An Analysis on the Role of Fixed Point Theory in Complete Cone Metric Spaces: A Review

INTRODUCTION

Fixed point theorems give conditions under which maps (single or multivalued) have arrangements. The theory itself is a wonderful blend of investigation, topology, and geometry. In the course of the most recent 80 years or so the theory of fixed points has been uncovered as an exceptionally amazing and significant instrument in the investigation of nonlinear marvels. Specifically, fixed point systems have been connected in such assorted fields as Biology, Chemistry, Economics, Engineering, Game Theory, and Physics. Fixed point theory assumes a significant job in functional examination; guess theory, differential conditions and applications, for example, limit esteem issues and so on. As of late, progressively fixed point results in cone metric spaces showed up. Topological inquiries in cone metric spaces were contemplated where it was demonstrated that each cone metric space is first countable topological space. Thus, progression is proportionate to consecutive congruity and smallness is identical to successive minimization. It merits referencing the spearheading work of Quilliot who presented the idea of generalized metric spaces. His methodology was effective and utilized by many. It is our conviction that cone metric spaces are a unique instance of generalized metric spaces. In this work, we present a metric sort structure in cone metric spaces and demonstrate that established verifications do convey indistinguishably in these metric spaces. This methodology recommends that any augmentation of realized fixed point result to cone metric spaces is excess. Besides the fundamental Banach space and the related cone subset are a bit much. Guang and Zhang (2007) as of late presented the idea of cone metric spaces and built up some fixed point theorems for contractive sort mappings in a normal cone metric space. Hence, some different creators contemplated the presence of points of fortuitous event and normal fixed points of mappings fulfilling a contractive kind condition in cone metric spaces. Thereafter, Rezapour and Hamlbarani (2008) demonstrated fixed point theorems of contractive kind mappings by precluding the presumption of normality in cone metric spaces. In this paper we get points of fortuitous event and regular fixed points for three self mappings fulfilling Jungck (1976) type contractive condition with the supposition of normality in cone metric spaces. First we review Jungck's theorem : THEOREM 1 : Let be a complete metric space. Give f a chance to be a constant self-map on X and g be any self-map on X that drives with f. Further given f and g a chance to fulfill and there exists a consistent such that for each At that point f and g have a novel regular fixed point. Sessa (1982) generalized the idea of commuting mappings by calling self mappings f,g on a metric space X, pitifully commuting if and just if for every one of the Clearly commuting mappings are weakly commuting yet banter isn't valid all in all. Subsequently, numerous authors obtained decent fixed point theorems by utilizing this idea. Along these lines Pant (1994) presented some less prohibitive ideas of

at their fortuitous event point. Jungck and Rhoades (1998), at that point characterized a couple (f,g ) of self mappings to be pitifully good on the off chance that they drive at their occurrence point (i.e. fgx = gfx at whatever point fx = gx). DEFINITION 1 : Let f and T act naturally maps of a nonempty set X. In the event that there exists such that fx = Tx then x is known as an incident point of f and T, while y = fx = Tx is known as a point of occurrence of f and T.

In this segment, we demonstrate two fundamental results and get the results of Azam. Arshad and Beg (2008) as conclusions. We begin with a lemma, which will be required in the continuation. LEMMA 1: Let X be a non-void set and the mappings have a one of a kind point of happenstance inX. On the off chance that (S,f ) and (T,f ) are pitifully perfect, at that point S, T and f have an extraordinary regular fixed point. THEOREM 2 : Let (X, d) be a cone metric space and the mappings satisfy: for each of the where In the event that and f(X) is a complete subspace of X , then S ,T and f have a novel point of fortuitous event . Besides on the off chance that (S,f ) and (T,f ) are pitifully perfect, at that point S, T and f have a special basic fixed point. In this theorem, in the event that we take x = y, at that point we will get S = T. Subsequently this theorem might be taken as a typical fixed point theorem for two maps S and f.

