The notion of $\rm BED$ algebras was introduced by Japanese mathematicians. Afterwards, some authors investigated the BED property of some commutative and semisimple Banach algebras. In this paper, assuming that $X$ is a Hausdorff locally compact space and $\mathcal{A}$ is a commutative and semisimple Banach algebra, we verify the BED property for $C_0(X,\mathcal{A})$. We first prove that $\mathcal{A}$ is a BED algebra, whenever $C_0(X,\mathcal{A})$ is so. Then, we verify the converse of this statement, and prove that $\widehat{\mathcal{A}}\subseteq C^0_{\operatorname{BSE}}(\Phi_{\mathcal{A}})$ implies that $\widehat{C_{0}(X,\mathcal{A})}\subseteq C^0_{\operatorname{BSE}}(\Phi_{C_{0}(X,\mathcal{A})})$. Finally, similar to the BSE case, we prove that if $X$ is discrete, then $C_0(X,\mathcal{A})$ is a BED algebra if and only if $\mathcal{A}$ is so.
In this paper, we consider the Diophantine equation $\prod_{i=1}^{m}(a_{i}^{k}-1)=y^{n}$, where $m\geq3$, $a_{1},a_{2},\dots,a_{m}$ are fixed integers which are pairwise coprime. We have proved that the equation has only finitely many solutions $(k,y,n)$ with $n>1$. We also study some special cases of equation when $m=3$ and $m=4$.
We analyze second-order linear differential equations whose coefficients contain a power of the independent variable and a general power of the natural logarithm of the independent variable together with periodic (or more general) functions. Such equations are a combination of known types of conditionally oscillatory linear equations. Applying the Riccati transformation and a new modification of the adapted Pr\"ufer angle, we prove an oscillation criterion for the studied equations which is new for several simple linear equations. The novelty is illustrated in examples of such equations.
The sequence $\{F_t\}_{t=0}^{\infty}$ refers to the Fibonacci sequence, defined by the recurrence formula $F_t=F_{t-1}+F_{t-2}$, for $t\geq 2$, with the initial conditions $F_0=0$ and $F_1=1$. In this paper, we establish that the only solution to the Diophantine equation $$ F_n=F_l^k (F_l^m - 1) $$ is $(n,l,k,m)=(6,3,3,1)$, where $n,m\geq 1$, $k\geq 2$, and $l\geq 3$.
In this paper, through the combination of the Nash--Moser iteration and the Saloff-Coste-type Sobolev inequality, we establish Cheng--Yau logarithmic gradient estimates for solutions to the following nonlinear elliptic equation $$ \Delta_f u+p\frac{|\nabla u|^2}{u}+\mu(x)u^{r}=0, $$ which is closely related to the gradient generalized $\tau$-quasi-Einstein metrics. Especially, we obtain the Liouville property and Harnack inequality for this equation when $\mu(x)$ is constant.
