An iterative algorithm for generalized nonlinear variational inclusions with relaxed strongly accretive mappings in Banach spaces

PII. S0161171204308069 http://ijmms.hindawi.com © Hindawi Publishing Corp.

AN ITERATIVE ALGORITHM FOR GENERALIZED NONLINEAR

VARIATIONAL INCLUSIONS WITH RELAXED STRONGLY

ACCRETIVE MAPPINGS IN BANACH SPACES

A. H. SIDDIQI and RAIS AHMAD

Received 7 August 2003

We use Nadler’s theorem and the resolvent operator technique form-accretive mappings to suggest an iterative algorithm for solving generalized nonlinear variational inclusions with relaxed strongly accretive mappings in Banach spaces. We prove the existence of solutions for our inclusions without compactness assumption and the convergence of the iterative sequences generated by the algorithm in real Banach spaces. Some special cases are also discussed.

2000 Mathematics Subject Classification: 49J40, 49J53, 47H06.

1. Introduction. Variational inequality theory provides us with a unified framework for dealing with a wide class of problems arising in elasticity, structural analysis, eco-nomics, physical and engineering sciences, and so forth, see, for example, [2,6,13,14] and references therein. A useful and important generalization of variational inequalities is a mixed type variational inequality containing nonlinear term. Due to the presence of the nonlinear term, the projection method cannot be used to study the existence and al-gorithm of solutions for the mixed type variational inequalities. In 1994, Hassouni and Moudafi [14] used the resolvent operator technique for maximal monotone mappings to study a class of mixed type variational inequalities with single-valued mappings called variational inclusion and developed a perturbed algorithm for finding approximate so-lutions to the mixed variational inequalities. Adly [1], Ahmad and Ansari [2], Ahmad et al. [3], Chang et. al. [9], Ding [11,12], and Huang [15,16,17] studied some important generalizations of variational inclusions in different directions. Very recently, Chang [8], Chang et al. [10], and Huang [17] introduced and studied some classes of set-valued variational inclusions in real Banach spaces. Also, Alber and Yao [5] introduced a new class of generalized multivalued covariational inequalities in Banach spaces.

2. Preliminaries and formulations. Throughout this paper, we assume thatE is a real Banach space whose norm is denoted by·,E∗is the topological dual ofE,CB(E) is the family of all nonempty, closed, and bounded subsets ofE,D(·,·)is the Hausdorff metric onCB(E)defined by

D(A, B)=max

sup x∈A

d(x, B),sup y∈B

d(A, y)

, (2.1)

whered(x, B)=infy∈Bd(x, y),d(A, y)=infx∈Ad(x, y),dis the metric onEinduced by the norm·,·,·is the duality pairing betweenEandE∗,D(T )is the domain of T, andJ:E→2E∗ _{is the normalized duality mapping defined by}

J(x)=f∈E∗:x, f = x·f, f = x, x∈E. (2.2)

We recall that theuniform convexity[5] of the spaceEmeans that for any given >0, there existsδ >0 such that for allx, y∈E,x ≤1,y<1, andx−y =, we have

x+y ≤2(1−δ). (2.3)

The function

δB()=inf

1−x+₂y:x =1,y =1,x−y =

(2.4)

is called themodulus of the convexityof the spaceE.

Theuniform smoothnessof the spaceEmeans that for any given >0, there exists δ >0 such that

x+y+x−y

2 −1≤y (2.5)

holds. The function

ρB(t)=sup

_x+y+x−y

2 −1 :x =1,y =t

(2.6)

is called themodulus of the smoothnessof the spaceE.

We observe that the spaceEis uniformly convex if and only ifδB() >0 for all >0, and it is uniformly smooth if and only if limt→0t−1ρB(t)=0. The following inequalities will be used for the proof of our main result.

Proposition_2.1[4]. _{LetEbe a uniformly smooth Banach space and letJ:E}→₂E∗ be the normalized duality mapping. Then for anyx, y∈E,

(i) x+y2_{≤ x}2_{+2y, j(x+y), for allj(x+y)∈J(x+y);}

(ii) x−y, j(x)−j(y) ≤2c2_{ρB(4x−y/c),wherec=(x}2_+y2_)/2.

Definition_2.2[5]. _LetJ:E→₂E∗_{be the normalized duality mapping. The mapping} f:E→Eis said to be

(1) strongly accretive if there exists a constantα >0 such that

(2) relaxed strongly accretive with respect to the set-valued mappingT:E→CB(E) if there exists a constantν≥0 such that

f (u)−f (v), j(x−y)≤ −νx−y2_{, (2.8)}

for allx, y∈E,u∈T (x),v∈T (y), andj(x−y)∈J(x−y).

Remark_2.3. _IfE=_{His a Hilbert space, then the notion of relaxed strongly accretive} coincides with that of relaxed Lipschitz (see [19]).

Definition2.4[17]. LetJ:E→2E∗ be the normalized duality mapping. The set-valued mappingA:D(A)⊂E→2E_{is said to be}

(1) accretive if for anyx, y∈D(A), there existsj(x−y)∈J(x−y)such that for allu∈A(x),v∈A(y),

u−v, j(x−y)≥0; (2.9)

(2) strongly accretive, k∈(0,1), if for any x, y ∈D(A), there existsj(x−y)∈ J(x−y)such that for allu∈A(x),v∈A(y),

u−v, j(x−y)≥kx−y2; (2.10)

(3) m-accretive ifAis accretive and(I+ρA)(D(A))=Efor every (equivalently, for some)ρ >0, whereI is the identity mapping (equivalently, ifAis accretive and (I+A)(D(A))=E).

Remark2.5[9]. IfE=His a Hilbert space, thenA:D(A)⊂H→2His anm-accretive mapping if and only ifA:D(A)⊂H→2H_{is a maximal monotone mapping.}

Lemma_2.6[17]. _{Let g:E}→_{E be a continuous andk-strongly accretive mapping.} ThengmapsEontoE.

For a given set-valued mappingT :E→CB(E), single-valued mappingsf , g:E→E, andm-accretive mappingA:E→2E_{, we consider the following problem:}

(i) GNVIP: findx∈E, w∈T (x)such that 0∈g(x)−f (w)+A(g(x)).

It is called thegeneralized nonlinear variational inclusion problem (GNVIP) in Banach spaces.

Now we present some particular cases of the GNVIP which show that our GNVIP is a more general and unified problem.

Special cases. _{(1) If E} =_{H is a Hilbert space and A}=_{∂ϕ, where ϕ: H}→_R∪ {+∞}is a proper convex lower semicontinuous function on H and ∂ϕ denotes the subdifferential of functionϕ, then GNVIP is equivalent to findingx∈Handw∈T (x) such thatg(x)∩dom(∂ϕ)= ∅and

which was introduced and studied by Ahmad and Ansari [2]. Ifϕ(x)=IK(x), the indi-cator function of a closed convex setKinHdefined by

IK(x)=   

0, x∈K,

+∞, otherwise, (2.12)

then problem (2.11) reduces to a problem considered by Verma [19].

(2) IfE=His a Hilbert space andA:H→2H_{is a maximal monotone mapping, then} byRemark 2.5,Ais also anm-accretive mapping. Thus problem GNVIP is equivalent to findingx∈Handw∈T (x)such that

0∈g(x)−f (w)+Ag(x) , (2.13)

which is a variant form of the problem considered by Huang [16].

(3) Ifg(x)−f (w)=w, then GNVIP is equivalent to findingx∈Handw∈T (x)such that

0∈w+Ag(x) , (2.14)

which was studied and introduced by Huang [15].

3. Iterative algorithms

Definition_3.1[7]. _LetA:D(A)⊂_E→₂E_{be anm-accretive mapping. For anyρ >0,} the mappingRA

ρ:E→D(A)associated withAdefined by

RA

ρ(u)=(I+ρA)−1(u), u∈E, (3.1)

is called the resolvent operator.

Definition 3.2. The resolvent operatorRA_{ρ is said to be a retraction onD(A) if} (I+ρA)−1_◦(I+ρA)−1_=(I+ρA)−1_.

Remark3.3. It is well known thatRA_ρ is a single-valued and nonexpansive mapping (see [8]).

To propose the algorithm for computing the approximate solution of GNVIP, we convert GNVIP into a fixed point problem.

Lemma_3.4. _{The pair(x, w)withx}∈_Eandw∈_{T (x)is a solution of GNVIP if and} only if(x, w)satisfies the following relation:

g(x)=RA ρ

g(x)−ρg(x)−f (w) , (3.2)

whereρ >0is a constant,RA

Proof. From the definition of the resolvent operatorRA_ρ associated withAand

re-lation (3.2), we have

g(x)=RρA

g(x)−ρg(x)−f (w)

=(I+ρA)−1_{g(x)−ρg(x)−f (w) ,} (3.3)

and therefore

g(x)−ρg(x)−f (w) ∈g(x)+ρAg(x) . (3.4)

The above relations hold if and only ifx∈Eandw∈T (x)such that

0∈g(x)−f (w)+Ag(x) . (3.5)

We now invoke Lemmas3.4and2.6and Nadler’s theorem [18] to suggest the follow-ing algorithm for solvfollow-ing GNVIP in the settfollow-ing of real Banach spaces.

Algorithm 3.5. Suppose that g:E→E is a continuous andk-strongly accretive mapping. For any givenx0∈E, we takew0∈T (x0)and we let

x1=x0−g

x0 +RAρ

gx0 −ρ

gx0 −f

w0

. (3.6)

Sincew0∈T (x0)∈CB(E), by Nadler [18], there existsw1∈T (x1)such that

w0−w1≤(1+1)D

Tx0 , T

x1

. (3.7)

Let

x2=x1−g

x1 +RAρ

gx1 −ρ

gx1 −f

w1

. (3.8)

Again by Nadler [18], there existsw2∈T (x2)such that

w1−w2≤

1+1

2

DTx1 , Tx2 . (3.9)

Continuing the above process inductively, we can obtain the sequences{xn}and{wn}

such that

xn+1=xn−gxn +RAρ

gxn −ρgxn −fwn ,

wn∈Txn , wn−wn+1≤

1+_n1₊₁

DTxn , Txn+1

, (3.10)

wheren=0,1,2, . . .andρ >0 is a constant.

Algorithm3.6. Suppose thatg:E→Eis a continuous and ak-strongly accretive mapping. For any given x0∈E, compute the sequence {xn}and {wn} by iterative scheme such that

gxn₊1 =RAρ

gxn −ρwn,

wn∈Txn , wn−wn₊1≤

1+ 1

n+1

DTxn , Txn₊1 ,

(3.11)

wheren=0,1,2, . . . .

4. Existence and convergence results. In this section, we prove the existence of a so-lution for GNVIP and the convergence of iterative sequences generated byAlgorithm 3.5.

Definition_4.1. _{The mappingg:E}→_{Eis said to be Lipschitz continuous if there} exists a constantδ >0 such that

g(x)−g(y)≤δx−y, ∀x, y∈E. (4.1)

Definition4.2. The set-valued mappingT :E→CB(E)is said to beD-Lipschitz continuous if there exists a constantη >0 such that

DT (x), T (y) ≤ηx−y, ∀x, y∈E. (4.2)

Theorem4.3. LetEbe a real Banach space with the module of smoothnessτB(t)≤Ct2 for someC >0. LetA:E→2E_{be anm-accretive mapping, letg, f:E→Ebe single-valued} mappings, and letT:E→CB(E)be a set-valued mapping. Suppose that the following conditions are satisfied:

(i) gis bothk-strongly accretive and Lipschitz continuous with constantδ >0; (ii) f is both relaxed strongly accretive with respect toT with constant ν≥0and

Lipschitz continuous with constantλ >0; (iii) T isD-Lipschitz continuous with constantη >0. If(I+ρA)−1_◦(I+ρA)−1_=(I+ρA)−1_and

ρ− 1+ν+P (1−2P )

1+2ν+64Cλ2_η2_−P2

<

(ν+1)+P (2P−1)2−4P (1−P )1+2ν+64Cλ2_η2_−P2 1+2ν+64Cλ2_η2_−P2 ,

1+ν > P (2P−1)+4P (1−P )1+2ν+64Cλ2_η2_−P2 _,

1+2ν+64Cλ2η2> P2 forP=1−2K+64Cλ2_<1 2,

(4.3)

then there exists a solution(x, w)of GNVIP, and the iterative sequences{xn}and{wn}

Proof. FromAlgorithm 3.5, we have

xn+1−xn=xn−xn−1−

gxn −gxn−1

+RA ρ

zxn −RA ρ

zxn−1 , (4.4)

wherez(xn)=g(xn)−ρ(g(xn)−f (wn)). Since the resolvent operatorRA

ρ is nonex-pansive, we have

RA ρ

zxn −RA ρ

zxn₋1

≤zxn −zxn₋1

=(1−ρ)gxn −gxn−1

+ρfwn −fwn−1

≤(1−ρ)xn−xn−1−

gxn −gxn−1

+(1−ρ)xn−xn₋1 +ρ

fwn −fwn₋1 .

(4.5)

From (4.4) and (4.5), we have

_{xn+1−xn≤xn−xn−1−}

gxn −gxn−1 +RAρ

zxn −RAρ

zxn−1

≤xn−xn−1−

gxn −gxn−1

+(1−ρ)xn−xn−1−

gxn −gxn−1

+(1−ρ)xn−xn₋1 +ρfwn −fwn−1

=(2−ρ)xn−xn−1−

gxn −gxn−1

+(1−ρ)xn−xn−1 +ρ

fwn −fwn−1 .

(4.6)

Sinceg isk-strongly accretive and Lipschitz continuous with constantδ >0, and by

Proposition 2.1, we have

xn−xn₋1−

gxn −gxn₋1 2

≤xn−xn−1 2

+2−gxn −gxn−1

, jxn−xn−1−

gxn −gxn−1

=xn−xn−1 2

+2−gxn −gxn−1

, jxn−xn−1

+2−gxn −gxn₋1

, jxn−xn₋1−

gxn −gxn₋1

−jxn−xn₋1 (4.7)

≤xn−xn₋12−2kxn−xn−12+4d2ρB

4gxn −gxn₋1 d

≤xn−xn−1 2

−2kxn−xn−1 2

+64Cgxn −gxn−1 2

≤1−2k+64Cδ2 xn−xn−1 2

.

Sincefis relaxed strongly accretive with respect toTand with constantν >0, and Lip-schitz continuous with constantλ >0, andT isD-Lipschitz continuous with constant η >0, and byProposition 2.1, we obtain

_(1−ρ)

xn−xn−1 +ρfwn −fwn−1 2

≤(1−ρ)2xn−xn−1 2

+2ρfwn −fwn₋1

, j(1−ρ)xn−xn₋1 +ρ

fwn −fwn₋1

=(1−ρ)2xn−xn−12

+2ρfwn −fwn−1

, j(1−ρ)xn−xn−1 +ρ

fwn −fwn−1

−j(1−ρ)xn−xn−1

+2ρfwn −fwn₋1 , j(1−ρ)xn−xn−1

=(1−ρ)2xn−xn−1 2

+4d2ρB

₄

d|ρ|f

wn −fwn−1

+2ρ(1−ρ)fwn −fwn₋1 , jxn−xn−1

≤(1−ρ)2xn−xn−1 2

−2ρ(1−ρ)νxn−xn−1 2

+64Cρ2_{fwn −fwn} −1

2

≤(1−ρ)2_{−2ρ(1−ρ)νxn−xn} −12

+64Cρ2_λ2

1+_n1 2

D2_{Txn , Txn} −1

≤

(1−ρ)2_{−2ρ(1−ρ)ν+64Cρ}2_λ2

1+1

n

2 η2

xn−xn₋12.

(4.9)

From (4.6), (4.7), and (4.8), it follows that

xn₊1−xn≤θnxn−xn−1, (4.10)

where

θn=(2−ρ)1−2k+64Cδ21/2

+

(1−ρ)2_{−2ρ(1−ρ)ν+64Cρ}2_λ2₁₊1 n

2 η2

1/2 .

(4.11)

Letting

we know thatθn→θasn→ ∞. From condition (4.3), it follows thatθ <1. Henceθn<1 fornsufficiently large. Consequently,{xn}is a Cauchy sequence, and thus, converges to somex∈E. ByAlgorithm 3.5and theD-Lipschitz continuity ofT, it follows that

wn−wn−1≤

1+_n+1₁

DTxn , Txn−1

≤

1+1

n

ηxn−xn₋1,

(4.13)

and hence{wn}is also a Cauchy sequence inE. Therefore, there existsw∈Esuch that wn→wasn→ ∞. Sinceg,f, andRA

ρ are continuous inE, we have

x=x−g(x)+RAρ

g(x)−ρg(x)−f (w) . (4.14)

Finally, we prove thatw∈T (x). In fact, sincewn∈T (xn)and

dwn, T (x) ≤max

dwn, T (x) , sup v∈T (x)

dTxn , v

≤max

sup y∈T (xn)

dy, T (x), sup v∈T (x)

dTxn , v

=DTxn , T (x) ,

(4.15)

we have

dw, T (x) ≤w−wn+dwn, T (x)

≤w−wn+DTxn , T (x)

≤w−wn+ηxn−x →0 (n→ ∞),

(4.16)

which implies thatd(w, T (x))=0. SinceT (x)∈CB(E), it follows thatw∈T (x). Then byLemma 3.4, we get the conclusion.

Remark 4.4. Huang [17] studied a particular case of our problem (GNVIP) in the setting of real Banach spaces using the resolvent operator technique form-acceretive mapping. By defining the resolvent operator to be a retraction mapping, we proved the existence of a solution of a more general problem than that considered by Huang [17].

The following result weakens the conditions offinTheorem 4.3.

Theorem4.5. LetEbe a real Banach space with the module of smoothnessτB(t)≤Ct2 for someC >0. LetA:E→2E_{be anm-accretive mapping, letg, f:E→Ebe single-valued} mappings, and letT:E→CB(E)be a set-valued mapping. Suppose that the following conditions are satisfied:

(1) gis bothk-strongly accretive and Lipschitz continuous with constantδ >0; (2) fis Lipschitz continuous with constantλ >0;

Then there exists a solution(x, w)of GNVIP and the iterative sequences{xn}and{wn}

generated byAlgorithm 3.5converge strongly tox,winE, respectively.

Proof. FromAlgorithm 3.5and using the fact that the resolvent operator is

nonex-pansive, we have

xn₊1−xn

=xn−gxn +RAρ

gxn −ρgxn −fwn

−xn−1−g

xn−1 +RρA

gxn−1 −ρ

gxn−1 −f

wn−1

≤xn−xn−1−

gxn −gxn−1

+(1−ρ)gxn −gxn₋1

+ρfwn −fwn₋1

≤xn−xn−1−gxn −gxn−1 +(1−ρ)gxn −gxn−1

+ρfwn −fwn−1 .

(4.17)

It follows from (4.8) that

xn−xn−1−

gxn −gxn−1 2

≤1−2k+64Cδ2 _xn−xn −1

2

. (4.18)

Also, it follows from the Lipschitz property of the corresponding functions that

_g

xn −gxn−1 =δxn−xn−1, f (wn)−fwn−1 =λwn−wn−1

≤λ

1+1 n

DTxn , Txn−1

≤λ

1+_n1

ηxn−xn₋1

=ληxn−xn−1 (asn → ∞).

(4.19)

From (4.8), (4.17), and (4.19), we have the following inequality:

_{xn+1−xn≤qxn−xn−1, (4.20)}

whereq=(1−2k+64Cδ2₎1/2_{(1−ρ)δ+ρλη and 0< q <1 by condition (4).} Conse-quently,{xn}is a Cauchy sequence inE. Then the result follows by using the same arguments ofTheorem 4.3.

Acknowledgment. _{A. H. Siddiqi would like to thank King Fahd University of}

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A. H. Siddiqi: Department of Mathematical Sciences, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia

E-mail address:[email protected]

Rais Ahmad: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India