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We establish a new fixed point theorem in the setting of convex b-metric spaces that ensures the existence of fixed point for Cirić contraction with the assumption . Also, the fixed point is approximated by Krasnoselskij iterative procedure. Moreover, we discuss the stability of fixed point for the aforesaid contraction. As a consequence, we develop a common fixed point and coincidence point result. Finally, we provide a number of examples to illustrate the findings presented here and incorporate these findings to solve an initial value problem.
In the field of fixed point theory, the most useful and widely applied fixed point theorem was proved by Stefan Banach [1] in 1922, where he ensured the existence of fixed point for a contraction defined on a complete metric space. In the literature, this result is also known as the Banach contraction principle. Moreover, as this result played a pivotal role in solving various real-life problems of nonlinear analysis, it has been extended by the researchers either by weakening the contractive condition or by enlarging the structure of the ambient space. In 1974, pursuing the former course of action, Cirić [2] weakened the contractive condition of Banach [1] by defining the notion of quasi contraction (also called Cirić contraction) and succeeded in obtaining a generalization of not only the Banach contraction principle but also the Kannan fixed point theorem and Chatterjea fixed point theorem existing in the literature. On the other hand, in lieu of extending this contraction principle, the notion of b-metric spaces was introduced by Bakhtin [3] in 1989. One can refer to [4,5,6,7,8] and references therein to learn more about this space. In the recent years, Chen et al. [9] introduced the notion of convex b-metric spaces by utilizing the concept of convex structure of Takahashi [10] in b-metric space which is given as under:
Definition 1
([9]).Let and (a real number). A mapping is said to be a b-metric if the following holds for every
1.
2.
3.
Further, a function (where I = [0,1]) is said to have convex structure on Ξ if
The triplet is called a convex b-metric space.
Additionally, they extended the Mann’s iterative algorithm in convex b-metric space and employed it to establish the Banach contraction principle in the framework of this newly introduced space. In 2022, Rathee et al. [11] extended this result by establishing a fixed point theorem for Cirić contraction which is stated as under:
Theorem 1
([11]).Suppose is a quasi-contraction, that is, Υ satisfies
for all and some , where is a complete convex b-metric space with . Let be a sequence defined by choosing an initial point with the property , where for each . If and for each , then Υ has a fixed point in Ξ that is unique.
The main aim of this work is to improve the above theorem by stretching the domain of constant k from to by motivating with the idea of Djafari-Rouhani and Moradi [12]. Furthermore, the fixed point is approximated by means of Krasnoselskij iteration, and then, we discuss the stability of the obtained fixed point. Moreover, some examples are presented to clarify the universality of the proven results over Theorem 1 as well as over the similar results existing in the literature.The obtained results can be utilized in various branch of mathematics, such as the theory of differential equation and integral equation, in numerical methods and in the theory of fractal. For example, we applied the main result of this paper to the initial value problem (27) and ensured that there is a unique solution to the given initial value problem. Hence, it can be said that these results can be helpful for solving real-life problems of nonlinear analysis, which can be formulated in any of the above-mentioned classes. Besides, as a consequence of main result, we obtained some coincidence and common fixed point theorem and hence the obtained results play a crucial role in the further development of fixed point theory.
2. Main Results
We start this section with the following lemma that is required in the sequel to assure the existence, approximation and stability of fixed point.
Lemma 1.
Let be a self mapping defined on , a complete b-metric space with parameter , such that for all and some , it satisfies
If , then the following statements are equivalent:
1.
Υ has a unique fixed point.
2.
Υ has approximate fixed point property, i.e., .
Proof.
(1) ⇒ (2)
Firstly, presume that a unique fixed point of , say , exists, i.e., . Then,
Thus, exhibits approximate fixed point property.
(2) ⇒ (1)
Conversely, presume that exhibits approximate fixed point property, i.e., . This indicates the existence of , a sequence in satisfying and by using (2) and triangle inequality for all , we have,
Now, since and , we are left with a Cauchy sequence as . Also, the space , being complete, proposes the existence of an element satisfying .
Again using triangle inequality
Taking limit as , we get
Also, consider
Now taking limit as , we get
Since , i.e., , we get
and . Thus has a fixed point in .
Now, if possible, let us consider two fixed points of , say and , exist and thus . By using inequality (2), we get
which is a contradiction since .
and thus , i.e., the fixed point is unique. □
Theorem 2.
Let be a self mapping defined on , a complete convex b-metric space with parameter such that for all and some , it satisfies
If , then Υ has approximate fixed point property.
Proof.
