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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
\[ u(t)=\int_0^1 k(t,s)g(s) f(s,u(s))ds, \]
where \(k\) is the corresponding Green’s function which is supposed to have certain positivity properties, and \(f:[0,1]\times [0,\infty)\to\mathbb R\) satisfies \(f(t,u)\geq -A\) 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:
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, Nemytskiĭ, Uryson, etc.)
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