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

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.TvrdĂ˝

MSC:

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

Citations:

Zbl 0375.47027
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