Solutions to nonlocal Neumann boundary value problems

Abstract

In this paper we study the nonlocal Neumann boundary value problem of the following form $$ u'' =f(t,u,u'),\quad u'(0)=0, \quad u'(1)=\int_{0 }^{1}u'(s)dg(s), $$ where $f:[0,1]\times\mathbb R^n\times\mathbb R^n\to\mathbb R^n$ and $g=\mbox{diag}(g_1,\ldots,g_n)$ with $g_i:[0,1]\to\mathbb R$, $i=1,\ldots,n$. The case when the function $f$ does not depend on $u'$ is also considered. The existence of solutions is obtained by means of the generalized Miranda theorem. The main results can be applied to many problems of this type depending on which conditions will be imposed upon the function $f$.

Keywords

• nonlocal boundary value problem
• boundary value problem at resonance
• neumann problem
• the miranda theorem
• $r_{\delta}$-sets