For every , we have
since . Thus the sequence is non-increasing and for
We let We need to prove that this . For this, let be a sequence such that , i.e., by (8), we have, for every and some ,
Now, being a complete convex b-metric space, defining leads to a well defined belonging to , where and we have,
Now suppose that and using inequality (11), we find that is finite. By definition of and , the inequality holds. We shall now prove that , which in turn, shall prove that .
For this, take lim sup as on both sides of inequality (11) and using the inequality , we must have,
If possible, suppose that . Then, by inequality (12), we get
which is a contradiction since , for and implying .
Thus our supposition is wrong, i.e., and, in turn, . Therefore, has approximate fixed point property. □
Let be a self mapping defined on , a convex b-metric space. We state the following Lemma to show the relation between the set of fixed points of the self mappings and defined by
Here, set of fixed points of the mappings and are denoted by and , respectively.
Lemma 2.
Let be a self mapping defined on , a convex b-metric space with parameter . Define another self mapping by
Then, for any ,
Proof.
By definition,
If , then
Now assume that and let a fixed point of , say , exists i.e., and therefore,
i.e., is a fixed point of .
Conversely, suppose that is a fixed point of , i.e., , then
Since , this implies that , i.e., . Therefore is a fixed point of . □
Lemma 1 and Theorem 2 imply the following result, extending the Cirić fixed point theorem in the case of Convex b-metric spaces by Rathee et al. [11].
Theorem 3.
Let be a self mapping defined on , a complete convex b-metric space with parameter such that for all and some , it satisfies
If , then
1.
A fixed point of Υ, say σ, exists that is unique.
2.
The sequence converges to σ for any that is obtained from the iterative procedure
3.
The error estimate
holds for
Proof.
With the given conditions, by Theorem 2, we arrive at the conclusion that has approximate fixed point property. By Lemma 1, a fixed point of , say , exists that is unique.
We observe that Krasnoselskij iterative procedure is nothing but the Picard iteration associated with and defined by , i.e.,
We shall now verify that the sequence is Cauchy. For this, consider the points and as and , respectively, in inequality (12).
This implies
In inequality (16), taking limit as and using condition (14), we get,
This shows that the aforementioned sequence is Cauchy and owing to completeness of the space , converges to some point, say . Now, consider the inequality (14),
Now taking limit as , we get,
Thus, , and therefore is a fixed point of . But by using Lemma 2, we must have
and , i.e., Fixed point of is , which is unique.
So, and thus obtained from the above iteration converges to .
Thus, Υ has approximate fixed point property and, hence, a unique fixed point exists which is equal to the limit of sequence obtained by applying Mann’s iteration, i.e., 0.
Remark 1.
If we take and , then and which yields
which is true for all and and therefore, Theorem 2 of Rathee et al. [11] does not guarantee the existence and uniqueness of a fixed point in this scenario. Thus, results provided by Theorem 3 extend the Cirić fixed point theorem proved by Rathee et al. [11].
Theorem 4.
Let be a self mapping defined on , a complete convex b-metric space with parameter such that a natural cardinal N exists for all and some , it satisfies
If , then
1.
A unique fixed point of Π, say σ exists that is unique.
2.
The sequence obtained from the iterative procedure
converges to σ for any .
Proof.
Applying Theorem 3 for the mapping , we obtain that has a unique fixed point, say . Also, we have
This shows that is a fixed point of . However, there is a unique fixed point of , . This implies that and thus, has a unique fixed point, .
Applying Theorem 3, we observe that the sequence obtained from the iterative procedure
converges to for any .
□
As far as approximation of fixed points is concerned, we prove that the convergence of every orbit of self mapping is to its unique fixed point that too for , even in the case of any complete b-metric space.
Theorem 5.
Let be a self mapping defined on a complete b-metric space with parameter such that for all and some , it satisfies
Then, if , a fixed point of Υ exists that is unique. Besides, the sequence of Picard iterates, for each , converges to this fixed point.
Proof.
Let the sequence be defined by
where is arbitrary in .
Preserving generality, assume that, for every , , as the result holds trivially if .
Now, we have
This implies
This shows that the aforementioned sequence is Cauchy and owing to completeness of the space, is convergent too. Let be its limit.
Consider now
Taking limit as , we get
since .
Thus, , i.e., and this proves that has a fixed point .
For uniqueness, let us suppose that has two distinct fixed points, say and , such that , then
which is a contradiction since . Thus, the fixed point so obtained is unique. □
Example 2.
