Existence and uniqueness of solution for abstract differential equations with state-dependent delayed impulses

Abstract

We study the existence and uniqueness of mild and classical solutions for a general class of abstract impulsive differential equations with state-dependent impulses. Some examples on partial differential equations are presented.

1 Introduction

In this paper, we study the existence and uniqueness of mild and classical solutions for a class of abstract impulsive differential equations of the form

$$\begin{aligned} u'(t)= & {} Au(t)+ f(t, u(\zeta (t,u(t)))), \,\, \quad t\in I_{i}= (t_{i},t_{i+1}], \,i=0,\ldots , N, \end{aligned}$$

(1.1)

$$\begin{aligned} u(t_{j}^{+})= & {} g_{j}(u(\sigma _{j}(u(t_{j}^{+})))), \,\,\,\, j=1,\ldots , N, \end{aligned}$$

(1.2)

$$\begin{aligned} u_{0}= & {} \varphi \in {\mathcal {B}}= C( I_{-1};X) , \,\,\, I_{-1}=[-p,0], \end{aligned}$$

(1.3)

where $A:D(A)\subset X\mapsto X$ is the generator of an analytic semigroup of bounded linear operators $(T(t))_{t\ge 0}$ on a Banach space $(X,\parallel \cdot \parallel )$, $0=t_{0}<t_{1}< t_{2}<\cdots<t_{N}< t_{N+1} =a$ are pre-fixed numbers and $f(\cdot )$, $g_{i}(\cdot )$, $\sigma _{i}(\cdot )$, $i=1,\ldots , N,$ are functions specified be later.

The study of state-dependent delay equations is motivated by applications and theory. Related ODEs on finite dimensional spaces we cite the early works by Driver [9, 10] and Aiello et al. [1], the survey by Hartung, Krisztin et al. [15], the papers by Hartung et.al. [16,17,18] and the references in these works. For the case PDEs and abstract differential equations with state-dependent delay, we mention [19, 20, 26, 36,37,38] and the recent interesting works by Krisztin and Rezounenko [25], Yunfei et al. [33], Kosovalic et al. [26, 27] and Hernandez et al. [24].

Concerning the theory of impulsive differential equations, their motivations and relevant developments, we cite the books by Bainov and Covachev [2], Lakshmikantham et al. [28], Samoilenko and Perestyuk [40] for the case of ordinary differential equations on finite dimensional space and Benchohra et al. [7] for abstract differential equations and partial differential equations. In addition, we cite the interesting papers [8, 11, 20,21,22,23, 29, 31, 34, 39, 43] and the references therein. Related differential equations with impulse at state-dependent moments and state-dependent delayed impulses, we refer the reader to [3,4,5,6, 13, 14, 30, 41].

Our work is motivated by the papers Hakl et al. [14] related partial differential equations with impulse at state-dependent moments and Li and Wu [30] on differential equations with state-dependent delayed impulses. Specifically, we study the existence and “uniqueness” of solutions for the problem (1.1)–(1.3) which is a highly not trivial problem since functions of the form $u\mapsto u(\zeta (\cdot ,u(\cdot )))$ are (in general) nonlinear and not Lipschitz on space of continuous or sectionally continuous functions. By noting that

$$\begin{aligned}&{{\parallel u(\zeta (\cdot ,u(\cdot ))) - v(\zeta (\cdot ,v(\cdot ))) \parallel _{C([-p,a];X )} }} \\&\quad \le (1+ [v]_{C_\mathrm{Lip}([-p,a];X)}[\zeta ]_{C_\mathrm{Lip}([0,a]\times X;[-p,a])}) \parallel u-v \parallel _{ C_\mathrm{Lip}([-p,a];X)}, \\&{{ \parallel u(\sigma _{i}(u(t_{i}^{+}))) - v(\sigma _{i}(v(t_{i}^{+}))) \parallel }} \\&\quad \le (1+ [v]_{C_\mathrm{Lip}([-p,a];X)}[\sigma _{i}]_{C_\mathrm{Lip}(X;[-p,a]})\parallel u-v \parallel _{C([-p,a];X)} , \end{aligned}$$

when the involved functions are Lipschitz, we study the existence of solutions on spaces of sectionally Lipschitz functions, a hard problem in the semigroup framework and in the general field of partial differential equations. In addition, we note that the Lipschizianity of $T(\cdot ) g_{i}(u(\sigma _{i}(u(t_{i}^{+}))))$ not depend on the Lipschizianity of $g_{i}(\cdot )$ and $u(\cdot )$, which introduce a extra difficulty in our studies.

This paper has four sections. The existence and uniqueness of a classical solution via the contraction mapping principle is proved in Theorems 2.1, 2.2 and Proposition 2.3. In Theorem 2.3 we prove the existence of a mild solution using the Schauder’s fixed point Theorem. The particular case in which $\sigma _{i}(\cdot )$ and (or) $\zeta (\cdot )$ have values in $[-p,0]$, is studied in Propositions 2.1 and 2.2. In the last section some examples on partial differential equations are presented.

We include now some notations and results used in this work. Let $(Z,\parallel \cdot \parallel _{Z})$ and $(W,\parallel \cdot \parallel _{W})$ be Banach spaces. We denote by ${\mathcal {L}}(Z,W)$ the space of bounded linear operators from Z into W endowed with operator norm denoted by $\parallel \cdot \parallel _{{\mathcal {L}}(Z,W)}$ and we write ${\mathcal {L}}(Z)$ and $\parallel \cdot \parallel _{{\mathcal {L}}(Z)}$ if $Z = W$. Moreover, if $X=Z=W$ we write simply $\parallel \cdot \parallel$ for the norms $\parallel \cdot \parallel _{ X}$ and $\parallel \cdot \parallel _{{\mathcal {L}}(X)}$. In addition, $B_{r}(z,Z)=\{y\in Z:\parallel y-z\parallel _{Z}\le r\}$.

Let $J\subset {\mathbb {R}}$ be a bounded interval. The spaces C(J, Z) and $C_\mathrm{Lip}(J,Z)$ and their norms denoted by $\parallel \cdot \parallel _{C (J,Z)}$ and $\parallel \cdot \parallel _{C_\mathrm{Lip}(J,Z)}$ are the usual. We only note that $\parallel \cdot \parallel _{C_\mathrm{Lip}(J;Z)}$ is given by $\parallel \cdot \parallel _{C_\mathrm{Lip}(J;Z)} =\parallel \cdot \parallel _{C(J;Z)} +[ \cdot ]_{C_\mathrm{Lip}(J;Z)}$ where $[ \zeta ]_{C_\mathrm{Lip}(J;Z)} =\sup _{t,s\in J, t\ne s }\frac{\parallel \zeta (s)-\zeta (t)\parallel _{Z}}{\mid t-s\mid }$.

The notation ${{\mathcal {P}}{\mathcal {C}}}(Z)$ is used for the space formed by all the bounded functions $u :[0, a] \mapsto Z$ such that $u(\cdot )$ is continuous at $t \ne t_{i},$ $u(t_{i}^{-})= u(t_{i})$ and $u(t_{i}^{+})$ exists for all $i = 1, \ldots , N$, provided with the norm $\parallel u\parallel _{{{\mathcal {P}}{\mathcal {C}}}(Z)}=\max _{i=0,1,\ldots ,N}\parallel u\parallel _{C((t_{i},t_{i+1}];Z)}$. In addition, ${{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(Z)$ represents the space of functions $u\in {{\mathcal {P}}{\mathcal {C}}}(Z)$ such that $u_{\mid _{(t_{i},_{i+1}]}}\in C_\mathrm{Lip}((t_{i},t_{i+1}];Z)$ for all $i=0,1,\ldots t_{N+1}$, endowed with the norm $\parallel u\parallel _{{{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(Z)}=\max _{i=0,\ldots , N} \parallel u_{\mid _{(t_{i},t_{i+1}]}}\parallel _{C_\mathrm{Lip}((t_{i},_{i+1}];Z)}$.

We use the symbol ${\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}(Z)$ for the set of all the functions $u:[-p,a]\mapsto Z$ such that $u_{\mid _{[-p,t_{1}]}}\in C([-p,t_{1}];Z)$ and $u_{\mid _{[0,a]}}\in {{\mathcal {P}}{\mathcal {C}}}(Z)$. In addition, ${\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(Z)$ is the space formed by all the functions $u:[-p,a]\mapsto Z$ such that $u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}(Z)$, $u_{\mid _{[-p,0]}}\in {C}_\mathrm{Lip}([-p,0];Z)$ and $u_{\mid _{[0,a]}}\in {\mathcal {P}}{\mathcal {C}}_\mathrm{Lip}(Z)$, endowed with the norm $\parallel u\parallel _{{\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(Z)}=\max \{\parallel u_{\mid _{I_{i}}}\parallel _{C_\mathrm{Lip}( I_{i};Z)} : i=-1,0,\ldots , N\} .$

For $u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}(Z)$ and $i\in \{-1, 0, 1, \cdots , N\}$, we use the notation ${\tilde{u}}_{i}$ for the function ${\tilde{u}}_{i} \in C([t_{i},t_{i+1}];Z)$ given by ${\widetilde{u}}_{i}(t) = u(t)$ for $t\in (t_{i},t_{i+1}]$ and ${\widetilde{u}}_{i}(t) =u(t_{i}^{+})$ for $t=t_{i}$. For $B\subseteq {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}(Z)$ and $i\in \{-1,0, 1, \cdots , N\}$, ${\widetilde{B}}_{i}$ is the set ${\widetilde{B}}_{i}=\{ {\tilde{u}}_{i}: u\in B \}.$ We note the following Ascoli–Arzela type criteria.

Lemma 1.1

A set $B\subseteq {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}(Z)}$ is relatively compact in ${\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}(Z)}$ if and only if each set ${\widetilde{B}}_{i}$ is relatively compact in $C([t_{i},t_{i+1}],Z)$.

