Banach空间中含m-增生算子方程的迭代问题
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摘要
本文的主要目的是研究形如z∈Sx+λAx的非线性算子方程的近似解问题。其中x∈D(A)，z∈X，λ＞0，A：D(A)X→2~X为m-增生算子，S：X→为连续的α-强增生算子。我们采用的是众所周知的Mann迭代方案和Ishikawa迭代方案。我们首先给出在一致光滑的Banach空间中满足某些条件下采用Mann迭代方案来逼近算子方程的唯一解的一些收敛结果，然后将Mann迭代方案推广到Ishikawa迭代方案情况。之后我们又给出了在一般的Banach空间中带误差的Ishikawa迭代方案和Mann迭代方案的相应的收敛结果。另外，作为应用，对给定的z∈X，我们研究了在S是连续的强增生算子的条件下算子方程的解序列{x_λ}(λ＞0)的收敛性。最后我们给出了预解算子J_λz当λ→0~+时的收敛性。
In this paPer，The main purPose 150 stuhe eovergenees of
well known iterative Proeesses，sueh as tlie Mann and the Ishikawa
sequenees，to appproximatehe:olutions of nonlinear funetional equa
tions ofeypeSx+Ax，D(A)，e，>O，where
A:DX、ZX 15 anmaccretive operaor ands:X*X
15 a eoinuous and。strongly aceretive one，fir，eablish some
eovergenee resul5 usinghe Mann proeess to approximae the unique
solution of equaion under some boundedness eondiions for uniformly
smooh Banaeh sPaees.Then we Prove thathese results are also true
underhe Ishikawa Proeess.also Provehathe Ishikawa and Mann
ierative Proee:ses with error:are alsorue with the Problem.n ad-
dition，for a given 2 inX，we inveigate the eonvergenee of the Pah
deseribed byhe solutions ofeion forhe case thats
15 eoinuous and strongly aeeretivethen esablish the eonvergenee
ofhe resolventas
In this paPer，The main purPose 150 stuhe eovergenees of
well known iterative Proeesses，sueh as tlie Mann and the Ishikawa
sequenees，to appproximatehe:olutions of nonlinear funetional equa
tions ofeypeSx+Ax，D(A)，e，>O，where
A:DX、ZX 15 anmaccretive operaor ands:X*X
15 a eoinuous and。strongly aceretive one，fir，eablish some
eovergenee resul5 usinghe Mann proeess to approximae the unique
solution of equaion under some boundedness eondiions for uniformly
smooh Banaeh sPaees.Then we Prove thathese results are also true
underhe Ishikawa Proeess.also Provehathe Ishikawa and Mann
ierative Proee:ses with error:are alsorue with the Problem.n ad-
dition，for a given 2 inX，we inveigate the eonvergenee of the Pah
deseribed byhe solutions ofeion forhe case thats
15 eoinuous and strongly aeeretivethen esablish the eonvergenee
ofhe resolventas
引文
[1] V.Barbu,Nonlinear semigroups and differential equations in Banach Spaces,Noordhoff, Leyden,1976.
[2] F.E.Browder,Nonlinear mappings of nonexpansive and accretive type in Banach Spaces, Bull.Amer.Math. Soc. 73 (1967) 875-882.
[3] M. M. Day, Normed linear spaces,3rd ed.,Springer, Berlin, 1973.
[4] K.S.Ha,J.S. Jung,Strong convergence theorems for accretive operators in Banach Spaces, J.Math.Anal.appl.147(1990)330-339.
[5] T.Kato,Nonlinear semigroups and evolution equations,J.Math.Soc. Japan 19 (1967)508-520.
[6] Y.Kobayashi,Difference approximation of Cauchy problems for quasidisspative operators and generation of nonlinear semigroups,J.Math. Soc.Japan 27 (1975)640-665.
[7] R.H.Martin Jr,A global existence theorem for antonomous differential equations in Banach Spaces,proc.Amer.Math. Soc. 26 (1970) 307-314.
[8] C.Morales,Surjectivity theorems for multi-valued mappings of accretive type,Comm,Math. Univ.Carolin.26(1985)397-413.
[9] S. Reich,Strong convergence theorems for resolvents of acrretive operators in Banach Spaces,J.Math.Anal.Appl.75(1980)287-292.
[10] X.L.Weng,Fixed point iteration for locally strictly pseudo-contractive mapping, Proc.Amer.Math.Soc. 113 (1991) 727-731.
[11] J.S. Jung, C.H. Morales,The Mann process for perturbed m-accretive operators in Banach Spaces,Nonl.Anal. 46(2001) 231-243.
[12] 张石生，φ-伪压缩映像的具误差的IShikawa和Mann迭代程序的收敛性问题，J．应用数学和力学，21(2000)1-10．
[13] S. Reich,Extension problems for accretive sets in Banach Spaces, J. Func.Anal. 26(1997)378-395.
[14] S.Reich,Product formulas nonlinear semigroups and accretive operators,J.Func.Anal. 36(1980)147-168.
