1. Introduction

The theory of differential equations with deviated argument is very important and significant branch of nonlinear analysis. It is worthwhile mentioning that differential equations with deviated argument appear often in investigations connected with mathematical physics, mechanics, engineering, economics and so on (cf. ^{[10, 11, 12]}, for example). One of the basic problems considered in the theory of differential equations with deviated argument is to establish convenient conditions guaranteeing the existence of solutions of those equations, we refer to some recent papers ^{[13, 14, 15, 16, 17]} and references.

Let E be a real Banach space with and let P be a cone in E. The partial order “” is introduced by cone P, i.e., if and only if A cone P is said to be normal if there exist a constant such that implies ; is called the normal constant of P. Recall that a cone P is said to be regular if every increasing and bounded in order sequence in E has a limit, i.e., implies as for some The regularity of P implies the normality of P. Let be the dual space of E, is called the dual cone. Obviously, if and only if for all Let where (a > 0) and denotes the Banach space of all continuous mapping with the norm It is clear that is a cone of the and so it defines a partial ordering in Obviously, the normality of P implies the normality of and the normal constants of and P are the same. For further details on cone theory, one can refer to ^{[3, 8, 9]}. Let

In this paper, we consider the solutions for the following initial value problems (IVP) of nonlinear second-order integro-differential equations of mixed type in ordered Banach spaces E,

(1.1)

where and

Let

For any let

The solutions for initial value problems (IVP) of nonlinear first-order integro-differential equations of mixed type in ordered Banach spaces have made considerable headway in recent years (see ^{[2, 6]}). But there has been little discussion for the solutions of (IVP) (1.1). In the special case where does not contain and the solutions for initial value problems (IVP) (1.1) in Banach spaces have some results (see ^{[1, 5]}). In another special case where f does not contain in ^[4], Su obtained some new results by using Mönch fixed point theorem and new comparison results.

In this paper, we first establish a new comparison theorem, and then, by requiring only a lower solution or an upper solution and some weaker conditions ,we investigate the existence of the minimal or maximal solutions of the (IVP) (1.1), where f contains , and delayed arguments under the conditions which are more extensive than those in ^{[1, 5]}.

2. Several Lemmas

The following comparison results and lemmas play an important role in this paper.

Lemma 1. (Comparison theorem) Assume that E is a Banach space, P is a cone in on J, and satisfies

(2.1)

where M, K, N, L are non-negative constants, and provided one of the following two conditions hold

(i)

(ii)

Then

Proof. For any let then

Thus, by (2.1) we have that

Let then and Hence, we have that

(2.2)

Now, we shall prove that

In the case of condition (i), if is not true , then there is a t0 such that Let then

If , then Then, by (2.2), we have So, is increasing in we have which contradicts

If then there exists a such that From (2.2), we have

Thus, we have that

Then, by we have which contradicts (i).

In the case of condition (ii) holding, let

and applying it to (2.2), by a similar process, we can obtain and so

Therefore, which implies that By the arbitrarily of we have

Lemma 1 is proved.

Lemma 2. ^[3] Let be countable and bounded, then and

Lemma 3. ^[3] Let be countable and equicontinuous, let then m(t) is continuous on J and

Lemma 4. ^{[2, 6]} Assume that satisfies

where are constants. Then provided one of the following two conditions holds

(i)

(ii)

3. Main Results

We list for convenience the following assumptions.

(H₁): (i) There exists satisfying

(ii) There exists satisfying

(H₂): (i) Whenever and

(ii) Whenever and

where M, K, N, L are non-negative constants and satisfy (i) or (ii) in Lemma 1.

