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 ςin 􏼈 􏼉 ∞ n�1 􏼈 􏼉 N i�1, the condition that 􏽐 N i�1 ςin � 1, for each n≥ 1. Furthermore, under appropriate conditions, the horizontal algorithm converges both weakly and strongly to a common fixed point of a finite family of type-one demicontractivemappings. 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.


Introduction
Let Y be a nonempty set and S: Y ⟶ Y be a mapping. A point y ∈ Y is called a fixed point of S if y � Sy. If S: Y ⟶ 2 Y is a multivalued mapping, then y is a fixed point of S if y ∈ Sy. y is called a strict fixed point of S if Sy � y . e set F(S) � y ∈ D(S): y ∈ Sy (respectively, F(S) � y ∈ D(S): y � Sy ) is called the set of fixed points of the multivalued (respectively, single-valued) mapping S, while the set F s (S) � y ∈ D(S): Sy � y is called the set of strict fixed points of S.
Let Y be a normed space. A subset K of Y is called proximinal if for each y ∈ Y, there exists k ∈ K such that ‖y − k‖ � inf ‖y − w‖: w ∈ K � d(y, K). (1) 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 Y by CB(Y), the family of all nonempty subsets of Y by 2 Y , and the family of all proximinal subsets of Y by P(Y), for a nonempty set Y.
Let D denote the Hausdorff metric induced by the metric d on Y, that is, for every A, B ∈ CB(Y), (2) Let Y be a normed space and S: D(S) ⊆ Y ⟶ 2 Y be a multivalued mapping on Y. S is called L − Lipschitzian if there exists L ≥ 0 such that, for all x, y ∈ D(S), In (3), if L ∈ [0, 1), then S is a contraction, while S is nonexpansive if L � 1. S is called quasi-nonexpansive if F(S) ≠ ∅ and for all p ∈ F(S), Clearly, every nonexpansive mapping with the nonempty fixed point set is quasi-nonexpansive. e multivalued mapping S is k-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 k ∈ [0, 1) such that given any pair x, y ∈ D(S) and u ∈ Sx, there exists v ∈ Sy satisfying ‖u − v‖ ≤ D(Sx, Sy) and If k � 1 in (5), then S is pseudo-contractive-type, while S is nonexpansive-type if k � 0. Every multivalued nonexpansive mapping S: D(S) ⊆ Y ⟶ P(Y) is nonexpansivetype. S is of type-one in the sense of Isiogugu et al. [4] if given any pair x, y ∈ D(S), then ‖u − v‖ ≤ D(Sx, Sy), for all u ∈ P S x, v ∈ P S y, (6) where P S x: � u ∈ Sx: ‖u − x‖ � d(x, Sx) { }. S 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 F(S) ≠ ∅ and for all p ∈ F(S) and x ∈ D(S), there exists k ∈ [0, 1) such that If k � 1 in (7), S is hemicontractive in the terminology of Naimpally and Singh [7] for single-valued hemicontractive, while S is quasi-nonexpansive if k � 0.
Furthermore, every multivalued k− strictly pseudocontractive-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 S: D(S) ⊆ H ⟶ H is called nonspreading in the sense of Kohsaka and Takahashi [8,9] Observe that if S is nonspreading and F(S) ≠ ∅, then S is quasi-nonexpansive. S is k-strictly pseudo-nonspreading in the sense of Osilike and Isiogugu [10] if there exists k ∈ [0, 1) such that for all x, y ∈ D(S). Clearly, every nonspreading mapping is k-strictly pseudo-nonspreading. If S is k− strictly pseudononspreading and F(S) ≠ ∅, then S 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 S i N i�1 , where N ∈ N (the set of nonnegative integers) (see, for example, [12][13][14][15][16][17][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]). e 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], eorem 3.1). Let K be a nonempty convex closed subset of a real Hilbert space H and S 1 , S 2 , . . . , S N be a finite family of nonexpansive mappings of Given points u, x 0 ∈ K, let x n be generated by where S n ≔ S n(modN) and ς n ⊂ (0, 1) satisfies n≥1 |ς n+N − ς n | < ∞. en, x n converges strongly to P F u, where P F : H ⟶ F is the metric projection.
e above algorithm of Bauschke was extended to approximate the family of more general class of strictly pseudocontractive 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. ey obtained the following theorem.
Theorem 2 (see [25], eorem 4.3). Let E be a real quniformly smooth Banach space which is also uniformly convex and K be a nonempty convex closed subset of E. Let N ≥ 1 be an integer, and for each be the sequence generated by the algorithm: Let ς n ∞ n�1 be a real sequence satisfying the conditions en, x n 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 S i N i�1 of k i -strictly pseudo-contractive mappings defined by where ς i n ∞ n�1 ⊆ (0, 1) for each i and N i�0 ς i n � 1 for each n (see, for example, [13] and references therein).
In [14], Iemoto and Takahashi studied the approximation of common fixed points of a nonexpansive self-mapping T and a nonspreading self-mapping S in a real Hilbert space. If T, S: C ⟶ C are, respectively, nonexpansive and nonspreading mappings, they considered the iterative scheme x n ∞ n�1 generated from arbitrary x 1 ∈ C by 2 Journal of Mathematics x n+1 � 1 − ς n x n + ς n β n Sx n + 1 − β n Tx n , n ≥ 1, where ς n and β n are suitable sequences in [0, 1]. ey proved the following main theorem: Theorem 3 (see [14], eorem 4.1). Let H be a real Hilbert space. Let C be a nonempty convex and closed subset of H. Let S be a nonspreading mapping of C into itself and T a nonexpansive mapping of C into itself such that F(T) ∩ F(S) ≠ ∅. Define a sequence x n ∞ n�1 in C as follows: . en, the following hold:

Motivated by the above result, Osilike and Isiogugu obtained the following result.
Theorem 4 (see [10], eorem 3.1.1). Let C be a nonempty convex closed subset of a real Hilbert space, and let T: C ⟶ C be a k-strictly pseudo-nonspreading mapping with a nonempty fixed point set F(T). Let β ∈ [k, 1), and let ς n ∞ n�1 be a real sequence in [0, 1) such that lim n⟶∞ ς n � 0. Let x n ∞ n�1 and z n ∞ n�1 be sequences in C generated for arbitrary x 1 ∈ C by x n+1 � ς n x n + 1 − ς n βx n +(1 − β)Tx n , n ≥ 1, x k , n ≥ 1.
We observed that all the existing iteration schemes for the approximation of a common fixed point of a finite family T 1 , T 2 , . . . , T N of mappings for N > 2, which are related to the parallel algorithm, require the condition that, for each n, N i�1 ς i n � 1 on the control sequences ς i n ∞ n�1 N i�1 . However, in real-life applications, if N 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 N i�1 ς i n � 1 on the control sequences for N � 2. Consequently, there is a need to extend the iteration schemes in [10,14] for N > 2.
Motivated by the above observations and the algorithms of Iemoto and Takahashi [14] and Osilike and Isiogugu [10], which do not require the imposition N i�1 ς i n � 1 on the control sequences for N � 2 and the need to extend the iteration schemes for N > 2, 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 ς i n ∞ n�1 N i�1 whose sum is unity. Second, to apply the result to construct a new (horizontal) algorithm which does not require the condition N i�1 ς i n � 1 on the finite family of the control sequences ς i n ∞ n�1 N i�1 . ird, 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 N � 2. 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.
e numerical examples and computations of the horizontal algorithm were also presented. e 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.

