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Integro-differential equations on unbounded domains in Banach spaces. (English) Zbl 0958.45014
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.

45N05Abstract integral equations, integral equations in abstract spaces
45G10Nonsingular nonlinear integral equations
45J05Integro-ordinary differential equations
45L05Theoretical approximation of solutions of integral equations
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