Inertial hybrid self-adaptive subgradient extragradient method for fixed point of quasi$-\phi-$nonexpansive multivalued mappings and equilibrium problem

Available online at www.atnaa.org Research Article

Inertial Hybrid Self-adaptive subgradient extragradient method for Fixed Point of Quasi−φ−nonexpansive multivalued mappings and Equilibrium problem

Murtala H. Harbau^a

aDepartment of Science and Technology Education, Bayero University, Kano, Nigeria.

Abstract

In this paper, we propose a new inertial self-adaptive subgradient extragradient algorithm for approximating common solution in the set of pseudomonotone equilibrium problems and the set of xed point of nite family of quasi−φ−nonexpansive multivalued mappings in real uniformly convex Banach spaces and uniformly smooth Banach spaces. Strong convergence of the iterative scheme is established. Our results generalizes and improves several recent results annouced in the literature.

Keywords: Pseudomonotone Equilibrium problem Inertial self adaptive hybrid method Multivalued quasi−φ−nonexpansive mapping Banach Spaces.

2010 MSC: Subject Classication 47H09, 47J25.

1. Introduction

Let E be a real Banach space and E^∗ be the dual of E. Let C be a nonempty closed and convex subset of E. The equilibrium problem is to nd ¯z ∈ C such that

g(¯z, y) ≥ 0 ∀ y ∈ C, (1)

where g : C × C → R is a bifunction with property g(x, x) = 0 ∀ x ∈ C. The equilibrium problem (1) was introduced by Blum and Oettli [5]. We denote by EP (g, C) to be the set solutions of equilibrium problem (1), i.e.

EP (g, C) = {¯z ∈ C : g(¯z, y) ≥ 0 ∀ y ∈ C}.

Email address: [email protected] (Murtala H. Harbau)

Received November 5, 2020; Accepted: June 21, 2021; Online: June 23, 2021.

Equilibrium problem (1) generalizes many important problems such as variational inequality problem, opti- mization problem, complementarity problem, xed point problem, see, for example, [5, 28].

A map T : E → E is said to be nonexpansive if ||T x−T y|| ≤ ||x−y|| ∀ x, y ∈ E . A point x ∈ E is said to be a xed point of T if x = T x. The set of xed points of T is denoted by F (T ), i.e F (T ) = {x ∈ E : x = T x}.

T is called quasi nonexpansive ||T x − z|| ≤ ||x − z|| ∀ x ∈ E, z ∈ F (T ) .

Let CB(E) be a family of nonempty closed and bounded subsets of E and T : E → CB(E) be a multivalued mapping. A point z ∈ E is called a xed point of T if z ∈ T z. We denote by F (T ) the set of all xed points of T i.e F (T ) = {z ∈ E : z ∈ T z}. A point z ∈ F (T ) is called an asymptotic xed point of T if there exists a sequence {xn} in E such that xn * z and lim

n→∞d(x_n, T x_n) = 0. The set of all asymptotic xed points of T is denoted by ˜F (T ).

A multi-valued mapping T : E → CB(E) is called relatively nonexpansive if F (T ) 6= ∅, F (T ) = ˜F (T ) and φ(z, p) ≤ φ(z, x) ∀x ∈ E, p ∈ T x, z ∈ F (T ).

T is said to be quasi-φ-nonexpansive if F (T ) 6= ∅ and φ(z, p) ≤ φ(z, x) ∀x ∈ E, p ∈ T x, z ∈ F (T ).

T is said to be closed if for any sequence {xn} in E with xn → x and {wn} ⊂ T (x_n) with wn → y, then y ∈ T (x).

Remark 1.1. Observe that from the above denitions, the class of quasi-φ-nonexpansive multi-valued map- pings contains the class of relatively nonexpansive multi-valued mappings which require a strong restriction F (T ) = F (T )˜ . Furthermore if E is a real Hilbert space H, the class of quasi-φ-nonexpansive mappings coincides with the class of quasi nonexpansive mappings which inturn contains the class of nonexpansive mappings.

Due to their importance, various methods have been imployed to approximate solutions of equilibrium and xed point problems (see, for example, [3, 18, 19, 35, 36] and the references contained therein). One of the common methods use is the proximal point method in which the convergence analysis has been considered when the bifunction g is monotone see [26]. However the proximal point method is not valid when the underlying bifunction g is pseudomonone see Wen, [41].

Another method use is the extragradient-like method [1, 17, 22, 23, 25, 34, 39] which involved two strongly convex optimization problem dened over the constrained set C and the Lipschitz-type condition imposed on the bifunction g. Moreover to solve the two strongly convex problem over the constrained set C in each iteration can be complicated especially if C is not simple. Motivated by this, Censor et al. [7] introduced a method called subgradient extragradient for approximating solutions of variational inequality problem in a real Hilbert space H, in which one projection was taken over constructed subpace which can easily be computed. Hieu [21] extended the subgradient extragradient method equilibrium problems in a real Hilbert spaces H, the author proposed the following algorithm;











x0 ∈ H

y_n=argmin{λf(xn, y) +¹₂kx_n− yk² : y ∈ C}, zn=argmin{λf(yn, y) +¹₂kx_n− yk²: y ∈ Tn}, xn+1= αnx0+ (1 − αn)zn, n ≥ 0,

(2)

where Tn= {v ∈ H : h(x_n−λw_n)−y_n, v −y_ni ≤ 0}, wn∂₂f (x_n, y_n)and λ, αnsatisfy the following conditions;

1. 0 < λ < min {_2c¹₁,_2c¹

2} 2. 0 < αn< 1, lim

n→∞α_n= 0, P∞

n=1α_n= +∞.

