#### Abstract

We establish some new existence theorems on the positive solutions for nonlinear integro-differential equations which do not possess any monotone properties in ordered Banach spaces by means of Banach contraction mapping principle and cone theory based on some new comparison results.

#### 1. Introduction

In this paper, we consider the existence of the unique positive solution and at least one positive solution for the following initial value problem (IVP) of the nonlinear integro-differential equation of mixed type in ordered Banach spaces : where , , , , , and is nonincreasing with the second variable , with is nondecreasing with the third variable . is a positive cone in ordered Banach spaces , and In (2), and , where , , denotes the set of nonnegative real numbers, and denotes the set of real numbers. In the special case when does not contain the third argument, Guo [1] proved the minimal and maximal solution of the following initial value problem: under some stronger conditions. He used the topological degree theory and the monotone iterative technique. When does not possess any monotone assumption, the problem of proving the existence results is an interesting and important question. The aim of the paper is to study this kind of problem. By means of Banach contraction mapping principle and cone theory based on some new comparison results, we obtain some new existence theorems of the solutions for the initial value problems of the nonlinear integro-differential equations of mixed type which does not possess any monotone properties in ordered Banach spaces. The new results obtained are quite general and are used to generalize, improve, and unify many recent results of [2–8]. Their results assert the existence of one solution, the minimal and maximal solution using monotone iterative technique with the stronger conditions. Our method is different from the method of mixed monotony, and the results obtained in the paper are new even if the space is finite dimensional.

Inspired and motivated greatly by the above work, the aim of the paper is to consider the existence of positive solutions for the boundary value problem (1) under simpler conditions. The main results of problem (1) are obtained by making use of new fixed point theorem. More specifically, in the proof of these theorems, we establish new comparison theorem and construct a special cone for strict set contraction operator. Our main results in essence improve and generalize the corresponding results of [1, 9–12]. Moreover, our method is different from those in [5, 10, 13].

The rest of the paper is organized as follows. In Section 2, we present some known results and introduce conditions to be used in the next section. The main theorems are formulated and proved in Section 3.

#### 2. Preliminaries and Lemmas

In this section, we will state some necessary definitions and preliminary results.

*Definition 1. *Let be a real Banach space. A nonempty closed set is called a cone if it satisfies the following two conditions:(1), implies ;(2), implies , where denotes the zero element of .

A cone is said to be solid if it contains interior points, . The positive cone is said to be generating if ; that is, every element can be represented in the form , where . Every cone with nonempty interior is generating. A cone induces a partial ordering in given by if . If and , we write ; if cone is solid and , we write .

*Definition 2. *A cone is said to be normal if there exists a positive constant such that , , , .

Proposition 3. *A positive cone is called normal if and only if there exists such that implies . The norm on is called monotone if implies . The cone is said to be normal if and only if there exists an equivalent monotone norm on .**For further details on cone theory, one can refer to [2, 14].**Let , where denotes the Banach space of all continuous mapping with norm . It is easy to see that is a cone in , and so it defines a partial order in by if and only if for all . Given a cone , the dual of is defined as , where is the dual of .**Let , . For any and , let
*

*Definition 4. *Let be a metric space and be a bounded subset of . The measure of noncompactness of is defined by

*Definition 5. *An operator is said to be completely continuous if it is continuous and compact. is called a -set-contraction if it is continuous, bounded, and for any bounded set , where denotes the measure of noncompactness of .

A -set-contraction is called a strict-set-contraction if . An operator is said to be condensing if it is continuous, bounded, and for any bounded set with .

Obviously, if is a strict-set-contraction, then is a condensing mapping, and if operator is completely continuous, then is a strict-set-contraction.

*Definition 6 (see [15]). *Let be a Banach space, and let . The mapping is said to have the mixed monotone property if is monotone nonincreasing in and is monotone nondecreasing in ; that is,

*Definition 7 (see [15]). *A couple point is said to be a coupled fixed point of the mapping if and .

Lemma 8 (see [2, 14]). *A positive cone is normal and generating if and only if there exists a positive constant and such that for all and .*

The following lemma plays a key role for improving the main results.

Lemma 9. *Let be a normal generate cone, and let be a nonlinear operator. If there exists a positive bounded linear operator such that and
**
then has a unique positive fixed point , and for any , let , .*

*Proof. *Since , then . Thus, there exists a natural number and such that
Since is generate, it follows from Lemma 8 that there exist , such that and
Then . Suppose that are not in the same line for any . Let , , . Then and , , , . It follows from (7) that
By virtue of (10) + (12) − (11), we have
We first define the operator as the following , . Let . Then . Since is a positive bounded linear operator, then . By induction, it is easy to see that
Since is arbitrary and is taken as in (8), we get
From and Banach contract principle, we know that has the unique positive fixed point in ; that is, is the unique positive solution of operator equation . And for any , let , we have .

