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Zbl 1200.34025
Webb, J.R.L.; Infante, Gennaro
Semi-positone nonlocal boundary value problems of arbitrary order.
(English)
[J] Commun. Pure Appl. Anal. 9, No. 2, 563-581 (2010). ISSN 1534-0392; ISSN 1553-5258/e

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 $f:[0,1]\times [0,\infty)\to\Bbb R$ satisfies $f(t,u)\ge -A$ 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.
[Irena Rach\uu nková (Olomouc)]
MSC 2000:
*34B18 Positive solutions of nonlinear boundary value problems
34B10 Multipoint boundary value problems
47H11 Degree theory
47H30 Particular nonlinear operators

Keywords: nonlocal boundary conditions; semi-positone; positive solution

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