General viscosity iterative method for a sequence of quasi-nonexpansive mappings

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1 Avalable onlne at J. Nonlnear Sc. Appl. 9 (2016), Research Artcle General vscosty teratve method for a sequence of quas-nonexpansve mappngs Cuje Zhang, Ynan Wang College of Scence, Cvl Avaton Unversty of Chna, Tanjn , Chna. Communcated by Y. H. Yao Abstract In ths paper, we study a general vscosty teratve method due to Aoyama and Kohsaka for the fxed pont problem of quas-nonexpansve mappngs n Hlbert space. Frst, we obtan a strong convergence theorem for a sequence of quas-nonexpansve mappngs. Then we gve two applcatons about varatonal nequalty problem to encourage our man theorem. Moreover, we gve a numercal example to llustrate our man theorem. c 2016 All rghts reserved. Keywords: Quas-nonexpansve mappng, varatonal nequalty, fxed pont, vscosty teratve method MSC: 47H10, 47J Introducton Throughout the present paper, let H be a real Hlbert space wth nner product, and norm. Let C be a nonempty closed convex subset of H and T : C C be a mappng. In ths paper, we denote the fxed-pont set of T by F x(t ). A mappng T s sad to be quas-nonexpansve, f F x(t ) and T x p x p for all x C and p F x(t ). We know that f T : C C s quas-nonexpansve, then F x(t ) s closed and convex (see [3] for more general results). A mappng T s sad to be nonexpansve, f T x T y x y for all x, y C. A mappng T s called demclosed at 0, f any sequence {x n } weakly converges to x, and f the sequence {T x n } strongly converges to 0, then T x = 0. The vscosty teratve method was proposed by Moudaf [11] frstly. Choose an arbtrary ntal x 0 H, the sequence {x n } s constructed by: x n+1 = ε n f(x n ) + 1 T x n, n 0, 1 + ε n 1 + ε n Correspondng author Emal addresses: cuje_zhang@126.com (Cuje Zhang), ynan_wang@163.com (Ynan Wang) Receved

2 C. Zhang, Y. Wang, J. Nonlnear Sc. Appl. 9 (2016), where T s a nonexpansve mappng and f s a contracton wth a coeffcent α [0, 1) on H, the sequence {ε n } s n (0, 1), such that: () lm n ε n = 0; () n=0 ε n = ; () lm n ( 1 ε n 1 ε n+1 ) = 0. Then lm n x n = x, where x C(C = F x(t )) s the unque soluton of the varatonal nequalty (I f)x, x x 0, x F x(t ). (1.1) Mangé consdered the vscosty teratve method for quas-nonexpansve mappngs n Hlbert space n [9]. Hs focus was on the followng algorthm: x n+1 = α n f(x n ) + (1 α n )T ω x n, where {α n } s a slow vanshng sequence, and ω (0, 1], T ω := (1 ω)i + ωt, T has two man condtons: () T s quas-nonexpansve; () I T s demclosed at 0. He proved the sequence {x n } converges strongly to the unque soluton of the varatonal nequalty (1.1). Tan and Jn consdered the followng teratve process n [13]: x n+1 = α n γf(x n ) + (I α n A)T ω x n, n 0, where the sequence {α n } satsfes certan condtons, ω (0, 1 2 ), T ω = (1 ω)i + ωt, and T s also satsfed the same condtons n Mangé [9]. Then they proved that {x n } converges strongly to the unque soluton of the varatonal nequalty: (γf A)x, x x 0, x F x(t ). Recently, Aoyama and Kohsaka consdered the followng general teratve method n [1]: x n+1 = α n f n (x n ) + (1 α n )S n x n, where f n s a θ-contracton wth respect to Ω = n=1 F x(s n) and {f n } s stable on Ω, and {S n } s a sequence of strongly quas-nonexpansve mappngs of C nto C. That s to say, S n s quas-nonexpansve and S n x n x n 0 whenever {x n } s a bounded sequence n C and x n p S n x n p 0 for some pont p Ω. Then they proved that f the sequence {α n } satsfes approprate condtons, {x n } converges strongly to the unque fxed pont of a contracton P Ω f 1. Many varous teratve algorthms have been studed and extended by many authors, especally about quas-nonexpansve mappngs (see [1, 4, 6 13, 15]). Motvated by the above results, we extend the teratve method to quas-nonexpansve mappngs. We consder the followng teratve process: x n+1 = α n f n (x n ) + =1 (α 1 α )S λn x n, (1.2) where S λn = (1 λ n )I + λ n S, and {S } =1 s a sequence of quas-nonexpansve mappngs. Under the approprate condtons, we establsh the strong convergence of the sequence {x n } generated by (1.2).

