Introduction

The asymptotic equilibrium problems of ordinary differential equations in a Banach space have been considered by several authors, Mitchell and Mitchell [3], Bay et al. [1], but the results for the asymptotic equilibrium of integro-differential equations with infinite delay still is not presented. In this paper, we extend the results in [1] to a class of integro-differential equations with infinite delay in a Hilbert space H which has the following form:

$$\begin{array}{l} {\left\{ \begin{array}{ll} \dfrac{\mathrm{d}x(t)}{\mathrm{d}t} = A(t)\left( x(t) +\int \limits _{-\infty }^{t}k(t - \theta )x(\theta )\mathrm{d}\theta \right) ,&\quad t \geqslant 0,\\ x(t)=\varphi (t), &\quad t \leqslant 0 \end{array}\right. } \end{array}$$
(1)

where \(A(t):H \rightarrow H\), \(\varphi\) in the phase space \(\fancyscript{B}\), and \(x_t\) is defined as

$$\begin{aligned} x_t(\theta ) = x(t + \theta ), \quad -\infty < \theta \leqslant 0. \end{aligned}$$

Preliminaries

We assume that the phase space \((\mathscr {B}, ||.||_{\fancyscript{B}})\) is a seminormed linear space of functions mapping \((-\infty , 0]\) into H satisfying the following fundamental axioms (we refer reader to [2])

(\(\mathrm{A}_1\)):

For \(a >0\), if x is a function mapping \((-\infty , a]\) into H, such that \(x \in \fancyscript{B}\) and x is continuous on [0, a], then for every \(t \in [0,a]\) the following conditions hold:

(i):

\(x_t\) belongs to \(\fancyscript{B}\);

(ii):

\(||x(t)|| \leqslant G||x_t||_{\fancyscript{B}}\);

(iii):

\(||x_t||_{\fancyscript{B}} \leqslant K(t)\sup \nolimits _{s\in [0,t]}||x(s)|| + M(t)||x_0||_\fancyscript{B}\)

where G is a possitive constant, \(K,M:[0,\infty ) \rightarrow [0, \infty )\), K is continuous, M is locally bounded, and they are independent of x.

(\(\mathrm{A}_2\)):

For the function x in (\(A_1\)), \(x_t\) is a \(\fancyscript{B}\)-valued continuous function for t in [0, a].

(\(\mathrm{A}_3\)):

The space \(\fancyscript{B}\) is complete.

Example 1

  1. (i)

    Let BC be the space of all bounded continuous functions from \((-\infty , 0]\) to H, we define \(C^0: = \{\varphi \in BC:\lim \nolimits _{\theta \rightarrow -\infty }\varphi (\theta ) =0\}\) and \(C^{\infty }:=\{\varphi \in BC:\lim \nolimits _{\theta \rightarrow -\infty }\varphi (\theta ) \ {\text {exists}}\ {\text{in}} \, H\}\) endowed with the norm

    $$\begin{aligned} ||\varphi ||_{\fancyscript{B}} = \sup \limits _{\theta \in (-\infty , 0]}||\varphi (\theta )|| \end{aligned}$$

    then \(C^0, C^{\infty }\) satisfies (\(\mathrm{A}_1\))–(\(\mathrm{A}_3\)). However, BC satisfies (\(\mathrm{A}_1\)) and (\(\mathrm{A}_3\)), but (\(\mathrm{A}_2\)) is not satisfied.

  2. (ii)

    For any real constant \(\gamma\), we define the functional spaces \(C_{\gamma }\) by

    $$\begin{aligned} C_{\gamma } = \left\{ \varphi \in C((-\infty ,0], X):\lim \limits _{\theta \rightarrow -\infty }e^{\gamma \theta }\varphi (\theta ) \ {\text {exists}} \, {\text {in}} \, H \right\} \end{aligned}$$

    endowed with the norm

    $$\begin{aligned} ||\varphi ||_{\fancyscript{B}} =\sup \limits _{\theta \in (-\infty , 0]}e^{\gamma \theta }||\varphi (\theta )||. \end{aligned}$$

    Then conditions (\(\mathrm{A}_1\))–(\(\mathrm{A}_3\)) are satisfied in \(C_{\gamma }\).

