New Fixed Point Theorems of Mixed Monotone Operators ()
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.
Share and Cite:
X. Du, "New Fixed Point Theorems of Mixed Monotone Operators,"
Applied Mathematics, Vol. 5 No. 3, 2014, pp. 352-357. doi:
10.4236/am.2014.53037.
Conflicts of Interest
The authors declare no conflicts of interest.
References
|
[1]
|
D. J. Guo and V. Lakskmikantham, “Coupled Fixed Points of Mixed Monotone Operator with Applications,” Nonlinear Analysis, Vol. 11, No. 5, 1987, pp. 623-632.
|
|
[2]
|
K. Deiming, “Nonlinear Functional Analysis,” Springer-Verlag, New York, 1985.
|
|
[3]
|
D. J. Guo and V. Lakskmikantham, “Nonlinear Problems in Abstract Cones,” Acadamic Press, Boston, 1988.
|
|
[4]
|
D. J. Guo, “Nonlinear Functional Analysis,” 5th Edition, Shandong Scientific Technical Publishers, Jinan, 2000.
|
|
[5]
|
D. J. Guo, “Fixed Points of Mixed Monotone Operators with Applications,” Journal of Mathematical Analysis and Applications, Vol. 31, 1988, pp. 215-224.
|
|
[6]
|
Y. X. Wu and Z. D. Liang, “Existence and Uniqueness of Fixed Points for Mixed Monotone Operators,” Nonlinear Analysis, Vol. 65, 2006, pp. 1913-1924.
|
|
[7]
|
Z. Q. Zhao, “Existence and Uniqueness of Fixed Points for Some Mixed Monotone Operators,” Nonlinear Analysis, Vol. 73, 2010, pp. 1481-1490.
|