Posts by Radu Precup

Abstract

In this work, we establish sufficient conditions for the existence of solutions for some functional integrodifferential equations with state-dependent delay in Banach spaces. We use $C_{0}$-semigroup theory and a fixed point approach based on Banach and Sadovskii’s fixed point theorems, nonlinear alternative for condensing maps, Bihari’s inequality and the technique of equivalent norms. Applications ares provided to a reaction-diffusion equation with state-dependent delay.

Authors

Sylvain Koumla
Département de Mathématiques, Faculté des Sciences et Techniques, Université Adam Barka d’Abéché, Abéché, Chad

Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania

Ngarkodje Ngarasta
Département de Mathématiques, Faculté des Sciences Exactes et Appliquées, Université de N’Djaména, N’Djaména, Chad

Keywords

Mild solution; Functional integrodifferential equation with state-dependent delay; C₀- $semigroup$ Nonlinear alternative; Condensing map

Paper coordinates

S. Koumla, R. Precup, N. Ngarasta, Existence results for some functional integrodifferential equations with state-dependent delay, Differ. Eq. Dyn. Syst., (2023). https://doi.org/10.1007/s12591-023-00661-y

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About this paper

Journal

Differential Equations and Dynamical Systems

Publisher Name

Springer International Publishing AG

DOI

https://doi.org/10.1007/s12591-023-00661-y

Print ISSN

09713514

Online ISSN

09746870

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Paper (preprint) in HTML form

Existence results for some functional integrodifferential equations with state-dependent delay

Sylvain Koumla, Radu Precup, Ngarkodje Ngarasta S. Koumla, Université Adam Barka d’Abeche, Faculté des Sciences et Techniques, Département de Mathématiques, Chad skoumla@gmail.com R. Precup, Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania & Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, 400110 Cluj-Napoca, Romania r.precup@math.ubbcluj.ro N. Ngarasta, Université de N’Djaména, Faculté des Sciences Exactes et Appliquées, Département de Mathématiques, Chad ngarkodje@gmail.com

Abstract.

In this work, we establish sufficient conditions for the existence of solutions for some functional integrodifferential equations with state-dependent delay in Banach spaces. We use $C_{0}$ -semigroup theory and a fixed point approach based on Banach and Sadovskii’s fixed point theorems, nonlinear alternative for condensing maps, Bihari’s inequality and the technique of equivalent norms. Applications ares provided to a reaction-diffusion equation with state-dependent delay.

Keywords and Phrases: mild solution, functional integrodifferential equation with state-dependent delay, $C_{0}$ -semigroup, nonlinear alternative, condensing map.

1. Introduction

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

\left\{\begin{array}[]{lll}\frac{d}{dt}\left[u(t)-G(t,u(t-\rho_{0}\left(t% \right)))\right]&=&A\left[u(t)-G(t,u(t-\rho_{0}\left(t\right)))\right]\vspace{% 2mm}\\ &+&\displaystyle\int_{0}^{t}g(t-s,u(s))ds+F(t,u(t-\rho(t,u(t)))),~{}~{}\text{% for}~{}~{}t\in[0,b]\vspace{2mm}\\ u(t)&=&\phi(t),\ \ \ \text{for }t\in[-r,0]\end{array}\right.

(1.1)

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

r=\sup_{\begin{subarray}{c}t\in\left[0,b\right]\\ z\in X\end{subarray}}\rho(t,z)<\infty,\ \ \ \ 0\leq\rho_{0}\left(t\right)\leq r.

We assume that

G\left(0,\phi\left(-\rho_{0}\left(0\right)\right)\right)=0

(1.2)

and we are interested into mild solutions of problem (1.1), in the space $C\left(\left[0,b\right],X\right)$ . 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

\left\{\begin{array}[]{l}u^{\prime}(t)=Au(t)+\displaystyle\int_{0}^{t}g(t-s,u(% s))ds+f(t),\quad t\geq 0\vspace{2mm}\\ u(0)=u_{0}\in X.\end{array}\right.

(1.3)

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.

2. Preliminaries

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

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

\left|T\left(t\right)\right|\leq Ne^{\omega t},\ \ \ t\geq 0,

(2.1)

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

\left|y\right|_{D\left(A\right)}=\left|y\right|_{X}+\left|Ay\right|_{X}

(2.2)

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

In regards with the abstract initial value problem

\left\{\begin{array}[]{l}u^{\prime}\left(t\right)=Au\left(t\right)+q\left(t% \right),\ \ \ t\in\left[0,b\right]\\ u\left(0\right)=v,\end{array}\right.

(2.3)

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

u\left(t\right)=T\left(t\right)v+\int_{0}^{t}T\left(t-s\right)q\left(s\right)% ds,\ \ \ t\in\left[0,b\right].

(2.4)

If $v\in D\left(A\right)$ and either $q\in C^{1}\left(\left[0,b\right],X\right)$ or $q\left(t\right)\in D\left(A\right)$ for all $t\in\left[0,b\right],$ then the function $u$ given by (2.4) is a strong solution of (2.3), i.e., $u\in C^{1}\left(\left[0,b\right],X\right)\cap C\left(\left[0,b\right],D\left(A% \right)\right),$ $u\left(0\right)=u_{0}$ and $u$ satisfies pointwise the differential equation.
Also recall the notion of (Kuratowski) measure of noncompactness,
$\mu(B)=\inf\left\{d>0:~{}~{}B~{}~{}\text{ can~{}be~{}covered~{}by~{}a~{}finite~{}number~{}of~{}set~{}of~{}diameter}<d% \right\}.$
Some basic properties of $\mu(.)$ are given in the following lemma.

Lemma 2.1 ([2]).

Let $E$ be a Banach space and $B,C\subseteq E$ be bounded sets. Then
$(i)$ $\mu(B)=0$ if and only if $B$ is relatively compact;
$(ii)$ $\mu(B)=\mu(\overline{B})=\mu(\overline{co}B),$ where $\overline{co}B$ is the closed convex hull of $B;$
$(iii)$ $\mu(B)\leq\mu(C)$ if $B\subseteq C;$
$(iv)$ $\mu(B+C)\leq\mu(B)+\mu(C);$
$(v)$ $\mu(B\cup C)\leq\max\left\{\mu(B),\mu(C)\right\};$
$(vi)$ $\mu(B(0,r))=2r,$ where $B(0,r)=\left\{x\in E:|x|\leq r\right\}$ and $\dim E=+\infty.$

Recall that a continuous map $P$ acting in $E$ is said to be condensing if $\ \mu\left(P\left(B\right)\right)<\mu\left(B\right)$ 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 $C\left(\left[0,b\right],X\right),$ primary endowed with the max norm

\left|u\right|_{\infty}=\left|u\right|_{C\left(\left[0,b\right],X\right)}=\max% _{t\in\left[0,b\right]}\left|u\left(t\right)\right|_{X}.

