psi$ $

In [24], Khan et al. established some fixed point theorems in complete and compact metric spaces by using altering distance functions. In [16] Gordji et al. described the notion of orthogonal set and orthogonal metric spaces. In [18] Gungor et al. established fixed point theorems on orthogonal metric spaces via altering distance functions. In [25] Lotfy et al introduced the notion of $\alpha_{*}$-$\psi$-common rational type mappings on generalized metric spaces with application to fractional integral equations. In [28] K. Royy et al. described the notion of Branciari $S_b$-metric space and related fixed point theorems with an application. In this paper, we introduce the notion of the common fixed point ($\alpha_*$-$\psi$-$\beta_{i}$)-contractive set-valued mappings on orthogonal Branciari $S_{b}$-metric space with the application of the existence of a unique solution to an initial value problem.

In this research article, we study the existence, uniqueness and Ulam-Hyers stability of solutions in connection to the generalized Caputo Langevin fractional differential equations in Banach Space. The existence, uniqueness, and stability in the sense of Ulam are established for the proposed system. Our approach is based on the technique of measure of noncompactness combined with the Monch fixed point theorem, the implementation Banach contraction principle fixed point theorem.  Moreover,  the Ulam--Hyers stability is discussed by utilizing the Urs's. Lastly, we deliver an example to check the efficiency and accuracy of the proposed methods.

In this manuscript, we introduce new notions of generalized ($\mathfrak{f^{*}}, \psi)$-contraction and utilize this concept to prove some fixed point results for lower semi-continuous $\psi$-mapping satisfying certain conditions in the frame of G-metric spaces. Our results improve the results of [6] and [8] by omitting the continuity condition of $F\in \Im$ with the aid of the $\psi$-fixed point. We give an illustrative example to help accessibility of the got results and to show the genuineness of our results. Also, many existing results in the frame of metric spaces are established. Moreover, as an application, we employ the achieved result to earn the existence and uniqueness criteria of the solution of a type of non-linear integral equation.

In this manuscript, we introduce generalized orthogonal ($\mathfrak{f^{*}}, \psi)$-contraction of kind (S) and use this concept to establish $\psi$-fixed point theorems in the frame of O-complete orthogonal metric space. Secondly, we introduce the new notion of generalized orthogonal ($\mathfrak{f^{*}}, \psi)$ expansive mapping and utilize the same to prove some fixed point results for surjective mapping satisfying certain conditions. Our results extend and improve the results of  [3] and [7] by omitting the continuity condition of $F\in \Im$ with the aid of $\psi$-fixed point. We also give an illustrative example which yields the main result. Also, many existing results in the frame of metric spaces are established.

In this manuscript, we introduce generalized ($\mathfrak{f^{*}},\psi)$-contraction of kind (S) and use this concept to establish $\psi$-fixed point theorems in the frame of complete metric space. Secondly, we introduce new notion of generalized($\mathfrak{f^{*}}, \psi)$ expansive mapping of kind (S) and utilize the same to prove some fixed point results for surjective mapping satisfying certain conditions. Our results improve the results of [8], [10] and [14] by omitting the continuity condition of $F\in \Im$ with the aid of $\psi$-fixed point. We also give an example which yields the main result. Also, many existing results in the frame of metric spaces are established.

The aim of this paper is to give sufficient conditions for $Psi$-instability of trivial solution of a nonlinear Lyapunov matrix differential equation with integral term as right side.

In this paper, we introduce the (G-$\psi$) contraction in a metric space by using a graph. Let $F,T$ be two multivalued mappings on $X.$ Among other things, we obtain a common fixed point of the mappings $F,T$ in the metric space $X$ endowed with a graph $G.$

We provide necessary and sucient conditions for -conditional as- ymptotic stability of the solution of a linear matrix Lyapunov system and sucient conditions for -conditional asymptotic stability of the solution of a rst order non-linear matrix Lyapunov system X0 = A(t)X + XB(t) + F(t;X).