Academic Editor: Hari M. Srivastava Received: 29 September 2016; Accepted: 6 February 2017; Published: 11 February 2017

Transcription

1 mathematics Article The Split Common Fixed Point Problem for a Family of Multivalued Quasinonexpansive Mappings and Totally Asymptotically Strictly Pseudocontractive Mappings in Banach Spaces Ali Abkar, Elahe Shahrosvand and Azizollah Azizi Department of Mathematics, Imam Khomeini International University, Qazvin 34149, Iran; kshahrosvand@yahoo.com (E.S.); azizi@sci.ikiu.ac.ir (A.A.) * Correspondence: abkar@sci.ikiu.ac.ir; Tel.: Academic Editor: Hari M. Srivastava Received: 29 September 2016; Accepted: 6 February 2017; Published: 11 February 2017 Abstract: In this paper, we introduce an iterative algorithm for solving the split common fixed point problem for a family of multi-valued quasinonexpansive mappings and totally asymptotically strictly pseudocontractive mappings, as well as for a family of totally quasi-φ-asymptotically nonexpansive mappings and k-quasi-strictly pseudocontractive mappings in the setting of Banach spaces. Our results improve and extend the results of Tang et al., Takahashi, Moudafi, Censor et al., and Byrne et al. Keywords: split common fixed point problem; totally asymptotically strictly pseudocontractive mapping; quasinonexpansive mapping; k-quasi-strictly pseudocontractive mapping MSC Classification: 47H05; 47H09; 47J25 1. Introduction Let H 1 and H 2 be two real Hilbert spaces and A : H 1 H 2 be a bounded linear operator. For nonlinear operators T : H 1 H 1 and U : H 2 H 2, the split fixed point problem (SFPP) is to find a point: x Fix(T) such that Ax Fix(U) (1) It is often desirable to consider the above problem for finitely many operators. Given n nonlinear operators T i : H 1 H 1 and m nonlinear operators U j : H 2 H 2, the split common fixed point problem (SCFPP) is to find a point: x n Fix(T i) such that Ax m j=1 Fix(U j) In particular, if T i = P Ci and U j = P Qj, then the SCFPP reduces to the multiple sets split feasibility problem (MSSFP); that is, to find x n C i, such that Ax m j=1 Q j, where {C i } n and {Q j} m j=1 are nonempty closed convex subsets in H 1 and H 2, respectively. In the Hilbert space setting, the split feasibility problem and the split common fixed point problem have been studied by several authors; see, for instance, [1 3]. In [4], Censor and Segal introduced the iterative scheme: x n+1 = U(I ρ n A (I T)A)x n which solves the problem (1) for directed operators. This algorithm was then extended to the case of quasinonexpansive mappings [5], as well as to the case of demicontractive mappings [6]. Recently, Takahashi in [7,8] extended the split feasibility problem in Hilbert spaces to the Banach space setting. Mathematics 2017, 5, 11; doi: /math

2 Mathematics 2017, 5, 11 2 of 18 Then, Alsulami et al. [1] established some strong convergence theorems for finding a solution of the split feasibility problem in Banach spaces. Using the shrinking projection method of [8], Takahashi proved the strong convergence theorem for finding a solution of the split feasibility problem in Banach spaces. In this direction, Byrne et al. [2] studied the split common null point problem for multi-valued mappings in Hilbert spaces. Consider finitely many multi-valued mappings F i : H 1 2 H 1, 1 i n, and B j : H 2 2 H 2, 1 j m, and let A j : H 1 H 2 be bounded linear operators. The split common null point problem is to find a point: z H 1 such that z ( n F 1 i 0) ( m j=1 A jb 1 j 0) Very recently, using the hybrid method and the shrinking projection method in mathematical programming, Takahashi et al. [9] proved two strong convergence theorems for finding a solution of the split common null point problem in Banach spaces. In [10], Tang et al. proved a theorem regarding the split common fixed point problem for a k-quasi-strictly pseudocontractive mapping and an asymptotical nonexpansive mapping. In this paper, motivated by [11], we use the hybrid method to study the split common fixed point problem for an infinite family of multi-valued quasinonexpansive mappings and an infinite family of L-Lipschitzian continuous and (k, {µ n }, {ξ n })-totally asymptotically strictly pseudocontractive mappings. Compared to the Theorem of Tang et al. [10], we remove an extra condition and present a strong convergence theorem, which is more desirable than the weak convergence. The point is that the authors of [10] considered a semi-compact mapping, that is a mapping T on a set X having the property that if {x n } is a bounded sequence in X such that Tx n x n tends to zero, then {x n } has a convergent subsequence. We will not assume that our mappings are semi-compact, and at the same time, we propose a different algorithm; instead, we impose some restrictions on the control sequences to get the strong convergence. We also present an algorithm for solving the split common fixed point problem for totally quasi-φ-asymptotically nonexpansive mappings and for k-quasi-strictly pseudocontractive mappings. Under some mild conditions, we establish the strong convergence of these algorithms in Banach spaces. As applications, we consider the algorithms for a split variational inequality problem and a split common null point problem. Our results improve and generalize the result of Tang et al. [10], Takahashi [12], Moudafi [5], Censor et al. [13] and Byrne et al. [2]. 2. Preliminaries Let E be a real Banach space and C be a nonempty closed convex subset of E. A mapping T : C C is said to be {k n }-asymptotically nonexpansive if there exists a sequence {k n } [1, ) with k n 1, such that: T n x T n y k n x y, x, y C, n 1 The mapping T : C C is said to be k-quasi-strictly pseudocontractive if F(T) = and there exists a constant k [0, 1], such that: Tx p 2 x p 2 + k x Tx 2 x C, p F(T) The mapping T : C C is said to be (k, {µ n }, {ξ n })-totally asymptotically strictly pseudocontractive if there exist a constant k [0, 1] and null sequences {µ n } and {ξ n } in [0, ) and a continuous strictly increasing function ζ : [0, ) [0, ) with ζ(0) = 0, such that for all x, y H and n 1: T n x T n y 2 x y 2 + k (x y) (Tx Ty) 2 + µ n ζ( x y ) + ξ n A Banach space E is said to be uniformly smooth if ρ E(t) t 0 as t 0, where ρ E (t) is the modulus of smoothness of E. Let q > 1; then, E is called q-uniformly smooth if there exists a constant c > 0,

