Abstract
These lectures center on the structure of real—valued Lipschitz functions, and their generalized derivatives on Banach spaces. We pay some attention to the role of measure and category and will try to illustrate a number of different techniques. These published notes are much more detailed and comprehensive than the lectures as given. Much of this development is based on recent joint work with Warren Moors (Wellington) and others. The exposition will be organized around the following interwoven themes.
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• Set—valued analysis. Minimal upper semi-continuous multifunctions (‘uscos and cuscos’) and related topological tools. Selections and single-valuedness. Relations to differentiability.
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• Measure and category as competing notions of smallness: Haar null sets and generic sets in Banach spaces. Other concepts of prevalence.
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• ‘Utility grade’ renorming theory and its application to the study of viscosity sub-derivatives and (partially) smooth variational principles. “Fuzzy” calculus and some equivalent reformulations.
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• The structure of Lipschitz functions. Especially the calculus of essentially smooth Lipschitz functions and the vector-lattice algebra they generate. Chain rules, and questions of integrability and representability.
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• Applications to and examples of distance functions. Minimality of distance functions and proximal normal formulae revisited. More general perturbation functions.
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• Convex functions and related sequences in Banach space. How properties of given Banach spaces are reflected in the convex functions they support. Conjugates and subdifferentials of eigenvalue functions.
Research was supported by NSERC and by the Shrum Endowment at Simon Fraser University
Research was supported by the National Science Foundation under grant DMS-9704203
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References
Arnold V. I., Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd ed., Springer, New York, 1988.
Aronszajn, N., Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), 147–190.
Aubin J. P., Frankowska H., Set-valued Analysis, Birkhäuser, Boston, 1990.
Aussel D., Corvellec J.-N., Lassonde M., Mean value property and subdifferential criteria for lower semicontinuous functions, Trans. Amer. Math. Soc. 347 (1995), 4147–4161.
Beer G., Borwein J. M., Mosco and slice convergence of level sets and graphs of linear functional, J. Math. Anal. Appl. 175 (1993), 53–67.
Beauzamy, B., Introduction to Banach Spaces and their Geometry, 2nd ed., North-Holland, Amsterdam 1985.
Benyamini Y., The uniform classification of Banach spaces, in: Texas Functional Analysis Seminar 1984–1985 (Austin, Tex.), Longhorn Notes, Univ. Texas Press, Austin, TX, 1985, 15–38.
Benoist J., Intégration du sous-différentiel proximal: un contre example, to appear in Canad. J. Math.
Bishop, E., Phelps, R. R., The support functionals of a convex set, in: Convexity (V. Klee, ed.) Proc. Sympos. Pure Math. 7, Amer. Math. Soc., Providence, RI, 1963, 27–35.
Borwein J. M., Convex relations in analysis and optimization, in: Generalized Concavity in Optimization and Economics ( S. Schaible and W. T. Ziemba, eds.), Academic Press, New York, 1981, 335–377.
Borwein J. M., Minimal cuscos and subgradients of Lipschitz functions, in Fixed Point Theory and its Applications, ( J-B. Baillon and M. Thera, eds.), Pitman Res. Notes Math. Ser., Longman, Essex, 1991, 57–82.
Borwein J. M., Fabian M., On convex functions having points of Gâteaux differentiability which are not points of Préchet differentiability, Canad. J. Math. 45 (1993), 1121–1134.
Borwein J. M., Fabian M., A note on regularity of sets and of distance functions in Banach space, J. Math. Anal. Appl. 182 (1994), 566–560.
Borwein J. M., Fitzpatrick S., Existence of nearest points in Banach spaces, Canad. J. Math. 61 (1989), 702–720.
Borwein J. M., Fitzpatrick S., A weak Hadamard smooth renorming of L1(Ω,μ), Canad. Math. Bull. 36 (1993), 407–413.
Borwein J. M., Fitzpatrick S., Weak* sequential compactness and bornological limit derivatives, J. Convex Anal. 2 (1995), 59–68 (Special Issue in Celebration of R.T. Rockafellar’s 60th Birthday. Part I).
