Abstract
In this paper, we introduce a new scheme and prove convergence results for nonexpansive mappings as well as for weak contractions in the frame of Banach spaces. Moreover, we prove analytically and numerically that the proposed scheme converges to a fixed point of a weak contraction faster than some known and leading schemes. Further, we prove that the new scheme is almost stable with respect to weak contraction. For supporting the main results, we give a couple of nontrivial numerical examples, and the visualization is shown by using the Matlab program. Finally, the solution of a nonlinear fractional differential equation is approximated by operating the main result of the paper.
1. Introduction
Throughout the paper, denotes the set of nonnegative integers. Let be a nonempty subset of a Banach space , is a mapping and . A mapping is said to be nonexpansive if for each ,
A self-map on is said to be a weak contraction [1] if for all a constant and some such that
Theorem 1 (see [1]). Let be a weak contraction with and some such that Then, has a unique fixed point, and Picard sequence converges to the fixed point.
But if we take an initial guess different from a fixed point in case of nonexpansive mapping, it is to be noted that the Picard iterative scheme fails to converge to the fixed points of such mappings; hence, we need some other iterative schemes. In the sequel, many authors gave the generalizations of nonexpansive mapping and proved existence and convergence results in linear space, e.g., see [2].
However, to find the fixed points of numerous nonlinear mappings is not an easy task. So, to overcome this kind of problem, several researchers constructed iterative schemes to approximate fixed points of mappings. A few of them are Picard-S [3], Thakur-New [4], and others [5–10].
Here, we consider some iterative schemes which are frequently used to approximate the fixed points of nonlinear mappings introduced by Picard [11], Mann [12], Ishikawa [13], Noor [14], and Agarwal et al. (S) [15], respectively, where the sequence is developed by an arbitrary point as follows: where , , and are sequences in .
Motivated by the previous work, we define a new iterative scheme for finding the fixed point of a weak contraction, where the sequence is developed iteratively by and where , , and are sequences in .
Our main focus is to consider those iterative schemes which save time when we approximate fixed points of mappings. Berinde [16] gave the following definitions about the rate of convergence of iterative schemes which is defined as follows:
Definition 2. Let and be two sequences of positive numbers that converge to and , respectively. Assume that (i)If , then converges to faster than to (ii)If , then and have the same rate of convergence
Definition 3. Consider and as two fixed point iterative schemes both converging to the same point of a mapping with error estimates If , then converges faster than .
Now, we discuss another concept related to iterative schemes called stability. Let be a theoretical sequence and an approximate sequence which is due to rounding errors and numerical approximation of functions. We say that the approximate sequence converges to the fixed point of mapping if and only if the given fixed point iterative scheme would be stable. Because of this fact, the concept of stability for a fixed point iterative scheme was coined by Ostrowski [17] which defined as follows.
Definition 4 (see [17]). Let be an iteration procedure, converging to a fixed point , which is said to be -stable or stable with respect to , if for , , we have , where is an approximate sequence in a subset of a Banach space .
In 1998, a weaker concept of stability, called almost stability, was coined by Osilike [18] which is defined as follows.
Definition 5 (see [18]). Let be an iteration procedure, converging to fixed point , which is said to be almost -stable or almost stable with respect to , if for , , we have .
Remark 6 (see [18]). It can be easily seen that any -stable iteration procedure is almost -stable, but reverse may fail.
Lemma 7 (see [19]). Let and and be any two sequences of nonnegative numbers satisfying , . If , then .
2. Preliminaries
Definition 8. A self-mapping on a Banach space is said to be demiclosed at , if for any sequence which converges weakly to , and if the sequence converges strongly to , then
Definition 9. A sequence in a normed space is said to be weakly convergent (denoted by ) if an element such that
Definition 10. A Banach space is said to satisfy Opial’s property [20] if for any in for all with .
Lemma 11 (see [21]). Let be a nonempty closed and convex subset of a uniformly convex Banach space and a nonexpansive mapping on . Then, is demiclosed at zero.
Lemma 12 (see [22]). Let be a uniformly convex Banach space and for all . Assume that and are two sequences in such that , , and holds, for some . Then, .
Definition 13 (see [23]). A mapping is said to satisfy property , if a nondecreasing mapping with and , , such that , .
3. Convergence Result for Weak Contractions
Throughout this section, we presume that is a nonempty closed and convex subset of a normed linear space and a weak contraction satisfying (3) with .
Theorem 14. Let be a sequence developed by new iterative scheme (9), then converges to a fixed point of .
Proof. From (9), for any ,
By using the fact that and , we have
Inductively, we get
Since , converges to .
Now, we prove almost stability of new iterative scheme (9) with respect to a weak contraction.
Theorem 15. Let be a sequence developed by iterative scheme (9), then is almost -stable.
Proof. Consider an approximate sequence of in . Suppose sequence defined by (9) is converging to a fixed point (by Theorem 14) and , . Now, we will prove that
Let , then by iterative scheme (9), we have
Since and and using (16), we get
Define , then
Since , by Lemma 7, we have . This implies , i.e., This shows that new iterative scheme (9) is almost -stable.
There is analytical comparison of the rate of convergence of iterative schemes with new iterative scheme (9) for weak contraction.
Theorem 16. Suppose that the sequence is introduced by Picard (4), by Mann (5), by Ishikawa (6), by Noor (7), by Agrawal (8), and by (9) iterative scheme which converges to the same point . Then iterative scheme (9) converges faster than all the schemes (4)–(8) to a fixed point of .
Proof. Using equation (15) of Theorem 14, we have From equation (7), we get It can be easily seen that , so we get Using (21), we obtain that Again, it can be easily seen that , so we get Using (23), we obtain that By using the fact that , we have Inductively, we get Let Then Thus, converges faster than to because , then as .
By applying a similar approach, we can also show that the rate of convergence of all the other leading iterative schemes to is slower than iterative scheme (9).
We embellish the following example to support our assertion.
Example 17. Let be a Banach space with respect to the norm and be a subset of . Let be defined by Then is a weak contraction satisfying (3) for , but is not a contraction mapping.
By Matlab 2015a, we exhibit that new iterative scheme (9) converges to a fixed point of the mapping faster than the iterative schemes Picard, Mann, Ishikawa, Noor, and S with initial point and control sequences , , and , , which can be easily seen in Tables 1 and 2 and Figure 1.
4. Convergence Results for Nonexpansive Mapping
Throughout this section, we presume that is a nonempty, closed, and convex subset of a uniformly convex Banach space and is a nonexpansive mapping. Now, we prove the following useful lemmas which are used to prove the next results of this section.
Lemma 18. Let be a sequence developed by new iterative scheme (9), then exists for all .
Proof. Suppose and . From (9), we have Using (31), we get which exhibit that is decreasing and bounded below. Therefore, exists.
Lemma 19. Let and be the iterative scheme developed by equation (9). Then .
Proof. Since exists by Lemma 18 and it is given that with . Presume that
By the inequalities (30) and (31), we get
respectively. Since is nonexpansive mapping, we have
Using (35), we get
Since
Now,
So that (33) and (40) give
by using Lemma 12 and Inequality (36) and (38), we have
Now,
which gives
using (36) and (45), we get
On the other hand, we have
Applying on both sides, we get
by using (34) and (48), we have
So,
Using Lemma 12 and Inequality (50), we get
Now, we prove a weak convergence result for nonexpansive mapping.
Theorem 20. Presume that enjoys Opial’s condition, then the sequence developed by iterative algorithm (9) converges weakly to a point of .
Proof. Let be a sequence with two subsequences and and and are two weak subsequential limits of and , respectively. From Lemmas 18 and 19, we get exists and , respectively. Now, we have to show that cannot have different weak subsequential limits in . Also, from Lemma 11, is demiclosed at . This implies that , i.e., , similarly . We have to show that .
Let on contrary , by Opial’s condition, we have
which is absurd, hence . Consequently, .
There is a strong convergence result for nonexpansive mapping.
Theorem 21. Let be the sequence developed by equation (9). Then if and only if converges to a point of , where .
Proof. If the sequence converges to a point , then it is obvious that .
Now, for the first part taking for any fixed point . From Lemma 18,
exists ; therefore, .
Now, our assertion is that is a Cauchy sequence in . Since , and for a given , there exists such that for all Precisely, . Therefore, there exists such that
Now, for ,
Thus, is a Cauchy in . Since is closed, for some . Now, implies ; hence, we get .
We now prove a strong convergence result by applying property .
Theorem 22. Let be a nonexpansive mapping with property . Then defined by (9) converges strongly to a fixed point of .
Proof. From equation (51) of Lemma 19, we have Using (57) and property , we get Since enjoy the conditions and , , then we obtain So by Theorem 21, we obtain the desired result.
5. An Illuminate Numerical Example
The purpose of this section is to present a numerical example to compare the rate of convergence for nonexpansive mapping.
Example 23. Suppose a Banach space with usual norm and let be a mapping defined as Then is nonexpansive, but not contraction.
Proof. Let and . Then Hence, is a nonexpansive mapping, but not contraction.
Now, by taking control sequences , , and with initial guess , , and , we can show that new iterative scheme (9) converges faster than all other leading iterative schemes which is shown in Tables 3 and 4 and Figure 2.
6. Application to Nonlinear Fractional Differential Equation
In recent years, many authors pointed out that derivatives and integrals of noninteger order are very suitable for the description of properties of various real materials, e.g., polymers. It has been shown that new fractional order models are more adequate than previously used integer order models. Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional derivatives in comparison with classical integer order models, in which such effects are in fact neglected. The advantages of fractional derivatives become apparent in modelling mechanical and electrical properties of real materials, as well as in the description of rheological properties of rocks, and in many other fields. Most nonlinear fractional differential equations have no exact solution, so the approximate solution or numerical solution may be a good choice [24]. Related to this topic, we may refer the readers to [25–27] and the references therein.
Consider the following fractional differential equation: where denotes the Caputo fractional derivative of order and is a continuous function.
In this section, we approximate the solution of problem (62) via new scheme (9) with which is a Banach space of continuous function from into endowed with the maximum norm.
Assume that for all and .
Theorem 24. Let and be an operator defined by , . Assume that the condition is satisfied. Then the new scheme (9) converges to a solution of the problem (62), say .
Proof. Observe that is a solution of (62) if and only if is a solution of the integral equation Now, let and . Using , we get Thus, for and for each , we get Thus, is nonexpansive mapping. Hence, iterative scheme (9) converges to the solution of (62).
Now, for the effectiveness of Theorem 24, we present the following example.
Example 25. The exact solution of problem (68) is given by The operator is defined by
Taking initial hypothesis , and , choose control sequences , and , . It is shown in Table 5 and Figure 3 that iterative scheme (9) converges to the exact solution of problem (68) for the operator constructed in (70).
7. Conclusion
In this paper, convergence and stability results of a new three step iterative scheme has been studied. Further, the solution of a fractional differential equation is approximated by applying Theorem 24. For nonlinear mappings, we compared the rate of convergence of remarkable iterative schemes analytically and numerically. To support the main result, we gave nontrivial examples.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.