International Journal of Mathematics and Mathematical Sciences 
Volume 21 (1998), Issue 3, Pages 499-505
doi:10.1155/S0161171298000714

Rapid convergence of approximate solutions for first order nonlinear boundary value problems

Alberto Cabada,1 Juan J. Nieto,1 and Seppo Heikkilä2

 1Departamento de Anàlise Matemhtica, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela 15706, Spain
2Department of Mathematical Sciences, University of Oulu, Oulu 57 90570, Finland
Received 20 August 1996

Abstract

In this paper we study the convergence of the approximate solutions for the following first order problem u′(t)=f(t,u(t));t∈[0,T],au(0)−bu(t0)=c,a,b≥0,t0∈(0,T]. Here f:I×ℝ→ℝ is such that ∂kf∂uk exists and is a continuous function for some k≥1. Under some additional conditions on ∂f∂u, we prove that it is possible to construct two sequences of approximate solutions converging to a solution with rate of convergence of order k.