Generalized System of Variational Inequalities in Banach Spaces

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1 Appl. Math. Inf. Sci. 8, No. 3, (2014) 985 Applied Mathematics & Information Sciences An International Journal Generalized System of Variational Ineualities in Banach Spaces Abdellah Bnouhachem 1,2,, Muhammad Aslam Noor 3, Mohamed Khalfaoui 4 Hafida Benazza 4 1 School of Management Science Engineering, Nanjing University, Nanjing, , P.R. China 2 Ibn Zohr University, ENSA, BP 1136, Agadir, Morocco 3 Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan 4 Ecole Supérieure de Technologie de Salé, Mohamed 5 University, Agdal-Rabat, Morocco Received: 9 May. 2013, Revised: 13 Sep. 2013, Accepted: 14 Sep Published online: 1 May Abstract: In this paper, we introduce consider a new system of extended general variational ineualities in Banach spaces. We establish the euivalence between the extended general variational ineualities the fixed point problems. We use this euivalent formulation to suggest some iterative methods for solving this new system. We prove the convergence analysis of the proposed iterative methods under some suitable conditions. Several special cases, which can be obtained from main results are discussed. The idea techniue of this paper may stimulate further research activities in these fields. Keywords: Sunny nonexpansive retraction, generalized variational ineualities system, - uniformly smooth Banach spaces, iterative algorithm, relaxed cocoercive mappings AMS Subject Classification: 49J40, 65N30 1 Introduction Variational ineuality complementarity problems are of fundamental importance in a wide range of mathematical applied sciences problems, such as mathematical programming, traffic engineering, economics euilibrium problems, see [1 18]. The ideas techniues of the variational ineualities are being applied in a variety of diverse areas of sciences proved to be productive innovative. It has been shown that this theory provides a simple, natural unified framework for a general treatment of unrelated problems. The fixed-point theory has played an important role in the development of various algorithms for solving variational ineualities. Using the projection operator techniue, one usually establishes an euivalence between the variational ineualities the fixed-point problem. This alternative euivalent formulation was used by Lions Stampacchia [6] to study the existence of a solution of the variational ineualities. Projection methods its variant forms represent important tools for finding the approximate solution of variational ineualities. We now have a variety of techniues to suggest analyze various iterative algorithms for solving variational ineualities the related optimization problems. In recent years, some new interesting problems, which are called the system of variational inclusions were introduced studied. Chang et al. [2], Huang Noor [5], Noor Noor [10], Noor [9], Verma [18,19,20] Yang et al. [22] introduced studied a system of variational inclusions involving four, three, two different nonlinear operators. Inspired motivated by research going on in this area, we introduce consider a new system of extended general variational ineualities involving six different nonlinear operators in Banach spaces. We establish the euivalence between this system of extended general variational ineualities the fixed point problems using the projection operator techniue. This euivalent formulation is used to suggest analyze some new iterative methods for solving the extended general variational ineualities. We also prove the convergence analysis of the proposed algorithm under some suitable mild conditions. Since this class of systems includes the system of variational ineualities involving four, three, Corresponding author babedallah@yahoo.com

