Topological analysis of doubly nonlocal boundary value problems
This is an interesting paper dealing with the perturbed Hammerstein integral equation of the form \[u(t)=\gamma(t)H(\varphi(u))+\lambda\int_0^1\left(A\left(\int_0^1|u(\xi)|^qd\xi\right)\right)^{-1}G(t,s)f(s,u(s))ds,\] with \(t\in(0,1)\), where \(q\geq1\) and the functions \(A:\mathbb{R}^+\to\mathbb{R},\) \(f:[0,1\times\to\mathbb{R}^+,\) \(G:[0,1]\times[0,1]\to\mathbb{R}^+,\) \(H:\mathbb{R}^+\to\mathbb{R}^+\) \(\gamma:[0,1]\to[0,1]\) are all continuous. The function \(\varphi\) has the linear integral form \[\int_0^1u(s)d\alpha(s),\] where \(\alpha\) is a monotone increasing function of bounded variation on the interval \([0,1].\) This equation unifies the integral forms produced by various nonlocal boundary value problems of second order, such as \[-M\left(\int_0^1(u(s))^q\right)u''(t)=\lambda f(t,u(t)),\] and \[-M\left(\int_0^1(u'(s))^q\right)u''(t)=\lambda f(t,u(t)),\] with \(t\in(0,1).\) In order to apply a standard fixed point method lying on the index on the boundary of some suitable sets, the author considers a specific cone that is the intersection of two obvious cones, to be used separately to discuss the two specific problems above. An application of these results is given explicitly.
scientific article; zbMATH DE number 7356819
scientific article