3 C. Zhang, Y. Wang, J. Nonlnear Sc. Appl. 9 (2016), Prelmnares We denote the strong convergence and the weak convergence of {x n } to x H by x n x and x n x, respectvely. Let f : C C be a mappng, Ω s a nonempty subset of C, and θ s a real number n [0, 1). A mappng f s sad to be a θ-contracton wth respect to Ω, f f(x) f(z) θ x z, x C, z Ω. f s sad to be a θ-contracton, f f s a θ-contracton wth respect to C. The followng lemmas are useful for our man result. Lemma 2.1 ([1]). Let Ω be a nonempty subset of C and f : C C a θ-contracton wth respect to Ω, where 0 θ < 1. If Ω s closed and convex, then P Ω f s a θ-contracton on Ω, where P Ω s the metrc projecton of H onto Ω. Lemma 2.2 ([1]). Let f : C C be a θ-contracton, where 0 θ < 1 and T : C C a quas-nonexpansve mappng. Then f T s a θ-contracton wth respect to F x(t ). Let D be a nonempty subset of C. A sequence {f n } of mappngs of C nto H s sad to be stable on D, f {f n (z) : n N} s a sngleton for every z D. It s clear that f {f n } s stable on D, then f n (z) = f 1 (z) for all n N and z D. Lemma 2.3 ([9]). Let T ω := (1 ω)i + ωt, wth T be a quas-nonexpansve mappng on H, F x(t ) φ, and ω (0, 1], q F x(t ). Then the followng statements are reached: () F x(t ) = F x(t ω ); () T ω s a quas-nonexpansve mappng; () T ω x q 2 x q 2 ω(1 ω) T x x 2 for all x H. Lemma 2.4 ([5]). Assume {s n } s a sequence of nonnegatve real numbers such that s n+1 (1 )s n + δ n, n 0, s n+1 s n η n + t n, n 0, where { } s a sequence n (0, 1), η n s a sequence of nonnegatve real numbers, and {δ n } and {t n } are two sequences n R such that: () n=0 = ; () lm n t n = 0; () lm k η nk = 0 mples lm sup k δ nk 0 for any subsequence {n k } {n}. Then lm n s n = 0. Lemma 2.5 ([10]). Assume A s a strongly postve lnear bounded operator on Hlbert space H wth coeffcent γ > 0 and 0 < ρ A 1. Then I ρa 1 ρ γ. 3. Man results In ths secton, we prove the followng strong convergence theorem. Theorem 3.1. Let H be a real Hlbert space, C a nonempty closed convex subset of H, {S n } a sequence of quas-nonexpansve mappngs of C nto C such that Ω = =1 F x(s ) s nonempty, and I S s demclosed at 0. Assume that {f n } s a sequence of mappngs of C nto C such that each f n s a θ-contracton wth