In this paper, $X_{1}$ is the domain of A endowed with the norm $\parallel x\parallel _{X_{1}}=\parallel x\parallel +\parallel Ax\parallel$ and $C_{0}, C_{1}$ are positive constants such that $\parallel AT(s)\parallel _{{\mathcal {L}}(X_{1},X)} \le C_{1}$, $\parallel T(s)\parallel \le C_{0}$ and $\parallel AT(t)\parallel \le \frac{ C_{1}}{t}$ for all $s\in [0,a]$ and $t\in (0,a]$.

Related the abstract Cauchy problem

$$\begin{aligned} u' (t) = Au(t) +\xi (t) , \quad t\in [a,b], \quad u(c)=x \in X, \end{aligned}$$

(1.4)

we note that the function $u\in C ([c,d];X)$ given by $u(t) = T(t-c)x +\int _{c}^{t}T(t-s) \xi (s) {\text {d}}s$, is called mild solution of (1.4). In addition, a function $v\in C ([c,d];X)$ is said to be a classical solution of (1.4) if $v\in C^{1}((c,d];X) \cap C ((c,d];X_{1} )$ and $v(\cdot )$ satisfies (1.4) on (c, d].

2 Existence of solutions

In this section we present some results on the existence of solution for (1.1)–(1.3). To begin, we introduce the followings concepts of solution.

Definition 2.1

A function $u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}(X)$ is called a mild solution of the problem (1.1)–(1.3) if $u_{0}=\varphi$, $u(t_{i}^{+})= g_{i}(u(\sigma _{i}(u(t_{i}^{+}))))$ for all $i=1,\ldots , N$ and

$$\begin{aligned} u(t)= & {} T(t) \varphi (0) + \int _{0}^{t}T(t-\tau )f(\tau ,u(\zeta (\tau ,u(\tau )))){\text {d}}\tau , \,\,\,\,\, t\in [0,t_{1}], \\ u(t)= & {} T(t-t_{i}) g_{i}(u(\sigma _{i}(u(t_{i}^{+})))) +\int _{t_{i}}^{t}T(t-\tau )f(\tau ,u(\zeta (\tau ,u(\tau )))) {\text {d}}\tau , \end{aligned}$$

for all $t\in (t_{i},t_{i+1}]$ and $i=1,\ldots , N.$

Definition 2.2

A function $u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}(X)$ is called a classical solution of (1.1)–(1.3) if $u_{0}=\varphi$, $u(t_{i}^{+})= g_{i}(u(\sigma _{i}(u(t_{i}^{+}))))$ for all $i=1,\ldots , N$ and $u(\cdot )$ satisfy (1.1).

In the remainder of this work, we assume that $(W,\parallel \cdot \parallel _{W})$ is Banach continuously embedded in $(X,\parallel \cdot \parallel )$ such that $A T(\cdot )\in L^{\infty }([0,a];{\mathcal {L}}(W,X))$. To prove our results, we introduce the following conditions.

${\mathbf{H}}_{\zeta }$ :

$\zeta \in C_\mathrm{Lip}( [0,a]\times X ;[-p,a])$ and there is a function $j:\{1,\ldots , N\}\mapsto \{-1,0,1,\ldots , N\}$ such that $\zeta \in C_\mathrm{Lip}( I_{i}\times X ;I_{j(i)})$ and $j(i)\le i$ for all $i\in \{1,\ldots , N\}$.

${\mathbf{H}}_{\sigma _{{\mathbf{i}}} }$ :

There is a function $q:\{1,\ldots , N\}\mapsto \{-1,0,1,\ldots , N\}$ such that $q(i)\le i$ and $\sigma _{i}\in C(X,I_{q(i)})$ for all $i\in \{1,\ldots , N\}$. Next we write $[\sigma _{i}]_{C_\mathrm{Lip}}$ in place $[\sigma _{i}]_{C_\mathrm{Lip}( X;I_{q(i)})}$.

${\mathbf{H_{g,X}^{W}}}$ :

$g_{i} \in C_\mathrm{Lip}(X; W)$ and ${\mathcal {C}}_{X,W}(g_{i})= \parallel g_{i}\parallel _{ C(X; W)}<\infty$ for every $i\in \{1,\ldots , N\}$. Next, $L_{Z,W}(g_{i})$ denotes the Lipschitz constant of $g_{i}(\cdot )$, $L_{Z,W}(g)=\max _{i=1,\ldots , N}L_{Z,W}(g_{i})$ and ${\mathcal {C}}_{Z,W}(g)=\max _{i=1,\ldots , N} {\mathcal {C}}_{Z,W}(g_{i})$.

${\mathbf{H_{g}}}$ :

$g_{i} \in C_\mathrm{Lip}(X;X)$ and ${\mathcal {C}}_{X}(g_{i})= \parallel g_{i}\parallel _{ C(X; X)}<\infty$ for all $i\in \{1,\ldots , N\}$. Next, $L_{g_{i}}$ is the Lipschitz constant of $g_{i}(\cdot )$, $L_{g}=\max _{i=1,\ldots , N}L_{g_{i}}$ and ${\mathcal {C}}_{X}(g)=\max _{i=1,\ldots , N} {\mathcal {C}}_{X}(g_{i})$.

${\mathbf{H_{f}}}$ :

$f\in C_\mathrm{Lip}([0,a]\times X; X)$ and $C_{X}(f)= \parallel f\parallel _{ C([0,a]\times X; X)}<\infty$. Next, $L_{f}$ denotes the Lipschitz constant of $f(\cdot )$.

Notations 1

Next, for convenience, we write $[\zeta ]_{C_\mathrm{Lip}}$ in place $[\zeta ]_{C_\mathrm{Lip}( [0,a]\times X ; [-p,a])}$, $b_{i}=t_{i+1}-t_{i}$, $b=\max _{i= 1,\ldots , N }b_{i}$, $i_{c}: W\mapsto X$ is the inclusion map and

$$\begin{aligned} \Lambda _{X,W}&= \max \{\parallel AT(\cdot ) \parallel _{L^{\infty }([0,b],{\mathcal {L}}(W,X))}, C_{0}{ \parallel i_{c}\parallel _{{\mathcal {L}}(W,X)}} \} \\ \Phi _{X,W}&= \Lambda _{X,W} {\mathcal {C}}_{X,W}(g) + C_{0}( C_{X}(f) + b L_{f} ) + [T(\cdot )\varphi (0)]_{C_\mathrm{Lip}([0,a];X)}\\&\quad + [\varphi ]_{C_\mathrm{Lip}([-p,0];X)} . \end{aligned}$$

The next useful result follows from the proof of [24, Lemma 1]. The proof is omitted.

Lemma 2.2

Assume that the conditions ${\mathbf{H}}_{\zeta }$, ${\mathbf{H}}_{\sigma _{{\mathbf{i}}}}$ are satisfied, $u,v\in {\mathcal {B}}{\mathcal {P}}{\mathcal {C}}_\mathrm{Lip}(X)$ and $u_{0}=v_{0}$. Then $u(\zeta (\cdot ,u(\cdot ))) \in {\mathcal {P}}{\mathcal {C}}_\mathrm{Lip}(X)$ and

$$\begin{aligned} { [ u(\zeta (\cdot ,u(\cdot ))) ]_{ {{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X)} }\le & {} [u]_{{\mathcal {B}}{\mathcal {P}}{\mathcal {C}}_\mathrm{Lip}(X)} [\zeta ]_{C_\mathrm{Lip}}(1 +[u_{\mid _{[0,a]}}]_{ {{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X)}) , \end{aligned}$$

(2.1)

$$\begin{aligned} \parallel u(\zeta (\cdot ,u(\cdot ))) - v(\zeta (\cdot ,v(\cdot ))) \parallel _{ {{\mathcal {P}}{\mathcal {C}}(X)}}\le & {} (1+ [v]_{ {\mathcal {B}}{\mathcal {P}}{\mathcal {C}}_\mathrm{Lip}(X)}[\zeta ]_{C_\mathrm{Lip}})\parallel u-v \parallel _{ {{\mathcal {P}}{\mathcal {C}}(X)}} , \end{aligned}$$

(2.2)

$$\begin{aligned} \parallel u(\sigma _{i}(u(t_{i}^{+}))) - v(\sigma _{i}(v(t_{i}^{+}))) \parallel\le & {} (1+ [v]_{ {{\mathcal {B}}{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X)}[\sigma _{i}]_{C_\mathrm{Lip}})\parallel u-v \parallel _{{{\mathcal {P}}{\mathcal {C}}(X)}} .\nonumber \\ \end{aligned}$$

(2.3)

We can prove now our first result.

Theorem 2.1

Assume that the conditions ${\mathbf{\mathbf H}}_{\zeta }, {\mathbf{H}}_{\sigma _{{\mathbf{i}}} }$, ${\mathbf{H}}_{{\mathbf{g,X}}}^{{\mathbf{W}}}$ and ${\mathbf{H}}_{{\mathbf{f}}}$ are satisfied, $T(\cdot )\varphi (0)\in C_\mathrm{Lip}([0,a];X)$, $\varphi \in C_\mathrm{Lip}([-p,0];X)$ and

$$\begin{aligned}&2 C_{0}b L_{f}( 1+ [\zeta ]_{C_\mathrm{Lip}} (1+ 2\Phi _{X,W}) )\nonumber \\&\quad +\,2 \Lambda _{X,W}L_{X,W}(g) (1+ 2\max _{i=1,\ldots ,N}[\sigma _{i}]_{C_\mathrm{Lip}} \Phi _{X,W}) <1. \end{aligned}$$

(2.4)

Then there exists a unique classical solution $u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X)$ of the problem (1.1)–(1.3).

