(张石生)

Center for General Education,China Medical University,Taichung 40402,Taiwan,China E-mail:changss2013@163.com

Jinfang TANG (唐金芳)

Department of Mathematics,Yibin University,Yibin 644007,China E-mail:jinfangt 79@163.com

Chingfeng WEN (温庆丰)

Department of Medical Research,Kaohsiung Medical University Hospital,Kaohsiung 80708,Taiwan,China E-mail:cfwen@kmu.edu.tw

Abstract In this article,we propose a new algorithm and prove that the sequence generalized by the algorithm converges strongly to a common element of the set of fixed points for a quasi-pseudo-contractive mapping and a demi-contraction mapping and the set of zeros of monotone inclusion problems on Hadamard manifolds.As applications,we use our results to study the minimization problems and equilibrium problems in Hadamard manifolds.

Key words Monotone inclusion problem;quasi-pseudo-contractive mapping;demi-contraction mapping;maximal monotone vector field;quasi-nonexpansive mappings;Hadamard manifold

1 Introduction

Rockafellar[1]considered the inclusion problem of finding

where

B

is a set-valued maximal monotone mapping de fined on a Hilbert space

H

.He developed an elegant method,known as the proximal point algorithm(PPA),to solve this inclusion problem.

During the last two decades,inclusion problem(1.1)has been extended and generalized in many directions because of its applications to different areas in science,engineering,management and the social sciences(see,for example,[2–9]and the references therein).

Recently,many convergence results attained by the proximal point algorithm have been extended from the classical linear spaces to the setting of manifolds(see,for examle,[10–14]).Li et al.[10]developed the proximal point method for problem(1.1)in the setting of Hadamard manifolds.Later,Li et al.[11]extended the Mann and Halpern iteration scheme for finding the fixed points of nonexpansive mappings from Hilbert spaces to Hadamard manifolds.Very recently,Ansari et al.[12]and Al-Homidan-Ansari-Babu[13]considered the problem of finding

in a Hadamard manifold,where

T

is a nonexpansive mapping,

B

is a set-valued maximal monotone mapping,and

A

is a single-valued continuous and monotone mapping.They proposed some Halpern-type and Mann-type iterative methods.They proved that,under suitable conditions,the sequence generated by the algorithm converges strongly to a common element of the set of fixed points of the mapping

T

and the set of solutions of the inclusion problem.

Motivated and inspired by the works in[5–8]and[11–14],in this article we consider the problem of finding

in the setting of Hadamard manifolds,where

S

is a quasi-pseudo-contractive mapping,

U

is a demi-contractive mapping,

B

is a set-valued maximal monotone mapping and

A

is a singlevalued and monotone mapping such that

A

+

B

is maximal monotone.We propose a new type of algorithm and prove that the sequences generalized by the algorithm converge strongly to a common element of problem(1.3).As applications we apply our results to study the minimization problems and equilibrium problems in Hadamard manifolds.

2 Preliminaries

The Riemannian distance d(

p,q

)is the minimal length over the set of all such curves joining

p

to

q

,which induces the original topology on

M

.A Riemannian manifold

M

is complete if,for any

p

∈

M

,all geodesics emanating from

p

are de fined for all

t

∈R.A geodesic joining

p

to

q

in

M

is said to be a minimal geodesic if its length is equal to d(

p,q

).A Riemannian manifold

M

equipped with Riemannian distance d is a metric space(

M,

d).By the Hopf-Rinow Theorem[15],if

M

is complete,then any pair of points in

M

can be joined by a minimal geodesic.Moreover,(

M,

d)is a complete metric space and bounded closed subsets are compact.

It is known that exp

tv

=

γ

(

t,p

)for each real number

t

.It is easy to see that exp0=

γ

(0

,p

)=

p

,where 0 is the zero tangent vector.Note that the exponential map expis differentiable on

T

M

for any

p

∈

M

.

De finition 2.2

A complete simply connected Riemannian manifold of non-positive sectional curvature is called a Hadamard Manifold.

Proposition 2.3

([15])Let

M

be a Hadamard manifold.Then,for any two points

x,y

∈

M

,there exists a unique normalized geodesic

γ

:[0

,

1]→

M

joining

x

=

γ

(0)to

y

=

γ

(1)which is in fact a minimal geodesic denoted by

The following inequalities can be proved easily:

Lemma 2.4

Let

M

be a finite dimensional Hadamard manifold.(i)Let

γ

:[0

,

1]→

M

be a geodesic joining

x

to

y

.Then we have

(From now on d(

x,y

)denotes the Riemannian distance).(ii)For any

x,y,z,u,w

∈

M

and

t

∈[0

,

1],the following inequalities hold:

Let

M

be a Hadamard manifold.A subset

C

⊂

M

is said to be geodesic convex if,for any two points

x

and

y

in

C

,the geodesic joining

x

to

y

is contained in

C

.In the sequel,unless otherwise speci fied,we always assume that

M

is a finite dimensional Hadamard manifold,and

C

is a nonempty,bounded,closed and geodesic convex set in

M

,and Fix(

S

)is the fixed point set of a mapping

S

.A function

f

:

C

→(−∞

,

∞]is said to be geodesic convex if,for any geodesic

γ

(

λ

)(0≤

λ

≤1)joining

x,y

∈

C

,the function

f

◦

γ

is convex,that is,

De finition 2.7

A mapping

S

:

C

→

C

is said to be(1)contractive if there exists a constant

k

∈(0

,

1)such that

If

k

=1,then

S

is said to be nonexpansive,and(2)quasinonexpansive if Fix(

S

)/=∅and

(3) firmly nonexpansive[18]if for all

x,y

∈

C

,the function

φ

:[0

,

1]→[0

,

∞]de fined by

is nonincreasing;

(4)

k

-demicontractive[19]if Fix(

S

)/=∅and there exists a constant

k

∈[0

,

1)such that

(5)quasi-pseudo-contractive if Fix(

S

)/=∅and

Proposition 2.8

([18])Let

S

:

C

→

C

be a mapping.Then the following statements are equivalent:(i)

S

is firmly nonexpansive;(ii)for any

x,y

∈

C

and

t

∈[0

,

1]

(iii)for any

x,y

∈

C

Lemma 2.9

If

S

:

C

→

C

is a firmly nonexpansive mapping and Fix(

S

)/=∅,then for any

x

∈

C

and

p

∈Fix(

S

),the following conclusion holds:

Proof

For given points

x

∈

C,p

∈Fix(

S

)and

Sx

,we consider a geodesic triangle△(

p,Sx,x

).By a comparison theorem for triangle([15]Proposition 4.5),we have

Since

S

:

C

→

C

is firmly nonexpansive,taking

y

=

p

in(2.9),we have

This,together with(2.11),shows that

The conclusion of Lemma 2.9 is proved.

Remark 2.10

From De finition 2.7 and Lemma 2.9,it is easy to see that if Fix(

S

)/=∅,then the following implications hold:

but the converse is not true.In fact,if Fix(

S

)/=∅and

S

is firmly nonexpansive,then,by(2.10),it is quasi-nonexpansive;therefore it is demicontractive and so it is quasi-pseudo-contractive.These show that the class of quasi-pseudo-contractive mappings is more general than the classes of quasinonexpansive mappings,firmly nonexpansive mappings and

k

-demicontractive mappings.In the sequel,we denote by X(

M

)the set of all set-valued vector fields

A

:

M

⇉

TM

such that

A

(

x

)⊂

T

M

for all

x

∈

M

,and we let the domain D(

A

)of

A

be de fined by D(

A

)={

x

∈

M

:

A

(

x

)/=∅}.

De finition 2.11

A set-valued vector field

A

∈X(

M

)on a Hadamard manifold

M

is said to be(1)monotone if,for any

x,y

∈D(

A

),

(2)maximal monotone if it is monotone and for all

x

∈D(

A

)and

u

∈

T

M

,the condition

implies

u

∈

A

(

x

);

is called the resolvent of

A

of order

λ>

0.

