relation by interchanging the roles of $v_{n_k}$ and $v_{*}$, so

by (H4). It is easy to see that

which is what we wished to show.

Step 5: A priori bounds on solutions. Let $y \in \lambda N(y)$ with $\lambda \in (0, 1]$. Then there is a $v \in S_{F, y}$ so that for each $t \in J_{m}$,

This implies by (H2) that, for each $t \in J_{m}$, we have

Thus,

Then by condition (3.5), there exists $C > 0$ such that $\|y\|_{\infty} \not = C$. Let $U = \{y \in C(J_{m}, {{\mathbb R}}):\|y\|_{\infty} < C\}$. The operator $N:\overline{U} \to {\mathcal P}(C(J_{m}, {{\mathbb R}}))$ is upper semi-continuous and completely continuous. From the choice of $U$, there is no $y \in \partial U$ such that $y \in \lambda N(y)$ for some $\lambda \in (0, 1]$. It then follows from the Leray-Shauder nonlinear alternative that $N$ has a fixed point $y \in \overline{U}$ that in turn is a solution of problem (3.8)-(3.9).

Step 6: A diagonalization process. First let ${{\mathbb N}}_{m} = {{\mathbb N}}^{*}-\{m\}$. For each $k \in {{\mathbb N}}$, let $y_k(t)$ be the solution of (3.8)-(3.9) whose existence is guaranteed by Steps 1-5 above, and set

For $m = 1$, there exists $v^1_k \in S_{F, u}$ such that

and

Now for $t_{1}$, $t_{2} \in J_{1}$ with $t_{1} < t_{2}$, we have

By the Arzelà-Ascoli Theorem, $\{u_k\}$ has a uniformly convergent subsequence, so there is a subset ${{\mathbb N}}_1$ of ${{\mathbb N}}$ and a function $z_1 \in C([1, T_1], {{\mathbb R}})$ such that

Now for $k \in {{\mathbb N}}_1$ and $m = 2$, we have

Also for $t_{1}$, $t_{2} \in J_{2}$ with $t_{1} < t_{2}$, there exists $v^2_{k} \in S_{F, u}$ such that

Again using the Arzelà-Ascoli Theorem, $\{u_k\}$ has a uniformly convergent subsequence, so there is a subset ${{\mathbb N}}_2$ of ${{\mathbb N}}_1$ and a function $z_2 \in C([1, T_1], {{\mathbb R}})$ such that

where $z_1 = z_2$ on $[1, T_1]$ since ${{\mathbb N}}_2 \subset {{\mathbb N}}_1$.

Proceeding inductively, we see that for $t_{1}, t_{2} \in J_{m}$ with $t_{1} < t_{2}$, there is $v^m_{k} \in S_{F, u}$, such that

and

Now, let $m \in {{\mathbb N}}$ with $s \leq T_{m}$, fix $t \in [1, \infty)$, and let $y(t) = z_{m}(t)$. Then $y \in C([1, \infty), {{\mathbb R}})$, $y(1) = y_{1}$, and $|y(t)|\leq M$ for $t \in [1, \infty)$.

Again for fixed $t\in [1, \infty)$ and $m\in {{\mathbb N}}$ with $s\leq T_{m}$, for $n \in {{\mathbb N}}_{m}$ there exists $v'_{n} \in S_{F, u}$ so that

as $n \to \infty$ through ${{\mathbb N}}_{m}$. Hence, there exists $v^{\prime} \in S_{F, u}$, such that

that is, there exists $v \in S_{F, y}$ such that

We can apply this method for each $t \in [1, T_{m}]$ and each $m \in {{\mathbb N}}$. Thus,

for each $m \in {{\mathbb N}}$. This completes the proof of the theorem.

3.2.   The nonconvex case

We now consider the case where right hand side of problem (1.1)-(1.2) is nonconvex valued. In this case the proof relies on the fixed point result contained in Theorem 2.3.

Theorem 3.6. In addition to condition (H4) assume that:

(H5) $F:J_{m} \times {{\mathbb R}} \to P_{cp}({{\mathbb R}})$ has the property that $F(\cdot, u) :J_{m} \to P_{cp}({{\mathbb R}})$ is measurable for each $u \in {{\mathbb R}}$.

If

then the problem (1.1)-(1.2) has at least one solution on $J$.

Remark 3.7. By (H5), we can see that $S_{F, y}$ is nonempty for each $y \in C(J_{m}, {{\mathbb R}})$, so $F$ has a measurable selection by [11,Theorem III.6].

Proof. We will show that $N$ satisfies the conditions of Theorem 2.3. Once again our proof will be given in steps.

Step 1: $N(y) \in P_{cl}(C(J_{m}, {{\mathbb R}}))$ for each $y \in C(J_{m}, {{\mathbb R}})$. Let $(y_{n})_{n \geq 0} \subset N(y)$ be such that $y_{n} \to \bar{y}$. Then, $\bar{y} \in C(J_{m}, {{\mathbb R}})$ and there exists $v_{n} \in S_{F, y}$, $n = 1, 2, \dots$ such that, for each $t \in J_{m}$,

From the fact that $F$ has compact values and condition (H4), passing if necessary to a subsequence, we can conclude that $v_{n}$ converges weakly to $v$ in $L_{w}^{1}(J_{m}, {{\mathbb R}})$ (the space endowed with the weak topology). Applying Mazur's theorem, this implies that $v_{n}$ is strongly convergent to $v$ and hence $v \in S_{F, y}$. Then, for $t \in J_{m}$,

Hence, $\bar{y} \in N(y)$.

Step 2: There exists $\gamma < 1$ such that $H_{d}(N(y), N(\bar{y})) \leq \gamma \|y-\bar{y}\|_{\infty}$ for $y$, $\bar{y} \in C(J_{m}, {{\mathbb R}})$. Let $y$, $\bar{y} \in C(J_{m}, {{\mathbb R}})$ and $h_{1} \in N(y)$. Then, there exists $v_{1} \in F(t, y(t))$ such that for each $t \in J_{m}$

From (H4) it follows that

Hence, there exists $w \in F(t, \bar{y}(t))$ such that

Consider $U:J_{m} \to \mathcal{P}({{\mathbb R}})$ given by

Since the multivalued operator $V(t) = U(t) \cap F(t, \bar{y}(t))$ is measurable, there exists a function $v_{2}(t)$ that is a measurable selection for $V$. So, $v_{2} \in F(t, \bar{y}(t))$, and for each $t \in J_{m}$,

Let us define, for $v_2 \in S_{F, \bar{y}}$,

Then, for each $t \in J_{m}$,

Thus,

An analogous relation obtained by interchanging the roles of $y$ and $\bar{y}$ gives

Hence, by (3.11), $N$ is a contraction, so by Theorem 2.3, $N$ has a fixed point $y$ that is a solution to (1.1)-(1.2).

Step 3: The diagonalization process. We can use a similar argument to the one used in Step 6 in the proof of Theorem 3.5. Thus

for each $m \in {{\mathbb N}}$. This proves the theorem.

4.   An example

We conclude this paper with an example illustrating our main result. We apply Theorem 3.5 to the fractional differential inclusion

We set

where $f_{1}, f_{2}: [1, T_{m}] \times {{\mathbb R}} \mapsto {{\mathbb R}}$. We assume that for each $t \in [1, T_{m}]$, $f_{1}(t, \cdot)$ is lower semi-continuous (i.e., the set $\{y \in {{\mathbb R}} : f_{1}(t, y) > \mu$} is open for each $\mu \in {{\mathbb R}})$, and assume that for each $t \in [1, T_{m}]$, $f_{2}(t, \cdot)$ is upper semi-continuous (i.e., the set $\{y \in {{\mathbb R}} : f_{2}(t, y) < \mu$} is open for each $\mu \in {{\mathbb R}})$. Assume that there exist $p \in C([1, T_{m}], {{\mathbb R}}^{+})$ and a continuous and nondecreasing function $\psi: [0, \infty) \mapsto (0, \infty)$ such that

It is clear that $F$ is compact and convex-valued and is upper semi-continuous. Finally, we assume that there exists a number $C > 0$ such that

Since all the conditions of Theorem 3.5 are satisfied, problem (4.1)-(4.2) has at least one bounded solution $y$ on $[1, \infty)$.

5.   Conclusions

In this paper we consider a boundary value problem for a fractional differential inclusion involving the Caputo-Hadamard type derivative of order $r \in (1, 2]$ on the infinite interval $[1, \infty)$. We give sufficient conditions for the existence of solutions in case the right hand side of the inclusion is convex valued and where it is not. In the convex valued case, the nonlinear alternative of Leray-Schauder type is used in the proof, and in the nonconvex case, the Covitz-Nadler fixed point theorem for multivalued contractions is applied. Due to the fact that our problem is on an infinite interval, a diagonalization method was needed to complete the proofs. This was the first time the diagonalization method has been applied to such problems.

Conflicts of interest

All authors declare no conflicts of interest in this paper.