an algorithm for finding common solutions of various problems in nonlinear operator theory

Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article Abstract In this paper, it is our aim to prove strong convergence of a new i

R E S E A R C H Open Access

An algorithm for finding common solutions

of various problems in nonlinear operator

theory

Eric U Ofoedu1, Jonathan N Odumegwu1, Habtu Zegeye2and Naseer Shahzad3*

* Correspondence:

nshahzad@kau.edu.sa

3 Department of Mathematics, King

Abdulaziz University, P.O Box 80203,

Jeddah, 21589, Saudi Arabia

Full list of author information is

available at the end of the article

Abstract

In this paper, it is our aim to prove strong convergence of a new iterative algorithm to

a common element of the set of solutions of a finite family of classical equilibrium problems; a common set of zeros of a finite family of inverse strongly monotone operators; the set of common fixed points of a finite family of quasi-nonexpansive mappings; and the set of common fixed points of a finite family of continuous pseudocontractive mappings in Hilbert spaces on assumption that the intersection of the aforementioned sets is not empty Moreover, the common element is shown to

be the metric projection of the initial guess on the intersection of these sets

MSC: 47H06; 47H09; 47J05; 47J25 Keywords: classical equilibrium problem; generalized mixed equilibrium problem;

η-inverse strongly monotone mapping; maximal monotone operator; nonexpansive mappings; real Hilbert space; pseudocontractive mappings; variational inequality problem

1 Introduction

Let H be a real Hilbert space A mapping T with domain D(T) and range R(T) in H is called

an L-Lipschitzian mapping (or simply a Lipschitz mapping) if and only if there exists L≥ 

such that for all x, y ∈ D(T),

Tx – Ty ≤ Lx – y.

If L ∈ [, ), then T is called strict contraction or simply a contraction; and if L = , then T

is called nonexpansive A point x ∈ D(T) is called a fixed point of an operator T if and only

if Tx = x The set of fixed points of an operator T is denoted by Fix(T), that is, Fix(T) := {x ∈ D(T) : Tx = x}.

A mapping T with domain D(T) and range R(T) in H is called a quasi-nonexpansive mapping if and only if Fix(T) = ∅ and for any x ∈ D(T), for any u ∈ Fix(T),

Tx – u ≤ x – u.

Every nonexpansive mapping with a nonempty fixed point set is quasi-nonexpansive The following examples show that the converse is not true

©2014 Ofoedu et al.; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction

Example . (see []) Let E = [–π, π] be a subspace of the set of real numbers R, endowed

with the usual topology Define T : E → E by Tx = x cos x for all x ∈ E Clearly, F(T) = {}.

Observe that

|Tx – | = |x| × | cos x| ≤ |x| = |x – |.

Thus, T is quasi-nonexpansive The mapping T is, however, not a nonexpansive mapping

since for x = π and y = π,

|Tx – Ty| =

π cos

π



 –π cos π

 = π.

But

|x – y| =

π –π

 =π

Example . (see [, ]) Let E = R be endowed with usual topology Define T : R → R by

Tx =



x

cos(x), x= ,

It is easy to see that F(T) = {} since for x = , Tx = x implies that x

cosx = x Thus, for any

x= , cos

x = , which is not possible So, F(T) = {} Next, observe that for any x ∈ R,

|Tx – | =

x ×cosx ≤|x| <|x| = |x – |.

So, the mapping T is quasi-nonexpansive Finally, we show that T is not nonexpansive To

see this, let x =π and y = π, then

|Tx – Ty| =

πcos



π



 – 

πcosπ



 =π But,

|x – y| =

π– 

π



 =π So,

|Tx – Ty| = 

π >



π =|x – y|.

The concept of quasi-nonexpansive mappings was essentially introduced by Diaz and Metcalf [] Although Examples . and . guarantee the existence of a

nonexpansive mapping which is not nonexpansive, we must note that a linear

quasi-nonexpansive mapping defined on a subspace of a given vector space is quasi-nonexpansive

on that subspace

Another important generalization of the class of nonexpansive mappings is the class of pseudocontractive mappings These mappings are intimately connected with the

impor-tant class of nonlinear accretive operators This connection will be made precise in what

follows

A mapping T with domain D(T) and range R(T) in H is called pseudocontractive if and only if for all x, y ∈ D(T), the following inequality holds:

x – y ≤( + r)(x – y) – r(Tx – Ty) (.)

for all r >  As a consequence of a result of Kato [], the pseudocontractive mappings can

also be defined in terms of the normalized duality mappings as follows: the mapping T is

called pseudocontractive if and only if for all x, y ∈ D(T), we have that

It now follows trivially from (.) that every nonexpansive mapping is pseudocontractive

We note immediately that the class of pseudocontractive mappings is larger than that of

nonexpansive mappings For examples of pseudocontractive mappings which are not

non-expansive, the reader may see []

To see the connection between the pseudocontractive mappings and the monotone

mappings, we introduce the following definition: a mapping A with domain D(A) and

range R(A) in E is called monotone if and only if for all x, y ∈ D(A), the following inequality

is satisfied:

(.)

The operator A is called η-inverse strongly monotone if and only if there exists η ∈ (, )

such that for all x, y ∈ D(A), we have that

It is easy to see from inequalities (.) and (.) that an operator A is monotone if and only

if the mapping T := (I – A) is pseudocontractive Consequently, the fixed point theory

for pseudocontractive mappings is intimately connected with the zero of monotone

map-pings For the importance of monotone mappings and their connections with evolution

equations, the reader may consult any of the references [, ]

Due to the above connection, fixed point theory of pseudocontractive mappings became

a flourishing area of intensive research for several authors

Let C be a closed convex nonempty subset of a real Hilbert space H with inner product (EP) for a bifunction f is to find u∗∈ C such that

f

u∗, y

The set of solutions for EP (.) is denoted by

EP(f ) =

u ∈ C : f (u, y) ≥ , ∀y ∈ C

The classical equilibrium problem (EP) includes as special cases the monotone inclusion

problems, saddle point problems, variational inequality problems, minimization

prob-lems, optimization probprob-lems, vector equilibrium probprob-lems, Nash equilibria in

noncoop-erative games Furthermore, there are several other problems, for example, the

comple-mentarity problems and fixed point problems, which can also be written in the form of

the classical equilibrium problem In other words, the classical equilibrium problem is a

unifying model for several problems arising from engineering, physics, statistics,

com-puter science, optimization theory, operations research, economics and countless other

fields For the past  years or so, many existence results have been published for various

equilibrium problems (see, e.g., [–]) Approximation methods for such problems thus

become a necessity

Iterative approximation of fixed points and zeros of nonlinear mappings has been stud-ied extensively by many authors to solve nonlinear mapping equations as well as

varia-tional inequality problems and their generalizations (see, e.g., [–]) Most published

results on nonexpansive mappings (for example) focus on the iterative approximation of

their fixed points or approximation of common fixed points of a given family of this class

of mappings

Some attempts to modify the Mann iteration method so that strong convergence is anteed have recently been made (we should recall that Mann iteration method only

guar-antees weak convergence (see, for example, Bauschke et al [])) Nakajo and Takahashi

[] formulated the following modification of the Mann iteration method for a

nonexpan-sive mapping T defined on a nonempty bounded closed and convex subset C of a Hilbert

space H:

⎧

⎪

⎪

⎨

⎪

⎪

⎩

x∈ C,

y n=α n x n+ ( –α n )Tx n,

C n={v ∈ C : y n – v≤ x n – v},

Q n={v ∈ C : x n – v, x– x n

x n+ = P C n ∩Q n (x), ∀n ∈ N,

(.)

where P C denotes the metric projection from H onto a closed convex subset C of H They

proved that if the sequence{α n}n≥is bounded away from , then{x n}n≥defined by (.)

converges strongly to P F(T) (x)

Formulations similar to (.) for different classes of nonlinear problems had been

pre-sented by Kim and Xu [], Nilsrakoo and Saejung [], Ofoedu et al [], Yang and Su

[], Zegeye and Shahzad [–]

In this paper, motivated by the results of the authors mentioned above, it is our aim to prove strong convergence of a new iterative algorithm to a common element of the set of

solutions of a finite family of classical equilibrium problems; a common set of zeros of a

finite family of inverse strongly monotone mappings; a set of common fixed points of a

finite family of quasi-nonexpansive mappings; and a set of common fixed points of a finite

family of continuous pseudocontractive mappings in Hilbert spaces on assumption that

the intersection of the aforementioned sets is not empty Moreover, the common element

is shown to be the metric projection of the initial guess on the intersection of these sets

Our theorems complement the results of the authors mentioned above and those of several

other authors

2 Preliminary

In what follows, we shall make use of the following lemmas

Lemma . (see, e.g., Kopecka and Reich []) Let C be a nonempty closed and convex

subset of a real Hilbert space Let x ∈ H and P C : H → C be the metric projection of H onto

C, then for any y ∈ C,

y – P C x+P C x – x≤ x – y

Lemma . Let C be a closed convex nonempty subset of a real Hilbert space H; and let

P C : H → C be the metric projection of H onto C Let x ∈ H, then x= P C x if and only if

z – x, x – x

Lemma . Let H be a real Hilbert space, then for any x, y ∈ H, α ∈ [, ],

α x + ( – α)y

=αx+ ( –α)y–α( – α)x – y

Lemma . (see Zegeye []) Let C be a nonempty closed convex subset of a real Hilbert

space H Let T : C → H be a continuous pseudocontractive mapping, then for all r >  and

x ∈ H, there exists z ∈ C such that



r



y – z, ( + r)z – x

≤ , ∀y ∈ C.

Lemma . (see Zegeye []) Let C be a nonempty closed convex subset of a real Hilbert

space H Let T : C → C be a continuous pseudocontractive mapping, then for all r >  and

x ∈ H, define a mapping F r : H → C by

F r x =



r



y – z, ( + r)z – x

≤ , ∀y ∈ C

 ,

then the following hold:

() F r is single-valued;

() F r is firmly nonexpansive type mapping, i.e., for all x, y ∈ H,

F r x – F r y≤ F r x – F r y, x – y

() Fix(F r ) is closed and convex; and Fix(F r ) = Fix(T) for all r > .

In the sequel, we shall require that the bifunction f : C × C → R satisfies the following

conditions:

(A) f (x, x) = , ∀x ∈ C;

(A) f is monotone in the sense that f (x, y) + f (y, x) ≤  for all x, y ∈ C;

(A) lim supt→ +f (tz + ( – t)x, y) ≤ f (x, y) for all x, y, z ∈ C;

(A) the function y → f (x, y) is convex and lower semicontinuous for all x ∈ C.

Lemma . (see, e.g., [, ]) Let C be a closed convex nonempty subset of a real Hilbert

space H Let f : C × C → R be a bifunction satisfying conditions (A)-(A), then for all

r >  and x ∈ H, there exists u ∈ C such that

f (u, y) +

Moreover, if for all x ∈ H we define a mapping G r : H→ C by

G r (x) =



u ∈ C : f (u, y) +

r



then the following hold:

() G r is single-valued for all r > ;

() G r is firmly nonexpansive, that is, for all x, z ∈ H,

G r x – G r z≤ G r x – G r z, x – z

() Fix(G r ) = EP(f ) for all r > ;

() EP(f ) is closed and convex.

Lemma . (see Ofoedu []) Let C be a nonempty closed convex subset of a real Hilbert

space H Let T : C → C be a continuous pseudocontractive mapping For r > , let F r : H→

C be the mapping in Lemma ., then for any x ∈ H and for any p, q > ,

F p x – F q x ≤|p – q|

p



F p x  + x

Lemma . (Compare with Lemma  of Ofoedu []) Let C be a closed convex nonempty

subset of a real Hilbert space H Let f : C × C → R be a bifunction satisfying conditions

(A)-(A) Let r >  and let G r be the mapping in Lemma ., then for all p, q >  and for

all x ∈ H, we have that

G p x – G q x ≤|p – q|

p



G p x  + x

3 Main results

Let C be a nonempty closed convex subset of a real Hilbert space H Let T, T, , T m:

C → C be m continuous pseudocontractive mappings; let S, S, , S l : C → C be l

con-tinuous quasi-nonexpansive mappings; let A, A, , A d : C → H be d γ j-inverse strongly

monotone mappings with constantsγ j ∈ (, ), j = , , , d; let f, f, , f t : C × C → R be

t bifunctions satisfying conditions (A)-(A) For all x ∈ E, i = , , , m, let

F i,r x :=



z ∈ C : y – z, T i z 

r



y – z, ( + r)z – x

≤ , ∀y ∈ C



and for all x ∈ E, h = , , , t, let

G h,r (x) =



u ∈ C : f h (u, y) +

r

 ,

then in what follows we shall study the following iteration process:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

x∈ C= C chosen arbitrarily,

z n = P C (x n–λ n A n+ x n),

y n=α n x n+ ( –α n )S n+ z n,

w n=ηm

i= β i F i,r n y n+ ( –η)t

h= ξ h G h,r n y n,

C n+={z ∈ C : w n – z ≤ x n – z},

x n+= C n+ (x), n≥ ,

(.)

where A n = A n(mod d) , S n = S n(mod l);{r n} ⊂ (, ∞) such that limn→∞r n = r> ;{α n}n≥ a

sequence in (, ) such that lim infn→∞α n( –α n) > ;{β i}m

i=,{ξ h}t h=⊂ (, ) such that

m

i= β i=  =t

h= ξ h; η ∈ (, ) and {λ n } is a sequence in [a, b] for some a, b ∈ R such that  < a < b <  γ , γ = min≤j≤d{γ j}

Lemma . Let C be a nonempty closed convex subset of a real Hilbert space H Let

T, T, , T m : C → C be m continuous pseudocontractive mappings; let S, S, , S l :

C → C be l continuous quasi-nonexpansive mappings; let A, A, , A d : C → H be d γ j

-inverse strongly monotone mappings with constants γ j ∈ (, ), j = , , , d; let f, f, , f t:

C × C → R be t bifunctions satisfying conditions (A)-(A) Let F :=m

i=Fix(Ti)∩

d

j= A–

j ()∩l

k=Fix(Sk)∩t

h= EP(f h)= ∅ Let {x n } be a sequence defined by (.), then

the sequence {x n } is well defined for each n ≥ .

Proof We first show that C n is closed and convex for each n∈ N∪{} From the definitions

of C n it is obvious that C nis closed Moreover, sincew n – z ≤ x n – z is equivalent to

 z, x n – w n n+w n≤ , it follows that C n is convex for each n∈ N ∪ {} Next,

we prove that F ⊂ C n for each n ∈ N ∪ {} From the assumption, we see that F ⊂ C= C.

Suppose that F ⊂ C k for some k ≥ , then for p ∈ F, we obtain that

w k – p =



η

m



i=

β i F i,r k y k+ ( –η)

m



h=

ξ h G h,r k y k – p







≤ y k – p = α k x k+ ( –α k )S k+ z k – p

≤ α k x k – p  + ( – α k)S k+ z k – p

≤ α k x k – p  + ( – α k)z k – p (.) Furthermore,

z k – p =P C (x k–λ k A k+ x k ) – p

≤ x k–λ k A k+ x k – p

=x k – p – λ k (A k+ x k – A k+ p)

=x k – p– λ k x k – p, A k+ x k – A k+ p k A k+ x k – A k+ p

≤ x k – p+λ k(λ k– γ )A k+ x k – A k+ p

≤ x – p (sinceλ < γ ).

z k – p ≤ x k – p. (.) Using (.) in (.) gives

w k – p ≤ x k – p.

So, p ∈ C k+ This implies, by induction, that F ⊂ C nso that the sequence generated by

(.) is well defined for all n≥  

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H Let

T, T, , T m : C → C be m continuous pseudocontractive mappings; let S, S, , S l :

C → C be l continuous quasi-nonexpansive mappings; let A, A, , A d : C → H be d γ j

-inverse strongly monotone mappings with constants γ j ∈ (, ), j = , , , d; let f, f, , f t:

C × C → R be t bifunctions satisfying conditions (A)-(A) Let F :=m

i=Fix(Ti)∩

d

j= A–

j ()∩l

k=Fix(Sk)∩t

h= EP(f h)= ∅ Let {x n } be a sequence defined by (.) Then

the sequence {x n}n≥converges strongly to the element of F nearest to x

Proof From Lemma ., we obtain that F ⊂ C n,∀n ≥  and x nis well defined for each

n ≥  From x n = P C n (x) and x n+ = P C n+ (x)∈ C n+ ⊂ C n, we obtain that

x n+ – x n , x n – x n – x ≤ x n+ – x

Besides, by Lemma .,

x n – p=P C n x– x ≤ x– p–x– x n≤ x– p

Thus, the sequence{x n – x}n≥is a bounded nondecreasing sequence of real numbers

So, limn→∞x n – x exists Similarly, by Lemma ., we have that for any positive integer

