Inertial iterative method for solving generalized equilibrium, variational inequality, and ﬁxed point problems of multivalued mappings in Banach space

We devise an iterative algorithm incorporating inertial techniques to approximate the shared solution of a generalized equilibrium problem, a ﬁxed point problem for a ﬁnite family of relatively nonexpansive multivalued mappings


Introduction
Consider a real Banach space E with dual E * , and let D be a nonempty closed convex subset of E. The normalized duality mapping J : E → 2 E * is defined by J(w) = {w 0 ∈ E * : w 0 , w = w 2 = w 0 2 } for all w ∈ E. The fixed point of a multivalued mapping S : D → 2 D , where 2 D denotes the power set of D, is a point w ∈ D such that w ∈ Sw.
If we set G(w 0 , v) = hw 0 , vw 0 , where h : D → D is a nonlinear mapping, then GEP(1.1)transforms into the variational inequality problem (VIP) hw 0 , vw 0 ≥ 0, ∀v ∈ D, (1.3) introduced by Hartmann and Stampacchia [12].The solution of (1.3) is denoted as Sol(VIP).The generalized equilibrium problem plays a pivotal role in various scientific and engineering domains, providing a natural and unified framework for problems in nonlinear analysis, optimization, economics, finance, game theory, physics, and engineering; see [1,27,29].
In 1973, Markin [19] introduced the fixed point problem (FPP) for multivalued nonexpansive mappings, which has found extensive applications in various fields such as convex optimization and control theory, as illustrated in [11,16,20,26].In 2011, Homaeipour et al. [13] presented an iterative algorithm involving relatively nonexpansive multivalued mapping S: Under certain conditions on the control sequence, Homaeipour et al. observed the convergence of the sequence {w n }.More recently, Zegeye et al. [32] investigated an iterative method to approximate the common solution of the equilibrium problem (EP) and the fixed point problem (FPP) for relatively nonexpansive multivalued mappings, providing a convergence analysis under appropriate parameters.Very recently, Taiwo et al. [28] introduced the following Halpern-S-iteration method: r n qu n , Ju n -Jw n ≥ 0 for all q ∈ D, They aimed to approximate the common solution of the EP and FPP for relatively nonexpansive multivalued mappings within uniformly convex and uniformly smooth Banach spaces.Moreover, they established strong convergence under appropriate conditions on the parameters.
An effective strategy for accelerating the convergence rate of iterative algorithms is to integrate an inertial term into the iterative scheme.This term γ n (snsn -1) serves as a powerful tool to enhance algorithm performance, showcasing favorable convergence characteristics.The concept of the inertial extrapolation method was initially introduced by Polyak [23] and inspired by an implicit discretization of a second-order-in-time dissipative dynamical system known as the "Heavy Ball with Friction." In 2008, Mainge [17] introduced the following inertial Krasnosel'skiǐ-Mann algorithm by integrating the Krasnosel'skiǐ-Mann algorithm with inertial extrapolation: He demonstrated that the sequence {s n } generated by the algorithm converges weakly to a fixed point of S under certain conditions on parameters.This has sparked growing interest among authors working in this area, as evidenced in works such as [2,4,[6][7][8][9]14].
Question: Could we apply the inertial technique involving projection method for solving GEP, VIP, and FPP for relatively nonexpansive multivalued mapping in the setting of a 2uniformly convex and uniformly smooth Banach space?
Explanations: Certainly!The inertial technique, when integrated with projection methods, is applicable to address the GEP, VIP, and FPP associated with relatively nonexpansive multivalued mappings in the context of a 2-uniformly convex and uniformly smooth Banach space.The inherent 2-uniform convexity and uniform smoothness properties of the Banach space create favorable conditions for the utilization of these techniques, leading to improved convergence behavior of iterative algorithms.
The inertial technique, characterized by its incorporation of an extrapolation term, is renowned for its capacity to expedite convergence in iterative approaches.When coupled with projection methods, it proves especially advantageous in solving complex problems involving multivalued mappings, equilibrium problems, variational inequalities, and fixed point problems.
Inspired by the contributions of Taiwo et al. [28], Zegeye et al. [32], Mainge [17], and Farid et al. [9], we present a novel iterative algorithm employing the inertial technique.This algorithm aims to determine the common solution of the generalized equilibrium problem (GEP), variational inequality problem (VIP), and fixed point problem (FPP) for relatively nonexpansive multivalued mappings.We delve into the strong convergence properties of our proposed method, highlighting specific aspects of our theorem.Additionally, we provide a computational analysis to underscore the significance of our findings and draw comparisons.The results presented in this paper serve to extend and unify numerous previously established outcomes in this particular research domain.
Our paper is organized as follows: In Sect.2, we offer basic concepts, essential lemmas, and underlying assumptions.Section 3 encompasses our main results, numerical analysis, and graphical presentations.In Sect.4, we delve into the interpretation of our results.

