On Over-Relaxed Proximal Point Algorithms for Generalized Nonlinear Operator Equation with (A,η,m)-Monotonicity Framework

Author(s)
Fang Li

Affiliation(s)

Department of Mathematics, Sichuan University of Science and Engineering, Zigong, China.

Department of Mathematics, Sichuan University of Science and Engineering, Zigong, China.

ABSTRACT

In this paper, a new class of over-relaxed proximal point algorithms for solving nonlinear operator equations with (A,η,m)-monotonicity framework in Hilbert spaces is introduced and studied. Further, by using the generalized resolvent operator technique associated with the (A,η,m)-monotone operators, the approximation solvability of the operator equation problems and the convergence of iterative sequences generated by the algorithm are discussed. Our results improve and generalize the corresponding results in the literature.

In this paper, a new class of over-relaxed proximal point algorithms for solving nonlinear operator equations with (A,η,m)-monotonicity framework in Hilbert spaces is introduced and studied. Further, by using the generalized resolvent operator technique associated with the (A,η,m)-monotone operators, the approximation solvability of the operator equation problems and the convergence of iterative sequences generated by the algorithm are discussed. Our results improve and generalize the corresponding results in the literature.

KEYWORDS

New Over-Relaxed Proximal Point Algorithm; Nonlinear Operator Equation with (A, η, m)-Monotonicity Framework; Generalized Resolvent Operator Technique; Solvability and Convergence

New Over-Relaxed Proximal Point Algorithm; Nonlinear Operator Equation with (A, η, m)-Monotonicity Framework; Generalized Resolvent Operator Technique; Solvability and Convergence

Cite this paper

F. Li, "On Over-Relaxed Proximal Point Algorithms for Generalized Nonlinear Operator Equation with (A,η,m)-Monotonicity Framework,"*International Journal of Modern Nonlinear Theory and Application*, Vol. 1 No. 3, 2012, pp. 67-72. doi: 10.4236/ijmnta.2012.13009.

F. Li, "On Over-Relaxed Proximal Point Algorithms for Generalized Nonlinear Operator Equation with (A,η,m)-Monotonicity Framework,"

References

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[2] R. P. Agarwal and R. U. Verma, “General Implicit Variational Inclusion Problems Based on A-Maximal (m)-Relaxed Monotonicity (AMRM) Frameworks,” Applied Mathematics and Computation, Vol. 215, No. 1, 2009, pp. 367-379. doi:10.1016/j.amc.2009.04.078

[3] R. P. Agarwal and R. U. Verma, “General System of (A,η)-Maximal Relaxed Monotone Variational Inclusion Problems Based on Generalized Hybrid Algorithms,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 2, 2010, pp. 238-251. doi:10.1016/j.cnsns.2009.03.037

[4] H. Y. Lan, “A Class of Nonlinear (A,η)-Monotone Operator Inclusion Problems with Relaxed Cocoercive Mappings,” Advances in Nonlinear Variational Inequalities, Vol. 9, No. 2, 2006, pp. 1-11.

[5] H. Y. Lan, “Approximation Solvability of Nonlinear Random (A,η)-Resolvent Operator Equations with Random Relaxed Cocoercive Operators,” Computers & Mathematics with Applications, Vol. 57, No. 4, 2009, pp. 624-632. doi:10.1016/j.camwa.2008.09.036

[6] H. Y. Lan, “Sensitivity Analysis for Generalized Nonlinear Parametric (A,η,m)-Maximal Monotone Operator Inclusion Systems with Relaxed Cocoercive Type Operators,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 74, No. 2, 2011, pp. 386-395.

[7] R. U. Verma, “Generalized Over-Relaxed Proximal Algorithm Based on A-Maximal Monotonicity Framework and Applications to Inclusion Problems,” Mathematical and Computer Modelling, Vol. 49, No. 7-8, 2009, pp. 1587-1594.

