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.


Introduction
In this paper, we have applied certain results due to Sahu [1] 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 and ⊆ a nonempty subset of . A mapping : → is said to be (see, e.g., [2]) (i) demicontinuous if whenever a sequence { } ⊂ converges strongly to ∈ it implies that the sequence { } converges weakly to ∈ ; (ii) hemicontinuous if whenever a sequence { } ⊂ converges stronly on a line to ∈ it implies that the sequence { } converges weakly to ∈ , that is, ( 0 + ) ⇀ 0 as → 0.
Asymptotic fixed point theory which has been studied by so many authors [1,[3][4][5][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][8][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 [1] 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.
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.

Preliminaries
Let be a nonempty subset of a Banach space and : → a nonlinear mapping. The mapping is said to be Next, let be a nonempty subset of a Banach space The infimum ( ) = sup{‖ − ‖/(‖ − ‖ + ); , ∈ , ̸ = } of constants for which (1) holds is called nearly Lipschitzian constant. Nearly Lipschitzian operators with sequences {( , ( ))} are classified in [1,2] as shown below: Examples and a short survey of these classes of nearly Lipschitzian operators are listed above, and related operators are illustrated in [1] (pp. 655-656) where it is remarked that if is bounded then the asymptotically nonexpansive mapping is a nearly nonexpansive mapping. Also, it is observed therein that a nearly asymptotically nonexpansive mapping reduces to asymptotically nonexpansive type if is bounded. For details authors are referred to Agarwal et al. [2] pp. 259-263, especially the bibliographic notes and remarks there in.

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 parameter in Lemma 3 below.

Lemma 3. Let be a normed linear space over, a scaler field F (F is real or complex). Then, for all distinct points , ∈
there exists ∈ R such that for all , ∈ F.
Proof. As mentioned above, the proof is a consequence of Archimedean property of real numbers that if and are positive real numbers then < for some ∈ N. Since ̸ = , we have Equation (3) follows from Archimedean property while boundedness is inferred from the fact that 1 = − for arbitrary , ∈ F.

Remark 4.
It is important to make the following observations.
(2) It is important to observe that if and were not distinct in Lemma 3 then = would be a valid and natural constraint. However, for = the problem is trivial.
Using the hypothesis together with the Root Test for convergence of series of real numbers, we obtain ∑ ∞ =1 ‖ − +1 ‖ < ∞ which means the sequence { ( , )} is a Cauchy sequence and so has a limit point * in .
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 let : → be hemicontinuous nearly Lipschitzian mapping. Suppose that = as → ∞ for some , ∈ . Then, is an element of Fix( ).
Proof. Consider the following operator S : N × → defined by Clearly, S restricted to reduces to the auxiliary operator above at the fixed point of S. We will show that given that is hemicontinuous then S is a selfmap of for all , that is, S(⋅, ) : → since is closed. Clearly, S restricted to and have common fixed point set, that is, Fix( ) = Fix( ) (provided has a fixed point) and ( + 1, ) = + ‖ +1 − ‖ = + . But from the last proof, we verified that is a continuous mapping on and has asymptotic fixed point * ∈ . Also, by hemicontinuity of and continuity of S the sequence = S( ; 0 ) converges strongly to which means that S( ; ) converges weakly to S( ; ) which means S is demicontinuous on .
By Lemma 1, we have that ∈ Fix(S) = Fix( ). Proof. By Lemma 6, the auxiliary operator given by = +‖ − ‖ is a selfmap of , and together with Lemma 5 we conclude that has a fixed point in which is also a fixed point of . To 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 S is demicontinuous contraction so that Theorem 2 applies.