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An algorithm for a common minimum-norm zero of a finite family of monotone mappings in Banach spaces

Abstract

We introduce an iterative process which converges strongly to a common minimum-norm point of solutions of a finite family of monotone mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

MSC:47H05, 47H09, 47H10, 47J05, 47J25.

1 Introduction

In many problems, it is quite often to seek a particular solution of the minimum-norm solution of a given nonlinear problem. In an abstract way, we may formulate such problems as finding a point x with the property

x Csuch that x = min x C { x } ,
(1.1)

where C is a nonempty closed convex subset of a real Hilbert space H. In other words, x is the (nearest point or metric) projection of the origin onto C,

x = P C (0),
(1.2)

where P C is the metric (or nearest point) projection from H onto C. For instance, the split feasibility problem (SFP), introduced in [1, 2], is to find a point

x Csuch thatA x Q,
(1.3)

where C and Q are closed convex subsets of Hilbert spaces H 1 and H 2 , respectively, and A is a linear bounded operator from H 1 to H 2 . We note that problem (1.3) can be extended to a problem of finding

xD(A)D(B)such thatx A 1 (0) B 1 (0),
(1.4)

where A:D(A) E and B:D(B) E are monotone mappings on a subset of a Banach space E. The problem has been addressed by many authors in view of the applications in image recovery and signal processing; see, for example, [35] and the references therein.

A mapping A:C E is said to be monotone if for each x,yC, the following inequality holds:

xy,AxAy0,
(1.5)

where C is a nonempty subset of a real Banach space E with E as its dual. A is said to be maximal monotone if its graph is not properly contained in the graph of any other monotone mapping. A mapping A:C E is said to be γ-inverse strongly monotone if there exists a positive real number γ such that

xy,AxAyγ A x A y 2 for all x,yC,
(1.6)

and it is called strongly monotone if there exists k>0 such that

xy,AxAyk x y 2 for all x,yC.
(1.7)

An operator A:CE is called accretive if there exists j(xy)J(xy) such that

A x A y , j ( x y ) 0for all x,yC,
(1.8)

where J is the normalized duality mapping from E into 2 E defined for each xE by

Jx:= { f E : x , f = x 2 = f 2 } .

It is well known that E is smooth if and only if J is single-valued, and if E is uniformly smooth, then J is uniformly continuous on bounded subsets of E (see [6]). A is called m-accretive if it is accretive and R(I+rA), the range of (I+rA), is E for all r>0; and an accretive mapping A is said to satisfy range condition if

D(A)C r > 0 R(I+rA)
(1.9)

for some nonempty closed convex subset C of a real Banach space H.

Clearly, the class of monotone mappings includes the class of strongly monotone and the class of γ-inverse strongly monotone mappings. However, we observe that accretive mappings and monotone mappings have different natures in Banach spaces more general than Hilbert spaces.

When A and B are maximal monotone mappings in Hilbert spaces, Bauschke et al. [7] proved that sequences generated from the method of alternating resolvents given by

{ x 2 n + 1 = J β A ( x 2 n ) , n 0 , x 2 n = J μ B ( x 2 n 1 ) , n 0 ,
(1.10)

where J μ A := ( I + μ A ) 1 is the resolvent of A, converge weakly to a point of A 1 (0) B 1 (0) provided that A 1 (0) B 1 (0) is nonempty. Note that strong convergence of these methods fails in general (see a counter example by Hundal [8]).

With regard to a finite family of m-accretive mappings, Zegeye and Shahzad [9] proved that under appropriate conditions, an iterative process of Halpern type defined by

x n + 1 = α n u+(1 α n ) S r n x n ,n0,
(1.11)

where α n (0,1) for all n0, u, x 0 H, S r := a 0 I+ a 1 J r 1 + a 2 J r 2 ++ a N J r N with J r i = ( I + r A i ) 1 for a i (0,1), i=0,1,,N, and i = 1 a i N =1, converges strongly to a point in i = 1 N A 1 (0) nearest to u, where { A i :i=1,2,,N} is the set of a finite family of m-accretive mappings in a strictly convex and reflexive (real) Banach space E which has a uniformly Gâteaux differentiable norm.

