ITERATIVE ALGORITHMS FOR FAMILIES OF VARIATIONAL INEQUALITIES FIXED POINTS AND EQUILIBRIUM PROBLEMS. Communicated by Heydar Radjavi. 1.

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1 Bulletin of the Iranian Matheatical Society Vol. 37 No. 1 (2011), pp ITERATIVE ALGORITHMS FOR FAMILIES OF VARIATIONAL INEQUALITIES FIXED POINTS AND EQUILIBRIUM PROBLEMS S. SAEIDI Counicated by Heydar Radjavi Abstract. We introduce an iterative algorith for finding a coon eleent of the set of fixed points for an infinite faily of nonexpansive appings, the set of solutions of the variational inequalities for a faily of α-inverse-strongly onotone appings and the set of solutions of a syste of equilibriu probles in a Hilbert space. We prove the strong convergence of the proposed iterative algorith to the unique solution of a variational inequality, which is the optiality condition for a iniization proble. Moreover, we apply our result to the proble of finding a coon fixed point of a faily of strictly pseudocontractive appings. 1. Introduction Let C be a closed convex subset of a Hilbert space H. Then, a apping S of C into itself is called nonexpansive if Sx Sy x y, for all x, y C. We denote by F ix(s) the set of fixed points of S. Recall that a apping B : C H is called α-inverse-strongly onotone [3] if there exists a positive real nuber α such that Bx By, x y α Bx By 2, x, y C. MSC(2010): Priary: 47H09; Secondary: 47H10, 47J20. Keywords: Coon fixed point, equilibriu proble, nonexpansive apping, variational inequality, viscosity approxiation. Received: 21 May 2009, Accepted: 11 February c 2011 Iranian Matheatical Society. 247

2 248 Saeidi It is easy to see that if B : C H is α-inverse-strongly onotone, then it is a 1 α-lipschitzian apping. Let B : C H be a apping. The classical variational inequality proble is to find u C such that Bu, v u 0, v C. The set of solutions of this variational inequality is denoted by V I(C, B). For finding an eleent of F ix(s) V I(C, B), Takahashi and Toyoda [27] introduced the following iterative schee: x n+1 = ɛ n x n + (1 ɛ n )SP C (x n λ n Bx n ), n 0, where, x 0 = x C, {ɛ n } is a sequence in (0, 1), and {λ n } is a sequence in (0, 2α). On the other hand, Moudafi [16] introduced the viscosity approxiation ethod for nonexpansive appings. Let f be a contraction on a Hilbert space H (i.e., f(x) f(y) l x y, x, y H and 0 l < 1). Starting with an arbitrary initial x 0 H, define a sequence {x n } recursively by (1.1) x n+1 = (1 ɛ n )Sx n +ɛ n f(x n ), n 0, where, {ɛ n } is a sequence in (0, 1). It is proved [16, 30] that, under certain appropriate conditions iposed on {ɛ n }, the sequence {x n } generated by (1.1) strongly converges to the unique solution x in F := F ix(s) of the variational inequality (I f)x, x x 0, x F. Iterative ethods for nonexpansive appings have recently been applied to solve convex iniization probles and variational inequalities; see, e.g., [4, 10,, 21, 28, 29, 31, 32]. A typical proble is to iniize a quadratic function over the set of the fixed points of a nonexpansive apping on a real Hilbert space H: (1.2) in x F 1 Ax, x x, u, 2 where, F is the fixed point set of a nonexpansive apping S on H and u is a given point in H. Assue A is strongly positive; that is, there is a constant γ > 0 with the property Ax, x γ x 2, for all x H.

3 Iterative algoriths for failies of variational inequalities 249 Xu [28] (see also [31]) proved that the sequence {x n }, defined by the iterative ethod (1.3) x n+1 = (I ɛ n A)Sx n +ɛ n u, n 0, with the initial guess x 0 H chosen arbitrarily, converges strongly to the unique solution of the iniization proble (1.2) provided that the sequence {ɛ n } satisfies certain conditions. Marino and Xu [15] cobined the iterative ethod (1.3) with the viscosity approxiation ethod (1.1) and considered the following general iterative ethod: (1.4) x n+1 = (I ɛ n A)Sx n +ɛ n γf(x n ), n 0, where, 0 < γ < γ/α. They proved that if the sequence {ɛ n } satisfies appropriate conditions, then the sequence {x n }, generated by (1.4), converges strongly to the unique solution of the variational inequality, (1.5) (A γf)x, x x 0, x F, which is the optiality condition for the iniization proble, (1.6) in x F 1 Ax, x h(x) 2 where, h is a potential function for γf (i.e., h (x) = γf(x), for x H). Finding an optial point in the intersection F of the fixed point sets of a faily of nonexpansive appings is a task frequently arising fro various areas of atheatical sciences and engineering. For exaple, the well-known convex feasibility proble reduces to finding a point in the intersection of the fixed point sets of a faily of nonexpansive appings; see, e.g., [1, 8]. A siple algorithic solution to the proble of iniizing a quadratic function over F is of an extree value in any applications including set theoretic signal estiation; see, e.g., [14, 33]. On the other hand, let C be a nonepty closed convex subset of H. Let F : C C R be a bifunction. The equilibriu proble for F is to deterine its equilibriu points, i.e., the set EP (F ) := {x C : F (x, y) 0 y C}. Let G = {F i } i I be a faily of bifunctions fro C C to R. The syste of equilibriu probles for G = {F i } i I is to deterine coon equilibriu points for G = {F i } i I, i.e., the set (1.7) EP (G) := {x C : F i (x, y) 0 y C i I}.

4 250 Saeidi Many probles in applied sciences, such as onotone inclusion probles, saddle point probles, variational inequality probles, iniization probles, Nash equilibria in noncooperative gaes, vector equilibriu probles, as well as certain fixed point probles reduce into finding soe eleent of EP (F ); see [2, 9, 11]. The forulation (1.7) extends this foralis to systes of such probles, covering, in particular, various fors of feasibility probles [9, 12]. Given any r > 0, the operator Jr F : H C, defined by Jr F (x) := {z C : F (z, y) + 1 y z, z x 0 y C}, r is called the resolvent of F (see [9, 12]). It is shown [5, 7] that under suitable hypotheses on F, Jr F : H C is single-valued and firly nonexpansive and satisfies F ix(jr F ) = EP (F ), r > 0. Using this result, Takahashi and Takahashi [26] introduced a viscosity approxiation ethod for finding a coon eleent of EP (F ) and F ix(s), where S is a nonexpansive apping. Starting with an arbitrary eleent x 1 in H, they defined the sequence {x n } recursively by (1.8) x n+1 = ɛ n f(x n )+(1 ɛ n )SJ F r n x n. They proved that, under certain appropriate conditions over ɛ n and r n, the sequences {x n } and {J F r n x n } both converge strongly to x = P F ix(s) EP (F ) f(x ). By cobining the schees (1.4) and (1.8), Plubtieng and Punpaeng [17] proposed the following iterative schee: (1.9) x n+1 = ɛ n γf(x n )+(I ɛ n A)SJ F r n x n. They proved that if the sequences {ɛ n } and {r n } of paraeters satisfy appropriate conditions, then the sequences {x n } and {J F r n x n } both converge strongly to the unique solution x F := F ix(s) EP (F ) of the variational inequality (1.5), being the optiality condition for the iniization proble (1.6). Note that the result in [15] is a particular case of this, corresponding to the choice F (x, y) = 0 (so that J F r n = I). Very recently, Colao et al. [6] proposed the following explicit schee with respect to W -appings for a finite faily of nonexpansive appings: (1.10) x n+1 = ɛ n γf(x n )+βx n +((1 β)i ɛ n A)W n J F r n x n.

5 Iterative algoriths for failies of variational inequalities 251 They proved, under weaker hypotheses, that both sequences {x n } and {J F r n x n } converge strongly to a point x F, which is the unique solution of the variational inequality (1.5). Here, otivated by Colao, et al. [6], Takahashi and Toyoda [27] and soe of our previous results [19, 20], we introduce an iterative algorith (Theore 3.1) for finding a coon eleent of the set of solutions of a syste of equilibriu probles EP (G) for a faily G = {F i : i = 1,..., M} of bifunctions, the set of solutions of variational inequalities V I(C, B j ) for a faily {B j : j = 1... N} of α-inverse-strongly onotone apping fro C into H and the set of fixed point for an infinite faily of nonexpansive appings ϕ = {S i : C C}, with respect to W -appings defined in [22]. We prove the strong convergence of the proposed iterative process to the unique solution of the variational inequality (1.5). Our results cover all previous schees specified by (1.1), (1.3), (1.4), (1.8), (1.9) and (1.10). Moreover, we apply our result to the proble of finding a coon fixed point of a faily of strictly pseudocontractive appings (Theore 3.2). 2. Preliinaries Let C be a nonepty closed and convex subset of H. Let F : C C R be a bifunction. The equilibriu proble for F is to deterine its equilibriu points, i.e., the set EP (F ) := {x C : F (x, y) 0 y C}. Given any r > 0, the operator J F r : H C, defined by Jr F (x) := {z C : F (z, y) + 1 y z, z x 0 y C}, r is called the resolvent of F ; See [9, 12]. Lea 2.1. ([9, 12]) Let C be a nonepty closed convex subset of H and F : C C R satisfy (A1) F (x, x) = 0, for all x C; (A2) F is onotone, i.e., F (x, y) + F (y, x) 0, for all x, y C. (A3) for all x, y, z C, li inf t 0 F (tz + (1 t)x, y) F (x, y);

6 252 Saeidi (A4) for all x C, y F (x, y) is convex and lower seicontinuous. Then, (1) Jr F is single-valued; (2) Jr F is firly nonexpansive, i.e., J F r x J F r y 2 J F r x J F r y, x y, for all x, y H; (3) F ix(j F r ) = EP (F ); (4) EP (F ) is closed and convex. Recall that the etric (nearest point) projection P C fro a Hilbert space H to a closed convex subset C of H is defined as follows: given x H, P C x is the only point in C with the property, x P C x = inf{ x y : y C}. It is known that P C is a nonexpansive apping and satisfies (2.1) P C x P C y 2 P C x P C y, x y, x, y H. P C is characterized as follows: y = P C x x y, y z 0, z C. In the context of the variational inequality proble, this iplies: (2.2) u V I(C, B) u = P C (u λbu), λ > 0. A set-valued apping T : H 2 H is said to be onotone if for all x, y H, f T x, and g T y, we have f g, x y 0. A onotone apping T : H 2 H is said to be axial if the graph G(T ) of T is not properly contained in the graph of any other onotone apping. It is known that a onotone apping is axial if and only if, for (x, f) H H, f g, x y 0, and (y, g) G(T ), we have f T x. Let B : C H be an inverse-strongly onotone apping and let N C v be the noral cone to C at v C, i.e., and define N C v = {w H : v u, w 0, u C}, T v = { Bv + NC v, v C,, v C. Then, T is axial onotone and 0 T v if and only if v V I(C, B) (see [13, 18]). The following lea is an iediate consequence of the inner product on H.

