Abstract
In this manuscript, we propose a novel three - step iteration scheme called - iteration to approximate the invariant points for the class of weak contractions in the sense of Berinde and obtain that - iteration strongly converges to one and only one fixed point for Berinde mappings. Proving ‘almost - stability’ of - iteration, we compare the - iterative scheme with other chief iterative algorithms, namely. Picard, Mann, Ishikawa, Noor, S, normal-S, Abbas, Thakur, Varat, Ullah and and claim that the framed iterative procedure converges to the invariant points of weak contractions at faster rate than other vital algorithms. Some numerical illustrations are adduced to strengthen our claim. We further ascertain data dependency through - iteration. Finally, we establish the solution of Caputo type fractional differential equation as an application and exhibit that - fixed point procedure converges to the solution of a fractional differential equation. The obtained results are not only new but, also, extend the scope of previous findings.
1. Introduction
Several nonlinear problems can be mathematically modelled via real valued self-mapping
Endowed with specific properties like continuity, contraction, nonexpansiveness etc. Acknowledging the work of Banach, mappings with contraction condition have proven to be fascinating in the theory of fixed points. But an obvious question arises:
What if the contraction condition on self - mapping is weaken?
Answering to this, Berinde in 2003 (see [1]) came up with a notion called weak contraction, sometimes termed as almost - contractions. He established that the class of these weak contractions is wider than contraction mappings, Zamfirescu mappings and developed the existence and uniqueness theorem for fixed points in the ambient domain.
Further, finding an analytical solution of (1) is not so straightforward that urges the researchers to numerical reckoning of fixed points. Owing to this limitation, researchers broached a number of approximation procedures to estimate fixed points of varying classes of nonlinear mappings, Picard [2], Mann [3], Ishikawa [4], [5], [6] are to name a few. Motivated by these works, we propose a new three - step approximation procedure called - iteration and prove that the suggested iterative scheme converges strongly to a unique fixed point for the class of Berinde’s weak contractions. Also, we establish almost - stability of our proposed algorithm in Subsection 3.1. In Section 4, we present that our algorithm has faster rate of convergence than certain other significant approximation procedures. Subsection 4.1, hooks our assertion of Section 4 numerically via few illustrations. In Section 5, we exhibit data dependency by - iteration. From application perspective, we obtain a solution of Caputo type fractional differential equation in the last section.
2. Brief Preliminaries
In this section, we present few requisite definitions, notations and results.Berinde in 2003 [7] came up with a broader class of contraction mappings called weak contractions, sometimes known as ‘almost contractions’, which can be defined as:
Definition 1. Let be a Banach space. A self - mapping on is called weak contraction if there exist a constant and a nonnegative constant , such thatIt is to be emphasized that weak contractions are different from contraction mappings (see Example 2.11 of [1]). He then manifests the subsequent theorem for the case of weak contraction mappings.
Theorem 1. The self - mapping on a Banach space agreeing (2) together with
ensues that the mapping has a unique fixed point in .
Our study incorporates a comparison of the various iterative procedures see Table 1 framed for the self - mapping , where is a nonempty subset of Banach space , and and are sequences in .
The following two definitions were accorded by Berinde [14] to size up the rate of convergence of two iterative procedures.
Definition 2. Let and be two positive sequences in reals, converging to and respectively. Suppose(i)If , then converges faster to as compare to converging to.(ii)If , then the rate of convergence of both and are coequal.
Definition 3. Assume and are two iterative procedures such that each of them is converging to the common point with error estimates:If , then converges faster than .
An iterative procedure, in general, is prone to errors that may cause harm to its aspect of convergence. To address this problem, the concept of stability of a fixed point procedure was introduced in 1967 by Ostrowski [15]. However, it was Harder et al. [16], who, using the idea of Ostrowski, clearly defined the concept for the first time in 1988 and established the stability of Picard iteration with respect to - contractions and Zamfirescu maps (see Theorems 1 and 2 of [16]) in the setting of metric space. He further showed that certain Mann iterations are stable with respect to Zamfirescu maps in the framework of normed linear space.