In 2009, Hu and Liu [10] also proved that under appropriate conditions, an iterative process of Halpern type defined by

x n + 1 = α n u+ δ n x n + γ n S r n x n ,n0,
(1.12)

where α n , δ n , γ n (0,1) with α n + δ n + γ n =1, for all n0, u= x 0 H, S r n := a 0 I+ a 1 J r n 1 + a 2 J r n 2 ++ a N J r n N with J r i = ( I + r A i ) 1 , for a i (0,1), i=0,1,,N, and i = 1 a i N =1, and { r n }(0,), for A i , i=1,2,,N, accretive mappings satisfying range condition (1.9), converges strongly to a point in i = 1 N A 1 (0) nearest to u in a strictly convex and reflexive (real) Banach space E which has a uniformly Gâteaux differentiable norm.

A natural question arises whether we can have the results of Zegeye and Shahzad [9]and Hu and Liu [10]for the class of monotone mappings or not, in Banach spaces more general than Hilbert spaces?

Let C be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space E. Let A i :C E for i=1,2,,N be continuous monotone mappings satisfying range condition (2.1) with F:= i = 1 N A i 1 (0).

It is our purpose in this paper to introduce an iterative scheme (see (3.1)) which converges strongly to the common minimum-norm zero of the family { A i ,i=1,2,,N}. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

2 Preliminaries

Let E be a normed linear space with dimE2. The modulus of smoothness of E is the function ρ E :[0,)[0,) defined by

ρ E (τ):=sup { x + y + x y 2 1 : x = 1 ; y = τ } .

The space E is said to be smooth if ρ E (τ)>0, τ>0, and E is called uniformly smooth if and only if lim t 0 + ρ E ( t ) t =0.

The modulus of convexity of E is the function δ E :(0,2][0,1] defined by

δ E (ϵ):=inf { 1 x + y 2 : x = y = 1 ; ϵ = x y } .

E is called uniformly convex if and only if δ E (ϵ)>0 for every ϵ(0,2].

Let C be a nonempty, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space E with dual E . A monotone mapping A is said to satisfy range condition if we have that

D(A)C r > 0 J 1 R(J+rA)
(2.1)

for some nonempty closed convex subset C of a smooth, strictly convex, and reflexive Banach space E. In the sequel, the resolvent of a monotone mapping A:C E shall be denoted by Q r A := ( J + r A ) 1 J for r>0. We know the following lemma.

Lemma 2.1 [11]

Let E be a smooth and strictly convex Banach space, C be a nonempty, closed, and convex subset of E, and AE× E be a monotone mapping satisfying (2.1). Let Q r n A be the resolvent of A for { r n }(0,) such that lim n r n =. If { x n } is a bounded sequence of C such that Q r n A x n z, then z A 1 (0).

Let E be a smooth Banach space with dual E . Let the Lyapunov function ϕ:E×ER, introduced by Alber [12], be defined by

ϕ(y,x)= y 2 2y,Jx+ x 2 for x,yE,
(2.2)

where J is the normalized duality mapping. If E=H, a Hilbert space, then the duality mapping becomes the identity map on H. We observe that in a Hilbert space H, (2.2) reduces to ϕ(x,y)= x y 2 for x,yH.

In the sequel, we shall make use of the following lemmas.

Lemma 2.2 [13]

Let E be a smooth and strictly convex Banach space, and C be a nonempty, closed, and convex subset of E. Let AE× E be a monotone mapping satisfying (2.1), A 1 (0) be nonempty and Q r A be the resolvent of A for some r>0. Then, for each r>0, we have that

ϕ ( p , Q r A x ) +ϕ ( Q r A x , x ) ϕ(p,x)

for all p A 1 (0) and xC.

Lemma 2.3 [14]

Let E be a smooth and strictly convex Banach space, C be a nonempty, closed, and convex subset of E, and T be a mapping from C into itself such that F(T) is nonempty and ϕ(p,Tx)ϕ(p,x) for all pF(T) and xC. Then F(T) is closed and convex.

Lemma 2.4 [15]

Let E be a real smooth and uniformly convex Banach space, and let { x n } and { y n } be two sequences of E. If either { x n } or { y n } is bounded and ϕ( x n , y n )0 as n, then x n y n 0 as n.

