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Fixed-point theorems in partially ordered metric spaces for operators with PPF dependence. (English) Zbl 1127.47049
Let $E$ be a complete metric space with a partial order, $T: C([a,b],E)\to E$ be a monotonically nondecreasing operator. A fixed point $\phi\in C([a,b],E)$ of $T$ means that there exists some $c\in [a,\,b]$ such that $T\phi= \phi(c)$. The present paper discusses the existence and uniqueness of the fixed points of $T$ under the conditions that $T$ is order-contractive and the fixed point equation $\phi(c)=T\phi$ has a lower solution. The obtained fixed point theorem is applied to a periodic boundary value problem of a delay ordinary differential equation, and a unique existence result for periodic solutions is obtained.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H07Monotone and positive operators on ordered topological linear spaces
47N20Applications of operator theory to differential and integral equations
54H25Fixed-point and coincidence theorems in topological spaces
54F05Linearly, generalized, and partial ordered topological spaces
Full Text: DOI
[1] Bernfeld, R. Stephen; Lakshmikantham, V.; Reddy, Y. M.: Fixed point theorems of operators with PPF dependence in a Banach space. Applicable analysis 6, 271-280 (1977) · Zbl 0375.47027
[2] T. Gnana Bhaskar, V. Lakshmikantham, Monotone iterative technique for functional differential equations with retardation and anticipation, Nonlinear Analysis, TMA (in press) · Zbl 1078.34032
[3] Nieto, J. J.; Rodriguez-Lopez, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, No. 3, 223-229 (2005)
[4] Ran, A. C. M.; Reurings, M. C. R.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. amer. Math. soc. 132, 1435-1443 (2003) · Zbl 1060.47056