Abstract

In this paper, we establish an existence theorem for a cubic Urysohn-Stieltjes integral equation in the Banach space $C([0,1])$. The equation under consideration is a general form of numerous integral equations encountered in the theory of radioactive transfer, in the kinetic theory of gases and in the theory of neutron transport. Our main tools are the measure of noncompactness (related to monotonicity) and a fixed point theorem due to Darbo.

Keywords: Cubic integral equation; Urysohn; Stieltjes; nondecreasing solutions; measure of noncompactness; Darbo's fixed point theorem.

MSC: 45G10, 47H30

Abstract

A topological space $X$ is called $G_{\delta}$-selectively (resp., $SG_{\delta}$-selectively) separable if for every sequence $\left( D_{n}: n\in\omega\right) $ of dense $G_\delta$ subsets of $X$, one can pick finite subsets $F_{n} \subset D_{n}$ such that $\bigcup_{ n\in\omega} F_{n} $ is dense (resp., dense and $G_\delta $). In this paper we introduce and study these kinds of spaces.

Keywords: Selectively separable; $R$-separable; $G_{\delta}$-selectively separable; $SG_{\delta}$-selectively separable.

MSC: 54C35, 54D65, 54E65

Abstract

The class of weakly stretch Finsler metrics contains the class of stretch metric. Randers metrics are important Finsler metrics which are defined as the sum of a Riemann metric and a 1-form. In this paper, we prove that every Randers metric with closed and conformal one-form is a weakly stretch metric if and only if it is a Berwald metric.

Keywords: Weakly stretch metric; Berwald metric; Randers metric.

MSC: 53B40, 53C60

Abstract

Let $V$ be a valuation ring such that $\mathrm{dim}(V)=0$ and the annihilator of each element in $V$ is finitely generated. In this paper it is proved that if $I$ is a finitely generated ideal in the polynomial ring $V[X]$, then there is a Gröbner basis for $I$. Also, an example of a zero-dimensional non-Noetherian valuation ring $R_M$ is presented, together with an example of finding a Gröbner basis for a certain ideal in a polynomial ring $R_M[X]$.

Keywords: Valuation ring; zero-dimensional ring; Gröbner basis.

MSC: 13P10, 13F30

Abstract

For the hyperspace of non-empty closed sets of a quantale-valued metric space, we define a quantale-valued convergence tower which generalizes the classical Wijsman convergence. We characterize this quantale-valued convergence tower by a quantale-valued neighbourhood tower and show that it is uniformizable. Finally we study compactness and completeness of the quantale-valued Wijsman convergence tower.

Keywords: Quantale-valued metric; probabilistic metric; quantale-valued convergence; hyperspace; Wijsman convergence; Wijsman topology.

MSC: 54A20, 54A40, 54B20, 54E35, 54E70

Abstract

In this article, we establish several infinite families of Ramanujan-type congruences modulo 16, 32 and 64 for $\overline{p}_o(n),$ the number of overpartitions of $n$ in which only odd parts are used.In this paper, we prove a coupled fixed point theorem, using the concept of simulation function, which generalizes the works of Bhaskar et al., Sintunavarat et al. and Zlatanov. The validity of main results is verified through interesting examples. As sequel we also prove that the theorem has a vital application in solving a system of nonlinear impulsive fractional stochastic differential equations.

Keywords: Coupled fixed point; Simulation function; Partially ordered metric space; Fractional stochastic differential equations.

MSC: 54H25, 47H10