Iterative Methods for Split Common Fixed Point Problems in Banach Spaces and Applications (Nonlinear Analysis and Convex Analysis)
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(2) 119. Q Fbrthermore,. if. .. D\cap A^{-1}Q. is. nonempty, then. z\in D\cap A^{-1}Q. equivalent. is. to. z=P_{D}(I- $\lambda$ A^{*}(I-P_{Q})A)z, where $\lambda$>0 and P_{D} is the metric projection of H_{1} onto D By using such results regarding nonlinear operators and fixed points, many authors have studied split feasibility problems and .. split. null point problems in Hilbert spaces, for instance, [1, 6, 8, 28]. article, motivated by split feasibility problems and split common null point problems. common. In this. in Hilbert spaces,. we. first introduce the concept of nonlinear operators in Banach spaces which. pseud‐contractions and generalized hybrid mappings in Hilbert spaces, and the metric projections and the metric resolvents in Banach spaces. Then we consider split common fixed point problems with the operators in Banach spaces. Using hybrid methods, Manns type iterations and Halperns type iterations, we prove weak and strong convergence theorems for finding solutions of split common fixed point problems in Banach spaces. Furthermore, using these results, we get well‐known and new results which are connected with split feasibility problems and split common null point problems in Banach spaces. covers. strict. Preliminaries. 2. Let E be. a. the value of. real Banach space with norm \Vert\cdot\Vert and let E^{*} be the dual space of E We denote at x\in E by \langle x, y^{*} }. When \{x_{n}\} is a sequence in E , we denote the strong .. y^{*}\in E^{*}. convergence of \{x_{n}\} to x\in E $\delta$ of convexity of E is defined. by by. x_{n}\rightarrow x and the weak convergence. $\delta$( $\epsilon$)=\displaystyle \inf\{1-\frac{\Vert x+y\Vert}{2}. :. \Vert x\Vert\leq 1, \Vert y\Vert\leq 1,. by. x_{n}\rightarrow x. .. The modulus. \Vert x-y\Vert\geq $\epsilon$\}. for every $\epsilon$ with 0\leq $\epsilon$\leq 2 A Banach space E is said to be uniformly convex if $\delta$( $\epsilon$)>0 for every $\epsilon$> O. A uniformly convex Banach space is strictly convex and reflexive. We also .. know that and. a. uniformly. \Vert x_{n}\Vert\rightarrow\Vert u\Vert imply. The. duality mapping. convex. Banach space has the Kadec‐Klee property, that is, x_{n}\rightarrow u. x_{n}\rightarrow u.. J from E into. 2^{E^{*}}. is defined. by. Jx=\{x^{*}\in E^{*} : \{x, x^{*}\rangle=\Vert x\Vert^{2}=\Vert x^{*}\Vert^{2}\} 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. convex. if and. only. if J is one‐to‐one.. Therefore, if. E is. 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 [18] convex. .. and. [19].. We know the. following result:.
