On Fixed Point Theorems of Mixed Monotone Operators
© X. Du and Z. Zhao. 2011
Received: 11 November 2010
Accepted: 4 January 2011
Published: 11 January 2011
We obtain some new existence and uniqueness theorems of positive fixed point of mixed monotone operators in Banach spaces partially ordered by a cone. Some results are new even for increasing or decreasing operators.
Mixed monotone operators were introduced by Guo and Lakshmikantham in  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, we obtain some new existence and uniqueness theorems of positive fixed point of mixed monotone operators in Banach spaces partially ordered by a cone. Some results are new even for increasing or decreasing operators.
Let the real Banach space be partially ordered by a cone of , that is, if and only if . is said to be a mixed monotone operator if is increasing in and decreasing in , that is, implies . Element is called a fixed point of if .
For all , the notation means that there exist and such that . Clearly, ~ is an equivalence relation. Given (i.e., and ), we denote by the set . It is easy to see that is convex and for all . If and , it is clear that .
2. Main Results
We divide the proof into 3 steps.
We prove (2.2).
Let . It is easy to see that and . Put , , , , . Similarly to Step 1, it follows that there exists such that , . By the uniqueness of fixed points of operator in , we get . And by induction, . Since is normal, we have , .
3. Concerned Remarks and Corollaries
This research was supported by the National Natural Science Foundation of China (nos. 10871116, 11001151), the Natural Science Foundation of Shandong Province (no. Q2008A03), the Doctoral Program Foundation of Education Ministry of China (no. 20103705120002), and the Youth Foundation of Qufu Normal University (nos. XJ200910, XJ03003).
- Guo DJ, Lakshmikantham V: Coupled fixed points of nonlinear operators with applications. Nonlinear Analysis: Theory, Methods & Applications 1987,11(5):623–632. 10.1016/0362-546X(87)90077-0MATHMathSciNetView ArticleGoogle Scholar
- Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.MATHGoogle Scholar
- Guo D: Nonlinear Functional Analysis. Shandong Scientific Technical, Jinan, China; 2000.Google Scholar
- Guo DJ: Fixed points of mixed monotone operators with applications. Applicable Analysis 1988,31(3):215–224. 10.1080/00036818808839825MATHMathSciNetView ArticleGoogle Scholar
- Wu Y, Liang Z: Existence and uniqueness of fixed points for mixed monotone operators with applications. Nonlinear Analysis: Theory, Methods & Applications 2006,65(10):1913–1924. 10.1016/j.na.2005.10.045MATHMathSciNetView ArticleGoogle Scholar
- Zhao Z: Existence and uniqueness of fixed points for some mixed monotone operators. Nonlinear Analysis: Theory, Methods & Applications 2010,73(6):1481–1490. 10.1016/j.na.2010.04.008MATHMathSciNetView ArticleGoogle Scholar
- Zhang Z, Wang K: On fixed point theorems of mixed monotone operators and applications. Nonlinear Analysis: Theory, Methods & Applications 2009,70(9):3279–3284. 10.1016/j.na.2008.04.032MATHMathSciNetView ArticleGoogle Scholar
- Wu Y: New fixed point theorems and applications of mixed monotone operator. Journal of Mathematical Analysis and Applications 2008,341(2):883–893. 10.1016/j.jmaa.2007.10.063MATHMathSciNetView ArticleGoogle Scholar
- Deimling K: Nonlinear Functional Analysis. Springer, Berlin, Germany; 1985:xiv+450.MATHView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.