4 C. Zhang, Y. Wang, J. Nonlnear Sc. Appl. 9 (2016), respect to Ω and {f n } s stable on Ω, where 0 θ < 1. Let {x n } be a sequence defned by x 1 C and x n+1 = α n f n (x n ) + =1 (α 1 α )S λn x n, for n N, where S λn = (1 λ n )I+λ n S, λ n (0, 1] and {λ n } satsfes 0 < lm nf n λ n lm sup n λ n < 1. Suppose that {α n } s a sequence n (0, 1] such that α 0 = 1, α n 0, n=1 α n = and {α n } s strctly decreasng. Then {x n } converges to ω Ω, where ω s the unque fxed pont of a contracton P Ω f 1. and Frst, we show some lemmas, then we prove Theorem 3.1. In the rest of ths secton, we set γ n = α 2 n f n (x n ) ω 2 +2α n = α n (1 + (1 2θ)(1 α n )), =1 (α 1 α ) S λn Lemma 3.2. {x n }, {S x n } and {f n (x n )} are bounded, and moreover, x n+1 ω α n f n (x n ) ω + =1 x n ω, f 1 (ω) ω. (α 1 α ) S λn x n ω, (3.1) and hold for every n N. x n+1 ω 2 (1 ) x n ω 2 +γ n, Proof. From Lemma 2.3, we know S λn s quas-nonexpansve and F x(s ) = F x(s λn ) for all N. Snce f n s a θ-contracton wth respect to Ω, S λn s quas-nonexpansve, ω Ω F x(s ) = F x(s λn ), and {f n } s stable on Ω, t follows that x n+1 ω = α n f n (x n ) + =1 (α 1 α )S λn x n ω α n ( f n (x n ) f n (ω) + f n (ω) ω ) + (α 1 α ) S λn x n ω =1 α n θ x n ω +α n f 1 (ω) ω +(1 α n ) x n ω = (1 α n (1 θ)) x n ω +α n (1 θ) f 1(ω) ω 1 θ for every n N. Thus, by the nducton on n, for every N, we have (3.2) S x n ω x n ω max{ x 1 ω, f 1(ω) ω }. 1 θ Therefore, t turns out that {x n } and {S x n } are bounded, and moreover, {f n (x n )} s also bounded. Equaton (3.1) follows from (3.2). By assumpton, for every N, t follows that S λn x n ω, f n (x n ) ω S λn x n ω f n (x n ) f n (ω) + S λn x n ω, f n (ω) ω θ x n ω 2 + S λn x n ω, f 1 (ω) ω, (3.3)

5 C. Zhang, Y. Wang, J. Nonlnear Sc. Appl. 9 (2016), and thus x n+1 ω 2 = α n (f n (x n ) ω) + =1 = α 2 n f n (x n ) ω α n =1 (α 1 α )(S λn x n ω) 2 =1 (α 1 α )(S λn x n ω) 2 (α 1 α )(S λn x n ω), f n (x n ) ω α 2 n f n (x n ) ω 2 +(1 α n ) 2 x n ω 2 + 2α n (α 1 α ) S λn x n ω, f n (x n ) ω =1 α 2 n f n (x n ) ω 2 +(1 α n ) 2 x n ω 2 +2α n (1 α n )θ x n ω 2 + 2α n (α 1 α ) S λn x n ω, f 1 (ω) ω =1 = (1 ) x n ω 2 +γ n for every n N. Lemma 3.3. The followng hold: 0 < 1 for every n N; 2α n (1 α n )/ 1/(1 θ) and 2α n / 1/(1 θ); α 2 n f n (x n ) ω 2 / 0; n=1 =. Proof. Snce 0 < α n 1 and 1 < 1 2θ 1, we know that 0 < α 2 n = α n (1 + ( 1)(1 α n )) < α n (1 + (1 α n )) = α n (2 α n ) 1. From α n 0 we have 2α n (1 α n )/ 1/(1 θ) and 2α n / 1/(1 θ). Snce {f n (x n )} s bounded and αn 2 α n = 1 + (1 2θ)(1 α n ) 0, t follows that αn 2 f n (x n ) ω 2 / 0. Fnally, we prove n=1 =. Suppose that 1 2θ 0. Then t follows that α n for every n N. Thus, n=1 =. Next, we suppose that 1 2θ < 0. Then > 2(1 θ)α n for every n N. Thus, n=1 2(1 θ) n=1 α n =. Ths completes the proof. Proof of Theorem 3.1. By Lemma 2.1, t mples that P Ω f 1 s a θ-contracton on Ω and hence t has a unque fxed pont on Ω. From Lemma 3.2, we know that x n+1 ω 2 (1 ) x n ω 2 +α 2 n f n (x n ) ω 2 + 2α n (α 1 α ) S λn x n ω, f 1 (ω) ω =1

