The Split Common Fixed Point Problem for New Classes of Nonlinear Operators in Banach Spaces (The structure of function spaces and its environment)

全文

(1)47. 数理解析研究所講究録第2041巻 2017年 47-56. The. Common Fixed Point Problem for New Classes. Split. of Nonlinear. Operators. in Banach. Spaces. 慶応義塾大学自然科学研究教育センター,高雄医学大学基礎科学センター高橋渉 (Wataru Takahashi). Keio Research and Education Center for Natural. Center for Fundamental. Science, Kaohsiung Email:. Abstract.. Medical. Sciences, Keio University, Japan and University, Kaohsiung 80702, Taiwan. wataru@is.titech.ac.jp;. wataru@a00.itscom.net. The aim of this article is to prove strong convergence theorems by the hybrid shrinking projection method for finding common fixed points of families of. method and the nonlinear. mappings in Banach spaces. We first deal with basic properties of new nonlinear particular, we prove that the common fixed point sets of new nonlinear mappings are closed and convex. Using these results and the hybrid method introduced by Nakajo and Takahashi [14], we prove a strong convergence theorem which solves the split common fixed point problem in two Banach Spaces. Furthermore, using the shrinking projection method introduced by Takahashi, Takeuchi and Kubota [28], we also prove another strong convergence theorem. Moreover, using these results, we obtain well‐known and new strong convergence new. mappings.. In. theorems in Hilbert spaces and Banach spaces. 2010 Mathematics. Subject Classification: 47\mathrm{H}10 Keywords and phrases: Maximal monotone mapping, hybrid method, shrinking projection method, generalized projection, generalized resolvent, split common fixed point problem.. Introduction. 1. Recently, Takahashi, follows: Let E be E and let $\eta$ and T : C\rightarrow E with. s. Wen and Yao. [29]. introduced. a new. class of nonlinear. F(T)\neq\emptyset. is called. ( $\eta$, s) ‐demigeneralized if,. for any x\in C and. 2\langle x-q, Jx-JTx\rangle\geq(1- $\eta$) $\phi$(x, Tx)+s $\phi$(Tx, x) where. F(T). mappings. as. smooth Banach space, let C be a nonempty, closed and convex subset of be real numbers with $\eta$\in(-\infty, 1) and s\in [0, \infty ), respectively. A mapping. a. is the set of fixed. points of T, J. is the. duality mapping. on. q\in F(T). ,. (1.1). ,. E and. $\phi$(x, y)=\Vert x\Vert^{2}-2\langle x, Jy\rangle+\Vert y\Vert^{2}, \forall x, y\in E. Observe the. that, in a Hilbert space H, mapping T is as follows:. 2\{x-qJx. $\phi$(x, y)=\Vert x-y\Vert^{2}. —JTx \} \geq. (1- $\eta$) $\phi$ ( x Tx), ,. for all x,. y\in H. .. If s=0 in. \forall x\in C, q\in F(T). .. (1.1),. then.

(2) 48. Such. ( $\eta$, 0) ‐demigeneralized mappings. a. are. important.. Hilbert space and let C be a nonempty, closed and real number with 0\leq k<1 A mapping U:C\rightarrow H is called Let H be. [5]. a. .. convex a. subset of H. k‐strict. .. Let k be. pseudo‐contraction. if. \Vert Ux-Uy\Vert^{2}\leq \Vert x-y\Vert^{2}+k\Vert x-Ux-(y-Uy)\Vert^{2}. for all x,. y\in C. .. If U is. a. k ‐strict. pseudo‐contraction and F(U)\neq\emptyset. then. ,. \Vert Ux-q\Vert^{2}\leq \Vert x-q\Vert^{2}+k\Vert x-Ux\Vert^{2} for all x\in C and. q\in F(U). From. .. this,. we. \Vert Ux-x\Vert^{2}+\Vert x-q\Vert^{2}+2 { Ux Therefore,. get that. we. —. x-q\rangle\leq \Vert x-q\Vert^{2}+k\Vert x-Ux\Vert^{2}.. x,. 2\{x-Ux, x-q\rangle \geq(1-k)\Vert x-Ux\Vert^{2}. (1.2). q\in F(U) Thus, from (1.2), a k‐strict pseudo‐contraction U with F(U)\neq\emptyset (k, 0) ‐demigeneralized. We also know that there exists such a mapping in a Banach space.. for all x\in C and is. have that. Let E be. operator. a. on. .. uniformly E. .. convex. and smooth Banach space and let B be a maximal monotone we consider the following equation. For each r>0 and x\in E ,. Jx\in Jx_{r}+rBx_{r}. J_{r}x Such a J_{r} is called the equation has a unique solution x_{r} We define J_{r} by x_{r} generalized resolvent of B The set of null points of B is defined by B^{-1}0=\{z\in E:0\in Bz\}. We know that B^{-1}0 is closed and convex; see [20]. The generalized resolvent has the following This. =. .. .. .. property: for. any x\in E and. q\in F(J_{r})=\{z\in E:0\in Bz\},. 2\langle J_{r}x-q, Jx-JJ_{r}x\}\geq 0. Then. we. get. 2\langle J_{r}x-x+x-q Jx—JJrx} \geq 0 ,. and hence 2 \{x-q , Jx. —. (1.3). JJ_{r}x\rangle. \geq 2 {x—Jrx,. Jx. —. JJ_{r}x\rangle= $\phi$(x, J_{r}x)+ $\phi$(J_{r}x, x). .. Thus, from (1.3), the generalized resolvent J_{r} with B^{-1}0\neq\emptyset is (0,1) ‐demigeneralized. In this article, we first deal with basic properties of new demigeneralized mappings. In particular, we prove that the common fixed point sets of new demigeneralized mappings are closed and convex. Using these results and the hybrid method introduced by Nakajo and Taka‐ hashi [14], we prove a strong convergence theorem which solves the split common fixed point problem in Banach Spaces. Furthermore, using the shrinking projection method introduced by Takahashi, Takeuchi and Kubota [28], we also prove another strong convergence theorem. Moreover, using these results, we obtain well‐known and new strong convergence theorems in Hilbert spaces and Banach spaces..