2 986 A. Bnouhachem et. al. : Generalized System of Variational Ineualities... two operators the classical variational ineualities as special cases, results obtained in this paper continue to hold for these problems. It is expected that these results may inspire motivate others to find novel innovative applications in various branches of pure applied sciences. The readers are encouraged to explore the novel innovative applications of the system of extended general variational ineualities its variant forms in different areas of pure applied sciences. 2 Preliminaries Throughout this article, let X be a real Banach space with its dual space X. We usually use, to denote the pairing between X X, 2 X denote the family of all the nonempty subsets of X. The generalized duality mapping J (x) : X 2 X is defined by J (x)={ f X : x, f = x, f = x 1 }, where >1 is a constant. In particular, J 2 = J is the usual normalized duality mapping. It is known that, in general, J = x 2 J 2, for all x 0, J (x) is single-valued if X is strictly convex. If X = H is a Hilbert space then J 2 becomes the identity mapping of H. Note that X is a uniformly smooth Banach space if only if J is singlevalued uniformly continuous on any bounded subset of X. Let K be a nonempty, closed convex subset of X. A mapping Q : X K is said to be sunny if Q(Q(x)+t(x Q(x)))=Q(x), x X, t 0. A mapping Q : X K is said to be a retraction or a projection if Q(x) = x; x K. If X is smooth then the sunny nonexpansive retraction of X onto K is uniuely decided (see [4]). Using J = x 2 J 2, replace J by J, it is easy to obtain the following results by Bruck [1], Goebel Reich [3]. Proposition 2.1 Let X be a -uniformly smooth Banach spaces let K be a nonempty subset of X. Let Q K : X K be a retraction let J be the generalized duality mapping on X. Then the following are euivalent: (a) Q K is sunny nonexpansive. (b) Q K (x) Q K (y) 2 x y,j (Q K (x) Q K (y)), x,y X. (c) x Q K (x),j (Q K (x) y) 0, y K. Lemma 2.1 ( See [21]) Let X be a real -uniformly smooth Banach space (>1), then there exists a constant c > 0 such that x+y x + y,j x +c y, x,y X. In particular, if X be a real 2-uniformly smooth Banach space, then there exists a constant c 2 > 0 such that x+y 2 x y,j 2 x +c 2 y 2, x,y X. Let K be a nonempty, closed convex subset of X. For given nonlinear operators T i (.,.) : X X X g i,h i : X X(i=1,2), we consider problem of finding (x,y ) X X :(h 1 (x ),h 2 (y )) K K such that ρt 1 (y,x )+h 1 (x ) g 1 (y ), J (g 1 (x) h 1 (x )) 0; x X,g 1 (x) K ρ > 0, ηt 2 (x,y )+h 2 (y ) g 2 (x ), (1) J (g 2 (x) h 2 (y )) 0; x X,g 2 (x) K η > 0. which is called the system of extended general variational ineualities. We now discuss some applications of the system of extended general variational ineualities (1). If X = H is a Hilbert space, J = I,h i = I, the identity operator, then problem (1) reduces to finding (x,y ) K K such that ρt 1 (y,x )+x g 1 (y ), g 1 (x) x 0; x H,g 1 (x) K ρ > 0, (2) ηt 2 (x,y )+y g 2 (x ),g 2 (x) y x H,g 2 (x) K η > 0, which were introduced studied by Noor Noor [10]. If g i = g(i = 1,2), then the system (2) is euivalent to finding (x,y ) K K such that ρt 1 (y,x )+x g(y ),g(x) x x H,g(x) K ρ > 0, ηt 2 (x,y )+y g(x ),g(x) y (3) x H,g(x) K η > 0, which was considered investigated by Noor [9]. If X = H is a Hilbert space, J = I, T i (x,y) = T i (x), h i = g i = g(i=1,2), then problem (1) reduces to finding (x,y ) H H :(g(x ),g(y )) K K such that ρt 1 (y )+g(x ) g(y ),g(x) g(x ) x H,g(x) K ρ > 0, ηt 2 (x )+g(y ) g(x ),g(x) g(y (4) ) x H,g 2 (x) K η > 0, which has been introduced studied by Yang et al. [22]. If X = H is a Hilbert space, x = y,j = I,T i (x,y) = T 1 (x)=t 2 (x)(i=1,2),h 1 = h 2,g 1 = g 2, then problem (1) reduces to finding x H : h 1 (x ) K K such that ρt 1 (x )+h 1 (x ) g 1 (x ),g 1 (x) h 1 (x ) 0; (5) x H : g 1 (x) K ρ > 0, which is called the extended general variational ineuality, introduced studied by Noor [11] in For the applications, numerical results, generalizations other aspects of extended general variational ineualities, see [11,12,13,14] the references therein.