De finition 2.13

A mapping

T

:

C

→

C

is said to be demiclosed at 0 if,for any sequence{

x

}⊂

C

such that

x

→

x

and d(

x

,Tx

)→0,then

x

∈Fix(

T

).

3 Main Results

First we give following Lemma,which will be needed in proving our main result:

Lemma 3.1

Let

M

be a Hadamard manifold and

T

:

C

→

C

be a mapping which is

L

-Lipschitzian(

L

≥1)and demiclosed at 0.Let

G

:

C

→

C

and

K

:

C

→

C

be two mappings de fined by

(1)Fix(

T

)=Fix(

T

◦

G

)=Fix(

K

);(2)

K

is also demiclosed at 0;(3)

K

:

C

→

C

is

L

-Lipschitzian;(4)In addition,if

T

:

C

→

C

is quasi-pseudo-contractive,then

K

:

X

→

X

is a quasinonexpansive mapping,that is,for any

x

∈

C

and

p

∈Fix(

K

)(=Fix(

T

)),

(5)In particular,in addition,if

T

:

C

→

C

is

k

-demi-contractive and

k

∈(0

,

1),then the mapping

W

:

C

→de fined by

has the following properties:

(a)Fix(

T

)=Fix(

W

);(b)

W

is

L

-Lipschitzian;(c)

W

is demiclosed at(0);(d)

W

is a quasi-nonexpansive mapping.

Proof

First we prove the conclusion(1)In fact,if

u

∈Fix(

T

),then

If

u

∈Fix(

T

◦

G

),then it follows from(2.4)that

If

u

∈Fix(

K

),then,from(2.4),we have

Simplifying,we have

Since

Lη<

1,this implies that

u

∈Fix(

T

).The conclusion(1)is proved.

Now we prove the conclusion(2)

Simplifying,we have

This implies that

By the assumption(1−

Lη

)

>

0 and d(

x

,Kx

)→0,this implies that d(

x

,Tx

)→0.Since

T

is demiclosed at 0,

x

∈Fix(

T

).Hence

x

∈Fix(

K

);that is,

K

is demiclosed at 0.

Next we prove the conclusion(3)

Since

T L

-Lipschitzian,for any

x,y

∈

C

it follows from(2.6)that

Similarly,from(2.6)and(3.4),we have

Now we prove the conclusion(4)

For any

p

∈Fix(

T

)and any

x

∈

X

,it follows from(2.5)that

Since

T

is quasi-pseudo-contractive,we have

From(2.5)we have

Substituting(3.6)and(3.7)into(3.5),after simplifying,we have

Finally we prove the conclusion(5)

It is easy to prove that

W

has the properties(a)–(c).Next we prove that

W

has the property(d).In fact,since Fix(

T

)=Fix(

W

),for any

p

∈Fix(

T

)=Fix(

W

)and

x

∈

C

it follows from(3.3),(2.5)and the de finition of

k

-demicontractive mapping that

The conclusion(d)is proved.Therefore the proof of Lemma 3.1 is completed.

In the sequel,we always assume that

(1)

M

is a finite dimensional Hadamard manifold and

C

is a nonempty closed and bounded geodesic convex subset of

M

;(2)

B

:

C

⇉

TM

is a set-valued maximal monotone mapping and

A

:

C

→

TM

is a single-valued and monotone mapping such that

A

+

B

is a set-valued maximal monotone vector field;

(4)

S

:

C

→

C

is a quasi-pseudo-contractive mapping,

U

:

C

→

C

is a

k

-demi-contractive mapping,

k

∈(0

,

1),and that

S

and

U

both are demiclosed at 0 and

L

-Lipschitzian,

L

≥1;(5)We can let

G

,K

:

C

→

C

and

K

:

C

→

C

be the mappings de fined by

Proof

(I)First we observe that by the assumptions of Theorem 3.2,Lemma 2.12 and Lemma 3.1,we have that

Since

K

is quasi-nonexpansive,from Lemma 2.4 and(3.11)we have that

Similarly,from Lemma 2.4 and(3.12),we have

This implies that

In fact,it follows from(3.13)that

Since

a

(1−

b

)

>

0,this implies that

From(3.16)and Lemma 2.4,we have

Furthermore,it follows from Lemma 2.9 and(3.14)that,for each

p

∈Ω,

The conclusion of(3.15)is proved.