(H₃): (i) There exists for any and satisfying

(ii) There exists for any and satisfying

(H₄): For any countable bounded equicontinuous set and

where are non-negative constants satisfying one of the following two conditions:

(i)

(ii)

Theorem 1. Let be a normal cone and on Assume that conditions and hold, then IVP(1.1) has a minimal solution in G. Moreover, there exist monotone increasing iterative sequence such that uniformly on where satisfying

(3.1)

Proof. First, for any it is easy to prove that (3.1) has a unique solution

Next, by(3.1), we have

(3.2)

(3.3)

By (3.3) and (H₁)(i), we have

and by Lemma 1, we can obtain That is

Suppose by (3.3) and we have

and so, by Lemma 1, we have That is and

From the above, by induction, it is not difficult to prove that

(3.4)

(3.5)

By (3.1), (3.4) and (H₃)(i), we know

(3.6)

and so, by (3.2), (3.5) and (3.6), we have

(3.7)

Then, let by the normality of P and (3.6) (3.7), we know that are bounded sequences in

For any by (H₂)(i) and (H₃)(i), it is easy to know that

is bounded. At the same, by (3.2) and (3.3), it is not difficult to show that are equicontinuous on

Let

and by the uniform boundedness of B(s) and uniform continuity of it is easy to show that (TB)(s), (SB)(s) are bounded and equicontinuous. Therefore, by Lemma 3, we have

(3.8)

(3.9)

then, from (3.1), (3.2), (3.8), (3.9), (H₄), Lemma 2 and Lemma 3, we know and

(3.10)

Similarly, we have

(3.11)

Let by (3.10), (3.11), we can get

where

Therefore, by Lemma 4 and the condition (i) or (ii) in (H₄), we see And so Hence Then are relatively compact sets in According to (3.4), (3.5) and the normality of P, we know , are convergent sequences respectively in Hence, there exists a that satisfies By taking limit in (3.1) as we have

so, is a solution of (IVP)(1.1) in G.

If there exist a and is also a solution of (IVP)(1.1) in G, then and

(3.12)

By (3.3), (3.12) and (H₂)(i), using induction, we can safely obtain

(3.13)

Letting in (3.13) and using the normality of P, we have That is, is a minimal solution of (IVP)(1.1) in G.

The proof of the theorem is complete.

Theorem 2. Let be a normal cone and on J. Assume that conditions (H1)(ii), (H2)(ii), (H3)(ii) and (H4) hold, then IVP(1.1) has a maximal solution in Q. Moreover, there exist monotone decreasing iterative sequence such that uniformly on where satisfying

(3.14)

Proof. The proof of Theorems 2 is almost the same as that of Theorem 1, so we omit it.

Theorem 3. Let be a regular cone and on J. Assume that conditions (H1)(i), (H2)(i) and (H3)(i) hold, then the results in Theorem 1 hold.

Proof. According to the proof of Theorems 1, we have (3.4), (3.5), by the regularity of P, we can obtain that uniformly on the rest of the proof is similar to the proof of Theorems 1.

Theorem 4. Let be a regular cone and on J. Assume that conditions (H1) (ii), (H2) (ii) and (H3)(ii) hold, then the results in Theorem 2 hold.

Proof. By using the similar method of the proof of Theorems 3, we can get the corresponding conclusion.

Remark 1. In (IVP)(1.1), if f does not contain the delayed argument and the differential argument then Theorem 1 implies the main results of ^{[2, 6]}, but the conditions in this paper are more extensive than those of ^{[2, 6]}. So the results presented in this paper generalize and unify the results of ^{[2, 6]}.

Remark 2. In paper ^[1], the author discussed the problem (IVP)(1.1) in which f does not contain and assumes the increase of Tu. Obviously in this paper, in the general case, we consider the second-order integro-differential equation in which f contains and weaken the increase of and we obtain the minimal and maximal solutions and the iteration sequence of (IVP) (1.1). Moreover, the conditions (H4) in this paper are more extensive than those in ^[1]. Therefore Theorem 1 improves and generalizes the results in ^[1].

Remark 3. We can see that Theorem 1 is suitable for any measure of non-compactness which is equal to the Kuratowski measure of non-compactness from the proof of Theorem 1.

Acknowledgment

The authors thanks the referee for his\her careful reading of the manuscript and useful suggestions.

Support

This work is supported by the NNSF of China (No.11501342) and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (No.2014135).

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