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Fixed point theorems for a class of mixed monotone operators with convexity
Fixed Point Theory and Applications volume 2013, Article number: 119 (2013)
Abstract
In this paper, we use partial order theory to study a class of mixed monotone operators with convexity, and we get the existence and uniqueness of fixed points without assuming the operator to be compact or continuous. Our results compliment the theory of mixed monotone operators in ordered Banach spaces.
MSC:47H10, 47H07.
1 Introduction and preliminaries
It is well known that mixed monotone operators were introduced by Guo and Lakshmikantham [1] in 1987. Thereafter many authors have investigated these kinds of operators in Banach spaces and obtained a lot of interesting and important results (see [1–18] and the references therein). Their study has not only important theoretical meaning but also wide applications in engineering, nuclear physics, biological chemistry technology, etc. That is, they are used extensively in nonlinear differential and integral equations. For a small sample of such work, we refer the reader to works [13, 15–21]. However, the mixed monotone operators considered in most of these papers are concave; see [2–4, 7–11, 15, 17, 18], for instance. To our knowledge, the fixed point results on mixed monotone operators with convexity are still very few. So it is worthwhile to investigate this operator. The purpose of this paper is to establish the existence and uniqueness of fixed points for a class of mixed monotone operators with convexity. We will use partial order theory to study the mixed monotone operator with convexity and get the existence and uniqueness of fixed points without assuming the operator to be compact or continuous. Our results compliment the theory of mixed monotone operators in ordered Banach spaces.
For the discussion of the following section, we state here some definitions and notations. For convenience of readers, we suggest that one refers to [1, 2, 17, 21–23] for details.
Suppose that is a real Banach space which is partially ordered by a cone , i.e., if and only if . If and , then we denote or . By θ we denote the zero element of E. Recall that a non-empty closed convex set is a cone if it satisfies (i) , ; (ii) , .
Further, P is called normal if there exists a constant such that, for all , implies ; in this case N is called the normality constant of P. If , the set is called the order interval between and . We say that an operator is increasing if implies .
is said to be a mixed monotone operator if is increasing in x and decreasing in y, i.e., , , , imply . The element is called a fixed point of A if .
2 Main results
In this section we consider the existence and uniqueness of fixed points for a class of mixed monotone operators in ordered Banach spaces.
Theorem 2.1 Let E be a real Banach space and let P be a normal cone in E. is a mixed monotone operator which satisfies the following:
(H1) for , , there exists such that
(H2) there exist , such that
Then A has a unique fixed point in . Moreover, constructing successively the sequences
for any initial values , we have , as .
Remark 2.1 (i) The operator A which satisfies (H1) can be called a mixed monotone operator with convexity; (ii) from (H1), we can get . In fact, for , , and thus . It follows from that ; (iii) from (2.1), we can get
Proof of Theorem 2.1 Let , . Then , , and
Construct successively the sequences
From (2.4), (2.5) and the mixed monotonicity of A, we have
Next we prove that
From (2.4),
Suppose that when , we have . Then, when , we obtain
By the induction method, we know that (2.7) holds. On the other hand, from (2.1),
Suppose that when , we have . Then, when , we obtain
By the induction method, we have
By (2.6)-(2.8), we get
Since P is normal, we have
Further,
Here N is the normality constant.
So, we can claim that and are Cauchy sequences. Because E is complete, there exist such that
By (2.6), we know that , and then
Further, (as ), and thus . Then we obtain
Let , then we get . That is, is a fixed point of A in .
In the following, we prove that is the unique fixed point of A in . Suppose that there is such that . Then we have . By the induction method and the mixed monotonicity of A, we have
Then from the normality of P, we have .
Moreover, constructing successively the sequences
for any initial values , we have , . Letting yields , as . □
Remark 2.2 Let be a constant , then Theorem 2.1 also holds.
Corollary 2.2 Let E be a real Banach space and let P be a normal cone in E. is a mixed monotone operator which satisfies (H2) and, for , , there exists a constant such that . Then A has a unique fixed point in . Moreover, constructing successively the sequences
for any initial values , we have , as .
From the proof of Theorem 2.1, we can easily obtain the following conclusion.
Corollary 2.3 Let E be a real Banach space and let P be a normal cone in E. is an increasing operator which satisfies the following:
(H3) for , , there exists a constant such that ;
(H4) there exist , such that , , .
Then A has a unique fixed point in . Moreover, constructing successively the sequence
for any initial value , we have as .
Remark 2.3 The operator A which satisfies (H3) is called α-convex. In 1977, A.J.B. Potter introduced the definition of an α-convex operator and showed that for , decreasing (−α)-convex mappings have contraction ratios less than or equal to α and gave the existence of solutions to the nonlinear eigenvalue problem . The method is based upon Hilbert’s projective metric. In [22] Guo studied the existence and uniqueness of fixed points for (−α)-convex operators. In [23], we obtained the existence and uniqueness of positive fixed points for α-convex operators by means of the properties of cone, concave operators and the monotonicity of set-valued maps. Here, the result on α-convex operators is new and the method is also new and different from previous ones.
Remark 2.4 If is an α-convex operator, then A is not a decreasing and constant operator. In fact, suppose that A is decreasing, then we have , , . Hence, , this is a contradiction. Suppose that A is a constant operator, that is, . Then , and thus , this is also a contradiction.
Theorem 2.4 Let E be a real Banach space and let P be a normal cone in E. is a mixed monotone operator which satisfies (2.1) and
(H5) there exist , such that
Then the operator equation has a unique solution in , where . Moreover, constructing successively the sequences
for any initial values , we have , as .
Proof Let . Then . Note that and from (2.3),
Set , . Then from the above inequalities, we have
Also, construct successively the sequences
From (2.9) and the mixed monotonicity of A, we have
Similar to the proof of Theorem 2.1, we can prove that
Further, by using the same method with the proof of Theorem 2.1, we can get the following conclusions: (i) B has a unique fixed point in ; (ii) for any initial values , constructing successively the sequences
we have , as . Therefore, the operator equation has a unique solution in . Moreover, constructing successively the sequences
for any initial values , we have , as . □
Corollary 2.5 Let E be a real Banach space and let P be a normal cone in E. is an increasing operator which satisfies (H3) and
(H6) there exist , such that , , .
Then the operator equation has a unique solution in , where . Moreover, constructing successively the sequence
for any initial value , we have as .
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Acknowledgements
The author was supported financially by the Youth Science Foundations of China (11201272) and Shanxi Province (2010021002-1).
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Zhai, C. Fixed point theorems for a class of mixed monotone operators with convexity. Fixed Point Theory Appl 2013, 119 (2013). https://doi.org/10.1186/1687-1812-2013-119
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DOI: https://doi.org/10.1186/1687-1812-2013-119