FIXED POINTS AND BEST APPROXIMATION IN MENGER CONVEX METRIC SPACES

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1 ARCHIVUM MATHEMATICUM BRNO Tomus , FIXED POINTS AND BEST APPROXIMATION IN MENGER CONVEX METRIC SPACES ISMAT BEG AND MUJAHID ABBAS Abstract. We obtai ecessary coditios for the existece of fixed poit ad approximate fixed poit of oexpasive ad quasi oexpasive maps defied o a compact covex subset of a uiformly covex complete metric space. We obtai results o best approximatio as a fixed poit i a strictly covex metric space. 1. Itroductio Noexpasive mappigs have bee studied extesively i recet years by may authors.the first fixed poit theorem of a geeral ature for oliear oexpasive mappigs i ocompact settig were proved idepedetly by Browder [8] ad Gohde [12]. Later o, Kirk [17] proved the same results uder slightly weaker assumptios. A fudametal problem i fixed poit theory of oexpasive mappigs is to fid coditios uder which a set has the fixed poit property for these mappigs. It is itimately coected with differetial equatios ad with the geometry of the Baach spaces.the iterplay betwee the geometry of Baach spaces ad fixed poit theory has bee very strog ad fruitful. I particular, geometrical properties play key role i metric fixed poit problems, see, for example, [2 5, 7, 19, 20] ad refereces metioed therei. These results use maily covexity hypothesis ad geometric properties of Baach spaces. These results were startig poit for a ew mathematical feild : the applicatio of the geometric theory of Baach spaces to fixed poit theory. The problem of proximity of subsets i ormed spaces has bee studied extesily, see, for example, [13, 15, 19, 20]. Aroszaj ad Paitchapakdi [2] ad Meger [18] defied the covexity structure o metric spaces through closed ball ad studied their properties. Khalil [16] further studied existece of fixed poits ad best approximatio i these covex metric spaces. This paper addresses the covexity structure of a metric space, usig geometric 2000 Mathematics Subject Classificatio: 47H09, 47H10, 47H19, 54H25. Key words ad phrases: fixed poit, covex metric space, uiformly covex metric space, strictly covex metric space, best approximatio, oexpasive map. Received December 21, 2003.

2 390 I. BEG, M. ABBAS properties of this space; we study fixed poits, approximate fixed poits ad structure of the set of fixed poits. We establish results o ivariat approximatio for the mappig defied o a class of o-covex sets i a covex metric space. 2. Prelimiaries We gather here some basic defiitios ad set out our termiology eeded i the sequel. We also preset some previously kow results about fixed poits. Defiitio 1 Meger [18]. Let X, d be a metric space. It is said to be Meger covex metric space if for every x, y i X, x y, 0 r dx, y, where B[x, r] = {y X : dx, y r}. B[x, r] B[y, dx, y r], Some of the properties of these spaces were itroduced by Blumethal [7]. Further results ca be foud i [6, 15]. A subset E of a covex metric space is called covex if for all x, y i E, B[x, r] B[y, dx, y r] E, 0 r dx, y. We ote that i geeral i a covex metric space, the set B[x, r] is ot covex. Defiitio 2. A covex metric space X is said to have a property A, if for every x, y X, the set B[x, 1 tdx, y] B[y, tdx, y] is a sigleto set for t [0, 1]. We deote this sigleto set by mx, y, t. Whe x = y the obviously mx, y, t = x. I a covex metric space havig property A, B[x, r] is a covex set. Defiitio 3. A covex metric space X is said to have property B, if for all x, y, z X ad t 0, 1. dmx, y, t, mz, y, t tdx, z, Defiitio 4. Let X be a covex metric space. It is said to be uiformly covex if it has property A ad for every give ε > 0, there exists δ = δε > 0 such that for all r > 0, ad x, y, z X with dz, x r, dz, y r ad dx, y rε, imply that d z, m x, y, 1 2 r 1 δε < r. Defiitio 5 Khalil [16]. A covex metric space X is said to be strictly covex, if x, y B[z, r] with x y, the B[x, 1 tdx, y] B[y, tdx, y] Bz, r, for all t 0, 1, ad all z X, r > 0. Where, Bz, r = {x X : dx, z < r}. It follows from above defiitio that uiformly covex metric space is strictly covex but coverse i geeral does ot hold. Defiitio 6. A subset F of a covex metric space X is called T-regular set if ad oly if T : F X ad mx, Tx, 1 2 F, for each x F.

