Algorithms for Common Solutions to Generalized Mixed Equilibrium Problems and Fixed Point Problems under Nonlinear Transformations in Banach Spaces

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https://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352

DOI: 10.4236/jamp.2019.711179 Nov. 5, 2019 2632 Journal of Applied Mathematics and Physics

Algorithms for Common Solutions to

Generalized Mixed Equilibrium Problems

and Fixed Point Problems under Nonlinear

Transformations in Banach Spaces

Yanlai Song, Xinhong Chen

College of Science, Zhongyuan University of Technology, Zhengzhou, China

Abstract

The purpose of this paper is to present a new iterative scheme for finding a common solution of the generalized mixed equilibrium problems with an in-finite family of inverse strongly monotone mappings and the fixed point problems of demimetric mappings under nonlinear transformations in Ba-nach spaces. Applications are also included. The results in this paper are the extension and improvement of the recent results in the literature.

Keywords

Fixed Point, Demimetric Mapping, Generalized Mixed Equilibrium Problems, Banach Space

1. Introduction

Let H be a real Hilbert space, C be a nonempty closed convex subset of H, T be a mapping on C and F T

( ) {

:= x∈C Tx: =x

}

. Let A C: →H be a nonlinear

mapping, ϕ:C→ be a function and F be a bifunction from C C× to ,

where  is the set of real numbers. Then, we consider the following generalized mixed equilibrium problem (for short, GMEP): finding x∈C such that

(

,

)

( )

, 0.

F x y +ϕ y −ϕ x + Ax y−x ≥ (1)

The set of solutions of the GMEP is denoted by GMEP

(

F, ,ϕ A

)

(see [1] and

the references therein). Here some special cases of the GMEP are stated as fol-lowings:

1) If A=0, then the GMEP becomes the following mixed equilibrium

prob-How to cite this paper: Song, Y.L. and Chen, X.H. (2019) Algorithms for Com-mon Solutions to Generalized Mixed Equi-librium Problems and Fixed Point Prob-lems under Nonlinear Transformations in Banach Spaces. Journal of Applied Mathe-matics and Physics, 7, 2632-2649.

https://doi.org/10.4236/jamp.2019.711179

Received: September 30, 2019 Accepted: November 2, 2019 Published: November 5, 2019

http://creativecommons.org/licenses/by/4.0/

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DOI: 10.4236/jamp.2019.711179 2633 Journal of Applied Mathematics and Physics lem (for short, MEP):

(

)

( )

findingx∈Csuch thatF x y, +ϕ y −ϕ x ≥0, ∀ ∈y C, (2)

which was studied by Ceng and Yao [2]. The set of solutions of the MEP is de-noted by MEP

(

F,ϕ

)

.

2) If ϕ =0 and A=0, then the GMEP becomes the following equilibrium

problem (for short, EP):

(

)

findingx∈Csuch thatF x y, ≥0, ∀ ∈y C. (3)

This general form of the EP was first considered by Nikaido and Isoda [3]. The MEP and EP play an important role in many fields, such as economics, physics, mechanics and engineering sciences. Also, the MEP and EP include many ma-thematical problems as particular cases, for example, mama-thematical programming problems, complementary problems, variational inequality problems, Nash equi-librium problems in noncooperative games, minimax inequality problems and fixed point problems. Because of their wide applicability, equilibrium problems and mixed equilibrium problems have been generalized in various directions for the past several years; see, for example, [2] [4]-[9].

3) If F=0 and ϕ =0, then the GMEP reduces to the following classical

vari-ational inequality problem (for short, VIP) [10]:

, 0, .

Ax y−x ≥ ∀ ∈y C (4)

Since the VIP inception by Stampacchia [10] in 1964, it has received much at-tention due to its applications in a large variety of problems arising in structural analysis, economics, optimization, operations research and engineering sciences. Using the projection technique, one can easily show that is equivalent to the fixed- point problem; see, [7] [8] [11] [12] and the references therein.

Motivated by Ceng and Yao [2], Nikaido and Isoda [3] and Stampacchia [10], Peng and Yao [1] introduced the GMEP, which can be viewed as development and extension of the MEP, the EP and the VIP. It shows that the GMEP has ap-plications in physics, economics, finance, transportation, network and structural analysis, therapy, image reconstruction, and elasticity. The GMEP includes special cases, MEPs, EPs, VIPs, fixed point problems, complementarity problems, opti-mization problems, Nash equilibrium problems in noncooperative games, etc (see e.g., [6] [7] [8] and the references contained in them). In other words, the GMEP is a unifying model for several problems arising in several areas of study. In general, the GMEP involves nonlinear equations and there are no known methods to obtain closed form solutions for them. Consequently, several me-thods are being deployed to approximate their solutions, assuming existence. A number of iterative methods have been utilized to solve equilibrium problems, generalized equilibrium problems and mixed equilibrium problems (see e.g., [2] [4] [5] [13] and the references therein).

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DOI: 10.4236/jamp.2019.711179 2634 Journal of Applied Mathematics and Physics problems such as zero point problems for monotone operators, convex feasibili-ty problems, convex minimization problems, variational inequalifeasibili-ty and equili-brium problems, and so on; see [14]-[19] for more details.

At the same time, to construct a mathematical model which is as close as possi-ble to a real complex propossi-blem, we often have to use more than one constraint. Solving such problems, we have to obtain some solution which is simultaneously the solution of two or more subproblems or the solution of one subproblem on the solution set of another subproblem. These subproblems can be given, for example, by two or more different variational inequality problems or two or more different fixed point problems. It is natural to construct a unified approach for these problems. In this direction, several authors have introduced some iter-ative schemes for finding a common solution of fixed-point problems for nonli-near mappings, equilibrium problems and variational problems; see, for example,

[1] [2] [9] [12] [19] [20] and the references therein.

Recently, Takahashi [21] introduced a broad class of nonlinear mappings in a Banach space called k-demimetric mapping. This class mapping contains the classes of generalized hybrid mappings, k-strict pseudo-contractions, firmly-quasi-nonexpansive mappings, quasi-nonexpansive mappings and demicontractive map-pings.

Definition 1.1 Let E be a smooth Banach space and let C be a nonempty, closed and convex subset of E. Let k be a real number with k∈ −∞

(

,1

)

. A map-ping T C: →E with F T

( )

≠ ∅ is called k-demimetric if, for any x∈C

and q∈F T

( )

,

(

)

1 2

, .

2

k

x−q J x Tx− ≥ − x Tx− (5)

We give an example of a k-demimetric mapping which is not pseudo-contractive, hence it is not strictly pseudo-contractive.

Example 1.2 ([22]) Let H be the real line and C= −

[

1,1

]

. Define T on C by

( )

2 1

sin 3

T x x

x  

= _{ }

  if x≠0 and T

( )

0 =0. Clearly, 0 is the only fixed point of

T. Also, for x∈C,

( )

2

( )

2 2 1 2 2 2 2

( )

2

0 sin | 0

3 3

x

T x T x x x x k T x x

x  

− = = _{ } ≤ ≤ ≤ − + −

  for any

[ )

0,1

k∈ . Thus T is demimetric.

In order to find a common solution of fixed point problems for an finite fam-ily of demimetric mappings and the variational inequality problems for a infinite family of inverse strongly monotone mappings in a Hilbert space, Takahashi [12]

recently introduced and studied the following iterative algorithm:

(

)

(

)

(

)

(

)

(

)

1 1 1

1 ,

1 , ,

M

n j j n n j n

N

n i i C n i n

n n n n C n n n n n n

z I T x

w P B x

x u P x z w n

ξ λ λ

σ η

δ δ α β γ

=

+

 = − +



 ₌ ₋



= + − + + ∀ ∈



∑

(4)

DOI: 10.4236/jamp.2019.711179 2635 Journal of Applied Mathematics and Physics where

{ }

Tj M_j₌₁:C→H is a finite family of kj-demimetric and demiclosed

map-pings, and

{ }

1:

N i i

B ₌ C→H is a finite family of µi-inverse strongly monotone mappings. Then he obtained a strong convergence theorem under some mild re-strictions on the parameters.

Very recently, Akashi and Takahashi [14] proposed the following Mann’s type iteration for finding a common solution of fixed-point problems for an infinite family of demimetric mappings without assuming that demimetric mappings are commutative:

(

)

(

)

(

)

(

)

1 1

1 ,

1 , ,

n j j n n j n n C n n n n

z I T x

x P x z n

ξ λ λ

α α

∞ = +

 = − +

 

= + − ∀ ∈



∑



where

{ }

Tj _j ₁:C H ∞

= → is an infinite family of kj-demimetric and demiclosed mappings. Then they obtained a weak convergence theorem under certain appropriate assumptions on the parameters.

Most very recently, Takahashi [15] also introduced the following iteration process for finding a common solution of fixed-point problems with an infinite family of demimetric mappings and the variational inequality problems with an infinite family of inverse strongly monotone mappings in a Hilbert space:

(

)

(

)

(

)

(

)

(

)

1 1 1

1 ,

1 , ,

n

n j j n n j n

n i i n i n

n n n n C n n n n n n

z I T x

w J B x

x u P x z w n

η

ξ λ λ

σ η

δ δ α β γ

∞ = ∞ =

+

 = − +



 = −



= + − + + ∀ ∈



∑



where

{ }

Tj _j ₁:C H ∞

= → is an infinite family of kj-demimetric and demiclosed mappings,

{ }

Bi i 1:C H

∞

= → is an infinite family of µi-inverse strongly mono-tone mappings. Then they obtained a strong convergence theorem under some mild restrictions on the parameters.

On other hand, in order to find a common solution of equilibrium problems and the set of fixed point problems with generalized hybrid mappings, Alizadeh and Moradlou [23] introduced the following Ishikawa-like iteration process by applying the hybrid projection method:

(

)

(

)

(

)

{

}

{

}

2 2

1

such that , , 0, ,

1 ,

: ,

: , 0 ,

,

n n

n n n n n

n

n n n n n

n n n n n

n n n

n C Q

u H f u y y u u x y H

r

y z Su

z x Sx

C u H y u x u

Q u H x u x x

x P x

α α

β β

+

 ∈ + − − ≥ ∀ ∈

 

 = + −

 = + −



 ₌ _∈ ₋ _≤ ₋



 ₌ _∈ ₋ ₋ _≤

 =

 

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DOI: 10.4236/jamp.2019.711179 2636 Journal of Applied Mathematics and Physics Motivated and inspired by Takahashi [12], Akashi and Takahashi [14], Taka-hashi [15], Alizadeh and Moradlou [23], we put forward two questions:

1) Can these corresponding results in [12] [14] [15] [23] in Hilbert spaces be extended to the framework of Banach spaces (for example, lp for 1< < ∞p )?

