A New Iteration Scheme for Approximating Common Fixed Points in Uniformly Convex Banach Spaces

Saleem, Naeem; Kalu Agwu, Imo; Ishtiaq, Umar; Jarad, Fahd

doi:https://doi.org/10.1155/2023/8332760

Journal of Mathematics

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Abstract Introduction and Preliminaries Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 8332760 | https://doi.org/10.1155/2023/8332760

A New Iteration Scheme for Approximating Common Fixed Points in Uniformly Convex Banach Spaces

Naeem Saleem,^1,2Imo Kalu Agwu,³Umar Ishtiaq,⁴and Fahd Jarad^5,6

Academic Editor: Ji Gao

Received20 Jul 2022

Revised02 Aug 2023

Accepted14 Nov 2023

Published12 Dec 2023

Abstract

In this paper, firstly, we introduce a method for finding common fixed point of -Lipschitzian and total asymptotically strictly pseudo-non-spreading self-mappings and -Lipschitzian and total asymptotically strictly pseudo-non-spreading non-self-mappings in the setting of a real uniformly convex Banach space. Secondly, the demiclosedness principle for total asymptotically strictly pseudo-non-spreading non-self-mappings is established. Thirdly, the weak convergence theorems of the proposed method to the common fixed point of the above mappings are proved. Our results improved, extended, and generalized some corresponding results in the literature.

1. Introduction and Preliminaries

Optimization theory (convex, nonconvex, and discrete) is an important field that has applications in almost every technical and nontechnical field, including wireless communication, networking, machine learning, security, transportation systems, finance (portfolio management), and operation research (supply chain and inventory). Numerous theoretical and practical areas, including variational and linear inequalities, approximation theory, nonlinear analysis, integral and differential equations and inclusions, dynamic systems theory, mathematics of fractals, mathematical economics (game theory, equilibrium problems, and optimization problems), mathematical modelling, and nonlinear analysis, rely on the fixed-point theory. Let be a Banach space (BS), the dual of , and is a closed and convex subset of . The mapping defined byis said to be normalized duality mapping.

Let be a nonlinear mapping. The symbols and will be used to denote the set of natural numbers, the set of real numbers, strong convergence (SC), weak convergence (WC), the set of fixed points of , and the set of common fixed points of , respectively.

Definition 1. Recall that(a)A mapping is said to be nonspreading if there exists such that, for all ,where and is the duality mapping on . Note that in real Hilbert spaces , the is an identity mapping and . Thus, in real Hilbert spaces, (2) is equivalent to In 2008, Kohasaka and Takahashi [1] established this class of mapping in a smooth, strictly convex, and reflexive Banach space (RBS).(b)A mapping is called asymptotically nonspreading (ANS) if there exists such that, for all , Naraghirad [2] established the class of ANS mapping as a generalization of the class of nonspreading mapping. In addition, he proved that if is a nonempty closed convex subset of a real BS and is an ANS mapping of , then has a fixed point.(c)A mapping is said to be uniformly Lipschitzian with the Lipschitz constant if(d)A mapping is called asymptotically strictly pseudo-non-spreading if there exist as and such thatwhere and . Observe that . In a real Hilbert space (see [3]), (6) is equivalent to

Remark 2. It is obvious from (4) and (7) that every ANS mapping is a subclass of the class of asymptotically strictly pseudo-non-spreading mapping with . Again, the class of -asymptotically strictly pseudo-non-spreading mappings is more general than the classes of -strictly pseudo-non-spreading mappings and -asymptotically pseudocontractions (see [4], for more detail).

Example 1 (see [4]). Let be a mapping defined byIt was shown in [4] that is -strictly pseudo-non-spreading (i.e., a mapping such that , there exists for which the inequality holds but not nonspreading.
Observe that for all integer , we haveClearly, is asymptotically strictly pseudo-non-spreading mapping (see [3] for details).

Example 2. Let with the usual norm defined byand be an orthogonal subspace of (i.e., , we have ). For each , define the mapping byThen, is asymptotically strictly pseudo-non-spreading mapping (see [5] for details).

Remark 3. In the above discussion, each of the mappings considered is from a subset of a given space into itself. However, there are so many real-life problems in which the domain of the mapping under consideration is taken into the whole space (and not its subset). When that happens, the aforementioned mappings and their generalizations (assuming self-mappings) become irrelevant. Consequently, there is a need to consider another set of mappings (called non-self-mappings) that will bridge this gap.
The following definition will be required in the sequel.

