Journal of Operators
Volume 2016 (2016), Article ID 5250394, 9 pages
http://dx.doi.org/10.1155/2016/5250394

Research Article

Fractals of Generalized -Hutchinson Operator in -Metric Spaces

Talat Nazir,1,2 Sergei Silvestrov,¹ and Xiaomin Qi¹

¹Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, 72123 Västerås, Sweden
²Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan

Received 29 April 2016; Accepted 22 June 2016

Academic Editor: Aref Jeribi

Copyright © 2016 Talat Nazir et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The aim of this paper is to construct a fractal with the help of a finite family of generalized -contraction mappings, a class of mappings more general than contraction mappings, defined in the setup of -metric space. Consequently, we obtain a variety of results for iterated function system satisfying a different set of contractive conditions. Our results unify, generalize, and extend various results in the existing literature.

1. Introduction and Preliminaries

Iterated function systems are method of constructing fractals and are based on the mathematical foundations laid by Hutchinson [1]. He showed that Hutchinson operator constructed with the help of a finite system of contraction mappings defined on Euclidean space has closed and bounded subset of as its fixed point, called attractor of iterated function system (see also [2]). In this context, fixed point theory plays significant and vital role to help in construction of fractals.

Fixed point theory is studied in environment created with appropriate mappings satisfying certain conditions. Recently, many researchers have obtained fixed point results for single and multivalued mappings defined on metrics spaces. Banach contraction principle [3] is of paramount importance in metrical fixed point theory with a wide range of applications, including iterative methods for solving linear, nonlinear, differential, integral, and difference equations. This initiated several researchers to extend and enhance the scope of metric fixed point theory. As a result, Banach contraction principles have been extended either by generalizing the domain of the mapping [4–10] or by extending the contractive condition on the mappings [11–15]. There are certain cases when the range of a mapping is replaced with a family of sets possessing some topological structure and consequently a single-valued mapping is replaced with a multivalued mapping. Nadler [16] was the first who combined the ideas of multivalued mappings and contractions and hence initiated the study of metric fixed point theory of multivalued operators; see also [17–19]. The fixed point theory of multivalued operators provides important tools and techniques to solve the problems of pure, applied, and computational mathematics which can be restructured as an inclusion equation for an appropriate multivalued operator.

The concept of metric has been generalized further in one to many ways. The concept of -metric space was introduced by Czerwik in [20]. Since then, several papers have been published on the fixed point theory of various classes of single-valued and multivalued operators in -metric space [20–30].

In this paper, we construct a fractal set of iterated function system, a certain finite collection of mappings defined on -metric space which induce compact valued mappings defined on a family of compact subsets of -metric space. We prove that Hutchinson operator defined with the help of a finite family of generalized -contraction mappings on a complete -metric space is itself generalized -contraction mapping on a family of compact subsets of . We then obtain a final fractal obtained by successive application of a generalized -Hutchinson operator in -metric space.

Definition 1. Let be a nonempty set and let be a given real number. A function is said to be a -metric if, for any , the following conditions hold:

() if and only if ,(),().

The pair is called -metric space with parameter .
If , then -metric space is metric spaces. But the converse does not hold in general [20, 21, 25].

Example 2 (see [31]). Let be a metric space, and , where is a real number. Then is -metric with .
Obviously conditions () and () of above definition are satisfied. If , then the convexity of the function implies

and hence holds. Thus, for each we obtain

So condition () of the above definition is satisfied and is -metric.
If (set of real numbers) and is the usual metric, then is -metric on with but is not a metric on .

Definition 3 (see [24]). Let be -metric space. Then a subset is called

(i)closed if and only if, for each sequence in which converges to an element , we have (i.e., ),(ii)compact if and only if for every sequence of elements of there exists a subsequence that converges to an element of ,(iii)bounded if and only if .

Let denote the set of all nonempty compact subsets of . For , let

where is the distance of a point from the set . The mapping is said to be the Pompeiu-Hausdorff metric induced by . If is a complete -metric space, then is also a complete -metric space.

For the sake of completeness, we state that the following lemma holds in -metric space [32].

Lemma 4. Let be -metric space. For all , the following hold:

()If , then ()()One has

The following lemmas from [20, 27, 28] will be needed in the sequel to prove the main result of the paper.

Lemma 5. Let be -metric space and denotes the set of all nonempty closed and bounded subsets of . For and , the following statements hold:

(1) is -metric space.(2) for all .(3)One has (4)For and , there is such that (5)For every and , there is such that (6)For every and , there is such that (7)For every and , there is such that implies (8) if and only if (9)For ,

Definition 6. Let be -metric space. A sequence in is called

()Cauchy if and only if, for , there exists such that for each one has ,()convergent if and only if there exists such that for all there exists such that for all one has . In this case one writes

It is known that a sequence in -metric space is Cauchy if and only if for all . A sequence is convergent to if and only if , and -metric space is said to be complete if every Cauchy sequence in is convergent in .

An et al. [21] studied the topological properties of -metric spaces and stated the following assertions:

()In -metric space is not necessarily continuous in each variable.()In -metric space , if is continuous in one variable then is continuous in other variables.()An open ball in -metric space is not necessarily an open set. An open ball is open if is continuous in one variable.

Wardowski [33] introduced another generalized contraction called -contraction and proved a fixed point result as interesting generalization of the Banach contraction principle in complete metric space (see also [34]).

Let be the collection of all continuous mappings that satisfy the following conditions:

() is strictly increasing, that is, for all such that implies that .()For every sequence of positive real numbers, and are equivalent.()There exists such that .

Definition 7 (see [33]). Let be a metric space. A self-mapping on is called -contraction if, for any , there exist and such that

whenever .
From () and (5), we conclude that

that is, every -contraction mapping is contractive, and in particular, every -contraction mapping is continuous.

Wardowski [33] proved that, in complete metric space , every -contractive self-map has a unique fixed point in and for every in a sequence of iterates converges to the fixed point of .

Let be the set of all mapping that is satisfying for all

Definition 8. Let be -metric space. A self-mapping on is called a generalized -contraction if, for any , there exist and such that

whenever .

Theorem 9. Let be -metric space and let be generalized -contraction. Then one has the following:

() maps elements in to elements in .()If, for any ,

 then is a generalized -contraction mapping on .

Proof. As generalized -contractive mapping is continuous and the image of a compact subset under is compact, we obtain

To prove (), let with . Since is a generalized -contraction, we obtain

Thus we have

Also

Now

Strictly increasing implies

Consequently, there exists a function with for all such that

Hence is a generalized -contraction.

Theorem 10. Let be -metric space and let be a finite family of generalized -contraction self-mappings on Define by

Then is a generalized -contraction on .

Proof. We demonstrate the claim for . Let be two -contractions. Take with . From Lemma 4 (iii), it follows that

Definition 11. Let be a metric space. A mapping is said to be a Ciric type generalized -contraction if, for and such that, for any , with , the following holds:

where

Theorem 12. Let be -metric space and let be a finite sequence of generalized -contraction mappings on If is defined by

then is a Ciric type generalized -contraction mapping on .

Proof. Using Theorem 10 with property , the result follows.

An operator in above theorem is called Ciric type generalized -Hutchinson operator.

Definition 13. Let be a complete -metric space. If , , are generalized -contraction mappings, then is called generalized -contractive iterated function system (IFS).
Thus generalized -contractive iterated function system consists of a complete -metric space and finite family of generalized -contraction mappings on

Definition 14. A nonempty compact set is said to be an attractor of the generalized -contractive IFS if

(a),(b)there is an open set such that and for any compact set , where the limit is taken with respect to the Hausdorff metric.

2. Main Results

We start with the following result.

Theorem 15. Let be a complete -metric space and let be a generalized -contractive iterated function system. Then the following hold:

(a)A mapping defined by

 is Ciric type generalized -Hutchinson operator.(b)Operator has a unique fixed point ; that is,

(c)For any initial set , the sequence of compact sets converges to a fixed point of .

