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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.

MSC:
 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 47H07 Monotone and positive operators on ordered topological linear spaces 47N20 Applications of operator theory to differential and integral equations 54H25 Fixed-point and coincidence theorems in topological spaces 54F05 Linearly, generalized, and partial ordered topological spaces
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References:
 [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