Strong Convergence Theorem of Split Equality Fixed Point for Nonexpansive Mappings in Banach Spaces

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1 Applied Mathematical Sciences, Vol. 12, 2018, no. 16, HIKARI Ltd Strong Convergence Theorem of Split Euality Fixed Point for Nonexpansive Mappings in Banach Spaces Xuejiao Zi, Zhaoli Ma 1, Yali Liu and Lili Yang Department of General Education The College of Arts and Sciences Yunnan Normal University, Kunming, Yunnan, , P.R. China Copyright c 2018 Xuejiao Zi, Zhaoli Ma, Yali Liu and Lili Yang. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The purpose of this paper is to propose an algorithm to solve the split euality fixed point problem of nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. The strong convergence theorem of the iterative scheme proposed in this paper is obtained without the assumption of semi-compactenss on the mappings. The results presented in the paper are new and extend some recent corresponding results. Mathematics Subject Classifications: 47H09, 47J25 Keywords: Split euality fixed point problem; Nonexpansive mappings; Strong convergence; Banach spaces 1 Introduction The split feasibility problem(sep) was originally put forward in finite-dimensional Hilbert spaces by Censor and Elfving [1] for modeling inverse problems which arise 1 Corresponding author. This work was supported by the Scientific Reserch Foundation of College of Arts and Sciences Yunnan Normal University (No. 17KJY030).

2 740 Xuejiao Zi, Zhaoli Ma, Yali Liu and Lili Yang from phase retrievals and in medical image reconstruction [2]. Its precise mathematical formulation is as follows: Let C and Q be nonempty closed and convex subsets of real Hilbert spaces H 1 and H 2 respectively. The split feasibility problem(sep) is to find a point x C such taht: x C and Ax Q, (1.1) where A : H 1 H 2 is a bounded linear operator. We use Ω to denote the set of solution of SEP (1.1), that is Ω = {x C : Ax Q}. The split feasibility problem was studied extensively due to it is an extremely powerful tool in various disciplines such as signal processing, image restoration, computer tomograph and radiation therapy treatment planning, for details see [3-5]. Recently, Byrne developed split feasibility problem (1.1) in the setting of infinite-dimensional Hilbert spaces, see also [6-8] and the references therein. If C and Q are the sets of fixed points of two nonlinear mappings, respectively, and C and Q are nonempty closed convex subsets, then the problem (1.1) is said to be a split common fixed point problem for the two nonlinear mappings. That is, the split common fixed point problem(scep) for mappings S and T is to find a point x H 1 with the property: x F (S) and Ax F (T ), (1.2) where S : H 1 H 1, T : H 2 H 2 are two nonlinear operators, F (S) and F (T ) denote the fixed point sets of S and T, respectively. In [9], Moudafi proposed split euality problem(sep) which is formulated as finding points x and y with the property: x C and y Q, such that Ax = By, (1.3) where C and Q are two nonempty closed convex subsets of H 1 and H 2, respectively, A : H 1 H 3, B : H 2 H 3 are two bounded linear operators and H 1, H 2, H 3 be real Hilbert spaces. When C := F (S), Q := F (T )(where S : H 1 H 3, T : H 2 H 3 be two nonlinear operators with nonempty fixed point sets F (S) and F (T )), then split euality problem(1.3) is called split euality fixed point problem(sefp), that is: F inding x F (S) and y F (T ), such that Ax = By, (1.4) which is a generalization of the split feasibility problem and the split common fixed point problem. It allows asymmetric and partial relations between the variables x and y. It includes many situations in the fields of science, such as decomposition methods for P DE s, applications in game theory and in intensity modulated radiation therapy(imrt) for further details, the interested reader is referred to [3-4,10]. We use Ω to denoted the set of solutions of the split euality fixed point problem(1.4), that is Ω = {(x, y ) : x F (s) : y F (T ) such that Ax = By }.

3 Strong convergence theorem of split euality fixed point 741 Concerning the strong and weak convergence theorems for finding a solution of split feasibility problems have been studied by many authors in Hilbert spaces(see, e.g.,[11-13]). There is, however, seldom results in the existing literature on split feasibility problems in the setting of two Banach spaces. In 2015, Takahashi and Yao [14] by using hybrid methods and under suitable conditions, obtained some strong and weak convergence theorems for the split feasibility problem and split common null point problem in the setting of one Hilbert space and one Banach space. Then, Tang et al.[15] proved a weak convergence theorem and a strong convergence theorem for split common fixed point problem involving a uasi-strict pseudo contractive mapping and an asymptotical nonexpansive mapping in the setting of two Banach spaces under the assumption of semi-compactenss on the mappings. Motivated by the works of Tang et al., Ma et al.[16] proposed the following iterative algorithm to approximate a solution of the split euality common fixed point problems of two asymptotically nonexpansive semigroups in the setting of two Banach spaces. For any given x 0 E 1 and y 0 E 2, the seuence {(x n, y n )} is defined as follows: z n J 3 (Ax n By n ) u n = S(t n )(x n γ n J1 1 A z n ) v n = T (t n )(y n + γ n J2 1 B z n ) (1.5) y n+1 = β n v n + (1 β n )(y n + γ n J2 1 B z n ) x n+1 = β n u n + (1 β n )(x n γ n J2 1 A z n ), Under some suitable conditions, strong and weak convergence theorems are established. In [17], Y.Shehu et al. suggested the following iterative scheme to study split feasibility problems and fixed point problems concerning left Bregman srongly relatively nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. { xn = Π C J E [J p 1 E 1 u n t n A J p E 2 (Au n P Q (Au n ))] u n+1 = Π C J E [α nj p 1 E 1 u + (1 α n )J p (1.6) E 1 (T x n )]. Very recently, Zhou et al.[18] constructed the following iterative scheme to solve split euality problem and proved the strong convergence in p-uniformly convex and uniformly smooth Banach spaces. For a fixed u C and a fixed v Q, the seuence {(x n, y n )} is generated by u n = J E [J p 1 E 1 x n λ n A J p E 3 (Ax n By n )] v n = J E [J p 2 E 2 y n + λ n B J p E 3 (Ax n By n )] x n+1 = Π C J E (β nj p 1 E 1 u + (1 β n )J p (1.7) E 1 u n ) y n+1 = Π Q J E (β nj p 2 E 2 v + (1 β n )J p E 2 v n ), In this paper, Motivated by works above, we propose the following new iterative algorithm to solve split euality fixed point problem(1.4) involved in left Bregman srongly nonexpansive mappings in p-uniformly convex and uniformly smooth Banach

