SOME NEW RESULTS FOR REICH TYPE MAPPINGS ON CONE b-METRIC SPACES OVER BANACH ALGEBRAS

The main purpose of this paper is to present some fixed point results concerning the generalized Reich type αadmissible mappings in cone b-metric spaces over Banach algebras. Our results are significant extensions and generalizations of resent results of N. Hussain at al. (2017) and many well-known results in abundant literature. We also gave an example that confirmed our results.


INTRODUCTION
The  concept of cone metric space was introduced by Huang and Zhang (2007).They supplanted the set of real numbers in metric space by a complete normed space and proved some fixed point results for different contractive conditions in such a space.
Recently, some scholars (see Du, 2010;Kadelburg et al., 2011) argued that extensions of fixed point results on metric space to cone metric space over complete normed spaces are redundand (not new results).In order to overcome this problem, Liu and Xu (2013) introduced the notion of a cone metric spaces over Banach algebras and proved that cone metric spaces over Banach algebras are not equivalent to metric spaces in terms of the existence of the fixed points of the generalized Lipschitz mappings.Very recently, Huang and Radenović (2016) introduced the notion of cone b-metric space over Banach algebras as a generalization of cone metric space over Banach algebra (see Xu & Radenović, 2014;Huang & Radenović, 2015b;Huang & Xu, 2013;Huang et al., 2017).On the other hand, Samet et al. (2012) introduced the notion of α-admissible mappings and proved some fixed point results that generalized several known results of metric spaces.Very recently, Malhottra et al. (2015;2017) used the concept of α-admissibility of mappings defined on cone metric space over Banach algebras and proved Banach and Kannan fixed point results for Lipschitz contractions in such spaces.In 2017, Hussaini et al. (2017) used the concept of α-admissibility of mappings defined on cone bmetric spaces over Banach algebras and proved Banach fixed point results for Lipschitz contractions in such spaces.The Reich contraction was introduced by Reich (1971) as a generalization of the well-known Banach contraction principle and Kannan contraction.In this work, we use the concept of α-admissibility of mappings defined on cone b-metric space over Banach algebras and proved Reich type fixed point theorems.We give an * Corresponding author: jelena.vujakovic@pr.ac.rs example to elucidate our results.Our results generalized the recent results of Malhotra et al. (2015;2017), Hussain et al. (2017), Nieto and Rodríguez-López (2005).

PRELIMINARIES
It In this section we recall some known definitions and results which will be used.
A real Banach algebra  is a real Banach space in which an operation of multiplication is defined in the following way: for all , , , x y z where  and e denote the zero and unit elements of Banach algebra  , respectively.For given cone   we define a partial ordering  with respect to  on following way: xy  if and only if yx   .It is well known that xy  stands for xy  and xy  , xy  stands for int yx   where int  means the interior of  .We say that  is a solid cone if int   .In further work, we always assume that  is a Banach algebra with a unit e ,  is a solid cone in  and  is the partial ordering with respect to  .Definition 2.1 (Huang & Zhang 2007;Liu & Xu, 2013) Let  be a nonempty set.Suppose that the mapping :

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x y z  .
Then ρ is called a cone metric on  and   ,   is called a cone metric space with a Banach algebra  .Definition 2.2 (Huang & Radenović, 2015a) Let  be a nonempty set, 1 s  be a constant and  a Banach algebra.Suppose that the mapping : Lemma 2.11 (Rudin, 1991) Let  be a Banach algebra with a unit e and 12 , kk Lemma 2.12 (Kadelburg & Radenović, 2013) Let E is a real Banach space with a solid cone  .

MAIN RESULTS
First we introduce the notion of α-admissible mapping and α-regularity in the setting of cone b-metric space over Banach algebra  .     be mappings.Then: Now we are able to define generalized Reich type contraction in cone b-metric spaces over Banach algebra  .
 be a cone b-metric space over Banach algebra  with coefficient 1 s  , let  be the underlying solid cone and : for all , Now, we shall show that generalized Reich type α-admissible contraction mappings on cone b-metric spaces over Banach algebra  has a fixed points.
over Banach algebra  with coefficient 1 s  ,  be the underlying solid cone and let : 1)  is a α-admissible mapping; 2) there exists 0 and the result is proved.Suppose now that and by induction we get Now, according to (1), ( 2) and (3), we have Adding up ( 4) and ( 5), we have where Note that, ( ) 1 2 v   .From Lemma 2.9 we conclude that 2ev  is invertible.Moreover, Hence, from (6), we have get where Then by Lemma 2.9 and Lemma 2.11, we have that   , by Lemma 2.9 and Lemma 2.11, it follows that es   is invertible.Moreover,

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Since   1 1 s   , by Remark 2.10, we have that From ( 7), we get ,, , for all i   .Hence, for , ij   with ij  , using ( 8) and ( 10), we obtain and by Lemma 2.6 there exists N   such that for any c   , with ii ii s e sv z z s v sv z z ii s e sv z z s v sv z z Hence, by combining ( 12) and ( 13), we obtain i i e sv sv z z s e sv sv z z ii s e sv z z s v sv z z Since by Lemma 2.9, 2e sv  is invertible, it follows from ( 14) that Hence, for any c   with c   , there exists N   such that This implies, based on Lemma 2.12, that . This completes the proof. Next example illustrates the above result.

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Example 3.4.Consider the algebra Define on  a multiplication in the usual way.Then,  is a Banach algebra with unite 1 e  .If then  is a solid cone which is not normal.
Suppose that and define a mapping :

s z y zy
Also, there exists 0 Similarly, ii i i i Adding up ( 17) and ( 18), we have where , for all i   .

Since
  1   , by Remark 2.10, it follows that   Therefore, based on Lemma 2.6, we conclude that for any  .Therefore, since  is continuous, all the conditions of Theorem 3.3 are satisfied, we conclude that the mapping  has a fixed point in  .This completes the proof. Now we deduce many existing results in the mention literature for metric, cone metric and cone b-matric spaces, as follows: Let :      be the function defined by   vv .

CONCLUSION
In this paper we introduced the notion of generalized Riech type α-admissible mappings on cone metric space over Banach algebra and we prove three fixed point theorems for those contractions.We notice that our results are actual generalization of the recent results of Liu and Xu (2013), Xu and Radenović (2015), Malhotra et al. (2015), Hussain et al. (2017), Riech (1971), Nieto and Rodríguez-López (2005) and many known results in the literature.
b-metric space over Banach algebra  with coefficients 1 s  ,  be the underlying solid cone, :     and :  .Then, by uniquieness of the limit, we have ** zy  .The theorem is thus proved.Thenext theorem is an ordered version of generalized Riech type contraction on cone b-metric space with Banach algebra.
Liu and Xu (2013)2.2inLiu and Xu (2013)are special cases of our Theorem 3.5 with normal cone  , with 1 s  , 23 vv  and 1    .