By the weighted ergodic function based on the measure theory, we study pseudo asymptotic behavior of mild solution for nonautonomous integrodifferential equations with nondense domain. The existence and uniqueness of

The study of pseudo asymptotic behavior of solution is one of the most interesting and important topics in the qualitative theory of differential equations. Much work has been done to investigate the existence of pseudo antiperiodic, pseudo periodic, pseudo almost periodic, and pseudo almost automorphic solution for differential equations [

Integrodifferential equations play a crucial role in qualitative theory of differential equations due to their application to physics, engineering, biology, and other subjects. This type of equations has received much attention in recent years and the general asymptotic behavior of solution is at present an active source of research.

In this paper, we study pseudo asymptotic behavior of solution to the following nonautonomous integrodifferential equations with nondense domain:

Some recent contributions on almost periodic, almost automorphic, pseudo almost periodic, and pseudo almost automorphic solution to integrodifferential equations of the form (

The paper is organized as follows. In Section

Let

A family of bounded linear operators

the map

An evolution family

the restriction

Exponential dichotomy is a classical concept in the study of long-time behaviour of evolution equations. If

If

For all

Let

Let

A function

A function

A function

A function

Next, we recall the

Let

Let

Let

Let

(i) If the measure

(ii) Let

Let

Let

Similarly as the proof of [

Let

Let

Let

for all bounded subsets

Let

This section is devoted to pseudo asymptotic behavior of mild solution to (

There exist constants

for

The evolution family

Consider

There exists a constant

There exists a constant

Before starting our main results, we recall the definition of the mild solution to (

A mild solution of (

Assume that

First, we show that

Note that

Assume that

Similarly as the proof of Lemma

(i) Note that

Let

Note that

By using Fubini’s theorem, one has

(ii) Note that

Let

(iii) Note that

By [

(iv) Note that

Similarly as the proof of [

In this subsection, we investigated the existence and uniqueness of pseudo almost automorphic mild solution of (

First, we introduce the concept of bi-almost automorphic function.

A function

Now, we make the following assumptions:

Assume that

Suppose

First, we show that (

For all

Next, define the operator

For any

Next, consider the following nonautonomous Volterra integrodifferential equations:

For the pseudo almost automorphy of (

Suppose

Let

For any

In this subsection, we investigated the existence and uniqueness of pseudo almost periodic mild solution of (

Similarly as the proof of [

Assume that

By Lemma

Suppose

Suppose

In this subsection, we investigated the existence and uniqueness of pseudo periodic (antiperiodic) mild solution of (

there exists

Assume that

Since

By Lemma

Suppose

Suppose

(i) If

(ii) If

In this section, we provide some examples to illustrate our main results.

Consider the heat equations with Dirichlet conditions

Note that

Let

Define a family of linear operators

Now, the following theorem is an immediate consequence of Theorem

Under the assumptions

For (

The author declares that there is no conflict of interests regarding the publication of this paper.

This research was supported by Zhejiang Provincial Natural Science Foundation of China under Grant no. LQ13A010015.