Singular discrete and continuous mixed boundary value problems

Singular discrete and continuous mixed boundary value problems

For each n ∈ N, n ≥ 2 we prove the existence of a positive solution of the singular discrete problem 1 h2 ∆uk−1 + f(tk, uk) = 0, k = 1, . . . , n− 1, ∆u0 = 0, un = 0, where T ∈ (0,∞), h = T n , tk = hk, f : [0, T ] × (0,∞) is continuous and has a singularity at x = 0. We prove that for n → ∞ the sequence of solutions of the above discrete problems converges to a solution y of the corresponding ...

Singular Discrete Higher Order Boundary Value Problems

We study singular discrete nth order boundary value problems with mixed boundary conditions. We prove the existence of a positive solution by means of the lower and upper solutions method and the Brouwer fixed point theorem in conjunction with perturbation methods to approximate regular problems. AMS subject classification: 39A10, 34B16.

Existence and Uniqueness of Solutions for Singular Higher Order Continuous and Discrete Boundary Value Problems

Positive Solutions for Singular Discrete Boundary Value Problems

where ∆ is the forward difference operator ∆xn = xn+1 − xn, {an} is a positive real sequence for n ≥ 1, g is a positive continuous function on N × (0,u0), 0 < u0 ≤ ∞, and Φp(u) = |u|p−2u with p > 1. The left-hand side in (1.1) is the one-dimensional discrete analogue of the p-Laplacian u→ div |∇u|p−2∇u that appears in searching for radial solutions of nonlinear partial equations modelling vario...

Discrete Boundary-value Problems

We consider multivariate sequences, deened by a partial linear recurrence equation together with appropriate boundary conditions. The domain of deenition of these sequences is the rst orthant of the integer lattice, restricted in some dimensions to an initial segment of the nonnegative integers. By means of the kernel method we obtain an explicit expression for the generating function of the so...