Weak and Strong Convergence Theorems for a Finite Family of Demimetric Mappings with Variational Inequality Problems in Hilbert Spaces (Nonlinear Analysis and Convex Analysis)

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(1)39. 数理解析研究所講究録第2065巻 2018年 39-50. Weak and Strong Convergence Theorems for a Finite Family of Demimetric Mappings with Variational Inequality Problems in Hilbert 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 Sciences, Keio University, Japan and Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80702, Taiwan Email: [email protected]; [email protected]. Abstract. In this article, using a new nonlinear mapping called demimetric, we prove weak and strong convergence theorenus for findinng a commou element of the set of commou fixed pomints of a fimite family of such new demimetnic mappings and the set of common solutions of variational inequality problems for a finite family of inverse strongly monotone mappings in a Hilbert space. Using the results, we obtain well‐known and new strong convergence theorems in a Hilbert space. 2010 Mathematics Subject Ơassificatiorc 47\mathrm{H}10 Keyworvls and phrases: Demimetric mapping, inverse strongly monotone mapping, Mann type iteration, Halpern type iteration, shrinking projection method, variational inequality problem, fixed point.. 1. Introduction Let. H. be a real Hilbert space and let C be a nonempty, closed and convex subset of. H.. For a mapping U : C\rightarrow H , we denote by F(U) the set of fixed points of U . Let k be a real number with 0\leq k<1 . A mapping U : C\rightarrow H is called a k‐strict pseudo‐contraction [5] if. \Vert Ux-Uy||^{2}\leq\Vert x-y\Vert^{2}+k\Vert x-Ux-(y-Uy)\Vert^{2} for all x\in C. x, y\in C . If U is a k‐stnct pseudo‐contraction and F(U)\neq\emptyset, then we have that, for and q\in F(U) ,. \Vert Ux-q||^{2}\leq||x-q\Vert^{2}+k\Vert x-Ux\Vert^{2}.. From \Vert Ux-q||^{2}=\Vert Ux-x\Vert^{2}+||x-q||^{2}+2\langle Ux-x,x-q\rangle , we have that. \Vert Ux-x\Vert^{2}+\Vert x-q\Vert^{2}+2\langle Ux-x,x-q\rangle\leq\Vert x-q\Vert^{2}+k\Vert x-Ux\Vert^{2}..

(2) 40. Therefore, we have that. 2\langle x-Ux,x-q)\geq(1-k)\Vert x-Ux\Vert^{2} for. Wx\in C. and q\in F(U) . A mapping. U. :. C\rightarrow H. (1.1). is called generalized hybrid [10] if there. exist a, $\beta$\in \mathbb{R} such that. $\alpha$\Vert Ux-Uy||^{2}+(1- $\alpha$)\Vert x-Uy\Vert^{2}\leq $\beta$||Ux-y\Vert^{2}+(1- $\beta$)||x-y\Vert^{2} for all x, y \in C . Such a mapping U is called ( $\alpha$, $\beta$) ‐genợahzed hybrid. Notice that the class of generalized hybrid mappings covers several well‐known mappings. For example, \mathrm{a} (1,0) ‐generalized hybrid mapping is nonexpansive, i.e., ||Ux-Uy||\leq||x-y \forall x,y\in C.. It is nonspreading [11, 12] for ư. =. 2 and $\beta$=1 , i.e.,. 2\Vert Ux-Uy\Vert^{2}\leq\Vert Ux-y\Vert^{2}+\Vert Uy-x\Vert^{2}, \forall x, y\in C. It is also hybrid [21] for $\alpha$=\displaystyle \frac{3}{2} and $\beta$=\displaystyle \frac{1}{2} , i.e.,. 3\Vert Ux-Uy\Vert^{2}\leq\Vert x-y\Vert^{2}+||Ux-y||^{2}+\Vert Uy-x\Vert^{2}, \forall x,y\in C. In general, nonspreading and hybnid mappings are not continuous; see [7]. If U is generalized hybrid and F(U)\neq\emptyset , then we have that, for x\in C and q\in F(U) ,. $\alpha$||q-Ux||^{2}+(1- $\alpha$)\Vert q-Ux\Vert^{2}\leq $\beta$\Vert q-x\Vert^{2}+(1- $\beta$)\Vert q-x||^{2} and hence \Vert Ux-q\Vert^{2}\leq\Vert x-q\Vert^{2} . From this, we have that. 2\langle x-q , x — Ux)\geq\Vert x-Ux\Vert^{2} .. (1.2). On the other hand, there exists such a mapping in a Banach space. Let E be a smooth Banach space and let B be a maximal monotone operator with B^{-1}0\neq\emptyset . Then, for the. metric resolvent J_{ $\lambda$} of B for. $\lambda$>0 ,. we have from [19] that, for any. x\in E. and q\in B^{-1}0,. \langle J_{ $\lambda$}x-q, J(x-J_{ $\lambda$}x)\rangle\geq 0. Then we get. \langle J_{ $\lambda$}x-x+x-q, J(x-J_{ $\lambda$}x))\geq 0 and hence. \langle x-q, J(x-J_{ $\lambda$}x)\rangle\geq||x-J_{ $\lambda$}x\Vert^{2} , where. J. is the duality mapping on. E.. (1.3). Motivated by (1.1), (1.2) and (1.3), Takahashi [23]. introduced a new nonlinear mapping as follows: Let E be a smooth Banach space, let C be a nonempty, closed and convex subset of E and let k be a real number with k\in(-\infty, 1) . \mathrm{A} mapping U : C\rightarrow E with F(U)\neq\emptyset is called k‐demmetric if, for any x\in C and q\in F(U) ,. 2\langle x-q, J(x-Ux))\geq(1-k)||x-Ux\Vert^{2}, where J is the duality mapping on E . Accordin\mathrm{g} to the definition, we get that a k‐stnct pseudo‐ contraction U with F(U)\neq\emptyset is k‐demmetnc, an ( $\alpha$, $\beta$) ‐generahzed hybrid mapping U with. F(U)\neq\emptyset. is. ‐demimetric and the metric resolvent J_{ $\lambda$} with. B^{-1}0\neq 0. is. (-1) ‐demimetric..

(3) 41. In this article, using this new nonlinear mapping called demimetric, we prove weak and strong convergence theorems for finding a common element of the set of common fixed pomints of a finite family of such ncw demimc,tric mappings and thc set of common sohitions of variational inequality problems for a finite family of inverse strongly monotone mappings in a Hilbert space. Using the results, we obtain well‐known and new strong convergence theorems in a \cdot. Hilbert space.. 2. Preliminaries Throughout this paper, let. \mathrm{N}. be the set of positive integers and let. \mathbb{R}. be the set of real. be a real Hilbert space with inner product ) and norm \Vert . When \{x_{n}\} is a sequence in H , we denote the strong convergence of \{x_{n}\} to x\in H by x_{n}\rightarrow x and the weak convergence by x_{n}\rightarrow x . We have from [20] that for any x,y\in H and $\lambda$\in \mathbb{R}, numbers. Let. H. ||x+y||^{2}\leq||x||^{2}+2\langle y,x+y\rangle ,. (2.1). \Vert $\lambda$ x+(1- $\lambda$)y\Vert^{2}= $\lambda$\Vert x||^{2}+(1- $\lambda$)\Vert y\Vert^{2}- $\lambda$(1- $\lambda$)\Vert x-y\Vert^{2} .. (2.2). Furthermore we have that for x, y,u,v\in H,. 2(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}.. (2.3). Let C be a nonempty, closed and convex subset of a Hilbert space H . A mapping T:C\rightarrow H is called nonexpansive if \Vert Tx-Ty\Vert\leq\Vert x-y|| for all x, y\in C. \mathrm{B}T:C\rightarrow H is nonexpansive, then F(T) is closed and convex; see [8, 20]. For a nonempty, closed and convex subset D of H , the nearest point projection of H onto D is denoted by P_{D} , that is, \Vert x-P_{D}x\Vert\leq\Vert x-y\Vert for all x\in H and y\in D . Such a mapping P_{D} is called the metnic projection of H onto D . We know. that the metric projection P_{D} is fimly nonexpansive; | P_{D}x-P_{D}y||^{2}\leq\langle P_{D}x-P_{D}y,x-y\rangle for an x, y\in H . Furthermore, \langle x-P_{D}x , y‐PDx) \leq 0 holds for \mathrm{a}\mathrm{n}_{X}\in H and y\in D ; see [18, 20].. Using this inequality and (2.3), we have that. \Vert P_{D}x-y\Vert^{2}+\Vert P_{D}x-x\Vert^{2}\leq||x-y\Vert^{2}, \forall x\in H, y\in D .. (2.4). Let H be a Hilbert space and let C be a nonempty, closed and convex subset of H . For $\alpha$>0, a mapping A:C\rightarrow H is called $\alpha$‐inverse strongly monotone if. \langle x-y,Ax-Ay\rangle\geq $\alpha$||Ax-Ay\Vert^{2}, \forall x, y\in C. If A is. $\alpha$-\dot{\mathrm{m}verse‐strongly. monotone and 0< $\lambda$\leq 2 $\alpha$ , then. In fact, we have that for all. x,. I- $\lambda$ A. :. C\rightarrow H. is nonexpansive.. y\in C,. \Vert(I- $\lambda$ A)x-(I- $\lambda$ A)y\Vert^{2}=\Vert x-y- $\lambda$(Ax-Ay)||^{2} =\Vert x-y\Vert^{2}-2 $\lambda$\langle x-y,Ax-Ay\rangle+$\lambda$^{2}||Ax-Ay\Vert^{2} \leq\Vert x-y\Vert^{2}-2 $\lambda \alpha$\Vert Ax-Ay\Vert^{2}+$\lambda$^{2}\Vert Ax-Ay||^{2} =||x-y||^{2}+ $\lambda$( $\lambda$-2 $\alpha$)||Ax-Ay||^{2} \leq||x-y\Vert^{2}..

