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A new generalization of the Banach contraction principle

Abstract

We present a new generalization of the Banach contraction principle in the setting of Branciari metric spaces.

1 Introduction

The fixed-point theorem, generally known as the Banach contraction principle, appeared in 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:X→X is a contraction map (i.e., d(Tx,Ty)≤λd(x,y) for all x,y∈X, 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 [2–9] and others. In other generalizations, the topology is weakened; see [10–23] 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 [25–30] and 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,v∈X. Various fixed point results were established on such spaces; see [10, 13, 17–20, 22] and references therein.

In this paper, we introduce a new type of contractive maps and we establish a new fixed-point theorem for such maps on the setting of generalized metric spaces.

2 Main results

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 ℓ∈(0,∞] such that lim t → 0 + θ ( t ) − 1 t r =ℓ.

Before we prove the main results, we recall the following definitions introduced in [11].

Definition 2.1 Let X be a non-empty set and d:X×X→[0,∞) be a mapping such that for all x,y∈X and for all distinct points u,v∈X, each of them different from x and y, one has

  1. (i)

    d(x,y)=0⟺x=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 2.2 Let (X,d) be a g.m.s., { x n } be a sequence in X and x∈X. 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 2.3 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 2.4 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 [31] (Lemma 1.10).

Lemma 2.1 Let(X,d)be a g.m.s., { x n }be a Cauchy sequence in(X,d), andx,y∈X. Suppose that there exists a positive integer N such that

  1. (i)

    x n ≠ x m , for all n,m>N;

  2. (ii)

    x n and x are distinct points in X, for all n>N;

  3. (iii)

    x n and y are distinct points in X, for all n>N;

  4. (iv)

    lim n → ∞ d( x n ,x)= lim n → ∞ d( x n ,y).

Then we havex=y.

We observe easily that if one of the conditions (ii) or (iii) is not satisfied, then the result of Lemma 2.1 is still valid.

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

Theorem 2.1 Let(X,d)be a complete g.m.s. andT:X→Xbe a given map. Suppose that there existθ∈Θandk∈(0,1)such that

x,y∈X,d(Tx,Ty)≠0⟹θ ( d ( T x , T y ) ) ≤ [ θ ( d ( x , y ) ) ] k .
(1)

Then T has a unique fixed point.

Proof Let x∈X be an arbitrary point in X. If for some p∈N, 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 n∈N. Now, from (1), for all n∈N, we have

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

1≤θ ( d ( T n x , T n + 1 x ) ) ≤ [ θ ( d ( x , T x ) ) ] k n ,for all n∈N.
(2)

Letting n→∞ in (2), we obtain

θ ( d ( T n x , T n + 1 x ) ) →1as n→∞,

which implies from ( Θ 2 ) that

lim n → ∞ d ( T n x , T n + 1 x ) =0.
(3)

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

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

Suppose that ℓ<∞. In this case, let B=ℓ/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 −ℓ|≤B,for 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 ≥ℓ−B=B,for 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 ℓ=∞. 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 ≥B,for 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 (2), 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 .
(4)

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,m∈N such that n≠m. Using (1), we obtain

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 .

Letting n→∞ in the above inequality and using ( Θ 2 ), we obtain

lim n → ∞ d ( T n x , T n + 2 x ) =0.
(5)

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 .
(6)

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

Case 1. If m>2 is odd, then writing m=2L+1, L≥1, using (4), for all n≥N, 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, L≥2, using (4) and (6), for all n≥N, 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 n≥N,m∈N.

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 z∈X such that T n x→z. On the other hand, observe that T is continuous, indeed, if Tx≠Ty, then we have from (1)

ln [ θ ( d ( T x , T y ) ) ] ≤kln [ θ ( d ( x , y ) ) ] ≤ln [ θ ( d ( x , y ) ) ] ,

which implies from ( Θ 1 ) that

d(Tx,Ty)≤d(x,y),for all x,y∈X.

From this observation, for all n∈N, we have

d ( T n + 1 x , T z ) ≤d ( T n x , z ) .