*Remark 10. *The results of Lemma 9 obtained are quite general and are used to improve, generalize, and unify many results of Chen [13], Guo [9], Krasnosel’skii and Zabreiko [16], Zhang [11], Su et al. [10], and Liu [12]. Not only do we obtain the existence and uniqueness of fixed points for mixed nonmonotone binary operators in ordered Banach spaces, but we also solve the difficult open problem that nonmonotone binary operator has unique fixed points under some weaker conditions.

Lemma 11 (see [17]). *Let be a Banach space and if is a countable set of strongly measurable functions such that there exists such that , , . Then is Lebesgue integrable on , and
*

Lemma 12 (see [1, 18]). *Let be bounded and equicontinuous; then is continuous on , and
*

Lemma 13 (see [2, 19]). *Let be a Banach space, closed and convex, and continuous with the further property that for some we have countable. implies that is relatively compact. Then has a fixed point in .*

Lemma 14. *Assume that satisfies
**
where and are constants. Then for provided one of the following two conditions hold:*(i)*;*(ii)*.*

*Proof. *Suppose that (i) holds. Let ; then , , for . From (18), we know that
Since , we have
Integrate from to ; noticing , we have
Hence,
We claim that
Otherwise, ; then we take in (22), we obtain , which contradicts assumption (i).

Therefore, from (22) and (23), we know that . Consequently, a.e. on .

Let us suppose now that (ii) holds. Then
It follows by integrating the above inequality that
which, by assumption (ii), implies that and so a.e. on . The proof of Lemma 14 is therefore complete.

Lemma 15 (see [2, 19]). * is relatively compact if and only if each element and are uniformly bounded and equicontinuous on .*

#### 3. Main Results

We are now in a position to prove our main results concerning the unique positive solution and at least one positive solution.

Theorem 16. *Suppose that is a normal solid cone whenever ; there exists , , such that
**
Then IVP (1) has the unique positive solution in .*

*Proof. *Define operator as the following:
Then is a solution of IVP (1) if and only if is a fixed point of ; that is, for any , . From (26) and (28), we know that
From (29) and (30), for any , we know that
where
Now we prove that , for . From (32), we know that
Denote . From (32) and (33), we get
By induction, for any natural number , we have
Therefore, .

Since is normal solid cone in and is generate, by virtue of Lemma 9, we know that has a unique positive fixed point in ; that is, IVP (1) has a unique positive solution in .

*Remark 17. *The conditions of Theorem 16 cannot be obtained by Chen and Zhuang [3], Lakshmikantham et al. [8], and Sun and Liu [20]. We obtain a unique positive solution of IVP (1). The conditions imposed on nonlinear term are sharper, and the result is new.

Denote , .

Theorem 18. *Assume that satisfying the following conditions.*(H_{1})* For any , is uniformly continuous on and
*(H_{2})* There exists such that
**for any bounded set , , and , with .**Then IVP (1) has at least one positive solution in .*

*Proof. *We first define the operator by the formula
It is easy to know that is a solution of IVP (1) if and only if is a fixed point of .

It follows from (H_{1}) that there exist and such that

Let
Then for any , we have
It follows from (38) and (39) that
Set . Then is a continuous operator from into . It is easy to see from (38), (H_{1}), and the normality of that is uniformly bounded and equicontinuous on .

Let be any countable subset satisfying for any . By applying Lemma 12, we get
where . Thus, by Ascoli-Arzela theorem, is relative compact in . It follows from Lemma 14 that has a fixed point which is a solution of IVP (1). This completes the proof of Theorem 18.

*Remark 19. *The conditions imposed on nonlinear term do not possess any monotone properties. The results of the paper cannot be obtained by making use of the fixed point theorems on decreasing and increasing operators or mixed monotone operators.

#### Conflict of Interests

The author declares no conflict of interests.

#### Acknowledgments

The author is very grateful to Professor Lishan Liu, Professor Tijun Xiao, and Professor R. P. Agarwal for their many valuable comments. The work is supported financially by the Foundation of Shanghai Natural Science (13ZR1430100) and the Foundation of Shanghai Municipal Education Commission (DYL201105).