[15] S. S. Chang,Y. J. cho, B. S. Jung, S. M. K ang, Iterative appoximations of fixed points and solutions for strongly accretive and strongly pseudocontractive mappings in Banach Spaces, J.Math. Anal.Appl. 224(1998) 149-165.
[16] L.Q.H,Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, J. Math.Anal. 259(2001) 18-24.
[17] Xu,Noor,Fixed point iterations for asymptotically nonexpansive mappings in Banach Spaces,J.Math.Anal.Appl.267(2002)444-453.
[18] C. E. Chidume, C. Moore,Fixed point iteration for pseudo-contractive mappings,Proc. Amer. Math. Soc. 127(1999) 1163-1170.
[19] Q.Liu,The convergence theorems of the sequence of Ishikawa iteration method for hemicontractive mappings,J.Math.Anal.Appl. 148 (1990)55-62.
[20] J. Schu,Approximating fixed points of Lipschitzian pseudo-contractive mappings, J.Math.Anal. 19(1993) 107-115.
[21] J.S.J,Y.J.C,H.Y.Z,Iterative methods with mixed errors for per-
turbed m-accretive operator equations in arbitrary Banach Spaces, J. Math. Comp. Mode. 85 (2002) 55-62.
[22] S.B. Nadler, Jr.,Multi-valued contraction mappings, Pacific J.Math. 30 (1969) 475-488.
[23] Lin.L.S,Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach Spaces,J.Math.Anal. Appl. 194(1995) 114-125.
[2] F.E.Browder,Nonlinear mappings of nonexpansive and accretive type in Banach Spaces, Bull.Amer.Math. Soc. 73 (1967) 875-882.
[3] M. M. Day, Normed linear spaces,3rd ed.,Springer, Berlin, 1973.
[4] K.S.Ha,J.S. Jung,Strong convergence theorems for accretive operators in Banach Spaces, J.Math.Anal.appl.147(1990)330-339.
[5] T.Kato,Nonlinear semigroups and evolution equations,J.Math.Soc. Japan 19 (1967)508-520.
[6] Y.Kobayashi,Difference approximation of Cauchy problems for quasidisspative operators and generation of nonlinear semigroups,J.Math. Soc.Japan 27 (1975)640-665.
[7] R.H.Martin Jr,A global existence theorem for antonomous differential equations in Banach Spaces,proc.Amer.Math. Soc. 26 (1970) 307-314.
[8] C.Morales,Surjectivity theorems for multi-valued mappings of accretive type,Comm,Math. Univ.Carolin.26(1985)397-413.
[9] S. Reich,Strong convergence theorems for resolvents of acrretive operators in Banach Spaces,J.Math.Anal.Appl.75(1980)287-292.
[10] X.L.Weng,Fixed point iteration for locally strictly pseudo-contractive mapping, Proc.Amer.Math.Soc. 113 (1991) 727-731.
[11] J.S. Jung, C.H. Morales,The Mann process for perturbed m-accretive operators in Banach Spaces,Nonl.Anal. 46(2001) 231-243.
[12] 张石生，φ-伪压缩映像的具误差的IShikawa和Mann迭代程序的收敛性问题，J．应用数学和力学，21(2000)1-10．
[13] S. Reich,Extension problems for accretive sets in Banach Spaces, J. Func.Anal. 26(1997)378-395.
[14] S.Reich,Product formulas nonlinear semigroups and accretive operators,J.Func.Anal. 36(1980)147-168.
[15] S. S. Chang,Y. J. cho, B. S. Jung, S. M. K ang, Iterative appoximations of fixed points and solutions for strongly accretive and strongly pseudocontractive mappings in Banach Spaces, J.Math. Anal.Appl. 224(1998) 149-165.
[16] L.Q.H,Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, J. Math.Anal. 259(2001) 18-24.
[17] Xu,Noor,Fixed point iterations for asymptotically nonexpansive mappings in Banach Spaces,J.Math.Anal.Appl.267(2002)444-453.
[18] C. E. Chidume, C. Moore,Fixed point iteration for pseudo-contractive mappings,Proc. Amer. Math. Soc. 127(1999) 1163-1170.
[19] Q.Liu,The convergence theorems of the sequence of Ishikawa iteration method for hemicontractive mappings,J.Math.Anal.Appl. 148 (1990)55-62.
[20] J. Schu,Approximating fixed points of Lipschitzian pseudo-contractive mappings, J.Math.Anal. 19(1993) 107-115.
[21] J.S.J,Y.J.C,H.Y.Z,Iterative methods with mixed errors for per-
turbed m-accretive operator equations in arbitrary Banach Spaces, J. Math. Comp. Mode. 85 (2002) 55-62.
[22] S.B. Nadler, Jr.,Multi-valued contraction mappings, Pacific J.Math. 30 (1969) 475-488.
[23] Lin.L.S,Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach Spaces,J.Math.Anal. Appl. 194(1995) 114-125.