Solvability of a multi-point boundary value problem of Neumann type

Chaitan P. Gupta, Sergei Trofimchuk
1999 Abstract and Applied Analysis  
Letf:[0,1]×ℝ2→ℝbe a function satisfying Carathéodory's conditions ande(t)∈L1[0,1]. Letξi∈(0,1),ai∈ℝ,i=1,2,...,m−2,0<ξ1<ξ2<⋯<ξm−2<1be given. This paper is concerned with the problem of existence of a solution for them-point boundary value problemx″(t)=f(t,x(t),x′(t))+e(t),0<t<1;x(0)=0,x′(1)=∑i=1m−2ai x′(ξi). This paper gives conditions for the existence of a solution for this boundary value problem using some new Poincaré type a priori estimates. This problem was studied earlier by Gupta,
more » ... er by Gupta, Ntouyas, and Tsamatos (1994) when all of theai∈ℝ,i=1,2,...,m−2, had the same sign. The results of this paper give considerably better existence conditions even in the case when all of theai∈ℝ,i=1,2,...,m−2, have the same sign. Some examples are given to illustrate this point.
doi:10.1155/s1085337599000093 fatcat:dbj57vovmvgalluuhp3yjmdgte