Vol. 41, No. 2, 1972

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Continuous dependence on parameters and boundary data for nonlinear two-point boundary value problems

Steven Kenyon Ingram

Vol. 41 (1972), No. 2, 395–408

Sufficient conditions are given for the continuous dependence of solutions to the two-point boundary value problem

x ′′ = f(t,x,x ′;μ)

x (a) = α  x(b) = β

on the boundary data and the parameter μ.

Previous results given by Gaines and Klaasen for continuous dependence on the boundary data have assumed continuity on f and uniqueness to two-point BVP’S. Klaasen has also shown assuming uniqueness to two-point BVP’S and the existence of a C2-solution to (1) (2) that there exist solutions x(t;α) to (1) with the boundary conditions

       ′         ′
x(a) = α  x(b) = β

for all (α) sufficiently close to (α,β). Furthermore, x(t;α) is a uniformly continuous function of (α) at (α,β) on [a,b]. This same result is shown to be valid under weaker uniqueness conditions. Sufficient conditions are also given for existence and continuous dependence on the parameter, μ, of solutions to (1)–(2).

Mathematical Subject Classification
Primary: 34A10
Received: 23 September 1970
Published: 1 May 1972
Steven Kenyon Ingram