Proof

Let $P:{\mathbb {R}}\mapsto {\mathbb {R}}$ be the polynomial given by

$$\begin{aligned} P(x)= & {} \Phi _{X,W} + (C_{0}b L_{f}(1+ [\zeta ]_{C_\mathrm{Lip}})+\Lambda _{X,W}L_{X,W}(g) -1)x\nonumber \\&(\Lambda _{X,W}L_{X,W}(g) \max _{i=1,\ldots ,N}[\sigma _{i}]_{C_\mathrm{Lip}} + C_{0}bL_{f} [\zeta ]_{C_\mathrm{Lip}}) x^{2} . \end{aligned}$$

(2.5)

From (2.4) and noting that $C_{0}b L_{f}(1+ [\zeta ]_{C_\mathrm{Lip}})+\Lambda _{X,W}L_{X,W}(g) <1$, we infer that $P(\cdot )$ has a root $R_{1}>0$ and there exists $R>0$ such that $P(R)<0$. From the definition of $P(\cdot )$, we get

$$\begin{aligned} \Phi _{X,W} + C_{0}b L_{f} [\zeta ]_{C_\mathrm{Lip}}( R+ R^{2})< & {} R, \end{aligned}$$

(2.6)

$$\begin{aligned} \Lambda _{X,W}L_{X,W}(g)(1+ \max _{i=1,\ldots ,N}[\sigma _{i}]_{C_\mathrm{Lip}} R) + C_{0}bL_{f} ( 1+ R[\zeta ]_{C_\mathrm{Lip}})< & {} 1. \end{aligned}$$

(2.7)

Let ${\mathcal {S}}(R)$ be the space ${\mathcal {S}}(R)= \{u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X) \, ;\, u_{0}=\varphi , \, [u_{\mid _{[0,a]}}]_{{{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X)}\le R\} ,$ endowed with the metric $d(u,v)=\parallel u-v\parallel _{{\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}} (X)}$ and $\Gamma : {\mathcal {S}}(R) \mapsto {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}} (X)$ be the map defined by $(\Gamma u)_{0} =\varphi$ and

$$\begin{aligned} \Gamma u(t)= & {} T(t)\varphi (0)+ \int _{0}^{t}T(t-s)f(s,u(\zeta (s ,u(s)))){\text {d}}s, \,\,\,\, t\in [0,t_{1}], \\ \Gamma u(t)= & {} T(t-t_{i}) g_{i}(u(\sigma _{i}(u(t_{i}^{+})))) + \int _{t_{i}}^{t}T(t-s)f(s,u(\zeta (s,u(s)))) {\text {d}}s, \,\,\,\, \end{aligned}$$

for $t\in (t_{i},t_{i+1}]$ and $i=1,\ldots , N.$

It’s easy to see that ${\mathcal {S}}(R)$ is closed in ${\mathcal {B}}{\mathcal {P}}{\mathcal {C}}(X)$ and that $\Gamma (\cdot )$ is well defined. Moreover, from Lemma 2.2, for $i\in \{1,\ldots ,i\}$, $t\in (t_{i},t_{i+1})$ and $h>0$ such that $t+h\in (t_{i},t_{i+1}]$, we get

$$\begin{aligned}&{{\parallel \Gamma u(t+h)- \Gamma u(t)\parallel }} \\&\quad \le \int _{t-t_{i}}^{t+h-t_{i}} \parallel A T(s) g_{i}(u(\sigma _{i}(u(t_{i}^{+}))))\parallel {\text {d}}s\\&\qquad + \int _{t_{i}}^{t_{i}+h} \parallel T(t+h-s) f(s,u(\zeta (s,u(s))))\parallel {\text {d}}s \\&\qquad + \int _{t_{i}}^{t} \parallel T(t-s) \parallel \parallel f(s+h, u(\zeta (s+h,u(s+h)))- f(s, u(\zeta (s,u(s))))\parallel {\text {d}}s \\&\quad \le \parallel AT(\cdot )\parallel _{L^{\infty }([0,b_{i}];{\mathcal {L}}(W,X))} h {\mathcal {C}}_{X,W}(g) + C_{0} {\mathcal {C}}_{X}(f) h \\&\qquad + \int _{t_{i}}^{t}\parallel T(t-s) \parallel L_{f}(1+ [ u(\zeta (\cdot ,u(\cdot )))]_{ C_\mathrm{Lip}(I_{i};X)}) h {\text {d}}s \\&\quad \le \parallel AT(\cdot )\parallel _{L^{\infty }([0,b_{i}];{\mathcal {L}}(W,X))} h {\mathcal {C}}_{X,W}(g) + C_{0}( {\mathcal {C}}_{X}(f) + L_{f}b ) h \\&\qquad +\, C_{0} bL_{f} [u]_{{\mathcal {B}}{\mathcal {P}}{\mathcal {C}}_\mathrm{Lip}(X)} [\zeta ]_{C_\mathrm{Lip}}( 1 + [u]_{{\mathcal {P}}{\mathcal {C}}_\mathrm{Lip}(X)}) h , \end{aligned}$$

which implies that $[ (\Gamma u)_{\mid _{I_{i}}} ]_{C_\mathrm{Lip}( I_{i};X)} \le \Phi _{X,W} + C_{0}b L_{f} [\zeta ]_{C_\mathrm{Lip}}( R+ R^{2})<R .$ In a similar way, we obtain that

$$\begin{aligned}{}[ (\Gamma u)_{\mid _{[0,t_{1}]}} ]_{C_\mathrm{Lip}([0,t_{1}];X)}\le & {} [T(\cdot )\varphi (0)]_{C_\mathrm{Lip}([0,a];X)} + C_{0} ( {\mathcal {C}}_{X}(f) + b L_{f}) \\&+\, C_{0}b L_{f} [\zeta ]_{C_\mathrm{Lip}}( R+ R^{2})) \le R. \end{aligned}$$

From the above and noting that $[\varphi ]_{C_\mathrm{Lip}([-p,0];X)}\le R$, we obtain that $[ \Gamma u]_{ {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X)}\le R$, which implies that $\Gamma$ is a ${\mathcal {S}}(R)$-valued function.

On the other hand, using (2.2), for $u,v \in {\mathcal {S}}(R)$, $i=1,\ldots , N$ and $t\in (t_{i},t_{i+1}]$ we have that

$$\begin{aligned}&\parallel \Gamma u (t)- \Gamma v(t)\parallel \le C_{0} { \parallel i_{c}\parallel _{{\mathcal {L}}(W,X)}} L_{X,W}(g) \parallel u(\sigma _{i}(u(t_{i}^{+}))) - v(\sigma _{i}(v(t_{i}^{+}))) \parallel \\&\qquad +{\ } C_{0} L_{f}\int _{t_{i}}^{t} \parallel u(\zeta (\cdot ,u(\cdot ))) - v(\zeta (\cdot ,v(\cdot ))) \parallel _{C(I_{i}; X )} {\text {d}}s \\&\quad \le \left( \Lambda _{X,W} L_{X,W}(g) (1+ R[\sigma _{i}]_{C_\mathrm{Lip}}) + C_{0}b L_{f} (1+ R[\zeta ]_{C_\mathrm{Lip}}) \right) d(u,v) . \end{aligned}$$

In addition, for $t\in [0,t_{1}]$ we note that $\parallel \Gamma u (t)- \Gamma v(t)\parallel \le C_{0}bL_{f} (1+ R[\zeta ]_{C_\mathrm{Lip}}) d(u,v) .$ From the above estimates we infer that

$$\begin{aligned} d( \Gamma u , \Gamma v) \le (\Lambda _{X,W}L_{X,W}(g)(1+ R\max _{i=1,\ldots ,N}[\sigma _{i}]_{C_\mathrm{Lip}}) + C_{0}bL_{f} ( 1+ R[\zeta ]_{C_\mathrm{Lip}}))d(u,v) . \end{aligned}$$

Thus, $\Gamma (\cdot )$ is a contraction and there exists a unique mild solution $u\in {\mathcal {S}}(R)$ of (1.1)–(1.3).

We prove now that $u(\cdot )$ is a classical solution. Let ${\widetilde{u}}_{i}$, $i=1,\ldots N,$ be defined as in the introduction. It is easy to see that ${\widetilde{u}}_{i}(\cdot )$ is the mild solution of the problem

$$\begin{aligned} w'(t)= & {} Aw(t)+ f(t, u(\zeta (t,u(t)))), \,\, \quad t\in I_{i}= [t_{i},t_{i+1}], \end{aligned}$$

(2.8)

$$\begin{aligned} w(t_{i})= & {} g _{i}(u(\sigma _{i}(u(t_{i}^{+})))). \end{aligned}$$

(2.9)

Since $f(\cdot , u(\zeta (\cdot ,u(\cdot ))))$ is Lipschitz on $I_{i}$ and the semigroup is analytic, from [35, Theorem 4.3.2] it follows that ${\widetilde{u}}_{i}$ is a classical solution of (2.8)–(2.9). The same argument prove that ${\widetilde{u}}_{0}$ is a classical solution of (2.8) on $[0,t_{1}]$ with initial condition $u(0)=\varphi (0)$. From the above, we obtain that $u(\cdot )$ is a classical solution of (1.1)–(1.3). $\square$

In the next result we establish the existence and uniqueness of a classical solution without to use condition ${\mathbf{H}}_{{\mathbf{g,X}}}^{{\mathbf{W}}}$. In place of this condition, we introduce the following one:

${\mathbf{H}}_{{{g}}_{{{i}}},\sigma _{{{j}}}}$ :

$\sigma _{i}\in C_\mathrm{Lip}(X,[-p,a])$ for all $i\in \{1,\ldots , N\}$, $\overline{\cup _{i=1}^{N}\sigma _{i}(X)}\subset \cup _{i=0}^{N}I_{i}\cup [-p,0]$, $g_{i}\in C(X_{1};X_{1})\cap C_\mathrm{Lip}(X;X)$ and there are constants $l_{g_{i}}, k_{g_{i}}$ such that $\parallel Ag_{i}(x)\parallel \le l_{g_{i}}r+ k_{g_{i}}$ for all $x\in B_{r}(0,X_{1})$, $i\in \{1,\ldots , N\}$ and every $r>0$.

Notations 2

If condition ${\mathbf{H}}_{{{g}}_{{{i}}},\sigma _{{{j}}}}$ is verified, we use the notations $l_{g}=\max _{i=1,\ldots }l_{g_{i}}$ and

$$\begin{aligned} \Upsilon&= C_{0}\max _{i=1,\ldots , N}k_{g_{i}} + { 2 C_{0} C_{X}(f) } + b(C_{0} +C_{1})L_{f} + [T(\cdot )\varphi (0)]_{C_\mathrm{Lip}([-p,0];X)} \\&\quad + \parallel \varphi \parallel _{C([-p,0];X_{1})} + [\varphi ]_{C_\mathrm{Lip}([-p,0];X)} . \end{aligned}$$

Theorem 2.2

Assume that the conditions ${\mathbf{H}}_{\zeta }$, ${\mathbf{H}}_{{{g}}_{{{i}}},\sigma _{{{j}}}}$, ${\mathbf{H}}_{\mathbf{g}}$ and ${\mathbf{H}}_{{\mathbf{f}}}$ are satisfied, X is a Hilbert space, A is self-adjoint, $T(\cdot )\varphi (0)\in C_\mathrm{Lip}([0,a];X)$, $\varphi \in C_\mathrm{Lip}([-p,0];X)\cap C([-p,0];X_{1})$ and

$$\begin{aligned} 2 bL_{f}( ( C_{0} +C_{1}) [\zeta ]_{C_\mathrm{Lip}} (1+ 2\Upsilon ) +C_{0}) +2 C_{0} (l_{g} + L_{g} (1+ 2\max _{i=1,\ldots ,N}[\sigma _{i}]_{C_\mathrm{Lip}} \Upsilon )) <1.\nonumber \\ \end{aligned}$$

(2.10)

Then there exists a unique classical solution $u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X)$ of the problem (1.1)–(1.3) such that $A{\widetilde{u}}_{i} \in C([t_{i},t_{i+1}];X )$ for all $i=1,\ldots , N$.

Proof

Let $P:{\mathbb {R}}\mapsto {\mathbb {R}}$ be the polynomial given by

$$\begin{aligned} P(x)= & {} \Upsilon + (b L_{f}( ( C_{0} +C_{1}) [\zeta ]_{C_\mathrm{Lip}}+C_{0}) + C_{0}(L_{g} +L_{G}) -1)x\nonumber \\&(C_{0}L_{g} \max _{i=1,\ldots ,N}[\sigma _{i}]_{C_\mathrm{Lip}} + bL_{f} ( C_{0} +C_{1}) [\zeta ]_{C_\mathrm{Lip}}) x^{2} . \end{aligned}$$

(2.11)

From (2.10) there exists $R>0$ such that $P(R)<0$ and

$$\begin{aligned} \Upsilon + C_{0}l_{g}R +{ ( C_{0} +C_{1})}b L_{f} [\zeta ]_{C_\mathrm{Lip}}( R+ R^{2})< & {} R, \end{aligned}$$

(2.12)

$$\begin{aligned} C_{0}L_{X,X}(g)(1+ R\max _{i=1,\ldots ,N}[\sigma _{i}]_{C_\mathrm{Lip}}) + C_{0}bL_{f} ( 1+ R[\zeta ]_{C_\mathrm{Lip}})< & {} 1. \end{aligned}$$

(2.13)

Let ${\mathcal {S}}(R)$ the space in the proof of Theorem 2.1 and ${\mathcal {S}}(\sigma _{i},R)$ be the space

$$\begin{aligned} {\mathcal {S}}(\sigma _{i},R) = \{u\in {\mathcal {S}}(R) : u(t)\in D(A)\,\,\hbox {and} \parallel Au(t) \parallel \le R, \, \, \, \forall \,\, t\in \cup _{i=1}^{N}\sigma _{i}(X) \} ,\nonumber \\ \end{aligned}$$

(2.14)

endowed with the metric $d(u,v)=\parallel u-v\parallel _{{{\mathcal {P}}{\mathcal {C}}} (X)}$. Let $\Gamma :{\mathcal {S}}(\sigma _{i},R) \mapsto {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}(X)$ be defined as in the proof of Theorem 2.1. Next we prove that $\Gamma$ is a contraction on ${\mathcal {S}}(\sigma _{i},R)$.

Let $u\in {\mathcal {S}}(\sigma _{i},R)$, $i\in \{1,\ldots ,N\}$, $t\in (t_{i},t_{i+1})$ and $h>0$ such that $t+h\in (t_{i},t_{i+1}]$. Arguing as in the proof of Theorem 2.1 and noting that $u(\sigma (u(t_{i}^{+})))\in X_{1}$, we see that

$$\begin{aligned}&\parallel \Gamma u(t+h)- \Gamma u(t)\parallel \\&\quad \le \int _{t-t_{i}}^{t+h-t_{i}} \parallel T(s) A g_{i}(u(\sigma _{i}(u(t_{i}^{+}))))\parallel {\text {d}}s \ + ( C_{0} ({\mathcal {C}}_{X}(f) + b L_{f} )h \\&\qquad +\, C_{0}b L_{f} [\zeta ]_{C_\mathrm{Lip}}( R+ R^{2} )h \le C_{0}( l_{g_{i}}R +k_{g_{i}}) h \\&\qquad +\, C_{0} ({\mathcal {C}}_{X}(f) + b L_{f} )h \\&\qquad +\, C_{0}b L_{f} [\zeta ]_{C_\mathrm{Lip}}( R+ R^{2} )h , \end{aligned}$$

and hence, $[ (\Gamma u)_{\mid _{I_{i}}} ]_{C_\mathrm{Lip}( I_{i};X)} \le \Upsilon + C_{0} l_{g_{i}}R+ C_{0}b L_{f} [\zeta ]_{C_\mathrm{Lip}}( R+ R^{2})\le R .$ In addition, it is easy to see that

$$\begin{aligned} {[} (\Gamma u)_{\mid _{[0,t_{1}]}} ]_{C_\mathrm{Lip}([0,t_{1}];X)}\le & {} [T(\cdot )\varphi (0)]_{C_\mathrm{Lip}([-p,0];Z)} + C_{0}({\mathcal {C}}_{X}(f) +b L_{f}) \\&+\,C_{0}b L_{f} [\zeta ]_{C_\mathrm{Lip}}( R+ R^{2})) \le R. \end{aligned}$$

From the above remarks we have that $[ (\Gamma u)_{\mid _{[0,a]}} ]_{ {{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X)}\le R$ which shows that $\Gamma u\in {\mathcal {S}}(R)$. In addition, arguing as in the proof of Theorem 2.1 it follows that

$$\begin{aligned} d( \Gamma u , \Gamma v) \le C_{0} (L_{g}(1+ R\max _{i=1,\ldots ,N}[\sigma _{i}]_{C_\mathrm{Lip}}) + bL_{f} ( 1+ R[\zeta ]_{C_\mathrm{Lip}}))d(u,v) . \end{aligned}$$

From the above remarks, we have that $\Gamma$ is a contraction on ${\mathcal {S}}(R)$.

Next we show that $\parallel A\Gamma u(t)\parallel \le R$ for all $t\in \cup _{j=1}^{N}\sigma _{j}(X)$. Let $t\in \cup _{j=1}^{N}\sigma _{j}(X)$ and assume that $t\in (t_{i},t_{i+1}]$ for $i\ge 1$. Using that $(T(t))_{t\ge 0}$ is analytic and that $u(\sigma (u(t_{i}^{+})))\in X_{1}$ and $\parallel A u(\sigma (u(t_{i}^{+}))) \parallel \le l_{g_{i}}R +k_{g_{i}}$, we note that

$$\begin{aligned} A\Gamma u(t)= & {} T(t-t_{i})A g_{i}(u(\sigma _{i}(u(t_{i}^{+}))))\\&+ \, { T(t-t_{i} )f(t, u(\zeta (t,u(t))))- f(t, u(\zeta (t,u(t))))}\\&+ \,\int _{ t_{i}}^{t} A T(t-s)(f(s, u(\zeta (s,u(s)))- f(t, u(\zeta (t,u(t)))) {\text {d}}s,\\ \parallel A\Gamma u(t)\parallel\le & {} C_{0}( l_{g_{i}}R +k_{g_{i}}) + 2C_{0}C_{X}(f) \\&+ \,\int _{0}^{t} \frac{C_{1}}{t-s} L_{f}(1+ [ u(\zeta (\cdot ,u(\cdot )))]_{ C_\mathrm{Lip}(I_{i};X)})(t-s) {\text {d}}s \\\le & {} C_{0}( l_{g_{i}}R +k_{g_{i}})+ 2C_{0} C_{X}(f) + bC_{1} L_{f} + C_{1} b L_{f} [\zeta ]_{C_\mathrm{Lip}}( R+ R^{2}), \end{aligned}$$

which implies that $\parallel A\Gamma u(t)\parallel \le \Upsilon + C_{0} l_{g_{i}}R + C_{1}b L_{f} [\zeta ]_{C_\mathrm{Lip}}( R+ R^{2})\le R .$ If $t\in I_{1}$ we see that

$$\begin{aligned} \parallel A\Gamma u(t)\parallel {\le } C_{0}\parallel A \varphi (0)\parallel + 2C_{0}C_{X}(f) + bC_{1} L_{f} + C_{1}b L_{f} [\zeta ]_{C_\mathrm{Lip}}( R+ R^{2}) {\le } R . \end{aligned}$$

Thus, $\parallel A\Gamma u(t)\parallel \le R$ for all $t\in \cup _{i=1}^{N}\sigma _{i}(X)$ and $\Gamma$ is a ${\mathcal {S}}(\sigma _{i},R)$-valued function.

