Generalized contractive conditions and single valued mapping in complete metric space

Introduction

In functional analysis the fixed point theory having incredible research field in applied mathematics. Also it has various applications to non-linear Sciences, Stefan Banach had proved one of the most famous result of fixed point theorem, which is the initial path in this direction of metric fixed point theory. A common fixed point theorem in metric space generally involves conditions of continuity, commutativity and contraction conditions with completeness. In 1976 G. Jungck [10] was the first mathematician who generalized the Banach contraction theorem by using commuting mappings and it has open problem that a pair of commuting and continuous self mapping in the interval [0,1] which has not a common fixed point. There are great numbers of generalizations of well-known Banach contraction mapping principle. M. Edelstein [01] extended and defined contractive mapping such as "A mapping T of a metric space ( , ) Xd into itself is said to be contractive if

( ( ), ( )) ( , ) d T x T y d x y 

x y X  A contraction mapping is always continuous and which has a unique fixed point. M. Edelstein [01] proved that if T is a contractive mapping on a compact metric space ( , ) Xd to itself then there exists a unique fixed point of T. Now we consider some important generalization of Banach contraction mapping principle. In 1969 D. W. Boyd and J. S. W. Wong [02] obtain the following generalization of contraction mapping theorem. In a complete metric space ( , ) Xd if: . Here we show that an is convergent. Assume that an >0 for all n > 0 then for all n > 1.

is monotonically decreasing and bounded below, so it is convergent.

Then by the upper semi continuity from the right of the function  we get

Let us assume that for each k, the smallest number kk mn 

for each equation ( 1) holds ( , )

On the other hand we have,

As k  we obtain the following condition Hence the function  satisfies all the conditions in above theorem except ( 2 ) 2   . And we observe that T1 has two fixed points -1 and 1, while T2 has no fixed points.

In the following result the continuity condition on  is replaced with another suitable condition.

Theorem 1.2

In a complete metric space ( , ) Xd let :

) Now in the following theorem we present a different kind of principle in which the contractive condition is imposed only at the first step.

Xd is a complete metric space and :

T X X  be a continuous mapping such that for some function :

is a contraction mapping then it is continuous and satisfies equation (I) inequality in above theorem 1.3. Since T is a contraction mapping then Adding ( , ( ))

to both the sides of the above inequality yields

Hence define the function :

, for all xX  . This gives us the basic inequality ( , ( )) ( ) ( ( ))

Single-valued Mapping

In this paper S be the complete metric space with the metric d, let R be the set of all real numbers, N be the set of positive integer and B(S) be the set of all nonempty bounded subset of S, CB(S) be the set of all nonempty bounded closed subset of S, CL(S) is the class of nonempty closed subset of S and K(S) be the set of all nonempty compact subset of S respectively. For any P, Q belongs to CB(S) then ( , ) s u p { ( , ) :

A single point x belongs to P we can write ( , ) ( , )

, and if P={x} and Q= {y} then we write ( , ( , )

Let CB(S) be the class of all nonempty bounded closed subset of S and H is the Hausdorff metric with respect to δ then ( , ) m a x s u p ( , ) , s u p ( , )

 where ( , ) in f ( , ). ( , ) m a x ( , ) , ( , ) 1 .

Where the set distance δ is not symmetric.

In 1989 Kaneko and Sessa [12] introduced the concept on compatible mapping of single valued and multi-valued mapping.

Definition 2.1 [04] , we know that compatible mappings are weakly compatible but converse is not true.

Example 2.2

Let the two single valued mappings , :

in the set S= [1, ∞) defined by



. Hence the mapping f and g has satisfies the common limit in the range with g.

Definition 2.3 In a metric space  

are said to be occasionally weakly compatible (OWC) if there exist a point t in S such that ft

 , is a function such that and ( 0 ) 0

ii) The pairs (T, f) and (U, g) are occasionally weakly compatible then f, g, t and U have a unique common fixed point in S.

Proof. Let , x y S  and the pairs (T, f) and (U, g) satisfy occasionally weakly compatible (OWC) property such that

Then we have to prove that fx g y  . Now if fx g y  then by using (i) condition we have . Therefore fx is a common fixed point of f, g, T and U moreover by the (i) condition we get

then assume that lm  is another common fixed point of f, g, T and U, hence from condition (i) we get

The pair (T, f) satisfies OWC property then f and T have a unique common fixed point in S. If T is a single-valued mapping then above corollary becomes as follows. ii)

The pair (T, f) satisfies the OWC property then f and T have a unique common fixed point in S. ,1

Then f(1) =T(1) =1 and fT(1) =1 =Tf (1) and so the pair (T, f) satisfies OWC property.

And for some J belongs to [0,1) if we define a function () t J t   for all   0, t  then all conditions in above corollary are satisfied and further the point 1 is a unique common fixed point of T and f.

Conclusion

In this paper we generalized contractictive condition and single valued mapping theorem in complete metric space.

Acknowledgement

AcknowledgementsThe authors are very grateful and wish to thank to an anonymous referee for their insightful reading the manuscript & valuable helpful suggestions which led to an improved preparation of the manuscript.

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