Preliminaries
Here we present a brief overview of some essential concepts that will be utilized in the subsequent discussion.The modulus of smoothness on the set D is represented by the mapping D : [0, ∞) → [0, ∞), defined as follows: If D (ϑ) > 0 for all ϑ > 0, then D is termed smooth, and it is uniformly smooth if and only if lim s→0 + D (s) s = 0.The strict convexity of D is characterized by the condition |w 1 +w 2 | 2 < 1 for all w 1 , w 2 ∈ U with w 1 = w 2 , where U = {w ∈ D : w = 1}.
The modulus of convexity on D is the map δ D : (0, 2] → [0, 1] defined as follows: A space E is uniformly convex if and only if δ D (ε) > 0 for all ε ∈ (0, 2].In the context of a space E, it is said to be p-uniformly convex if there exists a constant c p > 0 such that δ D (ε) ≥ c p for all ε ∈ (0, 2], as outlined in [30].For more detailed information, we refer to the cited source.
The Lyapunov function φ : E × E → R is defined by It is worth noting that the characterization of the metric projection on a subset of a Hilbert space as nonexpansive is specific to Hilbert spaces and is not readily applicable to more general Banach spaces.In addressing this limitation, Alber [3] introduced an operator in a Banach space known as the generalized projection, as further discussed in [24].
An element w 0 ∈ D is referred to as an asymptotic fixed point of S : D → D if there exists a sequence {w n } ⊂ D with w n w 0 such that lim n→∞ Sw nw n = 0. We denote the set of asymptotic fixed points as F(S).A map S is considered relatively nonexpansive if F(S) = F(S) = ∅ and φ(w 0 , Sw) ≤ φ(w 0 , w), ∀w ∈ D, w 0 ∈ F(S).
Consider N(D) = ∅ as a family of subsets of D, and CB(D) = ∅ as a family of closed bounded subsets of D. The Hausdorff metric, denoted as An element w 0 ∈ D is considered an asymptotic fixed point if there exists a sequence {w n } ⊂ D such that w n w 0 and lim n→∞ d(Sw n , w n ) = 0.
A map S is said to be relatively nonexpansive if F(S) = F(S) = ∅ and φ(w 0 , s) ≤ φ(w 0 , v) for all v ∈ D, s ∈ Sv, and w 0 ∈ F(S).It is worth noting that Homaeipour et al. [13] provided a counterexample for a relatively nonexpansive multivalued mapping that is not nonexpansive.

