Abstract
We establish optimal, in a sense, conditions under which, for arbitrary forcing terms from a suitable class, a linear inhomogeneous functional differential equation in a preordered Banach space possesses solutions satisfying a certain growth restriction.
Highlights
When studying mathematical models of various dynamic phenomena, it is often desirable to prove the existence of a solution satisfying the given initial or boundary conditions and to ensure that the solution in question possesses certain qualitative properties
We prove two theorems which provide conditions under which a linear functional differential equation has a unique absolutely continuous solution passing through a given point, having a given growth rate, representable as a uniformly convergent functional series with the coefficients defined recursively, and positive in a certain sense
We are interested in the solutions of the abstract functional differential equation u (t) = ( u)(t) + f (t), t ∈ [a, b], (1.1)
Summary
When studying mathematical models of various dynamic phenomena, it is often desirable to prove the existence of a solution satisfying the given initial or boundary conditions and to ensure that the solution in question possesses certain qualitative properties (e.g., has at least a prescribed number of zeroes on the given interval). We prove two theorems which provide conditions under which a linear functional differential equation has a unique absolutely continuous solution passing through a given point, having a given growth rate, representable as a uniformly convergent functional series with the coefficients defined recursively, and positive in a certain sense. Where φ : [a,b] → R is a given nonnegative continuous function possessing at least one zero on the interval [a,b].
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