(3) 49. Preliminaries. 2. Let E be. real Banach space with norm \Vert\cdot\Vert and let E^{*} be the dual space of E We denote y^{*} \in E^{*} at x\in E by \langle x, y^{*} }. When \{x_{n}\} is a sequence in E , we denote the strong. a. the value of. .. convergence of. \{x_{n}\}. to x\in E. x_{n}\rightarrow x and the weak convergence. by. by. x_{n}\rightarrow x. The modulus. .. $\delta$ of convexity of E is defined by. $\delta$( $\epsilon$)=\displaystyle \inf\{1-\frac{\Vert x+y\Vert}{2} : \Vert x\Vert \leq 1, \Vert y\Vert \leq 1, \Vert x-y\Vert \geq $\epsilon$\} 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}\}. \displaystyle \lim_{n\rightarrow\infty}\Vert x_{n}-y_{n}\Vert and. =0 holds. A a. uniformly. \Vert x_{n}\Vert\rightarrow\Vert u\Vert imply. convex. if and. if. only. $\delta$( $\epsilon$). >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 We also know that. uniformly convex. uniformly. and. convex. \displaystyle \lim_{n\rightar ow\infty}\Vert x_{n}+y_{n}\Vert=2,. Banach space is. strictly. convex. and reflexive.. Banach space has the Kadec‐Klee property, i.e., x_{n}\rightarrow u The duality mapping J from E into 2^{E^{*}} is defined by. convex. x_{n}\rightarrow u. .. Jx=\{x^{*}\in E^{*} : \langle x, x^{*}\}=\Vert x\Vert^{2}=\Vert x^{*}\Vert^{2}\} for every x \in E Let U differentiable if for each x, .. =. \{x \in E : \Vert x\Vert = 1\}. y\in U. ,. The. .. norm. of E is said to be Gâteaux. the limit. \displaystyle \lim_{t\rightar ow 0}\frac{\Vert x+ty\Vert-\Vert x\Vert}{t}. (2.1). exists. In this case, E is called smooth. We know that E is smooth if and only if J is a single‐ mapping of E into E^{*} The norm of E is said to be Fréchet differentiable if for each. valued. .. (2.1) is limit (2.1). uniformly for y\in U The norm of E is said to be uniformly uniformly for x y\in U We also know that E is reflexive if and only if J is surjective, and E is strictly convex if and only if J is one‐to‐one. Therefore, if E is a smooth, strictly convex 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 [19] and [20]. We know the following result. x\in U , the limit. attained. smooth if the. is attained. .. .. ,. .. Lemma 2.1. ([19]).. Let E be. a. smooth Banach space and let J be the x, y \in E. Furthermore, if E is. Then, \{x-y, Jx-Jy\rangle \geq 0 for all \langle x-y Jx—Jy \}=0 then x=y.. E.. ,. ,. Let E be. a. smooth Banach space and let J be the. $\phi$:E\times E\rightarrow \mathbb{R} by In the. duality mapping on strictly convex and. case. Let C be. duality mapping. $\phi$_{E}(x, y)=\Vert x\Vert^{2}-2\{x, Jy\rangle+\Vert y\Vert^{2}, \forall x, y\in E. when E is a. Banach space. on. .. E. .. Define. a. function. (2.2). clear, $\phi$_{E} is simply denoted by $\phi$. nonempty, closed and convex subset of a smooth, strictly convex and reflexive E Then we know that for any x\in E there exists a unique element z\in C such .. ,. that. $\phi$(z, x)=\displaystyle \min_{y\in C} $\phi$(y, x). ..

(4) 50. The C. mapping $\Pi$_{C} following. The. .. :. E\rightarrow C defined. of E onto. by z=$\Pi$_{C}x is called the generalized projection example, see [1, 2, 7].. well‐known results. For. are. Let E be a smooth, strictly convex and reflexive Banach space. Let C ([1, 2, 7 nonempty, closed and convex subset of E and let x\in E and z\in C. Then, the following. Lemma 2.2. be. a. conditions. are. equivalent:. (1) z=$\Pi$_{C}x ; (2) \langle z-y, Jx-Jz\rangle \geq 0, Let E be. a. Banach space and let A be a mapping of E into 2^{E^{*}} The effective domain of A that is, \mathrm{d}\mathrm{o}\mathrm{m}(A)=\{x\in E: Ax\neq\emptyset\} A multi‐valued mapping A on E ,. by \mathrm{d}\mathrm{o}\mathrm{m}(A). is denoted. .. is said to be monotone if monotone. graph. [20,. \forall y\in C.. operator A. \{x-y, u^{*}-v^{*}\}. on. E. .. The. 3.5.4].. y\in \mathrm{d}\mathrm{o}\mathrm{m}(A). u^{*}\in Ax and v^{*} \in Ay. \mathrm{A} graph is not properly contained in the following theorem is due to [4, 16]; see also x,. E is said to be maximal if its. on. of any other monotone operator. Theorem. \geq 0 for all. ,. ,. Let E be a uniformly convex and smooth Banach space and let J ([4, 16 duahty mapping of E into E^{*} Let A be a monotone operator of E into 2^{E^{*}} Then A maximal if and only if for any r > 0, R(J+rA) =E^{*} where R(J+rA) is the range of. Theorem 2.3 be the is. .. ,. J+rA.. Let E be. a. umiformly. Banach space with. convex. a. Gâteaux differentiable. norm. and let B. maximal monotone operator of E into 2^{E^{*}} For all x \in E and r > 0 , we consider the This following equation Jx\in Jx_{r}+rBx_{r} equation has a unique solution x_{r} We define J_{r} by. be. a. .. x_{r}=J_{r}x Such J_{r}, r>0 .. B is defined. Let E be E. .. called the. generalized. B^{-1}0=\{z\in E : 0\in Bz\}. by a. are. smooth and. strictly. convex. .. resolvents of B. .. The set of null. points of. We know that B^{-1}0 is closed and convex; see [20]. Banach space and let J be the duality mapping on. Let $\eta$ and s be real numbers with $\eta$\in(-\infty, 1) and s\in[0, \infty ). Then a mapping U : C\rightarrow E with F(U)\neq\emptyset is called ( $\eta$, s) ‐demigeneralized [29, 12] if, for any x\in C and q\in F(U) , .. 2 \{x-q , Jx. where. F(U). is the set of fixed. —. JUx\rangle\geq(1- $\eta$) $\phi$(x, Ux)+s $\phi$(Ux, x). ,. points of U.. Examples.. (1) be. a. [5]. if. Let H be. a. Hilbert space and let C be a nonempty, closed and convex subset of H Let k A mapping U : C\rightarrow H is called a k ‐strict pseud‐contraction .. real number with 0\leq k<1. \Vert for all x, y \in. C. .. Ux. —. If U is. Uy \Vert^{2}\leq a. demigenerahzed.. (2). Let H be. mapping. U. :. a. .. \Vert x-y\Vert^{2}+k\Vert x-Ux-(y-Uy)\Vert^{2}. k ‐strict. pseud‐contraction and F(U) \neq \emptyset. Hilbert space and let C be. C\rightarrow H is called. ,. then. U is. (k, 0)-. nonempty, closed and convex subset of H. \mathrm{A} generalized hybrid [8] if there exist $\alpha$, $\beta$\in \mathbb{R} such that a. $\alpha$\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} for all x, y \in C Such a mapping U is called ( $\alpha$, $\beta$) ‐generalized hybrid and F(U)\neq\emptyset , then U is (0,0) ‐demigeneralized. .. hybrid.. If U is. generalized.

(5) 51. (3). Let E be. closed and is. a. strictly. convex. convex, reflexive and smooth Banach space and let C be. subset of E. (0,1) ‐demigeneralized. (4) Let E be a uniformly. demigeneralized. The following lemma Lemma 2.4. ([29]).. is. $\Pi$_{C} be the generalized projection of E. a. onto C. .. nonempty, Then $\Pi$_{C}. and smooth Banach space and let B be a maximal Then the generalized resolvent J_{ $\lambda$} is. Let $\lambda$ > 0. .. mono‐. (0,1)-. .. important and crucial. Let E be. in the. proofs. of. our. main results.. smooth and. strictly convex Banach space and let C of E. Let $\eta$ be a real number with $\eta$\in(-\infty, 1) Let ( $\eta$, 0) ‐demigenerahzed mapping of C into E. Then F(U) is closed and convex.. nonempty, closed and an. convex. operator with B^{-1}0 \neq \emptyset. tone. Let. .. convex. a. subset. .. be. a. U be. Main Results. 3. In this. section, using the hybrid method, we prove a strong convergence theorem for finding split common fixed point problem for families of new nonlinear mappings in two Banach spaces. Let E be a Banach space and let C be a nonempty, closed and convex subset of E Let \{U_{n}\} be a sequence of mappings of C into E such that \mathrm{n}_{n=1}^{\infty}F(U_{n})\neq\emptyset The a. solution of the. .. .. \{U_{n}\} is said to satisfy the condition (I) if for any bounded sequence \{z_{n}\} of C \displaystyle \lim_{n\rightarrow\infty}\Vert z_{n}-U_{n}z_{n}\Vert=0 every weak cluster point of \{z_{n}\} belongs to \displaystyle \bigcap_{n=1}^{\infty}F(U_{n}). sequence. that. ,. Theorem 3.1. ([27]).. such. .. Let E and F be. uniformly. convex. and. and let J_{E} and J_{F} be the duality mappings on E and F , sequences of real numbers with $\tau$_{n}, $\eta$_{n} \in (-\infty, 1) and let. uniformly respectively.. smooth Banach spaces Let \{$\tau$_{n}\} and \{$\eta$_{n}\} be. \{t_{n}\} and \{\mathcal{S}_{n}\} be sequences of real numbers with t_{n}, s_{n}\in[0, \infty ). Let \{T_{n}\} be a family of ($\tau$_{n}, t_{n}) ‐demigenerahzed mappings of E into itself with \mathrm{n}_{n=1}^{\infty}F(T_{n})\neq\emptyset satisfying the condition (I) and let \{U_{n}\} be a family of ($\eta$_{n}, s_{n})demigeneralized mappings of. Let A. :. E\rightarrow F be. a. F into. itself. with. \displaystyle \bigcap_{n=1}^{\infty}F(U_{n}) \neq \emptyset satisfying. bounded linear operator such that. A\neq 0. of A. Suppose that \displaystyle \mathcal{F}=\mathrm{n}_{n=1}^{\infty}F(T_{n})\cap A^{-1}(\bigcap_{n=1}^{\infty}F(U_{n})) \neq\emptyset sequence generated by. where a,. the condition. and let A^{*} be the .. (I).. adjoint operator. Let x_{1} \in E and let. \{x_{n}\}. be. a. \left{bginary} z_=J{E^-1(}xnr_A{*JF}-U_{nAx\ y}=T_{nz,\C inE:2lagex_{}-z,J nE_{}\ geqrn(1-$ta_{})\phiF(Axn,U_{})+rsn$\phi_{F}(UAxn, \ D_{}=zinE:2\-,J_{}znEy\ragle q(1-$tu_{n})\phiE(z,y_{n})+t$\phiE(y_{n},z\ Q=inE:lagex_{}-z,J 1Ex_{n}\ragleq0, x_{n+1}=$\PiCcapD_{n}\Q^X1,foraln\imthb{N}, endary\ight.. b\in \mathbb{R}, \{r_{n}\}\subset (0, \infty). 0<a\leq r_{n}\leq Then the sequence. \{x_{n}\}. and. \{$\tau$_{n}\}, \{$\eta$_{n}\}\subset (-\infty, 1) satisfy. \displayst le\frac{1}2\VertA\Vert^{2}. converges. z_{0}= $\Pi$ 口 \infty {}_{n=1}F(T_{n})\cap A-1 ( 口誰 {}_{1}F(U_{n}))^{X}1.. and. the. 0<b\leq 1-$\tau$_{n}, 1-$\eta$_{n},. strongly. to z_{0}. \in. following inequalities \forall n\in \mathbb{N}.. \mathrm{n}_{n=1}^{\infty}F(T_{n}) \displaystyle \cap A^{-1}(\bigcap_{n=1}^{\infty}F(U_{n})). ,. where.

(6) 52. Next, using the shrinking projection method [28], we prove a strong convergence theorem finding a solution of the split common fixed point problem with families of mappings in. for. Banach spaces. For a sequence. \{C_{n}\}. of nonempty, closed and convex subsets of a Banach space E , define s‐Lin C_{n} w‐Lsn C_{n} as follows: x\in s‐Lin C_{n} if and only if there exists \{x_{n}\}\subset E such that converges strongly to x and x_{n}\in C_{n} for all n\in \mathbb{N} Similarly, y\in w‐Lsn C_{n} if and only \{x_{n}\} and. .. if there exist. weakly. subsequence \{C_{n_{i}} \} of \{C_{n}\} and a sequence \{y_{i}\}\subset E and y_{i}\in C_{n_{i}} for all i\in \mathbb{N} If C_{0} satisfies. to y. a. such that. \{y_{i}\}. converges. .. C_{0}=. s‐Lni C_{n}= w‐Lns C_{n}. (3.1). ,. \{C_{n}\} converges to C_{0} in the sense of Mosco [13] and we write C_{0} It is easy to show that if \{C_{n}\} is nonincreasing with respect to inclusion, then \displaystyle \mathrm{M}-\lim_{n\rightarrow\infty}C_{n} to in the sense of Mosco. For more details, see [13]. The following converges \{C_{n}\} lemma was proved by Ibaraki, Kimura and Takahashi [6]. it is said that. =. .. \displaystyle \bigcap_{n=1}^{\infty}C_{n}. ([6]). Let \{C_{n}\} be a. E be. exists and. nonempty, then for each x\in E, \{$\Pi$_{C_{n}}x\} converges strongly to $\Pi$_{C_{0}}x, the generalized projections of E onto C_{n} and C_{0} respectively.. Lemma 3.2 Let. norm.. \displaystyle \lim_{n\rightarrow\infty}C_{n} where. $\Pi$_{C_{n}. and. Theorem 3.3. $\Pi$_{C_{0}. a. smooth Banach space such that E^{*} has a Fréchet differentiable of nonempty, closed and convex subsets of E. If C_{0} =M-. sequence. are. ([27]).. ,. Let E and F be. uniformly. convex. and. and let J_{E} and J_{F} be the duality mappings on E and F , sequences of real numbers with $\tau$_{n}, $\eta$_{n} \in (-\infty, 1) and let. uniformly respectively.. \{t_{n}\}. and. smooth Banach spaces Let \{$\tau$_{n}\} and \{$\eta$_{n}\} be. \{s_{n}\}. be sequences. of. real. the condition. (I).. numbers with t_{n}, s_{n}\in [0, \infty ). Let \{T_{n}\} be a family of ($\tau$_{n}, t_{n}) ‐demigeneralized mappings of E into itself with \mathrm{n}_{n=1}^{\infty}F(T_{n})\neq\emptyset satisfying the condition (I) and let \{U_{n}\} be a family of ($\eta$_{n}, s_{n})-. demigeneralized mappings of. Let A. of. A.. \{x_{n}\}. :. E\rightarrow F be. Suppose be. a. F into. itself. with. \displaystyle \bigcap_{n=1}^{\infty}F(U_{n}) \neq \emptyset satisfying. bounded hnear operator such that A\neq 0 and let A^{*} be the adjoint operator that \mathcal{F}= \displaystyle \mathrm{n}_{n=1}^{\infty}F(T_{n})\cap A^{-1}(\bigcap_{n=1}^{\infty}F(U_{n})) \neq \emptyset For x_{1} \in E and C_{1} =E , let a. .. sequence. generated by. \left{bginary}l z_{=JE}^-1(_{xn}rA^{*(J_F}xn-{U_}Axn\ y_{}=Tnz,\ C_{n+1}=z\iC_{n:2lagex}-z,J_{En }z_{\ geqr_{n}(1-$\ta )phi$_{F}(Axn,U _{})+rns$\phi_{F}(UnAx, )\} cap{zinC_}:2\lagez_{n}-,JE _{}yn\ragle q(1-$\tau_{n})phi$E(z_{n},y)+t$\phi_{E}(yn,z\ x_{n+1}=$\PiC_{n+1}x,\foralnimthb{N}\ex） nd{ary}\ight.. where a, b\in \mathbb{R} and. \{r_{n}\}\subset (0, \infty) satisfy. 0<a\leq r_{n}\leq Then the sequence. \{x_{n}\}. \displayst le\frac{1}2\VertA\Vert^{2}. converges. z_{0}= $\Pi$ 口罷 {}_{1} F(T_{n})\cap A-1 ( 口盤 {}_{1}F(U_{n}))^{X}1\cdot. the. and. following inequalities. 0<b\leq 1-$\tau$_{n}, 1-$\eta$_{n},. strongly. to z_{0} \in. \forall n\in \mathbb{N}.. \displaystyle \mathrm{n}_{n=1}^{\infty}F(T_{n})\cap A^{-1}(\bigcap_{n=1}^{\infty}F(U_{n})). ,. where.

(7) 53. Applicationss. 4. In this. section, using Theorems 3.1 and 3.3, we get well‐known and new strong convergence are connected with the split common fixed point problem for families of demigeneralized mappings in Banach spaces. We know the following result obtained by Marino theorems which and Xu. [11].. Lemma 4.1. of H. Let If x_{n}\rightarrow z. ([11]).. k be. Let H be. a. Hilbert space and let C be a nonempty, closed and convex subset : C\rightarrow H be a k ‐strict pseudo‐contraction.. real number with 0\leq k<1 and U and x_{n}-Ux_{n}\rightarrow 0 , then z\in F(U) a. .. We also know the. result from. following. Plubtieng. and Takahashi. [15].. Let H_{1} and H_{2} be Hilbert spaces and let $\alpha$>0 Let G:H_{1} \rightarrow 2^{H_{1}} be a mapping and let J_{ $\lambda$} (I+ $\lambda$ G)^{-1} be the resolvent of G for $\lambda$ > 0 Let T:H_{2}\rightarrow H_{2} be an $\alpha$ ‐inverse strongly monotone mapping and let A:H_{1} \rightarrow H_{2} be a bounded linear operator. Suppose that G^{-1}0\cap A^{-1}(T^{-1}0) \neq\emptyset Let $\lambda$, r > 0 and z \in H_{1} Then the following are equivalent:. Lemma 4.2. ([15]).. .. maximal monotone. =. .. .. (i) z=J_{ $\lambda$}(I-rA^{*}TA)z ; (it) 0\in A^{*}TAz+Gz ; (iii) z\in G^{-1}0\cap A^{-1}(T^{-1}0) Using. .. .. Theorem 3.1 and Lemmas 4.1 and. 4.2,. we. obtain the. Theorem 4.3. Let H_{1} and H_{2} be Hilbert spaces. Let k\in[0 , H_{1} be an $\alpha$ ‐inverse strongly monotone mapping with T^{-1}0. 1). following and $\alpha$\in. \neq \emptyset. theorem.. (0, \infty). and let U. :. .. Let. H_{2}. T:H_{1}\rightarrow H_{2} be a operator of. \rightarrow. such that F(U) \neq \emptyset Let G be a maximal monotone H_{1} and J_{ $\lambda$} be the resolvent of G for $\lambda$>0 Define T_{n}=J_{$\lambda$_{n}}(I-$\lambda$_{n}T) for all n\in \mathbb{N} such that $\lambda$_{n} \in (0, \infty) and define U_{n}=$\alpha$_{n}I+(1-$\alpha$_{n})U for all n\in \mathbb{N} such that 0\leq $\alpha$_{n} < 1 and \displaystyle \sup_{n\in \mathbb{N} $\alpha$_{n} < 1 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 that T^{-1}0\cap G^{-1}0\cap A^{-1}F(U)\neq\emptyset Let x_{1}\in H_{1} and let \{x_{n}\} be a sequence generated by. k ‐strict. H_{1}. pseudo‐contraction. .. into. .. .. .. where. \left{bginary}{l z_n=x}-r{A^*(x_n}-U{Ax_n}),\ y{=T_n}z ,\ C_{n}=z\iH_{1}:2\xn-z,_{}n\geqr_{}(1-k)\VertAx_{n}-U Ax_{n}\Vert^2,\ D_{n}=z\iH_{1}:2\zn-,_{}yn\ragleq\Vrtz_{n}-y \Vert^{2},\ Q_{n}=z\iH_{1}:langex-z,_{1}n\geq0},\ x_{n+1}=PC \capD_{n} Q x_{1},\foralnimthb{N}, \endary}ight.. {rn}, \{$\lambda$_{n}\}\subset (0, \infty). and a, b\in \mathbb{R}. 0<a\leq r_{n}\leq. \displayst le\frac{1}2\VertA\Vert^{2}. satisfy the following inequalities and. \{x_{n}\} converges strongly z0=P_{$\tau$^{-1}0\cap G-1}0\cap A-1F(U)^{X}1. Then the sequence. Using. Theorem. 3.1,. we. to. 0<b\leq$\lambda$_{n}\leq 2 $\alpha$, a. get the following result. point. [24].. z_{0}. \forall n\in \mathbb{N}.. \in T^{-1}0\cap G^{-1}0\cap A^{-1}F(U). ,. where.

(8) 54. Theorem 4.4. ([24]).. Let E and F be. unOformly convex and uniformly smooth Banach spaces duality mappings on E and F respectively. Let G and B be maximal monotone operators ofE into E^{*} and F into F^{*} respectively. Let J_{ $\lambda$} and Q_{ $\mu$} be the generalized resolvents of G for $\lambda$>0 and B for $\mu$>0 respectively. Let A:E\rightarrow F be a bounded hnear 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 Let x_{1} \in E and let \{x_{n}\} be a sequence generated by and let J_{E} and J_{F} be the. ,. ,. ,. .. \left{bginary} z_=J{E^-1(}xnr_A{*JF}-Q_$\mu{nAx} y_=J{$\lambdn}z_, C{=\inE:2x_}-zte{）JEn_}z\ geqr{n$phi_F}(Ax,Q{\mu$n_})+r{\phi$F(Q_mu{n}Ax,_\ D{n}=ziE:2lage_{n}-,Jz Ey_{n}\ geq$phi_{E}(zn,y)+$\phi_{E}(ntex）z\ Q_{n}=iE:x-z,J_{}1Exn\ragleq0}, x_{n+1=$\PiC}capD_{n\Q^X}1,foraln\imthb{N}, edary\ight.. where a, b\in \mathbb{R} and. {rn}, \{$\lambda$_{n}\}, \{$\mu$_{n}\}\subset(0, \infty) satisfy. 0<a\displaystyle \leq r_{n}\leq\frac{1}{2\Vert A\Vert^{2} Then. \{x_{n}\}. converges. Similarly, using. strongly. Theorem. to. 3.3,. ,. and. the. 0<b\leq$\lambda$_{n}, $\mu$_{n},. z_{0}\in G^{-1}0\cap A^{-1}(B^{-1}0). we. have the. following inequalities. following. ,. where. \forall n\in \mathbb{N}.. z_{0}=$\Pi$_{G^{-1}0\cap A-1}(-1x.. results.. Theorem 4.5. Let H_{1} and H_{2} be Hilbert spaces. Let k\in [0 , H_{1} be an $\alpha$ ‐inverse strongly monotone mapping with T^{-1}0. 1). and. \neq \emptyset. $\alpha$\in(0, \infty). and let U. :. .. Let. H_{2}. T:H_{1}\rightarrow H_{2} be a operator of. \rightarrow. k ‐strict pseudo‐contraction such that F(U) \neq \emptyset Let G be a maximal monotone H_{1} into H_{1} and J_{ $\lambda$} be the resolvent of G for $\lambda$>0 Define T_{n}=J_{$\lambda$_{n}}(I-$\lambda$_{n}T) for all n\in \mathbb{N} such that $\lambda$_{n} \in (0, \infty) and define U_{n}=$\alpha$_{n}I+(1-$\alpha$_{n})U for all n\in \mathbb{N} such that 0\leq$\alpha$_{n} < 1 and \displaystyle \sup_{n\in \mathbb{N} $\alpha$_{n} < 1 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 T^{-1}0\cap G^{-1}0\cap A^{-1}F(U)\neq\emptyset For x_{1} \in H_{1} and C_{1}=H_{1} let \{x_{n}\} be a sequence generated by .. .. .. .. ,. where. \left{bginary}{l z_n}=x{-r_n}A^{*(xn-UAxn),\ y_{n}=T z_{n},\ C_{n+1}=\{zinC_{}:2\x_{n}-z, _{n}\ragle\qr_{n}(1-$\eta_{n})\VertAx_{n}-U Ax_{n}\Vert^{2}\ and2\{z_n}-, { y_n}\geq(1-$\tau_{n})\Vertz_{n}-y \Vert^{2}\, x_{n+1}=P_{Cn+1}x_{,\foraln\i mathb{N}, \end{ary}\ight.. {rn}, \{$\lambda$_{n}\}\subset (0, \infty). and a, b\in \mathbb{R}. 0<a\displaystyle \leq r_{n}\leq\frac{1}{2\Vert A\Vert^{2} Then the sequence. \{x_{n}\}. converges. and. strongly. z_{0}=P_{T^{-1}0\cap G^{-1}0\cap A^{-1}F(U)^{X}1}. Furthermore, using Theorem 3.3,. satisfy. we. to. the. following inequalities. 0<b\leq$\lambda$_{n}\leq 2 $\alpha$, a. point. z_{0} \in. \forall n\in \mathbb{N}.. T^{-1}0\cap G^{-1}0\cap A^{-1}F(U). get the following result. [17].. ,. where.

(9) 55. Theorem 4.6. ([17]).. Let E and F be. uniformly. and. convex. uniformly. 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 generalized resolvents. of G for. $\lambda$>0 and B. Let A:E\rightarrow F be. for $\mu$>0 respectively. ,. a. bounded hnear. 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_{=JE^-1}(_{xnrA^*}(J_{Fxn-}Q_{$\munAx}\ y_{n=J$lambd_{n}z,\ C+1}={z\inC_:2x{}-z,J_En{}z\ geqr_{n}$\phiF(Ax_{n},Q$\mu }Ax_{n)+r$\phi_{F}(Qmu$nAx_{}, )\ and2lgez_{}-,JEn_{}y\ragle q$\phi_{E}(zn,y)+$\phi_{E}(yn,z\ x_{+1}=$PiC_{n+1}x,\foralnimthb{N}, \endaryight.. where a, b\in \mathbb{R} and. {rn}, \{$\lambda$_{n}\}, \{$\mu$_{n}\}\subset (0, \infty) satisfy 0<a\leq r_{n}\leq. Then. \{x_{n}\}. converges. strongly. Acknowledgements.. to. \displayst le\frac{1}2\VertA\Vert^{2}. following inequalities \forall n\in \mathbb{N}.. 0<b\leq$\lambda$_{n}, $\mu$_{n},. z_{0}\in G^{-1}0\cap A^{-1}(B^{-1}0). ,. where. z_{0}=$\Pi$_{G^{-1}0\cap A^{-1}(B^{-1}0)^{X}1}.. partially supported by Grant‐in‐Aid Japan Society for the Promotion of Science.. The author. Research No. 15\mathrm{K}04906 from. and. the. was. for Scientific. References [1]. Y. I.. [3]. K.. Alber, Metric and generalized projections in Banach spaces: Properties and appli‐ cations, in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type (A. G. Kartsatos Ed Marcel Dekker, New York, 1996, pp. 15‐50. [2] Y. I. Alber and S. Reich, An iterative method for solving a class of nonlinear operator in Banach spaces, Panamer. Math. J. 4 (1994), 39‐54. Aoyama, F. Kohsaka and W. Takahashi, Three sive. [4]. (2009), F. E.. 175. [5]. mappings: Their relations and. continuous. generahzations 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.. Petryshyn,. Construction. of fixed points of. [6]. in Hilbert spaces, J. Math. Anal. Appl. 20 (1967), 197‐228. T. Ibaraki, Y. Kimura and W. Takahashi, Convegence theorems. [7]. S. Kamimura and W.. and maximal monotone operators in Banach spaces, Abstr. 621‐629.. Takahashi, Strong convergence of Optim. 13 (2002), 938‐945.. Banach space, SIAM J.. a. nonlinear. mappings. for generalized projections Appl. Anal. 2003:10 (2003),. proximal‐type algorithm. in. a.

(10) 56. [8]. P.. Kocourek, W. Takahashi and J.‐C. Yao, Fixed point theorems and weak convergence for generalized hybrid mappings in Hilbert spaces, Taiwanese J. Math. 14 (2010),. theorems. 2497‐2511.. [9]. F. Kohsaka and W.. Takahashi, Existence and approximation of fixed points of firmly nonexpansive‐type mappings in Banach spaces, SIAM J. Optim. 19 (2008), 824‐835. [10] F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. 166‐177.. [11]. G. Marino and H.‐K.. [12]. contractions in Hilbert spaces, J. Math. Anal. S. Matsushita, K. Nakajo and W. Takahashi,. Xu, Weak and strong. (Basel). 91. convergence theorems for strich Appl. 329 (2007), 336‐346.. (2008), pseudo‐. Strong convergence theorems obtained by a generalized projections hybrid method for families of mappings in Banach spaces, Nonlin‐ ear Appl. 73 (2010), 1466‐1480. U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. [13] Math. 3. (1969),. 510‐585.. [14]. K.. [16]. R. T.. [17]. S. Takahashi and W.. [18]. projection method in two Banach spaces, Linear Nonlinear Anal. 1 (2015), 297‐304. S. Takahashi and W. Takahashi, The spht common null point problem and the shrinking. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003), 372‐379. [15] S. Plubtieng and W. Takahashi, Generalized split feasibility problems and weak conver‐ gence theorems in Hilbert spaces, Linear Nonlinear Anal. 1. Rockafellar, On. Amer. Math. Soc. 149. the. maximality of. (1970),. sums. of. (2015),. 139‐158.. nonlinear monotone operators, Trans.. 75‐88.. Takahashi,. The. split. common. null. point problem and the shrinking. projection method in Banach spaces, optimization 65 (2016), 281‐287. W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000. W. Takahashi, Convex Analysis and Approximation of Fixed Points (Japanese), Yoko‐ hama Publishers, Yokohama, 2000. [21] W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J.. [19] [20]. Nonlinear Convex Anal. 11. [22]. W.. Takahashi,. The. (2010),. 79‐88.. split feasibility problem. spaces, J. Nonlinear Convex Anal. 16. [23] [24]. W.. Takahashi,. (2015), W.. The. W.. [27]. W.. (2015),. shrinking projection. method in Banach. 1449‐1459.. split. common. null point. problem. in Banach spaces, Arch. Math. 104. spht. common. null point. problem. in two Banach spaces, J. Nonlinear. 357‐365.. Takahashi,. The. Convex Anal. 16. [25]. and the. (2015),. 2343‐2350.. Takahashi, The split common fixed point problem and strong convergence theorems by hybrid methods in two Banach spaces, J. Nonlinear Convex Anal. 17 (2016), 1051‐1067. [26] W. Takahashi, The split common fixed point problem and the shrinking projection method in Banach spaces, J. Convex Anal. 24 Takahashi, The split common fixed. (2017),. to appear.. point problem for families of. new. nonlinear map‐. pings and hybrid methods in two Banach spaces, to appear. [28] W. Takahashi, Y. Takeuchi and R. Kubota, Strong convergence theorems by hybrid meth‐ ods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 341. [29]. (2008),. 276‐286.. W.. Takahashi, C.‐F. Wen and J.‐C. Yao, Strong convergence theorem by shrinking projec‐ tion method for new nonlinear mappings in Banach spaces and applications, optimization. 66. (2017),. 609‐621..

(11)