Borwein J. M., Fitzpatrick S., Characterization of Clarke subgradients among one-dimensional multifunctions, in: Proc. Optimization Miniconference II, Univ. Ballarat Press, 1995, 61–73.
Borwein J. M., Fitzpatrick S., Closed convex Haar null sets, unpublished manuscript, CECM Research Report 95–052 (1995).
Borwein J. M., Fitzpatrick S., Giles R., The differentiability of real functions on normed linear spaces using generalized subgradients, J. Math. Anal. Appl. 128 (1987) 512–534.
Borwein J. M., Fitzpatrick S., Kenderov P., Minimal convex usco and monotone operators on small sets, Canad. J. Math. 43 (1991), 461–476.
Borwein J. M., Fitzpatrick S., Vanderwerff J., Examples of convex functions and classifications of normed spaces, J. Convex Anal. 1 (1994), 61–73.
Borwein J. M., Girgensohn R., Wang X., On the Construction of Hölder and Proximal Subderivatives, CECM Research Report 97-091, to appear in Canad. Math. Bull.
Borwein J. M., Ioffe A., Proximal analysis in smooth spaces, Set-Valued Anal. 4 (1996), 1–24.
Borwein J. M., Jofré A., A nonconvex separation property in Banach spaces, CECM Research Report 97-103, to appear in ZOR: Math. Methods Oper. Res.
Borwein J. M., Lewis, A., Partially-finite convex programming, (I) & (II), Math. Programming (Ser. B) 57 (1992), 15–48, 49–84.
Borwein J. M., Lewis, A., Convex Analysis, Wiley, in press.
Borwein J. M., Moors W. B., Essentially smooth Lipschitz functions, J. Funct. Anal. 49 (1997), 305–351.
Borwein J. M., Moors W. B., A chain rule for essentially strictly differentiable functions, SIAM J. Optim. 8 (1998), 300–308.
Borwein J. M., Moors W. B., Null sets and essentially smooth Lipschitz functions, SIAM J. Optim. 8 (1998), 309–323.
Borwein J. M., Moors W. B., Lipschitz functions with minimal Clarke subdifferential mappings, in: Proc. Optimization Miniconference III (1996), Univ. of Ballarat Press, 1997, 5–12.
Borwein J. M., Moors W. B., Separable determination of integrability and minimality of the Clarke subdifferential mapping, CECM Research Report 97-102, to appear in Proc. Amer. Math. Soc.
Borwein J. M., Moors W. B., Shao, Y., Subgradient representation of multifunctions, CECM Research Report 97–090, J. Austral Math. Soc. Ser. B, volume in honour of B. Craven and B. Mond, 1998.
Borwein J. M., Moors W. B., Wang X., Lipschitz functions with prescribed derivatives and subdirevatives, Nonlinear Anal. 29 (1997), 53–63.
Borwein J. M., Mordukhovich B. S., Shao Y., On the equivalence of some basic principles in variational analysis, CECM Research Report 97–098.
Borwein J. M., Noll D., Second order differentiability of convex functions In Banach spaces, Trans. Amer. Math. Soc. 342 (1994), 43–81.
Borwein J. M., Preiss D., A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), 517–527.
Borwein J. M., Treiman J. S., Zhu Q. J., Necessary conditions for constrained optimization problems with semicontinuous and continuous data, Trans. Amer. Math. Soc. 350 (1998), 2409–2429.
Borwein J. M., Treiman J. S., Zhu Q. J., Partially smooth variational principles and applications, CECM Research Report 96-088 (1996), to appear In Nonlinear Anal.
Borwein J. M., Vanderwerff J., Banach spaces which admit support sets, Proc. Amer. Math. Soc. 124 (1996), 751–756.
Borwein J. M., Wang X., Lipschitz functions with maximal Clarke subgradients are generic, submitted to Proc. Amer. Math. Soc.
Borwein J. M., Zhu Q. J., Viscosity solutions and viscosity subderivatives In smooth Banach spaces with applications to metric regularity, SIAM J. Control Optim. 34 (1996), 1568–1591.
Borwein J. M., Zhu Q. J., Limiting convex examples for nonconvex subdifferential calculus, CECM Research Report, 97–097, to appear in J. Convex Anal.
Borwein J. M., Zhu Q. J., A survey of subdifferentials and their applications, CECM Research Report 98–105 (1998), to appear in Nonlinear Anal.
Choquet, G. Lectures on Analysis Vol. I, W. A. Benjamin, New York, 1969.
Christian B., Christensen J. P. R., Ressel P., Harmonic Analysis on Semigroups, Springer-Verlag, New York, 1984.
Christensen J. P. R., On sets of Haar measure zero in Abelian Polish groups, Israel J. Math. 13 (1972), 255–260.
Christensen J. P. R., Topological and Borel Structure, American Elsevier, New York, 1974.
Christensen J. P. R., Theorems of Namioka and R. E. Johnson type for upper semicon-tinuous and compact valued set-valued mappings, Proc. Amer. Math. Soc. 86 (1982), 649–655.
Clarke F. H., Necessary Conditions for Nonsmooth Problems in Optimal Control and the Calculus of Variations, Ph. D. thesis, Univ. of Washington, 1973.
Clarke F. H., Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1933, Russian edition MIR, Moscow, 1988. Reprinted as Classics Appl. Math. 5, SIAM, Philadelphia, PA, 1990.
Clarke F. H., Methods of Dynamic and Nonsmooth Optimization, CBMS-NSF Regional Conf. Ser. in Appl. Math., 57, SIAM, Philadelphia, PA, 1989.
Clarke F. H., Lecture Notes in Nonsmooth Analysis, unpublished.
Clarke F. H., Ledyaev Yu. S., Mean value inequalities, Proc. Amer. Math. Soc. 122 (1994), 1075–1083.
Clarke F. H., Ledyaev Yu. S., Mean value inequalities in Hilbert space, Trans. Amer. Math. Soc. 344 (1994), 307–324.
Clarke F. H., Ledyaev Yu. S., Stern R. J., Wolenski P. R., Nonsmooth Analysis and Control Theory, Grad. Texts in Math. 178, Springer-Verlag, New York, 1998.
Correa R., Jofré A., Tangentally continuous directional derivatives in Nonsmooth Analysis, J. Optim. Theory Appl. 61 (1) (1989), 1–21.
Correa R., Jofré A., Thibault L., Characterization of lower semicontinuous convex functions, Proc. Amer. Math. Soc. 116 (1992), 61–72.
Correa R., Jofré A., Thibault L., Subdifferential monotonicity as characterization of convex functions, Numer. Funct. Anal. Optim. 15 (1994), 531–535.
Correa R., Thibault L., Subdifferential Analysis of bivariate separately regular functions, J. Math. Anal. Appl. 148 (1990), 157–174.
Crandall M. G., Lions P.-L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1–42.
Crandall M. G., Evans L. C., Lions P.-L., Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), 487–502.
Day M. M., Normed Linear Spaces, 3rd ed., Springer-Verlag, New York, 1973.
Deville R., Godefroy G., Zizler V., Smoothness and Renormings in Banach Spaces, Pitman Monogr. Surveys Pure Appl. Math. 64, J. Wiley, New York, 1993.
Deviile R., Godefroy G., Zizler V., A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions, J. Funct. Anal. 111 (1993), 197–212.
Deville R., Haddad E. M. E., The subdifferential of the sum of two functions in Banach spaces, I. first order case, J. Convex Anal. 3 (1996), 295–308.
Deville R., Ivanov M., Smooth variational principles with constraints, Arch. Math. (Basel) 69 (1997), 418–426.
Diestel J., Sequences and Series in Banach Spaces, Springer-Verlag, New York, 1984.
Dudley, R. M., Real Analysis and Probability, Wadsworth, Belmont, 1989.
M. Edelstein M., Thompson A. C., Some results on nearest points and support properties of convex sets in c0, Pacific J. Math. 40 (1972), 553–560.
Ekeland I., On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353.
Evans, L. C., Gariepy, R. F. Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.
Fabian M., On class of subdifferentiability spaces of Ioffe, Nonlinear Anal. 12 (1988), 63–74.
Fabian M., Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss, Acta Univ. Carolin. — Math. Phys. 30 (1989), 51–56.