Remark 1

In this paper, we use the following acceptable hypotheses on K(t), M(t) in (\(A_1\))(iii) which were introduced by Hale and Kato [2] to estimate solutions as \(t \rightarrow \infty\),

  • (\(\gamma _1\)) \(K =K(t)\) is a constant for all \(t \geqslant 0\);

  • (\(\gamma _2\)) \(M(t) \leqslant M\) for all \(t \geqslant 0\) and some M.

Example 2

For the functional space \(C_{\gamma }\) in Example 1, the hypotheses (\(\gamma _1\)) and (\(\gamma _2\)) are satisfied if \(\gamma \geqslant 0\).

Definition 1

Equation (1) has an asymptotic equilibrium if every solution of it has a finite limit at infinity and, for every \(h_0 \in H\), there exists a solution x(t) of it such that \(x(t) \rightarrow h_0\) as \(t \rightarrow \infty\).

Main results

Now, we consider the asymptotic equilibrium of Eq. (1) which satisfies the following assumptions:

(\(\mathrm{M}_1\)):

A(t) is a strongly continuous bounded linear operator for each \(t \in \mathbb R^+\);

(\(\mathrm{M}_2\)):

A(t) is a self-adjoint operator for each \(t \in \mathbb R^+\);

(\(\mathrm{M}_3\)):

k satisfies

$$\begin{aligned} \int \limits _{0}^{+\infty }|k(\theta )|\mathrm{d}\theta = L <+\infty ; \end{aligned}$$

and

(\(\mathrm{M}_4\)):

There exists a constant \(T > 0\) such that

$$\begin{aligned} \sup \limits _{h \in S(0,1)}\int \limits _{T}^{\infty } ||A(t)h||\mathrm{d}t < q <\dfrac{1}{\kappa } \end{aligned},$$
(2)

herein S(0, 1) is a unit ball in H, \(\kappa = L(K+M) + 1,\) where K, M, L are given in (\(\gamma _1\)), (\(\gamma _2\)) and (\(M_3\)).

Theorem 1

If (\(\mathrm{M}_1\)), \((\mathrm{M}_2)\), \((\mathrm{M}_3)\) and \((\mathrm{M}_4)\) are satisfied, then Eq. (1) has an asymptotic equilibrium.

Proof

We shall begin with showing that all solutions of (1) has a finite limit at infinity. Indeed, Eq. (1) may be rewritten as

$$\begin{aligned} \dfrac{\mathrm{d}x(t)}{\mathrm{d}t} = A(t)\left( x(t) + \int \limits _{-\infty }^{0}k(-\theta )x_t(\theta )\mathrm{d}\theta \right) \end{aligned},$$

then for \(t \geqslant s \geqslant T\) we have

$$\begin{aligned} x(t) = x(s) + \int \limits _{s}^{t}A(\tau )\left( x(\tau ) + \int \limits _{-\infty }^{0}k(-\theta )x_{\tau }(\theta ) \mathrm{d}\theta \right) \mathrm{d} \tau \end{aligned}$$

and

$$\begin{aligned}&||x(t)|| \nonumber \\&\quad=\sup \limits _{h \in S(0,1)} \left| \left\langle x(s) + \int \limits _{s}^{t}A(\tau ) \left( x(\tau ) + \int \limits _{-\infty }^{0}k(-\theta )x_{\tau }(\theta ) {\rm d}\theta \right) {\rm d} \tau , h \right \rangle \right | \nonumber \\&\quad\leqslant ||x(s)|| + \sup \limits _{h \in S(0,1)}\int \limits _{s}^{t}\left| \left\langle x(\tau ) + \int \limits _{-\infty }^{0}k(-\theta )x_\tau (\theta ){\rm d}\theta , A(\tau )h \right \rangle\right |d\tau \nonumber \\&\quad\leqslant ||x(s)|| + q\left( (LK+1)\sup \limits _{\xi \in [0,t]}||x(\xi )|| +L M||\varphi ||_{\fancyscript{B}} \right ) \end{aligned}$$
(3)

