Initial value problems for the second order mixed monotone type of impulsive differential equations in Banach spaces
The authors consider initial value problems to the impulsive differential equation of the form \[ u''- f(x, u,u')=\theta,\quad x\in J,\quad x\neq x_i,\quad \Delta u|_{x=x_i}= I(u(x_i)),\quad \Delta u'|_{x= x_i}=\widetilde I(u(x_i)), \] \[ i= 1,2,\dots, m\quad u(0)= w_0,\quad u'(0)= w_1, \] where \(f\) is a continuous function of its arguments, \(J= [0,1]\), \(I_i\) and \(\widetilde I_i\) are impulsive sources (continuous functions) propagating the impulsive effect, \(0< x_1< x_2<\cdots< x_m< 1\), are given fixed points. The authors use the coupled fixed-point theorem for mixed monotone condensing operators to obtain an interesting existence and uniqueness theorem.
scientific article; zbMATH DE number 1497259