0 Fixed Point Theory and Applications

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

Dariusz Wardowski

Correspondence: wardd@math.uni. lodz.pl

Department of Nonlinear Analysis, Faculty of Mathematics and Computer Science, University of tödz, Banacha 22, 90-238 tödz, Poland

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-contraction, contractive mapping, fixed point, complete metric space

1 Introduction

Throughout the article denoted by R is the set of all real numbers, by R+ is the set of all positive real numbers and by N 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

Vx,yeX d(Tx, Ty) < kd(x, y), where k e (0,1), (1)

has a unique fixed point and for every x0 e X a sequence {T"x0}Be N is convergent to the fixed point. Some of the extensions weaken right side of inequality in the condition (1) by replacing l with a mapping, see e.g. [2,3]. In other results, the underlying space is more general, see e.g [4-7]. The Nadler's paper [8] started the invatigations concerning fixed point theory for set-valued contractions, see e.g. [9-20]. There are many theorems regarding asymptotic contractions, see e.g. [21-23], contractions of Meir-Keeler type [24], see e.g [19,23,25] and weak contractions, see e.g. [26-28]. 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: R+ ® R we introduce a new type of contraction called F-contraction and prove a new fixed point theorem concerning F-con-traction. 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.

Springer

© 2012 Wardowski; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 The result

Definition 2.1 Let F: R+ ® R be a mapping satisfying:

(F1) F is strictly increasing, i.e. for all a, b e R+ such that a < b, F (a) < F (b);

(F2) For each sequence {an}ne N of positive numbers limn®„ an = 0 if and only if limB®„ F (an) =

(F3) There exists k e (0, 1) such that lima®0+ a^F(a) = 0.

A mapping T: X ® X is said to be an F-contraction if there exists t >0 such that

Vx,yeX (d(Tx, Ty) > 0 ^ r + F(d(Tx, Ty)) < 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:

Fxample 2.1 Let F : R+ ® R be given by the formula F (a) = ln a. It is clear that F satisfies (F1)-(F3) ((F3) for any k e (0, 1)). Each mapping T: X ® X satisfying (2) is an F-contraction such that

d(Tx, Ty) < e-r d(x, y), for all x, y e X, Tx = Ty. (3)

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

Fxamp/e 2.2 If F(a) = ln a + a, a >0 then F satisfies (F1)-(F3) and the condition (2) is of the form

d(Tx'TY) ed{Tx,Ty)-d{x,y) < e-r / for all X/ e X/ Tx T (4)

d(x, y)

Example 2.3 Consider F(a) = — 1/s/a, a > 0. F satisfies (F1)-(F3) ((F3) for any k e (1/2, 1)). In this case, each F-contraction T satisfies

d(Tx, Ty) <-jd(.x7 y), for all x, y e X, Tx ^ Ty.

(1 + rVdCMO)

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

Example 2.4 Let F(a) = ln(a2 + a), a >0. Obviously F satisfies (F1)-(F3) and for F-contraction T, the following condition holds:

d(Tx, Ty )(d(Tx, Ty) + 1)

d(x, y)(d(x, y) + 1)

< e T, for all x, y e X, Tx = Ty.

Let us observe that in Examples 2.1-2.4 the contractive conditions are satisfied for x, y î 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(Tx, Ty) < d(x, y), for all x, y e X, Tx = Ty.

Thus every F-contraction is a continuous mapping.

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

Indeed, from Remark 2.1 we have G(d(Tx, Ty)) < G(d(x, y)) for all x, y e X, Tx * Ty. Thus, for all x, y e X, Tx * Ty we obtain

r + F2(d(Tx, Ty)) = r + Fi(d(Tx, Ty)) + G(d(Tx, Ty))

< Fi(d(x, y)) + G(d(x, y)) = F2(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 : X ® X be an F-con-traction. Then T has a unique fixed point x* e X and for every x0 e X a sequence {Tnx0}ne N is convergent to x*.

Proof. First, let us observe that T has at most one fixed point. Indeed, if x\, x*2 e X, Tx1 = x1 = x2 = Tx2, then we get

r < F(d(x1, x2)) - F(d(Tx1, Tx*)) = 0, which is a contradiction.

