New existence and uniqueness theorems of positive fixed points for mixed monotone operators with perturbation. (English) Zbl 1115.47044
The authors study the existence and uniqueness of positive solutions of the operator equation
\[
A(x,x)+Bx=x,\quad x\in E,
\]
where \(E\) is a Banach space ordered by a normal cone, \(A\) is a mixed monotone operator with convexity and concavity, and \(B\) is an affine operator. Some applications to the nonlinear integral equation
\[
\int_{\mathbb{R}^n} K(t,s)[f(x(s))+g(x(s))]\,ds= [1+G_1(t)]x(t)-G_2(t)x(t+\tau)-G_3(t),
\]
where \(t\), \(\tau\in \mathbb{R}^n\), are also given.
Reviewer: Mirosława Zima (Rzeszow)
MSC:
| 47H10 | Fixed-point theorems |
| 47H07 | Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces |
| 47N20 | Applications of operator theory to differential and integral equations |
References:
| [1] | Du, Y. H., Fixed points of increasing operators in ordered Banach spaces and applications, Appl. Anal., 38, 1-20 (1990) · Zbl 0671.47054 |
| [2] | Guo, D. J., Fixed points of mixed monotone operators with applications, Appl. Anal., 31, 215-224 (1988) · Zbl 0688.47019 |
| [3] | Guo, D. J.; Lakshmikantham, V., Coupled fixed points of nonlinear operators with applications, Nonlinear Anal. TMA, 1, 623-632 (1987) · Zbl 0635.47045 |
| [4] | Guo, D. J.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1998), Academic Press: Academic Press Boston |
| [5] | Li, F. Y.; Feng, J. F.; Shen, P. L., Fixed point theorems of some decreasing operators and applications, Acta Math. Sinica, 42, 2, 193-196 (1999), (in Chinese) · Zbl 1027.47517 |
| [6] | Li, K.; Liang, J.; Xiao, T. J., A fixed point theorem for convex and decreasing operators, Nonlinear Anal., 63, e209-e216 (2005) · Zbl 1159.47306 |
| [7] | (Nagel, R., One-parameter Semigroup of Positive Operators. One-parameter Semigroup of Positive Operators, Lecture Notes in Math., vol. 1184 (1986), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0585.47030 |
| [8] | Rudin, W., Functional Analysis (1991), McGraw-Hill: McGraw-Hill New York, p. 146 · Zbl 0867.46001 |
| [9] | Zhang, Z. T., New fixed point theorems of mixed monotone operators and applications, J. Math. Anal. Appl., 204, 307-319 (1996) · Zbl 0880.47036 |
| [10] | Zhang, Z. T., Fixed point theorems of mixed monotone operators and its applications, Acta Math. Sinica, 41, 6, 1121-1126 (1998), (in Chinese) · Zbl 1012.47019 |
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