Second order integro-differential equations of Volterra type on unbounded domains in Banach spaces
The author investigates the initial value problem for second-order integro-differential equations of Volterra type \[ u''= f(t,u,u', Tu),\quad t\in J, \] \[ u(0)= u_0,\quad u'(0)= u_1, \] where \(J= [0,+\infty)\), \((u_0, u_1)\in E\times E\), \(E\) being a real Banach space, \(f\in C\) \((J\times E\times E\times E,E)\) and \[ (Tu)(t)= \int^t_0 K(t,s) u(s) ds,\quad t\in J, \] \(K\in C(D, \mathbb{R}_+)\), \(D= \{(t,s)\in J\times J; s\leq t\}\), \(\mathbb{R}_+= \{t\in \mathbb{R}; t\geq 0\}\). For this purpose, the author establishes a comparison result and uses the upper and lower solutions.
second-order integro-differential equations of Volterra type
scientific article; zbMATH DE number 1462157