## 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)].

### MSC:

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

### Citations:

Zbl 0852.34018; Zbl 0853.34021
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