Existence and Attractivity Results for Fractional Differential Inclusions via Nocompactness Measures ^†

Abstract

In this paper, we use the concept of measure of nocompactness and fixed-point theorems to investigate the existence and stability of solutions of a class of Hadamard–Stieltjes fractional differential inclusions in an appropriate Banach space, and these results are proven under sufficient hypotheses. We also give an example to illustrate the obtained results.

1. Introduction

Integral and differential inclusions is generalization of integral and differential equations and inequalities, and became a basic tool in optimal control theory, stochastic processes and dynamical systems. This theory is located within the mainstream of nonlinear analysis—or to put it more precisely—multi-valued analysis. There exist numerous works focused in this direction, that is concerning the linear and nonlinear problems involving different types of fractional derivatives as well as integral we refer [1,2,3,4,5,6,7] and the references therein.

The theory of differential equations involving the Stieltjes derivative with respect to a non-decreasing function is a new class of differential equations and have many applications as a unified framework for dynamic equations on time scales and differential equations with impulses at fixed times. Such multivalued differential problems simply consist in replacing the usual derivatives by Stieltjes derivatives. Research on this subject can be found in [8,9,10,11,12] and the references therein.

In this work, we consider the following differential inclusion with an initial condition

$$\left\{\begin{array}{c}{\mathcal{D}}^{\gamma }\left(\frac{dw}{d\varphi }\right)\left(z\right)\in G(z,w\left(z\right));z\in (1,+\infty )\hfill \\ \frac{dw}{d\varphi }\left(1\right)=w_0;w\left(1\right)=w_1\hfill \end{array}\right.$$

(1)

where $w_0,w_1\in \mathbb{R}$, $\frac{dw}{d\varphi }$ is the Stieltjes derivative of w with respect to $\varphi$, ${\mathcal{D}}^{\gamma }$ is the Hadamard fractional derivative of order $0<\gamma <1$, $G:[1,+\infty )\times \mathbb{R}\to \mathcal{P}\left(\mathbb{R}\right)$ is multivalued map and $\mathcal{P}\left(\mathbb{R}\right)$ is the family of all nonempty subsets of $\mathbb{R}$. This paper contains some related ideas with those in.

2. Preliminaries

Assume that $\varphi :\mathbb{R}\to \mathbb{R}$ is monotone, nondecreasing and continuous from the left everywhere. $D_{\varphi }$ is the set of discontinuity points of $\varphi$. We denote by $\mathcal{B}\left(J\right)$ the Banach space of bounded functions on the interval $J=[1,+\infty )$ equipped with the norm of uniform convergence, and by ${\mathcal{B}}_{\varphi }\left(J\right)$ the subspace of bounded functions which are also $\varphi$-continuous on J.

Theorem 1. 

${\mathcal{B}}_{\varphi }\left(J\right)$ is a Banach space.

Let X and Y be two Banach spaces, we define

$${\mathcal{P}}_{cl,cv,cp}\left(X\right)=\left\{\Omega \in \mathcal{P}\left(X\right);\Omega \mathrm{is}\mathrm{closed}, \mathrm{convex}, \mathrm{compact}\right\}.$$

Theorem 2 

([13]). Let $G:X\to {\mathcal{P}}_{cl,cv}\left(Y\right)$ be a lower semicontinuous multifunction. Then G admits continuous selection.

Theorem 3. 

([14]). Let Ω be a nonempty, bounded, closed and convex subset of the Banach space X with ψ as a measure of nocompactness in it, and let $G:\Omega ⟶{\mathcal{P}}_{cl,cv}(\Omega )$ be closed. Assume that there exists a constant $k\in [0,1)$ such that $\psi \left(GA\right)\le k\psi \left(A\right)$ for any nonempty subset A of Ω. Then, G has a fixed point in the set Ω.

Remark 1. 

Let us denote by $FixG$ the set of all fixed points of the operator G which belong to Ω. The set $FixG$ belongs to the family ker ψ.

3. Main Results

Consider the function $\psi$ defined on the family ${\mathcal{M}}_{{\mathcal{B}}_{\varphi }\left(J\right)}$ by the formula

$$\psi (\Omega )=max\left\{{\omega }_{\varphi }(\Omega ),{\omega }_{\varphi }^{+}(\Omega )\right\}+\underset{z\to \infty }{lim\; sup}diam\Omega \left(z\right).$$

(2)

Theorem 4. 

The mapping ψ is a measure of nocompactness in the space ${\mathcal{B}}_{\varphi }\left(J\right)$.

For measures of nocompactness in the space of regulated functions see [15].

