Volume 39, Issue 3

In this study, we present the basic concepts of statistical convergence for double sequences on an arbitrary product time scale. Moreover, we investigate the connection between statistical convergence for double sequences and double Cesàro summability on a product time scale.

For N≥4{N\geq 4}, we let Ω be a smooth bounded domain of ℝN{\mathbb{R}^{N}}, Γ a smooth closed submanifold of Ω of dimension k , with 1≤k≤N-2{1\leq k\leq N-2}, and h a continuous function defined on Ω. We denote by ρΓ⁢(⋅):=dist⁡(⋅,Γ){\rho_{\Gamma}(\,\cdot\,):=\operatorname{dist}(\,\cdot\,,\Gamma)} the distance function to Γ. For σ∈(0,2){\sigma\in(0,2)}, we study the existence of positive solutions u∈H01⁢(Ω){u\in H^{1}_{0}(\Omega)} to the nonlinear equation -Δ⁢u+h⁢u=ρΓ-σ⁢u2*⁢(σ)-1 in ⁢Ω,-\Delta u+hu=\rho_{\Gamma}^{-\sigma}u^{2^{*}(\sigma)-1}\quad\text{in }\Omega, where 2*⁢(σ):=2⁢(N-σ)N-2{2^{*}(\sigma):=\frac{2(N-\sigma)}{N-2}} is the critical Hardy–Sobolev exponent. In particular, we prove the existence of solution under the influence of the local geometry of Γ and the potential h .

In this work, necessary and sufficient conditions are obtained such that every solution of nonlinear neutral first-order differential equations with several delays of the form (x⁢(t)+r⁢(t)⁢x⁢(t-τ))′+∑i=1mϕi⁢(t)⁢H⁢(x⁢(t-σi))=f⁢(t)\bigl{(}x(t)+r(t)x(t-\tau)\bigr{)}^{\prime}+\sum_{i=1}^{m}\phi_{i}(t)H\bigl{(}% x(t-\sigma_{i})\bigr{)}=f(t) is oscillatory or tends to zero as t→∞.{t\rightarrow\infty.} This problem is considered in various ranges of the neutral coefficient r . Finally, some illustrating examples are presented to show that feasibility and effectiveness of main results.

In this paper, we prove the general solution and generalized Hyers–Ulam stability of n -dimensional functional equations of the form ∑i=1i≠j≠knf⁢(-xi-xj-xk+∑l=1l≠i≠j≠knxl)=(n3-9⁢n2+20⁢n-126)⁢∑i=1nf⁢(xi),\sum_{\begin{subarray}{c}i=1\\ i\neq j\neq k\end{subarray}}^{n}f\biggl{(}-x_{i}-x_{j}-x_{k}+\sum_{% \begin{subarray}{c}l=1\\ l\neq i\neq j\neq k\end{subarray}}^{n}x_{l}\biggr{)}=\biggl{(}\frac{n^{3}-9n^{% 2}+20n-12}{6}\biggr{)}\sum_{i=1}^{n}f(x_{i}), where n is a fixed positive integer with ℕ-{0,1,2,3,4}\mathbb{N}-\{0,1,2,3,4\}, in a Banach space via direct and fixed point methods.