The pair , in Example 1, makes a complete b-metric space. If we take the sequence of Picard iterates, then
1.
for , we have
2.
and for , we have
If , then the sequence can be evaluated as in the above case. If , then . Continuing in a comparable manner, presume that , yielding
and
Hence for both the cases. Therefore, the sequence of Picard iterates converge to fixed point 0.
3. Stability of Fixed Point
This section is concerned with the stability results for fixed points of mappings satisfying the Cirić contraction.
Definition 2.
Let be a sequence of self mapping defined on a convex b-metric space. Then stability is nothing but a relation between the convergence of the sequence and their fixed points.
Theorem 6.
Let be a complete b-metric space with parameter and suppose be a sequence of self mappings such that for all and , it satisfies
Also, let be a self mapping satisfying
for all and , where . Let Υ has a fixed point ζ and for every n, be the fixed points of . Presuming pointwise and , then if .
Proof.
For every , we have
This implies
Taking limit as , we get
□
Theorem 6 can also be restated as
Theorem 7.
Let a sequence of self mappings be defined on a complete b-metric space with having fixed points and for all , satisfying
where and . Also for all , presume a self mapping satisfying
where and . If pointwise and , then is convergent if Υ has a fixed point ζ and in that case, as .
Proof.
We presume that a fixed point of map exists i.e., .
Now as proved in Theorem 6, as , i.e., is convergent sequence and as . □
Theorem 8.
Let a sequence of self mappings be defined on a complete b-metric space with having fixed points and for all , satisfying
where and . Also, for all , presume a self mapping satisfying
where and . If pointwise and , then a fixed point of Υ, say ζ, exists if is convergent and in that case, as .
Proof.
Presume that is convergent and as . Then,
Now taking limit as
Therefore, a fixed point of map exists. □
Theorem 7 and 8 can be combined to get the subsequent outcome:
Theorem 9.
Let a sequence of self mappings be defined on a complete b-metric space with having fixed points and for all , satisfying
where and . Also, let be a self mapping and for all , satisfying
where and . If pointwise and , then a fixed point of Υ, say ζ, exists if and only if is convergent and in that case, as .
By virtue of fixed points of sequence of self-mappings ; defined on a complete metric space and satisfying Círic contractive condition, we provide an approximation result for fixed points of self-mapping satisfying Círic contractive condition where pointwise.
Corollary 1.
Let a self mapping be defined on , a complete convex b-metric space with , such that for all and , it satisfies
where . Also suppose that a sequence of self mappings exists and for all , it satisfies
where is a sequence such that , . Presume pointwise and . Then, the sequence of fixed points of mappings is convergent and its limit is the fixed point ζ of Υ.
Proof.
By Theorem 3, has a fixed point which is unique.
By Theorem 5, have fixed points which are unique for all .
Finally, by Theorem 9, . □
Example 3.
In Example 1, we consider the sequence of self mappings such that
Thereafter, to prove that satisfies the inequality (24), the following four cases exist:
We observe that pointwise and . Also, the sequence of fixed points of given by is convergent and this sequence converges to 0 which is the fixed point of self mapping Υ.
4. Consequence
Presume self mappings defined on a non-empty set . For the mappings and I, a point for which is termed a coincidence point (common fixed point). Moreover, if the mappings and I commute at every coincidence point, then the mappings and I are termed weakly compatible.
Lemma 3.
Let Ξ be a nonempty set and be a self mapping defined on it. Then a subset of Ξ, say Θ, exists such that the mapping is one-to-one and .
Subsequently, a common fixed point theorem is obtained in continuation of the primary results established in the former section.
Theorem 10.
Let be self mappings defined on a convex b-metric space with parameter such that for all and , they satisfy
If , is complete and then a unique coincidence point of mappings Υ and I exists. Besides, if Υ and I are weakly compatible mappings, then a common fixed point of these mappings exists that is unique.
Proof.
By Lemma 3, a subset of , say , exists such that the mapping is one-to-one and . Further, let be another self mapping defined by . Then, since the mapping I is one-to-one, is clearly well defined. Thus, for all , we arrive that
As with , then is a Cirić contraction on . Besides, a unique point exists on account of Theorem 3 since is complete yielding implying . Thus, a coincidence point of mappings and I exists that is unique.
Let . Furthermore, let and I be weakly compatible mappings following . As a result,
which is true for if and hence a common fixed point of mappings and I exists that is unique. □
5. Application to Initial Value Problem
In this section, the existence of unique solution to an Initial Value Problem containing a differential equation of second order with two initial conditions is discussed.
where and are continuous functions in and is differentiable in .
