Abstract
In this article, we consider an extensive class of monotone nonexpansive mappings and introduce a new iteration algorithm to approximate the fixed point for monotone total asymptotically nonexpansive mappings in the framework of hyperbolic space. Faster convergence and stability results are proved for that iteration; also, fixed point is approximated numerically in a nontrivial example by using MATLAB.
1. Introduction
The concept of hyperbolic space was given by Reich and Shafrir [1] in , which is defined as a metric space that has a family of metric lines; for any two unique endpoints , there is a unique metric line in . This metric line works out a unique metric segment symbolize by , which is an isometric image of A unique point is denoted by which satisfieswhere Such a metric space is called a convex metric space, such thatfor all then is called hyperbolic metric space (abbreviated as H.M.S).
The class of hyperbolic spaces contains the normed spaces, CAT(0) spaces, and many others. There are many examples in literature which show that hyperbolic spaces are more general than Banach spaces; for details, see [2] and Example 1.1 of [3].
Recently, a new direction has been discovered dealing with the extension of the Banach Contraction Principle [4] to partially ordered metric spaces. The case of monotone nonexpansive mappings was first considered in [5]. After that, Dehaish and Khamsi [6] gave an analogue to Browder [7] and Göhde [8] fixed point theorems for monotone nonexpansive mappings. In 2018, Alfuraidan and Khamsi [9] extended Goebel and Kirk’s fixed point theorem [10] for asymptotically nonexpansive mappings to the case of monotone mappings. Multiple articles [11–14] can be found in the literature on fixed point of asymptotically nonexpansive mapping using multistep iterations and strong convergence analysis.
In 2016, Alber et al. [15] introduced the concept of total asymptotically nonexpansive mappings that generalizes the family of mapping that are the extension of asymptotically nonexpansive mappings. Example 1 of [16] and Example 3.1 of [4] show that total asymptotically nonexpansive mappings properly contain the asymptotically nonexpansive mappings.
In this article, we define monotone total asymptotically nonexpansive mappings (abbreviated as M.T.A.N.M) and also extend Alber’s fixed point theorem [10] for the respective class. In addition, this result also generalizes the results of Alfuraidan and Khamsi in hyperbolic space [9].
In Section 4, we introduce a new iteration scheme, prove fast convergence and stability results, and also provide comparison with some famous iterations listed as Banach [17], Mann [18], Ishikawa [19], Agarwal et al. [20], Noor [21], Abbas and Nazir [22], Vatan Two-step [23], an accelerated iteration [24] (by Chen and Wen [24]), and Thakur New [25]. Numerically, we compare the convergence of new iteration with these iterations in a nontrivial example.
2. Preliminaries
Let be a partially ordered (abbreviated as P.O) metric space, any two points are comparable whenever or
Definition 1. Consider be a partially ordered space and be a self map of , which is said to be(i)monotone or order preserving [5] if(ii)monotone Lipschitzian mapping [5] if is order preserving and there exist such thatIf , the mapping is said to be order preserving nonexpansive mapping(iii)monotone asymptotically nonexpansive mapping [9] if there exists a sequence for such thatfor every such that and are comparable
Now, we will define M.T.A.N.M in hyperbolic space.
Definition 2. Let be a hyperbolic metric space having a nonempty subset . A self map is monotone total asymptotically nonexpansive mapping on if there exist nonnegative sequences and with as , a strictly increasing continuous functionsuch thatand there exists a constant such that for thenfor every comparable elements
Example 1. Consider the real line as a hyperbolic metric space and be the subset of , , and be a mapping defined by . Suppose that there exist two nonnegative sequences and with and as and a strictly increasing continuous function with . Then, is M.T.A.N.M.
Example 2. Consider the real plane as a hyperbolic metric space. Let be defined asLet and be a mapping defined byThus, is the fixed point of Suppose two nonnegative sequences and with as , and a strictly increasing continuous function
Now, we consider the following cases with assumptions :
Case 1. If , then In this case, is monotone and satisfies all the conditions of total asymptotically nonexpansive mapping.
Case 2. If , then , , and Also,implies that is a M.T.A.N.M.
