We find the solution of the partial difference equation u(x, t + 1) = l u(x + 1, t)+ r u(x − 1, t) for x ∈ [1,m] subject to the absorbing boundary conditions at x = 0 and x = m+1. Green’s function will be determined using random walk techniques applying the reflective and inclusion-exclusion principles. AMS subject classification: 39A12, 37H10.

The authors consider the diierence equation () m yn ? pny n?k ] + qny (n+m?1) = 0 f(n)g is a sequence of integers with (n) n and limn!1 (n) = 1. They obtain results on the classiication of the set of nonoscillatory solutions of () and use a xed point method to show the existence of solutions having certain types of asymptotic behavior. Examples illustrating the results are included.

We show that the difference equation xn = f3(xn−1) f2(xn−2) f1(xn−3), n ∈ N0, where fi ∈ C[(0,∞),(0,∞)], i ∈ {1,2,3}, is periodic with period 4 if and only if fi(x) = ci/x for some positive constants ci, i ∈ {1,2,3} or if fi(x) = ci/x when i = 2 and fi(x) = cix if i ∈ {1,3}, with c1c2c3 = 1. Also, we prove that the difference equation xn = f4(xn−1) f3(xn−2) f2(xn−3) f1(xn−4), n ∈ N0, where fi ∈...

We investigate the global dynamics of solutions of two distinct competitive rational systems of difference equations in the plane. We show that the basins of attraction of different locally asymptotically stable equilibrium points are separated by the global stable manifolds of either saddle points or of non-hyperbolic equilibrium points. Our results give complete answer to Open Problem 1 posed...

We study differential-difference equation of the form d dx t(n+ 1, x) = f(t(n, x), t(n + 1, x), d dx t(n, x)) with unknown t(n, x) depending on continuous and discrete variables x and n. Equation of such kind is called Darboux integrable, if there exist two functions F and I of a finite number of arguments x, {t(n ± k, x)}k=−∞, { d dxk t(n, x) }∞ k=1 , such that DxF = 0 and DI = I, where Dx is ...

3 Analytic classification of admissible q-difference modules 10 3.1 Generalities on q-difference modules . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Simple objects in the category of admissible q-difference modules . . . . . . . 11 3.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Analytic vs formal classification . . . . . . . . . . . . ...

In this work we give sufficient conditions in order that the difference equation with advanced arguments x(n + 1) = A(n)x(n) + B(n)x(σ1(n)) + f(n, x(n), x(σ2(n))), σi(n) ≥ n + 1 has Φ−bounded solutions in the space `Φ of Φ−bounded sequences, are given.

The concept of derivative coordinate functions proved useful in the formulation of analytic fractal functions to represent smooth symmetric binary fractal trees [1]. In this paper we introduce a new geometry that defines the fractal space around these fractal trees. We present the canonical and degenerate form of this fractal space and extend the fractal geometrical space to R explicitly and R ...

This is a celebratory and pedagogical discussion of Sturm oscillation theory. Included is the discussion of the difference equation case via determinants and a renormalized oscillation theorem of Gesztesy, Teschl, and the author.