J. Nig. Soc. Phys. Sci. 1 (2019) 35–50 Journal of the Nigerian Society of Physical Sciences Review Strong Convergence Theorems for Split Common Fixed Point Problem of Bregman Generalized Asymptotically Nonexpansive Mappings in Banach Spaces Yusuf Ibrahim∗ Department of Mathematics, Saádatu Rimi College of Education, Kumbotso Kano, Nigeria Abstract In this paper, a new iterative scheme is introduced and also strong convergence theorems for solving split common fixed point problem for uniformly continuous Bregman generalized asymptotically nonexpansive mappings in uniformly convex and uniformly smooth Banach spaces are presented. The results are proved without the assumption of semicompactness property and or Opial condition. Keywords: Split common fixed point problem, Bregman distance, Generalized asymptotically nonexpansive mapping, Strong convergence, uniformly convex and uniformly smooth Banach space, Algorithm Article History : Received: 04 April 2019 Received in revised form: 30 April 2019 Accepted for publication: 11 May 2019 Published: 16 May 2019 c©2019 Journal of the Nigerian Society of Physical Sciences. All rights reserved. Communicated by: T. Latunde 1. Introduction Throughout this paper, E1 and E2 are uniformly convex and uniformly smooth Banach spaces. Let us make some conven- tions: we always use p, q ∈ (1,∞) as conjugate exponents so that 1p + 1 q = 1, where by q = p p−1 , pq = p + q, (p−1)(q−1) = 1. Furthermore, for real value a, b, we write a∨b = max{a, b} and a ∧ b = min{a, b} which is to be understood componentwise in case of sequences and pointwise in case of functions. Throughout this paper, let p, r ≥ 1, given sequences of nonempty closed convex subsets {Ci} p i=1 and {Qi} r i=1 of E1 and E2, respectively, Definition 1.1. [1] Multiple set split feasibility problem (MSSFP) ∗Corresponding author Tel. no: +2348062814778 Email address: danustazz@gmail.com (Yusuf Ibrahim) is formulated as finding a point x∗ ∈ E1 with the property x∗ ∈ p⋂ i=1 Ci and Ax ∗ ∈ r⋂ j=1 Q j. (1) Definition 1.2. [2] If in a MSSEP (1) p = r = 1, we get what is called the split feasibility problem (SFP), which is to find a point x∗ ∈ E, with the property x∗ ∈ C and Ax∗ ∈ Q. (2) Definition 1.3. [3] Split common fixed point problem (SCFPP) is formulated as finding a point x∗ ∈ E1, with the property x∗ ∈ p⋂ i=1 Fix(Ui) and Ax ∗ ∈ r⋂ j=1 Fix(T j), where each Ui : E1 −→ E1 (i = 1, 2, · · · p) and T j : E2 −→ E2 35 Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 35–50 36 ( j = 1, 2, · · ·r) are some (nonlinear) mappings. Definition 1.4. [3] two-set SCFPP is formulated as finding a point x∗ ∈ E1, with the property x∗ ∈ Fix(U) and Ax∗ ∈ Fix(T ) (3) where U : E1 −→ E1 and T : E2 −→ E2 are (nonlinear) mappings. As stated in [4], let E be a Banach space with norm ‖·‖. Let C be a nonempty closed convex subset of E and E∗ denote the dual space of E. Let B : C −→ E∗ be a nonlinear mapping and F : C × C −→ < be a bifunction. The generalized equilibrium problem is to find x ∈ C such that F(x, y) + 〈Bx, y − x〉 ≥ 0∀y ∈ C. (4) Now, let F(x, y) = f (x, y) + g(x, y) (5) where f, g : C × C −→ < are two bifunctions satisfying the following special properties (A1) − (A4), (B1) − (B3)and(C); (A1) − f (x, y) = 0,∀x ∈ C; (A2) − f is maximal monotone; (A3) −∀x, y, z ∈ Cwe have lim supn→0+ ( f (tz + (1 − t)x, y) ≤ f (x, y)); (A4) −∀x ∈ C, the functiony 7→ f (x, y)is convex and weakly lower semi-continuous; (B1) − g(x, x) = 0∀x ∈ C; (B2) − gis maximal monotone and monotone, and weakly upper semi-continuous in the first variable; (B3) − gis convex in the second variable; (C) − for fixedλ > 0and x ∈ C, there exists a bounded set K ⊂ Canda ∈ Ksuch that f (a, z) + g(z, a) + 1 λ (a − z, z − x) < 0∀x ∈ C\K. (6) The well known generalized mixed equilibrium problem is to find an x ∈ C such that f (x, y) + g(x, y) + 〈Bx, y − x〉 ≥ 0∀y ∈ C. (7) If B ≡ 0, the problem (7) reduces into mixed equilibrium problem for f and g, denoted by MEP( f, g), which is to find x ∈ C such that f (x, y) + g(x, y) ≥ 0,∀y ∈ C. (8) If g ≡ 0 and B ≡ 0, (7) reduces into equilibrium problem for f, denoted by EP(f), which is to find x ∈ C such that f (x, y) ≥ 0 ∀ y ∈ C. (9) In 1994, the SFP (2) was introduced by Censor and Elfving [1]. Using the CQ-algorithm for solving the SFP (2), in 2002 Byrne [2] proposed that which generates the new iterate as fol- lows choosing arbitrarily x1 ∈ H1, xn+1 = PC [xn −γA T (I − PQ)Axn] where γ ∈ (0, 2L ), L denotes the largest eigenvalue of the matrix AT A. In 2004, Yang [5] presented a relaxed CQ-algorithm for solving the SFP, where at n-th iteration, the projections onto C and Q were replaced with the halfspaces Cn and Qn, respec- tively. In 2005, Qu and Xiu [6] proposed a modified relaxed CQ- algorithm choosing arbitrarily x1 ∈ H1, xn+1 = PCn [xn −αn A T (I − PQn )Axn]. In 2007, Frank Schopfer[7] developed iterative methods for the solution of the SFP (2) in Banach spaces and also analyse stability and regularizing properties. these iterative methods are as follows. choosing arbitrarily x1 ∈ E1, xn+1 = J ∗ E1 (JE1 (xn) −µn A ∗JE2 (Axn − PQ(Axn)). The concept of SCFPP in finite dimensional Hilbert spaces, say H1 and H2, was first introduced by Censor and Segal in 2009[1], who invented an algorithm of the two-set SCFPP which generate a sequence {xn} according to the following iterative procedure: choosing arbitrarily x1 ∈ H1, xn+1 = U(xn + γA ∗(I − T )Axn), n ≥ 1 where the initial guess x0 ∈ H1 is chosen arbitrarily and 0 < γ < 1 ‖A‖2 and A : H1 −→ H2 as a bounded linear operator hav- ing A∗ as the adjoint operator of A. And U : H1 −→ H1 and T : H2 −→ H2 are (nonlinear) mappings. In 2010, Moudafi [8] proposed the following iteration method to approximate a SCFPP of demicontractive mappings in Hilbert spaces;  x1 ∈ H1 is arbitrarily chosen, un = xn + γA∗(I − T )Axn, xn+1 = αnun + (1 −αn)Uun, (10) where he proved that {xn} converges weakly to a split common fixed point x∗ ∈ Γ, where U : H1 −→ H1 and T : H2 −→ H2 are two demicontractive mappings, A : H1 −→ H2 is a bounded linear operator and H1 and H2 are two Hilbert spaces. Using the iterative scheme (10), in 2011, Moudafi[8] also obtained a weak convergence theorem for the split common 36 Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 35–50 37 fixed poin problem of quasi-nonexpansive mapping in Hilbert spaces. In 2012, Chang et al. [9] again, using (10) proved the weakly convergence of the sequence {xn} to the split common fixed point x∗ ∈ Γ of asymptotically quasi-nonexpansive map- ping in Hilbert spaces. But these authors could only obtain strong convergence theorem if those mappings and spaces are semi-compacts and Hilbert spaces, respectively. In 2015, Zhang et al [10] introduced