Case 3. If , then , and Now,
Hence, is M.T.A.N.M
Next, we have some definitions and lemmas that will be useful in the proof of the main result.
Definition 3 (see [26]). A hyperbolic space with metric is said to be uniformly convex if for any for every and for each
The function is called the modulus of uniform convexity of
A hyperbolic space satisfies the property [26]. If is nonincreasing sequence of nonempty, bounded, closed, and convex subset of , .
Definition 4 (see [27]). A bounded sequence is -convege to, if is the unique asymptotic centre of every subsequence of
Definition 5 (see [9]). A partially ordered hyperbolic metric space satisfies the monotone weak Opial condition if any sequence in which is monotone and weakly converges to , then the followingfor every such that or
Throughout in article, the order intervals are assumed to be closed and convex and any of the subsetsfor every
Lemma 6 [9]. Suppose be uniformly convex H.M.S, and be a subset of which is closed nonempty and convex. Let be a type function if a bounded sequence such thatfor any Since is hyperbolic space, is convex and continuous with distinctive minimum point such that
Moreover, if in is the minimizing sequence of , i.e.,then, strongly converges to
3. Main Result
Theorem 7. Let a uniformly convex P.O H.M.S be with nonempty convex closed bounded subset . Let be a continuous M.T.A.N.M on . Assume , such that . Then, has a fixed point.
Proof. Let be such thatBy the monotonicity of , we getfor each , and is a monotone increasing sequence. Also, the order intervals are closed and convex. So, we haveLet thenand the monotonicity of impliesfor every , i.e., Consider the type function produced by given as for any . Above lemma shows the occurrence of a unique such that . Since , we have , for every , which impliesAs is total asymptotically nonexpansive mapping, so , , when Hence,hence, is a minimizing sequence of . Using Lemma 6, converges to Since is continous, we havei.e., is a fixed point of ☐
The following corollary is the conclusion of Theorem 3.3 of [9].
Corollary 8. Let a uniformly convex P.O H.M.S be with nonempty convex closed bounded subset . Let be a continuous M.T.A.N.M on . Assume , such that . Then, has a fixed point.
The corollary given below is the consequence of Theorem 7 by replacing the continuity condition with weak Opial condition.
Corollary 9. Let be a uniformly convex P.O H.M.S, satisfying monotone weak Opial condition with a nonempty convex closed bounded subset . Let be a M.T.A.N.M on . Assume , such that . Then, has a fixed point.
4. Convergence Theorem and Stability Results
We introduce the new iteration scheme given below, let be a nonempty convex subset of a hyperbolic space , for , where , , and are sequences in , such that
Fastness and stability play an important role for an iteration process to be preferred on another iteration process, so now, we prove that new iteration is stable and has good speed of convergence than others. For faster convergence and new class of mapping on metric space introduced by Berinde [28], satisfyinghere, we will modify this mapping asfor all , where and
The following definitions and lemma will be helpful for faster convergence results given in [29].
Definition 10. Let and be two sequences, having convergent points and , respectively, then converges faster than if
Definition 11. Let and be two fixed point schemes that converge to the same fixed point . Ifwhere and are two sequences that converge to If converges faster than , then converges faster than to
Lemma 12. If is a real number such that and be a sequence such thatthen for any positive sequence satisfying
Lemma 13 (see [4]). Suppose , , and be sequences of nonnegative satisfying
If and , then exists.
