Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2011, Article ID 579819,10 pages doi:10.1155/2011/579819

Research Article

Invariant Points and t-Simultaneous Approximation

Sumit Chandok and T. D. Narang

Department of Mathematics, Guru Nanak Dev University, Amritsar-143005, India Correspondence should be addressed to Sumit Chandok, chansok.s@gmail.com Received 10 December 2010; Revised 18 March 2011; Accepted 19 April 2011 Academic Editor: S. M. Gusein-Zade

Copyright © 2011 S. Chandok and T. D. Narang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We generalize and extend Brosowski-Meinardus type results on invariant points from the set of best approximation to the set of e-simultaneous approximation. As a consequence some results on e-approximation and best approximation are also deduced. The results proved in this paper generalize and extend some of the known results on the subject.

1. Introduction and Preliminaries

Fixed point theory has gained impetus, due to its wide range of applicability, to resolve diverse problems emanating from the theory of nonlinear differential equations, theory of nonlinear integral equations, game theory, mathematical economics, control theory, and so forth. For example, in theoretical economics, such as general equilibrium theory, a situation arises where one needs to know whether the solution to a system of equations necessarily exists; or, more specifically, under what conditions will a solution necessarily exist. The mathematical analysis of this question usually relies on fixed point theorems. Hence finding necessary and sufficient conditions for the existence of fixed points is an interesting aspect.

Fixed point theorems have been used in many instances in best approximation theory. It is pertinent to say that in best approximation theory, it is viable, meaningful, and potentially productive to know whether some useful properties of the function being approximated is inherited by the approximating function. The idea of applying fixed point theorems to approximation theory was initiated by Meinardus [1]. Meinardus introduced the notion of invariant approximation in normed linear spaces. Brosowski [2] proved the following theorem on invariant approximation using fixed point theory by generalizing the result of Meinardus [1].

Theorem 1.1. Let T be a linear and nonexpansive operator on a normed linear space E. Let C be a T-invariant subset of E and x a T-invariant point. If the set PC (x) of best C-approximants to x is nonempty, compact, and convex, then it contains a T-invariant point.

Subsequently, various generalizations of Brosowski's results appeared in the literature. Singh [3] observed that the linearity of the operator T and convexity of the set PC (x) in Theorem 1.1 can be relaxed and proved the following.

Theorem 1.2. Let T : E ^ E be a nonexpansive self-mapping on a normed linear space E. Let C be a T-invariant subset of E and x a T-invariant point. If the set PC (x) is nonempty, compact, and star shaped, then it contains a T-invariant point.

Singh [4] further showed that Theorem 1.2 remains valid if T is assumed to be nonexpansive only on PC(x) U {x}. Since then, many results have been obtained in this direction (see Chandok and Narang [5, 6], Mukherjee and Som [7], Mukherjee and Verma [8], Narang and Chandok [9-11], Rao and Mariadoss [12], and references cited therein).

In this paper we prove some similar types of results on T-invariant points for the set of ¿-simultaneous approximation in a metric space (X, d). Some results on T-invariant points for the set of ¿-approximation and best approximation are also deduced. The results proved in the paper generalize and extend some of the results of [6, 8-13] and of few others.

Let G be a nonempty subset of a metric space (X, d) and let F be a nonempty bounded subset of X. For x e X, let dF(x) = sup{d(y,x) : y e F}, D(F,G) = inf{dF(x) : x e G} and PG(F) = {go e G : dF(g0) = D(F,G)}. An element go e PG(F) is said to be a best simultaneous approximation of F with respect to G.

For e > 0, we define Pc(e)(F) = {go e G : dF (go) < D(F, G) + e} = {go e G : supyeFd(y,go) < infgeGsupyeFd(y,g) + e}. An element go e Pc(e)(F) is said to be a e-simultaneous approximation of F with respect to G.

It can be easily seen that for e > o, the set PG(e) (F) is always a nonempty bounded set and is closed if G is closed.