Letting n→∞ in the above inequality, we get T n + 1 x→Tz. From Lemma 2.1, we obtain z=Tz, which is 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 n z , T n + 1 z ) ) ≤ [ θ ( d ( z , T z ) ) ] k n <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,u∈X 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. □

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

Corollary 2.1 Let(X,d)be a complete metric space andT:X→Xbe a given map. Suppose that there existθ∈Θandk∈(0,1)such that

x,y∈X,d(Tx,Ty)≠0⟹θ ( d ( T x , T y ) ) ≤ [ θ ( d ( x , y ) ) ] k .

Then T has a unique fixed point.

Observe that the Banach contraction principle follows immediately from Corollary 2.1. Indeed, if T is a Banach contraction, i.e., there exists λ∈(0,1) such that

d(Tx,Ty)≤λd(x,y),for all x,y∈X,

then we have

e d ( T x , T y ) ≤ [ e d ( x , y ) ] k ,for all x,y∈X.

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 Corollary 2.1. In the following example (inspired by [9]), we show that Corollary 2.1 is a real generalization of the Banach contraction principle.

Example Let X be the set defined by

X:={ τ n :n∈N},

where

Ï„ n := n ( n + 1 ) 2 ,for all n∈N.

We endow X with the metric d given by d(x,y):=|x−y| for all x,y∈X. It is not difficult to show that (X,d) is a complete metric space. Let T:X→X be the map defined by

T Ï„ 1 = Ï„ 1 ,T Ï„ n = Ï„ n − 1 ,for all n≥2.

Clearly, the Banach contraction is not satisfied. In fact, we can check easily that

lim n → ∞ d ( T τ n , T τ 1 ) d ( τ n , τ 1 ) =1.

Now, consider the function θ:(0,∞)→(1,∞) defined by

θ(t):= e t e t .

It is not difficult to show that θ∈Θ. We shall prove that T satisfies the condition (1), that is,

d(T τ n ,T τ m )≠0⟹ e d ( T τ n , T τ m ) e d ( T τ n , T τ m ) ≤ e k d ( τ n , τ m ) e d ( τ n , τ m ) ,

for some k∈(0,1). The above condition is equivalent to

d(T τ n ,T τ m )≠0⟹d(T τ n ,T τ m ) e d ( T τ n , T τ m ) ≤ k 2 d( τ n , τ m ) e d ( τ n , τ m ) .

So, we have to check that

d(T τ n ,T τ m )≠0⟹ d ( T τ n , T τ m ) e d ( T τ n , T τ m ) − d ( τ n , τ m ) d ( τ n , τ m ) ≤ k 2 ,
(7)

for some k∈(0,1). We consider two cases.

Case 1. n=1 and m>2. In this case, we have

d ( T τ 1 , T τ m ) e d ( T τ 1 , T τ m ) − d ( τ 1 , τ m ) d ( τ 1 , τ m ) = m 2 − m − 2 m 2 + m − 2 e − m ≤ e − 1 .

Case 2. m>n>1. In this case, we have

d ( T τ m , T τ n ) e d ( T τ m , T τ n ) − d ( τ m , τ n ) d ( τ m , τ n ) = m + n − 1 m + n + 1 e n − m ≤ e − 1 .

Thus, the inequality (7) is satisfied with k= e − 1 / 2 . Theorem 2.1 (or Corollary 2.1) implies that T has a unique fixed point. In this example Ï„ 1 is the unique fixed point of T.

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 2.2 Let(X,d)be a complete g.m.s. andT:X→Xbe a given map. Suppose that there existα,k∈(0,1)such that

2− 2 Ï€ arctan ( 1 [ d ( T x , T y ) ] α ) ≤ [ 2 − 2 Ï€ arctan ( 1 [ d ( x , y ) ] α ) ] k ,for all x,y∈X,Tx≠Ty.

Then T has a unique fixed point.

References

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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

    Google Scholar 

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

    Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Turinici, M: Functional contractions in local Branciari metric spaces (2012). arXiv:1208.4610v1 [math.GN]

    MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  31. 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.

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This project was supported by King Saud University, Deanship of Scientific Research, College of Science Research Center.

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Jleli, M., Samet, B. A new generalization of the Banach contraction principle. J Inequal Appl 2014, 38 (2014). https://doi.org/10.1186/1029-242X-2014-38

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