COMMON FIXED POINT UNDER JUNGCK CONTRACTIVE CONDITION IN A DIGITAL METRIC SPACE

Digital topology is concerned with the topological characteristics of digital image pictures or objects. A peculiar arrangement of non-negative numbers configures digital images. Digital image processing is a technique of dismantling the picture into its fundamental components and analyzing its various features with respect to component parts. In analyzing the fundamental segments of image pictures, the connected segments are separated out to ascertain the relationship of adjacency. During this process of tracking, coding and thinning, it is kept in mind that the connectedness peculiarity of the object remains unchanged. The features of the component subsets and their relationships can be detailed when the image is decomposed into its constituents. Some of the characteristics of these constituent points or subsets are depending on their positions. Thus, the primary topological features of digital images like connectedness, adjacency, etc. can be the basic clues for their


INTRODUCTION
In nonlinear analysis contraction principal plays a vital role. The wider applications of contraction mapping in the field of functional analysis and topology are notable. Contraction principle in a complete metric space (the idea of the metric space was propounded by M. Ferchet [1]) assures the subsistence of fixed points and their uniqueness as well. It was first introduced by Stephen Banach [3] in 1922. Banach fixed point theorem possesses a fixed point as a solution of a selfmapping using the convergence of the Cauchy sequences. Banach fixed point theorem extended the classical consequence of "a continuous map on a closed unit ball" bestowed by L.E.J. Brouwer [2] in 1912. J. Schauder [4] has given another extension for the same.
Number of authors has contributed to generalize fixed point theorem, thereafter, among them few notables are A. Tychonoff [5], S. Lefschetz [6], S. Kakutani [7], M. Edelstein [9], R. Kannan [10], etc. Some advances have further been made by the authors of references [32], [33], [34] to generalize the idea of contraction mapping using different type of contractions on general and digital metric spaces.
In 1942, K. Menger [8] used distribution function to introduce probabilistic metric space instead of general metric space. Similarly, some more spaces like d-complete topological spaces, Fcomplete metric spaces, G -metric spaces were further introduced by many authors later, as referred in [14], [18] and [25], respectively. To enrich the notion of fixed-point results based on contraction maps, probabilistic metric space in lieu of formal metric has been wielded by some 3069 COMMON FIXED POINT UNDER JUNGCK CONTRACTIVE CONDITION workers of the field, a particular example of the same has been shown in ref. [35].
Various spaces with their fixed-point results are studied under the theory of topological fixed points.
The geometrical situations which are dealt under the topology are not depending only on definite shape of the objects. The notion of general topology, in general, encompasses infinitely many points in a small arbitrary neighborhood of the certain point, whereas in digital topology some finite number of such points is considered in the neighborhood [11]. This could well be reconciled by the diagram shown below [32].
In image processing, general topology is not appropriate, as it assumes spaces which carry infinite number of points, even in its smallest neighborhood.
Hence, digital topology is taken in to account which is the topological properties of 2-dimensional (2D) and 3-dimensional (3D) digital images. It furnishes a solid mathematical basis for image thinning, object counting, contour filling and other image processing techniques. The 2D image array elements are denoted by pixels, whereas the same for 3D are known as voxels. In 2D-plane or 3D-space, every such pixel or voxel is related with the lattice points which are the points having integral coordinates. A pixel or voxel associated lattice point assumes the value 0 which stands for a black point or 1 which stands for a white point.
Digital topology was accorded as a tool of digital image processing by A. Rosenfield [11] [12].
Then after, the idea of digital fundamental group for a 3-dimentional image thinning was introduced by Kong [13]. Boxer [15] [17] [20] [21] [22] [23] studied various digital continuous mappings and forwarded the approach of digital topology. The concept of digital continuous function and features of fixed point in digital images was first proposed by Rosenfield [12].
Application of Lefchetz fixed point theorem for digital images was studied and fostered by Ege and Karaca [26] [27]. Digital metric space was defined by Ege and Karaca [28]. Also, they have given the proof of the Banach contraction theorem for digital images. Han [30] further improved the Banach fixed point theorem for digital metric spaces. Wielding a digital contraction defined on two mappings, a common coincidence point or a common fixed point has been procured by the authors in [32] [36].
Our interest in this paper is to further enrich the idea of digital topology by making use of some more contractions and their fixed common points in processing the digital images.

PRELIMINARIES
Prior to deal with the said purpose, it is obligatory to start with some basic points. Therefore, from above, we get For ∈ , number of points ∈ adjacent to can be depicted by ( , ), where ( , ) is not depending on . In short, we use = ( , ) Now, let us visualize this as follows: (і) When ∈ , that is, = 1, so will assume the value = 1. Therefore, we may write (іі) When, ∈ 2 , that is, = 2, so will assume the values = 1,2.
The generalization to n-D digital image can be shown by the following rule [19].

MAIN RESULTS
Our motive is to ascertain the applicability of contraction principal in digital topology particularly by establishing some common fixed point on Jungck contractive condition in digital metric space to widen the use of fixed-point theorems in digital topology. Then, and have a common fixed point in , which is unique.
This shows the convergence of the sequence,

CONCLUSION
Although, fixed point theorems have variety of uses but, in the field of science and technology it is not less than a boon. In processing of an image, fixed point theorems can be utilized to reduce 3077 COMMON FIXED POINT UNDER JUNGCK CONTRACTIVE CONDITION the size and enhance the quality of the image by making use of the contraction principle and its application. It is found that the quality of a compressed image is not so good every time even, it could not have the information same as the source image. Also, the memory of the data is sometimes very large. This can be another kind of headache for the storage of the data.
Nevertheless, compressions of digital image sources are the need of the hour. So, by making use of the contraction principle and common fixed point, as shown in the main result, we can reinforce the knowledge of digital topology especially, in digital image processing to have a compressed image without much redundancy.

ACKNOWLEDGEMENT
A number of researchers have paid their attention in obtaining and establishing some results under certain contractive conditions, the work of some of them are detailed by us while introducing the concept. This doesn't mean that the works done by others in this regard are not valuable and admissible. It is just to avoid the unnecessary comprehensiveness of the matter and to save the time of the reader as well. Still, we are thankful to all of them.
Authors would like to bestow their gratitude towards valuable ideas and constructive comments for further improvement of the paper and are eager to thank the previous scholars who have enriched the concept and thought of digital topology especially those who are cited.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.