the iterative scheme which guarantees the strong converges for SCFP of the asymp- totically nonexpansive mapping in Hilbert spaces, without as- sumption of semi-compactness. The sequence is defined as fol- lows; choosing arbitrarily x1 ∈ H1, C1 = H1, zn = xn + λA∗(T n2 − I)Axn, yn = αnzn + (1 −αn)T n1 (zn), Cn+1 = {v ∈ Cn : ‖yn − v‖ ≤ kn‖zn − v‖,‖zn − v‖ ≤ kn‖xn − v‖}, xn+1 = PCn+1 (x1), n ≥ 1 where A∗ denote the adjoint of A, λ ∈ (0, 1 ‖A∗‖2 ) and {αn} ⊂ (0,τ) ⊂ (0, 1) satisfies limn→∞ αn(1−αn) > 0, kn = max{k (1) n , k (2) n }, n ≥ 1. Takahashi [11], in 2015, also obtained a similar result for split common null point problem by using the following hybrid and shrinking projection methods, respectively. choosing arbitrarily x1 ∈ H1, zn = PC (xn − rA∗JF (Axn − PQ Axn)), yn = αn xn + (1 −αn)zn, Cn = {z ∈ H1 : ‖yn − z‖ ≤ ‖xn − z‖}, Qn = {z ∈ H2 : 〈xn − z, x1 − xn〉 ≥ 0} xn+1 = PCn∩Qn x1,∀n ∈ N, and  choosing arbitrarily x1 ∈ H1, zn = PC (xn − rA∗JF (Axn − PQ Axn)), Cn = {z ∈ H1 : ‖zn − z‖ ≤ ‖xn − z‖}∩ Cn, xn+1 = PCn+1 un+1,∀n ∈ N, where 0 ≤ αn ≤ a < 1 for some a ∈ R and 0 < r‖A‖2 < 2. In 2015, Tang et al.[12] introduced the split common fixed point problem (SCFP) for an asymptotical nonexpansive map- ping S and a τ−quasi-pseudocontractive mapping T in the set- ting of two Banach spaces, E1 and E2, by using the sequence {xn} defined as follows; choosing arbitrarily x1 ∈ E1, zn = xn + γJ−11 A ∗J2(T − I)Axn, xn+1 = (1 −αn)zn + αnS n(zn), n ≥ 1 where {αn} ⊂ (0, 1) with lim infn→∞ αn(1 −αn) > 0, γ ⊂ (0, min{1−2k 2 ‖A‖2 , 1−τ ‖A‖2 }), L = supn≥1{ln} and ∑ ∞ n=1(ln − 1) < ∞. In this paper, the modified algorithm of Zhang et al. for the solution of SCFP in Hilbert spaces is studied. Hence, the Bregman generalized asymptotically nonexpansive mapping is used to obtain the strong convergence for SCFPP (6) in uni- formly convex and uniformly smooth Banach spaces, without the assumption of semi-compactness property and or without the assumption of Opial condition. 2. Preliminaries Throughout this paper, the classes of Banach spaces we will deal with are [13]; Definition 2.1. A Banach space E is said to be uniformly con- vex, if δE (�) = inf{1 − ‖ 1 2 (x + y)‖;‖x‖ = ‖y‖ = 1,‖x − y‖ ≥ �, where 0 ≤ � ≤ 2 and 0 < δE (�) ≤ 1}. Definition 2.2. A Banach space E is said to be uniformly smooth, if limr→0( ρE (r) r ) = 0 where ρE (r) = 1 2 sup{‖x + y‖ + ‖x − y‖− 2 : ‖x‖ = 1,‖y‖ ≤ r; 0 ≤ r < ∞ and 0 ≤ ρE (r) < ∞}. Moreover, 1. ρE is continuous, convex and nondecreasing with ρE (0) = 0 and ρE (r) ≤ r 2. The function r 7→ ρE (r)r is nondecreasing and fulfills ρE (r) r > 0 for all r > 0 3. limr→0 ρE (r) r = 0. Throughout this paper, some important mappings we will be using are; Definition 2.3. [14] For each p > 1, let g(t) = t p−1 be a gauge function g : <+ −→ <+ such that g(0) = 0 and limn→∞ g(t) = ∞. we defined the ganeralized daulity map by Jp : E −→ 2E ∗ by Jg(t) = Jp(x) = {x ∗ ∈ E∗;〈x, x∗〉 = ‖x‖‖x∗‖,‖x∗‖ = g(‖x‖) = ‖x‖p−1}. The ganeralized daulity mapping has the following basic properties [7]. Lemma 2.1. For every x ∈ E the set J pE (x) is not empty and convex. Lemma 2.2. J pE (x) is homogeneous of degree p − 1, i.e. J pE (λx) = |λ| p−1 sgn(λ)J pE (x) ∀x ∈ E,λ ∈<. Lemma 2.3. If J pE∗(x) is the duality mapping of E ∗ with gauge function t 7→ tq−1 then x∗ ∈ J pE (x) iff x ∈ J q E∗(x ∗). Lemma 2.4. [7] In smooth Banach space, the Bregman dis- tance of x to y with respect to the function f (x) = 1p‖x‖ p is defined by 4p(x, y) = 1 q ‖x‖p −〈J p(x), y〉 + 1 p ‖y‖p. Definition 2.4. Let E be a smooth Banach space. Let 4p be a Bregman distance. A mapping T : E −→ E is said to be a Bregman generalized asymptotically nonexpansive with {kn} and {µn} if there exists nonnegative real sequences {kn} and {µn} with limn→∞ kn = 0 and limn→∞ µn = 0 such that 4p(T n(x), T n(y)) ≤ kn 4p (x, y) + 4p(x, y) + µn ∀ (x, y) ∈ E × E. 37 Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 35–50 38 Definition 2.5. [15] Let E be a reflexive, strictly convex and smooth Banach space. Then for every closed convex subset C ⊂ E and x ∈ E there exists a unique element PC (x) ∈ C such that ‖x−PC (x)‖ = miny∈C ‖x−y‖ is called the metric projection of x onto C. Moreover, if J p is a duality mapping of E, then x0 ∈ C is the metric projection of x onto C iff 〈J p(x0 − x), y − x0〉 ≥ 0 ∀y ∈ C. Definition 2.6. [7] Let E be a reflexive, strictly convex and smooth Banach space and J p be a duality mapping of E. Then for every closed convex subset C ⊂ E and x ∈ E there ex- ists a unique element ΠpC (x) ∈ C such that 4p(x, Π p C (x)) = miny∈C 4p(x, y). Π p C (x) is called the Bregman projection of x onto C, with respect to the function f (x) = 1p‖x‖ p. Moreover x0 ∈ C is the Bregman projection of x onto C iff 〈J p(x0) − J p(x), y − x0〉 ≥ 0 or equivalently 4p(x0, y) ≤4p(x, y) −4p(x, x0) f or every y ∈ C. Lemma 2.5. [7] The Bregman projection and the metric pro- jection are related via PC (x)− x = Π p C−x(0), ∀x ∈ E. Especially we have PC (0) = Π p C (0) and thus ‖Π p C (0)‖ = miny∈C ‖y‖. The Bregman distance has the following properties [7]. Lemma 2.6. ∀x, y ∈ E and {xn} ∈ E, 4p(x, y) ≥ 0 and 4p(x, y) = 0 ⇔ x = y. Lemma 2.7. ∀x, y ∈ E and λ ≥ 0 4p(−x,−y) = 4p(x, y) and 4p is positively homogeneous of degree p, i.e. 4p(λx,λy) = λp 4p (x, y). Lemma 2.8. 4p is continuous in both arguments and it is strictly convex, weakly lower semicontinuous and Gateaux differentiable with respect to the second variable with derivative ∂ ∂y 4p (x, y) = J p(y) − J p(x). Lemma 2.9. 4p satisfies the Three-point property that gener- alizes the ”Law of cosines”: 4p(x, y) = 4p(x, z) + 4p(z, y) −〈x − z, J py − J pz〉 Lemma 2.10. Let X be reflexive, smooth and strictly convex. Then for all x, y ∈ X the following holds: 4p(x, y) + 4p(y, x) = 〈J p(x) − J p(y), x − y〉 Lemma 2.11. Consider the following assertions; 1. limn→∞ ‖xn − x‖ = 0 2. limn→∞ ‖xn‖ = ‖x‖ and limn→∞〈J p(xn), x〉 = 〈J p(x), x〉 3. limn→∞4p(xn, x) = 0. The implication (1) =⇒ (2) =⇒ (3) are valid. If E is uniformly convex then the assertions are equivalent. Lemma 2.12. Let us write 4∗q(x ∗, y∗) = 1p‖x ∗‖ q E∗ − 〈J q E∗, y ∗〉 + 1 q‖y ∗‖ q E∗ for the Bregman distance on the dual space E ∗ with respect to the function f ∗(x∗) = 1q‖x ∗‖ q E∗. Then we have 4p(x, y) = 4 ∗ q(x ∗, y∗). f or x∗ = J pE (x) ⇔ JqE∗(x ∗) = x and y∗ = J pE (y) = y . ⇔ JqE∗(y ∗) = y. Lemma 2.13. ∏p C (x) maps bounded sets onto bounded sets; more precisely we have ‖Π p C (x)‖ ≤ (2 q−1 ‖x‖) ∨ (3‖ΠpC (0)‖)∀x ∈ E. Throughout this paper, some important characteristic in- equalities we will be using are; Lemma 2.14. [16] In the case of uniformly convex space E,with the duality map J p of E, ∀x, y ∈ E we have ‖x − y‖p = 〈J p(x − y), x − y〉 ≥ ‖x‖p − p〈J p(x), y〉 + σp(x, y). with σp(x, y) = pKp ∫ 1 0 (‖x − ty‖∨‖x‖)p t λE ( t‖y‖ 2(‖x − ty‖∨‖x‖) ) dt where by Kp = 4(2 + √ 4) min{ 1 2 p(p − 1) ∧ 1, ( 1 2 p ∧ 1)(p − 1), (p − 1)(1 − ( √ 3 − 1)q), 1 − (1 + (2 − √ 3)q)1−p}. Lemma 2.15. [16] In the case of uniformly smooth space, E,with the duality map J p of E, ∀x, y ∈ E we have ‖x − y‖p = 〈J p(x − y), x − y〉 ≤ ‖x‖p − p〈J p(x), y〉 + σ̄p(x, y). with σ̄p(x, y) = pG p ∫ 1 0 (‖x − ty‖∨‖x‖)p t ρE ( t‖y‖ 2(‖x − ty‖∨‖x‖) ) dt where by G p = 8 ∨ 64cK −1 p with Kp = 4(2 + √ 4) min{ 1 2 p(p − 1) ∧ 1, ( 1 2 p ∧ 1)(p − 1), (p − 1)(1 − ( √ 3 − 1)q), 1 − (1 + (2 − √ 3)q)1−p} and c = 4 λ0√ 1 + λ20 − 1 ∞∏ j=1 ( 1 + 15 2 j+2 λ0 ) 38 Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 35–50 39 with λ0 = √ 339 − 18 30 . Lemma 2.16. [17] Let E be a real uniformly convex Banach space. For arbitrary r > 1, let Br (0) = {x ∈ E : ‖x‖ ≤ r}. Then, there exists a continuous strictly increasing convex function g : [0,∞) −→ [0,∞), g(0) = 0 such that for every x, y ∈ Br (0), fx ∈ Jp(x), fy ∈ Jp(y), the following inequality hold: ‖λx+(1−λ)y‖p ≤ λ‖x‖p+(1−λ)‖y‖p−(λp(1−λ)+(1−λ)pλ)g(‖x−y‖) and, 〈x − y, fx − fy〉 ≥ g(‖x − y‖). Lemma 2.17. [17] Let E be a real uniformly smooth Banach space. For arbitrary r > 1, let Br (0) = {x ∈ E : ‖x‖ ≤ r}. Then, there exists a continuous strictly increasing convex function g : [0,∞) −→ [0,∞), g(0) = 0 such that for every x, y ∈ Br (0), fx ∈ Jq(x), fy ∈ Jq(y), the fol- lowing inequality hold: ‖λx+(1−λ)y‖q ≥ λ‖x‖q+(1−λ)‖y‖q−(λq(1−λ)+(1−λ)qλ)g(‖x−y‖) and, 〈x − y, fx − fy〉 ≤ g(‖x − y‖). 3. Main results Theorem 3.1. Let E1 and E2 be two uniformly convex and uniformly smooth Banach spaces, A : E1 −→ E2 is bounded and linear operator such that A(C), for C ⊂ E1, is closed and convex, U : E1 −→ E1 be a uniformly continuous Bregman generalized asymptotically nonexpansive operator with the se- quences {k(1)n } ⊂ [0,∞) and {µ (1) n } ⊂ [0,∞) satisfying lim n→∞ k(1)n = 0 and lim n→∞ µ (1) n = 0, respectively, and T : E2 −→ E2 be a uni- formly continuous Bregman generalized asymptotically nonex- pansive operator with the sequences {k(2)n } ⊂ [0,∞) and {µ (2) n } ⊂ [0,∞) satisfying lim n→∞ k(2)n = 0 and lim n→∞ µ (2) n = 0, respectively, and, for p, q ∈ (1,∞), ΠpAC : E2 −→ AC be a Bregman pro- jection onto a subset AC, and Fix(U) , φ and Fix(ΠpAC T ) , φ respectively, and (I −U) and (I −ΠpAC T ) be demiclosed at zero. Let x1 ∈ E1 be chosen arbitrarily and let C1 = E1 and the sequence {xn} be defined as follows; zn = J q E∗1 (J pE1 xn −λn A ∗J pE2 (I − Π p ACn T n)Axn), yn = J q E∗1 (αn J p E1 (zn) + (1 −αn)J p E1 (U n(zn))), Cn+1 = {v ∈ Cn : 4p(yn, v) ≤ [kn + 1] 4p (zn, v) + µn; 4p(zn, v) ≤4p(xn, v)}, xn+1 = Π p Cn+1 (x1), n ≥ 1 (11) where λn =  1 ‖A‖ 1 ‖J pE2 (I−ΠpACn T n )Axn‖ , xn , 0 1 ‖A‖p 〈J pE2 (I−ΠpACn T n )Axn,Axn−Π p ACn T n Axn〉p−1 ‖J pE2 (I−ΠpACn T n )Axn‖p , xn = 0 (12) and γ ∈ (0, 1) and τn = 1 ‖xn‖p−1 are chosen such that ρE∗1 (τn) = γ 2qGq‖A‖ × 〈J pE2 (I − Π p ACn T n)Axn, Axn − Π p ACn T n Axn〉 ‖xn‖p‖J p E2 (I − ΠpACn T n)Axn‖ , (13) and A∗ denote the adjoint of A, and {αn} ⊂ (0, 1) satisfies lim inf n→∞ αn(1 − αn) > 0, kn = (k (1) n ∨ k (2) n ), n ≥ 1. If Γ = {v ∈ Fix(U) : Av ∈ Fix(ΠpAC T )}, φ, then {xn} converges strongly to x∗ ∈ Γ. Proof: Assume that ‖λn A∗(J p E2 (I − ΠpACn T n)Axn)‖ = 0 ⇔ xn ∈ Γ = {xn ∈ Fix(U) : Axn ∈ Fix(Π p AC T )} , φ then from 11 we have that zn = xn and 4p(xn, zn) = 0 and the conclusion follows immediately. Now assume that ‖λn A∗(J p E2 (I −ΠpACn T n)Axn)‖, 0 then we will divide the proof into five steps. Step one: We first show that Cn is closed and convex, for any n ≥ 1. By induction hypothesis; since C1 = E1, so C1 is closed and convex. Assuming that Ck is closed and convex. We show that Ck+1 is closed and convex, for some k ≥ 0. We first show that Ck+1 is closed. Let {xm} be a sequence in Ck+1 such that xm → v as n →∞. We need to show that v ∈ Ck+1. From 11 and continuity of Bregman distance and the boundedness of {xm}, we have that 4p(zn, v) = lim m→∞ 4p (zn, xm) ≤ lim m→∞ 4p (xn, xm) = 4p(xn, v). (14) Similarly, we have that 4p(yn, v) = lim m→∞ 4p (yn, xm) ≤ lim m→∞ ([kn + 1] 4p (zn, xm) + µn) = [kn + 1](4p(xn, v) + µn. (15) From (14) and (15), we have that v ∈ Ck+1. Next we show that Ck+1 is convex. Let v1, v2 ∈ Ck+1 and t ∈ (0, 1); putting v = tv1 + (1 − t)v2, it suffices to show that v ∈ Ck+1. Let xk , 0, then 4p(zk, v) = 4p(zk, tv1 + (1 − t)v2) = 1 q ‖J pE1 xk −λk A ∗(J pE2 (I − Π p ACk T k)Axk)‖ q + 1 p ‖tv1 + (1 − t)v2‖ p −〈J pE1 xk −λk A ∗(J pE2 (I − Π p ACk T k)Axk), tv1 + (1 − t)v2〉 (16) 39 Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 35–50 40 = 1 q ‖J pE1 xk −λk A ∗(J pE2 (I − Π p ACk T k)Axk‖ q + 1 p ‖tv1 + (1 − t)v2‖ p −〈J pE1 xk, tv1 + (1 − t)v2〉 + 〈λk(J p E2 (I − ΠpACk T k)Axk, tv1 + (1 − t)v2〉. Where λk〈A ∗(J pE2 (I − Π p ACk T k )Axk, tv1 + (1 − t)v2〉 = λk〈(J p E2 (I − ΠpACk T k )Axk, tAv1 + (1 − t)Av2 + Π p ACk T k Axk − Π p ACk T k Axk〉 = λk〈(J p E2 (I − ΠpACk T k )Axk, tAv1 + (1 − t)Av2 − Π p ACk T k Axk〉 + λk〈(J p E2 (I − ΠpACk T k )Axk, Π p ACk T k Axk〉 = λk〈(J p E2 (I − ΠpACk T k )Axk, tAv1 + (1 − t)Av2 − Π p ACk T k Axk〉 + λk〈(J p E2 (I − ΠpACk T k )Axk, Π p ACk T k Axk − Axk + Axk〉 = λk〈(J p E2 (I − ΠpACk T k )Axk, tAv1 + (1 − t)Av2 − Π p ACk T k Axk〉 −λk〈(J p E2 (I − ΠpACk T k )Axk, Π p ACk T k Axk − Axk〉 + λk〈(J p E2 (I − ΠpACk T k )Axk, Axk〉 = λk〈(J p E2 (I − ΠpACk T k )Axk, tAv1 + (1 − t)Av2 − Axk + Axk − Π p ACk T k Axk〉 −λk〈(J p E2 (I − ΠpACk T k )Axk, Axk − Π p ACk T k Axk〉 + λk〈(J p E2 (I − ΠpACk T k )Axk, Axk〉 = λk〈(J p E2 (I − ΠpACk T k )Axk, (tAv1 + (1 − t)Av2 − Axk ) − (ΠpACk T k Axk − Axk )〉−λk〈(J p E2 (I − ΠpACk T k )Axk, Axk − Π p ACk T k Axk〉 + λk〈(J p E2 (I − ΠpACk T k )Axk, Axk〉 = −λk〈(J p E2 (ΠpACk T k − I)Axk, (tAv1 + (1 − t)Av2 − Axk ) − (ΠpACk T k Axk − Axk )〉−λk〈(J p E2 (I − ΠpACk T k )Axk, Axk − Π p ACk T k Axk〉 + λk〈(J p E2 (I − ΠpACk T k )Axk, Axk〉. By the assumption that A(Ck) is closed and convex and Lemma 2.5 and by the variational inequality for the Bregman projection of zero onto A(Ck) − Axk, as in Definition 2.6, we arrive at 〈(J pE2 (Π p ACk T k − I)Axk, (tAv1 + (1 − t)Av2 − Axk) −(ΠpACk T k Axk − Axk)〉 ≥ 0 and therefore, λk〈J p E2 (I − ΠpACk T k)Axk, tAv1 + (1 − t)Av2〉 ≤ λk〈J p E2 (I − ΠpACk T k)Axk, Axk〉 −λk〈J p E2 (I − ΠpACk T k)Axk, Axk − Π p ACk T k Axk〉. (17) In addition, from Lemma 2.15, we have that 1 q ‖J pE1 xk −λk A ∗(J pE2 (I − Π p ACk T k)Axk‖ q ≤ 1 q ‖J pE1 xk‖ q −λk〈xk, A ∗J pE2 (I − Π p ACk T k)Axk〉 + 1 q σ̄q(J p E1 xk,λk A ∗J pE2 (I − Π p ACk T k)Axk) = 1 q ‖J pE1 xk‖ q −λk〈Axk, J p E2 (I − ΠpACk T k)Axk〉 + 1 q σ̄q(J p E1 xk,λk A ∗J pE2 (I − Π p ACk T k)Axk) = 1 q ‖xk‖ p −λk〈Axk, J p E2 (I − ΠpACk T k)Axk〉 + 1 q σ̄q(J p E1 xk,λk A ∗J pE2 (I − Π p ACk T