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

\left|u\right|_{\theta}=\max_{t\in\left[0,b\right]}\left(\left|u\right|_{C% \left(\left[0,t\right],X\right)}e^{-\theta t}\right),

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

e^{-\theta b}\left|u\right|_{\infty}\leq\left|u\right|_{\theta}\leq\left|u% \right|_{\infty},

which proves the equivalence of the norms $\left|\cdot\right|_{\infty}$ and $\left|\cdot\right|_{\theta}$ on $C\left(\left[0,b\right],X\right).$ Note that the use of an equivalent norm $\ \left|\cdot\right|_{\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]).

Theorem 2.2 (Bihari inequality).

Assume that $f$ and $v$ are nonnegative continuous functions on $\ [0,\tau),$ and $h$ is a positive nondecreasing continuous function on $(0,+\infty)$ such that for all $\ t\in[0,\tau),$

v(t)\leq c_{0}+\int_{0}^{t}f(s)h(v(s))ds,

where $c_{0}>0$ is a constant. Then for all $\ t\in[0,\tau_{1}],$ one has

v(t)\leq\Phi^{-1}\left(\Phi(c_{0})+\int_{0}^{t}f(s)ds\right),

where

\Phi(x)=\int_{x_{0}}^{x}\frac{1}{h(s)}ds,\ ~{}~{}x\geq x_{0}>0,

$\Phi^{-1}$ is the inverse function of $\Phi,$ and

\tau_{1}=\sup\left\{t\in[0,\tau):\Phi(+\infty)\geq\Phi(c_{0})+\int_{0}^{t}f(s)% ds\right\}.

(2.5)

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

Theorem 2.3 (Nonlinear alternative).

Let $E$ be a Banach space and $\ \mathcal{D}\subset E$ be closed convex with $\ 0\in\mathcal{D}.$ Let $\ P:\mathcal{D}\rightarrow\mathcal{D}$ be a condensing map. Then either (i) $P$ has a fixed point, or (ii) the set $\ \Lambda=\{u\in\mathcal{D}:u=\lambda P(u),\ 0<\lambda<1\}$ is unbounded.

3. Main results

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

	$\displaystyle u\left(t\right)$	$\displaystyle=$	$\displaystyle T(t)\phi(0)+G(t,u(t-\rho_{0}\left(t\right)))+\int_{0}^{t}T(t-s)% \int_{0}^{s}g(s-\tau,u(\tau))d\tau ds$
			$\displaystyle+\int_{0}^{t}T(t-s)F(s,u(s-\rho(s,u(s))))ds,$

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

	$\displaystyle(Pu)(t)$	$\displaystyle=$	$\displaystyle T(t)\phi(0)+G(t,u(t-\rho_{0}\left(t\right)))+\int_{0}^{t}T(t-s)% \int_{0}^{s}g(s-\tau,u(\tau))d\tau ds$
			$\displaystyle+\int_{0}^{t}T(t-s)F(s,u(s-\rho(s,u(s))))ds\ \ \ \ \left(t\in% \left[0,b\right]\right),$

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

\left(Pu\right)\left(0\right)=T(0)\phi(0)+G(0,u(0-\rho_{0}\left(0\right)))=% \phi\left(0\right)+G\left(0,\phi\left(-\rho_{0}\left(0\right)\right)\right)=% \phi\left(0\right).

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

3.1. An existence and uniqueness result

Our first result gives the existence and uniqueness of the mild solution of (1.1), in the set $C_{\phi}\left(\left[0,b\right],X\right)$ assuming that the delay is not state-dependent.The result is obtained via Banach’s contraction principle with respect to an equivalent norm $\ \left|\cdot\right|_{\theta}\$ on $C\left(\left[0,b\right],X\right),$ with a suitable large enough number $\ \theta>0,$ and using global Lipschitz conditions on $\ F,G,g.$ Here are the hypotheses:

(H₁):

$F,G,g:[0,b]\times X\rightarrow X$ are continuous and Lipschitzian with respect to the second argument, that is, there are constants $\ L_{F},L_{G},L_{g}\geq 0\$ such that

$\displaystyle\left\|F(t,u)-F(t,v)\right\|_{X}$	$\displaystyle\leq$	$\displaystyle L_{F}\left\|u-v\right\|_{X},$
$\displaystyle\left\|G(t,u)-G(t,v)\right\|_{X}$	$\displaystyle\leq$	$\displaystyle L_{G}\left\|u-v\right\|_{X},$
$\displaystyle\left\|g(t,u)-g(t,v)\right\|_{X}$	$\displaystyle\leq$	$\displaystyle L_{g}\left\|u-v\right\|_{X},$

for all $t\in[0,b]$ and $u,v\in X.$

Theorem 3.1.

Assume that condition (H₁) holds and $\ L_{G}<1.$ In addition assume that $\rho\left(t,z\right)=\rho\left(t\right)$ for all $t\in\left[0,b\right],$ $z\in X$ . Then problem (1.1) has a unique mild solution $u\in C\left(\left[-r,b\right],X\right).$

Proof.

We look for a fixed point $u\in C_{\phi}\left(\left[0,b\right],X\right)$ of the operator $P$ given by (3). In order to apply Banach’s contraction principle, we need to prove that $P$ is a contraction on $C_{\phi}\left(\left[0,b\right],X\right)$ with respect the metric introduced by a suitable norm $\left|\cdot\right|_{\theta}$ on $C\left(\left[0,b\right],X\right).$ To show this, consider two arbitrary functions $u,v\in C_{\phi}\left(\left[0,b\right],X\right)$ and any $t\in\left[0,b\right].$ Using (2.1) and $\mathbf{(}$ H $\mathbf{{}_{1})}$ we have

$\displaystyle\left\|\left(Pu\right)\left(t\right)-\left(Pv\right)\left(t\right)% \right\|_{X}$	$\displaystyle\leq$	$\displaystyle L_{G}\left\|u-v\right\|_{C([0,t],X)}+NL_{g}e^{\omega b}\int_{0}^{t% }\int_{0}^{s}\left\|u-v\right\|_{C([0,\tau],X)}d\tau ds$
		$\displaystyle+\ NL_{F}e^{\omega b}\int_{0}^{t}\left\|u-v\right\|_{C([0,s],X)}ds$
	$\displaystyle=$	$\displaystyle L_{G}\left\|u-v\right\|_{C([0,t],X)}e^{-\theta t}e^{\theta t}$
		$\displaystyle+\ NL_{g}e^{\omega b}\int_{0}^{t}\int_{0}^{s}\left\|u-v\right\|_{C(% [0,\tau],X)}e^{-\theta\tau}e^{\theta\tau}d\tau ds$
		$\displaystyle+\ NL_{F}e^{\omega b}\int_{0}^{t}\left\|u-v\right\|_{C([0,s],X)}e^{% -\theta s}e^{\theta s}ds.$

It follows that

\left|\left(Pu\right)\left(t\right)-\left(Pv\right)\left(t\right)\right|_{X}% \leq\left(L_{G}e^{\theta t}+NL_{g}e^{\omega b}\int_{0}^{t}\int_{0}^{s}e^{% \theta\tau}d\tau ds+NL_{F}e^{\omega b}\int_{0}^{t}e^{\theta s}ds\right)\left|u% -v\right|_{\theta}.

