TIL:

Fixed Point And Variational Formulataion
For Functional Equation

by Dr. Khin Maung Win, M.A. (Yangon, Myanmar), Ph.D. (Caen, France),
Advisor, Mathematics Department, Yangon University. Paper received by TIL on 040706.

Abstract

Uing a  maximal  monotone  operator  for  multi-valued  mapping,

conditions for equivalence of functional equation, fixed point and

variational inequalities are given.

Introduction

We give the definitions of monotone and maximal monotone operators.

If  T is a maximal monotone operator on a real Hilbert space X to its dual, and

            f Î X, then

u Î u + l (Tu – f), l  ¹  0

is called a fixed point formulation of the functional equation, and

                        <  Tv – f, v – u > 0, " v Î X

is called a variational formulation.  Then the conditions for the equivalance of these formulations are given.

Monotone  Operators

Let X be a real Banach space and X* be its dual. <, > denotes the duality pairing between X and X*.

Let       T :        X ®    2^X*

U ®    Tu

We define graph of  T = G (T) = { (u, v) X × X* : v  Î Tu }

A subset T of X* × X is a monotone operator

If <v₁ – v₂, x₁ – x₂ > > 0 whenever (x₁, v₁) Î T and (x₂, v₂) Î T.

T is said be maximal monotone operator if T is monotone operator and there does not exist a monotone operator T¢ such that T Ì T¢ and T ¹ T¢ ;

i.e T¢  is monotone, G(T¢) É G(T) Þ G(T¢) = G(T)

Theorem

Let T be a multi-valued operator on real Hilbert space X into X*.

Suppose T is maximal monotone.

Then the following are equivalent.

(a)                f Î Tu

(b)               u is a fixed point u + l (Tu – f), l  ¹ 0

i.e         u Î u + l (Tu – f), l ¹ 0

(c)                < Tv – f, v – u > > 0, "v Î X

References

End of TIL file