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

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Introduction and Preliminaries
Optimization theory (convex, nonconvex, and discrete) is an important feld that has applications in almost every technical and nontechnical feld, including wireless communication, networking, machine learning, security, transportation systems, fnance (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 diferential 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 fxed-point theory.Let Z be a Banach space (BS), Z ⋆ the dual of Z, and ∅ ≠ D ⊂ Z is a closed and convex subset of Z. Te mapping J: Z ⟶ 2 Z ⋆ defned by is said to be normalized duality mapping.Let Γ: D ⟶ D be a nonlinear mapping.Te symbols N, R, ⟶ , ⇀, F(Γ) and F � ∩ N i�1 F(Γ i ) will be used to denote the set of natural numbers, the set of real numbers, strong convergence (SC), weak convergence (WC), the set of fxed points of Γ, and the set of common fxed points of Γ, respectively.Defnition 1. Recall that (a) A mapping Γ is said to be nonspreading if there exists j(s) ∈ J(s) such that, for all s ∈ D, ϕ(Γs, Γt) + ϕ(Γt, Γs) ≤ ϕ(Γs, t) + ϕ(Γt, s), (2) where ϕ(s, t) � ‖s‖ 2 − 2〈s, j(t)〉 + ‖t‖ 2 , for all s, t ∈ Z and J is the duality mapping on D. Note that in real Hilbert spaces (H), the J is an identity mapping and ϕ(ϖ, y) � ‖ϖ − y‖ 2 .Tus, in real Hilbert spaces, ( 2) is equivalent to ‖Γs − Γt‖ 2 ≤ ‖s − t‖ 2 + 2〈s − Γs, t − Γt〉.
In 2008, Kohasaka and Takahashi [1] established this class of mapping in a smooth, strictly convex, and refexive Banach space (RBS).(b) A mapping Γ is called asymptotically nonspreading (ANS) if there exists j(ϖ − y) ∈ J(ϖ − y) such that, for all s, t ∈ D, Naraghirad [2] established the class of ANS mapping as a generalization of the class of nonspreading mapping.In addition, he proved that if K is a nonempty closed convex subset of a real BS and Γ is an ANS mapping of K, then Γ has a fxed point.(c) A mapping Γ is said to be uniformly Lipschitzian with the Lipschitz constant L > 0 if � � � � ≤ L‖s − t‖, for all s, t ∈ D and n ∈ N. ( (d) A mapping Γ is called asymptotically strictly pseudonon-spreading if there exist where c � 1/2(1 − β) ∈ (0, 1), for allβ ∈ (0, 1) and σ n � 1/2 (1 + k n ).Observe that σ n ⟶ 1 as k n ⟶ 1 and n ⟶ ∞.
In a real Hilbert space (H) (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 β � 0 and k n � 1.Again, the class of k-asymptotically strictly pseudo-nonspreading mappings is more general than the classes of k-strictly pseudo-non-spreading mappings and k-asymptotically pseudocontractions (see [4], for more detail).
Observe that for all integer n ≥ 2, we have Clearly, Γ is asymptotically strictly pseudo-nonspreading mapping (see [3] for details)., for all s 1 , s 2 . . . ∈ Z, (10) and D � s � (s 1 , s 2 , . . ., s n , . ..)   be an orthogonal subspace of Z (i.e., for all ϖ, y ∈ D ⊂ Z, we have (〈s, t〉 � 0)).For each s � (s 1 , s 2 . . ., s n , . ..) ∈ D, defne the mapping Γ: D ⟶ D by Ten, Γ 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.
Te following defnition will be required in the sequel.

