In this paper, we are concerned with the existence of solutions to the semilinear system of abstract integrodifferential equations with nonlocal initial conditions, of the type

Here $n\geq 1,$ and for each $i\in I:=\left\{1,...,n\right\},$ the linear operator $-A_{i}:D(A_{i})\subseteq X_{i}\rightarrow X_{i}$ generates a $C_{0}$-semigroup of contractions $\left\{S_{i}(t);t\geq 0\right\}$ on a Banach space $\left(X_{i},\left|.\right|_{X_{i}}\right),$ $\tau\geq 0,$ $u\in C\left(\left[-\tau,T\right],X\right),$ where $X=X_{1}\times...\times X_{n},$ $u=\left(u_{1},...,u_{n}\right),$ and for each $t,$ $u_{t}$ is the restriction of $u$ to $\left[t-\tau,t\right]$ shifted to the interval $\left[-\tau,0\right],$ i.e., $u_{t}\in C\left(\left[-\tau,0\right],X\right)$ and

The nonlinear perturbations in equations are given by the continuous mappings $F_{i}$ from $[0,T]\times C\left(\left[-\tau,0\right],X\right)$ to $X_{i},$ $K_{i}$ from $\left[0,T\right]\times C\left(\left[-\tau,0\right],X\right)$ to $X_{i},$ and the nonlocal initial conditions are expressed by the continuous mappings $\alpha_{i}$ from $C\left(\left[-\tau,T\right],X\right)$ to $C\left(\left[-\tau,0\right],X_{i}\right).$

We note that the nonlocal initial conditions include in particular:

• •

  the initial condition:

  Report issue for preceding element

  $$u_{i}\left(t\right)=\varphi_{i}\left(t\right),\ \ \ t\ \in\left[-\tau,0\right],\ \ i=1,...,n$$

  where $\varphi=\left(\varphi_{1},...,\varphi_{n}\right)\in C\left(\left[-\tau,0\right],X\right)$ is given;

  Report issue for preceding element
• •

  linear multi-point conditions (linear nonlocal initial conditions of discrete type):

  Report issue for preceding element

  (1.3)

  $$u_{i}\left(t\right)=\varphi_{i}\left(t\right)+\sum_{j=1}^{m_{i}}a_{ij}\left(t\right)u_{i}\left(t+t_{ij}\right),\ \ \ t\in\left[-\tau,0\right],\ i=1,...,n,$$

  where $0<t_{ij}<t_{i,j+1}\leq T$ for $j=1,...,m_{i}$ and $i=1,...,n.$ The linear multi-point conditions include in particular the initial condition, and the periodicity condition

  Report issue for preceding element

  $$u_{i}\left(t\right)=u_{i}\left(T+t\right),\ \ \ t\in\left[-\tau,0\right],\ \ i=1,...,n;$$

• •

  linear nonlocal initial conditions of continuous type, given by integrals:

  Report issue for preceding element

  	$\displaystyle u_{i}\left(t\right)$	$\displaystyle=$	$\displaystyle\varphi_{i}\left(t\right)+\int_{0}^{T}k_{i}\left(t,s\right)u_{i}\left(t+s\right)ds$
  		$\displaystyle=$	$\displaystyle\varphi_{i}\left(t\right)+\int_{t}^{T+t}k_{i}\left(t,s-t\right)u_{i}\left(s\right)ds,\ \ \ t\in\left[-\tau,0\right],\ \ i=1,...,n.$

Starting with Volterra’s pioneering works on integrodifferential equations with delayed effects in population dynamics and materials with memory, the theory of delay differential equations has progressed continuously following the development of functional analysis and being stimulated by numerous applications in physics, chemistry, biology, medicine, economy, etc., see e.g., [23]), aimed to described evolution processes whose future states depend not only on the present, but also on the past history.

As concerns differential equations with nonlocal initial conditions of multi-point or integral type, we mention as some pioneering contributions, the papers of Cioranescu [15], Whyburn [42] and Conti [16]). Among further developments, we refer the readers to the works [2], [3], [7], [17], [21], [28], [29], [41], to the recent survey paper [35], and the references therein.

Parabolic problems with nonlocal initial conditions were considered in the papers of Kerefov [22], Vabishchevich [36], Chabrowski [14], Pao [33], Olmstead and Roberts [31], and Chapter 10 in [26], as nonlocal versions of some deterministic models from physics, mechanics, biology and medicine. Abstract evolution equations with nonlocal initial conditions were considered by Byszewski [11], Jackson [20], Lin and Liu [24]. For more recent contributions, we refer the readers to the papers [4], [6], [8], [10], [12], [19], [24], [25], [27], [30], [32], [39] and the recent monograph [9].

This paper has a double motivation. First, it is motivated by the second author’s recent paper [5], which mainly inspires the operator technique of proof, and secondly, by the paper of Webb [40] for the class of integrodifferential equations.

