AN APPLICATION OF FIXED POINT THEOREMS IN BEST APPROXIMATION THEORY

In this paper, we give an application of Jungck’s fixed point theorem to best approximation theory, which extends the results of Singh and Sahab et al.

Let X be a normed linear space.A mapping T X X is said to be contractwe on X (resp., on a subset C of X) if IITx-Tyll <_ IIx Yll for all x, y in X (resp., C).The set of fixed points of T on X is denoted by F(T).If is a point of X, then for 0 < a _< 1, we define the set Da of best (C, a)-approximants to consists of the points y in C such that Let D denote the set of best C-approximants to .For a 1, our definition reduces to the set D of best C-approximants to .A subset C of X is said to be starshaped with, respect to a point q E C if, for all x in C and all A 5 [0,1], Az + (1 A)q C. The point p is called the star-centre of C. A convex set is starshaped with respect to each of its points, but not conversely.For an example, the set C {0} [0,1] LI [1, 0] {0} is starshaped with respect to (0, 0) e C as the star-centre of C, but it is not convex.
In this paper, we give an application of Jungck's fixed point theorem to best approximation theory, which extends the results of Sahab et al. [9] and Singh [10].
By relaxing the linearity of the operator T and the convexity of D in the original statement of Brosowski [1], Singh [10] proved the following: Theorem 1.Let C be a T-invariant subset of a normed linear space X.Let T C C be a contractive operator on C and let F(T).If D c_ X is nonempty, compact and starshaped, then D f F(T) 0.
In the subsequent paper [11], Singh observed that only the nonexpansiveness of T on D' DU{} is necessary.Further, Hicks and Humphries [4] have shown that the assumption T" C C can be weakened to the condition T" OC C if y C, i.e., y E D is not necessarily in the interior of C, where OC denotes the boundary of C. Theorem 2. Let X be a Banach space.Let T, I X X be operators and C be a subset of X such that T" OC C and 5: F(T)f3 F(I).Further, suppose that T and I satisfy (1) for all x, y in D', I is linear, continuous on D and ITx TIx for all x in D. If D is nonempty, compact and starshaped with respect to a point q F(I) and I(D) D, then Df3F(T)f3F(I) .
Recall that two self-maps I and T of a metric space (X, d) with d(x, y) IIx-Yll for all x, y X are said to be compatible on X if .h_md(ITx., TIx.)(= .h_rnIlITz.Tlz.ll) 0 whenever there is a sequence {x.} in X such that Tx., Ix. t, as n oo, for some tin X ([6]- [8]).
We shall use N to denote the set of positive integers and CI(S) to denote the closure of a set S.
For our main theorem, we need the following: Proposition 3. [8] Let T and I be compatible self-maps of a metric space (X, d) with I being continuous.Suppose that there exist real numbers r > 0 and a (0,1) such that for all x, y X, d(Tx, Ty) <_ rd(Ix, Iy) + a max(d(Tx, Ix), d(Ty, Iy)}.
,Then Tw Iw for some w X if and only if A f3{Cl(T(Ko)) n N} # $, where for each go { x X" d(Tx, Ix) <_ -}.
On the other hand, using this proposition, Jungck [8] proved the following: Theorem 4. Let I and T be compatible self-maps of a closed convex subset C of a Banach space X.Suppose that I is continuous and linear with T(C) c_ I(C).If there exists an a e (0,1) such that for all x, y C, IITx T,II _< alllx lull + (1 a) max{llTx lxll, IITy-IulI}, (2) then I and T have a unique common fixed point in C.
By using this theorem, we extend Theorem 2 as in the following: Theorem 5. Let X be a Banach space.Let T, I X X be operators and C be a subset of X such that T" c9C C and 5: F(T)N F(1).Further, suppose that Tand I satisfy (2) for all x, y in D' Do t3 {5:} U E, where E {q X Ix=,Tx.q, {xo} C Do}, 0 < a < 1, I is linear, continuous on Do and T, I are compatible in D. If D is nonempty, cbmpact and convex, and I(D) Do, then D f F(T) f F(I) .
Suppose that w 6 A. Then for each n N, there exists y.T(Ko) such that d(w, I/o) < 1/n.
Consequently, for such n, we can and do choose x.K. such that d(w, Tx.) < 1In and so Tx.w.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation