Research Article | Open Access
Nearly Contraction Mapping Principle for Fixed Points of Hemicontinuous Mappings
We extend the application of nearly contraction mapping principle introduced by Sahu (2005) for existence of fixed points of demicontinuous mappings to certain hemicontinuous nearly Lipschitzian nonlinear mappings in Banach spaces. We have applied certain results due to Sahu (2005) to obtain conditions for existence—and to introduce an asymptotic iterative process for construction—of fixed points of these hemicontractions with respect to a new auxiliary operator.
In this paper, we have applied certain results due to Sahu  on nearly contraction mapping principle to obtain conditions for existence of fixed points of certain hemicontinuous mappings and introduced an asymptotic iterative process for construction of fixed points of these hemicontinuous mappings with respect to a new auxiliary operator. Our results are important generalizations and an extension of important and fundamental aspect of a branch of asymptotic theory of fixed points of non-Lipschitzian nonlinear mappings in real Banach spaces.
Let and be real Banach spaces anda nonempty subset of. A mappingis said to be (see, e.g., ) (i)demicontinuous if whenever a sequenceconverges strongly toit implies that the sequenceconverges weakly to;(ii)hemicontinuous if whenever a sequenceconverges stronly on a line toit implies that the sequenceconverges weakly to, that is, as
Asymptotic fixed point theory which has been studied by so many authors [1, 3–6] has a fundamental role in nonlinear functional analysis concerning existence and construction of fixed points of Lipschitzian mappings, -uniformly Lipschitzian mappings, and non-Lipschitzian mappings among other classes of operators (see, e.g., [5, 7–9]). A very important branch of the theory of asymptotic fixed point relates to the important class of asymptotically nonexpansive mappings which have been studied by various authors in specific types of Banach spaces.
Motivated by the need to relax continuity condition inherent in asymptotic nonexpansiveness of asymptotically nonexpansive mappings in certain applications, Sahu  considered and introduced the nearly contraction mapping principle into the study of asymptotic fixed point theory concerning nearly Lipschitzian mappings and obtained the following results among others.
Lemma 1. Letbe a nonempty subset of a Banach space, and letbe hemicontinuous. Suppose thatasfor someThen, is an element of, the set of fixed point of.
Theorem 2. Letbe a nonempty closed subset of a Banach spaceanda demicontinuous nearly Lipschitzian mapping with sequence. Suppose. Then, we have the following: (a)has unique fixed point(b)for each, the sequence converges strongly to;(c)for all.
The aim of this work is appling Lemma 1 to obtain conditions for existence and uniqueness of asymptotic fixed point of a new auxiliary operator and appling Theorem 2 on the auxiliary operator to obtain an extension and a generalization of Theorem 2 which is a fundamental extension of important classical and related results.
Letbe a nonempty subset of a Banach spaceanda nonlinear mapping. The mapping is said to be Lipschitzian if for each there exists a constantsuch that for all. A Lipschitzian mapping is called uniformly -Lipschitzian if for all and asymptotically nonexpansive if .
Next, let be a nonempty subset of a Banach spaceanda fixed sequence inwith as. A mapping is called nearly Lipschitzian mapping with respect to if for each there exists a constantsuch that The infimumof constants for which (1) holds is called nearly Lipschitzian constant. Nearly Lipschitzian operators with sequences are classified in [1, 2] as shown below: (a)nearly contraction iffor all(b)nearly nonexpansive iffor all(c)nearly asymptotically nonexpansive iffor all and (d)nearly uniformly -Lipschitzian iffor all(e)nearly uniformly -contraction iffor all.
Examples and a short survey of these classes of nearly Lipschitzian operators are listed above, and related operators are illustrated in  (pp. 655–656) where it is remarked that ifis bounded then the asymptotically nonexpansive mappingis a nearly nonexpansive mapping. Also, it is observed therein that a nearly asymptotically nonexpansive mapping reduces to asymptotically nonexpansive type ifis bounded. For details authors are referred to Agarwal et al.  pp. 259–263, especially the bibliographic notes and remarks there in.
3. Main Results
Our main results depend on Lemma 1 and the following new important inequality, needed in the sequel, which we shall prove using archimedean property. We are still sharpening an estimate for the parameterin Lemma 3 below.
