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Φ-Convex Functions Defined on Metric Spaces

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REFERENCES

  1. E. Asplund, “Farthest points in reflexive locally uniformly rotund Banach spaces,” Isr. J. Math. 4, 213–216 (1966).

    Google Scholar 

  2. E. Asplund, “Fréchet differentiability of convex functions,” Acta Math. 121, 31–47 (1968).

    Google Scholar 

  3. E. J. Balder, “An extension of duality-stability relations to nonconvex optimization problems,” SIAM J. Control Optimization 15, 329–343 (1977).

    Google Scholar 

  4. B. Choczewski, “Note on a functional-differential inequality,” In: Functional Equations: Results and Advances(Z. Daróczy and Zs. Páles, eds.), Kluwer Academic Publ., Dordrecht–Boston–London (2001), pp. 21–24.

    Google Scholar 

  5. B. Choczewski, R. Girgensohn, and Z. Kominek, “On S. Rolewicz's problem connected with Φ-subdifferentials in metric spaces,” SIAM Electr. Probl.(to appear).

  6. R. Correa, A. Jofré, and L. Thibault, “Subdifferential monotonicity as characterization of convex functions,” Numer. Funct. Anal. Optim. 15, 531–536 (1994).

    Google Scholar 

  7. L. Danzer, B. Grünbaum, and V. Klee, “Helly's theorem and its relatives,” In: Convexity, Proc. Symp. Pure Math. 7, Amer. Math. Soc., Providence (1963), pp. 101–180.

    Google Scholar 

  8. S. Dolecki and S. Kurcyusz, “On Φ-convexity in extremal problems,” SIAM J. Control Optimization 16, 277–300 (1978).

    Google Scholar 

  9. K. H. Elster and R. Nehse, “Zur Theorie der Polarfunktionale,” In: Math. Operationsforsch. und Stat. Ser. Optimization 5(1974), pp. 3–21.

    Google Scholar 

  10. K. Fan, “On the Krein–Milman theorem,” In: Convexity, Proc. Symp. Pure Math. 7, Amer. Math. Soc., Providence (1963), pp. 211–220.

    Google Scholar 

  11. P. C. Hammer, “Semispaces and the topology of convexity,” In: Convexity, Proc. Symp. Pure Math. 7, Amer. Math. Soc., Providence (1963), pp. 305–316.

    Google Scholar 

  12. A. Jofré, D. T. Luc, and M. Théra, “ε-Subdifferential and ε-monotonicity,” Nonlin. Anal. 33, 71–90 (1998).

    Google Scholar 

  13. A. Jourani, “Subdifferentiability and subdifferential monotonicity of γ-paraconvex functions,” Control Cyber. 25, 721–737 (1996).

    Google Scholar 

  14. V. Klee, “Convex sets in linear spaces,” Duke Math. J. 18, 443–466 (1951).

    Google Scholar 

  15. K. Kuratowski, “Sur l'operation Ade l'analyse situs,” Fundam. Math. 3, 182–199 (1922).

    Google Scholar 

  16. S. Kurcyusz, “Some remarks on generalized Lagrangians,” In: Proc. 7th IFIP Conf. Optimiz., Nice, September 1975, Springer-Verlag (1976).

  17. S. Kurcyusz, “On existence and nonexistence of Lagrange multipliers in Banach spaces,” J. Optimization Theory Appl. 20, 81–110 (1976).

    Google Scholar 

  18. S. S. Kutateladze and A. M. Rubinov, “Some classes of H-convex functions and sets,” Dokl. Akad. Nauk SSSR, Ser. Mat. 12, 665–668 (1971).

    Google Scholar 

  19. S. S. Kutateladze and A. M. Rubinov, “Minkowski duality and its applications,” Usp. Mat. Nauk 27, No. 3, 137–192 (1972).

    Google Scholar 

  20. S. S. Kutateladze and A. M. Rubinov, Minkowski Duality and Its Applications[in Russian], Nauka, Novosibirsk (1976).

  21. V. L. Levin, “Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem,” In: Set-Valued Analysis, Vol. 7 (1999), pp. 7–32.

    Google Scholar 

  22. D. T. Luc, H. V. Ngai, and M. Théra, “On ε-convexity and ε-monotonicity,” In: Calculus of Variation and Differential Equations(A. Ioffe, S. Reich, and I. Shapiro, Eds.), Chapman & Hall (1999), pp. 82–100.

  23. D. T. Luc, H. V. Ngai, and M. Théra, “Approximative convex functions,” J. Nonlin. Convex Anal. 1, 155–176 (2000).

    Google Scholar 

  24. S. Matsumura (Nakajima), “über convexe Kurven und Flächen,” Tohoku Math. J. 29, 227–230 (1928).

    Google Scholar 

  25. S. Mazur, “über konvexe Menge in lineare normierte Raümen,” Stud. Math. 4, 70–84 (1933).

    Google Scholar 

  26. K. Menger, “Untersuchen über allgemeine Metrik, I, II, III,” Math. Ann. 100, 75–163 (1928).

    Google Scholar 

  27. G. J. Minty, “On the monotonicity of the gradients of convex functions,” Pacif. J. Math. 14, 243–247 (1964).

    Google Scholar 

  28. D. Pallaschke and S. Rolewicz, Foundation of Mathematical Optimization, Math. Appl., Vol. 388, Kluwer Academic Publ., Dordrecht–Boston–London (1997).