3 FIXED POINTS AND BEST APPROXIMATION 391 Remark 7. Every covex set ivariat uder a map T i a covex metric space is a T- regular set. But a T-regular set eed ot be a covex set. Defiitio 8. Let X be a metric space ad T : X X, a poit x X is called a fixed poit of T, if it is the solutio of fuctioal equatio Tx = x. b ε-fixed poit of T if dx, Tx < ε for give ε > 0. Obviously fixed poit of T is ε-fixed poit for every give ε > 0 but coverse is ot true i geeral. Existece of ε-fixed poit of T for each ε > 0 sometimes guaratees the existece of fixed poit of T, for example, if a cotiuous map T defied o a closed subset F of metric space X with TF cotaied i some compact subset of X, has ε-fixed poit for each ε > 0, the T has a fixed poit. Defiitio 9. Let X, d be a metric space ad T : X X. The mappig T is called oexpasive if for all x, y X, d Tx, Ty dx, y. Defiitio 10. Let X, d be a metric space. A mappig T : X X is said to be quasi-oexpasive provided T has a fixed poit i X ad if x 0 is a fixed poit of T, the dtx, x 0 dx, x 0, for all x i X. Remark 11. The class of oexpasive mappigs is strictly cotaied i the class of quasi-oexpasive mappigs. We preset the followig example which supports the Remark 11. Example 12. Defie the self map T o Euclidea space R as Tx = xsi 1 x whe x 0 ad T0 = 0. Obviously 0 is the oly fixed poit of T ad T is quasi-oexpasive. Take x = 2 π, y = 2 3π, the, Tx Ty = 8 3π > 4 = x y, 3π gives that T is ot oexpasive. Defiitio 13. Let X, d be a metric space ad M X. For x X, we defie P M x = {z M : dx, z = dx, M}, where dx, M = if{dx, y, y M}. Ay z P M x is called poit of best approximatio for x from M. The set P M x φ, for ay compact subset M of a metric space X.If P M x is sigleto for each x i X the we may defie the earest poit projectio P : X M by assigig the poit of best approximatio for x from M to each x i X. It has bee proved [16] that P M x is covex for a closed covex set M i a covex metric space X havig property A. Defiitio 14. Let K be a subset of a complete covex metric space ad {f α : α K} be family of maps from [0, 1] ito K, havig property that for each α K we have f α 1 = α. Such a family is said to be a cotractive provided there exist a fuctio ϕ : 0, 1 0, 1 such that for all α, β K ad for all t 0, 1, we have df α t, f β t ϕtdα, β; b joitly cotiuous if t t 0 i [0, 1] ad α α 0 i K imply f α t f α0 t 0.

