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Positive solutions of nonlocal boundary value problems: a unified approach. (English) Zbl 1115.34028
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.
Reviewer: Ruyun Ma (Lanzhou)

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47H11 Degree theory for nonlinear operators
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiń≠, Uryson, etc.)
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