3 Mathematics 2017, 5, 11 3 of 18 such that ρ E (t) ct q for all t > 0. Throughout, J will stand for the duality mapping of E. We recall that a Banach space E is smooth if and only if the duality mapping J is single valued. Lemma 1. [14] If E is a two-uniformly smooth Banach space, then for each t > 0 and each x, y E: x + ty 2 x y, Jx + 2 ty 2 For a smooth Banach space E, Alber [15] defined: φ(x, y) = x 2 2 x, Jy + y 2, x, y E It follows that ( x y ) 2 φ(x, y) ( x + y ) 2 for each x, y E. Moreover, if we denote by Π C x the generalized projection from E onto a closed convex subset C in E, then we have: Lemma 2. [15] Let E be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of E. Then: (a) φ(x, Π C y) + φ(π C y, y) φ(x, y), for all x C and y E; (b) For x, y E, φ(x, y) = 0 if and only if x = y; (c) (d) For x, y, z E, φ(x, y) φ(x, z) + φ(z, y) + 2 x z, Jz Jy ; For x, y, z E, λ [0, 1], φ(x, J 1 (λjy + (1 λ)jz)) λφ(x, y) + (1 λ)φ(x, z). Lemma 3. [16] If E is a uniformly-smooth Banach space and r > 0, then there exists a continuous, strictly-increasing convex function g : [0, 2r] [0, ), such that g(0) = 0 and: φ(x, J 1 (λjy + (1 λ)jz)) λφ(x, y) + (1 λ)φ(x, z) λ(1 λ)g( Jy Jz ) for all λ [0, 1], x E and y, z B r = {u E : u r}. We denote by N(C), CB(C) and P(C) the collection of all nonempty subsets, nonempty closed bounded subsets and nonempty proximal bounded subsets of C, respectively. Let T : E N(E) be a multivalued mapping. An element x E is said to be a fixed point of T if x Tx. The set of fixed points of T is denoted by F(T). Definition 1. Let C be a closed convex subset of a smooth Banach space E and T : C N(C) be a multivalued mapping. We set: Φ(Tx, Tp) = max{ sup inf φ(y, q), sup inf φ(y, q)} q Tp y Tx y Tx q Tp We call T a quasinonexpansive multivalued mapping if F(T) = and: Φ(Tx, Tp) φ(x, p), p F(T), x C Definition 2. A multivalued mapping T is called demi-closed if lim n dist(x n, Tx n ) = 0 and x n w imply that w Tw. Let C be a nonempty closed convex subset of E and T := {T(s) : 0 s < } be a nonexpansive semigroup on C. We use Fix(T) to denote the common fixed point set of the semigroup T. It is well known that Fix(T) is closed and convex. A nonexpansive semigroup T on C is said to be uniformly asymptotically regular (u.a.r.) if for all h 0 and any bounded subset D of C: lim sup T(h)(T(t)x) T(t)x = 0 n x D

4 Mathematics 2017, 5, 11 4 of 18 For each h 0, define σ t (x) = 1 t t 0 T(s)xds. Then, lim t sup x D T(h)(σ t (x)) σ t (x) = 0 provided that D is a closed bounded convex subset of C. It is known that the set {σ t (x) : t > 0} is a u.a.r. nonexpansive semigroup; see [17]. A mapping T : E E is said to be α-averaged if T = (1 α)i + αs for some α (0, 1); here, I is the identity operator, and S : E E is a nonexpansive mapping (see [18]). It is known that in a Hilbert space setting, every firmly-nonexpansive mapping (in particular, a projection) is a 1 2 -averaged mapping (see Proposition 11.2 in the book [19]). Lemma 4. [20] (i) The composition of finitely many averaged mappings is averaged. In particular, if T i is α i -averaged, where α i (0, 1) for i = 1, 2, then the composition T 1 T 2 is α-averaged, where α = α 1 + α 2 α 1 α 2. (ii) If the mappings {T i } N are averaged and have a common fixed point, then N F(T i) = F(T 1 T N ). (iii) In case E is a uniformly-convex Banach space, every α-averaged mapping is nonexpansive. Lemma 5. [21] Let E be a uniformly-convex and smooth Banach space, and let {x n } and {y n } be two sequences in E. If φ(x n, y n ) 0 and either {x n } or {y n } is bounded, then x n y n 0. Lemma 6. [15] Let C be a nonempty closed convex subset of a smooth Banach space E and x E, then x 0 = Π C x if and only if for all y C, x 0 y, Jx Jx 0 0. Lemma 7. [22] Let E be a uniformly-convex Banach space, and let B r (0) = {x E : x r}, for r > 0, then there exists a continuous, strictly-increasing and convex function g : [0, ) [0, ) with g(0) = 0, such that, for any given sequence {x n } n=1 B r(0) and for any given sequence {α n } n=1 of positive numbers with n=1 a n = 1 and for any positive integers i, j with i < j : n=1 α n x n 2 α n x n 2 α i α j g( x i x j ). n=1 Lemma 8. [23] Let {α n } be a sequence in [0, 1], δ n and {γ n } be sequences in R, such that (i) n=1 α n =, (ii) lim sup n δ n 0 and (iii) γ n 0 and n=1 γ n <. If {a n } is a sequence of nonnegative real numbers, such that a n+1 (1 α n )a n + α n δ n + γ n, for each n 0, then lim n a n = 0. Lemma 9. [24] Let {s n } be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {s ni } of {s n }, such that s ni s ni+1 for all i 0. For every n N, define an integer sequence {τ(n)} as τ(n) = max{k n : s k < s k+1 }. Then, τ(n) and max{s τ(n), s n } s τ(n)+1. Lemma 10. [25] Let {λ n } and {γ n } be nonnegative and {α n } be positive real numbers, such that λ n+1 λ n α n λ n + γ n, n 0. Let for all n > 1, λ n α n c 1 and α n α. Then, λ n max{λ 1, K }, where K = (1 + α)c 1. Definition 3. (1) A mapping T : C C is said to be a k-quasi-strictly pseudocontractive mapping if there exists k [0, 1), such that Tx p 2 x p 2 + k x Tx 2, x C, p F(T). (2) A mapping T : C C is called quasinonexpansive if F(T) = ; and φ(p, Tx) φ(p, x) x C, p F(T). (3) A countable family of mappings {T i } : C C is said to be totally uniformly quasi-φ-asymptotically nonexpansive, if I = F(T i ) = and there exist nonnegative real sequences {µ n },{ν n } with µ n 0, ν n 0(as n ) and a strictly-increasing continuous function ζ : R + R + with ζ(0) = 0, such that φ(p, Ti nx) φ(p, x) + ν nζ(φ(p, x)) + µ n, n 1, i 1, x C, p I. (4) A mapping T : C C is said to be uniformly L-Lipschitzian continuous, if there exists a constant L > 0, such that T n x T n y L x y, x, y C, n 1. Lemma 11. [11] Let E be a real uniformly-smooth and uniformly-convex Banach space and C be a nonempty closed convex subset of E. Let T : C C be a closed and totally quasi-φ-asymptotically nonexpansive mapping