In case F = {p}, p e X, then elements of PG(p) are called best approximations to p in G and of PG(e)(p) are called e-approximation to p in G.

A sequence (yn) in G is called a e-minimizing sequence for F, if lim sup xeFd(x,yn) < D(F, G) + e. The set G is said to be ¿-simultaneous approximatively compact with respect to F if for every x e F, each e-minimizing sequence (yn) in G has a subsequence (yni) converging to an element of G.

Let (X,d) be a metric space. A continuous mapping W : X x X x [o, 1] ^ X is said to be a convex structure on X if for all x,y e X and 1 e [o, 1],

holds for all u e X. The metric space (X, d) together with a convex structure is called a convex metric space [14].

A convex metric space (X, d) is said to satisfy Property (I) [15] if for all x,y,p e X and

1 e [o, 1],

d(u,W(x,y,X)) < Xd(u,x) + (1 - X)d(u,y)

d(W(x,p,X),W(y,p,X)) < Xd{x,y).

A normed linear space and each of its convex subset are simple examples of convex metric spaces with W given by W(x,y,X) = Xx + (1 - X)y for x,y e X and 0 < X < 1. There are many convex metric spaces which are not normed linear spaces (see [14]). Property (I) is always satisfied in a normed linear space.

A subset K of a convex metric space (X, d) is said to be

(i) a convex set [14] if W(x,y,X) e K for all x,y e K and X e [0,1];

(ii) p-star shaped [16] where p e K, provided W(x,p, X) e K for all x e K and X e [0,1];

(iii) star shaped if it is p-star shaped for some p e K.

Clearly, each convex set is star shaped but not conversely. A self-map T on a metric space (X, d) is said to be

(i) contraction if there exists k,0 < k< 1 such that d(Tx, Ty) < kd(x, y) for all x,y e X;

(ii) nonexpansive if d(Tx, Ty) < d(x, y) for all x,y e X;

(iii) quasi-nonexpansive if the set F(T) of fixed points of T is nonempty and d(Tx,p) < d(x,p) for all x e X and p e F(T).

A nonexpansive mapping T on X with F (T)/ 0 is quasi-nonexpansive, but not conversely. A linear quasi-nonexpansive mapping on a Banach space is nonexpansive. But there exist continuous and discontinuous nonlinear quasi-nonexpansive mappings that are not nonexpansive.

2. Main Results

To start with, we prove the following proposition on e-simultaneous approximation which will be used in the sequel.

Proposition 2.1. Let F be a nonempty bounded subset of a metric space (X,d), and let C be a nonempty subset of X. If C is e-simultaneous approximatively compact with respect to F, then the set PC(e) (F) is a nonempty compact subset of C.

Proof. Since e> 0, PC(e)(F) is nonempty. We now show that PC(e)(F) is compact. Let (yn) be a sequence in PC(e)(F). Then limsupxeFd(x,yn) < D(F,C) + e, that is, (yn) is an e-minimizing sequence for C. Since C is e-simultaneous approximatively compact with respect to F, there is a subsequence (yni ) such that (ym) ^ y e C. Consider

sup d(x,y) = sup d(x, lim ym)

xeF xeF

= limsup d{x,yni) (2.1)

< D(F, C) + e.

This implies that y e PC(e)(F). Thus we get a subsequence (ym) of (yn) converging to an element y e Pc(s)(F). Hence Pc(s)(F) is compact. □

For F = {x}, we have the following result on the set of e-approximation.

Corollary 2.2 (see [9]). If C is an e-approximatively compact set in a metric space (X,d) then PC(e) (x) is a nonempty compact set.

For F = { x} and e = o, we have the following result on the set of best approximation.

Corollary 2.3 (see [1o]). Let C be a nonempty approximatively compact subset of a metric space (X,d), x e X, and PC be the metric projection of X onto C defined by PC(x) = {y e C : d(x,y)} = d(x, C). Then PC(x) is a nonempty compact subset of C.

We will be using the following result of Hardy and Rogers [17] in proving our first theorem.