(4) 42. Thus,. I- $\lambda$ A. :. C\rightarrow H. is nonexpansive; see [1, 16, 20] for more results of inverse‐strongly. monotone mappings. The variational inequalty problem for. A. :. C\rightarrow H. is to find a point. u\in C such that. \langle Au, x-\mathrm{u})\geq 0,. \forall x\in C.. (2.5). The set of solutions of (2.5) is denoted by VI(C,A) . We also have that, for any u=P_{C}(I-\mathrm{A}A)u if and only if \mathrm{u}\in VI(C,A) . In fact, let $\lambda$>0 . Then, for u\in C,. $\lambda$ > 0,. u=P_{C}(I- $\lambda$ A)u\Leftrightarrow\langle(I- $\lambda$ A)u-u, u-y)\geq 0, \forall y\in C \Leftrightarrow\langle- $\lambda$ Au, u-y\rangle\geq 0, \forall y\in C \Leftrightarrow(Au,u-y\rangle\leq 0, \forall y\in C \Leftrightarrow\langle Au,y-u\rangle\geq 0, \forall y\in C \Leftrightarrow \mathrm{u}\in VI(C, A) .. In the case when a Banach space E is a Hilbext spacp, the definition of a detmetric mapping is as follows: Let H be a Hilbert space and let C be a nonempty, closed and convex subset of H . Let k\in(-\infty, 1) . A mapping U : C\rightarrow H with F(U)\neq\emptyset i8 called k‐demimetric if, fer any x\in C and q\in F(U) ,. 2\langle x-q , x—Ux )\geq(1-k)\Vert x-Ux\Vert^{2}.. The following lemma which was essentially proved in [23] is important and crucial in the. proof of our main result. For the sake of \inftympleteness, we give the proof.. Lemma 2.1 ([23]). Let H be a Hilbert space and let C be a nonempty, dosed and convex subset of H. Let k be a real number with k\in(-\infty, 1) and let U be a k ‐demimehic mapping of C into H. Then F(U) is dosed and convex. Proof. Let us show that F(U) is closed. For a sequence \{q_{n}\} such that. we have from the definition of U that. q_{n}\rightarrow q. and q_{n}\in F(U) ,. 2\langle q-q_{n}, q-Uq\rangle\geq(1-k)\Vert q-Uq\Vert^{2}. From q_{n}\rightarrow q , we have 0\geq(1-k)||q-Uq||^{2} . From 1-k>0 , we have \Vert q-Uq\Vert^{2}=0 and hence q=Uq . This implies that F(U) is closed. Let us prove that F(U) is convex. Let p,q \in F(U) and set x $\alpha$ p+(1- $\alpha$)q , where $\alpha$\in[0 , 1 ] . Then we have =. 2\Vert x-Ux\Vert^{2}=2\langle x-Ux, x-Ux\rangle =2\langle $\alpha$ p+(1- $\alpha$)q-Ux,x-Ux\rangle =2( $\alpha$ p+(1- $\alpha$)q-( $\alpha$ Ux+(1- $\alpha$)Ux),x-Ux\rangle =2 $\alpha$\langle p-Ux,x-Ux\rangle+2(1- $\alpha$)\langle q-Ux,x-Ux\rangle =2 $\alpha$\langle p-x+x-Ux, x-Ux\rangle+2(1- $\alpha$)\langle q-x+x-Ux,x-Ux\rangle. \leq $\alpha$(k-1)\Vert x-Ux\Vert^{2}+2 $\alpha$\Vert x-Ux\Vert^{2} +(1- $\alpha$)(k-1)\Vert x-Ux\Vert^{2}+2(1- $\alpha$)\Vert x-Ux\Vert^{2} =(k-1)||x-Ux||^{2}+2||x-Ux||^{2} and hence 0 \leq (k-1)\Vert x-Ux||^{2} . We have from x=Ux . This means that \mathrm{F}(U) is convex.. 0 > k-1. that \Vert x-Ux\Vert \leq 0 and hence 口.

(5) 43. The following lemma is used in the proof of our main result.. Lemma 2.2 ([26]). Let H be a Hilbert space and let C be a nonempty, closed and convex subset of H. Let k\in ( -\infty, 1 ) and kt T be a k ‐demimetric mapping of Cinto H such that F(T) is nonempty. Let $\lambda$ be a real number with 0< $\lambda$\leq 1-k and define S=(1- $\lambda$)I+ $\lambda$ T. Then. S. is a guasi‐nonearpansive mapping of C into. Proof. It is obvious that F(T)=F(S) . Since. have that for any x\in C and z\in F(S) ,. T. H.. be a k‐demimetric mapping of C into. H,. we. 2\langle x-z,x-Sx\rangle=2\langle x-z,x-(1- $\lambda$)x- $\lambda$ Tx\rangle=2 $\lambda$\langle x-z,x-Tx\rangle. \displaystyle \geq $\lambda$(1-k)\Vert x-Tx\Vert^{2}=$\lambda$^{2}\frac{1-k}{ $\lambda$}\Vert x-Tx\Vert^{2} =\displaystyle \frac{1-k}{ $\lambda$}\Vert $\lambda$ x- $\lambda$ Tx\Vert^{2}=\frac{1-k}{ $\lambda$}\Vert x-Sx|^{2} \displaystyle \geq\frac{ $\lambda$}{ $\lambda$}\Vert x-Sx|^{2}=\Vert x-Sx\Vert^{2}. Then S is a 0‐demimetric mapping. Furthermore, we have from (2.3) that for any z\in F(S). x\in C. and. ,. \Vert x-Sx||^{2}\leq 2\langle x-z,x-Sx\rangle \Leftrightarrow\Vert x-Sx\Vert^{2}\leq\Vert x-Sx\Vert^{2}+\Vert x-z||^{2}-\Vert Sx-z\Vert^{2} \Leftrightarrow\Vert Sx-z\Vert^{2}\leq\Vert x-z\Vert^{2} \Leftrightarrow\Vert Sx-z\Vert\leq\Vert x-z Therefore, S is quasi‐nonexpansive.. 3. 口. Main Results. In this section, we first prove a weak convergence theorem of Mann’s type iteration for findming a common element of the set of common fixed points for a finite family of demimetric mappings $\varepsilon$\mathrm{A}\mathrm{i}\mathrm{d} the set of coiumon solutions of variatioual inequaJity problems for a fimite faiuily of inverse strongly monotone mappings in a Hilbert space. Let H be a Hilbert space and let C be a nonempty, closed and convex subset of H . A mappin\mathrm{g}U : C\rightarrow H is called demiclosed if, for a sequence \{x_{n}\} in C such that x_{n}\rightarrow w and x_{n}-Ux_{n}\rightarrow 0, w=Uw holds. For example, if C is a nonempty, closed and convex subset of H and T is a nonexpansive mapping of C of H,. then. T. is demiclosed; see [20].. Theorem 3.1 ([13]). Let H be a Hilbert space and let C be a nonempty, dosed and convex subset of H. Let \{k_{1}, \cdots, k_{M}\}\subset(-\infty, 1) and \{$\mu$_{1}, $\mu$_{N}\}\subset(0, \infty) . Let \{T_{j}\}_{j=1}^{M} be a finite family of k_{j} ‐demimetric and demiclosed mappings of C into H and let \{B_{i}\}_{i=1}^{N} be a finite family of $\mu$_{*} ‐inverse strongly monotone mappings of C into H. Assume that 寡. j=1MF(T_{\mathrm{j} )\displaystyle \cap(\bigcap_{i=1}^{N}VI(C, B_{i})) \neq\emptyset..

