INVARIANT APPROXIMATIONS, GENERALIZED I-CONTRACTIONS, AND R-SUBWEAKLY COMMUTING MAPS

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1 INVARIANT APPROXIMATION, GENERALIZED I-CONTRACTION, AND R-UBWEAKLY COMMUTING MAP NAEER HAHZAD Received 11 May 004 and in revised form 3 August 004 We present common fixed point theory for generalized contractive R-subweakly commuting maps and obtain some results on invariant approximation. 1. Introduction and preliminaries Let beasubsetofanormedspacex = (X, )andt and I self-mappings of X. Then T is called (1) nonexpansive on if Tx Ty x y for all x, y ; ()Inonexpansive on if Tx Ty Ix Iy for all x, y ; (3)I-contraction on if there exists k [0,1) such that Tx Ty k Ix Iy for all x, y. The set of fixed points of T (resp., I) is denoted by F(T) (resp.,f(i)). The set is called (4) p- starshaped with p if for all x, the segment [x, p] joining x to p is contained in (i.e., kx +(1 k)p for all x and all real k with 0 k 1); (5) convex if is p- starshaped for all p. The convex hull co() of is the smallest convex set in X that contains, and the closed convex hull clco() of is the closure of its convex hull. The mapping T is called (6) compact if clt(d) is compact for every bounded subset D of. The mappings T and I are said to be (7) commuting on if ITx = TIx for all x ; (8) R-weakly commuting on [7] if there exists R (0, ) suchthat TIx ITx R Tx Ix for all x. uppose X is p-starshaped with p F(I) and is both T- and I-invariant. Then T and I are called (8) R-subweakly commuting on [11] if there exists R (0, ) suchthat TIx ITx Rdist(Ix,[Tx, p]) for all x, where dist(ix,[tx, p]) = inf Ix z : z [Tx, p]}. Clearly commutativity implies R-subweak commutativity, but the converse may not be true (see [11]). The set P ( x) =y : y x =dist( x,)} is called the set of best approximants to x X out of, where dist( x,) = inf y x : y }. WedefineC I ( x) =x : Ix P ( x)} and denote by I 0 the class of closed convex subsets of X containing 0. For I 0, we define x =x : x x }. It is clear that P ( x) x I 0. In 1963, Meinardus [6] employed the chauder fixed point theorem to establish the existence of invariant approximations. Afterwards, Brosowski [] obtained the following extension of the Meinardus result. Copyright 005 Hindawi Publishing Corporation Fixed Point Theory and Applications 005:1 (005) DOI: /FPTA

2 80 Invariant approximations Theorem 1.1. Let T be a linear and nonexpansive self-mapping of a normed space X, X such that T(), and x F(T). IfP ( x) is nonempty, compact, and convex, then P ( x) F(T). ingh [15] observedthattheorem 1.1 is still true if the linearity of T is dropped and P ( x) is only starshaped. He further remarked, in [16], that Brosowski s theorem remains valid if T is nonexpansive only on P ( x) x}. Then Hicks and Humphries [5]improved ingh s result by weakening the assumption T() to T( ) ; here denotes the boundary of. On the other hand, ubrahmanyam [18] generalized the Meinardus result as follows. Theorem 1.. Let T be a nonexpansive self-mapping of X, a finite-dimensional T-invariant subspace of X,and x F(T). Then P ( x) F(T). In 1981, moluk [17] noted that the finite dimensionality of in Theorem 1. can be replaced by the linearity and compactnessoft. ubsequently, Habiniak [4] observed that the linearity of T in moluk s result is superfluous. In 1988, ahab et al. [8] established the following result which contains ingh s result as a special case. Theorem 1.3. Let T and I be self-mappings of a normed space X, X such that T( ), and x F(T) F(I).upposeT is I-nonexpansive on P ( x) x}, I is linear and continuous on P ( x), andt and I are commuting on P ( x). IfP ( x) is nonempty, compact, and p-starshaped with p F(I),andifI(P ( x)) = P ( x), then P ( x) F(T) F(I). Recently,Al-Thagafi [1] generalized Theorem 1.3 and proved some results on invariant approximations for commuting mappings. More recently, with the introduction of noncommuting maps to this area, hahzad [9, 10, 11, 1, 13, 14] further extended Al-Thagafi s results and obtained a number of results regarding best approximations. The purpose of this paper is to present common fixed point theory for generalized I-contraction and R- subweakly commuting maps. As applications, some invariant approximation results are also obtained. Our results extend, generalize, and complement those of Al-Thagafi [1], Brosowski [], Dotson Jr. [3], Habiniak [4], Hicks and Humphries [5], Meinardus [6], ahab et al. [8], hahzad [9, 10, 11, 1], ingh [15, 16], moluk [17], and ubrahmanyam [18].. Main results Theorem.1. Let be a closed subset of a metric space (X,d), and T and I R-weakly commuting self-mappings of such that T() I(). uppose there exists k [0,1) such that d(tx,ty) k max d(ix,iy),d(ix,tx),d(iy,ty), 1 [ ] } d(ix,ty)+d(iy,tx) (.1) for all x, y. If cl(t()) is complete and T is continuous, then F(T) F(I) is singleton.

