Strong convergence of iterative algorithm for a new system of generalized \(H(\cdots,\cdot)-\eta\)-cocoercive operator inclusions in Banach spaces. (English) Zbl 1364.47010

Alsulami, Saud M.; Naraghirad, Eskandar; Hussain, Nawab

Abstr. Appl. Anal. 2013, Article ID 540108, 10 p. (2013).

Summary: We introduce and study a new system of generalized \(H(\cdots,\cdot)-\eta\)-cocoercive operator inclusions in Banach spaces. Using the resolvent operator technique associated with \(H(\cdots,\cdot)-\eta\)-cocoercive operators, we suggest and analyze a new generalized algorithm of nonlinear set-valued variational inclusions and establish strong convergence of iterative sequences produced by the method. We highlight the applicability of our results by examples in function spaces.

MSC:

47J25	Iterative procedures involving nonlinear operators
47J22	Variational and other types of inclusions
47H06	Nonlinear accretive operators, dissipative operators, etc.

Keywords:

\(\eta\)-cocoercive operator; set-valued variational inclusions; strong convergence

References:

[1]	Adly, S., Perturbed algorithms and sensitivity analysis for a general class of variational inclusions, Journal of Mathematical Analysis and Applications, 201, 2, 609-630 (1996) · Zbl 0856.65077 · doi:10.1006/jmaa.1996.0277
[2]	Alotaibi, A.; Kumar, V.; Hussain, N., Convergence comparison and stability of Jungck-Kirk type algorithms for common fixed point problems, Fixed Point Theory and Applications, 2013 (2013) · Zbl 1476.47088 · doi:10.1186/1687-1812-2013-173
[3]	Ceng, L. C.; Husain, N.; Latif, A.; Yao, J. C., Strong convergence for solving general system of variational inequalities and fixed point problems in Banach spaces, Journal of Inequalities and Applications, 2013 (2013) · Zbl 1357.49029 · doi:10.1186/1029-242X-2013-334
[4]	Feng, H. R.; Ding, X. P., A new system of generalized nonlinear quasi-variational-like inclusions with \(A\)-monotone operators in Banach spaces, Journal of Computational and Applied Mathematics, 225, 2, 365-373 (2009) · Zbl 1161.65048 · doi:10.1016/j.cam.2008.07.048
[5]	Lan, H.-Y.; Cho, Y. J.; Verma, R. U., Nonlinear relaxed cocoercive variational inclusions involving \((A, \eta)\)-accretive mappings in Banach spaces, Computers and Mathematics with Applications, 51, 9-10, 1529-1538 (2006) · Zbl 1207.49011 · doi:10.1016/j.camwa.2005.11.036
[6]	Fang, Y.-P.; Huang, N.-J., \(H\)-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces, Applied Mathematics Letters, 17, 6, 647-653 (2004) · Zbl 1056.49012 · doi:10.1016/S0893-9659(04)90099-7
[7]	Wang, Z. B.; Ding, X. P., \(H \left(\left(., .\right), \eta\right)\)-accretive operators with an application for solving set-valued variational inclusions in Banach spaces, Computers and Mathematics with Applications, 59, 4, 1559-1567 (2010) · Zbl 1189.49027 · doi:10.1016/j.camwa.2009.11.003
[8]	Zou, Y.-Z.; Huang, N.-J., \(H(., .)\)-accretive operator with an application for solving variational inclusions in Banach spaces, Applied Mathematics and Computation, 204, 2, 809-816 (2008) · Zbl 1170.65043 · doi:10.1016/j.amc.2008.07.024
[9]	Jin, M.-M., Convergence and stability of iterative algorithm for a new system of \((A, \eta)\)-accretive mapping inclusions in Banach spaces, Computers and Mathematics with Applications, 56, 9, 2305-2311 (2008) · Zbl 1165.65352 · doi:10.1016/j.camwa.2008.03.053
[10]	Lan, H.-Y., Stability of iterative processes with errors for a system of nonlinear \((A, \eta)\)-accretive variational inclusions in Banach spaces, Computers and Mathematics with Applications, 56, 1, 290-303 (2008) · Zbl 1145.49302 · doi:10.1016/j.camwa.2007.12.007
[11]	Luo, X.-P.; Huang, N.-J., \( \left(H, \phi\right) - \eta -\) monotone operators in Banach spaces with an application to variational inclusions, Applied Mathematics and Computation, 216, 4, 1131-1139 (2010) · Zbl 1222.47079 · doi:10.1016/j.amc.2010.02.005