(IV)Now we prove that{

x

}converges strongly to some point in Ω.

This completes the proof of Theorem 3.2.

4 Applications

Throughout this section we assume that

M

is a finite dimensional Hadamard manifold,and that

C

is a bounded closed and geodesic convex subset of

M

.

4.1 Minimization problems on Hadamard manifolds

Let

f

:

M

→(−∞

,

+∞]be a proper,lower semicontinuous and geodesic convex function.Consider the minimization problem of finding a point

x

∈

M

such that

We denote by Ωthe solution set of the minimization problem(4.1),that is,

The subdifferential

∂f

(

x

)of

f

at

x

∈

M

[21]is de fined by

Lemma 4.1

([10])Let

f

:

M

→(−∞

,

+∞]be a proper,lower semicontinuous and geodesic convex function.Then,the subdifferential

∂f

of

f

is a maximal monotone vector field,and

From Lemma 4.1,we know that if

f

:

M

→(−∞

,

+∞]

,i

=1

,

2

,

is a proper,lower semicontinuous and geodesic convex function,and

∂f

is the subdifferential of

f

,so

∂

(

f

+

f

)=

∂f

+

∂f

and

∂

(

f

+

f

)is a maximal monotone vector field.Hence,from Theorem 3.2 and Lemma 4.1,we have the following result:

where{

β

}

,

{

δ

}⊂(0

,

1)such that 0

<a

≤

β

,δ

≤

b<

1

,

∀

n

≥0 and

K

and

K

are mappings de fined by(3.9).If

then the sequence{

x

}converges strongly to some point

x

∈Ω.In particular,if

S

=

U

=

I

(the identity mapping on

M

)and

f

=

f,f

=0,then the sequence{

x

}de fined by

converges strongly to a solution of minimization problem(4.1).

4.2 Equilibrium problems on Hadamard manifolds

Let

F

:

C

×

C

→R be a bifunction.We assume that the following conditions are satis fied:

The equilibrium problem(in short,EP)is to find

x

∈

C

such that

The solution set of equilibrium problem(4.5)is denoted by EP(F).

Lemma 4.4

([13])Let

C

be a nonempty closed and geodesic convex subset of a Hadamard manifold

M

.Let

F

:

C

×

C

→R be a bifunction satisfying the conditions(A1)–(A4).Let

H

:

M

⇉

TM

be a set-valued mapping de fined by

In Theorem 3.2,taking

B

=

H

,

A

=0 and

S

=

U

=

I

,the following result can be obtained from Theorem 3.2 immediately:

Theorem 4.5

Let

F

:

C

×

C

→R be a bifunction satisfying the conditions(A1)–(A4)and let

H

:

M

→

M

be the mapping de fined by(4.7).For any given

x

∈

C

,let{

x

}be the sequence de fined by

If

EP

(

F

)/=∅,then the sequence{

x

}converges strongly to a solution of equilibrium problem(4.5).

5 Conclusion

In this paper,an iterative algorithm to approximate a common element of the set of fixed points of a quasi-pseudo-contractive mapping and a demi-contraction mapping and the set of zeros of monotone inclusion problems on Hadamard manifolds has been proposed.Under suitable conditions,we proved that the sequence generated by the algorithm converges strongly to a common solution of problem(1.3).Since the quasi-pseudo-contractive mapping and the demicontractive mapping is more general than the nonexpansive mapping,firmly nonexpansive mapping and quasi-nonexpansive mapping,problem(1.3)studied in our paper is quite general.It includes many kinds of problems,such as convex optimization problems,the fixed point problem,variational inclusion problems,and equilibrium problems as its special cases.Therefore the results presented in the paper not only improve and generalize some recent results,but also provide a powerful tool for solving other problems related to(1.3).