非扩张映象和广义变分不等式的迭代收敛性
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摘要
引入更为一般的非扩张显式粘滞迭代算法,利用此迭代算法在Hilbert空间中建立了非扩张映象的公共不动点集与具有强单调映象的变分不等式解集的公共元素的强收敛定理,推广和改进了相关结果.
We introduce a general viscosity iterative sequence for nonexpansive mapping and establish the strong convergence theorems of viscosity approximation method for iterative sequence to find a common element of the set of common fixed points for nonexpansive mapping and the set of solutions of generalized variational inequalities with a strongly monotone mapping in Hilbert spaces. The corresponding results in some references were extended and improved.
We introduce a general viscosity iterative sequence for nonexpansive mapping and establish the strong convergence theorems of viscosity approximation method for iterative sequence to find a common element of the set of common fixed points for nonexpansive mapping and the set of solutions of generalized variational inequalities with a strongly monotone mapping in Hilbert spaces. The corresponding results in some references were extended and improved.
引文
[1]Noor M A.General variational inequalities[J].Applied Mathematics Letters,1988(1):119-121.
[2]Stampacchia G.Formes bilineaires coercivites surles ensembles convexes[J].Comptes Rendusde Academiedes Sciences,1964,258:4413-4416.
[3]徐永春,何欣枫,侯志彬,何震.Banach空间中Noor型变分不等式的广义投影法[J].数学物理学报,2010,30A(3):808-817.
[4]Moudafi A.Viscosity approximation methods for fixed-points problems[J].J Math Anal Appl,2000,241(1):46-55.
[5]Liu L S.Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces[J].J Math Anal Appl,1995,194(1):114-125.
[6]张丽娟,陈俊敏,侯志彬.非扩张映射和广义变分不等式的粘滞逼近法[J].数学学报,2010,53(4):691-698.
[7]Takahashi W,Toyoda M.Weak convergence theorems for nonexpansive mappings and monotone mappings[J].J Optim Theory Appl,2003,118:417-428.
[8]Chen J M,Zhang L J,Fan T G.Viscosity approximation methods for nonexpansive mappings and monotone mappings[J].J Math Anal Appl,2007,334:1450-1461.
[9]Qin X L,Kang S M,Shang M S.Strong convergence theorems of strict pseudo-contractions in Hilbert spaces[J].Czechoslovak Math J,2009,134(59):695-706.
[10]高雷阜,刘杰,高鼎.非扩张映射和广义变分不等式迭代算法[J].辽宁工程技术大学学报(自然科学版),2014,33(5):704-707.
[11]Zhang S Y,Song X G.Another note on a paper“Convergence theorem for the common solution for a finite family ofφ-strongly accretive operator equations”[J].Applied Math Comput,2015,258(1):367-371.
[12]Zhang S Y.Implicit iteration approximation for a finite family of asymptotically quasi-pseudocontractive type mappings[J].Bull Iranian Math Soc,2014,40(1):263-279.
[13]张树义.一致Lipschitz渐近φi-型拟伪压缩映象多步平行迭代算法的收敛性[J].系统科学与数学,2013,33(11):1233-1242.
[14]张树义.赋范线性空间中渐近拟伪压缩型映象不动点的修改的广义Ishikawa迭代逼近[J].应用数学学报,2011,34(5):886-894.
[15]张树义,宋晓光.非Lipschitz有限族集值广义渐近φ-半压缩映象的强收敛定理[J].系统科学与数学,2014,34(9):1051-1058.
[16]张树义,赵美娜,李丹.渐近半压缩映象具混合型误差的迭代收敛性[J].北华大学学报(自然科学版),2015,16(3):165-169.
[17]赵美娜,张树义,赵亚莉.有限族广义一致伪Lipschitz映象公共不动点的迭代收敛性[J].烟台大学学报(自然科学与工程版),2017,30(1):7-10.
[18]张树义,李丹,丛培根.增生算子零点的迭代逼近[J].北华大学学报(自然科学版),2017,18(2):1-7.
[19]赵美娜,张树义,郑晓迪.一类算子方程迭代序列的稳定性[J].轻工学报,2016,31(6):100-108.
[20]李丹,张树义,赵美娜.Φ-伪压缩映象迭代序列的收敛性与稳定性[J].烟台大学学报(自然科学与工程版),2017,30(2):79-88.
[21]张树义,李丹,林媛,丛培根.非自渐近非扩张型映象具误差的Reich-Takahashi粘滞迭代逼近[J].北华大学学报(自然科学版).2017,18(3):287-293.
[22]张树义,林媛,郑晓迪.强增生映像零点的迭代逼近[J].浙江师范大学学报(自然科学版),2017,40(2):127-129.
[23]赵美娜,张树义,赵亚莉.Banach空间中k-次增生算子方程解的迭代逼近[J].北华大学学报(自然科学版),2015,16(6):710-714.