[8] R. U. Verma, “General Over-Relaxed Proximal Point Algorithm Involving A-Maximal Relaxed Monotone Mappings with Applications,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 71, No. 12, 2009, pp. e1461-e1472. doi:10.1016/j.na.2009.01.184

[9] R. U. Verma, “A Hybrid Proximal Point Algorithm Based on the (A,η)-Maximal Monotonicity Framework,” Applied Mathematics Letters, Vol. 21, No. 2, 2008, pp. 142-147. doi:10.1016/j.aml.2007.02.017

[10] R. U. Verma, “A General Framework for the Over-Relaxed A-Proximal Point Algorithm and Applications to Inclusion Problems,” Applied Mathematics Letters, Vol. 22, No. 5, 2009, pp. 698-703. doi:10.1016/j.aml.2008.05.001

[11] M. A. Hanson, “On Sufficiency of Kuhn-Tucker Conditions,” Journal of Mathematical Analysis and Applications, Vol. 80, No. 2, 1981, pp. 545-550. doi:10.1016/0022-247X(81)90123-2

[12] M. Soleimani-Damaneh, “Generalized Invexity in Separable Hilbert Spaces,” Topology, Vol. 48, No. 2-4, 2009, pp. 66-79. doi:10.1016/j.top.2009.11.004

[13] M. Soleimani-Damaneh, “Infinite (Semi-Infinite) Problems to Characterize the Optimality of Nonlinear Optimization Problems,” European Journal of Operational Research, Vol. 188, No. 1, 2008, pp. 49-56. doi:10.1016/j.ejor.2007.04.026

[1] R. P. Agarwal and R. U. Verma, “Role of relative A-Maximal Monotonicity in Overrelaxed Proximal Point Algorithm with Applications,” Journal of Optimization Theory and Applications, Vol. 143, No. 1, 2009, pp. 1-15. doi:10.1007/s10957-009-9554-z

[2] R. P. Agarwal and R. U. Verma, “General Implicit Variational Inclusion Problems Based on A-Maximal (m)-Relaxed Monotonicity (AMRM) Frameworks,” Applied Mathematics and Computation, Vol. 215, No. 1, 2009, pp. 367-379. doi:10.1016/j.amc.2009.04.078

[3] R. P. Agarwal and R. U. Verma, “General System of (A,η)-Maximal Relaxed Monotone Variational Inclusion Problems Based on Generalized Hybrid Algorithms,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 2, 2010, pp. 238-251. doi:10.1016/j.cnsns.2009.03.037

[4] H. Y. Lan, “A Class of Nonlinear (A,η)-Monotone Operator Inclusion Problems with Relaxed Cocoercive Mappings,” Advances in Nonlinear Variational Inequalities, Vol. 9, No. 2, 2006, pp. 1-11.

[5] H. Y. Lan, “Approximation Solvability of Nonlinear Random (A,η)-Resolvent Operator Equations with Random Relaxed Cocoercive Operators,” Computers & Mathematics with Applications, Vol. 57, No. 4, 2009, pp. 624-632. doi:10.1016/j.camwa.2008.09.036

[6] H. Y. Lan, “Sensitivity Analysis for Generalized Nonlinear Parametric (A,η,m)-Maximal Monotone Operator Inclusion Systems with Relaxed Cocoercive Type Operators,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 74, No. 2, 2011, pp. 386-395.

[7] R. U. Verma, “Generalized Over-Relaxed Proximal Algorithm Based on A-Maximal Monotonicity Framework and Applications to Inclusion Problems,” Mathematical and Computer Modelling, Vol. 49, No. 7-8, 2009, pp. 1587-1594.

[8] R. U. Verma, “General Over-Relaxed Proximal Point Algorithm Involving A-Maximal Relaxed Monotone Mappings with Applications,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 71, No. 12, 2009, pp. e1461-e1472. doi:10.1016/j.na.2009.01.184

[9] R. U. Verma, “A Hybrid Proximal Point Algorithm Based on the (A,η)-Maximal Monotonicity Framework,” Applied Mathematics Letters, Vol. 21, No. 2, 2008, pp. 142-147. doi:10.1016/j.aml.2007.02.017

[10] R. U. Verma, “A General Framework for the Over-Relaxed A-Proximal Point Algorithm and Applications to Inclusion Problems,” Applied Mathematics Letters, Vol. 22, No. 5, 2009, pp. 698-703. doi:10.1016/j.aml.2008.05.001

[11] M. A. Hanson, “On Sufficiency of Kuhn-Tucker Conditions,” Journal of Mathematical Analysis and Applications, Vol. 80, No. 2, 1981, pp. 545-550. doi:10.1016/0022-247X(81)90123-2

[12] M. Soleimani-Damaneh, “Generalized Invexity in Separable Hilbert Spaces,” Topology, Vol. 48, No. 2-4, 2009, pp. 66-79. doi:10.1016/j.top.2009.11.004

[13] M. Soleimani-Damaneh, “Infinite (Semi-Infinite) Problems to Characterize the Optimality of Nonlinear Optimization Problems,” European Journal of Operational Research, Vol. 188, No. 1, 2008, pp. 49-56. doi:10.1016/j.ejor.2007.04.026