System of variational inclusions with ${(\emph{\textbf{H}},{\bm\phi})}$-$\bm\eta$-monotone operators in Banach spaces

Abstract

Abstract: This paper introduced a new system of variational inclusions with $(H,\phi)$-$\eta$-monotone operators in real uniformly smooth Banach spaces. By using the proximal mapping technique associated with $(H,\phi)$-$\eta$-monotone operators, we proved the existence and uniqueness of solution for this new system and construct a new iterative algorithm for approximating the solution of this system. The convergence of the iterative sequence generated by the iterative algorithm was also discussed. The results extend and improve some known results in the literature.

Key words: $(H,\phi)$-$\eta$-monotone operator, proximal mapping, system of variational inclusions, iterative algorithm, convergence

CLC Number:

ZHANG Chao. System of variational inclusions with ${(\emph{\textbf{H}},{\bm\phi})}$-$\bm\eta$-monotone operators in Banach spaces[J]. Journal of East China Normal University(Natural Sc, 2012, 2012(1): 74-83.

References

{1}

ZENG L C, GUU S M, YAO J C. Characterization of $H$-monotone

operators with applications to variational inclusions[J]. Comput

Math Appl, 2005, 50: 329-337.
{2}

LAN H Y, KIM J H, CHO Y J. On a new system of nonlinear $A$-monotone

multi-valued variational inclusions[J]. J Math Anal Appl, 2007, 327:

481-493.
{3}

VERMA R U. $A$-monotonicity and applications to nonlinear inclusion

problems[J]. J Appl Math Stochastic Anal, 2004, 17(2): 193-195.
{4}

VERMA R U. Generalized nonlinear variational inclusion problems

involving $A$-monotone mappings[J]. Appl Math Lett, 2006, 19(9):

960-963.
{5}

XIA F Q, HUANG N J. Variational inclusions with a general

$H$-monotone operator in Banach spaces[J]. Comput Math Appl, 2007,

54(1): 24-30.
{6}

DING X P, FENG H R. Algorithm for solving a new class of generalized

nonlinear implicit qusi-variational inclusions in Banach spaces[J].

Appl Math Comput, 2009, 208(2): 547-555.
{7}

FENG H R, DING X P. A new system of generalized nonlinear

quasi-variational-like inclusions with $A$-monotone operators in

Banach spaces[J]. J Comput Appl Math, 2009, 225(2): 365-373.
{8}

LOU J, HE X F, HE Z. Iterative methods for solving a system of

variational inclusions involving $H$-$\eta$-monotone operators in

Banach spaces[J]. Comput Math Appl, 2008, 55(7): 1832-1841.
{9}

DING X P, WANG Z B. System of set-valued mixed

quasi-variational-like inclusions involving $H$-$\eta$-monotone

operators in Banach spaces[J]. Appl Math Mech, 2009, 30(1): 1-12.
{10}

VERMA R U. Approximation solvability of a class of nonlinear

set-valued inclusions involving $(A, \eta)$-monotone mappings[J]. J

Math Anal Appl, 2008, 337(2): 969-975.
{11}

FANG Y P, HUANG N J, Thompson H B. A new system of variational

inclusions with $(H, \eta)$-monotone operators in Hilbert spaces[J].

J Comput Math Appl, 2005, 49: 365-374.
{12}

FANG Y P, HUANG N J. $H$-monotone operator and systems of

variational inclusions[J]. Commun Appl Nonlinear Anal, 2004, 11(1):

93-101.
{13}

FANG Y P, HUANG N J. $H$-monotone operator and resolvent operator

technique for variational inclusions[J]. Appl Math Comput, 2003,

145(2-3): 795-803.
{14}

LUO X P, HUANG N J. A new class of variational inclusions with

$B$-monotone operators in Banach spaces[J]. J Comput Appl Math,

2010, 233(8): 1888-1896.
{15}

PETERSHYN W V. A characterization of strictly convexity of Banach

spaces and other uses of duality mappings[J]. J Funct Anal, 1970, 6:

282-291.
{l6}

LUO X P, HUANG N J. $(H, \phi)$-$\eta$-monotone operators in Banach

spaces with an application to variational inclusions[J]. Appl Math

Comput, 2010, 216(4): 1131-1139.
{17}

HUANG N J, FANG Y P. A new class of general variational inclusions

involving maximal $\eta$-monotone mappings[J]. Publ Math Debrecen,

2003, 62(1-2): 83-98.

[1]	XU Xiao-guang, WANG Kai-rong. A class of conjugate gradient algorithm with sufficient descent property [J]. Journal of East China Normal University(Natural Sc, 2017, 2017(2): 44-51,60.
[2]	LI Xin, ZHANG Lu-Ming, CHAI Guang-Ying, . A linear compact scheme for the nonlinear Schr"odinger equation with wave operator [J]. Journal of East China Normal University(Natural Sc, 2016, 2016(3): 1-8.
[3]	HU Meng-Wei, HUANG Zhi-Gang, SUN Gui-Rong. An oscillation result for some higher order linear differential equations [J]. Journal of East China Normal University(Natural Sc, 2015, 2015(1): 75-83.
[4]	SUN Gui-rong. Relations between solutions of two classes of differential equations with entire function coefficients and functions of small growth [J]. Journal of East China Normal University(Natural Sc, 2014, 2014(1): 21-26.
[5]	LI Hui-bao, HU Xue-ping. Maximal Serfling inequalities for the partial sums of random variables sequence and its application [J]. Journal of East China Normal University(Natural Sc, 2013, 2013(6): 68-73.
[6]	MA Wei-yuan, LIU Hua. Weighted finite difference methods for two-sided space-time fractional diffusion equations [J]. Journal of East China Normal University(Natural Sc, 2012, 2012(3): 41-48,70.
[7]	CAO Wei;WANG Kai-rong. Global convergence of a new conjugate gradient method for modified Liu-Storey formula [J]. Journal of East China Normal University(Natural Sc, 2010, 2010(1): 44-51.
[8]	XU Xiang-jian;WANG Ming-hui;WEI Mu-sheng. Iterative algorithm for solving least Frobenius norm problem of an inconsistent matrix equation pair(English) [J]. Journal of East China Normal University(Natural Sc, 2008, 2008(3): 30-36.
[9]	LI Zhi-gang;ZHOU Xuan-wei. Stabilities for multiobjective programming under conical order perturbations(Chinese) [J]. Journal of East China Normal University(Natural Sc, 2008, 2008(2): 135-140.
[10]	LIU Can-wen;LIU Jie. Parallel Genetic Algorithm for Nonlinear Equations [J]. Journal of East China Normal University(Natural Sc, 2004, 2004(1): 29-34.