Guo, Dajun Integro-differential equations on unbounded domains in Banach spaces. (English) Zbl 0958.45014 Chin. Ann. Math., Ser. B 20, No. 4, 435-446 (1999). The author studies the existence of minimal and maximal solutions of the integro-differential equation \[ u^{(n)}(t) = f(t,u(t),u'(t),\ldots,u^{(n-1)}(t),\int_0^t k(t,s)u(s) ds), \] in an ordered Banach space when \(t\geq 0\) and \(u(0),\ldots u^{(n-1)}(0)\) are given. The kernel \(k\) is assumed to be nonnegative and continuous, and the function \(f\) is supposed to satisfy certain monotonicity conditions. The proofs use comparison principles and a monotone iterative technique. Reviewer: Gustaf Gripenberg (Hut) Cited in 2 Documents MSC: 45N05 Abstract integral equations, integral equations in abstract spaces 45G10 Other nonlinear integral equations 45J05 Integro-ordinary differential equations 45L05 Theoretical approximation of solutions to integral equations Keywords:integro-differential equation; minimal and maximal solutions; ordered Banach space; comparison principles; monotone iterative technique PDFBibTeX XMLCite \textit{D. Guo}, Chin. Ann. Math., Ser. B 20, No. 4, 435--446 (1999; Zbl 0958.45014) Full Text: DOI