New Iteration Scheme for Approximating a Common Fixed Point of a Finite Family of Mappings

Isiogugu, F. O.; Izuchukwu, C.; Okeke, C. C.

doi:https://doi.org/10.1155/2020/3287968

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Volume 2020 | Article ID 3287968 | https://doi.org/10.1155/2020/3287968

New Iteration Scheme for Approximating a Common Fixed Point of a Finite Family of Mappings

F. O. Isiogugu,^1,2C. Izuchukwu,¹and C. C. Okeke¹

Guest Editor: Aftab Hussain

Received28 Jul 2019

Accepted02 Apr 2020

Published01 May 2020

Abstract

We introduce a new algorithm (horizontal algorithm) in a real Hilbert space, for approximating a common fixed point of a finite family of mappings, without imposing on the finite family of the control sequences , the condition that , for each . Furthermore, under appropriate conditions, the horizontal algorithm converges both weakly and strongly to a common fixed point of a finite family of type-one demicontractive mappings. It is also applied to obtain some new algorithms for approximating a common solution of an equilibrium problem and the fixed point problem for a finite family of mappings. Our work is a contribution to ongoing research on iteration schemes for approximating a common solution of fixed point problems of a finite family of mappings and equilibrium problems.

1. Introduction

Let be a nonempty set and be a mapping. A point is called a fixed point of if . If is a multivalued mapping, then is a fixed point of if . is called a strict fixed point of if . The set (respectively, ) is called the set of fixed points of the multivalued (respectively, single-valued) mapping , while the set is called the set of strict fixed points of .

Let be a normed space. A subset of is called proximinal if for each , there exists such that

It is known that every convex closed subset of a uniformly convex Banach space is proximinal. We shall denote the family of all nonempty closed and bounded subsets of by , the family of all nonempty subsets of by , and the family of all proximinal subsets of by , for a nonempty set .

Let denote the Hausdorff metric induced by the metric on , that is, for every ,

Let be a normed space and be a multivalued mapping on . is called if there exists such that, for all ,

In (3), if , then is a contraction, while is nonexpansive if . is called quasi-nonexpansive if and for all ,

Clearly, every nonexpansive mapping with the nonempty fixed point set is quasi-nonexpansive. The multivalued mapping is -strictly pseudo-contractive-type of Isiogugu [1] using the terminology of Browder and Petryshen [2] for single-valued pseudo-contractive mapping and Markin [3] for the monotone operator if there exists such that given any pair and , there exists satisfying and

If in (5), then is pseudo-contractive-type, while is nonexpansive-type if . Every multivalued nonexpansive mapping is nonexpansive-type. is of type-one in the sense of Isiogugu et al. [4] if given any pair , thenwhere . is called a multivalued demicontractive in the sense of Isiogugu and Osilike [5] using the terminology of Hicks and Kubicek [6] for single-valued demicontractive if and for all and , there exists such that

If in (7), is hemicontractive in the terminology of Naimpally and Singh [7] for single-valued hemicontractive, while is quasi-nonexpansive if .

Furthermore, every multivalued strictly pseudo-contractive-type in the sense of [1] with the nonempty set of strict fixed points is demicontractive with respect to its set of strict fixed points.

A single-valued mapping is called nonspreading in the sense of Kohsaka and Takahashi [8, 9] if

Observe that if is nonspreading and , then is quasi-nonexpansive. is -strictly pseudo-nonspreading in the sense of Osilike and Isiogugu [10] if there exists such thatfor all . Clearly, every nonspreading mapping is -strictly pseudo-nonspreading. If is strictly pseudo-nonspreading and , then is demicontractive in the sense of [6] (see also [11]).

Several algorithms have been introduced by different authors for the approximation of common fixed points of finite family of mappings , where (the set of nonnegative integers) (see, for example, [12–18] and references therein). One of the motivations for this aspect of research is the well-known convex feasibility problem which is reducible to the problem of finding a point in the intersection of the set of fixed points of a family of nonexpansive mappings (see, for example, [19, 20]). The earliest of such algorithms was the cyclic algorithm introduced by Bauschke [12] using a Halpern-type iterative process considered in [21] for the approximation of a common fixed point of a finite family of nonexpansive self-mappings. He proved the following theorem.