The author proved strong convergence of the iterative sequence (2) to the solution of the equilibrium problem.

Recently, Dadashi et al. [14] used subgradient extragradient method to approximate solution of pseudomono- tone equilibrium problem in real Hilbert spaces.

One problem of the aforemention results was the computation of the Lipschitz constants c1, c₂ of the bifunc- tion f which sometimes is dicult to estimate. Motivated by this, very recently, Yang and Liu [42] introduced a new step size, in the subgradient extragradient method for pseudomonotone equilibrium problem and xed point of quasi nonexpansive mapping in a real Hilbert space. They proved strong convergence of the following iterative sequence without the prior knowledge of the Lipschitz-type constants of the bifunction f.

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x0∈ H yn=argmin

y∈C

{λ_nf (xn, y) +¹₂kx_n− yk²},

T_n= {v ∈ H : h(x_n− λ_nw_n) − y_n, v − y_ni ≤ 0}, zn=argmin

y∈Tn

{λ_nf (yn, y) +¹₂kx_n− yk²}, tn= αnx0+ (1 − αn)zn,

x_n+1= β_nz_n+ (1 − β_n)St_n, n ≥ 0

(3)

where S is quasi nonexpansive map, wn ∈ ∂₂f (x_n, y_n), λ0, µ ∈ (0, 1) and {αn}, {β_n} are real sequences satisfying some conditions and

λ_n+1= (

min{_{2(f (x}^{µ(||xn−yn||2+||zn−yn||2)}

n,zn)−f (xn,yn)−f (yn,zn)), λn}, f (x_n, zn) − f (xn, yn) − f (yn, zn) > 0,

λn, Otherwise.

They proved strong convergence of (3) to common point in the set of xed point of quasi nonexpansive mapping and set of pseudomonotone equilibrium problems.

Recently, inertial method which was introduced by Polyak [30] to speed up the rate of convergence of the iteration methods has been considerably attracting interest of reseachers, (see, for example, [4, 8, 9, 11, 12, 13, 16, 27, 29, 31, 37, 40] and the references contained therein).

Motivated by the above results, the purpose of this paper is to propose an inertial self-adaptive subgradi- ent extragradient algorithm for approximating common solution in the set of pseudomonotone equilibrium problem and the set of xed point of nite family of quasi−φ−nonexpansive multivalued mappings in real uniformly convex Banach spaces and uniformly smooth Banach spaces. The step size ηn is chosen self adaptively and estimates of Lipschizt-type constants are dispensed with.

2. Preliminaries

Let E be a real Banach space and E^∗ be the dual of E. Let C be a nonempty closed and convex subset of E. We denote by J : E → 2^E∗ the normalized duality mapping dened by

J (x) = {f^∗ ∈ E^∗: hx, f^∗i = kxk² = kf^∗k²},

where h., .i denotes the duality pairing between the element of E and that of E^∗. It is well known that J(x) is nonempty for each x ∈ E, see [36]. We denote weak and strong convergence by * and → respectively.

Let S(E) be a unit sphere centered at the origin. A Banach space is said to be strictly convex if k^x+y₂ k < 1, whenever x, y ∈ S(E) and x 6= y. The modulus of convexity of E is dened by

δE(t) = inf n

1 −1

2kx + yk : kxk = 1 = kyk, kx − yk ≥ o

, ∀ t ∈ [0, 2].

E is called uniformly convex if δE(t) ≥ 0 ∀ t ∈ [0, 2]and p-uniformly convex if there exists a constant cp > 0 such that δE(t) ≥ cpt^p∀ t ∈ [0, 2]. Note that every p-uniformly convex Banach space is uniformly convex and every uniformly convex is strictly convex and reexive. The modulus of smoothness ρE(τ ) : [0, ∞) → [0, ∞) is dened by

ρ_E(τ ) = supnkx + τ yk + kx − τ yk

2 − 1 : kxk = kyk = 1o .

E is said to be uniformly smooth if ^ρE_τ^{(τ )} → 0 as τ → 0 and E is q−uniformly smooth if there exists dq > 0 such that ρE(τ ) ≤ d_qτ^q. It is well known that if E is q−uniformly smooth, then q ≤ 2 and E uniformly smooth. Furthermore every uniformly smooth Banach space is smooth. We know that (see, for example, [10]) if E is smooth, strictly convex and reexive, then J is single-valued, one-to-one and onto respectively and J⁻¹ is also single-valued, one-to-one, onto and it is the duality mapping from E^∗ into E. In addition if E is uniformly smooth, then the norm on E is fr´echet dierentiable and J is uniformly norm-to-norm continuous on bounded subsets of E and E is uniformly smooth if and only if E^∗ is uniformly convex.

Let E be a smooth Banach space and C be a closed convex subset of E. The function φ : E ×E → R dened by

φ(x, y) = kxk²− 2hx, J yi + kyk², ∀ x, y ∈ E, (4) is called Lyapunov bifunction introduced by Alber [2], where J is the normalized duality mapping. Observe from the denition of φ in (4) above, we have that,

φ(x, y) = φ(x, z) + φ(z, y) + 2hx − z, J z − J yi, ∀x, y, z ∈ E, and (5)



kxk − kyk2

≤ φ(x, y) ≤

kxk + kyk2

, ∀x, y ∈ E (6)

Follwing Alber [2], the generalized projection ΠC : E → C is a mapping dened by Π_C(x) = arg min

y∈Cφ(y, x) ∀x ∈ E.