[24]林媛,张树义,李丹.Banach空间中渐近非扩张型映象Reich-Takahash迭代序列的收敛性[J].烟台大学学报(自然科学与工程版),2017,30(3):185-190.
[25]万美玲,张树义,郑晓迪.赋范线性空间中φ-强增生算子方程解的迭代收敛性[J].北华大学学报(自然科学版),2016,17(3),305-307.
[26]李丹,张树义,丛培根.φ-强增生算子方程解的Noor三步迭代收敛率的估计[J].鲁东大学学报(自然科学版),2017,33(3):193-199.
[2]Stampacchia G.Formes bilineaires coercivites surles ensembles convexes[J].Comptes Rendusde Academiedes Sciences,1964,258:4413-4416.
[3]徐永春,何欣枫,侯志彬,何震.Banach空间中Noor型变分不等式的广义投影法[J].数学物理学报,2010,30A(3):808-817.
[4]Moudafi A.Viscosity approximation methods for fixed-points problems[J].J Math Anal Appl,2000,241(1):46-55.
[5]Liu L S.Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces[J].J Math Anal Appl,1995,194(1):114-125.
[6]张丽娟,陈俊敏,侯志彬.非扩张映射和广义变分不等式的粘滞逼近法[J].数学学报,2010,53(4):691-698.
[7]Takahashi W,Toyoda M.Weak convergence theorems for nonexpansive mappings and monotone mappings[J].J Optim Theory Appl,2003,118:417-428.
[8]Chen J M,Zhang L J,Fan T G.Viscosity approximation methods for nonexpansive mappings and monotone mappings[J].J Math Anal Appl,2007,334:1450-1461.
[9]Qin X L,Kang S M,Shang M S.Strong convergence theorems of strict pseudo-contractions in Hilbert spaces[J].Czechoslovak Math J,2009,134(59):695-706.
[10]高雷阜,刘杰,高鼎.非扩张映射和广义变分不等式迭代算法[J].辽宁工程技术大学学报(自然科学版),2014,33(5):704-707.
[11]Zhang S Y,Song X G.Another note on a paper“Convergence theorem for the common solution for a finite family ofφ-strongly accretive operator equations”[J].Applied Math Comput,2015,258(1):367-371.
[12]Zhang S Y.Implicit iteration approximation for a finite family of asymptotically quasi-pseudocontractive type mappings[J].Bull Iranian Math Soc,2014,40(1):263-279.
[13]张树义.一致Lipschitz渐近φi-型拟伪压缩映象多步平行迭代算法的收敛性[J].系统科学与数学,2013,33(11):1233-1242.
[14]张树义.赋范线性空间中渐近拟伪压缩型映象不动点的修改的广义Ishikawa迭代逼近[J].应用数学学报,2011,34(5):886-894.
[15]张树义,宋晓光.非Lipschitz有限族集值广义渐近φ-半压缩映象的强收敛定理[J].系统科学与数学,2014,34(9):1051-1058.
[16]张树义,赵美娜,李丹.渐近半压缩映象具混合型误差的迭代收敛性[J].北华大学学报(自然科学版),2015,16(3):165-169.
[17]赵美娜,张树义,赵亚莉.有限族广义一致伪Lipschitz映象公共不动点的迭代收敛性[J].烟台大学学报(自然科学与工程版),2017,30(1):7-10.
[18]张树义,李丹,丛培根.增生算子零点的迭代逼近[J].北华大学学报(自然科学版),2017,18(2):1-7.
[19]赵美娜,张树义,郑晓迪.一类算子方程迭代序列的稳定性[J].轻工学报,2016,31(6):100-108.
[20]李丹,张树义,赵美娜.Φ-伪压缩映象迭代序列的收敛性与稳定性[J].烟台大学学报(自然科学与工程版),2017,30(2):79-88.
[21]张树义,李丹,林媛,丛培根.非自渐近非扩张型映象具误差的Reich-Takahashi粘滞迭代逼近[J].北华大学学报(自然科学版).2017,18(3):287-293.
[22]张树义,林媛,郑晓迪.强增生映像零点的迭代逼近[J].浙江师范大学学报(自然科学版),2017,40(2):127-129.
[23]赵美娜,张树义,赵亚莉.Banach空间中k-次增生算子方程解的迭代逼近[J].北华大学学报(自然科学版),2015,16(6):710-714.
[24]林媛,张树义,李丹.Banach空间中渐近非扩张型映象Reich-Takahash迭代序列的收敛性[J].烟台大学学报(自然科学与工程版),2017,30(3):185-190.
[25]万美玲,张树义,郑晓迪.赋范线性空间中φ-强增生算子方程解的迭代收敛性[J].北华大学学报(自然科学版),2016,17(3),305-307.
[26]李丹,张树义,丛培根.φ-强增生算子方程解的Noor三步迭代收敛率的估计[J].鲁东大学学报(自然科学版),2017,33(3):193-199.