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Defnition 4 (see [6]).Let Z be a BS and ⋌: then it is said to be a nonexpansive retraction (non-ER) of Z.
Note that if ⋌: Z ⟶ D is a retraction, then ⋌ 2 � ⋌.A retract of a Hausdorf 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 H � R n with an inner product 〈s, t〉 �  n i�1 s i t i and the usual norm ‖s‖ � ( n i�1 s Ten, ⋌ is a non-ER of H onto D.
Defnition 5. Let D be a nonempty, closed, and convex subset of a BS Z and Γ: D ⟶ Z a non-self-mapping.Ten, (1) Γ is said to be ANS non-self-mapping if there exists j(ϖ) ∈ J(ϖ) such that, for all ϖ, y ∈ D, (2) Γ is uniformly Lipschitzian with the Lipschitz constant (3) Γ is said to be strictly asymptotically pseudo-nonspreading non-self-mapping if there exist where c � 1/2(1 − β) ∈ (0, 1), for all β ∈ (0, 1) and Note that if Γ is a self-mapping, then ⋌ becomes the identity mapping so that (15) reduces to (7).Te above study of various nonlinear mappings is quite interesting.However, if there is no means to approximate their respective fxed 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 fxed points of diferent nonlinear mappings.Chidume and Adamu [7] attained convergence via their modifed iteration scheme for the common solution of split generalized mixed equality equilibrium and split equality fxed-point problems.Tianwan [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 fxed-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 fxed-point iteration method, Saleem et al. [12,13] proved several fxed-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 diferential 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 selfmappings 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 fnite family of total ANESMs and fnite family of total ANENSMs in real UCBS.Albert et al. [20] did work on the approximation of fxed point of nonexpansive mappings.Agwu et al. [18] proved the convergence of a threestep iteration scheme to the common fxed 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 efective iteration techniques for fxed-point problems are readily available in [22] and [23].
Chidume et al. [16] established the following iterative scheme: where α n is a sequence in (0,1), D is a nonempty closed convex subset of a real UCBS Z, and ⋌ is a non-ER of Z onto D and proved several SC and WC theorems for ANENSMs in the context of UCBSs.
For the papers studied, it was discovered that a lot of attention has been given to fxed-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 fxed point for a fnite family of total ANESMs and ANENSMs, and Chima [25] examined fxed 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-nonspreading 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 fxed point for a fnite 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-selfmapping in the setup of real UCBSs.Also, an afrmative answer is given to (1) and (2) in Question 6.

Relevant Preliminaries
In this section, we shall use the following defnitions, lemmas, and known results in order to prove the main theorems of this paper: given a BS Z whose dimension is greater than or equal to 2. Te mapping δ Z (ε): (0, 2] ⟶ (0, 2] represented by for all s, t ∈ Z, is called the modulus of convexity of Z.Note that if δ E (ε) > 0, for all ε ∈ (0, 2], then Z is called uniformly convex.We recall the following defnitions and lemmas which will be needed in what follows. Defnition 7 (see [27]).Let Z be a BS, Z ⋆ its dual and V � s { ∈ Z: ‖s‖ � 1}.If lim n ⟶ ∞ ‖s + xt‖ − ‖ϖ‖/t exists for all s, t ∈V, then Z is given the Gateaux diferentiable norm.

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Journal of Mathematics Defnition 8 (see [27]).If the limit in Defnition 7 exists and is attained uniformly for each s ∈V (and for ally ∈V), then Z is given the Frechet diferentiable norm (see [28]  Defnition 10 (see [5]).Let Γ: D ⟶ D be a nonlinear mapping.Ten, Γ is said to be demiclosed at 0, if, for any sequence s n  ∈ D, the condition that s n ⇀s ∈ D and Γs n ⟶ 0 implies Γs � 0. Defnition 11.Let Z be a real BS.If, for every sequence s n ∈ Z, s n ⇀s and ‖s n ‖ ⟶ ‖s‖ imply ‖ϖ n − ϖ‖ ⟶ 0. Ten, Z is given the Kadec-Klec property [30].
Lemma 17 (see [34]).Let Z be a real UCBS and ∅ ≠ D ⊂ Z bounded close and convex.Ten, there exists a strictly increasing continuous convex function ϕ: Lemma 18 (see [26]).If the sequence ϖ n   ∞ n�1 WC to ϖ, then there exists a sequence of convex combination as n ⟶ ∞.

