Open Access

Fixed points of a new type of contractive mappings in complete metric spaces

Fixed Point Theory and Applications20122012:94

https://doi.org/10.1186/1687-1812-2012-94

Received: 20 April 2012

Accepted: 7 June 2012

Published: 7 June 2012

Abstract

In the article, we introduce a new concept of contraction and prove a fixed point theorem which generalizes Banach contraction principle in a different way than in the known results from the literature. The article includes an example which shows the validity of our results, additionally there is delivered numerical data which illustrates the provided example.

MSC: 47H10; 54E50

Keywords

F-contractioncontractive mappingfixed pointcomplete metric space

1 Introduction

Throughout the article denoted by is the set of all real numbers, by + is the set of all positive real numbers and by is the set of all natural numbers. (X, d), (X for short), is a metric space with a metric d.

In the literature, there are plenty of extensions of the famous Banach contraction principle [1], which states that every self-mapping T defined on a complete metric space (X, d) satisfying
x , y X d ( T x , T y ) λ d ( x , y ) ,  where  λ ( 0 , 1 ) ,
(1)

has a unique fixed point and for every x0 X a sequence {T n x0}nis convergent to the fixed point. Some of the extensions weaken right side of inequality in the condition (1) by replacing λ with a mapping, see e.g. [2, 3]. In other results, the underlying space is more general, see e.g [47]. The Nadler's paper [8] started the invatigations concerning fixed point theory for set-valued contractions, see e.g. [920]. There are many theorems regarding asymptotic contractions, see e.g. [2123], contractions of Meir-Keeler type [24], see e.g [19, 23, 25] and weak contractions, see e.g. [2628]. There are also lots of different types of fixed point theorems not mentioned above extending the Banach's result.

In the present article, using a mapping F: + we introduce a new type of contraction called F-contraction and prove a new fixed point theorem concerning F-contraction. For the concrete mappings F, we obtain the contractions of the type known from the literature, Banach contraction as well. The article includes the examples of F-contractions and an example showing that the obtained extension is significant. Theoretical considerations that we support by computational data illustrate the nature of F-contractions.

2 The result

Definition 2.1 Let F: + be a mapping satisfying:

(F1) F is strictly increasing, i.e. for all α, β + such that α < β, F (α) < F (β);

(F2) For each sequence {α n }nof positive numbers limn→∞α n = 0 if and only if limn→∞F (α n ) = -∞;

(F3) There exists k (0, 1) such that limα→0+ α k F(α) = 0.

A mapping T: XX is said to be an F-contraction if there exists τ > 0 such that
x , y X ( d ( T x , T y ) > 0 τ + F ( d ( T x , T y ) ) F ( d ( x , y ) ) ) .
(2)

When we consider in (2) the different types of the mapping F then we obtain the variety of contractions, some of them are of a type known in the literature. See the following examples:

Example 2.1 Let F : + be given by the formula F (α) = ln α. It is clear that F satisfies (F1)-(F3) ((F3) for any k (0, 1)). Each mapping T : XX satisfying (2) is an F-contraction such that
d ( T x , T y ) e - τ d ( x , y ) , for all  x , y X , T x T y .
(3)

It is clear that for x, y X such that Tx = Ty the inequality d(Tx, Ty) ≤ e-τd(x, y) also holds, i.e. T is a Banach contraction [1].

Example 2.2 If F(α) = ln α + α, α > 0 then F satisfies (F1)-(F3) and the condition (2) is of the form
d ( T x , T y ) d ( x , y ) e d ( T x , T y ) - d ( x , y ) e - τ , for all x , y X , T x T y .
(4)
Example 2.3 Consider F ( α ) = - 1 / α , α > 0. F satisfies (F1)-(F3) ((F3) for any k (1/2, 1)). In this case, each F-contraction T satisfies
d ( T x , T y ) 1 ( 1 + τ d ( x , y ) ) 2 d ( x , y ) , for all  x , y X , T x T y .

Here, we obtained a special case of nonlinear contraction of the type d(Tx, Ty) ≤ α(d(x, y))d(x, y). For details see [2, 3].

Example 2.4 Let F(α) = ln(α2 + α), α > 0. Obviously F satisfies (F1)-(F3) and for F-contraction T, the following condition holds:
d ( T x , T y ) ( d ( T x , T y ) + 1 ) d ( x , y ) ( d ( x , y ) + 1 ) e - τ , for all x , y X , T x T y .

Let us observe that in Examples 2.1-2.4 the contractive conditions are satisfied for x, y X, such that Tx = Ty.

Remark 2.1 From (F1) and (2) it is easy to conclude that every F-contraction T is a contractive mapping, i.e.
d ( T x , T y ) < d( x y ), for all  x y X , T x T y .

Thus every F-contraction is a continuous mapping.

Remark 2.2 Let F1, F2 be the mappings satisfying (F1)-(F3). If F1(α) ≤ F2(α) for all α > 0 and a mapping G = F2 - F1 is nondecreasing then every F1-contraction T is F2-contraction.

Indeed, from Remark 2.1 we have G(d(Tx, Ty)) ≤ G(d(x, y)) for all x, y X, TxTy. Thus, for all x, y X, TxTy we obtain
τ + F 2 ( d ( T x , T y ) ) = τ + F 1 ( d ( T x , T y ) ) + G ( d ( T x , T y ) ) F 1 ( d ( x , y ) ) + G ( d ( x , y ) ) = F 2 ( d ( x , y ) ) .

Now we state the main result of the article.

Theorem 2.1 Let (X, d) be a complete metric space and let T : XX be an F-contraction. Then T has a unique fixed point x* X and for every x0 X a sequence {T n x0}nis convergent to x*.

Proof. First, let us observe that T has at most one fixed point. Indeed, if x 1 * , x 2 * X , T x 1 * = x 1 * x 2 * = T x 2 * , then we get
τ F ( d ( x 1 * , x 2 * ) ) - F ( d ( T x 1 * , T x 2 * ) ) = 0 ,

which is a contradiction.

In order to show that T has a fixed point let x0 X be arbitrary and fixed. We define a sequence {x n }n X, xn+1= Tx n , n = 0, 1, .... Denote γ n = d(xn+1, x n ), n = 0, 1, ....

If there exists n0 for which x n 0 + 1 = x n 0 , then T x n 0 = x n 0 and the proof is finished.

Suppose now that xn+1x n , for every n . Then γ n > 0 for all n and, using (2), the following holds for every n :
F ( γ n ) F ( γ n - 1 ) - τ F ( γ n - 2 ) - 2 τ F ( γ 0 ) - n τ .
(5)
From (5), we obtain limn→∞F(γ n ) = -∞ that together with (F2) gives
lim n γ n = 0 .
(6)
From (F3) there exists k (0, 1) such that
lim n γ n k F ( γ n ) = 0 .
(7)
By (5), the following holds for all n :
γ n k F ( γ n ) - γ n k F ( γ 0 ) γ n k ( F ( γ 0 ) - n τ ) - γ n k F ( γ 0 ) = - γ n k n τ 0 .
(8)
Letting n → ∞ in (8), and using (6) and (7), we obtain
lim n n γ n k = 0 .
(9)
Now, let us observe that from (9) there exists n1 such that n γ n k 1 for all nn1. Consequently we have
γ n 1 n 1 / k , for all n n 1 .
(10)

In order to show that {x n }nis a Cauchy sequence consider m, n such that m > nn1. From the definition of the metric and from (10) we get

d ( x m , x n ) γ m - 1 + γ m - 2 + + γ n < i = n γ i i = n 1 i 1 / k .

From the above and from the convergence of the series i = 1 1 / i 1 k we receive that {x n }nis a Cauchy sequence.

From the completeness of X there exists x* X such that limn→∞x n = x*. Finally, the continuity of T yields
d ( T x * , x * ) = lim n d ( T x n , x n ) = lim n d ( x n + 1 , x n ) = 0 ,

which completes the proof. □

Note that for the mappings F1(α) = ln(α), α > 0, F2(α) = ln(α) + α, α > 0, F1< F2 and a mapping F2 - F1 is strictly increasing. Hence, by Remark 2.2, we obtain that every Banach contraction (3) satisfies the contraction condition (4). On the other side in Example 2.5, we present a metric space and a mapping T which is not F1-contraction (Banach contraction), but still is an F2-contraction. Consequently, Theorem 2.1 gives the family of contractions which in general are not equivalent.

