Nontrivial solutions of boundary value problems for second-order functional differential equations (Q292633)

Let \(r>0\) and \(\psi:[-r,1[\to\mathbb{R}\) be a continuous function with \(\psi(s)=0\) for \(s\in [0,1].\) Also let \(K_0\) be the set of all functions \(u\in C([-r,1],\mathbb{R})\) with \(u=0\) on \([-r,0]\) and such that \(\min_{t\in J}u(t)\geq c\|u\|_{[-r,1]}\) and \(\alpha[u]\geq0\), where \(c>0\), \(J\) is a subinterval of \([0,1]\) and \(\alpha\) is a bounded linear functional defined on \(C([0,1],\mathbb{R}).\) This interesting paper deals with the existence of functions \(u\in \psi+K_0\) satisfying a delay integral equation of the form \[ u(t)=\psi(t)+\int_0^1k(t,s)g(s)F(s,u_s)ds+\gamma(t)\alpha[u], \quad t\in [-r,1]. \] In Section 4, under some additional conditions (as, e.g, \(\psi\) is nonnegative), it is shown that the problem has positive solutions. Due to the presence of the perturbation \(\psi\), the known methods based on the existence of fixed points in cones do not work. Here, the authors present an existence theorem of fixed points in affine cones and then they use facts from the degree theory. In Section 5 it is shown that the second order delay differential equation \[ -u''(t)=g(t)F(t,u_t),\quad t\in[0,1] \] admits a solution \(u\) satisfying the initial condition \(u(t)=\psi(t),\) \( t\in [-r,0]\), as well as the boundary conditions \[ u(0)=0, \text{ and } \beta u'(1)+u(\eta)=\alpha[u], \] where \(\beta>0\) and \(\eta\in(0,1).\) Ending the paper the authors apply the results to the autonomous equation \[ -u''(t)=\lambda |u(t|^{p-1}|u(t-r)|,\quad t\in [0,1], \] where \(p\geq 1,\) associated with the initial condition \(u(s)=\psi(s)\), where \(\|u\|_{[-r,0]}<1,\) and the boundary conditions \(u(0)=0,\) \(\frac{1}{4}u'(1)+u(\frac{1}{4})=0.\) The case \(\psi\geq 0\) is discussed separately.