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Remarks on the range properties of certain semilinear problems of Landesman-Lazer type. (English) Zbl 0993.34012

The authors consider the Dirichlet problem \[ u''(t)+ u(t)+ g(u'(t))= f(t),\quad u(0)= u(\pi)= 0. \] It is assumed that \(g\) is continuous, has finite limits \(g(+\infty)\), \(g(-\infty)\), and \(g(-\infty)< g(s)< g(+\infty)\) for all \(s\). The case of odd and increasing \(g\) with some asymptotic conditions is dealt with in more detail. The structure of the set of continuous functions \(f\) is studied for which the problem is solvable.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
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