Inertial KM-type extragradient scheme for solving a variational inequality and a hierarchical fixed point problems

We propose an inertial KM-type extragradient scheme to approximate a common solution of a variational inequality problem and a hierarchical fixed point problem for nonexpansive mappings. This scheme generalizes and unifies a number of known iterative schemes. Furthermore, we discuss the weak convergence for the proposed scheme. We also discuss an example to illustrate the main theorem.


Introduction
Let C be a nonempty convex and closed set in a real Hilbert space H and ·, · and · denote the inner product and induced norm on H. A mapping U : C → C is said to be nonexpansive if Uu -Uv ≤ uv , ∀u, v ∈ C. Note that if F(U) := {u ∈ C : Uu = u} = ∅ then set F(U) is convex and closed. Let F(U) = ∅. The subdifferential of a proper function g : H → (-∞, +∞] is the set-valued operator ∂g : H → 2 H defined by ∂g(u) = {w ∈ H : yu, w + g(u) ≤ g(y), ∀y ∈ H}. Let u ∈ H. Then g is subdifferential at u if ∂g(u) = ∅. The indicator function ψ C : H → (-∞, +∞] is given by ∞, otherwise. Note that ψ C is a convex function when C is a convex set. In 2006, Moudafi et al. [1] discussed the convergence of a scheme for the following hierarchical fixed point problem (in short, H-FPP): Findū ∈ F(U) such that ū -Vū,ūu ≤ 0, ∀u ∈ F(U), (1.1) ∂ψ F(U) (ū). Hence H-FPP(1.1) is equivalent to the variational inclusion: Findū ∈ F(U) such that 0 ∈ (I -V )ū + N F(U) (ū), (1.2) where the mapping I is identity on C and N F(U) (ū) denotes the normal cone to F(U) atū given by ∅, o t h e r w i s e .
If we set V = I, then is just F(U). Furthermore, we mention that H-FPP(1.1) is worth to study because it includes as special cases, the important problems such as the variational inequality on fixed point sets and hierarchical minimization problems; see Moudafi [2].
In 2007, Moudafi [2] proposed the following Krasnoselski-Mann (KM)-type scheme for solving H-FPP(1.1): For given u 0 ∈ C, where {α k } ⊂ (0, 1) and {σ k } ⊂ (0, 1). For further work related to scheme (1.3), see for example [1,[3][4][5][6][7]. In 2008, Mainge [8] introduced an inertial version of KM-type scheme by unifying the KM-type scheme and the inertial extrapolation, for approximating a fixed point of nonexpansive mappings and discussed the weak convergence. Recently, Bot et al. [9] derived some the convergence results of the following inertial KM-type scheme to approximate a fixed point of nonexpansive mapping U on H which generalize the results of Mainge [8]: for each k ≥ 1, where η k is a damping-type term and α k is a relaxation factor. Recently, the interest of studying inertial type algorithms has been increased due to their fast convergence. For further study of scheme (1.4) and its generalizations; see for example [10][11][12][13].
On the other hand, we consider the classical variational inequality (in short, VI): Find (1.5) introduced in [14] where h : H → H. The set of solutions of VI(1.5) is denoted by Sol(VI(1.5)). Note that the projected gradient scheme for solving VI(1.5) is where μ > 0 and P C is the metric projection onto C. In order to converge, this scheme requires the restrictive condition that h is inverse strongly (or strongly) monotone. To overcome this difficulty, Korpelevich [15] proposed an extragradient iterative scheme by where μ ∈ (0, 1 L ), where L > 0 is Lipschitz constant of h. Since then many researchers improved scheme (1.7) in various directions; see, e.g. [16][17][18][19][20][21][22][23][24] and the references therein. Note that the calculation of two projections onto C might affect the efficiency of such scheme. Therefore, Dong et al. [25] proposed the following inertial KM-type extragradient scheme for VI(1.5): where α, σ , δ > 0.
They proved the weak convergence theorem for scheme (1.8).
In this paper, we propose an inertial version of KM-type extragradient scheme by combining iterative schemes (1.3) and (1.8) to approximate a common solution of H-FPP(1.1) and VI(1.5). We prove a weak convergence theorem for the proposed scheme. Furthermore, we discuss an example to illustrate the main theorem. The theorems of the paper unify and generalize previously known corresponding theorems; see for example [2,8,9,[25][26][27]].

Preliminaries
We give some definitions and results of convex and nonlinear analysis, which will be used in the proof of the weak convergence theorem.
A mapping P C is called the metric projection of H onto C if for every point u ∈ H, there exists a unique point in C denoted by P C u such that Note that P C is nonexpansive and satisfies Moreover, P C u is characterized by the fact P C u ∈ C and [28] and demiclosed [29] on H.

Lemma 2.1 If a mapping U is nonexpansive on H then I -U is maximal monotone
Then the following hold: (a) lim k→∞ u ku exists for every u ∈ C;

Weak convergence theorem
We propose the following inertial KM-type extragradient scheme for solving H-FPP(1.1) and VI(1.5).
On the other hand, from v k = P C (Iμh)t k and v ∈ C, we get Since vu, whv ≥ 0, for all u ∈ C and v k i ∈ C, using the monotonicity of h, we have Since h is continuous, on taking the limit i → ∞ we have v -ū, w ≥ 0. Since G is maximal monotone, we haveū ∈ G -1 0 and henceū ∈ Sol(VI(1.5)) and thusū ∈ .  Concluding remark 4.1 In this paper, we considered a variational inequality problem (VI) and a hierarchical fixed point problem (H-FPP) in Hilbert space. We proposed an inertial version of Krasnoselski-Mann (KM)-type extragradient scheme (3.1) by combining the KM-type scheme (1.3) and an inertial version of the extragradient scheme (1.8) to approximate a common solution of H-FPP(1.1) and VI(1.5). Furthermore, we proved a weak convergence theorem for the proposed scheme (3.1). Finally, we discussed an example to illustrate Theorem 3.1.