5 Mathematics 2017, 5, 11 5 of 18 with nonnegative real sequences {µ n }, {ν n } and a strictly-increasing continuous function ζ : R + R +, such that µ n 0, ν n 0 and ζ(0) = 0. If µ 1 = 0, then the fixed point set of T is closed and convex. Lemma 12. [26] Let C be a nonempty closed convex subset of a real Banach space E, and let T : C C be a k-quasi-strictly pseudocontractive mapping. If F(T) =, then F(T) is closed and convex. 3. Main Results This section is devoted to the main results of this paper. Theorem 1. Let E 1 be a real uniformly-convex and two-uniformly-smooth Banach space with the best smoothness constant t satisfying 0 < t < 1 2, and let E 2 be a real smooth Banach space. Let A : E 1 E 2 be a bounded linear operator and A be its adjoint. Suppose T : E 2 E 2 is a uniformly L-Lipschitzian continuous and (k, {µ n }, {ξ n })-totally asymptotically strictly pseudocontractive mapping satisfying the following conditions: (1) n=1 µ n <, n=1 ξ n <, (2) {r n } is a real sequence in (0, 1), such that µ n = o(r n ), ξ n = o(r n ), lim r n = 0, n=1 r n =, (3) there exist constants M 0 > 0, M 1 > 0, such that ζ(λ) M 0 λ 2, λ > M 1. Let {S n } n=1 : E 1 CB(E 1 ) be a family of multivalued quasinonexpansive mappings, such that for each i 1, S i is demi-closed at zero, and for each p Fix(S i ), S i (p) = {p}. Suppose: Ω = { x } F(S i ) : Ax F(T) = and {x n } is the sequence generated by x 1 E 1 : u n = (1 r n )x n y n = J1 1 (α n J 1 u n + (1 α n )γa J 2 (T n I)Au n ) x n+1 = J1 1(β n,0j 1 y n + β n,i J 1 w n,i ) w n,i S i y n (2) where γ (0, 2 A 2 ); the sequences {α n }, {β n,i } (0, 1) satisfy the following conditions: (a) i=0 β n,i = 1, lim inf n β n,0 β n,i > 0, (b) lim n α n = 1, n=1 (1 α n) <, (1 α n ) = o(r n ). Then, {x n } converges strongly to an element of Ω. Proof. Since ζ is continuous, ζ attains its maximum in [0, M 1 ], and by assumption, ζ(λ) M 0 λ 2, λ > M 1. In either case, we have ζ(λ) M + M 0 λ 2, λ [0, ). Let p Ω, then: From (2) and Lemma 2(d,c), we have: φ(p, u n ) (1 r n )φ(p, x n ) + r n p 2 (3)

6 Mathematics 2017, 5, 11 6 of 18 φ(p, y n ) α n φ(p, u n ) + (1 α n )φ(p, J 1 1 (γa J 2 (T n I)Au n )) α n φ(p, u n ) + (1 α n )[φ(p, u n ) + φ(u n, J 1 1 (γa J 2 (T n I)Au n )) + 2 p u n, J 1 u n γa J 2 (T n I)Au n ] = φ(p, u n ) + (1 α n )[ u n 2 + γ 2 A 2 (T n I)Au n 2 2 u n, γa J 2 (T n I)Au n + 2 p u n, J 1 u n + 2 p u n, γa J 2 (T n I)Au n ] φ(p, u n ) + (1 α n )[ p 2 + γ 2 A 2 (T n I)Au n 2 2 u n, γa J 2 (T n I)Au n + 2 p u n, γa J 2 (T n I)Au n ] (4) From Lemma 1, we have: 2 u n, γa J 2 (T n I)Au n γa J 2 (T n I)Au n tu n 2 u n + γa J 2 (T n I)Au n 2 γ 2 A 2 (T n I)Au n 2 + u n 2 = γ 2 A 2 (T n I)Au n u n 1 2 p p 2 (5) γ 2 A 2 (T n I)Au n 2 + 4( 1 2 u n p p 2 ) = γ 2 A 2 (T n I)Au n u n p p 2 ) Since Ap F(T) and T is a totally quasi-asymptotically strictly pseudocontractive mapping, we obtain: u n p, γa J 2 (T n I)Au n = γ A(u n p), J 2 (T n I)Au n = γ A(u n p) + (T n I)Au n (T n I)Au n, J 2 (T n I)Au n = γ( T n A(u n ) Ap, J 2 (T n I)Au n (T n I)Au n 2 ) γ( 1 2 [ (Tn I)Au n t(t n Au n Ap) 2 Ap Au n 2 ] (T n I)Au n 2 ) γ( 1 2 [ (Tn I)Au n 2 + (T n Au n Ap) 2 (6) Substituting (5) and (6) into (4), we have: Ap Au n 2 ] (T n I)Au n 2 ) γ( 1 2 [ Au n Ap 2 + k (T n I)Au n 2 + µ n ζ( Au n Ap ) + ξ n ]) 1 2 ( (Tn I)Au n 2 + Ap Au n 2 ) = γ( k 1 2 (Tn I)Au n 2 + µ n 2 [M + M 0 Au n Ap 2 ] + ξ n 2 ) φ(p, y n ) α n φ(p, u n ) + (1 α n )φ(p, J 1 1 (γa J 2 (T n I)Au n )) φ(p, u n ) + (1 α n )[3 p 2 + 2γ 2 A 2 (T n I)Au n u n p 2 + γ(k 1) (T n I)Au n 2 + γµ n [M + M 0 A 2 u n p 2 ] + γξ n φ(p, u n ) + 3(1 α n ) p 2 γ(1 k 2γ A 2 ) (T n I)Au n 2 (7) + γµ n M + (γµ n M 0 A 2 + 2) u n p 2 + γξ n From Lemma 1 and the fact that 0 < t < 1 2, we have:

7 Mathematics 2017, 5, 11 7 of 18 φ(p, y n ) α n φ(p, u n ) + (1 α n )φ(p, J 1 1 (γa J 2 (T n I)Au n )) φ(p, u n ) + 3(1 α n ) p 2 γ(1 k 2γ A 2 ) (T n I)Au n 2 + γµ n M + (γµ n M 0 A 2 + 2) u n p 2 + γξ n φ(p, u n ) + 3(1 α n ) p 2 γ(1 k 2γ A 2 ) (T n I)Au n 2 (8) + γµ n M + (γµ n M 0 A 2 + 2)[ u n 2 p, Ju n + 2 tp 2 ] + γξ n φ(p, u n ) + 3(1 α n ) p 2 γ(1 k 2γ A 2 ) (T n I)Au n 2 + γµ n M + (γµ n M 0 A 2 + 2)φ(p, u n ) + γξ n Putting (3) and (8) into (2), we obtain: φ(p, x n+1 ) = φ(p, J1 1(β n,0j 1 y n + β n,0 φ(p, y n ) + = β n,0 φ(p, y n ) + β n,0 φ(p, y n ) + β n,i β n,i J 1 w n,i )) β n,i φ(p, w n,i ) inf φ(p, w n,i) t S i (p) β n,i Φ(p, w n,i ) = φ(p, y n ) φ(p, u n ) + 3(1 α n ) p 2 γ(1 k 2γ A 2 ) (T n I)Au n 2 (9) + γµ n M + (γµ n M 0 A 2 + 2)φ(p, u n ) + γξ n (1 r n )φ(p, x n ) + r n p 2 + 3(1 α n ) p 2 γ(1 k 2γ A 2 ) (T n I)Au n 2 + γµ n M + (γµ n M 0 A 2 + 2)((1 r n )φ(p, x n ) + r n p 2 ) + γξ n φ(p, x n ) (r n γµ n M 0 A 2 + 2)(1 r n )φ(p, x n ) + (3(1 α n ) + r n + µ n γm 0 A 2 r n ) p 2 + γµ n M + γξ n φ(p, x n ) (r n (γµ n M 0 A 2 + 2))(1 r n )φ(p, x n ) + σ n where σ n = (3(1 α n ) + r n + µ n γm 0 A 2 r n ) p 2 + µ n γm + γξ n. Since µ n = o(r n ), (1 α n ) = o(r n ) and ξ n = o(r n ), we may assume without loss of generality that there exist constants k 0 (0, 1) and M 2 > 0, such that for all n 1: µ n r n(1 k 0 + 2) 2 r n r n (1 r n )γm 0 A 2 and σ n r n M 2 Thus, we obtain: φ(p, x n+1 ) φ(p, x n ) r n k 0 φ(p, x n ) + σ n (10) According to Lemma 10, φ(p, x n+1 ) max{φ(p, x 1 ), (1 + k 0 )M 2 }. Therefore, {φ(p, x n )} and {x n } are bounded. Furthermore, the sequences {y n } and {u n } are bounded, as well. We now consider two cases. Case 1. Suppose that there exists n 0 N, such that {φ(p, x n )} n=n 0 is nonincreasing. Then, {φ(p, x n )} n=1 converges, and φ(p, x n ) φ(p, x n+1 ) 0 as n. Since E 1 is a uniformly smooth Banach space, it follows from Lemma 3 and Equations (8) and (10) that:

8 Mathematics 2017, 5, 11 8 of 18 φ(p, x n+1 ) φ(p, y n ) α n φ(p, u n ) + (1 α n )φ(p, J 1 1 (γa J 2 (T n I)Au n )) α n (1 α n )g( J 1 u n γa J 2 (T n I)Au n ) φ(p, u n ) + 3(1 α n ) p 2 γ(1 k 2γ A 2 ) (T n I)Au n 2 + γµ n M + (γµ n M 0 A 2 + 2)φ(p, u n ) + γξ n α n (1 α n )g( J 1 u n γa J 2 (T n I)Au n φ(p, x n ) (r n (γµ n M 0 A 2 + 2))φ(p, u n ) + (3(1 α n ) + r n ) p 2 + γξ n γ(1 k 2γ A 2 ) (T n I)Au n 2 α n (1 α n )g( J 1 u n γa J 2 (T n I)Au n ) φ(p, x n ) r n k 0 φ(p, x n ) + σ n γ(1 k 2γ A 2 ) (T n I)Au n 2 α n (1 α n )g( J 1 u n γa J 2 (T n I)Au n ) (11) Hence, from (10), we have: α n (1 α n )g( J 1 u n γa J 2 (T n I)Au n ) φ(p, x n ) φ(p, x n+1 ) r n k 0 φ(p, x n ) + σ n and: γ(1 k 2γ A 2 ) (T n I)Au n 2 φ(p, x n ) φ(p, x n+1 ) r n k 0 φ(p, x n ) + σ n Therefore, α n (1 α n )g( J 1 u n γa J 2 (T n I)Au n ) and γ(1 k 2γ A 2 ) (T n I)Au n 2 tend to zero as n. Since lim inf α n (1 α n ) > 0 and γ (0, 2 A 2 ), we obtain: J 1 u n γa J 2 (T n I)Au n 0 n (12) (T n I)Au n 2 0 n (13) Furthermore, we observe that J 1 y n J 1 u n = (1 α n ) J 1 u n γa J 2 (T n I)Au n 0. Since J 1 1 is uniformly norm-to-norm continuous on bounded subsets, we conclude that: Using (7) and Lemma 3 in (2), we have: lim y n u n = 0 (14) n φ(p, x n+1 ) = φ(p, J1 1(β n,0j 1 y n + β n,0 φ(p, y n ) + β n,i J 1 w n,i )) φ(p, y n ) β n,0 β n,i g( J 1 y n J 1 w n,i ) β n,i φ(p, w n,i ) β n,0 β n,i g( J 1 y n J 1 w n,i ) φ(p, u n ) + 3(1 α n ) p 2 γ(1 k 2γ A 2 ) (T n I)Au n 2 + γµ n M + (γµ n M 0 A 2 + 2) u n p 2 + γξ n β n,0 β n,i g( J 1 y n J 1 w n,i ) (15) It now follows from (3) and γ (0, 2 A 2 ) that: β n,0 β n,i g( J 1 y n J 1 w n,i ) φ(p, x n ) φ(p, x n+1 ) (r n (γµ n M 0 A 2 + 2))φ(p, u n ) + (3(1 α n ) + r n ) p 2 + γξ n φ(p, x n ) φ(p, x n+1 ) r n k 0 φ(p, x n ) + σ n From Condition (a), we have lim n g( J 1 y n J 1 w n,i ) = 0. Since g is continuous and g(0) = 0, we obtain lim n J 1 y n J 1 w n,i = 0. Since J1 1 is uniformly norm-to-norm continuous on bounded subsets, we have: lim y n w n,i = 0 i N (16) n which implies that lim n dist(y n, S i y n ) lim n y n w n,i = 0, i N. From (2), we obtain: J 1 x n+1 J 1 y n = (1 β n,0 ) J 1 y n J 1 w n,i 0 n

9 Mathematics 2017, 5, 11 9 of 18 Since J is uniformly norm-to-norm continuous on bounded subsets, we have: x n+1 y n 0 n (17) From (14), (17) and lim n r n = 0, we have: x n+1 x n x n+1 y n + y n u n + u n x n = x n+1 y n + y n u n + r n x n 0 n Consequently: u n+1 u n = (1 r n+1 )x n+1 (1 r n )x n ) r n+1 r n x n+1 + (1 r n ) x n+1 x n 0 Using the fact that T is uniformly L-Lipschitzian, we have: n (18) TAu n Au n TAu n T n+1 Au n + T n+1 Au n T n+1 Au n+1 + T n+1 Au n+1 Au n+1 + Au n+1 Au n From (13) and (18), we obtain: L Au n T n Au n + (1 + L) Au n+1 Au n + T n+1 Au n+1 Au n+1 L Au n T n Au n + (1 + L) A u n+1 u n + T n+1 Au n+1 Au n+1 (T I)Au n 0, n (19) Since {x n } is bounded, there exists a subsequence {x nj } of {x n }, such that x nj z. Using the fact that x nj z and y n x n 0, n, we have that y nj z. Similarly, u nj z, since u n x n 0, n. Now, we show that z Ω. Since y nj z and lim n dist(y n, S i (y n )) = 0 and by the demi-closedness of each S i, we have z i N F(S i ). On the other hand, since A is a bounded operator, it follows from u nj z that Au nj Az. Hence, from (13), we have TAu nj Au nj 0 as j. Since T is demi-closed at zero, we have that Az F(T). Hence, z Ω. Next, we prove that {x n } converges strongly to z. From (7), Lemma 1 and γ (0, 2 A 2 ), we have: φ(z, x n+1 ) φ(z, y n ) α n φ(z, u n ) + (1 α n )φ(z, J 1 1 (γa J 2 (T n I)Au n )) α n φ(z, u n ) + (1 α n )[φ(z, u n ) + φ(u n, J 1 1 (γa J 2 (T n I)Au n )) + 2 z u n, J 1 u n γa J 2 (T n I)Au n ] φ(z, u n ) + (1 α n )[ z 2 + u n z + z 2 + 2γ 2 A 2 (T n I)Au n 2 + γ(k 1) (T n I)Au n 2 + γµ n [M + M 0 A 2 u n z 2 ] + γξ n φ(z, u n ) + (1 α n )[ z 2 + u n z 2 + z u n z, Jz + 2γ 2 A 2 (T n I)Au n 2 + γ(k 1) (T n I)Au n 2 (20) + γµ n [M + M 0 A 2 u n z 2 ] + γξ n φ(z, u n ) + (1 α n )( u n z + 2 u n, J 1 z ) + µ n M + γξ n (1 r n )φ(z, x n ) 2r n x n z, J 1 z + (1 α n )( u n z + 2 x n, J 1 z ) + µ n M + γξ n (1 r n )φ(z, x n ) 2r n x n z, J 1 z + (1 α n )( u n z x n, J 1 z + µ n M + γξ n

10 Mathematics 2017, 5, of 18 where M > γ sup n 0 (M + M 0 A 2 u n z 2 ) > 0. It is clear that 2 u n z, z 0, n, and n=1 M µ n <, n=1 γξ n < and n=1 (1 α n)( u n z x n, J 1 z <. Now, using Lemma 8 in (20), we have φ(z, x n ) 0. Therefore, x n z as n. Case 2. Assume that there exists a subsequence {x nj } of {x n }, such that φ(z, x nj ) < φ(z, x nj +1), j N. By Lemma 9, there exists a nondecreasing sequence {τ(n)} of N, such that for all n n 0 (for some n 0 large enough) τ(n) as n and such that the following inequalities hold: φ(z, x n ) < φ(z, x τ(n)+1 ), φ(z, x τ(n) ) < φ(z, x τ(n)+1 ) By a similar argument as in Case 1, we obtain: φ(z, x τ(n)+1 ) (1 r τ(n) )φ(z, x τ(n) ) 2r τ(n) x τ(n) z, J 1 z + (1 α τ(n) )( u τ(n) z x τ(n), J 1 z ) + γµ τ(n) M + γξ τ(n) (21) and lim x τ(n) z, J 1 z = 0. Since φ(z, x τ(n) ) φ(z, x τ(n)+1 ), we have: r τ(n) φ(z, x τ(n) ) φ(z, x τ(n) ) φ(z, x τ(n)+1 ) 2r τ(n) x τ(n) z, J 1 z By our assumption that r τ(n) > 0, we obtain: + (1 α τ(n) )( u τ(n) z x τ(n), J 1 z ) + γµ τ(n) M + γξ τ(n) φ(z, x τ(n) ) 2r τ(n) x τ(n) z, J 1 z + (1 α τ(n) )( u τ(n) z x τ(n), J 1 z ) + γµ τ(n) M + γξ τ(n) which implies that lim n φ( x, x τ(n) ) = 0. It now follows from (21) that lim n φ( x, x τ(n)+1 ) = 0. Now, since φ( x, x n ) < φ( x, x τ(n)+1 ), we obtain that φ( x, x n ) 0. Finally, we conclude from Lemma 5 that {x n } converges strongly to x. Theorem 2. Let E 1 be a real uniformly-convex and two-uniformly-smooth Banach space with the best smoothness constant t satisfying 0 < t < 1 2, and let E 2 be a real smooth Banach space. Let A : E 1 E 2 be a bounded linear operator and A be its adjoint. Let T i : E 2 E 2 (i N) be an infinite family of k-quasi-strict pseudocontractive mappings and {S i } : E 1 E 1 be an infinite family of uniformly L i -Lipschitzian continuous and totally quasi-φ-asymptotically nonexpansive mappings. Let {x n } be the sequence generated by x 1 E 1 : u n = J1 1 (α n,0j 1 x n + α n,i(γa J 2 (T i I)Ax n )) y n,m = J1 1(β n J 1 x 1 + (1 β n )J 1 Smx n n ) C n+1 = {z C n : sup m 1 φ(z, y n,m ) β n φ(z, x 1 ) + (1 β n )(φ(z, x n ) + x n 2 + z 2 ) + ξ n } x n+1 = Π Cn+1 x 1 where ξ n = ν n sup z Ω ζ(φ(z, u n )) + µ n, γ (0, 2 A 2 ), and Π Cn+1 is the generalized projection of E onto C n+1 ; and the sequences {α n }, {β n,i } (0, 1) and satisfy the following conditions: (a) {β n } [0, 1] and lim n β n = 0 (b) {α n,i } [0, 1], i=0 α n,i = 1 and lim n α n,0 = 1 If Ω = {x m=1 F(S m) : Ax F(T i)} is nonempty and bounded and µ 1 = 0, then {x n } converges strongly to: Π Ω u. Proof. (I) Both Ω and C n, n 1, are closed and convex. (22)

11 Mathematics 2017, 5, of 18 We know from Lemma 11 and Lemma 12 that F(T i ) and F(S i ), i 1, are closed and convex. This implies that Ω is closed and convex. Again, by the assumption, C 1 = E 1 is closed and convex. Now, suppose that C n is closed and convex for some n 1. In view of the definition of φ, we have: C n+1 = {z C n : sup φ(z, y n,m ) β n φ(z, x 1 ) + (1 β n )(φ(z, x n ) + 2 z, J 1 x n ) + ξ n } m 1 = m 1 {z E 1 : φ(z, y n,m ) β n φ(z, x 1 ) + (1 β n )(φ(z, x n ) + 2 z, J 1 x n ) + ξ n } C n = m 1 {z E 1 : 2β n z, J 1 x 1 + 2(1 β n ) z, J 1 x n 2 z, y n,m β n x (1 β n ) x n 2 y n,m 2 + z 2 } C n from which, it follows that C n+1 is closed and convex. (II) Ω C n, n 1. It is clear that Ω E 1. Suppose that Ω C n for some n 1. Let u Ω C n, then we have: φ(u, u n ) = φ(u, J1 1(α n,0j 1 x n + α n,0 φ(u, x n ) + φ(u, x n ) + α n,i (γa J 2 (T i I)Ax n ))) α n,i φ(u, J 1 1 (γa J 2 (T i I)Ax n )) α n,i [φ(x n, J 1 1 (γa J 2 (T i I)Ax n ) + 2 u x n, J 1 x n γa J 2 (T i I)Ax n φ(u, x n ) + α n,i [ x n u x n, J 1 x n + γ 2 A 2 (T i I)Ax n 2 2 x n, J 1 1 (γa J 2 (T i I)Ax n ) + 2 u x n, γa J 2 (T i I)Ax n (23) From Lemma 1, we have: 2 x n, γa J 2 (T i I)Ax n γa J 2 (T i I)Ax n tx n 2 x n + γa J 2 (T i I)Ax n 2 γ 2 A 2 (T i I)Ax n 2 + x n 2 (24) Since Au F(T i) and T i is a k-quasi-strictly pseudocontractive mapping: x n u, γa J 2 (T i I)Ax n = γ A(x n u), J 2 (T i I)Ax n = γ A(x n u) + (T i I)Ax n (T i I)Ax n, J 2 (T i I)Ax n = γ( T i A(x n ) Au, J 2 (T i I)Ax n (T i I)Ax n 2 ) γ( 1 2 ( T i Ax n Au 2 + (T i I)Ax n 2 )) γ (T i I)Ax n 2 = γ 2 ( T i Ax n Au 2 (T i I)Ax n 2 ) (25) Substituting (24) and (25) into (23), we obtain: γ 2 ( Ax n Au 2 + (k 1) (T i I)Ax n 2 ) 1 2 x n u 2 + γ 2 (k 1) (T i I)Ax n 2

12 Mathematics 2017, 5, of 18 φ(u, u n ) α n,0 φ(u, x n ) + φ(u, x n ) + φ(u, x n ) + α n,i φ(u, J 1 1 (γa J 2 (T i I)Ax n )) α n,i [2 u, J 1 x n γ(1 k 2γ A 2 ) (T i I)Ax n 2 + x n u 2 ] α n,i ( x n 2 + u 2 ) γ(1 k 2γ A 2 ) (T i I)Ax n 2 (26) It now follows from Lemma 2(d) and Equation (22): φ(u, y n,m ) β n φ(u, x 1 ) + (1 β n )φ(u, S m n u n ) β n φ(u, x 1 ) + (1 β n )[φ(u, u n ) + ν n ζ(φ(u, u n )) + µ n ] β n φ(u, x 1 ) + (1 β n )[φ(u, u n ) + ν n sup ζ(φ(u, u n )) + µ n ] u Ω = β n φ(u, x 1 ) + (1 β n )(φ(u, u n ) + ξ n ) m 1 β n φ(u, x 1 ) + (1 β n )(φ(u, x n ) + α n,i ( x n 2 + u 2 ) + ξ n ) γ(1 2γ A 2 ) (T i I)Ax n 2 m 1 β n φ(u, x 1 ) + (1 β n )(φ(u, x n ) + α n,i ( x n 2 + u 2 ) + ξ n ) m 1 (27) Therefore, we have: sup m 1 φ(u, y n,m ) β n φ(u, x 1 ) + (1 β n )(φ(u, x n ) + α n,i ( x n 2 + u 2 ) + ξ n ) β n φ(u, x 1 ) + (1 β n )(φ(u, x n ) + x n 2 + u 2 + ξ n ) (28) This argument shows that u C n+1, and so, F C n+1. (III) {x n } converges strongly to some point p E 1. Since x n = Π Cn x 1, from Lemma 6, we have x n y, J 1 x 1 J 1 x n 0, y C n. Again, since Ω C n, we obtain x n u, J 1 x 1 J 1 x n 0, u Ω. It now follows from Lemma 2(a) that for each u Ω and each n 1: φ(x n, x 1 ) = φ(π Cn x 1, x 1 ) φ(u, x 1 ) φ(u, x n ) φ(u, x 1 ) (29) Therefore, {φ(x n, x 1 )} is bounded, and so is {x n }. Since x n = Π Cn x 1 and x n+1 = Π Cn +1x 1 C n+1 C n, we have φ(x n, x 1 ) φ(x n+1, x 1 ), n 1. This implies that {φ(x n, x 1 )} is nondecreasing. Hence, lim n φ(x n, x 1 ) exists. Since E is reflexive, there exists a subsequence x ni x n, such that x ni p (some point in E 1 ). Since C n is closed and convex and C n+1 C n, it follows that C n is weakly closed and p C n for each n 1. Now, in view of x ni = Π Cni x 1, we have φ(x ni, x 1 ) φ(p, x 1 ), n i 1. Since the norm. is weakly lower semicontinuous, we have: lim inf n i φ(x n i, x 1 ) = lim inf n i { x n i 2 + x x ni, J 1 x 1 } p 2 + x p, x 1 = φ(p, x 1 ) and so: φ(p, x 1 ) lim inf φ(x n n i i, x 1 ) lim sup φ(x ni, x 1 ) φ(p, x 1 ) n i This implies that lim ni φ(x ni, x 1 ) = φ(x 1, p ), and so, x ni p. Since x ni p and E 1 is uniformly convex, we obtain lim ni x ni = p. Now, the convergence of {φ(x n, x 1 )}, together

13 Mathematics 2017, 5, of 18 with lim ni φ(x ni, x 1 ) = φ(p, x 1 ), implies that lim n φ(x n, x 1 ) = φ(p, x 1 ). If there exists some subsequence {x nj } {x n }, such that x nj q, then from Lemma 2(a), we have: φ(p, q) = i.e., p = q, and so: lim φ(x n n i,n j i, x nj ) = lim φ(x n n i,n j i, Π Cj x 1 ) lim (φ(x n n i,n j i, x 1 ) φ(x nj, x 1 )) = φ(p, q) φ(p, q) = 0 By the way, it follows from from (26) that φ(u, u n ) is bounded, so: lim (φ(x n n i,n j i, x 1 ) φ(π Cj x 1, x 1 )) lim x n = p (30) n lim ξ n = lim {ν n sup ζ(φ(p, u n )) + µ n } = 0 (31) n n p Ω (IV) p Ω. Since x n+1 C n+1, from (28), (30) and (31): sup m 1 φ(x n+1, y n,m ) β n φ(x n+1, x 1 ) + (1 β n )[φ(x n+1, x n ) + α n,i( x n 2 + x n+1 2 ) + ξ n ] 0 (32) Since x n+1 C n+1, from (27) and (32) we have: γ(1 k 2γ A 2 ) (T i I)Ax n 2 β n φ(x n+1, x 1 ) + (1 β n )(φ(x n+1, x n ) Since γ (0, 2 A 2 ), we have: + α n,i ( x n x n 2 ) + ξ n ) φ(x n+1, y n,m ) 0 n (33) (T i I)Ax n 0 n (34) Since x n p, it follows from (32) and Lemma 5 that for each m 1: lim y n,m = p (35) n Since {x n } is a bounded sequence and {S m } m=1 is uniformly totally quasi-asymptotically nonexpansive, {Smx n n } m,n=1 is uniformly bounded. In view of β n 0 and (22), we conclude that for each m 1: J 1 y n,m J 1 Smx n n = lim β n J 1 x 1 J 1 S n n mx n = 0 (36) Since for each m 1, J 1 y n,m J 1 p, it follows that for each m 1, lim n J 1 S n mx n = J 1 p. Since J 1 is continuous on each bounded subset of E 1, for each m 1: lim n Sn mx n = p (37) On the other hand, by the assumption that for each m 1, S m is uniformly L m -Lipschitzian continuous, we have: Sm n+1 x n Smx n n Sm n+1 x n Sm n+1 x n+1 + Sm n+1 x n+1 x n+1 + x n+1 x n + x n Smx n n (L m + 1) x n+1 x n + Sm n+1 x n+1 x n+1 + x n Smx n n (38) From (37) and x n p, we have that lim n S n+1 m x n S n mx n = 0 and lim n S n+1 m x n = p, i.e., lim n S m S n mx n = p. In view of the closedness of S m, it follows that S m p = p, i.e., for each

14 Mathematics 2017, 5, of 18 m 1, p F(S m ). By the arbitrariness of m 1, we have p m=1 F(S m). On the other hand, since A is bounded, it follows from x ni p that Ax ni Ap. Hence, from (34), we have that: T i Ax ni Ax ni 0, i Since T i is demi-closed at zero, we have that Az F(T i ). Hence, z Ω. (V) Finally, p Π Ω x 1, and so, x n Π Ω x 1. Let w = Π Ω x 1. Since w Ω C n and x n = Π Cn x 1, we have φ(x n, x 1 ) φ(w, x 1 ), n 1. This implies that φ(p, x 1 ) = lim n φ(x n, x 1) φ(w, x 1 ). Since w = Π Ω x 1, it follows that p = w, and hence, x n p = Π Ω x 1. Corollary 1. Let E 1 be a real uniformly-convex and two-uniformly-smooth Banach space with the best smoothness constant t satisfying 0 < t < 1 2, and let E 2 be a real smooth Banach space. Let A : E 1 E 2 be a bounded linear operator and A be its adjoint. Let T : E 2 E 2 be a k-quasi-strict pseudocontractive mapping and T be demi-closed at zero. Let {S n } n=1 : E 1 CB(E 1 ) be a family of multivalued quasinonexpansive mappings, such that for each i 1, S i is demi-closed at zero. Assume that for each p Fix(S i ), S i (p) = {p}. Let {x n } be the sequence generated by x 1 E 1 : u n = (1 r n )x n y n = J1 1 (α n J 1 u n + (1 α n )γa J 2 (T I)Au n ) x n+1 = J1 1(β n,0j 1 y n + β n,i J 1 w n,i ) w n,i S i y n where γ (0, 2 A 2 ); the sequences {α n }, {β n,i } (0, 1) satisfy the following conditions: (a) i=0 β n,i = 1 and lim inf n β n,0 β n,i > 0, (b) lim n α n = 1, n=1 (1 α n) < and (1 α n ) = o(r n ). Then, {x n } converges strongly to an element of Ω. Proof. Since every k-quasi-strictly pseudocontractive mapping is clearly (k, 0, 0)-totally asymptotically strictly pseudocontractive, the result follows. Corollary 2. Let E 1 be a real uniformly-convex and two-uniformly-smooth Banach space with the best smoothness constant t satisfying 0 < t < 1 2, and let E 2 be a real smooth Banach space. Let A : E 1 E 2 be a bounded linear operator and A be its adjoint. Let T : E 2 E 2 be a uniformly L-Lipschitzian continuous and (k, {µ n }, {ξ n })-totally asymptotically strictly pseudocontractive mapping satisfying the following conditions: (a) n=1 µ n <, n=1 ξ n <, (b) {r n } is a real sequence in (0, 1), such that µ n = o(r n ), ξ n = o(r n ), lim r n = 0, n=1 r n =, (c) there exist constants M 0 > 0, M 1 > 0, such that ζ(λ) M 0 λ 2, λ > M 1. Let F = {S(t) : 0 t < } be a one-parameter nonexpansive semigroup on E 1. Suppose further that Ω = {x t 0 F(S(t)) : Ax F(T)} =, and {x n } is the sequence generated by x 1 E 1 : u n = (1 r n )x n y n = J1 1 (α n J 1 u n + (1 α n )γa J 2 (T n I)Au n ) x n+1 = J1 1(β n J 1 y n + (1 β n )( t 1 tn n 0 S(u)duJ 1y n ) where γ (0, 2 A 2 ); the sequence {α n } (0, 1), 0 < ɛ β n b < 1, and lim n α n = 1, n=1 (1 α n) < and (1 α n ) = o(r n ). Then, {x n } converges strongly to to an element of Ω.

15 Mathematics 2017, 5, of 18 Proof. Since {σ t (x) = 1 t from Corollary 1. t 0 S(u)xdu : t 0} is a u.a.r. nonexpansive semigroup, the result follows In the following, we shall provide an example to illustrate the main result of this paper. Example 1. Let C be the unit ball of the real Hilbert space l 2, and let T : C C be a mapping defined by: T(x 1, x 2,...) = (0, x 1, a 2 x 2, a 3 x 3,...) where {a i } is a sequence in (0, 1), such that i=2 a i = 1 2. It was shown in [27] that T is a (0, k n 1, ξ n ) totally asymptotically strictly pseudocontractive mapping and F(T) = {0}, where k n = 2 n i=2 a i. Let B be the unit interval in R, and let S i : B B be a mapping defined by: Then, Fix(S i) = {0} and: 1 S i (x) = 2 ix x [0, 1 2 ] 0 x ( 1 2, 1] S i x 0 = 1 2 i x 0 = 1 x x 2i Therefore, each S i is a quasinonexpansive mapping. Let A : B C be the linear operator defined by: A(x) = (0, x, a 2 x, a 3 a 2 x, a 4 a 3 a 2 x,...), x B R. Then, A is bounded and A = 1 + a (a 3a 2 ) 2 + (a 4 a 3 a 2 ) 2 +. It now follows that: A : C B, A (x 1, x 2, ) = x 2 + a 2 x 3 + a 3 a 2 x 4 + a 4 a 3 a 2 x 5 +. We now put, for n N, α n = 1 3, r n = 1 n, β n,0 = 1 2, β n,0 = 1 3 i and λ = 1 4 (1 + a (a n a 2 ) 2 ). Furthermore, we have: Ω = {x F(T) : Ax F(S i)} = {0} Now, all of the assumptions in Theorem 1 are satisfied. Let us consider the following numerical algorithm: T n (x 1, x 2,...) = (0, 0,..., 0, a n...a 2 x 2 1, a n+1...a 2 x 2,...) T n (Au n ) Au n = (0, u n, a 2 u n, a 3 a 2 u n,..., a n...a 2 u n, 0, 0,... A (T n (Au n ) Au n ) = u n (1 + a (a 3a 2 ) (a n...a 2 ) 2 ) y n = 1 6 u n = 1 6 (1 1 n )x n, x n+1 = 1 2 y n + x n+1 = 1 60 (1 1 n )x n By Theorem 1, the sequence {x n } converges to the unique element of Ω. 4. Application 1 3 i ( 1 2 i y n) = 1 10 y n Let E be a uniformly-smooth Banach space, E be the dual of E, J be the duality mapping on E and F : E 2 E be a multi-valued operator. Recall that F is called monotone if u v, x y 0, for any (x, u), (y, v) G(F), where G(F) = {(x, u) : x D(F), u F(x)}. A monotone operator F is said to be maximally monotone if its graph G(F) is not properly contained in the graph of any other monotone operator. For a maximally-monotone operator F : E 2 E and r > 0, we can define a single-valued operator:

16 Mathematics 2017, 5, of 18 J F r = (J + rf) 1 J : E E It is known that for any r > 0, J F r is firmly nonexpansive, and its domain is all of E, also 0 F(x) if and only if x Fix(J F r ). Theorem 3. Let E 1 be a real uniformly-convex and two-uniformly-smooth Banach space with the best smoothness constant t satisfying 0 < t < 1/ 2, and let E 2 be a real smooth Banach space and T : E 1 E 2 be a bounded linear operator. Let A : E 2 2 E 2 and B i : E 1 2 E 1, for i = 1, 2,..., be maximal monotone mappings, such that A 1 0 = and B 1 i 0 =. Suppose: Ω = {x E 1 : 0 B i(x) such that 0 A(Tx)} = Let {x n } be a sequence generated by x 0 E 1 and: u n = (1 r n )x n y n = J1 1 (α n J 1 u n + (1 α n )γt J 2 (Jr A Tu n Tu n ) x n+1 = J1 1(β n,0j 1 y n + β n,i J 1 J B i µ y n where r, µ > 0, γ (0, 2 T 2 ), and the sequences {α n }, {β n,i } (0, 1) satisfy the following conditions: (1) i=0 β n,i = 1 and lim inf n β n,0 β n,i > 0, (2) lim n α n = 1, n=1 (1 α n) < and (1 α n ) = o(r n ). Then, {x n } converges strongly to an element of Ω. Proof. Since J A r and J B i µ are nonexpansive, the result follows from Corollary 1. Remark 1. Set S i = J B i r in Corollary 1, where B i is a maximal monotone mapping, then Corollary 1 improves Theorem 4.2 of Takahashi et al. [12]. Moudafi [28] introduced the split monotone variational inclusion (SMVIP) in Hilbert spaces. We present the SMVIP in a Banach space. Let E 1 and E 2 be two real Banach spaces and J 1 and J 2 be the duality mapping of E 1 and E 2, respectively. Given the operators f : E 1 E 1, g : E 2 E 2, a bounded linear operator A : E 1 E 2 and two multi-valued mappings B 1 : E 1 2 E 1 and B 2 : E 2 2 E 2, the SMVI is formulated as follows: find a point x C such that 0 J 1 ( f (x)) + B 1 (x) and such that the point: y = A(x) E 2 solves 0 J 2 (g(y)) + B 2 (y) Note that if C and Q are nonempty closed convex subsets of E 1 and E 2, (resp.) and B 1 = N C and B 2 = N Q, where N C and N Q are normal cones to C and Q (resp.), then the split monotone variational inclusion problem reduces to the split variational inequality problem (SVIP), which is formulated as follows: find a point: x C such that J 1 ( f (x)), w x 0 for all w C and such that the point: y = Ax Q solves J 2 (g(y)), z y 0 for all z Q SVIP is quite general and enables the split minimization between two spaces in such a way that the image of a solution of one minimization problem, under a given bounded linear operator, is a solution of another minimization problem.

17 Mathematics 2017, 5, of 18 Let h : C E be an operator, and let C E. The operator h is called inverse strongly monotone with constant β > 0, or in brief (β ism), on E if: h(x) h(y), Jx Jy β h(x) h(y) 2, x, y C Remark 2. If h : E E is an α ism operator on E and B : E 2 E is a maximal monotone mapping, then Jλ B (I λh) is averaged for each λ (0, 2α). Theorem 4. Let E 1 be a real uniformly-convex and two-uniformly-smooth Banach space with the best smoothness constant t satisfying 0 < t < 1/ 2, and let E 2 be a real smooth Banach space and T : E 1 E 2 be a bounded linear operator. Let A : E 2 2 E 2 and, for i = 1, 2,..., B i : E 1 2 E 1 be maximal monotone mappings, such that A 1 0 = and B 1 i 0 = ; and that h : E 2 E 2 is an α ism operator and g i : E 1 E 1 is a γ i ism operator. Assume that ρ = α in f i N γ i > 0 and τ (0, 2ρ). Suppose SMVI: { x B 1 i 0 0 J 1 (g i (x)) + B i (x) i N Tx A J 2 (h(tx)) + A(Tx) has a nonempty solution set Ω. Let {x n } be a sequence generated by x 0 E 1 and: u n = (1 r n )x n y n = J1 1 (α n J 1 u n + (1 α n )γt J 2 ((Jr A (I τh) I)Tu n )) x n+1 = J1 1(β n,0j 1 y n + β n,i J 1 J B i µ (I τg i )y n ) where γ (0, 2 T 2 ); the sequences {α n }, {β n,i } (0, 1) satisfy the following conditions: (1) i=0 β n,i = 1 and lim inf n β n,0 β n,i > 0, (2) lim n α n = 1, n=1 (1 α n) < and (1 α n ) = o(r n ). Then, {x n } converges strongly to an element of Ω. Proof. The results follow from Remark 2, Lemma 4(iii) and Corollary 1. We mention in passing that the above theorem improves and extends Theorems 6.3 and 6.5 of [13] to Banach spaces. Indeed, we removed an extra condition and obtained a strong convergence theorem, which is more desirable than the weak convergence already obtained by the authors. Acknowledgments: We wish to thank the academic editor for his right choice of reviewers, and the anonymous reviewers for their comments and criticisms. Author Contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Conflicts of Interest: The authors declare no conflict of interest. References 1. Alsulami, S.M.; Latif, A.; Takahashi, W. Strong convergence theorems by hybrid methods for the split feasibility problem in Banach spaces. J. Nonlinear Convex Anal. 2015, 16, Byrne, C.; Censor, Y.; Gibali, A.; Reich, S. The split common null point problem. J. Nonlinear Convex Anal. 2012, 13, Masad, E.; Reich, S. A note on the multiple-set split feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 2008, 7, Censor, Y.; Segal, A. The split common fixed point problem for directed operators. J. Convex Anal. 2009, 16,

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