2) Can we extend corresponding results in [12] [14] [15] [23] from finding a solution of the fixed point problems of generalized hybrid mappings or a com-mon solution of the equilibrium problems and fixed point problems of genera-lized hybrid mappings to the more general and challenging problem for finding a common solution of the generalized mixed equilibrium problems and the fixed point problems of demimetric mappings under nonlinear transformations?

The purpose of this paper is to give the affirmative answers to these questions mentioned above. In this paper, we present a new iterative scheme for finding a common solution of the generalized mixed equilibrium problems and fixed point problems of demimetric mappings under nonlinear transformations in Banach spaces. Applications are also included. Our results improve essentially the cor-responding results in [12] [14] [15] [23]. Further, some other results are also improved; see [9] [11] [16] [17] [18] [20] [24] [25].

2. Preliminaries

We denote E the real Banach space, *

E the dual of E, I the identity mapping on

E, and  the set of positive integers. The expressions xn→x and xnx de-note the strong and weak convergence of the sequence

{ }

xn , respectively. The (normalized) duality mapping J from E to *

E is defined by

{

_* _* _* 2 _*2

}

: ,

Jx= x ∈E x x = x = x

for all x∈E, where ⋅ ⋅, denotes the duality product. If E is a Hilbert space,

then J =I, where I is the identity mapping on H.

The norm of a Banach space E is said to be Gâteaux differentiable if the limit

0

lim

t

x ty x

t

→

+ −

exists for all x y, on the unit sphere S E

( )

=

{

x∈E: x =1

}

. In this case, we say

that E is smooth.

A Banach space E is said to be strictly convex if x−y <2 whenever

( )

,

x y∈S E and x≠ y. It is known that if E is strictly convex, then the duality

mapping J is injective, that is, x y, ∈E and x≠y imply JxJy= ∅. It is

known that E is reflexive if and only if J is surjective. Therefore, if E is a smooth, strictly convex and reflexive Banach space, then J is a single-valued bijection, see

[26] for more details.

Definition 2.1 A mapping T C: →H is said to be:

1) nonexpansive if Tx Ty− ≤ x−y for all x y, ∈C;

2) contractive if there exists a constant v∈

( )

0,1 such that

, , ;

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DOI: 10.4236/jamp.2019.711179 2637 Journal of Applied Mathematics and Physics 3) β-demicontractive if there exists a constant β∈

[ )

0,1 such that

( )

2 2 2

, , .

Tx Ty− ≤ −x y +β x Tx− ∀ ∈x C y∈F T

We use ΠC to denote the collection of mappings T verifying the above in-equality. That is

{

: : is a contraction with constant

}

.

C T C H T v

Π = →

Let D be a nonempty subset of C. A sequence

{ }

Tn of mappings of C into H is said to be stable on D (see [27]) if

{

Tn

( )

x :n∈

}

is a singleton for every x∈D. It

is clear that if { }Tn is stable on D, then Tn

( )

x =T x1

( )

for all n∈ and

x∈D_.

Lemma 2.2 In a Hilbert space H, it holds for all x y, ∈H and λ∈

[ ]

0,1 that

(

)

2 2

(

)

2

(

)

2

1 1 1 ,

x y x y x y

λ + −λ =λ + −λ −λ −λ −

which can be extended to the more general situation: for all x x1, 2,,xn∈H,

[ ]

0,1

i

λ ∈ , and 1 1

n i i=λ =

∑

, we have

2 1 1 2 2

2 2

2 2 1 1 2 1

1

.

n n

n n i j i j

i j n

x x x

x x x x x

λ

λ λ

≤ ≤ ≤

+ + +

= + + + −

∑

−

 

Lemma 2.3 ([19]) Let

{ }

αn be a sequence of real numbers such that there exists a subsequence

{ }

ni of

{ }

n such that αni <αni+1 for all i∈. Then there

exists a nondecreasing sequence

{ }

mk ⊆ such that mk → ∞ and the follow-ing properties are satisfied for all (sufficiently large) numbers k∈:

1 and 1.

k k k

m m k m

α ≤α ₊ α ≤α ₊

In fact, mk =max

{

j≤k:αj<αj+1

}

.

Lemma 2.4 ([28]) Let

{ }

αn be a sequence of nonnegative numbers satisfying the property:

(

)

1 1 , ,

n n n bn n nc n

α + ≤ −γ α + +γ ∈

where

{ } { } { }

γ_n , b_n , c_n satisfy the restrictions:

1) n1γn , limn γn 0

∞

→∞

= = ∞ =

∑

,

2) bn 0, n1bn

∞ =

≥

∑

< ∞_,

3) lim supn→∞cn≤0.

Then, limn→∞αn =0.

Lemma 2.5 ([21]) Let E be a smooth, strictly convex and reflexive Banach space and let

η

_{be a real number with}η∈ −∞

(

,1

)

. Let U be an

η

-demimetric mapping of E into itself. Then F U

( )

is closed and convex.

Lemma 2.6 ([16]) Let PC:H→C be a metric projection from H on a

non-empty closed convex subset C of H. Given x∈H and z∈C, then z=P x_C if

and only if there holds the relation

, 0, .

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DOI: 10.4236/jamp.2019.711179 2638 Journal of Applied Mathematics and Physics Recall that a mapping A C: →H is said to be α-inverse-strongly

mono-tone (ism) if there exists a constant

α

>0 such that

2

, , , .

Ax−Ay x−y ≥α Ax−Ay ∀x y∈C

Lemma 2.7 ([29]) If A C: →H is α-ism and

λ

is any constant in

(

0, 2α

]

,

then the mapping I−

λ

A is nonexpansive.

For solving the generalized mixed equilibrium problem, let us assume that the bifunction F C C: × → and the nonlinear mapping ϕ:C→ satisfy the

fol-lowing conditions:

(A1) F x x

( )

, =0 for all x∈C;

(A2) F is monotone, i.e., F x y

(

,

)

+F y x

(

,

)

≤0 for all x y, ∈C;

(A3) for each fixed y∈C, xF x y

(

,

)

is weakly upper semicontinuous;

(A4) for each fixed x∈C, yF x y

(

,

)

is convex and lower semicontinuous;

(A5) for each x∈C and r>0, there exists a bounded subset D_x ⊆C and

x

y ∈C such that, for any z∈C D\ x,

(

) ( ) ( )

1

, x x x , 0;

F z y y z y z z x

r

ϕ ϕ

− − + − − <

(A6) C is a bounded set.

Lemma 2.8 [2] Let C be a nonempty, closed and convex subset of H and let

:

F C C× → be a bifunction satisfying (A1)-(A4). Let ϕ:C→

{ }

+∞ be

a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r>0 and x∈H, define a mapping Kr :H→C as follows:

( )

( ) ( ) ( )

1

: , , 0, .

r

K x z C F z y y z y z z x y C

r

ϕ ϕ

 

=_ ∈ + − + − − ≥ ∀ ∈ _

 

Then, the following conclusions hold:

1) For each x∈H, K_r

( )

x ≠ ∅ and K_r is single-valued;

2) Kr is a firmly nonexpansive mapping, i.e., for all x y, ∈H,

2

, ;

r r r r

K x−K y ≤ K x−K y x−y

3) F K

( )

_r =MEP F

(

,ϕ

)

;

4) MEP F

(

,ϕ

)

is closed and convex;

5) 2

,

s t s t s

s t

K x K x K x K x K x x

s −

− ≤ − − _{for all} s t, >0 and x∈H.

3. Main Results

Throughout the rest of this paper, we always assume the following: 1) H is a real Hilbert space, and C is a nonempty closed subspace of H; 2) E is a smooth, strictly convex and reflexive Banach space, and J is the dual-ity mapping on E;

3) F is a bifunction from C C× to  satisfying (A1)-(A4);

4) Kr is a mapping defined as in Lemma 2.8; 5)

{ }

Aj _j₁:C H

∞

= → is an infinite family of µj-ism mappings with

{

}

inf j: j

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DOI: 10.4236/jamp.2019.711179 2639 Journal of Applied Mathematics and Physics 6) ϕ:C→ is a lower semicontinuous and convex function with

restric-tions (B1) or (B2);

7) B H: →E is a bounded linear operator such that B≠0 and *

B is the

adjoint operator of B; 8)

{ }

Ti i1:E E

∞

= → is an infinite family of ki-demimetric and demiclosed map-pings with k=sup

{

k_i:i∈

}

<1;

9) 1

( )

(

)

1 1

: i B F Ti j GMEP F, ,ϕ Aj

∞ − ∞

= =

Γ =

_



_

≠ ∅;

10)

{ }

fn ⊂ ΠC is stable on Γ.

Theorem 3.1 For any x1∈C, define a sequence

{ }

xn as follows:

(

)

(

)

(

)

(

)

* 1

1 1

,

, ,

n

n i i i n

n j j r n j n n C n n n n n n n

z I B J I T B x

y K I r A x

x P f x y z n

τ σ

δ

α β γ

∞ = ∞

= +

 = − −



 = −



 = + + ∀ ∈



∑



(6)

where a b, ,τ∈

(

0,+∞

)

,

{ } { } { } ( )

αn , βn , γn ⊂ 0,1 and

{ }

rn ,

{ }

δj ,

{ } (

σ ⊂i 0,+∞

)

satisfy the following conditions:

(i) limn→∞αn =0 and n1αn

∞

= = ∞

∑

,

(ii) 0<lim infn→∞βn ≤lim supn→∞βn <1 and αn+βn+γn =1, (iii) 2

1

0 k

B

τ −

< < _,

(iv) j1

δ

j 1

∞ = =

∑

and i 1σi 1

∞ = =

∑

, (v) 0< ≤ ≤ <a rn b 2µ.

Then the sequence

{ }

xn generated by (6) converges strongly to a point z0∈ Γ,

where z0 =P f zΓ 1 0.

Proof. Set *

(

)

1 i i i

Tx=

∑

∞₌σ B J I−T Bx for all n∈ and x∈C. Then we can

prove that T is well defined. In fact, we have, for any i∈ and z∈ Γ,

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

2 2

* *

2

2 * *

2

2 2 2

2 2

2 2 2

2

2 2 2

2 ,

1

1 ,

i i

i i i

i

I B J I T B x z x z B J I T Bx

x z x z B J I T Bx B J I T Bx

x z Bx Bz J I T Bx B I T Bx

x z k Bx T Bx B Bx T Bx

x z k B Bx T Bx x z

τ τ

− − − = − − −

≤ − − − − + −

≤ − − − − − ≤ −

(7)

which implies

(

)

(

*

)

.

i

I−τB J I−T B x− ≤ −z x z (8)

Thus,

(

)

* 2 _.

i

B J I T Bx x z

τ

− ≤ −

Then we see the mapping *

(

)

1 i i i

Tx=

∑

∞₌σB J I−T Bx converges absolutely for

(9)

DOI: 10.4236/jamp.2019.711179 2640 Journal of Applied Mathematics and Physics Furthermore, define V xn j 1δjKrn

(

I r A xn j

)

∞ =

=

∑

− for all n∈ and x∈C.

Then we can prove that Vn is nonexpansive. Indeed, it follows that Krn

(

1−r An j

)

is nonexpansive from (v), Lemma 2.7 and Lemma 2.8(2). We obtain from Lem-ma 2.5 that, for any xˆ j1GMEP

(

F, ,ϕ Aj

)

∞ =

∈

_

,

(

)

(

)

ˆ ˆ ˆ ˆ .

n n

r n j r n j

K I−r A x ≤ K I−r A x− +x x ≤ − +x x x

Thus V xn j 1δjKrn

(

I r A xn j

)

∞ =

=

∑

− converges absolutely for each x∈C.

Since Krn

(

I−r An j

)

is nonexpansive, we have that

(

)

(

r_n n j

)

GMEP

(

, , j

)

F K I−r A = F ϕ A is closed and convex. Furthermore, we know

from Lemma 2.5 that F T

( )

i is closed and convex for each i∈. Therefore, we

have that 1

( )

(

)

1 i 1GMEP , , j

i B F T j F ϕ A

∞ − ∞

= =

Γ =



is nonempty, closed and convex (note that B is linear and continuous). Thus we have that PΓ is well defined.

We derive from Lemma 2.8 that

(

)

1

.

n

n j r n j n n

j

y z

δ

K I r A x z x z

∞

=

− =

∑

− − ≤ − ₍₉₎

Noting (8), we have

(

)

(

*

)

1

.

n i i n n

i

z z σ I τB J I T B x z x z

∞

=

− ≤

∑

− − − ≤ − ₍₁₀₎

It follows from (9), (10) and (v) that

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

1 1 1 1 1 1 1 1 1

max , .

1

n n n n n n n n

n n n n n

n n n n n n n n

n n n n n

n n n

n

x z f x z y z z z

f x z x z

f x f z f z z x z

v x z f z z x z

v x z f z z

f z z

x z

v

α β γ

α α

α α α

α α + − ≤ − + − + − ≤ − + − − ≤ − + − + − − ≤ − + − + − − = − − − + −  −    ≤ _ − _ −    

By induction, we obtain

1 1

max , , ,

1

n

f z z

x z x z n

v  −    − ≤ _ − _ ∀ ∈ −     

which gives that the sequence

{ }

xn is bounded, so are

{

f xn n

}

,

{ }

yn and

{ }

zn . We obtain from (7) that

(

)

(

)

(

)

(

)

(

)

(

)

2 2 * 1 2 * 1 2 2 2 1

2 2 2

1

1 .

n i i n

i

i i n

i

i n i n

i

n i n i n

i

z z I B J I T B x z

I B J I T B x z

x z k B Bx T Bx

τ σ

σ τ

σ τ τ

τ τ σ

(10)

DOI: 10.4236/jamp.2019.711179 2641 Journal of Applied Mathematics and Physics It follows from (9), (11) and Lemma 2.2 that

(

)

(

)

(

)

(

)

(

)

2 2

1

2 2 2

2

2 2 2

1

2 2 2

2 2

1

1 ,

n n n n n n n n

n n n n n n n n n n n

n n n n n n n n n

n n i n i n

i n n n n n n n n

n i n i n

i

x z f x z y z z z

f x z y z z z

y z

f x z x z y z

x z k B Bx T Bx

f x z x z y z

k B Bx T Bx

α β γ

β γ

α β β γ

γ τ τ σ

α β γ

γ τ τ σ

+

∞ =

− ≤ − + − + −

≤ − + − + −

− −

≤ − + − − −

 

+ _ − − − − − _

 

≤ − + − − −

− − − −

∑

which means that

(

)

2 2 2

1

2 2 2

1

.

n n n n n i n i n

i

n n n n n

y z k B Bx T Bx

f x z x z x z

β γ

γ τ

τ

σ

α

∞

=

+

− + − − −

≤ − + − − −

∑

(12)

