In this article, we introduce $(\mathrm{\Psi },\mathrm{\Phi })$-orthogonal interpolative contractions which generalize orthogonal interpolative contractions. We investigate different conditions on the functions $\mathrm{\Psi },\mathrm{\Phi }:(0,\mathrm{\infty })\to (-\mathrm{\infty },\mathrm{\infty })$ to show the existence of fixed-points of set-valued $(\mathrm{\Psi },\mathrm{\Phi })$-orthogonal interpolative contractions. Our fixed point results are improvement of several known results in literature. As an application, we resolve a fractional differential equation.

Topological indices (TIs) transform a molecular graph into a number. The TIs are a vital tool for quantitative structure activity relationship (QSAR) and quantity structure property relationship (QSPR). In this paper, we constructed two classes of Benes network: horizontal cylindrical Benes network and vertical cylindrical Benes network obtained by identification of vertices of first rows with last row and first column with last column of Benes network, respectively. We derive analytical close formulas for general Randić connectivity index, general Zagreb, first and the second Zagreb (and multiplicative Zagreb), general sum connectivity, atom-bond connectivity (), and geometric arithmetic index of the two classes of Benes networks. Also, the fourth version of and the fifth version of indices are computed for these classes of networks.

The focus of this paper is to present a new idea of fractional convex type contraction and establish some new results for such contraction under the improved approach of fractional convex type contractive condition in the context of $\mathcal{F}$ -complete $\mathcal{F}$ -metric space. The authors derive some results for Suzuki type contractions, orbitally T-complete and orbitally continuous mappings in $\mathcal{F}$ -metric spaces and obtain some consequences by using graphic contraction. The motivation of this paper is to observe the solution of fractional order differential equation with one of the boundary condition using fixed point technique in $\mathcal{F}$ -metric space.

The aim of this article is to design a new iteration process for solving certain fixed-point problems. In particular, we prove weak and strong convergence theorems for generalized nonexpansive mappings in the framework of uniformly convex Banach spaces. In addition, we discuss the stability of the solution under mild conditions. Further, we provide some numerical examples to indicate that the proposed method works properly.

In this manuscript, we coined pentagonal controlled fuzzy metric spaces and fuzzy controlled hexagonal metric space as generalizations of fuzzy triple controlled metric spaces and fuzzy extended hexagonal b-metric spaces. We use a control function in fuzzy controlled hexagonal metric space and introduce five noncomparable control functions in pentagonal controlled fuzzy metric spaces. In the scenario of pentagonal controlled fuzzy metric spaces, we prove the Banach fixed point theorem, which generalizes the Banach fixed point theorem for the aforementioned spaces. An example is offered to support our main point. We also presented an application to dynamic market equilibrium.

In this article, we present some fixed disc results for improving contractions in the frame work of F-metric space and to support the new results with examples. There are motivations for volterra Integral equation and findings of the existence of a solution to an initial value problem of second order differential equation.