Infante, Gennaro; Webb, J. R. L. Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations. (English) Zbl 1115.34026 Proc. Edinb. Math. Soc., II. Ser. 49, No. 3, 637-656 (2006). 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) Cited in 49 Documents MSC: 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 47H10 Fixed-point theorems 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) 45G10 Other nonlinear integral equations 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:Nonlocal boundary value problems; Hammerstein integral equations; fixed point index; nonzero solutions PDFBibTeX XMLCite \textit{G. Infante} and \textit{J. R. L. Webb}, Proc. Edinb. Math. Soc., II. Ser. 49, No. 3, 637--656 (2006; Zbl 1115.34026) Full Text: DOI