Compression–Expansion Fixed Point Theorems for Decomposable Maps and Applications to Discontinuous ϕ-Laplacian problems

  1. Rodríguez-López, Jorge 1
  1. 1 Universidade de Santiago de Compostela
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    Universidade de Santiago de Compostela

    Santiago de Compostela, España

    ROR https://ror.org/030eybx10

Revista:
Qualitative theory of dynamical systems

ISSN: 1575-5460

Ano de publicación: 2021

Volume: 20

Número: 3

Tipo: Artigo

DOI: 10.1007/S12346-021-00505-6 DIALNET GOOGLE SCHOLAR lock_openAcceso aberto editor

Outras publicacións en: Qualitative theory of dynamical systems

Resumo

In this paper, we prove new compression–expansion type fixed point theorems in cones for the so-called decomposable maps, that is, compositions of two upper semicontinuous multivalued maps. As an application, we obtain existence and localization of positive solutions for a differential equation with ϕ-Laplacian and discontinuous nonlinearity subject to multi-point boundary conditions. As far as we are aware, the existence results are new even in the classical case of continuous nonlinearities.

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Referencias bibliográficas

  • 1. Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18(4), 620–709 (1976)
  • 2. Avery, R.I.: A generalization of the Leggett–Williams fixed point theorem. Math. Sci. Res. Hot-Line 3, 9–14 (1999)
  • 3. Avery, R.I., Peterson, A.C.: Three positive fixed points of nonlinear operators on ordered Banach spaces. Comput. Math. Appl. 42, 313–322 (2001)
  • 4. Bereanu, C., Jebelean, P., Torres, P.J.: Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space. J. Funct. Anal. 264, 270–287 (2013)
  • 5. Bereanu, C., Mawhin, J.: Existence and multiplicity results for some nonlinear problems with singular φ-Laplacian. J. Differ. Equ. 243(2), 536–557 (2007)
  • 6. Bonanno, G., Iannizzotto, A., Marras, M.: On ordinary differential inclusions with mixed boundary conditions. Differ. Integral Equ. 30, 273–288 (2017)
  • 7. Bonanno, G., Jebelean, P., ¸Serban, C.: Three periodic solutions for discontinuous perturbations of the vector p-Laplacian operator. Proc. R. Soc. Edinb. Sect. A 147, 673–681 (2017)
  • 8. Cellina, A., Fryszkowski, A., Rzezuchowski, T.: Upper semicontinuity of Nemytskij operators. Ann. Mat. Pura Appl. 160(4), 321–330 (1991)
  • 9. Chinní, A., Di Bella, B., Jebelean, P., Precup, R.: A four-point boundary value problem with singular φ-Laplacian. J. Fixed Point Theory Appl. 21(66), 1–16 (2019)
  • 10. Cid, J. Á., Pouso, R. L.: Ordinary differential equations and systems with time-dependent discontinuity sets. Proc. R. Soc. Edinb. Sect. A, 134, 617–637 (2004)
  • 11. Couchouron, J.-F., Precup, R.: Homotopy method for positive solutions of p-Laplace inclusions. Topol. Methods Nonlinear Anal. 30, 157–169 (2007)
  • 12. Figueroa, R., Infante, G.: A Schauder-type theorem for discontinuous operators with applications to second-order BVPs. Fixed Point Theory Appl. 2016, 53 (2016)
  • 13. Fitzpatrick, P.M., Petryshyn, W.V.: Fixed point theorems and the fixed point index for multivalued mappings in cones. J. Lond. Math. Soc. 2(11), 75–85 (1975)
  • 14. Herlea, D.-R., O’Regan, D., Precup, R.: Harnack type inequalities and multiple solutions in cones of nonlinear problems. Z. Anal. Anwend. 39, 151–170 (2020)
  • 15. Herlea, D.-R., Precup, R.: Existence, localization and multiplicity of positive solutions to φ-Laplace equations and systems. Taiwan. J. Math. 20, 77–89 (2016)
  • 16. Hu, L.-G., Xu, J.: Positive solutions of nonhomogeneous boundary value problems for some nonlinear equation with φ-Laplacian. Bound. Value Probl. 2012, 130 (2012)
  • 17. Infante, G.: A short course on positive solutions of systems of ODEs via fixed point index. Lect. Notes Nonlinear Anal. 16, 93–140 (2017)
  • 18. Jebelean, P.,Mawhin, J., ¸Serban, C.: Periodic solutions for discontinuous perturbations of the relativistic operator. Bull. Sci. Math. 140, 99–117 (2016)
  • 19. Jebelean, P., ¸Serban, C.: Boundary value problems for discontinuous perturbations of singular φLaplacian operator. J. Math. Anal. Appl. 431, 662–681 (2015)
  • 20. Kryszewski, W., Maciejewski, M.: Degree for weakly upper semicontinuous perturbations of quasim-accretive operators. Phil. Trans. R. Soc. A 379, 20190377 (2021)
  • 21. Lan, K.Q.: Multiple positive solutions of semilinear differential equations with singularities. J. Lond. Math. Soc. 63, 690–704 (2001)
  • 22. Leggett, R.W., Williams, L.R.: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J. 28(4), 673–688 (1979)
  • 23. Pouso, R.L.: Schauder’s fixed-point theorem: new applications and a new version for discontinuous operators. Bound. Value Probl. 2012, 92 (2012)
  • 24. Precup, R.: Fixed point theorems for decomposable multi-valued maps and applications. Z. Anal. Anwend. 22, 843–861 (2003)
  • 25. Precup, R., Rodríguez-López, J.: Positive solutions for discontinuous problems with applications to φ-Laplacian equations. J. Fixed Point Theory Appl. 20(156), 1–17 (2018)
  • 26. Precup, R., Rodríguez-López, J.: Positive solutions for φ-Laplace equations with discontinuous statedependent forcing terms. Nonlinear Anal. Model. Control 24, 447–461 (2019)
  • 27. Precup, R., Rodríguez-López, J.: Fixed point index theory for decomposable multivalued maps and applications to discontinuous φ-Laplacian problems. Nonlinear Anal. 199, 111958 (2020)
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