implies

$$\begin{aligned} |||x(t)||| \leqslant ||x(s)|| + q\left( (LK+1)|||x(t)||| + LM||\varphi ||_{\fancyscript{B}} \right) \end{aligned}$$

or

$$\begin{aligned} |||x(t)||| \leqslant \dfrac{||x(s)|| + qLM||\varphi ||_{\fancyscript{B}}}{1 - q(LK+1)} \end{aligned}$$
(4)

where

$$\begin{aligned} |||x(t)||| = \sup \limits _{0 \leqslant \xi \leqslant t}||x(\xi )||. \end{aligned}$$

Now, we conclude that x(t) is bounded since

$$\begin{aligned} 0<q <\dfrac{1}{\kappa } = \dfrac{1}{L(K+M)+ 1} < \dfrac{1}{LK+1} \Rightarrow q(LK+1) < 1 \end{aligned}$$

and by (4).

Putting

$$\begin{aligned} M^* = \sup \limits _{t \in \mathbb R}||x(t)||, \end{aligned}$$

we have

$$\begin{aligned} {\begin{matrix} ||x(t) - x(s)|| &{}= \sup \limits _{h \in S(0,1)}\left| \left<x(t) - x(s), h\right>\right| \\ &{}\leqslant \sup \limits _{h \in S(0,1)}\int \limits _{s}^{t}\left| \left<A(\tau )\left( x(\tau ) + \int \limits _{-\infty }^{0}k(-\theta )x_{\tau }(\theta ) \mathrm{d}\theta \right) , h\right>\right| \mathrm{d}\tau \\ &{} \leqslant [M^*(LK+1) + LM||\varphi ||_{\fancyscript{B}}]\sup \limits _{h \in S(0,1)}\int \limits _{s}^{t}||A(\tau )h|| d \tau \rightarrow 0 \end{matrix}} \end{aligned},$$

as \(t \geqslant s \rightarrow +\infty\). That means all solutions of (1) have a finite limit at infinity. To complete the proof, it remains to show that for any \(h_0 \in H\), there exists a solution x(t) of (1) such that

$$\begin{aligned} \lim \limits _{t \rightarrow +\infty }x(t) = h_0. \end{aligned}$$

Indeed, let \(h_0\) be an arbitrary fixed element of H; we choose the initial function \(\varphi\) belongs to \(\fancyscript{B}\) such that \(\varphi (0) = h_0\) and \(||\varphi ||_{\fancyscript{B}} \leqslant ||h_0||\) and consider the functional

$$\begin{aligned}g_1(t,h) &= \langle h_0,h\rangle \\&\quad - \int \limits _{t}^{\infty }\left\langle A(\tau )\left( h_0 + \int \limits _{-\infty }^{\tau }k(\tau - \theta )x_0(\theta ){\rm d}\theta \right ),h \right\rangle{\rm d}\tau \end{aligned}$$

We have

$$\begin{aligned} {\begin{matrix} \left| g_1(t,h) \right|&\leqslant ||h_0||||h||+ \int \limits _{t}^{+\infty }\left\| x_0(\tau ) + \int \limits _{-\infty }^{\tau }k(\tau -\theta )x_0(\theta )\mathrm{d}\theta \right\| ||A(\tau )h||\mathrm{d}\tau . \end{matrix}} \end{aligned}.$$

Since \(x_0(\tau ) \equiv h_0,\) then

$$\begin{aligned} \left| g_1(t,h) \right| \leqslant ||h_0|| \left( ||h|| + q\kappa \right) . \end{aligned}$$

