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Zbl 1115.34028
Webb, J.R.L.; Infante, Gennaro
Positive solutions of nonlocal boundary value problems: a unified approach.
(English)
[J] J. Lond. Math. Soc., II. Ser. 74, No. 3, 673-693 (2006). ISSN 0024-6107; ISSN 1469-7750/e

The authors established existence of multiple solutions of nonlinear differential equations of the form $$-u''=g(t)f(t,u),$$ where $g$ and $f$ are nonnegative functions, subject to $$u(0)=\alpha [u], \qquad u(1)=\beta [u]$$ or other nonlocal boundary conditions. They study the problems via new results for a perturbed integral equations of the form $$u(t)=\gamma(t)\alpha[u]+\delta(t)\beta[u]+\int_G k(t,s)g(s)f(s, u(s))\,ds,$$ where $\alpha[u], \beta[u]$ are linear functionals given by Stieltjes integrals but not assumed to be positive for all positive $u$. $m$-point boundary value problems are special cases and they obtain sharp conditions on the coefficients, which allows some of them to have opposite signs.
[Ruyun Ma (Lanzhou)]
MSC 2000:
*34B18 Positive solutions of nonlinear boundary value problems
34B10 Multipoint boundary value problems
47H11 Degree theory
47H30 Particular nonlinear operators

Keywords: integral equations; fixed point index

Cited in: Zbl 1181.34025

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