# Existence results for fractional differential inclusions arising from real estate asset securitization and HIV models

- Bashir Ahmad
^{1}Email author and - Sotiris K Ntouyas
^{2}

**2013**:216

https://doi.org/10.1186/1687-1847-2013-216

© Ahmad and Ntouyas; licensee Springer 2013

**Received: **3 May 2013

**Accepted: **5 July 2013

**Published: **22 July 2013

## Abstract

This paper studies a new class of boundary value problems of nonlinear differential inclusions with Riemann-Liouville integral boundary conditions arising from real estate asset securitization and HIV models. Some new existence results are obtained by using standard fixed point theorems when the right-hand side of the inclusion has convex as well as non-convex values. Some illustrative examples are also discussed.

**MSC:**34A60, 34A08.

## Keywords

## 1 Introduction

where ${D}^{\alpha}$ is the standard Riemann-Liouville fractional derivative of order $2<\alpha \le 3$, $0<\beta <1$, $\alpha -\beta >2$, $F:[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$ is a multivalued map, $\mathcal{P}(\mathbb{R})$ is the family of all subsets of ℝ, ${\int}_{0}^{1}{D}^{\beta}x(s)\phantom{\rule{0.2em}{0ex}}dA(s)$ denotes the Riemann-Stieltjes integral, and *A* is a function of bounded variation.

The subject of fractional differential equations has evolved as an interesting and important field of research in view of its numerous applications in physics, mechanics, chemistry, engineering (like traffic, transportation, logistics *etc.*), and so forth [1–3]. The tools of fractional calculus have played a key role in improving the mathematical modeling of many real world processes based on classical calculus. The nonlocal characteristic of a fractional order differential operator distinguishes it from a classical integer-order differential operator. In fact, differential equations of arbitrary order are capable of describing memory and hereditary properties of some important and useful materials and processes. For some recent development on the topic, see [4–19] and the references cited therein.

On the other hand, the nonlocal condition given by a Riemann-Stieltjes integral is due to Webb and Infante in [20] and gives a unified approach to many BVPs. Since ${\int}_{0}^{1}{D}^{\beta}x(s)\phantom{\rule{0.2em}{0ex}}dA(s)$ covers a multipoint BVP and an integral BVP as special cases, the fractional differential equations with the Riemann-Stieltjes integral condition were extensively studied by many authors, see [21, 22].

*i.e.*, there are more degrees of freedom in the fractional-order models. Tao and Qian [24] discussed the existence and uniqueness of positive solutions for the following differential equation with nonlocal Riemann-Stieltjes integral condition arising from the real estate asset securitization:

where *λ* is a parameter, $1<\gamma <\alpha \le 2$, $1<\alpha -\beta <\gamma $, $0<\beta <1$ and *A*, *B* are functions of bounded variation.

In the present paper, we are motivated by some recent papers [18, 24, 29], which considered problem (1.1) with *F* being single-valued and provided results on the existence and nonexistence of positive solutions. Here we extend the results to cover the multivalued case.

We establish existence results for problem (1.1), 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 paper is organized as follows. In Section 2 we recall some preliminary facts that we need in the sequel, and in Section 3 we prove our main results. Examples illustrating the obtained results are presented in Section 4.

## 2 Preliminaries

In this section, we introduce notations, definitions and preliminary facts that will be used in the remainder of this paper.

Let $C([0,1])$ denote a Banach space of continuous functions from $[0,1]$ into ℝ with the norm $\parallel x\parallel ={sup}_{t\in [0,1]}|x(t)|$. Let ${L}^{1}([0,1],\mathbb{R})$ be the Banach space of measurable functions $x:[0,1]\to \mathbb{R}$ which are Lebesgue integrable and normed by ${\parallel x\parallel}_{{L}^{1}}={\int}_{0}^{1}|x(t)|\phantom{\rule{0.2em}{0ex}}dt$.

*A*,

*B*is defined by

where $d(A,b)={inf}_{a\in A}d(a;b)$ and $d(a,B)={inf}_{b\in B}d(a;b)$.