Huang and Zhang (2007) presented cone metric spaces which are speculations of metric spaces, and they stretched out Banach's constriction principle to such spaces, whereafter numerous creators contemplated fixed point theorems in cone metric spaces. As of late, Liu and Xu (2013) presented the thought of cone metric spaces over Banach algebras, which is a change of the idea of cone metric spaces over genuine Banach spaces, and demonstrated the presence of fixed points for mappings characterized on such spaces, and they gave an example that fixed point results in metric spaces and in cone metric spaces are not equal. 𝛽)- acceptable mappings and set up relating fixed point theorems in metric spaces. In the paper, we present the thoughts of C-class functions and cyclic (𝛼, 𝛽)- allowable mappings in Banach algebras. By utilizing such concepts,we present a newcontraction. We acquire another fixed point theorem and give an example to show fundamental result. At long last, we give utilizations of our principle result to cyclic mappings and feeble constriction type mappings in cone metric spaces over Banach algebras. A Banach space is known as a (genuine) Banach algebra (with unit) if there exists multiplication that has the followings properties: for all 1. (xy)z = x(yz); 2. x(y + z) = xy + xz and (x + y)z = xz + yz; 3. 4. there exists such that xe = ex = x; 5. 6. An element is called invertible if there exists such that xx-1 = x-1x = e. PROPOSITION :. Let be a Banach algebra, and let If the spectral radius of x is less than 1, i.e., Then where is the set of all invertible elements of si and REMARK : Let be a Banach algebra. Then the following are satisfied: 1. for all Consistent with Liu and Xu (2013), the following definitions will be needed in the sequel. Let be a Banach algebra. A subset P o f is called cone if the following conditions are satisfied: (1) P is a nonempty and closed subset of si and {0, e} c P; (2) whenever and

(3) (4)

Since the Banach Contraction Principles, a few kinds of speculation withdrawal Mappings on metric spaces have showed up. One such technique for speculation is altering the distances. Delbosco and Skof have set up Fixed Point Theorems for self maps of complete metric spaces by altering the distances between the points with the utilization of a positive genuine esteemed function. Huang and Zhag (2007) presented the idea of cone metric space by supplanting the arrangement of genuine numbers by an arranged Banach space and got some fixed point results. As of late Asadi and Soleimani demonstrated some Fixed Point results on cone metric space by utilizing altering separation function and the (ID) Property of somewhat requested cone metric space. We are giving some new results by presenting a vector esteemed function (Malhotra-Shukla altering function) in cone metric spaces which has similitude with altering function. It turns into the speculation of Altering Function in perspective on cone utilized instead of positive genuine numbers, just as the imperatives utilized for self map of cone metric spaces.

DEFINITION 1 ; Let E be a genuine Banach space and P be a subset of E. P is known as a cone if (a) P is a closed, nonempty and (b) (c) Given a cone we define a partial ordering ‖‖ in E by if We write to , for all implies PROPOSITION 2: Let P be a cone in a real Banach space E. If for and for some then a=0. PROPOSITION 3: Let P be a cone in a real Banach space E. If for and for all then a = 0. PROPOSITION 4 ; Let P be a cone in a real Banach space E. If a,b E E and and then and if and then DEFINITION 5: Let X be a nonempty set and E be a real Banach space. Suppose that the mapping satisfies (a) for all and if and only if x = y; (b) , for all (c) for all Then d is called a cone metric on X, and (X, d) is called a cone metric space. EXAMPLE 6: Let and such that where is a constant. Then (X, d) is a cone metric space. Henceforth unless otherwise indicated, P is a normal cone in real Banach space E and ‖‖ is partial ordering with respect to P.