    Google Scholar 

  29. R. R. Phelps, Convex Functions, Monotone Operators, and Differentiability, Lect. Notes Math., Vol. 1364, Springer-Verlag (1989).

  30. D. Preiss and L. Zajicek, “Stronger estimates of smallness of sets of Fréchet nondifferentiability of convex functions,” In: Proc. 11th Winter School, Suppl. Rend. Circ. Mat. Palermo, Ser. II 3(1984), pp. 219–223.

    Google Scholar 

  31. R. T. Rockafellar, Convex Analysis, Princeton Univ. Press (1970).

  32. R. T. Rockafellar, “On the maximal monotonicity of subdifferential mappings,” Pacif. J. Math. 33, 209–216 (1970).

    Google Scholar 

  33. R. T. Rockafellar, “Generalized directional derivatives and subgradients of nonconvex functions,” Can. J. Math. 32, 257–280 (1980).

    Google Scholar 

  34. S. Rolewicz, “On paraconvex multifunctions,” Oper. Res. Verf.(Methods Oper. Res.)31, 540–546 (1979).

    Google Scholar 

  35. S. Rolewicz, “On γ-paraconvex multifunctions,” Math. Jpn. 24, 293–300 (1979).

    Google Scholar 

  36. S. Rolewicz, “On conditions warranting Φ2-subdifferentiability,” Stud. Math. Progr. 14, 215–224 (1980).

    Google Scholar 

  37. S. Rolewicz, “On Asplund inequalities for Lipschitz functions,” Arch. Math. 61, 484–488 (1993).

    Google Scholar 

  38. S. Rolewicz, “On the globalization property,” Appl. Math. 22, 69–73 (1993).

    Google Scholar 

  39. S. Rolewicz, “On Mazur theorem for Lipschitz functions,” Arch. Math. 63, 535–540 (1994).

    Google Scholar 

  40. S. Rolewicz, “On Φ-differentiability of functions over metric spaces,” In: Topological Methods of Nonlinear Analysis(1995), pp. 229–236.

  41. S. Rolewicz, “On subdifferentials on nonconvex sets,” In: Different Aspects of Differentiability(D. Przeworska-Rolewicz, Ed.), Dissertationes Math., Vol. 340 (1995), pp. 301–308.

  42. S. Rolewicz, “Duality and convex analysis in the absence of linear structure,” Math. Jpn. 44, 165–182 (1996).

    Google Scholar 

  43. S. Rolewicz, “On approximation of functions on metric spaces,” Folia Math. Univ. Lodziensis 8, 99–108 (1996).

    Google Scholar 

  44. S. Rolewicz, “Locally monotone operators in spaces without linearity,” In: Lect. Notes Econ. Math. Syst., Vol. 452, Springer-Verlag (1997). pp. 292–297.

    Google Scholar 

  45. S. Rolewicz, “On uniformly Φ-convex functions and strongly monotone multifunctions,” Funct. Approx. 26, 231–238 (1998).

    Google Scholar 

  46. S. Rolewicz, “On α(·)-monotone multifunctions and differentiability of γ-paraconvex functions,” Stud. Math. 133, 29–37 (1999).

    Google Scholar 

  47. S. Rolewicz, “On cyclic α(·)-monotone multifunctions,” Stud. Math. 141, 263–272 (2000).

    Google Scholar 

  48. S. Rolewicz, “On α(·)-paraconvex and strongly α(·)-paraconvex functions,” Control Cyber. 29, 367–377 (2000).

    Google Scholar 

  49. S. Rolewicz, “On equivalence of Clarke, Dini, α(·)-subgradients and local α(·)-subgradients for strongly α(·)-paraconvex functions,” Optimization(to appear).

  50. S. Rolewicz, “On uniformly approximate convex and strongly α(·)-paraconvex functions,” Control Cyber.(to appear).

  51. A. M. Rubinov, Abstract Convexity and Global Optimization, Nonconvex Optimization and Its Applications, Vol. 44, Kluwer Academic Publ. (2000).

  52. A. M. Rubinov, “Abstract convexity: Examples and applications,” Optimization 47, 1–33 (2000).

    Google Scholar 

  53. I. Singer, Abstract Convex Analysis, Wiley (1997)

  54. V. P. Soltan, Introduction to Axiomatic Theory of Convexity[in Russian], Štiinca, Kishiniev (1984).

  55. V. P. Soltan and P. S. Soltan, “d-Convex functions,” Dokl. Akad. Nauk SSSR, Ser. Mat. 249, 555–568 (1979).

    Google Scholar 

  56. H. Tietze, “Ñber Konvexheit im kleinem und im grossen und Ñber gewisse der Punkten einer Menge zugeordnete Dimensionszahlen,” Math. Z. 28, 697–707 (1928).

    Google Scholar 

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Rolewicz, S. Φ-Convex Functions Defined on Metric Spaces. Journal of Mathematical Sciences 115, 2631–2652 (2003). https://doi.org/10.1023/A:1023296417861

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