Lemma 2.4. Let T be a mapping from a complete metric space (X, d) into itself satisfying

d(Tx,Ty) < a[d(x,Tx) + d(y,Ty)] + b[d(y,Tx) + d(x,Ty)] + cd(x,y), (2.2)

for any x,y e X, where a, b, and c are nonnegative numbers such that 2a + 2b + c < 1. Then T has a unique fixed point u in X. In fact for any x e X, the sequence {Tnx} converges to u.

Theorem 2.5. Let T be a continuous self-map on a complete convex metric space (X, d) with Property (I) and satisfying inequality (2.2), let C be a T-invariant subset of X, and let F be a nonempty bounded subset of X such that Tx = x for all x e F. If PC(e)(F) is compact, and star shaped, then it contains a T-invariant point.

Proof. Let z e PC(e)(F) be arbitrary. Then by (2.2), we have for all x e F

d(x, Tz) = d(Tx, Tz)

< a[d(x,Tx) + d(z,Tz)] + b[d(z,Tx) + d(x,Tz)] + cd(x,z)

= a[d(z,Tz)] + b[d(z, Tx) + d(x,Tz)] + cd(x,z) < a[d(z,x) + d(x,Tz)] + b[d(z,x) + d(x, Tx) + d(x, Tz)] + cd(x, z) = (a + b + c)d(x,z) + (a + b)[d(x, Tz)].

This gives

(1 - a - b)d(x, Tz) < (a + b + c)d(x, z) d(x, Tz) < d(x, z)

since 2a + 2b + c < 1. Therefore, using definition of PC(e)(F), we get

sup{d(x, Tz)} < sup{d(x, z)} < D(F, C) + e.

Hence Tz e PC(e)(F). Therefore T is a self-map on PC(e)(F).

Let q be the star-center of PC(e)(F). Define Tn : PC(e)(F) ^ PC(e)(F) as Tnx = W(Tx,q,Xn), x e PC(e)(F) where (yn) is a sequence in (0,1) such that Xn ^ 1. Consider

d(Tnx,Tny) = d(W(Tx, q, Xn), W(Ty, q, Xn))

< Xnd(Tx,Ty)

< Xn[a[d(x,Tx) + d(y,Ty)] + b[d(y,Tx) + d(x,Ty)] + cd(x,y)]

< a[d(x,Tx) + d(y,Ty)] + b[d(y,Tx) + d(x,Ty)] + cd(x,y),

where (2a+2b+c) < 1. Therefore by Lemma 2.4, each Tn has a unique fixed point zn in PC(e)(F). Since PC(e)(F) is compact, there is a subsequence (zm) of (zn) such that zm ^ zQ e PC(e)(F). We claim that Tz0 = z0. Consider zni = Tnizni = W(Tzni,q,Xm) ^ Tz0, as T is continuous. Thus z,H —Tzo and consequently, Tzc = zc, that is, € Pc(s)(F) is a T-invariant point. □

Since for an e-simultaneous approximatively compact subset C of a metric space (X, d) the set of e-simultaneous C-approximant is nonempty and compact (Proposition 2.1), we have the following result.

Corollary 2.6. Let T be a continuous self-map on a complete convex metric space (X, d) with Property (I) and satisfying inequality (2.2), let F be a nonempty bounded subset of X such that Tx = x for all x e F, and let C be a T-invariant subset of X. If C is e-simultaneous approximatively compact with respect to F and PC(e) (F) is star shaped, then it contains a T-invariant point.

Corollary 2.7 (see [8]). Let T be a continuous self-map on a Banach space X satisfying (2.2), let C be an approximatively compact and T-invariant subset of X. Let Txi = xi(i = 1,2) for some xi,x2 not in cl(C). If the set of best simultaneous C-approximants to x1 ,x2 is star shaped, then it contains a T-invariant point.