Case 1. Assume there exists some integer m>0 such that

{

x_n−z₀

}

is

de-creasing for all n≥m. In this case, we deduce that limn→∞ xn−z0 exists. From

(12), conditions (i), (ii), (iii) and (v), we deduce

2 1

lim i n i n 0

n _i σ Bx T Bx

∞

→∞ =

∑

− = (13) and

lim n n 0.

n→∞ y −z = (14) From (6) and (13), we get that

(

)

2

2 *

1 2 2

1

.

n n n i i n

i

i n i n

i

x z x I B J I T B x

Bx T Bx τ σ

τ σ

∞

= ∞

=

 

− = −_ − − _

 

≤ −

∑

Hence, we have

lim n n 0.

n→∞ x −z = (15) Since

{ }

xn is bounded, there exists a subsequence

{ }

xns of

{ }

xn satisfying

s

n

x x∈C. Without loss of generality, we may also assume

1 0 0 0 1 0 0 0

lim , lim sup , .

s

n n

s→∞ f z −z x −z = _n_→∞ f z −z x −z (16) Because B is bounded and linear, we see that Bxns Bx. This together with

(13) implies 1

( )

i

x∈B F T− for each i∈. And hence, x i ₁B F T1

( )

i

∞ − =

∈



_

.

Next let us prove that x i 1GMEP

(

F, ,ϕ Ai

)

∞ =

∈

(11)

DOI: 10.4236/jamp.2019.711179 2642 Journal of Applied Mathematics and Physics

(

)

(

)

0 0

1

2 1

, ,

1

. 2

n

n n n j n n r n j n

j

j r n j n n

j

x z x y x z x K I r A x

K I r A x x

δ

∞

= ∞

=

− − = − − −

≥ − −

∑

This together with (14) and (15) implies, for any i∈, that

(

)

lim 0.

n

r n j n n

n→∞ K I−r A x −x = (17) Consider a subsequence

{ }

rn_{s j} of

{ }

rns corresponding to the sequence

{ }

xns .

Since the subsequence

{ }

rn_{s j} of

{ }

rns is bounded, we have that there exists a

subsequence

{ }

rn_sk of

{ }

rn_{s j} such that limk→∞rn_sk =r. For such r, we have from Lemma 2.8 (5) that

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

.

sk sk

sk nsk sk nsk sk sk sk

nsk sk nsk sk sk sk

sk sk

nsk sk sk sk sk sk

r j n n

r j n r j n

r n j n n

r j n r n j n

n

r j n j n

r n j n n n j n

K I rA x x

K I rA x K I rA x

K I r A x x

K I rA x K I r A x

r r

K I rA x I rA x

r

K I r A x x r r A x

− −

≤ − − −

+ − −

+ − − −

−

≤ − − −

+ − − + −

On the other hand, since Kr and

(

I−rAj

)

are Lipscitz, noting (17), we in-fer for any j∈ that

(

)

lim _s _s 0.

k k

r j n n

k→∞ K I−rA x −x = (18) Therefore, we obtain x j1F K

(

r

(

I rAj

)

j1GMEP

(

F, ,ϕ Aj

)

∞ ∞

= =

∈ − =





.

It follows from (16) and Lemma 2.6 that

1 0 0 0

1 0 0 0 1 0 0 0

1 0 1 0 1 0

lim sup ,

lim , ,

, 0.

s

n n

n s

f z z x z

f z z x z f z z x z

f z P f z x P f z

→∞

Γ Γ

− −

= − − = − −

= − − ≤

 

(19)

Putting hn =αnf xn n+βnyn+γn nz for all n∈, we have from (6) that

1

n C n

x+ =P h . Since

{ }

fn is stable on Γ, we then get by (9), (10) and Lemma 2.6 that

(

)

(

)

2

1 0

0 0 0

0 1 0

0 0 1 0 0 1 0

, ,

,

n

C n n C n n C n n n n n n n n n

n n n n n n n n n

x z

P h h P h z h z P h z

f x y z z x z

y z z z x z f x z x z

α β γ

β γ α

+

+ +

−

= − − + − −

≤ + + − −

(12)

(

)

(

)

(

)

0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0

2

0 1 0 0 1 0

, ,

1

,

1 1 , .

n n n n n n

n n n n n n n n

n n n n n n

n n

n n n n

y z x z z z x z

f x f z x z f z z x z

x z x z v x z x z

f z z x z

v x z f z z x z

β γ

α α

α

α α

+ +

+

≤ − − + − −

+ − − + − −

≤ − − − + − −

+ − −

≤ − − − + − −

This together with Lemma 2.4 and (19) implies xn→z0 as n→ ∞.

Case 2: Suppose that there exists

{ }

ni of

{ }

n such that xni −z0 < xni+1−z0

for all i∈. Then by Lemma 2.3, there exists a nondecreasing sequence

{ }

m_j

in  such that

0 1 0 and 0 1 0 .

j j j

m m j m

x −z ≤ x + −z x −z ≤ x + −z (20)

Without loss of generality, there exists a subsequence

{ }

xm_jk of

{ }

xm_j such that xm_jk z for some z∈C and

1 0 0 0 1 0 0 0

lim , _j lim sup , _j .

k

m m

k→∞ f z −z x −z = _j_→∞ f z −z x −z

We show that

1 0 0 0

lim sup , 0,

j

m j

f z z x z

→∞ − − ≤ (21) where z0 =P f zΓ 1 0. To see this, we can first obtain z∈ Γ by a similar argument

as in Case 1. Therefore, we deduce that

1 0 0 0

1 0 0 0 1 0 1 0 1 0

lim sup ,

lim , , 0.

j

jk

m j

m k

f z z x z

f z z x z f z P f z z P f z

→∞

Γ Γ

→∞

− −

= − − = − − ≤ (22)

Like in Case 1, we can also get that

lim 0 and lim 0.

j j j j

m m m m

j→∞ y −z = j→∞ x −z = (23) Observing that

(

)

1

1 ,

j j

j j j j j j j j

j j j j j j j j j j

j j j j j j j j j j j j j

j j j j j j j j j j

m m

m m m m m m m m

m m m m m m m m m m

m m m m m m m m m m m m m

m m m m m m m m m m

x x

f x y z x

f x x y x z x

f x x y z z x z x

f x x z x y z

α β γ

α β β γ

α α β

+ −

≤ + + −

≤ − + − + −

≤ − + − + − + −

= − + − − + −

we then find from (23) and (i) that

1

lim 0.

j j

m m

j→∞ x + −x = (24)

Putting hmj =

α

mj f xmj mj +

β

mjymj +

γ

mjzmj for all j∈, we obtain by

(13)

(

)

(

)

2 1 0

0 0 0

0 1 0

0 0 1 0

0 1 0

, ,

,

j

j j j j j

j j j j j j j j

j j j j j

j j j j

m

C m m C m m C m

m m m m m m m m

m m m m m

m m m m

x z

P h h P h z h z P h z

f x y z z x z

y z z z x z

f x z x z

α β γ

β γ

α

+

−

= − − + − −

≤ + + − −

≤ − + − −

+ − −

(

)

0 1 0 0 1 0

0 1 0 0 0 1

0 0 0

0 1 0 0 1 0

0 0 1 0 0 0

, ,

,

1

,

j j j j j j

j j j j j j j j j

j j j

j j j j j j

j j j j j j j

m m m m m m

m m m m m m m m m

m m m

m m m m m m

m m m m m m m

y z x z z z x z

f x f z x z f z z x x

f z z x z

x z x z v x z x z

f z z x x f z z x z

β γ

α α

α

α α

+ +

+

≤ − − + − −

+ − − + − −

+ − −

≤ − − − + − −

+ − − + − −

(

)

(

)

(

)

(

)

0 1 0

1 0 0 1 1 0 0 0 2

1 0 1 0 0 1 1 0 0 0

1 1

,

1 1

, ,

j j j

j j j j j

j j

m m m

m m m m m

m m

v x z x z

f z z x x f z z x z

v x z f z z x x

f z z x z

α

α α

α

+

+ +

≤ − − − −

+ − − + − −

≤ − − − + − −

+ − −

which means that

2

1 0 1 0 0 1 1 0 0 0

1 1

, .

1 1

j j j j

m m m m

x z f z z x x f z z x z

v v

+ − ≤ ₋ − + − + ₋ − −

Noticing (22) and (24), we deduce

1 0

lim 0.

j

m

j→∞ x + −z = We can also obtain by (20) that

0 1 0

lim lim 0.

j

j m

j→∞ x −z ≤ j→∞ x + −z =

Consequently, we get xj→z0 as j→ ∞.

Remark 3.2 Theorem 3.1 extends, improves and develops Theorem 3.1 of Takahashi [12], Theorem 3.1 of Akashi and Takahashi [14], Theorem 3.1 of Ta-kahashi [15] and Theorem 3.1 of Alizadeh and Moradlou [23] in the following aspects:

• _{Theorem 3.1 improves and develops corresponding results in}_[23]_from

ge-neralized hybrid mappings to demimetric mappings;

• _{Theorem 3.1 extends, improves and develops corresponding results in}_[12]

[14] and [15] from finding a common solution of fixed-point problems and the variational inequality problems in Hilbert spaces to the more general and challenging problem for finding a common solution of the generalized mixed equilibrium problems and the null point problems in Banach spaces;

(14)

DOI: 10.4236/jamp.2019.711179 2645 Journal of Applied Mathematics and Physics given in [12] [14] [15] [23];

• _{The algorithm 6 is more advantageous and more flexible than the ones given}

in [12] [14] [15] [23]. Therefore, the new algorithm is expected to be widely applicable.

4. An Extension of Our Main Results

From Theorem 3.1, we deduce immediately the following results Corollary 4.1 Suppose 1

( )

(

)

1: _i₁B F Ti MEP F,ϕ

∞ ₋ =

Γ =



 ≠ ∅. For x1∈C,

define a sequence

{ }

xn as follows:

(

)

(

)

(

)

* 1

1

,

, ,

n

n i i i n

n r n

n C n n n n n n n

z I B J I T B x

y K x

x P f x y z n

τ σ

α β γ

∞ =

+

 = − −



 =



= + + ∀ ∈



∑



(25)

where τ∈

(

0,+∞

)

,

{ } { } { } ( )

α_n , β_n , γ_n ⊂ 0,1 and

{ } { } (

r_n , σ_i ⊂ 0,+∞

)

satisfy

the following conditions: 1) limn→∞αn =0 and n 1αn

∞

= = ∞

∑

,

2) 0<lim infn→∞βn ≤lim supn→∞βn <1 and αn+βn+γn =1,

3) 2

1

0 k

B

τ −

< < _and i₁σi 1

∞ = =

∑

.