6 C. Zhang, Y. Wang, J. Nonlnear Sc. Appl. 9 (2016), whch mples that = (1 ) x n ω 2 +α 2 n f n (x n ) ω 2 + 2α n (α 1 α ) λ n (S x n x n ), f 1 (ω) ω =1 + 2α n (α 1 α ) x n ω, f 1 (ω) ω, =1 x n+1 ω 2 (1 ) x n ω 2 + [ α 2 n f n (x n ) ω 2 + 2α n λ n (α 1 α ) x n S x n f 1 (ω) ω =1 + 2α n (1 α n ) x n ω, f 1 (ω) ω On the other hand, we obtan from Lemma 2.3 () that By usng (3.3), we have x n+1 ω 2 = α n (f n (x n ) ω) + =1 = α 2 n f n (x n ) ω α n =1 ]. (α 1 α )(S λn x n ω) 2 =1 (α 1 α )(S λn x n ω) 2 (α 1 α )S λn x n ω, f n (x n ) ω αn 2 f n (x n ) ω 2 +(1 α n ) 2 x n ω 2 (1 α n )λ n (1 λ n ) (α 1 α ) S x n x n 2 =1 + 2α n (α 1 α ) S λn x n ω, f n (x n ) ω. =1 (1 α n ) 2 x n ω 2 +2α n (α 1 α ) S λn x n ω, f n (x n ) ω =1 (1 α n ) 2 x n ω 2 +2α n (1 α n )θ x n ω 2 + 2α n (α 1 α ) S λn x n ω, f 1 (ω) ω ) =1 (1 ) x n ω 2 +2α n (1 α n ) x n ω f 1 (ω) ω. (3.4) (3.5) (3.6) Snce S λn s quas-nonexpansve, from (3.5) and (3.6), t follows that x n+1 ω 2 x n ω 2 +αn 2 f n (x n ) ω 2 +2α n (1 α n ) x n ω f 1 (ω) ω (1 α n )λ n (1 λ n ) (α 1 α ) S x n x n 2. =1 Suppose that M s a postve constant such that M sup{α n f n (x n ) ω 2 +2(1 α n ) x n ω f 1 (ω) ω, n N}.

7 C. Zhang, Y. Wang, J. Nonlnear Sc. Appl. 9 (2016), So we have Set x n+1 ω 2 x n ω 2 +α n M (1 α n )λ n (1 λ n ) s n = x n ω 2, t n = α n M, δ n = α2 n f n (x n ) ω 2 + 2α n λ n (α 1 α ) S x n x n 2. (3.7) =1 (α 1 α ) x n S x n f 1 (ω) ω =1 + 2α n (1 α n ) x n ω, f 1 (ω) ω, η n = (1 α n )λ n (1 λ n ) (α 1 α ) S x n x n 2. =1 Then (3.4) and (3.7) can be rewrtten as the followng forms, respectvely, s n+1 (1 )s n + δ n, s n+1 s n η n + t n. Fnally, we observe that the condton lm n α n = 0 and Lemma 3.3 mply lm n t n = 0 and n=1 =, respectvely. In order to complete the proof by usng Lemma 2.4, t suffces to verfy that lm η n k = 0, k mples lm sup δ nk 0, k for any subsequence {n k } {n}. In fact, for every N, f η nk 0 as k, then n k (1 α nk )λ nk (1 λ nk ) (α 1 α ) S x nk x nk 2 0. =1 And snce 0 < lm nf n λ n lm sup n λ n < 1, there exst λ > 0 and λ > 0, such that 0 < λ λ n λ < 1. Snce lm n α n = 0, there exst some postve nteger n 0 and α < 1, such that α n < α, when n > n 0, then (1 α)λ(1 λ)(α 1 α ) S x nk x nk 2 (1 α)λ(1 λ) (α 1 α ) S x nk x nk 2 Therefore, snce {α n } s strctly decreasng, t follows that n k =1 n k (1 α nk )λ nk (1 λ nk ) (α 1 α ) S x nk x nk 2 0. n k S x nk x nk 0 and (α 1 α ) S x nk x nk 2 0 =1 for every N. By usng the condton that I S s demclosed at 0, we obtan ω w (x nk ) F = =1 F x(s ). From Lemma 3.3, t turns out that lm sup k 2α nk (1 α nk ) k x nk ω, f 1 (ω) ω = 1 1 θ = 1 1 θ =1 lm sup x nk ω, f 1 (ω) ω k sup z ω, f 1 (ω) ω 0. z ω w(x nk )

8 C. Zhang, Y. Wang, J. Nonlnear Sc. Appl. 9 (2016), Snce lm n α n = 0, n k =1 (α 1 α ) S x nk x nk 2 0 and {f n (x n )}, {S x n } are bounded, t s easy to see that lm sup k δ nk 0. From Lemma 2.4, we conclude that x n ω. Remark 3.4. When S n = S, we can remove the followng condtons: α 0 = 1 and {α n } s strctly decreasng. In fact, the above condtons guarantee the coeffcents α 1 α greater than 0 for every N. The followng corollary s the drect consequence of Theorem 3.1. Corollary 3.5. Let H be a real Hlbert space, C a nonempty closed convex subset of H, S : C C a quas-nonexpansve mappng, such that F x(s) and I S s demclosed at 0. Assume that α n 0, n=1 α n =, and f n satsfes the same condtons of Theorem 3.1. Let {x n } be a sequence defned by x 1 C and x n+1 = α n f n (x n ) + (1 α n )S λn x n (3.8) for n N, where S λn = (1 λ n )I + λ n S, and {λ n } also satsfes the same condtons of Theorem 3.1. Then {x n } converges to ω Ω, where ω s the unque fxed pont of a contracton P Ω f 1. Remark 3.6. If f n = f and λ n = λ for all n N, (3.8) becomes the vscosty approxmaton process whch s ntroduced by Mangé (see [9]). 4. Applcaton to varatonal nequalty problem In ths secton, by applyng Theorem 3.1 and Corollary 3.5, frst we study the followng varatonal nequalty problem, whch s to fnd a pont x Ω, such that F (x ), x x 0, x Ω, (4.1) where Ω s a nonempty closed convex subset of a real Hlbert space H, and F : H H s a nonlnear operator. The problem (4.1) s denoted by V I(Ω, F ). It s well-known that V I(Ω, F ) s equvalent to the fxed pont problem (see, [7]). If the soluton set of V I(Ω, F ) s denoted by Γ, we know that Γ = F x(p Ω (I λf )), where λ > 0 s an arbtrary constant, P Ω s the metrc projecton onto Ω, and I s the dentty operator on H. Assume that, F s η-strongly monotone and L-Lpschtzan contnuous, that s, F satsfes the condtons F x F y, x y η x y 2, x, y Ω, F x F y L x y, x, y Ω. By usng Corollary 3.5, we obtan the followng convergence theorem for solvng the problem V I(Ω, F ). Theorem 4.1. Let F be η-strongly monotone and L-Lpschtzan contnuous wth η > 0, L > 0. Assume that S s a quas-nonexpansve operator wth Ω = F x(s), and I S s demclosed at 0. And {α n } s a sequence n (0, 1] such that α n 0, n=1 α n =. Let {x n } be a sequence defned by x 1 H and x n+1 = (I µα n F )S λn x n, (4.2) where S λn = (1 λ n )I + λ n S, λ n (0, 1], 0 < lm nf n λ n lm sup n λ n < 1, and 0 < µ < 2η L 2. Then {x n } converges strongly to the unque soluton of V I(Ω, F ). Proof. Set f n = (I µf )S λn for n N and θ = 1 2µη + µ 2 L 2. Note that (I µf )x (I µf )y 2 = x y 2 2µ x y, F x F y + µ 2 F x F y 2 x y 2 2µη x y 2 +µ 2 L 2 x y 2 = (1 µ(2η µl 2 )) x y 2.

9 C. Zhang, Y. Wang, J. Nonlnear Sc. Appl. 9 (2016), From 0 < µ < 2η L 2, we obtan that I µf s a θ-contracton. Snce S s quas-nonexpansve, from Lemma 2.3, S λn s quas-nonexpansve. By Lemma 2.2, f n s a θ-contracton wth respect to F x(s), and t s stable on Ω. Moreover, t follows from (4.2) that x n+1 = α n f n (x n ) + (1 α n )S λn x n for n N. Thus from Corollary 3.5, we have that {x n } converges strongly to ω = P F x(s) f 1 (ω) = P F x(s) (I µf )ω, whch s the unque soluton of V I(Ω, F ). Remark 4.2. The teraton (4.2) s called the hybrd steepest descent method, (see[2, 14] for more detals). Fnally, we study the followng varatonal nequalty problem, whch s to fnd a pont x F x(s), such that (γf A)x, x x 0, x F x(s), (4.3) where f s a α-contracton and A s strongly postve, that s, there exsts a constant γ > 0 such that Ax, x γ x 2 for all x H. Assume that 0 < γ < γ/α. The problem (4.3) s denoted by V IP, where x s the unque soluton of V IP, and we have x = P F x(s) (I A + γf)x. Theorem 4.3. Assume that S : H H s a quas-nonexpansve operator wth Ω = F x(s), and I S s demclosed at 0. Let {x n } be a sequence defned by x 1 H and x n+1 = α n γtf(x n ) + (I α n ta)s λn x n, n 0, (4.4) where S λn = (1 λ n )I + λ n S, and 0 < t < 1 A, {λ n} and {α n } satsfy the same condtons of Theorem 4.1. Then {x n } converges strongly to the unque soluton of the V IP. Proof. Set f n = tγf + (I ta)s λn. By usng Lemma 2.5, note that f n (x) f n (p) = (tγf + (I ta)s λn )x (tγf + (I ta)s λn )p tγα x p +(1 tγ) x p =(1 t( γ γα)) x p. From 0 < γ < γ/α, we obtan that f n s a θ-contracton wth respect to F x(s), and t s stable on F x(s). Moreover, t follows from (4.4) that x n+1 = α n f n (x n ) + (1 α n )S λn x n for n N. Thus from Corollary 3.5, we have that {x n } converges strongly to the unque soluton of V IP. Remark 4.4. Let ξ n = α n t, snce α n 0 and n=1 α n =, we have ξ n 0 and n=1 ξ n =, then (4.4) become that x n+1 = ξ n γf(x n ) + (I ξ n A)S λn x n, whch s ntroduced by Tan and Jn (see [13]). 5. Numercal example In ths secton, we gve an example to support Theorem 3.1. Example 5.1. In Theorem 3.1, we assume that H = R. Take f n (x) = x n, S x = x cos x, where x [ π, π]. Gven the parameter λ n = 3+2n 6n for every n N. By the defntons of S, we have n =1 F x(s ) = {0}. S s a quas-nonexpansve mappng snce, f x [ π, π] and q = 0, then S x q = S x 0 = x cos x x = x q.

10 C. Zhang, Y. Wang, J. Nonlnear Sc. Appl. 9 (2016), From Theorem 3.1, we can conclude that the sequence {x n } converges strongly to 0, as n. We can rewrte (1.2) as follows x n+1 = 1 n α nx n + =1 (α 1 α )( 4n 3 6n x n n 6n x n cos x n ). (5.1) Next, we gve the parameter α n has three dfferent expressons n (5.1), that s to say, we set α (1) n = 1 α n (2) = 1 2n+1, α(3) n = 1 n+1. Then, through takng a dstnct ntal guess x 1 = 3, by usng software Matlab, we obtan the numercal experment results n Table 1, where n s the teratve number, and the expresson of error we take x n+1 x n x n. n+1, n Table 1: The values of {x n}. α n (1) α n (2) α n (3) x n error x n error x n error From Table 1, we can easly see that wth teratve number ncreases, {x n } approaches to the unque fxed pont 0 and the errors gradually approach to zero. And wth the change of α n, the convergent speed of the sequence {x n } wll be changed, when α n = α n (3), the speed of the sequence {x n } s more faster than others, and when α n = α n (2) the convergent speed of the sequence {x n } become slower. Through ths example, we can conclude that our algorthm s feasble. Acknowledgment The frst author s supported by the Fundamental Scence Research Funds for the Central Unverstes (Program No k010). References [1] K. Aoyama, F. Kohsaka, Vscosty approxmaton process for a sequence of quasnonexpansve mappngs, Fxed Pont Theory Appl., 2014 (2014), 11 pages. 1, 2.1, 2.2 [2] A. Cegelsk, R. Zalas, Methods for varatonal nequalty problem over the ntersecton of fxed pont sets of quas-nonexpansve operators, Numer. Funct. Anal. Optm., 34 (2013), [3] W. G. Doson, Fxed ponts of quas-nonexpansve mappngs, J. Austral. Math. Soc., 13 (1972), [4] M. K. Ghosh, L. Debnath, Convergence of Ishkawa terates of quas-nonexpansve mappngs, J. Math. Anal. Appl., 207 (1997), [5] S. N. He, C. P. Yang, Solvng the varatonal nequalty problem defned on ntersecton of fnte level sets, Abstr. Appl. Anal., 2013 (2013), 8 pages. 2.4 [6] G. E. Km, Weak and strong convergence for quas-nonexpansve mappngs n Banach spaces, Bull. Korean. Math. Soc., 49 (2012), [7] D. Knderlehrer, G. Stampaccha, An ntroducton to varatonal nequaltes and ther applcatons, Pure and Appled Mathematcs, Academc Press, Inc. [Harcourt Brace Jovanovch, Publshers], New York-London, (1980). 4 [8] R. L, Z. H. He, A new teratve algorthm for splt soluton problems of quas-nonexpansve mappngs, J. Inequal. Appl., 2015 (2015), 12 pages.

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