Remark 2.1. (1) If E is a Hilbert space, then φ(y, x) = ky − xk², and the generalized projection reduces to metric projection PC of E onto C.

(2) If E is smooth and strictly convex, then φ(x, y) = 0 if and only if x = y ∀x, y ∈ E, see, for example, [36]

Denition 2.2. (see [6, 24]) The subdierential of f, ∂f is the mapping ∂f : E → 2^E∗ dened by

∂f (x) = {x^∗ ∈ E^∗ : f (y) − f (x) ≥ hy − x, x^∗i ∀ y ∈ E} for all x ∈ E.

Remark 2.3. It is known that if the function f is proper, lower semicontinuous and convex, then for each x ∈ D(f )the subdierential ∂f(x) is a nonempty closed convex set, where D(f) is the domain of f.

Denition 2.4. A bifunction g : C × C → R is said to be;

1. γ-strongly monotone on C if there exists γ > 0 such that

g(x, y) + g(y, x) ≤ −γ||x − y||² ∀ x, y ∈ C, 2. Monotone if

g(x, y) + g(y, x) ≤ 0 ∀ x, y ∈ C, 3. Pseudomonotone if

g(x, y) ≥ 0 ⇒ g(y, x) ≤ 0 ∀ x, y ∈ C,

It is clear from Denition 2.4, that (1) ⇒ (2) ⇒ (3). To solve the equilibrium problem, we assume the bifunction g : C × C → R satises the following conditions;

(D1) g(x, x) = 0 for every x ∈ C;

(D2) g(x, .) is convex, lower semicontinuous and subdierentiable on E;

(D3) g is pseudomonotone on C;

(D4) g is jointly continuous on E × C in the sense that if x ∈ E y ∈ C and {xn}, {y_n} are two sequences such that xn→ x, y_n→ y, then g(xn, yn) → g(x, y);

(D5) g(x, y) + g(y, z) ≥ g(x, z) − c1φ(y, x) − c2φ(z, y) ∀ x, y, z ∈ C and some c1, c2 > 0.

In the sequel we will need the following lemmas:

Lemma 2.5. [43] Let E be a real uniformly smooth and uniformly convex Banach space. Let T : E → 2^E be a closed quasi-φ-nonexpansive multivalued mapping, then F (T ) is closed and convex.

Lemma 2.6. [39] Assume the bifunction g satisties (D1)-(D4), then the set EP (g, C) of solutions of the equilibrium problems is closed and convex.

Lemma 2.7. [38] Let C be a nonempty subset of E and f : C → R be a convex and subdierentiable function, then f is minimized at x ∈ C if and only if

0 ∈ ∂f (x) + N_C(x), where NC(x) is the normal cone to C at x ∈ C, i.e.

N_C(x) = {ζ ∈ E^∗ : hy − x, ζi ≤ 0 ∀ y ∈ C}.

Lemma 2.8. [10] Let E be a reexive Banach space and f : E → R, g : E → R are two convex functions such that dom f ∩ dom g 6= ∅ and f is continuous, then

∂(f + g) = ∂f (x) + ∂g(x), ∀ x ∈ E.

Lemma 2.9. [2] Let E be a strictly convex, smooth and reexive Banach space and let K be a nonempty closed and convex subset of E. Let x ∈ E, then

φ(y, Π_Cx) + φ(Π_Cx, x) ≤ φ(y, x) ∀y ∈ C.

Lemma 2.10. [44] Let E be a uniformly convex Banach space and r > 0, then there exists a strictly increasing, continuous and convex function f : [0, 2r] → [0, +∞) such that f(0) = 0 and

N

X

i=1

αixi

2≤

N

X

i=1

αikx_ik²− α_iαjf (kxi− x_jk), where αi ∈ (0, 1), P^N_i=1α_i = 1 and xi∈ B_r(0), ∀i ∈ {1, 2, . . . , N },

Lemma 2.11. [26] Let E be a smooth and uniformly convex Banach space and let {xn} and {yn} be two sequences in E. If either {xn} or {yn} is bounded and φ(xn, y_n) → 0 as n → ∞, then xn− y_n → 0 as n → ∞.

3. Main Results

In this section we propose the following inertial hybrid self adaptive subgradient extragradient algorithm in a real uniformly convex Banach space E which is also uniformly smooth;

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

η1> 0, µ ∈ (0, 1), x0, x1 ∈ C₁= E, θ_n= x_n+ α_n(x_n− x_n−1),

y_n=argmin

y∈C

{η_ng(θ_n, y) +¹₂φ(y, θ_n)},

Γn= {z ∈ E : hJ θn− η_nwn− J y_n, z − yni ≤ 0}, z_n=argmin

y∈Γn

{η_ng(y_n, y) +¹₂φ(y, θ_n)}, un= J⁻¹(βnJ zn+ (1 − βn)[γn,0J θn+PN

i=1γn,iJ tn,i]), C_n+1 = {z ∈ C_n: φ(z, u_n) ≤ φ(z, θ_n)},

x_n+1= Π_Cn+1x₀, n ≥ 1

(7)

where wn∈ ∂₂g(θ_n, y_n), tn,i∈ T_iθ_n, Ti, i = 1, 2, 3, . . . , N are quasi−φ−nonexpansive multivalued mappings and

ηn+1= (

min{_2(g(θ^{µ(φ(yn,θn)+φ(zn,yn))}

n,zn)−g(θn,yn)−g(yn,zn)), ηn}, g(θ_n, zn) − g(θn, yn) − g(yn, zn) > 0.