μ,

x n+μ – x n =x n+μ – P C n x

≤ x n+μ – x–P C nx– x

=x n+μ – x–x n – x for all n≥ 

Since limn→∞x n – x exists, we have that limn→∞x n+μ – x n  =  and hence, {x n}n≥is

a Cauchy sequence in C Therefore, there exists x∗∈ C such that lim n→∞x n = x∗ Since

x n+ ∈ C n+, we have that

w n – x n+  ≤ x n – x n+

Thus,

lim

n→∞x n+ – w n =  (.)

and hencex n –w n  ≤ x n –x n+ +x n+ –w n  →  as n → ∞, which implies that w n → x∗

as n→ ∞

Next, we observe that for p ∈ F and using Lemma .,

y n – p =α n x n+ ( –α n )S n+ z n – p

=α n (x n – p) + ( – α n )(S n+ z n – p)

=α n x n – p  + ( – α n)S n+ z n – p–α n( –α n)x n – S n+ z n (.)

≤ α n x n – p+ ( –α n)z n – p–α n( –α n)x n – S n+ z n (.) But

z n – p≤ x n – p+λ n(λ n– γ )A n+ x n – A n+ p

So, using (.) in (.), we obtain that

y n – p≤ α n x n – p+ ( –α n)

x n – p+λ n(λ n– γ )A n+ x n –α n( –α n)x n – S n+ z n

=x n – p+ ( –α n)λ n(λ n– γ )A n+ x n

–α n( –α n)xn – S n+ z n (.) Moreover, we obtain that

w n – p =





η

m



i=

β i F i,r n y n+ ( –η)

m



h=

ξ h G h,r n y n – p









≤ y n – p (.) Using (.) in (.) we get that

w n – p≤ x n – p+ ( –α n)λ n(λ n– γ )A n+ x n

–α n( –α n)xn – S n+ z n (.) Now, using the fact thatλ n< γ , inequality (.) gives (for some constant M> ) that

α n( –α n)x n – S n+ z n  ≤ x n – p–w n – p≤ Mx n – w n (.) Hence, we obtain from inequality (.) that

x n – S n+ z n  →  as n → ∞. (.) Moreover, from (.) we obtain that

( –α )λ (γ – λ )A x ≤ x – p–w – p≤ M x – w ,

which yields that

lim

n→∞A n+ x n =  (.) Now,

x n – z n =x n – P C (x n–λ n A n+ x n)=P C x n – P C (x n–λ n A n+ x n)

≤ x n – x n+λ n A n+ x n  = λ n A n+ x n

≤ bA n+ x n (.)

It follows from (.) and (.) that

lim

n→∞x n – z n = ; (.)

and hence z n → x∗as n→ ∞

We now show that x∗∈l

k=Fix(Sk) Observe that from (.) and (.) we obtain that

S n+ z n – z n  ≤ S n+ z n – x n  + z n – x n  →  as n → ∞, (.)

so that

lim

n→∞S n+ z n = x∗ (.) Let{n σ}σ≥ ⊂ N be such that S n σ+= Sfor allσ ∈ N, then since z n σ → x∗asσ → ∞, we

obtain from (.), using the continuity of S, that

x∗= lim

σ→∞ S n σ+z n σ = lim

σ→∞ Sz n σ = Sx∗ Similarly, if{n j}j≥⊂ N is such that S n j+= Sfor all j∈ N, then we have again that

x∗= lim

j→∞S n j+z n j= lim

j→∞Sz n j = Sx∗

Continuing, we obtain that S k x∗= x∗, k = , , l Hence, x∗∈l

k= F(S k)

Next, we show that x∗∈d

j= A–j () Since A j isγ -inverse strongly monotone for j =

, , , d, we have that A jisγ-Lipschitz continuous Thus,

A n+ x n – A n+ x∗ ≤ 

γx n – x∗ → as n→ ∞ (.) Hence, from (.) and (.), we obtain that

A n+ x∗ ≤A n+ x n – A n+ x∗+A n+ x n  →  as n → ∞.

As a result, we get that

lim

n→∞A n+ x∗= 