## Solvability of a nonlinear two-point boundary value problem at resonance II.(English)Zbl 1054.34031

Summary: The author applies the Leray-Schauder continuation method to obtain existence theorems for solutions to $u''+ u+ g(x,u)= h\quad\text{in }(0,\pi),\;u(0)= u(\pi)= 0,\tag{i}$ in which the nonlinearity $$g$$ grows superlinearly in $$u$$ in one of directions $$u\to\infty$$ and $$u\to-\infty$$, and may grow sublinearly in the other, and to $-u''- u+ g(x,u)= h\quad\text{in }(0,\pi),\;u(0)= u(\pi)= 0,\tag{ii}$ in which the nonlinearity $$g$$ has no growth restriction in $$u$$ as $$| u|\to\infty$$. Let the $$L^1(0,\pi)$$ function $$h$$ satisfy $\int^\pi_0 g^-_\beta(x)\sin x\,dx< \int^\pi_0 h(x)\sin x\,dx= 0<\int^\pi_0 g^+_\alpha(x)\sin x\,dx,$ with $$\alpha,\beta\geq 0$$, $$g^+_\alpha(x)= \liminf_{u\to\infty}\, g(x,u)| u|^\alpha$$, and $$g^-_\beta= \limsup_{u\to-\infty}\, g(x,u)| u|^\beta$$.
For part I see [J. Differ. Equations 140, 1–9 (1997; Zbl 0887.34016)].

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations

Zbl 0887.34016
Full Text:

### References:

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