Main Results
Let Z a real normed space and ∅ ≠ D ⊂ Z be closed and convex.Let Γ i : D ⟶ Z be a fnite family of total asymptotically strictly pseudo-non-spreading non-self-mappings Journal of Mathematics and G i : D ⟶ D be a fnite family of total asymptotically strictly pseudo-non-spreading self-mappings.We defne an iterative scheme generated by ϖ n   n≥1 as follows: where Defnition 19.Let Z be an arbitrary BS and ∅ ≠ D ⊂ Z be closed and convex.Let Γ: D ⟶ Z be nonlinear mapping.Following the terminology of Alber et al. [20], Γ is called total asymptotically strictly pseudo-non-spreading if for every ϖ, y ∈ D, c ∈ (0, 1), and j(ϖ − y) ∈ J(ϖ − y), there exist sequences and a strictly increasing continuous function ϕ: R + ⟶ R + , R + denoting the set of positive real numbers, with ϕ(0) � 0 and lim If F(Γ) ≠ ∅ and q ∈ F(Γ), then (28) reduces to Lemma 20 (demiclosed principle for total asymptotically strictly pseudo-non-spreading non-self-maps).Let Z be a UCBS, ∅ ≠ D ⊂ Z be closed, convex, and bounded and Γ: D ⟶ Z be L-Lipschitz continuous and total asymptotically strictly pseudo-non-spreading mapping with ϕ: In fact, since ϖ n   n≥1 CW to ω, by Lemma 18 (see, e.g., [14]), we get that, for all n > 1, there exists a convex combination Since ϖ n − Γϖ n   converges to 0, it follows that for any positive integer m ≥ 1, and given any ϵ > 0, there exists Hence, for all n ≥ N 1 , using Defnition 19 and ⋌ is nonexpansive, we deduce, for any fxed k ≥ 1, utilizing the well-known inequality 6 Journal of Mathematics which holds for all ϖ, y ∈ E and for all j(ϖ + y) ∈ J(ϖ + y), we have From ( 31) and ( 33) and the condition on the function ϕ, we obtain In addition, Moreover, with the help of Lemma 17, and for all n ≥ N, there exists ϕ: [0, ∞) ⟶ [0, ∞) with ϕ(0) � 0 that is increasing function, and we obtain Equations ( 34), (35), and (37) imply that On the other hand, for any k ≥ 1, it follows that (using (34)) Taking lim sup k⟶∞ of both sides of (39), using (30) and for an arbitrary ϵ > 0, we deduce Tis completes the proof.

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Remark 21.Te result of Lemma 12 still holds true if λ � 1. Tus, Lemma 12 can as well serve as a proof for the demiclosedness principle for total asymptotically pseudocontractive non-self-mappings in UCBSs with λ � 1.

Lemma 22.
Let Z be a UCBS, ∅ ≠ D ⊂ Z be closed and convex, Γ i : D ⟶ Z be a fnite family of uniformly L ″ -Lipschitzian and total asymptotically strictly pseudo-non-spreading non-self-mappings with sequences μ n   n≥1 ,

fnite family of uniformly L ′ -Lipschitzian and total asymptotically strictly pseudo-non-spreading self-mappings with sequences c
Tere exist constants M ′ and M ″ and a strictly increasing and continuous functions ψ, ϕ: all q ∈ F, where ϖ n   n≥1 is as defned by (27).

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Proof.Set Suppose q∈ F is arbitrary, with the help of ( 27), we get By continuing in this manner, we obtain that 10