The following definition is based on the idea of Theorem 2 of Ostrowski [15].
Definition 4. Suppose, for a self - mapping defined on Banach space , we have:(i),(ii), with where is certain function,(iii)an approximate sequence of in ,(iv),Then the iterative procedure may be regarded as - stable on condition that:In 1990, Rhoades [17] extended the results of Harder et al. [16] and showed the stability for some fixed point iterations (Picard, Mann). Osilike [18] introduced the concept of weak stability called ‘almost-stability’ for iterative algorithms as follows:
Definition 5. Suppose, for a self - mapping defined on Banach space , we have:(i),(ii), with where is certain function,(iii)an approximate sequence of in ,(iv),Then the iterative procedure may be regarded as almost - stable if:Determination of fixed points is a challenging task, especially, in the case, when behaviour of an operator is not known. This propel the researchers to study about data dependency of fixed points. By data dependency for an iterative scheme, we mean, there exists an approximation by which one can find the fixed point of an unknown operator using the fixed point of a known operator. Markin [19] in 1973 was first to talk about continuous data dependency of fixed points. In , Rus and Muresan [20] studied data dependency of weakly Picard operators. Then, again Rus et.al. in 2001 [21] and 2003 [22], discussed data dependence of fixed points of some multivalued weakly Picard operators. Şoltuz [23] in 2004 established data dependence result for Mann- Ishikawa iteration scheme. He, together with Grosan [24] in 2008, demonstrated the result for Ishikawa iterative procedure for contractive - like mappings. In 2009, Olatinwo [25] documented couple of results on continuous data dependency of fixed points.
The following results will be needed in the sequel.
Definition 6 (see [1]). Suppose and be two self - operators defined on a nonvoid set then the operator may be called an approximate operator of if a positive constant exists such that
Lemma 1. [26] Let and be two nonnegative sequences in and , such that , for all . If , then .
We need the following result to prove data dependency which is according to Şoltuz et al. [24], modified by Ali et al. [13].
Lemma 2. Let there exists a positive integer for a sequence of positive reals, such thatwhere for all such that series is divergent and is nonnegative bounded sequence. This gives:Upcoming lemma is given by Şoltuz et al. [27] in 2007.
Lemma 3. For a sequence of positive reals satisfying following inequality,If and then .
3. Main Results
We begin with the introduction of a new three - step iteration called - iteration along these lines:
Let be a self - mapping where is nonvoid, closed and convex subset of Banach space . We define sequence in the following fashion:
- iterationwhere are sequences in .
We prove that - iteration defined by (12) converges strongly to fixed point for the class of weak contractions.
Theorem 2. For a Banach space and nonvoid, closed and convex set , let is Berinde weak contraction agreeing (3), then the sequence specified by - iteration (12) converges to unique fixed point of subject to the sequences generated by (12) obeys one of the following:
Proof. From Theorem 1, possess a unique fixed point say and using (3), we haveBy - iterative algorithm (12), we have,AndUsing (15) and (16), we getInductively, we getImplementing one of the condition ( 1)-( 3), we getBecause , consequently strongly converges to . □
3.1. Stability
This subsection embraces almost - stability of - iteration (12) in the sense of Osilike.
Theorem 3. For a Banach space and nonvoid, closed and convex set , let is Berinde weak contraction agreeing (3), then, iterative algorithm - iteration (12) with real sequences satisfying for some and for all is almost - stable.
Proof. Consider an arbitrary sequence . The sequence defined by - iteration (12) is and .
Now, we have to prove thatLet then by (18) and - iteration (12), we getDefine and , then . Therefore, we haveThus from Lemma 1, we get the result.
4. Comparison with Various Iterative Algorithms
In this section, we prove that - iteration (12) converges faster than other leading algorithms for the case of weak contraction mappings.
Theorem 4. For a Banach space and nonvoid, closed and convex 1set , let is Berinde weak contraction agreeing (3). Let all the iterative algorithms listed in Table 1 together with iterative algorithm converges to common fixed point, (say). Then, - iteration (12) converges to faster than those mentioned in Table 1.
