The Split Common Fixed Point Problem and the Hybrid Method in Banach Spaces (Nonlinear Analysis and Convex Analysis)
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(2) 128. In this. article, we consider the split common fixed point problem in Banach spaces. Using hybrid method in mathematical programming, we prove a strong convergence theorem for finding a solution of the split common fixed point problem in Banach spaces. Using this result, we get well‐known and new results which are connected with the split feasibility problem and the split common null point problem in Banach spaces. the. 2. Preliminaries this paper, we denote by \mathrm{N} the set of positive integers and by \mathbb{R} the set of real a real Hilbert space with inner product respectively. \{\cdot, \rangle and norm \Vert. Throughout. numbers. Let H be For x,. .. and $\lambda$\in \mathbb{R} ,. y\in H. we. have from. [24]. that. \Vert x+y\Vert^{2}\leq\Vert x\Vert^{2}+2\langle y, x+y\rangle ; \Vert $\lambda$ x+(1- $\lambda$)y\Vert^{2}= $\lambda$\Vert x\Vert^{2}+(1- $\lambda$)\Vert y\Vert^{2}- $\lambda$(1- $\lambda$)\Vert x-y\Vert^{2}. Furthermore. we. have that for x, y, u,. v\in H,. 2\langle x-y, u-v\rangle=\Vert x-v\Vert^{2}+\Vert y-u\Vert^{2}-\Vert x-u\Vert^{2}-\Vert y-v\Vert^{2}. Let C be. subset of a Hilbert space H The nearest point that is, \Vert x-P_{C}x\Vert\leq\Vert x-y\Vert for all x\in H and P_{C} by y\in C Such P_{C} is called the metric projection of H onto C We know that the metric projection P_{C} is firmly nonexpansive, i.e., a. nonempty, closed and. projection of. convex. H onto C is denoted. .. ,. .. .. \Vert P_{C}x-P_{C}y\Vert^{2}\leq\langle P_{C}x-P_{C}y, x-y\rangle for all x, y\in H Furthermore \langle x—Pcx, y-P_{C}x\rangle\leq 0 holds for all x\in H and y\in C ; see [22]. Let E be a real Banach space with norm \Vert\cdot\Vert and let E^{*} be the dual space of E We denote .. .. by \langle x, y^{*}\rangle When \{x_{n}\} is a sequence in E we denote the strong of to x\in E convergence by x_{n}\rightarrow x and the weak convergence by x_{n}\rightarrow x The modulus \{x_{n}\} $\delta$ of convexity of E is defined by the value of. y^{*}\in E^{*}. at x\in E. .. ,. .. $\delta$( $\epsilon$)=\displaystyle \inf\{1-\frac{\Vert x+y\Vert}{2}. :. \Vert x\Vert\leq 1, \Vert y\Vert\leq 1,. for every $\epsilon$ with 0\leq $\epsilon$\leq 2 A Banach space E is said to be every $\epsilon$>0 It is known that a Banach space E is uniformly .. .. sequences. \{x_{n}\}. and. \{y_{n}\}. \Vert x-y\Vert\geq $\epsilon$\} uniformly. convex. convex. if and. only. if. $\delta$( $\epsilon$)>0. for. if for any two. in E such that. \displaystyle \lim_{n\rightar ow\infty}\Vert x_{n}\Vert=\lim_{n\rightar ow\infty}\Vert y_{n}\Vert=1. and. \displaystyle \lim_{n\rightar ow\infty}\Vert x_{n}+y_{n}\Vert=2,. \displaystyle \lim_{n\rightarrow\infty}\Vert x_{n}-y_{n}\Vert=0 holds.. A uniformly convex Banach space is strictly convex and reflexive. uniformly convex Banach space has the Kadec‐Klee property, that is, x_{n}\rightarrow u and \Vert x_{n}\Vert\rightarrow\Vert u\Vert imply x_{n}\rightarrow u ; see [9]. The duality mapping J from E into 2^{E^{*}} is defined by We also know that. a. Jx=\{x^{*}\in E^{*} : \langle x, x^{*}\rangle=\Vert x\Vert^{2}=\Vert x^{*}\Vert^{2}\}.
