In this article, we apply one fixed point theorem in the setting of b-metric-like spaces to prove the existence of solutions for one type of Caputo fractional differential equation as well as the existence of solutions for one integral equation created in mechanical engineering.

Jaradat has proven some fixed point results using JS-G-contraction on $G-$metric spaces. Choudhury et al. were derived coupled fixed point theorems for the $G-$metric spaces. The purpose of this paper is to prove some coupled fixed point theorems of JS-G-contraction on G-metric spaces. Moreover, some example is presented to illustrate the validity of our results.

In this manuscript, we initiate the concept of rectangular $\alpha$-$G$-admissible mappings with respect to $\beta$ and we consider related type contractions in the setting of $G$-metric spaces. We provide some fixed point results. Also, some examples are given to support the obtained results. У цій статті ми вводимо поняття прямокутних $\alpha$-$G$-допустимих відображень по відношенню до функції $\beta$ і розглядаємо відповідного типу стиски у контексті $G$-метричних просторів. Ми доводимо деякі результати про нерухому точку. Також наведено деякі приклади для ілюстрації отриманих результатів.

In this manuscript, we provide some new results for the existence of fixed points for a certain contractive condition of Geraghty type in the setting of partially ordered $G$-metric space. Also, we provide an example to illustrate the usability of results. Our results generalize or extend many well known results in the literature.

Let d be a squarefree integer. We prove that(i) Pnμ(n)nd(n′) converges to zero, where n′ is the product of prime divisors of nwith ( dn ) = +1. We use the Prime Number Theorem.(ii) Q( dp )=+1(1 −1ps ) is not analytic at s=1, nor is Q( dp )=−1(1 −1ps ) .(iii) The convergence (i) leads to a proof that asymptotically half the squarefree ideals have an even number of prime ideal factors (analogue of Ramanujan’s assertion).

In this paper, we present the notation of G-F-Contractive mappings of Hardy-Rogers type and give some fixed point results of Hardy-Rogers type for self-mappings in complete G-metric spaces.