Non-Monotone Convergence Schemes

Abstract

We consider the second order BVP x″ = f (t, x, x′), x′(a) = A, x′(b) = B provided that there exist α and β (lower and upper functions) such that: β′ (α) < A < α′(a) and β′(b) < B < α′ (b). We consider monotone and non-monotone approximations of solutions to the Neumann problem. The results and examples are provided.

On Types of Solutions of the Second Order Nonlinear Boundary Value Problems

“…Applying the Arzela -Ascoli criterium (([5, Theorem 8.26, page 347]) let be a compact subset of R and let { } be a sequence of -dimensional vector functions that is uniformly bounded and equicontinuous on . Then there is a subsequence { } that converges uniformly on ) we can show (Theorem 2 in ([17])) that { } contains a subsequence which converges to * .The same type arguments show that { } also exists and subsequence of { } converges to * . Notice that < * by construction and, therefore, * ≤ * .…”

Types and Multiplicity of Solutions to Sturm–liouville Boundary Value Problem

“…Our study continues a series of papers devoted to two-point boundary value problems for the second order nonlinear differential equations [2,3,4]. This research is motivated by the papers of L. Jackson [6], H. Knobloch [7] and L. Erbe [5], who studied BVPs for equation (1.1) provided that there exist the so called lower and upper functions.…”

“…In articles [2,3,4] we investigated similar problems for the equation (1.1) in the cases of the Dirichlet and Neumann boundary conditions. This article is a continuation of research.…”