We make use of the function V:E× E R defined by

V ( x , x ) = x 2 2 x , x + x 2 for all xE and  x E,

studied by Alber [12]. That is, V(x, x )=ϕ(x, J 1 x ) for all xE and x E .

Lemma 2.5 [12]

Let E be a reflexive, strictly convex, and smooth Banach space with E as its dual. Then

V ( x , x ) +2 J 1 x x , y V ( x , x + y )

for all xE and x , y E .

Let E be a reflexive, strictly convex, and smooth Banach space, and let C be a nonempty, closed, and convex subset of E. The generalized projection mapping, introduced by Alber [12], is a mapping Π C :EC that assigns an arbitrary point xE to the minimizer, x ¯ , of ϕ(,x) over C, that is, Π C x= x ¯ , where x ¯ is the solution to the minimization problem

ϕ( x ¯ ,x)=min { ϕ ( y , x ) , y C } .
(2.3)

Lemma 2.6 [12]

Let C be a nonempty, closed, and convex subset of a real reflexive, strictly convex, and smooth Banach space E, and let xE. Then, yC,

ϕ(y, Π C x)+ϕ( Π C x,x)ϕ(y,x).

Lemma 2.7 [12]

Let C be a convex subset of a real smooth Banach space E. Let xE. Then x 0 = Π C x if and only if

z x 0 ,JxJ x 0 0,zC.

Lemma 2.8 [16]

Let E be a uniformly convex Banach space and B R (0) be a closed ball of E. Then there exists a continuous strictly increasing convex function g:[0,)[0,) with g(0)=0 such that

α 0 x 0 + α 1 x 1 + + α N x N 2 i = 0 N α i x i 2 α i α j g ( x i x j )

for α i (0,1) such that i = 0 N α i =1 and x i B R (0):={xE:xR} for some R>0.

Lemma 2.9 [17]

Let { a n } be a sequence of nonnegative real numbers satisfying the following relation:

a n + 1 (1 β n ) a n + β n δ n ,n n 0 ,

where { β n }(0,1) and { δ n }R satisfy the following conditions: lim n β n =0, n = 1 β n =, and lim sup n δ n 0. Then lim n a n =0.

Lemma 2.10 [18]

Let { a n } be sequences of real numbers such that there exists a subsequence { n i } of {n} such that a n i < a n i + 1 for all iN. Then there exists a nondecreasing sequence { m k }N such that m k , and the following properties are satisfied by all (sufficiently large) numbers kN:

a m k a m k + 1 and a k a m k + 1 .

In fact, m k is the largest number n in the set {1,2,,k} such that the condition a n a n + 1 holds.

3 Main result

We now prove the following theorem.

Theorem 3.1 Let C be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space E. Let A i :C E , for i=1,2,,N, be continuous monotone mappings satisfying (2.1). Assume that F:= i = 1 N A i 1 (0) is nonempty. Let { x n } be a sequence generated by

{ x 0 C , chosen arbitrarily , y n = Π C [ ( 1 α n ) x n ] , x n + 1 = J 1 ( β 0 J y n + i = 1 N β i J Q r n A i y n ) , n 0 ,
(3.1)

where α n (0,1), { β i } i = 1 N [c,d](0,1) and { r n }(0,) satisfy the following conditions: lim n α n =0, n = 1 α n =, i = 0 N β i =1, and lim n r n =. Then { x n } converges strongly to the minimum-norm point of .

Proof From Lemmas 2.2 and 2.3 we get that A i 1 (0) is closed and convex. Thus, Π F (0) is well defined. Let p= Π F (0). Then from (3.1), Lemma 2.6 and the property of ϕ, we get that

ϕ ( p , y n ) = ϕ ( p , Π C ( 1 α n ) x n ) ϕ ( p , ( 1 α n ) x n ) = ϕ ( p , J 1 ( α n J 0 + ( 1 α n ) J x n ) ) = p 2 2 p , α n J 0 + ( 1 α n ) J x n + α n J 0 + ( 1 α n ) J x n 2 p 2 2 α n p , J 0 2 ( 1 α n ) p , J x n + α n J 0 2 + ( 1 α n ) J x n 2 = α n ϕ ( p , 0 ) + ( 1 α n ) ϕ ( p , x n ) .
(3.2)

