New Developments in Fixed Point Theory and ApplicationsView this Special Issue
Research Article | Open Access
F. O. Isiogugu, C. Izuchukwu, C. C. Okeke, "New Iteration Scheme for Approximating a Common Fixed Point of a Finite Family of Mappings", Journal of Mathematics, vol. 2020, Article ID 3287968, 14 pages, 2020. https://doi.org/10.1155/2020/3287968
New Iteration Scheme for Approximating a Common Fixed Point of a Finite Family of Mappings
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.
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  using the terminology of Browder and Petryshen  for single-valued pseudo-contractive mapping and Markin  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.  if given any pair , thenwhere . is called a multivalued demicontractive in the sense of Isiogugu and Osilike  using the terminology of Hicks and Kubicek  for single-valued demicontractive if and for all and , there exists such that
Furthermore, every multivalued strictly pseudo-contractive-type in the sense of  with the nonempty set of strict fixed points is demicontractive with respect to its set of strict fixed points.
Observe that if is nonspreading and , then is quasi-nonexpansive. is -strictly pseudo-nonspreading in the sense of Osilike and Isiogugu  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  (see also ).
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  using a Halpern-type iterative process considered in  for the approximation of a common fixed point of a finite family of nonexpansive self-mappings. He proved the following theorem.
Theorem 1 (see , 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, ) and references therein.
In 2008, Zhang and Guo  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 , 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,  and references therein).
In , 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 , 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 , 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  and Osilike and Isiogugu  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  and Osilike and Isiogugu , 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  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.
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  if whenever a sequence weakly converges to , then it is the case thatfor all , .
Definition 4 (see ). Let be a normed space and be a multivalued map. is of type-one if given any pair , then
Lemma 1 (see ). 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:
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) that