[12]	Xu, Z.; Wang, Z., A Generalized mixed variational inclusion involving \((H \left(., .\right), \eta)\)-monotone operators in Banach spaces, Journal of Mathematics Research, 2, 3, 47-56 (2010) · Zbl 1219.47071
[13]	Verma, R. U., Approximation solvability of a class of nonlinear set-valued variational inclusions involving \((A, \eta)\)-monotone mappings, Journal of Mathematical Analysis and Applications, 337, 2, 969-975 (2008) · Zbl 1140.49008 · doi:10.1016/j.jmaa.2007.01.114
[14]	Lan, H.-Y., Nonlinear parametric multi-valued variational inclusion systems involving \((A, \eta )\)-accretive mappings in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications, 69, 5-6, 1757-1767 (2008) · Zbl 1357.49039 · doi:10.1016/j.na.2007.07.021
[15]	Lan, H.-Y., \((A, \eta)\)-accretive mappings and set-valued variational inclusions with relaxed cocoercive mappings in Banach spaces, Applied Mathematics Letters, 20, 5, 571-577 (2007) · Zbl 1136.47045 · doi:10.1016/j.aml.2006.04.025
[16]	Su, C.-L., Finger shape expansion and recognition by wavelet transform, Applied Mathematics and Computation, 190, 2, 1583-1592 (2007) · Zbl 1124.94005 · doi:10.1016/j.amc.2007.02.040
[17]	Lan, H.-Y.; Cui, Y.-S.; Fu, Y., New approximation-solvability of general nonlinear operator inclusion couples involving \((A, \eta, m)\)-resolvent operators and relaxed cocoercive type operators, Communications in Nonlinear Science and Numerical Simulation, 17, 4, 1844-1851 (2012) · Zbl 1272.47081 · doi:10.1016/j.cnsns.2011.09.005
[18]	Hussain, N.; Rafiq, A., On modified implicit Mann iteration method involving strictly hemicontractive mappings in smooth Banach spaces, Journal of Computational Analysis and Applications, 15, 5, 892-902 (2013) · Zbl 1283.47061
[19]	Hussain, N.; Rafiq, A.; Ciric, L. B., Stability of the Ishikawa iteration scheme with errors for two strictly hemicontractive operators in Banach spaces, Fixed Point Theory and Applications, 2012 (2012) · Zbl 1475.47057 · doi:10.1186/1687-1812-2012-160
[20]	Khan, S. H.; Rafiq, A.; Hussain, N., A three-step iterative scheme for solving nonlinear \(\phi \)-strongly accretive operator equations in Banach spaces, Fixed Point Theory and Applications, 2012 (2012) · Zbl 1475.47064 · doi:10.1186/1687-1812-2012-149
[21]	Ceng, L.-C.; Khan, A. R.; Ansari, Q. H.; Yao, J.-C., Viscosity approximation methods for strongly positive and monotone operators, Fixed Point Theory, 10, 1, 35-71 (2009) · Zbl 1196.47044
[22]	Ahmad, R.; Mursaleen, M., System of generalized \(H\)-resolvent equations and the corresponding system of generalized variational inclusions, Hacettepe Journal of Mathematics and Statistics, 41, 1, 33-45 (2012) · Zbl 1273.47100
[23]	Ahmad, R.; Dilshad, M.; Wong, M.-M.; Yao, J.-C., Generalized set-valued variational-like inclusions involving \(H(., .)-\eta \)-cocoercive operator in Banach spaces, Journal of Inequalities and Applications, 2012 (2012) · Zbl 1286.47040 · doi:10.1186/1029-242X-2012-149
[24]	Nadler, S. B., Multivalued contraction mappings, Pacific Journal of Mathematics, 30, 3, 457-488 (1969) · Zbl 0187.45002
[25]	Cioranescu, I., Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and Its Applications, 62 (1990), Dordrecht, The Netherlands: Kluwer Academic Publishers, Dordrecht, The Netherlands · Zbl 0712.47043 · doi:10.1007/978-94-009-2121-4
[26]	Petryshyn, W. V., A characterization of strict convexity of Banach spaces and other uses of duality mappings, Journal of Functional Analysis, 6, 2, 282-291 (1970) · Zbl 0199.44004 · doi:10.1016/0022-1236(70)90061-3
[27]	Ahmad, R.; Akram, M.; Yao, J. C., Generalized monotone mapping with an application for solving a variational inclusion problem, Journal of Optimization Theroy and Applications, 157, 2, 324-346 (2013) · Zbl 1268.49007 · doi:10.1007/s10957-012-0182-7