4 742 Xuejiao Zi, Zhaoli Ma, Yali Liu and Lili Yang spaces. For a fixed u E 1 and a fixed v E 2, the seuence {(x n, y n )} is generated by u n = Π C J E [J p 1 E 1 x n γ n A J p E 3 (Ax n By n )] v n = Π Q J E [J p 2 E 2 y n + γ n B J p E 3 (Ax n By n )] x n+1 = Π C J E (α nj p 1 E 1 u + (1 α n )(β n J p E 1 u n + (1 β n )J p (1.8) E 1 Su n )) y n+1 = Π Q J E (α nj p 2 E 2 v + (1 α n )(β n J p E 2 v n + (1 β n )J p E 2 T v n )), The strong convergence theorem of the iterative scheme proposed is obtained without the assumption of semi-compactenss on the mappings. The results presented in this paper are new and improve and extend [16-18] results. 2 Preliminaries Let E be a real Banach spaces and let 1 < 2 p with p modulus of convexity δ E : [0, 2] [0, 1] is defined by δ E (ɛ) = inf{1 x + y 2 : x = y = 1, x y ɛ}. = 1. The E is called uniformly convex if δ E (ɛ) > 0 for any ɛ (0, 2]; p-uniformly convex if there is a c p > 0 such that δ E (ɛ) c p ɛ p for any ɛ (0, 2]. The modulus of smoothness ρ E (τ) : [0, ) [0, ) is defined by x + τy + x τy ρ E (τ) = { 1 : x = y = 1}. 2 ρ E is called uniformly smooth if lim E (τ) = 0; E is called -uniformly smooth τ if there is a C > 0 so that ρ E (τ) C τ for any τ > 0. It is well know that if E is p-uniformly convex and uniformly smooth, then its dual space E is -uniformly smooth and uniformly convex. In this situation, the duality mapping J p E is one-toone, single-valued and satisfies J p E = (J E ) 1, where J E is the duality mapping of E. Definition 2.1. ([19]) The duality mapping J p E : E 2E J p E (x) = {x E : x, x = x p, x = x p 1 }. is defined by The duality mapping J p E is said to be weak-to-weak continuous if x n x J p E x n, y J p Ex, y, y E. Lemma 2.2. ([20]) Let x, y E. If E is -uniformly smooth, then there is a C > 0 such that x y x y, J E (x) + C y.

5 Strong convergence theorem of split euality fixed point 743 Definition 2.3. Given a Gâteaux differentiable convex function f : E R. The Bregman distance with respect to f is defined as: f (x, y) := f(y) f(x) f (x), y x, x, y E. It is well know that the duality mapping J p E is the derivative of the function f p(x) = 1 p x p. For sake of convenience, the fp (x, y) denotes by p (x, y), then the Bregman distance with respect to f p can be written as follows From the definition of p (, ), we get and fp (x, y) = 1 ( x p y p ) J p E x J p Ey, y. p (x, y) = p (x, z) + p (z, y) + z y, J p E x J p Ez, (2.1) p (x, y) + p (y, x) = x y, J p E x J p Ey. (2.2) for any x, y, z E. Likewise the definition of metric projection, the Bregman projection is defined as follows: Π C x = argmin y C p (x, y), x E, is the uniue minimizer of the Bregman distance (see[21]). The Bregman projection can also be characterized by a variational ineuality: from which one has J p E x J p E (Π Cx), z Π C x 0, z C, (2.3) p (Π C x, z) p (x, z) p (x, Π C x), z C. (2.4) Let C be a convex subset of int domf p, where f p = ( 1 p ) x p, 2 p < and let T be a self-mapping of C. A point p C is said to be an asymptotic fixed point of T if C contains a seuence {x n } n=1 which converges weakly to p and lim x n T x n = 0. The set of asymptotic fixed points of T is denoted by F (T ). Recalling that the Bregman distance is not symmetric, we define the following operators. Definition 2.4. A mapping T : C C with a nonempty asymptotic fixed point set is said to be: (i) left Bregman strongly nonexpansive (see[22]) with respect to a nonempty F (T ) if p (T x, p) p (x, p), x C, p F (T )

6 744 Xuejiao Zi, Zhaoli Ma, Yali Liu and Lili Yang and if whenever {x n } C is bounded, p F (T ) and it follows that lim ( p(x n, p) p (T x n, p)) = 0, lim p(x n, T x n ) = 0. According the Martin-Maruez et al.[22], a lef t Bregman strongly nonexpansive mapping T with respect to a nonempty F (T ) is called strictly left Bregman strongly nonexpansive mapping. (ii) An operator T : C int domf is said to be: left Bregman firmly nonexpansive(lbfne)if J p E (T x) J p E (T y), T x T y J p E (T x) J p E (T y), x y for any x, y C, or euivalently, p (T x, T y) + p (T y, T x) + p (x, T x) + p (y, T y) p (x, T y) + p (y, T x). It is known every left Bregman firmly nonexpansive mapping is left Bregman strongly nonexpansive with respect to F (T ) = F (T ). Lemma 2.5. ([23]) Let x, y E. If E is p-uniformly convex space, the metric and Bregman distance has the following relation: τ x y p p (x, y) x y, J p E x J p E y. where τ > 0 is some fixed number. Obviously, if {x n } and {y n } are both bounded seuences of a p-uniformly convex and uniformly smooth space E, then x n y n 0 as n implies that p (x n, y n ) 0 as n. Following [24,25], the function V p : E E [0, + ) associated with f p, which is defined by Then V p is nonnegative and V p ( x, x) = 1 x x, x + 1 p x p, x E, x E. V p ( x, x) = p (J p E ( x), x) = p(j E ( x), x), (2.5) for all x E and x E. Moreover, by the subdifferential ineuality, V p ( x, x) + ȳ, J E( x) x V p ( x + ȳ, x). (2.6) for all x E and x, ȳ E (see [16], [22]). In addition, V p is convex in the first variable. Thus, for all z E N p (J E ( t i J p E x i), z) i=1 N t i p (x i, z). (2.7) i=1

7 Strong convergence theorem of split euality fixed point 745 Lemma 2.6. ([20]) Let {a n } be a seuence of nonnegative real numbers satisfying the following ineuality: a n+1 (1 α n )a n + α n σ n + γ n, n 1, where, {α n }, {σ n }, {γ n } satisfy the following conditions: (i){α n } [0, 1], αn = ; (ii) lim sup σ n < 0; (iii)γ n 0 (n 1), γn <. Then a n 0 as n. Lemma 2.7. ([26]) Let {a n } be a seuence of real numbers such that there exists a subseuence {n i } of {n} such that a ni < a ni +1 for all i N. Then there exists a nondecreasing seuence {m k } N such that m k and the following properties are satisfied by all (sufficiently large) numbers k N: In fact, m k = max{j k : a j < a j+1 }. a mk a mk +1 and a k a mk Main Results Theorem 3.1. Let E 1, E 2 and E 3 be three p-uniformly convex and uniformly smooth Banach spaces. Let C and Q be nonempty closed and convex subsets of E 1 and E 2, respectively. Let A : E 1 E 3, B : E 2 E 3 be two bounded linear operators and A : E3 E1, B : E3 E2 be the adjoint operators of A and B, respectively. Let S be a lef t Bregman strongly nonexpansive mapping of C into itself such that F (S) = F (S), T be a left Bregman strongly nonexpansive mapping of Q into itself such that F (T ) = F (T ), For a fixed u E 1 and a fixed v E 2, the seuence {(x n, y n )} is generated by u n = Π C J E [J p 1 E 1 x n γ n A J p E 3 (Ax n By n )] v n = Π Q J E [J p 2 E 2 y n + γ n B J p E 3 (Ax n By n )] x n+1 = Π C J E (α nj p 1 E 1 u + (1 α n )(β n J p E 1 u n + (1 β n )J p (3.1) E 1 Su n )) y n+1 = Π Q J E (α nj p 2 E 2 v + (1 α n )(β n J p E 2 v n + (1 β n )J p E 2 T v n )), where the following conditions are satisfied: (i) {β n } (0, 1); (ii) lim α n = 0 and n=1 α n = ; (iii) 0 < 2L < γ n M, where M = max{( max{ C(λn A ), C(λn B ) }. C A ) 1 1, ( C B ) 1 1 }, and L = If the Ω is nonempty, then the seuence {(x n, y n )} converges strongly to a solution SEF P (1.4). Proof. Let (x, y) Ω, e n := Ax n By n, ε n := J E 1 [J p E 1 x n γ n A J p E 3 (Ax n By n )] and ω n := J E 2 [J p E 2 y n γ n B J p E 3 (Ax n By n )], n 1. From (3.1), we have

8 746 Xuejiao Zi, Zhaoli Ma, Yali Liu and Lili Yang p (u n, x) p (ε n, x) = p (J E 1 [J p E 1 x n γ n A J p E 3 (Ax n By n )], x) = 1 J p E 1 x n γ n A J p E 3 (Ax n By n ) J p E 1 x n, x +γ n J p E 3 (Ax n By n ), Ax + 1 p x p 1 J p E 1 x n γ n Ax n, J p E 3 e n + C(γn A ) J p E 3 e n J p E 1 x n, x + γ n J p E 3 e n, Ax + 1 p x p = 1 x n p J P E 1 x n, x + 1 p x p + γ n J p E 3 e n, Ax Ax n + C(γn A ) e n p = p (x n, x)+γ n J p E 3 e n, Ax Ax n + C(γn A ) e n p. (3.2) By similar steps as in (3.2), we have p (v n, y) p (ω n, y) p (y n, y)+γ n J p E 3 e n, By n By + C (γ n B ) e n p. (3.3) Adding(3.2) and (3.3), since Ax = By, and noting the assumption on condition(iii), we arrive at p (u n, x) + p (v n, y) p (ε n, x) + p (ω n, y) From (3.1) and (2.7), we have p (x n, x) + p (y n, y) γ n J p E 3 e n, e n + 2L e n p = p (x n, x) + p (y n, y) γ n e n p + 2L e n p = p (x n, x) + p (y n, y) (γ n 2L) e n p p (x n, x) + p (y n, y). (3.4) p (x n+1, x) p (J E 1 (α nj p E 1 u + (1 α n )(β n J p E 1 Su n )), x) α n p (u, x) + (1 α n )β n p (u n, x) + (1 α n )(1 β n ) p (Su n, x) α n p (u, x) + (1 α n )β n p (u n, x) + (1 α n )(1 β n ) p (u n, x) = α n p (u, x) + (1 α n ) p (u n, x). (3.5) Following similar process as in (3.5), we obtain Adding (3.5) and (3.6), we can get p (y n+1, y) α n p (v, y) + (1 α n ) p (v n, y). (3.6) p (x n+1, x)+ p (y n+1, y) (1 α n )[ p (u n, x)+ p (v n, y)]+α n [ p (u, x)+ p (v, y)].(3.7)

