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) 79 86 DOI: 10.1155/FPTA.005.79
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.
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,
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
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).
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).
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).
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, 318 33. [] B. Brosowski, Fixpunktsätze in der Approximationstheorie, Mathematica(Cluj)11 (34) (1969), 195 0 (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), 408 410. [4] L. Habiniak, Fixed point theorems and invariant approximations, J.Approx.Theory56 (1989), no. 3, 41 44. [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), 301 303 (German). [7] R. P. Pant, Common fixed points of noncommuting mappings, J. Math. Anal. Appl. 188 (1994), no., 436 440. [8].A.ahab,M..Khan,and.essa,A result in best approximation theory, J. Approx. Theory 55 (1988), no. 3, 349 351. [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, 39 45. [1], Noncommuting maps and best approximations, Rad. Mat.10 (001), no. 1, 77 83. [13], On R-subcommuting maps and best approximations in Banach spaces, Tamkang J. Math. 3 (001), no. 1, 51 53. [14], Remarks on invariant approximations, Int. J. Math. Game Theory Algebra 13 (003), no., 157 159. [15]. P. ingh, An application of a fixed-point theorem to approximation theory, J.Approx.Theory 5 (1979), no. 1, 89 90. [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. 389 394. [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., 165 17. Naseer hahzad: Department of Mathematics, King Abdul Aziz University, P.O. Box 8003, Jeddah 1589, audi Arabia E-mail address: nshahzad@kaau.edu.sa