- (i)
is

*convex*(*closed*)*valued*if $G(x)$ is convex (closed) for all $x\in X$; - (ii)
is

*bounded*on bounded sets if $G(\mathbb{B})={\bigcup}_{x\in \mathbb{B}}G(x)$ is bounded in*X*for all $\mathbb{B}\in {\mathcal{P}}_{b}(X)$ (*i.e.*, ${sup}_{x\in \mathbb{B}}\{sup\{|y|:y\in G(x)\}\}<\mathrm{\infty}$); - (iii)
is called

*upper semicontinuous*(*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 if, for each open set*N*of*X*containing $G({x}_{0})$, there exists an open neighborhood ${\mathcal{N}}_{0}$ of ${x}_{0}$ such that $G({\mathcal{N}}_{0})\subseteq N$; - (iv)
*G*is*lower semicontinuous*(*l.s.c.*) if the set $\{y\in X:G(y)\cap B\ne \mathrm{\varnothing}\}$ is open for any open set*B*in*E*; - (v)
is said to be

*completely continuous*if $G(\mathbb{B})$ is relatively compact for every $\mathbb{B}\in {\mathcal{P}}_{b}(X)$; - (vi)is said to be
*measurable*if, for every $y\in \mathbb{R}$, the function$t\mapsto d(y,G(t))=inf\{|y-z|:z\in G(t)\}$

- (vii)
*has a fixed point*if there is $x\in X$ such that $x\in G(x)$. The fixed point set of the multivalued operator*G*will be denoted by Fix*G*.

**Definition 2.1**A multivalued operator $N:X\to {\mathcal{P}}_{cl}(X)$ is called:

- (a)
*γ*-Lipschitz if and only if there exists $\gamma >0$ such that${H}_{d}(N(x),N(y))\le \gamma d(x,y)\phantom{\rule{1em}{0ex}}\text{for each}x,y\in X;$ - (b)
a contraction if and only if it is

*γ*-Lipschitz with $\gamma <1$.

**Definition 2.2**A multivalued map $F:[0,1]\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$ is said to be Carathéodory if

- (i)
$t\mapsto F(t,x)$ is measurable for each $x\in \mathbb{R}$;

- (ii)
$x\mapsto F(t,x)$ is upper semicontinuous for almost all $t\in [0,1]$.

*F*is called ${L}^{1}$-Carathéodory if

- (iii)for each $\alpha >0$, there exists ${\phi}_{\alpha}\in {L}^{1}([0,1],{\mathbb{R}}^{+})$ such that$\parallel F(t,x)\parallel =sup\{|v|:v\in F(t,x)\}\le {\phi}_{\alpha}(t)$

for all $\parallel x\parallel \le \alpha $ and for a.e. $t\in [0,1]$.

*F*by

We define the graph of *G* to be the set $Gr(G)=\{(x,y)\in X\times Y,y\in G(x)\}$ and recall two useful results regarding closed graphs and upper-semicontinuity.

**Lemma 2.3** [[30], Proposition 1.2]

*If* $G:X\to {\mathcal{P}}_{cl}(Y)$ *is u*.*s*.*c*., *then* $Gr(G)$ *is a closed subset of* $X\times Y$; *i*.*e*., *for every sequence* ${\{{x}_{n}\}}_{n\in \mathbb{N}}\subset X$ *and* ${\{{y}_{n}\}}_{n\in \mathbb{N}}\subset Y$, *if when* $n\to \mathrm{\infty}$, ${x}_{n}\to {x}_{\ast}$, ${y}_{n}\to {y}_{\ast}$ *and* ${y}_{n}\in G({x}_{n})$, *then* ${y}_{\ast}\in G({x}_{\ast})$. *Conversely*, *if* *G* *is completely continuous and has a closed graph*, *then it is upper semicontinuous*.

**Lemma 2.4** [31]

*Let*

*X*

*be a Banach space*.

*Let*$F:[0,T]\times \mathbb{R}\to {\mathcal{P}}_{cp,c}(X)$

*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* $C([0,1],X)\times C([0,1],X)$.

Let us recall some basic definitions of fractional calculus [1, 2].

**Definition 2.5**The Riemann-Liouville derivative of fractional order

*q*is defined as

provided the integral exists, where $[q]$ denotes the integer part of the real number *q*.

**Definition 2.6**The Riemann-Liouville fractional integral of order

*q*is defined as

provided the integral exists.

**Lemma 2.7** [2]

*For*$q>0$,

*the general solution of the fractional differential equation*${D}_{0+}^{q}x(t)=0$

*is given by*

*where* ${c}_{i}\in \mathbb{R}$, $i=1,2,\dots ,n$ ($n=[q]+1$).

**Lemma 2.8**[2]

- (1)
*If*$x\in {L}^{1}(0,1)$, $\nu >\sigma >0$,*then*${I}^{\nu}{I}^{\sigma}x(t)={I}^{\nu +\sigma}x(t),\phantom{\rule{2em}{0ex}}{D}^{\sigma}{I}^{\sigma}x(t)={I}^{\nu -\sigma}x(t),\phantom{\rule{2em}{0ex}}{D}^{\sigma}{I}^{\sigma}x(t)=x(t).$(2.1) - (2)
*If*$\nu >0$, $\sigma >0$,*then*${D}^{\nu}{t}^{\sigma -1}=\frac{\mathrm{\Gamma}(\sigma )}{\mathrm{\Gamma}(\sigma -\nu )}{t}^{\sigma -\nu -1}.$(2.2)

for some ${c}_{i}\in \mathbb{R}$, $i=1,2,\dots ,n$ ($n=[q]+1$).

and BVP (2.4) is equivalent to BVP (1.1).

**Lemma 2.9** [19]

*Given*$y\in {L}^{1}([0,1],\mathbb{R})$,

*then the unique solution of the problem*

*is given by*

*where*$G(t,s)$

*is the Green function of BVP*(2.5)

*and is given by*

**Lemma 2.10**

*For any*$t,s\in [0,1]$, $G(t,s)$

*satisfies*

Throughout the paper we always assume that the following holds.

(${\mathrm{H}}_{0}$) *A* is an increasing function of bounded variation such that $\mathcal{G}(s)\ge 0$ for $s\in [0,1]$ and $0\le \mathcal{C}<1$, where
is defined by (2.9).

**Lemma 2.11** [21]

*Let*$1<\alpha -\beta \le 2$.

*Assume that*(${\mathrm{H}}_{0}$)

*holds*.

*Then*$H(t,s)$

*satisfies*

We end this section by recalling two well-known fixed point theorems which will be used in the sequel, the nonlinear alternative of Leray-Schauder for multivalued maps and Covitz and Nadler fixed point theorem.

**Lemma 2.12** (Nonlinear alternative for Kakutani maps) [32]

*Let*

*E*

*be a Banach space*,

*let*

*C*

*be a closed convex subset of*

*E*,

*U*

*be an open subset of*

*C*

*and*$0\in U$.

*Suppose that*$F:\overline{U}\to {\mathcal{P}}_{c,cv}(C)$

*is an upper semicontinuous compact map*.

*Then either*

- (i)
*F**has a fixed point in*$\overline{U}$,*or* - (ii)
*there is a*$u\in \partial U$*and*$\lambda \in (0,1)$*with*$u\in \lambda F(u)$.

**Lemma 2.13** [33]

*Let* $(X,d)$ *be a complete metric space*. *If* $N:X\to {\mathcal{P}}_{cl}(X)$ *is a contraction*, *then* $FixN\ne \mathrm{\varnothing}$.

## 3 Existence results

Now we are in a position to present our main results. The methods used to prove the existence results are standard; however, their exposition in the framework of problem (1.1) is new.

### 3.1 Convex case

**Theorem 3.1** *Assume that* (${\mathrm{H}}_{0}$) *holds*. *In addition we assume that*:

(${\mathrm{H}}_{1}$) $F:[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathcal{P}(\mathbb{R})$ *is Carathéodory and has nonempty compact and convex values*;

*there exist two continuous nondecreasing functions*${\psi}_{i}:[0,\mathrm{\infty})\to (0,\mathrm{\infty})$, $i=1,2$,

*and a function*$p\in {L}^{1}([0,1],{\mathbb{R}}^{+})$

*such that*

*for each* $(t,x,y)\in [0,1]\times \mathbb{R}\times \mathbb{R}$;

*there exists a constant*$M>0$

*such that*

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

*Proof*Define the operator ${\mathrm{\Omega}}_{F}:C([0,1],\mathbb{R})\to \mathcal{P}(C([0,1],\mathbb{R}))$ by

We will show that ${\mathrm{\Omega}}_{F}$ satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As the first step, we show that ${\mathrm{\Omega}}_{F}$ *is convex for each* $x\in C([0,1],\mathbb{R})$. This step is obvious since ${S}_{F,x,y}$ is convex (*F* has convex values), and therefore we omit the proof.

*maps bounded sets*(

*balls*)

*into bounded sets in*$C([0,1],\mathbb{R})$. For a positive number

*ρ*, let ${B}_{\rho}=\{x\in C([0,1],\mathbb{R}):\parallel x\parallel \le \rho \}$ be a bounded ball in $C([0,1],\mathbb{R})$. Then, for each $h\in {\mathrm{\Omega}}_{F}(x)$, $x\in {B}_{\rho}$, there exists $v\in {S}_{F,x,y}$ such that

*maps bounded sets into equicontinuous sets of*$C([0,1],\mathbb{R})$. Let ${t}^{\prime},{t}^{\u2033}\in [0,1]$ with ${t}^{\prime}<{t}^{\u2033}$ and $y\in {B}_{\rho}$. For each $h\in {\mathrm{\Omega}}_{F}(y)$, we obtain

Obviously the right-hand side of the above inequality tends to zero independently of $y\in {B}_{\rho}$ as ${t}^{\u2033}-{t}^{\prime}\to 0$. As ${\mathrm{\Omega}}_{F}$ satisfies the above three assumptions, it follows by the Arzelá-Ascoli theorem that ${\mathrm{\Omega}}_{F}:C([0,1],\mathbb{R})\to \mathcal{P}(C([0,1],\mathbb{R}))$ is completely continuous.