3 Appl. Math. Inf. Sci. 8, No. 3, (2014) / If g = I, then problem (3) reduces of finding (x,y ) K K such that ρt 1 (y,x )+x y,x x x K ρ > 0, ηt 2 (x,y )+y x,x y (6) x K η > 0, which has been considered studied by Huang Noor [5]. If T 1 = T 2 = T, then problem (6) reduces of finding (x,y ) K K such that ρt(y,x )+x y,x x x K ρ > 0, ηt(x,y )+y x,x y (7) x K η > 0, The system (7) has been studied investigated by Chang et al. [2] Verma [19]. If T 1 = T 2 = T, g = I, then problem (4) reduces of finding (x,y ) K K such that ρt(y )+x y,x x x K ρ > 0, ηt(x )+y x,x y (8) x K η > 0, which has been introduced studied by Verma [18,20]. If x = y, then problem (8) collapses to finding x K. such that T(x ),x x 0, x K. (9) Ineuality of type (9) is called variational ineuality, which was introduced studied by Stampacchia [17] in The system of extended general variational ineualities (1) includes several classes of variational ineualities related optimization problems as special cases. For the recent research, see [1-18] the references therein. We now recall some well-known results concepts, which are needed. Definition 2.1 A mapping h : X X is called (a) r-strongly monotone if, there exists a constant r > 0 such that h(x) h(y),j (x y) r x y 2, x,y X; (b) (ξ, ς)-relaxed cocoercive if, there exist constants ξ,ς > 0 such that h(x) h(y),j (x y) ξ h(x) h(y) 2 +ς x y 2, x,y X; (c) γ-lipschitz continuous if, there exists a constant γ > 0 such that h(x) h(y) γ x y, x,y X. Definition 2.2 Let T : X X X g : X X. Then T is called (a) r-strongly monotone with respect to g if, there exists a constant r>0 such that T(x 1,y) T(x 2,y),J (g(x 1 ) g(x 2 )) r x 1 x 2 2, x 1,x 2,y X; (b) (ξ,ς)-relaxed cocoercive with respect to g if, there exist constants ξ,ς > 0 such that T(x 1,y) T(x 2,y),J (g(x 1 ) g(x 2 )) ξ T(x 1,y) T(x 2,y) 2 + ς x 1 x 2 2, x 1,x 2,y X; (c) γ-lipschitz continuous in the first variable if, there exists a constant γ > 0 such that T(x 1,y) T(x 2,y) γ x 1 x 2, x 1,x 2,y X. (d) γ-lipschitz continuous in the second variable if, there exists a constant γ > 0 such that T(x,y 1 ) T(x,y 2 ) γ y 1 y 2, x,y 1,y 2 X. 3 Main Results In this section, we first establish the euivalence between the system of the extended general variational ineuality (1) the fixed point problems. Then by using the obtained fixed point formulation, we construct a new iterative algorithm for solving the systems (1). Lemma 3.1 Let X be an smooth Banach space, for given nonlinear operators T i : X X X,g i,h i : X X(i = 1,2), ρ,η > 0. Then the point (x,y ) K K is a solution of (1) if only if h 1 (x )=Q K [g 1 (y ) ρt 1 (y,x )] h 2 (x )=Q K [g 2 (x ) ηt 2 (x,y )]. } (10) Proof. The first variational ineuality of (1) can be written as follows: [g 1 (y ) ρt 1 (y,x )] h 1 (x ),J (h 1 (x ) g 1 (x)) 0; x X,g 1 (x) K ρ > 0. We can deduce from Proposition 2.1 (c) that the above ineuality is euivalent to h 1 (x )=Q K [g 1 (y ) ρt 1 (y,x )]. Similar, the second variational ineuality of (1) is euivalent to h 2 (x )=Q K [g 2 (x ) ηt 2 (x,y )]. Lemma 3.1 implies that the system of extended general variational ineualities (1) the fixed point problems

4 988 A. Bnouhachem et. al. : Generalized System of Variational Ineualities... (10) are euivalent. This euivalent formulation plays a crucial role in developing the iterative methods for solving (1). We rewrite (10) in the following form: h 1 (x)=q K [z] h 2 (y)=q K [t] z=(α)z+α[g 1 (y) ρt 1 (y,x)] t =(α)t+ α[g 2 (x) ηt 2 (x,y)], which enables to suggest the following iterative method, Algorithm 3.1 Let T i,g i,h i (i = 1,2), be nonlinear operators, ρ, η > 0. For arbitrary chosen initial points z 0,t 0 K, compute the iterative seuence {(x k,y k } by using h 1 (x k )=Q K [z k ] h 2 (y k )=Q K [t k ] z k+1 =(α)z k + α[g 1 (y k ) ρt 1 (y k,x k )] t k+1 =(α)t k + α[g 2 (x k ) ηt 2 (x k,y k )], (11) where Q K is the sunny nonexpansive retraction of X onto K 0<α 1. We now establish the strongly convergence of the seuences generated by Algorithm 3.1. Theorem 3.1 Let T i,g i,h i (i = 1,2), be nonlinear operators, such that for each i = 1, 2, (a)t i is (κ i,θ i )-relaxed cocoercive with respect to g i γ i -Lipschitz continuous in the first variable, µ i -Lipschitz continuous in the second variable; (b)h i is (ξ i,ν i )-relaxed cocoercive δ i -Lipschitz continuous; (c)g i is σ i -Lipschitz continuous. If there exist constants ρ,η > 0 such that (σ 2 ηθ 2)+(ηκ 2 +C η )γ 2 < k 1 ρµ 1,k 1 + ρµ 1 < 1 ((σ 1 ρθ 1)+(ρκ 1 +C ρ )γ 1 ) < k 2 ηµ 2,k 2 + ηµ 2 < 1, (12) ν i < 1+(C + ξ i )δ i,(i=1,2), where k i = (ν i )+(C + ξ i )δ i,(i=1,2), then for arbitrarily chosen initial points x 0,y 0 K, x k y k obtained from Algorithm 3.1 converge strongly to x y respectively. Proof. It follows from (11) that z k+1 z k = (α)z k + α[g 1 (y k ) ρt 1 (y k,x k )] (α)z k 1 α[g 1 (y k 1 ) ρt 1 (y k 1,x k 1 )] (α) z k z k 1 +α g 1 (y k ) g 1 (y k 1 ) ρ(t 1 (y k,x k ) T 1 (y k 1,x k 1 )) (13) From T 1 µ 1 -Lipschitz continuous in the second variable, we have g 1 (y k ) g 1 (y k 1 ) ρ(t 1 (y k,x k ) T 1 (y k 1,x k 1 )) g 1 (y k ) g 1 (y k 1 ) ρ(t 1 (y k,x k ) T 1 (y k 1,x k )) + ρ T 1 (y k 1,x k ) T 1 (y k 1,x k 1 )) g 1 (y k ) g 1 (y k 1 ) ρ(t 1 (y k,x k ) T 1 (y k 1,x k )) + ρµ 1 x k x k 1 (14) Since T 1 is (κ 1,θ 1 )-relaxed cocoercive with respect to g 1 γ 1 -Lipschitz continuous in the first variable, g 1 is σ 1 -Lipschitz continuous, we can conclude that g 1 (y k ) g 1 (y k 1 ) ρ(t 1 (y k,x k ) T 1 (y k 1,x k )) = g 1 (y k ) g 1 (y k 1 ) ρ T 1 (y k,x k ) T 1 (y k 1,x k ), J (g 1 (y k ) g 1 (y k 1 )) + C ρ T 1 (y k,x k ) T 1 (y k 1,x k ) g 1 (y k ) g 1 (y k 1 ) + ρκ 1 T 1 (y k,x k ) T 1 (y k 1,x k ) ρθ 1 y k y k 1 + C ρ T 1 (y k,x k ) T 1 (y k 1,x k ) ( (σ 1 ρθ 1)+(ρκ 1 +C ρ )γ 1) y k y k 1 (15) Substituting (14) (15) in (13), we get z k+1 z k (α) z k z k 1 +α ((σ 1 ρθ 1) + (ρκ 1 +C ρ )γ 1 ) yk y k 1 + αρµ 1 x k x k 1. (16) In a similar way, we can prove that t k+1 t k (α) t k t k 1 + α ((σ 2 ηθ 2)+(ηκ 2 +C η )γ 2) x k x k 1 + αηµ 2 y k y k 1. (17) On the other h, by using (11), we find that x k x k 1 x k x k 1 (h 1 (x k ) h 1 (x k 1 )) + a k (z k ) Q k (x k 1 ) x k x k 1 (h 1 (x k ) h 1 (x k 1 )) + z k z k 1. (18) From (ξ 1,ν 1 )-relaxed cocoercive δ 1 -Lipschitz continuous of h 1, we have x k x k 1 (h 1 (x k ) h 1 (x k 1 )) x k x k 1 h 1 (x k ) h 1 (x k 1 ),J (x k x k 1 ) +C h 1 (x k ) h 1 (x k 1 ) x k x k 1 + ξ 1 h 1 (x k ) h 1 (x k 1 ) ν 1 x k x k 1 +C h 1 (x k ) h 1 (x k 1 ) ( (ν 1 )+(C + ξ 1 )δ 1) x k x k 1 (19)

5 Appl. Math. Inf. Sci. 8, No. 3, (2014) / Substituting (19) in (18), we get x k x k 1 1 z (ν z k 1 1 )+(C + ξ 1 )δ 1 (20) Similarly, we can prove that y k y k 1 1 t (ν t k 1. 2 )+(C + ξ 2 )δ 2 (21) From (16),(17), (20) (21), it follows that z k+1 z k ρµ 1 (α( )) (ν 1 )+(C + ξ 1 )δ 1 z k z k 1 + α ((σ 1 ρθ 1)+(ρκ 1 +C ρ )γ 1 ) (ν 2 )+(C + ξ 2 )δ 2 t k t k 1 (22) t k+1 t k ηµ 2 (α( )) (ν 2 )+(C + ξ 2 )δ 2 t k t k 1 ((σ + α 2 ηθ 2)+(ηκ 2 +C η )γ ) 2 (ν 1 )+(C + ξ 1 )δ 1 z k z k 1 (23) Now, we define on X X by (x,y) = x + y, for all (x,y) X X. From (22) (23), it follows that (z k+1,t k+1 ) (z k,t k ) (α) (z k,t k ) (z k 1,t k 1 ) + αλ (z k,t k ) (z k 1,t k 1 ) (24) where ((σ Λ = max ρµ 1+ 2 ηθ 2)+(ηκ 2 +C η )γ ) 2, (ν 1 )+(C + ξ 1 )δ 1 ηµ 2 + ((σ 1 ρθ 1)+(ρκ 1 +C ρ )γ 1 ) (ν 2 )+(C + ξ 2 )δ 2 In view of the condition (12), we know that 0 Λ < 1. Thus it follows from (24) that, for each k k 0, (z k+1,t k+1 ) (z k,t k ) (α(λ)) (z k,t k ) (z k 1,t k 1 ) (α(λ)) 2 (z k 1,t k 1 ) (z k 2,t k 2 ). (α(λ)) k k 0 (z k 0+1,t k 0+1 ) (z k 0,t k 0 ). Hence, for any m n>k 0, we have (z m,t m ) (z n,t n ) m 1 (z j+1,t j+1 ) (z j,t j ) j=n m 1 (α(λ)) j k 0 (z k0+1,t k0+1 ) (z k 0,t k 0 ). (25) j=n Since 1 α(1 Λ) < 1, it follows from (25) that (z m,t m ) (z n,t n ) = z m z n + t m t n 0, as n. Hence {z n },{t n } are both Cauchy seuences in K so there exist z,t K such that z n z t n t as n. From (20) (21) it follows that the seuences {x n } {y n } are also both Cauchy Cauchy seuences. Thus there exist x,y K such that x n x y n y as n. Since the mappings g 1,g 2,h 1,h 2 Q K are continuous, it follows from (11) that h 1 (x )=Q K [g 1 (y ) ρt 1 (y,x )] h 2 (x )=Q K [g 2 (x ) ηt 2 (x,y )]. Now, Lemma 3.1 guarantees that (x,y ) is a solution of the problem (1). This completes the proof. Theorem 3.2 Let T i,g i,h i (i=1,2) be nonlinear operators, such that for each i = 1, 2, (a)t i is θ i -strongly monotone with respect to g i γ i -Lipschitz continuous in the first variable, µ i -Lipschitz continuous in the second variable; (b)h i is ν i -strongly monotone δ i -Lipschitz continuous; (c)g i is σ i -Lipschitz continuous. If there exist constants ρ,η > 0 such that (σ 2 ηθ 2)+C η γ 2 < k ρµ 1, k 1 + ρµ 1 < 1 ((σ 1 ρθ 1)+C ρ γ 1 )<k 2 ηµ 2, k 2 + ηµ 2 < 1, ν i < 1+C δ i,(i=1,2), where k i = (ν i C δ i ),(i=1,2), then for arbitrarily chosen initial points x 0,y 0 K, x k y k obtained from Algorithm 3.1 converge strongly to x y respectively. 4 Conclusion A new system of extended general variational ineualities in Banach spaces is introduced studied. It is shown that the new system is euivalent to the system of fixed point problems. This alternative euivalent formulation is used to suggest an iterative method for solving system along with convergence criteria. Some special cases are studied. Results proved in this paper continue for these known new systems. The implementation comparison of these methods with other methods is a subject of the future research.

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7 Appl. Math. Inf. Sci. 8, No. 3, (2014) / Hafida Benazza graduate from Mohammed V-Agdal University. She obtained her Ph.D. degree in the Automatic Industrial Informatics (optional: Optimal Control) at Mohammadia School of Engineers in Currently, she is a Professor at High School of Technology in Sal, Mohammed V-Agdal University, Morocco.