Consider the following inclusion

$$u\left(z\right)\in \left(Fu\right)\left(z\right);z\in J.$$

(3)

Definition 1. 

The solution $u=u\left(z\right)$ of (3) is said to be globally attractive if for each solution $v=v\left(z\right)$ of (3) we have

$$\underset{z\to \infty }{lim}\left(u\left(z\right)-v\left(z\right)\right)=0.$$

In the case when this limit is uniform, i.e., when for each $ϵ>0$ there exists $T>1$ such that

$$\left|u\right(z)-v(z\left)\right|<ϵ,$$

(4)

for $z\ge T$, we will say that the solutions of (3) are uniformly globally attractive.

Remark 2. 

The kernel ker ψ consists of nonempty and bounded sets Ω such that functions from Ω are locally equiregulated on J, and for each $u,v\in \Omega$ and (4) hold.

Let $\frac{dw}{d\varphi }\left(z\right)=v\left(z\right)$, then ${\displaystyle w\left(z\right)=w_0+{\int }_1^{z}v\left(t\right)d\varphi \left(t\right)}$ and (1) can be written as

$$v\left(z\right)\in w_0+{\displaystyle \frac{1}{\Gamma \left(r\right)}}{\int }_1^z{\left(ln\frac{z}{t}\right)}^{\gamma -1}G\left(t,w_1+{\int }_1^{t}v\left(\tau \right)d\varphi \left(\tau \right)\right){\displaystyle \frac{dt}{t}},$$

(5)

Denote $\left|G\right(z,u\left)\right|=\left\{\right|u|;u\in F(z,u\left)\right\}$ and $\parallel G(z,u)\parallel =H\left(G\right(z,u),0)=sup\left\{\right|u|;u\in G(z,u\left)\right\}$. The differential inclusion (1) will be considered under the following assumptions:

$\left(H_1\right)$

There exist continuous and bounded functions $p,q:J\to {\mathbb{R}}_{+}$ such that

$$H\left(G\right(z,u),G(z,v\left)\right)\le p\left(z\right)|u-v|;z\in J,$$

for all $u,v\in \mathbb{R}$, and

$$\parallel G(z,0)\parallel =H\left(G\right(z,0),0)\le q\left(z\right);z\in J.$$

$\left(H_2\right)$

For every $(z,w)$ in $J\times \mathbb{R}$, $G(z,w)$ is a nonempty convex and closed subset of $\mathbb{R}$.

$\left(H_3\right)$

Assume that

$$p^{*}=\underset{z\in J}{sup}\left|\frac{w_1}{\Gamma \left(\gamma \right)}{\int }_1^z{\left(ln\frac{z}{t}\right)}^{\gamma -1}p\left(t\right)dt\right|<\infty ,$$

$$q^{*}=\underset{z\in J}{sup}\left|\frac{1}{\Gamma \left(\gamma \right)}{\int }_1^z{\left(ln\frac{z}{t}\right)}^{\gamma -1}q\left(t\right)dt\right|<\infty ,$$

$$p_{\varphi }=\underset{z\in J}{sup}\left|\frac{1}{\Gamma \left(\gamma \right)}{\int }_1^z{\left(ln\frac{z}{t}\right)}^{\gamma -1}p\left(t\right)\left[\varphi \left(t\right)-\varphi \left(1\right)\right]dt\right|<1,$$

Remark 3. 

As observed, G is lower semicontinuous, hence G admits continuous selection (Theorem 2).

Theorem 5. 

Under assumptions $\left(H_1\right)-\left(H_3\right)$, the differential inclusion (1) has at least one solution $u=u\left(z\right)$ in the space ${\mathcal{B}}_{\varphi }\left(J\right)$. Moreover, solutions of the differential inclusion (1) are globally attractive.

Proof. 

Consider the multivalued operator N defined on the space ${\mathcal{B}}_{\varphi }\left(J\right)$ in the following way:

$$N:{\mathcal{B}}_{\varphi }\left(J\right)\to \mathcal{P}\left({\mathcal{B}}_{\varphi }\left(J\right)\right);$$

such that, for each $u\in {\mathcal{B}}_{\varphi }\left(J\right)$,

$$\left(Nw\right)\left(z\right)=\left\{u\in {\mathcal{B}}_{\varphi }\left(J\right)|u\left(z\right)=w_0+{\displaystyle \frac{1}{\Gamma \left(r\right)}}{\int }_1^z{\left(ln\frac{z}{t}\right)}^{\gamma -1}g\left(t,w_1+{\int }_1^{t}w\left(\tau \right)d\varphi \left(\tau \right)\right){\displaystyle \frac{dt}{t}}\right\},$$