Corollary 2.8 (see [11]). Let T be a mapping on a metric space (X,d), let C be a T-invariant subset of X and x a T-invariant point. If PC (x) is a nonempty, compact set for which there exists a contractive jointly continuous family $ of functions and T is nonexpansive on PC(x) u{x} then PC(x) contains a T-invariant point.

Corollary 2.9. Let T be a mapping on a convex metric space (X, d) with Property (I),let C be an approximatively compact, p-star shaped, T-invariant subset of X and let x be a T-invariant point. If T is nonexpansive on PC(x) u {x}, then PC(x) contains a T-invariant point.

Corollary 2.10 (see [10, Theorem 4]). Let T be a quasi-nonexpansive mapping on a convex metric space (X, d) with Property (I), let C be a T-invariant subset of X, and let x be a T-invariant point. If PC(x) is nonempty, compact, and star shaped, and T is nonexpansive on PC(x), then PC(x) contains a T-invariant point.

Corollary 2.11 (see [10, Theorem 5]). Let T be a quasi-nonexpansive mapping on a convex metric space (X, d) with Property (I), let C be an approximatively compact, T-invariant subset of X, and let x be a T-invariant point. If PC (x) is star shaped and T is nonexpansive on PC (x), then PC(x) contains a T-invariant point.

Remark 2.12. Theorem 2.5 improves and generalizes Theorem 1 of Narang and Chandok [9] and of Rao and Mariadoss [12].

Definition 2.13. A subset K of a metric space (X,d) is said to be contractive if there exists a sequence (fn) of contraction mappings of K into itself such that fny ^ y for each y e K.

Theorem 2.14. Let T be a nonexpansive self-mapping on a metric space (X, d), let C be a T-invariant subset of X, and let F be a nonempty bounded subset of X such that Tx = x for all x e F. If the set PC(e) (F) is compact and contractive, then the set PC(e) (F) contains a T-invariant point.

Proof. Proceeding as in Theorem 2.5, we can prove that T is a self-map of PC(e) (F). Since PC(e)(F) is contractive, there exists a sequence (fn) of contraction mapping of PC(e)(F) into itself such that fnz ^ z for every z e PC(e) (F).

Clearly, fnT is a contraction on the compact set PC(e)(F) for each n and so by Banach contraction principle, each fnT has a unique fixed point, say zn in PC(e)(F). Now the compactness of PC(e)(F) implies that the sequence (zn) has a subsequence (zni) ^ z0 e D. We claim that z0 is a fixed point of T. Let e > o be given. Since z„ ^ zo and fnTz0 ^ Tz0, there exist a positive integer m such that for all n > m

d(zHl,z,:,)<-, d(fUlTzo,Tzo) < -. (2.7)

Again,

d(fniTzUl, fnj-o) < d(zUl,z0) < -. (2.8)

dtfnJzn^Tzo) < dfmTzmfmTzo) + d(fniTz0,Tz0)

ee < 2 + 2'

that is, d(fniTzni,Tz0) < e for all n > m and so fniTzni ^ Tz0. But fniTzni = zni ^ z0 and therefore Tzc = zc- □

Using Proposition 2.1 we have the following result.

Corollary 2.15. Let T be a nonexpansive self-mapping on a metric space (X, d), let C be a T-invariant subset of X, and let F be a nonempty bounded subset of X such that Tx = x for all x e F .If C is e-simultaneous approximatively compact with respect to F and the set PC(e) (F) is contractive, and T-invariant, then PC(e) (F) contains a T-invariant point.

Corollary 2.16 (see [9]). Let T be a self-mapping on a metric space (X,d), let G be a T-invariant subset of X, and let x be a T-invariant point. If the set D of e-approximant to x is compact, contractive and T is nonexpansive on D u {x}, then D contains a T-invariant point.

Remark 2.17. Theorem 2.14 also improves and generalizes the corresponding results of Brosowski [2], Mukherjee and Verma [8,13], Chandok and Narang [9], Rao and Mariadoss [12], and of Singh [3].