(3.3)

Since

\int_{0}^{t}e^{\theta s}ds=\frac{1}{\theta}\left(e^{\theta t}-1\right)\leq% \frac{1}{\theta}e^{\theta t},\ \ \ \int_{0}^{t}\int_{0}^{s}e^{\theta\tau}d\tau ds% \leq\frac{1}{\theta^{2}}e^{\theta t},

we deduce that

\left|\left(Pu\right)\left(t\right)-\left(Pv\right)\left(t\right)\right|_{X}% \leq\left[L_{G}+Ne^{\omega b}\left(\frac{L_{g}}{\theta^{2}}+\frac{L_{F}}{% \theta}\right)\right]e^{\theta t}\left|u-v\right|_{\theta}.

For $t\leq t_{1}\leq b,$ this inequality yields

\left|Pu-Pv\right|_{C\left(\left[0,t_{1}\right],X\right)}\leq\left[L_{G}+Ne^{% \omega b}\left(\frac{L_{g}}{\theta^{2}}+\frac{L_{F}}{\theta}\right)\right]e^{% \theta t_{1}}\left|u-v\right|_{\theta}

Dividing by $e^{\theta t_{1}}$ and taking the maximum for $t_{1}\in\left[0,b\right]$ gives

\left|Pu-Pv)\right|_{\theta}\leq\left[L_{G}+Ne^{\omega b}\left(\frac{L_{g}}{% \theta^{2}}+\frac{L_{F}}{\theta}\right)\right]\left|u-v\right|_{\theta}.

Therefore, in view of the assumption $L_{G}<1,$ for $\theta>0$ large enough that

L_{G}+Ne^{\omega b}\left(\frac{L_{g}}{\theta^{2}}+\frac{L_{F}}{\theta}\right)<1.

The operator $P$ is a contraction on $C_{\phi}\left(\left[0,b\right],X\right)$ with respect to the norm $\ \left|\cdot\right|_{\theta},$ and according to Banach’s fixed point theorem it has in $C_{\phi}\left(\left[0,b\right],X\right)$ a unique fixed point. ∎

3.2. Existence under compactness conditions

For the next results, instead of the Lipschitz conditions on $F$ and $g,$ we shall assume that semigroup $\left(T\left(t\right)\right)_{t\geq 0}$ is compact. The existence of solutions to problem (1.1) will be established via topological fixed point theorems involving condensing operators. For these results the delay can be state-dependent. We start with a result about the condensing property of our operator $P .$

We need the following conditions:

(H₂) (i):

The map $\ F:[0,b]\times X\rightarrow X$ is continuous and map bounded sets into bounded sets;

(ii):

The map $\ g:[0,b]\times X\rightarrow X$ is continuous and its primitive with respect to $t,$ that is the map $\widehat{g}:[0,b]\times X\rightarrow X\$ given by

\widehat{g}\left(t,x\right)=\int_{0}^{t}g\left(s,x\right)ds\ \ \ \left(t\in% \left[0,b\right],\ x\in X\right),

maps bounded sets into bounded sets;

(iii):

The map $\ G:\left[0,b\right]\times X\rightarrow X$ is continuous and satisfies

\left|G\left(t,u\right)-G\left(t,v\right)\right|\leq L_{G}\left|u-v\right|

for all $t\in\left[0,b\right]$ and $u,v\in X,$ and $L_{G}<1;$

(iv):

The semigroup $(T(t))_{t\geq 0}$ is compact.

Lemma 3.2.

Under assumption (H₂), the operator $\ P:C_{\phi}\left(\left[0,b\right],X\right)\rightarrow C_{\phi}\left(\left[0,b% \right],X\right)$ is condensing.

Proof.

The continuity of the operator $P$ follows directly from the continuity of $\rho,G,F$ and $g .$

Clearly, from (iii), the operator $\ P_{G}:C_{\phi}\left(\left[0,b\right],X\right)\rightarrow$ $C_{\phi}\left(\left[0,b\right],X\right)\$ given by

\left(P_{G}u\right)\left(t\right)=T(t)\phi(0)+G(t,u(t-\rho_{0}\left(t\right))),

is a contraction with Lipschitz constant $L_{G}.$ It remains to show that $P-P_{G}$ is completely continuous, i.e., it maps bounded sets into relatively compact sets. Let $D$ be any bounded subset of $C_{\phi}\left(\left[0,b\right],X\right)$ and let $l_{D}$ be a bound of $D,$ i.e., $\left|u\right|_{\infty}\leq l_{D}$ for all $u\in D.$ We have to prove that the set $\ \left(P-P_{G}\right)\left(D\right)\$ is (a) uniformly bounded, (b) uniformly equicontinuous and that (c) for each $t\in\left[0,b\right],$ the set $\ \left(P-P_{G}\right)\left(D\right)\left(t\right)\$ is relatively compact in $X .$ Without loss of generality, we assume that constant $w$ in (2.1) is positive.