Definition 2.1 A map
Lemma 2.1 [15] Consider a smooth uniformly convex Banach space E and two sequences Lemma 2.4 [3] In a reflexive strictly convex smooth Banach space E, with D being a nonempty closed convex subset of E, the following inequality holds for all w ∈ D and v ∈ E: Lemma 2.5 [3] In a reflexive strictly convex Banach space E, considering a nonempty closed convex subset D of a smooth Banach space E, and given w ∈ E and z ∈ D, we have the following equivalence: Lemma 2.6 [31] For a closed ball E R (0) of a uniformly convex Banach space E, there exists a continuous strictly increasing convex function g where Lemma 2.7 [18] Let {c n } be a sequence of real numbers that is nondecreasing at infinity.Then there exists a subsequence {c n i } of {c n } such that c n i < c n i+1 for all i ∈ N. Additionally, for a nondecreasing sequence Lemma 2.8 [22] Let {c n } be a sequence of nonnegative real numbers satisfying where γ n ∈ (0, 1) and ξ n ∈ R with lim Lemma 2.9 [25] Let E be a p-uniformly convex Banach space.Then the relation between the metric and Bregman distance is for all w, v ∈ E, where π p is a fixed positive number.Moreover, using Young's inequality, for all p, q > 1 such that 1 p + 1 q = 1, we have

Lemma 2.10 [30]
In a 2-uniformly convex Banach space E, Jw -Jv for all w, v ∈ E, where 0 < c ≤ 1; c is referred to as the 2-uniformly convex constant of E.

Main outcome
Let G, ε : D × D → R be bifunctions, and let h : E → E * be a nonlinear mapping.Let S i : D → CB(D), i = 1, 2, 3, . . ., N, represent a finite family of multivalued mappings.We now present our algorithm.
Iterative Steps: Iterate s n+1 using the following procedure: Step 1.For n ≥ 1 and γ > 0, given the iterates s n and s n-1 , choose γ n such that Step 2. Compute If t n = z n for some n ≥ 1, then stop and provide the solution to VIP(1.3).Otherwise, set n := n + 1.
Step 3. Compute where r n is defined in (2.4).Termination condition.If s n+1 = z n = t n and S i u n = u n for each i = 1, 2, 3, . . ., N, then stop.Otherwise, set n := n + 1 and move to Step 1.
We consider control parameters in our main theorem to be γ n , λ n ∈ (0, 1), r n ∈ (0, ∞), Proof We state that the sequence {s n } is bounded.Consider any q ∈ .Using properties of φ, we estimate Using Lemmas 2.4, 2.5, and 2.10, we compute Since z n = J -1 (Js n + γ n (Js n -Js n-1 )), by using Lemma 2.9 we estimate Using property (L3) of φ, we get Combining (3.5) and (3.6), we get Next, using (3.7), we compute Using induction, we get This concludes that {s n } is bounded, and consequently, {z n }, {t n }, {u n }, and {v n } are also bounded.
Next, we show that q ∈ and s n → q.Setting ρ Using the concept of S i and Lemmas 2.6, 2.11, and (3.9), we compute We are evaluating two scenarios outlined below.Case 1. Assume that for some m 0 ∈ N, φ(q, s n ) is nonincreasing for all n ≥ m 0 , and since φ(q, s n ) is bounded, it must be convergent.Therefore by utilizing (3.10) it follows that φ(s n , z n ) → 0 and φ(u n , v n ) → 0 as n → ∞.Additionally, according to (2.1), we obtain (3.12) Also, by (3.10), δ n,0 δ n,i g( Jv n -Jw n,i ) → 0 as n → ∞, which yields that Jv n -Jw n,i → 0, and thus by the uniform continuity of J -1 we have Using (3.1) and (3.2), we get which yields that Using (2.3) and Lemma 2.10, we get and by (3.15) By Lemma 2.1 Applying Lemmas 2.4 and 2.5, we compute which implies that Thus for each i = 1, 2, . . ., N, we have Let {ρ n i } be a subsequence of {ρ n } such that ρ n i ρ and sup n→∞ ρ nq, Jz n -Jq = lim i→∞ ρ n iq, Jz n i -Jq .Thus by (3.18), (3.20), and the concept of J we get Next, we show that q ∈ Sol(VIP(1.3)).Applying the concept of σ -ism mapping of h, by (3.15) and (3.12) we obtain lim n→∞ s n = q and q ∈ h -1 (0).Hence q ∈ Sol(VIP(1.3)).Further, we need to show that q ∈ Sol(GEP(1.1)).Since v n = r n u n , for each i = 1, 2, . . ., N, we get Let y s = (1s)q + sy, ∀s ∈ (0, 1].Since y ∈ D and q ∈ D, we get y s ∈ D, and hence Using the concept of ε and G, we have G(q, y s ) + ε(y s , q)ε(q, q) ≥ 0 ε(y s , q)ε(q, q) ≥ G(y s , q).
In a similar vein, we proceed to enumerate some corollaries derived from the implications of Theorem 3.1.This enumeration not only serves as a concise summary of the theoretical outcomes but also lays the groundwork for further exploration and application of the proposed iterative scheme in diverse mathematical and computational contexts.
For the computation and graphical representation of the proposed and Mainge algorithms, we use same initial points (s 0 , s 1 ), whereas for Homaeipur et al., we use s 1 .The stopping criterion for our computation is s n+1s n < 10 -10 .The computation and comparison graphs are shown in Table 1 and Figs.1-4, respectively.

