A Viscosity Method for Solving a General System of Finite Variational Inequalities for Finite Accretive Operators

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1 A Viscosity Method for Solving a General System of Finite Variational Inequalities for Finite Accretive Operators Phayap Katchang, Somyot Plubtieng and Poom Kumam Member, IAENG Abstract In this paper, we prove a strong convergence theorem for finding a common solution of a general system of finite variational inequalities for finite different inverse-strongly accretive operators and solutions of fixed point problems for a nonexpansive mapping in a Banach space by using the weak contraction Moreover, the above results are applied to find the solutions of zeros of accretive operators and the class of k- strictly pseudocontractive mappings Our results are extended and improved of some authors recent results of the literature works in involving this field Index Terms Inverse-strongly accretive operator, Fixed point, General system of finite variational inequalities, Sunny nonexpansive retraction, Weak contraction I INTRODUTION Let E be a real Banach space with norm and be a nonempty closed convex subset of E Let E be the dual space of E and, denote the pairing between E and E First, we recall the basic concept of mappings as shown in the following: T : is said a nonexpansive if T x T y x y for all x, y H We denote the set of fixed point of T by F T f : is called a weakly contractive if there exists ϕ : [0, [0, a continuous and strictly increasing function such that ϕ is positive on 0,, ϕ0 = 0, lim t ϕt = and x, y fx fy x y ϕ x y 1 For q > 1, the generalized duality mapping J q : E E is defined by J q x = f E : x, f = x q, f = x q 1 } for all x E In particular, if q =, the mapping J is called the normalized duality mapping, and usually written J = J A : E is said an accretive if there exists jx y Jx y such that Ax Ay, jx y 0 for all x, y This research was supported by the ommission on Higher Education, the Thailand Research Fund and the Rajamangala University of Technology Lanna Tak Grant no MRG P Katchang is with Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand, S Plubtieng is with the Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand and P Kumam is with the Department of Mathematics, Faculty of Science, King Mongkut s University of Technology Thonburi KMUTT, Bang Mod, Thrung Kru, Bangkok 10140, Thailand e- mails: pkatchang@hotmailcomp Katchang, somyotp@nuacths Plubtieng and poomkum@kmuttacth P Kumam A : E is said a β-strongly accretive if there exists a constant β > 0 such that Ax Ay, jx y β x y x, y A : E is said a β-inverse strongly accretive if, for any β > 0, Ax Ay, jx y β Ax Ay for all x, y The variational inequality problem is employed to find a point x and is defined by Ax, jy x 0, y where A : E is an accretive operator The general system of variational inequalities is used to find x, y and is defined by λay + x y, jx x 0, µbx + y x, jx y 3 0, for all x where λ and µ are two positive real numbers and A, B : E are two operators The general system of variational inequalities, we extend into the general system of finite variational inequalities is applied to find x 1, x,, x M and is defined by λ M A M x M + x 1 x M, jx x 1 0, λ M 1 A M 1 x M 1 + x M x M 1, jx x M 0, λ A x + x 3 x, jx x 3 0, λ 1 A 1 x 1 + x x 1, jx x 0,, 4 for all x where A l } M l=1 : E is a family of mappings, λ l 0, l 1,,, M} The set of solution of 4 is denoted by GSV I, A l II PRELIMINARIES We always assume that E is a real Banach space and is a nonempty closed convex subset of E Let D be a subset of and Q : D So Q is said to sunny if QQx + tx Qx = Qx, whenever Qx + tx Qx for any x and t 0 A subset D of is said a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction Q of onto D A mapping Q : is called a retraction if Q = Q If a mapping Q : is a retraction, then Qz = z for all z is in the range of Q Proposition II1 Let E be a smooth Banach space and let be a nonempty subset of E Let Q : E be a retraction and let J be the normalized duality mapping on E Then the

2 following are equivalent: i Q is sunny and nonexpansive; ii Qx Qy x y, JQx Qy, x, y E; iii x Qx, Jy Qx 0, x E, y Proposition II Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E and let T be a nonexpansive mapping of into itself with F T Then the set FT is a sunny nonexpansive retract of Lemma II3 Let E be a real -uniformly smooth Banach space with the best smooth constant K Then the following inequality holds: x + y x + y, Jx + Ky, x, y E Lemma II4 Let x n } and y n } be bounded sequences in a Banach space X and let β n } be a sequence in [0, 1] with 0 < lim inf β n lim sup β n < 1 Suppose x n+1 = y n + β n x n for all integers n 0 and lim sup x n+1 x n 0 Then, lim y n x n = 0 Lemma II5 Let a n } and b n } be two nonnegative real number sequences and α n } a positive real number sequence satisfying the conditions: n=1 α b n = and lim n αn = 0 Let the recursive inequality a n+1 a n α n ϕa n + b n, n 0 where ϕa is a continuous and strict increasing function for all a 0 with ϕ0 = 0 Then lim a n = 0 Lemma II6 Let E be a uniformly convex Banach space and B r 0 := x E : x r} be a closed ball of E Then there exists a continuous strictly increasing convex function g : [0, [0, with g0 = 0 such that λx+µy +γz λ x +µ y +γ z λµg x y for all x, y, z B r 0 and λ, µ, γ [0, 1] with λ+µ+γ = 1 Lemma II7 Let be a nonempty bounded closed convex subset of a uniformly convex Banach space E and let T be nonexpansive mapping of into itself If x n } is a sequence of such that x n x weakly and x n T x n 0 strongly, then x is s fixed point of T Lemma II8 Let be a nonempty closed convex subset of a real -uniformly smooth Banach space E Let the mapping A : E be β-inverse-strongly accretive Then, we have Proof Taking Q l = Q I λ l A l Q I λ A Q I λ 1 A 1, l 1,, 3,, M} and Q 0 = I, where I is the identity mapping on H Then we have Q = Q M For any x, y, we have Qx Qy = Q M x Q M y = Q I λ M A M Q M 1 x Q I λ M A M Q M 1 I λ M A M Q M 1 x Q M 1 y Q M 1 Q 0 x Q 0 y = x y Therefore Q is nonexpansive y x I λ M A M Q M 1 y Lemma III Let be a nonempty closed convex subset of a real smooth Banach space E Let Q be the sunny nonexpansive retraction from E onto Let A l : H be nonlinear mapping, where l 1,,, M} For x l, l 1,,, M}, x 1, x,, x M is a solution of problem 4 if and only if x 1 = Q I λ M A M x M x = Q I λ 1 A 1 x 1 x 3 = Q I λ A x 5 x M = Q I λ M 1 A M 1 x M 1, that is x 1 = Q I λ M A M Q I λ A Q I λ 1 A 1 x 1 Proof From 4, we rewrite as x 1 x M λ M A M x M, jx x 1 0, x M x M 1 λ M 1A M 1 x M 1, jx x M 0, x 3 x λ A x, jx x 3 0, x x 1 λ 1 A 1 x 1, jx x 0 6 for all x Using Proposition II1 iii, the system 6 equivalent to 5 Throughout this paper, the set of fixed points of the mapping Q is denoted by F Q I λax I λay x y +λλk β Ax Ay Theorem III3 Let E be a uniformly convex and - uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and be a nonempty If β λk, then I λa is nonexpansive closed convex subset of E Let S : be a nonexpansive III MAIN RESULT mapping and Q be a sunny nonexpansive retraction from Lemma III1 Let be a nonempty closed convex subset E onto Let A l : E be a β l -inverse-strongly accretive of a real -uniformly smooth Banach space E Let Q be such that β l λ l K where l 1,,, M} and K be the the sunny nonexpansive retraction from E onto Let the best smooth constant Let f be a weakly contractive of mapping A l : H be a β l -inverse-strongly accretive such into itself with function ϕ Suppose F := F Q F S that β l λ l K where l 1,,, M} If Q : be where Q is defined by Lemma III1 For arbitrary given a mapping defined by x 0 = x, the sequence x n } is generated by Qx = Q I λ M A M Q I λ A Q I λ 1 A 1 x, yn = Q I λ M A M Q I λ A Q I λ 1 A 1 x n, x n+1 = α n fx n + β n x n + γ n Sy n for all x, then Q is nonexpansive 7

3 where the sequences α n }, β n } and γ n } in 0, 1 satisfy α n +β n +γ n = 1, n 1 and λ l, l = 1,,, M are positive real numbers The following conditions: 1 lim α n = 0 and n=0 α n =, 0 < lim inf β n lim sup β n < 1 are satisfied Then x n } converges strongly to x 1 = Q F f x 1 and x 1, x,, x M is a solution of the problem 4 where Q F is the sunny nonexpansive retraction of onto F Proof First, we prove that x n } is bounded Let p F, and take Q l = Q I λ l A l Q I λ A Q I λ 1 A 1, for l 1,, 3,, M} and Q 0 = I, where I is the identity mapping on E Since Q is a nonexpansive, then Q l, l 1,, 3,, M} also We note that y n p = Q l x n Q l p x n p 8 It follows from 7 and 8, we also have x n+1 p = α n fx n + β n x n + γ n Sy n p α n fx n p + β n x n p + γ n Sy n p α n [ xn p ϕ x n p ] + α n fp p +β n x n p + γ n y n p x n p α n ϕ x n p + α n fp p max x 1 p, ϕ x 1 p, fp p } 9 This implies that x n } is bounded, so are fx n }, y n }, and Sy n } Next, we show that lim x n+1 x n = 0 Notice that = Q M x n+1 Q M x n = Q I λ M A M Q M 1 x n+1 Q I λ M A M Q M 1 x n I λ M A M Q M 1 x n+1 Q M 1 Q M 1 Q 0 x n+1 Q 0 x n = x n+1 x n x n+1 I λ M A M Q M 1 x n x n Setting x n+1 = z n + β n x n for all n 0, we see that z n = xn+1 βnxn 1 β n, then we have z n+1 z n = x n+ β n+1 x n+1 x n+ x n = fx n+1 + γ n+1 Sy n+1 α nfx n + γ n Sy n = fx n+1 + γ n+1 Sy n+1 fx n + fx n γ n+1sy n + γ n+1sy n α nfx n + γ n Sy n = fx n+1 fx n + γ n+1 Sy n+1 Sy n + α n fx n γ n+1 + γ n Sy n fx n+1 fx n + γ n+1 fx n + α n Sy n α [ ] n+1 x n+1 x n ϕ x n+1 x n + γ n+1 fx n Sy n x n+1 x n + γ n+1 fx n + Sy n x n+1 x n + fx n + Sy n x n+1 x n + x n+1 x n fx n + Sy n Therefore, z n+1 z n x n+1 x n x n+1 x n fx n + Sy n It follows from the conditions 1 and, which imply that lim sup z n+1 z n x n+1 x n 0 Applying Lemma II4, we obtain lim z n x n = 0 and also x n+1 x n = z n x n 0 as n Therefore, we have lim x n+1 x n = 0 10 Next, we show that lim Sy n y n = 0 Since p F, from Lemma II6, we obtain x n+1 p = α n fx n + β n x n + γ n Sy n p α n fx n p + 1 α n γ n x n p +γ n y n p

4 = α n fx n p + 1 α n x n p γ n x n p y n p = α n fx n p + 1 α n x n p γ n x n p y n p x n p + y n p α n fx n p + x n p γ n x n y n Therefore, we have γ n x n y n α n fx n p + x n p x n+1 p α n fx n p + x n p + x n+1 p x n x n+1 From the condition 1 and 10, this implies that x n y n 0 as n Now, we note that x n Sy n x n x n+1 + x n+1 Sy n = x n x n+1 + α n fx n + β n x n + γ n Sy n Sy n = x n x n+1 + α n fx n Sy n + β n x n Sy n x n x n+1 + α n fx n Sy n + β n x n Sy n Therefore, we get x n Sy n 1 x n x n+1 + α n fx n Sy n From the conditions 1, and 10, which imply that x n Sy n 0 as n Since Sy n y n Sy n x n + x n y n, it follows that lim Sy n y n = 0 Next, we prove that z F := F Q F S a First, we show that z F S To show this, we choose a subsequence y ni } of y n } Since y ni } is bounded, we have a subsequence y nij } of y ni } converging weakly to z We may assume without loss of generality that y ni z Since Sy n y n 0, we obtain Sy ni z Then, we can obtain z F Assuming that z / F S, we get Sz z From y ni z and Opial s condition, we obtain lim inf i y ni z < lim inf i y ni Sz lim inf i y ni Sy ni + Sy ni Sz lim inf i y ni z This is a contradiction Thus, we have z F S b Next, we show that z F Q From Lemma III1, we know that Q = Q M is a nonexpansive, it follows that y n Qy n = Q M x n Q M y n x n y n Thus lim y n Qy n = 0 Since Q is a nonexpansive, we get x n Qx n Therefore, we have x n y n + y n Qy n + Qy n Qx n x n y n + y n Qy n lim x n Qx n = 0 11 By Lemma II7 and 11, we have z F Q Therefore z F Next, we show that lim sup f I x 1, Jx n x 1 0, where x 1 = Q F f x 1 Since x n } is bounded, we can choose a sequence x ni } of x n } which x ni z, such that lim sup f I x 1, Jx n x 1 = lim i f I x 1, Jx ni x 1 1 Now, from 1, Proposition II1 iii and the weakly sequential continuity of the duality mapping J, we have lim sup f I x 1, Jx n x 1 From 10, it follows that = lim f I x 1, Jx ni x 1 i = f I x 1, Jz x lim sup f I x 1, Jx n+1 x Finally, we show that x n } converges strongly to x 1 = Q F f x 1 We compute that x n+1 x 1 = x n+1 x 1, Jx n+1 x 1 = α n fx n + β n x n + γ n Sy n x 1, Jx n+1 x 1 = α n fx n x 1 + β n x n x 1 +γ n Sy n x 1, Jx n+1 x 1 = α n fx n f x 1, Jx n+1 x 1 +β n x n x 1, Jx n+1 x 1 +γ n Sy n x 1, Jx n+1 x 1 ] α n [ x n x 1 ϕ x n x 1 x n+1 x 1 +β n x n x 1 x n+1 x 1 +γ n y n x 1 x n+1 x 1 α n x n x 1 x n+1 x 1 α n ϕ x n x 1 x n+1 x 1 +β n x n x 1 x n+1 x 1 +γ n x n x 1 x n+1 x 1 = x n x 1 x n+1 x 1 α n ϕ x n x 1 x n+1 x 1 = 1 x n x 1 + x n+1 x 1 α n ϕ x n x 1 x n+1 x 1 By 9 and x n+1 x 1 } bounded, there exist M > 0 such that x n+1 x 1 M, which imply that x n+1 x 1 x n x 1 α n Mϕ x n x 1 +α n f x 1 x 1, Jx n+1 x 1 15 Now, from 1 and applying Lemma II5 to 15, we get x n x 1 0 as n This completes the proof orollary III4 Let E be a uniformly convex and - uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and be a nonempty

5 closed convex subset of E Let S : be a nonexpansive mapping and Q be a sunny nonexpansive retraction from E onto Let A : E be a β-inverse-strongly accretive such that β λk where K be the best smooth constant Let f be a weakly contractive of into itself with function ϕ Let the sequences α n }, β n } and γ n } in 0, 1 satisfy α n + β n + γ n = 1, n 1 and satisfy the conditions 1 and in Theorem III3 Suppose F := F Q F S where Q is defined by Qx = Q I λaq I λa Q I λax, x, and λ be a positive real number For arbitrary given x 0 = x, the sequence x n } is generated by yn = Q I λaq I λa Q I λax n, x n+1 = α n fx n + β n x n + γ n Sy n 16 Then x n } converges strongly to x 1 = Q F f x 1, where Q F is the sunny nonexpansive retraction of onto F Proof Putting A = A M = A M 1 = = A = A 1, β = β M = β M 1 = = β = β 1 and λ = λ M = λ M 1 = = λ = λ 1 in Theorem III3, we can conclude the desired conclusion easily This completes the proof orollary III5 Let E be a uniformly convex and - uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and be a nonempty closed convex subset of E Let S : be a nonexpansive mapping and Q be a sunny nonexpansive retraction from E onto Let A l : E be a β l -inverse-strongly accretive such that β l λ l K where l 1, } and K be the best smooth constant Let f be a weakly contractive of into itself with function ϕ Let the sequences α n }, β n } and γ n } in 0, 1 satisfy α n + β n + γ n = 1, n 1 and satisfy the conditions 1 and in Theorem III3 Suppose F := F Q F S where Q is defined by Qx = Q I λ A Q I λ 1 A 1 x, x, and λ 1, λ are positive real numbers For arbitrary given x 0 = x, the sequence x n } is generated by yn = Q I λ A Q I λ 1 A 1 x n, 17 x n+1 = α n fx n + β n x n + γ n Sy n Then x n } converges strongly to x 1 = Q F f x 1 and x 1, x is a solution of the problem 3, where Q F is the sunny nonexpansive retraction of onto F Proof Taking M = in Theorem III3, we can conclude the desired conclusion easily This completes the proof IV SOME APPLIATIONS I Application to finding zeros of accretive operators In Banach space E, we always assume that E is a uniformly convex and -uniformly smooth Recall that, an accretive operator T is a m-accretive if RI + rt = E for each r > 0 We assume that T is a m-accretive and has a zero ie, the inclusion 0 T z is solvable The set of zeros of T is denoted by T 1 0, that T 1 0 = z DT : 0 T z} The resolvent of T, ie, Jr T = I +rt 1, for each r > 0 If T is a m-accretive, then Jr T : E E is a nonexpansive and F Jr T = T 1 0, r > 0 From the main result Theorem III3, we can conclude the following result immediately Theorem IV1 Let E be a uniformly convex and -uniformly smooth Banach space and be a nonempty closed convex subset of E Let A l : E be a β l -inverse-strongly accretive such that β l λ l K where l 1,,, M}, K be the -uniformly smoothness constant of E and T be an m-accretive mapping Let f be a weakly contractive of into itself with function ϕ and suppose the sequences α n }, β n } and γ n } in 0, 1 satisfy α n +β n +γ n = 1, n 1 Suppose F := T 1 0 M l=1 A 1 l 0 and λ l, l = 1,,, M, are positive real numbers The following conditions: i lim α n = 0 and n=0 α n =, ii 0 < lim inf β n lim sup β n < 1, are satisfied The sequence x n } is generated by x 0 = x and yn = Jr T I λ M A M I λ A I λ 1 A 1 x n, x n+1 = α n fx n + β n x n + γ n y n 18 Then x n } converges strongly to x 1 = Q F f x 1, where Q F is the sunny nonexpansive retraction of E onto F II Application to strictly pseudocontractive mappings Let E be a Banach space and let be a subset of E Recall that, a mapping T : is said a k-stricly pseudocontractive if there exist k [0, 1 and jx y Jx y such that T x T y, jx y x y 1 k I T x I T y 19 for all x, y Then 19 can be written in the following form I T x I T y, jx y 1 k I T x I T y 0 We know that, A is a 1 k inverse strongly monotone and A 1 0 = F T Theorem IV Let E be a uniformly convex and -uniformly smooth Banach space and be a nonempty closed convex subset of E Let S : be a nonepansive mapping and T l : be a k l -stricly pseudocontractive with λ l 1 k l K, l 1,,, M} Let f be a weakly contractive of into itself with function ϕ and suppose the sequences α n }, β n } and γ n } in 0, 1 satisfy α n + β n + γ n = 1, n 1 Suppose F := F S M l=1 F T l and let λ l, l = 1,,, M are positive real numbers The following conditions: i lim α n = 0 and n=0 α n =, ii 0 < lim inf β n lim sup β n < 1, are satisfied The sequence x n } is generated by x 0 = x and y n = 1 λ M + λ M T M 1 λ + λ T 1 λ1 + λ 1 T 1 xn, x n+1 = α n fx n + β n x n + γ n Sy n 1 Then x n } converges strongly to Q F, where Q F is the sunny nonexpansive retraction of E onto F

6 Proof Putting A l = I T l, l 1,,, M} Form 0, we get A l is a 1 k l inverse strongly accretive operator It follows that GSV I, A l = GSV I, I T l = F T l and M l=1 GSV I, I T l = F Q is the solution of problems 4 1 λ1 + λ 1 T 1 xn = Q 1 λ1 + λ 1 T 1 xn 1 λm + λ M T M 1 λ1 + λ 1 T 1 xn = Q 1 λm + λ M T M Q 1 λ1 + λ 1 T 1 xn Therefore, by Theorem III3, x n } converges strongly to some element x 1 of F V ONLUSION In this paper, motivated and inspired by the idea of eng et al [1], Katchang and Kumam [], Witthayarat et al [3] and Yao et al [4] We introduce a new iterative scheme with weak contraction for finding solutions of a new general system of finite variational inequalities 4 for finite different inverse-strongly accretive operators and solutions of fixed point problems for nonexpansive mapping in a Banach space onsequently, we obtain new strong convergence theorems for fixed point problems which solve the general system of variational inequalities 3 Moreover, using the above theorems, we can apply to find solutions of zeros of accretive operators and the class of k-strictly pseudocontractive mappings III Application to Hilbert spaces In a real Hilbert spaces H, by Lemma III, we obtain the following Lemma: Lemma IV3 x 1, x,, x M is a solution of λ M A M x M + x 1 x M, x x 1 0, λ M 1 A M 1 x M 1 + x M x M 1, x x M 0, λ A x + x 3 x, x x 3 0, λ 1 A 1 x 1 + x x 1, x x 0, for all x if and only if x 1 = P I λ M A M P I λ A P I λ 1 A 1 x 1 is a fixed point of the mapping P : which is defined by Px = P I λ M A M P I λ A P I λ 1 A 1 x for all x where P is a metric projection H onto It is well known that the smooth constant K = in Hilbert spaces From Theorem III3, we can obtain the following result immediately Theorem IV4 Let be a nonempty closed convex subset of a real Hilbert space H Let A l : H be a β l -inverse-strongly monotone mapping with λ l 0, β l, l 1,,, M} Let S : be a nonepansive mapping and f be a weakly contractive of into itself with function ϕ Suppose the sequences α n }, β n } and γ n } in 0, 1 satisfy α n + β n + γ n = 1, n 1 Assume that F := F P F S where P is defined by Lemma IV3 and λ l, l = 1,,, M, are positive real numbers The following conditions: i lim α n = 0 and n=0 α n =,, ii 0 < lim inf β n lim sup β n < 1, are satisfied For arbitrary given x 0 = x, the sequence x n } is generated by yn = P I λ M A M P I λ A P I λ 1 A 1 x n, x n+1 = α n fx n + β n x n + γ n Sy n 3 Then x n } converges strongly to x 1 = P F f x 1 and x 1, x,, x M is a solution of the problem AKNOWLEDGMENTS This research was partially supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education ommission Phayap Katchang was supported by the ommission on Higher Education, the Thailand Research Fund and the Rajamangala University of Technology Lanna Tak Grant no MRG REFERENES [1] L- eng, -Y Wang and J- Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Mathematical Methods of Operations Research, , [] P Katchang and P Kumam, An iterative algorithm for finding a common solution of fixed points and a general system of variational inequalities for two inverse strongly accretive operators, Positivity, , [3] U Witthayarat, Y J ho, and P Kumam, onvergence of an iterative algorithm for common solutions for zeros of maximal accretive operator with applications Journal of Applied Mathematics, Vol 01, Article ID , 17 pages [4] Y Yao, M A Noor, K I Noor, Y- Liou and H Yaqoob, Modified extragradient methods for a system of variational inequalities in Banach spaces, Acta Applicandae Mathematicae, ,