Definition 4 (see [6]). Let be a BS and a continuous mapping. Then, is called a retract of such that , . Further, if is nonexpansive, then it is said to be a nonexpansive retraction (non-ER) of . Note that if is a retraction, then . A retract of a Hausdorff space must be a closed subset. Every closed convex subset of a uniformly convex Banach space (UCBS) is a retract.

Example 3 (see [6]). Suppose with an inner product and the usual norm , then is a Hilbert space. Let . Define byThen, is a non-ER of onto .

Definition 5. Let be a nonempty, closed, and convex subset of a BS and a non-self-mapping. Then,(1) is said to be ANS non-self-mapping if there exists such that, for all ,(2) is uniformly Lipschitzian with the Lipschitz constant if(3) is said to be strictly asymptotically pseudo-non-spreading non-self-mapping if there exist with as and such that ,where and . Observe that . Note that if is a self-mapping, then becomes the identity mapping so that (15) reduces to (7).
The above study of various nonlinear mappings is quite interesting. However, if there is no means to approximate their respective fixed points, then the time spent in the study would be a waste. Over the years, several researchers have constructed varying iterative schemes to achieve approximate fixed points of different nonlinear mappings. Chidume and Adamu [7] attained convergence via their modified iteration scheme for the common solution of split generalized mixed equality equilibrium and split equality fixed-point problems. Thianwan [8] established a new iteration scheme for mixed-type asymptotically nonexpansive mappings in hyperbolic spaces. Taiwo et al. [9] studied a simple strong convergent method for solving split common fixed-point problems. Shehu [10] investigated an iterative approximation for zeros of the sum of accretive operators, and Suantai et al. [11] worked on nonlinear iterative methods for solving the split common null point problem in Banach spaces. Still on the construction of the fixed-point iteration method, Saleem et al. [12, 13] proved several fixed-point results, by utilizing some novel iterative methods, in the context of intuitionistic extended fuzzy b-metric-like spaces and uniformly convex Banach space, respectively. Saleem et al. [14], while working on graphical fuzzy metric spaces, employed a new iterative method with the graphical structure to solve fractional differential equations. Again, in 2006, Wang [15] generalized the scheme studied in [16] (see below) for the case of two asymptotically nonexpansive non-self-mappings (ANENSMs), which was subsequently improved to a hybrid mixed-type iterative scheme involving two asymptotically nonexpansive self-mappings ANESMs and two ANENSMs in [17], in UCBS. Agwu et al. [18] generalized the scheme studied in [17] to hybrid mixed-type iteration method involving three total ANESMs and three ANENSMs (which simultaneously included the scheme studied in [17]) in UCBS, and Agwu and Igbokwe [19] generalized the scheme in [18] to hybrid mixed-type iteration method involving finite family of total ANESMs and finite family of total ANENSMs in real UCBS. Albert et al. [20] did work on the approximation of fixed point of nonexpansive mappings. Agwu et al. [18] proved the convergence of a three-step iteration scheme to the common fixed points of mixed-type total asymptotically nonexpansive mappings in UCBSs. Acedo and Xu [21] gave iteration methods for strict pseudocontractions in Hilbert space. Other works concerning the formulation and implementation of effective iteration techniques for fixed-point problems are readily available in [22] and [23].
Chidume et al. [16] established the following iterative scheme:where is a sequence in (0,1), is a nonempty closed convex subset of a real UCBS , and is a non-ER of onto and proved several SC and WC theorems for ANENSMs in the context of UCBSs.
In [15], Wang generalized the iterative process (16) as follows:where are two ANENSMs and are real sequences in [0, 1) and proved several WC and SC theorems for ANENSMs.
In 2012, Guo et al. [17] generalized the iterative process (16) as follows:where are two ANESMs, are two ANENSMs, and are real sequences in [0, 1) and proved several WC and SC theorems for the mixed-type ANENSMs.
Recently, Saluja [24] generalized the iterative process (16) as follows:where are two total ANESMs, are two total ANENSMs, and are real sequences in [0, 1) and proved some weak SC theorems for the mixed-type ANENSMs.
For the papers studied, it was discovered that a lot of attention has been given to fixed-point results for asymptotically nonexpansive mappings and some of its generalizations (Wang [15] studied convergence behavior of two ANENSMs in UCBS, Guo et al. [17] examined convergence character of four (two self and two nonself) asymptotically nonexpansive mappings, Saluja [24] investigated convergence behavior of four (two self and two nonself) total asymptotically nonexpansive mappings, Agwu and Igbokwe [19] understudied the nature of fixed point for a finite family of total ANESMs and ANENSMs, and Chima [25] examined fixed point for total asymptotically pseudocontractive mappings in the setup of a real Hilbert space), and almost all the results were communicated in the setup of a real Hilbert space. It is worth mentioning that there are other nonlinear mappings (ANS and asymptotically strict pseudo-non-spreading mappings; see, for instance, [3, 5]) that share the same parents (asymptotically quasi-non-expansive and asymptotically demicontractive mappings) with asymptotically nonexpansive mappings and asymptotically strict pseudocontractive mappings. Unlike nonexpansive-type mappings and their various generalization, the ANS-type mappings (especially, the class of total asymptotically strictly pseudo-non-spreading non-self-mappings) have not received much attention in the setup of a real BS as compared to those of the mappings studied above, perhaps due to unavailability of some working instruments in this area. Consequently, the following questions become necessary.

