Measure of noncompactness and fractional integro-differential equations with state-dependent nonlocal conditions in Fréchet spaces

1.   Introduction

In this paper we discuss the existence of mild solutions defined on an unbounded interval for semilinear integro-differential equations of fractional order of the form

where $1 < \alpha < 2$ and $A: D(A)\subset E\to E$ is a closed linear operator, and $(E, \|\cdot\|)$ is a Banach space. The convolution integral in the equation is known as the Riemann–Liouville fractional integral, $f: {\mathbb R}_{+}\times {\mathcal{C}}\to E$, $\sigma:C([-r, +\infty), E)\to {\mathbb R}_{+}$, $G: {\mathbb R}_{+}\times C([-r, +\infty), E)\to {\mathcal{C}}$ and $\rho:{\mathbb R}_{+}\times{\mathcal{C}}\to{\mathbb R}_+$, are suitable functions. For any continuous function $y$ defined on $[-r, +\infty)$ and any $t\in[0, +\infty)$, we denote by $y_{t}$ the element of ${\mathcal{C}}$ defined by $y_t(\theta) = y(t+\theta)$ for $\theta \in [-r, 0]$.

In this paper we discuss some existence results for fractional integro-differential equations with state dependent nonlocal conditions. Often, proposed nonlocal conditions generalize several types of nonlocal conditions studied in the literature. For the importance of nonlocal conditions in different fields we refer to ^[11,12] and the references therein. In ^[22], Hernandez and O'Regan introduced a new type of nonlocal conditions, which they called state-dependent nonlocal conditions. Recently, in ^[21], Hernandez studied the existence of mild and strict solutions for a class of abstract differential equations with state-dependent delay.

The problem of existence of solutions of the Cauchy problem for fractional integro-differential equations has been studied in numerous works; we refer the reader to books by Abbas and Benchohra ^[1], Kilbas et al. ^[23], Lakshmikantham et al. ^[25], and to papers by Anguraj et al. ^[5], Balachandran et al. ^[7] and Benchohra and Litimein ^[9]. Cuevas et al. ^[13,14,15] studied S-asymptotically $\omega$-periodic solutions. Recently, Wang and Chen ^[34] considered a class of retarded integro-differential equations with nonlocal initial conditions where existence of solutions are given over the half-line $[0, \infty)$. Using the nonlinear alternative of Leray–Schauder type Agarwal et al. ^[2] studied the existence of mild solutions to a class of fractional order integro-differential equations with state-dependent delay.

In this paper we use a recent generalization of the classical Darbo fixed point theorem for Fréchet spaces associated with the concept of measures of noncompactness. It is well known that the measure of noncompactness provides an excellent tool for establishing the existence of solutions of nonlinear differential equations. More details are found in Abbas and Benchohra ^[1], Akhmerov et al. ^[3], ^[4], Banas and Goebel ^[8], Guo et al. ^[20], Olszowy ^[28,29,30], Olszowy and Wȩdrychowicz ^[31], and the references therein.

We derive some sufficient conditions for the existence of solutions of fractional integro-differential equations with state dependent nonlocal conditions in Fréchet spaces. The concept of measure of noncompactness in Fréchet spaces is applied to achieve our results.

The work is organized as follows. In Section 2, some preliminary facts are introduced which will be used throughout the following sections. The main results are presented in Section 3, where we prove existence of mild solutions for problems $(1.1)-(1.2)$. The last section is devoted to an illustrative example.

2.   Preliminaries

Let $I: = [0, T]$ where $T > 0$. A measurable function $y:I\to E$ is Bochner integrable if and only if $\|y\|$ is Lebesgue integrable.

By $B(E)$ we denote the Banach space of bounded linear operators from $E$ into $E$, with norm

Let $L^{1}(I, E)$ denote the Banach space of measurable functions $y:I\to E$ which are Bochner integrable normed by

Let $C(I, E)$ be the Banach space of continuous functions from $I$ into $E$ with the norm

The Laplace transformation of a function $f\in L^1({\mathbb R}_{+}, E)$ is defined by

if the integral is absolutely convergent for $Re(\lambda) > \omega.$ In order to define the mild solution of the problems $(1.1)-(1.2)$ we recall the following definition.