Theorem 1 (see [12], Theorem 3.1). Let be a nonempty convex closed subset of a real Hilbert space and be a finite family of nonexpansive mappings of into itself with with . Given points , let be generated bywhere and satisfies . Then, converges strongly to , where is the metric projection.

The above algorithm of Bauschke was extended to approximate the family of more general class of strictly pseudo-contractive mappings (see, for example, [22, 23]). Suantai et al. also considered similar algorithms (see, for example, [24]) and references therein.

In 2008, Zhang and Guo [25] considered a parallel iteration for approximating the common fixed points of a finite family of strictly pseudo-contractive mapping. They obtained the following theorem.

Theorem 2 (see [25], Theorem 4.3). Let be a real q-uniformly smooth Banach space which is also uniformly convex and be a nonempty convex closed subset of . Let be an integer, and for each , let be a -strictly pseudo-contractive mapping for some . Let . Assume the common fixed point set is nonempty. Assume also for each , is a finite sequence of positive numbers such that for all n and for all . Given , let be the sequence generated by the algorithm:Let be a real sequence satisfying the conditionsThen, converges weakly to a common fixed point of .

Motivated by the parallel algorithm above, many authors have considered in a real Hilbert space, another form of parallel algorithm for a finite family of -strictly pseudo-contractive mappings defined bywhere for each and for each (see, for example, [13] and references therein).

In [14], Iemoto and Takahashi studied the approximation of common fixed points of a nonexpansive self-mapping and a nonspreading self-mapping in a real Hilbert space. If are, respectively, nonexpansive and nonspreading mappings, they considered the iterative scheme generated from arbitrary bywhere and are suitable sequences in . They proved the following main theorem:

Theorem 3 (see [14], Theorem 4.1). Let be a real Hilbert space. Let be a nonempty convex and closed subset of . Let be a nonspreading mapping of into itself and a nonexpansive mapping of into itself such that . Define a sequence in as follows:for all , where , .
Then, the following hold:(i)If , , then converges weakly to .(ii)If and , then converges weakly to .(iii)If and , then converges weakly to .Motivated by the above result, Osilike and Isiogugu obtained the following result.

Theorem 4 (see [10], Theorem 3.1.1). Let be a nonempty convex closed subset of a real Hilbert space, and let be a -strictly pseudo-nonspreading mapping with a nonempty fixed point set . Let , and let be a real sequence in such that . Let and be sequences in generated for arbitrary byThen, converges weakly to , where .

We observed that all the existing iteration schemes for the approximation of a common fixed point of a finite family of mappings for , which are related to the parallel algorithm, require the condition that, for each , on the control sequences . However, in real-life applications, if is very large, it is very difficult or almost impossible to generate a family of such control sequences. Moreover, the computational cost of generating such a family of control sequences is very high and also takes a very long process. On the contrary, the algorithms of Iemoto and Takahashi [14] and Osilike and Isiogugu [10] do not require the imposition on the control sequences for . Consequently, there is a need to extend the iteration schemes in [10, 14] for .

Motivated by the above observations and the algorithms of Iemoto and Takahashi [14] and Osilike and Isiogugu [10], which do not require the imposition on the control sequences for and the need to extend the iteration schemes for , the aim of this work is first to study some possible linear combinations of the products of the elements of a family of sequence of real numbers whose sum is unity. Second, to apply the result to construct a new (horizontal) algorithm which does not require the condition on the finite family of the control sequences . Third, to prove that the new algorithm converges weakly and strongly to an element in the intersection of the set of fixed points of a countable finite family of multivalued type-one demicontractive mappings. We also show that our algorithm is an extension of the algorithm of Osilike and Isiogugu [10] when . Furthermore, the algorithm is applied to establish some new algorithms for the approximation of the common solution of an equilibrium problem and a fixed point problem for a finite family of type demicontractive mappings. The numerical examples and computations of the horizontal algorithm were also presented. The obtained results complement, extend, and improve many results on the iteration schemes for the approximation of common fixed points for a finite family of single-valued and multivalued mappings.

