Maximum, anti-maximum principles and monotone methods for boundary value problems for Riemann-Liouville fractional differential equations in neighborhoods of simple eigenvalues

• Autores: Paul W. Eloe, Jeffrey T. Neugebauer
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 25, Nº. 2, 2023, págs. 251-272
• Idioma: inglés
• DOI: 10.56754/0719-0646.2502.251
• Enlaces
  • Texto completo
• Resumen
  • español

    Resumen Se ha demostrado que, bajo hipótesis apropiadas, problemas de valor en la frontera de la forma Ly + λy = f, BCy = 0, donde L es un operador diferencial lineal ordinario o parcial y BC denota un operador lineal de frontera, entonces existe Λ > 0 tal que f ≥ 0 implica λy ≥ 0 para λ ∈ [−Λ, Λ] \ {0}, donde y es la única solución de Ly + λy = f, BCy = 0. Así, el problema de valor en la frontera satisface un principio del máximo para λ ∈ [−Λ, 0) y el problema de valor en la frontera satisface un anti-principio del máximo para λ ∈ (0, Λ]. En un resultado abstracto, entregaremos hipótesis apropiadas tales que los problemas de valor en la frontera de la forma Dα 0 y + βDα−1 0 y = f, BCy = 0 donde Dα 0 es un operador diferencial fraccionario de Riemann-Liouville de orden α, 1 < α ≤ 2, y BC denota un operador lineal de frontera, entonces existe B > 0 tal que f ≥ 0 implica βDα−1 0 y ≥ 0 para β ∈ [−B, B] \ {0}, donde y es la única solución de Dα 0 y + βDα−1 0 y = f, BCy = 0. Se entregan dos ejemplos en los cuales las hipótesis del teorema abstracto se satisfacen para obtener la propiedad de signo de βDα−1 0y. Las condiciones de frontera se eligen de tal forma de obtener también una propiedad de signo para βy con un análisis adicional. Se desarrolla una aplicación de métodos monótonos para ilustrar la utilidad del resultado abstracto.

  • English

    Abstract It has been shown that, under suitable hypotheses, boundary value problems of the form, Ly + λy = f, BCy = 0 where L is a linear ordinary or partial differential operator and BC denotes a linear boundary operator, then there exists Λ > 0 such that f ≥ 0 implies λy ≥ 0 for λ ∈ [−Λ, Λ] \ {0}, where y is the unique solution of Ly + λy = f, BCy = 0. So, the boundary value problem satisfies a maximum principle for λ ∈ [−Λ, 0) and the boundary value problem satisfies an anti-maximum principle for λ ∈ (0, Λ]. In an abstract result, we shall provide suitable hypotheses such that boundary value problems of the form, Dα 0 y + βDα−1 0 y = f, BCy = 0 where Dα 0 is a Riemann-Liouville fractional differentiable operator of order α, 1 < α ≤ 2, and BC denotes a linear boundary operator, then there exists B > 0 such that f ≥ 0 implies βDα−1 0 y ≥ 0 for β ∈ [−B, B] \ {0}, where y is the unique solution of Dα 0 y+βDα−1 0y = f, BCy = 0. Two examples are provided in which the hypotheses of the abstract theorem are satisfied to obtain the sign property of βDα−1 0y. The boundary conditions are chosen so that with further analysis a sign property of βy is also obtained. One application of monotone methods is developed to illustrate the utility of the abstract result.

• Referencias bibliográficas
  • Alziary, B.,Fleckinger, J.,Takáč, P.. (1999). An extension of maximum and anti-maximum principles to a Schrödinger equation in Rn. J. Differential...
  • Arcoya, D.,Gámez, J. L.. (2001). Bifurcation theory and related problems: anti-maximum principle and resonance. Comm. Partial Differential...
  • Bai, Z.,Lü, H.. (2015). Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl.....
  • Barteneva, I. V.,Cabada, A.,Ignatyev, A. O.. (2003). Maximum and anti-maximum principles for the general operator of second order with variable...
  • Cabada, A.,Cid, J. Á.. (2012). On comparison principles for the periodic Hill’s equation. J. Lond. Math. Soc. (2). 86. 272
  • Cabada, A.,Cid, J. Á. (2018). Maximum Principles for the Hill’s Equation. Academic Press. London, England.
  • Cabada, A.,Cid, J. Á,Tvrdý, M.. (2010). A generalized anti-maximum principle for the periodic on-dimensional p−Lapacian with sign changing...
  • Campos, J.,Mawhin, J.,Ortega, R.. (2008). Maximum principles around an eigenvalue with constant eigenfunctions. Commun. Contemp. Math.. 10....
  • Clément, Ph.,Peletier, L. A.. (1979). An anti-maximum principle for second-order elliptic operators. J. Differential Equations. 34. 218
  • Clément, Ph.,Sweers, G.. (2000). Uniform anti-maximum principles. J. Differential Equations. 164. 118
  • Diethelm, K.. (2010). The analysis of fractional differential equations. Springer-Verlag. Berlin, Germany.
  • Eloe, P.,Jonnalagadda, J.. (2020). Quasilinearization applied to boundary value problems at resonance for Riemann-Liouville fractional differential...
  • Eloe, P.,Neugebauer, J.. Maximum and anti-maximum principles for boundary value problems for ordinary differential equations in neighborhoods...
  • Hess, P.. (1981). An antimaximum principle for linear elliptic equations with an indefinite weight function. J. Differential Equations. 41....
  • Infante, G.,Pietramala, P.,Tojo, F. A. F.. (2016). Nontrivial solutions of local and nonlocal Neumann boundary value problems. Proc. Roy....
  • Kilbas, A. A.,Srivastava, H. M.,Trujillo, J. J.. (2006). Theory and Applications of Fractional Differential Equations. Elsevier Science B.V.....
  • Mawhin, J.. (2009). Nonlinear elliptic partial differential equations. American Mathematical Society. Providence, RI, USA.
  • Pinchover, Y.. (1999). Maximum and anti-maximum principles and eigenfunctions estimates via perturbation theory of positive solutions of elliptic...
  • Protter, M. H.,Weinberger, H.. (1967). Maximum Principles in Differential Equations. Prentice-Hall, Inc.. Englewoods Cliffs, NJ, USA.
  • Samko, S. G.,Kilbas, A. A.,Marichev, O. I.. (1993). Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science...
  • Takáč, P.. (1996). An abstract form of maximum and anti-maximum principles of Hopf’s type. J. Math. Anal. Appl.. 201. 339
  • Zhang, M.. (2010). Optimal conditions for maximum and antimaximum principles of the periodic solution problem. Bound. Value Probl.. 2010.
  • Webb, J. R. L.. (2019). Initial value problems for Caputo fractional equations with singular nonlinearities. Electron. J. Differential Equations....
  • Zhang, S. Q.. (2000). The existence of a positive solution for a nonlinear fractional differential equation. J. Math. Anal. Appl.. 252. 804