ηn, Otherwise

Observe, it is obvious from (7) that C ⊆ Γn.Also using algorithm (7), we have the following Lemmas:

Lemma 3.1. The sequence {ηn} is a monotone nonincreasing and has a lower bound minn

µ

2 max{c1,c2}, η1o, Proof. It is clear that {ηn}is a monotone nonincreasing sequence. By condition (D5), we get

µ(φ(yn, θn) + φ(zn, yn))

2(g(θn, zn) − g(θn, yn) − g(yn, zn)) ≥ µ(φ(yn, θn) + φ(zn, yn))

2(c1φ(yn, θn) + c2φ(zn, yn)) ≥ µ 2 max{c1, c2}. Hence {ηn} has a lower bound minn

µ

2 max{c1,c2}, η1o. Consequently the lim

n→∞ηn exists.

Lemma 3.2. Let yn be dened as in algorithm (7). Then ∀ n ≥ 1 and y ∈ C we have η_ng(θ_n, y) − η_ng(θ_n, y_n) ≥ hy − y_n, J θ_n− J y_ni.

Proof. Let n ≥ 0 and y ∈ C, then by Lemma 2.7 and Lemma 2.8, we get 0 ∈ ηn∂2g(θn, yn) +1

2∇₁φ(yn, θn) + NC(yn).

Therefore there exists w ∈ ∂2g(θ_n, y_n)and ¯w ∈ N_C(y_n) such that

0 = ηnw + J yn− J θ_n+ ¯w. (8)

Since w ∈ ∂2g(θn, yn), then

g(θ_n, y) ≥ g(θ_n, y_n) + hy − y_n, wi. (9) Using (8) and Denition of NC(y_n), we get

hy − y_n, −ηnw − J yn+ J θni ≤ 0, so that

ηnhy − y_n, wi ≥ hy − yn, J θn− J y_ni. (10) Hence by (9) and (10), we obtain

ηng(θn, y) − ηng(θn, yn) ≥ hy − yn, J θn− J y_ni.

Lemma 3.3. Let C be a nonempty closed convex subset of real uniformly convex and uniformly smooth Banach space E. Let Ti : E → 2^E, i = 1, 2, 3, . . . , N be nite family of quasi−φ−nonexpansive multivalued mappings. Assume g satises (D1)-(D5) and F = EP (g, C) ∩ (∩^N_i=1F (Ti)) 6= ∅. Let {θn}, {y_n} and {zn} be dened as in algorithm (7), then

φ(x^∗, z_n) ≤ φ(x^∗, θ_n) − (1 − η_n ηn+1

µ)φ(y_n, θ_n) − (1 − η_n ηn+1

µ)φ(z_n, y_n).

Proof. Let x^∗ ∈ F, then from Denition of zn, Lemma 2.7 and Lemma 2.8, we get 0 ∈ η_n∂₂g(y_n, z_n) +1

2∇₁φ(z_n, θ_n) + N_Γn(z_n).

Hence 0 = ηnw¯n+ J zn− J θ_n+ ¯w for some ¯wn∈ ∂₂g(yn, zn) and ¯w ∈ NΓn(zn), i.e.

¯

w = −J zn− η_nw¯n+ J θn. (11)

From Denition of normal cone NΓn(zn), we have

hy − z_n, ¯wi ≤ 0 ∀ y ∈ Γn. (12)

By (11) and (12), we obtain

ηnhy − z_n, ¯wni ≥ hy − z_n, J θn− J z_ni ∀ y ∈ Γ_n. Since x^∗∈ F ⊂ EP (g, C) ⊂ C ⊂ Γ_n⊂ E, we have

ηnhx^∗− z_n, ¯wni ≥ hx^∗− z_n, J θn− J z_ni. (13) On the other hand since ¯wn∈ ∂₂g(yn, zn), we have

g(yn, y) − g(yn, zn) ≥ hy − zn, ¯wni ∀ y ∈ E. (14) Therefore, combining (13) and (14), we obtain

ηn(g(yn, x^∗) − g(yn, zn)) ≥ hx^∗− z_n, J θn− J z_ni. (15) As g is pseudomonotone, we have g(yn, x^∗) ≤ 0. Thus,

−2η_ng(yn, zn)) ≥ 2hx^∗− z_n, J θn− J z_ni − 2η_ng(yn, x^∗)

≥ 2hx^∗− z_n, J θn− J z_ni. (16)

Since wn∈ ∂₂g(θ_n, y_n), then

g(θn, y) − g(θn, yn) ≥ hy − yn, wni ∀ y ∈ E.

