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Further generalizations of the Banach contraction principle

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Abstract

We establish a new fixed point theorem in the setting of Branciari metric spaces. The obtained result is an extension of the recent fixed point theorem established in Jleli and Samet (J. Inequal. Appl. 2014:38, 2014).

1 Introduction

The fixed point theorem, generally known as the Banach contraction principle, appeared in an explicit form in Banach’s thesis in 1922 [1], where it was used to establish the existence of a solution to an integral equation. Since then, because of its simplicity and usefulness, it has become a very popular tool in solving existence problems in many branches of mathematical analysis. This principle states that if (X,d) is a complete metric space and T:XX is a contraction map (i.e., d(Tx,Ty)λd(x,y) for all x,yX, where λ(0,1) is a constant), then T has a unique fixed point.

The Banach contraction principle has been generalized in many ways over the years. In some generalizations, the contractive nature of the map is weakened; see [29] and others. In other generalizations, the topology is weakened; see [1023] and others. In [24], Nadler extended the Banach fixed point theorem from single-valued maps to set-valued contractive maps. Other fixed point results for set-valued maps can be found in [2530] and the references therein.

In 2000, Branciari [11] introduced the concept of generalized metric spaces, where the triangle inequality is replaced by the inequality d(x,y)d(x,u)+d(u,v)+d(v,y) for all pairwise distinct points x,y,u,vX. Various fixed point results were established on such spaces; see [10, 13, 1720, 22, 3133] and the references therein.

We recall the following definitions introduced in [11].

Definition 1.1 Let X be a non-empty set and d:X×X[0,) be a mapping such that for all x,yX and for all distinct points u,vX, each of them different from x and y, one has

  1. (i)

    d(x,y)=0x=y;

  2. (ii)

    d(x,y)=d(y,x);

  3. (iii)

    d(x,y)d(x,u)+d(u,v)+d(v,y).

Then (X,d) is called a generalized metric space (or for short g.m.s).

Definition 1.2 Let (X,d) be a g.m.s, { x n } be a sequence in X and xX. We say that { x n } is convergent to x if and only if d( x n ,x)0 as n. We denote this by x n x.

Definition 1.3 Let (X,d) be a g.m.s and { x n } be a sequence in X. We say that { x n } is Cauchy if and only if d( x n , x m )0 as n,m.

Definition 1.4 Let (X,d) be a g.m.s. We say that (X,d) is complete if and only if every Cauchy sequence in X converges to some element in X.

The following result was established in [17] (see also [34]).

Lemma 1.1 Let (X,d) be a g.m.s and { x n } be a Cauchy sequence in (X,d) such that d( x n ,x)0 as n for some xX. Then d( x n ,y)d(x,y) as n for all yX. In particular, { x n } does not converge to y if yx.

We denote by Θ the set of functions θ:(0,)(1,) satisfying the following conditions:

( Θ 1 ) θ is non-decreasing;

( Θ 2 ) for each sequence { t n }(0,), lim n θ( t n )=1 if and only if lim n t n = 0 + ;

( Θ 3 ) there exist r(0,1) and (0,] such that lim t 0 + θ ( t ) 1 t r =;

( Θ 4 ) θ is continuous.

Recently, Jleli and Samet [35] established the following generalization of the Banach fixed point theorem in the setting of Branciari metric spaces.

Theorem 1.1 (Jleli and Samet [35])

Let (X,d) be a complete g.m.s and T:XX be a given map. Suppose that there exist θΘ and k(0,1) such that

x,yX,d(Tx,Ty)0θ ( d ( T x , T y ) ) [ θ ( d ( x , y ) ) ] k .

Then T has a unique fixed point.

Note that the condition ( Θ 4 ) is not supposed in Theorem 1.1.

The aim of this paper is to extend the result given by Theorem 1.1.

2 Result and proof

Now, we are ready to state and prove our main result.

Theorem 2.1 Let (X,d) be a complete g.m.s and T:XX be a given map. Suppose that there exist θΘ and k(0,1) such that

x,yX,d(Tx,Ty)0θ ( d ( T x , T y ) ) [ θ ( M ( x , y ) ) ] k ,
(1)

where

M(x,y)=max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) } .
(2)

Then T has a unique fixed point.

