Contractive-Like Mapping Principles in Ordered Metric Spaces and Application to Ordinary Differential Equations

Existence of fixed point in partially ordered sets has been considered recently in 1–17 . Tarski’s theorem is used in 9 to show the existence of solutions for fuzzy equations and in 11 to prove existence theorems for fuzzy differential equations. In 2, 6, 7, 10, 13 some applications to ordinary differential equations and to matrix equations are presented. In 3– 5, 17 some fixed point theorems are proved for a mixedmonotone mapping in a metric space endowed with partial order and the authors apply their results to problems of existence and uniqueness of solutions for some boundary value problems. In the context of ordered metric spaces, the usual contraction is weakened but at the expense that the operator is monotone. The main tool in the proof of the results in this context combines the ideas in the contraction principle with those in themonotone iterative technique 18 . Let S denote the class of the class of the functions β : 0,∞ → 0, 1 which satisfies the condition