Fixed point problems for generalized contractions with applications

Abstract

In this paper, we investigate the conditions on the control mappings psi,phi : (0, infinity) -> R that guarantee the existence of the fixed points of the mapping T : X -> P(X) satisfying the following inequalities: psi (H (Tx, Ty)) <= phi (d(x, y)) for all x,y epsilon X, provided that H (Tx, Ty) > 0, and psi(H(Tx, Ty)) <= phi (A(x, y)) for all x, y epsilon X, provided that H(Tx, Ty) > 0, where A (x, y) = max{d(x,y), d (x, Tx), d (y, Ty), (d (x, Ty) + d (Tx, y))/2}, and (X, d) is a metric space. The obtained fixed point results improve many earlier results on the set-valued contractions. As an application, we consider the existence of the solutions of an FDE.