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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),...,u (n-1) (t), 0 t k(t,s)u(s)ds),

in an ordered Banach space when t0 and u(0),...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.

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