4 392 I. BEG, M. ABBAS 3. Fixed poits ad approximate fixed poits I this sectio we study fixed poit ad ε-fixed poit of oexpasive ad quasi oexpasive mappigs defied o a compact covex subset of a metric space. Theorem 15. Let F be a oempty compact covex subset of a covex complete metric space X havig properties A ad B, the ay oexpasive self map T o F has a fixed poit. Proof. Fix y F, cosider the mappig T : F F defied by T x = m Tx, y, 1. The d T x, T z = d m Tx, y, 1, m Tz, y, 1 1 dx, z. It implies that T is a cotractio for each positive iteger, Baach cotractio priciple [12] further implies that for each there exists x F for which Tx = x. Now the sequece {x } has a coverget subsequece {x j } covergig to x 0. It follows by the cotiuity of T that Tx j Tx 0. Now d x j, Tx 0 = d T x j, Tx 0 = d m Tx j, y, 1, Tx 0. Whe, {x j } Tx 0. Now the result follows by the uiqueess of the limit. Remark 16. Theorem 15, geeralizes [11, Theorem 3.1]. The special feature of our ext theorem is that i this theorem we relaxe the coditio of covexity o the domai of defiitio of the mappig. It ot oly improves Theorem 15 but also cotais the geeralizatio of Theorem 1 [9]. Theorem 17. Let F be a compact subset of a complete covex metric space X. Suppose there exists a cotractive, joitly cotiuous family of maps associated with F, the ay oexpasive mappig T of F ito itself has a fixed poit i F. Proof. For each = 1, 2, 3,.... Let k = +1 [0, 1]. Defie a mappig T from F ito itself by T x = f Tx k for all x F. Sice TF F ad k < 1, each T is well defied map from F ito itself. Moreover, for each ad all x, y F, we have d T x, T y = d f Tx k, f Ty k φk d Tx, Ty φk dx, y. So for each, T is a cotractio mappig o F. Baach cotractio priciple [12] further implies that each T has uique fixed poit x F. Now compactess of F gives a subsequece {x j } of {x } such that x j x 0 i F. Sice T j x j = x j, we have T j x j x 0. By the cotiuity of T, Tx j Tx 0, also by the joit cotiuity of the family, we have T j x j = f Txj k j f Tx01 = Tx 0.

5 It gives that x 0 is the fixed poit of T. FIXED POINTS AND BEST APPROXIMATION 393 Now we preset the theorem which will drop the coditio of compactess o the domai of the fuctio ad thus further improves the Theorem 15. Moreover, it exteds well kow Schauder s fixed poit theorem for oexpasive mappig to covex metric spaces. Theorem 18. Let F be a oempty closed covex subset of a covex metric space havig properties A ad B. Let T : F F be a oexpasive ad TF a subset of a compact subset of F, the T has a fixed poit. Proof. Let x 0 F, for = 2, 3, 4,..., defie T x = m x 0, Tx, 1. Covexity of F implies that T is a self map o F for each. Now d T x, T y = d m x 0, Tx, 1, m x 0, Ty, 1 1 d Tx, Ty 1 dx, y. Baach cotractio priciple [12] i turs implies T has a uique fixed poit x i F for each. Sice TF lies i a compact subset of F. The sequece {x } i F has a subsequece {x j } such that Tx j x 0, where x 0 F. Now x j = T x j = m x 0, Tx j, 1 x 0, as. By the cotiuity of T, the result follows immediately. Now we prove the theorem that gives ecessary ad sufficiet coditios for the covergece of iterative sequece Baach type to the fixed poit of quasioexpasive mappig i a covex metric space settig. Theorem 19. Let F be a closed subset of a covex complete metric space X ad T : F X be a quasi-oexpasive mappig. Suppose there exists a poit x 0 F with x = T x 0 F. The the sequece {x } coverges to the fixed poit of T if ad oly if lim x, Fix T = 0, where FixT stads for the set of fixed poits of F. Proof. As d x, Fix T = sup x FixT dx, x. Thus lim x, FixT = 0, immediately follows by the covergece of sequece {x } to the fixed poit of T. For the coverse, let ε > 0, the there exists 0 such that for all 0, we have d x, FixT < ε 2. Take the poit p i Fix T ad m, 0, we have dx, x m dx, p + dp, x m = d T x 0, p + d p, T m x 0 2d T 0 x 0, p 2dx 0, p 2d x 0, Fix T, give {x } is a Cauchy sequece i F. As F is a closed subset of a complete metric space. So there exists a poit u i F such that x u. Hece d u, FixT = 0, Fix T is closed due to quasi-oexpasiveess of T, we have u Fix T.

