Nonexistence and existence of multiple positive solutions for superlinear three-point boundary value problems via index theory
The paper is devoted to the three point boundary value problem \[ x''+ f(t,x,x')=0, \quad t\in (0,1), \] \[ x'(0)=0, \qquad x(1)=\alpha x(\eta), \] where \(f\) is nonnegative and continuous, \(\eta\in (0,1)\) and \(0<\alpha\eta<1\). Nonexistence of positive solutions is proved when \(f\) is superlinear in \(x'\) at \(0\). On the other hand, if \(f\) is superlinear in \(x'\) at \(\infty\), then, using topological fixed point index in an appropriate cone, the authors prove the existence of at least two positive solutions.
scientific article; zbMATH DE number 5357860
scientific article