Inertial Method for Solving Pseudomonotone Variational Inequality and Fixed Point Problems in Banach Spaces

: In this paper, we introduce a new iterative method that combines the inertial subgradient extragradient method and the modiﬁed Mann method for solving the pseudomonotone variational inequality problem and the ﬁxed point of quasi-Bregman nonexpansive mapping in p -uniformly convex and uniformly smooth real Banach spaces. Under some standard assumptions imposed on cost operators, we prove a strong convergence theorem for our proposed method. Finally, we perform numerical experiments to validate the efﬁciency of our proposed method.


Introduction
Let C be a nonempty subset of a real Banach space E with the norm ||.|| and the duality space E * .We denote the value of y * ∈ E * at x ∈ E by x, y * .For any nonlinear operator A : C → E * , According to Stampacchia [1], the variational inequality problem (VIP) is defined as follows: We use V I(C, A) to represent the solution set of (1).The study of VIP originates from solving a minimization problem involving infinite-dimensional functions and variational calculus.As an analytical application of mechanics to the solution of partial differential equations in infinite-dimensional spaces, Hartman and Stampacchia [2] initiated the systematic study of VIP in 1964.In 1966, Stampacchia [1] demonstrated the first VIP existence and uniqueness solution.In 1979, Smith [3] originally used VIP to solve variational inequality problems in finite-dimensional spaces when he formulated the traffic assignment problem.He was unaware that his formulation was an exact variational inequality problem before Dafermos [4] realized it in 1980 while working on traffic and equilibrium problems.Since then, a variety of VIP models have been used in real-world settings.These models have a rich theoretical mathematics, some intriguing crossovers between various fields, and several significant applications in engineering and economics.Furthermore, variational inequalities give us a tool for a wide range of issues in mathematical programming, such as nonlinear systems of equation, issues with optimization, and fixed point theorems.Numerous real-world "equilibrium" problems systematically employ variational inequalities (see [5]).
There are a number of well-known techniques for resolving variational inequalities.The regularized method and the projection method are two prominent and general approaches to solving VIPs.Numerous methods have been considered and put forth to solve the VIP (1) problem based on these directives.The extragradient method, which Korpelevich [6] first proposed and which was later expanded upon due to the strong assumption of his result, uses two projections on the underlying feasible closed and convex set over each iteration.This can have an impact on the computational effectiveness of the method.There are ways to circumvent these problems.The first is the subgradient extragradient technique, Algorithm 1, first proposed by Censor et al. [7].This method substitutes a projection onto a particular constructible half-space for the second projection onto C.They use the following approach: where τ ∈ (0, 1 L ).We are aware that several authors have studied iterative methods for solving variational inequality problems and fixed points of nonexpansive and quasinonexpansive mappings, as well as their generalizations, in real Hilbert spaces (see, for instance [7,8] and the references therein).Bregman [9] developed methods using the Bregman distance function D f in (2) rather than the norm when constructing and investigating feasibility and optimization problems.This approach was used to navigate problems that arise when the useful illustrations of nonexpansive operators in Hilbert spaces H, such as the metric projection P C onto a nonempty, closed, and convex subset C of H, are no longer nonexpansive in Banach spaces.This led to the development of a growing body of research on approximating solutions to problems involving variational inequality, fixed points, and other issues (see, e.g., [10,11] and the references therein).
Recently, Ma et al. [12] developed the following Algorithm 2, known as the modified subgradient extragradient method, for solving variational inequality and fixed point problems in the context of Banach space: . If e n ≡ f n and Te n = e n , then stop: Let n := n + 1 and return to Step 1.
where P C is the generalized projection on E, J is the duality mapping, A : E → E * is the pseudomonotone mapping, and T is the nonexpansive mapping.It was proven that the sequence {x n } generated by Algorithm 2 converges strongly to a point x * ∈ V I(C, A) ∩ F(T), where x * = P V I(C,A)∩F(T) x 0 , under some mild conditions, in 2-uniformly convex real Banach spaces.For more information on the common solution of VIP and fixed point problems in real Banach spaces, which is more general than Hilbert spaces, the reader may refer to any of the following recent papers: [13,14].Motivated by the above results, this paper investigates the strong convergence of the inertial subgradient extragradient method for solving the pseudomonotone variational inequality problem and the fixed point problem of quasi-Bregman nonexpansive mapping in p-uniformly convex and uniformly smooth real spaces.We demonstrate that, under a number of suitable conditions placed on the parameters, the suggested method strongly converges to a point in V I(C, A) ∩ (F(T)).Finally, we offer a few numerical experiments that support our main finding in comparison to previous published papers.

Preliminaries
Let 1 < q ≤ 2 ≤ p < ∞, where 1  p + 1 q = 1 .Consider E to be a real normed space with dual E * and S := {x ∈ E : ||x|| = 1}.If for any x, y in S with x = y, λ ∈ (0, 1); then E is (i) = 0; if a positive real integer C q exits such that ρ E (τ) ≤ C q τ q for any τ > 0, E is referred to as being q-uniformly smooth.As a result, each and every q-uniformly smooth space is uniformly smooth.If and only if the dual, E * , is p-uniformly convex, then E is q-uniformly smooth, see [15].It is widely known that L p , p , and W m p are 2-uniformly convex and q-uniformly smooth for 1 ≤ q < 2; 2-uniformly smooth and p-uniformly convex for 2 ≤ p < ∞ (see [16]).The expression, defines the generalized duality mapping J p E from E to 2 E * .The mapping J 2 E = J is frequently referred to as the normalized duality mapping in the case where p = 2.It is common knowledge that on bounded subsets of E, J p E is norm-to-norm uniformly continuous if E is uniformly smooth.It follows that J p E is single-valued if E is smooth.It is wellknown that if the duality mapping J q E * from E * to E is injective and sujective, then E is reflexive and strictly convex with a strictly convex dual, and J p E J q E * = I E * (identity map in E * ) (see [17]), thus, we have the generalized duality mapping We recall the following definitions, which were introduced in [18].For any closed unit ball B in E with radius r > 0, we have rB = {u ∈ E : ||u|| ≤ r}.If ρ r (t) > 0 for every r, t > 0, and ρ r : [0, ∞) → [0, ∞) express as ρ r (t) = inf x,y∈rB,||x−y||=t,δ∈(0,1) for all t ≥ 0 then, a function f : E → R is said to be uniformly convex on bounded sets.The ρ r function is also known as the gauge of uniform convexity of f , and is well known and nondecreasing.The following lemma, which is widely known, if f is uniformly convex, is crucial for the verification of our main result.

Lemma 1 ([19]
).Let E be a Banach spance and f : E → R a uniformly convex function on bounded subsets of E. If r > 0 and δ j ∈ (0, 1) where ρ r is its gauge of uniform convexity of f , for each j, k ∈ {0, The Bregman distance in relation to f is given by Let f p (x) := 1 p ||x|| in particular.The derivative of the function f p is the generalized duality mapping J p E from E to 2 E * .Consequently, the Bregman distance with regard to f p is described by The three-point identity, a crucial property of the Bregman distance, is defined as: Due to the lack of symmetry, the Bregman distance is not a metric in the traditional sense, but it does possess some distance-like characteristics.If E is a p-uniformly convex space, then the Bregman distance function ∆ p and the metric function satisfy the relation shown below (see [20]), which proves to be extremely helpful in the demonstration of our result: let τ p > 0 be any fixed constant.
for all x, y ∈ E. Additionally, for q > 1 and 1 p + 1 q = 1, recall from Young's inequality, that Let E be a smooth and strictly convex real Banach space and C a nonempty, closed, and convex subset of E. The Bregman projection operator in the sense of Bregman [9] is The Bregman projection is described in the following way [21]: With respect to Bregman function ∆ p , we obtain The Bregman projection in terms of f 2 and the metric projection are identical in Hilbert spaces, but otherwise they are different.More significantly, in Banach spaces, the metric projection cannot share the same property, ( 9), as the Bregman projection.
If E is smooth, strictly convex, and reflexive Banach space.We defined the function with 1 p + 1 q = 1 (see [22]).It is well known that V p is nonnegative, and with respect to the Bregman function, we also have Furthermore, V p satisfies the following inequality: Additionally, in the second variable and for all z ∈ E; V p is convex, that is where [23][24][25]).We also need the nonlinear operators, which are introduced below.If C is a nonempty subset of E, a Banach space, and T : denotes the set of fixed point of T.An element q in C is asymptotic fixed point of T, if for any sequence {x n } in C, converges weakly to q such that lim n→∞ ||Tx n − x n || = 0. We describe the set set of asymptotic fixed point of T by F(T).

Definition 1 ([26]
).Let C be a nonempty subset of a real Banach space E that is uniformly smooth and p-uniformly convex ((0 < p < ∞)).Let T : C → E be a mapping with F(T) = ∅, then T is said to be: Next, we outline a few ideas about the monotonicity of an operator.
Definition 3. Let E be a Banach space that has E * as its dual.The operator A : E → E * is referred to as: where L ≥ 0 and p ∈ [1, ∞) are two constants.(m2) monotone, if Ax − Ay, x − y ≥ 0 for all x, y ∈ E; (m3) pseudomonotone, if for all x, y ∈ E, Ax, y − x ≥ 0 =⇒ Ay, y − x ≥ 0; (m4) weakly sequentially continuous if for any {x n } in E such that x n x implies Ax n Ax.
It is clear that (m2) ⇒ (m3); the example that follows demonstrates that the implication's converse is not generally true.Let A(x) = 1 − x for all x ∈ E := [0, 1].Then, A is pseudomonotone but not monotone.
When demonstrating the strong convergence of our sequence, the following result is helpful:

Lemma 2 ([27]
).Let {a n } be a nonnegative sequence of real numbers, and {α n } a real sequence of numbers in (0, 1), with

Main Results
For the purpose of solving pseudomonotone variational inequality and fixed point problems, in this section, we formulate Algorithm 3, combining a modified inertial Manntype method with a subgradient extragradient algorithm.For the convergence of the method, we require the following conditions: Assumption 1. (C1) E is a p−uniformly convex real Banach space which is also uniformly smooth and C is a nonempty, closed, and convex subset of E. (C2) A : C → E * is pseudomonotone and (p − 1) − L−Lipschitz continuous on E. (C3) A is weakly sequentially continuous; that is, for any {x n } ⊂ E, we have x n x * , which implies Ax n Ax * .
We know the following lemma, which was essentially proved in [13], is important and crucial in the proof of our main result.Lemma 4 ([13], Lemma 3.4).Let {y n } and {w n } be two sequences formulated in Algorithm 3. If there exists a subsequence {y n s } of {y n } that converges weakly to a point z ∈ E and lim We demonstrate that the Algorithm 3 converges strongly under the assumptions (C1)-(C6) based on the analysis described above and Lemma 4.
Theorem 1. Suppose that Assumption 1 holds.Then, the sequence {x n } defined by Algorithm 3 converges strongly to the unique solution of the Γ.
)), then using ( 11), ( 12), ( 15) and ( 18), we obtain For any > 0 such that ∈ 0, pτ p 2 p−1 , there exists a natural number N such that for all n ≥ N, we obtain Using ( 20) and ( 21), it follows that Next, using Lemma 2 and ( 23), it remains to show that lim sup holds and, from (22), we denotes {Υ n s } as follows: thus, from (22), we obtain Hence, lim sup s→∞ Υ n s ≤ 0, which implies that lim s→∞ Υ n s = 0.It follows from ( 24) that and By the property of ρ r , we obtain and, since J q E * is uniformly continuous on a bounded subset of E * , we obtain lim Additionally, using ( 5) and ( 25), we obtain lim With J p E being uniformly norm-to-norm continuous on bounded sets, we also have However, we understand from the definition that y n := Moreover, from ( 28) and ( 30), since J q E * is also uniformly continuous, we obtain from (30) that lim s→∞ and from ( 16), we obtain ||J || and with (26), since δ n in (0, 1) for all n ≥ 1, we obtain Thus, from (31), we obtain By uniform continuity of J q E * on a bounded subset of E * , we conclude, respectively, from (31),we obtain lim and Since {x n s } is bounded, it follows that there exists a subsequence {x n s k } of {x n s } that converges weakly to some point z in E. By using (33), we obtain v n s z; from (27) and Definition 2, we conclude that z ∈ F(T).Furthermore, from (32), we obtain that y n s z.This together with lim s→∞ ||y n s − w n s || = 0 in (28) and Lemma 4, we conclude that z ∈ V I(C, A), therefore z ∈ Γ.Finally, using σ as a zero point in C, it follows from the definition of the Bregman projection that lim sup We know from (23), that Hence, combining (34), (35), and together with Lemma 2, we conclude that lim n→∞ ∆ p (x n , x * ) = 0, and together with the fact that τ p ||x n − x * || p ≤ ∆ p (x n , x * ), we obtain x n → x * as n → ∞.This complete the proof.
We obtain the following corollary from Theorem 1 by setting T = 0 in Algorithm 3.
Next, if, in Algorithm 3, we assume that A = 0, we obtain the following corollary: Corollary 2. Let E be a p-uniformly convex and uniformly smooth real Banach space with sequentially continuous duality mapping J p E * .Let T : E → E be a quasi-Bregman nonexpansive mapping such that F(T) = ∅.Suppose {δ n } is a sequence in (a, b) for some 0 < a < b and {µ n } is a positive sequence in 0, n=0 be a sequence generated in Algorithm 4 as follows: Algorithm 4: First Modified Inertial Mann-type Method Initialization: Choose x 0 , x 1 ∈ E to be arbitrary, θ ∈ (0, τ p ), µ ∈ (0, τ p ) and λ 1 > 0. Iterative Steps: Calculate x n+1 as follows: (Step1) Given the iterates x n−1 and x n for each n ≥ 1, θ > 0, choose θ n such that 0 ≤ θ n ≤ θn , where Then, {x n } ∞ n=0 converges strongly to a point p ∈ F(T).
Proof.We observe that the necessary assertion is provided by the method of proof of Theorem 1.
Let B : E → E * be a set-valued mapping with domain D(B) = {x ∈ E : B(x) = ∅} and range R(B) = {x * ∈ E * : x * ∈ B(x)}, and the graph of B is given as Gra(B) := {(x, x * ) ∈ E × E * : x * ∈ Bx}.Then B is said to be monotone if x * − y * , x − y ≥ 0 whenever (x, x * ), (y, y * ) ∈ Gra(B), and B is said to be maximal monotone if its graph is not contained in the graph of any other monotone operator on E. Let B : E → 2 E * be a mapping.Additionally, B is called a monotone mapping if, for any x, y ∈ domB, we have B is called maximal if B is monotone and the graph of B is not properly contained in the graph of any other monotone operator.It is known that if B is maximal monotone, then the set B −1 (0) := {u ∈ E : 0 ∈ B(u)} is closed, and convex.The resolvent of B is the operator Res B σ : E → 2 E defined by It is known that Res B σ is single-valued, Bregman firmly nonexpansive, and F( [28,29]).Since every Bregman firmly nonexpansive is quasi-Bregman nonexpansive, from Corollary 2, we obtain the following result as a special case: Corollary 3. Let E be a p-uniformly convex and uniformly smooth real Banach space with sequentially continuous duality mapping J p E .Let B : E → 2 E * be a maximal monotone with B −1 (0) = 0. Suppose {δ n } be a sequence in (a, b) for some 0 < a < b and {µ n } is a positive sequence in 0, pτ p 2 p−1 , where τ p is defined in (5), µ n = •(α n ), where α n is a sequence in (0, 1) such that lim n→∞ α n = 0 and ∑ ∞ n=1 α n = ∞.Let {x n } ∞ n=0 be a sequence generated in Algorithm 5 as follows: Algorithm 5: Second Modified Inertial Mann-type Method Initialization: Choose x 0 , x 1 ∈ E to be arbitrary, θ ∈ (0, τ p ), µ ∈ (0, τ p ) and λ 1 > 0.