Proof. Part (a) follows from Theorem 12. For parts (b) and (c), we proceed as follows. Let be an arbitrary element in If , then the proof is finished. So we assume that . Define

for
We may assume that for all If not, then for some implies and this completes the proof. Take for all . From (18), we have

where

In case , we have

a contradiction as . Therefore and we have

Thus is decreasing and hence convergent. We now show that By property of , there exists with such that for all . Note that

gives which together with () implies that . By (), there exists such that

Thus we have

On taking limit as we obtain that . Hence There exists such that for all and hence for all For with , we have

By the convergence of the series , we get as . Therefore is a Cauchy sequence in Since is complete, we have as for some
In order to show that is the fixed point of , we on the contrary assume that Pompeiu-Hausdorff weight assigned to and is not zero. Now

where

Now we consider the following cases:

(1)If , then, on taking lower limit as in (32), we have

 a contradiction as for all .(2)When , then, by taking lower limit as , we obtain

 which gives a contradiction.(3)In case , we get

 a contradiction as .(4)If , then, on taking lower limit as , we have

 a contradiction as is strictly increasing map.(5)When , then

 which gives a contradiction.(6)In case , then, on taking lower limit as in (32), we get

 a contradiction.(7)Finally if , then, on taking lower limit as , we have

 a contradiction.

Thus, is the fixed point of .
To show the uniqueness of fixed point of , assume that and are two fixed points of with being not zero. Since is -contraction map, we obtain that

where

that is,

a contradiction as . Thus has a unique fixed point .

Remark 16. In Theorem 15, if we take the collection of all singleton subsets of , then clearly . Moreover, consider for each , where for any ; then the mapping becomes

With this setting we obtain the following fixed point result.

Corollary 17. Let be a complete -metric space and let be a generalized iterated function system. Let be a mapping defined as in Remark 16. If there exist some and such that, for any with , the following holds:

where

then has a unique fixed point in Moreover, for any initial set , the sequence of compact sets converges to a fixed point of .

Corollary 18. Let be a complete -metric space and let be iterated function system where each for is a contraction self-mapping on . Then defined in Theorem 15 has a unique fixed point in Furthermore, for any set , the sequence of compact sets converges to a fixed point of .

Proof. It follows from Theorem 10 that if each for is a contraction mapping on , then the mapping defined by

is contraction on . Using Theorem 15, the result follows.

Corollary 19. Let be a complete -metric space and let be an iterated function system. Suppose that each for is a mapping on satisfying

for all , , where Then the mapping defined in Theorem 15 has a unique fixed point in . Furthermore, for any set , the sequence of compact sets converges to a fixed point of .

Proof. Take , in Theorem 10; then each mapping for on satisfies

for all , , where Again from Theorem 10, the mapping defined by

satisfies

for all and . Using Theorem 15, the result follows.

Corollary 20. Let be a complete -metric space and let (, ) be iterated function system. Suppose that each for is a mapping on satisfying

for all , , where . Then the mapping defined in Theorem 15 has a unique fixed point in . Furthermore, for any set , the sequence of compact sets converges to a fixed point of .

Proof. By taking , , in Theorem 10, we obtain that each mapping for on satisfies

for all , , where . Again it follows from Theorem 10 that the mapping defined by

satisfies

for all , . Using Theorem 15, the result follows.

Corollary 21. Let be a complete -metric space and let be iterated function system. Suppose that each for is a mapping on satisfying

for all , , where Then the mapping defined in Theorem 15 has a unique fixed point Furthermore, for any set , the sequence of compact sets converges to a fixed point of .

Proof. Take , , in Theorem 10, and then each mapping for on satisfies

where Again it follows from Theorem 10 that the mapping defined by

satisfies

for all , . Using Theorem 15, the result follows.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

Talat Nazir and Xiaomin Qi are grateful to the Erasmus Mundus Project FUSION for supporting the research visit to Mälardalen University, Sweden, and to the Division of Applied Mathematics at the School of Education, Culture and Communication for creating excellent research environment.

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