Fixed points of generalized cyclic contractions without continuity and application to fractal generation

. In this paper, we deﬁne a generalized cyclic contraction and prove a unique ﬁxed point theorem for these contractions. An illustrative example is given, which shows that these contraction mappings may admit the discontinuities and also that an existing result in the literature is effectively generalized by the theorem. We apply the ﬁxed point result for generation of fractal sets through construction of an iterated function system and the corresponding Hutchinsion–Barnsley operator. The above construction is illustrated by an example. The study here is in the context of metric spaces.


Introduction and mathematical preliminaries
Contractions of various kinds appear in a large way in metric fixed point theory.In fact, metric fixed point theory is widely held to have originated in the work of Banach [5] published in 1922, where the notion of contraction and the famous contraction mapping principle was introduced.In the following hundred years, various classes of mappings satisfying different types of contractive inequalities have been studied in the context of fixed point theory.The handbook [17] provides a comprehensive description of this development till the year 2000.Some of the more recent references are noted in [4,9,15,23,28,29].Contraction mappings are well known for their applications [7,24].One such domain of application is in Hutchinson-Barnsley's theory, where families of contractions are utilized for the generation of fractals [20], which are sets characterized by self-similarity appearing in different domains of mathematics, physics and computer science [10,27].It is a method originally proposed by Hutchinson [12] and further elaborated by Barnsley [6] in which a finite family of Banach contractions was used for the generation of fractal sets in arbitrary metric spaces.In later research the theory has been extended, modified and applied in the framework of different mathematical spaces for obtaining fractals.Some instances of these works are in references [3,8,22,25,26].
In this paper, we primarily introduce a cyclic contraction by generalizing the wellknown θ-contraction.Further, it is illustrated that the class of cyclic contractions introduced here may contain discontinuous functions.We establish a unique fixed point theorem for these cyclic contractions, which is an actual generalization of certain existing results as demonstrated through an example.
In the second part of the present research, we construct an iterated function system (IFS) by utilizing a finite family of the cyclic contractions mentioned above.Then, by an application of the fixed point theorem proved here to the Hutchinson-Barnsley operator constructed out of the IFS, we establish the existence of fractal sets.Further, the iterations leading to the fractal sets are also obtained.The above method of generating fractals is demonstrated.
Throughout the paper, we use the following notations: N n will denote the set of first n natural numbers, CB(X) will denote the set of all nonempty, closed and bounded subsets, while K(X) will denote the set of compact subsets of the metric space (X, d), respectively.
A self-mapping T : X → X on a metric space (X, d) is said to be a contraction mapping if there exists a constant k ∈ [0, 1) such that, for every x, y ∈ X, d(T x, T y) kd(x, y). ( The Banach contraction principle [5,17] guarantees the existence of a unique fixed point for contraction mappings in a complete metric space.A remarkable fixed point result was established by Jleli et al. [15] for the generalized contractions given through the condition that where θ : (0, ∞) → (1, ∞) is a function with certain properties.The contraction defined in ( 2) is called θ-contraction because of its dependence on the function θ.
Below are certain notions associated with Hutchinson-Barnsley's theory for fractal generation, which is used in the subsequent discussion.Definition 1. (See [6].)Let (X, d) be a metric space.The mapping h : where is a metric on K(X) and called the Hausdorff metric induced by d.
Definition 2. (See [6].)An iterated function system (IFS) consists of a complete metric space (X, d) and a finite set of contraction mappings T Definition 3. (See [6].)Let X be a metric space, and let {T i , i ∈ N N } be a finite set of mappings on X.The Hutchinson-Barnsley operator F : K(X) → K(X) is defined as where F is the corresponding Hutchinson-Barnsley operator.
In Hutchinson-Barnsley's theory, IFS is the basic instrument for generation of fractal sets.It has many versions in which different types of contractions have been utilized.Some instances of these works are noted in [8,22,25].
Let Θ denotes the class of functions θ : (0, ∞) → (1, ∞) satisfying the following conditions: Definition 5. Let (X, d) be a metric space, m be a positive integer, and let A 1 , A 2 , . . ., A m by nonempty subsets of X. Suppose T : (ii) there exists θ ∈ Θ and k ∈ (0, 1) such that 2 Fixed point result Theorem 2. Let (X, d) be a complete metric space, m be a positive integer, and let A 1 , A 2 , . . ., A m by nonempty closed subsets of X. Suppose T : A i be a generalized cyclic θ-contraction with respect to some θ ∈ Θ .Then m i=1 A i is nonempty, and T has a unique fixed point x * ∈ m i=1 A i .Further, the sequence {x n }, where x n+1 = T x n converges to x * for any initial choice of x n , which shows that x n is a fixed point of T .So, we assume that d(x n+1 , x n ) = 0 for all n.For any n 0, there exists i(l) ∈ {1, 2, . . ., m} such that Taking limit as n → ∞ in the above inequality, since 0 < k < 1, we get Then, by the property (Θ1 ) of the function θ, Next, we show that {x n } is a Cauchy sequence.If not, then there exists > 0 for which we can find two subsequences Let n(k) be the smallest integer with n(k) > m(k) satisfying (5).Then https://www.journals.vu.lt/nonlinear-analysis Then we have Taking limit as n → ∞ in (6), we get Also, Taking limit as n → ∞ in (8) and using ( 4), ( 7), we get Taking limit as n → ∞ in (10) and using (Θ2), ( 7), ( 9), we get which is a contraction as 0 < k < 1.This shows that {x n } is a Cauchy sequence and hence convergent in the complete metric space (X, d).Suppose x n → x * as n → ∞.Also, from the cyclic representation of {A i ; i = 1, 2, . . ., m} it is possible to construct a subsequence of {x n } from each A i , which converges to x * .Therefore, Then T is a generalized cyclic θ-contraction with the choice of θ(t) = e t and k = 1/2.Therefore, by Theorem 2, T has a unique fixed point x * = 0.
Remark 1.If we take A i = X for all i in Theorem 2, then Eq. ( 2) is satisfied for all x, y ∈ X.Therefore, generalized cyclic θ-contraction is a generalization of a theorem in [13].
It can be noted in Example 1 that T is not a contraction in the sense of Imdad et al. [13] as for x = 3/2 and y = 7/4, we have d(T x, T y) = d(x, y).So, there does not exist any k ∈ (0, 1) such that Eq. ( 2) is satisfied.Hence Theorem 2 is a nontrivial generalization of the above mentioned result of [13], and therefore, is also a proper generalization of the main theorem in [1] in turn.
Also, θ-contraction is continuous, whereas generalized cyclic θ-contraction mappings can be discontinuous as can be concluded from the observation that in Example 1 the mapping T is discontinuous at x = 3/2.
A similar type of result is established in the work [16], where the function θ is assumed continuous in addition to assumptions (Θ1)-(Θ3).Theorem 2 is derived here with θ satisfying only (Θ1) and (Θ2), which is without (Θ1) and (Θ3) utilized in the above mentioned work.

