Strong convergence of a proximal-type algorithm for an occasionally pseudomonotone operator in Banach spaces

R E S E A R C H

Open Access

Strong convergence of a proximal-type

algorithm for an occasionally

pseudomonotone operator in Banach spaces

Hemant Kumar Pathak

_{and Yeol Je Cho}

*_{Correspondence: [email protected]} 2_{Department of Mathematics}

Education and the RINS, Gyeongsang National University, Chinju, 660-701, Korea Full list of author information is available at the end of the article

Abstract

It is known that the proximal point algorithm converges weakly to a zero of a maximal monotone operator, but it fails to converge strongly. Then, in (Math. Program.

87:189-202, 2000), Solodov and Svaiter introduced the new proximal-type algorithm to generate a strongly convergent sequence and established a convergence property for the algorithm in Hilbert spaces. Further, Kamimura and Takahashi (SIAM J. Optim. 13:938-945, 2003) extended Solodov and Svaiter’s result to more general Banach spaces and obtained strong convergence of a proximal-type algorithm in Banach spaces. In this paper, by introducing the concept of an occasionally pseudomonotone operator, we investigate strong convergence of the proximal point algorithm in Hilbert spaces, and so our results extend the results of Kamimura and Takahashi.

MSC: 47H05; 47J25

Keywords: proximal point algorithm; monotone operator; maximal monotone operator; pseudomonotone operator; occasionally pseudomonotone operator; maximal pseudomonotone operator; maximal occasionally pseudomonotone operator; Banach space; strong convergence

1 Introduction

LetHbe a real Hilbert space with inner product·,·, and letT:H→H be a maximal monotone operator (or a multifunction) onH. We consider the classical problem:

Findx∈Hsuch that

∈Tx. (.)

A wide variety of the problems, such as optimization problems and related fields, min-max problems, complementarity problems, variational inequalities, equilibrium problems and fixed point problems, fall within this general framework. For example, ifTis the subd-ifferential∂f of a proper lower semicontinuous convex functionf :H→(–∞,∞), thenT

is a maximal monotone operator and the equation ∈∂f(x) is reduced tof(x) =min{f(z) :

z∈H}. One method of solving ∈Txis the proximal point algorithm. LetIdenote the identity operator onH. Rockafellar’s proximal point algorithm generates, for any starting pointx=x∈H, a sequence{xn}inHby the rule

xn+= (I+rnT)–xn, ∀n≥, (.)

where{rn}is a sequence of positive real numbers. Note that (.) is equivalent to

∈Txn++



rn

(xn+–xn), ∀n≥.

This algorithm was first introduced by Martinet [] and generally studied by Rockafellar [] in the framework of Hilbert spaces. Later, many authors studied the convergence of (.) in Hilbert spaces (see Agarwalet al.[], Brezis and Lions [], Choet al.[], Cholamjiak

et al. [], Güler [], Lions [], Passty [], Qinet al. [], Songet al. [], Solodov and Svaiter [], Wei and Cho [] and the references therein). Rockafellar [] proved that, if T–_{=∅andlim inf}

n→∞rn> , then the sequence {xn} generated by (.) converges

weakly to an element ofT–_{. Further, Rockafellar [] posed an open question of whether}

the sequence{xn}generated by (.) converges strongly or not. This question was solved

by Güler [], who introduced an example for which the sequence{xn}generated by (.)

converges weakly, but not strongly.

On the other hand, Kamimura and Takahashi [, ], Solodov and Svaiter [] one decade ago modified the proximal point algorithm to generate a strongly convergent se-quence. In , Solodov and Svaiter [] introduced the following algorithm{xn}:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈H,

 =vn+_r_n(yn–xn), vn∈Tyn,

Hn={z∈E:z–yn,vn ≤},

Wn={z∈E:z–xn,x–xn ≤},

xn+=PHn∩Wnx, ∀n≥.

(.)

To explain how the sequence{yn}is generated, we formally state the above algorithm as

follows.

Choose anyx∈Handσ∈[, ). At iterationn, havingxn, choosern>  and find (yn,vn),

an inexact solution of  =vn+_r_n(yn–xn),vn∈Tynwith toleranceσ. DefineHnandWnas

in (.). Takexn+=PHn∩Wnx. Note that at each iteration, there are two subproblems to be

solved: find an inexact solution of the proximal point subproblem and find the projection ofxontoHn∩Wn, the intersection of two half-spaces. By a classical result of Minty [],

the proximal subproblem always has an exact solution, which is unique. Notice that com-puting an approximate solution makes things easier. Hence, this part of the method is well defined. Regarding the projection step, it is easy to prove thatHn∩Wnis never empty, even

when the solution set is empty. Therefore, the whole algorithm is well defined in the sense that it generates an infinite sequence{xn}and an associated sequence of pairs{(yn,vn)}.

In , Kamimura and Takahashi [] extended Solodov and Svaiter’s result to more general Banach spaces like the spacesLp _{( <p<∞) by further modifying the}

proximal-point algorithm (.) in the following form in a smooth Banach spaceE: ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈E,

 =vn+_r_n(J(yn) –J(xn)), vn∈Tyn,

Hn={z∈E:z–yn,vn ≤},

Wn={z∈E:z–xn,J(x) –J(xn) ≤},

xn+=PHn∩Wnx, ∀n≥,

to generate a strongly convergent sequence. They proved that if T–_{ = ∅ and}

lim infn→∞rn> , then the sequence{xn}generated by (.) converges strongly to a point

P_T–_x_.

In this paper, by introducing the concept of an occasionally pseudomonotone operator, we investigate strong convergence of the proximal point algorithm in Hilbert spaces, and so our results extend the results of Kamimura and Takahashi.

2 Preliminaries and definitions

LetEbe a real Banach space with norm · , and letE*denote the dual space ofE. Let

x,fdenote the value off ∈E*_{atx∈E. Let{x}

n}be a sequence inE. We denote the strong

convergence of{xn}tox∈Ebyxn→xand the weak convergence byxnx, respectively. Definition . A multivalued operatorT :E→E*

with domainD(T) ={z∈E:Tz=∅} and rangeR(T) ={Tz:z∈D(T)}is said to bemonotoneifx–x,y–y ≥ for anyxi∈

D(T) andyi∈Txi,i= , . A monotone operatorTis said to bemaximalif its graphG(T) = {(x,y) :y∈Tx}is not properly contained in the graph of any other monotone operator.

Definition . A multivalued operatorT:E→E*

with domainD(T) and rangeR(T) is said to bepseudomonotone(see also Karamardian []) ifx–x,y ≥ impliesx–

x,y ≥ for anyxi∈D(T) andyi∈Txi,i= , .

It is obvious that each monotone operator is pseudomonotone, but the converse is not true.

We now introduce the concept of occasionally pseudomonotone as follows.

Definition . A multivalued operatorT:E→E*_{is said to beoccasionally}

pseudomono-toneif, for anyxi∈D(T), there existyi∈Txi,i= , , such thatx–x,y ≥ implies x–x,y ≥.

It is clear that every monotone operator is pseudomonotone and every pseudomonotone operator is occasionally pseudomonotone, but the converse implications need not be true. To this end, we observe the following examples.

Example . LetE=R_andT:E→E*_{be a multi-valued operator defined by}

Tx={y=Arx:r∈R}, ∀x∈E,

where

Ar=

⎡ ⎢ ⎣

  –

 –r 

  

⎤ ⎥ ⎦.

Then for anyx= (x() ,x ()  ,x

()

 )T,x= (x() ,x ()  ,x

()

 )TinR, ify=Arxandy=Arx, then

we have

x–x,y–y= –r

Thus, ifr≤, thenT is monotone. However, ifr> , thenT is neither monotone nor pseudomonotone. Indeed, forx= (, , ), then we havey=Arx= (, –r, ),x= (, , )

andx–x,y= ≥, butx–x,y= –r< .

Further, we see that T is occasionally pseudomonotone. To effect this, for anyx=

(x()_ ,x()_ ,x()_ )T_andx

= (x() ,x ()  ,x

()

 )TinR, ifyi=Axi,i= , , then we have x–x,y= ≥ =⇒ x–x,y= ≥.

Example . The rotation operator onR_{given by}

A=

 

– 

is monotone and hence it is pseudomonotone. Thus, it follows thatAis also occasionally pseudomonotone.

Maximality of pseudomonotone and occasionally pseudomonotone operators are de-fined as similar to maximality of a monotone operator. We denote by L[x,x] the ray

passing throughx,x.

A Banach space E is said to bestrictly convexifx+y_ <  for allx,y∈E withx=

y=  andx=y. It is also said to beuniformly convexiflim_n→∞xn–yn=  for any two

sequences{xn},{yn}inEsuch thatxn=yn=  andlimn→∞xn+y_n= .

It is known that a uniformly convex Banach space is reflexive and strictly convex. The spacesandL_{are neither reflexive nor strictly convex. Note also that a reflexive Banach}

space is not necessarily uniformly convex. For example, consider a finite dimensional Ba-nach space in which the surface of the unit ball has a ‘flat’ part. We note that such a BaBa-nach space is reflexive because of finite dimension. But the ‘flat’ portion in the surface of the ball makes it nonuniformly convex. It is also well known that a Banach spaceEis reflexive if and only if every bounded sequence of elements ofEcontains a weakly convergent sequence.

LetU={x∈E:x= }. A Banach spaceEis said to besmoothif the limit

lim

t→

x+ty–x

t (.)

exists for allx,y∈U. It is also said to beuniformly smoothif the limit (.) is attained uniformly forx,y∈U.

It is well known that the spacesp,Lp_andWm_{(Sobolev space) ( <p<∞,mis a positive}

integer) are uniformly convex and uniformly smooth Banach spaces. For anyp∈(,∞), the mappingJp:E→E

*

defined by

Jpx=

f∈E*:x,f=f · x,f=xp–, ∀x∈E,

is called theduality mappingwith the gauge functionϕ(t) =tp–_{. In particular, forp= , the}

duality mappingJwith gauge functionϕ(t) =tis called thenormalized duality mapping.

The following proposition gives some basic properties of the duality mapping.

Proposition . Let E be a real Banach space.For <p<∞,the duality mapping Jp:E→

() Jp(x)=φfor allx∈EandD(Jp) =E,whereD(Jp)denotes the domain ofJp; () Jp(x) =xp–·Jxfor allx∈Ewithx= ;

() Jp(αx) =αp–·Jxfor allα∈[,∞); () Jp(–x) = –Jp(x);

() xp_–yp_{≥px–y,jfor allx,y∈Eandj∈J} py; () IfEis smooth,thenJpis norm-to-weak*continuous;

() IfEis uniformly smooth,thenJpis uniformly norm-to-norm continuous on each

bounded subset ofE;

() Jpis bounded,i.e.,for any bounded subsetA⊂E,Jp(A)is a bounded subset inE*; () Jpcan be equivalently defined as the subdifferential of the functionalψ(x) =p–xp

(Asplund []),i.e.,

Jp(x) =∂ψ(x) =

f ∈E*:ψ(y) –ψ(x)≥ y–x,f,∀y∈E;

() Eis a uniformly smooth Banach space(equivalently,E*_{is a uniformly convex}

Banach space)if and only ifJpis single-valued and uniformly continuous on any

bounded subset ofE(see,for instance,Xu and Roach[],Browder[]). Proposition . Let E be a real Banach space, and let Jp :E→E

*

,  <p<∞,be the duality mapping.Then for any x,y∈E,

x+yp≤ xp+py,jp, ∀jp∈Jp(x+y).

Proof It is a straightforward consequence of the assertion () of Proposition . applied forxandx+y. Alternatively, from Proposition .(), it follows thatJp(x) =∂ψ(x)

(subdif-ferential of the functionalψ(x)), whereψ(x) =p–_xp_{. Also, it follows from the definition}

of the subdifferential ofψthat

ψ(x) –ψ(x+y)≥x– (x+y),jp

, ∀jp∈Jp(x+y).

Now, substitutingψ(x) byp–xp, we have

x+yp≤ xp+py,jp, ∀jp∈Jp(x+y).

This completes the proof.

Remark . IfEis a uniformly smooth Banach space, it follows from Proposition .() thatJp( <p<∞) is a single-valued mapping. We now define the functions,φ:E×E→

Rby

(x,y) =xp–px–y,Jp(y)

–yp, ∀x,y∈E,

andφis the support function satisfying the following condition:

It is obvious from the definition ofand Proposition .() that

(x,y)≥, ∀x,y∈E. (.)

Also, we see that

(x,y) =xp–px,Jp(y)

+ (p– )yp

≥ xp–pxJp(y)+ (p– )yp

=xp–pxyp–+ (p– )yp. (.)

In particular, forp= , we have(x,y)≥(x–y)_.

Further, we can show the following two propositions.

Proposition . Let E be a smooth Banach space,and let{yn},{zn}be two sequences in E.

If(yn,zn)→,then yn–zn→.

Proof It follows from(yn,zn)→ that

φ(yn,zn)→, yn–zn≤ yn–zn →

because of (.) and (.). Therefore, if {zn} is bounded, then {yn} (and also if {yn} is

bounded, then{zn}) is also bounded andyn–zn→. This completes the proof. Proposition . Let E be a reflexive,strictly convex and smooth Banach space,let C be a nonempty closed convex subset of E and let x∈E.Then there exists a unique element x∈C

such that

(x,x) =inf

(z,x) :z∈C. (.)

Proof SinceEis reflexive andzn → ∞implies(zn,x)→ ∞, there existsx∈Csuch

that(xo,x) =inf{(z,x) :z∈C}. SinceEis strictly convex, · pis a strictly convex

func-tion, that is,

λx+ ( –λ)x p

<λxp+ ( –λ)xp

for allx,x∈Ewithx=x, ≤p<∞andλ∈(, ). Then the function(·,y) is also

strictly convex. Therefore,x∈Cis unique. This completes the proof.

For each nonempty closed convex subsetCof a reflexive, convex and smooth Banach spaceE, we define the mappingRCofEontoCbyRCx=x, wherexis defined by (.). For

the casep= , it is easy to see that the mapping is coincident with the metric projection in the setting of Hilbert spaces. In our discussion, instead of the metric projection, we make use of the mappingRC. Finally, we prove two results concerning Proposition . and

the mappingRC. The first one is the usual analogue of a characterization of the metric

Proposition . Let E be a smooth Banach space,let C be a convex subset of E,let x∈E and x∈C.Then

(x,x) =inf

(z,x) :z∈C (.)

if and only if

z–x,Jp(x) –Jp(x)

≥, ∀z∈C. (.)

Proof First, we show that (.)⇒(.). Letz∈Candλ∈(, ). It follows from(x,x)≤ (( –λ)x+λz,x) that

≤( –λ)x+λxp–p

( –λ)x+λz–x,Jp(x)

–xp–xp+p

x–x,Jp(x)

+xp

=( –λ)x+λz p

–xp–pλ

z–x,Jp(x)

≤pλz–x,Jp

( –λ)x+λz

–pλz–x,Jp(x)

=pλz–x,Jp

( –λ)x+λz

–Jp(x)

,

which implies

z–x,Jp

( –λ)x+λz

–Jp(x)

≥.

Takingλ↓, sinceJpis norm-to-weak* continuous, we obtain

z–x,Jp(x) –Jp(x)

≥,

which shows (.).

Next, we show that (.)⇒(.). For anyz∈C, we have

(z,x) –(x,x) =zp–p

z–x,Jp(x)

–xp

–xp+p

x–x,Jp(x)

+xp

=zp–xp–p

z–x,Jp(x)

≥pz–x,Jp(x)

–pz–x,Jp(x)

=pz–x,Jp(x) –Jp(x)

≥,

which proves (.). This completes the proof.

Proposition . Let E be a reflexive,strictly convex and smooth Banach space,let C be a nonempty closed convex subset of E,let x∈E and RCx∈C with

Then we have

(y,RCx) +(RCx,x)≤(y,x), ∀y∈L[x,RCx]∩C.

Proof It follows from Proposition . that

(y,x) –(y,RCx) –(RCx,x)

=yp–py–x,Jp(x)

–xp+y–x–yp

+py–RCx,Jp(RCx)

+RCxp–y–RCx–RCxp

+pRCx–x,Jp(xP

+xp–RCx–x

= –py–x,Jp(x)

+py–RCx,Jp(RCx)

+pRCx–x,Jp(x)

=py–RCx,Jp(RCx) –Jp(x)

≥, ∀y∈L[x,RCx]∩C.

This completes the proof.

3 Main results

Throughout this section, unless otherwise stated, we assume that T :E→E* _{is a}

oc-casionally pseudomonotone maximal monotone operator. In this section, we study the following algorithm{xn}in a smooth Banach spaceE, which is an extension of (.):

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈E,

 =vn+_r_n(Jp(yn) –Jp(xn)), vn∈Tyn,

Hn={z∈E:z–yn,vn ≤},

Wn={z∈E:z–xn,Jp(xn) –Jp(xn) ≤},

xn+=RHn∩Wnxn, ∀n≥,

(.)

where{rn}is a sequence of positive real numbers.

First, we investigate the condition under which the algorithm (.) is well defined. Rock-afellar [] proved the following theorem.

Theorem . Let E be a reflexive,strictly convex and smooth Banach space,and let T:

E→E*be a monotone operator.Then T is maximal if and only if R(Jp+rT) =E*for all

r> .

By the appropriate modification of arguments in Theorem ., we can prove the follow-ing.

Theorem . Let E be a reflexive,strictly convex and smooth Banach space,and let T:

E→E* _{be an occasionally pseudomonotone operator.Then T is maximal if and only if}

R(Jp+rT) =E*for all r> .

Proposition . Let E be a reflexive,strictly convex and smooth Banach space.If T–_=∅,

then the sequence generated{xn}by(.)is well defined.

Proof From the definition of the sequence{xn}, it is obvious that bothHn andWnare

closed convex sets. Letw∈T–_{. From Theorem ., there exists (y}

,v)∈E×E*such

that

 =v+



r

Jp(y) –Jp(x)

, v∈Ty.

SinceT is occasionally pseudomonotone andy–w, = ≥, fromTw, it follows

that

y–w,v ≥

for somev∈Ty. It follows thatw∈H. On the other hand, it is clear thatw∈W=E.

Thenw∈H∩Wand sox=RH∩Wxis well defined. Suppose thatw∈Hn–∩Wn–is well defined for somen≥. Again, by Theorem ., we obtain (yn,vn)∈E×E*such that

 =vn+



rn

Jp(yn) –Jp(xn)

, vn∈Tyn.

Then sinceT is occasionally pseudomonotone andyn–w, = ≥, fromTw, it

follows that

yn–w,vn ≥

for somevn∈Tyn, and sow∈Hn. It follows from Proposition . that

w–xn,Jp(x) –Jp(xn)

=w–RHn–∩Wn–x,Jp(x) –Jp(RHn–∩Wn–x)

≤,

which impliesw∈Wn. Therefore,w∈Hn∩Wnand hencexn–=RHn∩Wnxis well defined.

Then by induction, the sequence{xn}generated by (.) is well defined for eachn≥.

This completes the proof.

Remark . From the above proof, we obtain

T–⊂Hn∩Wn, ∀n≥.

Now, we are ready to prove our main theorem.

Theorem . Let E be a reflexive,strictly convex and uniformly smooth Banach space.If T–_{=∅,φsatisfies the condition(.)and{r}

n} ⊂(,∞)satisfieslim infn→∞rn> ,then

Proof It follows from the definition ofWn+and Proposition . thatxn+=RWn+x. Fur-ther, fromx∈L(xn,RWn+x)∩Wn–and Proposition ., we have

(xn,RWn+x) +(RWn+x,x)≤(xn,x) and hence

(xn,xn+) +(xn+,x)≤(xn,x). (.)

Since the sequence{(xn,x)}is monotone decreasing and bounded below by , it follows

thatlim infn→∞(xn,x) exists and, in particular,{(xn,x)}is bounded. Then by (.), {xn}is also bounded. This implies that there exists a subsequence{xni}of{xn}such that

xniwfor somew∈E.

Now, we show thatw∈T–_{. It follows from (.) that(x}

n,xn+)→. On the other

hand, we have

(RHnxn,xn) –(yn,xn) =RHnxn

p_–pR

Hnxn–xn,Jp(xn)

–xnp

+RHnxn–xn–yn

p_+py

n–xn,Jp(xn)

+xnp–yn–xn

=RHnxnp–ynp+p

yn–RHnxn,Jp(xn)

+RHnxn–yn ≥pRHnxn–yn,Jp(yn)

+pyn–RHnxn,Jp(xn)

+RHnxn–yn

=pyn–RHnxn–yn,Jp(xn) –Jp(yn)

+RHnxn–yn.

SinceRHnxn∈Hnand  =vn+_r_n(Jp(yn) –Jp(xn)), it follows that

yn–RHnxn–ynJp(xn) –Jp(yn)

≥,

and so(RHnxn,xn)≥(yn,xn). Further, sincexn+∈Hn, we have

(xn+,xn)≥(RHnxn,xn), which yields that

(xn+,xn)≥(RHnxn,xn)≥(yn,xn).

Then it follows from(xn,xn+)→ that(yn,xn)→. Consequently, by Proposition .,

we haveyn–xn→, which impliesyniw. Moreover, sinceJis uniformly norm-to-norm continuous on bounded subsets andlim infn→∞rn> , we obtain

vn= –



rn

Jp(yn) –Jp(xn)

→.

It follows fromvn∈Tynwithvn→ andyniwthat lim

Then, sinceT is occasionally pseudomonotone, it follows thatz–w,z=  for some

z∈Tz. Therefore, from the maximality ofT, we obtainw∈T–_{. Letw}*_∈R

T–_x_. Now, fromxn+=RHn∩Wnxandw*∈T–⊂L(xn,RWn+x)∩Hn∩Wn, we have

(xn+,x)≤

w*,x

.

Then we have

xn,w*

=(xn,x) +

x,w*

–pxn,x,Jp

w*–Jp(x)

+xn–w*–xn–x–x–w* ≤w*,x

+x,w*

–pxn–x,Jp

w*–Jp(x)

+xn–w*–xn–x–x–w*,

which yields

lim sup

i→∞

xni,w*

≤w*,x

+x,w*

–pw–x,Jp

w*–Jp(x)

+w–w*–w–x–x–w*.

Thus, from Proposition ., we have

w*,x

+x,w*

–pw–x,Jp

w*–Jp(x)

+w–w*–w–x–x–w*

=pw–w*,Jp(x) –Jp

w*

≤.

Then we obtain lim sup_i→∞(xni,w

*_{)≤, and hence (x} ni,w

*_{)→. It follows from}

Proposition . thatxni →w*. This means that the whole sequence{xn} generated by (.) converges weakly tow*and each weakly convergent subsequence of{xn}converges

strongly to w*_{. Therefore,{x}

n} converges strongly to w*∈RT–_x. This completes the

proof.

4 An application

Letf :E→(–∞,∞] be a proper convex lower semicontinuous function. Then the subd-ifferential∂f off is defined by

∂f(x) =v∈E*:f(y) –f(x)≥ y–x,v,∀y∈E.

Using Theorem ., we consider the problem of finding a minimizer of the functionf.

(,∞)satisfieslim infn→∞rn> and{xn}is the sequence generated by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈E,

yn=argminz∈E{f(z) +prnz

p_– 

rnz,Jp(xn)},  =vn+_r_n(Jp(yn) –Jp(xn)), vn∈Tyn,

Hn={z∈E:z–yn,vn ≤},

Wn={z∈E:z–xn,Jp(xn) ≤},

xn+=RHn∩Wnx, ∀n≥.

(.)

If(∂f)–_{=∅,then the sequence{x}

n}generated by(.)converges strongly to the minimizer

of f.

Proof Sincef:E→(–∞,∞] is a proper convex lower semicontinuous function, by Rock-afelllar [], the subdifferential∂f off is a maximal monotone operator and so it is also an occasionally pseudomonotone maximal operator. We also know that

yn=argmin z∈E

f(z) + 

prn

zp– 

rn

z,Jp(xn)

is equivalent to the following:



rn

Jp(z) –Jp(xn)

∈∂f(yn), ∀z∈E.

This implies that

∈∂f(yn) +



rn

Jp(yn) –Jp(xn)

.

Thus, we have vn∈∂f(yn) such that  =vn+_r_n(Jp(yn) –Jp(xn)). Therefore, using

Theo-rem ., we get the conclusion. This completes the proof.

5 Concluding remarks

We presented a modified proximal-type algorithm with the varied degree of rate of the convergence depending upon the choice ofp( <p<∞) for an occasionally pseudomono-tone operator, which is a generalization of a monopseudomono-tone operator, to extend Kamimura and Takahashi’s result to more general Banach spaces which are not necessarily uniformly con-vex like locally uniformly Banach spaces. As an application, we consider the problem of finding a minimizer of a convex function in a more general setting of Banach spaces than what Kamimura and Takahashi have considered.

Competing interests

The authors declare that they have no competing interests. Authors’ contributions

All authors read and approved the final manuscript. Author details

1_{School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, C.G. 492010, India.}2_{Department of}

Acknowledgements

The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant number: 2012-0008170).

Received: 9 May 2012 Accepted: 24 September 2012 Published: 24 October 2012 References

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doi:10.1186/1687-1812-2012-190