(6) 44. For any x_{1}=x\in C , define \{x_{n}\} as follows:. \left{bgin{ary}l z_{n}=\sum_{j=1}^M$\xi_{j}(1-$\lambd$_{n})I+$\lambd$_{n}Tj)x_{n},\ w_{n}=\sum_{i=1}^N$\sigma$_{i}PC(I-$\eta_{n}Bi)x_{n},\ x_{n+1}=P_{C($\alph$_{\mathfrk{n}x_ +$\beta_{n}z +$\gam $_{n}w ), \end{ary}\ight.. where \{$\lambda$_{n}\}, \{$\eta$_{n}\}\subset(0,\infty) , \{$\xi$_{1}, \cdots , $\xi$_{M}\}, \{$\sigma$_{1_{i}} \cdots , $\sigma$_{N}\}\subset(0,1) , \{$\alpha$_{n}\}, \{$\beta$_{n}\}, \{$\gamma$_{n}\}\subset(0,1) and b, c\in \mathbb{R} satisfv the following conditions:. a,. (1) 0<a\displaystyle \leq$\lambda$_{n}\leq\min\{1-k_{1}, \cdots, 1-k_{M}\}, 0<b\leq$\eta$_{n}\leq 2\mathrm{m}\dot{\mathrm{m} \{$\mu$_{1}, \cdots : $\mu$_{N}\} ; (2) \displaystyle \sum_{j=1}^{M}$\xi$_{j}=1 and \displaystyle \sum_{i=1}^{N}$\sigma$_{i}=1 ; (S) 0<c\leq$\alpha$_{n},$\beta$_{n},$\gamma$_{n}<1 and $\alpha$_{n}+$\beta$_{n}+$\gamma$_{n}=1. Then the seợuence \{x_{n}\} converges weakly to a point z_{0}\displaystyle \in\bigcap_{j=}^{M}{}_{1}F(T_{\mathrm{j} )\cap(\bigcap_{i=1}^{N}VI(C,B_{i}. where. z_{0}=\displaystyle \lim_{n\rightar ow\infty}P_{\bigcap_{\mathrm{j}=1}^{M}F(T_{\mathrm{j} )\cap(\mathrm{n}_{:=1}^{N}VI(C,B_{:}) ^{X_{n} }.. Next, we prove a strong convergence theorenJ of Halpern’s type iteration for finding a com‐ mon element of the set of common fixed points for a fimite family of demimetric mappings and the \mathrm{s}\mathrm{t}^{1},\mathrm{t} of common solutions of variational inequalIty problemg for a finite family of inverse strongly monotone mappings in a Hilbert space.. Theorem 3.2 ([24]). Let H be a Hilbert space and let C be a nonempty, dosed and convex subset of H. Let \{k_{1}, \cdots, k_{M}\}\subset(-\infty, 1) and \{$\mu$_{1}, \cdots, $\mu$_{N}\}\subset(0, \infty) . Let \{T_{j}\}_{j=1}^{M} be afinite family of k_{j} ‐demimetric and demiclosed mappings of C into H and let \{B_{i}\}_{:=1}^{N} be a finite famaty of $\mu$_{i} ‐inverse strongly monotone mappings of C into H. Assume that 寡. j=1MF(T_{j})\displaystyle \cap(\bigcap_{i=1}^{N}VI(C, B_{i}))\neq\emptyset.. Let \{u_{n}\} be a sequence in C such that generated by. u_{n}\rightarrow u .. For x_{1}=x\in C, let \{x_{n}\}\subset C be a sequence. \left{\begin{ar y}{l z_{n}=\sum_{\ athrm{j}=1^{M}$\xi_{\'u} cdot(1-$\lambda$_{n})I+$\lambda$_{n}T j)x_{n},&\ w_{n}=\sum_{=1}^N$\sigma$_{i}P C(I-$\eta$_{\mathfrk{n}B_{i)x n},&\ x_{n+1}=$\delta$_{n}u +(1-$\delta$_{n})(P_{C}($\alph$_{n}x +$\beta$_{n}z +$\gam $_{n}w &\foraln\i mathrm{N}, \end{ar y}\right.. where \{$\lambda$_{n}\}, \{$\eta$_{n}\}\subset(0, \infty) , \{$\xi$_{1}, \cdots , $\xi$_{M}\}, \{$\sigma$_{1}, \cdots, $\sigma$_{N}\}, \{$\alpha$_{n}\}, \{$\beta$_{n}\}, \{$\gamma$_{n}\}, \{$\delta$_{n}\}\subset(0,1) and b,c\in \mathbb{R} satisfy the following conditions:. a,. (1) 0<a\leq$\lambda$_{n}\leq \mathrm{m}\mathrm{n}\{1-k_{1}, \cdots , 1-k_{M}\}, 0<b\leq$\eta$_{n}\leq 2 min \{$\mu$_{1}, \cdots ; /$\iota$_{N}\} ; (2) \displaystyle \sum_{\mathrm{j}=1}^{M}$\xi$_{j}=1 and \displaystyle \sum_{i=1}^{N}$\sigma$_{l}=1 ; (3) 0<c\leq$\alpha$_{n}, $\beta$_{n},$\gamma$_{n}<1 and $\alpha$_{n}+$\beta$_{n}+$\gamma$_{n}=1 ; (4) \mathrm{h}\mathrm{m}\rightarrow\infty^{$\delta$_{n} =0 and \displaystyle \sum_{i=1}^{\infty}$\delta$_{n}=\infty. Then the sequence \{x_{n}\} converges s\hslash vngly to a point z_{0}\displaystyle \in\bigcap_{j=1}^{M}F(T_{j})\cap(\bigcap_{i=1}^{N}VI(C,B_{i}. 勧. where. =P_{\mathrm{n}_{\mathrm{j}=1}^{M}F(T_{\mathrm{j} )\cap(\text{寡_{}=1}^{\dot{N} VI(C,B:) }u.. Usming the hybrid method by Nakajo and Takahashi [17], we can \mathrm{d}\infty prove a strong conver‐. geuce theorern for finding a cointoon element of the set of conimon fixed points for a finite. family of demimetric mappings and the set of common solutions of variational inequality problemg for a fimite family of mvvpxse strongly monotone mappings in a Hilbert space.. Theorem 3.3 ([2]). Let H be a Hilbert space and let C be a nonempty, dosed and convex subset of H. Let \{k\mathrm{i}, \cdots, k_{M}\}\subset ( -\infty, 1 ) and \{$\mu$_{1}, \cdots , /i_{N}\}\subset(0,\infty) . Let \{T_{j}\}_{j=1}^{M} be a finite.