Proof. For , using Theorems 1 and 2 and inequality (18), we getSymbolising Noor iteration by , from [8],ThereforeAlso, we getIn accordance with the Ratio test, (26) implies is convergent, which says .
It validates that converges faster than to .
If normal-S - iterative algorithm [9] be denoted by , thenInductively, we getThereforeHence, converges faster than to .
If represents Abbas iterative scheme, then as shown in Theorem 3 of [10], we haveAccordinglyThis gives converges faster than to .
Considering Theorem 3 of [11] by Thakur et al. and using to mark the corresponding iterative scheme, we getWhich establishes that converges faster than to .
As proved by Sintunavarat et al. ([12], Theorem 2)where denotes Varat’s iterative algorithm.
Therefore,Hence, converges faster than to .
Theorem 4 of [6] by Ullah et al. elucidates thatwhere signifies AK iteration, which further givesConsequently, converges faster than to .
Let stands for iteration. Theorem 1 of [13] conveys,ThereforeHence, converges even faster than to .
Also, in [12], it is shown that Varat iterative scheme has faster rate of convergence than Picard, Mann, Ishikawa and S - iteration. Henceforth Equations (26) - (38) confirms - iteration converges faster than all the approximation procedures enlisted in Table 1.
4.1. Comparison Results
In this section, we give some examples to strengthen the above claim that - iteration converges faster than various interesting algorithms existing in the literature.
Example 1. Let us take Banach space equipped with usual norm. We define a self map on a subset of Banach space by:It can be eventually establish that is a weak contraction in sense of Berinde with unique fixed point . Choosing control sequences , initial guess , using Matlab R 2019(b) software, we prove our claim (see Figure 1 and Table 2, Table 3 for your reference).
Example 2. Let and be same as in Example 1: we define byHere is a weak contraction with unique fixed point . Through Matlab 2019(b)software, it is shown that - iteration converges faster than other iterative algorithms by selecting control sequences starting with guess value (see Figure 2 and Table 4, Table 5 for your reference).
Remark 4.1. It can be easily infer from the above two examples that in the category of weak contractions, Picard iterative scheme has good speed of convergence when compared to Mann and Ishikawa iterative schemes (see Table 6).
5. Data Dependency
The present section marks the data dependency result for - iteration when applied to weak contractions. Here, instead of computing the fixed point of an operator, we approximate the operator with a given weak contraction operator for which it is possible to compute the fixed point and therefore, approximate the fixed point of an initial operator.
Theorem 5. Let be a weak contraction with an approximate operator of it and be a sequence governed by - iterative procedure (12) for . Define sequence for as follows:where are sequences in satisfying , , for all and . If and such that as , thenwhere is a fixed constant.
Proof. ConsiderLet us takeIt follows from (43), (44) and (45),Since , therefore, . Also, , which gives, and . We getDefineThus, equation (46) can be rewritten asThis calls for the application of lemma (2.3), which givesUsing Theorem 2 and given hypotheses, we getThe essence of Theorem 5 can be realised from the following example which has been inspired from [13].
Example 3. Let us take Banach space equipped with usual norm. We define a self map on a subset of by:It can be observed that is a weak contraction in the sense of Berinde with and a unique fixed point .
We further define another self map on as follows:Here is a unique fixed point lying in the domain of . Using MATLAB 2019(b) software, we get . Therefore, if we choose , then as per Berinde’s definition of an approximate operator (Definition 6), is an approximate operator of . Moreover, distance between fixed points of and is .
For , choosing in (36) and using (48), we obtain approximated fixed point of the operator as visualized from Table 7.
However, using Theorem 5, we computeFrom here, we can conclude that without knowing the fixed point of and in fact, without calculating it, we can approximate its fixed point by directly applying Theorem 5.
6. Application to Nonlinear Caputo Type Fractional Differential Equation
Fractional derivatives have been used through and through for mathematical modelling over recent years. For instance, numerous real world problems, signal processing, image restoration, determination of fluid level, biological algorithms, traffic flow, telecommunication etc., are being modelled in terms of nonlinear fractional order differential equations, majority of which do not possess exact solution and call for the approximate numerical solutions. Caputo et al. in [28] developed a novel fractional differential operator with a special feature of non singular kernel having exponential decay due to which it has attracted many researchers (see [29‐31]).where denotes Caputo type fractional differential equation, is continuous and is real valued with .