(a) Since $F$ maps bounded sets into bounded sets, there is a constant $C_{1}$ depending on $l_{D}$ such that $\ \left|F\left(s,u\left(s-\rho\left(s,u\left(s\right)\right)\right)\right)% \right|\leq C_{1}\$ for all $s\in\left[0,b\right]$ and $u\in D.$ Then

\left|\int_{0}^{t}T(t-s)F(s,u(s-\rho(s,u(s))))ds\right|\leq C_{1}bNe^{\omega b}

(3.4)

for all $t\in\left[0,b\right]$ and $u\in D.$ Similarly, since $\widehat{g}$ maps bounded sets into bounded sets, there is a constant $C_{2}$ depending on $l_{D}$ such that $\ \left|\int_{0}^{s}g(s-\tau,u(\tau))d\tau\right|\leq C_{2}\$ for all $s\in\left[0,b\right]$ and $u\in D.$ Hence

\left|\int_{0}^{t}T(t-s)\int_{0}^{s}g(s-r,u(r))drds\right|\leq C_{2}bNe^{% \omega b}

(3.5)

for all $t\in\left[0,b\right]$ and $u\in D.$ Now (3.4) and (3.5) show that $\ \left(P-P_{G}\right)\left(D\right)\$ is uniformly bounded on $\left[0,b\right].$

(b) To prove the uniform equicontinuity on $\left[0,b\right]$ of the set of functions $\ \left(P-P_{G}\right)\left(D\right),\$ take any $\ t,t^{\prime}\in\left[0,b\right].$ Assume $t^{\prime}<t.$ We have

			$\displaystyle\left\|\int_{0}^{t}T(t-s)F(s,u(s-\rho(s,u(s))))ds-\int_{0}^{t^{% \prime}}T(t^{\prime}-s)F(s,u(s-\rho(s,u(s))))ds\right\|$
		$\displaystyle\leq$	$\displaystyle\left\|\int_{0}^{t}T(t-s)F(s,u(s-\rho(s,u(s))))ds-\int_{0}^{t^{% \prime}}T(t-s)F(s,u(s-\rho(s,u(s))))ds\right\|$
			$\displaystyle+\left\|\int_{0}^{t^{\prime}}\left(T(t-s)-T\left(t^{\prime}-s% \right)\right)F(s,u(s-\rho(s,u(s))))ds\right\|$
		$\displaystyle\leq$	$\displaystyle C_{1}Ne^{\omega b}\left\|t-t^{\prime}\right\|+\int_{0}^{t^{\prime}% }\left\|T(t-s)-T\left(t^{\prime}-s\right)\right\|\left\|F(s,u(s-\rho(s,u(s))))% \right\|ds$
		$\displaystyle\leq$	$\displaystyle C_{1}Ne^{\omega b}\left\|t^{\prime}-t\right\|+C_{1}\int_{0}^{t^{% \prime}}\left\|T(t-s)-T\left(t^{\prime}-s\right)\right\|ds.$

Since

T\left(t-s\right)-T\left(t^{\prime}-s\right)=T\left(t^{\prime}-s\right)\left(T% \left(t-t^{\prime}\right)-I\right),

and $\ T\left(\xi\right)-I\rightarrow 0$ as $\xi\rightarrow 0,\$ one has that for any $\ \varepsilon>0,$ there is a $\ \delta_{\varepsilon}>0$ with

\left|T(t-s)-T\left(t^{\prime}-s\right)\right|\leq Ne^{\omega b}\varepsilon,

if $\ \left|t-t^{\prime}\right|\leq\delta_{\varepsilon}.$ Then

\int_{0}^{t^{\prime}}\left|T(t-s)-T\left(t^{\prime}-s\right)\right|ds\leq bNe^% {\omega b}\varepsilon

and so

			$\displaystyle\left\|\int_{0}^{t}T(t-s)F(s,u(s-\rho(s,u(s))))ds-\int_{0}^{t^{% \prime}}T(t^{\prime}-s)F(s,u(s-\rho(s,u(s))))ds\right\|$
		$\displaystyle\leq$	$\displaystyle C_{1}Ne^{\omega b}\left(1+b\right)\varepsilon=:\widetilde{C}_{1}% \varepsilon,\ \ \ \text{for \ }\left\|t-t^{\prime}\right\|\leq\delta_{% \varepsilon}.$

Analogously, there is a constant $\widetilde{C}_{2}>0$ with

			$\displaystyle\left\|\int_{0}^{t}T(t-s)\int_{0}^{s}g(s-r,u(r))drds-\int_{0}^{t^{% \prime}}T(t^{\prime}-s)\int_{0}^{s}g(s-r,u(r))drds\right\|$
		$\displaystyle\leq$	$\displaystyle\widetilde{C}_{2}\varepsilon,\ \ \ \text{for \ }\left\|t-t^{\prime% }\right\|\leq\delta_{\varepsilon}.$

Since two constants $\ \widetilde{C}_{1}$ and $\ \widetilde{C}_{2}$ only depend on $\ l_{D},$ we may infer that $\ \left(P-P_{G}\right)\left(D\right)\$ is uniformly equicontinuous in $\left[0,b\right].$

(c) The set $\left(P-P_{G}\right)\left(D\right)\left(t\right)$ is relatively compact in $X,$ for every $t\in\left[0,b\right].$ For $t=0,~{}$ it reduces to the $\left\{0\right\},$ so it is compact. Let $0<t\leq b.$ We need to show that the sets

	$\displaystyle\mathcal{M}_{1}$	$\displaystyle:$	$\displaystyle=\left\{\int_{0}^{t}T(t-s)F(s,u(s-\rho(s,u(s))))ds:\ u\in D\right% \},\$
	$\displaystyle\mathcal{M}_{2}$	$\displaystyle:$	$\displaystyle=\left\{\int_{0}^{t}T(t-s)\int_{0}^{s}g(s-\tau,u(\tau))d\tau ds:% \ u\in D\right\}$

are relatively compact in $X .$ Now, let $t>0$ and $t>\epsilon>0.$ Then

			$\displaystyle\int_{0}^{t}T(t-s)F(s,u(s-\rho(s,u(s))))ds$
		$\displaystyle=$	$\displaystyle\int_{0}^{t-\epsilon}T(t-s)F(s,u(s-\rho(s,u(s))))ds\vspace{2mm}+% \int_{t-\epsilon}^{t}T(t-s)F(s,u(s-\rho(s,u(s))))ds\vspace{2mm}$
		$\displaystyle=$	$\displaystyle T(\epsilon)\int_{0}^{t-\epsilon}T(t-\epsilon-s)F(s,u(s-\rho(s,u(% s))))ds+\int_{t-\epsilon}^{t}T(t-s)F(s,u(s-\rho(s,u(s))))ds\vspace{2mm}.$

Using (3.4) we have

\ \ \left|\int_{0}^{t-\epsilon}T(t-\epsilon-s)F(s,u(s-\rho(s,u(s))))ds\right|% \leq C_{1}bNe^{\omega b},

which together with the compactness of $\ T\left(\varepsilon\right)\$ gives

\mu\left(\left\{T(\epsilon)\int_{0}^{t-\epsilon}T(t-\epsilon-s)F(s,u(s-\rho(s,% u(s))))ds,\ \ u\in D\right\}\right)=0.\vspace{1mm}\newline

Next using (3.4) we have

\left|\int_{t-\epsilon}^{t}T(t-s)F(s,u(s-\rho(s,u(s))))ds\right|\leq% \varepsilon Ne^{\omega b}C_{1},

which based on Lemma 2.1 (vi) gives

\mu\left(\left\{\int_{t-\epsilon}^{t}T(t-s)F(s,u(s-\rho(s,u(s))))ds\vspace{2mm% }:\ u\in D\right\}\right)\leq 2\varepsilon Ne^{\omega b}C_{1}.