(7) 45. family of k_{j} ‐demimetrnc and demidosed mappings of C into H and let \{B_{l}\}_{i=1}^{N} be a finite family of $\mu$;‐inverse strongly monotone mappings of C into H. Assume that 寡. \displaystyle \mathrm{j}=1MF(T_{j})\cap(\bigcap_{i=1}^{N}VI(C, B_{i}) \neq\emptyset.. Let x_{1}\in C . Let \{x_{n}\} be a sequence generated by. \left{bginary} z_=\sum{j1^M}$xi_(-\lambd{n})I+$ a_T{jxn},\ w=sum_{i1}^N$\gaP_{C(I-et$n}Bi)x_{,\ ym=$alph_{n}x+\bet$z_{n}gam w,\ C_{n}=zi:Verty-\lqx_{n}z\ Q=iC:langex_{1$\ot}-z, n)geq0\ x_{+1}=PCmathrn\cpQ_{}x1,foraln\imth{N}, edary\ight.. wheoe \{$\lambda$_{n}\}, \{7h\}\subset(0,\infty) , \{$\xi$_{1}, \cdots,$\xi$_{M}\}, \{$\sigma$_{1:}\ldots, $\sigma$_{N}\}\subset(0,1) , {an}, \{$\beta$_{n}\}, \{$\gamma$_{n}\}\subset(0,1) and a,. b, c\in \mathbb{R} satisfy the following conditions:. (1) 0<a\leq$\lambda$_{l1}\leq \mathrm{m}\mathrm{n}\{1-k_{1}, \cdots, 1-k_{M}\}, 0<b\leq$\eta$_{n}\leq 2\mathrm{m}\dot{\mathrm{m}}\{$\mu$_{1}, \cdots $\mu$_{N}\} ; (2) \displaystyle \sum_{j=1}^{M}$\xi$_{j}=1 and \displaystyle \sum_{\dot{\mathrm{a} =1}^{N}$\sigma$_{i}=1 ; (S) 0<c\leq$\alpha$_{\mathfrak{n}},$\beta$_{n}, $\gamma$_{n}<1 and $\alpha$_{n}+$\beta$_{n}+$\gamma$_{n}=1. Then the sequence \{x_{n}\} converges strongly to a point z_{0}\displaystyle \in\bigcap_{\mathrm{j}=1}^{M}F(T_{j})\cap(\bigcap_{=1}^{N}VI(C,B_{i} z_{0}=. 島弘 {}_{1}F(T_{j})\cap(\text{寡_{}=1}^{N}Vi(C,B_{:})) 餌1. where. \cdot. Usming the shrinking projection method [25], we finally prove a strong convergence theorem. for \mathrm{f}\mathrm{i}\mathrm{n}\mathrm{d}_{i\dot{\mathrm{m}\mathrm{g} a common element of the set of common fixed points for a finite family of demimetnic mappings and the set of common solutions of variational inequality problems for a fimite family of inverse strongly monotone mappings in a Hilbert space.. Theorem 3.4 ([26]). Let subset of H. Let. H. be a Hilbert space and let. \{k_{1}, \cdots, k_{M}\}\subset(-\infty, 1). and. C. be a nonempty, closed and convex. \{$\mu$_{1}, . . , $\mu$_{N}\}\subset(0,\infty) .. Let. \{T_{j}\}_{j=1}^{M}. be a finite. family of k_{j} ‐demimeirnc and demiclosed mappings of C into H and let \{B_{i}\}_{i=1}^{N} be a finite family of $\mu$_{\dot{2} ‐inverse strongly monotone mappings of C into H. Assume that 寡. M{}_{j=1}F(T_{j})\displaystyle \cap(\bigcap_{i=1}^{N}VI(C, B_{i}))\neq\emptyset.. Let x_{1}\in C and C_{1}=C . Let \{x_{n}\} be a sequence generated by. \left{bginary}l z_{=\sumj1}^M$\xi_{(1-lambd$_{n})I+\lambd$_{n}Tj)x,\ w_{n}=sum\dot$ia}=1^{N\sigma$_}P{C(I-\eta$_n}B{i)x,\ y_n}=$alph{x_n}+$\beta{z_n}+$\gma{w_n},\ C+1={z\in_}:Verty{n-z\ leqVrtx_{n}-z\ +1=P_{Cn}x1,\foralnimth{N}, \endaryight.. where \{$\lambda$_{n}\}, \{$\eta$_{n}\}\subset(0, \infty) , \{$\xi$_{1}, \cdots , $\xi$_{M}\}, \{$\sigma$_{1_{j}} \cdots, $\sigma$_{N}\}\subset(0,1) , \{$\alpha$_{n}\}, \{$\beta$_{n}\}, \{$\gamma$_{n}\}\subset(0,1) and b,c\in \mathbb{R} satisfy the following conditions:. a,. (1) 0<a\leq$\lambda$_{n}\leq \mathrm{m}\mathrm{n}\{1-k_{1}, \cdots, 1-k_{M}\}, 0<b\leq$\eta$_{n}\leq 2\mathrm{m}\dot{\mathrm{m} \{$\mu$_{1}, \cdots $\mu$_{N}\} ; (2) \displaystyle \sum_{\mathrm{j}=1}^{M}$\xi$_{j}=1 and \displaystyle \sum_{1=1}^{N}$\sigma$_{\dot{*} =1 ; (S) 0<c\leq$\alpha$_{n},$\beta$_{n},$\gamma$_{n}<1 and $\alpha$_{n}+$\beta$_{n}+$\gamma$_{n}=1..