Fractal generation
Let {A i } m i=1 be a collection of nonempty subsets of a metric space (X, d), and let T : we define the operator generated by the continuous mapping T as T : The above construction is possible since T is cyclic.Lemma 2. If A is closed subset of the complete metric space (X, d), then K(A) is a closed subset of the complete metric space (K(X), h).
In the following, we obtain a theorem, which is an application of Theorem 2 under the additional condition that θ is nondecreasing.
i=1 is a collection of nonempty subsets of a metric space (X, d) and T : ) is also a generalized cyclic θ-contraction in the metric space (K(X), h), where θ ∈ Θ is nondecreasing. https://www.journals.vu.lt/nonlinear-analysis This implies that T (C) ∈ K(A i+1 ).Therefore, Let A ∈ K(A i ) and B ∈ K(A i+1 ) for some i ∈ N m .We have to show Since x 0 ∈ A is arbitrary and θ is nondecreasing, Also, Similarly, we can show that θ(D( T (B), This completes the proof.
i=1 is a collection of nonempty subsets of a metric space (X, d) and T n : Nonlinear Anal.Model.Control, 29(Online First):1-12, 2024 S. Roy et al.

We have to show that θ(h(F
θ max = max

Now,
log max Since log is injective, Hence the proof.
It is also easy to check that T 2 is a generalized cyclic θ-contraction.Also, it is noted that both T 1 and T 2 are continuous.Note that for x = 15/4 and y = 4, we have d(T 1 x, T 1 y) = d(T 2 x, T 2 y) = d(x, y) = 1/4.Therefore, both T 1 and T 2 do not satisfy (1) with some k ∈ (0, 1).Hence they are not Banach contractions.Also inequality (3) is not satisfied for all x, y ∈ A 1 ∪ A 2 .Since both T 1 and T 2 are continuous generalized cyclic θ-contraction, by Theorem 4, the IFS {(A 1 ∪ A 2 ); T 1 , T 2 } admits a fractal set, that is, there exists a set A such that A = F (A), where F is the Hutchinson-Barnsley operator.
Here the set A is similar to a Cantor set for [2,3] with 8 subintervals and retaining first and last subintervals at each stage.We have shown the first four iterations of the same in Fig. 1.

Conclusion
The unique fixed point Theorem 2 proved here is an actual generalization of a result in [13], which in turn is a generalization of some other results including the Banach contraction mapping principle.Thus, in effect, we have been able to generalize the contraction mapping principle through our theorem, which also applies to certain functions with discontinuities.The second part of our paper is a contribution to the Hutchinson-Barnsley's theory.It is our perception that there are large scopes of research towards the goal of fractal generation by the construction of IFS through other types of cyclic contractions as well.Such efforts are supposed to be taken up in our future work.

Theorem 4 .
Let (X, d) be a complete metric space and {A i } m i=1 be a collection of nonempty closed subsets of X.Let T n :m i=1 A i → m i=1 A i be continuous generalized cyclic θ-contractions for each n ∈ N N .Then the Hutchinson-Barnsley operator F : m i=1 K(A i ) → m i=1 K(A i) has a unique fixed point A ∈ K(X), and the limit lim n→∞ F n (B) = A for any B ∈ m i=1 K(A i ), which is the fractal generated by the IFS { Then the theorem follows by an application of Theorem 2. https://www.journals.vu.lt/nonlinear-analysisExample 2. Let X = R be equipped with the usual metric d

F 4 Figure 1 .
Figure 1.Attractor of the IFS in Example 2.