On a nonlocal boundary value problem at resonance (Q5945759)

Let \(E\) be the interval \([0,1]\) of \(\mathbb{R}\) and let \(N:C^1(E,\mathbb{R}) \rightarrow L^1(E,\mathbb{R})\) be a continuous operator. The authors study the existence of solutions to the initial value problem NEWLINE\[NEWLINELx(t)=x''(t)=Nx(t), \text{ a.a.t. in \(E\) and } x(0)=0, \tag{1}NEWLINE\]NEWLINE that satisfies the restriction of the form NEWLINE\[NEWLINETx =0, \tag{2}NEWLINE\]NEWLINE with \(T:C^1(E,\mathbb{R})\rightarrow \mathbb{R}\) being continuous and linear. NEWLINENEWLINENEWLINEIf \(\text{Ker}(L)\) intersects \(\text{Ker}(T)\) at the origin, one has the nonresonance case. NEWLINENEWLINENEWLINEIn this paper, the authors are interested in the case of resonance, namely, when \(\text{Ker}(L) \subseteq \text{Ker}(T)\), thus \(L\) being noninvertible in \(\text{Ker}(L)\), and they prove an existence result on the problem at resonance (1), (2), based on the coincidence degree theory of \textit{J. Mawhin} [Furi, Massimo (ed.) et al., Topological methods for ordinary differential equations. Lectures given at the 1st session of the Centro Internzionale Matematico Estimvo (C.I.M.E.) held in Montecatini Terme, Italy, June 24--July 2, 1991. Berlin: Springer-Verlag. Lect. Notes Math., 1537, 74-142 (1993; Zbl 0798.34025)].