Iterative Procedure for Finite Family of Total Asymptotically Nonexpansive Maps (TAN)
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In this paper, CQ Algorithms for iterative approximation of a common fixed point of a finite family of nonlinear maps were introduced and sufficient conditions for the strong convergence of this process to a common fixed point of the family of Total asymptotically Nonexpansive maps (TAN) were proved.
Preliminaries
Let H be a normed space, K be a nonempty closed convex subset of H and be a map. The mapping T is called asymptotically nonexpansive mapping if and only if there exists a sequence , with such that for all
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] as a generalisation of nonexpansive mappings. As further generalisation of class of nonexpansive mappings, Alber et al. [2] introduced the class of total asymptotically nonexpansive mappings, where a mapping is called total asymptotically nonexpansive (TAN) if and only if there exist two sequences , with and nondecreasing continuous function with such that for all
In Ofoedu and Nnubia [3], an example to show that the class of asymptotically nonexpansive mappings is properly contained in the class of total asymptotically nonexpansive mappings was given. The class of asymptotically nonexpansive type mappings includes the class of mappings which are asymptotically nonexpansive in the intermediate sense. These classes of mappings had been studied extensively by several authors (see e.g., [4]–[9]).
A map T is said to satisfies condition B if there exists strictly increasing, continuous, such that for all where and .
Lemma 1.1 Takahashi [10]
Let be sequences of nonnegative numbers satisfying the conditons: as and . Suppose that where is a strictly increasing function with Then as .
Lemma 1.2 Moore and Nnoli [11]
Let K be closed convex nonempty subset of a real Hilbert Space H. Let be a sequence in H, and be such that and then converges strongly to
Lemma 1.3 Ofoedu and Nnubia [9, p. 703]
Let E be a reflexive Banach space with weakly continuous normalized duality mapping. Let K be a closed convex subset of E and a uniformly continuous total asymptotically nonexpansive mapping with bounded orbits. Then I − T is demiclosed at zero.
Proposition 1.1 Ofoedu and Nnubia [9, p. 704]
Let H be a real Hilbert space, let K be a nonempty closed convex subset of H and let be m uniformly continuous total asymptotically nonexpansive mappings from K into itself with sequences such that and with function satisfying for some constants Let and and, . Suppose that then is closed and convex.
Proposition 1.2 Nnubia and Bishop [6, p. 74]
Let K be a nonempty subset of a real normed space E and be m total asymptotically nonexpansive mappings, then there exist sequences , with and nondecreasing continuous function with such that for all
Main Result
Proposition 2.1 Result
Suppose that there exist constants such that then T is total asymptotically nonexpansive if and such that
Proof
Suppose T is total asymptotically nonexpansive, that is, let T be such that
Since is continuous, it follows that attains its maximum (say ) on the interval moreover, whenever . Thus,
So, we have, where and Thus completing the proof.
Theorem 2.1 Let H, K, , and F be as in Propostion 1.1, then generated iteratively by: converges to where and
Proof
Let .
So that Hence, . For .
Let , we show that . Now, is the projection of onto . then (i)
Since, then So, and hence
Now,
Hence, and since , then
In particular,
Since,
So,
Then, .
Since is bounded, then exists.
Thus,
Observe that by (7)
Now,
so that
Hence, and hence,
Thus,
Now, , we have and since , we obtain , so that . Hence, and . Similarly, and i(n + 1 − N) = i(n + 1). Using this we obtain but, so that by hypothesis, (9), (8) and the uniform countinuity of we have:
Furthermore,
So that using (7) and (11) we have,
Now, let be arbitrarily choosen, then,
By uniform continuity of and from (8) and (13) we have that
Observe that , put in another way, Hence,
Since is demiclosed at 0 is bounded and H is reflexive,
so, and such that, as . Since, as then and so
Let arbitrary. Then and as So that since is demiclosed at Hence, Moreover, where Then by the lemma 1.2 converges strongly to (that is the common fixed point nearest to ).
Theorem 2.2 Let H, K, , and F be as in Theorem 2.1, then generated iteratively by converges to where
Proof
Let .
So,
So, where so, and hence So, and hence, . Following the argument as in Theorem 2.1, we have that and
Now, but so, so that and so
Hence, and hence,
Thus, .
Thus, , so that by uniform continuity of , . Now, since is demiclosed at 0, the same argument in Theorem 2.1 completes the Proof.
Conclusion
CQ Algorithms for iterative approximation of a common fixed point of a finite family of nonlinear maps were introduced and sufficient conditions for the strong convergence of this process to a common fixed point of the family of Total asymptotically Nonexpansive maps were proved. Our iterative processes generalise some of the existing ones, our theorems improve, generalise and extend several known results and our method of proof is of independent interest.
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