It follows from Riesz representation theorem that there exists an element \(x_1(t)\) in H, such that

$$\begin{aligned} g_1(t,h) = \langle x_1(t) , h \rangle \end{aligned}$$

and

$$\begin{aligned} ||x_1(t)|| \leqslant ||h_0|| \left( 1 + q\kappa \right) . \end{aligned}$$

Now, we consider the functional

$$\begin{aligned}&g_2(t,h) = \langle h_0,h\rangle \\&\quad - \int \limits _{t}^{+\infty }\left\langle A(\tau )\left( x_1(t) + \int \limits _{-\infty }^{\tau }k(\tau - \theta )x_1(\theta ) {\rm d}\theta \right), h \right\rangle {\rm d}\tau . \end{aligned}$$

By an argument analogous to the previous one, we get

$$\begin{aligned} \left| g_2(t,h) \right| \leqslant ||h_0||[ ||h|| + q\kappa + (q\kappa )^2 ] \end{aligned}$$

and there exists an element \(x_2(t)\) in H, such that

$$\begin{aligned} g_2(t,h) = \langle x_2(t), h\rangle \end{aligned}$$

with

$$\begin{aligned} ||x_2(t)|| \leqslant ||h_0||(1 + q\kappa + (q\kappa )^2). \end{aligned}$$

Continuing this process, we obtain the linear continuous functional

$$\begin{aligned}g_n(t,h) &= \langle h_0, h\rangle \nonumber \\&\quad- \int \limits _{t}^{+\infty }\left\langle A(\tau )\left( x_{n-1}(t) + \int \limits _{-\infty }^{\tau }k(\tau - \theta )x_{n-1}(\theta ) {\rm d}\theta \right ), h \right \rangle{\rm d}\tau \end{aligned}$$
(5)

and \(x_n(t) \in H\) such that

$$\begin{aligned} g_n(t,h) = \langle x_n(t), h\rangle \end{aligned}$$

satisfies the following estimate

$$\begin{aligned} ||x_n(t)|| \leqslant (1 + q\kappa +(q\kappa )^2 +\cdots +(q\kappa )^n)||h_0|| \leqslant \dfrac{||h_0||}{1 - q\kappa }. \end{aligned}$$

Futhermore,

$$\begin{aligned} \left\| x_n(t) - x_{n-1}(t) \right\| \leqslant ||h_0||(q\kappa )^n. \end{aligned}$$

This inequality shows that \(\{x_n(t)\}\) is uniformly convergent on \([T, +\infty )\) since \(q \kappa < 1\). Put

$$\begin{aligned} x(t) = \lim \limits _{n \rightarrow +\infty }x_n(t). \end{aligned}$$

In (5), let \(n \rightarrow +\infty ,\) we have

$$\begin{aligned}\langle x(t), h\rangle &= \langle h_0, h\rangle \nonumber \\&\quad - \int \limits _{t}^{+\infty }\left\langle A(\tau )\left( x(t) + \int \limits _{-\infty }^{\tau }k(\tau - \theta )x(\theta ) {\rm d}\theta \right ), h \right\rangle {\rm d}\tau \end{aligned}$$
(6)

and since

$$\begin{aligned} {\begin{matrix}&\left| \langle x_n(t), h_0\rangle \right| < \int \limits _{T}^{+\infty }\left\| x_{n-1}(\tau ) + \int \limits _{-\infty }^{\tau }k(\tau - \theta )x_{n-1}(\theta ) \mathrm{d}\theta \right\| \left\| A(\tau )h\right\| \mathrm{d}\tau \end{matrix}} \end{aligned}$$

or

$$\begin{aligned} \left| \langle x_n(t), h_0\rangle \right| \leqslant \dfrac{||h_0||q}{1-q\kappa } \end{aligned},$$

we have \(x_n(t) \rightarrow h_0\) as \(q \rightarrow 0\), which means that there exists a solution of (1) converging to \(h_0\). The theorem is proved.