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.


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
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