Fabian M., Zajíček L., Zizler V., On residuality of the set of rotund norms on a Banach space, Math. Ann. 258 (1982), 349–351.
Finet C., Godefroy G., Biorthogonal systems and big quotient spaces, Contemp. Math. 85 (1989), 87–110.
Giles J. R., Convex Analysis with Applications in Differentiation of Convex Functions, Res. Notes in Math. 58, Pitman, Boston-London-Melbourne, 1982.
Giles J. R., Kenderov P. S., Moors W. B., Sciffer S. D., Generic differentiability of convex functions on the dual of a Banach space, Pacific J. Math. 172 (1996), 413–433.
Giles J. R., Moors W. B., A continuity property related to Kuratowski’s index of non-compactness, it relevance to the Drop property, and its implications for differentiability theory, J. Math. Anal. Appl. 178 (1993), 247–268.
Giles J. R., Moors W. B., Generic continuity of minimal set-valued mappings, J. Austral. Math. Soc. Ser. A 63 (1997), 1–25.
Helgason S., Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.
Hiriart-Urruty J. B., Tangent cones, generalized gradients and mathematical programming In Banach spaces, Math. Oper. Res. 4 (1979), 79–97.
Holmes R. B., Geometric Functional Analysis and Its Applications, Springer, New York, 1975.
Ioffe A. D., Nonsmooth analysis: differential calculus of nondifferentiable mappings, Tram. Amer. Math. Soc. 266 (1981), 1–56.
Ioffe A. D., Calculus of Dini subdifferentials of functions and contingent derivatives of set-valued maps, Nonlinear Anal. 8 (1984), 517–539.
Ioffe A. D., Approximate subdifferentials and applications. I: The finite dimensional theory, Trans. Amer. Math. Soc. 281 (1984), 389–416.
Ioffe A. D., Approximate subdifferentials and applications. II: Functions on local convex spaces, Mathematika 33 (1986), 111–128.
Ioffe A. D., Approximate subdifferentials and applications III: The metric theory, Mathematika 36 (1989), 1–38.
Ioffe A. D., Proximal analysis and approximate subdifferentials, J. London Math. Soc. 41 (1990), 175–192.
Ioffe A. D., Fuzzy principles and characterization of trustworthiness, preprint.
Ioffe A. D., Rockafellar T. R., The Euler and Weierstrass conditions for nonsmooth variational problems, Calc. Var. Partial Differential Equations 4 (1996), 59–87.
James, R. C., Super-reflexive spaces with bases, Pacific J. Math. 41 (1972), 409–419.
Janin, R., Sur des multiapplications qui sont des gradients geńeŕaliseś, C.R. Acad. Sci. Paris Sér. I Math. 294 (1982), 115–117.
Jokl L., Minimal convex-valued weak* usco correspondences and the Radon-Nikodym property, Comment. Math. Univ. Carolin. 28 (1987), 353–375.
Jofre A., Thibault L., D-representation of subdifferentials of directionally Lipschitzian functions, Proc. Amer. Math. Soc. 110 (1990), 117–123.
Kempisty S., Sur les fonctions quasicontinues, Fund. Math. 19 (1932), 184–197.
Kenderov P. S., Orihuela, On a generic factorization theorem, Mathematika 42 (1995), 56–66.
Konjagin S. V., On approximation property of closed sets in Banach spaces and the characterization of strongly convex spaces, Soviet Math. Dokl. 21 (1980), 418–422.
Krasovskii N. N., Subbotin A. I., Game-Theoretical Control Problems, Springer-Verlag, New York, 1990.
Kruger A. Y., Mordukhovich B. S., Extremal points and Euler equations in nonsmooth optimization, Dokl. Akad. Nauk. BSSR 24 (1980), 684–687 (Russian).
Larman D. G., Phelps R. R., Gâteaux differentiability of convex functions on Banach spaces, J. London Math.Soc. 20 (1979), 115–127.
Lassonde M., First-order rules for nonsmooth constrained optimization, preprint.
Levy A. B., Poliquin R. A., Characterizing the single-valuedness of multifunctions, Set-Valued Anal. 5 (1997), 351–364.
