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Zbl 0622.34015
Gupta, Chaitan P.
Two-point boundary value problems involving reflection of the argument.
(English)
[J] Int. J. Math. Math. Sci. 10, 361-371 (1987). ISSN 0161-1712; ISSN 1687-0425/e

The author studies the following problems involving reflection of the argument $$(1)\quad \ddot x(t)+f(x(t))\dot x(t)+g(t,x(t),x(- t))=e(t),\quad x(-1)=x(1)=0$$ $$(2)\quad \ddot x(t)+g(t,x(t),x(- t))=e(t),\quad \dot x(-1)=\dot x(1)=0$$ where f is continuous, $e\in L\sp 1(-1,1)$ and g satisfies (i) Carathéodory's conditions $$(ii)\quad \lim\sb{\vert x\vert \to +\infty}\sup \vert \frac{g(t,x,y)}{x}\vert \le \Gamma (t)=\Gamma\sb 0(t)+\Gamma\sb 1(t)+\Gamma\sb{\infty}(t)\quad with\quad \Gamma\sb 0(t)\le \pi\sp 2/4.$$ The asymptotic behavior of g(t,x,y)/x is related to the first eigenvalue $\pi\sp 2/4$ of the linear problem corresponding to (1) and to the first two eigenvalues 0 and $\pi\sp 2/4$ of the linear problem corresponding to (2).
[A.Boucherif]
MSC 2000:
*34B10 Multipoint boundary value problems

Keywords: second order differential equation; Wirtinger's inequality; reflection of the argument; Carathéodory's conditions; eigenvalue

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