Definition 2.1. Let $A$ be a closed and linear operator with a dense domain $D(A)$ defined on a Banach space $E.$ We call $A$ the generator of a solution operator if there exists $\omega > 0$ and a strongly continuous function $S :{\mathbb R}_{+} \to B(E)$ such that

and

In this case, $S(t)$ is called the solution operator generated by $A.$

The following result is a direct consequence of [26, Proposition 3.1 and Lemma 2.2].

Proposition 2.2. Let $\{S(t)\}_{t\geq0}\subset B(E)$ be the solution operator with generator $A.$ Then the following conditions are satisfied:

a) $S(t)$ is strongly continuous for $t\geq 0$ and $S(0) = I.$

b) $S(t)D(A)\subset D(A)$ and $AS(t)x = S(t)Ax$ for all $x\in D(A), \ t\geq0.$

c) For every $x\in D(A)$ and $t\geq 0,$

d) Let $x\in D(A).$ Then $\int_0^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}S(s)xds\in D(A)$ and

Remark 2.3. The concept of a solution operator, as defined above, is closely related to the concept of a resolvent family (see Prüss ^[33]). Because of the uniqueness of the Laplace transform, in the border case $\alpha = 1$, the family $S(t)$ corresponds to a $C_0$ semigroup (see ^[18]), whereas in the case $\alpha = 2$, a solution operator corresponds to the concept of a cosine family (see ^[6]).

More information on the $C_0-$semigroups and sine families can be found in ^[18,19,32].

Definition 2.4. A solution operator $\{S(t)\}_{t > 0}$ is called uniformly continuous if

Let $C({\mathbb R}_{+})$ be the Fréchet space of all continuous functions $\nu$ from ${\mathbb R}_{+}$ into $E$, equipped with the family seminorms

and the distance

We recall the following definition of the notion of a sequence of measures of noncompactness ^[16,17].

Definition 2.5. Let ${\mathcal{M}}_{X}$ be the family of all nonempty and bounded subsets of a Fréchet space $X$. A family of functions $\{\mu_{n}\}_{n\in{\mathbb N}}$ where $\mu_{n}:{{\mathcal{M}}_{X}}\to [0, \infty)$ is said to be a family of measures of noncompactness in the real Fréchet space $X$ if it satisfies the following conditions for all $B, B_{1}, B_{2}\in {\mathcal{M}}_{X}$:

(a) $\{\mu_{n}\}_{n\in{\mathbb N}}$ is full; that is, $\mu_{n}(B) = 0$ for $n\in{\mathbb N}$ if and only if $B$ is precompact,

(b) $\mu_{n}(B_{1})\leq \mu_{n}(B_{2})$ for $B_{1}\subset B_{2}$ and $n\in{\mathbb N},$

(c) $\mu_{n}(Conv B) = \mu_{n}(B)$ for $n\in{\mathbb N},$

(d) If $\{B_{i}\}_{i = 1}^\infty$ is a sequence of closed sets from ${\mathcal{M}}_{X}$ such that $B_{i+1}\subset B_{i}, i = 1, \ldots,$ and if $\lim_{i\to \infty}\mu_{n}(B_{i}) = 0$, for each $n\in{\mathbb N}$, then the intersection set $B_{\infty}: = \cap_{i = 1}^{\infty} B_{i}$ is nonempty

Some properties:

(e) We say the family of measures of noncompactness $\{\mu_n\}_{n\in {\mathbb N}}$ is homogenous if $\mu_{n}(\lambda B) = |\lambda|\mu_{n}(B);$ for $\lambda \in {\mathbb R}$ and $n\in {\mathbb N}$.

(f) If the family $\{\mu_n\}_{n\in {\mathbb N}}$ satisfies the condition $\mu_{n}(B_1 + B_2)\leq \mu_{n}(B_1)+ \mu_{n}(B_2),$ for $n\in {\mathbb N}$, it is called subadditive.

(g) The family $\{\mu\}_{n\in {\mathbb N}}$ is sublinear if both conditions $(e)$ and $(f)$ hold.