Definition 2.18. For each bounded subset G of a metric space (X,d), the Kuratowski's measure of noncompactness of G, a[G] is defined as

A mapping T : X ^ X is called condensing if for all bounded sets G c X, a[T(G)] < a[G].

We will be using the following result of [18] on fixed points of nonexpansive condensing maps.

Lemma 2.19. Let X be a complete contractive metric space with contractions {fn}. Let C be a closed bounded subsets of X and T : C ^ C is nonexpansive and condensing, then T has a fixed point in C.

Using the above lemma and Theorem 2.5, we now prove the following result.

Theorem 2.20. Let (X,d) be a complete, contractive metric space with contractions fn. Let G be a closed and bounded subset of X and let F be a nonempty bounded subset of X. If T is a nonexpansive and condensing self-map on X such that Tx = x for all x e F, then PG(e) (F) has a T-invariant point.

Proof. As G is closed and bounded, PG(e)(F) is nonempty, closed and bounded. Using Theorem 2.5, we can prove that T is a self-map of PG(e)(F). Now a direct application of Lemma 2.19, gives a T-invariant point in PG^ (F). □

Corollary 2.21 (see [8, Theorem 3.1]). Let X be a complete, contractive metric space with contractions fn. Let G be a closed and bounded subset of X. IfT is a nonexpansive and condensing self-map on X such that Tx1 = x1 and Tx2 = x2 for some x1,x2 e X, and D = PG(x1,x2) is nonempty, then it has a T-invariant point.

Corollary 2.22 (see [12, Theorem 4]). Let X be a complete, contractive metric space with contractions fn. Let G be a closed and bounded subset of X. If T is a nonexpansive and condensing self-map on X such that Tx = x for some x e X, and PG(x) is nonempty, then it has a T-invariant point.

Definition 2.23. A mapping T on a metric space (X, d) is called a Kannan mapping [19] if there exists a e (0,1/2) such that

for all x, y e X.

Kannan [19] proved that if X is complete, then every Kannan mapping has a unique fixed point.

For e-simultaneous approximation, we have the following result.

a[G] = inf{e > 0 : G is covered by a finite number of closed balls centered at points of X of radius < e .

(2.10)

d(Tx, Ty) < a[d(x, Tx) + d(y, Ty)]

(2.11)

Theorem 2.24. Let G be a nonempty subset of a complete metric space (X, d) and let F be a nonempty bounded subset of X. Let T be a self-map on X with Tx = x for all x e F and Tm satisfies,

d(Tmy,Tmz) < + d(z,Tmz)],

(2.12)

for some positive integer m, all y,z e X and some fixed o < a < 1/2. If D = PG(e) (F) is compact, then it has a unique fixed point of T.

Proof. As Tx = x, Tnx = x for all positive integers n. Let yo e D. Then, for o < a < 1/2,

d(x,Tmyo) = d(Tmx,Tmy0)

< a[d(x,Tmx) + d(y0,Tmy0)] = ad(yo, Tnyo)

< a[d(y0,x) + d(x,Tmy0)],

(2.13)

which implies that

d(x,V"y0) < Y—^d(y0,x).

(2.14)

Further, we have

sup d(x,Tmy0) < sup{d{y0,x)} < D(F,G) + e.

xeF xeF

(2.15)

Therefore, Tmy0 e D, Tm(D) c D. Since Tm satisfies the conditions of Kannan map, Tm has a unique fixed point x0 in D. Now, Tm(Tx0) = T(Tmx0) = Tx0, implies that Tx0 is a fixed point of Tm. But the fixed point of Tm is unique and equals x0. Therefore Tx0 = x0 and hence x0 is a unique fixed point of T in D. □

Corollary 2.25. Let F be a nonempty bounded subset of a complete metric space (X, d) and G a subset of X. Let G be e-simultaneous approximatively compact with respect to F, and T a self map on X with Tx = x for all x e F, and Tm satisfies

d(Tmy,Tmz) < a[d{y,Tmy) + d(z,Tmz)],

(2.16)

for some positive integer m, all y,z e X and some fixed o < a < 1/2 then D = PG(e)(F) has a unique fixed point of T.

