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Solvability of a nonlinear two point boundary value problem at resonance. (English) Zbl 0887.34016

The paper deals with the solvability of the boundary value problem at resonance: \[ u''+k^2u +g(x,u)=h \quad \text{on} \quad (0,\pi),\;u(0)= u(\pi)=0. \] The main result is the following theorem:
Let \(g:(0,\pi) \times\mathbb{R}\to\mathbb{R}\) be a Carathéodory function such that \(|g(x,u) |\leq p(x) |u|+b(x)\) a.e. \(x\in (0,\pi)\), \(|u|\geq r_0>0\), where \(p\), \(b\in L^1(0, \pi;\mathbb{R}_+)\) and \(|p|_1< 2k(k+1) \tan \pi/(2(k+1))\). If there exist \(c\), \(d\in L^1(0,\pi)\) such that for a.e. \(x\in(0,\pi)\) and \(u\geq r_0\), \(c(x)\leq g(x,u)\), and for a.e. \(x\in (0,\pi)\) and \(u\leq-r_0\), \(g(x,u)\leq d(x)\), then the problem is solvable for any \(h\in L^1(0,\pi)\) provided that \(\int^\pi_0 hv dx <\int_{v>0} g_+(x)v(x) dx+\int_{v<0} g_-(x) v(x)dx\), where \(g_+ (x) =\liminf_{u\to\infty} g(x,u)\), \(g_-(x)= \limsup_{u\to-\infty} g(x,u)\) and \(v(x)= \alpha\sin kx\) for \(\alpha\in\mathbb{R}\setminus \{0\}\).

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
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