Volume 2009, Article ID 290713,13pages doi:10.1155/2009/290713
Research Article
Perturbed Iterative Approximation of
Solutions for Nonlinear General
A
-Monotone
Operator Equations in Banach Spaces
Xing Wei,
1Heng-you Lan,
2and Xian-jun Zhang
21College of Mathematics and Software Science, Sichuan Normal University, Chengdu,
Sichuan 610066, China
2Department of Mathematics, Sichuan University of Science & Engineering, Zigong,
Sichuan 643000, China
Correspondence should be addressed to Heng-you Lan,[email protected]
Received 2 January 2009; Accepted 19 March 2009
Recommended by Ram U. Verma
We introduce and study a new class of nonlinear generalA-monotone operator equations with multivalued operator. By using Alber’s inequalities, Nalder’s results, and the new proximal mapping technique, we construct some new perturbed iterative algorithms with mixed errors for solving the nonlinear generalA-monotone operator equations and study the approximation-solvability of the nonlinear operator equations in Banach spaces. The results presented in this paper improve and generalize the corresponding results on strongly monotone quasivariational inclusions and nonlinear implicit quasivariational inclusions.
Copyrightq2009 Xing Wei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
LetXbe a real Banach space with the topological dual space ofX∗, letx, ybe the pairing betweenx∈ X∗andy∈ X, let 2X∗denote the family of all subsets ofX∗,and letCBXdenote the family of all nonempty closed bounded subsets ofX. We denote by thez, xzxfor allx ∈ Xand z ∈ X∗. Let f : X → X∗,T : X → 2X∗, g : X → X,and A : X → X∗ be nonlinear operators, and letM : X → 2X∗ be a generalA-monotone operator such that gX∩domM·/∅. We will consider the following nonlinear generalA-monotone operator equation with multivalued operator.
Findx∈ Xsuch thatu∈Txand
where∈0,1is a constant andPMA AρM−1is the proximal mapping associated with the generalA-monotone operatorMdue to Cui et al.1.
It is easy to see that the problem1.1is equivalent to the problem of findingx ∈ X such that
gx∈PMAAgx−ρfx Tx. 1.2
Example 1.1. If≡1, then the problem1.1is equivalent to findingx∈ Xsuch thatu∈Tx and
gx PMAAgx−ρfx u. 1.3
Based on the definition of the proximal mappingPA
M,1.3can be written as
0∈fx uMgx. 1.4
Example 1.2. IfT : X → X∗ is a single-valued operator, then a special case of the problem 1.3is to determine elementx∈ Xsuch that
gx−PMAAgx−ρQx0, 1.5
where Q : X → X∗ is defined by Qx fx Tx for allx ∈ X. The problem1.5
was studied by Xia and Huang2whenMis a generalH-monotone mapping. Further, the problem1.5 was studied by Peng et al. 3 if g I, the identity operator, andM is a multivalued maximal monotone mapping.
Example 1.3. Ifg I,T :X → X∗, andN : X × X → X∗are single-valued operators, and fx Tx Nx, xfor allx ∈ X, then the problem1.3reduces to finding an element x∈ Xsuch that
x−PMAAx−ρNx, x0, 1.6
which was considered by Verma4,5.
We note that for appropriate and suitable choices of , g, A, M, f, T, and X, it is easy to see that the problem 1.1 includes a number of quasivariational inclusions, generalized quasivariational inclusions, quasivariational inequalities, implicit quasivariational inequalities, complementarity problems, and equilibrium problems studied by many authors as special cases; see, for example,4–7and the references therein.
inequalities, variational inclusions, complementarity problems, and equilibrium problems by using the resolvent operator technique, which is a very important method to find solutions of variational inequality and variational inclusion problems; see, for example,1–15and the references therein.
On the other hand, Verma 4,5introduced the concept of A-monotone mappings, which generalizes the well-known general class of maximal monotone mappings and originates way back from an earlier work of the Verma 7. Furthermore, motivated and inspired by the works of Xia and Huang 2, Cui et al. 1 introduced first a new class of generalA-monotone operators in Banach spaces, studied some properties of general A -monotone operator, and defined a new proximal mapping associated with the general A -monotone operator.
