Let $X,Y$ be complex vector spaces. Recently, Park and Th.M. Rassias showed that if a mapping $f:X\to Y$ satisfies $$\begin{array}{c}\hfill f(x+iy)+f(x-iy)=2f\left(x\right)-2f\left(y\right)\end{array}$$ for all $x$, $y\in X$, then the mapping $f:X\to Y$ satisfies $f(x+y)+f(x-y)=2f\left(x\right)+2f\left(y\right)$ for all $x$, $y\in X$. Furthermore, they proved the generalized Hyers-Ulam stability of the functional equation () in complex Banach spaces. In this paper, we will adopt the idea of Park and Th. M. Rassias to prove the stability of a quadratic functional equation with complex involution via fixed point method.

In the paper, conditions are obtained, in terms of coefficient functions, which are necessary as well as sufficient for non-oscillation/oscillation of all solutions of self-adjoint linear homogeneous equations of the form $$\Delta \left(p_{n-1}\Delta y_{n-1}\right)+q{y}_n=0,n\ge 1,$$ where $q$ is a constant. Sufficient conditions, in terms of coefficient functions, are obtained for non-oscillation of all solutions of nonlinear non-homogeneous equations of the type $$\Delta \left(p_{n-1}\Delta y_{n-1}\right)+q_{n}g\left(y_n\right)=f_{n-1},n\ge 1,$$ where, unlike earlier works, $f_n\ge 0$ or $\le 0$ (but $\neg \equiv 0)$ for large $n$. Further, these results are used to obtain...

This paper aims at discussing asymptotic behaviour of nonoscillatory solutions of the forced fractional difference equations of the form $$\begin{array}{cc}\hfill {\Delta }^{\gamma }u\left(\kappa \right)& +\Theta [\kappa +\gamma ,w(\kappa +\gamma \left)\right]\hfill \\ \hfill =& \Phi (\kappa +\gamma )+{\rm Y}(\kappa +\gamma )w^{\nu }(\kappa +\gamma )+\Psi [\kappa +\gamma ,w(\kappa +\gamma )],\kappa \in {\mathbb{N}}_{1-\gamma },\hfill \\ \hfill u_0=& c_0,\hfill \end{array}$$ where ${\mathbb{N}}_{1-\gamma }=\{1-\gamma ,2-\gamma ,3-\gamma ,\cdots \}$, $0<\gamma \le 1$, ${\Delta }^{\gamma }$ is a Caputo-like fractional difference operator. Three cases are investigated by using some salient features of discrete fractional calculus and mathematical inequalities. Examples are presented to illustrate the validity of the theoretical results.

We consider the implicit discretization of Nagumo equation on finite lattices and show that its variational formulation corresponds in various parameter settings to convex, mountain-pass or saddle-point geometries. Consequently, we are able to derive conditions under which the implicit discretization yields multiple solutions. Interestingly, for certain parameters we show nonuniqueness for arbitrarily small discretization steps. Finally, we provide a simple example showing that the nonuniqueness...

The paper discusses basics of calculus of backward fractional differences and sums. We state their definitions, basic properties and consider a special two-term linear fractional difference equation. We construct a family of functions to obtain its solution.