Some problems in integrodifferential equations of Volterra type. (English) Zbl 0598.45015

Lakshmikantham, V.

J. Integral Equations 10, Suppl., 137-146 (1985).

Using the method of upper and lower solutions, monotone iterative techniques and a contraction mapping theorem for operators whose domain and range are different Banach spaces [see S. R. Bernfeld, the author and Y. M. Reddy, Appl. Anal. 6, 271-280 (1977; Zbl 0375.47027)] the author proves two existence theorems for the system (1) \(u'=f(t,u,Tu)\), \(u(0)=u(2\pi)\), where \((Tu)(t)=\int^{t}_{0}K(t,s)u(s)ds\). Furthermore, the extension of the Lyapunov method for the equation (1) is discussed. A general comparison theorem which enables to consider a unified stability theory for this equation is proved.

Reviewer: M.Tvrdý

MSC:

45J05	Integro-ordinary differential equations
45N05	Abstract integral equations, integral equations in abstract spaces

Keywords:

periodic boundary value problem; upper and lower solutions; monotone iterative techniques; Banach spaces; existence; Lyapunov method; comparison; stability

Citations:

Zbl 0375.47027