(3) 120. ([18]).. Lemma 2.1. Let E be. smooth Banach space and let J be the all x, y\in E. Furthernore, if E is. a. Then, \langle x-y, Jx-Jy\rangle\geq 0 for \langle x-y Jx—Jy \}=0 then x=y.. E.. ,. ,. Let C be space E. .. nonempty, closed and. a. Then. strictly. a. convex. and reflexive Banach. C. Let E be. nonempty, closed and. conditions. subset of. .. onto. ([18]).. Lemma 2.2. convex. 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 such a mapping P_{C} the metric. we. \Vert x-z\Vert\leq\Vert x-y\Vert. projection of E a. duality mapping on strictly convex and. 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 A is denoted 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. Let E be. a. .. A monotone operator A on E is said to be maximal if its graph is not properly contained in graph of any other monotone operator on E The following theorem is due to Browder. the. [4];. .. [19,. also. see. ([4]).. Theorem 2.3 the. Theorem. 3.5.4].. Let E be. duality mapping of if and only if for. maximal. Let E be. a. uniformly. any. uniformly. a. E into E^{*}. Let A be. .. and smooth Banach space and let J be operator of E into 2^{E^{*}} Then A is. convex a. monotone. r>0, R(J+rA)=E^{*} where R(J+rA). convex. ,. Banach space with. a. is the range. Gâteaux differentiable. norm. of J+rA. and let A. maximal monotone operator of E into 2^{E^{*}} For all x\in E and r>0 , we consider the following equation 0\in J(x_{r}-x)+rAx_{r} This equation has a unique solution x_{r} We define be. a. .. J_{r} by x_{r}=J_{r}x Such J_{r}, r>0 .. A is defined. are. .. called the metric resolvents of A. by A^{-1}0=\{z\in E:0\in Az\}. The set of null. A^{-1}0 is closed and. We know that. .. .. convex;. \{C_{n}\}. points of see. [19].. of nonempty, closed and convex subsets of a Banach space E , define s‐Lin C_{n} and w‐Lsn C_{n} as follows: x\in s‐Lin C_{n} if and only if there exists \{x_{n}\}\subset E such that \{x_{n}\} converges strongly to x and x_{n}\in C_{n} for all n\in \mathrm{N} Similarly, y\in w‐Lsn C_{n} if and only For. a. sequence. .. if there exist. weakly. subsequence \{C_{n_{i}}\}. a. to y and. y_{i}\in C_{n_{i}}. of. \{C_{n}\}. for all i\in \mathrm{N} If .. C_{0}=. and. C_{0}. a. sequence. satisfies. \{y_{i}\}\subset E. such that. \{y_{i}\}. converges. s‐Lni C_{n}= w‐Lns C_{n},. \{C_{n}\} converges to C_{0} in the sense of Mosco [14\mathrm{J} and we write C_{0}= It is easy to show that if \{C_{n}\} is nonincreasing with respect to inclusion, then in the sense of Mosco. For more details, see [14]. The following converges to. it is said that. \displaystyle \mathrm{M}-\lim_{n\rightarrow\infty}C_{n}. \{C_{n}\}. lemma. .. \displaystyle \bigcap_{n=1}^{\infty}C_{n}. was. proved by Tsukada [30].. ([30]). Let E be a uniformly convex Banach space. Let \{C_{n}\} be a sequence of nonempty, closed and convex subsets of E. If C_{0}=M-\displaystyle \lim_{n\rightarrow\infty}C_{n} exists and nonempty, then for each x\in E, \{P_{C_{n}}x\} converges strongly to P_{C_{0}}x where P_{C_{n}} and P_{C_{0}} are the mertic projections of E onto C_{n} and C_{0} respectively. Lemma 2.4. ,. ,.
(4) 121. Iterative Results. 3. Let E be. by Hybrid. Methods. smooth, strictly convex and reflexive Banach space and let $\eta$ be a real number $\eta$\in(-\infty, 1) Then a mapping U:E\rightarrow E with F(U)\neq\emptyset is called $\eta$‐demimetric [22] if,. with. a. .. for any x\in E and. q\in F(U). ,. \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. a. points of U.. examples. of $\eta$ ‐demimetric. mappings from [22].. Hilbert space and let k be a real number with 0\leq k<1 Let U be of H into itself such that F(U)\neq\emptyset Then U is k ‐demimetric. .. pseud‐contraction [5]. (2). Let H be. is called. a. a. Hilbert space and let C be a nonempty subset of H if there exist $\alpha$, $\beta$\in \mathbb{R} such that. U is called. mapping. then U is 0 ‐demimetric.. (3). strict. .. A. mapping U : C\rightarrow H. generalized hybrid [10]. $\alpha$\Vert Ux-Uy\Vert^{2}+(1- $\alpha$)\Vert x-Uy\Vert^{2}\leq $\beta$\Vert Ux-y\Vert^{2}+(1- $\beta$)\Vert x-y\Vert^{2}, Such. a. .. Let E be. closed and. a. If U is. generalized hybrid. and. convex, reflexive and smooth Banach space and let C be. strictly. convex. ( $\alpha$, $\beta$) ‐generalized hybrid.. \forall x, y\in H.. subset of E. .. Let. P_{C} be the metric projection of. E onto C. .. a. F(U)\neq\emptyset, nonempty,. Then P_{C} is. (-1) ‐demimetric. (4) 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. know. ([22]).. important result for demimetric mappings. Let E be. real number with. F(U). an. in. a. smooth, strictly. and reflexive Banach space.. Lemma 3.1. be. we. .. is closed and. a. smooth, strictly. $\eta$\in(-\infty, 1). .. Let U be. an. convex. and. reflexive. $\eta$ ‐demimetric. Banach space and let $\eta$ E into itself. Then. mapping of. convex.. Using the hybrid methods in mathematical programming, we prove two strong convergence finding a solution of the split common fixed point problem in Banach spaces.. theorems for Let E be. Banach space and let D be a nonempty, closed and convex subset of E. \mathrm{A} D\rightarrow E is called demiclosed if for a sequence \{x_{n}\} in D such that x_{n}\rightarrow p and mapping x_{n}-Ux_{n}\rightarrow 0, p=Up holds. The following theorems are proved by Takahashi [23]. a. U. :. Theorem 3.2. ([23]).. Let E and F be. J_{E} and J_{F} be the duality mappings. on. uniformly E and F ,. convex. and smooth Banach spaces and let Let $\tau$ and $\eta$ be real numbers. respectively.. with $\tau$, $\eta$\in(-\infty, 1) Let T : E\rightarrow E be a $\tau$ ‐demimetric and demiclosed mapping and let U : F\rightarrow F be an $\eta$ ‐demimetric and demiclosed mapping with F(U)\neq\emptyset Let A:E\rightarrow F be .. .. a. bounded linear operator such that. A\neq 0. and let A^{*} be the. adjoint operator of A. Suppose.
(5) 122. that. F(T)\cap A^{-1}F(U)\neq\emptyset. where. where. Let x_{1}\in E and let. .. \{x_{n}\}. be. a. sequence. generated by. \left{bginary}{l z_n=x}-rJ{E^1A*}J_{F(xn-UA),\ y_{n}=Tz,\ C_{n}=z\iE:langez_{}-,JE(xnz_{})\geq0, D_{n}=\ziE:2langez_{}-,JB(ny_{})\ragleq(1-$\tau)Verz_{n}-y\Vert^{2}, Q_n=\{ziE:langex_{}-z,JE(1x_{n})\ragleq0\}, x_{n+1=PC}\capD_{n Q}x_{1,\foralnimthb{N}, \endary}ight.. 0<2r\Vert A\Vert^{2}\leq(1- $\eta$) Then \{x_{n}\} z_{1}=P_{F(T)F(U)^{X}1}\cap A-1. .. ([23]).. Theorem 3.3. J_{E} and J_{F} be. the. Let E and F be. duality mappings. on. converges. uniformly E and F ,. strongly. convex. to. a. point. z_{1}\in F(T)\cap A^{-1}F(U). ,. and smooth Banach spaces and let Let $\tau$ and $\eta$ be real numbers. respectively.. with $\tau$, $\eta$\in(-\infty, 1) Let T : E\rightarrow E be a $\tau$ ‐demimetric and demiclosed mapping and let U : F\rightarrow F be an $\eta$ ‐demimetric and demiclosed mapping with F(U)\neq\emptyset Let A : E\rightarrow F be .. .. 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 For x_{1}\in E and C_{1}=E , let \{x_{n}\} be a sequence generated by a. .. where where. \left{begin{ary}l z_{n}=x -rJ_{E}^-1A{*}J_F(Ax_{n}-UAxn),\ y_{n}=Tz_{n},\ C_{n+1}=\{zinC_{}:\z_{n}-,J_{E}(xn-z_{})\geq0\} and2\{z_n}-,J_{E}(zn-y_{})\geq(1-$\tau)Vertz_{n}-y \Vert^{2}\, x_{n+1}=P_{Cn+1}x_{,\foraln\i mathb{N}, \end{ary}\ight.. 0<2r\Vert A\Vert^{2}\leq(1- $\eta$) Then \{x_{n}\} z_{1}=P_{F(T)\cap A-1}x.. Using with the obtained. .. converges. strongly. to. a. point. z_{1}\in F(T)\cap A^{-1}F(U). ,. Theorems 3.2 and. 3.3, we get strong convergence theorems which are connected split common fixed point problems in Banach spaces. We know the following result by Marino and Xu [13]; see also [27].. Lemma 3.4. ([13]).. of H and k be. Let H be. a. Hilbert space, let C be. a. nonempty, closed and. real number with 0\leq k<1 Let U:C\rightarrow H be If x_{n}\rightarrow z and x_{n}-Ux_{n}\rightarrow 0 , then z\in F(U) a. .. a. k ‐strict. convex. subset. pseudo‐contraction.. .. H_{1} and H_{2} be Hidbert spaces. Let k be a real number with k\in[0 1). Let nonexpansive mapping and let U : H_{2}\rightarrow H_{2} be a k ‐strict pseud‐contraction with F(U)\neq\emptyset Let A:H_{1}\rightarrow H_{2} 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_{1} and let \{x_{n}\} be a sequence generated by. Theorem 3.5. Let. T:H_{1}\rightarrow H_{1} be. ,. a. .. .. \left{bginary}l z_{=xn}-rA^*(_{ UAxn),\ y_{}=Tzn,\ C_{}=zinH1:\lagez_{n}-,x \rangleq0},\ D_{n=ziH1}:2\langez_{-,n}y\ragleqVrtz_{n}-y\Vert^{2}, Q_n=\{ziH1}:langex_{-z,1}n)\geq0}, x_{n+1=PC}\capD_{n Q}x_{1,\foralnimthb{N}, \endaryight..
(6) 123. where. 0<2r\Vert A\Vert^{2}\leq(1-k). Then. .. where z_{1}=P-1.. \{x_{n}\}. converges. strongly. to. point. a. z_{1}\in F(T)\cap A^{-1}F(U). ,. Theorem 3.6. Let E and F be. uniformly convex and smooth Banach spaces and let J_{E} and respectively. Let C and D be nonempty, closed and convex subsets of E and F respectively. Let P_{C} and P_{D} be the metric projections of E onto C and F onto D respectively. Let A : E\rightarrow F be a bounded linear operator such that A\neq 0 and let A^{*} be the adjoint operator of A. Suppose that C\cap A^{-1}D\neq\emptyset For x_{1}\in E and C_{1}=E let \{x_{n}\} be a sequence generated by J_{F} be the duality mappings. on. E and F ,. ,. ,. .. where. ,. \left{bginary}{l z_n}=x{-rJ_E}^{1A*}J_{F(Axn}-P_{DAxn}),\ y_{n}=PCz_{n},\ C_{n+1}=\zinC_{}:\langez_{}-,JE(x_{n}-z )\geq0}\ and\lgez_{n}-,JE(z_{n}-y )\rangle q\Vertz_{n}-y \Vert^{2}\, x_{n+1}=P_{Cn+1}x_{,\foraln\i mathr{N}, \end{ary}\ight.. 0<r\Vert A\Vert^{2}\leq 1. \{x_{n}\}. Then. .. converges. strongly. to. point z_{1}\in C\cap A^{-1}D. a. ,. where. z_{1}=P_{C\cap A^{-1}D^{X}1}. Theorem 3.7. Let E and F be. uniformly convex and smooth Banach spaces and let J_{E} and J_{F} be the duality mappings on E and F respectively. Let G and B be maximal monotone operators of E into E^{*} and F into F^{*} respectively. Let J_{ $\lambda$} and Q_{ $\mu$} be the metric resolvents of G for $\lambda$>0 and B for $\mu$>0 respectively. Let A:E\rightarrow F be a bounded linear operator such that A\neq 0 and let A^{*} be the adjoint operator of A. Suppose that G^{-1}0\cap A^{-1}(B^{-1}0)\neq\emptyset For x_{1}\in E and C_{1}=E let \{x_{n}\} be a sequence generated by ,. ,. ,. .. ,. \left{bginary}{l z_n}=x{-rJ_E}^{1A*}J_{F(Axn}-Q_{$\mu}Ax_{n),\ y_{n}=J$\lambd$}z_{n,\ C_{n+1}=\zinC_{}:\langez_{}-,JE(x_{n}-z )\rangle q0\} and\lgez_{n}-,JE(z_{n}-y )\rangle q\Vertz_{n}-y \Vert^{2}\, x_{n+1}=P_{Cn+1}x_{,\foralni\mathr{N}, \end{ary}\ight.. 0<r\Vert A\Vert^{2}\leq 1 and $\lambda$, $\mu$> O. Then the sequence \{x_{n}\} z_{1}\in G^{-1}0\cap A^{-1}(B^{-1}0) where z_{1}=P_{G^{-1}0\cap A-1}(B-1x. where. converges. strongly. to. a. point. ,. Iterative Results. 4. In this the. split. section, common. Theorem 4.1. we. by. first prove. fixed. ([24]).. Mann and a. Halpern. Iterat. weak convergence theorem in Banach spaces.. point problem Let H be. a. [24]. Hilbert space and let F be. a. ons. of Manns type iteration for. smooth, strictly. convex. and. smooth Banach space. Let J_{F} be the duality mapping on F and let $\eta$ be a real number with $\eta$\in (-\infty, 1) Let T : H\rightarrow H be a nonexpansive mapping and let U : F\rightarrow F be an $\eta$ ‐demimetric .. and demiclosed that. A\neq 0. mapping with F(U)\neq\emptyset Let A : H\rightarrow F be adjoint operator of A. Suppose .. and let A^{*} be the. x_{1}=x\in H define. bounded linear operator such F(T)\cap A^{-1}F(U)\neq\emptyset For any a. ,. x_{n+1}=$\beta$_{n}x_{n}+(1-$\beta$_{n})T(I-rA^{*}J_{F}(A-UA))x_{n}, \forall n\in \mathbb{N},. ..
(7) 124. where. \{$\beta$_{n}\}\subset[0 1 ]. and. ,. r\in(0, \infty) satisfy. the. and. 0<a\leq$\beta$_{n}\leq b<1. for. some. a,. b\in \mathbb{R}. \{x_{n}\}. Then. .. following:. converges. 0<r\Vert AA^{*}\Vert<(1- $\eta$). weakly. to. point. a. z_{0}=\displaystyle \lim_{n\rightarrow\infty}P_{F(\mathrm{T})\cap A^{-1}F(U)^{X}n}. Next,. we. common. z_{0}\in F(T)\cap A^{-1}F(U). a strong convergence theorem [24] of Halperns type point problem in Banach spaces.. prove. fixed. ,. where. iteration for the. split. ([24]). Let H be a Hilbert space and let F be a smooth, strictly convex and smooth Banach space. Let J_{F} be the duality mapping on F and let $\eta$ be a real number with $\eta$\in (-\infty, 1) Let T : H\rightarrow H be a nonexpansive mapping and let U : F\rightarrow F be an $\eta$ ‐demimetric. Theorem 4.2. .. and demiclosed. mapping. F(U)\neq\emptyset. with. A\neq 0. be. sequence in H such that u_{n}\rightarrow u. a. Let A. .. :. H\rightarrow F be. a. bounded linear operator such Let \{u_{n}\}. adjoint operator of A. Suppose F(T)\cap A^{-1}F(U)\neq\emptyset. that. and let A^{*} be the. For x_{1}=x\in H , let. .. \{x_{n}\}\subset H. be. a. .. sequence. generated. by. x_{n+1}=$\beta$_{n}x_{n}+(1-$\beta$_{n})($\alpha$_{n}u_{n}+(1-$\alpha$_{n})T(x_{n}-rA^{*}J_{F}(I-U)Ax_{n})) for. all n\in \mathrm{N} , where. r\in(0, \infty) \{$\alpha$_{n}\}\subset(0,1) ,. 0<r\Vert AA^{*}\Vert<(1- $\eta$) for. some. a, b\in \mathbb{R}. .. Then. and. \displaystyle\sum_{n=1}^{\infty}$\alpha$_{n}=o. \displaystyle \lim_{n\rightar ow\infty}$\alpha$_{n}=0,. ,. \{x_{n}\}. strongly. converges. \{$\beta$_{n}\}\subset(0,1) satisfy. to. point. a. z_{0}=P_{F(T)\cap A^{-1}F(U)}u.. and. 0<a\leq$\beta$_{n}\leq b<1. z_{0}\in F(T)\cap A^{-1}F(U). Using Theorems 4.1 and 4.2, we get weak and strong convergence theorems split common fixed point problems in Banach spaces. We following result from Takahashi, Yao and Kocourek [29]; see also [10]. nected with the. Lemma 4.3. of H. ([29]).. and let U. Let H be. C\rightarrow H be. :. a. Hilbert space, let C be. a. generalized hybrid. If x_{n}\rightarrow z. which. ,. where. are con‐. also know the. nonempty, closed and convex subset and x_{n}-Ux_{n}\rightarrow 0 then z\in F(U) .. ,. H_{1} and H_{2} be Hilbert spaces. Let k be a real number with k\in[0 1). Let nonexpansive mapping with F(T)\neq\emptyset and let U:H_{2}\rightarrow H_{2} be a k ‐strict pseud‐contraction with F(U)\neq\emptyset Let A : H_{1}\rightarrow H_{2} be a bounded hnear operator such that A\neq 0 and let A^{*} be the adjoint operator of A. Suppose F(T)\cap A^{-1}F(U)\neq\emptyset For any x_{1}=x\in H_{1} define Theorem 4.4. Let. T:H_{1}\rightarrow H_{1} be. ,. a. .. .. ,. x_{n+1}=$\beta$_{n}x_{n}+(1-$\beta$_{n})T(I-rA^{*}J_{F}(A-UA))x_{n}, \forall n\in \mathbb{N}, where. \{$\beta$_{n}\}\subset[0 1 ]. and. ,. r\in(0, \infty) satisfy. the. 0<a\leq$\beta$_{n}\leq b<1 for. some. a,. b\in \mathbb{R}. .. Then. \{x_{n}\}. converges. following:. and. 0<r\Vert AA^{*}\Vert<(1-k). weakly. to. a. z_{0}=\displaystyle \lim_{n\rightarrow\infty}P_{F(T)\cap A^{-1}F(U)^{X}n}. Theorem 4.5. Let H be. a. Hilbert space and let F be. a. point. z_{0}\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 A. :. H\rightarrow F be. a. bounded linear operator such that. A\neq 0.
(8) 125. adjoint operator of A. Suppose C\cap A^{-1}D\neq\emptyset. and let A^{*} be the. H such that u_{n}\rightarrow u. .. For x_{1}=x\in H , let. \{x_{n}\}\subset H. be. a. .. sequence. Let. \{u_{n}\}. be. a. sequence in. generated by. x_{n+1}=$\beta$_{n}x_{n}+(1-$\beta$_{n})($\alpha$_{n}u_{n}+(1-$\alpha$_{n})P_{C}(x_{n}-rA^{*}J_{F}(I-P_{D})Ax_{n})) for. r\in(0, \infty) \{$\alpha$_{n}\}\subset(0,1). all n\in \mathrm{N} , where. ,. 0<r\Vert AA^{*}\Vert<2, for. a, b\in \mathbb{R}. Then. .. \{x_{n}\}. \displayst le\sum_{n=1}^{\infty}$\alpha$_{n}=\infty. \displaystyle \lim_{n\rightar ow\infty}$\alpha$_{n}=0,. converges. \{$\beta$_{n}\}\subset(0,1) satisfy. and. strongly. to. a. and. 0<a\leq$\beta$_{n}\leq b<1. point z_{0}\in C\cap A^{-1}D where z_{0}=P_{C\cap A^{-1}D}u. ,. Theorem 4.6. 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 T and B be manmal monotone operators of H into H and F into F^{*} , respectively. Let Q_{ $\mu$} be the resolvent of T for $\mu$>0 and let J_{ $\lambda$} be. the metric resolvent. such that Let. \{u_{n}\}. A\neq 0 be. a. of B for. $\lambda$>0 ,. and let A^{*} be the. respectively. Let A:H\rightarrow F be a bounded linear operator adjoint operator of A. Suppose T^{-1}0\cap A^{-1}(B^{-1}0)\neq\emptyset.. sequence in H such that u_{n}\rightarrow u. .. For x_{1}=x\in H , let. generated by. \{x_{n}\}\subset H. be. a. sequence. x_{n+1}=$\beta$_{n}x_{n}+(1-$\beta$_{n})($\alpha$_{n}u_{n}+(1-$\alpha$_{n})Q_{ $\mu$}(x_{n}-rA^{*}J_{F}(I-J_{ $\lambda$})Ax_{n})) for. r\in(0, \infty) \{$\alpha$_{n}\}\subset(0,1). all n\in \mathrm{N} , where. ,. 0<r\Vert AA^{*}\Vert<2, for. a, b\in \mathbb{R}. some. .. Then. \displaystyle \lim_{n\rightar ow\infty}$\alpha$_{n}=0,. \{x_{n}\}. converges. \{$\beta$_{n}\}\subset(0,1) satisfv. and. \displayst le\sum_{n=1}^{\infty}$\alpha$_{n}=\infty strongly. to. a. and. point. z_{0}=P_{T^{-1}0\cap A-1}(-1.. 0<a\leq$\beta$_{n}\leq b<1. z_{0}\in T^{-1}0\cap A^{-1}(B^{-1}0). ,. where. References [1]. S. M. Alsulami and W. monotone. [2]. (2014), K.. The. split common null point problem for maximal applications, J. Nonlinear Convex Anal. 15. 793‐808.. Aoyama,. K.. (2007),. Y.. Kimura,. W. Takahashi and M.. Toyoda, Approximation of. (2009), 175. F. Kohsaka and W.. mappings:. F. E.. a. Takahashi,. generalizations of firmly nonexpan‐ properties, J. Nonlinear Convex Anal. 10. 131‐147.. Browder, Nonlinear maximal. (1968),. monotone. operators in Banach spaces, Math. Ann.. 89‐113.. F. E. Browder and W. V.. [6]. in Hilbert spaces, J. Math. Anal. Appl. 20 (1967), 197‐228. C. Byrne, Y. Censor, A. Gibali and S. Reich, The split common null. Petryshyn, Construction of fixed points of nonlinear mappings. Nonlinear Convex Anal. 13 Y. Censor and T.. product. fixed. Three. Their relations and continuous. [5]. [7]. common. Banach space, Nonlinear Anal.. 2350‐2360.. Aoyama,. sive. [4]. Takahashi,. in Hilbert spaces and. points of a countable family of nonexpansive mappings in 67. [3]. mappings. (2012),. point problem, J.. 759‐775.. Elfving, A multiprojection algorithm using Bregman projections Algorithms 8 (1994), 221‐239.. space, Numer.. in. a.
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