lower semi-continuous function

The basic purpose of this article is to introduce the concept of R - λ - G-contraction by using R-functions, lower semi-continuous functions and digraphs and discuss the existence and uniqueness of points of coincidence and common fixed points for a pair of self-mappings satisfying such contractions in the setting of metric spaces endowed with a graph. As some consequences of our results, we obtain several recent results in metric spaces and partial metric spaces.

Upper and lower semi-continuous functions are important in many areas and play a key role in optimization theory. This paper characterizes the lower and upper semi-continuity of $L^{p}$-space functions. We prove that a function $\vartheta:\mathcal L\rightarrow \overline{\mathbb R}$ is lower semi-continuous if and only if each convergent Moore-Smith sequence  $\{q_{j}\}_{j\in \mathbb N}$ converging to $q\in \mathcal L$ implies that $\int_{\mathcal L} \vartheta(q)d\mu\leq\liminf \int_{\mathcal L}\vartheta(q_{j})d\mu, \forall q\in \mathcal L$. We further show that the sum of any two proper lower semi-continuous functions is lower semi-continuous and the product of a lower semi-continuous function by a positive scalar gives a lower semi-continuous function and the case of upper semi-continuous functions follows analogously. Additionally, we prove that for a function in an $L^p$-space L if $\vartheta(\varphi)=\int_{\mathcal L}\varphi d\mu$ such that $\varphi$ is measurable with respect to a Borel measure $\mu$, then $\vartheta$ is upper semi-continuous.

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.