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Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations. (English) Zbl 1115.34026
Let \(G\subset \mathbb R^n\) be compact. The authors establish existence results of nonzero solutions of integral equations of the form \[ u(t)=\gamma(t)\alpha[u]+\int_G k(t,s)f(s, u(s))\,ds, \] where \(\alpha[u]\) is a positive functional and \(f\) is positive, while \(k\) and \(\gamma\) may change sign. Applying abstract results for integral equations, they investigated the existence of multiple nonzero solutions of the equation \[ -u''=f(t,u) \] subject to one of the following sets of nonlocal boundary conditions under suitable conditions:
(1) \(u'(0)+u(0)=0, \beta u'(1)+u(\eta)=0\);
(2) \(u(0)=\alpha [u], \;\;\beta u'(1)+u(\eta)=0\), where \(\eta\in (0,1)\). They also show that solutions of the BVPs lose positivity as a parameter decreases.
Reviewer: Ruyun Ma (Lanzhou)

MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47H10 Fixed-point theorems
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiń≠, Uryson, etc.)
45G10 Other nonlinear integral equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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