Then

\mu\left(\left\{\int_{0}^{t}T(t-s)F(s,u(s-\rho(s,u(s))))ds:\ u\in D\right\}% \right)\leq 2\varepsilon Ne^{\omega b}C_{1}

and letting $\varepsilon\rightarrow 0$ we obtain

\mu\left(\left\{\int_{0}^{t}T(t-s)F(s,u(s-\rho(s,u(s))))ds:\ u\in D\right\}% \right)=0,

that is the set $\mathcal{M}_{1}$ is relatively compact in $X .$ The proof of the relative compactness of $\mathcal{M}_{2}$ is analogous and we omit it. ∎

3.2.1. Existence via Sadovskii’s fixed point theorem

Our second existence result is obtained via Sadovskii’s fixed point theorem assuming a linear growth of $F$ and $g .$ Namely.

(H₃):

There are constants $\ L_{F},L_{g}\geq 0$ and $\ c_{F},c_{g}\geq 0$ such that

	$\displaystyle\left\|F(t,u))\right\|_{X}$	$\displaystyle\leq$	$\displaystyle L_{F}\left\|u\right\|_{X}+c_{F},$		(3.6)
	$\displaystyle\left\|g(t,u))\right\|_{X}$	$\displaystyle\leq$	$\displaystyle L_{g}\left\|u\right\|_{X}+c_{g},$

for all $t\in[0,b]$ and $u\in X.$

Theorem 3.3.

Assume that conditions (H₂) and (H₃) hold. Then problem (1.1) has at least one mild solution $u\in C\left(\left[-r,b\right],X\right).$

Proof.

From Lemma 3.2, the operator $P$ is condensing. In order to apply Sadovskii’s theorem it suffices to find a closed bounded and convex subset of $\ C_{\phi}\left(\left[0,b\right],X\right)$ which is invariated by $P .$ More precisely we shall prove the existence of two numbers $\ \theta,\ R>0$ such that

\left|u\right|_{\theta}\leq R\ \ \ \text{implies\ \ \ }\left|Pu\right|_{\theta% }\leq R.

(3.7)

We start from the following estimate immediately derived from (3.1) (take $v=0$ ) by using (3.6):

\left|\left(Pu\right)\left(t\right)\right|_{X}\leq\left(L_{G}e^{\theta t}+NL_{% g}e^{\omega b}\int_{0}^{t}\int_{0}^{s}e^{\theta\tau}d\tau ds+NL_{F}e^{\omega b% }\int_{0}^{t}e^{\theta s}ds\right)\left|u\right|_{\theta}+c,

where $\ c=\left|G\left(\cdot,0\right)\right|_{\infty}+Nb^{2}e^{\omega b}c_{g}+Nbe^{% \omega b}c_{F}.\$ Then

\left|\left(Pu\right)\left(t\right)\right|_{X}\leq\left[L_{G}+Ne^{\omega b}% \left(\frac{L_{g}}{\theta^{2}}+\frac{L_{F}}{\theta}\right)\right]e^{\theta t}% \left|u\right|_{\theta}+c,

whence

\left|Pu)\right|_{\theta}\leq\left[L_{G}+Ne^{\omega b}\left(\frac{L_{g}}{% \theta^{2}}+\frac{L_{F}}{\theta}\right)\right]\left|u\right|_{\theta}+c.

Therefore, in view of the assumption $L_{G}<1,$ for $\theta>0$ large enough that

L_{G}+Ne^{\omega b}\left(\frac{L_{g}}{\theta^{2}}+\frac{L_{F}}{\theta}\right)<1,

one has (3.7) for $\ R:=c\left(1-L_{G}-Ne^{\omega b}\left(\frac{L_{g}}{\theta^{2}}+\frac{L_{F}}{% \theta}\right)\right)^{-1}.$ Thus Sadovskii’s fixed point theorem applies to the condensing operator $P$ restricted to the closed bounded and convex set

\mathcal{D}:=\left\{u\in C_{\phi}\left(\left[0,b\right],X\right):\ \left|u% \right|_{\theta}\leq R\right\}.

∎

3.2.2. Existence via nonlinear alternative

One can relax the growth conditions on $F$ and $g$ as shows the following hypothesis:

(H₄):

There exist continuous functions $\ \eta_{F},\ \eta_{g}:\left[0,b\right]\rightarrow\mathbb{R}_{+}$ and nondecreasing continuous functions $\ h_{F},h_{g}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ such that

	$\displaystyle\left\|F(t,u)\right\|_{X}$	$\displaystyle\leq$	$\displaystyle\eta_{F}\left(t\right)h_{F}\left(\left\|u\right\|_{X}\right),$
	$\displaystyle\left\|g(t,u))\right\|_{X}$	$\displaystyle\leq$	$\displaystyle\eta_{g}\left(t\right)h_{g}\left(\left\|u\right\|_{X}\right),$

for all $t\in[0,b]$ and $u\in X.$

Theorem 3.4.

Assume that conditions (H₂) and (H₄) hold. In addition assume that

\int_{c_{0}}^{+\infty}\frac{ds}{(h_{F}+h_{g})(s)}>Mb,

(3.8)

where

c_{0}:=\frac{Ne^{\omega b}\left|\phi(0)\right|+\left|G\left(\cdot,0\right)% \right|_{\infty}}{1-L_{G}}\quad\text{and}\quad M:=\frac{Ne^{wb}}{1-L_{G}}\max% \left\{|\eta_{g}|_{L^{1}\left(0,b\right)},\ |\eta_{F}|_{L^{\infty}\left(0,b% \right)}\right\}.

Then problem (1.1) has at least one mild solution $u\in C([-r,b],X).$

Proof.

