Bot, Radu Ioan; Grad, Sorin-Mihai; Wanka, Gert Weaker constraint qualifications in maximal monotonicity. (English) Zbl 1119.47051 Numer. Funct. Anal. Optimization 28, No. 1-2, 27-41 (2007). The authors give a sufficient condition (involving the Fitzpatrick function) for the maximal monotonicity of the operator \(S+T\), where \(S:X\to 2^{X^*}\), \(T:X\to 2^{X^*}\) are two maximal monotone operators and \(X\) is a reflexive Banach space.In a nonreflexive Banach space \(Y\), a condition for the Brézis–Haraux-type approximation of the range of the sum of two subdifferentials \(\partial f\) and \(\partial g\) (here, \(f,g:Y\to\overline{\mathbb R}\) are proper, convex, lower-semicontinuous functions) is also presented, extending and correcting a result of H.Riahi [Proc.Am.Math.Soc.124, No.11, 3333–3338 (1996; Zbl 0865.47037)]. Reviewer: Rodica Luca Tudorache (Iaşi) Cited in 1 ReviewCited in 6 Documents MSC: 47H05 Monotone operators and generalizations 42A50 Conjugate functions, conjugate series, singular integrals 90C25 Convex programming Keywords:Brézis-Haraux-type approximation; Fitzpatrick function; maximal monotone operator; subdifferential Citations:Zbl 0865.47037 PDFBibTeX XMLCite \textit{R. I. Bot} et al., Numer. Funct. Anal. Optim. 28, No. 1--2, 27--41 (2007; Zbl 1119.47051) Full Text: DOI References: [1] DOI: 10.1016/S0924-6509(09)70252-1 · doi:10.1016/S0924-6509(09)70252-1 [2] Attouch H., Serdica – Mathematical Journal 22 pp 165– (1996) [3] Borwein J.M., J. Convex Anal. 13 pp 561– (2006) [4] DOI: 10.1016/j.na.2005.09.017 · Zbl 1087.49026 · doi:10.1016/j.na.2005.09.017 [5] Burachik R.S., J. Convex Anal. 12 pp 279– (2005) [6] DOI: 10.1007/BF01418765 · Zbl 0159.43901 · doi:10.1007/BF01418765 [7] DOI: 10.1307/mmj/1029005463 · Zbl 0935.47033 · doi:10.1307/mmj/1029005463 [8] S. Fitzpatrick(1988).Representing monotone operators by convex functions In: Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), Proceedings of the Centre for Mathematical Analysis 20, Australian National University, Canberra, pp.59–65. [9] Fitzpatrick S., J. Convex Anal. 8 pp 423– (2001) [10] DOI: 10.1016/0022-247X(71)90119-3 · Zbl 0228.47040 · doi:10.1016/0022-247X(71)90119-3 [11] Hiriart-Urruty J.-B., Convex Analysis and Optimization (London, 1980) pp 43– (1982) · Zbl 0446.26006 [12] Martinez-Legaz J.E., J. Nonlinear Convex Anal. 2 pp 243– (2004) [13] DOI: 10.1137/S1052623498340448 · Zbl 0963.90070 · doi:10.1137/S1052623498340448 [14] Pennanen T., J. Nonlinear Convex Anal. 2 pp 193– (2001) [15] DOI: 10.1016/j.na.2004.05.018 · Zbl 1078.47008 · doi:10.1016/j.na.2004.05.018 [16] Phelps R.R., Extracta Mathematicae 12 pp 193– (1997) [17] DOI: 10.1090/S0002-9939-96-03314-X · Zbl 0865.47037 · doi:10.1090/S0002-9939-96-03314-X [18] Rockafellar R.T., Pacific J. Math. 33 pp 209– (1970) [19] DOI: 10.1090/S0002-9947-1970-0282272-5 · doi:10.1090/S0002-9947-1970-0282272-5 [20] Simons S., Minimax and Monotonicity (1998) · Zbl 0922.47047 · doi:10.1007/BFb0093633 [21] DOI: 10.1006/jmaa.1996.0135 · Zbl 0863.47034 · doi:10.1006/jmaa.1996.0135 [22] Simons S., J. Nonlinear Convex Anal. 6 pp 1– (2005) [23] Verona A., J. Convex Anal. 7 pp 115– (2000) [24] DOI: 10.1007/0-387-24276-7_66 · doi:10.1007/0-387-24276-7_66 [25] DOI: 10.1142/9789812777096 · doi:10.1142/9789812777096 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.