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Singular quadratic functionals of one dependent variable. (English) Zbl 0838.34036
The authors consider the singular quadratic functionals over $$W^{1,2}_{\text{loc}}([a, +\infty[)$$, $I(u)= \alpha u^2(a)+ \liminf_{b\to +\infty} \Biggl[\gamma u^2(b)+ \int^b_\alpha (r(s)u^{\prime 2}(s)- p(s) u^2(s))ds\Biggr]$ subjected to boundary conditions $\lim_{b\to +\infty} D\Biggl({u(a)\atop u(b)}\Biggr)= 0,$ where $$D$$ is a $$2\times 2$$ matrix; $$r^{- 1}$$, $$p\in L_{\text{loc}}([a,+ \infty[)$$, $$r\geq 0$$, $$\alpha, \gamma\in {\mathcal R}$$. In particular they discuss necessary and sufficient conditions for the nonnegativity of $$I$$ either in the case of free end points or under boundary conditions of different types. For recent connected results see the first author and P. Zezza [Comment. Math. Univ. Carolinae 33, 411-425 (1992; Zbl 0779.49026)].

##### MSC:
 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations