992752075 height=28.4049996376038  />Then the Ishikawa iteration process denoted by in a geodesic space X is as under:

(1.2)

We say that is well-defined if

2. Fixed Point Approximation

Following the investigations of Hussain and Khamsi [8], the existence of the fixed point of pointwise asymptotic nonexpansive map can not be achieved without its bounded domain. We shall follow them for the purpose. We start with proving the following lemma.

Lemma 2.1. Let C be a nonempty, bounded, closed and convex set in a geodesic space X and let Let be such that the sequence in (1.2) is well defined. If the set is quasiperiodic and

(2.1)

then

Proof. Set and

From

we have

(2.2)

Also

(2.3)

Using (2.1) and (2.2) in (2.3), we have

(2.4)

Since

(2.5)

therefore taking on both the sides of inequality (2.5) and using (2.1) and (2.4), we get

and hence

Similarly

That is,

Remark 2.2. Lemma 2.1 extends the corresponding Lemma 3 of Khan and Takahashi [22] from Lipschitzian to non-Lipschitzian maps.

Lemma 2.3. Let be a nonempty, bounded, closed and convex subset of a Hadamard space and let

. Let for some

and be such that the sequence in is well-defined. If the set is quasiperiodic and then  

Proof. Let Then use (CN) inequality (1.1) for the scheme (1.2) to have

Since is bounded, there exists such that for some Therefore the above inequality becomes

(2.6)

From (2.6), the following two inequalities are obtained

(2.7)

and

(2.8)

Now, we prove that

(2.9)

First assume Then there exists a subsequence(use the same notation for subsequence as for the sequence) of and such that.

From (2.7), it follows that

Since and so there exists such that for all

Hence the above inequality reduces to

(2.10)

Let be any positive integer. Then from (2.10), we have

(2.11)

Letting in (2.11), we get

a contradiction.

Hence

Consequently, we have

(2.12)

Following the similar procedure of proof with (2.8), we conclude

(2.12.1)

Since

therefore with the help of (2.2) and (2.12), we get

Finally, Lemma 2.1 appeals that

(2.13)

Let be a bounded sequence in a metric space X. For define The asymptotic radius of is given by:

A bounded sequence in is regular if for every subsequence of

The asymptotic center of a bounded sequence with respect to is defined

If the asymptotic center is taken with respect to then it is simply denoted by

A bounded sequence in X. is said to be regular if for every subsequence of Recall that a sequence converges weakly to w (written as) if and only if where C is a closed and convex subset containing the bounded sequence Moreover, a sequence (in X.) Δ-converges to if x is the unique asymptotic center of for every subsequence of In this case, we write and x is called Δ-limit of

In a Banach space setting, Δ-convergence coincides with weak convergence. A connection between weak convergence and Δ-convergence in geodesic spaces is characterized in the following lemma due to Nanjaras and Panyanak [26].

Lemma 2.4. ([26], Proposition 3.12). Let be a bounded sequence in a space and let be a closed and convex subset of and contains. Then 1) implies that

2) the converse of (1) is true if is regular.

Next, we state the demiclosed principle in spaces due to Hussain and Khamsi [8] needed in the next convergence theorem.

Lemma 2.5. Let be a nonempty, bounded, closed and convex set in a space X. and be a pointwise asymptotic nonexpansive map. Let be a sequence in such that and Then

Next, we prove our weak convergence theorem.

Theorem 2.6. Let C be a nonempty, bounded, closed and convex set in a Hadamard space X. and let

. Let for some

and be such that the sequence in is well defined. If the set is quasiperiodic and then converges weakly to a point in

Proof. Let be the weak -limit set of given by

Since C is a nonempty bounded closed convex subset of a Hadamard space, there exists a subsequence of such that as and vice versa. This shows that As

and (by Lemma 2.1), therefore by Lemma 2.5, That is, Next, we follow the idea of Chang et al. [14]. For any there exists a subsequence of such that

(2.14)

Hence from (2.12) and (2.14), it follows that

(2.15)

Now from (1.2), (2.14) and (2.15), we get that

(2.16)

Also from (2.12) and (2.14), we have that

(2.17)

Again from (1.2), (2.14) and (2.17), we conclude that

Continuing in this way, by induction, we can prove that, for any

By induction, one can prove that converges weakly to as in fact gives that as

Remark 2.7. If is regular in a geodesic space, then is Δ-convergent.

Our strong convergence theorem is as follows. We do not use the rate of convergence condition namely

in its proof.

Theorem 2.8. Let C be a nonempty, bounded, closed and convex set in a Hadamard space X and let

. Let for some

and be such that the sequence in is well defined. If the set is quasiperiodic, and either or is semi-compact (completely continuous), then converges strongly to a point in F.

Proof. Let S be semi-compact. As , there exists a subsequence of such that

Using in (2.13) and continuity of and, we obtain that The rest of the proof follows by replacing with in Theorem 2.6 and we, therefore, omit the details.

Finally, we state a theorem due to Nanjaras and Panyanak [26] proved in Hadamard spaces in which rate of convergence condition is necessary for Δ-convergence of the sequence.

Theorem 2.9. Let C be a nonempty, bounded, closed and convex set in a Hadamard space X and let with a sequence for which Suppose that and is a sequence in for some. Then the sequence, Δ-converges to a fixed point of T.

We pose the following open question.

Open question: Does Theorem 2.6 hold if we replace weak convergence by Δ-convergence?

REFERENCES

  1. K. K. Tan and H. K. Xu, “Fixed Point Iteration Processes for Asymptotically Nonexpansive Mapping,” Proceedings of the American Mathematical Society, Vol. 122, No. 3, 1994, pp. 733-739. doi:10.1090/S0002-9939-1994-1203993-5
  2. M. Bridson and A. Haefliger, “Metric Spaces of Non-Positive Curvature,” Springer-Verlag, Berlin, 1999.
  3. S. Dhompongsa and B. Panyanak, “On Δ-Convergence Theorems in CAT(0) Spaces,” Computers & Mathematics with Applications, Vol. 56, No. 10, 2008, pp. 2572-2579. doi:10.1016/j.camwa.2008.05.036
  4. F. Bruhat and J. Tits, “Groupes Réductifs Sur Un Corps Local. I. Données Radicielles Valuées,” Institut des Hautes Études Scientifiques Publications Mathématiques, Vol. 41, No. 1, 1972, pp. 5-251. doi:10.1007/BF02715544
  5. W. A. Kirk, “A Fixed Point Theorem in CAT(0) Spaces and R-Trees,” Fixed Point Theory and Applications, Vol. 4, 2004, pp. 309-316.
  6. W. A. Kirk and H. K. Xu, “Asymptotic Pointwise Contractions,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 69, No. 12, 2008, pp. 4706-4712. doi:10.1016/j.na.2007.11.023
  7. K. Goebel and W. A. Kirk, “A Fixed Point Theorem for Asymptotically Nonexpansive Mappings,” Proceedings of the American Mathematical Society, Vol. 35, 1972, pp. 171-174. doi:10.1090/S0002-9939-1972-0298500-3
  8. N. Hussain and M. A. Khamsi, “On Asymptotic Pointwise Contractions in Metric Spaces,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 71, No. 10, 2009, pp. 4423-4429. doi:10.1016/j.na.2009.02.126
  9. C. W. Groetsch, “A Note on Segmenting Mann Iterates,” Journal of Mathematical Analysis and Applications, Vol. 40, No. 2, 1972, pp. 369-372. doi:10.1016/0022-247X(72)90056-X
  10. W. R. Mann, “Mean Value Methods in Iteration,” Proceedings of the American Mathematical Society, Vol. 4, No. 3, 1953, pp. 506-510. doi:10.1090/S0002-9939-1953-0054846-3
  11. S. Ishikawa, “Fixed Points by a New Iteration Method,” Proceedings of the American Mathematical Society, Vol. 44, No. 1, 1974, pp. 147-150. doi:10.1090/S0002-9939-1974-0336469-5
  12. S. C. Bose, “Weak Convergence to the Fixed Point of an Asymptotically Nonexpansive,” Proceedings of the American Mathematical Society, Vol. 68, No. 3, 1978, pp. 305-308. doi:10.1090/S0002-9939-1978-0493543-4
  13. R. Bruck, T. Kuczumow and S. Reich, “Convergence of Iterates of Asymptotically Nonexpansive Mappings in Banach Spaces with the Uniform Opial Property,” Collectanea Mathematica, Vol. 65, No. 2, 1993, pp. 169-179.
  14. S. Chang, Y. J. Cho and H. Zhou, “Demiclosed Principle and Weak Convergence Problems for Asymptotically Nonexpansive Mappings,” Journal of the Korean Mathematical Society, Vol. 38, 2001, pp. 145-1260.
  15. H. Fukhar-ud-din and S. H. Khan, “Convergence of TwoStep Iterative Scheme with Errors for Two Asymptotically Nonexpansive Mappings,” International Journal of Mathematics and Mathematical Sciences, Vol. 2004, No. 37, 2004, pp. 1965-1971. doi:10.1155/S0161171204308161