6 394 I. BEG, M. ABBAS Theorem 20. Let F be a closed subset of a covex metric space X The a compact cotiuous map T : F X, has a fixed poit if ad oly if T has ε-fixed poit for each ε > 0. Proof. Assume that T has ε-fixed poit for each ε > 0. Now for each positive iteger, let x be 1 -fixed poit, that is d x, Tx < 1. Sice T is a compact, TF is cotaied i a compact subset D of X. For a sequece {x } i F, we have a subsequece {x j } ad a poit u i F such that Tx j u as j gives x j u, as j. Now closedess of F ad cotiuity of T give Tx j Tu, u F, ad this together with argumet used i Theorem 18 imply the result. 4. Best approximatio I this sectio we preset applicatio of fixed poit theorems to approximatio theory i a covex metric space settig. Theorem 21. Let M be a closed covex subset of a uiformly covex complete metric space X. If P M x is sigleto for each x X, the the earest poit projectio P : X M is cotiuous. Proof. Let the sequece {x } coverges to x i X ad P M x = {z}. For the sake of simplicity, deote Px by u. Now {u } is a Cauchy sequece i M, if ot, the there are positive real umbers ε ad subsequeces {u k } ad {u mk } such that m k > k ad du k, u mk ε for all k. Put a k = u k, b k = u mk ad = max{dx, a k, dx, b k }. Note that dx, M as k. Now dx, a k, dx, b k ad da k, b k ε M Mk k. It gives d x, m a k, b k, δ ε dak, b k 1 δ. Also δ ε 1 dx,m, lettig k, δ ε 0 ad ε ca ot be positive. Thus {Px } is a Cauchy sequece i M ad therefore coverges to a poit z i M, as dx, z = dx, M, ad z = Px. Theorem 22. Let F be a bouded T-regular subset of a uiformly covex complete metric space X. The either Tx = x for all x i F or there exists a poit x 0 i F such that dx 0, F < diamf. Proof. Suppose for some x F, x Tx, let d x, Tx = ε. Now for ay y i F, d y, Tx diamf ad dy, x diamf. As F is a T-regular set, so m x, Tx, 2 1 F. Now by uiform covexity of X, there exists positive real umber δε with d y, m x, Tx, 1 1 δε diamf, 2 which i tur implies d F, m x, Tx, 1 1 δε diamf, 2

7 FIXED POINTS AND BEST APPROXIMATION 395 ad the result follows. We prove the followig pair of propositios eeded for Theorem 25. Propositio 23. Let X be a strictly covex metric space, u X ad M a subset of X. If x, y P M u with x y. The mx, y, t / M, where t 0, 1. Proof. If mx, y, t M, the x, y P M u gives dx, u d u, mx, y, t, ad dy, u d u, mx, y, t. Sice X is strictly covex metric space, so we arrive at a cotradictio. Hece mx, y, t / M, 0 < t < 1. Propositio 24. Let M be ay subset of a strictly covex metric space X ad T : M M. If P M u is a oempty T-regular set for ay u X, the each poit of P M u is a fixed poit of T. Proof. Suppose for some x i P M u, we have Tx x. By the Propositio 23, m x, Tx, 2 1 / M. Hece m x, Tx, 1 2 / PM u. Sice P M u is a T-regular set, therefore x = Tx must hold. Thus each best M-approximatio of u is a fixed poit of T. Theorem 25. Let M be a oempty closed ad T-regular subset of a strictly covex metric space X, where T is a compact mappig ad u be a poit i M. Suppose that d Tx, u dx, u for all x i M. The each x i M, which is best approximatio to u, is a fixed poit of T. Proof. Let r = du, M, the there is a miimizig sequece {y } i M such that lim du, y = r. Clearly {y } is a bouded sequece. Sice T is a compact, cl{ty } is a compact subset of M ad so {Ty } has a coverget sequece {Ty k } with lim Ty k = x i M. Now r du, x = lim k d u, Ty k lim k du, y k = limdu, y = r. Thus x P M u. Also, if y P M u ad r d Ty, u dy, u = r imply that Ty P M u. Therefore y P M u gives d Ty, u = r. Now strict covexity of X implies r d m y, Ty, 2 1, u < r. Thus m y, Ty, 1 2 PM u. The result ow follows by usig Propositio 23. Theorem 26. Let M be a oempty closed ad T-regular subset of strictly covex metric space X, where T : X X is a compact mappig. If u is a fixed poit of T i X \ M, ad d Tx, Ty αdx, y + β dx, Tx + d y, Ty + γ d x, Ty + d y, Tx, for all x, y X, where α, β ad γ are real umbers with α + 2β + γ 1. The each best approximatio i M to u is a fixed poit of T.