Conclusions
In conclusion, our investigation has yielded several key findings.The proposed algorithm, presented in this work, demonstrates strong convergence to a solution in 2-uniformly convex and uniformly smooth real Banach space setting with relatively nonexpansive multivalued mapping.The theoretical results are supported by numerical experiments, where we employed Matlab R2015(a) for computation and compared our findings with existing algorithms, particularly those proposed by Homaeipour et al. and Mainge.The use of consistent initial points and a specified stopping criterion allowed for a fair comparison across different algorithms.The results presented in Table 1 and Figs.1-4 showcase the effectiveness of our approach in terms of convergence behavior.These findings contribute to the ongoing research in optimization algorithms and provide valuable insights into the applicability of the proposed method in various contexts.

Corollary 3 . 1 Corollary 3 . 2
Let E be a 2-uniformly convex and uniformly smooth real Banach space with dual space E * , and let D be a nonempty closed convex subset of E. Consider bifunctions G, ε : D × D → R that satisfy Assumption 2.1, and let h : E → E * be a σ -ism mapping, where σ ∈ (0, 1).Additionally, let S : D → CB(D) be a relatively nonexpansive multivalued mapping.Suppose := F(S) ∩ Sol(GEP(1.1))∩ Sol(VIP(1.3))= ∅.Then the sequence {s n } generated by Algorithm 3.1 converges strongly to x * ∈ , where x * = s 0 .Continuing in the same vein, we explore further implications and consequences arising from the conditions established in Theorem 3.1 when G and ε are specifically assumed to be zero.Let E bedenote a 2-uniformly convex and uniformly smooth real Banach space, with dual space E * , and let D be a nonempty, closed, and convex subset of E. Consider bifunctions G, ε : D × D → R that satisfy Assumption 2.1, and let h : E → E * be a σ -ism mapping, where σ ∈ (0, 1).Additionally, let S : D → CB(D) be a relatively nonexpansive multivalued mapping.Suppose := ∩_i = 1 N F(S i ) ∩ Sol(VIP(1.3))= ∅.Then, the sequence {s n } generated by Algorithm 3.1 converges strongly to x * ∈ , where x * = s 0 .Remark 3.1 If E is a Hilbert space H, then E * = H, J = I, the identity mapping, φ(u, v) = uv 2 , ∀u, v ∈ E, c = 1, 2-uniformly convex constant of E, D = P D , the metric projection onto D, and a relatively nonexpansive mapping is nonexpansive.These simplifications result from the specific properties and structures of Hilbert spaces, making certain operations and concepts more straightforward.
[13]a 2.2[21]Let D be a nonempty closed convex subset of a real Banach space E, and let h be a monotone hemicontinuous mapping from D into E * .Then the solution set of the VIP (1.3), denoted as VIP(D, h) = Sol(VIP(1.3)),isclosedand convex.Lemma 2.3[13]Consider a strictly convex and smooth Banach space E and a nonempty closed convex subset D of E. Let S : D → CB(D) be a relatively nonexpansive multivalued mapping.Then the fixed point set F(S) is closed and convex.

Table 1
Comparison of Algorithms