In this study, we investigate the existence and multiplicity of solutions for a fractional discrete $p-$ Laplacian equation on $ \mathbb{Z} $, via the mountain pass lemma and Ekeland’s variational principle. Under suitable hypotheses on functions $V$ and $f$, we prove that this equation admits at least two nonnegative and nontrivial homoclinic solutions when the real parameter $\lambda >0$ is large enough.

This paper is devoted to the stability of a viscoelastic problem with Balakrishn\^{a}n-Taylor damping and time delay involving variable-exponent nonlinearity. Under some assumptions on the relaxation function, we establish the general decay estimate for the energy via suitable Lyapunov functionals. The problem considered is novel and meaningful because the presence of the flutter panel equation and the spillover problem with memory and variable exponents time delay control. Our result generalizes and improves previous conclusion in the literature.

This paper is concerned with the existence and multiplicity of solutions for the following variable $s(\cdot)$-order fractional $p(\cdot)$-Kirchhoff type problem \begin{equation*} \left\{\begin{array}{ll} M\left(\displaystyle\iint_{\mathbb R^{2N}}\frac{1}{p(x,y)}\displaystyle{\frac{|v(x)-v(y)|^{p(x,y)}}{|x-y|^{N+p(x,y)s(x,y)}}}dxdy\right)(-\Delta)^{s(\cdot)}_{p(\cdot)}v(x)+|v(x)|^{\overline{p}(x)-2}v(x) =\mu g(x,v)\ \ {\rm in}~\mathbb{R}^{N},\\ v\in W^{s(\cdot),p(\cdot)}(\mathbb{R}^{N}), \end{array}\right. \end{equation*} where $N>p(x,y)s(x,y)$ for any $(x,y)\in\mathbb{R}^{N}\times\mathbb{R}^{N}$, $(-\Delta)^{s(\cdot)}_{p(\cdot)}$ is a variable $s(\cdot)$-order $p(\cdot)$-fractional Laplace operator with $s(\cdot):\mathbb R^{2N}\to(0,1)$ and $p(\cdot):\mathbb R^{2N}\to(1,\infty)$, $\overline{p}(x)=p(x,x)$ for $x\in\mathbb{R}^{N}$, and $M$ is a continuous Kirchhoff-type function, $g(x,v)$ is a Carath\’{e}odory function, $\mu>0$ is a parameter. We obtain that there are at least two distinct solutions for the above problem by applying the generalized abstract critical point theorem. Under the weaker conditions, we also show the existence of one solution and infinitely many solutions by using the mountain pass lemma and fountain theorem, respectively. In particular, the new compact embedding result of the space $ W^{s(\cdot),p(\cdot)}(\mathbb{R}^{N})$ into $L^{q(\cdot)}_{a(x)}(\mathbb{R}^{N})$ will be used to overcome the lack of compactness in $\mathbb{R}^N$. The main feature and difficulty of this paper is the presence of a double non-local term involving two variable parameters.