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On the convergence of sequence of maximal monotone operators of type (D) in Banach spaces

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Abstract

Within the setting of general real Banach spaces, we prove that the sequence of maximal monotone operators of type (D) graphically converges provided, their corresponding class of representative functions converge epigraphically. Moreover, we provide a condition to guarantee that the lower limit of a sequence of maximal monotone operators of type (D) is a maximal monotone operator of type (D) in real Banach spaces.

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References

  1. Aftabizadeh, A.R., Pavel, N.H.: Nonlinear boundary value problems for some ordinary and partial differential equations associated with monotone operators. J. Math. Anal. 156, 535–557 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. Apreutesei, N.C.: Second order differential equations on half-line associated with monotone operators. J. Math. Anal. 223, 472–493 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Attouch, H., Baillon, J.-B., Thera, M.: Variational sum of monotone operators. J. Convex Anal. 1(1), 1–29 (1994)

    MathSciNet  MATH  Google Scholar 

  4. Brézis, H.: Monotone operators, nonlinear semigroups, and applications, In: Proceedings of the International Congress Mathematicians, Vancouver, pp. 249-255 (1974)

  5. Brézis, H., Haraux, A.: Image d’une somme d’op\(\acute{e}\)rateurs monotones et applications. Isr. J. Math. 23, 165–186 (1976)

    Article  MATH  Google Scholar 

  6. Brézis, H., Crandall, M.G., Pazy, A.: Perturbations of nonlinear maximal monotone sets in banach spaces. Commun. Pure Appl. Math. XXIII, 123–144 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bueno, O., García, Y., Marques Alves, M.: Lower limit of type (D) monotone operators in general Banach spaces. J. Convex Anal. 23(2), 333–345 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Calvert, B.D., Gupta, C.P.: Nonlinear elliptic boundary value problems in \(L^p\) spaces and sums of ranges of accretive operators. Nonlinear Anal. Theory Methods Appl. 2, 1–26 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fitzpatrick, S.: Representing monotone operators by convex functions, In: Workshop/Mini-conference on Functional Analysis and Optimization (Canberra 1988). Proceedings of the Centre for Mathematical Analysis, Australian National University, Canberra, Australia, vol. 20, pp. 59–65 (1988)

  10. García, Y., Lassonde, M.: Representable monotone operators and limits of sequence of maximal monotone operators. Set Valued Anal. 20, 61–73 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. García, Y., Lassonde, M.: Lower limit of maximal operators driven by their representative functions, Optim. Lett. (2018). https://doi.org/10.1007/s11590-018-1279-1

  12. Gossez, J.-P.: On a convexity property of the range of a maximal monotone operator. Proc. AMS 55, 359–360 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gossez, J.-P.: Operateurs monotones non lineaires dans les espaces de Banach nonreflexivs. J. Math. Anal. Appl. 34, 371–395 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gupta, C.P.: Sum of ranges of operators and applications. In: Nonlinear Systems and Applications, Academic Press, New York, pp. 547–559 (1997)

  15. Gupta, C.P., Hess, P.: Existence theorems for nonlinear non-coercieve operator equations and nonlinear elliptic boundary value problems. J. Differ. Equ. 22, 305–313 (1976)

    Article  MATH  Google Scholar 

  16. Marques Alves, M., Svaiter, B.F.: On the surjectivity properties of perturbations of maximal monotone operators in non-reflexive Banach spaces. J. Convex Anal. 18(1), 209–226 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Marques Alves, M., Svaiter, B.F.: Brondsted–Rockafellar property and maximality of monotone operators representable by convex functions in non-reflexive Banach spaces. J. Convex Anal. 15(4), 693–706 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Marques Alves, M., Svaiter, B.F.: A new old class of maximal monotone operators. J. Convex Anal. 16(4), 881–890 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Martinez-LegaZ, J.E., Svaiter, B.F.: Monotone operators representable by l.s.c. convex functions. Set Valued Anal. 2(2), 21–46 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Minty, G.J.: Monotone networks. Proc. R. Soc. Lond. 257, 194–212 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pennanen, T., Revalski, J.P., Thera, M.: Variational composition of a monotone operator and a linear mapping with applications to elliptic PDE’s with singular coefficients. J. Funct. Anal. 198(1), 84–105 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Revalski, J.P., Thera, M.: Generalized sums of monotone operators. C. R. Acad. Sci. Paris Ser. I Math. 329(11), 979–984 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Revalski, J.P., Thera, M.: Variational and extended sums of monotone operators, In: Illposed Variational Problems and Regularization Techniques (Trier, 1998). Lecture Notes in Economics and Mathematics Systems, vol. 477, Springer, Berlin, pp. 229–246 (1999)

  24. Rockafellar, R.T., Wets, R.J-B : Variational analysis. Grundlehren der mathematischen Wissenschaften (1998)

  25. Simons, S.: The range of a monotone operator. J. Math. Anal. Appl. 199, 176–201 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Simons, S.: From Hahn–Banach to Monotonicity. Springer, Berlin (2008)

    MATH  Google Scholar 

  27. Zeidler, E.: Nonlinear Functional Analysis and Its Applications: Part 2 A: Linear Monotone Operators and Part 2 B: Nonlinear Monotone Operators. Springer, Berlin (1990)

    MATH  Google Scholar 

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  1. Department of Mathematics, NIT Rourkela, Rourkela, India

    S. R. Pattanaik & D. K. Pradhan

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  1. S. R. Pattanaik
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  2. D. K. Pradhan
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Correspondence to S. R. Pattanaik.

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Pattanaik, S.R., Pradhan, D.K. On the convergence of sequence of maximal monotone operators of type (D) in Banach spaces. Positivity 23, 1009–1020 (2019). https://doi.org/10.1007/s11117-019-00648-6

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