2. Preliminaries

In the sequel, we shall need the following definitions and lemmas.

Definition 1 (see, e.g., [26–27]). Let be a Banach space and be a multivalued mapping. is at if for any sequence, such that converges weakly to and a sequence with for all such that strongly converges to . Then, (i.e., ).

Definition 2. A Banach space is said to satisfy Opial’s condition [28] if whenever a sequence weakly converges to , then it is the case thatfor all , .

Definition 3 (see [29]). A multivalued mapping is said to satisfy condition (1) (see, for example, [29]) if there exists a nondecreasing function with and for all such that

Definition 4 (see [4]). Let be a normed space and be a multivalued map. is of type-one if given any pair , then

Lemma 1 (see [30]). Let and be sequences of nonnegative real numbers satisfying the following relation:If , then exists.

3. Main Results

Let be a nonempty convex and closed subset of a real Hilbert space . Suppose that , is a countable finite family of mappings , and we consider the horizontal iteration process generated from arbitrary for the finite family of mappings , using a finite family of the control sequences as follows: For N = 2, For N = 3,

For arbitrary but finite ,

We now present the following results which are very useful in establishing our convergence theorems.

Proposition 1. Let be a countable subset of the set of real numbers , where is an arbitrary integer. Then, the following holds:

Proof. For ,We assume it is true for N and prove for N+1.

Remark 1. Proposition 1 holds if is replaced with .

Proposition 2. Let be a countable subset of the set of real numbers , where is a fixed nonnegative integer and is any integer with . Then, the following holds:

Proof. For , , and , the proofs follow from Remark 1 and Proposition 1, respectively.
We assume it is true for and . Now, for and ,

Proposition 3. Let , and be arbitrary elements of a real Hilbert space . Let be a fixed nonnegative integer and be such that . Let and be a countable finite subset of and , respectively. Define

Then,where , , and .

Proof. Using the well-known identity,which holds for all and for all , we prove by (i) direct computation and (ii) induction.
Observe that, for , . Consequently, by the direct computation, we haveTherefore, it holds for from direct computation.
Since induction holds for a fixed and each from direct computation, then it is true for . Thus, to prove by induction, we then assume that it is true for and prove for and . Fromwe have thatObserve thatAlso,Furthermore,It then follows from (34–37) thatWe now apply Propositions 2 and 3 to prove the following weak and strong convergence theorems for type-one demicontractive mappings.

Theorem 5. Let be a nonempty convex and closed subset of a real Hilbert space . Suppose that , is a countable finite family of type-one demicontractive mappings from into the family of all proximinal subsets of with contractive coefficients for each . Suppose that and for each , is weakly demiclosed at zero; then, the sequence of the horizontal algorithm defined byconverges weakly to , where for each and is a countable finite family of real sequences in [0, 1] satisfying the following:(i); , for each i.(ii), .(iii).Also, if, in addition, is L-Lipschitzian and satisfies condition (1) for each , then converges strongly to .

Proof. Setting , , , , and in Proposition 3, we obtainApplying type-one demicontractive condition on each , we obtainConsequently, if we set k = 1 in Proposition 2, we obtainFurthermore, condition (i) on the control sequences implies that exists; hence, is bounded. Similarly, conditions (ii) and (iii) imply that . Finally, the demiclosedness property of each , boundedness of , uniqueness of the limit of a weakly convergent sequence, and Opial property of a real Hilbert space guarantee the weak convergence of to . Also, since is L-Lipschitzian and satisfies condition (1) for each i, it then follows from standard argument that converges strongly to .

Remark 2. If and we set and , for all , , and , we obtainwhich was considered by Osilike and Isiogugu [10].

