The aim of this paper is investigating the existence one or more weak solutions coupled quasilinear elliptic system gradient type \[ (P)\qquad \left\{ \begin{array}{ll} - {\rm div} (A(x, u)\vert\nabla u\vert^{p_1 -2} \nabla u) + \frac{1}{p_1}A_u (x, u\vert^{p_1} = G_u(x, u, v) &\hbox{ in $\Omega$,}\\[5pt] (B(x, v)\vert\nabla v\vert^{p_2 +\frac{1}{p_2}B_v(x, v\vert^{p_2} G_v\left(x, v\right) u v...

The purpose of this paper is to investigate the following invariance equation involving two 2-variable generalized Bajraktarević means, i.e., we aim solve functional $$f^{-1}\Bigl(\frac{p_1(x)f(x) +p_2(y)f(y)}{p_1(x)+p_2(y)}\Bigr)+g^{-1}\Bigl(\frac{q_1(x)g(x) +q_2(y)g(y)}{q_1(x)+q_2(y)}\Bigr)=x + y \ (x,y\in I),$$ where I a nonempty open real interval and $$f,g \colon \to\mathbb{R}$$ are contin...

Graph theory is a branch of algebra that growing rapidly both in concept and application studies. This graph can be used chemistry, transportation, cryptographic problems, coding theory, design communication network, etc. There currently bridge between graphs algebra, especially an algebraic structures, namely algebra. One researchs on formed by prime ring elements or called over R. The commuta...

Let $t>0$ be a real number and $G$ graph. We say is $t$-tough if for every cutset $S$ of $G$, the ratio $|S|$ to components $G-S$ at least $t$. Determining toughness an NP-hard problem arbitrary graphs. The Toughness Conjecture Chv\'atal, stating that there exists constant $t_0$ such $t_0$-tough graph with three vertices hamiltonian, still open in general. A called $(P_2\cup P_3)$-free it do...

It is shown that every measurable partition ${A_1,..., A_k}$ of $\mathbb{R}^3$ satisfies $$\sum_{i=1}^k||\int_{A_i} xe^{-\frac12||x||_2^2}dx||_2^2\le 9\pi^2.\qquad(*)$$ Let ${P_1,P_2,P_3}$ be the partition of $\mathbb{R}^2$ into $120^\circ$ sectors centered at the origin. The bound is sharp, with equality holding if $A_i=P_i\times \mathbb{R}$ for $i\in {1,2,3}$ and $A_i=\emptyset$ for $i\in \{4...

We study bilinear rough singular integral operators $$\mathcal {L}_{\Omega }$$ associated with a function $$\Omega $$ on the sphere $$\mathbb {S}^{2n-1}$$ . In recent work of Grafakos et al. (Math Ann 376:431–455, 2020), they showed that is bounded from $$L^2\times L^2$$ to $$L^1$$ , provided \in L^q(\mathbb {S}^{2n-1})$$ for $$4/3<q\le \infty mean value zero. this paper, we provide generalizat...