(3) 129. for every x\in E Let U=\{x\in E : \Vert x\Vert=1\} differentiable if for each x, y\in U , the limit .. .. The. norm. of E is said to be Gâteaux. \displaystyle \lim_{t\rightar ow 0}\frac{\Vert x+ty\Vert-\Vert x\Vert}{t} exists. In the case, E is called smooth. We know that E is smooth if and only if J is a single mapping of E into E^{*} We also know that E is reflexive if and only if J is surjective,. valued. .. and E is. strictly. if and. convex. only. if J is one‐to‐one.. if E is. Therefore,. a. smooth, strictly. and reflexive Banach space, then J is a single‐valued bijection and in this case, the inverse mapping J^{-1} coincides with the duality mapping J_{*} on E^{*} For more details, see [22] convex. .. and. [23].. following. We know the. Lemma 2.1. Let E be. \langle x-y, Jx-Jy\rangle\geq 0 for. smooth Banach space and let J be the duality mapping on E. Then, y\in E Furthermore, if E is strictly convex and \langle x-y, Jx-Jy\rangle=. a. all x,. 0 , then x=y.. Let C be space E. .. result:. .. nonempty, closed and. a. Then. convex. subset of. a. strictly. convex. and reflexive Banach. know that for any x\in E , there exists a unique element z\in C such that for all y\in C Putting z=P_{C}x , we call P_{C} the metric projection of E onto. we. \Vert x-z\Vert\leq\Vert x-y\Vert. .. C.. Lemma 2.2 a. ([22]).. Let E be. nonempty, closed and. conditions. a. convex. smooth, strictly convex and reflexive Banach space. Let C be of E and let x_{1}\in E and z\in C. Then, the following. subset. equivalent. are. (1) z=P_{C}x_{1} ; (2) \langle z-y, J(x_{1}-z)\rangle\geq 0,. \forall y\in C.. Banach space and let A be a mapping of E into 2^{E^{*}} The effective domain of by \mathrm{d}\mathrm{o}\mathrm{m}(A) that is, \mathrm{d}\mathrm{o}\mathrm{m}(A)=\{x\in E: Ax\neq\emptyset\} A multi‐valued mapping A on E is said to be monotone if \langle x-y, u^{*}-v^{*} ) \geq 0 for all x, y\in \mathrm{d}\mathrm{o}\mathrm{m}(A) u^{*}\in Ax and v^{*}\in Ay. A monotone operator A on E is said to be maximal if its graph is not properly contained in the graph of any other monotone operator on E The following theorem is due to Browder Let E be. a. A is denoted. ,. .. ,. ,. .. [4];. see. also. [23,. 3.5.4].. Theorem. ([4]). Let E be a uniformly duality mapping of E into E^{*} Let A be maximal if and only if for any r>0,. Theorem 2.3 the. .. convex a. and smooth Banach space and let J be operator of E into 2^{E^{*}} Then A is. monotone. R(J+rA)=E^{*}, where. R(J+rA). Let E be. be. a. a. is the range. uniformly. of J+rA.. convex. Banach space with a Gâteaux differentiable 2^{E^{*}} For all x\in E and r>0 ,. maximal monotone operator of E into. norm. we. and let A. consider the. following equation. 0\in J(x_{r}-x)+rAx_{\mathrm{r}}. This. equation has. a. unique solution. x_{r}. .. We define J_{r} by x_{r}=J_{r}x points of A is defined by. the metric resolvents of A The set of null .. .. Such. J_{r}, r>0. are. called. A^{-1}0=\{z\in E:0\in Az\}.. We know that A^{-1}0 is closed and convex; see [23]. Let E be a smooth, strictly convex and reflexive Banach space and let $\eta$ be a real number with $\eta$\in(-\infty, 1) Then a mapping U : E\rightarrow E with F(U)\neq\emptyset is called $\eta$‐demimetric [27] if, ..
(4) 130. q\in F(U). for any x\in E and. ,. \displaystyle \langle x-q, J(x-Ux)\rangle\geq\frac{1- $\eta$}{2}\Vert x-Ux\Vert^{2}, F(U). where. is the set of fixed. We know. Examples.. (1). Let H be. points of U. of $\eta$ ‐demimetric. examples. mappings from [27].. Hilbert space and let k be a real number with 0\leq k<1 Let U be pseud‐contraction [5] of H into itself such that F(U)\neq\emptyset Then U is k‐demimetric. a. .. a. strict. .. (2). Let E be. closed and. a. convex, reflexive and smooth Banach space and let C be. strictly. subset of E. convex. .. Let. P_{C} be the metric projection of E. onto C. .. a. nonempty, P_{C} is. Then. (-1) ‐demimetric. (3) Let E be a uniformly convex and smooth Banach space and let B be a maximal monotone operator with B^{-1}0\neq\emptyset Let $\lambda$>0 Then the metric resolvent J_{ $\lambda$} is (-1) ‐demimetric. .. Furthermore, convex. a. ([27]).. an. important. Let E be. real number with. F(U)\dot{u}. 