Here again, Lemma 3.2 guarantees that the operator $P$ is condensing. In order to apply the nonlinear alternative, Theorem 2.3, we have to find a priori bounds of solutions, more exactly to show that the set

\Lambda=\left\{u\in C_{\phi}([0,b],X):\ u=\lambda P(u),\ 0<\lambda<1\right\}~{% }\text{ }

is bounded. Let $u\in\Lambda.$ Then for some $\lambda\in\left(0,1\right),$ one has

	$\displaystyle u\left(t\right)$	$\displaystyle=$	$\displaystyle\lambda T(t)\phi(0)+\lambda G(t,u(t-\rho_{0}\left(t\right)))+% \lambda\int_{0}^{t}T(t-s)\int_{0}^{s}g(s-\tau,u(\tau))d\tau ds\vspace{1mm}$
			$\displaystyle+\ \lambda\int_{0}^{t}T(t-s)F(s,u(s-\rho(s,u(s))))ds\ \ \ \left(t% \in\left[0,b\right]\right).$

Let $\ \nu:\left[0,b\right]\rightarrow\mathbb{R}_{+}$ be defined by

\nu(t):=\max\left\{\left|u\right|_{\mathcal{C}\left[0,t\right]},\left|\phi% \right|_{\mathcal{C}\left[-r,0\right]}\right\}.\vspace{0.2cm}\newline

Clearly $\nu$ is nondecreasing and

\left|u\left(t-\rho_{0}\left(t\right)\right)\right|\leq v\left(t\right),\ \ \ % \left|u(s-\rho(s,u(s)))\right|\leq\nu\left(t\right)\ \ \ \text{for\ all \ }s% \in[0,t],\text{ }t\in\left[0,b\right].

The problem is to find a bound of function $\nu$ independent of $u$ and $\lambda.$ Let $\xi$ be such that

\nu\left(\xi\right)=\left|\phi\right|_{\mathcal{C}\left[-r,0\right]}\ \ \ % \text{and\ \ }\ \nu\left(t\right)>\left|\phi\right|_{\mathcal{C}\left[-r,0% \right]}\ \ \ \text{for all\ }t\in(\xi,b].

Obviously, if such a point $\xi$ does not exist, then $\ \left|u\right|_{C\left[0,b\right]}\leq\left|\phi\right|_{\mathcal{C}\left[-r% ,0\right]}\$ and the bound for $u$ is $\ \left|\phi\right|_{\mathcal{C}\left[-r,0\right]}.$ Let $\ t\in(\xi,b]$ be any point, hence $\ \left|u\right|_{\mathcal{C}\left[0,t\right]}>$ $\left|\phi\right|_{\mathcal{C}\left[-r,0\right]},$ and let $\ t_{0}\in[0,t]\$ be a point with $~{}\ \left|u\right|_{\mathcal{C}\left[0,t\right]}=\left|u(t_{0})\right|.$ From (3.2.2), since $\lambda\in\left(0,1\right),$ one finds

$\displaystyle v\left(t\right)$	$\displaystyle=$	$\displaystyle\left\|u\left(t_{0}\right)\right\|\leq\|T(t_{0})\phi(0)\|+\|G(t_{0},u(% t_{0}-\rho_{0}\left(t_{0}\right)))\|$
		$\displaystyle+\int_{0}^{t_{0}}\|T(t_{0}-s)\|\int_{0}^{s}\|g(s-\tau,u(\tau))\|d\tau ds% \vspace{1mm}$
		$\displaystyle+\int_{0}^{t_{0}}\|T(t_{0}-s)\|\|F(s,u(s-\rho(s,u(s))))\|ds$
	$\displaystyle\leq$	$\displaystyle Ne^{wb}\|\phi(0)\|+L_{G}v\left(t_{0}\right)+\left\|G\left(\cdot,0% \right)\right\|_{\infty}+Ne^{\omega b}\left\|\eta_{g}\right\|_{L^{1}\left(0,b% \right)}\int_{0}^{t_{0}}h_{g}\left(\nu\left(s\right)\right)ds$
		$\displaystyle+Ne^{wb}\|\eta_{F}\|_{L^{\infty}\left(0,b\right)}\int_{0}^{t_{0}}h_% {F}(\nu(s))ds.$

If we use $\ \nu\left(t_{0}\right)\leq\nu\left(t\right),$ we deduce that

	$\displaystyle\left(1-L_{G}\right)v\left(t\right)$	$\displaystyle\leq$	$\displaystyle Ne^{wb}\|\phi(0)\|+\left\|G\left(\cdot,0\right)\right\|_{\infty}+Ne^% {\omega b}\left\|\eta_{g}\right\|_{L^{1}\left(0,b\right)}\int_{0}^{t_{0}}h_{g}% \left(\nu\left(s\right)\right)ds$
			$\displaystyle+Ne^{wb}\|\eta_{F}\|_{L^{\infty}\left(0,b\right)}\int_{0}^{t_{0}}h_% {F}(\nu(s))ds,$

whence

v\left(t\right)\leq c_{0}+M\int_{0}^{t}\left(h_{g}\left(\nu\left(s\right)% \right)+h_{F}(\nu(s))\right)ds\ \ \ \left(t\in(\xi,b]\right).

Now we use Bihari’s inequality, Theorem 2.2, with $\ f=M,$ $h=h_{g}+h_{F}\$ and

\Phi(x):=\int_{x_{0}}^{x}\frac{1}{\left(h_{g}+h_{F}\right)\left(s\right)}ds.

In order to have the bound of $\ \nu\$ on the whole interval $\left[0,b\right],$ we need $\ \tau_{1}$ given by (2.5) to be $\geq b.$ This is guaranteed by our assumption (3.8). Therefore, Theorem 2.2 applies and implies that there exists a constant $\ c\$ such that $\ \nu(t)\leq c\$ for all $t\in[0,b],$ and $\ c\$ does not depend on the solution $\ u$ and $\ \lambda.$

Finally, Theorem 2.3 guaranteed that $P$ has a fixed point in $C_{\phi}([0,b],X).\quad$ ∎

4. Applications

To apply our previous results, we consider the following problem related to a partial functional integrodifferential equation:

\left\{\begin{array}[]{l}\frac{\partial}{\partial t}\left[u(t,x)-G\left(t,u% \left(t-\rho_{0}\left(t\right)\right)\right)\left(x\right)\right]=\frac{% \partial^{2}}{\partial x^{2}}\left[u(t,x)-G\left(t,u\left(t-\rho_{0}\left(t% \right)\right)\right)\left(x\right)\right]\\ +\displaystyle\ \int_{0}^{t}g\left(t-s,\ u(s,\cdot)\right)\left(x\right)ds+F% \left(t,u\left(t-\rho\left(t,u\left(t,\cdot\right)\right),\cdot\right)\right)% \left(x\right)\ \ \ \text{for}~{}0\leq t\leq b,\ x\in[0,1]\vspace{2mm}\\ u(t,0)=u(t,1)=0,\quad 0\leq t\leq b,\vspace{2mm}\\ u(t,x)=\phi(t,x)\ \ ~{}\text{for}\quad-r\leq t\leq 0~{}\ \text{and}\quad x\in[% 0,1],\end{array}\right.