8 396 I. BEG, M. ABBAS Proof. For x X, cosider Tx, Tu αdx, u + β d x, Tx + du, Tu + γ d x, Tu+d u, Tx = αdx, u + βd x, Tx + γ dx, u + d Tu, Tx αdx, u + β dx, u + dtu, Tx + γ dx, u+dtu, Tx = α + β + γdx, u + β + γd Tu, Tx. Thus d Tx, Tu α+β+γ 1 β γ dx, u, ad so d Tx, Tu dx, u. The result follows by Theorem 25. Refereces [1] Aksoy, A. G. ad Khamsi, M. A., Nostadard methods i fixed poit theory, Spriger, New York, Berli, [2] Aroszaj, N. ad Paitchpakdi, P., Extesio of uiformly cotiuous trasformatios ad hyper covex metric spaces, Pacific J. Math , [3] Ayerbe Toledao, J. M., Domiguez Beavides, T. ad Lopez Acedo, G., Measures of ocompactess i metric fixed poit theory, Birkhauser, Basel, [4] Beg, I. ad Azam, A., Fixed poits of asymptotically regular multivalued mappigs, J. Austral. Math. Soc. Ser. A , [5] Beg, I. ad Azam, A., Commo fixed poits for commutig ad compatible maps, Discuss. Math. Differetial Icl , [6] Berard, A., Characterizatio of metric spaces by the use of their mid sets itervals, Fud. Math , 1 7. [7] Blumethal, L. M., Distace Geometry, Claredo Press, Oxford, [8] Browder, F. E., Noexpasive oliear operators i Baach spaces, Proc. Nat. Acad. Sci. U.S.A , [9] Dotso, W. G., O fixed poits of oexpasive mappigs i o covex sets, Proc. Amer. Math. Soc , [10] Goeble, K. ad Kirk, W. A., Topics i metric fixed poit theory, Cambridge Stud. Adv. Math. 28, Cambridge Uiversity Press, Lodo, [11] Goeble, K. ad Reich, S., Uiform covexity, hyperolic geometry, ad oexpasive mappigs, Marcel Dekker, Ic. New York ad Basel [12] Gohde, D., Zum Prizip der kotraktive Abbildug, Math. Nachr , [13] Habiiak, L., Fixed poit theorem ad ivariet approximatio, J. Approx. Theory , [14] Hadzic, O., Almost fixed poit ad best approximatio theorems i H-Spaces, Bull. Austral. Math. Soc , [15] Khalil, R., Extreme poits of the uit ball of Baach spaces, Math. Rep. Toyama Uiv , [16] Khalil, R., Best approximatio i metric spaces, Proc. Amer. Math. Soc , [17] Kirk, W. A., A fixed poit theorem for mappigs which do ot icrease distaces, Amer. Math. Mothly ,

9 FIXED POINTS AND BEST APPROXIMATION 397 [18] Meger, K., Utersuchuge über allegemeie Metrik, Math. A , [19] Prus, B. ad Smarzewski, R. S., Strogly uique best approximatio ad ceters i uiformly covex spaces, J. Math. Aal. Appl , [20] Veeramai, P., O some fixed poit theorems o uiformly covex Baach spaces, J. Math. Aal. Appl , Ismat Beg Departmet of Mathematics, Lahore Uiversity of Maagemet Scieces Lahore, Pakista ibeg@lums.edu.pk Mujahid Abbas Departmet of Mathematics, Covermet Post Graduate College Sahiwal, Pakista