Proof Let xX be an arbitrary point in X. If for some pN we have T p x= T p + 1 x, then T p x will be a fixed point of T. So, without restriction of the generality, we can suppose that d( T n x, T n + 1 x)>0 for all nN. Now, from (1), for all nN, we have

θ ( d ( T n x , T n + 1 x ) ) [ θ ( M ( T n 1 x , T n x ) ) ] k ,
(3)

where from (2)

M ( T n 1 x , T n x ) = max { d ( T n 1 x , T n x ) , d ( T n 1 x , T T n 1 x ) , d ( T n x , T T n x ) } = max { d ( T n 1 x , T n x ) , d ( T n 1 x , T n x ) , d ( T n x , T n + 1 x ) } = max { d ( T n 1 x , T n x ) , d ( T n x , T n + 1 x ) } .
(4)

If M( T n 1 x, T n x)=d( T n x, T n + 1 x), then inequality (3) turns into

θ ( d ( T n x , T n + 1 x ) ) [ θ ( d ( T n x , T n + 1 x ) ) ] k ,

which implies that

ln [ θ ( d ( T n x , T n + 1 x ) ) ] kln [ θ ( d ( T n x , T n + 1 x ) ) ] ,

that is a contradiction with k(0,1). Hence, from (4) we have M( T n 1 x, T n x)=d( T n 1 x, T n x), and inequality (3) yields

θ ( d ( T n x , T n + 1 x ) ) [ θ ( d ( T n 1 x , T n x ) ) ] k [ θ ( d ( T n 2 x , T n 1 x ) ) ] k 2 [ θ ( d ( x , T x ) ) ] k n .

Thus we have

1θ ( d ( T n x , T n + 1 x ) ) [ θ ( d ( x , T x ) ) ] k n for all nN.
(5)

Letting n in (5), we obtain

θ ( d ( T n x , T n + 1 x ) ) 1as n,
(6)

which implies from ( Θ 2 ) that

lim n d ( T n x , T n + 1 x ) =0.

From condition ( Θ 3 ), there exist r(0,1) and (0,] such that

lim n θ ( d ( T n x , T n + 1 x ) ) 1 [ d ( T n x , T n + 1 x ) ] r =.

Suppose that <. In this case, let B=/2>0. From the definition of the limit, there exists n 0 N such that

| θ ( d ( T n x , T n + 1 x ) ) 1 [ d ( T n x , T n + 1 x ) ] r |Bfor all n n 0 .

This implies that

θ ( d ( T n x , T n + 1 x ) ) 1 [ d ( T n x , T n + 1 x ) ] r B=Bfor all n n 0 .

Then

n [ d ( T n x , T n + 1 x ) ] r An [ θ ( d ( T n x , T n + 1 x ) ) 1 ] for all n n 0 ,

where A=1/B.

Suppose now that =. Let B>0 be an arbitrary positive number. From the definition of the limit, there exists n 0 N such that

θ ( d ( T n x , T n + 1 x ) ) 1 [ d ( T n x , T n + 1 x ) ] r Bfor all n n 0 .

This implies that

n [ d ( T n x , T n + 1 x ) ] r An [ θ ( d ( T n x , T n + 1 x ) ) 1 ] for all n n 0 ,

where A=1/B.

Thus, in all cases, there exist A>0 and n 0 N such that

n [ d ( T n x , T n + 1 x ) ] r An [ θ ( d ( T n x , T n + 1 x ) ) 1 ] for all n n 0 .

Using (5), we obtain

n [ d ( T n x , T n + 1 x ) ] r An ( [ θ ( d ( x , T x ) ) ] k n 1 ) for all n n 0 .

Letting n in the above inequality, we obtain

lim n n [ d ( T n x , T n + 1 x ) ] r =0.

Thus, there exists n 1 N such that

d ( T n x , T n + 1 x ) 1 n 1 / r for all n n 1 .
(7)

Now, we shall prove that T has a periodic point. Suppose that it is not the case, then T n x T m x for every n,mN such that nm. Using (1), we obtain

θ ( d ( T n x , T n + 2 x ) ) [ θ ( M ( T n 1 x , T n + 1 x ) ) ] k ,
(8)

where from (2)

M ( T n 1 x , T n + 1 x ) =max { d ( T n 1 x , T n + 1 x ) , d ( T n 1 x , T n x ) , d ( T n + 1 x , T n + 2 x ) } .
(9)

Since θ is non-decreasing, we obtain from (8) and (9)

θ ( d ( T n x , T n + 2 x ) ) [ max { θ ( d ( T n 1 x , T n + 1 x ) ) , θ ( d ( T n 1 x , T n x ) ) , θ ( d ( T n + 1 x , T n + 2 x ) ) } ] k .
(10)

Let I be the set of nN such that

u n : = max { θ ( d ( T n 1 x , T n + 1 x ) ) , θ ( d ( T n 1 x , T n x ) ) , θ ( d ( T n + 1 x , T n + 2 x ) ) } = θ ( d ( T n 1 x , T n + 1 x ) ) .

