On sum of monotone operator of type (FPV) and a maximal monotone operator

Year 2016, Volume: 4 Issue: 4, 188 - 197, 31.12.2016

D. K. Pradhan S. R. Pattanaik

https://izlik.org/JA63NA85RF

Abstract

Keywords

Sum problem , Fitzpatrick function , maximal monotone operator

References

• H. H. Bauschke, X. Wang and L. Yao, An answer to S. Simons’ question on the maximal monotonicity of the sum of a maximal monotone linear operator and a normal cone operator, Set-Valued Var. Anal. 17 (2009) 195-201.
• H. H. Bauschke, X. Wang and L. Yao, On the maximal mono tonicity of the sum of a maximal monotone linear relation and the subdifferential operator of a sublinear function, Proceedings of the Haifa Workshop on Optimization Theory and Related Topics. Contemp. Math., Amer. Math. Soc., Providence, RI 568 (2012) 19-26.
• J. M. Borwein, Maximality of sums of two maximal monotone operators in general Banach space, P. Am. Math. Soc. 135 (2007) 3917-3924.
• J. M. Borwein and L. Yao, Maximality of the sum of a maximally monotone linear relation and a maximally monotone operator, Set-Valued Var Anal. 21 (2013) 603-616.
• J. M. Borwein and L. Yao, Structure theory for maximally monotone operators with points of continuity, J. Optim Theory Appl. 157 (2013) 1-24 http://dx.doi.org/10.1007/s10957-012-0162-y.
• J.M. Borwein and L. Yao, Sum theorems for maximally monotone operators of type (FPV), J. Aust. Math. Soc. 97 (2014) 1-26.
• S. Fitzpatrick, Representing monotone operators by convex functions, in Work- shop/Miniconference on Functional Analysis and Optimization (Canberra 1988), Proceedings of the Centre for Mathematical Analysis, Australian National University, Canberra, Australia, 20 (1988) 59-65.
• R.R. Phelps, Convex Functions, Monotone Operators and Differentiability, 2nd Edition, Springer-Verlag, 1993.
• R.T. Rockafellar, Local boundedness of nonlinear, monotone operators, Mich. Math. J. 16 (1969) 397-407.
• R.T. Rockafellar, On the maximality of sums of nonlinear monotone operators, T. Am. Math. Soc. 149 (1970) 75-88.
• R. Rudin, Functional Analysis, Second Edition, McGraw-Hill, 1991.
• S. Simons, Minimax and Monotonicity, Springer-Verlag, 1998.
• S. Simons, From Hahn-Banach to Monotonicity, Springer-Verlag, 2008.
• M.D. Voisei, The sum and chain rules for maximal monotone operators, Set-Valued Var. Anal. 16 (2008) 461-476.
• L. Yao, The sum of a maximally monotone linear relation and the subdifferential of a proper lower semicontinuous convex function is maximally monotone, Set-Valued Var. Anal. 20 (2012) 155-167.
• L. Yao, Maximality of the sum of the subdifferential operator and a maximally monotone operator, arXiv: 1406.7664v1[math.FA] 30 Jun 2014, http://arxiv.org/pdf/1406.7664.pdf.
• C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing, 2002.