7 Iterative algoriths for failies of variational inequalities 253 Lea 2.2. For all x, y H, we have x + y 2 x y, x + y. Lea 2.3. ([29]) Assue {a n } is a sequence of nonnegative real nubers such that a n+1 (1 γ n )a n + δ n, n 0, where, {γ n } is a sequence in (0, 1) and {δ n } is a sequence in R such that (i) n=1 γ n = ; (ii) li sup n δ n /γ n 0 or n=1 δ n <. Then, li n a n = 0. Lea 2.4. ([23]) Let {x n } and {z n } be bounded sequences in a Banach space X and let {β n } be a sequence in [0, 1] with 0 < li inf n β n and li sup n β n < 1. Suppose for all integers n 0 and Then, li n x n y n = 0. x n+1 = β n x n + (1 β n )z n, li sup( z n+1 z n x n+1 x n ) 0. n Definition 2.5. Let {S i : C C} be an infinite faily of nonexpansive appings and {µ i } be a nonnegative real sequence with 0 µ i < 1, i 1. For any n 1, define a apping W n : C C as follows: U n,n+1 := I, U n,n := µ n S n U n,n+1 + (1 µ n )I, U n,n 1 := µ n 1 S n 1 U n,n + (1 µ n 1 )I,. U n,k := µ k S k U n,k+1 + (1 µ k )I, U n,k 1 := µ k 1 S k 1 U n,k + (1 µ k 1 )I,. U n,2 := µ 2 S 2 U n,3 + (1 µ 2 )I, (2.3) W n := U n,1 = µ 1 S 1 U n,2 + (1 µ 1 )I. Such a apping W n is nonexpansive fro C to C and it is called the W -apping, generated by S n, S n 1,..., S 1 and µ n, µ n 1,..., µ 1.

8 254 Saeidi The concept of W -appings was introduced in [24, 25]. It is now a ain tool in studying convergence of iterative ethods to coon fixed points of nonlinear appings. Lea 2.6. ([22]) Let C be a nonepty closed convex subset of a Hilbert space H, ϕ = {S i : C C} be an infinite faily of nonexpansive appings with F ix(ϕ) := i=1 F ix(s i), and {µ i } be a real sequence such that 0 < µ i b < 1, i 1. Then, (1) W n is nonexpansive and F ix(w n ) = n i=1 F ix(s i), for each n 1; (2) for each x C and for each positive integer k, li n U n,k x exists; (3) the apping W : C C, defined by W x := li n W nx = li n U n,1x, is a nonexpansive apping satisfying F ix(w ) = F ix(ϕ) and it is called the W -apping, generated by S 1, S 2,... and µ 1, µ 2,... ; (4) if K is any bounded subset of C, then li sup W x W n x = 0. n x K Lea 2.7. ([15]) Assue A is a strongly positive linear bounded operator on a Hilbert space H with coefficient γ > 0 and 0 < ρ A 1. Then, I ρa 1 ργ. 3. Main Results The following is our ain result. Theore 3.1. Let C be a nonepty closed convex subset of a Hilbert space H, ϕ = {S i : C C} be an infinite faily of nonexpansive appings, G = {F j : j = 1,..., M} be a finite faily of bifunctions fro C C into R which satisfy (A1)-(A4), {B k : k = 1... N} be a finite faily of α-inverse-strongly onotone appings fro C into H, and F := N k=1 V I(C, B k) F ix(ϕ) EP (G). Let A be a strongly positive bounded linear operator with coefficient γ, f be an l-contraction on H, for soe 0 < l < 1, {ɛ n } be a sequence in (0, 1), {λ k,n } N k=1 be sequences in [a, b] with 0 < a b < 2α, {r j,n} M j=1 be sequences in (0, ) and γ and β be two real nubers such that 0 < β < 1 and 0 < γ < γ/l. Assue:

9 Iterative algoriths for failies of variational inequalities 255 (C1) li n ɛ n = 0; (C2) n=1 ɛ n = ; (D1) li inf n r j,n > 0, for every j {1,..., M}; (D2) li n r j,n /r j,n+1 = 1, for every j {1,..., M}; (D3) li n λ k,n λ k,n+1 = 0, for every k {1,..., N}. For every n N, let W n be the W -apping defined by (2.3). If {x n } is the sequence generated by x 1 H and n 1, (3.1) z n = J F M rm,n... J F 2 r 2,n J F 1 r 1,n x n, y n = P C (I λ N,n B N )... P C (I λ 2,n B 2 )P C (I λ 1,n B 1 )z n, x n+1 = ɛ n γf(x n ) + βx n + ((1 β)i ɛ n A)W n y n, then {x n } and {J F k x n } M k=1 converge strongly to x F := N k=1 V I(C, B k) F ix(ϕ) EP (G), which is the unique solution of the variational inequality (1.5). Equivalently, we have P F (I A + γf)x = x. Proof. Since ɛ n 0, we shall assue that ɛ n (1 β) A 1 and 1 ɛ n (γ lγ) > 0. Observe that if u = 1, then ((1 β)i ɛ n A)u, u = (1 β) ɛ n Au, u (1 β ɛ n A ) 0. By Lea 2.7, we have (3.2) (1 β)i ɛ n A 1 β ɛ n γ. Moreover, by taking and J k n := J F k... J F 2 r 2,n J F 1 r 1,n, k {1,..., M}, J 0 n := I, P k n := P C (I λ k,n B k )... P C (I λ 2,n B 2 )P C (I λ 1,n B 1 ), k {1,..., N}, we write the schee (3.1) as: P 0 n := I, (3.3) x n+1 = ɛ n γf(x n )+βx n +((1 β)i ɛ n A)W n P N n J M n x n. We shall divide the proof into several steps. Step 1. The sequence {x n } is bounded.

10 256 Saeidi Proof of Step 1. Let v F. Since, for each k {1,..., M}, J F k nonexpansive, we have J M n x n v = J M n x n J M n v x n v. On the other hand, since for each k {1,... N}, the apping B k : C H is an α-inverse-strongly onotone apping, for any x, y C and λ k,n [a, b] [0, 2α], we have (I λ k,n B k )x (I λ k,n B k )y 2 = (x y) λ k,n (B k x B k y) 2 = x y 2 2λ k,n x y, B k x B k y + λ 2 k,n B kx B k y 2 x y 2 + λ k,n (λ k,n 2α) B k x B k y 2 x y 2, which iplies that P C (I λ k,n B k ) is nonexpansive, and consequently that the appings P k n are nonexpansive. Fro Lea 2.6 and (2.2), we have P N n v = v = W n v. It follows that x n+1 v = ((1 β)i ɛ n A)(W n P N n J M n x n W n P N n J M n v) ɛ n γ(f(x n ) f(v)) + ɛ n (γf(v) Av) + β(x n v) γf(v) Av (1 ɛ n (γ lγ)) x n v + ɛ n (γ lγ), γ lγ which gives x n v ax{ x 1 v, γf(v) Av }, n 1. γ lγ is Step 2. Let {ω n } be a bounded sequence in H. Then, (3.4) li n J k n+1w n J k n ω n = 0, for every k {1,..., M}. Proof of Step 2. Fro [6, Step 2], we have that (3.5) li n J F k +1 ω n J F k ω n = 0, for for every k {1,..., M}. Note that for every k {1,..., M}, we have Jn k = J F k Jn k 1. So, Jn+1w k n Jn k ω n J F k +1 Jn+1 k 1 w n J F k Jn+1 k 1 w n + J F k J F k 1 r k 1,n+1 Jn+1 k 2 w n J F k J F k 1 r k 1,n Jn+1 k 2 w n J F k J F k 1 r k 1,n... Jr F3 3,n Jr F2 2,n+1 Jr F1 1,n+1 ω n J F k J F k 1 r k 1,n... Jr F3 3,n Jr F2 2,n Jr F1 1,n+1 ω n

11 Iterative algoriths for failies of variational inequalities J F k J F k 1 r k 1,n... Jr F3 3,n Jr F2 2,n Jr F1 1,n+1 ω n J F k J F k 1 r k 1,n... Jr F3 3,n Jr F2 2,n Jr F1 1,n ω n J F k +1 Jn+1 k 1 w n J F k Jn+1 k 1 w n + J F k 1 r k 1,n+1 Jn+1 k 2 w n J F k 1 r k 1,n Jn+1 k 2 w n + + Jr F2 2,n+1 Jr F1 1,n+1 ω n Jr F2 2,n Jr F1 1,n+1 ω n + Jr F1 1,n+1 ω n Jr F1 1,n ω n = k j=1 J Fj r j,n+1 (J j 1 n+1 w n) J Fj r j,n (J j 1 n+1 w n). Now, apply (3.5) to conclude (3.4). Step 3. Let {ω n } be a bounded sequence in C. Then, li P C(I λ k,n+1 B k )w n P C (I λ k,n B k )ω n = 0, n and li n Pk n+1w n Pnω k n = 0, for every k {1,..., N}. Proof of Step 3. Since {ω n } is bounded and B k, for k {1,..., N}, is a Lipschitzian apping, we know that Now, L := sup{ B k ω n } <. n P C (I λ k,n+1 B k )w n P C (I λ k,n B k )ω n (I λ k,n+1 B k )w n (I λ k,n B k )ω n = λ k,n+1 λ k,n B k ω n λ k,n+1 λ k,n L 0, as n. Fro this and applying a technique siilar to that used in proof of Step 2, it is easy to prove the second assertion. Step 4. li n x n+1 x n = 0. Proof of Step 4. Define a sequence {z n } by z n = (x n+1 βx n )/(1 β) so that x n+1 = βx n + (1 β)z n. Now, copute (3.6) z n+1 z n = 1 1 β (x n+2 βx n+1 (x n+1 βx n ) = 1 1 β γ(ɛ nf(x n+1 ) ɛ n f(x n ))+((1 β)i ɛ n+1 A)W n+1 P N n+1j M n+1x n+1 ((1 β)i ɛ n A)W n Pn N Jn M x n = γ 1 β (ɛ nf(x n+1 ) ɛ n f(x n )) 1 1 β (ɛ n+1aw n+1 Pn+1J N n+1x M n+1

12 258 Saeidi ɛ n AW n P N n J M n x n ) + W n+1 P N n+1j M n+1x n+1 W n P N n J M n x n. Since {x n } is bounded and by (3.6), we have for soe big enough constant K > 0, (3.7) z n+1 z n W n+1 P N n+1j M n+1x n+1 W n P N n J M n x n +K(ɛ n+1 +ɛ n ) W n+1 P N n+1j M n+1x n+1 W n+1 P N n+1j M n+1x n + W n+1 Pn+1J N n+1x M n W n+1 Pn+1J N n M x n + W n+1 Pn+1J N n M x n W n+1 Pn N Jn M x n + W n+1 P N n J M n x n W n P N n J M n x n + K(ɛ n+1 + ɛ n ) x n+1 x n + J M n+1x n J M n x n + P N n+1j M n x n P N n J M n x n + W n+1 P N n J M n x n W n P N n J M n x n + K(ɛ n+1 + ɛ n ). Now, since ɛ n 0 and by Steps 2, 3 and Lea 2.6, we iediately conclude fro (3.7) that li sup( z n+1 z n x n+1 x n ) n li sup{ Jn+1x M n Jn M x n + Pn+1J N n M x n Pn N Jn M x n n + W n+1 Pn N Jn M x n W n Pn N Jn M x n + K(ɛ n+1 + ɛ n )} 0. Apply Lea 2.4 to get li n x n+1 x n = (1 β) li n x n z n = 0. Step 5. li n Jn k x n Jn k+1 x n = 0, k {0, 1,..., M 1}. Proof of Step 5. Let v F and k {0, 1,..., M 1}. Since J F k+1 r k+1,n is firly nonexpansive, we obtain v J k+1 n x n 2 = J F k+1 r k+1,n v J F k+1 r k+1,n J k n x n 2 J F k+1 r k+1,n J k n x n v, J k n x n v = 1 2 ( J F k+1 r k+1,n J k n x n v 2 + J k n x n v 2 J k n x n J F k+1 r k+1,n J k n x n 2 ). It follows that (3.8) J k+1 n x n v 2 x n v 2 J k n x n J k+1 n x n 2. Set t n = γf(x n ) AW n Pn N Jn M x n and let λ > 0 be a constant such that λ > sup{ t n, x k v }. n,k Using Lea 2.2 and noting that. 2 is convex, we derive, using (3.8), (3.9) x n+1 v 2 = (1 β)(w n P N n J M n x n v) + β(x n v) + ɛ n t n 2