Then the sequence

{ }

xn generated by (25) converges strongly to a point

0 1

z ∈ Γ , where

1

0 0

z =P fz_Γ .

Corollary 4.2 Let

{ }

Ti i1:C H

∞

= → be an infinite family of directed and demic-losed mappings. For x1∈C, define a sequence

{ }

xn as follows:

(

)

(

)

(

)

(

)

* 1

1 1

,

, ,

n

n i i i n

n i j r n j n n C n n n n n n n

z I B J I T B x

y K I r A x

x P f x y z n

τ σ

δ

α β γ

∞ = ∞

= +

 = − −



 ₌ ₋



 = + + ∀ ∈



∑



(26)

where τ∈

(

0,+∞

)

,

{ } { } { } ( )

α_n , β_n , γ_n ⊂ 0,1 and

{ } { } (

r_n , σ_i ⊂ 0,+∞

)

satisfy

the following conditions: 1) limn→∞αn =0 and n 1αn

∞

= = ∞

∑

,

2) 0<lim infn→∞βn ≤lim supn→∞βn <1 and αn+βn+γn =1,

3) 2

1 0

B τ

< < _,

4) j1

δ

j 1

∞ = =

∑

and i1σi 1

∞ = =

∑

, 5) 0< ≤ ≤ <a rn b 2µ.

Then the sequence

{ }

xn generated by (26) converges strongly to a point

0

z ∈ Γ, where z0=P f zΓ 1 0.

(15)

5. Numerical Examples

In this section, we discuss the direct application of Theorem 3.1 on a typical ex-ample on a real line.

Example 5.1 Let C=H = =E  with the inner product defined by x y, =xy

for all x y, ∈ and the standard norm ⋅. Let Ti:→, Ai:→,

:

M  → and fn:→ be defined by

1 1 1

, , 4 and , , .

1

i i n

i i

T x x A x x Mx x f x x x i

i i n

+ +

= − = = = ∀ ∈ ∈

+  

It is easy to check that Γ =

{ }

0 , T_i is an infinite family of 1

1 2+ i -demime-

tric and demiclosed mappings,

{ }

Ai i1

∞

= is an infinite family of ₁

i i

+ -ism

map-pings, and fn is

1

2-contractive on H and stable on Γ. Let ϕ =0 for all x∈,

then we see

ϕ

satisfies (A5). Letting 3

2

Bx= x for all x∈, we then see B is a bounded linear operator

with its adjoint *

B =B. Note that * 3

2

B = B = . Define a bifunction

:

F C C× → by

( )

2 2

, 2 3 .

F z y =y + zy− z

We then find that F satisfies (A1)-(A4). So, by Lemma 2.8, we have Kr

( )

x is nonempty and single-valued for each x∈C. Hence, for any r>0, there exists

z∈C such that

( )

( ) ( )

1

, , 0, ,

F z y y z y z z x y C

r

ϕ ϕ

+ − + − − ≥ ∀ ∈

which is equivalent to

(

)

(

)

2 2 2

2 3 0, .

ry + rz+ −z x y+ zx− rz −z ≥ ∀ ∈y C

After solving the above inequality, we get

1 4

x z

r =

+ , i.e. r 1 4

x K x

r =

+ .

Let us choose 1

9

τ = _, 1

6

n

α = _, 1

3

n

n n

β = + _, 4 3

6

n

n n

γ = − _, 2 1

4

n

n r

n −

= _{, and} 1

2

i i i

δ =σ = _{(choosing other values of these variables arbitrarily which satisfy}

the conditions of Theorem 3.1, the same convergence result also can be ob-tained). Then τ , αn, βn, γn, rn, δi and σi satisfy all the conditions of Theorem 3.1. Then (6) can be rewrite as

(

)

(

)

1

1 4 ln 2 ,

9 9

2 1 1 1 ln 2

3 1 4

1 1 4 3

, .

6 1 3 6

n n

n n n n

z x

n n

y x

n n

x x y z n

n n n n

+

 ₌ ₋ 

 _ _



 −

 ₌  ₋ ₊ 

 _ _

−  



 ₊ ₋

= + + ∀ ∈



+

(16)

DOI: 10.4236/jamp.2019.711179 2647 Journal of Applied Mathematics and Physics It is not hard to estimate that

1 1

,

2 2

n n n

x₊ ≤ x ≤≤ x

which shows xn→ ∈ Γ0 .

6. Conclusion

The present work has been aimed to theoretically establish a new iterative scheme for finding a common solution of the generalized mixed equilibrium problems with an infinite family of inverse strongly monotone mappings and the fixed point problems of demimetric mappings under nonlinear transformations in Ba-nach spaces. Our results can be viewed as improvement, supplementation, devel-opment and extension of the corresponding results in some references to a great extent.

Acknowledgements

This research was supported by the Key Scientific Research Projects of Higher Education Institutions in Henan Province (20A110038).

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this pa-per.

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