k)Axk) (18) and 1 q σ̄q(J p E1 xk,λk A ∗J pE2 (I − Π p ACk T k)Axk) = Gq ∫ 1 0 (‖J pE1 xk − tλk A ∗J pE2 (I − Π p ACk T k)Axk‖∨‖J p E1 xk‖)q t × ρE∗  t‖λk A∗J pE2 (I − ΠpACk T k)Axk‖(‖J pE1 xk − tλk A∗J pE2 (I − ΠpACk T k)Axk‖∨‖J pE1 xk‖)  dt, (19) f or every t ∈ [0, 1]. But ‖J pE1 xk−tλk A ∗J pE2 (I − Π p ACk T k)Axk‖ ≤ ‖xk‖ p−1 + ‖λk A ∗J pE2 (I − Π p ACk T k)Axk‖. From (12), suppose that λk = τk ‖A‖ ‖xk‖p−1 ‖J pE2 (I − Π p ACn T n)Axn‖ then we have that ‖J pE1 xk − tλk A ∗J pE2 (I − Π p ACk T k)Axk‖ ≤ 2‖xk‖ p−1 and that ‖xk‖ p−1 ≤ ‖J pE1 xk − tλk A ∗J pE2 (I − Π p ACk T k)Axk‖∨‖J p E1 xk‖ ≤ 2‖xk‖ p−1. (20) From Definition 2.2 (2), (12) and (20), we have that ρE∗1  t‖λk A∗J pE2 (I − ΠpACk T k )Axk‖(‖J pE1 xk − tλk A∗J pE2 (I − ΠpACk T k )Axk‖∨‖J pE1 xk‖)  ≤ ρE∗1  t‖λk A∗J pE2 (I − ΠpACk T k )Axk‖ ‖xk‖p−1  = ρE∗1 (tτk ). (21) Substituting Eqs. (20) and (21) into (19), and since t ≤ τk and ρE∗1 is nondecreasing function then, we have that 1 q σ̄q(J p E2 xk,λk A ∗J pE2 (I − Π p ACk T k)Axk) ≤ 2qGq‖xk‖ ( p−1)q ∫ τk 0 ρE∗1 (t) t dt ≤ 2qGq‖xk‖ p ∫ τk 0 ρE∗1 (τk) τk dt = 2qGq‖xk‖ pρE∗1 (τk). (22) 40 Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 35–50 41 Substituting Eqs. (22) into (18), we have that 1 q ‖J pE1 xk −λk A ∗J pE2 (I − Π p ACk T k)Axk‖ q ≤ 1 q ‖xk‖ p −λk〈Axk, J p E2 (I − ΠpACk T k)Axk〉 + 2 qGq‖xk‖ pρE∗1 (τk) = 1 q ‖J pE1 xk‖ q −λk〈(J p E2 (I − ΠpACk T k)Axk, Axk〉 + 2qGq‖xk‖ pρE∗1 (τk). (23) Substituting (23) and (17) into (16), we have that 4p(zk, tv1 + (1 − t)v2) ≤ 1 q ‖xk‖ p + 1 p ‖tv1 + (1 − t)v2‖ p −〈J pE1 xk, tv1 + (1 − t)v2〉 + 2 qGq‖xk‖ pρE∗1 (τk) −λn〈(J p E2 (I − ΠpACk T k)Axk), Axk − Π p ACk T k Axk〉 = 4p(xk, tv1 + (1 − t)v2) + 2 qGq‖xk‖ pρE∗1 (τk) −λn〈(J p E2 (I − ΠpACk T k)Axk), Axk − Π p ACk T k)Axk〉. (24) Substituting (13) and (12) into (24), we have that 4p(zk, tv1 + (1 − t)v2) ≤4p(xk, tv1 + (1 − t)v2) + γ〈J pE2 (I − Π p ACk T k)Axk, Axk − Π p ACk T k Axk〉 ‖A‖‖J pE2 (I − Π p ACk T k)Axk)‖ (25) − 〈J pE2 (I − Π p ACk T k)Axk, Axk − Π p ACk T k)Axk〉 ‖A∗‖‖J pE2 (I − Π p ACk T k)Axk‖ = 4p(xk, tv1 + (1 − t)v2) − [1 −γ] × 〈J pE2 (I − Π p ACk T k)Axk, Axk − Π p ACk T k)Axk〉 ‖A‖‖J pE2 (I − Π p ACk T k)Axk)‖ . (26) Therefore, 4p(zk, tv1 + (1 − t)v2) ≤4p(xk, tv1 + (1 − t)v2). (27) Let xk = 0, we have 4p(xk, tv1 + (1 − t)v2) = 1 p ‖tv1 + (1 − t)v2‖ p (28) and from (28), we have 4p(zk, tv1 + (1 − t)v2) = 1 q ‖λk A ∗J pE2 (I − Π p ACk T k)Axk‖ q + 4p(xk, tv1 + (1 − t)v2) + λk〈J p E2 (I − ΠpACk T k)Axk), tAv1 + (1 − t)Av2〉. (29) Substituting (17) into (29), we have that 4p(zk, tv1 + (1 − t)v2) ≤ 1 q ‖λk A ∗J pE2 (I − Π p ACk T k)Axk‖ q + 4p(xk, tv1 + (1 − t)v2) + λn〈J p E2 (I − ΠpACk T k)Axk, Axk〉 −λn〈J p E2 (I − ΠpACk T k)Axk, Axk − Π p ACk T k)Axk〉. (30) But, from (12), we have that 1 q ‖λk A ∗J pE2 (I − Π p ACk T k)Axk‖ q = 1 q 1 ‖A‖p 〈J pE2 (I − Π p ACk T k)Axk, Axk − Π p ACk T k)Axk〉p ‖J pE2 (I − Π p ACk T k)Axk‖p . (31) Substituting (31) into (30), we have that 4p(zk, tv1 + (1 − t)v2) ≤ 1 q 1 ‖A‖p 〈J pE2 (I − Π p ACk T k)Axk, Axk − Π p ACk T k)Axk〉p ‖J pE2 (I − Π p ACk T k)Axk‖p + 4p(xk, tv1 + (1 − t)v2) + λk〈J p E2 (I − ΠpACk T k)Axk, Axk〉 −λn〈J p E2 (I − ΠpACk T k)Axk, Axk − Π p ACk T k)Axk〉 ≤ (1 − 1 p ) 1 ‖A‖p 〈J pE2 (I − Π p ACk T k)Axk, Axk − Π p ACk T k)Axk〉p ‖J pE2 (I − Π p ACk T k)Axk‖p + 4p(xk, tv1 + (1 − t)v2) + λk〈A ∗J pE2 (I − Π p ACk T k)Axk, xk〉 − 1 ‖A‖p 〈J pE2 (I − Π p ACk T k)Axk, Axk − Π p ACk T k)Axk〉p ‖J pE2 (I − Π p ACk T k)Axk‖p ×〈J pE2 (I − Π p ACk T k)Axk, Axk − Π p ACk T k)Axk〉 ≤ (1 − 1 p ) 1 ‖A‖p 〈J pE2 (I − Π p ACk T k)Axk, Axk − Π p ACk T k)Axk〉p ‖J pE2 (I − Π p ACk T k)Axk‖p + 4p(xk, tv1 + (1 − t)v2) + λk‖J p E2 (I − ΠpACk T k)Axk‖‖Axk‖ − 1 ‖A‖p 〈J pE2 (I − Π p ACk T k)Axk, Axk − Π p ACk T k)Axk〉p ‖J pE2 (I − Π p ACk T k)Axk‖p ×〈J pE2 (I − Π p ACk T k)Axk, Axk − Π p ACk T k)Axk〉 = 4p(xk, tv1 + (1 − t)v2) − 1 p 1 ‖A‖p 〈J pE2 (I − Π p ACk T k)Axk, Axk − Π p ACk T k)Axk〉p ‖J pE2 (I − Π p ACk T k)Axk‖p . This implies that 4p(zk, tv1 + (1 − t)v2) ≤4p(xk, tv1 + (1 − t)v2). (32) In addition, it follows from Eq. (11), Definition 2.4 and Lemma 2.16 that 4p(yk, tv1 + (1 − t)v2) = 4p(J q E∗1 (αk J p E1 (zk) + (1 −αk)J p E1 (U k(zk)), tv1 + (1 − t)v2) = 1 q ‖αk J p E1 (zk) + (1 −αk)J p E1 (U k(zk)‖ q −〈αk J p E1 (zk) + (1 −αk)J p E1 (U k(zk), tv1 + (1 − t)v2〉 + 1 p ‖tv1 + (1 − t)v2‖ p = 1 q ‖αk J p E1 (zk) + (1 −αk)J p E1 (U k(zk)‖ p −αk〈J p E1 zk, tv1 + (1 − t)v2〉 41 Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 35–50 42 − (1 −αk)〈J p E1 (U k(zk)), tv1 + (1 − t)v2〉 + 1 p ‖tv1 + (1 − t)v2‖ p ≤ 1 q αk‖zk‖ p + 1 q (1 −αk)‖U kzk‖ p − (αqk (1 −αk) + (1 −αk) qαk) × g(‖zk − U kzk‖) −αk〈J p E1 (zk), tv1 + (1 − t)v2〉 − (1 −αk)〈J p E1 (U k(zk)), tv1 + (1 − t)v2〉 + 1 p ‖tv1 + (1 − t)v2‖ p = αk ( 1 p ‖tv1 + (1 − t)v2‖ p −〈J pE1 (zk), tv1 + (1 − t)v2〉 + 1 q ‖zk‖ p ) + (1 −αk) ( 1 p ‖tv1 + (1 − t)v2‖ p −〈J pE1 (U k(zk)), tv1 + (1 − t)v2〉 + 1 q ‖U kzk‖ p ) − (αqk (1 −αk) + (1 −αk) qαk)g(‖zk − U kzk‖) = αk 4p (zk, tv1 + (1 − t)v2) + (1 −αk) 4p (U k(zk), tv1 + (1 − t)v2) − (α p k (1 −αk) + (1 −αk) pαk) × g(‖zk − U kzk‖) ≤ αk 4p (zk, tv1 + (1 − t)v2) + (1 −αk)(kk 4p (zk, tv1 + (1 − t)v2) + 4p(zk, tv1 + (1 − t)v2) + µk) − (α q k (1 −αk) + (1 −αk) qαk)g(‖zk − U kzk‖) ≤ kk 4p (zk, tv1 + (1 − t)v2) + 4p(zk, tv1 + (1 − t)v2) + µk − (αqk (1 −αk) + (1 −αk) qαk)g(‖zk − U kzk‖) ≤ [kk + 1] 4p (zk, tv1 + (1 − t)v2) + µk. (33) From (27), (32) and (33), we have that v ∈ Ck+1. Hence, Ck+1 is convex. Therefore Cn is closed and convex for each n ∈ N Step two: We prove Γ ⊂ Cn for any n ≥ 1. Clearly Γ ⊂ C1. Now assuming that Γ ⊂ Cn for some n ≥ 1. Let v ∈ Γ then from (11) and Lemma 2.15, having considered xn , 0, we get 4p(zn, v) = 1 q ‖J pE1 xn −λn A ∗(J pE2 (I − Π p ACn T k)Axn)‖ q + 1 p ‖v‖p −〈J pE1 xn −λn A ∗(J pE2 (I − Π p ACn T n)Axn), v〉 (34) = 1 q ‖J pE1 xn −λn A ∗(J pE2 (I − Π p ACn T n)Axn‖ q + 1 p ‖v‖p −〈J pE1 xn, v〉 + 〈λn(J p E2 (I − ΠpACn T n)Axn, v〉. Where λn〈A ∗(J pE2 (I − Π p ACn T n)Axn, v〉 = λn〈(J p E2 (I − ΠpACn T n)Axn, Av + Π p ACn T n Axn − Π p ACn T n Axn〉 = λn〈(J p E2 (I − ΠpACn T n)Axn, Av − Π p ACn T n Axn〉 + λn〈(J p E2 (I − ΠpACn T n)Axn, Π p ACn T n Axn〉 = λn〈(J p E2 (I − ΠpACn T n)Axn, Av − Π p ACn T n Axn〉 + λn〈(J p E2 (I − ΠpACn T n)Axn, Π p ACn T n Axn − Axn + Axn〉 = λn〈(J p E2 (I − ΠpACn T n)Axn, Av − Π p ACn T n Axn〉 −λn〈(J p E2 (I − ΠpACn T n)Axn, Π p ACn T n Axn − Axn〉 + λn〈(J p E2 (I − ΠpACn T n)Axn, Axn〉 = λn〈(J p E2 (I − ΠpACn T n)Axn, Av − Axn + Axn − Π p ACn T n Axn〉 −λn〈(J p E2 (I − ΠpACn T n)Axn, Axn − Π p ACn T n Axn〉 + λn〈(J p E2 (I − ΠpACn T n)Axn, Axn〉 = λn〈(J p E2 (I − ΠpACn T n)Axn, (Av − Axn) − (Π p ACn T n Axn − Axn)〉 −λn〈(J p E2 (I − ΠpACn T n)Axn, Axn − Π p ACn T n Axn〉 + λn〈(J p E2 (I − ΠpACn T n)Axn, Axn〉 = −λn〈(J p E2 (ΠpACn T n − I)Axn, (Av − Axn) − (Π p ACn T n Axn − Axn)〉 −λn〈(J p E2 (I − ΠpACn T n)Axn, Axn − Π p ACn T n Axn〉 + λn〈(J p E2 (I − ΠpACn T n)Axn, Axn〉. By the assumption that A(Cn) is closed and convex and Lemma 2.5 and by the variational inequality for the Bregman projection of zero onto A(Cn) − Axn, as in Definition 2.6, we arive at 〈(J pE2 (Π p ACn T n − I)Axn, (Av − Axn) − (Π p ACn T n Axn − Axn)〉 ≥ 0 and therefore, λn〈J p E2 (I − ΠpACn T n)Axn, Av〉 ≤ λn〈J p E2 (I − ΠpACn T n)Axn, Axn〉 −λn〈J p E2 (I − ΠpACn T n)Axn, Axn − Π p ACn T n Axn〉. (35) In addition, from Lemma 2.15, we have that 1 q ‖J pE1 xn −λn A ∗(J pE2 (I − Π p ACn T n)Axn‖ q ≤ 1 q ‖J pE1 xn‖ q −λn〈xn, A ∗J pE2 (I − Π p ACn T n)Axn〉 + 1 q σ̄q(J p E1 xn,λn A ∗J pE2 (I − Π p ACn T n)Axn) = 1 q ‖J pE1 xn‖ q −λn〈Axn, J p E2 (I − ΠpACn T n)Axn〉 + 1 q σ̄q(J p E1 xn,λn A ∗J pE2 (I − Π p ACn T n)Axn) = 1 q ‖xn‖ p −λn〈Axn, J p E2 (I − ΠpACn T n)Axn〉 + 1 q σ̄q(J p E1 xn,λn A ∗J pE2 (I − Π p ACn T n)Axn) (36) and 1 q σ̄q(J p E1 xn,λn A ∗J pE2 (I − Π p ACn T n)Axn) = Gq ∫ 1 0 (‖J pE1 xn − tλn A ∗J pE2 (I − Π p ACn T n)Axn‖∨‖J p E1 xn‖)q t × ρE∗  t‖λn A∗J pE2 (I − ΠpACn T n)Axn‖(‖J pE1 xn − tλn A∗J pE2 (I − ΠpACn T n)Axn‖∨‖J pE1 xn‖)  dt, (37) f or every t ∈ [0, 1]. 42 Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 35–50 43 But ‖J pE1 xn−tλn A ∗J pE2 (I − Π p ACn T n)Axn‖ ≤ ‖xn‖ p−1 + ‖λn A ∗J pE2 (I − Π p ACn T n)Axn‖. From (12), suppose that λn = τn ‖A‖ ‖xn‖p−1 ‖J pE2 (I − Π p ACn T n)Axn‖ then we have ‖J pE1 xn − tλn A ∗J pE2 (I − Π p ACn T n)Axn‖ ≤ 2‖xn‖ p−1 and that ‖xn‖ p−1 ≤ ‖J pE1 xn − tλn A ∗J pE2 (I − Π p ACn T n)Axn‖∨‖J p E1 xn‖ ≤ 2‖xn‖ p−1. (38) From Definition 2.2(2), we have that from Eqs. (38) and (13) ρE∗1  t‖λn A∗J pE2 (I − ΠpACn T n)Axn‖(‖J pE1 xn − tλn A∗J pE2 (I − ΠpACn T n)Axn‖∨‖J pE1 xn‖)  ≤ ρE∗1  t‖λn A∗J pE2 (I − ΠpACn T n)Axn‖ ‖xn‖p−1  = ρE∗1 (tτn). (39) Substituting Eqs. (38) and (39) into (37), and since t ≤ τn and ρE∗1 is nondecreasing then, we have that 1 q σ̄q(J p E2 xn,λn A ∗J pE2 (I − Π p ACn T n)Axn) ≤ 2qGq‖xn‖ (p−1)q ∫ τ 0 ρE∗1 (t) t dt ≤ 2qGq‖xn‖ p ∫ τ 0 ρE∗1 (τn) τn dt ≤ 2qGq‖xn‖ pρE∗1 (τn). (40) Substituting Eq. (40) into (36), we have that 1 q ‖J pE1 xn −λn A ∗J pE2 (I − Π p ACn T n)Axn‖ q ≤ 1 q ‖xn‖ p −λn〈Axn, J p E2 (I − ΠpACn T n)Axn〉 + 2 qGq‖xn‖ pρE∗1 (τn) = 1 q ‖J pE1 xn‖ q −λn〈(J p E2 (I − ΠpACn T n)Axn, Axn〉 + 2qGq‖xn‖ pρE∗1 (τn). (41) Substituting Eqs. (35) and (41) into (34), we have that 4p(zn, v) ≤ 1 q ‖xn‖ p + 1 p ‖v‖p −〈J pE1 xn, v〉 + 2 qGq‖xn‖ pρE∗1 (τn) −λn〈(J p E2 (I − ΠpACn T n)Axn), Axn − Π p ACn T n)Axn〉 = 4p(xn, v) + 2 qGq‖xn‖ pρE∗1 (τn) −λn〈(J p E2 (I − ΠpACn T n)Axn), Axn − Π p ACn T n)Axn〉. (42) Substituting (13) and (12) into (42), we have that 4p(zn, v) ≤4p(xn, v) + γ〈J pE2 (I − Π p ACn T n)Axn, Axn − Π p ACn T n)Axn〉 ‖A‖‖J pE2 (I − Π p ACn T n)Axn)‖ − 〈J pE2 (I − Π p ACn T n)Axn, Axn − Π p ACn T n)Axn〉 ‖A∗‖‖J pE2 (I − Π p ACn T n)Axn‖ = 4p(xn, v) − [1 −γ] × 〈J pE2 (I − Π p ACn T n)Axn, Axn − Π p ACn T n)Axn〉 ‖A∗‖‖J pE2 (I − Π p ACn T n)Axn‖ . (43) Therefore, 4p(zn, v) ≤4p(xn, v). (44) For xn = 0 we have 4p(xn, v) = 1 p ‖v‖p (45) and from (45), we have 4p(zn, v) = 1 q ‖λn A ∗J pE2 (I − Π p ACn T n)Axn‖ q + 4p(xn, v) + λn〈J p E2 (I − ΠpACn T n)Axn), Av〉. (46) Substituting Eq. (35) into Eq. (46), we have that 4p(zn, v) ≤ 1 q ‖λn A ∗J pE2 (I − Π p ACn T n)Axn‖ q + 4p(xn, v) + λn〈J p E2 (I − ΠpACn T n)Axn, Axn〉 −λn〈J p E2 (I − ΠpACn T n)Axn, Axn − Π p ACn T n)Axn〉 (47) But, from (13), we have that 1 q ‖λn A ∗J pE2 (I − Π p ACn T n)Axn‖ q = 1 q 1 ‖A‖p 〈J pE2 (I − Π p ACn T n)Axn, Axn − Π p ACn T n)Axn〉p ‖J pE2 (I − Π p ACn T n Axn‖p . (48) Substituting (48) into (47), we have that 4p(zn, v) (49) ≤ 1 q 1 ‖A‖p 〈J pE2 (I − Π p ACn T n)Axn, Axn − Π p ACn T n Axn〉p ‖J pE2 (I − Π p ACn T n)Axn‖p + 4p(xn, v) + λn〈J p E2 (I − ΠpACn T n)Axn, Axn〉 −λn〈J p E2 (I − ΠpACn T n)Axn, Axn − Π p ACn T n Axn〉 ≤ (1 − 1 p ) 1 ‖A‖p 〈J pE2 (I − Π p ACn T n)Axn, Axn − Π p ACn T n Axn〉p ‖J pE2 (I − Π p ACn T n)Axn‖p 43 Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 35–50 44 + 4p(xn, v) + λn〈A ∗J pE2 (I − Π p ACn T n)Axn, xn〉 − 1 ‖A‖p 〈J pE2 (I − Π p ACk T n)Axn, Axn − Π p ACn T n Axn〉p ‖J pE2 (I − Π p ACn T n)Axn‖p ×〈J pE2 (I − Π p ACn T n)Axn, Axn − Π p ACn T n Axn〉 ≤ (1 − 1 p ) 1 ‖A‖p 〈J pE2 (I − Π p ACn T n)Axn, Axn − Π p ACn T n Axn〉p ‖J pE2 (I − Π p ACn T n)Axn‖p + 4p(xn, v) + λn‖A ∗J pE2 (I − Π p ACn T n)Axn‖‖xn‖ − 1 ‖A‖p 〈J pE2 (I − Π p ACn T n)Axn, Axn − Π p ACn T n Axn〉p ‖J pE2 (I − Π p ACn T n)Axn‖p ×〈J pE2 (I − Π p ACn T n)Axn, Axn − Π p ACn T n Axn〉 = 4p(xn, v) − 1 p ) 1 ‖A‖p 〈J pE2 (I − Π p ACn T n)Axn, Axn − Π p ACn T n Axn〉p ‖J pE2 (I − Π p ACn T n)Axn‖p = 4p(xn, v) − 1 p ) 1 ‖A‖p × 〈J pE2 (I − Π p ACn T n)Axn, Axn − Π p ACn T n)Axn〉p ‖J pE2 (I − Π p ACn T n)Axn‖p (50) This implies that 4p(zn, v) ≤4p(xn, v). (51) In addition, it follows from (11), Definition 2.4, Lemma 2.16 and p ∈ (1,∞) that 4p(yn, v) = 4p(J q E∗1 (αn J p E1 (zn) + (1 −αn)J p E1 (U n(zn))), v) = 1 q ‖αn J p E1 zn + (1 −αn)J p E1 U zn‖ q −〈αn J p E1 zn + (1 −αn)J p E1 U nzn, v〉 + 1 p ‖v‖p = 1 q ‖αn J p E1 zn + (1 −αn)J p E1 U nzn‖ p −αn〈J p E1 zn, v〉 − (1 −αn)〈J p E1 (U n(zn)), v〉 + 1 p ‖v‖p ≤ 1 q αn‖zn‖ p + 1 q (1 −αn)‖U nzn‖ p − (αqn(1 −αn) + (1 −αn) qαn) × g(‖zn − U nzn‖) −αn〈J p E1 (zn), v〉− (1 −αn)〈J p E1 (U n(zn)), v〉 + 1 p ‖v‖p = αn( 1 p ‖v‖p −〈J pE1 (zn), v〉 + 1 q ‖zn‖ p) + (1 −αn)( 1 p ‖v‖p −〈J pE1 (U n(zn)), v〉 + 1 q ‖U nzn‖ p) − (αqn(1 −αn) + (1 −αn) qαn)g(‖zn − U nzn‖) = αn 4p (zn, v) + (1 −αn) 4p (U n(zn), v) − (αpn (1 −αn) + (1 −αn) pαn) × g(‖zn − U nzn‖) ≤ αn 4p (zn, v) + (1 −αn)(kn 4p (zn, v) + 4p(zn, v) + µn) − (αqn(1 −αn) + (1 −αn) qαn)g(‖zn − U nzn‖) ≤ kn 4p (zn, v) + 4p(zn, v) + µn − (αqn(1 −αn) + (1 −αn) qαn)g(‖zn − U nzn‖) (52) ≤ [kn + 1] 4p (zn, v) + µn. (53) From Eqs. (44), (51) and (53), we have that v ∈ Cn and Γ ⊂ Cn for any n ≥ 1. Step three: we show that {xn} is Cauchy sequence. For v ∈ Cn, and xn+1 = Π p Cn+1 (x1) ∈ Cn+1 ⊂ Cn, then from Definition 2.6, we have that 0 ≤ 〈J pE1 Π p Cn+1 (x1) − J p E1 (x1), v − Π p Cn+1 (x1)〉 = 〈J pE1 Π p Cn+1 (x1), v〉− 〈J p E1 (x1), v〉 − 〈J pE1 Π p Cn+1 (x1), Π p Cn+1 (x1)〉 + 〈J p E1 (x1), Π p Cn+1 (x1)〉. This implies that |〈J pE1 Π p Cn+1 (x1), Π p Cn+1 (x1)〉| ≤ |〈J p E1 Π p Cn+1 (x1), v〉 − 〈J pE1 (x1), v〉 + 〈J p E1 (x1), Π p Cn+1 (x1)〉|. That is ‖Π p Cn+1 (x1)‖ p ≤ ‖Π p Cn+1 (x1)‖ p−1 ‖v‖ + ‖x1‖ p−1 ‖v‖ + ‖x1‖ p−1 ‖Π p Cn+1 (x1)‖. (54) Observe that if ‖Π p Cn+1 (x1)‖ p−1 < 2‖x1‖ p−1 then we have ‖ΠpCn+1 (x1)‖ < 2 q−1 ‖x1‖ (55) and so we are done. Otherwise, let tx1 = ‖ΠpCn+1 (x1)‖ ‖x1‖  p−1 ≥ 2. For x1 , 0 and using (54) we have that ‖Π p Cn+1 (x1)‖(‖Π p Cn+1 (x1)‖ p−1 −‖x1‖ p−1) ≤ ‖v‖(‖ΠpCn+1 (x1)‖ p−1 + ‖x1‖ p−1) and so ‖Π p Cn+1 (x1)‖ ≤ ‖v‖ ‖Π p Cn+1 (x1)‖p−1 + ‖x1‖p−1 ‖Π p Cn+1 (x1)‖p−1 −‖x1‖p−1 . (56) From (56), since the function h(tx1 ) = tx1 +1 tx1−1 is decreasing for tx1 > 1 we arive at ‖Π p Cn+1 (x1)‖ ≤ min v∈Cn+1 ‖v‖h(tx1 ) ≤ min v∈Cn+1 ‖v‖h(2) = 3 min v∈Cn+1 ‖v‖. (57) If x1 = 0 then from (54), we have that ‖Π p Cn+1 (x1)‖ ≤ min v∈Cn+1 ‖v‖. (58) From (55), (57) and (58), we have that xn+1 = Π p Cn+1 (x1) is 44 Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 35–50 45 bounded. For any n ≥ 1, by Definition 2.6, we have that 0 ≤4p(J p E1 Π p Cn (x1), Π p Cn+1 (x1)) ≤4p(x1, Π p Cn+1 (x1)) −4p(x1, Π p Cn (x1)) (59) From (59), we have that 4p(x1, Π p Cn (x1)) ≤4p(x1, Π p Cn+1 (x1)). Thus {4p(x1, Π p Cn (x1))} is nondecreasing. Therefore by bound- edness of ΠpCn (x1), we have that limn→∞ 4p (x1, Π p Cn (x1)) exists. Let m, n ∈ N m < n. From xn = Π p Cn (x1) ⊂ Cn and (59), we have that 4p(Π p Cn (x1), Π p Cm (x1)) ≤4p(x1, Π p Cm (x1)) −4p(x1, Π p Cn (x1)). (60) Since lim n→∞ 4p (x1, Π p Cn (x1)) exists, it follows from (60) that 4p(Π p Cm (x1), Π p Cn (x1)) → 0 as m, n →∞. Therefore {ΠpCn (x1)} is a Cauchy sequence. Step four: we will show that lim n→∞ 4p(zn, Uzn) = 0 and lim n→∞ 4p (Axn, Π p ACn T n Axn) = 0. Let xn+1 = Π p Cn+1 (x1) ⊂ Cn+1 ⊂ Cn. From 11, we have that 4p(zn, v) ≤4p(xn, v) ⇔ 1 p ‖zn − v‖ p ≤ 1 p ‖xn − v‖ p. Therefore {zn} is bounded as {xn} is bounded. Hence, from Lemma 2.9 and 2.10, we have 4p(zn, xn) = 4p (zn, xn+1) + 4p(xn+1, xn) −〈zn − xn+1, J p xn − J p xn+1〉 ≤4p(xn, xn+1) + 4p(xn+1, xn) −〈zn − xn+1, J p xn − J p xn+1〉 = 〈J p xn+1 − J p xn, xn+1 − xn,〉 − 〈zn − xn+1, J p xn − J p xn+1〉 → 0 as n →∞. (61) Now, for xn , 0 for all n ∈ N, from (43), we have 4p(zn, v) ≤4p(xn, v) − [1 −γ]‖Axn − Π p ACn T n Axn‖ p−1. This implies that ‖Axn − Π p ACn T n Axn‖ ≤ ( 4p(xn, v) −4p(zn, v) [1 −γ] ) 1 p−1 =  1 q (‖zn‖ p −‖xn‖p) + 〈J p E1 zn − J p E1 xn, v〉 [1 −γ]  1 p−1 and therefore we have that lim n→∞ ‖Axn − Π p ACn T n Axn‖ = 0. (62) For xn = 0 for all n ∈ N, from (49), we have 4p(zn, v) ≤4p(xn, v) − 1 p 〈J pE2 (I − Π p ACn T n)Axn, Axn − Π p ACn T n Axn〉p ‖A‖p‖J pE2 (I − Π p ACn T n)Axn)‖p . T his implies that ‖Axn − Π p ACn T n Axn‖ ≤ p 1 p2−p ‖A‖p ( 4p(xn, v) −4p(zn, v) ) 1 p2−p = p 1 p2−p ‖A‖p ( 1 q (‖zn‖ p −‖xn‖ p) + 〈J pE1 zn − J p E1 xn, v〉 ) 1 p2−p . Therefore, we have that lim n→∞ ‖Axn − Π p ACn T n Axn‖ = 0. (63) By Lemma 2.11, we have that lim n→∞ 4p (Axn, Π p ACn T n Axn) = 0. (64) On the other hand, since 4p (yn, xn) = 4p(yn, xn+1) + 4p(xn+1, xn) −〈yn − xn+1, J p xn − J p xn+1〉 ≤ [1 + kn] 4p (zn, xn+1) + 〈J p xn+1 − J p xn, xn+1 − xn,〉 − 〈zn − xn+1, J p xn − J p xn+1〉 + µn ≤ [1 + kn] 4p (xn, xn+1) + 〈J p xn+1 − J p xn, xn+1 − xn,〉 − 〈zn − xn+1, J p xn − J p xn+1〉 + µn → 0 as n →∞ (65) and 4p(yn, zn) = 4p(yn, xn) + 4p(xn, zn) −〈yn − xn, J pzn − J p xn〉→ 0 as n →∞. (66) Now, from (52), we have that 4p(yn, v) ≤ [kn + 1] 4p (zn, v) + µn (αpn (1 −αn) + (1 −αn) pαn)g(‖zn − U nzn‖). This implies, From (44), we have that g(‖zn − U nzn‖) ≤ [kn + 1] 4p (zn, v) + µn −4p(yn, v) (αpn (1 −αn) + (1 −αn)pαn) = 1 q ([kn + 1]‖zn‖ p −‖yn‖p) + 〈J pyn − J pzn, v〉− kn〈J pzn, v〉 (αpn (1 −αn) + (1 −αn)pαn) . + kn p ‖v‖ p + µn (αpn (1 −αn) + (1 −αn)pαn) . Since αn(1 − αn) > 0, and g is continuous, strictly increasing and convex function, then we have that lim n→∞ g‖zn − U n(zn)‖ = 0 45 Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 35–50 46 and lim n→∞ ‖zn − U n(zn)‖ = 0. (67) By Lemma 2.11, we have that lim n→∞ 4p (zn, U nzn) = 0. (68) Since T and U are continuous, then from (64) and (68), we have that lim n→∞ 4p (Axn, Π p AC T Axn) = 0 (69) and lim n→∞ 4p (zn, Uzn) = 0. (70) Step five: we will show that {xn} converges strongly to an el- ement of Γ. Since {xn} is Cauchy sequence, we may assume that xn −→ x∗, from (61) we have zn −→ x∗, which implies that zn ⇀ x∗. So it follows from (70) and the demicloseness of (I−U) at zero that x∗ ∈ Fix(U). In addition, since A is bounded linear operator, we have that lim n→∞ ‖Axn−Ax∗‖ = 0. Hence, it fol- lows from (69), and demicloseness of (I − ΠpAC T ) at zero that Ax∗ ∈ Fix(ΠpAC T ). This means that x ∗ ∈ Γ and {xn} converges strongly to x∗ ∈ Γ. The proof is completed. In theorem 3.1, as U = T , E1 = E2 = E and A = I, we have the following result. Corollary 3.1.1. Let E be a uniformly convex and uniformly smooth Banach spaces, T : E −→ E be a uniformly contin- uous Bregman generalized asymptotically nonexpansive opera- tor with the sequences {kn} ⊂ [0,∞) and {µn} ⊂ [0,∞) satisfying lim n→∞ kn = 0 and lim n→∞ µn = 0, respectively, and, for p, q ∈ (1,∞), Π p C : E −→ C be a Bregman projection onto a subset C ∈ E, and Fix(T ) , φ and (I − T ) be demiclosed at zero. Let x1 ∈ E be chosen arbitrarily and let C1 = E. Define a sequence {xn} as follows; zn = J q E∗1 (J pE1 xn −λn J p E2 (I − ΠpCn T n)xn), yn = J q E∗1 (αn J p E1 (zn) + (1 −αn)J p E1 (T n(zn))), Cn+1 = {v ∈ Cn : 4p(yn, v) ≤ [kn + 1] 4p (zn, v) + µn; 4p(zn, v) ≤4p(xn, v)}, xn+1 = Π p Cn+1 (x1), n ≥ 1 (71) where λn =  1 ‖J pE2 (I−ΠpCn T n )xn‖ , xn , 0 〈J pE2 (I−ΠpCn T n )xn,xn−Π p Cn T n xn〉p−1 ‖J pE2 (I−ΠpCn T n )xn‖p , xn = 0 and γ ∈ (0, 1) and τn = 1 ‖xn‖p−1 are chosen such that ρE∗1 (τn) = γ 2qGq × 〈J pE2 (I − Π p Cn T n)xn, xn − Π p Cn T n xn〉 ‖xn‖p‖J p E2 (I − ΠpCn T n)xn‖ , and {αn} ⊂ (0, 1) satisfies lim inf n→∞ αn(1 − αn) > 0. If Fix(T ) = {v ∈ E : v = T v} , φ, then {xn} converges strongly to x∗ ∈ Fix(T ). In theorem 3.1, when U and T are two Bregman symptotically nonexpansive mappings, the following result holds. Corollary 3.1.2. Let E1 and E2 be two uniformly convex and uniformly smooth Banach spaces, A : E1 −→ E2 is bounded and linear operator, such that A(C), for C ⊂ E1, is closed and convex, U : E1 −→ E1 be a uniformly continuous Breg- man asymptotically nonexpansive operator with the sequence {k(1)n } ⊂ [1,∞) satisfying limn→∞ k (1) n = 1, and T : E2 −→ E2 be a uniformly continuous Bregman asymptotically nonex- pansive operator with the sequence {k(2)n } ⊂ [1,∞) satisfying limn→∞ k (2) n = 1, respectively and, for p, q ∈ (1,∞), Π p AC : E2 −→ AC be a Bregman projection onto a subset AC, and Fix(U) , φ and Fix(ΠpAC T ) , φ respectively, and (I − U) and (I − ΠpAC T ) be demiclosed at zero. Let x1 ∈ E1 be chosen arbi- trarily and let C1 = E1. Define a sequence {xn} as follows; zn = J q E∗1 (J pE1 xn −λn A ∗J pE2 (I − Π p ACn T n)Axn), yn = J q E∗1 (αn J p E1 (zn) + (1 −αn)J p E1 (U n(zn))), Cn+1 = {v ∈ Cn : 4p(yn, v) ≤ kn 4p (zn, v) ≤ kn 4p (xn, v)}, xn+1 = Π p Cn+1 (x1), n ≥ 1 (72) where λn =  1 ‖A‖ 1 ‖J pE2 (I−ΠpACn T n )Axn‖ , xn , 0 1 ‖A‖p 〈J pE2 (I−ΠpACn T n )Axn,Axn−Π p ACn T n Axn〉p−1 ‖J pE2 (I−ΠpACn T n )Axn‖p , xn = 0 and γ ∈ (0, 1) and τn = 1 ‖xn‖p−1 are chosen such that ρE∗1 (τn) = γ 2qGq‖A‖ × 〈J pE2 (I − Π p ACn T n)Axn, Axn − Π p ACn T n Axn〉 ‖xn‖p‖J p E2 (I − ΠpACn T n)Axn‖ , and A∗ denote the adjoint of A, and {αn} ⊂ (0, 1) satisfies lim inf n→∞ αn(1 − αn) > 0, kn = (k (1) n ∨ k (2) n ), n ≥ 1. If Γ = {v ∈ Fix(U) : Av ∈ Fix(ΠpAC T )}, φ, then {xn} converges strongly to x∗ ∈ Γ. In theorem 3.1, when U and T are two φ-asymptotically nonex- pansive mappings, the following result holds. Corollary 3.1.3. Let E1 and E2 be two uniformly convex and uniformly smooth Banach spaces, A : E1 −→ E2 is bounded and linear operator, such that A(C), for C ⊂ E1, is closed and convex, U : E1 −→ E1 be a uniformly continuous φ- asymptotically nonexpansive operator with the sequence {k(1)n } ⊂ [1,∞) satisfying limn→∞ k (1) n = 1, and T : E2 −→ E2 be a uniformly continuous φ-asymptotically nonexpansive operator with the sequence {k(2)n } ⊂ [1,∞) satisfying limn→∞ k (2) n = 1, respectively and, for p, q ∈ (1,∞), ΠpAC : E2 −→ AC be a generalized projection onto a subset AC, and Fix(U) , φ and Fix(ΠpAC T ) , φ respectively, and (I − U) and (I − Π p AC T ) be demiclosed at zero. Let x1 ∈ E1 be chosen arbitrarily and let 46 Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 35–50 47 C1 = E1. Define a sequence {xn} as follows; zn = J−1E∗1 (JE1 xn −λn A ∗JE2 (I − Π p ACn T n)Axn), yn = J−1E∗1 (αn JE1 (zn) + (1 −αn)JE1 (U n(zn))), Cn+1 = {v ∈ Cn : φ(yn, v) ≤ knφ(zn, v) ≤ knφ(xn, v)}, xn+1 = ΠCn+1 (x1), n ≥ 1 (73) where λn =  1 ‖A‖ 1 ‖JE2 (I−Π p ACn T n )Axn‖ , xn , 0 1 ‖A‖p 〈JE2 (I−ΠACn T n )Axn,Axn−ΠACn T n Axn〉 ‖JE2 (I−ΠACn T n )Axn‖ , xn = 0 and γ ∈ (0, 1) and τn = 1 ‖xn‖ are chosen such that ρE∗1 (τn) = γ 2G‖A‖ × 〈JE2 (I − ΠACn T n)Axn, Axn − ΠACn T n Axn〉 ‖xn‖‖JE2 (I − ΠACn T n)Axn‖ , and A∗ denote the adjoint of A, and {αn} ⊂ (0, 1) satisfies lim inf n→∞ αn(1 − αn) > 0, kn = (k (1) n ∨ k (2) n ), n ≥ 1. If Γ = {v ∈ Fix(U) : Av ∈ Fix(ΠAC T )}, φ, then {xn} converges strongly to x∗ ∈ Γ. In theorem 3.1, when E1 and E2 are real Hilbert spaces H1 and H2, respectively, we have the following result. Corollary 3.1.4. Let H1 and H2 be two real Hilbert spaces, A : H1 −→ H2 is bounded and linear operator, U : H1 −→ H1 be a uniformly continuous generalized asymptotically nonexpansive operator with the sequences {k(1)n } ⊂ [0,∞) and {µ (1) n } ⊂ [0,∞) satisfying lim n→∞ k(1)n = 0 and lim n→∞ µ (1) n = 0, respectively, and T : H2 −→ H2 be a uniformly continuous generalized asymptoti- cally nonexpansive operator with the sequences {k(2)n } ⊂ [0,∞) and {µ(2)n } ⊂ [0,∞) satisfying lim n→∞ k(2)n = 0 and lim n→∞ µ (2) n = 0, re- spectively, and (I − U) and (I − T ) be demiclosed at zero. Let x1 ∈ H1 be chosen arbitrarily and let C1 = H1 and the sequence {xn} be defined as follows; zn = xn −λn A∗(I − T n)Axn), yn = αnzn + (1 −αn)U n(zn), Cn+1 = {v ∈ Cn : ‖yn − v‖ ≤ [kn + 1]‖zn − v‖ + µn; ‖zn − v‖ ≤ [kn + 1]‖xn − v‖ + µn}, xn+1 = PCn+1 (x1), n ≥ 1 (74) where λn ∈ (0, 1 ‖A∗‖2 ) and A ∗ denote the adjoint of A, and {αn} ⊂ (0, 1) satisfies lim inf n→∞ αn(1−αn) > 0, and kn = (k (1) n ∨k (2) n ), n ≥ 1. If Γ = {v ∈ Fix(U) : Av ∈ Fix(T )} , φ, then {xn} converges strongly to x∗ ∈ Γ. In theorem 3.1, when U and T are two nonexpansive map- pings, the following result holds. Corollary 3.1.5. Let E1 and E2 be two uniformly convex and uniformly smooth Banach spaces, A : E1 −→ E2 is bounded and linear operator, such that A(C), for C ⊂ E1, is closed and convex, U : E1 −→ E1 be a uniformly continuous Bregman nonexpansive operator, and T : E2 −→ E2 be a uniformly con- tinuous Bregman nonexpansive operator, and, for p, q ∈ (1,∞), Π p AC : E2 −→ AC be a Bregman projection onto a subset AC, and Fix(U) , φ and Fix(ΠpAC T ) , φ respectively, and (I − U) and (I − ΠpAC T ) be demiclosed at zero. Let x1 ∈ E1 be chosen arbitrarily and let C1 = E1 and the sequence {xn} be defined as follows; zn = J q E∗1 (J pE1 xn −λn A ∗J pE2 (I − Π p ACn T )Axn), yn = J q E∗1 (αn J p E1 (zn) + (1 −αn)J p E1 (U(zn))), Cn+1 = {v ∈ Cn : 4p(yn, v) ≤4p(zn, v) ≤4p(xn, v)}, xn+1 = Π p Cn+1 (x1), n ≥ 1 (75) where λn =  1 ‖A‖ 1 ‖J pE2 (I−ΠpACn T )Axn‖ , xn , 0 1 ‖A‖p 〈J pE2 (I−ΠpACn T )Axn,Axn−Π p ACn T Axn〉p−1 ‖J pE2 (I−ΠpACn T )Axn‖ p , xn = 0 and γ ∈ (0, 1) and τn = 1 ‖xn‖p−1 are chosen such that ρE∗1 (τn) = γ 2qGq‖A‖ × 〈J pE2 (I − Π p ACn T )Axn, Axn − Π p ACn T Axn〉 ‖xn‖p‖J p E2 (I − ΠpACn T )Axn‖ , and A∗ denote the adjoint of A, and {αn} ⊂ (0, 1) satisfies lim inf n→∞ αn(1−αn) > 0. If Γ = {v ∈ Fix(U) : Av ∈ Fix(Π p AC T )}, φ, then {xn} converges strongly to x∗ ∈ Γ. 4. Application to the mixed equilibrium problem Lemma 4.1. [4] Let E be a reflexive, strictly convex and smooth Banach space, and let C be a nonempty closed convex subset of E. Let f, g : C × C −→ < be two bifunctions which satisfy the conditions (A1) − (A4), (B1) − (B3)and(C), in (6), then for every x ∈ E and r > 0, there exists a unique point z ∈ C such that f (z, y) + g(z, y) + 1 r 〈y − z, jz − jx〉 ≥ 0∀y ∈ C} In Reich, S, Sabach, S (2010) [18], when f (x) = 1p‖x‖ p then we have the following Lemma. Lemma 4.2. Let E be a reflexive, strictly convex and smooth Banach space, and let C be a nonempty closed convex subset of E. Let f, g : C × C −→ < be two bifunctions which satisfy the conditions (A1) − (A4), (B1) − (B3)and(C), in (6), then for every x ∈ E and r > 0, define a mapping S r : E −→ C as follows; S r (x) = {x ∈ C : f (z, y)+g(z, y)+ 1 r 〈y−z, J pE z−J p E x〉 ≥ 0∀y ∈ C} then the following hold; 1. S r is a single-valued; 2. S r is a Bregman firmly nonexpansive-type mapping, i.e. ∀x, y ∈ E〈S r x−S r y, J p E S r x−J p E S r y〉 ≤ 〈S r x−S r y, J p E x−J p E y〉; or equivalently 4p(S r x, S r y) + 4p(S r y, S r x) + 4p(S r x, x) + 4p(S r y, y) ≤ 4p(S r x, y) + 4p(S r y, x) 47 Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 35–50 48 3. F(S r ) = MEP( f, g); 4. MEP( f, g) is closed and convex; 5. for all x ∈ E and for all v ∈ F(S r ), 4p(v, S r x)+4p(S r x, x) ≤ 4p(v, x). Theorem 4.1. Let E1 and E2 be two uniformly convex and uni- formly smooth Banach spaces, A : E1 −→ E2 is bounded and linear operator, such that A(C), for C ⊂ E1, is closed and con- vex, T : E2 −→ E2 be a uniformly continuous Bregman nonex- pansive operator, and, for p, q ∈ (1,∞), ΠpAC : E2 −→ AC be a Bregman projection onto a subset AC, and f, g : C×C −→< be two bifunctions which satisfy the conditions (A1) − (A4), (B1) − (B3) and (C), in (6), for C × C ⊂ E1 × E1, assume that C = MEP( f, g) , φ and Fix(T ) , φ, taking C1 = E1, the sequence {xn} is defined as follows; choosing arbitrarily x1 ∈ E1, zn = J q E∗1 (J pE1 xn −λn A ∗J pE2 (I − Π p ACn T n)Axn), f (S r (xn), b) + g(S r (xn), b) + 1 rn 〈b − S r (xn), J p E1 (S r (xn)) −J pE1 (xn)〉 ≥ 0 ∀b ∈ E1 yn = J q E∗1 (αn J p E1 (zn) + (1 −αn)J p E1 (S r (zn))), Cn+1 = {v ∈ Cn : 4p(yn, v) ≤4p(zn, v) ≤4p(xn, v)} xn+1 = Π p Cn+1 (x1), n ≥ 1, (76) where λn =  1 ‖A‖ 1 ‖J pE2 (I−ΠpACn T n )Axn‖ , xn , 0 1 ‖A‖p 〈J pE2 (I−ΠpACn T n )Axn,Axn−Π p ACn T n Axn〉p−1 ‖J pE2 (I−ΠpACn T n )Axn‖p , xn = 0 and γ ∈ (0, 1) and τn = 1 ‖xn‖p−1 are chosen such that ρE∗1 (τn) = γ 2qGq‖A‖ × 〈J pE2 (I − Π p ACn T n)Axn, Axn − Π p ACn T n Axn〉 ‖xn‖p‖J p E2 (I − ΠpACn T n)Axn‖ , and A∗ denote the adjoint of A, and {αn} ⊂ (0, 1) satisfies lim inf n→∞ αn(1−αn) > 0. If Γ = {v ∈ MEP( f, g) : Av ∈ Fix(Π p AC T )}, φ, then {xn} converges strongly to x∗ ∈ Γ. Proof: It follows from the lemma 4.2 that Fix(S r ) = MEP( f, g) is nonempty, closed and convex and S r is a firmly nonexpansive mapping. Meanwhile S r is assume to be contin- uous. Hence all conditions in corollary 3.1.5 are satisfied. The conclusion of theorem 4.1 can be directly obtained from corol- lary 3.1.5. Let E1 and E2 be two uniformly convex and uniformly smooth Banach spaces. Let C and Q be nonempty closed convex sub- sets of E1 and E2, respectively. A : E1 −→ E2 is bounded and linear operator. Assume that f, g : C ×C −→< be two bifunc- tions and f ′, g′ : Q × Q −→< be another two bifunctions. The split mixed equilibrium problem (SMEP) is to find an element v ∈ C such that f (v, y) + g(v, y) ≥ 0∀y ∈ C (77) and such that Av ∈ Q solve f (Av, Ay) + g(Av, Ay) ≥ 0∀Ay ∈ Q (78) Let ω = {v ∈ MEP( f, g) : Av ∈ MEP( f ′, g′)} denote the solu- tion set of the SMEP. Corollary 4.1.1. Let E1 and E2 be two uniformly convex and uniformly smooth Banach spaces. Let C and Q be nonempty closed convex subsets of E1 and E2, respectively. A : E1 −→ E2 is bounded and linear operator, such that A(C) is closed and convex, and, such that A(C), for C ⊂ E1, is closed and convex, for p, q ∈ (1,∞), ΠpAC : E2 −→ AC be a Bregman projection onto a subset AC. Assume that f, g : C × C −→ < be two bifunctions and f ′, g′ : Q × Q −→ < be another two bifunctions which satisfy the conditions (A1) − (A4), (B1) − (B3) and (C), in (6), for C×C ⊂ E1×E1, assuming MEP( f, g) , φ and MEP(ΠpAC, f ′, g′) , φ, respectively, taking C1 = E1, the sequence {xn} is defined as follows; choosing arbitrarily x1 ∈ E1, f (S f +gr (xn), b) + g(S f +g r (xn), b) + 1 rn 〈b − S f +gr (xn), J p E1 (S f +gr (xn)) −J pE1 (xn)〉 ≥ 0 ∀b ∈ E1 f (S f ′+g′ r (Axn), b) + g(S f ′+g′ r (Axn), b) + 1 rn 〈b − S f ′+g′ r (Axn), J p E2 (S f ′+g′ r (Axn)) −J pE2 (Axn)〉 ≥ 0 ∀ b ∈ E2 zn = J q E∗1 (J pE1 (xn) −λA ∗(J pE2 (I − Π p AC S f ′+g′ rn )Axn)), yn = J q E∗1 (αn J p E1 (zn) + (1 −αn)J p E1 (S f +gr (zn)), Cn+1 = {v ∈ Cn : 4p(yn, v) ≤4p(zn, v) ≤4p(xn, v)} xn+1 = ∏p Cn+1 (x1), n ≥ 1, (79) where λn =  1 ‖A‖ 1 ‖J pE2 (I−ΠpAC S f′+g′ rn )Axn‖ , xn , 0 1 ‖A‖p 〈J pE2 (I−ΠpAC S f′+g′ rn )Axn ),Axn−Π p AC S f′+g′ rn Axn〉 p−1 ‖J pE2 (I−ΠpAC S f′+g′ rn )Axn‖ p , xn = 0 and γ ∈ (0, 1) and τn = 1 ‖xn‖p−1 are chosen such that ρE∗1 (τn) = γ 2qGq‖A‖ 〈JE2 (I − Π p AC S f ′+g′ rn )Axn, Axn − Π p AC S f ′+g′ rn Axn〉 ‖xn‖p‖J p E2 (I − ΠpAC S f ′+g′ rn )Axn‖ and A∗ denote the adjoint of A, and {rn} ⊂ (0, 1), and {αn} ⊂ (0, 1) satisfies limn→∞ αn(1 − αn) > 0, n ≥ 1. If ω = {v ∈ MEP( f, g) : Av ∈ MEP(ΠpAC, f ′, g′)} , φ, then {xn} converges strongly to x∗ ∈ ω. 5. An Example In this section we discuss how to apply Theorem 3.1 on the following example. Let E1 = E2 = C1 = (−∞,∞), A = A∗ = I, p = q = 2, and αn = 1 2n . Define U : E1 −→ E1 by U(x) = x, x ∈ (−∞, 0] = B11 2 x, x ∈ (0,∞) = B2, A : E1 −→ E2 by A(x) = x, x ∈ (−∞,∞), 48 Yusuf Ibrahim / J. Nig. Soc. Phys. Sci. 1 (2019) 35–50 49 T : E2 −→ E2 by T (x) = x, x ∈ (−∞, 0] = D11 4 x, x ∈ (0,∞) = D2, Π 2 C : E2 −→ C by Π 2 C (x) = 0, x ∈ (−∞, 0)x, x ∈ [0,∞), Π 2 C T : E2 −→ C by Π 2 C T (x) =  x, x ∈ {0} 0, x ∈ (−∞, 0) 1 4 x, x ∈ (0,∞). Now we check whether or not the mappings U and T are uni- formly continuous Bregman generalized asymptotically nonex- pansive. To do this it suffices to check for each map the follow- ing four cases. Case I: if x, y ∈ B1 then 42(U n(x), U n(y)) = 42(x, y) ≤ [ 1 2n + 1] 42 (x, y) + 1 2n ; (80) Case II: if x, y ∈ B2 then 42(U n(x), U n(y)) = 42 ( 1 2n x, 1 2n y) = 1 22n 42 (x, y) ≤ [ 1 2n + 1] 42 (x, y) + 1 2n ; (81) Case III: if x ∈ B2 and y ∈ B1, then 42(U n(x), U n(y)) = 42( 1 2n x, y) = 1 22n 1 2 |x|2 − 1 2n 〈x, y〉 + 1 2 |y|2 ≤ 1 2 |x|2 −〈x, y〉 + 1 2 |y|2 = 42(x, y) ≤ [ 1 2n + 1] 42 (x, y) + 1 2n ; (82) Case IV: if x ∈ B1 and y ∈ B2, then 4p(U n(x), U n(y)) = 42(x, 1 2n y) = 1 2 |x|2 − 1 2n 〈x, y〉 + 1 2 1 22n |y|2 ≤ 1 2 |x|2 −〈x, y〉 + 1 2 |y|2 = 42(x, y) ≤ [ 1 2n + 1] 42 (x, y) + 1 2n . (83) Since it is clear that U is uniformly continuous, then from (80), (81), (82) and (83) U is a uniformly continuous Bregman gen- eralized asymptotically nonexpansive mapping. Similarly, it is also clear that T is a uniformly continuous Bregman general- ized asymptotically nonexpansive mapping. Next, we simplify the scheme as follows. Clearly Fix(U) = (−∞, 0], Fix(T ) = (−∞, 0] and Fix(Π2C T ) = {0} such that Γ = {x ∈ Fix(U); Ax ∈ Fix(Π2C T )} = {0}. Now we have that λn =  1 |xn| , xn ∈ (−∞, 0) 4 3xn , xn ∈ (0,∞) 1, xn ∈ {0}. zn =  xn + 1, xn ∈ (−∞, 0) xn − 1, xn ∈ (0,∞) 0, xn ∈ {0}, yn =  xn + 1, xn ∈ (−∞, 0) ( 22n − 1 22n )(xn − 1), xn ∈ (0,∞) 0, xn ∈ {0}, Cn+1 = {v ∈ Cn;  1 2 |xn + 1 − v| 2 ≤ [ 12n + 1] 1 2 |xn + 1 − v| 2 + 12n ; 1 2 |xn + 1 − v| 2 ≤ 1 2 |xn − v| 2, xn ∈ (−∞, 0) 1 2 |( 2 2n − 1 22n )(xn − 1) − v| 2 ≤ [ 12n + 1] 1 2 |xn − 1 − v| 2 + 12n ; 1 2 |xn − 1 − v| 2 ≤ 1 2 |xn − v| 2, xn ∈ (0,∞) 1 2 |0 − v| 2 ≤ [ 12n + 1] 1 2 |0 − v| 2 + 12n , xn ∈ {0}}, xn+1 = Π 2 Cn+1 (x1), n ≥ 1. Now, take the initial point x1 = 0.5 from positive real numbers, the numerical experiment result is given below; z1 = −0.5; y1 = −0.375; C2 = (−∞, 0]; x2 = 0; z2 = 0; y2 = 0; x3 = x2 = 0. This implies that zero is in the solution set Γ. Similarly, from the negative real numbers, let x1 = −0.5 then we arrive at the same fixed point zero as follows; z1 = 0.5; y1 = 0.5; C2 = [0,∞); x2 = 0; z2 = 0; y2 = 0; x3 = x2 = 0. 6. Conclusion In summary, the validity of the proposed scheme is provided by step one and two in the manuscript. Strong convergence of the proposed sequence is achieved in step three. Approximate fixed point sequence of the proposed mappings is given by step four. Application of the demicloseness principle for the pro- posed mappings is discussed in step five. Furthermore, in this paper, the strong convergence for SCFPP of Bregman general- ized asymptotically nonexpansive mapping is obtained in uni- formly convex and uniformly smooth Banach spaces, without the assumption of semi-compactness property and or without the assumption of Opial condition. This infer that, the main re- sult presented here generalizes and extends that of Zhang et al. [10] and the references therein. 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