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Set-Valued Variational Inclusions in Banach Spaces
Journal of Mathematical Analysis and Applications 248, 438᎐454 Ž2000. doi:10.1006rjmaa.2000.6919, available online at http:rrwww.idealibrary.com on
Set-Valued Variational Inclusions in Banach Spaces S. S. Chang 1 Department of Mathematics, Sichuan Uni¨ ersity, Chengdu, Sichuan 610064, People’s Republic of China Submitted by William F. Ames Received April 20, 1999
The purpose of this paper is to introduce and study a class of set-valued variational inclusions without compactness condition in Banach spaces. By using Michael’s selection theorem and Nadler’s theorem, an existence theorem and an iterative algorithm for solving this kind of set-valued variational inclusions in Banach spaces are established and suggested. 䊚 2000 Academic Press Key Words: variational inclusion; variational inequality; m-accretive mapping; accretive mapping; Ishikawa iterative sequence; Mann iterative sequence.
1. INTRODUCTION In recent years, variational inequalities have been extended and generalized in different directions, using novel and innovative techniques. Useful and important generalizations of variational inequalities are variational inclusions. Recently, in w18, 19x, Noor et al. introduced and studied the following class of set-valued variational inclusion problems in a Hilbert space H. For a given maximal monotone mapping A : H ª H, a nonlinear mapping N Ž⭈ , ⭈ . : H = H ª H, set-valued mappings T, V : H ª C Ž H ., and a single-valued mapping g : H ª H, find u g H, w g T Ž u., y g V Ž u. such that
g N Ž w, y . q A Ž g Ž u . . ,
Ž 1.1.
where C Ž H . denotes the family of all nonempty compact subsets of H. 1 This paper was done while the author visited Korea and was supported by the Korean Science and Engineering Foundation and National Natural Science Foundation of China.
438 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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For a suitable choice of the mappings T, V, g, A, N, a number of known and new variational inequalities, variational inclusions, and related optimization problems introduced and studied by Noor w19᎐24x can be obtained from Ž1.1.. Inspired and motivated by the results in Noor w18, 19x, the purpose of this paper is to introduce and study a class of more general set-valued variational inclusions without the compactness condition in Banach spaces. By using the famous Michael’s selection theorem w15x and Nadler’s theorem w17x an existence theorem and an approximate theorem for solving the set-valued variational inclusions in Banach spaces are established and suggested. The results presented in this paper generalize, improve, and unify the corresponding results of Noor w18᎐24x, Ding w7x, Huang w9, 10x, Kazmi w12x, Jung and Morales w11x, Hassouni and Moudafi w8x, Zeng w26x, and Chang et al. w3, 4x.
2. PRELIMINARIES Throughout this paper, we assume that E is a real Banach space, EU is the topological dual space of E, CB Ž E . is the family of all nonempty closed and bounded subsets of E, ² ⭈ , ⭈ : is the dual pair between E and EU , DŽ⭈ , ⭈ . is the Hausdorff metric on CB Ž E . defined by D Ž A, B . s max sup d Ž x, B . , sup d Ž A, y . ,
½
xgA
5
ygB
A, B g CB Ž E . .
U
DŽT . denotes the domain of T, and J : E ª 2 E is the normalized duality mapping defined by J Ž x . s f g EU : ² x, f : s 5 x 5 ⭈ 5 f 5 , 5 f 5 s 5 x 5 4 ,
x g E.
DEFINITION 2.1. Let A : DŽ A. ; E ª 2 E be a set-valued mapping and : w0, ⬁. ª w0, ⬁. be a strictly increasing function with Ž0. s 0. Ž1. The mapping A is said to be accretive if, for any x, y g DŽ A., there exists jŽ x y y . g J Ž x y y . such that
² u y ¨ , j Ž x y y .: G 0 for all u g Ax, ¨ g Ay. Ž2. The mapping A is said to be -strongly accretive if, for any x, y g DŽ A., there exists jŽ x y y . g J Ž x y y . such that for any u g Ax, ¨ g Ay,
² u y ¨ , j Ž x y y .: G Ž 5 x y y 5 . 5 x y y 5 .
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Especially, if Ž t . s kt, 0 - k - 1, then the mapping A is said to be k-strongly accretive. Ž3. The mapping A is said to be m-accretive if A is accretive and Ž I q D A.Ž DŽ A.. s E for all D ) 0, where I is the identity mapping. Ž4. The mapping A is said to be -expansive if for any x, y g DŽ A. and for any u g Ax, ¨ g Ay, 5 u y ¨ 5 G Ž 5 x y y 5. . Remark. It is easy to see that if A is -strongly accretive, then A is -expansive. DEFINITION 2.2. Let T, F : E ª CB Ž E . be two set-valued mappings, A : DŽ A. ; E ª 2 E an m-accretive mapping, g : E ª DŽ A. a single-valued mapping, and N Ž⭈ , ⭈ . : E = E ª E a nonlinear mapping. For any given f g E and ) 0 we consider the following problem. Find q g E, w g T Ž q ., ¨ g F Ž q . such that f g N Ž w, ¨ . q A Ž g Ž q . . .
Ž 2.1.
This problem is called the set-valued variational inclusion problem in Banach spaces. A number of problems arising in pure and applied sciences can be reduced to study this kind of variational inclusion problem Žsee, for example, w6, 18, 19, 25x.. Next we consider some special cases of problem Ž2.1.. Ž1. If E s H is a Hilbert space and A : DŽ A. s H ª H is a maximal monotone mapping, then the problem Ž2.1. is equivalent to finding q g H, w g Tq, ¨ g F Ž q . such that f g N Ž w, ¨ . q A Ž g Ž q . . .
Ž 2.2.
This problem was introduced and studied in Noor w18x and Noor et al. w19x under some additional conditions. Ž2. If g ' I, then the problem Ž2.1. is equivalent to finding q g DŽ A., w g Tq, ¨ g F Ž q . such that f g N Ž w, ¨ . q A Ž q . .
Ž 2.3.
This problem seems to be a new one. We will study this in another paper. Ž3. If g s I, F s 0, T s I, S : E ª E is a single-valued mapping and N Ž x, y . s Sx for all Ž x, y . g E = E, then the problem Ž2.1. is equivalent to finding q g DŽ A. such that f g Sq q Aq. This problem was introduced and studied in Jung and Morales w11x.
Ž 2.4.
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Ž4. If E s H is a Hilbert space, s 1, and A s ⭸ , the subdifferential of a proper convex lower semi-continuous functional : H ª ⺢ j q⬁4 , then the problem Ž2.1. is equivalent to finding q g H, w g Tq, ¨ g F Ž q . such that
² N Ž w, ¨ . y f , x y g Ž q .: G Ž g Ž q . . y Ž x . ,
for all x g H. Ž 2.5.
This problem is called the generalized set-valued mixed variational inequality, which was introduced and studied by Noor et al. w19x. Recently, this problem with N being some special case was also considered in the setting of Banach spaces Žsee Chang w3x.. Ž5. If H is a Hilbert space, T, F, M : H ª 2 H are three set-valued mappings, and m, S, G : H ª H are three single-valued mappings, K Ž z . s mŽ z . q K, where K is a closed convex subset of E, N Ž x, y . s Sx q Gy and
Ž x . s IK Ž z . Ž x . s
½
0, q⬁,
if x g K Ž z . , if x f K Ž z . ,
then the problem Ž2.5. is equivalent to finding q g H, w g T Ž q ., ¨ g F Ž q ., z g M Ž q . such that g Ž q. g K Ž q. ,
² Sw q G¨ y f , x y g Ž q .: G 0, x g K Ž z . . Ž 2.6.
This problem is called the generalized strongly nonlinear implicit quasivariational inequality, which was studied in Huang w10x. Summing up the above arguments, it shows that for a suitable choice of the mapping T, F, A, g, N and the space E, one can obtain a number of known and new classes of variational inequalities, variational inclusions, and the corresponding optimization problems from the set-valued variational inclusions problem Ž2.1.. Furthermore, these types of variational inclusions enable us to study many important problems arising in mathematical, physical, and engineering sciences in a general and unified framework. DEFINITION 2.3. Let T, F : E ª 2 E be two set-valued mappings, N Ž⭈ , ⭈ . : E = E ª E a nonlinear mappings, and : w0, ⬁. ª w0, ⬁. a strictly increasing function with Ž0. s 0.
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Ž1. The mapping x ¬ N Ž x, y . is said to be -strongly accretive with respect to the mapping T, if for any x 1 , x 2 g E there exists jŽ x 1 y x 2 . g J Ž x 1 y x 2 . such that for any u1 g Tx 1 , u 2 g Tx 2
² N Ž u1 , y . y N Ž u 2 , y . , j Ž x 1 y x 2 .: G Ž 5 x 1 y x 2 5 . 5 x 1 y x 2 5 for all y g E. Ž2. The mapping y ¬ N Ž x, y . is said to be accretive with respect to the mapping F, if for any y 1 , y 2 g E there exists jŽ y 1 y y 2 . g J Ž y 1 y y 2 . such that for any ¨ 1 g Ty 1 , ¨ 2 g Ty 2
² N Ž x, ¨ 1 . y N Ž x, ¨ 2 . , j Ž y1 y y 2 .: G 0 for all x g E. DEFINITION 2.4. Let T : E ª CB Ž E . be a set-valued mapping and DŽ⭈ , ⭈ . be the Hausdorff metric on CB Ž E .. T is said to be -Lipschitzian continuous if, for any x, y g E, D Ž Tx, Ty . F 5 x y y 5 , where ) 0 is a constant. The following lemmas play an important role in proving our main results. U
LEMMA 2.1 w1, 2x. Let E be a real Banach space and J : E ª 2 E be the normalized duality mapping. Then for any x, y g E, the following inequality holds, 5 x q y 5 2 F 5 x 5 2 q 2² y, j Ž x q y .: , for all jŽ x q y . g J Ž x q y .. LEMMA 2.2. Let E be a real smooth Banach space, T, F : E ª 2 E two set-¨ alued mappings, and N Ž⭈ , ⭈ . : E = E ª E a nonlinear mapping satisfying the following conditions: Ž1. the mapping x ¬ N Ž x, y . is -strongly accreti¨ e with respect to the mapping T ; Ž2. the mapping y ¬ N Ž x, y . is accreti¨ e with respect to the mapping F. Then the mapping S : E ª 2 E defined by Sx s N Ž Tx , Fx . is -strongly accreti¨ e.