9 Strong convergence theorem of split euality fixed point 747 Inserting (3.4) into (3.7) yields p (x n+1, x) + p (y n+1, y) (1 α n )[ p (x n, x) + p (y n, y)] (1 α n )(γ n 2L) e n p + α n [ p (u, x) + p (v, y)] (1 α n )[ p (x n, x) + p (y n, y)] +α n [ p (u, x) + p (v, y)]. (3.8) Now, by setting Γ n (x, y) = p (x n, x) + p (y n, y), it follows from (3.8) that Γ n+1 (x, y) (1 α n )Γ n (x, y) + α n ( p (u, x) + p (v, y)). max{γ n (x, y), ( p (u, x) + p (v, y))} max{γ 1 (x, y), ( p (u, x) + p (v, y))}. (3.9) Therefore, {Γ n (x, y)} is bounded. Since p (x n, x) Γ n (x, y), p (y n, y) Γ n (x, y) and We know that p (x n, x), p (y n, y) are bounded. Conseuently {x n }, {y n }, {u n }, {v n }, {Su n }, {T v n } are bounded. Now we divide the rest of the proof into two case. Case 1. Let (x, y) Ω. Assume that Γ n (x, y) is monotonically decreasing. Obviously {Γ n (x, y)} is convergent as n. From (3.8) and condition (ii) we know that 0 (1 α n )(γ n 2L) e n p (1 α n )[ p (x n, x) + p (y n, y)] [ p (x n+1, x) so, e n p 0 as n, which implies that By (3.4) and (3.7), we have + p (y n+1, y)] + α n [ p (u, x) + p (v, y)] p (u n+1, x) + p (v n+1, y) [ p (u n, x) + p (v n, y)] = (1 α n )Γ n (x, y) Γ n+1 (x, y) + α n [ p (u, x) + p (v, y)] Γ n (x, y) Γ n+1 (x, y) + α n [ p (u, x) + p (v, y)], lim Ax n By n = 0. (3.10) p (x n+1, x) + p (y n+1, y) [ p (u n, x) + p (v n, y)] α n [ p (u n, x)+ p (v n, y)]+α n [ p (u, x)+ p (v, y)] 0, n. (3.11) Let s n := J E 1 (β nj p E 1 Su n ), l n := J E 2 (β nj p E 1 v n + (1 β n )J p E 1 T v n ), then x n+1 = Π C J E 1 (α nj p E 1 u + (1 α n )J p E 1 s n ), y n+1 = Π C J E 2 (α nj p E 2 v + (1 α n )J p E 2 l n ), Further, we obtain that p (x n+1, x) + p (y n+1, y)

10 748 Xuejiao Zi, Zhaoli Ma, Yali Liu and Lili Yang and p (J E 1 (α nj p E 1 u + (1 α n )J p E 1 s n ), x) + p (J E 2 (α nj p E 2 v + (1 α n )J p E 2 l n ), y) α n p (u, x)+(1 α n ) p (s n, x)+α n p (v, y)+(1 α n ) p (l n, y), (3.12) p (s n, x) = p (J E (β nj p 1 E 1 Su n ), x) β n p (u n, x) + (1 β n ) p (Su n, x) β n p (u n, x) + (1 β n ) p (u n, x) = p (u n, x). (3.13) Similarly p (l n, x) p (v n, y). (3.14) from (3.4), (3.11), (3.12), (3.13), (3.14), we have 0 p (u n, x) + p (v n, y) [ p (s n, x) + p (l n, y)] = p (u n, x) + p (v n, y) [ p (u n+1, x) + p (v n+1, y)] +[ p (u n+1, x) + p (v n+1, y)] [ p (s n, x) + p (l n, y)] p (u n, x) + p (v n, y) [ p (u n+1, x) + p (v n+1, y)] +[ p (x n+1, x) + p (y n+1, y)] [ p (s n, x) + p (l n, y)] p (u n, x) + p (v n, y) [ p (u n+1, x) + p (v n+1, y)] +[α n p (u, x) + (1 α n ) p (s n, x) + α n p (v, y) + (1 α n ) p (l n, y)] [ p (s n, x) + p (l n, y)] = p (u n, x) + p (v n, y) [ p (u n+1, x) + p (v n+1, y)] +α n [ p (u, x) + p (v, y) p (s n, x) p (l n, y)] = { p (u n+1, x) + p (v n+1, y) [ p (u n, x) + p (v n, y)} +α n [ p (u, x)+ p (v, y) p (s n, x) p (l n, y)] 0, as n. (3.15) In addition p (s n, x) + p (l n, y) β n [ p (u n, x) + p (v n, y)] +(1 β n )[ p (Su n, x) + p (T v n, y)] = [ p (u n, x) + p (v n, y)] (1 β n )[ p (u n, x) + p (v n, y)] +(1 β n )[ p (Su n, x) + p (T v n, y)] Therefore, it follows from (3.15), (3.16) that = p (u n, x) + p (v n, y) + (1 β n )[ p (Su n, x) p (u n, x) + p (T v n, y) p (v n, y)]. (3.16) (1 β n )[ p (u n, x) p (Su n, x) + p (v n, y) p (T v n, y)]