(8) 46. Then the sequence \{x_{n}\} converges strongly to a point z_{0}\displaystyle \in\bigcap_{j=1}^{M}F(T_{j})\cap(\bigcap_{i=1}^{N}VI (C,Bi)) , where 掬. 4. =P_{\bigcap_{g=1}^{M}F(T_{\dot{f} )\cap(\bigcap_{=1}^{N}VI(C,B_{l}) ^{X_{1} }.. Applicatíonss. Jn this section, we apply Theorems 3.1, 3.2, 3.3 and 3.4 to obtain well‐known and new strong convergence theorems in Hilbert spaces. We know the following lemmas obtained by Marino. and Xu [15] and Kocourek, Takahashi and Yao [10]; see also [27, 28]. Lemma 4.1 ([15, 27. Let. H. be a Hilbert space and let. C. be a nonempty, closed and convex. subset of H. Let k be a real number with 0\leq k< 1 and U : contaction. If x_{n}\rightarrow z and x_{n}-Ux_{n}\rightarrow 0 , then z\in F(U) .. Lemma 4.2 ([10, 28. subset of H and let z\in F(U). U. :. Let. H. C\rightarrow H. be a Hibert space, let. C. C\rightarrow H. be a. k ‐strict. pseudo‐. be a nonempty, closed and convex. be generalized hybrid. If x_{n}\rightarrow z and x_{n}-Ux_{n}\rightarrow 0 , then. .. Using Theorem 3.1, we obtain the following weak convergence results. Theorem 4.3. Let H be a Hilbert space and let C be a nonempty, closed and convex subset of H. Let \{$\mu$_{1}, . . ; $\mu$_{N}\}\subset(0,\infty) . Let \{B_{i}\}_{\dot{*}=1}^{N} be a finite family of $\mu$_{i} ‐inverse strongly monotone mappings of C into H. Assume that \mathrm{n}_{:=1}^{N}VI(C,B_{i})\neq\emptyset . For any x_{1}=x\in C , define \{x_{n}\} as follows:. \left\{ begin{ar y}{l w_{n}=\sum_{i=1}^{N}$\sigma$_{i}P_{C}(I-$\eta$_{n}B_{i})x_{n},\ x_{n+1}=$\alpha$_{n}x_{n}+$\gam a$_{n}w_{n}, \end{ar y}\right. wheoe \{$\eta$_{n}\}. \subset. (0,\infty) , \{$\sigma$_{1}, \cdots , $\sigma$_{N}\}. following conditions:. \subset. (0,1) , {an}, \{$\gamma$_{n}\}. \subset. (0,1) and b,c. \in \mathbb{R}. satisfy the. (1) 0<b\leq$\eta$_{n}\leq 2\mathrm{m}\dot{\mathrm{m} \{$\mu$_{1}, \cdots , $\mu$_{N}\} ; (2) \displaystyle \sum_{i=1}^{N}$\sigma$_{i}=1 ; (S) 0<c\leq$\alpha$_{n},$\gamma$_{n}<1 and $\alpha$_{n}+$\gamma$_{n}=1. Then \{x_{n}\} converges weakly to z_{0}\displaystyle \in\bigcap_{i=1}^{N}VI(C,B_{l}) , wheoe. zo=\displaystyle \lim_{n\rightarrow\infty}P_{\bigcap_{:=1}^{N}VI(C,B_{:})}x_{n}.. Proof. The identity mapping is a \displayte\frac{1}2 ‐demimetric mapping of C into H . Putting T_{j}=I for all I. j\in\{1, \cdots, M\}. and $\lambda$_{n}=\displaystyle \frac{1}{2} for all n\in \mathrm{N} in Theorem 3.1, we have that z_{n}=x_{n} for all Furthermore, replacing $\beta$_{n}+$\gamma$_{n} by $\gamma$_{n} , we have the desired result hom Th\infty \mathrm{r}\mathrm{e}\mathrm{m}3.1 .. n\in \mathrm{N}.. ロ. Theorem 4.4. Let H be a Hilbert space and let C be a nonempty, closed and convex subset of H. Let \{T_{j}\}_{j=1}^{M} be a finite family of,qeneralized hybrid mappings of C into be a finite family of nonexpansive mappings of C into H. Assume that. 寡HIF(乃) \cap ( \cap:N 1F(の) =. \neq\emptyset .. For any x_{1}=x\in C , define \{x_{n}\} as follows:. \left{begin{ary}l z_{n}=\sum_{j=1}^M$\xi_{j}(1-$\lambd$_{n})I+$\lambd$_{n}Tj)x_{n},\ w_{n}=\sum_{i=1}^N$\sigma$_{i}PC(1-$\eta_{n})I+$\eta_{n}U:)x_{n},\ x_{n+1}=P_{C($\alph$_{n}x +$\beta_{n}z +$\gam $_{n}w ), \end{ary}\ight.. H. and let \{U_{i}\}_{i=1}^{N}.

