Dr. Smita Nair

Study of Fixed point theorem in G metric space and Dislocated metric space

Minor Research Project
Funded by U.G.C.
F. No. MS 21/ 102054/ xii/14-15 CRO
2015 - 2017


One of the significant works in field of mathematics that has been proven to be an asset for mathematicians worldwide, busy in solving problems of various kinds is fixed point theory. The theory of fixed point is one of the most powerful tools of modern mathematics. Fixed point theory is a beautiful mixture of analysis, topology & geometry. It concerns itself with very simple and basic mathematical settings. Fixed point theory has always been exciting in itself and its applications in new areas because of its interdisciplinary approach.

A point is often called fixed point when it remains invariant, irrespective of the type of transformation it undergoes. A point x of a space X is called a fixed point of a function
f: X→ X, if f(x) = x.
Thus it is obvious that the presence or absence of fixed point is an intrinsic property of function. However many necessary and/or sufficient conditions for the existence of such points involves a mixture of algebraic order theoretic or topological properties of mapping or its domain and range.
Many mathematical problems, originating from various branches of mathematics, can be equivalently formulated as fixed point problems. Fixed point theorems provide sufficient conditions under which there exists a fixed point for a given function, and thus allow us to guarantee the existence of a solution of the original problem. The fixed point theory has been applied to show the existence and uniqueness of the solution of differential equation, integral equation. Because of the wide variety of uses, fixed point theorems are of great interest many disciplines such as mathematics, engineering, physics, economics, game theory, biology and chemistry etc.

The origin of fixed point theory lies in the method of successive approximations used for proving existence of solutions of differential equation which formally started in the beginning of twentieth century as an important part of analysis. If one looks into the annals of history the study of fixed point theory began in 1912 with a theorem given by famous Dutch mathematician L. E. Y. Brouwer (1881-1966). It is one of most famous and important theorem on the topological fixed point property. It can be formulated as ”The closed unit ball Bn ϵ Rnhas the topological fixed point property.”He also proved the fixed point theorems for a square, a sphere and their n-dimensional counterparts. Brouwer’s theorem has many applications in analysis, differential equation and generally in proving all kinds of so-called existence theorems for many types of equations. Its discovery had a tremendous influence in the development of several branches of mathematics, especially algebraic topology. But the abstraction of this classical theory is the pioneering work of the great polish mathematician Stefan Banach published in 1922 which provides a constructive method to find the fixed points of a map.  This theorem foreshadowed the work of Brower. Banach contraction principle (BCP) states that “A contraction mapping on a completemetric space has a unique fixed point”. Banach used the idea of shrinking (contracting) map to obtain this fundamental theorem. Banach proved the theorem using the concept of Lipschitz mappings. A lipschitizan mapping with a lipschitz constant k less than 1 is called contraction. The theorem Banach contraction principle (BCP) is treated as one of the main tools for both the theoretical and the computational aspects in mathematical sciences.  There have been numerous generalizations and extensions of this theorem in the literature due to its simplicity and constructive approach.
Beginning with simple contractive type mapping, the interest now shifted to newer type of mappings and existence of their fixed point. In 1976Jungck obtained an important generalisation of (BCP) in the form of common fixed point theorem for commuting pair of maps. Sessa  introduced the concept of weakly commuting maps and proved common fixed point theorem for weakly commuting mappings. According to Sessa “Let A and B be mappings from a metric space(X,d) into itself. Then A and B are weakly commuting on X if
                              d(ABx,BAx) ≤ d(Ax, Bx) for all x ϵ X.”
This concept was further improved and generalized by Jungck with the notion of weakly compatible mappings. He defined it as, “Let A, B be self- mappings of a metric space (X,d) into itself. Then A and B are said to be compatible mappings on X if
where {xn} is a sequence in X such that
 =   = t for some t in X.”
The concept of dislocated metric space was introduced by P. Hitzler in which the self distance of points necessarily need not to be zero. They also generalized famous Banach’s contraction principle in dislocated metric space. Dislocated metric space play a vital rule in topology, logical programming and electronic engineering. Dislocated metric space is defined as
Let X be a non empty set and d: X×X→ [0,) be a function satisfying the following conditions
2. d(x,y)=d(y,x) =0 implies x=y
3. d(x,y)d(x,z)+d(z,y) for all x,y,z X
Then (X,d) is called dislocated metric space”
In 2003 Mustafa and Sims [58] proved that most of the claims concerning the topological properties of D metric were incorrect. In order to repair these drawbacks they gave a more appropriate notion of generalized metric called G metric. Mustafa provided many examples of G metric spaces in [58] and developed some of their properties. He proved that G metric spaces are provided with a Hausdroff topology which allows us to consider among other topological notions, convergent sequence, limits, Cauchy sequence, continuous mappings, completeness and compactness. “A G metric space is a pair (X.G) where X is a non empty set and G: X×X×X→ [0,) is a function such that for all x,y,z,a  X the following conditions are fulfilled
1.G(x,y,z)=0 if x=y=x
2. G(x,x,y)>0 for all x,yX with xy
3. G(x,x,y) G(x, y,z) for all x,y X with xy
4. G(x,y,z)=G(x,z,y)= ------ (symmetry in all 3)
5. G(x,y,z)  G (x,a,a) + G (a,y,z)       ( triangle inequality)
In such a case the function G is called a G metric on X. Then (X,G) is a G metric space “.In this case G(x,y,z) can be interpreted as the perimeter of the triangle of vertices x,y,z.
In this project we present some theorems based on finding fixed points for mappings in G metric and dislocated metric space. We have proved six theorems which are extension and generalisation of the results that have already been proved in metric and 2 metric spaces. The first theorem is based on concept of weakly compatible mappings. The theorem is  extension  of the results proved in 2 metric space. It makes use of two self mappings defined on complete  G metric space  and satisfying the compatibility conditions. The second result consist of finding common fixed point for six self maps defined on complete G metric space. The mapping used satisfy weak compatibility conditions. The third and fourth  theorem proved  uses the concept of compatibility for finding fixed point  for a pair of self  maps satisfying rational inequality. The fifth theorem is  generalization of the common fixed theorem of Jain and Bajaj    in  dislocated metric space using the concept of intimate mappings .  The sixth theorem uses chatterjia type contraction mapping .