11 Strong convergence theorem of split euality fixed point 749 So Obvious and Hence and p (u n, x) + p (v n, y) [ p (s n, x) + p (l n, y)] 0 as n, p (u n, x) p (Su n, x) + p (v n, y) p (T v n, y) 0 as n. By definition 2.4, we obtain that which implies that p (u n, x) p (Su n, x) 0, p (v n, y) p (T v n, y) 0. lim [ p(u n, x) p (Su n, x)] = 0, lim [ p(v n, y) p (T v n, y)] = 0. lim p(su n, u n ) = 0 and lim p (T v n, v n ) = 0, (3.17) lim Su n u n = 0 and lim T v n v n = 0. (3.18) Since ε n := J E 1 [J p E 1 x n γ n A J p E 3 (Ax n By n )], then we have Hence, we obtain 0 J p E 1 ε n J p E 1 x n γ n A J p E 3 (Ax n By n ) ( C A ) 1 1 A Ax n By n M A Ax n By n 0, n. lim J p E 1 ε n J p E 1 x n = 0. (3.19) Since J E is also norm-to-norm uniformly continuous on bounded subsets of 1 E 1, we have lim ε n x n = 0. (3.20) Similarly, Since ω n := J E 2 [J p E 2 x n γ n B J p E 3 (Ax n By n )], we can get lim J p E 2 ω n J p E 2 y n = 0. (3.21) Since J E2 have is also norm-to-norm uniformly continuous on bounded subsets of E 2, we lim ω n y n = 0. (3.22)

12 750 Xuejiao Zi, Zhaoli Ma, Yali Liu and Lili Yang Furthermore, we have from (2.4) that p (ε n, u n ) = p (ε n, Π C ε n ) p (ε n, x) p (Π C ε n, x) = p (ε n, x) p (u n, x), (3.23) by the same way, p (ω n, v n ) p (ω n, ȳ) p (v n, ȳ). (3.24) From (3.23), (3.24), (3.4) and (3.7), we obtain p (ε n, u n ) + p (ω n, v n ) p (ε n, x) p (u n, x) + [ p (ω n, ȳ) p (v n, ȳ)] = [ p (ε n, x) + p (ω n, ȳ)] [ p (u n, x) + p (v n, ȳ)] [ p (x n, x) + p (y n, ȳ)] [ p (u n, x) + p (v n, ȳ)] α n 1 [ p (u, x) + p (v, ȳ)] +(1 α n 1 )[ p (u n 1, x) + p (v n 1, ȳ)] [ p (u n, x) + p (v n, ȳ)] α n 1 [ p (u, x) + p (v, ȳ)] +[ p (u n 1, x) + p (v n 1, ȳ)] [ p (u n, x) + p (v n, ȳ)] = α n 1 [ p (u, x) + p (v, ȳ)] {[ p (u n, x) + p (v n, ȳ)] [ p (u n 1, x) + p (v n 1, ȳ)]}. Following condition(ii) and (3.11), we have lim [ p(ε n, u n ) + p (ω n, v n )] = 0, so lim p(ε n, u n ) = 0 and lim p (ω n, v n ) = 0, (3.25) which implies that lim ε n u n = 0 and lim ω n v n = 0. (3.26) Hence and x n u n x n ε n + ε n u n 0, n, (3.27) y n v n y n ω n + ω n v n 0, n. (3.28)

13 Strong convergence theorem of split euality fixed point 751 Since {x n } is bounded, there exists a subseuence {x nj } of x n that converges weakly to x, from (3.27) we get u nj x. From (2.3) and Lemma(2.5), we have that p (x, Π C x ) J p E 1 x J p E 1 Π C x, x Π C x = J p E 1 x J p E 1 Π C x, x x nj + J p E 1 x J p E 1 Π C x, x nj u nj + J p E 1 x J p E 1 Π C x, u nj Π C x J p E 1 x J p E 1 Π C x, x x nj + J p E 1 x J p E 1 Π C x, x nj u nj. As j, we have p (x, Π C x )=0, i.e. x C. Similarly, we can obtain y Q. By (3.18) and F (S) = F (S), we have x F (S). In the same way, By the boundedness of y n, there exists a subseuence {y nj } of {y n } that converges weakly to y, from (3.28) we get v nj y, by (3.18) and F (T ) = F (T ), we have y F (T ). On the other hand, since A and B are bounded linear operators, we know that Ax n Ax and By n By, also by the weakly lower semi-continuous property of the norm and (3.10), we get Ax By lim inf Ax n By n = 0. So, Ax = By, this implies (x, y ) Ω. Next, we prove that {(x n, y n )} converges strongly to (x, y ). Now, from (2.6) and (3.1) we have p (x n+1, x ) p (J E 1 (α nj p E 1 u + (1 α n )(β n J p E 1 Su n )), x ) = V p (α n J p E 1 u + (1 α n )(β n J p E 1 Su n ), x ) V p (α n J p E 1 u + (1 α n )(β n J p E 1 Su n ) α n (J p E 1 x ), x ) α n (J p E 1 x ), J E 1 [α nj p E 1 u + (1 α n )(β n J p E 1 Su n )] x = V p (α n J p E 1 x +(1 α n )(β n J p E 1 Su n ), x ) +α n J p E 1 x, ρ n x = p (J E 1 (α nj p E 1 x + (1 α n )(β n J p E 1 Su n ), x ) +α n J p E 1 x, ρ n x α n p (x, x ) + (1 α n )β n p (u n, x ) +(1 α n )(1 β n ) p (Su n, x ) + α n J p E 1 x, ρ n x (1 α n ) p (u n, x ) + α n J p E 1 x, ρ n x, (3.29) where ρ n := J E 1 [α nj p E 1 u + (1 α n )(β n J p E 1 u n +(1 β n )J p E 1 Su n )]. From (3.1), we have p (ρ n, u n ) = p (J E 1 [α nj p E 1 u + (1 α n )(β n J p E 1 Su n )], u n ) α n p (u, u n ) + (1 α n )β n p (u n, u n ) +(1 α n )(1 β n ) p (Su n, u n ) 0, n, (3.30)