(9) 47. where \{$\lambda$_{n}\}, \{ln\}\subset(0, \infty) , \{$\xi$_{1}, \cdots , $\xi$_{M}\}, \{$\sigma$_{1}, \cdots , $\sigma$_{N}\}\subset(0,1) , {ưn}, \{$\beta$_{\mathfrak{n} \}, \{$\gamma$_{n}\}\subset(0,1) and a,. b, c\in \mathbb{R} satisfy the following conditions:. (1) 0<a\leq$\lambda$_{n}\leq 1, 0<b\displaystyle \leq l\int n\leq 1 ; (2) \displaystyle \sum_{j=1}^{M}$\xi$_{j}=1 and \displaystyle \sum_{*=1}^{N}$\sigma$_{i}=1 ; (3) 0<c\leq$\alpha$_{n}, $\beta$_{n},$\gamma$_{n}<1 and $\alpha$_{n}+$\beta$_{n}+$\gamma$_{n}=1.. Then the sequence \{x_{n}\} converges weakly to a point .. z_{0} \in. z=\displaystyle \lim_{n}P.\cdot. \displaystyle \bigcap_{j=1}^{M}F(T_{j})\cap(\bigcap_{\dot{ $\iota$}=1}^{N}F(U_{i}) , where. Proof. Since T_{j} is generalized hybrid, T_{j} is 0-\mathrm{d}\mathrm{e}\mathrm{m}\dot{\mathrm{m} etric. RMhemore, aom Lemma 4.2 T_{j} is demiclosed. Since U_{i} is nonexpansive, B_{i}=I-U_{i} is a \mathrm{Z}1 ‐inverse strongly monotone mapping.. We also have. \mathrm{b}\mathrm{o}\mathrm{m}\cap^{\dot{N} {}_{=\mathrm{L} F(U_{i})\neq\emptyset 寡. that. i=1NVI(C, I-U_{i}) =\displaystyle \bigcap_{i=1}^{N}F(P_{C}U_{i})= 寡 N{}_{i=1}F(U_{i}) . \square. Therefore, we have the desired result from Theorem 3.1.. Using Theorem 3.2, we can prove a strong convergence theorem for a finite family of strict pseudo contractions in a Hilbert space. \cdot. Theorem 4.5. Let H be a Hilbert space and let C be a nonempty, closed and convex subset of H. Let \{k_{1}, . . ., k_{M}\}\subset[0 , 1) and let \{T_{\mathrm{j} \}_{j=1}^{M} be a finite family of k_{\mathrm{j} ‐strict pseudo‐contractions. of C into H. Let {uly,} be a sequence in. C. such that u_{n}\rightarrow u . Assume that. For any x_{1}=x\in C , define \{x_{n}\} as follows:. \displaystyle \bigcap_{\mathrm{j}=}^{M}{}_{1}F(T_{j})\neq\emptyset.. \left\{ begin{ar y}{l z_{n}=\sum_{j=1}^{M}$\xi$_{j}(1-$\lambda$_{n})I+$\lambda$_{n}T_{\mathrm{j})x_{n},\ x_{n+1}=$\delta$_{n}u_{$\eta$}+(1-$\delta$_{n})(P_{C}($\alpha$_{n}x_{n}+$\beta$_{n}z_{n} \end{ar y}\right. where a, c\in \mathbb{R}, \{$\lambda$_{n}\}\subset(0, \infty) , \{$\xi$_{1}, \cdots , $\xi$_{M}\}\subset(0,1) and \{$\alpha$_{n}\}, \{$\beta$_{n}\}, \{$\beta$_{n}\}\subset(0,1) satisfy the following conditions:. (1) 0<a\displaystyle \leq$\lambda$_{n}\leq\min\{1-k_{1}, \cdots, 1-k_{M}\} ; (2) \displaystyle \sum_{j=1}^{M}$\xi$_{j}=1 ; (S) 0<c\leq l_{n}, $\beta$_{n}<1 and u_{n}+$\beta$_{n}=1 ; (4) 1\dot{\mathrm{u} u\rightarrow\infty^{$\delta$_{n} =0 and \displaystyle \sum_{i=1}^{\infty}$\delta$_{n}=\infty. Then \{x_{n}\} converges strongly to. z_{0}\displaystyle \in\bigcap_{j=1}^{M}F(T_{j}) ,. where. z_{0}=P_{\mathrm{n}_{\mathrm{j}=1}^{M}F(T_{j})}u.. Proof. Since T_{\mathrm{j} is a k_{j} ‐strict pseud‐contraction of into H such that F(T_{j})\neq\emptyset, T_{\mathrm{j} is k_{j^{-}} demimetric. Furthermore, \mathrm{h}\mathrm{o}\mathrm{m} Lemma 4.1, T_{j} is demiclosed. RMhermore, if B_{i}=0 for all i\in\{1, \cdots, N\} in Theorem 3.2, then B_{i} is a 1-\dot{\mathrm{m}verse strongly monotone mapping. Putting 1 for all n\in \mathrm{N} in Theorem 3.2, we have that w_{n} =x_{n} for \mathrm{a}\mathrm{U}n\cdot\in N. Furthermore, $\eta$_{n} replaceing $\beta$_{n}+$\gamma$_{n} by $\beta$_{n} . we have the desired result \mathrm{f}_{\mathrm{r} \mathrm{o}\mathrm{m} Theorem 3.2. ロ C. =. Using Theorem 3.3, we prove a strong convergence theorem for a finite family of minverse strongly monotone mappings in a Hilbert space. Theorem 4.6. Let H be a Hilbert space and let C be a nonempty, closed and convex subset of H. Let \{$\mu$_{1}, \cdots : $\mu$_{N}\}\subset(0, \infty) . Let \{B_{i}\}_{i=1}^{N} be a finite family of $\mu$_{\dot{$\tau$} ‐inverse strongly monotone mappings of C into H. Assume $\theta$_{1}at\displaystyle \bigcap_{\dot{*}=1}^{N}VI(C,B_{i})\neq\emptyset . Let x_{1}\in C . Let \{x_{n}\} be a sequence.