Inspired and motivated by the research works going on this field, the purpose of this paper is to introduce the new class of nonlinear generalA-monotone operator equation with multivalued operator. By using Alber’s inequalities, Nalder’s results, and the new proximal mapping technique, some new perturbed iterative algorithms with mixed errors for solving the nonlinear generalA-monotone operator equations will be constructed, and applications of generalA-monotone operators to the approximation-solvability of the nonlinear operator equations in Banach spaces will be studied. The results presented in this paper improve and extend some corresponding results in recent literature.
2. Preliminaries
In this paper, we will use the following definitions and lemmas.
Definition 2.1. LetA : X → X∗,g : X → X, andf : X → Xbe single-valued operators. Then
iAisr-strongly monotone, if there exists a positive constantrsuch that
Ax−Ay, x−y ≥rx−y2, ∀x, y∈ X; 2.1
iiAisτ-Lipschitz continuous, if there exists a constantτ >0 such that
Ax−A
y≤τx−y2, ∀x, y∈ X; 2.2
iiig isk-strongly accretive if for anyx, y ∈ X, there existjx−y∈ Jx−yand a positive constantksuch that
jx−y, gx−gy≥kx−y2, 2.3
where the generalized duality mappingJq :X → 2X
∗
is defined by
Jqx f∗∈ X∗:
ivfisγ, μ-relaxed cocoercive with respect toA, if for allx, y∈ X, there exist positive constantsγandμsuch that
Ax−Ay, fx−fy ≥ −γfx−fy2μx−y2. 2.5
Example 2.2see 11, 12. 1Consider an r-strongly monotone and hence r-expanding operatorT :X → X. ThenT isrr2,1-relaxed cocoercive with respect toI.
2 Very m-cocoercive operator is m-relaxed cocoercive, while each r-strongly monotone mapping isrr2,1-relaxed cocoercive with respect toI.
Remark 2.3. The notion of the cocoercivity is applied in several directions, especially to solving variational inequality problems using the auxiliary problem principle and projection methods5, while the notion of the relaxed cocoercivity is more general than the strong monotonicity as well as cocoercivity. Several classes of relaxed cocoercive variational inequalities have been studied in4,5.
Definition 2.4. A multivalued operatorM:X → 2X∗is said to be imaximal monotone if, for anyx∈ X,u∈Mx,
u−v, x−y≥0 implies y∈ X, v∈My; 2.6
iim-relaxed monotone if, for anyx, y ∈ X,u∈Mx, andv ∈My, there exists a positive constantmsuch that
u−v, x−y ≥ −mx−y2; 2.7
iiiξ-H-Lipschitz continuous, if there exists a constantξ >0 such that
HMx, My≤ξx−y, ∀x, y∈ X, 2.8
whereH : 2X×2X → −∞,∞∪ {∞}is the Hausdorffpseudometric, that is,
HD, E max
sup
x∈D
inf
y∈E
x−y,sup
x∈E
inf
y∈D
x−y, ∀D, E∈2X∗
. 2.9
Note that if the domain ofH is restricted to closed bounded subsetsCBX, then H is the Hausdorffmetric.
Definition 2.5. A single-valued operatorg:X → X∗is said to be
icoercive if
lim x → ∞
gx, x
iihemicontinuous if, for any fixedx, y, z ∈ X, the function t → gxty, z is continuous at 0.
We remark that the uniform convexity of the spaceXmeans that for any givenε >0, there existsδ > 0 such that for allx, y ∈ X,x ≤ 1,y ≤ 1 andx−y εensure the inequalityxy ≤21−δ. The function
δXε inf
1−xy
2 :x1,y1,x−yε
2.11
is called the modulus of the convexity of the spaceX.
The uniform smoothness of the spaceXmeans that for any givenε > 0, there exists δ >0 such that1/2xyx−y −1≤εyholds. The function X:0,∞ → 0,∞
defined by
Xt sup
1
2xyx−y−1 :x ≤1,y≤t
2.12
is called the modulus of the smoothness of the spaceX.
We also remark that the space Xis uniformly convex if and only if δXε > 0 for all ε > 0, and it is uniformly smooth if and only if limt→0 Xt/t 0. Moreover, X is uniformly convex if and only if Xis uniformly smooth. In this case,X is reflexive by the Milman theorem. A Hilbert space is uniformly convex and uniformly smooth. The proof of the following inequalities can be found, for example, in page 24 of Alber16.
Lemma 2.6. LetXbe a uniformly smooth Banach space, and letJbe the normalized duality mapping fromXintoX∗. Then, for allx, y∈ X, we have
ixy2≤ x22y, Jxy;
iix−y, Jx−Jy ≤2d2 X4x−y/d, whered
x2y2/2.
Definition 2.7. LetXbe a Banach space with the dual spaceX∗,A:X → X∗be a nonlinear operator, andM : X → 2X∗ be a multivalued operator. The mapMis said to be general A-monotone ifMism-relaxed monotone andRAρM X∗holds for everyρ >0.
This is equivalent to stating thatMis generalA-monotone if.
iMism-relaxed monotone;
iiAρMis maximal monotone for everyρ >0.
Remark 2.8. 1If m 0, that is,M is 0-relaxed monotone, then the general A-monotone operators reduce to generalH-monotone operatorssee, e.g.,1,2.
Example 2.9. Let Xbe a reflexive Banach space with the dual space X∗,M : X → 2X∗ a maximal monotone mapping, andA : X → Xa bounded, coercive, hemicontinuous, and relaxed monotone mapping. Then for any givenρ > 0, it follows from Theorem 3.1 in page 401 of Guo10thatAρMX X∗. This shows thatMis a generalA-monotone operator.
Example 2.10 see 4. Let X be a reflexive Banach space with X∗ its dual, and let A : X → X∗ be r-strongly monotone. Letf : X → Rbe locally Lipschitz such that ∂f ism -relaxed monotone. Then ∂f is A-monotone, which is equivalent to stating that A∂f is pseudomonotoneand in fact, maximal monotone.
Lemma 2.11see1. LetXbe a reflexive Banach space with the dual spaceX∗, letA:X → X∗
be a nonlinear operator, and letM:X → 2X∗be a generalA-monotone operator. Then the proximal mappingPA
Mis
i 1/r −ρm-Lipschitz continuous whenA is r-strongly monotone withr > m and
ρ∈0, r/m;
ii 1/ρm-Lipschitz continuous ifAis a strictly monotone operator andMis anm-strongly monotone operator.
3. Perturbed Algorithms and Convergence
Now we will consider some new perturbed algorithms for solving the nonlinear generalA -monotone operator equation problem1.1or1.2by using the proximal mapping technique associated with the general A-monotone operators and the convergence of the sequences given by the algorithms.
Lemma 3.1. Let,g,A,M,f, andT be the same as in1.1. Then the following propositions are equivalent.
1 x, uis a solution of the problem1.1, wherex∈ Xandu∈Tx.
2xis the fixed-point of the functionFdefined by
Fx
u∈Tu
x−gx PMAAgx−ρfx u, 3.1
whereρ >0is a constant.
3 x, u, zis a solution of the following equation system:
gx−PMAz 0,
zAgx−ρfx u, 3.2
Lemma 3.2see17. Let{an},{bn},and{cn}be three nonnegative real sequences satisfying the following condition. There exists a natural numbern0such that
an1≤1−tnanbntncn, ∀n≥n0, 3.3
wheretn∈0,1,∞n0tn∞,limn→ ∞bn0,∞n0cn<∞. Thenan → 0 n → ∞.
Algorithm 3.3.
Step 1. Choose an arbitrary initial pointx0∈ X. Step 2. Take any{un} ⊂ {Txn} ⊂ Xforn0,1,2, . . . .
Step 3. Choose sequences{αn},{dn},{en},and{ωn}such that forn ≥ 0,{αn},{βn}are two
sequences in0,1and∞i0αi∞;{dn},{en},and{ωn}are error sequences inXto take into
account a possible inexact computation of the operator point, which satisfies the following conditions:
idndndn;
iilimn→ ∞dnlimn→ ∞en0;
iii∞n0dn<∞, ∞n0ωn<∞.
Step 4. Let{xn, zn, un} ⊂ X × X∗× Xsatisfy
xn1 1−αnxnαn
xn−gxn PMAzn
αndnωn,
znA
gxn
−ρfxn un
en,
3.4
whereρ >0 is a constant.
Step 5. Ifxn,zn,un,ωn,dn, anden n 0,1,2, . . .satisfy3.4to sufficient accuracy, stop;
otherwise, setk:k1 and return toStep 2.
Algorithm 3.4. For anyx0∈ X,u0∈Tx0, compute the iterative sequence{xn}by
xn1 1−αnxnαn xn−gxn PMAzn
αndnωn,
znA
gxn
−ρfxn un
en, ∀un∈Txn,
n0,1,2, . . . ,
3.5
whereρ,αn,dn,ωn, andenare the same as inAlgorithm 3.3.
Theorem 3.5. LetXbe a uniformly smooth Banach space with Xt≤Ct2for someC >0, and let X∗ be the dual space ofX. LetA : X → X∗ber-strongly monotone andτ-Lipschitz continuous,
g1x A◦gx Agxfor allx∈ X. If, in addition, there exist constantsρ >0andm∈0, r such that
k1−2δ64Cσ2<1, τσ > r1−k,
hξm1−k<8π√C, ρ <min
r m,
r1−k h
,
μπ2> γτ2σ2rh1−k
64Cπ2μ2−h2τ2σ2−r21−k2,
ρ−
μπ2−γτ2σ2−rh1 −k
64Cπ2−h2
<
μπ2−γτ2σ2−rh1−k2−64Cπ2−h2τ2σ2−r21 −k2
64Cπ2−h2 ,
3.6
then the following results hold:
1the solution set of the problem1.1is nonempty;
2the iterative sequence {xn, un} generated by Algorithm 3.3converges strongly to the solutionx∗, u∗of the problem1.1.
Proof. Setting a multivalued functionF: X → 2Xto be the same as3.1, then we can prove thatFis a multivalued contractive operator.
In fact, for anyx,x∈ Xand anya∈Fx, there existsu∈Txsuch that
ax−gx PA M
Agx−ρfx u. 3.7
Note thatTx ∈CBX; it follows from Nadler’s result18that there existsu∈Tx such that
u−u ≤HTx, Tx. 3.8
Letting
bx−gx PMAAgx −ρfx u, 3.9
continuity of f, and the τ-Lipschitz continuity of A, Lemma 2.6, and the inequality 3.8
imply that
x−x−gx−gx 2
≤ x−x2−2jx−x, gx −gx
2jx−x−gx−gx −jx−x, −gx−gx
≤ x−x2−2δx−x24d2 X
4gx−gx d
≤1−2δ64Cσ2x−x2,
u−u ≤HTu, Tu ≤ξx−x, Agx−Agx −ρfx−fx 2
≤Agx−Agx 2−2ρjAgx−Agx , fx−fx
2jAgx−Agx −ρfx−fx −jAgx−Agx −ρfx−fx
≤τ2σ2x−x2−2−γAgx−Agx 2μfx−fx 2
4d2 X
4ρfx−fx d
≤τ2σ2−2ρμπ22ργτ2σ264Cρ2π2x−x2.
3.10
Thus, it follows from3.4andLemma 2.11that
a−b ≤x−x−gx−gx
PMAAgx−ρfx u−PMAAgx −ρfx u
≤x−x−gx−gx ρ
r−ρmu−u
1
r−ρmA
gx−Agx −ρfx−fx
≤θx−x,
3.11
where
θ1−2δ64Cσ2ρξ
τ2σ2−2ρμπ22ργτ2σ264Cρ2π2
It follows from condition3.6thatθ <1. Hence, from3.11, we get
da, Fx inf
b∈Fxa−b ≤θx−x. 3.13
Sincea∈Fxis arbitrary, we obtain supa∈Fxda, Fx ≤θx−x. By using same argument, we can prove supb∈FxdFx, b≤θx−x. It follows from the definition of the Hausdorff metricH onCBXthat
HFx, Fx ≤θx−x, ∀x, x∈ X, 3.14
and soFis a multivalued contractive mapping. By a fixed-point theorem of Nadler18, the definition ofFand3.2, now we know thatFhas a fixed-pointx∗, that is,x∗ ∈Fx∗,and there existsu∗∈Tx∗such that
gx∗ PA M
Agx∗−ρfx∗ u∗. 3.15
Hence, it follows fromLemma 3.1thatx∗, u∗is a solution of the problem1.1, that is, the solution set of the problem1.1is nonempty.
Next, we prove the conclusion2. Letx∗, u∗be a solution of problem1.1. Then for alln≥0, we have
x∗ 1−αnx∗αn x∗−gx∗ PMAz∗
, z∗Agx∗−ρfx∗ u∗. 3.16
FromAlgorithm 3.3, the assumptions of the theorem 3.5 and Lemma 2.11, it follows that
zn−z∗ ≤A
gxn
−Agx∗−ρfxn−fx∗ρun−u∗en, 3.17
xn1−x∗ ≤1−αnxn−x∗αnxn−x∗−
gxn−gx∗
αnPMAzn−PMAz∗αndnωn
≤1−αnxn−x∗αnxn−x∗−
gxn−gx∗
αn
r−ρmzn−z ∗α
ndndnωn
.
3.18
Combining3.17and3.18, we obtain
xn1−x∗
≤1−αnxn−x∗θαnxn−x∗
αn
r−ρmenαnd
ndnωn
≤1−αn1−θxn−x∗αn
dn 1 r−ρmen
dnωn
,
whereθis the same as in3.11. Sinceθ <1, we know that 1−θ >0 and3.19implies
un1−u∗
≤1−αn1−θxn−x∗αn1−θ·
1 1−θ
dn 1 r−ρmen
dnωn
.
3.20
Sinceni0αi ∞, it follows fromLemma 3.2that the sequencexnstrongly converges tox∗.
Byun∈Txn,u∗∈Tu∗, and theH-Lipschitz continuity ofT, we obtain
un−u∗ ≤HSu n, Su∗≤ξun−u∗, yn−y∗≤HTu n, Tu∗≤ζun−u∗.
3.21
Thus, {un} is also strongly converges to u∗. Therefore, the iterative sequence {xn, un}
generated byAlgorithm 3.3converges strongly to the solutionx∗, u∗of the problem1.1
or1.2. This completes the proof.
Based on Theorem 3.3 in2, we have the following comment.
Remark 3.6. If ≡ 1, g is k-strongly accretive and δ-Lipschitz continuous, A is a strictly monotone ands-Lipschitz continuous operator,Mis a generalH-monotone andβ-strongly monotone operator,T is a single-valued operator andQ fT isα-Lipschitz continuous, andρ >0 is some constant such that
ρ > sδ
β−α−β√1−2k64Cδ2,
1−2k64Cδ2αβ−1<1, 3.22
then3.6holds.
Theorem 3.7. Assume that A,T,f,g,M,and X are the same as in Theorem 3.5. If there exist constantsρ >0andm ∈ 0, rsuch that
k1−2δ64Cσ2<1, τσ > r1−k,
hξm1−k<8π√C, ρ <min
r m,
r1−k h
,
μπ2> γτ2σ2rh1−k
64Cπ2μ2−h2τ2σ2−r21−k2,
ρ−
μπ2−γτ2σ2−rh1 −k
64Cπ2−h2
<
μπ2−γτ2σ2−rh1 −k2−64Cπ2−h2τ2σ2−r21 −k2
64Cπ2−h2 ,
3.23
then there existsx∗ ∈ Xsuch thatx∗ is a solution of the problem1.3, and the iterative sequence
Remark 3.8. Ifαnωnen≡0 forn≥0 inAlgorithm 3.4,TandQas the same in the problem
1.5, then the results of Theorem 3.4 obtained by Xia and Huang2also hold. For details, we can refer to1,2,4,5.
Remark 3.9. If dn 0 or en 0 or ωn 0 n ≥ 0 in Algorithms 3.3 and 3.4, then the
conclusions of Theorems3.5and3.7also hold, respectively. The results of Theorems3.5and
3.7improve and generalize the corresponding results of2,4–9,15,17. For other related works, we refer to1–16and the references therein.
Acknowledgments
The authors are grateful to Professor Ram U. Verma and the referees for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China 60804065, the Scientific Research Fund of Sichuan Provincial Education Department2006A106, 07ZB151, and the Sichuan Youth Science and Technology Foundation08ZQ026-008.
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