3. know. we. result for demimetric. mappings. in. a. smooth, strictly. and reflexive Banach space.. Lemma 2.4 be. .. closed and. a. smooth, strictly. $\eta$\in(-\infty, 1). .. Let U be. an. convex. and. reflexive. Banach space and let $\eta$ of E into itself. Then. $\eta$ ‐demimetric mapping. convex.. \mathrm{M}\mathrm{a}|\mathrm{n} result and |\mathrm{t}\mathrm{s}. Applications. In this. section, using the demimetric operators, we prove a strong convergence theorem for a solution of the split common fixed point problem in Banach spaces. Let E be a Banach space and let C be a nonempty, closed and convex subset of E A mapping U:C\rightarrow E is called demiclosed if, for a sequence \{x_{n}\} in C such that x_{n}\rightarrow p and x_{n}-Ux_{n}\rightarrow 0, p=Up holds. The following theorem was proved by Hojo and Takahashi [11].. finding. .. Theorem 3.1. reflexive. (-\infty, 1). ([11]).. Let H be. a. Hilbert space and let F be a smooth, strictly convex and mapping on F and let $\eta$ be a real number with $\eta$\in. Banach space. Let J_{F} be the duality Let T:H\rightarrow H be a nonexpansive. mapping and let U : F\rightarrow F be an $\eta$ ‐demimetric mapping with F(U)\neq\emptyset Let A:H\rightarrow F be a bounded linear operator such that A\neq 0 and let A^{*} be the adjoint operator of A. Suppose that F(T)\cap A^{-1}F(U)\neq\emptyset Let x_{1}\in H and let \{x_{n}\} be a sequence generated by .. and demiclosed. .. .. where. \{$\alpha$_{n}\}\subset[0. ,. 1 ] and. \left{bginary}{l z_n=T(x{}-$\lambd_{n}A^*JF(x_{n}-UA \ y_{n}=$alph_{n}x+(1-$\alph_{n})z,\ C_{n}=z\iH:Verty_{n}-z\ leqVrtx_{n}-z\ D={z\inH:x_}-z,{1n\ragleq0\}, x_{n+1=PC}\capD_{nx1},\foralnimthr{N}, \endary}ight.. \{$\lambda$_{n}\}\subset(0, \infty) satisfy. 0\leq$\alpha$_{n}\leq a<1. ,. and. the conditions such that. 0<b\leq$\lambda$_{n}\Vert A\Vert^{2}\leq c<(1- $\eta$).
(5) 131. for. some. a,. b, c\in \mathbb{R}. .. Then. \{x_{n}\}. converges. strongly. to. a. point. z_{1}=P_{F(T)\cap A^{-1}F(U)^{X}1}.. z_{1}\in F(T)\cap A^{-1}F(U). ,. where. Theorem. 3.1, we get well‐known and new strong convergence theorems which are con‐ split common fixed point problems in Banach spaces. We know the following result obtained by Marino and Xu [15]; see also [30].. Using. nected with the. Lemma 3.2. of H. Let If x_{n}\rightarrow z. ([15]).. k be and. a. Let H be. a. Hilbert space and let C be a nonempty, closed and convex subset 0\leq k<1 and U:C\rightarrow H be a k ‐strict pseudo‐contraction.. real number with. x_{n}-Ux_{n}\rightarrow 0 then z\in F(U) ,. .. Theorem 3.3. Let H_{1} and H_{2} be Hilbert spaces. Let k be a real number with k\in[0 , 1). Let T:H_{1}\rightarrow H_{1} be a nonexpansive mapping and let U : H_{2}\rightarrow H_{2} be a k ‐strict pseud‐contraction such that be. a. F(U)\neq\emptyset. A:H_{1}\rightarrow H_{2} be a bounded linear operator such that A\neq 0 and let adjoint operator of A. Suppose that F(T)\cap A^{-1}F(U)\neq\emptyset Let x_{1}\in H and let \{x_{n}\} sequence generated by. A^{*} be the. where. .. Let. .. \{$\alpha$_{n}\}\subset[0 1 ]. and. ,. \left{bginary}l z_{n=T(x}-$\lambd_{n}A^*(x-U_{n}\ y=$alph_{n}x+(1-$\alph_{n})z,\ C_{n}=z\iH:Verty_{n}-z\ leqVrtx_{n}-z\ D={z\inH:lagex_{n}-z,1 \rangleq0\}, x_{n+1=PC}\capD_{nx1},\foralnimthb{N}, \endary}ight.. \{$\lambda$_{n}\}\subset(0, \infty) satisfy. 0\leq$\alpha$_{n}\leq a<1. for. some. a,. b, c\in \mathbb{R}. .. Then. \{x_{n}\}. and. ,. converges. the conditions such that. 0<b\leq$\lambda$_{n}\Vert A\Vert^{2}\leq c<(1-k) strongly. to. a. z_{1}=P_{F(T)\cap AF(U)^{X}1}-1. Theorem 3.4. Let H be. a. Hilbert space and let F be. a. point. z_{1}\in F(T)\cap A^{-1}F(U). smooth, strictly. convex. and. ,. where. reflexive. Banach space. Let J_{F} be the duality mapping on F. Let C and D be nonempty, closed and convex subsets of H and F , respectively. Let P_{C} and P_{D} be the metric projections of H onto. C and F onto D ,. respectively.. Let T:H\rightarrow H be. a. nonexpansive mapping, let A:H\rightarrow F be of A. Suppose. bounded linear operator such that A\neq 0 and let A^{*} be the adjoint operator that C\cap A^{-1}D\neq\emptyset Let x_{1}\in H and let \{x_{n}\} be a sequence generated by a. .. where. \{$\alpha$_{n}\}\subset[0 1 ] ,. and. \left{bginary}l z_{n=PC}(x-$\lambd_{n}A^*JF(x_{n}-PDA \ y_{n}=mathr _{n}x+(1-$\alph_{n})z,\ C_{n}=ziH:\Verty_{n}-z \leqVrtx_{n}-z\ D={inH:\lagex_{n}-z,1 \rangleq0},\ x_{n+1=PC}\capD_{nx1},\foralnimthr{N}, \endaryight. \{$\lambda$_{n}\}\subset(0, \infty) satisfy. 0\leq$\alpha$_{n}\leq a<1. for. Then a, b, c\in \mathbb{R} z_{0}=P_{C\cap A^{-1}D}x_{1}.. some. where. .. ,. the sequence. the conditions such that. and. 0<b\leq$\lambda$_{n}\Vert A\Vert^{2}\leq c<2. \{x_{n}\}. converges. strongly. to. a. point z_{0}\in C\cap A^{-1}D,.
(6) 132. Theorem 3.5. Let H be. a. Hilbert space and let F be. a. uniformly. convex. and smooth Banach. space. Let J_{F} be the duality mapping on F. Let A and B be maximal monotone operators of H into H and F into F^{*} , respectively. Let J_{ $\lambda$} be the resolvent of A for $\lambda$>0 and let Q_{ $\mu$} be. the metric resolvent. of B for $\mu$>0 respectively. Let T:H\rightarrow F adjoint operator of T. Suppose x_{1}\in H and let \{x_{n}\} be a sequence generated by. such that Let. where. T\neq 0. ,. and let $\tau$* be the. \{$\alpha$_{n}\}\subset[0 1 ] ,. and. some. a,. T^{-1}(B^{-1}0). \{$\lambda$_{n}\}\subset(0, \infty) satisfy. where. bounded linear operator. A^{-1}0\cap T^{-1}(B^{-1}0)\neq\emptyset.. and. ,. the conditions such that. 0<b\leq$\lambda$_{n}\Vert T\Vert^{2}\leq c<2. Then the sequence \{x_{n}\} z_{0}=P_{A^{-1}0\cap T^{-1}(B-1}x.. b, c\in \mathbb{R}. ,. be. \left{bginary}l z_{=J$\lambd}(x_{n-$\lambd_{n}T^*JF(x_{n}-Q$\muTx_{n} y=$\alph_{n}x+(1-$\alph_{n})z,\ C_{n}=ziH:\Verty_{n}-z \leqVrtx_{n}-z\ D={inH:\lagex_{n}-z,1 \}geq0, x_{n+1}=PC\capD_{n}x1,\foralnimthr{N}, \endaryight.. 0\leq$\alpha$_{n}\leq a<1. for. a. that. .. converges. strongly. to. a. point z_{0}\in A^{-1}0\cap. References [1]. S. M. Alsulami and W. monotone. [2] [3]. (2014),. [6]. 793‐808.. 67. (2007),. K.. Aoyama,. 2350‐2360. F. Kohsaka and W.. mappings:. (2009), F. E. 175. [5]. split common null point problem for maximal applications, J. Nonlinear Convex Anal. 15. K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal.. sive. [4]. mappings. The. Takahashi,. in Hilbert spaces and. Three. Takahashi,. Their relations and continuous. generalizations of firmly. nonexpan‐ 10. properties, J. Nonlinear Convex Anal.. 131‐147.. Browder, Nonlinear maximal. (1968),. monotone. operators. in Banach spaces, Math. Ann.. 89‐113.. F. E. Browder and W. V.. Petryshyn, Construction of fixed points of nonlinear mappings. in Hilbert spaces, J. Math. Anal. Appl. 20 (1967), 197‐228. C. Byrne, Y. Censor, A. Gibali and S. Reich, The split common null. Nonlinear Convex Anal. 13. [7]. Y. Censor and T.. [9]. I.. (2012),. point problem, J.. 759‐775.. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms 8 (1994), 221‐239. [8] Y. Censor and A. Segal, The split common fixed‐point problem for directed operators, J. Convex Anal. 16. (2009),. 587‐600.. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht, 1990. [10] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957‐ 961.. [11]. M.. Hojo. and W.. Takahashi,. The. split. common. in Banach spaces, Linear Nonlinear Anal.. 16,. fixed point problem to appear.. and the. hybrid. method.
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