Lewis A. S., Convex analysis on the Hermitian matrices, SIAM J. Optim. 6 (1996), 164–177.
Lewis A. S., Nonsmooth analysis of eigenvalues, preprint.
Li Y., Shi S., A generalization of Ekeland’s ε-variational principle and of its Borwein-Preiss’ smooth variant, to appear in J. Math. Anal. Appl.
Lindenstrauss J., Tzafriri L., Classical Banach Spaces II: Function Spaces, Springer-Verlag, Berlin, 1979.
Mankiewicz, P. On the differentiability of Lipschitz mappings in Fréchet spaces, Studia Math. 45 (1973), 15–29.
Matouskova E., Stegall C., A characterization of reflexive spaces, Proc. Amer. Math. Soc. 124 (1996), 1083–1090.
Michel P., Penot J. P., Calcul sous-différential pour des fonctions Lipschitziennes et non-Lipschiziennes, C. R. Acad. Sci. Paris Sér. I Math. 298 (1985), 269–272.
Mifflin R., Semismooth and semiconvex functions in Optimization, SIAM J. Control. Optim. 15 (1977), 959–972.
Minty, G., Monotone (nonlinear) operators in Hilbert spaces, Duke Math. J. 29 (1962), 341–346.
Moors W. B., A characterisation of minimal subdifferential mappings of locally Lipschitz functions, Set-Valued Anal. 3 (1995), 129–141.
Mordukhovich B. S., Maximum principle in problems of time optimal control with nonsmooth constraints, J. Appl. Math. Mech. 40 (1976), 960–969.
Mordukhovich B. S., Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Soviet Math. Dokl. 22 (1980), 526–530.
Mordukhovich B. S., Approximation Methods in Problems of Optimization and Control, Nauka, Moscow, 1988 ( Russian; English transl. to appear in Wiley-Interscience).
Mordukhovich B. S., Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions, Trans. Amer. Math. Soc. 340 (1993), 1–35.
Mordukhovich B. S., Generalized differential calculus for nonsmooth and set-valued mappings, J. Math. Anal. Appl. 183 (1994), 250–288.
Mordukhovich B. S., Shao Y., Extremal characterizations of Asplund spaces, Proc. Amer. Math. Soc. 124 (1996), 197–205.
Oxtoby J. C., Measure and Category, Springer Verlag, New York-Berlin-Tokyo, 1971.
Phelps R. R., Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math. 1364, Springer Verlag, New York, Berlin, Tokyo, 1988, 2nd ed. 1993.
Phelps R. R., Gaussian null sets and differentiability of Lipschitz maps on Banach spaces, Pacific J. Math. 77 (1978), 523–531.
Poliquin R. A., Integration of subdifferentials of nonconvex functions, Nonlinear Anal. 17 (1991), 385–398.
Poliquin R. A., A characterization of proximal subgradient set-valued mappings, Canad. Math. Bull. 36 (1993), 116–122.
Preiss D., Fréchet derivatives of Lipschitzian functions, J. Funct. Anal. 91 (1990), 312–345.
Preiss D., Zajiček L., Stronger estimates of smallness of sets of Fréchet nondifferentiability of convex functions, Proc. 11th Winter School, Rend. Circ. Mat. Palermo (2) Suppl. 3 (1984), 219–223.
Qi L., The maximal normal operator space and integration of subdifferentials of non-convex functions, Nonlinear Anal. 13 (1989), 1003–1011.
Rockafellar R. T., Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
Rockafellar R. T., On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209–216.
Rockafellar R. T., Favorable classes of Lipschitz-continuous functions in subgradient optimization, in: Progress in Nondifferentiable Optimization ( E. Nurminski, ed.), IIASA Collaborative Proc. Ser., Internat. Inst. Appl. Systems Anal., Laxenburg, 1982, 125–144.
Rockafellar R. T., Extensions of subgradients and its applications to optimization, Nonlinear Anal. 9 (1985), 665–698.
Rockafellar R. T., Wets R. J-B., Variational Analysis, Springer, New York, 1997.
Rudin W., Functional Analysis, McGraw-Hill, New York, 1973.