If |I|<, then there exists NN such that for every nN,

max { θ ( d ( T n 1 x , T n + 1 x ) ) , θ ( d ( T n 1 x , T n x ) ) , θ ( d ( T n + 1 x , T n + 2 x ) ) } = max { θ ( d ( T n 1 x , T n x ) ) , θ ( d ( T n + 1 x , T n + 2 x ) ) } .

In this case, we obtain from (10)

1θ ( d ( T n x , T n + 2 x ) ) [ max { θ ( d ( T n 1 x , T n x ) ) , θ ( d ( T n + 1 x , T n + 2 x ) ) } ] k

for all nN. Letting n in the above inequality and using (6), we get

θ ( d ( T n x , T n + 2 x ) ) 1as n.

If |I|=, we can find a subsequence of { u n }, that we denote also by { u n }, such that

u n =θ ( d ( T n 1 x , T n + 1 x ) ) for n large enough.

In this case, we obtain from (10)

1 θ ( d ( T n x , T n + 2 x ) ) [ θ ( d ( T n 1 x , T n + 1 x ) ) ] k [ θ ( d ( T n 2 x , T n x ) ) ] k 2 [ θ ( d ( x , T 2 x ) ) ] k n

for n large enough. Letting n in the above inequality, we obtain

θ ( d ( T n x , T n + 2 x ) ) 1as n.
(11)

Then in all cases, (11) holds. Using (11) and the property ( Θ 2 ), we obtain

lim n d ( T n x , T n + 2 x ) =0.

Similarly, from condition ( Θ 3 ), there exists n 2 N such that

d ( T n x , T n + 2 x ) 1 n 1 / r for all n n 2 .
(12)

Let N=max{ n 0 , n 1 }. We consider two cases.

Case 1. If m>2 is odd, then writing m=2L+1, L1, using (7), for all nN, we obtain

d ( T n x , T n + m x ) d ( T n x , T n + 1 x ) + d ( T n + 1 x , T n + 2 x ) + + d ( T n + 2 L x , T n + 2 L + 1 x ) 1 n 1 / r + 1 ( n + 1 ) 1 / r + + 1 ( n + 2 L ) 1 / r i = n 1 i 1 / r .

Case 2. If m>2 is even, then writing m=2L, L2, using (7) and (12), for all nN, we obtain

d ( T n x , T n + m x ) d ( T n x , T n + 2 x ) + d ( T n + 2 x , T n + 3 x ) + + d ( T n + 2 L 1 x , T n + 2 L x ) 1 n 1 / r + 1 ( n + 2 ) 1 / r + + 1 ( n + 2 L 1 ) 1 / r i = n 1 i 1 / r .

Thus, combining all the cases, we have

d ( T n x , T n + m x ) i = n 1 i 1 / r for all nN,mN.

From the convergence of the series i 1 i 1 / r (since 1/r>1), we deduce that { T n x} is a Cauchy sequence. Since (X,d) is complete, there is zX such that T n xz as n. Without restriction of the generality, we can suppose that T n xz for all n (or for n large enough). Suppose that d(z,Tz)>0, using (1), we get

θ ( d ( T n + 1 x , T z ) ) [ θ ( M ( T n x , z ) ) ] k for all nN,

where

M ( T n x , z ) =max { d ( T n x , z ) , d ( T n x , T n + 1 x ) , d ( z , T z ) } .

Letting n in the above inequality, using ( Θ 4 ) and Lemma 1.1, we obtain

θ ( d ( z , T z ) ) [ θ ( d ( z , T z ) ) ] k <θ ( d ( z , T z ) ) ,

which is a contradiction. Thus we have z=Tz, which is also a contradiction with the assumption: T does not have a periodic point. Thus T has a periodic point, say z, of period q. Suppose that the set of fixed points of T is empty. Then we have

q>1andd(z,Tz)>0.

Using (1), we obtain

θ ( d ( z , T z ) ) =θ ( d ( T q z , T q + 1 z ) ) [ θ ( z , T z ) ] k q <θ ( d ( z , T z ) ) ,

which is a contradiction. Thus, the set of fixed points of T is non-empty, that is, T has at least one fixed point. Now, suppose that z,uX are two fixed points of T such that d(z,u)=d(Tz,Tu)>0. Using (1), we obtain

θ ( d ( z , u ) ) =θ ( d ( T z , T u ) ) [ θ ( d ( z , u ) ) ] k <θ ( d ( z , u ) ) ,

which is a contradiction. Then we have one and only one fixed point. □

3 Some consequences

We start by deducing the following fixed point result.