  16. H. Fukhar-ud-din and S. H. Khan, “Convergence of Iterates with Errors of Asymptotically Quasi-Nonexpansive Mappings and Applications,” Journal of Mathematical Analysis and Applications, Vol. 328, No. 2, 2007, pp. 821-829. doi:10.1016/j.jmaa.2006.05.068
  17. H. Fukhar-ud-din and A. R. Khan, “Approximating Common Fixed Points of Asymptotically Nonexpansive Maps in Uniformly Convex Banach Spaces,” Computers & Mathematics with Applications, Vol. 53, No. 9, 2007, pp. 1349-1360. doi:10.1016/j.camwa.2007.01.008
  18. H. Fukhar-ud-din and A. R. Khan, “Convergence of Implicit Iterates with Errors for Mappings with Unbounded Domain in Banach Spaces,” International Journal of Mathematics and Mathematical Sciences, Vol. 10, 2005, pp. 1643-1653. doi:10.1155/IJMMS.2005.1643
  19. H. Fukhar-ud-din, A. R. Khan, D. O’Regan and R. P. Agarwal, “An Implicit Iteration Scheme with Errors for a Finite Family of Uniformly Continuous Mappings,” Functional Differential Equations, Vol. 14, No. 3-4, 2007, pp. 245-256.
  20. S. H. Khan and H. Fukhar-ud-din, “Weak and Strong Convergence of a Scheme with Errors for Two Nonexpansive Mappings,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 61, No. 8, 2005, pp. 1295-1301. doi:10.1016/j.na.2005.01.081
  21. A. R. Khan and N. Hussain, “Iterative Approximation of Fixed Points of Nonexpansive Maps,” Scientiae Mathematicae Japonicae, Vol. 54, No. 3, 2001, pp. 503-511.
  22. S. H. Khan and W. Takahashi, “Approximating Common Fixed Points of Two Asymptotically Nonexpansive Mappings,” Scientiae Mathematicae Japonicae, Vol. 53, No. 1, 2001, pp. 143-148.
  23. W. M. Kozlowski, “Fixed Point Iteration Processes for Asymptotic Pointwise Nonexpansive Mappings in Banach Spaces,” Journal of Mathematical Analysis and Applications, Vol. 377, No. 1, 2011, pp. 43-52. doi:10.1016/j.jmaa.2010.10.026
  24. T. Laokul and B. Panyanak, “Approximating Fixed Points of Nonexpansive Mappings in CAT(0) Spaces,” International Journal of Mathematics and Mathematical Sciences, Vol. 3, No. 25-28, 2009, pp. 1305-1315.
  25. W. Laowang and B. Panyanak, “Approximating Fixed Points of Nonexpansive Nonself Mappings in CAT(0) Spaces,” Fixed Point Theory and Applications, 2010, 11 pages, Article ID: 367274.
  26. B. Nanjaras and B. Panyanak, “Demiclosed Principle for Asymptotically Nonexpansive Mapping in CAT(0) Spaces,” Fixed Point Theory and Applications, 2010, 14 pages, Article ID: 268780.
  27. M. A. Noor and B. Xu, “Fixed Point Iterations for Asymptotically Nonexpansive Mappings in Banach Spaces,” Journal of Mathematical Analysis and Applications, Vol. 267, No. 2, 2002, pp. 444-453. doi:10.1006/jmaa.2001.7649
  28. Z. Opial, “Weak Convergence of Successive Approximations for Nonexpansive Mappings,” Bulletin of the American Mathematical Society, Vol. 73, 1967, pp. 591-597. doi:10.1090/S0002-9904-1967-11761-0
  29. G. B. Passty, “Construction of Fixed Points for Asymptotically Nonexpansive Mappings,” Proceedings of the American Mathematical Society, Vol. 84, 1982, pp. 212- 216. doi:10.1090/S0002-9939-1982-0637171-7
  30. B. E. Rhoades, “Fixed Point Iterations for Certain Nonlinear Mappings,” Journal of Mathematical Analysis and Applications, Vol. 183, No. 1, 1994, pp. 118-120. doi:10.1006/jmaa.1994.1135
  31. J. Schu, “Iterative Construction of Fixed Points of Asymptotically Nonexpansive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 158, No. 2, 1991, pp. 407-413. doi:10.1016/0022-247X(91)90245-U
  32. J. Schu, “Weak and Strong Convergence to Fixed Points of Asymptotically Nonexpansive Mappings,” Bulletin of the Australian Mathematical Society, Vol. 43, No. 1, 1991, pp. 153-159. doi:10.1017/S0004972700028884

Journal Menu >>