3 Naseer hahzad 81 Proof. Let x 0 and let x 1 be such that Ix 1 = Tx 0. Inductively, choose x n so that Ix n = Tx n 1. This is possible since T() I(). Notice d ( ) ( ) Ix n+1,ix n = d Txn,Tx n 1 k max d ( ) ( ) ( ) Ix n,ix n 1,d Ixn,Tx n,d Ixn 1,Tx n 1, 1 [ ( ) ( )] } d Ixn,Tx n 1 + d Ixn 1,Tx n = k max d ( ) ( ) Ix n,ix n 1,d Ixn,Tx n, d ( } (.) ) 1 Ix n 1,Tx n 1, d( ) Ix n 1,Tx n k max d ( ) ( ) Ix n,ix n 1,d Ixn,Tx n, 1 [ ( ) ( )] } d Ixn 1,Ix n + d Ixn,Tx n kd ( ) Ix n,ix n 1 for all n. This shows that Ix n } is a Cauchy sequence in. Consequently, Tx n } is a Cauchy sequence. The completeness of cl(t()) further implies that Tx n y and so Ix n y as n.incet and I are R-weakly commuting, we have This implies that ITx n Ty as n.now d ( TIx n,itx n ) Rd ( Txn,Ix n ). (.3) d ( Tx n,ttx n ) k max d ( Ix n,itx n ),d ( Ixn,Tx n ),d ( ITxn,TTx n ), 1 [ ( ) ( )] } d Ixn,TTx n + d ITxn,Tx n. Taking the limit as n,weobtain d ( y,ty ) k max d(y,ty),d(y, y),d(ty,ty), 1 [ ] } d(y,ty)+d(ty, y) = kd(y,ty), (.4) (.5) which implies y = Ty.inceT() I(), we can choose z such that y = Ty = Iz. ince d ( TTx n,tz ) k max d ( ITx n,iz ),d ( ) ( ) ITx n,ttx n,d Iz,Tz, 1 [ ( d ITxn,Tz ) + d ( )] } (.6) Iz,TTx n,

4 8 Invariant approximations taking the limit as n yields d(ty,tz) kd(ty,tz). (.7) This implies that Ty= Tz. Therefore, y = Ty= Tz = Iz. Using the R-weak commutativity of T and I,weobtain d(ty,iy) = d(tiz,itz) Rd(Tz,Iz) = 0. (.8) Thus y =Ty=Iy.Clearlyy is a unique common fixed point of T and I.Hence F(T) F(I) is singleton. Theorem.. Let be a closed subset of a normed space X, andt and I continuous selfmappings of such that T() I(). upposei is linear, p F(I), is p-starshaped, and cl(t()) is compact. If T and I arer-subweakly commuting and satisfy Tx Ty max Ix Iy,dist ( Ix,[Tx, p] ),dist ( Iy,[Ty, p] ), 1 [ ( ) ( )] } (.9) dist Ix,[Ty, p] + dist Iy,[Tx, p] for all x, y, then F(T) F(I). Proof. Choose a sequence k n } [0,1) such that k n 1asn. Define, for each n, a map T n by T n (x) = k n Tx+(1 k n )p for each x.theneacht n is a self-mapping of. Furthermore, T n () I()foreachn since I is linear and T() I(). Now the linearity of I and the R-subweak commutativity of T and I imply that T n Ix IT n x = kn TIx ITx k n Rdist ( Ix,[Tx, p] ) k n R T n x Ix for all x. This shows that T n and I are k n R-weakly commuting for each n.also T n x T n y = kn Tx Ty k n max Ix Iy,dist ( Ix,[Tx, p] ),dist ( Iy,[Ty, p] ), 1 [ ( ) ( )] } dist Ix,[Ty, p] + dist Iy,[Tx, p] k n max Ix Iy, Ix T n x, Iy T n y, 1 [ Ix T n y + Iy T n x ]} (.10) (.11) for all x, y. Now Theorem.1 guarantees that F(T n ) F(I) =x n } for some x n. The compactness of cl(t()) implies that there exists a subsequence x m } of x n } such