Letting y = zn we obtain

2η_n(g(θ_n, z_n) − g(θ_n, y_n)) ≥ 2η_nhz_n− y_n, w_ni. (17) Observe as zn∈ Γ_n, we get

2hzn− y_n, J θn− J y_ni ≤ 2η_nhz_n− y_n, wni. (18) Combining (16), (17), (18) and (5), we obtain

2η_n

g(θ_n, z_n) − g(θ_n, y_n) − g(y_n, z_n)

≥ 2hz_n− y_n, J θ_n− J y_ni + 2hx^∗− z_n, J θ_n− J z_ni

= −2hz_n− y_n, J y_n− J θ_ni − 2hx^∗− z_n, J z_n− J θ_ni

= −(φ(z_n, θn) − φ(zn, yn) − φ(yn, θn))

−(φ(x^∗, θn) − φ(x^∗, zn) − φ(zn, θn))

= φ(z_n, y_n) + φ(y_n, θ_n) − φ(x^∗, θ_n) + φ(x^∗, z_n). (19) Thus, from (19) we have

φ(x^∗, z_n) ≤ φ(x^∗, θ_n) − φ(z_n, y_n) − φ(y_n, θ_n) + 2ηn



g(θn, zn) − g(θn, yn) − g(yn, zn)

 .

From the Denition of ηn we obtain

φ(x^∗, zn) ≤ φ(x^∗, θn) − φ(zn, yn) − φ(yn, θn) + 2η_n

η_n+1η_n+1

g(θ_n, z_n) − g(θ_n, y_n) − g(y_n, z_n)

≤ φ(x^∗, θ_n) − φ(z_n, y_n) − φ(y_n, θ_n) + η_n

ηn+1



µ(φ(y_n, θ_n) + φ(z_n, y_n))

= φ(x^∗, θ_n) −

1 − η_n η_n+1µ

φ(y_n, θ_n) −

1 − η_n η_n+1µ

φ(z_n, y_n).

Theorem 3.4. Let C be a nonempty closed convex subset of real uniformly convex and uniformly smooth Banach space E. Let Ti : E → 2^E, i = 1, 2, 3, . . . , N be nite family of closed quasi−φ−nonexpansive multivalued mappings. Assume g satises (D1)-(D5) and F = EP (g, C) ∩ (∩^Ni=1F (Ti)) 6= ∅. Let {αn}, {β_n} and {γn,i}be real sequences such that αn, β_n∈ (0, 1), γn,i∈ (, 1−)for some  ∈ (0, 1) and γn,0+PN

i=1γ_n,i= 1. Then the sequence {xn} generated by (7) converges strongly to p^∗= ΠFx0.

Proof. The proof is divided in to steps;

Step 1: We show F = EP (g, C) ∩ (∩^N_i=1F (T_i))is closed and convex. By Lemma 2.5, ∩^N_i=1F (T_i) is closed and convex and by Lemma 2.6, EP (g, C) is closed and convex, therefore F = EP (g, C) ∩ (∩^N_i=1F (T_i))is closed and convex.

Step 2: Here we show Cn, ∀n ≥ 1is closed and convex;

Observe C1 = C is closed and convex. Assume Cn is closed and convex for some n > 1, then φ(z, un) ≤ φ(z, θn)

is equivalent to

2hz, J θn− J u_ni ≤ kθ_nk²− ku_nk².

Thus, we obtain Cn+1 is closed and convex and therefore Cn is closed and convex ∀n ≥ 1. This shows that the iterative sequence generated by (7) is well dened.

Step 3: We show F ⊂ Cn ∀n ≥ 1.

It is clear that F ⊂ C = C1. Suppose F ⊂ Cn for some n > 1. Then for any x^∗∈ F ⊂ C_n, we have φ(x^∗, u_n) = φ(x^∗, J⁻¹(β_nJ z_n+ (1 − β_n)[γ_n,0J θ_n+

N

X

i=1

γ_n,iJ t_n,i]))

= kx^∗k²− 2hx^∗, βnJ zn+ (1 − βn)γn,0J θn+ (1 − βn)

N

X

i=1

γn,iJ tn,ii

+kβ_nJ z_n+ (1 − β_n)[γ_n,0J θ_n+

N

X

i=1

γ_n,iJ t_n,i]k²

≤ kx^∗k²− 2β_nhx^∗, J zni − 2(1 − β_n)γn,0hx^∗, J θni

−2(1 − β_n)

N

X

i=1

γn,ihx^∗, J tn,ii + β_nkJ z_nk²

+(1 − β_n)kγ_n,0J θ_n+

N

X

i=1

γ_n,iJ t_n,ik²

≤ kx^∗k²− 2β_nhx^∗, J z_ni − 2(1 − β_n)γ_n,0hx^∗, J θ_ni

−2(1 − β_n)

N

X

i=1

γ_n,ihx^∗, J t_n,ii + β_nkJ z_nk²

+(1 − βn)γn,0kJ θ_nk²+ (1 − βn)

N

X

i=1

γn,ikJ t_n,ik²

= β_nφ(x^∗, z_n) + (1 − β_n)γ_n,0φ(x^∗, θ_n) + (1 − β_n)

N

X

i=1

γ_n,iφ(x^∗, t_n,i).

Since tn,i∈ T_iθ_n and Ti, i = 1, 2, 3, . . . , N are quasi−φ−nonexpansive multivalued mappings, we obtain

φ(x^∗, un) ≤ β_nφ(x^∗, zn) + (1 − βn)γn,0φ(x^∗, θn) + (1 − βn)

N

X

i=1

γn,iφ(x^∗, θn)

= β_nφ(x^∗, z_n) + (1 − β_n)φ(x^∗, θ_n).