Lemma 14 (see [4]). Suppose be a uniformly convex H.M.S. Let be such thatwhere with Then, we get
Lemma 15 (see [26]). Let be a P.O hyperbolic space. Let be a nonempty convex and closed subset of . Let be a monotone mapping. Let , such that or Then, the sequence in then(a) or (b) or , provided that -converge to
Lemma 16. Let a uniformly convex P.O H.M.S be with nonempty convex closed bounded subset . Let be a M.T.A.N.M with If the sequence is defined by with or Then, the following holds
Proof. Let By the above lemma , as is monotoneNow, using Definition 2 and after simplification, we getAgain using Definition 2 and (40), we getConsiderUsing Definition 2, (40) and (41), we getwhere and Using Lemma 13, exist for
For part , we have to show thatthe proof resembles to Theorem 2.1 of [4].☐☐
Theorem 17. Let a uniformly convex P.O H.M.S be with convex closed bounded and nonempty subset . Let be a continuous M.T.A.N.M on , with If is defined by with or If or for , then -converges to a fixed point of
Proof. By Lemma 16,sequence is bounded and Let be any subsequence of for , such that -converges to By Lemma 15, we haveNow, we have to show that every -convergent subsequence of has a unique -limit in Let and be two subsequences of -converging to and , respectively. By Lemma 16, is bounded andWe claim that , and produced by isFrom Theorem 7, , same for By the definition of -convergence and Lemma 6, we havewhich is contradiction, unless ☐
Theorem 18. Let a uniformly convex P.O H.M.S be with nonempty convex closed bounded subset . Let be a continuous M.T.A.N.M on with If is defined by with or If or for , then converges to a fixed point of if and only if
Proof. If converges to a fixed point of , thenConversely, consider From Lemma 16therefore, exists. As , so we getNow, we prove that is a Cauchy sequence in . For such that for all in particular , so a fixed point such thatFor Hence, is a Cauchy sequence in closed subset of ; therefore, it converges in such thatAs we have , since is closed so
Now, we prove that newly proposed iteration scheme is faster than Thakur New [25] for a mapping defined in in hyperbolic metric space.☐
Theorem 19. Let be a P.O H.M.S. Let be a nonempty convex closed bounded subset of , and be a mapping satisfying (30) with Let be defined by (28), and defined in [25], then converges faster than
Proof. Let Now, using (28) and (30), we haveAgain, using (28), (30), and (58), we haveFurther, considerNow, using (30) and then (28), we getLetand calculated in Theorem 3.1 of Thakur New [25]. Then,Hence, converges faster than ☐
Now, we will prove the stability result for this we have the following definition by [30].
Definition 20. Let be any arbitrary sequence, then iteration sequence converging to unique fixed point is said to be -stable if for , we have
Theorem 21. Let be a P.O H.M.S. Let be a nonempty convex closed bounded subset of , and be a mapping satisfying with Let be defined by , satisfying , then the iteration is -stable.
Proof. Let be any arbitrary sequence, the sequence defined by (28) converging to unique fixed point , andWe have to prove thatLet , then applying , we getConversely, let , we haveHence, it is -stable.☐
We have a nontrivial example for M.T.A.N.M, and fixed point is numerically approximated by using MATLAB.
Example 3. Let be a hyperbolic space. Define a relation asLet be defined asLet and be a mapping defined byAs is the fixed point of we have to show that is monotone for this considerthenthat isNext,Hence, is M.T.A.N.M with
The rate of convergence of Mann, Ishikawa, Agarwal, Noor, Abbas, Thakur, and accelerated and new iterations for the mapping defined in Example 2 is given below. Table 1 shows the convergence behavior of Mann, Ishikawa, Agarwal, and Noor iterations with new iteration for the parameters and with the initial values and New iteration requires less number of iterations for convergence.
Table 2 shows the convergence of Abbas, Thakur, and accelerated iteration with new iteration for the same initial values, parameters, and tolerance.
The following figures show the convergence behavior of different iterations with new iteration in Figures 1–3.
All iterations converges to Comparison shows that new iteration requires the least number of iterations for convergence. Table 3 shows that different parameters have an effect on iterations and by changing the initial values, new iteration not only converges faster but also stable than other iterations
5. Conclusions
In the present article, the concept of monotone asymptotically nonexpansive mapping has been generalized to monotone total asymptotically nonexpansive mapping in the framework of hyperbolic space. New iteration has been introduced to approximate the fixed point for that mapping. We proved the existence of fixed point, faster convergence, and stability results for new iteration. We also constructed a nontrivial example to approximate the fixed point numerically and compare the convergence result of new iteration with some well-known iterations by using MATLAB.
By relaxing the condition of monotonicity, we can also achieve some similar results presented in recent articles [31, 32] by using the proposed iteration.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors have no conflict of interests regarding the publication of this paper.
Acknowledgments
The research was supported by the National Natural Science Foundation of China (Grant Nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485) and the Taif University Researchers Supporting Project number (TURSP-2020/77), Taif University, Taif, Saudi Arabia.