Necessary and sufficient conditions for the existence of quasi-symmetric positive solutions of singular boundary value problem
The paper is concerned with the problem \[ \begin{cases} (\phi(u''))''(t)=f(t,u(t),u'(t),u''(t)), &t\in(0,1)\\ u^{(2i)}(0)=0,\;u^{(2i)}(\eta)=u^{(2i)}(1), &i=0,1, \end{cases} \] where \(\eta\in(0,1)\) and \(\phi: \mathbb{R}\longrightarrow\mathbb{R}\) is an odd, \(C^1\) increasing homeomorphism, and bounded by separate-variable functions. The nonlinearity \(f:(0,1)\times[0,+\infty)\times\mathbb{R}\times(-\infty,0]\longrightarrow[0,+\infty)\) is continuous and satisfies some growth conditions in terms of the last three variables. The main result of the paper is to prove that quasi-symmetric positive solutions, i.e. such that \(u(t)=u(1+\eta-t)\) for \(t\in[\eta,1]\) exist if and only if \[ 0<\int_0^1F(s)ds<\infty, \] where \(F(t)=f(t,\omega(t),1-2t/(1+\eta),-\phi^{-1}(\omega(t)))\) and \(\omega(t)=\min\{t,1+\eta-t\}.\) The proof employs the upper and lower solution method together with the Schauder fixed point theorem. To this end, the definition of upper and lower quasi-symmetric solutions is given.
scientific article; zbMATH DE number 5563067