(h) We say that the family of measures $\{\mu_n\}_{n\in {\mathbb N}}$ has the maximum property if

(i) The family of measures of noncompactness $\{\mu_n\}_{n\in {\mathbb N}}$ is said to be regular if and only if the conditions (a), (g) and (h) hold; (full sublinear and has the maximum property).

Example 2.6. Let $X = C({\mathbb R}_{+})$. For $B\in {\mathcal{M}}_{X}, x\in B, n \in {\mathbb N}$ and $\varepsilon > 0$, let us denote by $\omega^n(x, \varepsilon)$, for $n \in {\mathbb N}$, the modulus of continuity of the function $x$ on the interval $[0, n]$; that is

Further, let us put

and

The family of mappings $\{\beta_n\}_{n\in{\mathbb N}}$ where $\beta_n : {\mathcal{M}}_{X}\to[0, \infty)$, satisfies the conditions $(a)-(d)$ from Definition 2.5.

Definition 2.7. A nonempty subset $B\subset X$ is said to be bounded if for $n \in {\mathbb N}$, there exists $M_n > 0$ such that

Lemma 2.8. ^[10] If $Y$ is a bounded subset of a Banach space $X$, then for each $\varepsilon > 0$ there is a sequence $\{y_k\}_{k = 1}^\infty \subset Y$ such that

where $\mu$ is a Kuratowskii measure of noncompactness on $X$.

Lemma 2.9. ^[27] If $\{u_k\}_{k = 1}^\infty \subset L^1({\mathbb R}_{+}, E)$ is uniformly integrable, then $\mu (\{u_k(\cdot)\}_{k = 1}^\infty)$ is measurable and

where $\mu$ is a Kuratowskii measure of noncompactness on $E$.

Definition 2.10. Let $\Omega$ be a nonempty subset of a Fréchet space $X$, and let $A:\Omega\to X$ be a continuous operator which transforms bounded subsets into bounded ones. One says $A$ satisfies the Darbo condition with constants $\{k_n\}_{n\in {\mathbb N}}$ with respect to a family of measures of noncompactness $\{\mu_n\}_{n\in {\mathbb N}}$, if

for each bounded set $B\subset \Omega$ and $n\in {\mathbb N}$. If $k_{n} < 1$, $n\in{\mathbb N}$ then $A$ is called a contraction with respect to $\{\mu_n\}_{n\in {\mathbb N}}$.

In the sequel we will make use of the following generalization of the classical Darbo fixed point theorem for Fréchet spaces.

Theorem 2.11. ^[16,17] Let $\Omega$ be a nonempty, bounded, closed, and convex subset of a Fréchet space $X$ and let $V:\Omega\to \Omega$ be a continuous mapping. Suppose that $V$ is a contraction with respect to a family of measures of noncompactness $\{\mu_n\}_{n\in {\mathbb N}}$. Then $V$ has at least one fixed point in the set $\Omega$.

3.   Existence of mild solutions

In this section, we present the main results for the global existence of solutions for our problem. Let us start by defining what we mean by mild solution of the problems $(1.1)-(1.2)$.

Definition 3.1. A function $y\in C([-r, +\infty), E)$ is said to be a mild solution of $(1.1)-(1.2)$ if $y_{0} = G(\sigma(y), y)$ for all $t\in [-r, 0]$, and $y$ satisfies the integral equation,

Let us introduce the following hypotheses:

$(H1)$ There exists a constant $M > 1$ such that

$(H2)$ The function $t\longmapsto f(t, y)$ is measurable on ${\mathbb R}_{+}$ for each $y\in {\mathcal{C}}$, and the function $y\longmapsto f(t, y)$ is continuous on ${\mathcal{C}}$ for a.e. $t\in {\mathbb R}_{+}$.

$(H3)$ There exists a function $p\in L_{loc}^{1}({\mathbb R}_{+}, {{\mathbb R}}_+)$ and a continuous nondecreasing function $\psi :{\Bbb R}_+\to [0, \infty)$ such that

$(H4)$ For each bounded set $B\subset {\mathcal{C}}$ and for each $t\in [0, n], \ n\in{\mathbb N}$, we have

where $\mu$ is a measure of noncompactness on the Banach space $E$.