First we shall convert this Initial Value Problem (27) into Voltera Integral Equation of the second kind.
Lemma 4.
The Initial Value Problem (27) is equivalent to Voltera Integral Equation of the second kind
where is continuous and .
Proof.
Integrating first equation of (27) from to and using remaining two initial conditions, we have,
Integrating again from to ,
Thus, we have integral equation of the kind
□
Thus, the Initial Value Problem (27) is equivalent to the Voltera Integral Equation of the second kind (28)
where , say, and .
Define by
Define as
Additionally, presume a self mapping defined as
Then, existence of unique fixed point of map implies the existence and uniqueness of solution of Voltera integral Equation of the second kind (28) and hence, the Initial Value Problem (27).
Lemma 5.
Suppose and define by
Define the convex structure as
Then, is a convex b-metric space with parameter .
Proof.
We perceive that
as
Also, for , we have
Thus, for , is convex b-metric space. □
Theorem 11.
Suppose that
for all ; and some where and
Then, a unique solution exists for Voltera Integral Equation of the second kind (28).
Proof.
Consider
and thus all the hypothesis of Theorem 3 are satisfied for implying Voltera Integral Equation of the second kind (28) and hence, the Initial Value Problem (27) has a solution that is unique. □
6. Discussion, Conclusions and Open Problems
In the framework of convex b-metric spaces, we established a fixed point theorem as an extension of the main result of Rathee et al. [11] that guarantees the availability of fixed point for Cirić contraction. Additionally, the Krasnoselskij iterative process is used for approximating the fixed point. Furthermore, we discussed about the fixed point’s stability for the prior mentioned contraction. We constructed a common fixed point and coincidence point result as a consequence. Finally, we provided several examples to highlight the conclusions drawn here and use these conclusions to solve an initial value problem. Following open problems may be worked upon in future:
Rathee et al. [11] ensured the existence of fixed point for Cirić contraction for the constant . In addition, we extended the same for . Is it viable to further relax the condition for ?
Besides, we proved that the fixed points so obtained are stable for . Can the hypothesis be eased?
Author Contributions
Investigation, S.R., A.K. (Anshuka Kadyan), M. (Minakshi) and A.K. (Anil Kumar); Methodology, G., S.R., M. (Mahima), A.K. (Anshuka Kadyan), M. (Minakshi) and A.K. (Anil Kumar); Writing—original draft, S.R., A.K. (Anshuka Kadyan), M. (Minakshi) and A.K. (Anil Kumar); Writing—review & editing, G. and M. (Mahima); Software, G. and M. (Mahima); Writing and typesetting using LaTeX, G. and M. (Mahima). All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Data Availability Statement
Data sharing not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
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Gazal; Rathee, S.; Mahima; Kadyan, A.; Minakshi; Kumar, A.
Existence, Approximation and Stability of Fixed Point for Cirić Contraction in Convex b-Metric Spaces. Axioms2023, 12, 646.
https://doi.org/10.3390/axioms12070646
AMA Style
Gazal, Rathee S, Mahima, Kadyan A, Minakshi, Kumar A.
Existence, Approximation and Stability of Fixed Point for Cirić Contraction in Convex b-Metric Spaces. Axioms. 2023; 12(7):646.
https://doi.org/10.3390/axioms12070646
Chicago/Turabian Style
Gazal, Savita Rathee, Mahima, Anshuka Kadyan, Minakshi, and Anil Kumar.
2023. "Existence, Approximation and Stability of Fixed Point for Cirić Contraction in Convex b-Metric Spaces" Axioms 12, no. 7: 646.
https://doi.org/10.3390/axioms12070646
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Gazal; Rathee, S.; Mahima; Kadyan, A.; Minakshi; Kumar, A.
Existence, Approximation and Stability of Fixed Point for Cirić Contraction in Convex b-Metric Spaces. Axioms2023, 12, 646.
https://doi.org/10.3390/axioms12070646
AMA Style
Gazal, Rathee S, Mahima, Kadyan A, Minakshi, Kumar A.
Existence, Approximation and Stability of Fixed Point for Cirić Contraction in Convex b-Metric Spaces. Axioms. 2023; 12(7):646.
https://doi.org/10.3390/axioms12070646
Chicago/Turabian Style
Gazal, Savita Rathee, Mahima, Anshuka Kadyan, Minakshi, and Anil Kumar.
2023. "Existence, Approximation and Stability of Fixed Point for Cirić Contraction in Convex b-Metric Spaces" Axioms 12, no. 7: 646.
https://doi.org/10.3390/axioms12070646
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.