A class of generalized $(\psi ,\alpha ,\beta )$ —weak contraction is introduced and some fixed-point theorems in a framework of partially ordered metric spaces are proved. The main result of this paper is applied to a first-order ordinary differential equation to find its solution.

Our aim in this communication is to present a new type of contraction and common fixed point results for non-continuous self mappings without using the compatibility and commutative property. Moreover, we have found common fixed point without using the Cauchy's criterion for convergence. Some illustrative examples with graphical representation and remarks are given to justify that the contraction introduced in the main result is new and unique, which will be useful for future researches.

In this paper, we define a generalized C‐ condition and prove a fixed point theorem for weak (ψ,β)‐contraction in partially ordered metric spaces. Our result extends the result of Suzuki [Proc. of the 8th Int. Conf. of Fixed Point Theory and Its Appl. 2007; 65: 751 – 761.] and Gupta and Mani [Functional Analysis: Theory, Methods and Applications. 2017; 3: 26 – 34]. As an application, existence and uniqueness of solution of first order periodic boundary value problem has been given in support of our finding.

This paper establishes a new type of space, modified intuitionistic fuzzy soft metric space (MIFSMS). Basic properties and topological structures are defined in the setting of this new notion with valid examples. Moreover, we have given some new results along with suitable examples to show their validity. An application for finding the solution of an integral equation is also given by utilizing our newly developed results.