VARIATIONAL INCLUSIONS IN BANACH SPACES
443 U
Proof. Since E is smooth, the normalized duality mapping J : E ª 2 E is single-valued. For any given x 1 , x 2 g E and for any u i g Sx i , i s 1, 2, there exists wi g Tx i and ¨ i g Fx i such that u i s N Ž wi , ¨ i ., i s 1, 2. By conditions Ž1. and Ž2. we have
² u1 y u 2 , J Ž x 1 y x 2 .: s² N Ž w1 , ¨ 1 . y N Ž w 2 , ¨ 2 . , J Ž x 1 y x 2 .: s² N Ž w 1 , ¨ 1 . y N Ž w 2 , ¨ 1 . , J Ž x 1 y x 2 .: q² N Ž w 2 , ¨ 1 . y N Ž w 2 , ¨ 2 . , J Ž x 1 y x 2 .: G Ž 5 x1 y x 2 5. 5 x1 y x 2 5 . This implies that the mapping S s N ŽT Ž⭈., F Ž⭈.. is -strongly accretive. LEMMA 2.3 ŽMichael w15x.. Let X and Y be two Banach spaces, T : X ª 2 Y a lower semi-continuous mapping with nonempty closed con¨ ex ¨ alues. Then T admits a continuous selection; i.e., there exists a continuous mapping h : X ª Y such that hŽ x . g Tx for each x g X. LEMMA 2.4. Let E be a real uniformly smooth Banach space and T : E ª 2 E be a lower semi-continuous m-accreti¨ e mapping. Then the following conclusions hold. Ž1. T admits a continuous and m-accreti¨ e selection; Ž2. In addition, if T is also -strongly accreti¨ e, then T admits a continuous, m-accreti¨ e and -strongly accreti¨ e selection. Proof. Ž1. It is well known that if E is a uniformly smooth Banach space and T : E ª 2 E is a m-accretive mapping, then for each x g E, T Ž x . is nonempty closed and convex Žsee, for example, Deimling w5, p. 293x.. By Lemma 2.3, T admits a continuous selection h : E ª E such that hŽ x . g T Ž x . for all x g E. Next we prove that h : E ª E is m-accretive. In fact, since T : E ª 2 E is accretive, for any x, y g E and for any u g Tx, ¨ g Ty,
² u y ¨ , J Ž x y y .: G 0. In particular, letting u s hŽ x . g Tx, ¨ s hŽ y . g Ty, we have
² h Ž x . y h Ž y . , J Ž x y y .: G 0. This implies that h : E ª E is a continuous accretive mapping. By a well-known result due to Martin w14x, h is a continuous m-accretive selection.
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S. S. CHANG
Ž2. In addition, if T is also -strongly accretive, then the selection h : E ª E given in Ž1. is also -strongly accretive. In fact, for any x, y g E and for any u g Tx, ¨ g Ty, we have
² u y ¨ , J Ž x y y .: G Ž 5 x y y 5 . 5 x y y 5 .
Ž 2.7.
Letting u s hŽ x . g Tx, ¨ s hŽ y . g Ty, from Ž2.7. we have
² h Ž x . y h Ž y . , J Ž x y y .: G Ž 5 x y y 5 . 5 x y y 5 . This implies that h is -strongly accretive. This completes the proof of Lemma 2.4. LEMMA 2.5 ŽNadler w17x.. Let E be a complete metric space, T : E ª CB Ž E . be a set-¨ alued mapping. Then for any gi¨ en ⑀ ) 0 and for any gi¨ en x, y g E, u g Tx, there exists ¨ g Ty such that d Ž u, ¨ . F Ž 1 q ⑀ . D Ž Tx, Ty . . LEMMA 2.6 w13x. Let E be a uniformly smooth Banach space and A : DŽ A. ; E ª 2 E be an m-accreti¨ e and -expansi¨ e mapping, where : w0, ⬁. ª w0, ⬁. is a strictly increasing function with Ž0. s 0. Then A is surjecti¨ e. We now invoke Michael’s selection theorem and Nadler’s theorem to suggest the following algorithms for solving the set-valued variational inclusion Ž2.1.. Let ␣ n4 , n4 be two sequences in w0, 1x, f g E be any given point, and ) 0 be any given positive number. For given x 0 g E, u 0 g T Ž x 0 ., z 0 g F Ž x 0 ., take y0 g Ž 1 y  0 . x 0 q  0 Ž f q x 0 y N Ž u0 , z0 . y AŽ g Ž x 0 . . and w 0 g Ty 0 ,
¨ 0 g Fy 0 .
Define x1 g Ž 1 y ␣ 0 . x 0 q ␣ 0 Ž f q y0 y N Ž w0 , ¨ 0 . y AŽ g Ž y0 . . . . Since u 0 g Tx 0 , z 0 g Fx 0 , by Nadler’s theorem, there exist u1 g Tx 1 , z1 g Fx 1 such that 5 u 0 y u1 5 F Ž 1 q 1 . D Ž T Ž x 0 . , T Ž x 1 . . , 5 z 0 y z1 5 F Ž 1 q 1 . D Ž F Ž x 0 . , F Ž x 1 . . .
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For x 1 g E, u1 g Tx 1 , z1 g Fx 1 , define y 1 g Ž 1 y  1 . x 1 q  1 Ž f q x 1 y N Ž u1 , z 1 . y A Ž g Ž x 1 . . . . Since w 0 g Ty 0 , ¨ 0 g Fy 0 , again by Nadler’s theorem, there exist w 1 g Ty 1 , ¨ 1 g Fy 1 such that 5 w 0 y w1 5 F Ž 1 q 1. D Ž T Ž y 0 . , T Ž y1 . . , 5 ¨ 0 y ¨ 1 5 F Ž 1 q 1. D Ž F Ž y 0 . , F Ž y1 . . . Define x 2 g Ž 1 y ␣ 1 . x 1 q ␣ 1 Ž f q y1 y N Ž w1 , ¨ 1 . y AŽ g Ž y1 . . . . Continuing in this way, we can obtain the following algorithms: ALGORITHM 2.1. For any given x 0 g E, u 0 g T Ž x 0 ., and z 0 g F Ž x 0 ., compute the sequences x n4 , yn4 , u n4 , z n4 , wn4 , and ¨ n4 by the iterative schemes such that
Ž i.
x nq 1 g Ž 1 y ␣ n . x n q ␣ n Ž f q yn y N Ž wn , ¨ n . y Ag Ž yn . . ,
Ž ii .
yn g Ž 1 y n . x n q n Ž f q x n y N Ž u n , z n . y Ag Ž x n . . ,
Ž iii .
u n g Tx n , 5 u n y u nq1 5 F 1 q
Ž iv.
z n g Fx n , 5 z n y z nq1 5 F 1 q
Ž v.
wn g T Ž yn . , 5 wn y wnq1 5 F 1 q
Ž vi.
¨ n g F Ž yn . , 5 ¨ n y ¨ nq1 5 F 1 q
ž
ž
1 nq1 1
ž
ž
/
nq1
/
D Ž Tx n , Tx nq1 . ,
D Ž Fx n , Fx nq1 . ,
1 nq1 1
nq1
/
/
Ž 2.8.
D Ž Tyn , Tynq1 . ,
D Ž Fyn , Fynq1 . ,
n s 0, 1, 2, . . . . The sequence x n4 defined by Ž2.8., in the sequel, is called the Ishikawa iterative sequence. In Algorithm 2.1, if n s 0 for all n G 0, then yn s x n . Take wn s u n , z n s ¨ n for all n G 0 and we obtain the following
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ALGORITHM 2.2. For any given x 0 g E, w 0 g Tx 0 , ¨ 0 g Fx 0 compute the sequences x n4 , wn4 , ¨ n4 by iterative schemes such that x nq 1 g Ž 1 y ␣ n . x n q ␣ n Ž f q x n y N Ž wn , ¨ n . y Ag Ž x n . . , wn g Tx n , ¨ n g Fx n ,
5 wn y wnq1 5 F 1 q
ž
5 ¨ n y ¨ nq1 5 F 1 q
ž
1 nq1 1
nq1
/
/
D Ž Tx n , Tx nq1 . ,
Ž 2.9.
D Ž Fx n , Fx nq1 .
n G 0. The sequence x n4 defined by Ž2.9., in the sequel, is called the Mann iterative sequence.