Question 6. (1)Is it possible to develop a demiclosedness principle for total asymptotically strict pseudo-non-spreading mappings in the setup of a real BS?(2)Can one construct an independent mixed-type iterative scheme for the approximation of a common fixed point for a finite family of certain nonlinear mappings?Motivated and inspired by the works of Ma and Wang [5] and Wojtaszczyk [26], inadequate iteration method for the class ANS-type mappings and the indispensable nature of weak convergence theorems in applications, in this paper, we study a new independent mixed-type iteration scheme (27) and then provide some WC theorems of this new iterative scheme (27) for mixed-type total asymptotically strictly pseudo-non-spreading self-mapping and total asymptotically strictly pseudo-non-spreading non-self-mapping in the setup of real UCBSs. Also, an affirmative answer is given to (1) and (2) in Question 6.

2. Relevant Preliminaries

In this section, we shall use the following definitions, lemmas, and known results in order to prove the main theorems of this paper: given a BS whose dimension is greater than or equal to 2. The mapping (] represented byfor all , is called the modulus of convexity of . Note that if , then is called uniformly convex.

We recall the following definitions and lemmas which will be needed in what follows.

Definition 7 (see [27]). Let be a BS, its dual and . If exists , then is given the Gateaux differentiable norm.

Definition 8 (see [27]). If the limit in Definition 7 exists and is attained uniformly for each (and ), then is given the Frechet differentiable norm (see [28] for more details). Consequent upon this, we have, where functional at is the pairing between and and is an increasing function defined on such that .

Definition 9. The BS is given Opial condition [29] if, for any sequence for each , it follows that and equivalently for all with . Whereas Hilbert spaces and all spaces satisfy Opial conditions, the space with does not satisfy the Opial condition.

Definition 10 (see [5]). Let be a nonlinear mapping. Then, is said to be demiclosed at 0, if, for any sequence , the condition that and implies .

Definition 11. Let be a real BS. If, for every sequence , and imply . Then, is given the Kadec–Klec property [30].

Lemma 12 (see [31, 32]). Let be a real BS. Then, for all ,

Lemma 13 (see [33]). Let the sequences and satisfying the inequality:If , then(1)(2)ln particular, if subsequence which converges strongly .

Lemma 14 (see [30]). Let be a UCBS and for each . Suppose that and are sequences in such thathold for some . Then, .

Lemma 15 (see [30]). Let be a real RBS such that its dual has the Kadec–Klec property. Let be a bounded sequence in and (where denotes the set of all weak subsequential limits of ). Suppose exists for all . Then, .

Lemma 16 (see [30]). Let be a real UCBS and be convex. Then, there exists a strictly increasing continuous convex function such that for each Lipschitzian mapping with the Lipschitz constant ,for all and for all .

Lemma 17 (see [34]). Let be a real UCBS and bounded close and convex. Then, there exists a strictly increasing continuous convex function with such that for any Lipschitzian mapping with Lipschitz constant and elements in and any nonnegative numbers with , the following inequality holds:

Lemma 18 (see [26]). If the sequence WC to , then there exists a sequence of convex combination , and , such that . as .

