In this work, we study the existence of solutions for the following functional integrodifferential equation with state-dependent delay,

where $A:D(A)\subset X\to X$ is the infinitesimal generator of a $C_0$-semigroup ${(T(t))}_{t\ge 0}$ on a Banach space $X;$ $F,G,g:[0,b]\times X\to X$ are given functions; $\varphi \in C([-r,0],X);$ ${\rho }_0:[0,b]\to {ℝ}_{+};\rho$ is a positive bounded continuous function on $[0,b]\times X,$ and $r$ is the maximal delay defined by

We assume that

and we are interested into mild solutions of problem (1.1), in the space $C([0,b],X)$. The results are based on Banach and Sadovskii fixed point theorems, the nonlinear alternative for condensing maps, Bihari’s inequality and on the technique of equivalent norms.

In [23], the author investigated the existence and regularity of solutions to the following integrodifferential equation

Recently, Ezzinbi and Koumla [9] considered equation (1.3) with finite and infinite delay. They obtained their result via Banach’s fixed point theorem. Koumla and al. [14, 15] have considered a class of perturbed semilinear neutral functional integrodifferential equations with finite delay, under some approaches that are commonly used: the contraction mapping principle and Schauder’s fixed point theorem.

Functional differential equations with state-dependent delay appear frequently in applications as models of real processes and for this reason the study of this type of equations has received great attention in recent years. For the theory of differential equations with state-dependent delay and its applications, we refer the reader to the handbook by Cañada et al. [5] and the papers [1, 6, 8, 9, 10, 12, 13, 17]. The study of partial differential equations with state-dependent delay has been initiated recently. For the case when $A$ generates a $C_0$-semigroup, we refer to Hernandez et al. [13]. In the case where $g=0,$ problem (1.1) has been studied by several authors. Thus, Belmekki et al. [3] established the existence of solutions of problem (1.1) when $g=0$ and$G=0.$ The authors proved the existence of solutions by using a Leray-Schauder type alternative and the semigroup theory. In [16], the author uses fixed point arguments, namely the Rothe type fixed point theorem to establish the controllability of a class of retarded semilinear systems. The solutions are understood in the sense of mild solutions too.

Our work is mainly motivated by papers [14] and [23]. We generalize the method used in [3] and [14] to derive the existence of mild solutions of (1.1). The obtained results represent a generalization and a continuation of those from papers [3, 14].

The organization of this work is as follows: in Section 2, we collect some useful notions and results; in Section 3, we give our main results: an existence and uniqueness result based on Banach’s contraction principle and the technique of equivalent norms; an existence result based on Sadovskii’s fixed point theorem; and an existence result based on the nonlinear alternative for condensing maps. Finally, in Section 4 we present applications to a reaction-diffusion integrodifferential equation with state-dependent delay.

In this section, we recall some notions and results that we need in the following.

Throughout the paper, $X$ is a Banach space, $A:D\left(A\right)\subset X\to X$ is a linear operator which generates a $C_0$-semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ on $X.$ For more details, we refer to [18, 22]. Recall that for such a semigroup, there exist $N>0$ and $\omega \in ℝ$ such that

where $\left|T\left(t\right)\right|$ is the norm of the bounded linear operator $T\left(t\right).$

The linear space $D(A)$ equipped with the graph norm defined by

is a Banach space. In this sense, we may speak about the space $C([0,b],D\left(A\right)).$

In regards with the abstract initial value problem

where $v\in X$ and $q\in C([0,b],X),$ we have the following definition (see [18, p 486]): by the mild solution of problem (2.3), one means the function $u\in C([0,b],X)$ given by

If $v\in D\left(A\right)$ and either $q\in C^1([0,b],X)$ or $q\left(t\right)\in D\left(A\right)$ for all $t\in [0,b],$ then the function $u$ given by (2.4) is a strong solution of (2.3), i.e., $u\in C^1([0,b],X)\cap C([0,b],D\left(A\right)),$ $u\left(0\right)=u_0$ and $u$ satisfies pointwise the differential equation.
Also recall the notion of (Kuratowski) measure of noncompactness,

Some basic properties of $\mu (.)$ are given in the following lemma.

Recall that a continuous map $P$ acting in $E$ is said to be condensing if for each bounded subset $B$ of the domain of $P,$ with $\mu \left(B\right)>0.$ For example, the sum of a contraction map with a completely continuous map is condensing.

In this paper we shall use the technique of equivalent norms related to the space $C([0,b],X),$ primary endowed with the max norm

As in [4], the space $C([0,b],X)$ can be equipped with an equivalent norm defined by

for some $\theta >0.$ It is easy to check that $|\cdot {|}_{\theta }$ is a norm on $C([0,b],X)$ and that

which proves the equivalence of the norms $|\cdot {|}_{\mathrm{\infty }}$ and $|\cdot {|}_{\theta }$ on $C([0,b],X).$ Note that the use of an equivalent norm $|\cdot {|}_{\theta }$ with a suitable large enough $\theta >0$ is extremely convenient when dealing with Volterra type equations (see [15, 14, 19, 20]).

We conclude this preliminary section by stating two basic results that are used in this paper. The first one is Bihari’s inequality (see, e.g., [21]).

Another tool in our approach is the nonlinear alternative for condensing maps (see, e.g., [11, p. 133]).

In view of (2.4), by a mild solution of problem (1.1) we mean the function $u\in C([-r,b],X)$ with $u\left(t\right)=\varphi \left(t\right)$ for $t\in [-r,0],$ such that

for all $t\in [0,b].$ Thus, $u\in C([-r,b],X)$ is a mild solution of (1.1) if it is a fixed point of the operator $P:C_{\varphi }([0,b],X)\to C_{\varphi }([0,b],X)$ given by

where $C_{\varphi }([0,b],X)=\{u\in C([0,b],X):u\left(0\right)=\varphi \left(0\right)\}$ and it is understood that $u\left(t\right)=\varphi \left(t\right)$ for $t\in [-r,0].$ Note that in view of (1.2), the operator $P$ is well-defined since

Also note that it is a causal operator in the sense of Corduneanu [7], which makes possible to use the technique of equivalent norms.