Abstract
In this paper, we propose the generalized M-iteration process for approximating the fixed points from Banach spaces to hyperbolic spaces. Using our new iteration process, we prove -convergence and strong convergence theorems for the class of mappings satisfying the condition and the condition which is the generalization of Suzuki generalized nonexpansive mappings in the setting of hyperbolic spaces. Moreover, a numerical example is given to present the capability of our iteration process and the solution of the integral equation is also illustrated using our main result.
1. Introduction
The concept of nonexpansive mappings can be defined in many general setting of metric spaces. Suzuki [1] introduced the concept of generalized nonexpansive single-valued mappings which are called Suzuki generalized nonexpansive mappings or the condition and received some fixed-point results and convergence results for such mappings in Banach spaces. Also, the new conditions for single-valued mappings are defined by GarcĂa-Falset et al. [2], called the condition and the condition which are weaker than a nonexpansive mapping and stronger than a quasi-nonexpansive mapping. Moreover, the class of mappings satisfying the condition and the condition is larger than the class of mappings satisfying the condition .
Let be a self-mapping on a nonempty subset of a Banach space and and are real sequences in for all . Mann [3] iteration process was determined by the following method:
The following is called the S-iteration process defined by Agarwal et al. [4]:
Next, Gursoy et al. [5] introduced the Picard-S-iteration process. By providing an example, they presented that Picard-S-iteration process converges faster than all Mann, Picard, Noor, Ishikawa, CR, SP, S, , Abbas, and Normal-S involving Two-step Mann iteration processes. The Picard-S-iteration process is defined as
Recently in 2018, Ullah et al. [6] introduced a new iterative process called the M-iteration process defined as follows:
They proved that M-iteration process can be used to approximate the fixed point of Suzuki generalized nonexpansive mappings and obtain weak convergence and strong convergence results on Banach spaces. They also presented that the M-iteration process converges faster than Picard-S-iteration and S-iteration processes by providing an instance.
Motivated by above, we introduce a generalized M-iteration process and use our iterative scheme for proving some -convergence and strong convergence theorems for mappings satisfying the condition and the condition which is the generalization of Suzuki generalized nonexpansive mappings (the condition ()) in the setting of hyperbolic spaces. Numerically, we give an example of a mapping satisfying the condition and the condition , which does not satisfy the condition . Finally, we compare the speed of convergence of our generalized M-iteration process with M-iteration process of Ullah et al. [6].
2. Preliminaries
For this research, we discuss on the setting of hyperbolic spaces which was introduced by Kohlenbach [7], containing normed linear spaces and convex subsets and Hadamard manifolds [8], CAT(0) spaces in the sense of Gromov [9], and the Hilbert ball equipped with the hyperbolic metric [8].
A hyperbolic space is a triple where is a metric space and such that (W1) , (W2) , (W3) , (W4) ,for all and .
First we recall some definitions, lemmas, and propositions that will be used in the next part.
A hyperbolic space is called uniformly convex [10] if for all , , and there exists such that , , and . Then, we have
A mapping which provides for a given and is well known as a modulus of uniform convexity of . We call as a monotone if it decreases with (for a fixed ), i.e., for any given and for any , we have .
A nonempty subset of a hyperbolic space is said to be convex if for any and .
If and , then we use the notion for . In [10], it is remarked that any normed space is a hyperbolic space, with . Hence, the class of uniformly convex hyperbolic spaces is a natural generalization of uniformly convex Banach spaces.
Let be a nonempty subset of metric space . If , then is said to be a fixed point of a mapping . The set of all fixed points of is denoted by .
Definition 1. (see [11]). A mapping is said to be(i)Nonexpansive if for all ;(ii)Quasi-nonexpansive if and for all and .Suzuki [1] defined the notion of Suzuki generalized nonexpansive mappings which is called the condition . Such mappings are weaker than nonexpansive mappings and stronger than quasi-nonexpansive mappings.
Definition 2. (see [1]). A mapping is called to satisfy the condition iffor all .
Falset et al. [2] introduced the definition of generalized nonexpansive mappings as follows.
Definition 3. (see [2]). A mapping is called to satisfy the condition provided thatWe say that satisfies the condition whenever satisfies the condition for some .
Proposition 1. (see [12]). Let be a mapping which satisfies the condition . Then, satisfies the condition for some .
There is an example presented that there exists a mapping which satisfies the condition , but it fails to satisfy the condition . Then, the mappings satisfying the condition are more generalized than mappings satisfying the condition .
Example 1. (see [13]). Let , and under the supremum norm, consider a nonempty subset of defined byFor any , associate a function defined by . It is easy to verify that satisfies the condition but does not satisfy the condition .