Moreover, from (3.1), Lemma 2.8, Lemma 2.2 and (3.2), we get that

ϕ ( p , x n + 1 ) = ϕ ( p , J 1 ( β 0 J y n + i = 1 N β i J Q r n A i y n ) ) = p 2 2 p , β 0 J y n + i = 1 N β i J Q r n A i y n + β 0 J y n + i = 1 N β i J Q r n A i y n 2 p 2 2 β 0 p , J y n 2 i = 1 N β i p , J Q r n A i y n + β 0 y n 2 + i = 1 N β i Q r n A i y n 2 β 0 β i g ( J y n J Q r n A i y n ) = β 0 ϕ ( p , y n ) + i = 1 N β i ϕ ( p , Q r n A i y n ) β 0 β i g ( J y n J Q r n A i y n ) β 0 ϕ ( p , y n ) + ( 1 β 0 ) ϕ ( p , y n ) β 0 β i g ( J y n J Q r n A i y n ) ϕ ( p , y n ) β 0 β i g ( J y n J Q r n A i y n ) ϕ ( p , y n )
(3.3)
α n ϕ(p,0)+(1 α n )ϕ(p, x n )
(3.4)

for each i{1,2,,N}. Thus, by induction,

ϕ(p, x n + 1 )max { ϕ ( p , 0 ) , ϕ ( p , x 0 ) } ,n0,

which implies that { x n } and hence { y n } are bounded. Now let z n =(1 α n ) x n . Then we note that y n = Π C z n . Using Lemma 2.6, Lemma 2.5 and the property of ϕ, we obtain that

ϕ ( p , y n ) ϕ ( p , z n ) = V ( p , J z n ) V ( p , J z n α n ( J 0 J p ) ) 2 z n p , α n ( J 0 J p ) = ϕ ( p , J 1 ( α n J p + ( 1 α n ) J x n ) ) 2 α n z n p , J p α n ϕ ( p , p ) + ( 1 α n ) ϕ ( p , x n ) 2 α n z n p , J p = ( 1 α n ) ϕ ( p , x n ) 2 α n z n p , J p ( 1 α n ) ϕ ( p , x n ) 2 α n z n p , J p .
(3.5)

Furthermore, from (3.3) and (3.5) we have that

ϕ ( p , x n + 1 ) ( 1 α n ) ϕ ( p , x n ) 2 α n z n p , J p β 0 β i g ( J y n J Q r n A i y n )
(3.6)
(1 α n )ϕ(p, x n )2 α n z n p,Jp.
(3.7)

Now, following the method of proof of Lemma 3.2 of Maingé [18], we consider two cases as follows.

Case 1. Suppose that there exists n 0 N such that {ϕ(p, x n )} is nonincreasing for all n n 0 . In this situation, {ϕ(p, x n )} is convergent. Then from (3.6) we have that

β 0 β i g ( J y n J Q r n A i y n ) 0,
(3.8)

which implies, by the property of g, that

J y n J Q r n A i y n 0as n,
(3.9)

and hence, since J 1 is uniformly continuous on bounded sets, we obtain that

y n Q r n A i y n 0as n,
(3.10)

for each i{1,2,,N}.

Furthermore, Lemma 2.6, the property of ϕ and the fact that α n 0, as n, imply that

ϕ ( x n , y n ) = ϕ ( x n , Π C z n ) ϕ ( x n , z n ) = ϕ ( x n , J 1 ( α n J 0 + ( 1 α n ) J x n ) ) α n ϕ ( x n , 0 ) + ( 1 α n ) ϕ ( x n , x n ) α n ϕ ( x n , 0 ) + ( 1 α n ) ϕ ( x n , x n ) 0 as  n ,
(3.11)

and hence from Lemma 2.4 we get that

x n y n 0, x n z n 0as n.
(3.12)

Since { z n } is bounded and E is reflexive, we choose a subsequence { z n i } of { z n } such that z n i z and lim sup n z n p,Jp= lim i z n i p,Jp. Then from (3.12) we get that

y n i zas i.
(3.13)

Thus, from (3.10) and Lemma 2.1, we obtain that z A i 1 (0) for each i{1,2,,N} and hence z i = 1 N A i 1 (0).

Therefore, by Lemma 2.7, we immediately obtain that lim sup n z n p,Jp= lim i z n i p,Jp=zp,Jp0. It follows from Lemma 2.9 and (3.7) that ϕ(p, x n )0 as n. Consequently, from Lemma 2.4 we obtain that x n p.

Case 2. Suppose that there exists a subsequence { n i } of {n} such that

ϕ(p, x n i )<ϕ(p, x n i + 1 )

for all iN. Then, by Lemma 2.10, there exists a nondecreasing sequence { m k }N such that m k , ϕ(p, x m k )ϕ(p, x m k + 1 ), and ϕ(p, x k )ϕ(p, x m k + 1 ) for all kN. Then, from (3.6) and the fact that α n 0, we obtain that

g ( J y m k J Q r m k A i y m k ) 0as k,

for each i{1,2,,N}. Thus, following the method of proof of Case 1, we obtain that y m k Q r m k A i y m k 0, x m k y m k 0, x m k z m k 0 as k, and hence we obtain that

lim sup k z m k p,Jp0.
(3.14)

Then from (3.7) we have that

ϕ(p, x m k + 1 )(1 α m k )ϕ(p, x m k )2 α m k z m k p,Jp.
(3.15)

Now, since ϕ(p, x m k )ϕ(p, x m k + 1 ), inequality (3.15) implies that

α m k ϕ ( p , x m k ) ϕ ( p , x m k ) ϕ ( p , x m k + 1 ) 2 α m k z m k p , J p 2 α m k z m k p , J p .

In particular, since α m k >0, we get

ϕ(p, x m k )2 z m k p,Jp.

Then from (3.14) we obtain ϕ(p, x m k )0 as k. This together with (3.15) gives ϕ(p, x m k + 1 )0 as k. But ϕ(p, x k )ϕ(p, x m k + 1 ) for all kN, thus we obtain that x k p. Therefore, from the above two cases, we can conclude that { x n } converges strongly to p, which is the common minimum-norm zero of the family { A i ,i=1,2,,N}, and the proof is complete. □

We would like to mention that the method of proof of Theorem 3.1 provides the following theorem.

Theorem 3.2 Let C be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space E. Let A i :C E , for i=1,2,,N, be continuous monotone mappings satisfying (2.1). Assume that F:= i = 1 N A i 1 (0) is nonempty. Let { x n } be a sequence generated by

{ u = x 0 C , chosen arbitrarily , y n = Π C J 1 ( α n J u + ( 1 α n ) J x n ) , x n + 1 = J 1 ( β 0 J y n + i = 1 N β i J Q r n A i y n ) , n 0 ,
(3.16)

where α n (0,1), { β i } i = 1 N [c,d](0,1), and { r n }(0,) satisfy lim n α n =0, n = 1 α n =, i = 0 N β i =1, and lim n r n =. Then { x n } converges strongly to Π F (u).

If in Theorem 3.1, N=1, then we get the following corollary.

Corollary 3.3 Let C be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space E. Let A:C E be a continuous monotone mapping satisfying (2.1). Assume that A 1 (0) is nonempty. Let { x n } be a sequence generated by

{ x 0 C , chosen arbitrarily , y n = Π C [ ( 1 α n ) x n ] , x n + 1 = J 1 ( β J y n + ( 1 β ) J Q r n A y n ) , n 0 ,
(3.17)

where α n (0,1), β(0,1), and { r n }(0,) satisfy lim n α n =0, n = 1 α n =, and lim n r n =. Then { x n } converges strongly to the minimum-norm element of A 1 (0).

We remark that if A is a maximal monotone mapping, then A 1 (0) is closed and convex (see [6] for more details). The following lemma is well known.

Lemma 3.4 [19]

Let E be a smooth, strictly convex, and reflexive Banach space, let C be a nonempty closed convex subset of E, and let AE× E be a monotone mapping. Then A is maximal if and only if R(J+rA)= E for all r>0.

We note from the above lemma that if A is maximal, then it satisfies condition (2.1) and hence we have the following corollary.