4. Applications

We now present the application of Theorem 5 in the construction of algorithms for approximating a common solution of an equilibrium problem and fixed point problem.

For solving the equilibrium problems for a bifunction , let us assume that satisfies the following conditions: (A1): for all (A2): is monotone, that is, , for all (A3): for each , (A4): for each , is convex and lower semicontinuous

Lemma 2 (see [31]). Let be a nonempty convex closed subset of a real Hilbert space and , a bifunction satisfying (A1)–(A4). Let and . Then, there exists such that

Lemma 3 (see [32]). Let be a nonempty convex closed subset of a real Hilbert space . Assume that satisfies (A1)–(A4). Let and . Define byThen, the following hold:(1) is single valued.(2) is firmly nonexpansive, that is, for any , .(3).(4) is convex and closed.

Lemma 4 (see [33]). Let be a nonempty convex closed subset of a real Hilbert space and , a bifunction satisfying (A1)–(A4). Let and . Then, for all and ,

Lemma 5. Let be a real Hilbert space, and let be a nonempty convex closed subset of . Let be the convex projection onto . Then, convex projection is characterized by the following relations:(i), for all .(ii).(iii).

Motivated by Algorithm 19 of Isiogugu et al. [34], we obtain the following result using a selection of Algorithm 4.2 above in the sense of [34].

Theorem 6. Let be a nonempty convex closed subset of a real Hilbert space , , a bifunction satisfying (A1)–(A4) and be such that is type-one -strictly pseudo-contractive-type mappings, and is weakly demiclosed at for each . Suppose that . Let be a sequence generated from arbitrary as follows:

Algorithm 1. where for each and is a finite family of real sequences in [0, 1] for each satisfying(i); , for each i.(ii), .(iii). Also, if, in addition, satisfies condition (1) for each ,(iv) for some .Then, converges strongly to .

Proof. It then follows that exists; hence, is bounded. Also,Thus, from (i), (ii), and (iii), we have that , for all . Furthermore, . Consequently, which implies that is a Cauchy sequence in . Also, since is convex and closed, converges strongly to some . From the Opial condition of and the demiclosedness property of , we have that , for all .
The remaining part of the proof is similar to the method of [34], Theorem 20. Therefore, it is omitted.

5. Examples

We present the numerical computation of the iteration scheme of Theorem 5.

Let with the usual norm on and partial order “” on , . Observe that is a convex closed linear total ordered subset of with if and only if for all . Denote the order interval by , and let be a countable infinite family of mappings and define for each and by

Clearly, for each , (I). (II). (III) , for all . (IV) . (V) . (VI) . (VII) . It then follows from (V) and (VII) that (VIII) . Also, from (V) and (VI), we obtain that (IX) , where is defined by .

In summary, for each , we have from (III), (VIII), and (IX) that is type-one demicontractive mapping with contraction coefficient and satisfies condition (1).

Observe that . Therefore, if we set and define bythen(i); .(ii), .(iii).

Table 1 and Figure 1 show the sequences for N = 5 and N = 10. The values are rounded up to 9 decimal places.

(a)

(b)

(c)

(d)

6. Conclusion

A horizontal iteration scheme for the approximation of a common fixed point of a finite family of mappings is introduced in a real Hilbert space. This algorithm does not require the imposition of sum = 1 on the control sequences. Its applicability in developing other algorithms is demonstrated in Algorithm 1. Furthermore, its computability is also exhibited in our numerical computations presented in Section 5.

Data Availability

All data generated or analyzed during this study are included in this published article.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

The first and third authors acknowledge with thanks the bursary and financial support from Department of Science and Technology and National Research Foundation, Republic of South Africa, Center of Excellence in Mathematical and Statistical Sciences (DST-NRF COE-MaSS) (Postdoctoral and Doctoral Bursary) respectively. Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS. This research was supported by CoE-MaSS BA2018-012.

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Copyright © 2020 F. O. Isiogugu 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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