(10) 48. generated by. where b,c \in \mathbb{R}, \{$\eta$_{n}\} follounng conditions:. \subset. \left{bginary}{l w_n=\sum{i1}^N_$\sigma}P_{C(IB)xn},\ y_{=$alph_{n}x+$\gam _{n}w,\ C_{n}=z\i:Verty_{n}-z\ leqVrtx_{n}-z\ Q={z\inC:lagex_{n}-z,1 \rangleq0\}, x_{n+1=PC}\capQ_{nx1},\foralnimthr{N}, \endary}ight. (0, \infty) , \{$\sigma$_{1}, \cdots , $\sigma$_{N}\}. \subset. (0,1) and \{$\alpha$_{n}\}, \{$\gamma$_{n}\}. \subset. (0,1) satisfy the. (1) 0<b\leq$\eta$_{n}\leq 2 min \{$\mu$_{1}, \cdots , $\mu$_{N}\} ; (2) \displaystyle \sum_{i=1}^{N}$\sigma$_{\dot{*} =1 ; (S) 0<c\leq$\alpha$_{n},$\gamma$_{n}<1 and $\alpha$_{n}+$\gamma$_{n}=1. Then \{x_{n}\} converges strongly to. Proof. The identity mapping. I. z_{0}\displaystyle \in\bigcap_{\dot{*}=1}^{N}VI(C,B_{:}) , where z_{0}=P_{\mathrm{n}_{:=1}^{N}VI(C,B_{:})}x_{1}.. is a \displayt e\frac{1}2 ‐demmetnc mapping of. all j\in \{1, \cdots, M\} and $\lambda$_{n}= \displayt e\frac{1}2 for n\in \mathrm{N} . Furthermore, replacee $\beta$_{n}+$\gamma$_{r $\iota$} by. C. into. H.. Putting Tj=I for. in we have that z_{n}=x_{n} for all $\gamma$_{n} . Thus, we have the desired result from Theorem. \mathrm{a}\mathrm{U}n\in \mathrm{N}. \mathrm{T}\mathrm{h}\infty \mathrm{r}\mathrm{e}\mathrm{m}3.3 ,. 3.3.. 口. Usirig Theorem 3.4, we prove a strong convergence theorem for a finite family of generaliAed hybnid mappings and a finite family of inverse strongly monotone mappings in a Hilbert space.. Theorem 4.7. Let H be a Hilbert space and let C be a nonempby, closed and convex subset of H. Let \{$\mu$_{1}, \cdots, $\mu$_{N}\}\subset(0,\infty) . Let \{T_{j}\}_{j=1}^{M} be a finite family of generalized hybrid mappings of C into H and let \{B_{i}\}_{i=1}^{N} be a finite family of $\mu$_{i} ‐inverse strongly monotone mappings of C into H. Assume that. \displaystyle \bigcap_{j=1}^{M}F(T_{j})\cap(\bigcap_{i=1}^{N}VI(C,B_{i}) \neq\emptyset. Let x_{1}\in C and C_{1}=C . Let \{x_{n}\} be a sequence generated by. \left{bginary}l z_{=\sum athr{j}=1^M$\xi_{j}(1-lambd$_{n})I+\lambd$_{n}Tj)x,\ w_{n}=sum1^{\dotN}$sigma_{PC}(IB)xn,\ y_{}=$alphnx_{}+$\betanz_{}+$\gam_{n}w,\ C_{n+1}=z\iC_{n:|y}-z\leqx_{n +1}=P_{Cnx1},\foralnimth{N}, \endaryight.. wheoe \{$\lambda$_{n}\}, \{$\eta$_{n}\}\subset(0,\infty) , \{$\xi$_{1}, \cdots, $\xi$_{M}\}, \{$\sigma$_{1_{i}} \cdots, $\sigma$_{N}\}\subset(0,1) , \{$\alpha$_{n}\}, \{$\beta$_{n}\}, \{$\gamma$_{n}\}\subset(0,1) and b, c\in \mathbb{R} satisfy the following conditions:. a,. (1) 0<a\leq$\lambda$_{n}\leq 1, 0<b\leq$\eta$_{n}\leq 2\mathrm{m}\mathrm{n}\{$\mu$_{1}, \cdots , $\mu$_{N}\} ; (2) \displaystyle \sum_{j=1}^{M}$\xi$_{j}=1 and \displaystyle \sum_{i=1}^{N} $\sigma$:=1 ; (3) 0<c\leq$\alpha$_{n},$\beta$_{n},$\gamma$_{n}<1 and a_{n}+$\beta$_{n}+$\gamma$_{n}=1. Then the sequence \{x_{n}\} converges strongly to a point z_{0}\displaystyle \in\bigcap_{j=1}^{M}F(T_{j})\cap(\bigcap_{\dot{*}=1}^{N}VI(C,B_{1}) , where. z_{0\text{寡_{}=1}^{N}VI(C,B_{:}) ^{X_{1} }=P_{\mathrm{n}_{j=1}^{M}F(T_{j})\cap(\prime}.. Proof. Since Tj is a generalized hybrid mapping of C into H such that F(T_{j})\neq\emptyset, \mathrm{h}\mathrm{o}\mathrm{m}(1.2) , Tj is 0‐‐demimetric. Bmhemore, from Lemma 4.2, Tj is demiclosed. Therefore, we have the. desired result from Theorem 3.4.. 口.

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