Journal of Applied Mathematics and Physics, 2014, 2, 170175 Published Online April 2014 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.4236/jamp.2014.25021 How to cite this paper: Khan, A.R. and Fukharuddin, H. (2014) Common Fixed Point Iterations of Generalized Asymptoti cally QuasiNonexpansive Mappings in Hyperbolic Spaces. Journal of Applied Mathematics and Physics, 2, 170175. http://dx.doi.org/10.4236/jamp.2014.25021 Common Fixed Point Iterations of Generalized Asymptotically QuasiNonexpansive Mappings in Hyperbolic Spaces A. R. Khan, H. Fukharuddin Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia Email: arahim@kfupm.edu.sa, hfdin@kfupm.edu.sa Received Dec emb er 2013 Abstract We introduce a general iterative method for a finite family of generalized asymptotically quasi nonexpansive mappings in a hyperbolic space and study its strong convergence. The new iterative method includes multistep iterative method of Khan et al. [1] as a special case. Our results are new in hyperbolic spaces and generalize many known results in Banach spaces and CAT(0) sp aces, simultaneousl y. Keywords Hyperbolic Space, General Iterative Method, Generali zed Asymptot ically Qu asiNonex pansi ve Mapping, Common Fixed Point, S tron g Conve rgenc e 1. Introduction Let be a nonempty subset of a metric space and be a mapping. Throughout this paper, we assume that , the set of fixed points of is nonempty and The mapping is: 1) asymptotically nonexpansive if there exists a sequence of real numbers in with such that ( ) ( ) ,1(, ) nn n dTxTyud xy≤+ for all and 2) asymptotically quasi nonexpansive if there exists a sequence of real numbers in with such that ( ) ( ) ,1(, ) nn dTxpu dxp≤+ for all and 3) generalized asymptotically quasinonexpansive if there exist two se quences of real numbers and in with such that ( ) ( )( ) ,, , nnn dTxpdxp udxpc≤+ + for all and (iv) uniformly Lipschitzian if there exists a constant such that for all and (v)
A. R. Khan, H. Fukharuddin uniformly Lipschitzian if there are constants such that for all and and (vi) semicompact if for any sequence in wit h , there exists a subsequence of such that Let be a metric space. Suppose that there exists a family of metric segments such that any two points in are endpoints of a unique metric segment is an isometric image of the real line interval ). We shall denote by the unique point of which satisfies ( ,)(1)(,)and(,)( ,)for[0,1].dxzdxy dzydxyJ α αα =−= ∈= Such metric spaces are usually called convex metric spaces [2] [3]. One can easily deduce that , and from the definition of a convex metric space [2]. A convex metric space is hyperbolic if ((1) ,(1))(,)(1) ( ,)dxy zwdxzdyw αααααα ⊕−⊕−≤+− for all and . For , the hyperbolic inequality reduces to convex structure [3]. ( ) ( ) ()() () 1,, 1,.dxyzd xzdyz αα αα ⊕− ≤+− (1.1) A nonempty subset of a convex metric space is convex if for all and Normed spaces and their subsets are linear hyperbolic spaces while spaces [4][6] qualify for the criteria of nonlinear hyperbolic spaces [2] [7]. A convex metric space is uniformly convex [7] if ()() () () 1 11 ,inf 1,:,,,,,0, 22 rdaxyd axrdayrdxyr r δε ε = −⊕≤≤≥> for any and . From now onwards we assume that is a uniformly convex hyperbolic space with the property that for every , there exists depending on and such that for any . We now translate the iterative method (1.3) [1] from normed space setting to the more general setup of hy perbolic space as follows: (1.2) whe r e ( ) ( ) ( ) ( ) ( ) 0 111 01 2 2212 1 the identity mapping 1 1 1 n n n nnn n n nnn n rnrn rrn rn UI U xaTUxax Uxa TUxax U xaTUxax − = = ⊕− = ⊕− = ⊕− and is a family of generalized asymptotically quasinonexpansive selfmappings of , i.e. , ( ) ()() ,1 , n iiini in dTxpu dxpc≤+ + for all and and are sequences in with and for each . The purpose of this paper is to: 1) establish convergence of iterative method (1.2) to a common fixed point of a finite family of generalized asymptotically quasinonexpansive mappings on a hyperbolic space(uniformly convex hyperbolic space). Our work is a significant generalization of the corresponding results in Banach spaces and spaces. In the sequel, we assume that 2. Convergence Theorems in Hyperbolic Space Lemma 2.1. Let be a nonempty, closed and convex subset of a hyperbolic space . Then, for the sequence
A. R. Khan, H. Fukharuddin in (1.2), there are sequences and in satisfying such that 1) ()()() 1 ,1, , nnn n dxpv dx p ξ + ≤+ + for all and all 2) ()() 11 ,( ,), nmn n n dxpMdx p ξ ∞ += ≤+ ∑ for all and Proof. (a) Let and for all Since for each , therefore . Now we have ()( ) ( ) ()() ( ) ()()() ()()()()() 111 01111 1 111111 1 ,1 ,1,, 1,1, 1,1,. nn nnnnnn nnnnn nnnnn nnnnnn n dUxpdaTUxaxpa dxpadTxp adx paudx pcudx pcvdx pc =⊕− ≤−+ ≤− ++ +≤+ +≤++ Assume that ()()() () 1 1 ,1 ,1 kk kn nnnnin k i dUxpv dxpvc − = ≤+++ ∑ holds for some Consider ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )() ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )( )( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( )( ) 11 1111 1 11 111 111 11 111 1 ,1 ,1,, 1 ,1, 1,1 , 1 ,11 nn nkknnnnkkn n kn knknknkn nkn n knkn knknkn nkn n knknkn knkn k n nn knknkn kn dUxpdaTUxaxpadxp adTUxp adxp audUxp ac adxp acaudUxp adx pacavv ++ ++++ + ++ +++ +++ ++ +++ + =⊕−≤−+ ≤−+ ++ ≤−++ + ≤−++++ ()( ) ( ) ( ) ()() ( ) ( ) ( )( ) ()() ( ) ( ) ( )()( ) 1 1 11 11 11 1 1 1 1 1 1 1 ,1 11 ,11 ,1 1 ,1 k nn in k kk nn nnn nin knknkn knkn kk n k i k i k i nn in dx pvc a vdxpavcavdxpavc vdx pvc − + +− = + = + = ++ +++ + ++ ≤−+++++ ++ ≤+ ++ ∑ ∑ ∑ By mathematical induction, we have ( ) () () () 1 1 ,1, 1,1. jj j jn nnnnin i dUxpvdxpvcjr − = ≤+ ++≤≤ ∑ (2.1) Now, by (1.2) and (2.1), we obtain ( ) ( ) ( ) ( ) ( ) ( ) ()() ( ) ( ) ( ) ()() ()()( )()()( ) ( )()( ) 11 1 1 1 12 1 1 1 , 1, ,1 , 1,1, 1 1,11, 1 ,11 n nrn rrn rn n rn rnrnn rn rnrnnrnrn n rn r rr rnrnnnninrn rnrnn i r rr rn nnrn nininr i dxpd aTUxaxp adTUxpa dxp audUxpcadxp auvdx pvcacadx p avdxpav cca +− − − − −− = − = = ⊕− ≤ +− ≤++ +− ≤ ++++++− ≤++ ++− ∑ ∑ ()() ( )()( ) () () ( ) ( )() () ()()() () 1 1 1 1 11 1 1 , 1 1,1 11 1 1,1 ! 1,1 1,, nn rr r rn rnnnrnnin i rr r k rn rnnnrnnin ki r r rr nnninnnn i dx p aavdxpav c rrr k aavdx pavc k vdx pvcvdx p ξ − = − − = = − = ≤−++ ++ −… −+ =−+++ + ≤+ ++≤+ + ∑ ∑∑ ∑ Where ( ) 1 1 supsup 1, rr n nin i M MvMc ξ − = ==+= ∑ and . (b) We know that for . Thus, by part (a), we have