(4.1)

where $\ 0\leq\rho_{0},\rho\leq r.$ Here $X=C_{0}[0,1]$ is the Banach space of the continuous real functions defined on $[0,1]$ vanishing at $0$ and $1,$ equipped with the uniform norm topology. Also $A:D(A)\subset X\rightarrow X$ is defined as

D(A)=C_{0}[0,1]\cap C^{2}[0,1],\vspace{2mm}\ \ \ Au=u^{\prime\prime}.

It is known that $A$ generates a strongly continuous semigroup on $X$ which is compact. If we let $\ u\left(t\right)=u\left(t,\cdot\right)$ and $\ \phi\left(t\right)=\phi\left(t,\cdot\right),$ then problem (4.1) appears as a particular case of the abstract problem (1.1).

Next we particularize the maps $\ F,G$ and $g\$ to illustrate our results, Theorems 3.1, 3.3 and 3.4.

Theorem 4.1.

:

(a):

Let

G\left(t,u\right)=\alpha\left(t\right)u,\ \ \ F\left(t,u\right)=\beta\left(t% \right)\sin u,\ \ \ g\left(t,u\right)=\gamma\left(t\right)u,

for $\ t\in\left[0,b\right],$ $u\in C_{0}[0,1],$ where $\ \alpha,\beta,\gamma\in C\left[0,b\right].$ Then, with a state-independent delay, problem (4.1) has a unique solution.

(b):

Let

G\left(t,u\right)=\alpha\left(t\right)u,\ \ F\left(t,u\right)=\beta\left(t% \right)\frac{\left|u\right|^{p}u}{1+\left|u\right|^{p}},\ \ \ g\left(t,u\right% )=\gamma\left(t\right)u\cos u+\gamma_{0}\left(t\right),

for $\ t\in\left[0,b\right],$ $u\in C_{0}[0,1],$ where $\ \alpha,\alpha_{0},\beta,\gamma,\gamma_{0}\in C\left[0,b\right].$ If $\ \left|\alpha\right|_{L^{\infty}\left(0,b\right)}<1,$ the problem (4.1) has at least one solution.

(c):

Let

G\left(t,u\right)=\alpha\left(t\right)\sin u,\ \ \ F\left(t,u\right)=\beta% \left(t\right)\left(\left|u\right|^{p}+2\arctan u\right),\ \ \ g\left(t,u% \right)=\gamma\left(t\right)\left|u\right|^{q},

for $\ t\in\left[0,b\right],$ $u\in C_{0}[0,1],$ where $\ \alpha,\beta,\beta_{0},\gamma\in C\left[0,b\right],$ and $\ p,q\geq 1.$ If $\ \left|\alpha\right|_{L^{\infty}\left(0,b\right)}<1,$ then problem (4.1) has at least one solution provided that $b$ is sufficiently small.

Proof.

(a) All the assumptions of Theorem 3.1 are fulfilled. In this case, one has

L_{G}=\left|\alpha\right|_{\infty},\ \ \ L_{F}=\left|\beta\right|_{\infty}\ \ % \ \text{and\ \ }L_{g}=\left|\gamma\right|_{\infty}.

(b) The assumptions of Theorem 3.3 hold with

L_{G}=\left|\alpha\right|_{\infty},\ \ \ L_{F}=\left|\beta\right|_{\infty},\ % \ \ c_{F}=0,\ \ \ L_{g}=\left|\gamma\right|_{\infty}\ \ \ \text{and\ \ }c_{g}=% \left|\gamma_{0}\right|_{\infty}.

The result follows from Theorem 3.3.

L_{G}=\left|\alpha\right|_{\infty},\ \ \ \eta_{F}\left(t\right)=\left|\beta% \left(t\right)\right|,\ \ \ \eta_{g}\left(t\right)=\left|\gamma\left(t\right)% \right|,\ \ \ h_{F}\left(s\right)=s^{p}+\pi\ \ \ \text{and\ \ }h_{g}\left(s% \right)=s^{q}\ \left(s\in\mathbb{R}_{+}\right).

The length $b$ of the interval $\left[0,b\right]$ has to be small enough that condition (3.8) is satisfied. ∎

Acknowledgement. This work was supported by the SCAC (Le Service de Coopération et d’Action Culturelle) via SSHN (Séjour Scientifique de Haut Niveau de l’Ambassade de France au Tchad).
This is the opportunity for the first author to express his sincere gratitude to the members of the Department of Mathematics from Babeş-Bolyai University of Cluj-Napoca for their warm welcome and collaboration.

The authors thank the anonymous referees for their useful comments and bibliographic suggestions.

References

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[2] Banas S. and Goebel K., Mesure of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, (1980).
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[1] Alia, M., Ezzinbi, K., Koumla, S., Mild solutions for some partial functional integrodifferential equations with state-dependent delay. Discuss. Math. Differ. Incl. Control Optim. 37, 173–186 (2017) Article MathSciNet MATH Google Scholar
[2] Banas, S., Goebel, K., Measure of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, New York (1980) MATH Google Scholar
[3] Belmekki, M., Benchohra, M., Ezzinbi, K., Existence results for some partial functional differential equations with state-dependent delay. Appl. Math. Lett. 24, 1810–1816 (2011) Article MathSciNet MATH Google Scholar
[4] Bolojan, O., Precup, R., Hybrid delay evolution systems with nonlinear constraints. Dynam. Syst. Appl. 27, 773–790 (2018) Google Scholar
[5] Cañada, A., Drabek, P., Fonda, A.: Handbook of Ordinary Differential Equations. Elsevier, Oxford (2006) MATH Google Scholar
[6] Cao, Y., Fan, J., Gard, T.C., The effects of state-dependent time delay on a stage-structured population growth model. Nonlinear Anal. 19(2), 95–105 (1992) Article MathSciNet MATH Google Scholar
[7] Corduneanu, C., Functional Equations with Causal Operators. Taylor and Francis, London (2002) Book MATH Google Scholar
[8] Domoshnitsky, A., Drakhlin, M., Litsyn, E.: One equations with delay depending on solution. Nonlinear Anal. 49(5), 689–701 (2002) Article MathSciNet MATH Google Scholar
[9] Ezzinbi, K., Koumla, S., An abstract partial functional integrodifferential equations. Adv. Fixed Point Theory 6(4), 469–485 (2016) Google Scholar
[10] Ezzinbi, K., Koumla, S., Sene, A., Existence and regularity for some partial functional integrodifferential equations with infinite delay. J. Semigroup Theory Appl. 2016, 6 (2016) Google Scholar
[11] Granas, A., Dugundji, J., Fixed Point Theory. Springer, New York (2003) Book MATH Google Scholar
[12] Hartung, F., Krisztin, T., Walther, O.H., Wu, J., Functional differential equations with state-dependent delay. In: Canada, A., Drabek, P., Fonda, A. (eds.) Handbook of Differential Equations. Ordinary Differential Equations, pp. 435–545. Elsevier, North Holland (2006) Chapter Google Scholar
[13] Hernandez, E., Prokopcsyk, A., Ladeira, L., A note on partial functional differential equations with state-dependent delay. Nonlinear Anal. Real World Appl. 7(4), 510–519 (2006) Article MathSciNet MATH Google Scholar
[14] Koumla, S., Precup, R., Integrodifferential evolution systems with nonlocal initial conditions. Stud. Univ. Babeş -Bolyai Math. 65, 93–108 (2020) Article MathSciNet MATH Google Scholar
[15] Koumla, S., Precup, R., Sene, A.: Existence results for some partial neutral functional integrodifferential equations with bounded delay. Turk. J. Math. 43, 1809–1822 (2019) Article MATH Google Scholar
[16] Louihi, M., Hbid, M.L., Arino, O., Semigroup properties and the Crandall Liggett approximation for a class differential equations with state-dependent delay. J. Differ. Equ. 181, 1–30 (2002) Article MathSciNet MATH Google Scholar
[17] Pazy, A., Semigroup of Linear Operators and Applications to Partial Differential Equations. Springer, New-York (1983) Book MATH Google Scholar
[18] Precup, R., The nonlinear heat equation via fixed point principles. Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convex. 4, 111–127 (2006) MATH Google Scholar
[19] Precup, R., Linear and Semilinear Partial Differential Equations. De Gruyter, Berlin (2013) MATH Google Scholar
[20] Vrabie, I.I., Vrabie, I.I.: $C_{0}$ Semigroups and Applications. Elsevier, Amsterdam (2003) MATH Google Scholar
[21] Webb, F.G., An abstract semilinear Volterra integrodifferential equation. Proc. Am. Math. Soc. 69, 255–260 (1978) Article MathSciNet MATH Google Scholar
[22] Yuming, Q., Integral and Discrete Inequalities and Their Applications. Birkhäuser, Basel (2016) MATH Google Scholar

