Quadruple Fixed Point Theorem for Partially Ordered Metric Space with Application to Integral Equations

In this paper, two theorems have been established. The first theorem says the existences of a quadruple fixed point theorem in partially ordered metric space for nonlinear contraction mapping which is \((\alpha )\)-admissible and satisfies the mixed monotone property. The second result is proved for non-continuous mapping in addition to some other conditions. A suitable example of nonlinear contraction mapping validates the result. Moreover, an application to the integral equation is also presented.