Corollary 3.1 Let (X,d) be a complete g.m.s and T:XX be a given map. Suppose that there exists λ(0,1) such that

d(Tx,Ty)λmax { d ( x , y ) , d ( x , T x ) , d ( y , T y ) } for all x,yX.
(13)

Then T has a unique fixed point.

Proof From (13), we have

e d ( T x , T y ) [ e max { d ( x , y ) , d ( x , T x ) , d ( y , T y ) } ] λ for all x,yX.

Clearly the function θ:(0,)(1,) defined by θ(t):= e t belongs to Θ. So, the existence and uniqueness of the fixed point follows from Theorem 2.1. □

The following fixed point result established in [11] is an immediate consequence of Corollary 3.1.

Corollary 3.2 Let (X,d) be a complete g.m.s and T:XX be a given map. Suppose that there exists λ(0,1) such that

d(Tx,Ty)λd(x,y)for all x,yX.

Then T has a unique fixed point.

The following fixed point result established in [34] is an immediate consequence of Corollary 3.1.

Corollary 3.3 Let (X,d) be a complete g.m.s and T:XX be a given map. Suppose that there exist λ,μ0 with λ+μ<1 such that

d(Tx,Ty)λd(x,Tx)+μd(y,Ty)for all x,yX.

Then T has a unique fixed point.

The following fixed point result is also an immediate consequence of Corollary 3.1.

Corollary 3.4 Let (X,d) be a complete g.m.s and T:XX be a given map. Suppose that there exist λ,μ,ν0 with λ+μ+ν<1 such that

d(Tx,Ty)λd(x,y)+μd(x,Tx)+νd(y,Ty)for all x,yX.

Then T has a unique fixed point.

We note that Θ contains a large class of functions. For example, for

θ(t):=2 2 π arctan ( 1 t α ) ,0<α<1,t>0,

we obtain from Theorem 2.1 the following result.

Corollary 3.5 Let (X,d) be a complete g.m.s and T:XX be a given map. Suppose that there exist α,k(0,1) such that

2 2 π arctan ( 1 [ d ( T x , T y ) ] α ) [ 2 2 π arctan ( 1 [ M ( x , y ) ] α ) ] k for all x,yX,TxTy,

where M(x,y) is given by (2). Then T has a unique fixed point.

Finally, since a metric space is a g.m.s, from Theorem 2.1 we deduce immediately the following result.

Corollary 3.6 Let (X,d) be a complete metric space and T:XX be a given map. Suppose that there exist θΘ and k(0,1) such that

x,yX,d(Tx,Ty)0θ ( d ( T x , T y ) ) [ θ ( M ( x , y ) ) ] k ,

where M(x,y) is given by (2). Then T has a unique fixed point.

References

  1. 1.

    Banach S: Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales. Fundam. Math. 1922, 3: 133–181.

  2. 2.

    Boyd DW, Wong JSW: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20: 458–464. 10.1090/S0002-9939-1969-0239559-9

  3. 3.

    Ćirić L: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 1974,45(2):267–273.

  4. 4.

    Kirk WA: Fixed points of asymptotic contractions. J. Math. Anal. Appl. 2003, 277: 645–650. 10.1016/S0022-247X(02)00612-1

  5. 5.

    Meir A, Keeler E: A theorem on contraction mappings. J. Math. Anal. Appl. 1969, 28: 326–329. 10.1016/0022-247X(69)90031-6

  6. 6.

    Rakotch E: A note on contractive mappings. Proc. Am. Math. Soc. 1962, 13: 459–465. 10.1090/S0002-9939-1962-0148046-1

  7. 7.

    Reich S: Fixed points of contractive functions. Boll. Unione Mat. Ital. 1972, 5: 26–42.

  8. 8.

    Suzuki T: Fixed point theorem for asymptotic contractions of Meir-Keeler type in complete metric spaces. Nonlinear Anal. 2006, 64: 971–978. 10.1016/j.na.2005.04.054

  9. 9.

    Wardowski D: Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 94

  10. 10.

    Di Bari C, Vetro P: Common fixed points in generalized metric spaces. Appl. Math. Comput. 2012,218(13):7322–7325. 10.1016/j.amc.2012.01.010

  11. 11.

    Branciari A: A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces. Publ. Math. (Debr.) 2000, 57: 31–37.

  12. 12.

    Cherichi M, Samet B: Fixed point theorems on ordered gauge spaces with applications to nonlinear integral equations. Fixed Point Theory Appl. 2012., 2012: Article ID 13

  13. 13.

