Taiwanese Journal of Mathematics


N. H. Chieu and N. T. Q. Trang

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Sufficient conditions for a norm-to-weak$^*$ continuous mapping $f: X \rightarrow X^*$ being monotone or submonotone are established by its Fréchet and normal coderivatives, where $X$ is an Asplund space with its dual space $X^*$. Under some additional assumptions, they are also necessary conditions. Among other things, we obtain a criterion for the monotonicity of continuous mappings which extends the following classical result: a differentiable mapping $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is monotone if and only if for each $x \in \mathbb{R}^n$ the Jacobian matrix $\nabla F(x)$ is positive semi-definite; see [22, Proposition 12.3]. As a by-product, sufficient conditions for a function being convex or approximately convex are given.

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Taiwanese J. Math., Volume 16, Number 1 (2012), 353-365.

First available in Project Euclid: 18 July 2017

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Zentralblatt MATH identifier

Primary: 49K40: Sensitivity, stability, well-posedness [See also 90C31] 49J40: Variational methods including variational inequalities [See also 47J20] 49J52: Nonsmooth analysis [See also 46G05, 58C50, 90C56] 49J53: Set-valued and variational analysis [See also 28B20, 47H04, 54C60, 58C06]

monotonicity submonotonicity convexity coderivative positive semi-definiteness


Chieu, N. H.; Trang, N. T. Q. CODERIVATIVE AND MONOTONICITY OF CONTINUOUS MAPPINGS. Taiwanese J. Math. 16 (2012), no. 1, 353--365. doi:10.11650/twjm/1500406545. https://projecteuclid.org/euclid.twjm/1500406545

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  • D. Aussel, J.-N. Corvellec and M. Lassonde, Mean value property and subdifferential criteria for lower semicontinuous functions, Trans. Amer. Math. Soc., 347 (1995), 4147-4161.
  • H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North Holland, Amsterdam, 1973.
  • N. H. Chieu, T. D. Chuong, J.-C. Yao and N. D. Yen, Characterizing convexity of a function by its Fréchet and limiting second-order subdifferentials, Set-Valued Var. Anal., 19 (2011), 75-96.
  • N. H. Chieu and N. Q. Huy, Second-order subdifferentials and convexity of real-valued functions, Nonlinear Anal., 74 (2011), 154-160.
  • N. H. Chieu and J.-C. Yao, Characterization of convexity for a piecewise $C^2$ function by the limiting second-order subdifferential, Taiwanese J. Math., 15 (2011), 31-42.
  • I. Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems, Kluwer Academic Publishers, Dordrecht, Boston, London, 1990.
  • A. Daniilidis and P. Georgiev, Approximate convexity and submonotonicity, J. Math. Anal. Appl., 291 (2004), 292-301.
  • A. Daniilidis, F. Jules and M. Lassonde, Subdifferential characterization of approximate convexity: the lower semicontinuous case, Math. Program. Ser. B, 116 (2009), 115-127.
  • P. Georgiev, Submonotone Mappings in Banach Spaces and Applications, Set-Valued Anal., 5 (1997), 1-35.
  • A. B. Levy, R. A. Poliquin and R. T. Rockafellar, Stability of locally optimal solutions, SIAM J. Optim., 10 (2000), 580-604.
  • B. S. Mordukhovich, Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problmes, Soviet Math. Dokl., 22 (1980), 526-530.
  • B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Vol. II: Applications, Springer, Berlin, 2006.
  • B. S. Mordukhovich, N. M. Nam and N. D. Yen, Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming, Optimization, 55 (2006), 685-708.
  • B. S. Mordukhovich and J. V. Outrata, On second-order subdifferentials and their applications, SIAM J. Optim., 12 (2001), 139-169.
  • B. S. Mordukhovich and Y. Shao, Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc., 348 (1996), 1235-1280.
  • H. V. Ngai, D. T. Luc and M. Théra, Approximate convex functions, J. Nonlinear Convex Anal., 1 (2000), 155-176.
  • H. V. Ngai and J.-P. Penot, Approximately convex functions and approximately monotonic operators, Nonlinear Anal., 66 (2007), 547-564.
  • R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math., 1364, Springer, Berlin, 1993.
  • R. A. Poliquin and R. T. Rockafellar, Tilt stability of a local minimum, SIAM J. Optim., 8 (1998), 287-299.
  • R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math., 17 (1966), 497-510.
  • R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33 (1970), 209-216.
  • R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1998.
  • J. E. Spingarn, Submonotone subdifferentials of Lipschitz functions, Trans. Amer. Math. Soc., 264 (1981), 77-89.
  • D. Zagrodny, Approximate mean value theorem for upper subderivatives, Nonlinear Anal., 12 (1988), 1413-1428.