$\displaystyle\left\|F(t,u)-F(t,v)\right\|_{X}$	$\displaystyle\leq$	$\displaystyle L_{F}\left\|u-v\right\|_{X},$
$\displaystyle\left\|G(t,u)-G(t,v)\right\|_{X}$	$\displaystyle\leq$	$\displaystyle L_{G}\left\|u-v\right\|_{X},$
$\displaystyle\left\|g(t,u)-g(t,v)\right\|_{X}$	$\displaystyle\leq$	$\displaystyle L_{g}\left\|u-v\right\|_{X},$

$\displaystyle\left\|\left(Pu\right)\left(t\right)-\left(Pv\right)\left(t\right)% \right\|_{X}$	$\displaystyle\leq$	$\displaystyle L_{G}\left\|u-v\right\|_{C([0,t],X)}+NL_{g}e^{\omega b}\int_{0}^{t% }\int_{0}^{s}\left\|u-v\right\|_{C([0,\tau],X)}d\tau ds$
		$\displaystyle+\ NL_{F}e^{\omega b}\int_{0}^{t}\left\|u-v\right\|_{C([0,s],X)}ds$
	$\displaystyle=$	$\displaystyle L_{G}\left\|u-v\right\|_{C([0,t],X)}e^{-\theta t}e^{\theta t}$
		$\displaystyle+\ NL_{g}e^{\omega b}\int_{0}^{t}\int_{0}^{s}\left\|u-v\right\|_{C(% [0,\tau],X)}e^{-\theta\tau}e^{\theta\tau}d\tau ds$
		$\displaystyle+\ NL_{F}e^{\omega b}\int_{0}^{t}\left\|u-v\right\|_{C([0,s],X)}e^{% -\theta s}e^{\theta s}ds.$

			$\displaystyle\left\|\int_{0}^{t}T(t-s)F(s,u(s-\rho(s,u(s))))ds-\int_{0}^{t^{% \prime}}T(t^{\prime}-s)F(s,u(s-\rho(s,u(s))))ds\right\|$
		$\displaystyle\leq$	$\displaystyle\left\|\int_{0}^{t}T(t-s)F(s,u(s-\rho(s,u(s))))ds-\int_{0}^{t^{% \prime}}T(t-s)F(s,u(s-\rho(s,u(s))))ds\right\|$
			$\displaystyle+\left\|\int_{0}^{t^{\prime}}\left(T(t-s)-T\left(t^{\prime}-s% \right)\right)F(s,u(s-\rho(s,u(s))))ds\right\|$
		$\displaystyle\leq$	$\displaystyle C_{1}Ne^{\omega b}\left\|t-t^{\prime}\right\|+\int_{0}^{t^{\prime}% }\left\|T(t-s)-T\left(t^{\prime}-s\right)\right\|\left\|F(s,u(s-\rho(s,u(s))))% \right\|ds$
		$\displaystyle\leq$	$\displaystyle C_{1}Ne^{\omega b}\left\|t^{\prime}-t\right\|+C_{1}\int_{0}^{t^{% \prime}}\left\|T(t-s)-T\left(t^{\prime}-s\right)\right\|ds.$

	$\displaystyle\left\|F(t,u))\right\|_{X}$	$\displaystyle\leq$	$\displaystyle L_{F}\left\|u\right\|_{X}+c_{F},$		(3.6)
	$\displaystyle\left\|g(t,u))\right\|_{X}$	$\displaystyle\leq$	$\displaystyle L_{g}\left\|u\right\|_{X}+c_{g},$

	$\displaystyle\left\|F(t,u)\right\|_{X}$	$\displaystyle\leq$	$\displaystyle\eta_{F}\left(t\right)h_{F}\left(\left\|u\right\|_{X}\right),$
	$\displaystyle\left\|g(t,u))\right\|_{X}$	$\displaystyle\leq$	$\displaystyle\eta_{g}\left(t\right)h_{g}\left(\left\|u\right\|_{X}\right),$

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Existence results for some functional integrodifferential equations with state-dependent delay

Abstract.

1. Introduction

2. Preliminaries

Lemma 2.1 ([2]).

Theorem 2.2 (Bihari inequality).

Theorem 2.3 (Nonlinear alternative).

3. Main results

3.1. An existence and uniqueness result

Theorem 3.1.

Proof.

3.2. Existence under compactness conditions

Lemma 3.2.

Proof.

3.2.1. Existence via Sadovskii’s fixed point theorem

Theorem 3.3.

Proof.

3.2.2. Existence via nonlinear alternative

Theorem 3.4.

Proof.

4. Applications

Theorem 4.1.

Proof.

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