Solution and stability of a fixed point problem for mappings without continuity

In this paper by taking into account three trends prevalent in metric fixed point theory, namely, use of control functions instead of contraction constants, consideration of relational structure in the metric space and fixed point studies of discontinuous functions, we formulate and solve a new problem in relational metric fixed point theory. Our result extends the well known result of Kannan. The theorems are illustrated with examples. Further the proble is shown to have Hyers-Ulam-Rassias stability property. We make an application of our main result to a problem of a nonlinear integral equation.