This paper is concerned with the existence of mild and strong solutions on
the interval

The purpose of this paper is to study the existence of mild and strong solutions for the following neutral evolution problem with nonlocal initial conditions:

Equation (

Recently, Liang et al. [

There is a

Motivated by the works in [

The present work is organized as follows. Section

Throughout this paper,

The following assertions hold:

For every

If

Let

(H1) There exist

(H2) The function

For each

For each

(H3) The function

A continuous function

To see the existence of mild solution of nonlocal problem (

Assume that hypotheses (H1)–(H3) are satisfied, and, in addition, there holds the following inequality:

Suppose, on the contrary, that, for each

By Lemma

Assume that hypotheses (H1)–(H4) are satisfied. If

Observe that, for

Assume that hypotheses (H1)–(H4) are satisfied, and, in addition, the following is given.

Then the problem (

Let

For the compactness of

By the fixed-point theorem of Sadovskiĭ [

For the main results in this section, we introduce a family of nonlocal neutral problems as follows. Firstly, we define, for each

Suppose that (H1)–(H4) are satisfied. Then, for any

Suppose that, hypotheses (H1)–(H4) are satisfied. Then, problem (

Choose a decreasing sequence

Now, for each

We will consider the case more generally; that is, the nonlocal condition

Suppose that, hypotheses (H1) and (H2) are satisfied, and, in addition, there hold the following hypotheses.

Let

A mild solution

In the following, we establish a result of a strong solution for (

Let

Let

The following result is an immediate corollary of Theorems

Suppose that the hypotheses (H7)–(H9), and (H11) are satisfied, and in addition, there holds the following hypotheses.

If

In the last section, our existence results will be applied to solve the following system:

The operator

We need the following assumptions to solve (

(A1) The function

(A2) The function

For each

For each

There is an

(A3) The functions

Assumptions (

The functions

(a) By the definition of

(b) By the part of (c) of assumption (A2) and Hölder’s inequality, we have

Now, we define

If

(a) This follows since

(b) This is clear from the proof of part (a).

(c) Let

Theorem

If

Theorem

(A4)

Assume that assumptions (A2)–(A4) are satisfied and the function

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported partly by the National Science Council of the Republic of China.