Vol. 49, No. 1, 1973

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Hyperpolynomial approximation of solutions of nonlinear integro-differential equations

Athanassios G. Kartsatos and Edward Barry Saff

Vol. 49 (1973), No. 1, 117–125
Abstract

Consider the integro-differential equation

                   ∫
′            t
U (x ) ≡ x + A(t,x )+ a F(t,s,x(s)) ds = T(t),t ∈ [a,b]
(*)

subject to the initial condition

x(a) = h.
(**)

Then a problem in approximation theory is whether a solution x(t) of ((),(∗∗)) can be approximated, uniformly on [a,b], by a sequence of polynomials Pn, which satisfy () and minimize the expression ||T() U(Pn), where ∥⋅∥ is a certain norm. It is shown here that such a sequence of minimizing polynomials, or, more generally, hyperpolynomials, exists with respect to the Lp-norm (1 < p ) and converges to x(t), uniformly on [a,b], under the mere assumption of existence and uniqueness of x(t).

Mathematical Subject Classification 2000
Primary: 45J05
Milestones
Received: 21 July 1972
Published: 1 November 1973
Authors
Athanassios G. Kartsatos
Edward Barry Saff