New Nonlinear Conditions and Inequalities for the Existence of Coincidence Points and Fixed Points

Let us begin with some basic definitions and notation that will be needed in this paper. Throughout this paper, we denote by N and R, the sets of positive integers and real numbers, respectively. Let X, d be a metric space. For each x ∈ X, and A ⊆ X, let d x,A infy∈Ad x, y . Denote by N X the class of all nonempty subsets of X, C X the family of all nonempty closed subsets ofX, and CB X the family of all nonempty closed and bounded subsets of X. A function H : CB X × CB X → 0,∞ , defined by