In the present paper, we generalize the results of Karapinar and Khojasteh [7], Karapinar and Fulga [6] in super metric space by using the control function and weakly compatible mappings.
Fixed points are the points which remain invariant under a map or transformation. Fixed points give us the idea of points that are not moved by the transformation. Geometrically, the fixed points of a curve are the point of intersection of the curve with the line y = x. A map can have one fixed point, two fixed points, infinitely many fixed points and no fixed point. The mapping f∶ R → R defined by f(x) = 3x, for all x ∈ R has a unique fixed point x = 0. The mapping f∶ R → R defined by f(x) = x^2, for all x ∈ R has a two fixed points x = 0 and x=1. The identity mapping has infinitely many fixed point where as the translation mapping has no fixed point.
Metric fixed point theory involves the study of fixed points depending on the mapping conditions on the spaces under consideration. There is a revolution in metric fixed point theory with the escalation of the Banach contraction principle and this principle is popularly known as the contraction principle, which states that “every contraction mapping on a complete metric space has a unique fixed point."