Existence, well-posedness of coupled fixed points and application to nonlinear integral equations

Abstract

We investigate a fixed point problem for coupled Geraghty type contraction in a metric space with a binary relation. The role of the binary relation is to restrict the scope of the contraction to smaller number of ordered pairs. Such possibilities have been explored for different types of contractions in recent times which has led to the emergence of relational fixed point theory. Geraghty type contractions arose in the literatures as a part of research seeking the replacement contraction constants by appropriate functions. Also coupled fixed point problems have evoked much interest in recent times. Combining the above trends we formulate and solve the fixed point problem mentioned above. Further we show that with some additional conditions such solution is unique. Well-posedness of the problem is investigated. An illustrative example is discussed. The consequences of the results are discussed considering \(\alpha\)-dominated mappings and graphs on the metric space. Finally we apply our result to show the existence of solution of some system of nonlinear integral equations.