13 Iterative algoriths for failies of variational inequalities 259 So, (1 β)(w n P N n J M n x n v) + β(x n v) 2 + 2ɛ n t n, x n+1 v (1 β) W n P N n J M n x n v 2 + β x n v 2 + 2λ 2 ɛ n. x n+1 v 2 (1 β) Jn k+1 x n v 2 + β x n v 2 + 2λ 2 ɛ n (1 β)( x n v 2 Jn k x n Jn k+1 x n 2 ) + β x n v 2 + 2λ 2 ɛ n = x n v 2 (1 β) J k n x n J k+1 n x n 2 + 2λ 2 ɛ n. It follows, by Step 4 and condition (C1), that J k n x n J k+1 n x n β ( x n v 2 x n+1 v 2 + 2λ 2 ɛ n ) 1 1 β (2λ x n x n+1 + 2λ 2 ɛ n ) 0, as n. Step 6. li n x n W n P N n J M n x n = 0. Proof of Step 6. Observe that x n W n P N n J M n x n x n x n+1 + x n+1 W n P N n J M n x n = x n x n+1 + ɛ n (γf(x n ) AW n P N n J M n x n ) + β(x n W n P N n J M n x n ) x n x n+1 +ɛ n (γ f(x n ) + AW n P N n J M n x n )+β x n W n P N n J M n x n. It follows fro Step 4 that x n W n P N n J M n x n 1 1 β ( x n x n+1 +ɛ n (γ f(x n ) + AW n P N n J M n x n ) 0, as n. Step 7. li n P k nj M n x n P k+1 n J M n x n = 0, k {0, 1,..., N 1}. Proof of Step 7. Since {B k : k = 1... N} are α-inverse-strongly onotone, by the assuptions iposed on {λ n }, for given v F and k {0, 1,..., N 1}, we have W n P N n J M n x n v 2 P k+1 n J M n x n v 2 = P C (I λ k+1,n B k+1 )P k nj M n x n P C (I λ k+1,n B k+1 )v 2 (I λ k+1,n B k+1 )P k nj M n x n (I λ k+1,n B k+1 )v 2 P k nj M n x n v 2 + λ k+1,n (λ k+1,n 2α) B k+1 P k nj M n x n B k+1 v 2 x n v 2 + a(b 2α) B k+1 P k nj M n x n B k+1 v 2.

14 260 Saeidi Thus, by (3.9), we have x n+1 v 2 (1 β) W n P N n J M n x n v 2 + β x n v 2 + 2λ 2 ɛ n So, (1 β){ x n v 2 + a(b 2α) B k+1 P k nj M n x n B k+1 v 2 } + β x n v 2 + 2λ 2 ɛ n = (1 β)a(b 2α) B k+1 P k nj M n x n B k+1 v 2 + x n v 2 + 2λ 2 ɛ n. (1 β)a(2α b) B k+1 P k nj M n x n B k+1 v 2 x n v 2 x n+1 v 2 + 2λ 2 ɛ n x n x n+1 ( x n v + x n+1 v ) + 2λ 2 ɛ n. Since 0 < β < 1, and x n x n+1 0, we obtain: (3.10) B k+1 P k nj M n x n B k+1 v 0 (n ). Again fro (2.1) and the fact that I λ k+1,n B k+1 is nonexpansive, we have Pn k+1 Jn M x n v 2 = P C (I λ k+1,n B k+1 )PnJ k n M x n P C (I λ k+1,n B k+1 )v 2 (PnJ k n M x n λ k+1,n B k+1 PnJ k n M x n ) (v λ k+1,n B k+1 v), Pn k+1 Jn M x n v = 1 2 { (Pk njn M x n λ k+1,n B k+1 PnJ k n M x n ) (v λ k+1,n B k+1 v) 2 + Pn k+1 Jn M x n v 2 (PnJ k n M x n λ k+1,n B k+1 PnJ k n M x n ) (v λ k+1,n B k+1 v) (Pn k+1 Jn M x n v) 2 } 1 2 { Pk njn M x n v 2 + Pn k+1 Jn M x n v 2 PnJ k n M x n Pn k+1 Jn M x n λ k+1,n (B k+1 PnJ k n M x n B k+1 v) 2 } = 1 2 { Pk njn M x n v 2 + Pn k+1 Jn M x n v 2 PnJ k n M x n Pn k+1 Jn M x n 2 +2λ k+1,n PnJ k n M x n Pn k+1 Jn M x n, B k+1 PnJ k n M x n B k+1 v λ 2 k+1,n B k+1pnj k n M x n B k+1 v 2 }. This iplies: P k+1 n J M n x n v 2 P k nj M n x n v 2 P k nj M n x n P k+1 n J M n x n 2 +2λ k+1,n P k nj M n x n P k+1 n J M n x n, B k+1 P k nj M n x n B k+1 v λ 2 k+1,n B k+1p k nj M n x n B k+1 v 2. Then, by (3.9), we have x n+1 v 2 (1 β) W n P N n J M n x n v 2 + β x n v 2 + 2λ 2 ɛ n (1 β) P k+1 n J M n x n v 2 + β x n v 2 + 2λ 2 ɛ n (1 β){ P k nj M n x n v 2 P k nj M n x n P k+1 n J M n x n 2

