## 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

### Keywords:

boundary value problem; resonance; Carathéodory function
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