14 752 Xuejiao Zi, Zhaoli Ma, Yali Liu and Lili Yang which implies that lim ρ n u n = 0, and by (3.27) we have ρ n x n ρ n u n + u n x n 0, as n, so lim ρ n x n = 0. (3.31) Since x nj x, so we have ρ nj x. Furthermore, lim sup J p E 1 x, ρ n x = lim J p E j 1 x, ρ nj x 0. (3.32) Submitting (3.2) into (3.29), we obtain p (x n+1, x ) (1 α n ) p (u n, x ) + α n J p E 1 x, ρ n x Similary, we have (1 α n ) p (x n, x ) + (1 α n )γ n J p E 3 e n, Ax Ax n +(1 α n ) C (λ n A ) e n p + α n J p E 1 x, ρ n x. (3.33) p (y n+1, y ) (1 α n ) p (y n, y ) + (1 α n )γ n J p E 3 e n, By n By +(1 α n ) C (γ n B ) e n p + α n J p E 2 v J p E 2 y, η n y, (3.34) where η n := J E 2 [α nj p E 2 v + (1 α n )(β n J p E 2 v n + (1 β n )J p E 2 T v n )]. Adding (3.33) and (3.34), we get Γ n+1 (x, y ) = p (x n+1, x ) + p (y n+1, y ) (1 α n )Γ n (x, y ) (1 α n )(γ n 2L) e n p +α n [ J p E 1 x, ρ n x + J p E 2 v J p E 2 y, η n y ] (1 α n )Γ n (x, y ) + α n [ J p E 1 x, ρ n x + J p E 2 v J p E 2 y, η n y ]. (3.35) By Lemma2.6, we know that lim Γ n (x, y ) = 0, which implies then lim [ p(x n, x ) + p (y n, y )] = 0, (3.36) lim x n x = 0 and lim y n y = 0, (3.37) That is the seuence {(x n, y n )} converges strongly to a solution (x, y ) Ω. Case 2. Assume that {Γ n ( x, ȳ)} is not a monotonically decreasing seuence. Let τ : N N be a mapping for all n n 0 by τ(n) := max{k N : k n, Γ k Γ k+1 }.

15 Strong convergence theorem of split euality fixed point 753 Obviously, τ(n) is a non-decreasing seuence such that τ(n) as n and 0 Γ τ(n) Γ τ(n)+1, n n 0. (3.38) The same as in case 1, we have the following eualities or ineualities lim Ax τ(n) By τ(n) = 0, lim A J p E 3 (Ax τ(n) By τ(n) ) = 0, lim B J p E 3 (Ax τ(n) By τ(n) ) = 0, lim x τ(n)+1 x τ(n) = 0, lim y τ(n)+1 y τ(n) = 0, lim Su τ(n) u τ(n) = 0, lim T v τ(n) v τ(n) = 0 and lim sup J p E 1 x, ρ τ(n) x 0, lim sup J p E 2 v J p E 2 ȳ, η τ(n) ȳ 0. Since {(x τ(n), y τ(n) )} is bounded, there exists a subseuence of {(x τ(n), y τ(n) )} which converges weakly to (x, y ) Ω. From (3.35), we know that Γ τ(n)+1 (x, y ) (1 α τ(n) )Γ τ(n) (x, y ) From (3.38), we have i.e. Γ τ(n) (x, y ) Γ τ(n)+1 (x, y ) +α τ(n) [ J p E 1 x, ρ τ(n) x + J p E 2 v J p E 2 y, η τ(n) y ]. (1 α τ(n) )Γ τ(n) (x, y ) +α τ(n) [ J p E 1 x, ρ τ(n) x + J p E 2 v J p E 2 y, η τ(n) y ]. Γ τ(n) (x, y ) J p E 1 x, ρ τ(n) x + J p E 2 v J p E 2 y, η τ(n) y. Then from (3.32) we have lim sup Γ τ(n) (x, y ) 0. Thus, lim p (x τ(n), x ) = 0, lim p (y τ(n), y ) = 0. So lim x τ(n) x = 0, lim y τ(n) y = 0. It follows from lim x τ(n)+1 x τ(n) = 0 and lim y τ(n)+1 y τ(n) = 0 that lim x τ(n)+1 x = 0, lim y τ(n)+1 y = 0. Now, by Lemma 2.5, we have that 0 Γ τ(n)+1 (x, y ) x τ(n)+1 x, J p E 1 x τ(n)+1 J p E 1 x + y τ(n)+1 y, J p E 1 y τ(n)+1 J p E 1 y x τ(n)+1 x J p E 1 x τ(n)+1 J p E 1 x + y τ(n)+1 y J p E 2 y τ(n)+1 J p E 2 y 0, n.

16 754 Xuejiao Zi, Zhaoli Ma, Yali Liu and Lili Yang For n n 0, it is easy to see that Γ τ(n) (x, y ) Γ τ(n)+1 (x, y ) if n τ(n)(that is, τ(n) < n), because Γ j (x, y ) Γ j+1 (x, y ) for τ(n)+1 j n. As a conseuence, we obtain for all n n 0, 0 Γ n (x, y ) max{γ τ(n) (x, y ), Γ τ(n)+1 (x, y )} = Γ τ(n)+1 (x, y ). Hence, lim Γ n (x, y ) = 0. That is {(x n, y n )} converges strongly to (x, y ). This completes the proof. 4 Application 4.1 Application to split variational inclusion problem Suppose that A : E 1 E 3, B : E 2 E 3 be two bounded linear operators. Let M : E 1 2 E 1 and N : E2 2 E 2 be maximal monotone operators. Let us consider the following so called split variational inclusion problem: find x M 1 (0), y N 1 (0) such that Ax = By. (4.1) Given a maximal monotone operator M : E 1 2 E 1, from [27], we known that the associated resolvent mapping, Jµ M := (I + µm) 1 x, x E 1 is left Bregman strongly nonexpansive and 0 M(x ) if and only if x is a fixed point of F (Jµ M ). Taking S = Jµ M, T = Jν N in Theorem 3.1, we have the following result. Theorem 4.1. Let E 1, E 2 and E 3 be three p-uniformly convex and uniformly smooth Banach spaces. Let A : E 1 E 3, B : E 2 E 3 be two bounded linear operators and A : E3 E1, B : E3 E2 be the adjoint operators of A and B, respectively. Let M : E 1 2 E 1 and N : E2 2 E 2 be maximal monotone operators and suppose that the solution set of problem (4.1) is denoted by Ω. The seuence {(x n, y n )} is generated by u n = Π C J E [J p 1 E 1 x n γ n A J p E 3 (Ax n By n )] v n = Π Q J E [J p 2 E 2 y n + γ n B J p E 3 (Ax n By n )] x n+1 = Π C J E (α nj p 1 E 1 u + (1 α n )(β n J p E 1 Jµ M (4.2) u n )) y n+1 = Π Q J E (α nj p 2 E 2 v + (1 α n )(β n J p E 2 v n + (1 β n )J p E 2 Jν N v n )), where the following conditions are satisfied: (i) {β n } (0, 1); (ii) lim α n = 0 and n=1 α n = ; (iii) 0 < 2L < γ n R, where R = max{( max{ C(λn A ), C(λn B ) }. C A ) 1 1, ( C B ) 1 1 }, and L = Then, the seuence {(x n, y n )} converges strongly to (x, y ) in the solution set Ω of (4.1).

17 Strong convergence theorem of split euality fixed point Application to split Euilibrium problem Let E 1, E 2 and E 3 be three p-uniformly convex and uniformly smooth Banach spaces. Let C and Q be nonempty closed and convex subsets of E 1 and E 2, respectively. Let A : E 1 E 3, B : E 2 E 3 be two bounded linear operators. Let F : C C R, H : Q Q R be bi-functions. Let us consider the following so called split Euilibrium problem: find x C, y Q such that F (x, z) 0, H(y, µ) 0, z C, µ Q and Ax = By. (4.3) To solve problem(4.3), we assume the following conditions are satisfied (A1) F (x, x) = 0, x C and H(x, x) = 0, x Q; (A2) F and H are monotone, that is, F (x, y)+f (y, x) 0, x, y C and H(x, y)+ H(y, x) 0, x, y Q; (A3) For all x, y, z C, lim t 0 F (tz + (1 t)x, y) F (x, y) and For all x, y, z Q, lim t 0 H(tz + (1 t)x, y) H(x, y); (A4) For each x C, the function y F (x, y) is convex and lower semi-continuous and For each x Q, the function y H(x, y) is convex and lower semicontinuous. Let us define the resolvent mappings T F µ, µ > 0 and T H ν, ν > 0 as T F µ (x) = {z C : F (z, y) + 1 µ y z, J p E 1 z J p E 1 x 0, y C}. and T H ν (x) = {z Q : H(z, y) + 1 ν y z, J p E 2 z J p E 2 x 0, y Q}. It is well know that the resolvent mapping T F µ and T H ν are left Bregman firmly nonexpansive (hence left Bregman strongly nonexpansive). Furthermore, it is know that if x, y solves problem (4.3), then x = T F µ x and y = T H ν y. The following result can be directly obtained from Theorem 3.1. Theorem 4.2. Let E 1, E 2 and E 3 be three p-uniformly convex and uniformly smooth Banach spaces. Let C and Q be nonempty closed and convex subsets of E 1 and E 2, respectively. Suppose that F : C C R, H : Q Q R be bi-functions. Let A : E 1 E 3, B : E 2 E 3 be two bounded linear operators and A : E 3 E 1, B : E 3 E 2 be the adjoint operators of A and B, respectively. Suppose that the solution set of problem (4.3) is denoted by Ω. The seuence {(x n, y n )} is

18 756 Xuejiao Zi, Zhaoli Ma, Yali Liu and Lili Yang generated by u n = Π C J E [J p 1 E 1 x n γ n A J p E 3 (Ax n By n )] v n = Π Q J E [J p 2 E 2 y n + γ n B J p E 3 (Ax n By n )] x n+1 = Π C J E (α nj p 1 E 1 u + (1 α n )(β n J p E 1 Tµ F u n )) y n+1 = Π Q J E (α nj p 2 E 2 v + (1 α n )(β n J p E 2 v n + (1 β n )J p E 2 Tν H v n )), where the following conditions are satisfied: (i) {β n } (0, 1); (ii) lim α n = 0 and n=1 α n = ; (iii) 0 < 2L γ n N, where N = max{( max{ C(λn A ), C(λn B ) }. C A ) 1 1, ( C B ) 1 (4.4) 1 }, and L = Then, the seuence {(x n, y n )} converges strongly to (x, y ) in the solution set Ω of (4.4). References [1] Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), [2] C. Byrne, Iterative obliue projection onto convex sets and the split feasibility problem, Inverse Probl., 18 (2002), [3] Y. Censor, T. Bortfels, A. Trofimov, A unified approach for inversion problem in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), [4] Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The multiple-sets split feasiblility problem and its applications for inverse problems, Inverse Problems, 21 (2005), [5] Y. Censor. A. Motova. A. Segal, Perturbed projections ans subgradient projections for the multiple-sets split feasibility problem, J. Math. Anal. Appl., 327 (2007), [6] H. K. Xu, A variable Krasonsel, skii-mann algorithm and multiple-ser split feasibility problem, Inverse Problems, 22 (2006), [7] H. K. Xu, Iterative methods for the split feasibility problem in infinitedimensional Hilbert spaces, Inverse Problems, 26 (2010), Article ID

19 Strong convergence theorem of split euality fixed point 757 [8] B. Qu, N. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problems, 21 (2005), [9] A. Moudafi, A relaxed alternating CQ-algorithms for convex feasibility problems, Nonlinear Anal. Theory Methods Appl., 79 (2013), [10] H. Attouch, J. Bolte, P. Redont, A. Soubeyran, Alternating proximal algorithms for weakly coupled convex minimization problems, Applications to dynamical games and PDEs, J. Convex Anal., 15 (2008), [11] A. Aleyner, S. Reich, Block-iterative algorithms for solving convex feadibility problems in Hilbert and in Banach spaces, Journal of Mathematical Analysis and Applications, 343 (2008), [12] Q. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, 20 (2004), [13] X. F. Zhang, L. Wang, Z. L. Ma and L. J. Qin, The strong convergence theorems for split common fixed point problem of asymptotically nonexpansive mappings in Hilbert spaces, Journal of Ineualityes and Applications, 2015 (2015), no [14] W. Takahashi, J. C. Yao, Strong convergence theorems by hybrid method for the split common null point problem in Banach spaces, Fixed point theorems and Applications, 2015 (2015), [15] J. F. Tang, S. S. Chang, L. Wang, X. R. Wang, On the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in Banach spaces, J. Ineual. Appl., 2015 (2015), [16] Z. L. Ma, L. Wang, On the split euality common fixed point problem for asymptotically nonexpansive semigroups in Banach spaces, J. Nonlinear Sci. Appl., 9 (2016), [17] Y. Shehu, O. S. Iyiola, C. D. Enyi, An iterative algorithem fof solving split feasibility problems and fixed point problems in Banach spaces, Numer Algor., 72 (2015),

20 758 Xuejiao Zi, Zhaoli Ma, Yali Liu and Lili Yang [18] Z. Zhou, S. J. He, J. Niu, J. Q. Zhang, On the strong convergence of split euality problems in Banach spaces, International Mathematical Forum, 13 (2018), [19] Y. Shehu, Iterative methods for split feasibility problems in certain Banach spaces, Journal of Nonlinear and Convex Analysis, 16 (2015), no. 12. [20] H. K. Xu, Ineualities in Banach spaces with applications, Nonlinear Analysis: Theory, Methods and Applications, 16 (1991), [21] F. Schöpfer, Iterative Regularization Method for the Solution of the Split Feasibility Problem in Banach Spaces, PhD Thesis, Universitat des Saarlandes, Saabrüchen, [22] V. Martín-Máruez, S. Reich, S. Sabach, Right Bregman nonexpansive operators in Banach spaces, Nonlinear Analysis: Theory, Methods and Applications, 75 (2012), [23] F. Schöpfer, T. Schuster, A. K. Louis, An iterative regularization method for the solution of the split feasibility problem in Banach spaces, Inverse Problems, 24 (2008), [24] Y. I. Alber, Metric and generalized projection operator in Banach spaces, Properties and applications, in: Theory and Applications Operators of Accretive and Monotone Type, Vol. 178, Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 1996, [25] Y. Censor, A. Lent, An iterative row-action method for interval convex programming. J. Optim. Theory Appl., 34 (1981), [26] P. E. Maingé, Strong convergence of projected sub-gradient methods for nonsmooth and non-strictly convex minimization, Set-Valued Anal., 16 (2008), [27] F. U. Ogbuisi, O. T. Mewomo, Iterative solution of split variational inclusion problem in a real Banach space, African Mathematical, 28 (2016), Received: June 1, 2018; Published: June 26, 2018