Remarks on semilinear problems with nonlinearities depending on the derivative. (English) Zbl 1033.34021

Let \(g\in C([a, b]\times \mathbb{R}^n;\mathbb{R}^n)\), \(\overline f\in\mathbb{R}^n\), and \(\widetilde f\in C([a, b];\mathbb{R}^n)\) with \(\int^b_a\widetilde f(t)\,dt= 0\). The authors consider the second-order differential equation \[ u''(t)+ g(t, u'(t))=\overline f+\widetilde f(t),\quad t\in (a,b), \] with either Neumann or periodic type boundary conditions at \(a\), \(b\), and investigate the range of each respective problem for a fixed \(\widetilde f\). Of special interest is the asymptotic behavior of the range when \(n= 1\). The authors extend earlier results by A. Canada and P. Drábek [SIAM J. Math. Anal. 27, 543–557 (1996; Zbl 0852.34018)] and J. Mawhin [Acta Math. Inform. Univ. Ostrav. 2, 61–69 (1994; Zbl 0853.34021)].


34B15 Nonlinear boundary value problems for ordinary differential equations
34L30 Nonlinear ordinary differential operators
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