The motivation behind this paper is to demonstrate some basic fixed point results in cone metric spaces. Aage and Salunke (2009) demonstrated two theorems on cone metric spaces. We demonstrate that they are trifling. We likewise give an example with that impact and make reasonable alterations in the speculation of the theorems to make the end viable. For this reason we present the idea of S-cone metric spaces and demonstrate a theorem on fixed points of a self map on a S-cone metric space. Supporting example to the theorem is additionally given. Huang and Zhang (2007) presented the idea of cone metric spaces and some fixed point theorems for contractive mappings were demonstrated in these spaces.The results in Huang and Zhang (2007) were generalized by Sh.Rezapour and R.Hamlbarani in Rezapour Sh and Hamlbarani R (2008). Thusly Abbas and Jungck (2008), and

purpose, We begin with fundamental definitions on cone metric spaces. DEFINITION 1: Let E be a genuine Banach space and P a subset of E. P is known as a cone if (I) P is shut, non-void and (ii) and non-negative genuine numbers an and b. (iii) . DEFINITION 2 : We characterize a fractional requesting on E regarding P and by if and just if We will compose if denotes the interior of P. We signify by the standard on E. The cone P is called normal if there is a number with the end goal that for every one of the infers

...(1)

The least positive number K fulfilling (1) is known as the normal steady of P. DEFINITION 3 : A cone P is called ordinary if each expanding succession which is limited from above is focalized. That is, on the off chance that is an arrangement with the end goal that for some at that point there is to such an extent that We see that a cone P is ordinary if and just if each diminishing succession which is limited from beneath is focalized.

In this area we demonstrate that results of Aage and Salunke (2009) are trifling. We likewise present the thought of a S-cone metric space and build up a fixed point theorem in a S-cone metric space pursued by supporting examples.

THEOREM 1 :

Let (X, d) be a complete cone metric space and P a normal cone with normal steady K. Assume that the mappings fulfill:

...(2)

At that point f and g have an interesting basic fixed point in X. In the above theorem, it very well may be effectively seen that f = g by taking x = y. Further, on the off chance that and at that point f is the personality map, and on the off chance that at that point this diminishes to Banach constriction principle in cone metric spaces. Presently we present the idea of a S-cone metric space and get our principle fixed point theorem.

1. Aage C.T. and Salunke J. N. (2009). On common fixed points for contractive type mappings in Cone metric spaces , Bulletin of Math-Analy and Appl. Vol 1, Issue 3 pp. 10 – 15. 2. Abbas M. and Jungck G. : ―Common fixed point results for non-commuting mappings without continuity in cone metric spaces‖, J.Math.Anal.Appl.341 pp. 416-420. 3. Akaber Azam, Muhammad Arshad and Ismat Beg (2009). Common Fixedpoint theorems in cone metric spaces, J. Nonlinear Sci.Appl., no. 4, pp. 204-213 4. Akbar Azam, Muhammad Arshad and Ismat Beg: (2008). Common fixed points of two maps in cone metric spaces, Rendiconti del circolo matematico di Palermo, 57 pp. 433 – 441. 5. H. Liu andS. Xu (2013). ―Conemetric spaceswithBanachalgebras and fixed point theorems of generalized Lipschitz mappings,‖ Fixed PointTheory and Applications, vol. article no. 320, 6. Huang Long – Guang, Zhan Xian (2007). ―Cone metric spaces and fixed point theorems of contractive mapping.‖ J.Math.Anal.Appl. 332 pp. 1468 - 1476. 7. Jungck G. (1976). Commuting maps and fixed points, Amer. Math. Monthly 83, pp. 261- 263. 8. Jungck G. and Rhoades B.E. (1998). Fixed points for set valued functions without continuity, Indian J.Pure Appl. Math., 29(3) pp. 227 – 238. spacess, arXive, 1102, 4019 Vl [Maths FA] 10. Pant R. P. (1994) Common fixed points of noncommuting mappings, J.Math. Anal. Appl.188 pp. 436-440. . 11. Rezapour Sh and Hamlbarani R. (2008). Some notes on the paper ―cone metric spaces and fixed point theorems of contractive mappings‖, J.Math, Anal. Appl., 345 pp. 719 – 724. 12. S. Chandok, K. Tas, and A. H. Ansari (2016). ―Some Fixed Point Results for TAC-Type Contractive MappingsContractive Mappings,‖ Journal of Function Spaces, vol. 2016, Article ID 1907676, 6 pages. 13. Sessa S (1982). On a weak commutativity condition of mappings in fixed point considerations, Publ. Inst. Math. 32, pp. 149-153.

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