$(H5)$ For each $n\in {\mathbb N},$ there exists $L_n > 0$ such that

$(H6)$ For each $n\in {\mathbb N},$ there exists $K_n > 0$ such that

for any bounded $B\subset C([-r, \infty), E)$.

$(H7)$ For each $n\in {\mathbb N},$ there exists $R_{n} > 0$ such that

where for $n\in {\mathbb N},$

Define on $C([-r, \infty), E)$ the family of measures of noncompactness by

and $D(t) = \{v(t)\in E; v\in D\}, \ t\in [-r, n].$

Remark 3.2. Notice that if the set $D$ is equicontinuous, then $\omega_0^n(D) = 0.$

Theorem 3.3. Assume $(H1)-(H7)$ are satisfied, and for each $n\in {\mathbb N}$,

Then the problems $(1.1)-(1.2)$ has at least one mild solution.

Proof. Consider the operator $N: C([-r, +\infty), E)\to C([-r, +\infty), E)$ defined by

Clearly, the fixed points of the operator $N$ are solutions of the problems $(1.1)-(1.2).$ We define the ball

For any $n\in {\mathbb N},$ and each $y\in B_{R_{n}}$ and $t\in[0, n],$ by $(H1)-(H3), (H5)$ and $(H7),$ we have

Thus

This proves that $N$ transforms the ball $B_{R_{n}}$ into itself.

We shall show that the operator $N:B_{R_{n}}\to B_{R_{n}}$ satisfies all the assumptions of Theorem $2.11$.

Step 1: $N:B_{R_{n}}\to B_{R_{n}}$ is continuous.

Let $\{y^{k}\}_{k\in {\mathbb N}}$ be a sequence such that $y^{k}\to y$ in $B_{R_{n}}$. Then for each $t\in[0, n],$ we have

Since $y^{k}\longrightarrow y$ as $k\longrightarrow \infty$, the Lebesgue dominated convergence theorem implies that

Thus $N$ is continuous.

Step 2: $N(B_{R_{n}})$ is bounded.

Since $N(B_{R_{n}})\subset B_{R_{n}}$ and $B_{R_{n}}$ is bounded, then $N(B_{R_{n}})$ is bounded.

Step 3: For each equicontinuous subset $D$ of $B_{R_{n}}, \mu_{n}(N(D))\leq l_{n}\mu_{n}(D)$.

From Lemmas 2.8 and 2.9, for any equicontinuous set $D\subset B_{R_{n}}$ and any $\epsilon > 0$, there exists a sequence $\{y^{k}\}_{k = 1}^{\infty}\subset D$, such that for all $t\in [0, n]$, we have

Since $\epsilon > 0$ is arbitrary, then

Thus

As a consequence of Steps 1 to 3, together with Theorem 2.11, we can conclude that $N$ has at least one fixed point in $B_{R_{n}}$ which is a mild solution of problems $(1.1)-(1.2).$

4.   An Example

We consider the following fractional integro-differential equation with state dependent delay

where $1 < \mu < 2$, $\eta\in C({\mathbb R}, [0, r])$, $\alpha\in C({\mathbb R}, {\mathbb R})$, $\sigma\in C((C[-r, +\infty), E), [0, +\infty))$, $Q$ is a continuous function from $[0, +\infty)$ to ${\mathbb R}$ and $L_{\xi}$ stands for the operator with respect to the spatial variable $\xi$ which is given by

Consider $E = L^{2}([0, \pi], {{\mathbb R}})$ and the operator $A: = L_{\xi}:D(A)\subset E\to E$ with domain

Clearly $A$ is densely defined in $E$ and is sectorial. Hence $A$ is a generator of a solution operator on $E$.

Set

Thus, under the above definitions the problem (4.1) can be represented by the problems $(1.1)-(1.2)$. Furthermore, we can check that the assumptions of Theorem 3.3 hold. Consequently, Theorem 3.3 implies that the problem (4.1) has at least one mild solution on $[-r, +\infty)$.

Acknowledgement

The authors are grateful to the referees for the careful reading of the paper and for their helpful remarks.

Conflict of interest

The authors declare no conflict of interest.