Journal of Applied Sciences

Year: 2006 | Volume: 6 | Issue: 8 | Page No.: 1854-1857
DOI: 10.3923/jas.2006.1854.1857

Approximation of Fixed Points of Certain Linear Pseudocontractive Map by a Stochastic Iterative Method

A.C. Okoroafor and B.O. Osu

Abstract: Let S = be a compact subset of Rⁿ and let T: D(T with domain D(T) be positive linear pseudocontractive map induced by a positive irreducible matrix. We establish, the strong convergence of a recursive stochastic approximation algorithm of Robbins-Monroe type to the unique fixed point of T.

Keywords: irreducible matrix, Linear pseudocontractive maps and stochastic fixed point iteration

INTRODUCTION

Let Rⁿ be n-dimensional Euclidean space. If T and T^*are mappings with domains

and with values in Rⁿ, then such T is called strongly Pseudocontractive if there exist α<1 such that the inequality

holds for all x,y∈D(T) while T^* is said to be strongly accretive if

Let M be n x n matrix with positive elements

We assume that the iterates

are defined and finite and that for each pair of indices i,j there exists an integer t>0 depending in general on i and j such that

We define

where M is considered as an operator acting on column vectors having positive entries.

Then elementary calculation yields

so that clearly, T is a pseudocontractive mapping. Hence by the fixed point theorem of Browder (1967) there is a unique fixed point x₀∈S with x₀ = Tx₀. A number of results recurringly used in Economics concern the theorem about sI-T, s>0, where I is identity matrix (Kemeny and Snell, 1960; Solow, 1952). For instance, a well known result relevant in simple direct discussion of linear models of production in econometrics is that the equation (sI-T)x = y has a unique positive solution given by x = (sI-T)^-1y for every y, s>r (Solow, 1952).

Sometimes the condition Tx≤x is imposed as a fundamental assumption so that there exist a y≥0 such that Tx = x+c. This assumption is relevant to the productivity in linear models (Gale, 1960). It is shown in Pruitt (1964) that if M is irreducible and sMx≤x for some s>0 and non trivial x, then x_j>0 ∀ j.

Thus if s = 1, the problem of finding x* such that

is equivalent to finding x*>0 such that Tx* = x*.

The firm connection between the pseudocontractive mappings and the accretive operators is that a mapping T is pseudocontractive if and only if I-T is accretive (Browder, 1967; Chidume, 1987) so that any solution of the equation T*x = 0 is also a solution of Tx = x. This leads us to consider a stochastic sequence {x_n} defined by

whose limit is the solution of the equation T*x = 0 where T* = 1-T.

Chidume (1987) has proved the

Theorem 1: Suppose S a nonempty closed bounded and convex subset of L_p p≥2 and T:S→S is a lipschitz strongly pseudocontractive mapping of S into itself.

Let {ρ^k} be a real sequence satisfying:

The main concern of this study is to prove strong convergence with probability one of the stochastic sequence to the fixed point of T.

PRELIMINARIES

We proceed now with some background in notation and definitions. Here, as in the sequel, || x || denotes the Euclidean norm of x∈Rⁿ and given the points x and y in Rⁿ, ⟨x, y⟩ is the usual inner product in Rⁿ.

For the purpose of this study a random variable will be a numerically valued observable X in a probability space (Ω, f, Ρ). The probability measure P is characterized by the singleton probabilities

Let E denote the expectation operator. The expectation of the observation is by definition,

If E||x||<∞ then Ez is defined by the requirement that E(a,z) = (a,Ez) for the random element z and a fixed a in Rⁿ, where ||z||,⟨z₁,z₂⟩, ⟨a, z ⟩ are real valued observable random variables in the usual sense.

In order to solve the problem by successive approximation it is convenient to transform the equation to the form

so that ∂f(x)/∂x = Tx and since T is positive definite, there exists a minimum of f in Rⁿ and every minimizing sequence coverages to the unique minimum of f at

We shall now obtain the estimate of the gradient mapping

using the following approximation theory

APPROXIMATION THEORY

We assume that for each set of values x₁, x₂, …, x_n, using functions values y(x₁), y(x₂), ….y(x_n) as observations, there are uncertainties associated with these observations

This can be modelled by adding a white noise process

which are independently distributed about zero with common variance σ².

The resulting model for the stochastic process is then given by

for a giving tj∈Rn. From this it follows that if f is transformed to a linear function of x with coefficients

the most efficient method of fitting is the method of least square, which yields un biased estimate

having the least possible variance.

The natural Taylor’s expansion of a quadratic function of f about a point X^* is given by

where x_c is on the line segment between x and x_c and H(x_c) is the Hessian of f at x_c so that for a fixed k and j = 1,…, m.

If y(x₁), y(x₂), …, y(x_m) are real-valued independent observable random variables performed on x₁, x₂, …, x_m chosen in the neighborhood of a fixed x^k

then

is identifiable with (15).

and

Then for a fixed t_j in Rⁿ satisfying

An easy calculation yields

Thus by the condition (19), tj linearizes f so that the relationship between y_j and t_j, t_j = x_j-x^k is adequately approximated by the model

and the least squares approximation

exists and is adequate for approximating

such that

for each k and yields a minimum Euclidean distance between the true and the estimated gradient vector

The stochastic gradient type recursive sequence

is suggested, where {ρ^k} is a sequence of positive scalar to be specified and {d^k} is a sequence of independent and identically distributed random vectors.

This procedure is a way of stochastically solving the equation

The stochastic algorithm is thus given as follows:

Let x^k be giving an estimate of x^*

CONVERGENCE THEOREM

Theorem 2: Let {ρ^k} be a real sequence satisfying

Then the stochastic sequence defined by x⁰∈D(T), x^k+1 = x^k-ρ^kd^k remains in D(t) and converges strongly to the fixed point of T almost surely

Proof

so that {q_j}is a sequence of independent identically distributed random variables with Eq_j =0 for each j.

Thus the sequence of partial sums

is a martingale. But

Thus bythe hypothesis . Hence by martingale convergence theorem(Whittle,1976),

we have So that

Hence by Throrem 1, the unique fixed point x^*such that

where

satisfying

is limit of the stochastic sequence x^j+1 = x^j-ρ^jd^j as j→∞ since x^j∈D(T) ∀j and S is compact, S contains unique fixed point satisfying

Thus, the unique fixed point x^* of T is the limit of every successive approximate

for an arbitrary non-zero starting x^o∈S

REFERENCES