15 Iterative algoriths for failies of variational inequalities λ k+1,n PnJ k n M x n Pn k+1 Jn M x n, B k+1 PnJ k n M x n B k+1 v λ 2 k+1,n B k+1pnj k n M x n B k+1 v 2 } + β x n v 2 + 2λ 2 ɛ n x n v 2 (1 β) PnJ k n M x n Pn k+1 Jn M x n 2 +2(1 β)λ k+1,n PnJ k n M x n Pn k+1 Jn M x n, B k+1 PnJ k n M x n B k+1 v +2λ 2 ɛ n, which iplies: (1 β) P k nj M n x n P k+1 n J M n x n 2 x n v 2 x n+1 v 2 + 2λ 2 ɛ n +2(1 β)λ k+1,n P k nj M n x n P k+1 n J M n x n B k+1 P k nj M n x n B k+1 v. Hence, it follows fro Step 4 and (3.10) that P k nj M n x n P k+1 n J M n x n 0. Step 8. The weak ω-liit set of {x n }, ω w (x n ), is a subset of F. Proof of Step 8. Let z 0 ω w (x n ) and let {x n } be a subsequence of {x n } weakly converging to z 0. Fro steps 5 and 7, we also obtain that for all k {1,..., M}, and J k n x n z 0, P k n J M n x n z 0, for all k {1,..., N}. We need to show that z 0 F. First, we prove z 0 N i=1 V I(C, B i). For this purpose, let k {1,..., N} and T k be the axial onotone apping defined by { Bk x + N T k x = C x, x C;, x C. For any given (x, u) G(T k ), u B k x N C x. Since PnJ k n M x n C, by the definition of N C, we have (3.11) x P k nj M n x n, u B k x 0. On the other hand, since PnJ k n M we have So, x n = P C (P k 1 n J M n x n λ k,n B k P k 1 n J M n x n ), x P k nj M n x n, P k nj M n x n (P k 1 n J M n x n λ k,n B k P k 1 n J M n x n ) 0. x PnJ k n M x n, Pk njn M x n Pn k 1 Jn M x n + B k Pn k 1 Jn M x n 0. λ k,n By (3.11) and the α-inverse onotonicity of B k, we have x P k n J M n x n, u x P k n J M n x n, B k x

16 262 Saeidi x P k n J M n x n, B k x x Pn k Jn M x n, Pk n Jn M x n Pn k 1 Jn M x n λ k,n = x P k n J M n x n, B k x B k P k n J M n x n + B k P k 1 n J M n x n + x P k n J M n x n, B k P k n J M n x n B k P k 1 n J M n x n x Pn k Jn M x n, Pk n Jn M x n Pn k 1 Jn M x n λ k,n x P k n J M n x n, B k P k n J M n x n B k P k 1 n J M n x n x Pn k Jn M x n, Pk n Jn M x n Pn k 1 Jn M x n. λ k,n Since PnJ k n M x n Pn k 1 Jn M x n 0, Pn k Jn M x n z 0 and {B k : k = 1,..., N} are Lipschitz continuous, we have li x Pk n Jn M x n, u = x z 0, u 0. Again, since T k is axial onotone, then 0 T k z 0. This shows that z 0 V I(C, B k ). Fro this, it follows: (3.12) z 0 N i=1v I(C, B i ). Note that by (A2) and given y C and k {0, 1,..., M 1}, we have 1 y Jn k+1 x n, Jn k+1 x n Jn k x n F k+1 (y, Jn k+1 x n ). r n,k+1 Thus, (3.13) y Jn k+1 x n, J n k+1 x n Jn k x n F k+1 (y, Jn k+1 r x n ). n,k+1 By condition (A4), F i (y,.), i, is lower seicontinuous and convex, and thus weakly seicontinuous. Step 5 and condition li inf n r n,j > 0 iply that Jn k+1 x n Jn k x n 0, r n,k+1 in nor. Therefore, letting in (3.13), yields: F k+1 (y, z 0 ) li F k+1 (y, J k+1 n x n ) 0,

17 Iterative algoriths for failies of variational inequalities 263 for all y C and k {0, 1,..., M 1}. Replacing y with y t := ty + (1 t)z 0 with t (0, 1) and using (A1) and (A4), we obtain: 0 = F k+1 (y t, y t ) tf k+1 (y t, y) + (1 t)f k+1 (y t, z 0 ) tf k+1 (y t, y). Hence, F k+1 (ty + (1 t)z 0, y) 0, for all t (0, 1) and y C. Letting t 0 + and using (A3), we conclude F k+1 (z 0, y) 0, for all y C and k {0,..., M 1}. Therefore, (3.14) z 0 M EP (F k ) = EP (G). k=1 We next show z 0 i=1 F ix(s i). By Lea 2.6, we have, for every z C, (3.15) W n z W z, and F ix(w ) = i=1 F ix(t i). Assue that z 0 F ix(w ). Then, z 0 W z 0. Fro Opial s property of Hilbert space, (3.12), (3.14), (3.15) and Step 6, we have li inf x n z 0 < li inf x n W z 0 li inf ( x n W n Pn N Jn M x n + W n P N n J M n x n W n P N n J M n z 0 + W n z 0 W z 0 ) li inf ( x n z 0 + W n z 0 W z 0 ) = li inf x n z 0. This is a contradiction. Therefore, z 0 belongs to F ix(w ) = i=1 F ix(s i). Step 9. Let x be the unique solution of the variational inequality, (3.16) (γf A)x, x x 0, x F. Then, (3.17) li sup (γf A)x, x n x 0. n Proof of Step 9. Lea 2.3 of Marino and Xu [15] guarantees that P F (γf +(I A)) has a unique fixed point x, which is the unique solution of (3.16). Let {x nk } be a subsequence of {x n } such that (3.18) li (γf A)x, x nk x = li sup (γf A)x, x n x. k n

18 264 Saeidi Without loss of generality, we can assue that {x nk } weakly converges to soe z C. By Step 8, z F. Thus, cobining (3.16) and (3.18), we get li sup (γf A)x, x n x (γf A)x, z x 0, n as required. Step 10. The sequences {x n } and {J F k x n } M k=1 converge strongly to x. Proof of Step 10. Taking u n = W n P N n J M n x n, n 1, we have u n x x n x. By using leas 2.2 and 2.7, we have x n+1 x 2 = [((1 β)i ɛ n A)(u n x ) + β(x n x )] +ɛ n (γf(x n ) Ax ) 2 ((1 β)i ɛ n A)(u n x ) + β(x n x ) 2 +2ɛ n γf(x n ) Ax, x n+1 x (1 β) ((1 β)i ɛ na) (u n x ) 2 + β x n x 2 1 β +2ɛ n γ f(x n ) f(x ), x n+1 x + 2ɛ n γf(x ) Ax, x n+1 x (1 β)i ɛ na 2 u n x 2 + β x n x 2 1 β +2ɛ n γl x n x x n+1 x + 2ɛ n γf(x ) Ax, x n+1 x (1 β)i ɛ na 2 x n x 2 + β x n x 2 1 β +ɛ n γl( x n x 2 + x n+1 x 2 ) + 2ɛ n γf(x ) Ax, x n+1 x ( ((1 β) γɛ n) 2 + β + ɛ n γl) x n x 2 1 β +ɛ n γl x n+1 x 2 + 2ɛ n γf(x ) Ax, x n+1 x. It follows that x n+1 x 2 (1 2(γ lγ)ɛ n 1 lγɛ n ) x n x 2 + ɛ n (2 γf(x ) Ax, x n+1 x + γ2 ɛ n 1 lγɛ n 1 β x n x 2 ). Now, fro conditions (C1) and (C2), Step 8 and Lea 2.3, we get x n x 0; naely, x n x, in nor. Finally, noticing J F k x n

19 Iterative algoriths for failies of variational inequalities 265 x x n x, we have that, for all k {1,..., M}, J F k x n x, in nor. A apping T : C H is called strictly pseudocontractive on C if there exists k with 0 k < 1 such that T x T y 2 x y 2 + k (I T )x (I T )y 2, for all x, y C. If k = 0, then T is nonexpansive. Put B = I T, where, T : C H is a strictly pseudocontractive apping with k. It is known that B is 1 k 2 -inverse-strongly onotone and B 1 (0) = F ix(t ). Now, using Theore 3.1, we state a strong convergence theore for a faily of strictly pseudocontractive appings as follows. Theore 3.2. Let C be a nonepty closed convex subset of a Hilbert space H, ϕ = {S i : C C} be an infinite faily of nonexpansive appings, G = {F j : j = 1,..., M} be a finite faily of bifunctions fro C C into R which satisfy (A1)-(A4), ψ = {T j : j = 1... N} be a finite faily of strictly pseudocontractive appings with 0 k < 1 fro C into C, and F := F ix(ϕ) EP (G) F ix(ψ). Let A be a strongly positive bounded linear operator with coefficient γ, f be an l-contraction on H for soe 0 < l < 1, {ɛ n } be a sequence in (0, 1), {λ j,n } N j=1 be sequences in [a, b], with 0 < a b < 1 k, {r j,n } M j=1 be sequences in (0, ) and γ and β be two real nubers such that 0 < β < 1 and 0 < γ < γ/l. Assue: (C1) li n ɛ n = 0; (C2) n=1 ɛ n = ; (D1) li inf n r j,n > 0, for every j {1,..., M}; (D2) li n r j,n /r j,n+1 = 1, for every j {1,..., M}; (D3) li n λ j,n λ j,n+1 = 0, for every j {1,..., N}. For every n N, let W n be the W -apping defined by (2.3). If {x n } is the sequence generated by x 1 H and n 1, z n = J F M rm,n... J F 2 r 2,n J F 1 r 1,n x n, y n = ((1 λ N,n )I + λ N,n T N )... ((1 λ 2,n )I + λ 2,n T 2 ) ((1 λ 1,n )I + λ 1,n T 1 )z n, x n+1 = ɛ n γf(x n ) + βx n + ((1 β)i ɛ n A)W n y n, then {x n } and {J F k x n } M k=1 converge strongly to x F := F ix(ϕ) EP (G) F ix(ψ), which is the unique solution of the variational inequality (1.5). Equivalently, we have P F (I A + γf)x = x.

20 266 Saeidi Proof. Put B j = I T j, for every j {1,..., N}. Then, B j is 1 k 2 - inverse-strongly onotone. We have that F ix(t j ) is the solution set of V I(C, B j ); i.e., F ix(t j ) = V I(C, B j ). Therefore, by Theore 3.1, the result follows. Reark 3.3. We ay put v n = P C (I λ N,n (I T N ))... P C (I λ 2,n (I T 2 ))P C (I λ 1,n (I T 1 ))u n, in the schee of Theore 3.2, and obtain a schee for failies of nonself strictly pseudocontractive appings. References [1] H. H. Bauschke and J. M. Borwein, On projection algoriths for solving convex feasibility probles, SIAM Rev. 38 (1996) [2] E. Blu and W. Oettli, Fro optiization and variational inequalities to equilibriu probles, Math. Student 63 (1994) [3] F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear appings in Hilbert space, J. Math. Anal. Appl. 20 (1967) [4] L. C. Ceng, H. K. Xu and J. C. Yao, A hybrid steepest-descent ethod for variational inequalities in Hilbert spaces, Appl. Anal. 87 (2008) [5] L. C. Ceng and J. C. Yao, A hybrid iterative schee for ixed equilibriu probles and fixed point probles, J. Coput. Appl. Math. 214 (2008) [6] V. Colao, G. Marino and H. K. Xu, An iterative ethod for finding coon solutions of equilibriu and fixed point probles, J. Math. Anal. Appl. 344 (2008) [7] P. L. Cobettes, Quasi-Fejérian analysis of soe optiization algoriths, in Inherently Parallel Algoriths for Feasibility and Optiization and Their Applications (D. Butnariu, Y. Censor, and S. Reich, eds.), , Stud. Coput. Math., 8, North-Holland, Asterda, [8] P. L. Cobettes, The foundations of set theoretic estiation, Proc. IEEE 81 (1993) [9] P. L. Cobettes and S. A. Hirstoaga, Equilibriu prograing in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005) [10] F. Deutsch and I. Yaada, Miniizing certain convex functions over the intersection of the fixed point sets of nonexpansive appings, Nuer. Funct. Anal. Opti. 19 (1998) [11] A. Göpfert, H. Riahi, C. Taer and C. Zălinescu, Variational Methods in Partially Ordered Spaces, Springer-Verlag, New York, [12] N. Hadjisavvas and H. Khatibzadeh, Maxial onotonicity of bifunctions, Optiization 59 (2010) [13] H. Iiduka and W. Takahashi, Strong convergence theores for nonexpansive appings and inverse-strongly onotone appings, Nonlinear Anal. 61 (2005)

21 Iterative algoriths for failies of variational inequalities 267 [14] A. N. Iuse and A. R. De Pierro, On the convergence of Han s ethod for convex prograing with quadratic objective, Math. Prograing Ser. B 52 (1991) [15] G. Marino and H. K. Xu, A general iterative ethod for nonexpansive appings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) [16] A. Moudafi, Viscosity approxiation ethods for fixed-points probles, J. Math. Anal. Appl. 241 (2000) [17] S. Plubtieng and R. Punpaeng, A general iterative ethod for equilibriu probles and fixed point probles in Hilbert spaces, J. Math. Anal. Appl. 336 (2007) [18] R. T. Rockafellar, On the axiality of sus of nonlinear onotone operators, Trans. Aer. Math. Soc. 149 (1970) [19] S. Saeidi, Iterative algoriths for finding coon solutions of variational inequalities and systes of equilibriu probles and fixed points of failies and seigroups of nonexpansive appings, Nonlinear Anal. 70 (2009) [20] S. Saeidi, Iterative ethods for equilibriu probles, variational inequalities and fixed points, Bull. Iranian. Math. Soc. 36 (2010) [21] S. Saeidi, Modified hybrid steepest-descent ethods for variational inequalities and fixed points, Math. Coput. Modelling 52 (2010) [22] K. Shioji and W. Takahashi, Strong convergence to coon fixed points of infinite nonexpansive appings and applications, Taiwanese J. Math. 5 (2001) [23] T. Suzuki, Strong convergence of Krasnoselskii and Mann s type sequences for one-paraeter nonexpansive seigroups without Bochner integrals, J. Math. Anal. Appl. 305 (2005) [24] W. Takahashi, Weak and strong convergence theores for failies of nonexpansive appings and their applications, Ann. Univ. Mariae Curie-Sklodowska Sect. A 51 (1997) [25] W. Takahashi and K. Shioji, Convergence theores for nonexpansive appings and feasibility probles, Math. Coput. Modelling 32 (2000), [26] S. Takahashi and W. Takahashi, Viscosity approxiation ethods for equilibriu probles and fixed point probles in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) [27] W. Takahashi and M. Toyoda, Weak convergence theores for nonexpansive appings and onotone appings, J. Opti. Theory Appl. 118 (2003) [28] H. K. Xu, An iterative approach to quadratic optiization, J. Opti. Theory Appl. 116 (2003) [29] H. K. Xu, Iterative algoriths for nonlinear operators, J. London Math. Soc. (2) 66 (2002) [30] H. K. Xu, Viscosity approxiation ethods for nonexpansive appings, J. Math. Anal. Appl. 298 (2004) [31] I. Yaada, The hybrid steepest descent ethod for the variational inequality proble over the intersection of the fixed point sets of nonexpansive appings, in Inherently Parallel Algoriths in Feasibility and Optiization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, eds., North-Holland, Asterda, 2001,

22 268 Saeidi [32] I. Yaada, N. Ogura, Y. Yaashita, and K. Sakaniwa, Quadratic optiization of fixed points of nonexpansive appings in Hilbert space, Nuer. Funct. Anal. Opti. 19 (1998) [33] D. C. Youla, Matheatical theory of iage restoration by the ethod of convex projections, in H. Stark, ed., Iage Recovery Theory and Applications, Acadeic Press, Florida, 1987, S. Saeidi Departent of Matheatics, University of Kurdistan, P.O.Box , Sanandaj 416, Iran Eail: and shahra