To finish the proof, we prove that ${\mathcal {S}}(\sigma _{i},R)$ is a closet subset of ${\mathcal {S}}(R)$. Let $(u_{n})_{n\in {\mathbb {N}}}$ be a sequence in ${\mathcal {S}}(\sigma _{i},R)$ and $u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}} (X)$ such that $u_{n}\rightarrow u$ as $n\rightarrow \infty$. Let $t\in \cup _{i=1}^{N}\sigma _{i}(X)$. Since $(Au_{n}(t))_{n\in {\mathbb {N}}}$ is bounded, there exists $w\in X$ such that $<Au_{n}(t), z>\rightarrow <w, z>$ as $n\rightarrow \infty$ for all $z\in X$. In particular, for $v\in X_{1}$ we have that $<Au_{n}(t),v>=<u_{n}(t),Av>\rightarrow <u(t),Av>$ as $n\rightarrow \infty$, which implies that $<w, v> = <u(t),Av>$ for all $v\in X_{1}$. Using that A is self-adjoint, we obtain that $u(t)\in X_{1}$, $Au(t)=w$ and $\parallel Au(t)\parallel =\parallel w\parallel \le \liminf _{n\rightarrow \infty } \parallel Au_{n}(t)\parallel \le R$, which completes the proof that ${\mathcal {S}}(\sigma _{i},R)$ is closed.

From the above it follows that $\Gamma$ is a contraction on ${\mathcal {S}}(\sigma _{i},R)$ and there exists a unique mild solution $u\in {\mathcal {S}}(\sigma _{i},R)$. The fact that $u(\cdot )$ is a classical solution follows from the proof of Theorem 2.1. $\square$

The next result consider the case where $\sigma _{i}(X)\subset [-p,0]$ for all $i=1,\ldots ,N$. The proof use the ideas in the proof of Theorem 2.1 and we include a short proof for completeness.

Proposition 2.1

Let conditions ${\mathbf{H}}_{\mathbf{g}}$ and ${\mathbf{H}}_{\mathbf{f}}$ be holds. Assume $\zeta \in C_\mathrm{Lip}([0,a]\times X;[-p,a])$, $\sigma _{i}\in C_\mathrm{Lip}(X;[-p,0])$ for all $i=1,\ldots ,N$, $T(\cdot )\varphi (0)\in C_\mathrm{Lip}([0,a];X)$, $\varphi \in C_\mathrm{Lip}([-p,0];X)$, $g_{i}(\varphi (\cdot ))\in C([-p,0];W)$ for all $i=1,\ldots ,N$ and

$$\begin{aligned} 2C_{0}b L_{f}(1+ [\zeta ]_{C_\mathrm{Lip}} (1+ 2\Phi _{X,W,\varphi })) + 2 L_{g}\Psi _{\varphi ,\sigma _{i},g_{i}} <1. \end{aligned}$$

(2.15)

where ${ \Phi _{X,W,\varphi }} = \Phi _{X,W}\max _{i=1,\ldots ,N}\parallel g_{i}(\varphi (\cdot ))\parallel _{C([-p,0];W)} + C_{0} ( {\mathcal {C}}_{X}(f) + bL_{f}) + [\varphi ]_{C_\mathrm{Lip}([-p,0];X)}$ $+[T(\cdot )\varphi (0)]_{C_\mathrm{Lip}([-p,0];X)}$ and $\Psi _{\varphi ,\sigma _{i},g_{i}}= C_{0}[\varphi ]_{C_\mathrm{Lip}([-p,0];X)}\max _{i=1,\ldots , N}[\sigma _{i}]_{C_\mathrm{Lip}}$. Then there exists a unique classical solution $u \in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X)$ of the problem (1.1)–(1.3).

Proof

Let $P:{\mathbb {R}}\mapsto {\mathbb {R}}$ be given by $P(x) = \Phi _{X,W,\varphi } + (C_{0}b L_{f}(1+ [\zeta ]_{C_\mathrm{Lip}})+ L_{g}\Psi _{\varphi ,\sigma _{i},g_{i}}-1)x + C_{0}bL_{f} [\zeta ]_{C_\mathrm{Lip}} x^{2} .$ From (2.15) there exists $R>0$ such that $P(R)<0$. Let ${\mathcal {S}}(R)$ be defined as in the proof of Theorem 2.1 and $\Gamma : {\mathcal {S}}(R) \mapsto {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}} (X)$ be the map given by $\Gamma u_{0} =\varphi$ and

$$\begin{aligned} \Gamma u(t)= & {} T(t)\varphi (0)+ \int _{0}^{t}T(t-\tau )f(\tau ,u(\zeta (\tau ,u(\tau )))){\text {d}}\tau , \,\,\,\, t\in [0,t_{1}], \\ \Gamma u(t)= & {} T(t-t_{i}) g_{i}(\varphi (\sigma _{i}(u(t_{i}^{+})))) + \int _{t_{i}}^{t}T(t-s)f(s,u(\zeta (s,u(s)))) {\text {d}}s, \,\,\,\, t\in (t_{i},t_{i+1}]. \end{aligned}$$

Arguing as in the proof of Theorem 2.1, for $i\in \{1,\ldots ,i\}$, $t\in (t_{i},t_{i+1})$ and $h>0$ such that $t+h\in (t_{i},t_{i+1}]$, it is easy to see that

$$\begin{aligned} \parallel \Gamma u(t+h)- \Gamma u(t)\parallel\le & {} \Phi _{X,W}\max _{j=1,\ldots ,N,} \parallel g_{j}(\varphi (\cdot ))\parallel _{C([-p,0];W)} h \\&+\, ( C_{0} {\mathcal {C}}_{X}(f) + b L_{f}) h+ C_{0}b L_{f} [\zeta ]_{C_\mathrm{Lip}}( R+ R^{2}) h , \end{aligned}$$

which implies (from the definition of $P(\cdot )$) that $[ (\Gamma u)_{\mid _{I_{i}}} ]_{C_\mathrm{Lip}( I_{i};X)} \le R .$ Similarly, we have that

$$\begin{aligned} {[} (\Gamma u)_{\mid _{[0,t_{1}]}} ]_{C_\mathrm{Lip}([0,t_{1}];X)}\le & {} [T(\cdot )\varphi (0)]_{C_\mathrm{Lip}([-p,0];X)} + C_{0} ( {\mathcal {C}}_{X}(f) + b L_{f}) \\&+ \, C_{0}b L_{f} [\zeta ]_{C_\mathrm{Lip}}( R+ R^{2}) \\\le & {} \Phi _{X,W,\varphi }+ C_{0}b L_{f} [\zeta ]_{C_\mathrm{Lip}}( R+ R^{2})\le R. \end{aligned}$$

From the above, $[ (\Gamma u)_{\mid _{[0,a]}} ]_{ {{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X)}\le R,$ which proves that $\Gamma$ is a ${\mathcal {S}}(R)$-valued function.

On the other hand, for $u,v \in {\mathcal {S}}(R)$, $i=1,\ldots , N$, $t\in (t_{i},t_{i+1}]$ and $s\in [0,t_{1}]$ we get

$$\begin{aligned} \parallel \Gamma u (t)- \Gamma v(t)\parallel\le & {} ( C_{0} L_{g}[\varphi ]_{C_\mathrm{Lip}([-p,0];X)}\max _{j=1,\ldots , N}[\sigma _{j}]_{C_\mathrm{Lip}}\\&+ \, C_{0}b L_{f} (1+ R[\zeta ]_{C_\mathrm{Lip}}) ) d(u,v) ,\\ \parallel \Gamma u (s)- \Gamma v(s)\parallel\le & {} C_{0}bL_{f}( 1+ R[\zeta ]_{C_\mathrm{Lip}} )d(u,v) , \end{aligned}$$

which allows us infer that $\Gamma$ is a contraction and there exists a unique mild solution $u\in {\mathcal {S}}(R)$ of the problem (1.1)–(1.3). The fact that $u(\cdot )$ is a classical solution follows from the proof of Theorem 2.1. $\square$

In the next result, we assume that the functions $\zeta (\cdot )$ and $\sigma _{i}(\cdot )$ have values in $[-r,0]$.

Proposition 2.2

Suppose that the conditions ${\mathbf{H}}_{\mathbf{g}}$, ${\mathbf{H}}_{{\mathbf{f}}}$ are satisfied, $\varphi \in C_\mathrm{Lip}([-p,0];X)$, $\sigma _{i}\in C_\mathrm{Lip}(X;[-p,0])$ for all $i=1,\ldots ,N$, $\zeta \in C_\mathrm{Lip}([0,a]\times X;[-p,0])$ and

$$\begin{aligned} C_{0} [\varphi ]_{C_\mathrm{Lip}([-p,0];X)}( L_{g} \max _{j=1,\ldots , N}[\sigma _{j}]_{C_\mathrm{Lip}} + b L_{f} [\zeta ]_{C_\mathrm{Lip}}) < 1. \end{aligned}$$

Then there exists a unique mild solution $u\in {{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X)$ of (1.1)–(1.3).

Proof

Let $\Gamma : {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}} (X) \mapsto {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}} (X)$ be defined as in the proof of Theorem 2.1, but using $f(\tau ,\varphi (\zeta (\tau ,u(\tau ))))$ in place $f(\tau ,u(\zeta (\tau ,u(\tau ))))$. In this case, for $u,v\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X)$ we see that

$$\begin{aligned} \parallel \Gamma u - \Gamma v\parallel _{C((t_{i},t_{i+1}];X)}\le & {} C_{0} L_{g}[\varphi ]_{C_\mathrm{Lip}([-p,0];X)}\max _{j=1,\ldots , N}[\sigma _{j}]_{C_\mathrm{Lip}} d(u,v) \\&+ \, C_{0}b L_{f}[\varphi ]_{C_\mathrm{Lip}([-p,0];X)}[\zeta ]_{C_\mathrm{Lip}} d(u,v) ,\\ \parallel \Gamma u - \Gamma v\parallel _{C([0,t_{1}];X)}\le & {} C_{0}b L_{f}[\varphi ]_{C_\mathrm{Lip}([-p,0];X)}[\zeta ]_{C_\mathrm{Lip}} d(u,v) , \end{aligned}$$

which allows us to conclude that $\Gamma$ is a contraction. $\square$

Next, we discuss briefly the case in which the functions $f(\cdot )$ and $g_{i}(\cdot )$ are locally bounded and (or) locally Lipschitz. For sake of clarity, we include the next conditions.

${\mathcal {H}}_{g, X}^{W}$ :

For all $i=1,\ldots ,N$, there is ${L}_{X,W}(g_{i},\cdot )\in C({\mathbb {R}}; {\mathbb {R}})$ such that $\parallel g_{i}(x)-g_{i}(y)\parallel _{W}\le {L}_{X,W}(g_{i},r)\parallel x-y\parallel$ for all $x,y\in B_{r}(0,X)$ and every $r>0$. Next, ${L}_{X,W}(g,r)= \max _{i=1,\ldots , N} L_{X,W}(g_{i},r)$ and ${\mathcal {C}}_{X,W}(g_{i},r)=\parallel g_{i}\parallel _{C(B_{r}(0,X);W)}$.

${\mathcal {H}}_{f}$ :

There is ${L}_{f} \in C({\mathbb {R}}; {\mathbb {R}})$ such that $\parallel f(t,x)-f(s,y)\parallel \le {L}_{f}(r)(\mid t-s\mid + \parallel x-y\parallel )$ for all $x,y\in B_{r}(0,X)$, $t,s\in [0,a]$ and $r>0$. Next, for $r>0$ we use the notation ${\mathcal {C}}_{X}(f,r)=\parallel f\parallel _{C([0,a]\times B_{r}(0,X);X)}$.

${\mathcal {H}}_{\mathbf{g}}$ :

There are functions ${L}_{g_{i}} \in C({\mathbb {R}}; {\mathbb {R}})$ such that $\parallel g_{i}(x)-g_{i}(y)\parallel \le {L}_{g_{i}}(r)\parallel x-y\parallel$ for all $x,y\in B_{r}(0,X)$ and $r>0$. Next, $L_{g}(r){=}\max _{i=1,\ldots , N}L_{g_{i}}(r)$, ${\mathcal {C}}_{X}(g)(r)=\max _{i=1,\ldots , N} {\mathcal {C}}_{X}(g_{i})(r)$ and ${\mathcal {C}}_{X}(g_{i},r)=\parallel g_{i}\parallel _{C(B_{r}(0,X);X)}$.

Notations 3

For $r>0$, we define $\Phi _{X,W}(r) = \Lambda _{X,W} {\mathcal {C}}_{X,W}(g,r) + C_{0}( C_{X}(f,r) + b L_{f}(r)) + [T(\cdot )\varphi (0)]_{C_\mathrm{Lip}([-p,0];X)} + [\varphi ]_{C_\mathrm{Lip}([-p,0];X)} .$

The proof of Proposition 2.3 follows from the proof of Theorem 2.1.

Proposition 2.3

Let conditions ${\mathbf{H}}_{\zeta }, {\mathbf{H}}_{\sigma _{{\mathbf{i}}} }$, ${\mathcal {H}}_{g,X}^{W}$ and ${\mathcal {H}}_{f}$ be holds. Suppose that $T(\cdot )\varphi (0)\in C_\mathrm{Lip}([0,a];X)$, $\varphi \in C_\mathrm{Lip}([-p,0];X)$ and there is $r>0$ such that (2.4) is satisfied with $L_{f}(r),$ $\Phi _{X,W} (r)$ and $L_{X,W}(g,r)$ in place $L_{f}, \Phi _{X,W}$ and $L_{X,W}(g)$, and

$$\begin{aligned} \max \{C_{0} (\max \{ \parallel \varphi (0)\parallel , \parallel i_{c}\parallel _{{\mathcal {L}}(W,X)} {\mathcal {C}}_{X,W}(g,r)\} + b {\mathcal {C}}_{X}(f,r)),\parallel \varphi \parallel _{C([-p,0];X)}\}\le r. \end{aligned}$$

Then there exists a unique classical solution $u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X)$ of the problem (1.1)–(1.3).

Proof

Let $P:{\mathbb {R}}\mapsto {\mathbb {R}}$ be defined as in the proof of Theorem 2.1, but using $L_{f}(r), \Phi _{X,W} (r)$ and $L_{X,W}(g,r)$ in place $L_{f}, \Phi _{X,W}$ and $L_{X,W}(g)$. Arguing as in the proof of Theorem 2.1 we infer that there exists $R>0$ such that

$$\begin{aligned} \Phi _{X,W} (r) + C_{0}b L_{f}(r) [\zeta ]_{C_\mathrm{Lip}}( R+ R^{2})< & {} R, \nonumber \\\end{aligned}$$

(2.16)

$$\begin{aligned} \Lambda _{X,W}L_{X,W}(g,r)(1+ R\max _{i=1,\ldots ,N}[\sigma _{i}]_{C_\mathrm{Lip}}) + C_{0}bL_{f}(r) ( 1+ R[\zeta ]_{C_\mathrm{Lip}})< & {} 1.\nonumber \\ \end{aligned}$$

(2.17)

Let ${\mathcal {S}}(R)$ be the space in the proof of Theorem 2.1 and ${\mathcal {S}}(r,R)= \{u\in {\mathcal {S}}(R) : \, \parallel u \parallel _{ {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}} (X)}\le r \} ,$ endowed with the metric $d(u,v)=\parallel u-v\parallel _{{\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}} (X)}$. Let $\Gamma : {\mathcal {S}}(r,R) \mapsto {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}} (X)$ be defined as in the proof of Theorem 2.1.

From the proof of Theorem 2.1 we infer that $\Gamma$ is a contraction on ${\mathcal {S}}( R)$. Moreover, for $t\in I_{i}$ with $i\ge 0$ it is easy to see that

$$\begin{aligned} \parallel \Gamma u(t)\parallel \le C_{0} \max \{\parallel \varphi (0)\parallel , \parallel i_{c}\parallel _{{\mathcal {L}}(W,X)}{\mathcal {C}}_{X,W}(g,r)\} + C_{0}b {\mathcal {C}}_{X}(f,r) \le r, \end{aligned}$$

which implies that $\parallel \Gamma u \parallel _{ {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}} (X)}\le r$ since $r>\parallel \varphi \parallel _{C([-p,0];X)}$. Thus, $\Gamma$ is a contraction on ${\mathcal {S}}( r,R)$ and there exists a unique mild solution $u\in {\mathcal {S}}(r,R)$ of (1.1)–(1.3). Finally, from [35, Theorem 4.3.2] we infer that $u(\cdot )$ is a classical solution. $\square$

Corollary 2.1

Assume that the conditions ${\mathbf{H}}_{\zeta }, {\mathbf{H}}_{\sigma _{{\mathbf{i}}} }$, ${\mathcal {H}}_{g,X}^{W}$ and ${\mathcal {H}}_{f}$ are satisfied, the functions $L_{f}(\cdot ),$ ${\mathcal {C}}_{X}(f,\cdot )$, $L_{X,W}(g,\cdot )$ and ${\mathcal {C}}_{X,W}(g,\cdot )$ are non-decreasing, $\varphi \in C_\mathrm{Lip}([-p,0];X)$, $T(\cdot )\varphi (0)\in C_\mathrm{Lip}([0,a];X)$, $\limsup _{r\rightarrow \infty }\frac{1}{r} C_{0} (\parallel i_{c}\parallel _{{\mathcal {L}}(W,X)} {\mathcal {C}}_{X,W}(g,r) + b {\mathcal {C}}_{X}(f,r)) < 1$ and

$$\begin{aligned}&{{ 2 C_{0}b \limsup _{r\rightarrow \infty } L_{f}(r)\frac{1}{r}( 1+ [\zeta ]_{C_\mathrm{Lip}} (1+2 \Phi _{X,W}(r)) }} \nonumber \\&\quad +{\ }2 \Lambda _{X,W}\limsup _{r\rightarrow \infty }L_{X,W}(g,r)\frac{1 }{r}(1+ 2\max _{i=1,\ldots ,N}[\sigma _{i}]_{C_\mathrm{Lip}} \Phi _{X,W}(r)) <1 .\nonumber \\ \end{aligned}$$

(2.18)

Then there exists a unique classical solution $u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X)\cap B_{r}(0, {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}} (X))$ of (1.1)–(1.3).

We establish now, without proof, a result similar to Theorem 2.2 for the case where $f(\cdot )$ satisfy the condition ${\mathcal {H}}_{f}$.

Proposition 2.4

Suppose the conditions ${\mathbf{H}}_{\zeta }$, ${\mathbf{H}}_{g_{i},\sigma _{j}}$, ${\mathcal {H}}_{g}$ and ${\mathcal {H}}_{f}$ be holds, X is a Hilbert space, A is self-adjoint, $T(\cdot )\varphi (0)\in C_\mathrm{Lip}([0,a];X)$ and $\varphi \in C_\mathrm{Lip}([-p,0];X)\cap C([-p,0];X_{1})$. If there is $r>0$ such that the inequality (2.10) is valid with $L_{f}(r)$, $L_{g}(r)$, ${\mathcal {C}}_{X}(f,r)$ and ${\mathcal {C}}_{X}(g,r)$ in place $L_{f}$, $L_{g}$, ${\mathcal {C}}_{X}(f)$ and ${\mathcal {C}}_{X}(g)$, and $C_{0}( \max \{ \parallel \varphi (0)\parallel ,{\mathcal {C}}_{X}(g,r)\} + b {\mathcal {C}}_{X}(f,r))\le r,$ then there exists a unique classical solution $u \in B_{r}(0,{\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}} (X))\cap {{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X))$ of (1.1)–(1.3).

To complete this section, we study the existence of solution using the Schauder’s fixed point Theorem. The next lemma follows from the proof of [32, Proposition 4.2.1].

Lemma 2.3

Let $\alpha \in (0,1)$, $\xi \in L^{\infty }([b,c];X)$ and $v:[b,c]\mapsto X$ be the function defined by $v(t)=\int _{b}^{t}T(t-s)\xi (s){\text {d}}s$. Then ${ [v]_{ C^{\alpha }([b,c];X) } } \le \parallel \xi \parallel _{L^{\infty }([b,c];X)}({ (c-b)^{1-\alpha }}C_{0}+ \frac{C_{1}}{\alpha (1-\alpha )})$.

Theorem 2.3

Assume that the conditions ${\mathbf{H}}_{\zeta }$ and ${\mathbf{H}}_{\sigma _{{\mathbf{i}}} }$ are satisfied, there is a Banach space $(Y,\parallel \cdot \parallel _{Y}) \hookrightarrow (X,\parallel \cdot \parallel )$ such that $\parallel T(t)- I\parallel _{{\mathcal {L}}(Y,X)} \rightarrow 0$ as $t\rightarrow 0$, $g_{i} \in C(X;Y)$ for all i, $f\in C([0,a]\times X;X)$, the functions $g_{i}(\cdot ),f(\cdot )$ are bounded and $(T(t))_{t\ge 0}$ is compact. Then there exists a mild solution of the problem (1.1)–(1.3).

Proof

Let ${\mathcal {C}}_{X,Y}(g)=\max _{i=1,\ldots , N}\parallel g_{i}\parallel _{ C(X; Y)}$, $C_{X}(f)= \parallel f\parallel _{ C([0,a]\times X; X)}$ and $\alpha \in (0,1)$. Let ${\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X)=\{ u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}(X):\, u_{0}=\varphi \}$ endowed with the metric $d(u,v)=\parallel u- v\parallel _{{{\mathcal {B}}{\mathcal {P}}{\mathcal {C}}}(X)}$ and $\Gamma : {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X) \mapsto {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}(X)$ be defined as in the proof of Theorem 2.1.

It is easy to prove that $\Gamma$ is continuous. Next, using Lemma 1.1, we show that $\Gamma$ is completely continuous.

Let $i\in \{1,\ldots , N\}$. From Lemma 2.3, for $t\in (t_{i},t_{i+1})$, $h>0$ with $t+h\in (t_{i},t_{i+1}]$, we get

$$\begin{aligned}&{{\parallel \Gamma u(t+h)- \Gamma u(t)\parallel }}\\&\quad \le \parallel (T(t+h{ -t_{i}})-T(t{ -t_{i}}))g_{i}(u(\sigma _{i}(u(t_{i}^{+}))))\parallel \\&\qquad + \parallel \int _{t_{i}} ^{t+h} T(t+h-s) f(s, u(\zeta (s,u(s)))) {\text {d}}s- \int _{t_{i}} ^{t} T(t-s) f(s, u(\zeta (s,u(s)))){\text {d}}s \parallel \\&\quad \le \parallel (T(t+h{ -t_{i}})-T(t{ -t_{i}})) \parallel _{{\mathcal {L}}(Y,X)} {\mathcal {C}}_{X,Y}(g) + {\mathcal {C}}_{X}(f)\left( a^{1-\alpha } C_{0}+ \frac{C_{1}}{\alpha (1-\alpha )}\right) h^{\alpha } , \end{aligned}$$

which shows that $\{(\Gamma u)_{\mid _{I_{i}}} :u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X) \}$ is right equicontinuous at $t\in (t_{i},t_{i+1})$. A similar argument prove that $\{(\Gamma u)_{\mid _{I_{i}}} :u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X) \}$ is left equicontinuous at $t=t_{i+1}$, which implies that $\{(\Gamma u)_{\mid _{I_{i}}} :u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X) \}$ is equicontinuous on $I_{i}$. In addition, for $u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X)$ and $0<h<\delta$ we note that

$$\begin{aligned}&{{\parallel \widetilde{ \Gamma u } (t_{i}+h)- \widetilde{ \Gamma u } (t_{i})\parallel }} \\&\quad = \parallel (T(h)-I)g_{i}(u(\sigma _{i}(u(t_{i}^{+}))))\parallel + \int _{t_{i}} ^{t_{i}+h} T(t_{i}+h-s) f(s, u(\zeta (s,u(s)))) {\text {d}}s\parallel \\&\quad \le \parallel T(h)-I\parallel _{{\mathcal {L}}(Y,X)} {\mathcal {C}}_{X,Y}(g) + {\mathcal {C}}_{X}(f)\left( a^{1-\alpha } C_{0}+ \frac{C_{1}}{\alpha (1-\alpha )}\right) h^{\alpha }, \end{aligned}$$

which proves that $\widetilde{ \Gamma {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X) }_{i}= \{(\widetilde{ \Gamma u})_{i} : u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X) \}$ is right equicontinuous at $t_{i}$. From the above it follows that $\{\widetilde{(\Gamma u)}_{{i}} :u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X) \}$ is equicontinuous on $I_{i}$.

We prove now that $\{(\widetilde{ \Gamma u})_{i} (t) :u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X) \}$ is relatively compact in X for all $t\in [t_{i},t_{i+1}]$. Since the semigroup is compact, $(Y,\parallel \cdot \parallel _{Y}) \hookrightarrow (X,\parallel \cdot \parallel )$ and $g_{i}(\cdot )$ is bounded with values in Y, we have that $U= \{ g_{j}(u(\sigma _{j}(u(t_{j}^{+})))):u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X),\,j=1,\ldots ,N\}$ is relatively compact in X. For $t\in (t_{i}, t_{i+1}]$ and $0<\varepsilon <t-t_{i}$, we note that

$$\begin{aligned} (\widetilde{ \Gamma u})_{i}(t)= & {} T(t-t_{i})U+ T(\varepsilon )\int _{t_{i}}^{t-\varepsilon }T(t-\varepsilon -s) f(s, u(\zeta (s,u(s)))) {\text {d}}s \\&+ \int _{t-\varepsilon }^{t}T(t-s) f(s, u(\zeta (s,u(s)))) {\text {d}}s \\&\in T(t-t_{i})U+ T(\varepsilon ) C_{0}(t-\varepsilon -t_{i}) {\mathcal {C}}_{X}(f) B_{1}(0,X) + \varepsilon C_{0}{\mathcal {C}}_{X}(f) B_{1}(0,X), \end{aligned}$$

and hence, $\{(\widetilde{ \Gamma u})_{i} (t) :u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X) \} \subset K_{\varepsilon }+ D_{\varepsilon },$ where $K_{\varepsilon }$ is relatively compact and the diameter of $D_{\varepsilon }$ converges to zero as $\varepsilon \rightarrow 0$. This prove that the set $\Gamma {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X) (t)$ is relatively compact in X. Moreover, since $\widetilde{ \Gamma {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X) }_{i} (t_{i})= \{ g_{i}(u(\sigma _{i}(u(t_{i}^{+})))):u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X)\} \subset {\overline{U}},$ we obtain that $\widetilde{ \Gamma {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X) }(t_{i})$ is relatively compact in X. From the above remarks we have that $(\widetilde{ \Gamma {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X) })_{i}$ is relatively compact in $C([t_{i},t_{i+1}];X)$. Moreover, the same argument also prove that $(\widetilde{\Gamma {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X)})_{1} =\{ (\Gamma u)_{\mid _{[0,t_{1}]}} : u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X))\}$ is relatively compact in $C([0,t_{1}];X)$.

From the above and Lemma 1.1, it follows that $\Gamma$ is completely continuous and noting that the functions $f(\cdot )$ and $g_{i}(\cdot )$ are bounded, we infer that there exists $r>0$ such that $\Gamma ( {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X)) \subset B_{r}( 0, {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X) )$. Thus, $\Gamma$ is completely continuous from $B_{r}(0, {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X) )$ into $B_{r}(0, {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X) )$ and there exists a mild solution $u\in B_{r}(0, {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_{\varphi }(X) )$ of (1.1)–(1.3). $\square$

3 Examples

In this section, $X= L^{2}(\Omega ;{\mathbb {R}})$ or $X= C(\Omega ;{\mathbb {R}})$, $\Omega \subset {\mathbb {R}}^{n}$ is a open set with smooth boundary and $A:D(A)\subset X\mapsto X$ is the realization of an second order strongly elliptic operator. Next, we assume that $(T(t))_{t\ge }$ is the analytic semigroup generated by A, $D(A)=\{u\in L^{2}(\Omega ):Au\in L^{2}(\Omega )\}$ if $\Omega ={\mathbb {R}}^{n}$ and $D(A)=W^{2,2}(\Omega )\cap W^{2,1}_{0}(\Omega )$ if $\Omega$ is bounded. For sake of simplicity, we suppose that the conditions ${\mathbf{H}}_{\zeta }$ and ${\mathbf{H}}_{\sigma _{{\mathbf{i}}} }$ are satisfies, $0\in \rho (A)$, $\varphi \in C_\mathrm{Lip}([-p,0];X)$ and $T(\cdot )\varphi (0)\in C_\mathrm{Lip}([0,a];X)$. In addition, $X_{1}$ is the domain of A endowed with the norm $\parallel x\parallel _{X_{1}}= \parallel Ax\parallel$ and $C_{0},C_{1}$ are the constants in the introduction.

