Transformation techniques and positive solutions on a parameter for three-point impulsive differential equations
The authors consider the three-point boundary value problem for ODE with impulses at fixed times \[ \begin{aligned} & u''(t) + \lambda h(t) f(u(t)) = 0, \quad t \in (0,1), \;t \neq t_k,\\ & u(t_k^+) - u(t_k) = d_k u(t_k), \quad k = 1,\ldots,n,\\ & u'(0) = 0, \quad u(1) = \alpha u(\eta), \end{aligned} \] where \(\lambda > 0\), \(h \in L^p([0,1])\), \(p \in [1,\infty]\), \(f\) is continuous and nonnegative, \(0< t_1 <\ldots < t_n < 1\), \(\alpha >0\), \(d_k > -1\). The authors transform the problem into a non-impulsive problem. Using the eigenvalue and fixed point theory they obtain sufficient conditions for the existence of a positive solution of the BVP according to the values of parameter \(\lambda\).
scientific article; zbMATH DE number 6859501
scientific article