In this work, the model equation with space fractional-order (FO) is used to investigate the nonlinear ion acoustic shock wave excitations (NIASWEs) in an unmagnetized collisionless weakly relativistic plasma having inertial relativistic ions fluid with viscous effects, inertial-less non-thermal electrons and inertial-less Boltzmann positrons. To do it, the Korteweg-de Vries Burgers equation (KdVBE) is derived from the considered fluid model equations by implementing the standard reductive perturbation method. Accordingly, such equation is converted to space fractional KdVBE via Agrawal’s variational principle with the help of the beta fractional derivative and its properties. The exact analytical solutions of KdVBE with space FO are determined via the modified Kudryashov method. The influence of space fractional and other related plasma parameters on NIASWEs are investigated. The outcomes would be useful to understand the nature of shocks with the presence of non-local or local space in many astrophysical and space environments (especially in the relativistic wind of pulsar magnetosphere, polar regions of neutron stars, etc.) and further laboratory verification.

In this paper, we generalize the notion of an implicativity discussed in $BCK$ -algebras, and apply it to some groupoids and $BCK$ -algebras. We obtain some relations among those axioms in the theory of groupoids.

In this paper, we introduce and study Golden-like statistical manifolds. We obtain some basic inequalities for curvature invariants of statistical submanifolds in Golden-like statistical manifolds. Also, in support of our definition, we provide a couple of examples.

In this paper, we introduce the notion of a $BV$-algebra, and we show that a $BV$-algebra is logically equivalent to several algebras, i.e., $BM$-algebras, $BT$-algebras, $BO$-algebras and 0-commutative B-algebras. Moreover, we show that a $BV$-algebra with $\left(F\right)$ is logically equivalent to several algebras, and we show some relationships between a $BV$-algebra with $\left(F\right)$ and several related algebras.

In this note we investigate the general solution for the quartic functional equation of the form $ \begin{align*} \left( 3n+4 \right)f\left( \sum\limits_{i = 1}^{n}{{{x}_{i}}} \right)+\sum\limits_{j = 1}^{n}{f\left( -n{{x}_{j}}+\sum\limits_{i = 1,i\ne j}^{n}{{{x}_{i}}} \right)}& = \left( {{n}^{2}}+2n+1 \right)\sum\limits_{i = 1,i\ne j\ne k}^{n}{f\left( {{x}_{i}}+{{x}_{j}}+{{x}_{k}} \right)}\\ &-\frac{1}{2}\left( 3{{n}^{3}}-2{{n}^{2}}-13n-8 \right)\sum\limits_{i = 1,i\ne j}^{n}{f\left( {{x}_{i}}+{{x}_{j}} \right)} \\ & +\frac{1}{2}\left( {{n}^{3}}+2{{n}^{2}}+n \right)\sum\limits_{i = 1,i\ne j}^{n}{f\left( {{x}_{i}}-{{x}_{j}} \right)}\\ &+\frac{1}{2}\left( 3{{n}^{4}}-5{{n}^{3}}-7{{n}^{2}}+13n+12 \right)\sum\limits_{i = 1}^{n}{f\left( {{x}_{i}} \right)} \\ \end{align*} $ $\left(n\in \mathbb{N}, \, \, n > 4 \right)$ and also investigate the Hyers-Ulam stability of the quartic functional equation in random normed spaces using the direct approach and the fixed point approach.