The time dependent Caputo type fractional differential operator is defined bywhere is a normalization function satisfying . The following lemma is due to [31].
Lemma 4. The initial value problem (IVP)Possess following integeral solutionLet and be the space of continuous functions defined on . Let the Banach space , with norm .
By Lemma 4, IVP (55) can be expressed asLet us define the operator byThen the solution of IVP (55) is equivalent to the fixed point operator .
We prove the following theorem.
Theorem 6. For , let be continuous function such that:(i(ii)Then the IVP (55) has unique solution which is fixed point of weak contraction .
Proof. We denote fixed point operator bywhere .
For , we haveThus is a weak contraction. In view of Theorem 1, we get has a unique fixed point which is the solution of IVP (55).
Losada et al. [31] proved the following theorem in context of nonlinear Caputo type fractional differential equation.
Theorem 7. For , suppose the following conditions hold:
(CM1) is a continuous function such that there exists satisfying,
for all.
(CM2) .
Then the IVP (55) has unique solution (59) on .
We now prove that - iteration converges to the solution of IVP (55)
Theorem 8. If the conditions are satisfied, then for , the sequence generated by - iterative procedure converges to the solution say of IVP (55) on , for some .
Proof. Let us define the operator by:Which can be rewritten as:where
In view of Theorem 7, the solution of IVP (55) is fixed point of operator .
Let be a sequence defined by - iterative procedure on fixed point operator . We shall establish that as .
Define
Using condition , we have:Using (65) and condition (CM1), we get:Due to condition , we obtain that:If we define , then in similar fashion as above, we can obtain:Therefore using condition ,Likewise, we retrieve:Since , which gives and .
Therefore (70) becomes:Define:where such that .
Therefore, (71) can be viewed as:Executing Lemma 3, we get - iteration converges to the solution of IVP (55).
7. Conclusion
This article is endowed with introduction of a new three - step iterative technique called - iteration which converges strongly to the invariant point of almost - contraction mappings in Banach space. Followed by almost - stability of approximation procedure, we presented that this novel technique has fast rate of convergence as compare to other vital procedures for the class of weak contractions which can be visualized from two numerical illustrations, namely. Example 1 and Example 2 presented here. We further established data dependency of our algorithm and demonstrated that Caputo type fractional differential equation possess a solution which, under certain circumstances, is a fixed point of Berinde contractions. Lastly, we showed that - iterative procedure converges to the solution of fractional differential equation.
Nevertheless, few obvious questions are engrossed in the field:(1)Can one define an iterative technique which is even faster than iterative procedure for the class of weak contractions in the setting of Banach space?(2)Does - iteration strongly converges to the fixed point of weak contractions in the framework of a space weaker than Banach space (quasi-Banach space or metric space, for instance)?(3)Does - iterative algorithm converges for certain other class of mappings like quasi - nonexpansive or enriched contractions ?
Data Availability
No data were used to support this study.
Additional Points
This article is embraced with following key findings and implications: (i) Proposing a new fixed point approximation procedure namely -iteration which strongly converges to single fixed point of weak contractions. (ii) Stability discussion of - iteration followed by analytical and graphical contrast of - iteration with other leading algorithms and establishing that - iteration has faster rate of convergence than those taken into consideration via numerical illustrations executed through MATLAB R2019(b) software. (iii) Data Dependency result for - iteration. (iv) Finding solution of Caputo type fractional differential equation under certain conditions and manifesting that the new iterative procedure converges to the solution of fractional differential equation.
Conflicts of Interest
The authors have no conflicts of interest to declare that are relevant to the content of this article.
Authors’ Contributions
All authors have made equal contribution towards the preparation of this article.
Acknowledgments
The authors are thankful to the learned referee for valuable suggestions. The second author is thankful to NBHM, DAE for the research grant 02011/11/2020/NBHM(RP)/R&D-II/7830.