Remarks 2.26. Theorem 2.24 extends and generalizes Theorem 3.2 of Mukherjee and Verma [8] and Theorem 5 of Rao and Mariadoss [12] from the set of best simultaneous approximation and best approximation, respectively, to e-simultaneous approximation.

For e> o, we define Rg(£)(F) = {go e G : supgeGd(g,go) + e < infgeGsupyeFd(y,g)}. An element go e RG(e) (F) is said to be a e-simultaneous coapproximation of F with respect to G.

A mapping T : X ^ X satisfies condition (A) (see [13]) if d(Tx,y) < d(x,y) for all x, y e X.

We now prove a result for T-invariant points from the set of e-simultaneous coapproxi-mations.

Theorem 2.27. Let T be a self-map satisfying condition (A) and inequality (2.2) on a convex metric space (X, d) satisfying Property (I), let G be a subset of X, and let F be a nonempty bounded subset of X such that RG(e) (F) is compact and star shaped. Then RG(e) (F) contains a T-invariant point.

Proof. Let g0 e RG(e)(F). Consider

d(Tgo,g) + e < d(go,g)

e < infsup d(y,g),

geG yeF

(2.17)

and so Tg0 e RG(e)(F), that is, T : RG(e)(F) ^ RG(e)(F). Since RG(e)(F) is star shaped, there exists p e RG(e)(F) such that W(z,p,X) e Rg(s)(F) for all z e Rg(s)(F), X e [0,1]. Let (kn),0 < kn < 1, be a sequence of real numbers such that kn ^ 1 as n ^ to. Define Tn as Tn(z) = W(Tz,p,kn), z e RG(e)(F). Since T is a self-map on RG(e)(F) and RG(e)(F) is star shaped, each Tn is a well defined and maps RG(e)(F) into RG(e)(F). Moreover,

d(Tny,Tnz) = d(W(Ty,p,kn),W(Tz,p,kn))

< knd(Ty,Tz) (2.18)

< kn[a[d(y,Ty) + d(z,Tz)] + b[d(z,Ty) + d(y,Tz)] + cd(y,z)],

where kn[2a + 2b + c] < 1. So by Lemma 2.4 each Tn has a unique fixed point xn e RG(e)(F), that is, Tnxn = xn for each n. Since RG(e)(F) is compact, (xn) has a subsequence xni ^ x e RG(e)(F). Now, we claim that Tx = x. As d(xni,Tx) < d(xn,x). On letting n ^ to, we have x,n -H- Tx. Therefore Tx = x, that is, x is T-invariant, hence the result. □

For F = { x} , x e X, we have the following result on the set of e-coapproximation.

Corollary 2.28 (see [9, Theorem 4]). Let T be a self-map satisfying condition (A) on a convex metric space (X, d) satisfying Property (I), let G be a subset of X such that RG(e) (x) is nonempty compact, star shaped, and let T be nonexpansive on RG(e) (x). Then there exists a g0 e RG(e) (x) such that Tg0 = g0.

Remarks 2.29. (i) Theorem 2.27 also improves and generalizes Theorem 4.1 of Mukherjee and Verma [13] from the set of best approximation to e-simultaneous approximation.

(ii) By taking F = {x1,x2}, x1,x2 e X, the set PG(e)(F) (respectively, RG(e)(F)) is the set of e-simultaneous approximation (respectively, e-simultaneous coapproximation) to the pair of points x1 ,x2 and so the results of this paper generalize and extend the corresponding results proved in [6].

Acknowledgments

The authors are thankful to the learned referee for careful reading and very valuable suggestions leading to an improvement of the paper. The research work of the second author has been supported by the UGC India under the Emeritus Fellowship.

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19] R. Kannan, "Some results on fixed points. II," The American Mathematical Monthly, vol. 76, pp. 405-408, 1969.

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