There are several aspects in the present paper which are mixed together requiring a laboured technique of proof and yielding to a very general result:

$\blacktriangleright\$The use of the notion of support of a nonlocal initial condition and of a corresponding split norm. Throughout the paper, by $\left[-\tau,T_{0}\right]$ we shall denote the support of the nonlocal initial condition, that is the smallest subinterval $\left[-\tau,T_{0}\right]$ of $[-\tau,T]$ with $T_{0}\geq 0$ such that

Here by $\left.u\right|_{\left[-\tau,T_{0}\right]}$ we mean the restriction of the function $u$ to the interval $\left[-\tau,T_{0}\right].$ Physically, this means that the evolution of a process is subjected to some constraints until a given moment of time $T_{0},$ and becomes free of any constraints after that moment.

The notion of support of a nonlocal initial condition was first used in the papers [7] and [8], and used after in [29], [2], [12], [4], [5]. As explained in these papers, and as we shall see in the following, stronger conditions on nonlinearities have to be asked on the support subinterval, compared to those required on the rest of the interval. Mathematically, the integral equation equivalent to the nonlocal initial problem is of Fredholm type on the support interval, and of Volterra type on the rest of the interval. This makes useful to consider a split norm on the functional space where the problem is studied. Thus, in connection with the delay system (1.1) and with the support $\left[-\tau,T_{0}\right]$ of the nonlocal initial condition, on a space of the type $C\left(\left[-\tau,T\right],E\right),$ where $\left(E,\left|\cdot\right|_{E}\right)$ is a Banach space, we shall consider the split norm

where $\left|u\right|_{C\left([-\tau,T_{0}],E\right)}$ is the usual max norm

while for any $\theta>0,$ $\left|u\right|_{C_{\theta}\left([T_{0}-\tau,T],E\right)}$ is the Bielecki type norm on $C\left([T_{0}-\tau,T],E\right),$

In particular, when there is no a delay, i.e., when $\tau=0,$ the norm (1.4) reduces to the split norm previously considered in [7], [2], [28] and [29].

$\mathbf{\blacktriangleright}$ The hybrid character of the system. The system is split into to subsystems: the first $m$ equations for which Lipschitz conditions are assumed to guarantee that the corresponding integral operators are contractive, and the last $n-m$ equations $\left(0\leq m\leq n\right)$ for which only at most linear growth conditions are required on the nonlinear terms, but in return, the compactness of the semigroups of operators is assumed to insure the compactness of the integral operators. In this way the proof will be a perfect illustration of Krasnoselskii’s fixed point theorem for a sum of a compact map and a contraction, more exactly of its vector version of Viorel [37].

$\mathbf{\blacktriangleright}$ The presence of integral terms. There is not only the bounded delay in the equations of system (1.1), but also cumulative integral terms which bring into the equations the whole history of the process. Such kind of equations arise from mathematical modeling of many real processes with memory from physics, biology and economics. These cumulative terms play a special role in the split analysis on two intervals as discussed previously.

For the treatment of systems we use the vector approach based on vector-valued metrics and norms, and matrices instead of constants.

Let us make the convention that the elements of $\mathbb{R}^{n}$ are seen as column vectors. By a vector-valued metric on a set $E$ we mean a mapping $d:E\times E\rightarrow\mathbb{R}_{+}^{n}$ such that $d(x,y)=0$ if and only if $x=y;$ $\ d(x,y)=d(y,x)$ for all $x,y\in E$ and $d(x,y)\leq d(x,z)+d(z,y)$ for all $x,y,z\in E.$ Here by $\leq\$ we mean the natural componentwise order relation of $\ \mathbb{R}^{n},$ more exactly, if $\ r,s\in\mathbb{R}^{n},$ $r=(r_{1},...,r_{n}),$ $s=(s_{1},...,s_{n}),$ then by $r\leq s$ one means that $r_{i}\leq s_{i}$ for $i=1,...,n.$ A set $E$ together with a vector-valued metric $d$ is called a generalized metric space. For such a space, the notions of Cauchy sequence, convergence, completeness, open and closed set, are similar to those in usual metric spaces.

Similarly, a vector-valued norm on a linear space $E,$ is defined as being a mapping $\left\|\cdot\right\|:E\rightarrow\mathbb{R}_{+}^{n}$ with $\left\|x\right\|=0$ only for $x=0;\$ $\left\|\lambda x\right\|=\left|\lambda\right|\left\|x\right\|$ for $x\in E,$ $\lambda\in\mathbb{R},$ and $\left\|x+y\right\|\leq\left\|x\right\|+\left\|y\right\|$ for every $x,y\in E.$ To any vector-valued norm $\left\|.\right\|$ one can associate the vector-valued metric $d\left(x,y\right):=\left\|x-y\right\|.$ A linear space $E$ endowed with a vector-valued norm $\left\|\cdot\right\|$ is called a generalized Banach space if $E$ is complete with respect to the associated vector-valued metric $d.$

If $\left(E,d\right)$ is a generalized metric space with $d$ taking values in $\mathbb{R}^{n},$ we say that a mapping $\Gamma:E\rightarrow E$ is a generalized contraction (in Perov’s sense) if there exists a square matrix $M$ of size $n$ with nonnegative entries such that its powers $M^{k}$ tend to the zero matrix $0$ as $k\rightarrow\infty,$ and

Such a matrix is said to be a Lipschitz matrix. Notice that for a matrix $M$ the property $M^{k}\rightarrow 0$ as $k\rightarrow\infty$ is equivalent to the fact that the spectral radius $\rho\left(M\right)$ of the matrix $M$ is less than one. The role of matrices with spectral radius less than one in the study of operator systems was pointed out in [34], in connection with several abstract principles from nonlinear functional analysis.

For generalized contractions, the following extension of Banach’s contraction principle holds.

In this paper we use the following generalization of Theorem 2.1, a vector version of Krasnoselskii’s fixed point theorem for a sum of two operators, owed to Viorel [37].

The following obvious proposition will be used in the proof of the main result.

We conclude this preliminary section by a result about the compactness of the solution operator associated to a non-homogenous evolution equation [1].

For other basic notions and results of semigroup theory we mention the books [13], [18] and [38].

Looking for mild solutions to the problem (1.1), with $u_{i}\in C\left(\left[-\tau,T\right],X_{i}\right)$ for $i=1,...,n$ we are led in a standard way to the following integral system

Our assumptions are given differently for two sets of indices,

where $\ 0\leq m\leq n,$ and it is understood that $\ I_{1}=\emptyset$ if $m=0,$ and $\ I_{2}=\emptyset$ if $m=n.$ Let $p>1$ be any fixed number.

The hypotheses are:

(H${}_{\text{0}}$) (a) For each $i\in I_{1},$ the linear operator $-A_{i}:D(A_{i})\subset X_{i}\rightarrow X_{i}$ generates a

$C_{0}$-semigroup of contractions on the Banach space $X_{i}.$

(b) For each $i\in I_{2},$ the linear operator $-A_{i}:D(A_{i})\subset X_{i}\rightarrow X_{i}$ generates a compact

$C_{0}$-semigroup of contractions on the Banach space $X_{i}.$

(H${}_{\text{1}}$) (a) For each $i\in I_{1},\$ $K_{i}:\left[0,T\right]\times C([-\tau,0],X)\rightarrow X_{i},$ is continuous, and there

exist $a_{ij}\in C(\left[0,T\right],\mathbb{R}_{+})$ for $j\in I,$ such that

for all $u,v\in C\left(\left[-\tau,0\right],X\right)$ and $t\in[0,T].$

(b) For each $i\in I_{2},\$ $K_{i}:\left[0,T\right]\times C([-\tau,0],X)\rightarrow X_{i},$ is continuous, and there exist

$d_{i},\ a_{ij}\in C\left(\left[0,T\right],\mathbb{R}_{+}\right)$ for all $j\in I,$ such that

for all $u\in C\left(\left[-\tau,0\right],X\right)$ and $t\in[0,T].\vskip 6.0pt plus 2.0pt minus 2.0pt$

(H${}_{\text{2}}$) (a) For each $i\in I_{1},\$ $F_{i}:[0,T]\times C\left(\left[-\tau,0\right],X\right)\rightarrow X_{i}$ is continuous and there

exists $b_{ij}\in C\left(\left[0,T\right],\mathbb{R}_{+}\right)$ for all $j\in I,$ such that

for $u,v\in$ $C([-\tau,0],X)$ and $t\in\left[0,T\right].$

(b) For each $i\in I_{2},\$ $F_{i}:[0,T]\times C\left(\left[-\tau,0\right],X_{i}\right)\rightarrow X_{i}$ is continuous and there exist

$f_{i},\ b_{ij}\in C\left(\left[0,T\right],\mathbb{R}_{+}\right)\$for all $\ j\in I,$ such that

for all $u\in$ $C([-\tau,0],X)$ and $t\in\left[0,T\right].\vskip 6.0pt plus 2.0pt minus 2.0pt$

(H${}_{\text{3}}$) For each $i\in I,\$ $\alpha_{i}:C\left(\left[-\tau,T\right],X\right)\rightarrow C\left(\left[-\tau,0\right],X_{i}\right)$ and there exist $c_{ij}\in\mathbb{R}_{+}\$

for all $j\in I,$ such that

for all $u,v\in C\left(\left[-\tau,T\right],X\right).$