3. Main Results

Let a real normed space and be closed and convex. Let be a finite family of total asymptotically strictly pseudo-non-spreading non-self-mappings and be a finite family of total asymptotically strictly pseudo-non-spreading self-mappings. We define an iterative scheme generated by as follows:where .

Definition 19. Let be an arbitrary BS and be closed and convex. Let be nonlinear mapping. Following the terminology of Alber et al. [20], is called total asymptotically strictly pseudo-non-spreading if for every , , and , there exist sequences and a strictly increasing continuous function , denoting the set of positive real numbers, with such thatIf , then (28) reduces to

Lemma 20 (demiclosed principle for total asymptotically strictly pseudo-non-spreading non-self-maps). Let be a UCBS, be closed, convex, and bounded and be -Lipschitz continuous and total asymptotically strictly pseudo-non-spreading mapping with and the sequences such that as . Then, is demiclosed at zero.

Proof. Suppose WC to and SC to 0. We show that . It is clear that is bounded. Hence, there exists such that is a closed ball in with center 0 and radius . Thus, is nonempty closed bounded and convex subset in .
claimed as . In fact, since CW to , by Lemma 18 (see, e.g., [14]), we get that, for all , there exists a convex combinationSince converges to 0, it follows that for any positive integer , and given any , there exists such thatHence, , using Definition 19 and is nonexpansive, we deduce, for any fixed , utilizing the well-known inequalitywhich holds for all and for all , we haveFrom (31) and (33) and the condition on the function , we obtainIn addition,Moreover, with the help of Lemma 17, and , there exists with that is increasing function, and we obtainsince .
Thus,Equations (34), (35), and (37) imply thatOn the other hand, for any , it follows that (using (34))Taking of both sides of (39), using (30) and for an arbitrary , we deduceThat is, . By the continuity of , we getThis completes the proof.

Remark 21. The result of Lemma 12 still holds true if . Thus, Lemma 12 can as well serve as a proof for the demiclosedness principle for total asymptotically pseudocontractive non-self-mappings in UCBSs with .

Lemma 22. Let be a UCBS, be closed and convex, be a finite family of uniformly -Lipschitzian and total asymptotically strictly pseudo-non-spreading non-self-mappings with sequences and be a finite family of uniformly -Lipschitzian and total asymptotically strictly pseudo-non-spreading self-mappings with sequences . Let be real sequences such that . Suppose . If the following conditions are satisfied,(i)(ii)(iii)There exist constants and a strictly increasing and continuous functions with such that

Then, both exist for all , where is as defined by (27).

Proof. Set . Then, . Suppose is arbitrary, with the help of (27), we getBy continuing in this manner, we obtain thatFollowing the same method as above, we getIn general,where .
Also, for , we obtain the following estimation using (27):By following the same method as above for , we getand in general,In addition, using (27) and Lemma 12, we obtain, for all , thatFrom (45) and (49) and the fact that , we obtainSince each , for , is total asymptotically strictly pseudo-non-spreading mappings, the last inequality becomesSet . Then, we obtain from (51) thatObserve thatThus,Since , it follows that . Consequently, for any , there exists a natural number such that . Without loss of generality, let so that . LetThus, when , we haveBy condition (ii), and .
From (51)–(53), we haveAgain, from (57) and Lemma 13, it follows that exists so that there exists a constant such that .
By utilizing the infimum for all in (57), we obtainMoreover, and , by utilizing Lemma 13 that exists. This completes the proof.

Lemma 23. Let be as stated in Lemma 13. If, in addition to the assumptions of Lemma 13, the following conditions are satisfied:(a),(b),then and .