Simon B., Trace Ideals and Their Applications, Cambridge University Press, 1979.
Simons S., The least slope of a convex function and the maximal monotonicity of its subdifferential, J. Optim. Theory Appl. 71 (1991), 127–136.
Stegall C., A class of topological spaces and differentiation of functions between Banach spaces, in: Proc. Conf. on Vector Measures and Integral Representations of Operators (W. Ruess, ed.), Vorlesungen Fachbereich Math. Univ. Essen 10, 1983, 63–77.
Stromberg K. R., An Introduction to Classical Real Analysis, Wadsworth International Mathematics Series, 1981.
Sussmann H. J., A strong version of the maximum principle under weak hypotheses, in: Proc. 33rd IEEE Conf. Decision and Control (Lake Buena Vista, FL, 1994).
Sussmann H. J., Multidifferential calculus: chain rule, open mapping and transversal intersection theorems, to appear in: Proc. IFIP Conf. on Optimal Control: Theory Algorithms, and Applications (Gainesville, 1997) (W. Hager, ed.).
Thibault L., On Generalized differentials and subdifferentials of Lipschitz vector-valued functions, Nonlinear Anal. 6 (1982), 1037–1053.
Thibault L., A note on the Zagrodny mean value theorem, Optimization 35 (1995), 127–130.
Thibault L., Zagrodny D., Integration of subdifferentials of lower semicontinuous functions on Banach spaces, J. Math. Anal. Appl. 189 (1995), 33–58.
Treiman J. S., Characterization of Clarke’s tangent and normal cones in finite and infinite dimensions, Nonlinear Anal. 7 (1983), 771–783.
Treiman J. S., Clarke’s Gradients and epsilon-subgradients in Banach spaces, Trans. Amer. Math. Soc. 294 (1986), 65–78.
Treiman J. S., Finite dimensional optimality conditions: B-gradients, J. Optitn. Theory Appl. 62 (1989), 139–150.
Treiman J. S., Optimal control with small generalized gradients, SIAM J. Control Optim. 28 (1990), 720–732.
Valadier M., “Entraînement unilatéral, lignes de descente, fonctions lipschitziennes non pathologiques,” C. R. Acad. Sci. Paris Sér. I. Math. 308 (1989), 241–244.
Vanderwerff J. Private communication. September, 1996.
Vanderwerff J., Zhu Q. J., A limiting example for the local “fuzzy” sum rule in nonsmooth analysis, CECM Research Report 96-083 (1996), to appear in Proc. Amer. Math. Soc.
Vlasov L. P., Almost convex and Chebychev sets, Mat. Zametki 8 (1970), 545–550 (Russian); Math. Notes 8 (1970), 776–779.
Wang X. F., Distance function on any Hilbert space are δ convex, preprint.
Warga J., Derivate containers, inverse functions, and controllability, in Calculus of Variations and Control Theory (D.L. Russell, ed.), Math. Res. Center Univ. Wisconsin Publ. 36, Academic Press, New York, 1976, 13–45.
Warga J., Fat homeomorphisms and unbounded derivate containers, J. Math. Anal. Appl. 81 (1981), 545–560.
L. Zajíček, On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function, Comment. Math. Univ. Carolin. 38 (1997), 329–336.
Zizler V., Some notes on various rotundity and smoothness properties of separable Banach spaces, Comment Math. Univ. Carolin. 10 (1967), 195–206.
Zhu Q. J., The Clarke-Ledyaev mean value inequality in smooth Banach spaces, Nonlinear Anal. 32 (1998), 315–324.
Zhu Q. J., Subderivatives and their applications, in: Proc. Internat. Conf. on Dynamical Systems and Differential Equations (Springfield, MO, 1996).
Zhu Q. J., The equivalence of several basic theorems for sub differentials, CECM Research Report 97-093 (1997), to appear in Set-Valued Analysis.
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Borwein, J.M., Zhu, Q.J. (1999). Multifunctional and functional analytic techniques in nonsmooth analysis. In: Clarke, F.H., Stern, R.J., Sabidussi, G. (eds) Nonlinear Analysis, Differential Equations and Control. NATO Science Series, vol 528. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4560-2_2
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