To begin, we study the impulsive problem

$$\begin{aligned} u' (t,\xi )= & {} A u(t)(\xi ) + \beta _{1} (t,\xi , u(\zeta (u(t))-t,\xi )) + \beta _{2} (t) u(\zeta (u(t)) ,\xi ) , \,\, t\in I_{i}, \, \xi \in \Omega , \end{aligned}$$

(3.1)

$$\begin{aligned} u(t_{i}^{+},\xi )= & {} \int _{{\mathbb {R}}^{n}}{\mathcal {L}}_{i}(\xi ,y)u( \sigma (u(t_{i}^{+})),y){\text {d}}y, \end{aligned}$$

(3.2)

$$\begin{aligned} u(\theta , \xi )= & {} \varphi (\theta ,\xi ),\quad \theta \in [-p,0], \end{aligned}$$

(3.3)

where $\Omega = {\mathbb {R}}^{n}$, $X= L^{2}(\Omega ;{\mathbb {R}})$, $0=t_{0}< \cdots < t_{N+1} =a$ are pre-fixed, $I_{i}= (t_{i},t_{i+1}]$, $\beta _{1}\in C_\mathrm{Lip}([0,a]\times {\mathbb {R}};{\mathbb {R}})$, $\beta _{1}(\cdot )$ is bounded, $\beta _{2} \in C_\mathrm{Lip}([-p,a]; {\mathbb {R}})$ and ${\mathcal {L}}_{i}, A {\mathcal {L}}_{i} \in L^{2}(\Omega \times \Omega ,{\mathbb {R}})$. In addition, we assume that there is $\gamma \in L^{p}(\Omega )$ such that

$$\begin{aligned} \mid \beta _{1}(t,\xi ,x)-\beta _{1}(s,\xi ,y) \mid \le \gamma (\xi )(\mid t-s\mid + \mid x-y\mid ) , \quad \forall \,\, t,s\in [0,a], \xi ,x, y \in {\mathbb {R}}^{n}. \end{aligned}$$

To represent this problem in the form (1.1)–(1.3) we define the functions $g_{i}(\cdot )$ and $f(\cdot )$ by $g_{i}(t,x)(\xi ) = \int _{{\mathbb {R}}^{n}}{\mathcal {L}}(\xi ,y)x(y){\text {d}}y$ and $f(t,x)(\xi ) {=} \beta _{1} (t,\xi ,x(\xi ))+ \beta _{2} (t) x(\xi )$. It is easy to see that $\parallel A g_{i}(x)\parallel \le \parallel A{\mathcal {L}}_{i}\parallel _{L^{2}(\Omega \times \Omega ;{\mathbb {R}})}\parallel x\parallel$ and

$$\begin{aligned} \parallel f(t,x)- f(s,y)\parallel\le & {} [ \gamma ]_{C_\mathrm{Lip}}\mid t-s\mid + \parallel \gamma \parallel _{C(\Omega )}\parallel x-y\parallel \\&+\,[\beta _{2}]_{C_\mathrm{Lip}([0,a];{\mathbb {R}})}\parallel x\parallel \mid t-s\mid + \parallel \beta _{2}\parallel _{C(\Omega )}\parallel x-y\parallel . \end{aligned}$$

Thus, we can apply Proposition 2.3 with $L_{f}(r)= \parallel \gamma \parallel _{C_\mathrm{Lip}(\Omega )} + [\beta _{2}]_{C_\mathrm{Lip}}r + \parallel \beta _{2}\parallel _{C(\Omega )}$, $C_{X}(f,r)= \parallel \beta _{1}\parallel _{C([0,a]\times \Omega \times \Omega ;{\mathbb {R}})}+ \parallel \beta _{2}\parallel _{C(\Omega )}r$, $L_{X,X_{1}}(g_{i})=\parallel A{\mathcal {L}}_{i}\parallel _{L^{2}(\Omega \times \Omega ;{\mathbb {R}})}$, $C_{X,X_{1}}(g,r)= \max _{i=1,\ldots ,n} \parallel A{\mathcal {L}}_{i}\parallel _{L^{2}(\Omega \times \Omega ;{\mathbb {R}})} r$ and $L_{X,X_{1}}(g)=\sup _{i=1,\ldots ,n} \parallel A{\mathcal {L}}_{i}\parallel _{L^{2}(\Omega \times \Omega ;{\mathbb {R}})}$.

In the next result, we adopt the above notations and the notations in Remark 1. In addition, we say that $u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}(X)$ is a classical solution of (3.1)–(3.3) if $u(\cdot )$ is a classical solution of the associate problem (1.1)–(1.3) and we adopt a similar (for mild and classical solutions) in the following examples.

Proposition 3.5

If $\max \{ \parallel \varphi \parallel _{C([-p,0];X)}, C_{0} b {\mathcal {C}}_{X}(f,r) + C_{1} {\mathcal {C}}_{X,W}(g,r)\} \le r$ and

$$\begin{aligned}&{{ 2 C_{0}b L_{f}(r)\frac{1}{r}( 1+ [\zeta ]_{C_\mathrm{Lip}} (1+2 \Phi _{X,W}(r)) }} \\&\quad +\,2 \Lambda _{X,W}L_{X,W}(g,r)\frac{1 }{r}(1+ 2\max _{i=1,\ldots ,N}[\sigma _{i}]_{C_\mathrm{Lip}} \Phi _{X,W}(r)) <1, \end{aligned}$$

for some $r>0$, then there exists a unique classical solution $u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X)$ of (3.1)–(3.3).

We study now the problem

$$\begin{aligned} u' (t,x)= & {} A u(t)(x) + \int _{0}^{t} \beta _{1} (s, u(\zeta (u(t))-t,x )){\text {d}}s , \,\,\, \, x\in \Omega ,\,t\in I_{i}= (t_{i},t_{i+1}], \nonumber \\\end{aligned}$$

(3.4)

$$\begin{aligned} u(t_{i}^{+},x)= & {} \alpha _{i}u( \sigma _{i}(u(t_{i}^{+})),x), \end{aligned}$$

(3.5)

$$\begin{aligned} u(\theta , x)= & {} \varphi (\theta ,x),\quad \theta \in [-p,0], x\in \Omega , \end{aligned}$$

(3.6)

where $\Omega$ is bounded, $\beta _{1}\in C_\mathrm{Lip}([0,a]\times {\mathbb {R}};{\mathbb {R}})$ and $\beta _{1}(\cdot )$ is bounded.

To apply Theorem 2.2, we assume $X=L^{2}(\Omega )$, the condition ${\mathbf{H}}_{g_{i},\sigma _{j}}$ is satisfied and we define $g_{i}(\cdot )$ and $f(\cdot )$ by $f(t,x)(\xi ) = \int _{0}^{t} \beta _{1} (\tau ,x(\xi )) {\text {d}}\tau$ and $g_{i}(t,x)(\xi )= \alpha _{i} x(\xi ).$ From the above,

$$\begin{aligned} \parallel f(t,x)- f(s,y)\parallel\le & {} \parallel \beta _{1}\parallel _{C([0,a]\times {\mathbb {R}};{\mathbb {R}} )} \mid t-s\mid + b [\beta _{1}]_{C_\mathrm{Lip}([0,a]\times {\mathbb {R}};{\mathbb {R}} )}\parallel x-y\parallel , \, \\ \parallel g_{i}(x )- g_{i}(y)\parallel\le & {} \mid \alpha _{i}\mid \parallel x-y\parallel ,\,\,\,\, \parallel A g_{i}(z )\parallel \le \mid \alpha _{i}\mid \parallel Az\parallel , \end{aligned}$$

for $t,s\in [0,a]$, $x,y\in X$ and $z\in D(A)$, and the conditions in Theorem 2.2 are satisfied with $L_{f} =(1+b) \parallel \beta _{1}\parallel _{C_\mathrm{Lip}([0,a]\times {\mathbb {R}};{\mathbb {R}} )}$, $C_{X}(f)= a\parallel \beta _{1}\parallel _{C([0,a]\times {\mathbb {R}};{\mathbb {R}})}$, $l_{g_{i}}= \mid \alpha _{i}\mid$, $k_{g_{i}}=0$, $L_{g}=\max _{i=1,\ldots ,N} \mid \alpha _{i}\mid$ and $\Upsilon = 2 C_{0} C_{X}(f) + b(C_{0} +C_{1})L_{f} + [T(\cdot )\varphi (0)]_{C_\mathrm{Lip}([-p,0];X)} + [\varphi ]_{C_\mathrm{Lip}([-p,0];X)} .$ The next result follows from Theorem 2.2.

Proposition 3.6

Under the above conditions and notations, if the inequality (2.10) is verified, then there exists a unique classical solution $u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X)$ of (3.4)–(3.6).

We complete this section studying a problem motivated by equations arising in population dynamics. Consider the problem

$$\begin{aligned} u' (t,x)= & {} A u(t)(x) + \alpha u(t,x)(1-u(t,x)) , \,\,\, \, x\in \Omega ,\,t\in I_{i}= (t_{i},t_{i+1}], \end{aligned}$$

(3.7)

$$\begin{aligned} u(t_{i}^{+},x)= & {} \alpha _{i}u( \sigma _{i}(u(t_{i}^{+})),x), \end{aligned}$$

(3.8)

$$\begin{aligned} u(\theta , x)= & {} \varphi (\theta ,x),\quad \theta \in [-p,0]. \end{aligned}$$

(3.9)

To treat this problem, we assume $X= C(\Omega ;{\mathbb {R}})$ and $\alpha ,\alpha _{i}\in {\mathbb {R}}$ and we define $g_{i}(\cdot )$ and $f(\cdot )$ by $g_{i}(t,x)(\xi )= \alpha _{i} x(\xi )$ and $f(t,x)(\xi ) = \alpha x(\xi )(1- x(\xi ))$. It is trivial to see that

$$\begin{aligned} \parallel f(t,x)- f(s,y)\parallel\le & {} \mid \alpha \mid (1+2r)\parallel x-y\parallel , \,\,\,\, \parallel f(t,x)\parallel \le \mid \alpha \mid r(1+r), \\ \parallel g_{i}(x )- g_{i}(y)\parallel\le & {} \mid \alpha _{i}\mid \parallel x-y\parallel \,\,\hbox {and}\,\, \parallel A g_{i}(z )\parallel \le \mid \alpha _{i}\mid \parallel Az\parallel , \end{aligned}$$

for all $t,s\in [0,a]$, $x,y\in B_{r}(0;X)$ and $z\in D(A)$. From Proposition 2.4, we get.

Proposition 3.7

Suppose that there is $r>\parallel \varphi \parallel _{C([-p,0];X)}$ such that the inequality (2.10) is verified with $L_{f}(r)$ in place $L_{f}$ and $C_{0} ( \parallel \varphi (0)\parallel + b \mid \alpha \mid (1+2r)) + C_{0}(\max _{i=1,\ldots ,N}l_{ g_{i}}r+ k_{g_{i}}) < r.$ Then there exists a unique classical solution $u\in {\mathcal {B}}{{\mathcal {P}}{\mathcal {C}}}_\mathrm{Lip}(X)$ of (3.7)–(3.9).