Definition 4. (see [2]). Let be a mapping and . Then, is said to satisfy the condition if for all Of course, it follows that the condition is the special case for in the condition .
On the contrary, the notion of a mapping satisfying the condition was introduced by Senter and Dotson [14].
Definition 5. Let said to satisfy the condition , if there exists a nondecreasing function satisfying and for all , such thatWe need the following definition of convergence in hyperbolic spaces [15] which is called -convergence. It plays an essential role in the principle results. In order to define -convergence, we need the following concepts.
Let be a bounded sequence in a hyperbolic space . We can define a function byAn asymptotic radius of a bounded sequence with respect to a nonempty subset of is determined and denoted byAn asymptotic center of a bounded sequence with respect to a nonempty subset of is determined and denoted byThe sequence in is said to -converge to if is a unique asymptotic center of for every subsequence of . In this case, we write - and call the - of .
Next, we review some definitions and lemmas.
Lemma 1. (see [16]). Let be a complete uniformly convex hyperbolic space with a monotone modulus of uniform convexity . Then, every bounded sequence in has a unique asymptotic center with respect to any nonempty closed convex subset of .
Lemma 2. (see [17]). Let be a complete uniformly convex hyperbolic space with a monotone modulus of uniform convexity . Let and be a sequence in for some . If and are sequences in such that , , and for some . Then,
Lemma 3. (see [18]). Let be a nonempty closed convex subset of a complete uniformly convex hyperbolic space with a monotone modulus of uniform convexity . Let be a mapping which satisfies the condition for some and the condition . Suppose that is a bounded sequence in such that . Then, has a fixed point.
As a consequence of Theorem 3.2 [18], we receive the following lemma.
Lemma 4. Let be a nonempty closed convex subset of a complete uniformly convex hyperbolic space with a monotone modulus of uniform convexity . Let be a mapping which satisfies the condition for some . Then, is closed.
3. Main Results
For this section, we construct an iteration process in hyperbolic spaces named as “generalized M-iteration process,” as follows:for all , , , and are real sequences in .
Remark that if we take , then we obtain that (15) reduces to (4).
The following lemmas play crucial role in proving the main theorems of this section.
Throughout in this paper, let be a complete uniformly convex hyperbolic space with a monotone modulus of uniform convexity and be a nonempty closed convex subset of .
Lemma 5. Let be a nonempty closed convex subset of and be a mapping which satisfies the condition for some with . For an arbitrary chosen , a sequence is defined by (15). Then, exists for each .
Proof. Assume that and . By hypothesis, we obtain that satisfies the condition for some . SoThus using (15), we obtain thatUsing (15) together with (17), we getAgain, using (15) and (18), we obtain thatThis presents that is bounded below and decreasing. Then, exists for each .
Lemma 6. Let be a nonempty closed convex subset of and be a mapping which satisfies the condition for some and the condition . For an arbitrary chosen , a sequence is defined by (15). Then, if and only if is bounded and .
Proof. Assume that , and let . So, it follows from Lemma 5, is bounded. Next, we will indicate that . Since satisfies the condition for some and , for each we obtain thatTherefore,From Lemma 5, we achieve exists for all . Assume that . If , thenLetting limit as on both sides of the above estimate, we haveIf , thenTaking lim sup as on the above inequality,It follows from (18) and (19) that . By taking lim inf as on both sides, we obtain thatThe relations (25) and (26) imply thatSince , we get thatIt follows from Lemma 2 through (27) and (28) thatConversely, suppose that is bounded and . Then, it follows from Lemma 3 that has a fixed point, so we have which is nonempty.
We now establish the -convergence theorem for generalized nonexpansive mappings in hyperbolic spaces.
Theorem 1. Let be a nonempty closed convex subset of and be a mapping which satisfies the condition for some and the condition with . Let be a sequence defined by (15). Then, -converges to a fixed point of .
Proof. It follows from Lemma 6 that is a bounded sequence. Thus, has a -convergent subsequence. Now, we are going to show that every -convergent subsequence of has a unique -limit in . Let and be -limits of the subsequence and of , respectively. From Lemma 1, we have and . By Lemma 6, we obtain that and . Next, we prove that and are fixed points of and reunique should be are unique. Since satisfies the condition , there exists such thatLetting lim sup as on both sides of the above inequality, we getThe uniqueness of the asymptotic center implies . Thus, is a fixed point of . Similarly, we also have as a fixed point of . Lastly, we show that . Suppose , and so by the uniqueness of an asymptotic center, we haveThis is a contradiction. Thus, . Then, -converges to a fixed point of .
Next, we present some strong convergence theorems.
Theorem 2. Let be a nonempty closed convex subset of and be a mapping which satisfies the condition for some and the condition with . Let be a sequence defined by (15). Then, converges strongly to a fixed point of if and only if .
Proof. Assume that converges strongly to . So, . Because , therefore .
Conversely, suppose that . It follows from Lemma 5, and we get for all . Thus, . Therefore, exists. From the assumption of our theorem, , so we have . Next, we prove that is a Cauchy sequence in . Let for each . Since , for any given , there is such thatIn particular, . Then, there exists such that .
For any , we getThis shows that is a Cauchy sequence in . Since is a closed subset of a complete hyperbolic space , is complete. Then, must converge to a point in . Let . Now, we show that is a fixed point of . Since satisfies the condition , there exists such thatLetting , it follows from Lemma 6 that . We have . Then, is a fixed point of . Hence, converges strongly to a point in .
Theorem 3. Let be a nonempty closed convex subset of and be a mapping which satisfies the condition for some and the condition with . If a sequence is determined by (15) and satisfies condition , then converges strongly to a fixed point of .
Proof. From Lemma 5, we have exists for each and for some . Let . If , we obtain the desired result. Assume that . From Lemma 6, we obtain that . Since satisfies condition , we haveTherefore,From is a nondecreasing mapping satisfying and for each , we have . By applying Theorem 2, we obtain the desired result.
4. Numerical Example
First, we present an example of the convergence for a mapping satisfying the condition and the condition , but it fails to satisfy the condition . Next, we give an example to compare the speed of convergence of our generalized M-iteration process with M-iteration process of Ullah et al. [6].
Example 2. Let with metric and . Define by for each and . Thus, is a complete uniformly hyperbolic space with a monotone modulus of uniform convexity and is a nonempty compact convex subset of . For a given , let defined byBy [2], it is shown that the mapping satisfies the condition but it fails to satisfy the condition whenever . Moreover, satisfies the condition for some . For an initial point and , , and , we have , , and and belonging in . Set the stop parameter to , where 0 is a fixed point of . By using MATLAB, we compute the convergence of the generalized M-iteration (15). The numerical experiments of all iterations for approximating the fixed point 0 are given in Table 1, and the convergence of an iterative sequence generated by generalized M-iteration process is shown in Figure 1.
Example 3. Let a mapping be defined byBy the proof of [6], it is shown that a mapping satisfies the condition . Then, also satisfies the condition for some and the condition for some . For an initial value and , , , and , , and belong in . Set the stop parameter to , where 1 is a fixed point of . We compare our iteration process (15) with M-iteration process (4). Numerically, we can easily see that the sequence generated by the generalized M-iteration converges faster than the sequence generated by the M-iteration process as given in Table 2 and Figure 2.
5. Applications to Integral Equations
In this section, we prove the existence of a solution of a nonlinear quadratic integral equation taken from Allahari et al. [19].
Let be the set of all continuous functions defined on and defined by
We can see that is a metric space and a hyperbolic space with the modulus of uniform convexity.
Let be the set of functions which satisfy the following condition:(i) is nondecreasing and for all .
Consider the nonlinear quadratic equation as follows:where , , and . From now on, let be defined by
Suppose that the following conditions hold: (A1) is continuous; (A2) is continuous, and there exist and such that for all and for all ,  (A3) is continuous at for every and measurable at for all such that and ; (A4) ; (A5) The sequence defined as in (15) is bounded and .
Theorem 4. Under assumptions (A1)–(A5), the integral equation (41) has a solution in .
Proof. Let such thatfor some . Using (A2), we obtain thatSince is nondecreasing, we haveFrom (A3), we obtain thatTherefore,This implies that satisfies the condition for .
Let . We obtain thatThis implies thatfor some . Then, satisfies the condition . By Lemma 3 and (A5), we get that . Therefore, all assumptions in Theorem 1 are satisfied. Hence, -converges to a fixed point of in . This implies that the integral equation (41) has a solution in .
6. Conclusion Remark
Our space is a uniformly convex hyperbolic space which is a natural generalization of a uniformly convex Banach space. Since our mappings are more general than mappings of Ullah et al. [6] and the sequence generated by the generalized M-iteration process (15) converges faster than the sequence generated by the M-iteration process (4), our results improve the results of Ullah et al. [6].
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
The first author was supported by the Science Achievement Scholarship of Thailand (SAST). The third author would like to express her deep thanks to Naresuan University for supporting this research under grant R2563C002.