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

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

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 {Tnx0}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+ αkF(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,yX,TxTy.

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,yX,TxTy.

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 yX,TxTy.

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 {Tnx0}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

τF ( 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 ) -τF ( γ n - 2 ) -2τF ( γ 0 ) -nτ.
(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 nτ0.
(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

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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.

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Correspondence to Dariusz Wardowski.

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Wardowski, D. Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl 2012, 94 (2012). https://doi.org/10.1186/1687-1812-2012-94

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Keywords

  • F-contraction
  • contractive mapping
  • fixed point
  • complete metric space
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