By Lemma 3.3, we get φ(x^∗, u_n) ≤ β_nh

φ(x^∗, θ_n) −

1 − η_n η_n+1µ

φ(y_n, θ_n) −

1 − η_n η_n+1µ

φ(z_n, y_n)i +(1 − β_n)φ(x^∗, θ_n)

= φ(x^∗, θ_n) − β_n

1 − η_n ηn+1

µ

φ(y_n, θ_n) − β_n

1 − η_n ηn+1

µ

φ(z_n, y_n).

Since lim

n→∞

ηn

ηn+1µ = µand 0 < µ < 1, then there exists a natural number N0such that 0 < _η^η_n+1ⁿ µ < 1 ∀n ≥ N₀. Thus, ∀n ≥ N0,we have

φ(x^∗, un) ≤ φ(x^∗, θn), which implies x^∗ ∈ C_n+1, that is F ⊂ Cn+1. Hence F ⊂ Cn∀n ≥ 1. Step 4: We prove {xn}is Cauchy sequence.

Since xn+1= Π_Cn+1x₀ ∈ C_n+1⊂ C_n ∀n ≥ 1, then

φ(xn, x0) = φ(ΠCnx0, x0) ≤ φ(xn+1, x0), ∀n ≥ 1 (20) Also by Lemma 2.9, we obtain

φ(xn, x0) = φ(ΠCnx0, x0) ≤ φ(x^∗, x0) − φ(x^∗, xn)

≤ φ(x^∗, x0), ∀n ≥ 1. (21)

From (20) and (21), it follows that lim

n→∞φ(xn, x0) exists. This implies {φ(xn, x0)}is bounded and from (6) we have that {xn} is bounded. Observe from Lemma 2.9

φ(xn+1, xn) = φ(xn+1, Π_Cnx0) ≤ φ(xn+1, x0) − φ(xn, x0). (22) Therefore

n→∞limφ(x_n+1, x_n) = 0.

By Lemma 2.11, we get

n→∞limkx_n+1− x_nk = 0. (23)

From (22) and any m, n ∈ N with m > n, we obtain

φ(x_m, x_n) = φ(x_m, Π_Cnx₀) ≤ φ(x_m, x₀) − φ(x_n, x₀). (24)

Since the lim

n→∞φ(x_n, x₀) exists, we get

m,n→∞lim φ(x_m, x_n) = 0.

Again by Lemma 2.11 we get

m,n→∞lim kx_m− x_nk = 0. (25)

It follows from (25) that the sequence {xn} is Cauchy in C ⊂ E.

Step 5: We prove lim

n→∞kθ_n− t_n,ik = 0, lim

n→∞kz_n− y_nk = 0, lim

n→∞ky_n− θ_nk = 0. Observe from the scheme (7),

kθ_n− x_nk = α_nkx_n− x_n−1k.

Therefore from (23), we obtain

n→∞limkθ_n− x_nk = 0. (26)

Also from (23) and (26), we get

n→∞limkx_n+1− θ_nk = 0. (27)

Now,

φ(x_n+1, θ_n) = kx_n+1k²− 2hx_n+1, J θ_ni + kθ_nk²

= kx_n+1k²− 2hx_n+1− θ_n, J θni − kθ_nk²

= (kxn+1k − kθ_nk)(kx_n+1k + kθ_nk) − 2hx_n+1− θ_n, J θni

≤ kx_n+1− θ_nk(kx_n+1k + kθ_nk) + 2|hx_n+1− θ_n, J θ_ni|

≤ kx_n+1− θ_nk(kx_n+1k + kθ_nk) + 2kx_n+1− θ_nkkJ θ_nk.

Since {xn}, {θ_n} are bounded and the duality mapping J is uniformly norm-norm continuous on bounded subsets of E, it follows from (27) that

n→∞limφ(x_n+1, θ_n) = 0. (28)

From the scheme (7), xn+1 = Π_Cn+1x₀∈ C_n+1⊂ C_n. Hence φ(xn+1, un) ≤ (xn+1, θn).

Therefore from (28), we obtain

n→∞limφ(x_n+1, u_n) = 0 and consequently by Lemma 2.11, we obtain

n→∞limkx_n+1− u_nk = 0. (29)

From (27) and (29), we get

n→∞limku_n− θ_nk = 0. (30)

Observe that

φ(x^∗, θn) − φ(x^∗, un) = kθ_nk²− ku_nk²− 2hx^∗, J θn− J u_ni

≤ kθ_n− u_nk(kθ_n+ unk) + 2kx^∗kkJ θ_n− J u_nk.

From (30) and norm-to-norm uniform continuity of J on bounded sets, we obtain

n→∞lim(φ(x^∗, θn) − φ(x^∗, un)) = 0. (31)

Again, from the scheme (7)

φ(x^∗, u_n) = φ(x^∗, J⁻¹(β_nJ z_n+ (1 − β_n)[γ_n,0J θ_n+

N

X

i=1

γ_n,iJ t_n,i]))

= kx^∗k²− 2hx^∗, βnJ zn+ (1 − βn)γn,0J θn+ (1 − βn)

N

X

i=1

γn,iJ tn,ii

+kβ_nJ z_n+ (1 − β_n)[γ_n,0J θ_n+

N

X

i=1

γ_n,iJ t_n,i]k²

≤ kx^∗k²− 2β_nhx^∗, J zni − 2(1 − β_n)γn,0hx^∗, J θni

−2(1 − β_n)

N

X

i=1

γn,ihx^∗, J tn,ii + β_nkJ z_nk²

+(1 − β_n)kγ_n,0J θ_n+

N

X

i=1

γ_n,iJ t_n,ik². (32)