Lemma 3. Letbe a normed linear space over, a scaler field ( is real or complex). Then, for all distinct pointsthere exists such that for all.
Proof. As mentioned above, the proof is a consequence of Archimedean property of real numbers that ifandare positive real numbers then for some . Since , we have Equation (3) follows from Archimedean property while boundedness is inferred from the fact that for arbitrary .
Remark 4. It is important to make the following observations. (1) If then (3) reduces to as verified below: since, if on the contrary (4) is not satisfied then from (3) we have which end up with a contradiction demonstrated below. Suppose. Settingsuch that yields whenever such that which is a contradiction. (2) It is important to observe that ifandwere not distinct in Lemma 3 thenwould be a valid and natural constraint. However, for the problem is trivial.
Lemma 5. Letbe a nonempty subset of a Banach space, and letbe a nearly Lipschitzian map with sequencesuch that Then the auxiliary operator defined byhas a fixed point in.
Proof. Given that where is a nearly Lipschitzian map with sequence , we have
This gives which yields where
Using the hypothesis
together with the Root Test for convergence of series of real numbers, we obtain which means the sequence is a Cauchy sequence and so has a limit pointin.
We are left to show that the limitofis a fixed point of, for all . To achieve this, it suffices to prove thatis continuous which follows an application of Lemma 3, namely. Let, then for some positive real number . So given any , we have such that whenever for some . Therefore, is continuous in and so .
To apply Lemma 1, we need its extension for hemicontinuous mappings given in the following form.
Lemma 6. Let be a nonempty subset of a Banach space, and letbe hemicontinuous nearly Lipschitzian mapping. Suppose that as for some . Then, is an element of .
Proof. Consider the following operator defined by
Clearly, restricted to reduces to the auxiliary operator above at the fixed point of . We will show that given that is hemicontinuous then is a selfmap of for all , that is, since is closed.
Clearly, restricted to and have common fixed point set, that is, (providedhas a fixed point) and . But from the last proof, we verified thatis a continuous mapping onand has asymptotic fixed point . Also, by hemicontinuity of and continuity of the sequence converges strongly to which means that converges weakly to which means is demicontinuous on.
By Lemma 1, we have that.
Theorem 7. Letbe a nonempty closed subset of a Banach spaceanda hemicontinuous nearly Lipschitzian mapping with sequence . Suppose . Then, we have the following: (a)has unique fixed point(b)for each, the sequenceconverges strongly to(c) for all where and .
Proof. By Lemma 6, the auxiliary operator given by is a selfmap of , and together with Lemma 5 we conclude thathas a fixed point inwhich is also a fixed point ofTo prove (a), we are left to show that the fixed point is unique. The proof of uniqueness and for (b) and (c) follow from the fact that is demicontinuous contraction so that Theorem 2 applies.
- D. R. Sahu, “Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces,” Commentationes Mathematicae Universitatis Carolinae, vol. 46, no. 4, pp. 653–666, 2005.
- R. P. Agarwal, D. O'Regan, and D. R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Springer Science+Business, New York, NY, USA, 2009.
- R. E. Bruck, “On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces,” Israel Journal of Mathematics, vol. 38, no. 4, pp. 304–314, 1981.
- K. Goebel and W. A. Kirk, “A fixed point theorem for transformations whose iterates have uniform Lipschitz constant,” Studia Mathematica, vol. 47, pp. 135–140, 1973.
- K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambidge, UK, 1990.
- D. R. Sahu and J. S. Jung, “Fixed-point iteration processes for non-lipschitzian mappings of asymptotically quasi-nonexpansive type,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 33, pp. 2075–2081, 2003.
- T. D. Benavides, G. L. Acedo, and H. K. Xu, “Weak uniform normal structure and iterative fixed points of nonexpansive mappings,” Colloquium Mathematicum, vol. 68, no. 1, pp. 17–23, 1995.
- F. E. Browder, “Fixed point theorems for noncompact mappings in Hilbert spaces,” Proceedings of the National Academy of Sciences of the United States of America, vol. 53, no. 6, pp. 1272–1276, 1965.
- K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonex- pansive mappings,” Proceedings of the American Mathematical Society, vol. 35, no. 1, pp. 171–174, 1972.
Copyright © 2013 Xavier Udo-utun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.