Preliminaries
In the sequel, we shall need the following definitions and lemmas.
Definition 1 (see, e.g., [26][27]). Let Y be a Banach space and S: D(S) ⊆ Y ⟶ 2 Y be a multivalued mapping. I − S is weakly demiclosed at zero if for any sequence, x n ∞ n�1 ⊆ D(S) such that x n converges weakly to p and a sequence y n with y n ∈ Sx n for all n ∈ N such that x n − y n strongly converges to zero. en, p ∈ Sp (i.e., 0 ∈ (I − S)p).

Definition 2.
A Banach space Y is said to satisfy Opial's condition [28] if whenever a sequence x n weakly converges to x ∈ Y, then it is the case that for all y ∈ Y, y ≠ x.
Definition 3 (see [29]). A multivalued mapping S: C ⟶ P(C) is said to satisfy condition (1) (see, for example, [29]) if there exists a nondecreasing function Definition 4 (see [4]). Let Y be a normed space and S: D(S) ⊆ Y ⟶ 2 Y be a multivalued map. S is of type-one if given any pair x, y ∈ D(S), then

Journal of Mathematics
Lemma 1 (see [30]). Let a n and c n be sequences of nonnegative real numbers satisfying the following relation: a n+1 ≤ a n + c n , ∀n ∈ N. (20) If c n < ∞, then lim n⟶∞ a n exists.

Main Results
Let K be a nonempty convex and closed subset of a real Hilbert space H. Suppose that S i N i�1 , N ≥ 2 is a countable finite family of mappings S i : K ⟶ K, and we consider the horizontal iteration process generated from arbitrary x 1 for the finite family of mappings S i N i�1 , using a finite family of the control sequences ς i n ∞ n�1 N i�1 as follows: For arbitrary but finite N ≥ 2, We now present the following results which are very useful in establishing our convergence theorems.
is an arbitrary integer. en, the following holds: Proof. For N � 2, We assume it is true for N and prove for N+1.
en, the following holds: Proof. For k � 0, N � 1, and k � 1, N � 2, the proofs follow from Remark 1 and Proposition 1, respectively. We assume it is true for k and N. Now, for k and N + 1, en, 4 Journal of Mathematics Proof. Using the well-known identity, which holds for all x, y ∈ H and for all t ∈ [0, 1], we prove by (i) direct computation and (ii) induction.
Observe that, for Consequently, by the direct computation, we have erefore, it holds for k, N from direct computation. Since induction holds for a fixed k and each N from direct computation, then it is true for k, N � 1, 2, 3. us, to prove by induction, we then assume that it is true for k, N and prove for k and N + 1. From we have that Observe that Journal of Mathematics (37) It then follows from (34-37) that We now apply Propositions 2 and 3 to prove the following weak and strong convergence theorems for type-one demicontractive mappings.
Also, if, in addition, S i is L-Lipschitzian and satisfies condition (1) for each i, then x n converges strongly to q ∈ ∩ N i�1 F(S i ).
Proof. Setting x n+1 � y, x n � t, p � u, k � 1, and y n,N ∈ S N x n � v in Proposition 3, we obtain Applying type-one demicontractive condition on each S i , we obtain 8

Journal of Mathematics
Consequently, if we set k � 1 in Proposition 2, we obtain Furthermore, condition (i) on the control sequences implies that lim n⟶∞ ‖x n − p‖ exists; hence, x n is bounded. Similarly, conditions (ii) and (iii) imply that lim n⟶∞ ‖x n − y n,i ‖0, i � 1, 2, . . . , N. Finally, the demiclosedness property of each (I − S i ), boundedness of x n , uniqueness of the limit of a weakly convergent sequence, and Opial property of a real Hilbert space guarantee the weak convergence of x n to q ∈ ∩ N i�1 F(S i ). Also, since S i is L-Lipschitzian and satisfies condition (1) for each i, it then follows from standard argument that x n converges strongly to q ∈ ∩ N i�1 F(S i ).