Example 2.5 Consider the sequence {S n }nas follows:
S 1 = 1 , S 2 = 1 + 2 , S n = 1 + 2 + + n = n ( n + 1 ) 2 , n ,
Let X = {S n : n } and d(x, y) = |x - y|, x, y X. Then (X, d) is a complete metric space. Define the mapping T : XX by the formulae:
T ( S n ) = S n - 1 for n > 1 , T ( S 1 ) = S 1 .
First, let us consider the mapping F1 defined in Example 2.1. The mapping T is not the F1-contraction in this case (which actually means that T is not the Banach contraction). Indeed, we get
lim n d ( T ( S n ) , T ( S 1 ) ) d ( S n , S 1 ) = lim n S n - 1 - 1 S n - 1 = 1 .

On the other side taking F2 as in Example 2.2, we obtain that T is F2-contraction with τ = 1. To see this, let us consider the following calculations:

First, observe that
m , n [ T ( S m ) T ( S n ) ( ( m > 2 n = 1 ) ( m > n > 1 ) ) ] .
For every m , m > 2 we have
d ( T ( S m ) , T ( S 1 ) ) d ( S m , S 1 ) e d ( T ( S m ) , T ( S 1 ) ) - d ( S m , S 1 ) = S m - 1 - 1 S m - 1 e S m - 1 - S m = m 2 - m - 2 m 2 + m - 2 e - m < e - m < e - 1 .
For every m, n , m > n > 1 the following holds
d ( T ( S m ) , T ( S n ) ) d ( S m , S n ) e d ( T ( S m ) , T ( S n ) ) - d ( S m , S n ) = S m - 1 - S n - 1 S m - S n e S n - S n - 1 + S m - 1 - S m = m + n - 1 m + n + 1 e n - m < e n - m e - 1 .
Clearly S1 is a fixed point of T. To see the computational data confirming the above calculations the reader is referred to Table 1.
Table 1

The comparison of Banach contraction condition with F-contraction condition

n

xn

C F 1 ( S 1 , S n )

C F 2 ( S 1 , S n )

3

378

0.91629

3.91629

4

351

0.58779

4.58779

5

325

0.44183

5.44183

6

300

0.35667

6.35667

7

276

0.30010

7.30010

8

253

0.25951

8.25951

9

231

0.22884

9.22884

10

210

0.20479

10.20479

11

190

0.18540

11.18540

12

171

0.16942

12.16942

13

153

0.15600

13.15600

14

136

0.14458

14.14458

15

120

0.13473

15.13473

16

105

0.12615

16.12615

17

91

0.11861

17.11861

18

78

0.11192

18.11192

19

66

0.10595

19.10595

20

55

0.10059

20.10059

21

45

0.09575

21.09575

22

36

0.09135

22.09135

23

28

0.08734

23.08734

24

21

0.08367

24.08367

25

15

0.08030

25.08030

26

10

0.07719

26.07719

27

6

0.07431

27.07431

28

3

0.07164

28.07164

29

1

0.06916

29.06916

30

1

0.06684

30.06684

3 ×104

1

6.66667 ×10-5

30000.00007

n → ∞

T 1 = 1

tends to 0

τ = 1

The generated iterations start from a point x0 = S29 = 435. C F (S1, S n ) denotes F(d(S1, S n )) - F(d(T(S1), T(S n )))

Declarations

Acknowledgements

The author is very grateful to the reviewers for their insightful reading the manuscript and valuable comments. This article was financially supported by University of Łódź as a part of donation for the research activities aimed in the development of young scientists.

Authors’ Affiliations

(1)
Department of Nonlinear Analysis, Faculty of Mathematics and Computer Science, University of Łódź

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Copyright

© Wardowski; licensee Springer. 2012

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