    Das P: A fixed point theorem on a class of generalized metric spaces. Korean J. Math. Sci. 2002, 9: 29–33.

  14. 14.

    Frigon M: Fixed point results for generalized contractions in gauge spaces and applications. Proc. Am. Math. Soc. 2000, 128: 2957–2965. 10.1090/S0002-9939-00-05838-X

  15. 15.

    Janković S, Kadelburg Z, Radenović S: On cone metric spaces: a survey. Nonlinear Anal. 2011, 74: 2591–2601. 10.1016/j.na.2010.12.014

  16. 16.

    Khamsi MA, Kozlowski WM, Reich S: Fixed point theory in modular function spaces. Nonlinear Anal. 1990,14(11):935–953. 10.1016/0362-546X(90)90111-S

  17. 17.

    Kirk WA, Shahzad N: Generalized metrics and Caristi’s theorem. Fixed Point Theory Appl. 2013., 2013: Article ID 129

  18. 18.

    Lakzian H, Samet B:Fixed points for (ψ,φ)-weakly contractive mappings in generalized metric spaces. Appl. Math. Lett. 2012,25(5):902–906. 10.1016/j.aml.2011.10.047

  19. 19.

    Samet B: Discussion on: A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, by A. Branciari. Publ. Math. (Debr.) 2010,76(4):493–494.

  20. 20.

    Sarama IR, Rao JM, Rao SS: Contractions over generalized metric spaces. J. Nonlinear Sci. Appl. 2009,2(3):180–182.

  21. 21.

    Tarafdar E: An approach to fixed point theorems on uniform spaces. Trans. Am. Math. Soc. 1974, 191: 209–225.

  22. 22.

    arXiv: 1208.4610v1

  23. 23.

    Vetro C: On Branciari’s theorem for weakly compatible mappings. Appl. Math. Lett. 2010,23(6):700–705. 10.1016/j.aml.2010.02.011

  24. 24.

    Nadler SB Jr.: Multi-valued contraction mappings. Pac. J. Math. 1969, 30: 475–488. 10.2140/pjm.1969.30.475

  25. 25.

    Berinde M, Berinde V: On a general class of multi-valued weakly Picard mappings. J. Math. Anal. Appl. 2007, 326: 772–782. 10.1016/j.jmaa.2006.03.016

  26. 26.

    Ćirić L: Multi-valued nonlinear contraction mappings. Nonlinear Anal. 2009, 71: 2716–2723. 10.1016/j.na.2009.01.116

  27. 27.

    Markin JT: A fixed point theorem for set-valued mappings. Bull. Am. Math. Soc. 1968, 74: 639–640. 10.1090/S0002-9904-1968-11971-8

  28. 28.

    Mizoguchi N, Takahashi W: Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 1989, 141: 177–188. 10.1016/0022-247X(89)90214-X

  29. 29.

    Naidu SVR: Fixed point theorems for a broad class of multimaps. Nonlinear Anal. 2003, 52: 961–969. 10.1016/S0362-546X(02)00146-3

  30. 30.

    Zhang CK, Zhu J, Zhao PH: An extension of multi-valued contraction mappings and fixed points. Proc. Am. Math. Soc. 2000, 128: 2439–2444. 10.1090/S0002-9939-99-05318-6

  31. 31.

    Karapınar E: Fixed points results for alpha-admissible mapping of integral type on generalized metric spaces. Abstr. Appl. Anal. 2014., 2014: Article ID 141409

  32. 32.

    Karapınar E: Discussion on contractions on generalized metric spaces. Abstr. Appl. Anal. 2014., 2014: Article ID 962784

  33. 33.

    La Rosa V, Vetro P: Common fixed points for α - ψ - φ -contractions in generalized metric spaces. Nonlinear Anal., Model. Control 2014,19(1):43–54.

  34. 34.

    Jleli M, Samet B: The Kannan’s fixed point theorem in a cone rectangular metric space. J. Nonlinear Sci. Appl. 2009,2(3):161–167.

  35. 35.

    Jleli M, Samet B: A new generalization of the Banach contraction principle. J. Inequal. Appl. 2014., 2014: Article ID 38

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Acknowledgements

The first and third authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project No: RG-1435-034.

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Correspondence to Bessem Samet.

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All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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Jleli, M., Karapınar, E. & Samet, B. Further generalizations of the Banach contraction principle. J Inequal Appl 2014, 439 (2014) doi:10.1186/1029-242X-2014-439

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Keywords

  • fixed point
  • Branciari metric space
  • existence
  • uniqueness