ABSTRACT

Mixed monotone operator is an important nonlinear operator. It exists extensively in the research of nonlinear differential and integral equations. Generally, the research of mixed monotone operators in partially ordered Banach spaces requires compactness, continuity or concavity-convexity of the operators. In this paper, without any compact and continuous assumption, we obtain some new existence and uniqueness theorems of positive fixed point of e-concave-convex mixed monotone operators in Banach spaces partially ordered by a cone, which extends the existing corresponding results.

Keywords:Mixed Monotone Operators; Cone; Partial Order

1. Introduction

Mixed monotone operators were introduced by Dajun Guo and V. Lakshmikantham in [1] in 1987. Thereafter, many authors have investigated these kinds of operators in Banach spaces and obtained a lot of interesting and important results. They are used extensively in nonlinear differential and integral equations. In this paper, without any compact and continuous assumption, we obtained some new existence and uniqueness theorems of positive fixed point of e-concave-convex mixed monotone operators in Banach spaces partially ordered by a cone.

Let the real Banach space be partially ordered by a cone of, i.e., iff. is said to be a mixed monotone operator if is increasing in and decreasing in, i.e., , implies.. Element is called a fixed point of iff.

Recall that cone is said to be solid if the interior is nonempty and we denote if. is normal if there exists a positive constant N, such that implies, is called the normal constant of.

For all, the notation means that there exist and, such that. Clearly, is an equivalence relation. Given, we denote by the set. It is easy to see that is convex and for all. If and, it is clear that.

All the concepts discussed above can be found in [2-4]. For more facts about mixed monotone operators and other related concepts, the reader could refer to [5-7] and some of the reference therein.

2. Main Results

In this section, we present our main results. To begin with, we give the definition of e-concave-convex operators.

Definition 2.1. We say an operator is an e-concave-convex operators if there exist one positive function such that

(2.1)

Remark 2.1. (2.1) implies that

(2.2)

Theorem 2.1. Let be a normal cone of, and let be a mixed monotone and e-concave-convex operator. In addition, suppose that there exist (Since, we can choose a sufficiently small such that) such that

hold, where. Then has exactly one fixed pint x^* in. Moreover, constructing successively the sequence

for any initial, we have

(2.3)

Proof. We divide the proof into 3 steps.

Step 1. We prove A has a fixed point in.

Construct successively the sequences

.

It follows from and the mixed monotonicity of that

. (2.4)

Let

.

Thus we have, and then

.

Therefore, , i.e., is increasing with.

Suppose as. Then. Indeed, suppose to the contrary that. By, (2.1) and the mixed monotonicity of, we have

.

Therefore

(2.5)

By (2.1) and the mixed monotonicity of, we know that, and thus

exists. We distinguish two cases.

Case 1:. In this case we know that So, we have

, which is a contradiction.

Case 2:. In this case, it is easy to see that. For convenience’s sake, let. Since, there is a nonincreasing subsequence of such that

. Without loss of generality, we may still use to stand for. From (2.5), we obtain

.

Hence

which is also a contradiction. Thus.

For any natural number we have

Since is normal, we have,. Here N is the normality constant.

So and are Cauchy sequences. Because E is complete, there exist such that, By (2.4) we know that and

.

Further

And thus. Let, we obtain

Let, we get. That is, x^* is a fixed point of in.

Step 2. We prove that x^* is the unique fixed point of in.

In fact, suppose is another fixed point of in. Since, there exist positive numbers such that. Let

. (2.6)

Evidently,. We now prove. If otherwise,. From (2.1), we obtain

Which contradicts the definition of. Hence, thus. Therefore A has a unique fixed point x^* in.

Step 3. We prove (2.3).

For any, we can choose a small number such that

(2.7)

Let

(2.8)

Using (2.7), (2.8) and the mixed monotonicity of A, we have

(2.9)

Let

. (2.10)

From (2.8) and (2.9), we obtain

(2.11)

In what follows, we will prove that. If not, that is, , then from (2.8) and (2.11), we have

(2.12)

. (2.13)

Let. It follows from (2.11), (2.12) and (2.13) that

, which is a contradiction. That is.

From (2.9) and (2.11), we have

.

Thus,

.

By using the normality of, we know that (2.2) holds.

3. Concerned Remarks and Corollaries

Using Theorem 2.1, we have the following corollaries.

Corollary 3.1. Let P be a normal cone of E, and let be a mixed monotone and e-concaveconvex operator. In addition, suppose that there exist (Since, we can choose a sufficiently small, such that), such that

hold, where. Then A has exactly one fixed pint x^* in. Moreover, constructing successively the sequence

for any initial, we have

Corollary 3.2. Let P be a normal cone of E, and let be a mixed monotone and e-concaveconvex operator. In addition, suppose that

there exist such that

for all, there exist, such that hold.

Then has exactly one fixed pint in. Moreover, constructing successively the sequence

for any initial, we have

Corollary 3.3. Let P be a normal cone of, and let be a mixed monotone and e-concaveconvex operator. In addition, suppose that, are monotone on and. Then a necessary and sufficient condition for to have exactly one fixed pint in is that holds. Moreover, constructing successively the sequence

for any initial, we have

Proof. Corollary 3.1 ensures the sufficiency of Corollary 2.3, so we have only to prove the necessity of Corollary 2.3.

Suppose that x^* is the unique fixed point of in. For any, let. It follows from (1.1), (1.2) and the mixed monotonicity of A that

Therefore, holds.

Corollary 3.4. Let be a normal cone of, and let be a mixed monotone operator. In addition, suppose that, there exists such that

(3.1)

holds. Then a necessary and sufficient condition for to have exactly one fixed pint in is that holds. Moreover, constructing successively the sequence

for any initial, we have

Corollary 3.5. Let be a normal cone of, and let be a mixed monotone operator with property (2.1). In addition, suppose that holds. Then has exactly one fixed pint x^* in.

Proof. In fact, by corollary 2.4, we have only to prove that holds.

For any, it follows from that there exists, such that

. Let. Following from (3.1) and the mixed monotonicity of A, we get

The Corollary 3.5 is thus proved.

Acknowledgements

The author thanks the referee for his valuable comments and suggestions. This paper was supported by SRFDP (NO. 20103705120002).

REFERENCES