5 Naseer hahzad 83 that x m y as m. By the continuity of T and I,wehavey F(T) F(I). Hence F(T) F(I). The following corollaries extend and generalize [3, Theorem 1] and [4,Theorem4]. Corollary.3. Let be a closed subset of a normed space X,andT and I continuous selfmappings of such that T() I(). upposei is linear, p F(I), is p-starshaped, and cl(t()) is compact. If T and I are R-subweakly commuting and T is I-nonexpansive on, then F(T) F(I). Corollary.4. Let be a closed subset of a normed space X, andt and I continuous self-mappings of such that T() I(). upposei is linear, p F(I), is p-starshaped, and cl(t()) is compact. If T and I are commuting and satisfy (.9) forallx, y, then F(T) F(I). Let D R,I ( x) = P ( x) G R,I ( x),where G R,I ( x) = x : Ix x (R + 1)dist( x,) }. (.1) Theorem.5. Let T and I be self-mappings of a normed space X with x F(T) F(I) and X such that T( ). upposei is linear on D R,I ( x), p F(I), D R,I ( x) is closed and p-starshaped, clt(d R,I ( x)) is compact, and I(D R,I ( x)) = D R,I ( x). IfT and I are R- subweakly commuting and continuous on D R,I ( x) and satisfy, for all x D R,I ( x) x}, Ix I x if y = x, Tx Ty max Ix Iy,dist ( Ix,[Tx, p] ),dist ( Iy,[Ty, p] ), 1 [ ( ) ( )] } dist Ix,[Ty, p] + dist Iy,[Tx, p] if y D R,I ( x), (.13) then P ( x) F(T) F(I). Proof. Let x D R,I ( x). Then x (see [1]) and so Tx since T( ).Now Tx x = Tx T x Ix I x = Ix x =dist( x,). (.14) This shows that Tx P ( x). From the R-subweak commutativity of T and I, it follows that ITx x = ITx T x R Tx Ix + I x I x (R + 1)dist( x,). (.15) This implies that Tx G R,I ( x). Consequently, Tx D R,I ( x)andsot(d R,I ( x)) D R,I ( x) = I(D R,I ( x)). Now Theorem. guarantees that P ( x) F(T) F(I). Theorem.6. Let T and I be self-mappings of a normed space X with x F(T) F(I) and X such that T( ) I(). upposei is linear on D R,I ( x), p F(I), D R,I ( x) is closed and p-starshaped, clt(d R,I ( x)) is compact, and I(G R,I ( x)) D R,I ( x) I(D R,I ( x)) D R,I ( x). IfT and I are R-subweakly commuting and continuous on D R,I ( x) and satisfy, for all x D R,I ( x) x},(.13), then P ( x) F(T) F(I).

6 84 Invariant approximations Proof. Let x D R,I ( x). Then, as in Theorem.5, Tx D R,I ( x), that is, T(D R,I ( x)) ( x). Also (1 k)x + k x x < dist( x,)forallk (0,1). This implies that x D R,I (see [1]) and so T(D R,I ( x)) T( ) I(). Thus we can choose y such that Tx = Iy.inceIy = Tx P ( x), it follows that y G R,I ( x). Consequently, T(D R,I ( x)) I(G R,I ( x)) P ( x). Therefore, T(D R,I ( x)) I(G R,I ( x)) D R,I ( x) I(D R,I ( x)) D R,I ( x). Now Theorem. guarantees that P ( x) F(T) F(I). Remark.7. Theorems.5 and.6 remain valid when D R,I ( x) = P ( x). If I(P ( x)) ( x) = P ( x). Consequently, Theo- P ( x), then P ( x) C( x) I G R,I rem.5 contains Theorem 1.3 as a special case. ( x) (see [1]) and so D R,I The following result includes [1, Theorem 4.1] and [4, Theorem 8]. It also contains the well-known results due to moluk [17] and ubrahmanyam [18]. Theorem.8. Let T be a self-mapping of a normed space X with x F(T) and I 0 such that T( x ). IfclT( x ) is compact and T is continuous on x and satisfies for all x x x} x x if y = x, Tx Ty max x y,dist ( x,[tx,0] ),dist ( y,[ty,0] ), 1 [ ( ) ( )] } dist x,[ty,0] + dist y,[tx,0] if y x, then (i) P ( x) is nonempty, closed, and convex, (ii) T(P ( x)) P ( x), (iii) P ( x) F(T). Proof. (i) We may assume that x.ifx \ x,then x > x. Notice that (.16) x x x x > x dist ( x, x ). (.17) Consequently, dist( x, x ) = dist( x,) x. Also z x =dist( x,clt( x )) for some z clt( x ). Thus dist ( x, x ) ( )) ( )) dist ( x,clt x dist ( x,t x Tx x = Tx T x (.18) x x for all x x. This implies that z x =dist( x,) andsop ( x) isnonempty.furthermore, it is closed and convex. (ii) Let y P ( x). Then Ty x = Ty T x y x =dist( x,). (.19) This implies that Ty P ( x)andsot(p ( x)) P ( x).

