Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces
Volume 2018, Article ID 6783748, 5 pages
https://doi.org/10.1155/2018/6783748
Research Article

Uniform Boundedness Principle for Nonlinear Operators on Cones of Functions

1Faculty of Mechanical Engineering, University of Ljubljana, Aškerčeva 6, SI-1000 Ljubljana, Slovenia
2Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia

Correspondence should be addressed to Aljoša Peperko; is.jl-inu.sf@okrepep.asojla

Received 29 December 2017; Revised 7 March 2018; Accepted 15 March 2018; Published 23 April 2018

Academic Editor: Adrian Petrusel

Copyright © 2018 Aljoša Peperko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. B. Conway, A Course in Functional Analysis, vol. 96 of Graduate Texts in Mathematics, Springer-Verlag, 2nd edition, 1990. View at MathSciNet
  2. Y. A. Abramovich and C. D. Aliprantis, An Invitation to Operator Theory, American Mathematical Society, Providence, RI, USA, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  3. K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, 2000. View at MathSciNet
  4. C. D. Aliprantis and R. Tourky, Cones and Duality, vol. 84 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  5. J. Mallet-Paret and R. D. Nussbaum, “Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index,” Journal of Fixed Point Theory and Applications, vol. 7, no. 1, pp. 103–143, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. B. Lins and R. Nussbaum, “Denjoy-Wolff theorems, Hilbert metric nonexpansive maps and reproduction-decimation operators,” Journal of Functional Analysis, vol. 254, no. 9, pp. 2365–2386, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. C. D. Aliprantis and O. Burkinshaw, Positive Operators, Springer, Dordrecht, Netherlands, 2006, Reprint of the 1985 original.
  8. A. C. Zaanen, Riesz spaces II, vol. 30 of North-Holland Mathematical Library, North Holland, Amsterdam, 1983. View at MathSciNet
  9. A. Peperko, “Bounds on the joint and generalized spectral radius of the Hadamard geometric mean of bounded sets of positive kernel operators,” Linear Algebra and its Applications, vol. 533, pp. 418–427, 2017. View at Publisher · View at Google Scholar · View at Scopus
  10. R. Drnovšek and A. Peperko, “Inequalities on the spectral radius and the operator norm of hadamard products of positive operators on sequence spaces,” Banach Journal of Mathematical Analysis, vol. 10, no. 4, pp. 800–814, 2016. View at Publisher · View at Google Scholar · View at Scopus
  11. A. Peperko, “Inequalities on the spectral radius, operator norm and numerical radius of the Hadamard weighted geometric mean of positive kernel operators, 2016,” https://arxiv.org/abs/1612.01767.
  12. J. Appell, E. De Pascale, and A. Vignoli, Nonlinear Spectral Theory, Walter de Gruyter GmbH and Co. KG, Berlin, Germany, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  13. W. Wnuk, Banach Lattices with Order Continuous Norms, Polish Scientific Publishers PWN, Warszawa, Poland, 1999.
  14. C. D. Aliprantis, D. J. Brown, and O. Burkinshaw, Existence and Optimality of Competitive Equilibria, Springer-Verlag, Berlin, Germany, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  15. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I and II, Springer, 1996, A reprint of the 1977 and 1979 editions.
  16. M. de Jeu and M. Messerschmidt, “A strong open mapping theorem for surjections from cones onto Banach spaces,” Advances in Mathematics, vol. 259, pp. 43–66, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. V. Müller and A. Peperko, “On the bonsall cone spectral radius and the approximate point spectrum,” Discrete and Continuous Dynamical Systems- Series A, vol. 37, no. 10, pp. 5337–5354, 2017. View at Publisher · View at Google Scholar · View at Scopus
  18. V. Müller and A. Peperko, “Lower spectral radius and spectral mapping theorem for suprema preserving mappings,” https://arxiv.org/abs/1712.00340.
  19. J. Mallet-Paret and R. D. Nussbaum, “Eigenvalues for a class of homogeneous cone maps arising from max-plus operators,” Discrete and Continuous Dynamical Systems - Series A, vol. 8, no. 3, pp. 519–562, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  20. V. N. Kolokoltsov and V. P. Maslov, Idempotent analysis and Its applications, vol. 401 of Mathematics and Its Applications, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  21. M. Akian, S. Gaubert, and R. D. Nussbaum, A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones, 2011, https://arxiv.org/abs/1112.5968.
  22. M. K. Fijavž, A. Peperko, and E. Sikolya, “Semigroups of max-plus linear operators,” Semigroup Forum, vol. 94, no. 2, 2017. View at Publisher · View at Google Scholar · View at Scopus
  23. B. Andreianov, M. Kramar Fijavž, A. s. Peperko, and E. Sikolya, “Erratum to: Semigroups of max-plus linear operators,” Semigroup Forum, vol. 94, no. 2, pp. 477–479, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  24. R. B. Bapat, “A max version of the Perron-Frobenius theorem,” Linear Algebra and Its Applications, vol. 275, no. 276, pp. 3–18, 1998. View at Google Scholar
  25. P. Butkovic, Max-Linear Systems: Theory and Algorithms, Springer-Verlag, London, UK, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  26. V. Müller and A. Peperko, “On the spectrum in max algebra,” Linear Algebra and its Applications, vol. 485, pp. 250–266, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  27. V. Müller and A. Peperko, “Generalized spectral radius and its max algebra version,” Linear Algebra and its Applications, vol. 439, no. 4, pp. 1006–1016, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  28. N. Guglielmi, O. Mason, and F. Wirth, “Barabanov norms, Lipschitz continuity and monotonicity for the max algebraic joint spectral radius, 2017,” https://arxiv.org/abs/1705.02008.
  29. L. Pachter and B. Sturmfels, Algebraic Statistics for Computational Biology, L. Pachter and B. Sturmfels, Eds., Cambridge University Press, New York, NY, USA, 2005.
  30. H. m. Brezis and F. E. Browder, “Nonlinear integral equations and systems of Hammerstein type,” Advances in Mathematics, vol. 18, no. 2, pp. 115–147, 1975. View at Publisher · View at Google Scholar · View at MathSciNet
  31. H. A. Salem, “On the nonlinear Hammerstein integral equations in Banach spaces and application to the boundary value problem of fractional order,” Mathematical and Computer Modelling, vol. 48, no. 7-8, pp. 1178–1190, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. I. A. Ibrahim, “On the existence of solutions of functional integral equation of Urysohn type,” Computers & Mathematics with Applications, vol. 57, no. 10, pp. 1609–1614, 2009. View at Publisher · View at Google Scholar · View at Scopus
  33. E. I. Berezhnoj and E. I. Smirnov, “Countable Semiadditive Functionals and the HardyLittlewood Maximal Operator,” Studies in Mathematical Sciences, vol. 7, no. 2, pp. 55–60, 2013. View at Google Scholar
  34. T. Iida, “The boundedness of the Hardy-Littlewood maximal operator and multilinear maximal operator in weighted Morrey type spaces,” Journal of Function Spaces, Article ID 648251, 8 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  35. M. Mastylo and C. Pérez, “The Hardy-Littlewood maximal type operators between Banach function spaces,” Indiana University Mathematics Journal, vol. 61, no. 3, pp. 883–900, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  36. C. Niu, Z. Liu, and P. Wang, “Two-weight norm inequality for the one-sided Hardy-Littlewood maximal operators in variable LEBesgue spaces,” Journal of Function Spaces, Article ID 1648281, 8 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet