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Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. (English) Zbl 1220.54025
The purpose of this paper is to present, in the context of an ordered metric space, some fixed point theorems for continuous, nondecreasing mappings which satisfy Caristi type conditions with perturbed metrics. For example, the following theorem is proved.
Theorem 2.1. Let \((X,\leq)\) be a partially ordered set and suppose that there exists a metric \(d\) in \(X\) such that \((X,d)\) is a complete metric space. Let \(f:X\to X\) be a continuous and nondecreasing mapping such that \[ \psi(d(f(x),f(y)))\leq \psi(d(x,y))-\phi(d(x,y))\text{ for all }x,y\in X,\;x\geq y, \] where \(\psi,\phi:\mathbb{R}_+\to \mathbb{R}_+\) are some altering distance functions. If there exists \(x_0\in X\) with \(x_0\leq f(x_0)\), then \(f\) has a fixed point in \(X\).
As an application, an existence result for a first-order periodic problem for a differential equation is given.

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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