Applications
We now present the application of eorem 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 F: C × C ⟶ R, let us assume that F satisfies the following conditions: h↦F(g, h) is convex and lower semicontinuous Lemma 2 (see [31]). Let C be a nonempty convex closed subset of a real Hilbert space H and F: C × C ⟶ R, a bifunction satisfying (A1)-(A4). Let r > 0 and g ∈ H. en, there exists z ∈ C such that Lemma 3 (see [32]). Let C be a nonempty convex closed subset of a real Hilbert space H. Assume that F: C × C ⟶ R satisfies (A1)-(A4). Let r > 0 and g ∈ H. Define T r : H ⟶ 2 C by en, the following hold: (1) T r is single valued.
(2) T r is firmly nonexpansive, that is, for any g, h ∈ H, ‖T r g − T r h‖ 2 ≤ 〈T r g − T r h, g − h〉.

(3) F(T r ) � EP(F). (4) EP(F) is convex and closed.
Lemma 4 (see [33]). Let C be a nonempty convex closed subset of a real Hilbert space H and F: C × C ⟶ R, a bifunction satisfying (A1)-(A4). Let r > 0 and g ∈ H. en, for all g ∈ H and p ∈ F(T r ), Lemma 5. Let H be a real Hilbert space, and let C be a nonempty convex closed subset of H. Let P C be the convex projection onto C. en, convex projection is characterized by the following relations: 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 C be a nonempty convex closed subset of a real Hilbert space H, f: C × C ⟶ R, a bifunction satisfying (A1)-(A4) and T i N i�1 be such that T i : C ⟶ P(C) is type-one λ i -strictly pseudo-contractive-type mappings, and (I − T i ) is weakly demiclosed at zero for each i � 1, 2, . . . , N.
Let x n be a sequence generated from arbitrary x 0 ∈ C as follows: 1 − ς n,j y n,N , u n ∈ K such that F u n , y + 1 r n 〈y − u n , u n − y n 〉 ≥ 0, ∀y ∈ K, where y n,i ∈ T i x n for each i and ς n,i ∞ n�1 N i�1 is a finite family of real sequences in [0, 1] for each i satisfying Also, if, in addition, T i satisfies condition (1) for each i, (iv) r n ⊂ [a, ∞) for some a > 0.
en, x n converges strongly to p ∈ F .
It then follows that lim n⟶∞ ‖x n − p‖ exists; hence, x n is bounded. Also, us, from (i), (ii), and (iii), we have that lim n⟶∞ ‖x n − y n,i ‖ � 0, for all i � 1, 2,.. ., N. Furthermore, lim n⟶∞ ‖x n − u n ‖ � 0. Consequently, lim n⟶∞ ‖x n+1 − x n ‖ 2 � lim n⟶∞ ‖1/2(x n − u n )‖ 2 � 0 which implies that x n is a Cauchy sequence in K. Also, since K is convex and closed, x n converges strongly to some q ∈ K. From the Opial condition of H and the demiclosedness property of T i , we have that q ∈ T i q, for all i − 1, 2,... , N. e remaining part of the proof is similar to the method of [34], eorem 20. erefore, it is omitted.

Examples
We present the numerical computation of the iteration scheme of eorem 5.
Let H � (R m , ‖.‖, ≤ ) R m with the usual norm "‖.‖" on R m and partial order " ≤ " on R, Observe that (C, ‖.‖, ≤ ) is a convex closed linear total ordered subset of R n with a ≤ b if and only if a t ≤ b t for all t � 1, 2, 3, . . . , m. Denote the order interval a ≤ x ≤ b by [a, b], and let S i ∞ i�1 be a countable infinite family of mappings and S i : C ⟶ CB(C) define for each i and x ∈ C by Clearly, for each i, (III) ‖u − v‖ � (3i/(2i + 1))‖x − y‖ ≤ D(S i x, S i y), for all u ∈ P S i x, v ∈ P S i y.

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. is 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
e authors declare that there are no conflicts of interest.