On a nonlocal boundary value problem at resonance. (English) Zbl 1002.34057

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 \[ Lx(t)=x''(t)=Nx(t), \text{ a.a.t. in \(E\) and } x(0)=0, \tag{1} \] that satisfies the restriction of the form \[ Tx =0, \tag{2} \] with \(T:C^1(E,\mathbb{R})\rightarrow \mathbb{R}\) being continuous and linear.
If \(\text{Ker}(L)\) intersects \(\text{Ker}(T)\) at the origin, one has the nonresonance case.
In 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 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)].


34K10 Boundary value problems for functional-differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations


Zbl 0798.34025
Full Text: DOI


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