Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions. (Q2810145)

In this interesting paper, the authors study the existence and localization of solutions of the boundary value problem (BVP) NEWLINE\[NEWLINEu''(t)=f(t,u(t),u'(t)),\quad u(0)=0,\quad u'(1)=\int_0^1u'(s)\, dg(s),\tag{1}NEWLINE\]NEWLINE where \(f\) is continuous and \(g\) is increasing with \(\int_0^1\, dg(s)=1\). The BVP (1) is resonant in the sense that the corresponding linear homogeneous BVP NEWLINE\[NEWLINEu''(t)=0,\quad u(0)=0,\quad u'(1)=\int_0^1u'(s)\, dg(s),NEWLINE\]NEWLINE has a nontrivial solution if and only if \(\int_0^1\, dg(s)=1\).NEWLINENEWLINE The authors introduce the concept of a couple \((\sigma,\tau)\) of upper and lower slopes for the BVP (1). The authors prove, via Schauder's fixed point theorem, (see Theorem 4.1) that the existence of a couple of upper and lower slopes imply the existence and localization of a solution for the BVP (1). In Section 6 the authors illustrate how to find a couple of upper and lower slopes. In the last Section, the authors discuss multiplicity results for the BVP (1). Some useful examples are given in order to illustrate the theoretical results.