In this paper, we find the solution of the following quadratic functional equation $n{\sum }_{1\le i<j\le n}Q\left(x_i-x_j\right)={\sum }_{i=1}^{n}Q\left({\sum }_{j\ne i}{x}_j-(n-1)x_i\right)$, which is derived from the gravity of the n distinct vectors $x_1,\cdots ,x_n$ in an inner product space, and prove that the stability results of the $\mathcal{A}$-quadratic mappings in $\mu$-complete convex fuzzy modular ∗-algebras without using lower semicontinuity and $\beta$-homogeneous property. Full article

We prove that the Lebesgue space of variable exponent $L^{p(·)}\left(\mathsf{\Omega }\right)$ is modularly uniformly convex in every direction provided the exponent p is finite a.e. and different from 1 a.e. The notion of uniform convexity in every direction was first introduced by Garkavi for the case of a norm. The contribution made in this work lies in the discovery of a modular, uniform-convexity-like structure of $L^{p(·)}\left(\mathsf{\Omega }\right)$ , which holds even when the behavior of the exponent $p(·)$ precludes uniform convexity of the Luxembourg norm. Specifically, we show that the modular $\rho \left(u\right)=\underset{\mathsf{\Omega }}{\int }\left|u\left(x\right)\right|dx$ possesses a uniform-convexity-like structure even if the variable exponent is not bounded away from 1 or ∞. Our result is new and we present an application to fixed point theory. Full article

In this paper, an extension of Darbo’s fixed point theorem via $\theta$ -F-contractions in a Banach space has been presented. Measure of noncompactness approach is the main tool in the presentation of our proofs. As an application, we study the existence of solutions for a system of integral equations. Finally, we present a concrete example to support the effectiveness of our results. Full article

We show that the limits for dynamical systems of self-similar groups are eventually conjugate if, and only if, there is an isomorphism between their Deaconu groupoid preserving cocycles. For limit solenoids of self-similar groups, we show that the conjugacy of limit solenoids is equivalent to existence of isomorphism between the Deaconu groupoids of limit solenoid preserving cocycles. Full article

Kannan maps have inspired a branch of metric fixed point theory devoted to the extension of the classical Banach contraction principle. The study of these maps in modular vector spaces was attempted timidly and was not successful. In this work, we look at this problem in the variable exponent sequence spaces ${\ell }_{p(·)}$ . We prove the modular version of most of the known facts about these maps in metric and Banach spaces. In particular, our results for Kannan nonexpansive maps in the modular sense were never attempted before. Full article