7 Naseer hahzad 85 (iii) Theorem. guarantees that P ( x) F(T) since clt(p ( x)) clt( x )and clt( x )iscompact. Theorem.9. Let I and T be self-mappings of a normed space X with x F(I) F(T) and I 0 such that T( x ) I().upposethatI is linear, Ix x = x x for all x, cli( x ) is compact and I satisfies, for all x, y x, Ix Iy max x y,dist ( x,[ix,0] ),dist ( y,[iy,0] ), 1 [ ( ) ( )] } dist x,[iy,0] + dist y,[ix,0]. (.0) If I and T are R-subweakly commuting and continuous on x and satisfy, for all x x x}, and p F(I), Ix I x if y = x, Tx Ty max Ix Iy,dist ( Ix,[Tx, p] ),dist ( Iy,[Ty, p] ), 1 [ ( ) ( )] } dist Ix,[Ty, p] + dist Iy,[Tx, p] if y x, (.1) then (i) P ( x) is nonempty, closed, and convex, (ii) T(P ( x)) I(P ( x)) P ( x), (iii) P ( x) F(I) F(T). Proof. From Theorem.8, (i) follows immediately. Also, we have I(P ( x)) P ( x). Let y T(P ( x)). ince T( x ) I() andp ( x) x, there exist z P ( x) andx 1 such that y = Tz = Ix 1.Furthermore,wehave Ix 1 x = Tz T x Iz I x z x =d( x,). (.) Thus x 1 C( x) I = P ( x) and so (ii) holds. ince, by Theorem.8, P ( x) F(I), it follows that there exists p P ( x)suchthat p F(I). Hence (iii) follows from Theorem.. The following corollary extends [1, Theorem 4.(a)] to a class of noncommuting maps. Corollary.10. Let I and T be self-mappings of a normed space X with x F(I) F(T) and I 0 such that T( x ) I(). upposethati is linear, Ix x = x x for all x, cli( x ) is compact, and I is nonexpansive on x.ifi and T are R-subweakly commuting on x and T is I-nonexpansive on x x}, then (i) P ( x) is nonempty, closed and convex, (ii) T(P ( x)) I(P ( x)) P ( x),and (iii) P ( x) F(I) F(T).

8 86 Invariant approximations Acknowledgment The author would like to thank the referee for his suggestions. References [1] M. A. Al-Thagafi, Common fixed points and best approximation, J. Approx. Theory 85 (1996), no. 3, [] B. Brosowski, Fixpunktsätze in der Approximationstheorie, Mathematica(Cluj)11 (34) (1969), (German). [3] W.G.DotsonJr.,Fixed point theorems for non-expansive mappings on star-shaped subsets of Banach spaces,j.london Math.oc. ()4(197), [4] L. Habiniak, Fixed point theorems and invariant approximations, J.Approx.Theory56 (1989), no. 3, [5] T. L. Hicks and M. D. Humphries, A note on fixed-point theorems, J. Approx. Theory34 (198), no. 3, 1 5. [6] G. Meinardus, Invarianz bei linearen Approximationen, Arch. Rational Mech. Anal. 14 (1963), (German). [7] R. P. Pant, Common fixed points of noncommuting mappings, J. Math. Anal. Appl. 188 (1994), no., [8].A.ahab,M..Khan,and.essa,A result in best approximation theory, J. Approx. Theory 55 (1988), no. 3, [9] N. hahzad, A result on best approximation, Tamkang J. Math. 9 (1998), no. 3, 3 6. [10], Correction to: A result on best approximation, Tamkang J. Math. 30 (1999), no., 165. [11], Invariant approximations and R-subweakly commuting maps, J. Math. Anal. Appl.57 (001), no. 1, [1], Noncommuting maps and best approximations, Rad. Mat.10 (001), no. 1, [13], On R-subcommuting maps and best approximations in Banach spaces, Tamkang J. Math. 3 (001), no. 1, [14], Remarks on invariant approximations, Int. J. Math. Game Theory Algebra 13 (003), no., [15]. P. ingh, An application of a fixed-point theorem to approximation theory, J.Approx.Theory 5 (1979), no. 1, [16], Application of fixed point theorems in approximation theory, Applied Nonlinear Analysis (Proc. Third Internat. Conf., Univ. Texas, Arlington, Tex, 1978) (V. Lakshmikantham, ed.), Academic Press, New York, 1979, pp [17] A. moluk, Invariant approximations, Mat. tos.(3) 17 (1981), 17 (Polish). [18] P. V. ubrahmanyam, An application of a fixed point theorem to best approximation,j.approximation Theory 0 (1977), no., Naseer hahzad: Department of Mathematics, King Abdul Aziz University, P.O. Box 8003, Jeddah 1589, audi Arabia address: nshahzad@kaau.edu.sa