Fixed Point Theorems with Applications to the Solvability of Operator Equations and Inclusions on Function Spaces
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Renu Chugh, Preety Malik, Vivek Kumar, "On a New Faster Implicit Fixed Point Iterative Scheme in Convex Metric Spaces", Journal of Function Spaces, vol. 2015, Article ID 905834, 11 pages, 2015. https://doi.org/10.1155/2015/905834
On a New Faster Implicit Fixed Point Iterative Scheme in Convex Metric Spaces
Abstract
The purpose of this paper is to consider a new implicit iteration and study its strong convergence, stability, and data dependence. It is proved through numerical examples that newly introduced iteration has better convergence rate than well known implicit Mann iteration as well as implicit Ishikawa iteration and implicit iterations converge faster as compared to corresponding explicit iterations. Applications of implicit iterations to RNN (Recurrent Neural Networks) analysis are also presented.
1. Introduction
In recent years, numerous papers have been published on explicit iterations in various spaces [1–6], but there are very few works on implicit iterations (regarding convergence rate and data dependence) [7–13]. Implicit iterations have an advantage over explicit iterations for nonlinear problems as they provide better approximation of fixed points and are widely used in many applications when explicit iterations are inefficient. Approximation of fixed points in computer oriented programs by using implicit iterations can reduce the computational cost of the fixed point problems. The study of stability of iterations enjoys a celebrated place in applied sciences and engineering due to chaotic behavior of functions in discrete dynamics and other numerical computations. Data dependence of fixed points is a related and new issue which has been studied by many authors; see [4, 14] and references therein. In computational mathematics, it is of theoretical and practical importance to compare the convergence rate of iterations and to find out, if possible, which one of them converges more rapidly to the fixed point. Recent works in this direction are [1, 3, 4, 15–17]. In concrete, a fixed point iteration is valuable from a numerical point of view and is useful for applications if it satisfies the following requirements:(a)it converges to fixed point of an operator;(b)it is stable;(c)it is faster as compared to other iterations existing in the literature;(d)it shows data dependence results.Motivated by the fact that threestep iterations give better approximation than onestep and twostep iterations [18], we define a new and more general threestep implicit iteration which satisfies the above requirements.
Let be a nonempty convex subset of a convex metric space and let be a given mapping. For the real sequences , , and in , Noor iteration [19] in convex metric spaces can be written asPutting in , we get well known Ishikawa iteration [20, 21] in convex metric spaces:Putting in , we get well known Mann iteration [21, 22] in convex metric spaces:
For , we define the following iteration, namely, implicit Noor iteration in convex metric spaces:where and are sequences in .
Equivalence form of iteration in linear space can be written asPutting in , we get well known implicit Ishikawa iteration [23]:Putting in , we get well known implicit Mann iteration [2, 6, 11, 24]:
Zamfirescu operators [25] are the most general contractive like operators which have been studied by several authors, satisfying the following condition: for each pair of points in at least one of the following is true:where , , and are nonnegative constants satisfying , , and .
operators are equivalent to the following contractive condition:The contractive condition (3) implieswhere [see [15]].
In [5], Rhoades used the following more general contractive condition than (4): there exists such thatIn [26], Osilike used a more general contractive definition than those of Rhoades: there exist , such that
We use the contractive condition due to Imoru and Olatinwo [27], which is more general than (6): there exist and a monotone increasing function with , such thatAlso, we use the following definitions and lemmas to achieve our main results.
Definition 1 (see [23]). A map is a convex structure on if for all and . A metric space together with a convex structure is known as convex metric space and denoted by . A nonempty subset of a convex metric space is convex if for all and .
All normed spaces and their subsets are the examples of convex metric spaces. But there are many examples of convex metric spaces which are not embedded in any normed space (see [23, 28]). After that several authors extended this concept in many ways; one such convex structure is hyperbolic space which was introduced by Kohlenbach [29] as follows.
Definition 2 (see [29]). A hyperbolic space is a metric space together with a convexity mapping satisfying(W1),(W2),(W3),(W4) for all and .Evidently every hyperbolic space is a convex metric space but the converse may not true. For example, if we take , and define for , then is a convex metric space but not a hyperbolic space.
The stability of explicit as well as implicit iterations has extensively been studied by various authors [4, 7, 21, 27, 30–32] due to its increasing importance in computational mathematics, especially due to revolution in computer programming. The concept of stability in convex metric space setting was given by Olatinwo [21].
Definition 3 (see [21]). Let be a convex metric space and let a selfmapping.
Let be the sequence generated by an iterative scheme involving which is defined by where is the initial approximation and is some function having convex structure such that . Suppose that converges to a fixed point of . Let be an arbitrary sequence and set . Then, the iteration (9) is said to be stable with respect to if and only if implies .
Lemma 4 (see [4, 15]). If is a real number such that and is a sequence of positive numbers such that , then for any sequence of positive numbers satisfyingone has .
Definition 5 (see [15]). Suppose and are two real convergent sequences with limits and , respectively. Then is said to converge faster than if
Definition 6 (see [15]). Let and be two fixed point iterations that converge to the same fixed point on a normed space such that the error estimates are available, where and are two sequences of positive numbers (converging to zero). If converge faster than , then one says that converge faster to than .
Definition 7 (see [14]). Let be two operators on . One says is approximate operator of if, for all and for a fixed , one has .
Lemma 8 (see [4, 14]). Let be a nonnegative sequence for which there exists such that, for all , one has the following inequality:where , for all , , and .
Then, .
Having introduced the implicit Noor iteration , we use it to prove the results concerning convergence, stability, and convergence rate for contractive condition (7) in convex metric spaces. Also, data dependence result of the same iteration is proved in hyperbolic spaces. Moreover, applications of implicit iterations in RNN analysis will be discussed in the last section.
2. Convergence and Stability Results of New Implicit Iteration in Convex Metric Spaces
Theorem 9. Let be a nonempty closed convex subset of a convex metric space and let be a quasicontractive operator satisfying (7) with . Then, for , the sequence defined by , with , converges to the fixed point of .
Proof. Using and (7), we have, for ,which further impliesAgain from , we have the following estimates:which givesimplying thatUsing (14)–(18), we arrive atIf we take , thenand henceSimilarly, with ease we can show thatUsing (21)–(23), (19) becomesBut ; hence (24) yields . Therefore converges to .
Theorem 10. Let be a nonempty closed convex subset of a convex metric space and let be a quasicontractive operator satisfying (7) with . Then, for , the sequence defined by with , , is stable.
Proof. Suppose that is an arbitrary sequence, , where , and let .
Then, using (7) we havewhich implies and thereforeUsing (21), (27) becomesNow, using (22) and (23), we have the following estimates:Using and , we have . Hence using Lemma 4, together with estimates (29), (28) yields .
Conversely, if we let , then using contractive condition (7), it is easy to see that .
Therefore, the iteration is stable.
Remark 11. As contractive condition (7) is more general than those of (2)–(6), the convergence and stability results for implicit iteration using contractive conditions (2)–(6) can be obtained as special cases.
Remark 12. As implicit Mann iteration and implicit Ishikawa iteration are special cases of implicit iteration , results similar to Theorems 9 and 10 hold for these iterations as well.
3. Convergence Rate of Implicit Iterations
Theorem 13. Let be a nonempty closed convex subset of a convex metric space and let be a quasicontractive operator satisfying (7) with . Then, for , the sequence defined by with , , converges faster than implicit Mann and implicit Ishikawa iterations to the fixed point of . Moreover, implicit iterations converge faster than the corresponding explicit iterations.
Proof. For implicit Mann iteration , we havewhich further yieldand soSimilarly, for implicit Ishikawa and implicit Noor iterations, we have the estimates (33) and (34), respectively, as follows:Also, for explicit Mann iteration , we have For explicit Ishikawa iteration , we have Using (37), (36) becomesSimilarly, for explicit Noor iteration , we haveNow, using (21) and (22), we obtainSimilarly using (21)–(23), we getKeeping in mind Berinde’s Definitions 5 and 6, inequalities (21), (32), and (35) yield that implicit Mann iteration converges faster than explicit Mann iteration , inequalities (33), (38), and (40) yield that implicit Ishikawa iteration converges faster than explicit Ishikawa iteration , and inequalities (34), (39), and (41) yield that implicit Noor iteration converges faster than explicit Noor iteration .
Moreover, again using Berinde’s Definitions 5 and 6 withand inequalities (32)–(34), we conclude that decreasing order of convergence speed of implicit iterations is as follows: implicit Noor, implicit Ishikawa, and implicit Mann iteration.
Example 14. Let , , , , , and for , ; then for implicit Mann iteration , we havewhich further impliesand soAlso, for implicit Ishikawa iteration , we havewith Using (47), (46) becomesSimilarly, for implicit Noor iteration , we haveUsing (48) and (49), we havewithHence . Therefore using Definition 5, implicit Noor iteration converges faster than the implicit Ishikawa iteration to the fixed point .
Similarly, using (45) and (48), we haveThat is, implicit Ishikawa iteration converges faster than the implicit Mann iteration to the fixed point .
Now, we compare implicit iterations with their corresponding explicit iterations.
For explicit Mann iteration , we haveSimilarly, for explicit Ishikawa and explicit Noor iterations, we have following estimates (54) and (55), respectively:Using (45) and (53), we arrive atwithHence . Therefore using Definition 5, implicit Mann iteration converges faster than the corresponding explicit Mann iteration to the fixed point .
Similarly, using (48) and (54), we havewithwhich implies that . That is, implicit Ishikawa iteration converges faster than the corresponding explicit Ishikawa iteration to the fixed point .
Again, similarly using (49) and (55), we havewithwhich implies that . That is, implicit Noor iteration converges faster than the explicit Noor iteration to the fixed point .
Using computer programs in C++, the convergence speed of various iterations is compared and observations are listed in Table 1 by taking initial approximation , and , . The table reveals that implicit Noor iteration has better convergence rate as compared to implicit Mann iteration as well as implicit Ishikawa iteration and implicit iterations converge faster than corresponding explicit iterations.

4. Data Dependence of Implicit Iterations in Hyperbolic Spaces
Theorem 15. Let be a mapping satisfying (7). Let be an approximate operator of as in Definition 7, and let , be two implicit iterations associated to and defined by respectively, where , and are real sequences in satisfying , , . Let and , then, for , we have the following estimate:
Proof. Using Definition 2, iteration , and iteration (62), we have the following estimates:which further giveswithUsing (66) and (67), we havewithUsing (69) and (70) and simplifying, we get the following estimate:Using estimates (65), (68), and (71), we arrive atKeeping in mind inequalities (21)–(23), (72) reduces toPutting , , and , the above inequality becomesNow from Theorem 9, we have , and since is continuous, hence .
Therefore, using Lemma 8, (74) yields
Remark 16. Putting in and (62), data dependence result of implicit Mann iteration can be proved easily on the same lines as in Theorem 15.
Remark 17. Putting in and (62), data dependence result of implicit Ishikawa iteration can be proved easily on the same lines as in Theorem 15.
5. Applications
5.1. Implicit Iterations in RNN (Recurrent Neural Networks) Analysis
Neutral networks are a class of nonlinear functions approximations and stable state is achieved in recurrent autoassociative neural networks using iterations. Here we analyze the convergence speed of implicit iterations in recurrent network and many important results will be studied for decreasing and increasing functions. The achieved results possess multifaceted real line applications and in particular can be helpful to design the inner product kernel of support vector machine with faster convergence rate. (For details about RNN and SVM please refer to [33].)
5.1.1. Decreasing Function
The function defined by is a decreasing function with fixed point . By taking initial approximation , , the comparison of convergence of implicit iterations to the fixed point is listed in Table 2.

5.1.2. Increasing Function
The function defined by is an increasing function with fixed point . By taking initial approximation , , the comparison of convergence of implicit iterations to the fixed point is listed in Table 3.

From computational study of convergence of implicit iterations to the fixed point of decreasing as well as increasing functions, the following observations are made.
Decreasing Function. For , , implicit Noor iteration converges in 3 iterations, implicit Ishikawa iteration converges in 4 iterations, and implicit Mann iteration converges in 8 iterations.
For , , implicit Noor, implicit Ishikawa, and implicit Mann iterations converge in 2, 3, and 7 iterations, respectively.
Increasing Function. For , , implicit Noor iteration converges in 7 iterations, implicit Ishikawa iteration converges in 9 iterations, and implicit Mann iteration converges in 14 iterations.
For , , implicit Noor, implicit Ishikawa, and implicit Mann iterations converge in 6, 8, and 13 iterations, respectively.
6. Conclusions
(i)The speed of implicit iterations depends on parameters , , and .(ii)In every case the newly introduced threestep implicit Noor iteration has better convergence rate as compared to other iterations and hence has a good potential for further applications in various disciplines of science.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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Copyright © 2015 Renu Chugh 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.