3. EXISTENCE THEOREM OF SOLUTIONS FOR SET-VALUED VARIATIONAL INCLUSION In this section, we shall establish an existence theorem of solutions for set-valued variational inclusion Ž2.1.. We have the following result. THEOREM 3.1. Let E be a real uniformly smooth Banach space, T, F : E ª CB Ž E . and A : DŽ A. ; E ª 2 E three set-¨ alued mappings, g : E ª DŽ A. a single-¨ alued mapping, and N Ž⭈ , ⭈ . : E = E ª E a single-¨ alued continuous mapping satisfying the following conditions: Ži. A( g : E ª 2 E is m-accreti¨ e; Žii. T : E ª CB Ž E . is -Lipschitzian continuous; Žiii. F : E ª CB Ž E . is -Lipschitzian continuous, where and are positi¨ e constants; Živ. the mapping x ¬ N Ž x, y . is -strongly accreti¨ e with respect to the mapping T, where : w0, ⬁. ª w0, ⬁. is a strictly increasing function with Ž0. s 0; Žv. the mapping y ¬ N Ž x, y . is accreti¨ e with respect to the mapping F. Then for any gi¨ en f g E, ) 0, there exist q g E, w g T Ž q ., ¨ g F Ž q . which is a solution of the set-¨ alued ¨ ariational inclusion Ž2.1.. Proof. It follows from conditions Živ., Žv., and Lemma 2.2 that the mapping S : E ª 2 E defined by Sx s N Ž Tx, Fx . ,
xgE
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is -strongly accretive. Since N Ž⭈ , ⭈ . is continuous, T and F both are Lipschitzian continuous, and so S is continuous and accretive. By Morales w16x, S is m-accretive and -strongly accretive. By Lemma 2.4Ž2., S admits a continuous, -strongly accretive and m-accretive selection h : E ª E such that h Ž x . g S Ž x . s N Ž Tx, Fx .
for all x g E.
Next we consider the variational inclusion f g h Ž x . q AŽ g Ž x . . ,
) 0.
Ž 3.1.
By the assumption A( g is m-accretive and h : E ª E is continuous and -strongly accretive. Hence by Kobayashi w13, Theorem 5.3x, h q A( g is m-accretive and -strongly accretive. Therefore it is also an m-accretive and -expansive mapping. By Lemma 2.6, h q A( g : E ª 2 E is surjective. Therefore for given f and ) 0, there exists a unique q g E such that f g h Ž q . q AŽ g Ž q . . ; N Ž T Ž q . , F Ž q . . q AŽ g Ž q . . . ŽThe uniqueness of q is a direct consequence of the -strongly accretiveness of h q A( g.. Consequently, there exist w g Tq and ¨ g Fq such that f g N Ž w, ¨ . q Ag Ž q . . . This completes the proof.
4. APPROXIMATE PROBLEM OF SOLUTIONS FOR SET-VALUED VARIATIONAL INCLUSION In Theorem 3.1, under some conditions, we have proved that there exist q g E, w g Tq, and ¨ g Fq which is a solution of set-valued variational inclusion Ž2.1.. In this section we shall study the approximate problem of solutions for variational inclusion Ž2.1.. We have the following result: THEOREM 4.1. Let E, T, F, A, g, N be as in Theorem 3.1. Let ␣ n4 , n4 be two sequences in w0, 1x satisfying the following conditions: Ži. ␣ n ª 0; n ª 0; Žii. Ý⬁ns 0 ␣ n s ⬁.
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If the ranges RŽ I y N ŽT Ž⭈., F Ž⭈... and RŽ A( g . both are bounded, then for any gi¨ en x 0 g E, u 0 g Tx 0 , z 0 g Fx 0 the iterati¨ e sequences x n4 , wn4 , and ¨ n4 defined by Ž2.8. con¨ erge strongly to the solution q, w, ¨ of set-¨ alued ¨ ariational inclusion Ž2.1. which is gi¨ en in Theorem 3.1, respecti¨ ely. Proof. In Ži. and Žii. of Ž2.8. choose h n g Ag Ž x n ., k n g Ag Ž yn . such that x nq 1 s Ž 1 y ␣ n . x n q ␣ n Ž f q yn y N Ž wn , ¨ n . y k n . , yn s Ž 1 y  n . x n q  n Ž f q x n y N Ž u n , z n . y h n . .
Ž 4.1.
Let pn [ f q yn y N Ž wn , ¨ n . y k n , rn [ f q x n y N Ž u n , z n . y h n . Then Ž4.1. can be rewritten as x nq 1 s Ž 1 y ␣ n . x n q ␣ n pn , yn s Ž 1 y  n . x n q  n r n .
Ž 4.2.
Since the ranges RŽ I y N ŽT Ž⭈., F Ž⭈..., RŽ A( g . are bounded, let M s sup 5 w y q 5 : w g Ž f q x y N Ž Tx, Fx . y Ag Ž x . . , x g E 4 q5 x 0 y q 5 - ⬁. This implies that 5 pn y q 5 F M, 5 rn y q 5 F M,
for all n G 0.
Since 5 x 0 y q 5 F M, we have 5 y0 y q 5 s Ž 1 y 0 . Ž x 0 y q . q 0 Ž r0 y q . F Ž 1 y 0 . 5 x 0 y q 5 q 0 5 r0 y q 5 F M. This implies that 5 x1 y q 5 F Ž 1 y ␣ 0 . 5 x 0 y q 5 q ␣ 0 5 p0 y q 5 F M ; 5 y 1 y q 5 F Ž 1 y  1 . 5 x 1 y q 5 q  1 5 r 1 y q 5 F M.
Ž 4.3.
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By induction we can prove that 5 x n y q 5 F M,
Ž 4.4.
5 yn y q 5 F M. On the other hand, by Ž4.3., Ž4.4., and Lemma 2.1 we have 5 x nq 1 y q 5 2 s Ž 1 y ␣ n . Ž x n y q . q ␣ n Ž pn y q .
2
2 F Ž 1 y ␣n . 5 xn y q 5 2
q 2 ␣ n² pn y q, J Ž x nq1 y q .: 2 s Ž 1 y ␣ n . 5 x n y q 5 2 q 2 ␣ n² pn y q, J Ž yn y q .:
q 2 ␣ n² pn y q, J Ž x nq1 y q . y J Ž yn y q .: .
Ž 4.5.
Now we consider the third term on the right side of Ž4.5.. Since
Ž x nq 1 y q . y Ž yn y q . s 5 x nq 1 y yn 5 s Ž 1 y ␣ n . Ž x n y yn . q ␣ n Ž pn y yn . F Ž 1 y ␣ n . n 5 x n y rn 5 q ␣ n 5 pn y q 5 q 5 yn y q 5 4 F Ž 1 y ␣ n . n 5 x n y q 5 q 5 rn y q 5 4 q ␣ n 5 pn y q 5 q 5 yn y q 5 4 F 2 Ž Ž 1 y ␣ n . n q ␣ n . ⭈ M ª 0, U
by the uniform continuity of J : E ª 2 E , we have J Ž x nq 1 y q . y J Ž yn y q . ª 0
Ž n ª ⬁. .
Since pn y q4 is bounded, this implies that
␦n [ ² pn y q, J Ž x nq1 y q . y J Ž yn y q .: ª 0
Ž n ª ⬁ . . Ž 4.6.
Next we consider the second term on the right side of Ž4.5.. Since wn g Tyn , ¨ n g Fyn , k n g Ag Ž yn ., this implies that N Ž wn , ¨ n . q k n g N Ž T Ž ⭈ . , F Ž ⭈ . . q Ag Ž ⭈ . Ž yn . . Again since q g E is a solution of the variational inclusion, f g h Ž q . q Ag Ž q . ; N Ž T Ž ⭈ . , F Ž ⭈ . . q Ag Ž ⭈ . Ž q . .
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S. S. CHANG
This shows that f is a point of w N ŽT Ž⭈., F Ž⭈.. q Ag Ž⭈.xŽ q .. By the assumptions of Theorem 4.1, N ŽT Ž⭈., F Ž⭈.. q Ag Ž⭈. : E ª 2 E is -strongly accretive, hence we have
² f y Ž N Ž wn , ¨ n . q k n . , J Ž yN y q .: s y² N Ž wn , ¨ n . q k n y f , J Ž yn y q .: F y Ž 5 yn y q 5 . 5 yn y q 5 . Therefore we have 2 ␣ n² pn y q, J Ž yn y q .: s2 ␣ n² f q yn y N Ž wn , ¨ n . y k n y q, J Ž yn y q .: s 2 ␣ n² yn y q, J Ž yn y q .: y² N Ž wn , ¨ n . q k n y f , J Ž yn y q .: 4 F 2 ␣ n 5 yn y q 5 2 y Ž 5 yn y q 5 . 5 yn y q 5 4 .
Ž 4.7.
Substituting Ž4.6. and Ž4.7. into Ž4.5. we have 5 x nq 1 y q 5 2 F Ž 1 y ␣ n . 2 5 x n y q 5 2 q 2 ␣ n 5 yn y q 5 2 y Ž 5 yn y q 5 . 5 yn y q 5 4 q 2 ␣ n ␦ n .
Ž 4.8. Next we make an estimation for 5 yn y q 5 2 . We have 5 yn y q 5 2 s Ž 1 y  n . Ž x n y q . q  n Ž r n y q .