*has a closed graph*. Let ${y}_{n}\to {y}_{\ast}$, ${h}_{n}\in {\mathrm{\Omega}}_{F}({y}_{n})$ and ${h}_{n}\to {h}_{\ast}$. Then we need to show that ${h}_{\ast}\in {\mathrm{\Omega}}_{F}({y}_{\ast})$. Associated with ${h}_{n}\in {\mathrm{\Omega}}_{F}({y}_{n})$, there exists ${v}_{n}\in {S}_{F,{y}_{n}}$ such that for each $t\in [0,1]$,

for some ${v}_{\ast}\in {S}_{F,{y}_{\ast}}$.

*M*such that $\parallel y\parallel \ne M$. Let us set

Note that the operator ${\mathrm{\Omega}}_{F}:\overline{U}\to \mathcal{P}(C([0,1],\mathbb{R}))$ is upper semicontinuous and completely continuous. From the choice of *U*, there is no $y\in \partial U$ such that $y\in \lambda {\mathrm{\Omega}}_{F}(y)$ for some $\lambda \in (0,1)$. Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 2.12), we deduce that ${\mathrm{\Omega}}_{F}$ has a fixed point $y\in \overline{U}$, which is a solution of problem (1.1). This completes the proof. □

### 3.2 Non-convex case

In this subsection, we study the case when *F* is not necessarily convex-valued.

**Definition 3.2** Let *A* be a subset of $[0,1]\times \mathbb{R}$. *A* is $\mathcal{L}\otimes \mathcal{B}$ measurable if *A* belongs to the *σ*-algebra generated by all sets of the form $\mathcal{J}\times \mathcal{D}$, where
is Lebesgue measurable in $[0,1]$ and
is Borel measurable in ℝ.

**Definition 3.3** A subset
of ${L}^{1}([0,1],\mathbb{R})$ is decomposable if, for all $u,v\in \mathcal{A}$ and measurable $\mathcal{J}\subset [0,1]=J$, the function $u{\chi}_{\mathcal{J}}+v{\chi}_{J-\mathcal{J}}\in \mathcal{A}$, where ${\chi}_{\mathcal{J}}$ stands for the characteristic function of
.

**Theorem 3.4** *Assume that* (${\mathrm{H}}_{0}$), (${\mathrm{H}}_{2}$), (${\mathrm{H}}_{3}$) *and the following condition holds*:

*is a nonempty compact*-

*valued multivalued map such that*

- (a)
$(t,x,y)\mapsto F(t,x,y)$

*is*$\mathcal{L}\otimes \mathcal{B}\otimes \mathcal{B}$*measurable*, - (b)
$(x,y)\mapsto F(t,x,y)$

*is lower semicontinuous for each*$t\in [0,1]$.

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

*Proof* It follows from (${\mathrm{H}}_{2}$), (${\mathrm{H}}_{4}$) and Lemma 4.1 of [34] that *F* is of l.s.c. type. Then, from the selection theorem due to Bressan and Colombo [35] for lower semicontinuous maps with decomposable values, there exists a continuous function $f:A{C}^{2}([0,1],\mathbb{R})\to {L}^{1}([0,1],\mathbb{R})$ such that $f(x)\in \mathcal{F}(x)$ for all $x\in C([0,1],\mathbb{R})$.

*x*is a solution to problem (1.1). In order to transform problem (3.1) into a fixed point problem, we define the operator ${\overline{\mathrm{\Omega}}}_{F}$ as

It can easily be shown that ${\overline{\mathrm{\Omega}}}_{F}$ is continuous and completely continuous and satisfies all the conditions of the nonlinear alternative of Leray-Schauder type for single-valued maps [32]. The remaining part of the proof is similar to that of Theorem 3.1. So we omit it. This completes the proof. □

Now we prove our second existence result for problem (1.1) with a non-convex-valued right-hand side by applying a fixed point theorem for a multivalued map due to Covitz and Nadler [33].

**Theorem 3.5** *Assume that the following conditions hold*:

(${\mathrm{H}}_{5}$) $F:[0,1]\times \mathbb{R}\times \mathbb{R}\to {\mathcal{P}}_{cp}(\mathbb{R})$ *is such that* $F(\cdot ,x,y):[0,1]\to {\mathcal{P}}_{cp}(\mathbb{R})$ *is measurable for each* $x\in \mathbb{R}$;

(${\mathrm{H}}_{6}$) ${H}_{d}(F(t,x,y),F(t,\overline{x},\overline{y}))\le m(t)(|x-\overline{x}|+|y-\overline{y}|)$ *for almost all* $t\in [0,1]$ *and* $x,\overline{x},y,\overline{y}\in \mathbb{R}$ *with* $m\in {L}^{1}([0,1],{\mathbb{R}}^{+})$ *and* $d(0,F(t,0,0))\le m(t)$ *for almost all* $t\in [0,1]$.

*Then boundary value problem*(1.1)

*has at least one solution on*$[0,1]$

*if*

*Proof*We transform boundary value problem (1.1) into a fixed point problem. Define the operator ${\mathrm{\Omega}}_{F}:C([0,1],\mathbb{R})\to \mathcal{P}(C([0,1],\mathbb{R}))$ by

We show that the operator ${\mathrm{\Omega}}_{F}$ satisfies the assumptions of Lemma 2.13. The proof will be given in two steps.

*Step 1*. ${\mathrm{\Omega}}_{F}(x)$ is nonempty and closed for every $v\in {S}_{F,x}$. Note that since the set-valued map $F(\cdot ,x(\cdot ),y(\cdot ))$ is measurable with the measurable selection theorem (

*e.g.*, [[36], Theorem III.6]), it admits a measurable selection $v:I\to \mathbb{R}$. Moreover, by the assumption (${\mathrm{H}}_{5}$), we have

*i.e.*, $v\in {L}^{1}([0,1],\mathbb{R})$ and hence *F* is integrably bounded. Therefore, ${S}_{F,y}\ne \mathrm{\varnothing}$.

*F*has compact values, we pass onto a subsequence (if necessary) to obtain that ${v}_{n}$ converges to

*v*in ${L}^{1}([0,1],\mathbb{R})$. Thus, $v\in {S}_{F,y}$ and for each $t\in [0,1]$, we have

Hence, $u\in {\mathrm{\Omega}}_{F}(x)$.

*Step 2*. The multivalued map ${\mathrm{\Omega}}_{F}(x)$ is a contraction. We show that there exists $\delta <1$ such that

Since the multivalued operator $U(t)\cap F(t,\overline{x}(t),\overline{y}(t))$ is measurable (Proposition III.4 in [36]), there exists a function ${v}_{2}(t)$ which is a measurable selection for *U*. So, ${v}_{2}(t)\in F(t,\overline{x}(t),\overline{y}(t))$ and for each $t\in [0,1]$, we have $|{v}_{1}(t)-{v}_{2}(t)|\le m(t)(|x(t)-\overline{x}(t)|+|y(t)-\overline{y}(t)|)$.

*x*and $\overline{x}$, we obtain

Since ${\mathrm{\Omega}}_{F}$ is a contraction, it follows by Lemma 2.13 that ${\mathrm{\Omega}}_{F}$ has a fixed point *x* which is a solution of (1.1). This completes the proof. □

## 4 Examples

In this section, we give two examples to show the applicability of our results.

**Example 4.1**Consider the following fractional boundary value problem:

Thus ${\mathcal{G}}_{A}(s)\ge 0$, $0\le \mathcal{C}<1$ and $A(s)$ is increasing, so (${\mathrm{H}}_{0}$) holds.

*F*is convex, compact-valued and is of Carathéodory type. Let $p(t)=1$ and ${\psi}_{1}(|x|)=3$, ${\psi}_{2}(|y|)=10$, we get for $t\in [0,1]$, $x,y\in \mathbb{R}$

and hence (${\mathrm{H}}_{2}$) holds.

Clearly, all the conditions of Theorem 3.1 are satisfied. Hence the conclusion of Theorem 3.1 applies to problem (4.1).

**Example 4.2**Consider the following fractional boundary value problem:

and ${\ell}_{1},{\ell}_{2}\in {L}^{1}([0,1],{\mathbb{R}}^{+})$.

Hence all the assumptions of Theorem 3.5 are satisfied and by the conclusion of it, BVP (4.3) has at least one solution on $[0,1]$.

## Authors’ information

Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) - Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.

## Declarations

### Acknowledgements

The authors thank the reviewers for their useful comments. This paper was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

## Authors’ Affiliations

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