Remark 1.
The following are considered: (a) Theorem 1 improves, extends, and generalizes the corresponding results [12,13,[30][31][32][33] in the sense that either our method requires an inertial term to improve the convergence rate and/or the space considered is more general.(b) We observe that the result in Corollary 1 improves, and extends the results in [7,[34][35][36] from Hilbert space to a p-uniformly convex and uniformly smooth real Banach space as well as from solving the monotone variational inequality problem to the pseudomonotone variational inequality problem.(c) Corollary 3 improves, and extends the corresponding results of Wei et al. [37], Ibaraki [38], and Tianchai [39] in the sense that our iterative method does not require computation of C n+1 for each n ≥ 1 or the class of mappings considered in our corollary is more general and inertial in our method, which aids in increasing the convergence rate of the sequence generated by the method.

Numerical Example
In this section, we intend to demonstrate the efficiency of our Algorithm 3 with the aid of numerical experiments.Furthermore, we compare our iterative method with the methods of Censor et al. [7] (Algorithm 1) and Ma et al. [12] . Define metric projection P C as follows: , then, A is pseudomonotone and uniformly continuous mapping (see [40]) and let T(x(t)) = t 0 x(s)ds, for all x ∈ L 2 [0, 1], t ∈ [0, 1], then T is nonexpansive mapping.For the control parameters, we use α n = 1 5n+2 , δ n = 1 2 − α n , µ n = α n n 0.01 and θ n = θn .We define the sequence TOL n := ||x n+1 − x n || 2 and apply the stopping criterion TOL n < ε for the iterative processes because the solution to the problem is unknown.ε is the predetermined error.Here, the terminating condition is set to ε = 10 −5 .For the numerical experiments illustrated in Figure 1 and Table 1 below, we take into consideration the resulting cases.

Conclusions
The paper has proposed a new inertial subgradient extragradient method with the modified Mann algorithm for solving the Lipschitz pseudomonotone variational inequality problem and the fixed point of quasi-Bregman nonexpansive mapping in p-uniformly convex and uniformly smooth real Banach spaces.Under some suitable conditions imposed on parameters, we have proved the strong convergence of the algorithms.The efficiency of the proposed algorithm has also been illustrated by numerical experiments in comparison with other existing methods.
Bregman firmly nonexpansive if, for all x, y ∈ C Ty) + ∆ p (Ty, Tx) + ∆ p (Tx, x) + ∆ p (Ty, y) ≤ ∆ p (Tx, y) + ∆ p (Ty, x).The well known demiclosedness principle plays an important role in our main result.Assume that C is a nonempty, closed, convex subset of a uniformly convex Banach space E and that T : C → C is a nonlinear mapping.Then, T is called demiclosed at 0; if {x n } is a sequence in C such that x nx and lim n→∞ ||x n − Tx n || = 0, then x = Tx.