A. R. Khan, H. Fukharuddin () ()( ) ()( ) () () () () ( ) ( ) ( ) ( ) () 11 1 11 1 1 2212 1 11 1 1 11 1 1 ,1 , exp , exp , exp , exp,, , where r nmnm nmnm nm nmnm nmnmnmnm nm nmnmnm ini i inin in ini ni ii i dx pdxp rdx p rrdx p rvdx pv rvdx pM dx p M νξ νξ ν νξξ ξ ξξ ++−+−+− +− +−+− +−+−+−+− +− +−+− +− = =+= ∞∞ ∞ = == ≤+ + ≤+ ≤ +++ ≤+ ≤+=+ ∑ ∑∑ ∑∑ ∑ ( ) 1 exp . i i rv ∞ = =∑ Theorem 2.2. Let be a nonempty, closed and convex subset of a complete hyperbolic space . Then the sequence in (1.2) converges strongly to a point in if and only if , where . Proof. We only prove the sufficiency. By Lemma 2.1 (a), we have ()()() 1 ,1,for all and 1. r nnn n dxpvdx ppFn ξ + ≤++∈ ≥ Therefore, ( ) () () ( ) ( ) ( ) ( ) 11 , 111!, rk nnn n k dxFrrrkk vdxF ξ += ≤+−… −++ ∑ As so ()() ( ) ( ) 11 1 1! rk n nk rrrkk v ∞ = = − …− +<∞ ∑∑ . Now in Lemma 2.1 (a), so by Lemma 1.1 [1] and , we get that . Let From the proof of Lemma 2.1 (b), we have ()() () ( ) ( ) 11 ,, ,1, nm nnmnni in dxxdxFdxFM dxFM ξ ++ ∞ = ≤+≤++ ∑ (2.2) Since and , therefore there exists a natural number such that and for all . So for all integers we obtain from (2.2) that ( ) () () 11 1 1 ,1 . 2 21 nmn dx xMMM M εε ε + ≤++ = + Thus, is a Cauchy sequence in and so converges to . Finally, we show that . For any , there exists natural number such that ()( ) , inf,3 npFn dxFdx p ε ∈ = < and for all . There must exist such that for all , in particular, and Hence () () ( ) ** 11 ,, , nn dpqdxpdx q ε ≤ +< . Since is arbitrary, therefore That is, Theorem 2.3. Let be a nonempty, closed and convex subset of a complete convex metric space , If for the sequence in (1.2), and one of the mappings is semicompact, then converges strongly to . Proof. Let be semicompact for some Then there exists a subsequence of such that Hence ( )() ,lim ,0. j li li n dpTpdx Tx →∞ = = Thus and so by Theorem 2.2, converges strongly to a common fixed point of the family of mappings.
A. R. Khan, H. Fukharuddin 3. Results in a Uniformly Convex Hyperbolic Space Lemma 3.1.Let be a nonempty, closed and convex subset of a uniformly convex hyperbolic space . Then, for the sequence in (1.2) with for some , we have exists for all (b) for each . Proof. (a) Let and for all .By Lemma 1.1 [1] and Lemma 2.1 (a), it follows that exists. Assume that (3.1) (b) The inequality (2.1) together with (3.1) gives that ( ) limsup, ,1. njn n dU xpcjr →∞ ≤ ≤≤ (3.2) Note that () () ( ) ( ) () ( ) ( ) () ()( ) ( ) ( )()() ( ) ()() ( ) ()( ) () ( ) ( ) ()( ) () 11 1 1 12 1 1 12 1 , ,1, 1,1, 11,1 , 1,1,1 , n nrnnrn rnrnn rn rnnnrnrn n rn n rnnrnnrn rnrnn rnr nrn n rnnrnnrn rnrnn rnr nrn r dxpdUxpdaTUxa xp avdUxpcadxp avdaTUxaxpaca dxp avadTUxpadxpaca dxp a +− − − −− − − −− − = =⊕− ≤++ +− =+⊕−++ − ≤ ++−++− ≤ ( ) ( ) ( ) ( ) ( ) ( ) () () ( ) () ( ) () ( ) ( ) () () () () ( )( ) 22 12 1 22 11 11 1 12 1,11 , 11 1,11, 11 1. nnnrnn n rnrnrn rnnrnnrn rn rn rr rj rj innjn ninnn ij ij rr rjrj rj in nin n jnrnnrn jn ij ij av dUxpaav dxp aavca vc av dUxpav dxp a vca vcavc −− − −− −− =+=+ − −− + =+=+ ++− + ++ ++ ≤ ++−+ +++++…+ + ∏∏ ∏∏ and therefore, we have ( )()()( ) ( ) ( ) 11 1 ,, ,, 1 jn nn rn rj rjrj njn njn rj n c dx pdxpc dxpdUxp c v δ δδ δ + − −−+ + − ≤−+++++ + Hence , (3.3) Using (3.2) and (3.3), we have . That is, ( ) ( ) ( ) 1 li 1,m n njn jnjnn jn d aTUxaxpc →∞ − ⊕− = for This together with (3.1), (3.2) and Lemma 2.5 [8] gives that ( ) ( ) 1 lm ,0i n n jnn jn dTUx x →∞ − = for (3.4) If , we have by (3.4), . In case , we observe that ( ) ( ) ( )()() ( ) ( ) ( )() ( ) 11 11211 2 ,, ,1,0 nn nnnjnnjnn jnjnj njnjnjn dxUxdxaTU xaxadTU xx −− −−−−−− =⊕− ≤→ (3.5)
