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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


or other nonlocal boundary conditions. They study the problems via new results for a perturbed integral equations of the form

u(t)=γ(t)α[u]+δ(t)β[u]+ G k(t,s)g(s)f(s,u(s))ds,

where α[u],β[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.

34B18Positive solutions of nonlinear boundary value problems for ODE
34B10Nonlocal and multipoint boundary value problems for ODE
47H11Degree theory (nonlinear operators)
47H30Particular nonlinear operators