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Semi-positone nonlocal boundary value problems of arbitrary order. (English) Zbl 1200.34025

The authors develop a new unifying theory, which allows to study the existence of multiple positive solutions for semi-positone problems for differential equations of arbitrary order with a mixture of local and nonlocal boundary conditions. The nonlocal boundary conditions are quite general, they involve positive linear functionals on the space C[0,1], given by Stieltjes integrals. These general BVPs are studied via a Hammerstein integral equation of the form

u(t)= 0 1 k(t,s)g(s)f(s,u(s))ds,

where k is the corresponding Green’s function which is supposed to have certain positivity properties, and f:[0,1]×[0,) satisfies f(t,u)-A for some A>0. The proofs are based on fixed point index results. Examples of a second order and a fourth order problem are presented. Here, the authors determine explicit values of constants that appear in the theory.

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