(6)

where g is a selctor of G. For each w in ${\mathcal{B}}_{\varphi }\left(J\right)$, and for each function u in $Nw$, u is $\varphi$-continuous on J. Next, let us take an arbitrary function $w\in {\mathcal{B}}_{\varphi }\left(J\right)$, $u\in Nu$ and fixing $z\in J$, we obtain

$$\left|u\left(z\right)\right|\le |w_1|+p^{*}+\parallel w\parallel p_{\varphi }+q^{*}.$$

So, the operator N is well defined. We take

$$\zeta =\frac{|w_1|+{\displaystyle \frac{p^{*}}{\Gamma \left(\gamma \right)}}+{\displaystyle \frac{q^{*}}{\Gamma \left(\gamma \right)}}}{1-p_{\varphi }}.$$

We deduce that the operator $N:B_{\zeta }⟶\mathcal{P}\left(B_{\zeta }\right)$. Further, the set $Nu$ is closed and convex in $B_{\zeta }$. Now, we take a nonempty $\Omega \subset B_{\zeta }$ for $T>1$, $z_1,z_2\in [1,T]$ with $z_1<z_2$ and $|\varphi \left(z_2\right)-\varphi \left(z_1\right)|\le ϵ$ for each $ϵ>0$. Fixing arbitrarily $w\in \Omega$ and $u\in Nw$, we obtain

$${\omega }_{\varphi }(N\Omega )=0.$$

(7)

For $z_0\in J\cap D_{\varphi }$, fix $z\in (z_0,z_0+ϵ)$, we have

$${\omega }_{\varphi }^{+}(N\Omega )=0.$$

(8)

From (7) and (8), we infer that the set $N\Omega$ is equiregulated on J. Further, for $w,w^{\prime }\in \Omega$, $u\in Nw$ and $u^{\prime }\in N{w}^{\prime }$, and an arbitrary fixed $z\in [1,T]$, we obtain

$$\underset{z\to \infty }{lim\; sup}diamN\Omega \left(z\right)\le p_{\varphi }\underset{z\to \infty }{lim\; sup}diam\Omega \left(z\right).$$

(9)

Consequently, in view of (7)–(9), we deduce that

$$\psi (N\Omega )\le p_{\varphi }\psi (\Omega ).$$

Then, by Theorem 3, the operator N has at least one fixed point in $B_{\zeta }$ which is a solution of differential inclusion (1). Moreover, taking into account the fact that the set $FixN\in$ ker $\psi$ (Remark 1) and the characterization of sets belonging to ker $\psi$ (Remark 2), we conclude that all solutions of (1) are globally attractive in the sense of Definition 1. □

4. Example

We consider the following differential inclusion:

$$\left\{\begin{array}{c}{\mathcal{D}}^{\gamma }\left(\frac{dw}{d\varphi }\right)\left(z\right)\in \left[\frac{e^{-z}}{\left|u\right(z\left)\right|+z},\frac{(z-1)e^{-z}}{\left|u\right(z\left)\right|+z}\right]\subset \mathbb{R};z>1,\hfill \\ \frac{dw}{d\varphi }\left(1\right)=w_0;w\left(1\right)=w_1.\hfill \end{array}\right.$$

(10)

It is clear that (10) can be written as the differential inclusion (1), where $\varphi \left(\tau \right)=arctan\left(\right[\tau \left]\right)$ (the symbol $\left[\tau \right]$ indicates the integer value of $\tau$). Let us show that conditions $\left(H_1\right)-\left(H_3\right)$ hold. For $z\in J$ and $u,v\in \mathbb{R}$, we have

$$H(F(z,u),F(z,v))\le (z-1)e^{-z}|u-v|.$$

So $p\left(z\right)=(z-1)e^{-z}$, $F(z,0)=\left[inf\left\{\frac{e^{-z}}{z},\frac{(z-1)}{z}{e}^{-z}\right\},max\left\{\frac{e^{-z}}{z},\frac{(z-1)}{z}{e}^{-z}\right\}\right]$, hence

$$\parallel F(z,0)\parallel =max\left\{\frac{e^{-z}}{z},\frac{(z-1)}{z}{e}^{-z}\right\}=q\left(z\right).$$

Notice that the functions $p,q$ are continuous and bounded on J. Next, for a fixed $z\in J$, we have $p_{\varphi }<1$. This estimate shows that $p^{*}$ and $q^{*}$ are finite quantities. Consequently, from Theorem 5, the differential inclusion (10) has at least solution in the space ${\mathcal{B}}_{\varphi }\left(J\right)$ and solutions of (10) are globally attractive.