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Non-resonance of the first two eigenvalues of a quasilinear problem. (Non-résonance entre les deux premières valeurs propres d’un problème quasi-linéaire.) (French) Zbl 0920.34030
If $$\lambda_k(p)$$ is the $$k$$th eigenvalue of the problem $-(\phi_p(u'))'= \lambda\phi_p(u),\quad a<x<b,\quad u(a)= u(b)= 0,$ and sign $$sf(s)\to+\infty$$ when $$| s|\to+\infty$$, $\lambda_1(p)< \limsup_{s\to\pm\infty} {pF(s)\over| s|^p},\;\limsup_{s\to\infty} {f(s)\over\phi_p(s)}\leq \lambda_2(p)\text{ and }\limsup_{s\to-\infty} {f(s)\over\phi_p(s)}< \lambda_2(p),$ then the problem $-(\phi_p(u'))'= f(u)+ h(x),\quad a< x<b,\quad u(a)= u(b)= 0$ has at least one solution for each $$h\in L^1(a,b)$$. Here $$F(s)= \int^s_0 f(t)dt$$, $$f\in C(\mathbb{R},\mathbb{R})$$ and $$\phi_p(s)= | s|^{p-2}s$$, $$p>1$$.
##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations
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