In order to show that T has a fixed point let x0 e X be arbitrary and fixed. We define a sequence {xn}ne N c X, xn+1 = Txn, n = 0, 1, .... Denote gn = d(xn+1, xn), n = 0, 1, ....

If there exists n0 e N for which xno+i = xno, then Tx„0 = x„0 and the proof is finished. Suppose now that xn+1 * xn, for every n e N. Then gn >0 for all n e N and, using (2), the following holds for every n e N:

F(Yn) < F(yn-i) - r < F(yn-2) - 2r < ... < F(yo) - nr. (5)

From (5), we obtain limn®„ F(gn) = that together with (F2) gives

lim Yn = 0. (6)

From (F3) there exists k e (0, 1) such that

lim YnfeF(Yn) = 0. (7)

By (5), the following holds for all n e N:

YnkF(Yn) - YnkF(Y0) < Ynk(F(Y0) - nr) - YnkF(Y0) = -Ynknr < 0. (8)

Letting n ® in (8), and using (6) and (7), we obtain

lim nYnk = 0. (9)

Now, let us observe that from (9) there exists n1 e N such that nYnk < 1 for all n > n1. Consequently we have

Yn < for all n>ni. (10)

In order to show that {xn}ne N is a Cauchy sequence consider m, n e N such that m > n > n1. From the definition of the metric and from (10) we

getd(.rm/ Xn) < Ym-l + Km-2 + • • • + Yn < Yi < H -rn^-

i=n i=n I

From the above and from the convergence of the series \j\\ we receive that {xn}neN is a Cauchy sequence.

From the completeness of X there exists x* e X such that limn®„ xn = x*. Finally, the continuity of T yields

d(Tx*, x) = lim d(Txn, xn) = lim d(xn+i, xn) = 0,

n—>to n—>TO

which completes the proof. □

Note that for the mappings Fi(a) = ln(a), a >0, F2(a) = ln(a) + a, a >0, Fi < 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.

Fxamp/e 2.5 Consider the sequence {Sn}ne N as follows:

51 = 1,

52 = 1+2,

n(n + 1)

Sn = 1 + 2 +••• + « = —-n e N,

Let X = {Sn : n e N} and d(x, y) = |x - y|, x, y e X. Then (X, d) is a complete metric space. Define the mapping T : X ® X by the formulae:

T(Sn) = Sn—1 for n > 1,

T(Si) = S1.

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

n—TO d(Sn, S1) n—TO Sn — 1

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

First, observe that

Vm,neN [T(Sm) = T(Sn) ^ ((m > 2 A n =1) V (m > n > 1))].

For every m e N, m >2 we have

d(r(Sm),r(Si)) dfTfs„,iTfs1ii-dfs„„s1i = ~ l^,„_1_s,„ d(S

m S1) m-

m2 + m - 2

For every m, n e N, m > n >1 the following holds

m Vm < e-'n < e'1.

d(r(Sm)/ T(Sn)) cd(T(Sm),T(Sn))-d(Sm,sn) = sm-i csn-sn_i+sm_i-s,

d(Sm, Sn) Sm Sn

m + n — 1

m + n +1

en-m < en-m < e-1.

Table 1 The comparison of Banach contraction condition with F-contraction condition

n__Cfi (Si, Sn )_Cf2(SI, Sn)

3 378 0.91629 3.91629

4 35' 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

r 190 0.18540 11.18540

12 17- 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 x104 1 6.66667 x10-5 30000.00007

n ® <x T1 = 1 tends to 0 > T = 1

The generated iterations start from a point x0 = S29 = 435. CF(S1, Sn) denotes F(d(S1, Sn)) - F(d(7(S1), T(Sn))]

Clearly Si is a fixed point of T. To see the computational data confirming the above calculations the reader is referred to Table 1.

Acknowledgements

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

Competing interests

The author declares that he has no competing interests.

Received: 20 April 2012 Accepted: 7 June 2012 Published: 7 June 2012

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doi:10.1186/1687-1812-2012-94

Cite this article as: Wardowski: Fixed points of a new type of contractive mappings in complete metric spaces.

Fixed Point Theory and Applications 2012 2012:94.

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