Multiple positive solutions for singular periodic boundary value problems of impulsive differential equations in Banach spaces
The authors consider the following singular periodic boundary value problem with impulsive effects in a Banach space \[ \begin{aligned} & -u''(t)+M^2 u(t)=f(t, u(t)),\,\,\, t\in [0,2\pi], t\neq t_k,\\ &\Delta u|_{t=t_{k}}=I_{k}(u(t_{k})),\\ &\Delta u'|_{t=t_{k}}=-\bar I_{k}(u(t_{k})),\,\,\, k=1,2,\dots, m,\\ &u(0)=u(2\pi), \,\,\, u'(0)=u'(2\pi), \end{aligned} \] where \(M>0\) is constant, \(f(t,u)\) may be singular at \(t=0,\) and/or \(t=2\pi,\) \(f\in C[(0,2\pi)\times P, P],\) \(I_k, \bar I_k\in C[P,P],\) \(\Delta u|_{t=t_{k}}=u(t_{k}^+)-u(t_k^-),\) \(\Delta u'|_{t=t_{k}}=u'(t_{k}^+)-u'(t_k^-)\) and \(E\) is a Banach space partially ordered by a cone \(P\) of \(E.\) The existence of multiple positive solutions is proved by means of the fixed point index theory of strict set contraction operators. An example illustrating the main results is also presented.
scientific article; zbMATH DE number 5997801