**Turkish Journal of Analysis and Number Theory**

## The Solutions of Initial Value Problems for Second-order Integro-differential Equations with Delayed Arguments in Banach Spaces

School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, P. R. China### Abstract

By using the partial order method and some new comparison results, the maximal or minimal solution of the initial value problem for nonlinear second order integro-differential equations with delayed arguments in Banach spaces are investigated. In this paper, we require only a lower solution or an upper solution and some weaker conditions presented here, and we extend and improve some recent results (see [1-11]).

**Keywords:** second-order integro-differential equation, delayed arguments, measure of non-compactness, solution, monotone iterative technique

Received June 24, 2015; Revised November 26, 2015; Accepted December 04, 2015

**Copyright**© 2015 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Tingting Guan. The Solutions of Initial Value Problems for Second-order Integro-differential Equations with Delayed Arguments in Banach Spaces.
*Turkish Journal of Analysis and Number Theory*. Vol. 3, No. 6, 2015, pp 154-159. http://pubs.sciepub.com/tjant/3/6/3

- Guan, Tingting. "The Solutions of Initial Value Problems for Second-order Integro-differential Equations with Delayed Arguments in Banach Spaces."
*Turkish Journal of Analysis and Number Theory*3.6 (2015): 154-159.

- Guan, T. (2015). The Solutions of Initial Value Problems for Second-order Integro-differential Equations with Delayed Arguments in Banach Spaces.
*Turkish Journal of Analysis and Number Theory*,*3*(6), 154-159.

- Guan, Tingting. "The Solutions of Initial Value Problems for Second-order Integro-differential Equations with Delayed Arguments in Banach Spaces."
*Turkish Journal of Analysis and Number Theory*3, no. 6 (2015): 154-159.

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### 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 *t*0* ** *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.

(*H*_{1}): (*i*) There exists* ** *satisfying

(*ii*) There exists* ** *satisfying

(*H*_{2}): (*i*) Whenever* ** *and

(ii) Whenever and

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

(*H*_{3}): (*i*) There exists* ** *for any* ** *and* ** *satisfying* ** *

(*ii*) There exists* ** *for any* ** *and* ** *satisfying

(*H*_{4}): 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 (*H*_{1})(*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 (*H*_{3})(*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 (*H*_{2})(*i*) and (*H*_{3})(*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),* *(*H*_{4}), 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 (*H*_{4}), 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 (*H*_{2})(*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 (*H*1)(*ii*)*,** *(*H*2)(*ii*)*,** *(*H*3)(*ii*) and (*H*4) 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 (*H*1)(*i*)*, *(*H*2)(*i*) and (*H*3)(*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 (*H*1) (*ii*)*, *(*H*2) (*ii*) and (*H*3)(*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 (*H*4) 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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