A notion for convex structures on metric space, first introduce by Takahashi [1] and has become an important part of the theory of fixed points and nonlinear analysis. This framework can be generalized in the present paper by introduces r-convex metric spaces notion, i.e., the metric spaces endowed with an r-convex structures. We construct and establish a fixed - points theorem on r-convex metric space on the basis of this new proposed structure. Some of the basic findings regarding the fixed points of the r-convex framework are rigorously established and addressed. We use the theoretical discoveries to prove that solutions to a set of integral equations exist. Moreover, graphic example is provide to demonstrates validity and applicability for the derived findings.