Proof. Set . Then, . With the help of arbitrary exists by Lemma 22. Now, assume that , then using the fact that , we getAlso, we haveFurthermore,Therefore, from (59)–(61) and Lemma 22, we obtainFrom (62) and condition (ii) (i.e., ) of Lemma 23, we getAgain, from (27), we getUsing (59) and (64), condition (b), Lemma 14, and following the same technique as above, we obtainMoreover, from (54), we getSimilarly, we obtain from (54) thatNow, from (65) and (66) and the inequalitywe obtain thatAgain, (48) and (74) and utilizing , we obtainAlso, from (71) and (72) and the inequalitywe obtain thatObserve thatEquations (68), (71), (74), and (75) imply thatFurthermore,so that from (69), (74), and (77), we getObserve thatso that, using (71) and (78), we haveThus, from condition (a) of Lemma 14 and (80), we obtainAgain, observe thatso that from (71) and (80), we getAlso,so that from (71) and (81), we getNow, we estimate as follows:From (71), (74), (81), and (82), we obtainNext, observe thatBy (80) and (81), again, observe thatso that from (69), (74), and (88), we get

Lemma 24. Let be as stated in Lemma 13. Under the conditions of Lemma 13, , the exists for all , where is the sequence defined by (27).

Proof. Clearly, exists for all (by Lemma 13), and hence, is bounded. Let exists for all . Then, and exist by Lemma 22. It remains, therefore, to prove Lemma 24 for . Now, for all , defineThen, it follows that . Now, from (57) of Lemma 22, we see thatwhere and with and . Since , it follows that as . SetThen, it follows from the standard argument that exists; i.e., exists for all . This completes the proof.

Lemma 25. Let be as stated in Lemma 13. If, in addition to the assumptions of Lemma 13, has Frechet differentiable norm, then, for all , the limit exists, where is the sequence defined by (27). If denotes the set of all weak subsequential limits of , then for all and for all .

Proof. Set with and in (21). Then, we haveFrom for some , we getThat is,If , then exists for all and for . This completes the proof.

Theorem 26. Let be as stated in Lemma 13. If, in addition to the assumptions of Lemma 13, has Frechet differentiable norm, then the sequence is defined by (27) WC to a common fixed point in .

Proof. With the help of Lemma 25, , . Hence, . Thus, . Therefore, WC to a common fixed point of . This completes the proof.

Theorem 27. Let be as stated in Lemma 13. If, in addition to the assumptions of Lemma 13, the space of has the Kadec–Klec property and the mappings and for , where is an identity mapping, are demiclosed at zero, then the sequence described by (27) WC to a common fixed point in .

Proof. By Lemma 14, is bounded and is reflexive, there exists a subsequence of which WC to some . With the help of Lemma 14, we deduce and . By the assumptions, the mappings and for , where is an identity mapping, are demiclosed at zero, boundedness of , and the uniqueness of the limit of the weakly convergence sequence follows that the sequence WC to . This completes the proof.

Theorem 28. Let be as stated in Lemma 13. If, in addition to the assumptions of Lemma 13, satisfies Opial’s condition and the mappings and for , where denotes the identity mapping, are demiclosed at zero, then the sequence defined by (27) WC to a common fixed point in .

Proof. Suppose . With the help of Lemma 22, the sequence exists and is convergent and is bounded. By utilizing Lemma 23, we deduce that and . Finally, the demiclosed property of each boundedness of , the uniqueness of the limit of the weakly convergence sequence, and the Opial property of the underlying space follows that the sequence weakly converges to .

Remark 29. The following iteration techniques are immediate consequences of our newly constructed iteration scheme:(1)If in (27), we havewhere .(2)If in (97), we havewhere .(3)If in (98), where is an identity map on , we havewhere .(4)If in (27), we getwhere .(5)If in (99), where is an identity map on , we havewhere .(6)If in (100), we get

4. Conclusion

In this manuscript,(1)We established a new fixed-point algorithm for approximating the common fixed point of finite families of -Lipschitzian and total asymptotically strictly pseudo-non-spreading self-mappings and -Lipschitzian and total asymptotically strictly pseudo-non-spreading non-self-mappings in the setup of a real UCBS(2)We introduce a new type of nonlinear mapping called total asymptotically strictly pseudo-non-spreading self-mappings in the setup of UCBS(3)Demiclosedness principle for total asymptotically strictly pseudo-non-spreading self-mappings and several WC results were obtained using our newly constructed iteration scheme in the setup of a real UCBS(4)A slight modification of our iteration scheme resulted in several well-known iteration schemes currently existing in the literature, see, for instance, (97)–(102)(5)Our WC results improve, generalize, and extend several well-known WC results from the setup of real Hilbert spaces to those of real UCBSs

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2023 Naeem Saleem 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.

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