A. R. Khan, H. Fukharuddin Since is uniformly Lipschitzian, therefore the inequality () () () ( ) ( ) ( ) ( ) () ( ) , 111 1 , ,,,, n nnnn jnnjnj nj nnnnjnn jn jnjnjn dTxxdTxTU xdTU xxLdxU xdTU xx γ −−− − ≤+ ≤+ together with (3.4) and (3.5) gives that ( ) lim , 0. n jnn n dTx x →∞ = Hence, for (3.6) Note that ( ) ( ) ( ) ( ) ( ) ( ) 111 , ,1,0. nn nnnrn rnnrnnrnnrn rn rn dxxdxaTU xax adxTU x +−− =⊕− ≤→ Let us observe that: ( ) ( ) ()() ( ) ( ) () ( ) ( ) 111 1 1 111 1 1 111 ,, ,,, , ,,,. nnn n njnnnnjnjnjnjnjn nn nnnjnnnjnn dxTxdxxdxTxdTxTx dTxTx dx xdxTxLdxxLdTx x γ γ +++ + + +++ + + +++ ≤+ ++ ≤++ + So by uniformly Lipschitzian property of , (3.5) and (3.6), we get ( ) , 0,1lim nn jn d xTxjr →∞ = ≤≤ . Theorem 3.2. Under the hypotheses of Lemma 3.1, assume that, for some , is semicompact for some positive integer m. Then in (1.2), converges strongly to a point in . Proof. Fix and suppose is semicompact for some By Lemma 3.1 (b), we obtain ()() ()() ( ) ( ) ( ) 1 122 ,,,, , ,1(, )0. mmmmm jnnjnj nj njnjnjnjnn jn njn n dTxxdT xTxdTx TxdTxTxdTxx d TxxmLdTxx γ − −− ≤++++ ≤ +−→ Since is bounded and is semicompact, has a convergent subsequence such that . Hence, by Lemma 3.1 (b), we have ( ) ( ) , lim,0, jj inn in d qTqdxTxiI →∞ == ∈ . Thus and so by Theorem 2.2, converges strongly to a common fixed point of the family . Acknowledgements The author A. R. Khan is grateful to KACST for supporting the research project ARP3234. The author H. Fukharuddin acknowledges King Fahd University of Petroleum & Minerals for supporting research project IN121037. References [1] Khan, A.R., Domlo, A.A. and Fukharuddin, H. (2008) Common Fixed Points Noor Iteration for a Finite Family of Asymptotically QuasiNonexpansive Mappings in Banach Space. Journal of Mathematical Analysis and Applications. 341, 111. http://dx.doi.org/10.1016/j.jmaa.2007.06.051 [2] Menger, K. (1928) Untersuchungenüberallgemeine Metrik. Mathematische Annalen, 100, 75163. http://dx.doi.org/10.1007/BF01448840 [3] Takahashi, W. (1970) A Convexity in Metric Spaces and Nonexpansive Mappings. Kodai. Math Sem. Rep., 22, 142149. http://dx.doi.org/10.2996/kmj/1138846111 [4] Bridson, M. and Haefliger, A. (1999) Metric Spaces of NonPositive Curvature. SpringerVerlag, Berlin, Heidelberg, New York. http://dx.doi.org/10.1007/978366212494 9 [5] Fukharuddin, H. (2013) Strong Convergence of an Ishikawatype Algorithm inCAT (0) Spaces. Fixed Point Theory and Applications, 2013, 207. [6] Khan, A.R., Khamsi, M.A. and Fukharuddin, H. (2011) Strong Convergence of a General Iteration Scheme in CAT(0) Spaces, Nonlinear Anal. 74, 783791. http://dx.doi.org/10.1016/j.na.2010.09.029 [7] Goebel, K. and Reich, S. (1984) Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Series of Monographs and Textbooks in Pure and Applied Mathematics, Dekker, New York. [8] Khan, A.R., Fukharuddin, H. and Khan, M.A.A. (2012) An Implicit Algorithm for Two Finite Families of Nonex pansive Maps in Hyperbolic Spaces. Fixed Point Theory and Applications, 2012, 54.
