In this paper, we present a Pata-type fixed point theorem in modular spaces which generalizes and improves some old results. As an application, we study the existence of solutions of integral equations in modular function spaces.

In present paper we introduce the concept of a new g-monotone mapping and define the notions of n-fixed point and n-coincidence point and prove some related theorems for nonlinear contractive mappings in partially ordered complete metric spaces. Our results are generalization of the main results of Lakshmikantham andĆirić (Nonlinear Anal. 70:4341-4349, 2009) and include several recent developments. Moreover, we give an example to support our results. MSC: Primary 47H10; secondary 54H25; 34B15

The Sturm–Liouville differential equation is an important tool for physics, applied mathematics, and other fields of engineering and science and has wide applications in quantum mechanics, classical mechanics, and wave phenomena. In this paper, we investigate the coupled hybrid version of the Sturm–Liouville differential equation. Indeed, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with multi-point boundary coupled hybrid condition. Furthermore, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with an integral boundary coupled hybrid condition. We give an application and some examples to illustrate our results.

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