2
2 F Ž 1 y n . 5 x n y q 5 2 q 2 n² rn y q, J Ž yn y q .: 2 F Ž 1 y  n . 5 x n y q 5 2 q 2  n 5 r n y q 5 ⭈ 5 yn y q 5
F 5 x n y q 5 2 q 2 n M 2 .
Ž 4.9.
Substituting Ž4.9. into Ž4.8. and simplifying, we have 5 x nq 1 y q 5 2 F Ž 1 y ␣ n . 2 5 x n y q 5 2 q 2 ␣ n 5 x n y q 5 2 q 2 n M 2 4 y 2 ␣ n Ž 5 yn y q 5 . 5 yn y q 5 q 2 ␣ n ␦ n s 5 x n y q 5 2 y ␣ n Ž 5 yn y q 5 . 5 yn y q 5 y ␣ n Ž 5 yn y q 5 . 5 yn y q 5 y4n M 2 y ␣ n 5 x n y q 5 2 y 2 ␦n 4 .
Ž 4.10.
VARIATIONAL INCLUSIONS IN BANACH SPACES
451
Let
s inf nG 0 5 yn y q 5 . Next we prove that s 0. Suppose the contrary, ) 0. Then we have 5 yn y q 5 G ) 0 for all n G 0. It follows from Ž4.10. that 5 x nq 1 y q 5 2 F 5 x n y q 5 2 y ␣ n Ž . 4 y ␣ n Ž . y 4n M 2 y ␣ n M 2 y 2 ␦n 4 . Ž 4.11. Since n ª 0, ␣ n ª 0, and ␦n ª 0, there exists n 0 such that for n G n 0
Ž . ⭈ y 4n M 2 y ␣ n M 2 y 2 ␦n ) 0. Therefore from Ž4.11. we have 5 x nq 1 y q 5 2 F 5 x n y q 5 2 y ␣ n Ž . 4
for all n G n 0 ,
␣ n Ž . 4 F 5 x n y q 5 2 y 5 x nq1 y q 5 2
for all n G n 0 .
i.e.,
Hence for any m G n 0 we have m
Ý
␣ n Ž . 4 F 5 x n 0 y q 5 2 y 5 x mq1 y q 5 2
nsn 0
F 5 x n0 y q 5 2 . Letting m ª ⬁, we have ⬁s
⬁
Ý
␣ n Ž . 4 F 5 x n 0 y q 5 2 .
nsn 0
This is a contradiction. Therefore s 0, and so there exists a subsequence yn 4 ; yn4 such that yn ª q, i.e., j j yn j s Ž 1 y n j . x n j q n j rn j ª q. Since n ª 0, ␣ n ª 0, and rn4 , pn4 both are bounded, this implies that x n j ª q, and so x n jq1 s Ž 1 y ␣ n j . x n j q ␣ n j pn j ª q.
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S. S. CHANG
Hence yn jq1 s Ž 1 y n jq1 . x n jq1 q ␣ n jq1 rn jq1 ª q. By induction, we can prove that x n jq1 ª q
yn jqi ª q
and
for all i G 0.
This implies that xn ª q
yn ª q.
and
Since T is -Lipschitzian and F is -Lipschitzian, it follows from Žiii. and Živ. in Ž2.8. that 5 u n y u nq1 5 F 1 q
ž
1 nq1
F 1q
ž
/
D Ž Tx n , Tx nq1 .
1 nq1
/
5 x n y x nq1 5 ,
and 5 z n y z nq1 5 F 1 q
ž
1 nq1
F 1q
ž
/
1 nq1
D Ž Fx n , Fx nq1 .
/
5 x n y x nq1 5 .
This implies that the sequences u n4 , z n4 both are Cauchy sequences. By the same way we know that wn4 , ¨ n4 are also Cauchy sequences. Therefore there exist uU , wU , ¨ U , zU g E such that u n ª uU wn ª wU ¨n ª ¨U z n ª zU
Ž n ª ⬁. .
Next we prove that uU s wU s w ;
zU s ¨ U s ¨ .
VARIATIONAL INCLUSIONS IN BANACH SPACES
453
In fact, since d Ž wU , Tq . F d Ž wU , wn . q d Ž wn , Tq . F d Ž wU , wn . q D Ž Tyn , Tq . F d Ž wU , wn . q 5 yn y q 5 ª 0, this implies that wU g Tq. Similarly we can prove that uU g Tq. Again since d Ž zU , Fq . F d Ž zU , z n . q d Ž z n , Fq . F d Ž zU , z n . q D Ž Fx n , Fq . F d Ž zU , z n . q 5 x n y q 5 ª 0, this implies that zU g Fq. Similarly we can prove that ¨ U g Fq. Next we prove that wU s uU s w. In fact, we have d Ž wU , w . F d Ž wU , wn . q d Ž wn , w . F d Ž wU , wn . q D Ž Tyn , Tq . F d Ž wU , wn . q 5 yn y q 5 ª 0. This implies that wU s w. Again since d Ž uU , w . F d Ž uU , u n . q d Ž u n , w . F d Ž uU , u n . q D Ž Tx n , Tq . F d Ž uU , u n . q 5 x n y q 5 ª 0, this implies that uU s w. Similarly we can prove that zU s ¨ U s ¨ . Summing up the above arguments we know that the sequences x n4 , wn4 , and ¨ n4 defined by Ž2.8. converge strongly to the solution Ž q, w, ¨ . of Ž2.1., respectively. This completes the proof.
REFERENCES 1. Shih-sen Chang, Some problems and results in the study of nonlinear analysis, Nonlinear Anal. 30 Ž1997., 4197᎐4208. 2. Shih-sen Chang, Y. J. Cho, J. S. Jung, and S. M. Kang, Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces, J. Math. Anal. Appl. 224 Ž1998., 249᎐165.
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S. S. CHANG
3. Shih-sen Chang, On the Mann and Ishikawa iterative approximation of solutions to variational inclusions with accretive type mappings, Comput. Math. Appl. 37 Ž1999., 17᎐24. 4. Shih-sen Chang, X. Z. Yuan, F. Wang, and X. Long, The study of algorithms and convergence for generalized multi-valued quasi-variational inclusions, Topol. Methods Nonlinear Anal., to appear. 5. K. Deimling, ‘‘Nonlinear Functional Analysis,’’ Springer-Verlag, Berlin, 1985. 6. V. F. Demyanov, G. E. Stavroulakis, L. N. Polyakova, and P. D. Panagiotopoulos, ‘‘Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics,’’ Kluwer Academic, DordrechtrNorwell, MA, 1996. 7. X. P. Ding, Iterative process with errors of nonlinear equations involving m-accretive operators, J. Math. Anal. Appl. 209 Ž1997., 191᎐201. 8. A. Hassouni and A. Moudafi, A perturbed algorithm for variational inclusions, J. Math. Anal. Appl. 185 Ž1994., 706᎐712. 9. N. J. Huang, On the generalized implicit quasi-variational inequalities, J. Math. Anal. Appl. 216 Ž1997., 197᎐210. 10. N. J. Huang, Generalized nonlinear variational inclusions with noncompact valued mappings, Appl. Math. Lett. 9, No. 3 Ž1996., 25᎐29. 11. J. S. Jung and C. H. Morales, The Mann process for perturbed m-accretive operators in Banach spaces, submitted for publication. 12. K. R. Kazmi, Mann and Ishikawa type perturbed iterative algorithms for generalized quasi-variational inclusions, J. Math. Anal. Appl. 209 Ž1997., 572᎐584. 13. Y. Kobayashi, Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups, J. Math. Soc. Japan 27 Ž1975., 640᎐665. 14. R. H. Martin, Jr., A global existence theorem for autonomous differential equations in Banach spaces, Proc. Amer. Math. Soc. 26 Ž1970., 307᎐314. 15. E. Michael, Continuous solutions, I, Ann. Math. 63 Ž1956., 361᎐382. 16. C. Morales, Surjectivity theorems for multi-valued mappings of accretive type, Comment. Math. Uni¨ . Carolin. 26 Ž1985., 397᎐413. 17. S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30 Ž1969., 475᎐488. 18. M. A. Noor, Generalized set-valued variational inclusions and resolvent equations, J. Math. Anal. Appl. 228 Ž1998., 206᎐220. 19. M. A. Noor, K. I. Noor, and Th. M. Rassias, Set-valued resolvent equations and mixed variational inequalities, J. Math. Anal. Appl. 220 Ž1998., 741᎐759. 20. M. A. Noor, Auxiliary principle for generalized mixed variational-like inequalities, J. Math. Anal. Appl. 215 Ž1997., 75᎐85. 21. M. A. Noor, Numerical methods for monotone mixed variational inequalities, Ad¨ . Nonlinear Variat. Inequal. 1 Ž1998., 51᎐79. 22. M. A. Noor, An implicit method for mixed variational inequalities, Appl. Math. Lett. 11, No. 4 Ž1998., 109᎐113. 23. M. A. Noor, Sensitivity analysis of quasi-variational inequalities, J. Optim. Theory Appl. 95, No. 2 Ž1997., 399᎐407. 24. M. A. Noor, Generalized multi-valued quasi-variational inequalities, II, Comput. Math. Appl. 35, No. 5 Ž1997., 63᎐78. 25. P. D. Panagiotoupoulos and G. E. Stavroulakis, New types of variational principles based on the notion of quasi-differentiability, Acta Mech. 94 Ž1992., 171᎐194. 26. L. C. Zeng, Iterative algorithm for finding approximate solutions to completely generalized strongly nonlinear quasi-variational inequality, J. Math. Anal. Appl. 201 Ž1996., 180᎐191.
Set-Valued Variational Inclusions in Banach Spaces S. S. Chang 1 Department of Mathematics, Sichuan Uni¨ ersity, Chengdu, Sichuan 610064, People’s Republic of China Submitted by William F. Ames Received April 20, 1999
The purpose of this paper is to introduce and study a class of set-valued variational inclusions without compactness condition in Banach spaces. By using Michael’s selection theorem and Nadler’s theorem, an existence theorem and an iterative algorithm for solving this kind of set-valued variational inclusions in Banach spaces are established and suggested. 䊚 2000 Academic Press Key Words: variational inclusion; variational inequality; m-accretive mapping; accretive mapping; Ishikawa iterative sequence; Mann iterative sequence.
1. INTRODUCTION In recent years, variational inequalities have been extended and generalized in different directions, using novel and innovative techniques. Useful and important generalizations of variational inequalities are variational inclusions. Recently, in w18, 19x, Noor et al. introduced and studied the following class of set-valued variational inclusion problems in a Hilbert space H. For a given maximal monotone mapping A : H ª H, a nonlinear mapping N Ž⭈ , ⭈ . : H = H ª H, set-valued mappings T, V : H ª C Ž H ., and a single-valued mapping g : H ª H, find u g H, w g T Ž u., y g V Ž u. such that
g N Ž w, y . q A Ž g Ž u . . ,
Ž 1.1.
where C Ž H . denotes the family of all nonempty compact subsets of H. 1 This paper was done while the author visited Korea and was supported by the Korean Science and Engineering Foundation and National Natural Science Foundation of China.
438 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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VARIATIONAL INCLUSIONS IN BANACH SPACES
For a suitable choice of the mappings T, V, g, A, N, a number of known and new variational inequalities, variational inclusions, and related optimization problems introduced and studied by Noor w19᎐24x can be obtained from Ž1.1.. Inspired and motivated by the results in Noor w18, 19x, the purpose of this paper is to introduce and study a class of more general set-valued variational inclusions without the compactness condition in Banach spaces. By using the famous Michael’s selection theorem w15x and Nadler’s theorem w17x an existence theorem and an approximate theorem for solving the set-valued variational inclusions in Banach spaces are established and suggested. The results presented in this paper generalize, improve, and unify the corresponding results of Noor w18᎐24x, Ding w7x, Huang w9, 10x, Kazmi w12x, Jung and Morales w11x, Hassouni and Moudafi w8x, Zeng w26x, and Chang et al. w3, 4x.
2. PRELIMINARIES Throughout this paper, we assume that E is a real Banach space, EU is the topological dual space of E, CB Ž E . is the family of all nonempty closed and bounded subsets of E, ² ⭈ , ⭈ : is the dual pair between E and EU , DŽ⭈ , ⭈ . is the Hausdorff metric on CB Ž E . defined by D Ž A, B . s max sup d Ž x, B . , sup d Ž A, y . ,
½
xgA
5
ygB
A, B g CB Ž E . .
U
DŽT . denotes the domain of T, and J : E ª 2 E is the normalized duality mapping defined by J Ž x . s f g EU : ² x, f : s 5 x 5 ⭈ 5 f 5 , 5 f 5 s 5 x 5 4 ,
x g E.
DEFINITION 2.1. Let A : DŽ A. ; E ª 2 E be a set-valued mapping and : w0, ⬁. ª w0, ⬁. be a strictly increasing function with Ž0. s 0. Ž1. The mapping A is said to be accretive if, for any x, y g DŽ A., there exists jŽ x y y . g J Ž x y y . such that
² u y ¨ , j Ž x y y .: G 0 for all u g Ax, ¨ g Ay. Ž2. The mapping A is said to be -strongly accretive if, for any x, y g DŽ A., there exists jŽ x y y . g J Ž x y y . such that for any u g Ax, ¨ g Ay,
² u y ¨ , j Ž x y y .: G Ž 5 x y y 5 . 5 x y y 5 .
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S. S. CHANG
Especially, if Ž t . s kt, 0 - k - 1, then the mapping A is said to be k-strongly accretive. Ž3. The mapping A is said to be m-accretive if A is accretive and Ž I q D A.Ž DŽ A.. s E for all D ) 0, where I is the identity mapping. Ž4. The mapping A is said to be -expansive if for any x, y g DŽ A. and for any u g Ax, ¨ g Ay, 5 u y ¨ 5 G Ž 5 x y y 5. . Remark. It is easy to see that if A is -strongly accretive, then A is -expansive. DEFINITION 2.2. Let T, F : E ª CB Ž E . be two set-valued mappings, A : DŽ A. ; E ª 2 E an m-accretive mapping, g : E ª DŽ A. a single-valued mapping, and N Ž⭈ , ⭈ . : E = E ª E a nonlinear mapping. For any given f g E and ) 0 we consider the following problem. Find q g E, w g T Ž q ., ¨ g F Ž q . such that f g N Ž w, ¨ . q A Ž g Ž q . . .
Ž 2.1.
This problem is called the set-valued variational inclusion problem in Banach spaces. A number of problems arising in pure and applied sciences can be reduced to study this kind of variational inclusion problem Žsee, for example, w6, 18, 19, 25x.. Next we consider some special cases of problem Ž2.1.. Ž1. If E s H is a Hilbert space and A : DŽ A. s H ª H is a maximal monotone mapping, then the problem Ž2.1. is equivalent to finding q g H, w g Tq, ¨ g F Ž q . such that f g N Ž w, ¨ . q A Ž g Ž q . . .
Ž 2.2.
This problem was introduced and studied in Noor w18x and Noor et al. w19x under some additional conditions. Ž2. If g ' I, then the problem Ž2.1. is equivalent to finding q g DŽ A., w g Tq, ¨ g F Ž q . such that f g N Ž w, ¨ . q A Ž q . .
Ž 2.3.
This problem seems to be a new one. We will study this in another paper. Ž3. If g s I, F s 0, T s I, S : E ª E is a single-valued mapping and N Ž x, y . s Sx for all Ž x, y . g E = E, then the problem Ž2.1. is equivalent to finding q g DŽ A. such that f g Sq q Aq. This problem was introduced and studied in Jung and Morales w11x.
Ž 2.4.
VARIATIONAL INCLUSIONS IN BANACH SPACES
441
Ž4. If E s H is a Hilbert space, s 1, and A s ⭸ , the subdifferential of a proper convex lower semi-continuous functional : H ª ⺢ j q⬁4 , then the problem Ž2.1. is equivalent to finding q g H, w g Tq, ¨ g F Ž q . such that
² N Ž w, ¨ . y f , x y g Ž q .: G Ž g Ž q . . y Ž x . ,
for all x g H. Ž 2.5.
This problem is called the generalized set-valued mixed variational inequality, which was introduced and studied by Noor et al. w19x. Recently, this problem with N being some special case was also considered in the setting of Banach spaces Žsee Chang w3x.. Ž5. If H is a Hilbert space, T, F, M : H ª 2 H are three set-valued mappings, and m, S, G : H ª H are three single-valued mappings, K Ž z . s mŽ z . q K, where K is a closed convex subset of E, N Ž x, y . s Sx q Gy and
Ž x . s IK Ž z . Ž x . s
½
0, q⬁,
if x g K Ž z . , if x f K Ž z . ,
then the problem Ž2.5. is equivalent to finding q g H, w g T Ž q ., ¨ g F Ž q ., z g M Ž q . such that g Ž q. g K Ž q. ,
² Sw q G¨ y f , x y g Ž q .: G 0, x g K Ž z . . Ž 2.6.
This problem is called the generalized strongly nonlinear implicit quasivariational inequality, which was studied in Huang w10x. Summing up the above arguments, it shows that for a suitable choice of the mapping T, F, A, g, N and the space E, one can obtain a number of known and new classes of variational inequalities, variational inclusions, and the corresponding optimization problems from the set-valued variational inclusions problem Ž2.1.. Furthermore, these types of variational inclusions enable us to study many important problems arising in mathematical, physical, and engineering sciences in a general and unified framework. DEFINITION 2.3. Let T, F : E ª 2 E be two set-valued mappings, N Ž⭈ , ⭈ . : E = E ª E a nonlinear mappings, and : w0, ⬁. ª w0, ⬁. a strictly increasing function with Ž0. s 0.
442
S. S. CHANG
Ž1. The mapping x ¬ N Ž x, y . is said to be -strongly accretive with respect to the mapping T, if for any x 1 , x 2 g E there exists jŽ x 1 y x 2 . g J Ž x 1 y x 2 . such that for any u1 g Tx 1 , u 2 g Tx 2
² N Ž u1 , y . y N Ž u 2 , y . , j Ž x 1 y x 2 .: G Ž 5 x 1 y x 2 5 . 5 x 1 y x 2 5 for all y g E. Ž2. The mapping y ¬ N Ž x, y . is said to be accretive with respect to the mapping F, if for any y 1 , y 2 g E there exists jŽ y 1 y y 2 . g J Ž y 1 y y 2 . such that for any ¨ 1 g Ty 1 , ¨ 2 g Ty 2
² N Ž x, ¨ 1 . y N Ž x, ¨ 2 . , j Ž y1 y y 2 .: G 0 for all x g E. DEFINITION 2.4. Let T : E ª CB Ž E . be a set-valued mapping and DŽ⭈ , ⭈ . be the Hausdorff metric on CB Ž E .. T is said to be -Lipschitzian continuous if, for any x, y g E, D Ž Tx, Ty . F 5 x y y 5 , where ) 0 is a constant. The following lemmas play an important role in proving our main results. U
LEMMA 2.1 w1, 2x. Let E be a real Banach space and J : E ª 2 E be the normalized duality mapping. Then for any x, y g E, the following inequality holds, 5 x q y 5 2 F 5 x 5 2 q 2² y, j Ž x q y .: , for all jŽ x q y . g J Ž x q y .. LEMMA 2.2. Let E be a real smooth Banach space, T, F : E ª 2 E two set-¨ alued mappings, and N Ž⭈ , ⭈ . : E = E ª E a nonlinear mapping satisfying the following conditions: Ž1. the mapping x ¬ N Ž x, y . is -strongly accreti¨ e with respect to the mapping T ; Ž2. the mapping y ¬ N Ž x, y . is accreti¨ e with respect to the mapping F. Then the mapping S : E ª 2 E defined by Sx s N Ž Tx , Fx . is -strongly accreti¨ e.
VARIATIONAL INCLUSIONS IN BANACH SPACES
443 U
Proof. Since E is smooth, the normalized duality mapping J : E ª 2 E is single-valued. For any given x 1 , x 2 g E and for any u i g Sx i , i s 1, 2, there exists wi g Tx i and ¨ i g Fx i such that u i s N Ž wi , ¨ i ., i s 1, 2. By conditions Ž1. and Ž2. we have
² u1 y u 2 , J Ž x 1 y x 2 .: s² N Ž w1 , ¨ 1 . y N Ž w 2 , ¨ 2 . , J Ž x 1 y x 2 .: s² N Ž w 1 , ¨ 1 . y N Ž w 2 , ¨ 1 . , J Ž x 1 y x 2 .: q² N Ž w 2 , ¨ 1 . y N Ž w 2 , ¨ 2 . , J Ž x 1 y x 2 .: G Ž 5 x1 y x 2 5. 5 x1 y x 2 5 . This implies that the mapping S s N ŽT Ž⭈., F Ž⭈.. is -strongly accretive. LEMMA 2.3 ŽMichael w15x.. Let X and Y be two Banach spaces, T : X ª 2 Y a lower semi-continuous mapping with nonempty closed con¨ ex ¨ alues. Then T admits a continuous selection; i.e., there exists a continuous mapping h : X ª Y such that hŽ x . g Tx for each x g X. LEMMA 2.4. Let E be a real uniformly smooth Banach space and T : E ª 2 E be a lower semi-continuous m-accreti¨ e mapping. Then the following conclusions hold. Ž1. T admits a continuous and m-accreti¨ e selection; Ž2. In addition, if T is also -strongly accreti¨ e, then T admits a continuous, m-accreti¨ e and -strongly accreti¨ e selection. Proof. Ž1. It is well known that if E is a uniformly smooth Banach space and T : E ª 2 E is a m-accretive mapping, then for each x g E, T Ž x . is nonempty closed and convex Žsee, for example, Deimling w5, p. 293x.. By Lemma 2.3, T admits a continuous selection h : E ª E such that hŽ x . g T Ž x . for all x g E. Next we prove that h : E ª E is m-accretive. In fact, since T : E ª 2 E is accretive, for any x, y g E and for any u g Tx, ¨ g Ty,
² u y ¨ , J Ž x y y .: G 0. In particular, letting u s hŽ x . g Tx, ¨ s hŽ y . g Ty, we have
² h Ž x . y h Ž y . , J Ž x y y .: G 0. This implies that h : E ª E is a continuous accretive mapping. By a well-known result due to Martin w14x, h is a continuous m-accretive selection.
444
S. S. CHANG
Ž2. In addition, if T is also -strongly accretive, then the selection h : E ª E given in Ž1. is also -strongly accretive. In fact, for any x, y g E and for any u g Tx, ¨ g Ty, we have
² u y ¨ , J Ž x y y .: G Ž 5 x y y 5 . 5 x y y 5 .
Ž 2.7.
Letting u s hŽ x . g Tx, ¨ s hŽ y . g Ty, from Ž2.7. we have
² h Ž x . y h Ž y . , J Ž x y y .: G Ž 5 x y y 5 . 5 x y y 5 . This implies that h is -strongly accretive. This completes the proof of Lemma 2.4. LEMMA 2.5 ŽNadler w17x.. Let E be a complete metric space, T : E ª CB Ž E . be a set-¨ alued mapping. Then for any gi¨ en ⑀ ) 0 and for any gi¨ en x, y g E, u g Tx, there exists ¨ g Ty such that d Ž u, ¨ . F Ž 1 q ⑀ . D Ž Tx, Ty . . LEMMA 2.6 w13x. Let E be a uniformly smooth Banach space and A : DŽ A. ; E ª 2 E be an m-accreti¨ e and -expansi¨ e mapping, where : w0, ⬁. ª w0, ⬁. is a strictly increasing function with Ž0. s 0. Then A is surjecti¨ e. We now invoke Michael’s selection theorem and Nadler’s theorem to suggest the following algorithms for solving the set-valued variational inclusion Ž2.1.. Let ␣ n4 , n4 be two sequences in w0, 1x, f g E be any given point, and ) 0 be any given positive number. For given x 0 g E, u 0 g T Ž x 0 ., z 0 g F Ž x 0 ., take y0 g Ž 1 y  0 . x 0 q  0 Ž f q x 0 y N Ž u0 , z0 . y AŽ g Ž x 0 . . and w 0 g Ty 0 ,
¨ 0 g Fy 0 .
Define x1 g Ž 1 y ␣ 0 . x 0 q ␣ 0 Ž f q y0 y N Ž w0 , ¨ 0 . y AŽ g Ž y0 . . . . Since u 0 g Tx 0 , z 0 g Fx 0 , by Nadler’s theorem, there exist u1 g Tx 1 , z1 g Fx 1 such that 5 u 0 y u1 5 F Ž 1 q 1 . D Ž T Ž x 0 . , T Ž x 1 . . , 5 z 0 y z1 5 F Ž 1 q 1 . D Ž F Ž x 0 . , F Ž x 1 . . .
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VARIATIONAL INCLUSIONS IN BANACH SPACES
For x 1 g E, u1 g Tx 1 , z1 g Fx 1 , define y 1 g Ž 1 y  1 . x 1 q  1 Ž f q x 1 y N Ž u1 , z 1 . y A Ž g Ž x 1 . . . . Since w 0 g Ty 0 , ¨ 0 g Fy 0 , again by Nadler’s theorem, there exist w 1 g Ty 1 , ¨ 1 g Fy 1 such that 5 w 0 y w1 5 F Ž 1 q 1. D Ž T Ž y 0 . , T Ž y1 . . , 5 ¨ 0 y ¨ 1 5 F Ž 1 q 1. D Ž F Ž y 0 . , F Ž y1 . . . Define x 2 g Ž 1 y ␣ 1 . x 1 q ␣ 1 Ž f q y1 y N Ž w1 , ¨ 1 . y AŽ g Ž y1 . . . . Continuing in this way, we can obtain the following algorithms: ALGORITHM 2.1. For any given x 0 g E, u 0 g T Ž x 0 ., and z 0 g F Ž x 0 ., compute the sequences x n4 , yn4 , u n4 , z n4 , wn4 , and ¨ n4 by the iterative schemes such that
Ž i.
x nq 1 g Ž 1 y ␣ n . x n q ␣ n Ž f q yn y N Ž wn , ¨ n . y Ag Ž yn . . ,
Ž ii .
yn g Ž 1 y n . x n q n Ž f q x n y N Ž u n , z n . y Ag Ž x n . . ,
Ž iii .
u n g Tx n , 5 u n y u nq1 5 F 1 q
Ž iv.
z n g Fx n , 5 z n y z nq1 5 F 1 q
Ž v.
wn g T Ž yn . , 5 wn y wnq1 5 F 1 q
Ž vi.
¨ n g F Ž yn . , 5 ¨ n y ¨ nq1 5 F 1 q
ž
ž
1 nq1 1
ž
ž
/
nq1
/
D Ž Tx n , Tx nq1 . ,
D Ž Fx n , Fx nq1 . ,
1 nq1 1
nq1
/
/
Ž 2.8.
D Ž Tyn , Tynq1 . ,
D Ž Fyn , Fynq1 . ,
n s 0, 1, 2, . . . . The sequence x n4 defined by Ž2.8., in the sequel, is called the Ishikawa iterative sequence. In Algorithm 2.1, if n s 0 for all n G 0, then yn s x n . Take wn s u n , z n s ¨ n for all n G 0 and we obtain the following
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S. S. CHANG
ALGORITHM 2.2. For any given x 0 g E, w 0 g Tx 0 , ¨ 0 g Fx 0 compute the sequences x n4 , wn4 , ¨ n4 by iterative schemes such that x nq 1 g Ž 1 y ␣ n . x n q ␣ n Ž f q x n y N Ž wn , ¨ n . y Ag Ž x n . . , wn g Tx n , ¨ n g Fx n ,
5 wn y wnq1 5 F 1 q
ž
5 ¨ n y ¨ nq1 5 F 1 q
ž
1 nq1 1
nq1
/
/
D Ž Tx n , Tx nq1 . ,
Ž 2.9.
D Ž Fx n , Fx nq1 .
n G 0. The sequence x n4 defined by Ž2.9., in the sequel, is called the Mann iterative sequence.
3. EXISTENCE THEOREM OF SOLUTIONS FOR SET-VALUED VARIATIONAL INCLUSION In this section, we shall establish an existence theorem of solutions for set-valued variational inclusion Ž2.1.. We have the following result. THEOREM 3.1. Let E be a real uniformly smooth Banach space, T, F : E ª CB Ž E . and A : DŽ A. ; E ª 2 E three set-¨ alued mappings, g : E ª DŽ A. a single-¨ alued mapping, and N Ž⭈ , ⭈ . : E = E ª E a single-¨ alued continuous mapping satisfying the following conditions: Ži. A( g : E ª 2 E is m-accreti¨ e; Žii. T : E ª CB Ž E . is -Lipschitzian continuous; Žiii. F : E ª CB Ž E . is -Lipschitzian continuous, where and are positi¨ e constants; Živ. the mapping x ¬ N Ž x, y . is -strongly accreti¨ e with respect to the mapping T, where : w0, ⬁. ª w0, ⬁. is a strictly increasing function with Ž0. s 0; Žv. the mapping y ¬ N Ž x, y . is accreti¨ e with respect to the mapping F. Then for any gi¨ en f g E, ) 0, there exist q g E, w g T Ž q ., ¨ g F Ž q . which is a solution of the set-¨ alued ¨ ariational inclusion Ž2.1.. Proof. It follows from conditions Živ., Žv., and Lemma 2.2 that the mapping S : E ª 2 E defined by Sx s N Ž Tx, Fx . ,
xgE
VARIATIONAL INCLUSIONS IN BANACH SPACES
447
is -strongly accretive. Since N Ž⭈ , ⭈ . is continuous, T and F both are Lipschitzian continuous, and so S is continuous and accretive. By Morales w16x, S is m-accretive and -strongly accretive. By Lemma 2.4Ž2., S admits a continuous, -strongly accretive and m-accretive selection h : E ª E such that h Ž x . g S Ž x . s N Ž Tx, Fx .
for all x g E.
Next we consider the variational inclusion f g h Ž x . q AŽ g Ž x . . ,
) 0.
Ž 3.1.
By the assumption A( g is m-accretive and h : E ª E is continuous and -strongly accretive. Hence by Kobayashi w13, Theorem 5.3x, h q A( g is m-accretive and -strongly accretive. Therefore it is also an m-accretive and -expansive mapping. By Lemma 2.6, h q A( g : E ª 2 E is surjective. Therefore for given f and ) 0, there exists a unique q g E such that f g h Ž q . q AŽ g Ž q . . ; N Ž T Ž q . , F Ž q . . q AŽ g Ž q . . . ŽThe uniqueness of q is a direct consequence of the -strongly accretiveness of h q A( g.. Consequently, there exist w g Tq and ¨ g Fq such that f g N Ž w, ¨ . q Ag Ž q . . . This completes the proof.
4. APPROXIMATE PROBLEM OF SOLUTIONS FOR SET-VALUED VARIATIONAL INCLUSION In Theorem 3.1, under some conditions, we have proved that there exist q g E, w g Tq, and ¨ g Fq which is a solution of set-valued variational inclusion Ž2.1.. In this section we shall study the approximate problem of solutions for variational inclusion Ž2.1.. We have the following result: THEOREM 4.1. Let E, T, F, A, g, N be as in Theorem 3.1. Let ␣ n4 , n4 be two sequences in w0, 1x satisfying the following conditions: Ži. ␣ n ª 0; n ª 0; Žii. Ý⬁ns 0 ␣ n s ⬁.
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S. S. CHANG
If the ranges RŽ I y N ŽT Ž⭈., F Ž⭈... and RŽ A( g . both are bounded, then for any gi¨ en x 0 g E, u 0 g Tx 0 , z 0 g Fx 0 the iterati¨ e sequences x n4 , wn4 , and ¨ n4 defined by Ž2.8. con¨ erge strongly to the solution q, w, ¨ of set-¨ alued ¨ ariational inclusion Ž2.1. which is gi¨ en in Theorem 3.1, respecti¨ ely. Proof. In Ži. and Žii. of Ž2.8. choose h n g Ag Ž x n ., k n g Ag Ž yn . such that x nq 1 s Ž 1 y ␣ n . x n q ␣ n Ž f q yn y N Ž wn , ¨ n . y k n . , yn s Ž 1 y  n . x n q  n Ž f q x n y N Ž u n , z n . y h n . .
Ž 4.1.
Let pn [ f q yn y N Ž wn , ¨ n . y k n , rn [ f q x n y N Ž u n , z n . y h n . Then Ž4.1. can be rewritten as x nq 1 s Ž 1 y ␣ n . x n q ␣ n pn , yn s Ž 1 y  n . x n q  n r n .
Ž 4.2.
Since the ranges RŽ I y N ŽT Ž⭈., F Ž⭈..., RŽ A( g . are bounded, let M s sup 5 w y q 5 : w g Ž f q x y N Ž Tx, Fx . y Ag Ž x . . , x g E 4 q5 x 0 y q 5 - ⬁. This implies that 5 pn y q 5 F M, 5 rn y q 5 F M,
for all n G 0.
Since 5 x 0 y q 5 F M, we have 5 y0 y q 5 s Ž 1 y 0 . Ž x 0 y q . q 0 Ž r0 y q . F Ž 1 y 0 . 5 x 0 y q 5 q 0 5 r0 y q 5 F M. This implies that 5 x1 y q 5 F Ž 1 y ␣ 0 . 5 x 0 y q 5 q ␣ 0 5 p0 y q 5 F M ; 5 y 1 y q 5 F Ž 1 y  1 . 5 x 1 y q 5 q  1 5 r 1 y q 5 F M.
Ž 4.3.
VARIATIONAL INCLUSIONS IN BANACH SPACES
449
By induction we can prove that 5 x n y q 5 F M,
Ž 4.4.
5 yn y q 5 F M. On the other hand, by Ž4.3., Ž4.4., and Lemma 2.1 we have 5 x nq 1 y q 5 2 s Ž 1 y ␣ n . Ž x n y q . q ␣ n Ž pn y q .
2
2 F Ž 1 y ␣n . 5 xn y q 5 2
q 2 ␣ n² pn y q, J Ž x nq1 y q .: 2 s Ž 1 y ␣ n . 5 x n y q 5 2 q 2 ␣ n² pn y q, J Ž yn y q .:
q 2 ␣ n² pn y q, J Ž x nq1 y q . y J Ž yn y q .: .
Ž 4.5.
Now we consider the third term on the right side of Ž4.5.. Since
Ž x nq 1 y q . y Ž yn y q . s 5 x nq 1 y yn 5 s Ž 1 y ␣ n . Ž x n y yn . q ␣ n Ž pn y yn . F Ž 1 y ␣ n . n 5 x n y rn 5 q ␣ n 5 pn y q 5 q 5 yn y q 5 4 F Ž 1 y ␣ n . n 5 x n y q 5 q 5 rn y q 5 4 q ␣ n 5 pn y q 5 q 5 yn y q 5 4 F 2 Ž Ž 1 y ␣ n . n q ␣ n . ⭈ M ª 0, U
by the uniform continuity of J : E ª 2 E , we have J Ž x nq 1 y q . y J Ž yn y q . ª 0
Ž n ª ⬁. .
Since pn y q4 is bounded, this implies that
␦n [ ² pn y q, J Ž x nq1 y q . y J Ž yn y q .: ª 0
Ž n ª ⬁ . . Ž 4.6.
Next we consider the second term on the right side of Ž4.5.. Since wn g Tyn , ¨ n g Fyn , k n g Ag Ž yn ., this implies that N Ž wn , ¨ n . q k n g N Ž T Ž ⭈ . , F Ž ⭈ . . q Ag Ž ⭈ . Ž yn . . Again since q g E is a solution of the variational inclusion, f g h Ž q . q Ag Ž q . ; N Ž T Ž ⭈ . , F Ž ⭈ . . q Ag Ž ⭈ . Ž q . .
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S. S. CHANG
This shows that f is a point of w N ŽT Ž⭈., F Ž⭈.. q Ag Ž⭈.xŽ q .. By the assumptions of Theorem 4.1, N ŽT Ž⭈., F Ž⭈.. q Ag Ž⭈. : E ª 2 E is -strongly accretive, hence we have
² f y Ž N Ž wn , ¨ n . q k n . , J Ž yN y q .: s y² N Ž wn , ¨ n . q k n y f , J Ž yn y q .: F y Ž 5 yn y q 5 . 5 yn y q 5 . Therefore we have 2 ␣ n² pn y q, J Ž yn y q .: s2 ␣ n² f q yn y N Ž wn , ¨ n . y k n y q, J Ž yn y q .: s 2 ␣ n² yn y q, J Ž yn y q .: y² N Ž wn , ¨ n . q k n y f , J Ž yn y q .: 4 F 2 ␣ n 5 yn y q 5 2 y Ž 5 yn y q 5 . 5 yn y q 5 4 .
Ž 4.7.
Substituting Ž4.6. and Ž4.7. into Ž4.5. we have 5 x nq 1 y q 5 2 F Ž 1 y ␣ n . 2 5 x n y q 5 2 q 2 ␣ n 5 yn y q 5 2 y Ž 5 yn y q 5 . 5 yn y q 5 4 q 2 ␣ n ␦ n .
Ž 4.8. Next we make an estimation for 5 yn y q 5 2 . We have 5 yn y q 5 2 s Ž 1 y  n . Ž x n y q . q  n Ž r n y q .
2
2 F Ž 1 y n . 5 x n y q 5 2 q 2 n² rn y q, J Ž yn y q .: 2 F Ž 1 y  n . 5 x n y q 5 2 q 2  n 5 r n y q 5 ⭈ 5 yn y q 5
F 5 x n y q 5 2 q 2 n M 2 .
Ž 4.9.
Substituting Ž4.9. into Ž4.8. and simplifying, we have 5 x nq 1 y q 5 2 F Ž 1 y ␣ n . 2 5 x n y q 5 2 q 2 ␣ n 5 x n y q 5 2 q 2 n M 2 4 y 2 ␣ n Ž 5 yn y q 5 . 5 yn y q 5 q 2 ␣ n ␦ n s 5 x n y q 5 2 y ␣ n Ž 5 yn y q 5 . 5 yn y q 5 y ␣ n Ž 5 yn y q 5 . 5 yn y q 5 y4n M 2 y ␣ n 5 x n y q 5 2 y 2 ␦n 4 .
Ž 4.10.
VARIATIONAL INCLUSIONS IN BANACH SPACES
451
Let
s inf nG 0 5 yn y q 5 . Next we prove that s 0. Suppose the contrary, ) 0. Then we have 5 yn y q 5 G ) 0 for all n G 0. It follows from Ž4.10. that 5 x nq 1 y q 5 2 F 5 x n y q 5 2 y ␣ n Ž . 4 y ␣ n Ž . y 4n M 2 y ␣ n M 2 y 2 ␦n 4 . Ž 4.11. Since n ª 0, ␣ n ª 0, and ␦n ª 0, there exists n 0 such that for n G n 0
Ž . ⭈ y 4n M 2 y ␣ n M 2 y 2 ␦n ) 0. Therefore from Ž4.11. we have 5 x nq 1 y q 5 2 F 5 x n y q 5 2 y ␣ n Ž . 4
for all n G n 0 ,
␣ n Ž . 4 F 5 x n y q 5 2 y 5 x nq1 y q 5 2
for all n G n 0 .
i.e.,
Hence for any m G n 0 we have m
Ý
␣ n Ž . 4 F 5 x n 0 y q 5 2 y 5 x mq1 y q 5 2
nsn 0
F 5 x n0 y q 5 2 . Letting m ª ⬁, we have ⬁s
⬁
Ý
␣ n Ž . 4 F 5 x n 0 y q 5 2 .
nsn 0
This is a contradiction. Therefore s 0, and so there exists a subsequence yn 4 ; yn4 such that yn ª q, i.e., j j yn j s Ž 1 y n j . x n j q n j rn j ª q. Since n ª 0, ␣ n ª 0, and rn4 , pn4 both are bounded, this implies that x n j ª q, and so x n jq1 s Ž 1 y ␣ n j . x n j q ␣ n j pn j ª q.
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S. S. CHANG
Hence yn jq1 s Ž 1 y n jq1 . x n jq1 q ␣ n jq1 rn jq1 ª q. By induction, we can prove that x n jq1 ª q
yn jqi ª q
and
for all i G 0.
This implies that xn ª q
yn ª q.
and
Since T is -Lipschitzian and F is -Lipschitzian, it follows from Žiii. and Živ. in Ž2.8. that 5 u n y u nq1 5 F 1 q
ž
1 nq1
F 1q
ž
/
D Ž Tx n , Tx nq1 .
1 nq1
/
5 x n y x nq1 5 ,
and 5 z n y z nq1 5 F 1 q
ž
1 nq1
F 1q
ž
/
1 nq1
D Ž Fx n , Fx nq1 .
/
5 x n y x nq1 5 .
This implies that the sequences u n4 , z n4 both are Cauchy sequences. By the same way we know that wn4 , ¨ n4 are also Cauchy sequences. Therefore there exist uU , wU , ¨ U , zU g E such that u n ª uU wn ª wU ¨n ª ¨U z n ª zU
Ž n ª ⬁. .
Next we prove that uU s wU s w ;
zU s ¨ U s ¨ .
VARIATIONAL INCLUSIONS IN BANACH SPACES
453
In fact, since d Ž wU , Tq . F d Ž wU , wn . q d Ž wn , Tq . F d Ž wU , wn . q D Ž Tyn , Tq . F d Ž wU , wn . q 5 yn y q 5 ª 0, this implies that wU g Tq. Similarly we can prove that uU g Tq. Again since d Ž zU , Fq . F d Ž zU , z n . q d Ž z n , Fq . F d Ž zU , z n . q D Ž Fx n , Fq . F d Ž zU , z n . q 5 x n y q 5 ª 0, this implies that zU g Fq. Similarly we can prove that ¨ U g Fq. Next we prove that wU s uU s w. In fact, we have d Ž wU , w . F d Ž wU , wn . q d Ž wn , w . F d Ž wU , wn . q D Ž Tyn , Tq . F d Ž wU , wn . q 5 yn y q 5 ª 0. This implies that wU s w. Again since d Ž uU , w . F d Ž uU , u n . q d Ž u n , w . F d Ž uU , u n . q D Ž Tx n , Tq . F d Ž uU , u n . q 5 x n y q 5 ª 0, this implies that uU s w. Similarly we can prove that zU s ¨ U s ¨ . Summing up the above arguments we know that the sequences x n4 , wn4 , and ¨ n4 defined by Ž2.8. converge strongly to the solution Ž q, w, ¨ . of Ž2.1., respectively. This completes the proof.
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3. Shih-sen Chang, On the Mann and Ishikawa iterative approximation of solutions to variational inclusions with accretive type mappings, Comput. Math. Appl. 37 Ž1999., 17᎐24. 4. Shih-sen Chang, X. Z. Yuan, F. Wang, and X. Long, The study of algorithms and convergence for generalized multi-valued quasi-variational inclusions, Topol. Methods Nonlinear Anal., to appear. 5. K. Deimling, ‘‘Nonlinear Functional Analysis,’’ Springer-Verlag, Berlin, 1985. 6. V. F. Demyanov, G. E. Stavroulakis, L. N. Polyakova, and P. D. Panagiotopoulos, ‘‘Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics,’’ Kluwer Academic, DordrechtrNorwell, MA, 1996. 7. X. P. Ding, Iterative process with errors of nonlinear equations involving m-accretive operators, J. Math. Anal. Appl. 209 Ž1997., 191᎐201. 8. A. Hassouni and A. Moudafi, A perturbed algorithm for variational inclusions, J. Math. Anal. Appl. 185 Ž1994., 706᎐712. 9. N. J. Huang, On the generalized implicit quasi-variational inequalities, J. Math. Anal. Appl. 216 Ž1997., 197᎐210. 10. N. J. Huang, Generalized nonlinear variational inclusions with noncompact valued mappings, Appl. Math. Lett. 9, No. 3 Ž1996., 25᎐29. 11. J. S. Jung and C. H. Morales, The Mann process for perturbed m-accretive operators in Banach spaces, submitted for publication. 12. K. R. Kazmi, Mann and Ishikawa type perturbed iterative algorithms for generalized quasi-variational inclusions, J. Math. Anal. Appl. 209 Ž1997., 572᎐584. 13. Y. Kobayashi, Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups, J. Math. Soc. Japan 27 Ž1975., 640᎐665. 14. R. H. Martin, Jr., A global existence theorem for autonomous differential equations in Banach spaces, Proc. Amer. Math. Soc. 26 Ž1970., 307᎐314. 15. E. Michael, Continuous solutions, I, Ann. Math. 63 Ž1956., 361᎐382. 16. C. Morales, Surjectivity theorems for multi-valued mappings of accretive type, Comment. Math. Uni¨ . Carolin. 26 Ž1985., 397᎐413. 17. S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30 Ž1969., 475᎐488. 18. M. A. Noor, Generalized set-valued variational inclusions and resolvent equations, J. Math. Anal. Appl. 228 Ž1998., 206᎐220. 19. M. A. Noor, K. I. Noor, and Th. M. Rassias, Set-valued resolvent equations and mixed variational inequalities, J. Math. Anal. Appl. 220 Ž1998., 741᎐759. 20. M. A. Noor, Auxiliary principle for generalized mixed variational-like inequalities, J. Math. Anal. Appl. 215 Ž1997., 75᎐85. 21. M. A. Noor, Numerical methods for monotone mixed variational inequalities, Ad¨ . Nonlinear Variat. Inequal. 1 Ž1998., 51᎐79. 22. M. A. Noor, An implicit method for mixed variational inequalities, Appl. Math. Lett. 11, No. 4 Ž1998., 109᎐113. 23. M. A. Noor, Sensitivity analysis of quasi-variational inequalities, J. Optim. Theory Appl. 95, No. 2 Ž1997., 399᎐407. 24. M. A. Noor, Generalized multi-valued quasi-variational inequalities, II, Comput. Math. Appl. 35, No. 5 Ž1997., 63᎐78. 25. P. D. Panagiotoupoulos and G. E. Stavroulakis, New types of variational principles based on the notion of quasi-differentiability, Acta Mech. 94 Ž1992., 171᎐194. 26. L. C. Zeng, Iterative algorithm for finding approximate solutions to completely generalized strongly nonlinear quasi-variational inequality, J. Math. Anal. Appl. 201 Ž1996., 180᎐191.