Corollary 3.5 Let C be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space E. Let A i :C E , i=1,2,,N, be maximal monotone mappings. Assume that F:= i = 1 N A i 1 (0) is nonempty. Let { x n } be a sequence generated by

{ x 0 C , chosen arbitrarily , y n = Π C [ ( 1 α n ) x n ] , x n + 1 = J 1 ( β 0 J y n + i = 1 N β i J Q r n A i y n ) , n 0 ,
(3.18)

where α n (0,1), { β i } i = 1 N [c,d](0,1) and { r n }(0,) satisfy lim n α n =0, n = 1 α n =, i = 0 N β i =1 and lim n r n =. Then { x n } converges strongly to the minimum-norm element of .

If in Corollary 3.5, N=1, then we get the following corollary.

Corollary 3.6 Let C be a nonempty, closed and convex subset of a smooth and uniformly convex real Banach space E. Let A:C E be a maximal monotone mapping. Assume that A 1 (0) is nonempty. Let { x n } be a sequence generated by

{ x 0 C , chosen arbitrarily , y n = Π C [ ( 1 α n ) x n ] , x n + 1 = J 1 ( β J y n + ( 1 β ) J Q r n A y n ) , n 0 ,
(3.19)

where α n (0,1), β(0,1), and { r n }(0,) satisfy lim n α n =0, n = 1 α n =, and lim n r n =. Then { x n } converges strongly to the minimum-norm element of A 1 (0).

If E=H, a real Hilbert space, then E is uniformly convex and smooth real Banach space. In this case, J=I, identity map on H, and Π C = P C , projection mapping from H onto C. Furthermore, (2.1) reduces to (1.9). Thus, the following corollaries hold.

Corollary 3.7 Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let A i :C E , for i=1,2,,N, be continuous monotone mappings satisfying (1.9). Assume that F:= i = 1 N A i 1 (0) is nonempty. Let { x n } be a sequence generated by

{ x 0 C , chosen arbitrarily , y n = P C [ ( 1 α n ) x n ] , x n + 1 = β 0 y n + i = 1 N β i Q r n A i y n , n 0 ,
(3.20)

where Q r A := ( I + r A ) 1 , α n (0,1), { β i } i = 1 N [c,d](0,1), and { r n }(0,) satisfy lim n α n =0, n = 1 α n =, i = 0 N β i =1, and lim n r n =. Then { x n } converges strongly to the minimum-norm element of .

Corollary 3.8 Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let A i :CH, i=1,2,,N, be maximal monotone mappings. Assume that F:= i = 1 N A i 1 (0) is nonempty. Let { x n } be a sequence generated by

{ x 0 C , chosen arbitrarily , y n = P C [ ( 1 α n ) x n ] , x n + 1 = β 0 y n + i = 1 N β i Q r n A i y n , n 0 ,
(3.21)

where Q r A := ( I + r A ) 1 , α n (0,1), { β i } i = 1 N [c,d](0,1), and { r n }(0,) satisfy lim n α n =0, n = 1 α n =, i = 0 N β i =1, and lim n r n =. Then { x n } converges strongly to the minimum-norm element of .

4 Application

In this section, we study the problem of finding a minimizer of a continuously Fréchet differentiable convex functional which has minimum-norm in Banach spaces. The following is deduced from Corollary 3.6.

Theorem 4.1 Let E be a uniformly convex and uniformly smooth real Banach space. Let f i be a continuously Fréchet differentiable convex functional on E, and let f i be maximal monotone with F:= i = 1 N ( f i ) 1 (0), where ( f i ) 1 (0)={zE: f i (z)= min y E f i (y)}, for i=1,2,,N. Let { x n } be a sequence generated by

{ x 0 C , chosen arbitrarily , y n = Π C [ ( 1 α n ) x n ] , x n + 1 = J 1 ( β 0 J y n + i = 1 N β i J ( J + r n f i ) 1 J y n ) , n 0 ,
(4.1)

where α n (0,1), { β i } i = 1 N [c,d](0,1), and { r n }(0,) satisfy lim n α n =0, n = 1 α n =, i = 0 N β i =1, and lim n r n =. Then { x n } converges strongly to the minimum-norm element of .

Remark 4.2 Theorem 3.1 provides convergence scheme to the common minimum-norm zero of a finite family of monotone mappings which improves the results of Bauschke et al. [7] to Banach spaces more general than Hilbert spaces. We also note that our results complement the results of Zegeye and Shahzad [9] and Hu and Liu [10] which are convergence results for accretive mappings.

References

  1. Byrne C: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 2002, 18(2):441–453. 10.1088/0266-5611/18/2/310

    MathSciNet  Article  Google Scholar 

  2. Censor Y, Elfving T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 1994, 8(2–4):221–239.

    MathSciNet  Article  Google Scholar 

  3. Qin X, Kang JI, Cho YJ: On quasi-variational inclusions and asymptotically strict pseudo-contractions. J. Nonlinear Convex Anal. 2010, 11: 441–453.

    MathSciNet  Google Scholar 

  4. Takahashi S, Takahashi W, Toyoda M: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 2010, 147: 27–41. 10.1007/s10957-010-9713-2

    MathSciNet  Article  Google Scholar 

  5. Zhang M: Iterative algorithms for common elements in fixed point sets and zero point sets with applications. Fixed Point Theory Appl. 2012., 2012: Article ID 21

    Google Scholar 

  6. Takahashi W: Nonlinear Functional Analysis. Kindikagaku, Tokyo; 1988. (Japanese)

    Google Scholar 

  7. Bauschke HH, Combettes PL, Reich S: The asymptotic behavior of the composition of two resolvents. Nonlinear Anal. 2005, 60(2):283–301.

    MathSciNet  Article  Google Scholar 

  8. Hundal H: An alternating projection that does not converge in norm. Nonlinear Anal. 2004, 57: 35–61. 10.1016/j.na.2003.11.004

    MathSciNet  Article  Google Scholar 

  9. Zegeye H, Shahzad N: Strong convergence theorems for a common zero of a finite family of m -accretive mappings. Nonlinear Anal. 2007, 66: 1161–1169. 10.1016/j.na.2006.01.012

    MathSciNet  Article  Google Scholar 

  10. Hu L, Liu L: A new iterative algorithm for common solutions of a finite family of accretive operators. Nonlinear Anal. 2009, 70: 2344–2351. 10.1016/j.na.2008.03.016

    MathSciNet  Article  Google Scholar 

  11. Aoyama K, Kohsaka F, Takahashi W: Proximal point method for monotone operators in Banach spaces. Taiwan. J. Math. 2011, 15(1):259–281.

    MathSciNet  Google Scholar 

  12. Alber Y: Metric and generalized projection operators in Banach spaces: properties and applications. Lecture Notes in Pure and Appl. Math. 178. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos AG. Dekker, New York; 1996:15–50.

    Google Scholar 

  13. Kamimura S, Kohsaka F, Takahashi W: Weak and strong convergence theorems for maximonotone operators in a Banach spaces. Set-Valued Anal. 2004, 12: 417–429. 10.1007/s11228-004-8196-4

    MathSciNet  Article  Google Scholar 

  14. Matsushita S, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. J. Approx. Theory 2005, 134: 257–266. 10.1016/j.jat.2005.02.007

    MathSciNet  Article  Google Scholar 

  15. Kamimura S, Takahashi W: Strong convergence of proximal-type algorithm in a Banach space. SIAM J. Optim. 2002, 13: 938–945. 10.1137/S105262340139611X

    MathSciNet  Article  Google Scholar 

  16. Zegeye H, Ofoedu EU, Shahzad N: Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings. Appl. Math. Comput. 2010, 216: 3439–3449. 10.1016/j.amc.2010.02.054

    MathSciNet  Article  Google Scholar 

  17. Xu HK: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc. 2002, 65: 109–113. 10.1017/S0004972700020116

    Article  Google Scholar 

  18. Maingé PE: Strong convergence of projected subgradient methods for nonsmooth and non-strictly convex minimization. Set-Valued Anal. 2008, 16: 89–912.

    Article  Google Scholar 

  19. Rockafellar RT: Monotone operators and proximal point algorithm. Trans. Am. Math. Soc. 1970, 194: 75–88.

    MathSciNet  Article  Google Scholar 

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Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The second author acknowledges with thanks DSR for financial support.

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Correspondence to Naseer Shahzad.

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Zegeye, H., Shahzad, N. An algorithm for a common minimum-norm zero of a finite family of monotone mappings in Banach spaces. J Inequal Appl 2013, 566 (2013). https://doi.org/10.1186/1029-242X-2013-566

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Keywords

  • accretive mappings
  • maximal monotone mappings
  • proximal point algorithm
  • resolvent mappings