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Following the same method as above, we get In general, where Also, for i � 1, we obtain the following estimation using (27): By following the same method as above for i� 2, we get and in general, In addition, using (27) and Lemma 12, we obtain, for all q∈ F and i � 1, 2, . . ., m, that From ( 45) and (49) and the fact that 2ab ≤ a 2 + b 2 , we obtain Since each Γ i : D ⟶ Z and each G i : D ⟶ D, for i � 1, 2, . . ., m, is total asymptotically strictly pseudo-nonspreading mappings, the last inequality becomes Ten, we obtain from (51) that Observe that Tus, Tus, when n ≥ n 0 , we have 0 51)-(53), we have (57) Again, from (57) and Lemma 13, it follows that lim n ⟶ ∞ ‖ϖ n − q‖ exists so that there exists a constant Q such that ‖ϖ n − q‖ ≤ Q.
By utilizing the infmum for all q ∈ F in (57), we obtain and F be as stated in Lemma 13.If, in addition to the assumptions of Lemma 13, the following conditions are satisfed: Also, we have Furthermore, Terefore, from (59)-(61) and Lemma 22, we obtain From (62) and condition (ii Again, from (27), we get Journal of Mathematics Using (59) and (64), condition (b), Lemma 14, and following the same technique as above, we obtain Moreover, from (54), we get Similarly, we obtain from (54) that lim inf Now, from (65) and (66) and the inequality we obtain that Again, (48) and (74) and utilizing Also, from (71) and (72) and the inequality we obtain that Observe that 16 Journal of Mathematics Equations ( 68), ( 71), (74), and (75) imply that so that from (69), (74), and (77), we get Observe that so that, using (71) and (78), we have Tus, from condition (a) of Lemma 14 and (80), we obtain Again, observe that so that from (71) and (80), we get Also, so that from (71) and (81), we get lim Journal of Mathematics Now, we estimate ‖ϖ n − Γ i ϖ n ‖ as follows: From ( 71), (74), (81), and (82), we obtain Next, observe that By ( 80) and (81), again, observe that 18 Journal of Mathematics so that from (69), (74), and (88), we get and F be as stated in Lemma 13.Under the conditions of Lemma 13,forallξ , where ϖ n   is the sequence defned by (27) by Lemma 22.It remains, therefore, to prove Lemma 24 for u ∈ (0, 1).Now, for all ϖ ∈ D, defne Ten, it follows that ϖ n+1 � V n ϖ n , V n ξ � ξ, for all ξ∈ F. Now, from (57) of Lemma 22, we see that Ten, it follows from the standard argument that lim n ⟶ ∞ a n (u) exists; i.e., lim and F be as stated in Lemma 13.If, in addition to the assumptions of Lemma 13, E has Frechet diferentiable norm, then, for all ξ i , J ξ j ∈ F, i, j � 1, 2, . . ., m; i < j, the limit lim n ⟶ ∞ 〈(ϖ n , J(ξ i − ξ j )〉 exists, where ϖ n   is the sequence defned by (27).If ω ω s n   denotes the set of all weak subsequential limits of ϖ n  ,
Hence, ‖η ⋆ − ξ⋆‖ 2 � 〈η ⋆ − ξ ⋆ , J(η ⋆ − ξ ⋆ )〉 � 0. Tus, p ⋆ � q ⋆ .Terefore, ϖ n   WC to a common fxed point of F. Tis completes the proof.N i�1 and F be as stated in Lemma 13.If, in addition to the assumptions of Lemma 13, the space Z ⋆ of Z has the Kadec-Klec (KK) property and the mappings I − G i and I − Γ i for i � 1, 2, . . ., m∈ N, where I is an identity mapping, are demiclosed at zero, then the sequence ϖ n   described by (27) WC to a common fxed point in F.
Proof.By Lemma 14, ϖ n   is bounded and Z is refexive, there exists a subsequence ϖ n k   of ϖ n   which WC to some η ⋆ ∈ K.With the help of Lemma 14, we deduce lim By the assumptions, the mappings I − G i and I − Γ i for i � 1, 2, . . ., m∈ N, where I is an identity mapping, are demiclosed at zero, boundedness of ϖ n  , and the uniqueness of the limit of the weakly convergence sequence follows that the sequence ϖ n   WC to q ⋆ ∈ F. Tis completes the proof.

Conclusion
In this manuscript, (1) We established a new fxed-point algorithm for approximating the common fxed point of fnite families of L-Lipschitzian and total asymptotically strictly pseudo-non-spreading self-mappings and L-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-nonspreading 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 modifcation 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

Example 2 .
Let Z � l 2 with the usual norm ‖.‖ defned by

□ Theorem 26 .
Let Z, D, Γ i   N i�1 , G i   N i�1and F be as stated inLemma 13.If, in addition to the assumptions of Lemma 13, Z has Frechet diferentiable norm, then the sequence ϖ n