TY - JOUR

T1 - A numerical verification method for the existence of weak solutions for nonlinear boundary value problems

AU - Nakao, Mitsuhiro T.

PY - 1992/3/1

Y1 - 1992/3/1

N2 - In this paper, we consider a numerical technique which enables us to verify the existence of solutions for nonlinear two point boundary value problems (BVP). We formulate the problem as a fixed point of a Newton-like operator and present a verification algorithm by computer based on Sadovskii's fixed point theorem. In order to compute the Newton type iterative sequences in infinite dimensional space, two concepts, the rounding and the rounding error, are introduced. Also some numerical examples are illustrated. We emphasize that, since the present method uses the weak formulation of BVP and the finite element approximations instead of the existing validation techniques, it is aslo applicable, in principle, to the BVP for partial differential equations as it stands.

AB - In this paper, we consider a numerical technique which enables us to verify the existence of solutions for nonlinear two point boundary value problems (BVP). We formulate the problem as a fixed point of a Newton-like operator and present a verification algorithm by computer based on Sadovskii's fixed point theorem. In order to compute the Newton type iterative sequences in infinite dimensional space, two concepts, the rounding and the rounding error, are introduced. Also some numerical examples are illustrated. We emphasize that, since the present method uses the weak formulation of BVP and the finite element approximations instead of the existing validation techniques, it is aslo applicable, in principle, to the BVP for partial differential equations as it stands.

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U2 - 10.1016/0022-247X(92)90129-2

DO - 10.1016/0022-247X(92)90129-2

M3 - Article

AN - SCOPUS:38249013605

VL - 164

SP - 489

EP - 507

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -