Generalized strongly set valued nonlinear complementarity problems

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GENERALIZED STRONGLY SET-VALUED NONLINEAR

COMPLEMENTARITY PROBLEMS

NAN-JING HUANG and YEOL JE CHO

(Received 7 October 1997)

Abstract.In this paper, we introduce a new class of generalized strongly set-valued non-linear complementarity problems and construct new iterative algorithms. We show the existence of solutions for this kind of nonlinear complementarity problems and the con-vergence of iterative sequences generated by the algorithm. Our results extend some recent results in this ﬁeld.

Keywords and phrases. Complementarity problem, strongly monotone and Lipschitz con-tinuous mappings, strongly monotone andH-Lipschitz continuous set-valued mappings, algorithm.

1991 Mathematics Subject Classiﬁcation. 90C33, 47H04.

1. Introduction. The complementarity theory, which was introduced by Lemke [11], Cottle, and Dantzig [6] in the early 1960s and later developed by others, plays an im-portant and fundamental role in the study of a wide class of problems arising in mechanics, physics, control and optimization, economics and transportation equilib-rium, contact problems in elasticity, ﬂuid ﬂow through porous media, and many other branches of mathematical and engineering sciences [1, 2, 3, 4, 5, 7, 8, 9, 10, 13, 14, 15]. In particular, the set-valued quasi-(implicit)complementarity problems, considered and studied by Chang and Huang [2, 3], are important among the generalizations of the complementarity problems. In [14], Noor introduced and studied some new classes of nonlinear complementarity problems for single-valued mappings inRn_and, in [4], Chang and Huang introduced and studied some new nonlinear complementarity problems for compact-valued fuzzy mappings and set-valued mappings which include many kinds of complementarity problems, considered by Chang [1], Cottle et al. [7], Isac [9], and Noor [13, 14], as special cases.

In this paper, we introduce and study a new class of generalized strongly set-valued nonlinear complementarity problems and construct new iterative algorithms. We also discuss the existence of solutions for this kind of nonlinear complementarity prob-lems and the convergence of iterative sequences generated by the algorithm. Our results improve and develop some results in [4, 13, 14].

2. Preliminaries. LetRn_{be the Euclidean space endowed with norm}_·_{and inner} product(·,·), respectively. In the sequel, we use the following notations:

2Rn

= {A:A⊂Rn_and_A_{is nonempty}_}_, _(2.1)

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|x| =|x1|,|x2|,...,|xn| for allx∈Rn_. _(2.3) LetF, G, Q:Rn_→₂Rn

be three set-valued mappings and f, g:Rn _→_Rn _{be two} single-valued mappings. Now, we consider the following problem.

Findu∈Rn_,_x_∈_F(u)_,_y_∈_G(u)_{, and}_z_∈_Q(y)_{such that}

u≥0, f (x)+g(z)≥0, u,f (x)+g(x)=0. (2.4)

This problem is called the generalized strongly set-valued nonlinear complementar-ity problem.

IfQis the identity mapping onRn_{, then problem (2.4) is equivalent to the following.} Findu∈Rn_,_x_∈_F(u)_and_y_∈_G(u)_{such that}

u≥0, f (x)+gy≥0, u,f (x)+gy=0. (2.5)

This problem is called the generalized set-valued nonlinear complementarity problem. IfF is the identity mapping onRn_{, then problem (2.5) is equivalent to the following.} Findu∈Rn_and_y_∈_G(u)_{such that}

u≥0, f (u)+gy≥0, u,f (u)+gy=0, (2.6)

which was considered by Chang and Huang in [4].

If F =G is the identity mapping on Rn_{, then problem (2.5) is equivalent to the} following.

Findu∈Rn_{such that}

u≥0, f (u)+g(u)≥0, u,f (u)+g(u)=0, (2.7)

which was considered by Noor [14].

IfF =Gis the identity mapping onRn_and_f_≡₀_,_{then problem (2.5) is equivalent} to the following.

Findu∈Rn_{such that}

u≥0, g(u)≥0, u,g(u)=0, (2.8)

which was considered by Karamardian [10], Fang [8], and Noor [13]. Obviously, problem (2.4) can be written as follows:

Findu∈Rn_,_x_∈_F(u)_,_y_∈_G(u)_and_z_∈_Q(y)_{such that}

u≥0, v=f (x)+g(z)≥0, (u,v)=0. (2.9)

We now consider the following equalities:

u=1 2

|w|+w, v=λρ−1|w|−w, (2.10) whereλ,ρ >0 are constants. Clearly,u≥0 andv≥0. From (2.5) and (2.9), it follows that problem (2.9) is equivalent to the following.

Findw∈Rn_,_x_∈_F(₁_/₂₍_|_w_|+_w))_,_y_∈_G(₁_/₂₍_|_w_|+_w))_{, and}_z_∈_Q(y)_{such that}

w=1 2

|w|+w−1 2λρ

f (x)+g(z), (2.11)

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3. Algorithms. Based on the formulations in Section 2, we now construct the new algorithms for the generalized strongly set-valued nonlinear complementarity prob-lem (2.4).

LetF,G,Q:Rn_→_CB₍_Rn₎_{be three set-valued mappings and let}_f_,_g_:_Rn_→_Rn _be two single-valued mappings. For anywo∈Rn, let

x0∈F ₁

2

|w0|+w0, y0∈G ₁

2

|w0|+w0, z0∈Qy0 (3.1)

and

w1=1₂(1−λ)|w0|+w0+1₂λ|w0|+w0−ρf (x0)+g(z0), (3.2)

whereλ,ρ >0 are constants.

Sincex0∈F(1/2(|w0|+w0)), y0∈G(1/2(|w0| +w0))and z0∈Q(y0), by Nadler [12], there existx1∈F(1/2(|w1|+w1)),y1∈G(1/2(|w1|+w1))andz1∈Q(y1)such that

x0−x1 ≤(1+1)H

F1₂|w0|+w0,F ₁

2

|w1|+w1, (3.3) y0−y1 ≤(1+1)HG1₂|w0|+w0,G

1 2

|w1|+w1, (3.4) z1−z0 ≤(1+1)HQy1,Qy0, (3.5)

whereH(·,·)denotes the Hausdorﬀ metric on CB(Rn_). Thus, by induction, we can obtain the following algorithm:

Algorithm3.1. LetF,G,Q:Rn_→_CB₍_Rn₎_{be three set-valued mappings and let}

f,g:Rn_→_Rn _{be two single-valued mappings. For any} _w

0∈Rn, we can construct sequences{wn},{xn},{yn}, and{zn}inRn_{as follows:}

xn∈F

1 2

|wn|+wn, yn∈G

1 2

|wn|+wn, zn∈Qyn,

xn−xn+1 ≤

1+_n₊1₁

H

F1₂|wn|+wn,F ₁

2

|wn+1|+wn+1, yn−yn+1 ≤

1+_n1₊₁

H

G1₂|wn|+wn,G ₁

2

|wn+1|+wn+1, (3.6) zn−zn+1 ≤

1+ 1

n+1

HQyn,Qyn+1,

wn+1=1₂(1−λ)|wn|+wn+1₂λ

|wn|+wn−ρf (xn)+g(zn)

forn=0,1,2,..., whereλ,ρ >0 are constants.

From Algorithm 3.1, we can obtain the following algorithms.

Algorithm3.2. LetF,G:Rn_→_CB₍_Rn₎_{be two set-valued mappings and let}_f_,_g_: Rn_→_Rn_{be two single-valued mappings. For any}_w

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{wn},{xn}, and{yn}inRn_{as follows:}

xn∈F ₁

2

|wn|+wn, yn∈G ₁

2

|wn|+wn,

xn−xn+1 ≤

1+_n₊1₁

H

F1₂|wn|+wn,F ₁

2

|wn+1|+wn+1, yn−yn+1 ≤

1+_n1₊₁

H

G1₂|wn|+wn,G ₁

2

|wn+1|+wn+1,

wn+1=1₂(1−λ)|wn|+wn+1₂λ

|wn|+wn−ρf (xn)+g(zn)

(3.7)

Algorithm3.3[4]. LetG:Rn_→_CB₍_Rn₎_{be a set-valued mapping and let}_f_,_g_: Rn_→_Rn_{be two single-valued mappings. For any}_w

0∈Rn, we can construct sequences {wn}and{yn}inRn_{as follows:}

yn∈G ₁

2

|wn|+wn, (3.8)

_y_n₋_y_n₊₁_≤₁₊ 1

n+1

HG1 2

|wn|+wn,G

1 2

|wn+1|+wn+1, (3.9)

wn+1=1₂(1−λ)|wn|+wn+1₂λ

|wn|+wn−ρ

f1₂|wn|+wn+gyn (3.10) forn=0,1,2,..., whereλ,ρ >0 are constants.

Algorithm3.4. [14] Letf,g:Rn_→_Rn _{be two single-valued mappings. For any}

w0∈Rn, we can construct sequences{wn}and{yn}inRnas follows.

yn=1₂|wn|+wn, (3.11)

wn+1=12(1−λ)

|wn|+wn +1₂λ|wn|+wn−ρ

f1₂|wn|+wn+gyn

(3.12)

Algorithm3.5. [15] Letg:Rn_→_Rn_{be a single-valued mapping. For any}_w 0∈Rn, we can construct sequences{wn}and{yn}inRn_{as follows.}

yn=1₂|wn|+wn, (3.13)

wn+1=12(1−λ)

|wn|+wn+12λ

|wn|+wn−ρgyn (3.14) forn=0,1,2,..., whereλ,ρ >0 are constants.

4. Existence and convergence. In this section, we show the existence of solutions for the generalized strongly set-valued nonlinear complementarity problem (2.4) and the convergence of the iterative sequences constructed by Algorithm 3.1. We ﬁrst give some deﬁnitions.

Deﬁnitions4.1.

(1) A mappingf:Rn_→_Rn_{is said to be}_{strongly monotone}_{if there exists a constant}

α >0 such that

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for allu,v∈Rn_.

(2) A mappingf:Rn_→_Rn_{is said to be}_{Lipschitz continuous}_{if there exists a constant}

β >0 such that

_{f (u)}₋_{f (v)}_≤_β_u₋_v _(4.2)

for allu,v∈Rn_{. The number}_β_in₍₂₎_{is called}_{Lipschitz constant}_. It is easy to see thatα≤β.

Deﬁnitions4.2.

(1) A set-valued mappingF :Rn_→_CB₍_Rn₎_{is said to be} _{strongly monotone with}

respect to the mappingf:Rn_→_Rn_{if there exists a constant}_{α >}_{0 such that}

f (x)−fy,u−v≥αu−v2 _(4.3)

for allu,v∈Rn_,_x_∈_F(u)_and_y_∈_F(v)_.

(2) A set-valued mappingF :Rn_→_CB₍_Rn₎_{is said to be}_H_{-Lipschitz continuous}_if there exists a constantβ >0 such that

HF(u),F(v)≤βu−v (4.4)

for allu,v∈Rn_{. The number}_β_in₍₂₎_{is called the}_H_{-Lipschitz constant}_. Now, we give our main theorems in this paper.

Theorem4.1. Suppose thatf,g:Rn_→_Rn_{are Lipschitz continuous with Lipschitz}

constantsδandξ, respectively, andF,G,Q:Rn_→_CB₍_Rn₎_are_H_{-Lipschitz continuous}

withH-Lipschitz constantsβ,η,ν, respectively, andFis strongly monotone with respect tofwith strongly monotone constantα. If

0< ρ < 4(α−ξνη)

(δβ)2₋_(ξνη)2, ρξνη <2, ξνη <min{α,δβ}, (4.5)

then there existu,x,y,z∈Rn_{which solve problem (2.4). Furthermore, it follows that}

1 2

|wn|+wn →u, xn →x, yn →y, zn →z as n→ ∞, (4.6)

where{wn},{xn},{yn}, and{zn}inRn_{are the sequences generated by Algorithm 3.1.} Proof. By Algorithm 3.1, we have

wn+1−wn

=1₂(1−λ)|wn|+wn+12λ

|wn|+wn−ρfxn+gzn −1₂(1−λ)|wn−1|+wn−1−12λ

|wn−1|+wn−1−ρfxn−1+gzn−1 ≤(1−λ)wn−wn−1+₂1λρgzn−gzn−1

+λ1 2

|wn|+wn−1₂|wn−1|+wn−1−1₂ρfxn−fxn−1.

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SinceF,G, andQareH-Lipschitz continuous andf andgare Lipschitz continuous, from (3.6), it follows that

_f

xn−fxn−1≤δxn−xn−1 ≤δ

1+_n1

HF1₂|wn|+wn,F ₁

2

|wn−1|+wn−1 ≤δ1+1

n

β|wn|+wn

2 −

|wn−1|+wn−1 2

≤δ

1+_n1

βwn−wn−1

(4.8)

and _g

zn−gzn−1 ≤ξzn−zn−1≤ξ

1+_n1

HQyn,Qyn−1 ≤ξν1+1

n

yn−yn−1

≤ξν1+1

n

2

HG1 2

|wn|+wn,G

1 2

|wn−1|+wn−1

≤ξν

1+_n1

2

ηwn−wn−1.

(4.9)

Further, from the strong monotonicity ofF with respect tof and (4.8), we have

1₂|wn|+wn−1₂|wn−1|+wn−1−1₂ρfxn−fxn−1 2

≤

1−αρ+1₄ρ2_δ2₁₊1

n

2

β2_w

n−wn−12.

(4.10)

Thus, it follows, from (4.7), (4.8), (4.9), and (4.10), that _w_n₊₁₋_w_n

≤

1−λ+1₂λρξη

1+_n1

+λ 1−αρ+1₄ρ2_δ2_β2₁₊1

n

2

wn−wn−1 =θnwn−wn−1,

(4.11)

where

θn=1−λ+1₂λρξνη

1+_n1 2

+λ 1−αρ+1₄ρ2_δ2_β2

1+_n1 2

. (4.12)

Letting

θ=1−λ+1

2λρξνη+λ 1−αρ+ 1

4ρ2δ2β2, (4.13)

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we can suppose thatwn→wasn→ ∞. (4.8) and (4.9) imply that{xn},{yn}, and{zn} are also Cauchy sequences inRn_{. Let}_x

n→x,yn→y, andzn→zasn→ ∞. Letting

u=1₂(|w|+w), we get 1

2

|wn|+wn →u, xn →x, yn →y, zn →z asn → ∞. (4.14)

Now, we prove thatx∈F(u),y∈G(u), andz∈Q(y). In fact, we have

d(x,F(u))=inf{x−z:z∈F(u)} ≤x−xn+dxn,F(u) ≤x−xn+H

F1₂wn+wn,F(u)

≤x−xn+β1₂wn+wn−u,

(4.15)

and henced(x, F(u))=0. This implies thatx∈F(u). Similarly, we havey∈G(u)

andz∈Q(y). This completes the proof.

From Theorem 4.1, we get the following theorem.

Theorem4.2. Suppose thatf,g:Rn_→_Rn_{are Lipschitz continuous with Lipschitz}

constantsδandξ, respectively, andF,G:Rn_→_CB₍_Rn₎_are_H_{-Lipschitz continuous with}

H-Lipschitz constantsβandη, respectively, andFis strongly monotone with respect to

fwith strongly monotone constantα. If

0< ρ <_(δβ)4(α₂₋−ξη)_(ξη)₂, ρξη <2, ξη <min{α,δβ}, (4.16)

then there existu,x,y∈Rn_{which solve problem (2.5). Furthermore, it follows that}

1

2wn+wn →u, xn →x, yn →y asn → ∞, (4.17)

where{wn},{xn}, and{yn}inRn_{are the sequences generated by Algorithm 3.2.} Theorem4.3[4]. Letf,g, andGbe the same as in Theorem 4.2. Further, suppose thatf is strongly monotone with strongly monotone constantα. If

0< ρ <_δ4₂(α₋−_(ξη)ξη)₂, ρξη <2, ξη < α, (4.18)

then there existu,y∈Rn_{which solve problem (2.6), and}

1

2(|wn|+wn) →u, yn →y, asn → ∞, (4.19)

where{wn}and{yn}are two sequences generated by Algorithm 3.3.

Theorem4.4[14]. Letf andgbe the same as in Theorem 4.3. If

0< ρ <4_δ(α₂₋−_ξξ)₂ , ρξ <2, ξ < α, (4.20)

then there existsu∈Rn_{which is a solution of problem (2.7), and} 1

2(|wn|+wn)→uas

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Theorem4.5[15]. Suppose thatg:Rn_→_Rn _{is strongly monotone and Lipschitz}

continuous with strongly monotone constantαand Lipschitz constant δ. If 0< ρ <

4α/δ2_{, then there exists}_u_∈_Rn_{which is a solution of problem (2.8), and}1

2(|wn|+wn)→

uasn→ ∞, where{wn}is the sequence generated by Algorithm 3.5.

Acknowledgement. The second author was supported by the academic research fund of Ministry of Education, Republic of Korea, 1997, Project No. BSRI-97-1405.

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Huang: Department of Mathematics, Sichuan University, Chengdu, Sichuan610064, China