We study a Hamiltonian system that has a superquadratic potential and is asymptotic to an autonomous system. In particular, we show the existence of a nontrivial solution homoclinic to zero. Many results of this type rely on a convexity condition on the nonlinearity, which makes the problem resemble in some sense the special case of homogeneous (power) nonlinearity. This paper replaces that con...

This article initiates the study of nonlinear elliptic partial difference equations (PdE) on graphs. We seek solutions u : V → R to the semilinear elliptic difference equation −Lu + f(u) = 0 on a graph G = (V,E), where L is the (negative) Laplacian on the graph G. We extend techniques used to prove existence theorems and derive numerical algorithms for the partial differential equation (PDE) ∆u...

ed sign out of two iconically derived signs. This is furthered by Japanese kokuji, invented to represent Japanese words using compound Chinese characters. Other SS combinations are 働 meaning EFFORT, made from PERSON, 人 , and MOVE, 動, and the visually iconic 峠 meaning MOUNTAIN PASS, made by combining mountain, 山, with UP, 上, and DOWN, 下, which create a visual “pass” between them (all examples fr...

and Applied Analysis 3 Definition 2.2. Let f be any real-valued function and ν ∈ 0, 1 . The left discrete fractional difference and the right discrete fractional difference operators are, respectively, defined as tΔaf t ΔtΔ − 1−ν a f t 1 Γ 1 − ν Δ t ν−1 ∑ s a t − s − 1 −ν f s , t ≡ a − ν 1 mod1 , bΔt f t −ΔbΔ 1−ν t f t 1 Γ 1 − ν −Δ b ∑ s t 1−ν s − t − 1 −ν f s , t ≡ b ν − 1 mod1 . 2.2 Definitio...

<p style='text-indent:20px;'>This paper is concerned with the existence of nontrivial solutions for quasilinear elliptic equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \notag -\Delta u-\Delta(|u|^{2\alpha})|u|^{2\alpha-2}u+V(x)u = |u|^{q-2}u,\quad x\in \mathbb{R}^{N}, \end{equation} $\en...

Imagine v valleys and all cities placed in this valleys. Cost of traveling from town to town in valley is very small (negligible), let us say equal to 1. When salesmen wants to go to other valley he has to pass mountain chain – this is of a huge cost, let us say equal to 1000. Salesman lives in one of towns in valley A. Optimal solution must then contain 4 mountain chains crossing (from A to B,...