A new iterative algorithm for the sum of infinite m-accretive mappings and infinite \(\mu_{i}\)-inversely strongly accretive mappings and its applications to integro-differential systems

R E S E A R C H

Open Access

A new iterative algorithm for the sum of

infinite

m

-accretive mappings and infinite

μ

_i

-inversely strongly accretive mappings and

its applications to integro-differential systems

Li Wei

1*

_{and Ravi P Agarwal}

2,3

*_{Correspondence:}

[email protected]

1_{School of Mathematics and}

Statistics, Hebei University of Economics and Business, Shijiazhuang, 050061, China Full list of author information is available at the end of the article

Abstract

A new three-step iterative algorithm for approximating the zero point of the sum of an infinite family ofm-accretive mappings and an infinite family of

μ

i-inversely

strongly accretive mappings in a realq-uniformly smooth and uniformly convex

Banach space is presented. The computational error in each step is being considered. A strong convergence theorem is proved by means of some new techniques, which extend the corresponding work by some authors. The relationship between the zero

point of the sum of an infinite family ofm-accretive mappings and an infinite family of

μ

i-inversely strongly accretive mappings and the solution of one kind variational

inequalities is investigated. As an application, an integro-differential system is

exemplified, from which we construct an infinite family ofm-accretive mappings and

an infinite family of

μ

i-inversely strongly accretive mappings. Moreover, the iterative sequence of the solution of the integro-differential systems is obtained.

MSC: 47H05; 47H09; 47H10

Keywords: m-accretive mapping;

μ

i-inversely strongly accretive mapping; contraction; retraction; integro-differential systems

1 Introduction and preliminaries

Throughout this paper, we assume thatEis a real Banach space with norm·andE∗is the dual space ofE. We use ‘→’ and ‘’ (or ‘w–lim’) to denote strong and weak convergence either inEor inE∗, respectively. We denote the value off ∈E∗atx∈Ebyx,f.

A Banach spaceEis said to be uniformly convex if, for eachε∈(, ], there existsδ>  such that

x=y= , x–y ≥ε ⇒ x+y



≤ –δ.

A Banach spaceEis said to be smooth if

lim t→

x+ty–x t

exists for eachx,y∈ {z∈E:z= }.

In addition, we define a functionρE: [, +∞)→[, +∞) called the modulus of

smooth-ness ofEas follows:

ρE(t) =sup

 

x+y+x–y–  :x,y∈E,x= ,y ≤t

.

It is well known thatEis uniformly smooth if and only if ρE(t)

t →, ast→. Letq> 

be a real number. A Banach spaceE is said to beq-uniformly smooth if there exists a positive constantCsuch thatρE(t)≤Ctq. It is obvious thatq-uniformly smooth Banach

space must be uniformly smooth.

An operatorB:E→E∗is said to be monotone [] ifu–v,Bu–Bv ≥, for allu,v∈ D(B). The monotone operatorBis said to be maximal monotone if the graph ofB,G(B), is not contained properly in any other monotone subset ofE×E∗. An operatorB:E→E∗is said to be coercive iflimn→∞xn_xn,Bxn= +∞for{xn} ⊂D(B) such thatlimn→∞xn= +∞.

A single-valued mapping F:D(F) =E→E∗ is said to be hemi-continuous [] ifw–

limt→F(x+ty) =Fx, for anyx,y∈E. A single-valued mappingF:D(F) =E→E∗is said to

be demi-continuous [] ifw–limn→∞Fxn=Fx, for any sequence{xn}strongly convergent

toxinE.

Following from [] or [], the functionhis said to be a proper convex function onEif

his defined fromEonto (–∞, +∞],his not identically +∞such thath(( –λ)x+λy)≤ ( –λ)h(x) +λh(y), wheneverx,y∈Eand ≤λ≤.his said to be strictly convex ifh(( –

λ)x+λy) < ( –λ)h(x) +λh(y), for all  <λ<  andx,y∈E withx= y,h(x) < +∞and

h(y) < +∞. The functionh:E→(–∞, +∞] is said to be lower-semi-continuous onE if

lim infy→xh(y)≥h(x), for anyx∈E. Given a proper convex functionhonEand a point

x∈E, we denote by∂h(x) the set ofx∗∈E∗such thath(x)≤h(y) +x–y,x∗for anyy∈E. Such elementsx∗are called subgradients ofhatx, and∂h(x) is called the subdifferential ofhatx.

Forq> , the generalized duality mappingJq:E→E

∗

is defined by

Jqx:=

f ∈E∗:x,f=xq,f=xq–, x∈E.

In particular,J:=Jis called the normalized duality mapping andJq(x) =xq–J(x) forx=

. It is well known that ifEis smooth, thenJqis single-valued. IfEis reduced to the Hilbert

spaceH, thenJq≡Iis the identity mapping. It can be seen from [] that the normalized

duality mappingJhas the following properties:

(i) ifEis uniformly smooth, thenJis norm-to-norm uniformly continuous on each

bounded subset inE;

(ii) the reflexivity ofEand strict convexity ofE∗imply thatJis single-valued, monotone, and demi-continuous.

In the following, we still denote byJandJqthe single-valued normalized duality mapping

and the single-valued generalized duality mapping.

For a mappingT:D(T)E→E, we useFix(T) to denote the fixed point set of it; that is,Fix(T) :={x∈D(T) :Tx=x}.

LetT:D(T)E→Ebe a mapping. ThenTis said to be

() non-expansive if

() k-Lipschitz if there existsk> such that

Tx–Ty ≤kx–y for∀x,y∈D(T);

in particular, if <k< , thenT is called a contraction and ifk= , thenT reduces to a non-expansive mapping;

() accretive if, for allx,y∈D(T), there existsjq(x–y)∈Jq(x–y)such that

Tx–Ty,jq(x–y)

≥;

() μ-inversely strongly accretive if, for allx,y∈D(T), there existsjq(x–y)∈Jq(x–y)

such that

Tx–Ty,jq(x–y)

≥μTx–Tyq

for someμ> ;

() m-accretive ifTis accretive andR(I+λT) =Efor∀λ> ;

() strongly positive(see []) ifD(T) =EwhereEis a real smooth Banach space and there existsγ > such that

Tx,Jx ≥γx for∀x∈E;

in this case,

aI–bT= sup x≤

(aI–bT)x,J(x),

whereIis the identity mapping anda∈[, ],b∈[–, ];

() demiclosed atpif whenever{xn}is a sequence inD(T)such thatxnx∈D(T)and

Txn→pthenTx=p;

() strongly accretive if, for allx,y∈D(T), there existsj(x–y)∈J(x–y)such that

Tx–Ty,j(x–y)≥ x–y

for some > .

For the accretive mappingA, we useN(A) to denote the set of zero points of it; that is,

N(A) :={x∈D(A) :Ax= }. IfAis accretive, then we can define, for eachr> , a single-valued mappingJA

r :R(I+rA)→D(A) byJrA:= (I+rA)–, which is called the resolvent of

A[]. It is well known thatJA

r is non-expansive andN(A) =Fix(JrA).

LetCbe a nonempty, closed and convex subset ofEandQbe a mapping ofEontoC. ThenQis said to be sunny [] ifQ(Q(x) +t(x–Q(x))) =Q(x), for allx∈Eandt≥.

A mappingQofEintoE is said to be a retraction [] ifQ_{=Q. If a mappingQis a}

retraction, thenQ(z) =zfor everyz∈R(Q), whereR(Q) is the range ofQ.

Many practical problems can be reduced to finding zeros of the sum of two accretive op-erators; that is, ∈(A+B)x. Forward-backward splitting algorithms, which have recently received much attention to many mathematicians, were proposed by Lions and Mercier [], by Passty [], and, in a dual form for convex programming, by Han and Lou [].

The classical forward-backward splitting algorithm is given in the following way:

xn+= (I+rnB)–(I–rnA)xn, n≥. ()

Based on iterative algorithm (), much work has been done for findingx∈Hsuch that

x∈N(A+B), whereAandBareμ-inversely strongly accretive mapping andm-accretive mapping defined in the Hilbert space H, respectively. In , Weiet al. extended the related work from the Hilbert space to the real smooth and uniformly convex Banach space and presented the following iterative algorithm with errors []:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C chosen arbitrarily, yn=QC[( –αn)(xn+en)],

zn= ( –βn)xn+βn[ayn+

N

i=aiJrn,iAi(yn–rn,iBiyn)],

xn+=γnηf(xn) + (I–γnT)zn, n≥,

()

whereCis a nonempty, closed, and convex sunny non-expansive retract ofE,QCis the

sunny non-expansive retraction ofEontoC,{en} ⊂Eis the error sequence,{Ai}Ni= is a

finite family ofm-accretive mappings and{Bi}Ni=is a finite family ofμ-inversely strongly

accretive mappings.T :E→Eis a strongly positive linear bounded operator with coef-ficientγ andf :E→Eis a contraction with coefficientk∈(, ).Jrn,iAi = (I+rn,iAi)–, for

i= , , . . . ,N,N_m=am= ,  <am< , form= , , , . . . ,N. Then{xn}is proved to

con-verge strongly top∈ N

i=N(Ai+Bi), which solves the variational inequality

(T–ηf)p,J(p–z)

≤,

for∀z∈N_i=N(Ai+Bi) under some conditions.

The implicit midpoint rule (IMR) is one of the powerful numerical methods for solving ordinary differential equations, which is extensively studied recently by Alghamdiet al.

They presented the following implicit midpoint rule for approximating the fixed point of a non-expansive mapping in a Hilbert space in []:

x∈H, xn+= ( –αn)xn+αnT

xn+xn+



, n≥, ()

whereTis non-expansive fromHtoH. IfFix(T)=∅, then they proved that{xn}converges

weakly top∈Fix(T), under some conditions.

Inspired by the work in [] and [], we shall present the following iterative algorithm with errors in a realq-uniformly smooth and uniformly convex Banach space:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈C chosen arbitrarily, yn=QC[( –αn)(xn+en)],

zn=δnyn+βni∞=aiJrn,iAi[yn_+zn–rn,iBi(yn+_zn)] +ζnen,

xn+=γnηf(xn) + (I–γnT)zn+en, n≥,

where C is a nonempty, closed, and convex sunny non-expansive retract of E, QC is

the sunny non-expansive retraction ofEontoC,{e_n},{e_n}, and{e_n} are three error se-quences, Ai :C→E ism-accretive and Bi :C →E isμi-inversely strongly accretive,

wherei∈N+_{.T :E→E is a strongly positive linear bounded operator with coefficient} γ andf :E→Eis a contraction with coefficientk∈(, ).Jrn,iAi = (I+rn,iAi)–, fori∈N+.

∞

m=am = ,  <am < , form∈N+. δn+βn+ζn≡, for n≥. More details of

it-erative algorithm (A) will be presented in Section . Then {xn} is proved to converge

strongly top∈ ∞

i=N(Ai+Bi), which is also a solution of one kind of variational

in-equality.

Our main contributions are:

(i) A new three-step iterative algorithm is designed by combining the ideas of famous iterative algorithms such as proximal methods, Halpern methods, convex

combination methods, viscosity methods, and implicit midpoint methods.

(ii) The assumption that ‘the duality mappingJis weakly sequentially continuous’ or

‘Jis weakly sequentially continuous at zero’ in most of the existing related work is deleted.

(iii) ‘Biisμ-inversely strongly accretive’ in most of the related work is replaced by ‘Biis

μi-inversely strongly accretive’, which makes the discussion more general.

Moreover, the design of the iterative algorithm is extended from finite case of the

sum ofm-accretive mappings andμ-inversely strongly accretive mappings to the

infinite case.

(iv) The discussion is undertaken in the frame of a realq-uniformly smooth and

uniformly convex Banach space, which is more general than that in a Hilbert space. (v) In each step of the three-step iterative algorithm, computational error is being

considered - that is, we consider three error sequences{e_n},{e_n}, and{e_n}. (vi) All of the three sequences{yn},{zn}, and{xn}constructed in the new iterative

algorithm are proved to be strongly convergent to the zero point of the sum of an infinite family ofm-accretive mappings and an infinite family ofμi-inversely

strongly accretive mappings.

(vii) The connection between the zero point of the sum ofm-accretive mappings and

μi-inversely strongly accretive mappings and the solution of one kind variational

inequalities is being set up.

(viii) In Section , the application of the main result in Section  to one kind integro-differential systems is demonstrated, from which we can see the connections between variational inequalities, integro-differential equations, and iterative algorithms.

Next, we list some results we need in the sequel.

Lemma (see []) Let E be a Banach space and C be a nonempty closed and convex subset

of E.Let f :C→C be a contraction.Then f has a unique fixed point u∈C.

Lemma (see []) Let E be a real uniformly convex Banach space,C be a nonempty,

closed, and convex subset of E and T :C→E be a non-expansive mapping such that

Lemma (see []) In a real Banach space E,the following inequality holds:

x+y≤ x+ y,j(x+y), ∀x,y∈E,

where j(x+y)∈J(x+y).

Lemma (see []) Let{an}and{cn}be two sequences of nonnegative real numbers

satis-fying

an+≤( –tn)an+bn+cn, ∀n≥,

where {tn} ⊂ (, ) and {bn} is a number sequence. Assume that

∞

n=tn = +∞, lim sup_n→∞bn_tn ≤,and∞_n=cn< +∞.Thenlimn→∞an= .

Lemma (see []) Let E be a Banach space and let A be an m-accretive mapping.For

λ> ,μ> ,and x∈E,one has

J_λAx=J_μA

μ λx+

 –μ

λ

J_λAx

,

where JλA= (I+λA)–and JμA= (I+μA)–.

Lemma (see []) Let E be a real Banach space and let C be a nonempty,closed,and

convex subset of E.Suppose A:C→E is a single-valued mapping and B:E→E_is

m-accretive.Then

Fix(I+rB)–(I–rA)=N(A+B) for∀r> .

Lemma (see []) Assume T is a strongly positive bounded operator with coefficientγ > 

on a real smooth Banach space E and <ρ≤ T–_{.ThenI–ρT ≤ –ργ.}

Lemma  (see []) Let E be a real strictly convex Banach space and let C be a

nonempty closed and convex subset of E.Let Tm:C→C be a non-expansive mapping

for each m≥. Let {am} be a real number sequence in (,) such that

∞

m=am = .

Suppose that ∞_m=Fix(Tm)=∅. Then the mapping ∞m=amTm is non-expansive with

Fix(∞_m=amTm) =

∞

m=Fix(Tm).

Lemma (See []) Let C be a nonempty,closed,and convex subset of a real q-uniformly

smooth Banach space E with constant Kq. Let the mapping A:C→E be aμ-inversely

strongly accretive mapping.Then the following inequality holds:

(I–rA)x– (I–rA)yq≤ x–yq–rqμ–Kqrq–

Ax–Ayq.

2 Strong convergence theorems

Lemma  Let E be a real uniformly smooth and uniformly convex Banach space and C

be a nonempty,closed,and convex sunny non-expansive retract of E,and let QC be the

sunny non-expansive retraction of E onto C.Let f :E→E be a fixed contractive mapping

with coefficient k∈(, ),T :E→E be a strongly positive linear bounded operator with

coefficientγ and U:C→C be a non-expansive mapping.Suppose that <η< _γ_k and

Fix(U)=∅.If for each t∈(, ),define Tt:E→E by

Ttx:=tηf(x) + (I–tT)UQCx, ()

then Tthas a fixed point xt,for each <t≤ T–,which is convergent strongly to the fixed

point of U,as t→.That is,limt→xt=p∈Fix(U).Moreover,psatisfies the following

variational inequality:for∀z∈Fix(U),

(T–ηf)p,J(p–z)

≤. ()

Proof Copying Steps  to  of Lemma  in [], we have the following results:

(a) Ttis a contraction, for <t<T–.

(b) Tthas a unique fixed pointxt.

(c) {xt}is bounded, for <t<T–.

(d) xt–UQCxt→, ast→.

(e) If the inequality () has a solution, then the solution must be unique.

Finally, we are to show thatxt→p∈Fix(U), ast→, which satisfies the variational

inequality ().

Assumetn→. Setxn:=xtnand defineμ:E→Rby

μ(x) =LIMxn–x, x∈E,

whereLIMis the Banach limit onl∞. Let

K=

x∈E:μ(x) =min

x∈ELIMxn–x

_.

It is easily seen thatK is a nonempty, closed, convex, and bounded subset ofE. Since

xn–UQCxn→, forx∈K,

μ(UQCx) =LIMxn–UQCx≤LIMxn–x=μ(x),

it follows thatUQC(K)⊂K; that is,Kis invariant underUQC. Since a uniformly smooth

Banach space has the fixed point property for non-expansive mappings,UQChas a fixed

point, sayp, inK. That is,UQCp=p∈C, which ensures thatp=Upfrom the

defi-nition ofUand thenp∈Fix(U). Sincepis also a minimizer ofμoverE, it follows that,

fort∈(, )

≤μ(p+ηtf(p) –tTp) –μ(p)

t

=LIMxn–p–ηtf(p) +tTp

_–x

=LIMxn–p–ηtf(p) +tTp,J(xn–p–ηtf(p) +tTp)–xn–p



t

=LIM xn–p,J

xn–p–ηtf(p) +tTp

+tTp–ηf(p),J

xn–p–ηtf(p) +tTp

–xn–p

/t.

SinceEis uniformly smooth, then by lettingt→, we find the two limits above can be interchanged and obtain

LIMηf(p) –Tp,J(xn–p)

≤. ()

Sincext–p=t(ηf(xt) –Tp) + (I–tT)(UQCxt–p), then

xt–p=tηf(xt) –Tp,J(xt–p)

+ (I–tT)(UQCxt–p),J(xt–p)

≤tηf(xt) –f(p),J(xt–p)

+tηf(p) –Tp,J(xt–p)

+I–tTxt–p

≤ –t(γ –ηk)xt–p+tηf(p) –Tp,J(xt–p)

. Therefore,

xt–p≤



γ –ηk ηf(p) –Tp,J(xt–p)

.

Hence by ()

LIMxn–p≤



γ–ηkLIMηf(p) –Tp,J(xn–p)

≤,

which implies thatLIMxn–p= , and then there exists a subsequence which is still

denoted by{xn}such thatxn→p.

Next, we shall show thatpsolves the variational inequality ().

Sincext=tηf(xt) + (I–tT)UQCxt, (T–ηf)xt= –_t(I–tT)(I–UQC)xt. For∀z∈Fix(U),

(T–ηf)xt,J(xt–z)

= –

t (I–tT)(I–UQC)xt,J(xt–z)

= –

t (I–UQC)xt– (I–UQC)z,J(xt–z)

+ T(I–UQC)xt,J(xt–z)

= –

t[xt–z _{– UQ}

Cxt–UQCz,J(xt–z)

+ T(I–UQC)xt,J(xt–z)

≤ T(I–UQC)xt,J(xt–z)

.

Taking the limits on both sides of the above inequality,(T–ηf)p,J(p–z) ≤ since xn→pandJis uniformly continuous on each bounded subsets ofE.

Thuspsatisfies the variational inequality ().

Now assume there exists another subsequence{xm}of{xt}satisfyingxm→q. Then

at zero, thenq=UQCq, which ensures thatq∈Fix(U). Repeating the above proof, we

can also know thatqsolves the variational inequality (). Thusp=qby using the result

of (e).

Hencext→p, ast→, which is the unique solution of the variational inequality ().

This completes the proof.

Theorem  Let E be a real q-uniformly smooth Banach space with constant Kqand also

be a uniformly convex Banach space.Let C be a nonempty,closed,and convex sunny

non-expansive retract of E, and QC be the sunny non-expansive retraction of E onto C. Let

f :E→E be a contraction with coefficient k∈(, ),T:E→E be a strongly positive linear

bounded operator with coefficientγ.Let Ai:C→E be m-accretive mappings,Bi:C→E be

μi-inversely strongly accretive mappings,for i∈N+.Let D:=

∞

i=N(Ai+Bi)=∅.Suppose

 <η<_γ_k.Suppose{αn},{δn},{βn},{ζn},{γn} ⊂(, ),and{rn,i} ⊂(, +∞)for i∈N+.

Sup-pose{ai}∞i=⊂(, )with ∞

i=ai= ,{en} ⊂C,and{en},{en} ⊂E are three error sequences.

Let{xn}be generated by the iterative algorithm(A).Further suppose that the following

con-ditions are satisfied:

(i) ∞_n=αn< +∞;

(ii) ∞_n=γn=∞,γn→,asn→ ∞and

∞

n=|γn–γn–|< +∞;

(iii) ∞_n=|rn+,i–rn,i|< +∞, <ε≤rn,i≤(q_Kqμi) 

q–_,forn≥_andi∈N+_;

(iv) δn+βn+ζn≡,forn≥,

∞

n=|δn–δn–|< +∞, ∞

n=|βn–βn–|< +∞, ∞

n=

ζn

βn < +∞,andβn→,asn→ ∞;

(v) ∞_n=e_n< +∞,∞_n=e_n< +∞,∞_n=e_n< +∞.

Then three sequences{xn},{yn}, and{zn}converge strongly to the unique elementp∈ D, which satisfies the following variational inequality: for∀y∈D,

(T–ηf)p,J(p–y)

≤. ()

Proof We shall split the proof into seven steps.

Step .{xn}is well defined.

In fact, it suffices to show that{zn}is well defined.

Fort,s,r∈(, ) andt+s+r≡, defineUt,s,r:C→CbyUt,s,rx:=tu+sU(u+_x) +rv, where

U:C→Cis non-expansive forx,u,v∈C. Then

Ut,s,rx–Ut,s,ry ≤s

u+x

 –

u+y



≤ s

x–y.

ThusUt,s,ris a contraction, which ensures from Lemma  that there existsxt,s,r∈Csuch

thatUt,s,rxt,s,r=xt,s,r. That is,xt,s,r=tu+sU(u+xt,s,r_ ) +rv.

SinceJrn,iAi(I–rn,iBi) is non-expansive in view of Lemma  and

_∞

i=ai= ,

_∞

i=aiJrn,iAi(I–

rn,iBi) is non-expansive, which implies that{zn}is well defined, and then{xn}is well

de-fined.

Step .D:=∞_i=N(Ai+Bi) =Fix(

∞

i=aiJrn,iAi(I–rn,iBi)).

Lemma  implies that N(Ai+Bi) =Fix(Jrn,iAi(I–rn,iBi)), where i∈N+. Then Lemma 

ensures that∞_i=N(Ai+Bi) =

∞ i=Fix(J

Ai

rn,i(I–rn,iBi)) =Fix(

∞

i=aiJrn,iAi(I–rn,iBi)).

∀p∈D, we see that, forn≥,

yn–p ≤( –αn)xn–p+ ( –αn)en+αnp. ()

Therefore, forp∈Dandn≥, we have

zn–p ≤δnyn–p+βn

∞

i= aiJrn,iAi

yn+zn

 –rn,iBi

yn+zn



–p

+ζnen–p ≤

δn+

βn



yn–p+

βn

 zn–p+ζne

n–p ≤

 –βn 

yn–p+

βn

 zn–p+ζne

n–p,

which implies that

zn–p ≤ yn–p+

ζn

 –βn

e_n–p≤ yn–p+ en+

ζn

 –βn

p. ()

Noticing () and (), using Lemma , we have, forn≥,

xn+–p ≤γnηkxn–p+γnηf(p) –Tp+ ( –γnγ)zn–p+en ≤ –γn(γ –kη)

xn–p+γnηf(p) –Tp

+e_n+ e_n+e_n+αnp+

ζn

 –βn

p. ()

By using the inductive method, we can easily get the following result from ():

xn+–p ≤max

x–p,

ηf(p) –Tp

γ–kη

+

n

k=

e_k+ 

n

k= e_k

+

n

k=

e_k+p _n

k= αk+

n

k=

ζk

 –βk

.

Therefore, from assumptions (i), (iv), and (v), we know that{xn}is bounded. Then{yn}

and{zn}are bounded in view of () and (), respectively.

Letun,i= (I–rn,iBi)(yn+_zn), then{un,i} is bounded in view of Lemma , forn≥ and

i∈N+_.

Since∞_i=aiJrn,iAiun,i ≤

∞

i=aiJrn,iAiun,i–

∞

i=aiJrn,iAi(I–rn,iBi)p+p ≤ yn+_zn –p+ pin view of Step , then{∞_i=aiJrn,iAiun,i}is bounded. Moreover, we can easily know that {f(xn)},{Tzn},{Bi(yn+_zn)}, and{Jrn,iAiun,i}are all bounded, forn≥ andi∈N+.

Set M = sup{un,i,xn,Tzn,yn,f(xn),JrAin,iun,i,

∞

i=aiJrAin,iun,i,Bi( yn+zn

 ) : n≥,i∈N+}. ThenMis a positive constant.

In fact, ifrn,i≤rn+,i, then, using Lemma ,

J_rn+,iAi un+,i–Jrn,iAiun,i

≤ un+,i–un,i+

rn+,i–rn,i

ε J

Ai

rn+,iun+,i–un,i ≤ un+,i–un,i+ 

rn+,i–rn,i

ε M

_{. ()}

Ifrn+,i≤rn,i, then imitating the proof of (), we have

JAi

rn+,iun+,i–J Ai

rn,iun,i≤ un+,i–un,i+ 

rn,i–rn+,i

ε M

_{. ()}

Combining () and (), we have, forn≥ andi∈N+_, J_rn+,iAi un+,i–Jrn,iAiun,i≤ un+,i–un,i+ |

rn,i–rn+,i|

ε M

_{. ()}

Then in view of Lemma 

un+,i–un,i=

(I–rn+,iBi)

yn++zn+

 –

yn+zn



+|rn,i–rn+,i|

Bi

yn+zn



≤yn+–yn



+zn+–zn 

+|rn,i–rn+,i|M. ()

In view of () and (), we have

zn+–zn

≤δn+yn+–yn+|δn+–δn|yn+|βn+–βn|

∞ i=

aiJrn,iAiun,i

+βn+ ∞ i=

aiJrn+,iAi un+,i– ∞

i=

aiJrn,iAiun,i

+ζn+en+–ζnen ≤

δn++

βn+



yn+–yn+|βn+–βn|M+|δn+–δn|M+

βn+

 zn+–zn +

 +

ε

βn+|rn,i–rn+,i|M+ζn+en+–ζnen,

which implies that

zn+–zn ≤

δn++βn+_

 –βn+



yn+–yn+

|βn+–βn|M

 –βn+

+|δn+–δn|M

 –βn+

+( +



ε)βn+|rn,i–rn+,i|M

 –βn+

+ζn+e

n+–ζnen

 –βn+

≤ yn+–yn+ |βn+–βn|M+ |δn+–δn|M

+ 

 +

ε

On the other hand,

yn+–yn ≤( –αn+)xn+–xn+|αn+–αn|xn

+ ( –αn+)en+–en+|αn+–αn|en. ()

Thus in view of () and (), we have, forn≥,

xn+–xn

≤γnηf(xn) –f(xn–)+η|γn–γn–|f(xn–)

+I–γnTzn–zn–+|γn–γn–|Tzn–+en –en–

≤γnηkxn–xn–+η|γn–γn–|f(xn–)+ ( –γnγ)zn–zn–

+|γn–γn–|Tzn–+en–en–

≤ –γn(γ –ηk)

xn–xn–+ ( +η)M|γn–γn–|+en –en–

+ ( –γnγ)

M|αn–αn–|+ M|βn–βn–|+ M|δn–δn–|

+ M

 +

ε

|rn,i–rn–,i|+en+ en–+ en+ en–

. ()

Using Lemma , we have from ()limn→∞xn+–xn= .

Step . lim_n→∞yn –zn = , limn→∞

∞

i=aiJrn,iAi(I – rn,iBi)(yn+_zn) – zn =  and limn→∞∞i=aiJrn,iAi(I–rn,iBi)yn–yn= .

Since both{xn}and{Tzn}are bounded andγn→, asn→+∞,

xn+–zn=γn

ηf(xn) –Tzn

+e_n →, asn→+∞.

In view of Step ,xn–zn→, asn→+∞. Sinceαn→,yn–QCxn ≤αnxn+ ( –

αn)en →, asn→+∞. Therefore

yn–zn=yn–QCzn=yn–QCxn+QCxn–QCzn→, asn→+∞.

Sinceδn+βn+ζn≡,βn→, asn→ ∞, and{

∞

i=aiJrn,iAi(I–rn,iBi)(yn+_zn)}is bounded,

zn–

∞

i=

aiJrn,iAi(I–rn,iBi)

yn+zn



≤δn

yn–

∞

i=

aiJrn,iAi(I–rn,iBi)

yn+zn



+ζn

en–

∞

i=

aiJrn,iAi(I–rn,iBi)

yn+zn



asn→+∞. Using the above facts, we have ∞ i=

aiJrn,iAi(I–rn,iBi)yn–yn

≤ ∞ i=

aiJrn,iAi(I–rn,iBi)yn– ∞

i=

aiJrn,iAi(I–rn,iBi)

yn+zn

 + ∞ i=

aiJrn,iAi(I–rn,iBi)

yn+zn



–zn

+zn–yn →, asn→ ∞.

Step .lim sup_n→+∞ηf(p) –Tp,J(xn+–p) ≤, wherep∈D, which is the unique

solution of the variational inequality ().

Since∞_i=aiJrn,iAi(I–rn,iBi) :C→Cis non-expansive, using Lemma , we know that

there existszt such thatzt=tηf(zt) + (I–tT)

∞

i=aiJrn,iAi(I–rn,iBi)QCztfort∈(,T–).

Moreover,zt→p∈D, ast→, which is the unique solution of the variational inequality

().

Sincezt ≤ zt–p+p,{zt}is bounded, ast→. Using Lemma , we have

zt–yn

=

zt–

∞

i=

aiJrn,iAi(I–rn,iBi)yn+ ∞

i=

aiJrn,iAi(I–rn,iBi)yn–yn



≤zt– ∞

i=

aiJrn,iAi(I–rn,iBi)yn

 +  _∞ i=

aiJrn,iAi(I–rn,iBi)yn–yn,J(zt–yn)

=

tηf(zt) + (I–tT) ∞

i=

aiJrAn,ii(I–rn,iBi)QCzt– ∞

i=

aiJrAn,ii(I–rn,iBi)yn

 +  _∞ i=

aiJrn,iAi(I–rn,iBi)yn–yn,J(zt–yn)

≤ ∞

i=

aiJrn,iAi(I–rn,iBi)QCzt– ∞

i=

aiJrn,iAi(I–rn,iBi)yn



+ t

ηf(zt) –T ∞

i=

aiJrn,iAi(I–rn,iBi)QCzt,J

zt–

∞

i=

aiJrn,iAi(I–rn,iBi)yn

+ 

_∞

i=

aiJrn,iAi(I–rn,iBi)yn–yn,J(zt–yn)

≤ zt–yn

+ t

ηf(zt) –T ∞

i=

aiJrn,iAi(I–rn,iBi)QCzt,J

zt–

∞

i=

aiJrn,iAi(I–rn,iBi)yn

+  ∞ i=

aiJrn,iAi(I–rn,iBi)yn–yn

which implies that t T ∞ i=

aiJrAin,i(I–rn,iBi)QCzt–ηf(zt),J

zt–

∞

i=

aiJrAin,i(I–rn,iBi)yn

≤ ∞

i=

aiJrn,iAi(I–rn,iBi)yn–yn

zt–yn.

So, lim_t→lim sup_n→+∞T∞_i=aiJrn,iAi(I – rn,iBi)QCzt – ηf(zt),J(zt –

∞

i=aiJrn,iAi(I –

rn,iBi)yn) ≤ in view of Step .

Sincezt →p, _∞

i=aiJrn,iAi(I–rn,iBi)QCzt→

_∞

i=aiJrn,iAi(I–rn,iBi)QCp=p, ast→.

Noticing the following fact:

Tp–ηf(p),J

p–

∞

i=

aiJrn,iAi(I–rn,iBi)yn

=

Tp–ηf(p),J

p–

∞

i=

aiJrn,iAi(I–rn,iBi)yn

–J

zt– ∞

i=

aiJrn,iAi(I–rn,iBi)yn

+

Tp–ηf(p),J

zt– ∞

i=

aiJrn,iAi(I–rn,iBi)yn

=

Tp–ηf(p),J

p–

∞

i=

aiJrn,iAi(I–rn,iBi)yn

–J

zt– ∞

i=

aiJrn,iAi(I–rn,iBi)yn

+

Tp–ηf(p) –T

∞

i=

aiJrAn,ii(I–rn,iBi)QCzt+ηf(zt),

J

zt– ∞

i=

aiJrn,iAi(I–rn,iBi)yn

+ T ∞ i=

aiJrn,iAi(I–rn,iBi)QCzt–ηf(zt),J

zt–

∞

i=

aiJrn,iAi(I–rn,iBi)yn

,

we havelim sup_n→+∞Tp–ηf(p),J(p– ∞

i=aiJrn,iAi(I–rn,iBi)yn) ≤.

Since Tp–ηf(p),J(p–xn+)=Tp–ηf(p),J(p–xn+) –J(p– ∞

i=aiJrn,iAi(I–

rn,iBi)yn) + Tp – ηf(p),J(p – ∞i=aiJrn,iAi(I – rn,iBi)yn) and xn+ – ∞i=aiJrn,iAi(I –

rn,iBi)yn→ in view of Step , thenlim supn→∞ηf(p) –Tp,J(xn+–p) ≤.

Step .xn→p, asn→+∞, wherep∈Dis the same as that in Step .

LetM=sup{( –αn)(xn+en) –p,xn–p,Mp,en–p:n≥}. By using

Lemma  again, we have

yn–p

≤( –αn)xn–p+ ( –αn)en–αnp,J

( –αn)

xn+en

–p

Since

zn–p≤δnyn–p+βn

yn+zn

 –p

+ζnen–p 

≤

δn+

βn



yn–p+ βn

 zn–p

_+ζ

nen–p 

,

combining (), we have

zn–p

≤ yn–p+ ζnen–p

≤( –αn)xn–p+ ( –αn)en–αnp,J

( –αn)

xn+en

–p

+ ζnen–p. ()

Using () and Lemma , we have, forn≥,

xn+–p

=γn

ηf(xn) –Tp

+ (I–γnT)(zn–p) +en



≤( –γnγ)zn–p+ γnηf(xn) –Tp,J(xn+–p)

+ e_n,J(xn+–p)

≤( –γnγ)( –αn)xn–p+ en,J(xn+–p)

+ γnηf(xn) –f(p),J(xn+–p) –J(xn–p)

+ γnηf(xn) –f(p),J(xn–p)

+ γnηf(p) –Tp,J(xn+–p)

+ ( –γnγ)( –αn)en,J

( –αn)

xn+en

–p

– αn( –γnγ)p,J

( –αn)

xn+en

–p

+ ( –γnγ)ζnen–p 

≤ –γn(γ – ηk)

xn–p+ Men+en+ ( –γnγ)(αn+ζn)

+ γn ηf(p) –Tp,J(xn+–p)

+ηxn–pxn+–xn

. ()

Letδ()n =γn(γ – ηk),δn()= γn[ηf(p) –Tp,J(xn+–p)+ηxn–pxn+–xn],

δ()n = M[en+en+ ( –γnγ)(αn+ζn)]. Then () can be simplified asxn+–p≤

( –δn())xn–p+δ()n +δ()n .

From the assumptions (i), (ii), (iv), and (v), the results of Steps , , and  and Lemma , we know thatxn→p, asn→+∞.

Combine the result of Step ,yn→pandzn→p, asn→ ∞.

This completes the proof.

Remark  The assumptions imposed on the real number sequences in Theorem  are reasonable if we takeαn=_n,γn=_n,δn=  –_n –_nn_+,βn=_nn_+, andζn=_n forn≥. Remark  Three sequences{xn},{yn}, and{zn}are proved to be strongly convergent to

family ofμi-inversely strongly accretive mappings. The strongly convergent pointpis

the unique solution of a variational inequality.

Remark  Compared to the previous work, the computational error is considered in each step and the work on finding zero point of the sum of a finite family ofm-accretive mappings and an finite family ofμ-inversely strongly accretive mapping is extended to the infinite case. Compared to the work in [], the construction ofznin the iterative algorithm

(A) is implicit and a differentBicorresponds to a differentμi, which makes the iterative

algorithm (A) more general. Moreover, the assumption that ‘the normalized duality map-pingJis weakly sequentially continuous at zero’ is deleted.

Remark  Ife_n=e_n=e_n ≡, then iterative algorithm (A) becomes an accurate one.

Remark  IfC≡E, then the iterative algorithm (A) becomes the following one:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈E,

yn= ( –αn)(xn+en),

zn=δnyn+βn

∞

i=aiJrn,iAi[yn+_zn–rn,iBi(yn+_zn)] +ζnen,

xn+=γnηf(xn) + (I–γnT)zn+en, n≥.

3 Integro-differential systems and iterative algorithms

In this section, we have five purposes: () based on one kind nonlinear integro-differential system, construct an infinite family ofm-accretive mappings and an infinite family ofμi

-inversely strongly accretive mappings; () prove that under some conditions, the nonlinear integro-differential systems discussed exist solutions; () show the connections between the solution of the integro-differential systems and the zero point of the sum of an infinite family ofm-accretive mappings and an infinite family ofμi-inversely strongly accretive

mappings; () construct the iterative approximate sequence of the solution of the integro-differential systems; () demonstrate the relationship between the solution of the nonlin-ear integro-differential systems and the solution of one kind variational inequalities.

3.1 Discussion of integro-differential systems

We shall study the following nonlinear integro-differential systems involving the general-izedpi-Laplacian:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∂u(i)(x,t)

∂t –div[(C(x,t) +|Du

(i)_|₎pi_–_Du(i)_{] +ε|u}(i)_|ri–_u(i)

+g(x,u(i)_,Du(i)_{) +a}∂ ∂t

u(i)dx=f(x,t), (x,t)∈×(,T),

–ϑ, (C(x,t) +|Du(i)|)pi–_Du(i) ∈_βx(u(i)_{), (x,t)}∈×_(,T),

u(i)_{(x, ) =u}(i)_{(x,T), x∈,i∈N}+_,

()

whereis a bounded conical domain of a Euclidean spaceRN (N≥),is the boundary ofwith∈C_{[] andϑdenotes the exterior normal derivative to.·,·and| · |}

de-note the Euclidean inner-product and Euclidean norm inRN_{, respectively.T is a positive}

constant.Du(i)= (∂u(i) ∂x,

∂u(i) ∂x , . . . ,

∂u(i)

∂xN) andx= (x,x, . . . ,xN)∈.βxis the subdifferential of

∞ i=Vi :=

∞

i=Lpi(,T;W,pi()), f(x,t) ∈ ∞

i=Wi :=

∞ i=L

max{pi,p_i}_(,T;Lmax{pi,p_i}₍₎₎

andg:×RN+_{→Rare given functions.}

Our discussion of () is based on the following assumptions, some of which can be found in [–].

Assumption  {pi}∞i= is a real number sequence with NN+ <pi< +∞,{μi}∞i= is any real

number sequence in (, ] and {ri}∞i= is a real number sequence satisfying NN+ <ri ≤

min{pi,pi}< +∞. pi +



p_i=  and



ri+



r_i=  fori∈N

+_.

Assumption  Green’s formula is available.

Assumption  For eachx∈,ϕx=ϕ(x,·) :R→Ris a proper, convex and lower-

semi-continuous function andϕx() = .

Assumption  ∈βx() and for eacht∈R, the functionx∈→(I+λβx)–(t)∈Ris

measurable forλ> .

Assumption  Suppose thatg:×RN+_{→Rsatisfies the following conditions:}

(a) Carathéodory’s conditions; (b) growth condition:

g(x,r, . . . ,rN+)

max{pi,p_i}

≤hi(x,t) pi

+bi|r|pi,

where(r,r, . . . ,rN+)∈RN+,hi(x,t)∈Wi, andbiis a positive constant, fori∈N+;

(c) monotone condition:gis monotone in the following sense:

g(x,r, . . . ,rN+) –g(x,t, . . . ,tN+)

≥(r–t),

for allx∈and(r, . . . ,rN+), (t, . . . ,tN+)∈RN+.

Assumption  Fori∈N+_{, letV}∗

i denote the dual space ofVi. The norm inVi, · Vi, is

defined by

u(x,t)_Vi=

T



u(x,t)pi_W,_pi()dt

 pi

, u(x,t)∈Vi.

Lemma (see []) For i∈N+_{,define the operator B}

i:Vi→Vi∗by w,Biu=

T



C(x,t) +|Du|pi

–

 _{Du,Dwdx dt+ε}

T



|u|ri–uw dx dt,

for u,w∈Vi.Then Biis maximal monotone and coercive,where i∈N+.

Lemma (see []) For i∈N+_{,define the function}

i:Vi→R by

i(u) = T



ϕx

u|(x,t)

d(x)dt,

for u(x,t)∈Vi. Theniis a proper, convex and lower-semi-continuous mapping on Vi.

Lemma  (see []) For i ∈N+_{, define S}

i : D(Si) = {u(x,t)∈ Vi : ∂∂ut ∈Vi∗,u(x, ) =

u(x,T)} →V_i∗by

Siu=

∂u

∂t +a

∂ ∂t

u dx.

Then Si is linear maximal monotone operator possessing a dense domain in Vi, where

i∈N+_.

Definition  Fori∈N+_{, define a mappingA}

i:Wi→Wias follows:

D(Ai) =

u∈Wi|there exists anf ∈Wisuch thatf ∈Biu+∂i(u) +Siu

. Foru∈D(Ai), we setAiu={f ∈Wi|f ∈Biu+∂i(u) +Siu}.

Proposition  The mapping Ai:Wi→Wiis m-accretive,where i∈N+.

Proof Similar to the proof of Lemmas . and . in [] or the proof of Proposition . in

[], we haveR(I+λAi) =Wi, for∀λ> .

LetJi:Wi→Wi∗denote the generalized duality mapping. Then, foru(x,t)∈Wi,

Jiu=

⎧ ⎨ ⎩

|u|pi–_{sgnu, p}

i≥, |u|pi–_{sgnu,  <pi< .}

In fact, if pi ≥ , then u,Jiu =

T



|u|pidx dt = u

pi

Wi and JiuWi∗ =

(_T|u|(pi–)pi_{dx dt)}  p_i

=u

pi p_i Wi =u

pi–

Wi . Thus Jiu=|u|pi–sgnu, ifpi≥. Similarly,

Jiu=|u|p

i–_{sgnu, if  <pi< .}

By using a similar method as that of Proposition . in [], we can prove that for any

u,v∈D(Ai),Aiu–Aiv,Ji(u–v) ≥. ThusAiis accretive. The result follows. This

com-pletes the proof.

Remark  Noticing Proposition , an infinite family ofm-accretive mappings{Ai}∞i=is

constructed.

Definition  DefineCi:D(Ci) =Lmax{pi,p

i}_(,T;W,max{pi,pi}₍₎₎⊂_Wi→_Wiby

(Ciu)(x,t) =g(x,u,Du) –f(x,t)

for∀u(x,t)∈D(Ci) andf(x,t) is the same as that in (), wherei∈N+.

Lemma  The mapping Ci:D(Ci)⊂Wi→Wiis continuous and strongly accretive.If,

further assume that g(x,r, . . . ,rN+)≡r,then Ciisμi-inversely strongly accretive,where

i∈N+.

Proof Similar to Proposition . in [], we know that foru∈D(Ci), x→g(x,u,Du) is

measurable on, and thenCiis everywhere defined and continuous.

Case .pi≥. From assumption , we know that

Ciu–Civ,!Ji(u–v)

=

T



g(x,u,Du) –g(x,v,Dv)u–v–_Wipi|u–v|pi–sgn(u–v)dx dt

≥ u–v–_Wpi

i T



|u–v|pidx dt=u–v_Wi,

where!Ji:Wi→Wi∗is the normalized duality mapping, which implies thatCiis strongly

accretive.

If, furthermore,g(x,r, . . . ,rN+)≡r, since{μi} ⊂(, ], then we have

Ciu–Civ,Ji(u–v)

=

T



|u–v|pidx dt=Ciu–CivWipi ≥μiCiu–CivpWii ,

whereJi:Wi→Wi∗is the generalized duality mapping in Proposition , which implies

thatCiisμi-inversely strongly accretive.

Case .  <pi< . Similar to Case , the result follows.

This completes the proof.

Remark  Noticing Lemma , we have constructed an infinite family ofμi-inversely

strongly accretive mappings.

Lemma ([, ]) ()If w(x,t)∈∂i(u),then w(x,t)∈∂βx(u),a.e.on×(,T). () ϕ,∂i(u) ≡,∀ϕ∈C∞(,T;).

Lemma ([]) Let E be a smooth Banach space,let A:D(A)⊂E→E_{be an m-accretive}

mapping, and S:D(S)⊂E→E be a continuous and strongly accretive mapping with

D(A)⊂D(S).Then,for any z∈E, the equation z∈Sx+λAx has a unique solution xλ,

λ> .

Theorem  For f(x,t)∈∞_i=Wi,there exists unique u(i)∈Wisatisfying the following:

(a) ∂u(i)_∂(_tx,t)–div[(C(x,t) +|Du(i)_|₎pi–_Du(i)_{] +ε}|_u(i)|ri–_u(i)_+g(x,u(i)_,Du(i)_{) +}

a_∂∂_tu(i)dx=f(x,t),(x,t)∈×(,T);

(b) –ϑ, (C(x,t) +|Du(i)|)pi–_Du(i) ∈_βx(u(i)_(x,t)),_(x,t)∈×_(,T);

(c) u(i)_{(x, ) =u}(i)_{(x,T),x∈,wherei∈N}+_.

Proof Using Proposition , Lemmas  and , we know that forθ ∈Wi, there exists

uniqueu(i)_(x,t)∈D(A

i)⊂Wisuch that

θ=Ciu(i)+Aiu(i). ()

Then, forϕ∈C_∞(,T;), we have

ϕ,θ= ϕ,Ciu(i)

+ ϕ,Biu(i)

+ ϕ,∂i

u(i)+ ϕ,Siu(i)