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Bounded and zero-convergent solutions of a class of Stieltjes integro- differential equations. (English) Zbl 0739.45005

Consider the equation (1) \(G(p(t)x'(t))=C+\int^ t_ a f(x(s))d\sigma(s)\), \(t\geq a\), where \(p,\sigma: [a,\infty)\to R\) are locally of bounded variation with \(p>0\), \(p^{-1}\in L^ 1_{loc}\) and \(\sigma\) nondecreasing. Let \(f,G\in C(R;R)\) satisfy \(xf(x)>0\), \(xG(x)>0\), \(x\neq 0\), with \(| G(x)|\to\infty\) as \(| x|\to\infty\), \(f\) nondecreasing and \(G\) increasing. Under these conditions one has that the set \(X\) of nonconstant solutions of (1) can be written as \(X=A\cup B\) where \(A=\{\exists t_ x\) such that \(x(t)x'(t)>0\) for \(t>t_ x\}\), \(B=\{\exists t_ x\) such that \(x(t)x'(t)\leq 0\) for \(t>t_ x\}\). The authors give conditions under which solutions in \(A\) are bounded. In addition, the structure of \(B\) is analyzed in detail.
Reviewer: S.O.Londen (Espoo)

MSC:

45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
45D05 Volterra integral equations
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