Since {θn}is bounded, tn,i∈ T_iθ_n, i = 1, 2, . . . , N and Ti are quasi−φ−nonexpansive multivalued mappings, it follows that {tn,i} is bounded for each i ∈ {1, 2, . . . , N}. Let r = max

1≤i≤Nsup

n≥1

{kθ_nk, kt_n,ik}. Since E is uniformly smooth, then E^∗ is unifromly convex, therefore, from (32) and Lemma 2.10, we have

φ(x^∗, u_n) ≤ kx^∗k²− 2β_nhx^∗, J z_ni − 2(1 − β_n)γ_n,0hx^∗, J θ_ni

−2(1 − β_n)

N

X

i=1

γ_n,ihx^∗, J t_n,ii + β_nkJ z_nk²

+(1 − βn)γn,0kJ θ_nk²+ (1 − βn)

N

X

i=1

γn,ikJ t_n,ik²

−(1 − β_n)γ_n,0γ_n,if (kJ θ_n− J t_n,ik)

= β_nφ(x^∗, z_n) + (1 − β_n)γ_n,0φ(x^∗, θ_n) + (1 − β_n)

N

X

i=1

γ_n,iφ(x^∗, t_n,i)

−(1 − β_n)γn,0γn,if (kJ θn− J t_n,ik)

≤ β_nφ(x^∗, zn) + (1 − βn)φ(x^∗, θn) − (1 − βn)γn,0γn,if (kJ θn− J t_n,ik).

By Lemma 3.3, we obtain

φ(x^∗, u_n) ≤ φ(x^∗, θ_n) − β_n

1 − η_n ηn+1

µ

φ(y_n, θ_n) − β_n

1 − η_n ηn+1

µ

φ(z_n, y_n)

−(1 − β_n)γn,0γn,if (kJ θn− J t_n,ik). (33)

From (31), (33) and condition γn,i∈ (, 1 − ), we obtain

n→∞limf (kJ θn− J t_n,ik) = 0, ∀ i ∈ {1, 2, . . . , N }.

By the property of f, we get

n→∞limkJ θ_n− J t_n,ik = 0, ∀ i ∈ {1, 2, . . . , N }.

Since J⁻¹ uniformly norm-to-norm continuous on bounded subsets of E^∗, we have

n→∞limkθ_n− t_n,ik = 0, ∀ i ∈ {1, 2, . . . , N }. (34)

Also form (33), we have

n→∞limφ(yn, θn) = 0, lim

n→∞φ(zn, yn) = 0.

By Lemma 2.11, we obtain

n→∞limky_n− θ_nk = 0, lim

n→∞kz_n− y_nk = 0. (35)

Hence step 5 is proved.

Since {xn} is Cauchy and E is reexive Banach space, there exists p^∗ ∈ Esuch that xn→ p^∗ as n → ∞.

As C is closed, we have p^∗∈ C.

Step 6: We show p^∗ ∈ F = EP (g, C) ∩ (∩^N_i=1F (T_i)). Since xn→ p^∗ as n → ∞, then from (26),we have

n→∞limkθ_n− p^∗k = 0. (36)

From (34) and (36), we obtain

n→∞limkt_n,i− p^∗k = 0, ∀ i ∈ {1, 2, . . . , N }. (37) Since tn,i∈ T_iθnfor each i ∈ {1, 2, . . . , N}, then from (36), (37) and closedness of Ti, we have p^∗∈ F (T_i) ∀ i ∈ {1, 2, . . . , N }, i.e. p^∗ ∈ ∩^N_i=1F (T_i).

On the other hand

ky_n− p^∗k ≤ ky_n− θ_nk + kθ_n− p^∗k.

Therefore, using (35) and (36) we obtain

n→∞limky_n− p^∗k = 0. (38)

From Lemma 3.2, we have

η_ng(θ_n, y) − η_ng(θ_n, y_n) ≥ hy − y_n, J θ_n− J y_ni ∀ y ∈ C. (39) Since lim

n→∞ηn> min

n µ

2 max{c1,c2}, η0

o

> 0, then from (39), (38), (36), conditions (D1) and (D4), we obtain g(p^∗, y) ≥ 0, ∀ y ∈ C, i.e. p^∗∈ EP (g, C).

Step 7: Finally, we show p^∗ = ΠFx₀. Let ¯y = ΠFx0, then since p^∗ ∈ F, we have

φ(¯y, x0) ≤ φ(p^∗, x0). (40)

From the scheme (7), xn= Π_Cnx₀. Since ¯y ∈ F ⊂ Cn, we have φ(x_n, x₀) ≤ φ(¯y, x₀).

Also since φ(., y) is continuous and xn→ p^∗ as n → ∞, we obtain

φ(p^∗, x0) ≤ φ(¯y, x0). (41)

From (40) and (41), we have φ(p^∗, x₀) = φ(¯y, x₀).Thus, p^∗ = ¯y = ΠFx₀. This compeletes the proof.

