Journal of Linear and Topological Algebra
Volume:12 Issue: 1, Winter 2023

Frame theory has been rapidly generalized and various generalizations have been developed. In this paper, we present a brief survey of the frames in Hilbert $C^{\ast}$-modules, including frames, $\ast$-frames, g-frames, $\ast$-g-frames, $\ast$-$K$-$g$-frame, operator frame and $\ast$-$K$-operator frame in Hilbert $C^{\ast}$-modules. Various proofs are given for some results. We will also provide some new results. Moreover, non-trivial examples are presented.

‎In the present work‎, ‎Banach and Kannan integral type contractions in metric spaces endowed with a graph are considered and‎ ‎the existence and uniqueness of best proximity points for mappings satisfying in these contractions are proved‎.

‎The distance eigenvalues of a connected graph $G$ are the eigenvalues of its distance matrix $D(G)$‎. ‎A graph is called distance integral if all of its distance eigenvalues are integers‎. ‎In this paper‎, ‎we introduce some new classes of distance integral graphs‎. ‎In particular‎, ‎we show that if $n,k$ are integers such that $n \geq 3k >0$‎, ‎then the bipartite Kneser graph $H(n,k)$ is distance integral‎. ‎Moreover‎, ‎we determine the distance spectrum of $H(n,k)$‎. ‎Also‎, ‎we show that every distance regular integral graph is distance integral‎.

In this paper, for extending the concepts of $p$-frame and controlled frame for Hilbert spaces, we will introduce the concept of controlled $pg$-frames in Hilbert spaces. Then, we present characterizations of controlled $pg$-frames and some results of frames in the view of controlled $pg$-frames.

In this paper, we present some fixed point results for cyclic weak $\phi$-contractions in $\omega$-complete modular metric spaces and $\omega$-compact modular metric spaces, respectively. Some results for contractions that have zero cyclic properties are also provided.

We investigate the mechanisms of dynamic technique to rational type contraction in the context of partially ordered metric spaces and obtain coupled fixed points in this article. Our derived results extend and generalize some prominent outcomes in the literature. At last, we have produced an example and an application for a system of integral that preserve the main results.

‎In this work‎, ‎we first introduce the concept of weighted sequence space $m_\omega(\Delta_{\mathfrak{v}}^{\varsigma}‎, ‎\psi,q)$‎. ‎Then‎, ‎we construct a Hausdorff measure of noncompactness on this sequence space‎. ‎Furthermore‎, ‎by employing this measure of noncompactness we discuss the solvability of an infinite system of nonlinear third-order differential equations with initial conditions in the weighted sequence space $m_\omega(\Delta_{\mathfrak{v}}^{\varsigma}‎, ‎\psi,q)$‎. ‎Eventually‎, ‎we demonstrate an example to show the usefulness of the obtained result‎.