Observe that if E is a real Hilbert space, then by Remark 2.1(1) algorithm (7) reduces to the following































η1 > 0, µ ∈ (0, 1), x0, x1 ∈ C₁ = H θn= xn+ αn(xn− x_n−1),

y_n=argmin

y∈C

{η_ng(θ_n, y) +¹₂kθ_n− yk²}, Γ_n= {z ∈ H : hθ_n− η_nw_n− y_n, z − y_ni ≤ 0}, zn=argmin

y∈Γn

{η_ng(yn, y) +¹₂kθ_n− yk²}, un= βnzn+ (1 − βn)[γn,0θn+PN

i=1γn,itn,i], Cn+1= {z ∈ Cn: kun− zk² ≤ kθ_n− zk²}, x_n+1 = P_Cn+1x₀, n ≥ 1

(42)

where wn∈ ∂₂g(θn, yn), tn,i∈ T_iθn, Ti, i = 1, 2, 3, . . . , N are quasi nonexpansive multivalued mappings and

η_n+1= (

min{_2(g(θ^{µ(kyn−θnk2+kzn−ynk2)}

n,zn)−g(θn,yn)−g(yn,zn)), η_n}, g(θ_n, z_n) − g(θ_n, y_n) − g(y_n, z_n) > 0.

η_n, Otherwise

Using (42), Theorem 3.4 reduces to the following Corollary;

Corollary 3.5. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Ti : H → 2^H, i = 1, 2, 3, . . . , N be nite family of closed quasi nonexpansive multivalued mappings. Assume g satises (D1)- (D5) and F = EP (g, C) ∩ (∩^N_i=1F (T_i)) 6= ∅. Let {αn}, {β_n} and {γn,i} be real sequences such that αn, β_n∈ (0, 1), γn,i ∈ (, 1 − ) for some  ∈ (0, 1) and γn,0+P_N

i=1γ_n,i = 1. Then the sequence {xn} generated by (42) converges strongly to p^∗ = PFx₀.

Remark 3.6. Theorem 3.4 extends the results of Yang and Liu [42] from Hilbert space to real uniformly convex and uniformly smooth Banach spaces and from single valued quasi nonexpansive mappings to nite family of multivalued quas−φ−nonexpansive mappings.

4. Numerical example

In this section, we demonstrate Theorem 3.4

Let E = R with k.k = |.| and hx, yi = xy. Let C = [−40, 40] and for i = 1, 2, 3, 4, let Ti : R → 2^Rbe dened by Tix = [_i+3^x ,^x_i]. It is clear that 0 ∈ ∩⁴_i=1F (T_i). Let p ∈ Tix, then p = ax for some a, _i+3¹ ≤ a ≤ ^x_i and

φ(0, p) = |0|²− 2h0, pi + |p|²

= |p|²= |ax|²= a²|x|²

≤ 1

i²|x|²

≤ |x|²

= |0|²− 2h0, xi + |x|²

= φ(0, x).

Thus, Ti is quasi−φ−nonexpansive multivalued mapping for each i ∈ {1, 2, 3, 4}.

Dene g(x, y) = y²+ 6xy − 7x². It is easy to see 0 ∈ EP (g, C). Also g satises (D1), (D2) with ∂2g(x, y) = 2y + 6x, (D3) and (D4). If φ(x, y) = (x − y)², then

g(x, y) + g(y, z) = z²+ 6xy + 6yz − 7x²− 6y²

= z²+ 6xy − 7x²+ 6yz − 6y²

= g(x, z) − 3(y − x)²− 3(z − y)²+ 3(z − x)²

= g(x, z) − 3φ(y, x) − 3φ(z, y) + 3φ(z, x)

≥ g(x, z) − 3φ(y, x) − 3φ(z, y).

Thus, g satises (D5) with c1 = c₂ = 3. Furthermore if ηn= ²₅, µ = ₁₀⁷ , α_n= ₁₀³, β_n= _3n+2ⁿ ,  = ₁₀¹ , γ_n,0= γn,1 = γn,2 = γn,3 = γn,4 = ¹₅, then , µ, ηn, αn, βn and γn,i satisfy all the conditions of Theorem 3.4.

Therefore scheme 7 takes the following form;



























θ_n= x_n+ α_n(x_n− x_n−1), yn= ^1−6η_2η ⁿ

n+1θn,

Γn= {z ∈ R : hJθn− η_nwn− J y_n, z − yni ≤ 0}, zn= ^θn_2η^−6ηnyn

n+1 ,

θn

i+3 ≤ t_n,i≤ ^θ_iⁿ, i = 1, 2, 3, 4 un= βnzn+ (1 − βn)[¹₅θn+¹₅P4

i=1tn,i]), Cn+1= {z ∈ Cn: z ≤ ^θn+u₂ ⁿ},

xn+1= ΠCn+1x0= ^θn+u₂ ⁿ

(43)

Using (43) the numerical results using MATLAB is given in Figure 1 and Figure 2.

Number of iterations:n

0 10 20 30 40 50 60

Value of the sequence:x(n)

-15 -10 -5 0 5 10

15 Case 1

Figure 1: Convergence process of {(xn} with initial points x0 = 15, x₁ = −10

Number of iterations:n

0 10 20 30 40 50 60

Value of the sequence:x(n)

-40 -30 -20 -10 0 10 20 30

40 Case 2

Figure 2: Convergence process of {(xn} with initial points x0 = −35, x1 = 25

5. Conclusion

We studied an inertial hybrid self-adaptive subgradient extragradient algorithm in a real uniformly convex Banach space which is also uniformly smooth. Strong convergence Theorem was proved to approximate solutions of pseudomonotone equilibrium problems and xed points of quasi−φ−nonexpansive multivalued mappings. Numerical example was presented to show that our iteratative scheme is implementable.

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