Mild solutions of a class of semilinear fractional integro-differential equations subjected to noncompact nonlocal initial conditions

• Autores: Abdeldjalil Aouane, Smail Djebali, Mohamed Aziz Taoudi
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 22, Nº. 3, 2020, págs. 361-377
• Idioma: inglés
• DOI: 10.4067/S0719-06462020000300361
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• Resumen
  • español

    Resumen En este artículo, probamos la existencia de soluciones leves de una clase de ecuaciones integro-diferenciales fraccionales semilineales de orden β ∈ (1, 2] con condiciones nocompactas iniciales no-locales. Asumimos que la parte lineal genera una familia coseno de orden fraccional β arbitrariamente fuertemente continua, mientras que el término no-lineal de forzamiento es de tipo Carathéodory y satisface algunas condiciones de crecimiento bastante generales. Nuestro enfoque combina el teorema de punto fijo de Monch con algunos resultados recientes sobre la medida de no-compacidad de operadores integrales. Nuestras conclusiones mejoran y generalizan muchos trabajos anteriores relacionados. Se provee un ejemplo para ilustrar los resultados principales.

  • English

    Abstract In this paper, we prove the existence of mild solutions of a class of fractional semilinear integro-differential equations of order β ∈ (1, 2] subjected to noncompact initial nonlocal conditions. We assume that the linear part generates an arbitrarily strongly continuous β-order fractional cosine family, while the nonlinear forcing term is of Carath´eodory type and satisfies some fairly general growth conditions. Our approach combines the Monch fixed point theorem with some recent results regarding the measure of noncompactness of integral operators. Our conclusions improve and generalize many earlier related works. An example is provided to illustrate the main results.

• Referencias bibliográficas
  • Araya, D.,Lizama, C.. (2008). Almost automorphic mild solutions to fractional differential equations. Nonlinear Analysis. 69. 3692
  • Baleanu, D.,Diethelm, K.,Scalas, E.,Trujillo, J. J.. (2012). Fractional Calculus Models and Numerical Methods. World Scientific Publishing....
  • Banas, J.,Goebel, K.. (1980). Measure of Noncompactness in Banach Spaces. Marcel Dekker Inc.. New York.
  • Bazhlekova, E.. (2001). Fractional Evolution Equations in Banach Spaces. University Press Facilities, Eindhoven University of Technology.
  • Das, S.. (2008). Functional Fractional Calculus for System Identification and Controls. Springer-Verlag. Berlin.
  • Delbosco, D.,Rodino, L.. (1996). Existence and uniqueness for a fractional differential equation. J. Math. Anal. Appl.. 204. 609
  • Ezzinbi, K.,Ghnimi, S.,Taoudi, M. A.. (2019). Existence results for some nonlocal partial integrodifferential equations without compactness...
  • Gafiychuk, V.,Datsko, B.,Meleshko, V. V.. (2019). Mathematical modeling of time fractional reaction-diffusion systems. J. Comput. Appl. Math.....
  • Heinz, H. P.. (1983). On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions....
  • Hilfer, R.. (2000). Applications of Fractional Calculus in Physics. World Scientific. Singapore.
  • Ji, S.,Li, G.. (2013). Solutions to nonlocal fractional differential equations using a noncompact semigroup. Electron. J. Differ. Equ.. 2013....
  • Kamenskii, M.,Obukhovskii, V.,Zekka, P.. (2001). Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. De Gruyter....
  • Li, K.,Peng, J.,Gao, J.. (2012). Existence results for semilinear fractional differential equations via Kuratowski measure of noncompactness....
  • Li, K.,Peng, J.,Gao, J.. (2013). Nonlocal fractional semilinear differential equations in separable Banach spaces. Electron. J. Differential...
  • Li, K.,Peng, J.,Gao, J.. (2013). Controllability of nonlocal fractional differential systems of order α ∈ (1, 2] in Banach spaces. Rep. Math....
  • Matar, M. M.. (2009). Existence and uniquness of solutions to fractional semilinear mixed VolterraFredholm integrodifferential equations with...
  • Mönch, H.. (1980). Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal.....
  • Mophou, G. M.. (2010). Almost automorphic solutions of some semilinear fractional differential equations. Int. J. Evol. Equ.. 5. 109
  • Muslim, M.. (2009). Existence and approximation of solutions to fractional differential equations. Mathematical and Computer Modelling. 49....
  • Qin, H.,Zuo, X.,Liu, J.,Liu, L.. (2015). Approximate controllability and optimal controls of fractional dynamical systems of order 1 <...
  • Shu, X. B.,Wang, Q.. (2012). The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions...
  • Tarasov, V. E.. (2010). Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles. Fields and Media. Springer, New...
  • Travis, C. C.,Webb, G. F.. (1977). Compactness, regularity, and uniform continuity properties of strongly continuous cosine families. Houston...
  • Travis, C. C.,Webb, G. F.. (1978). Cosine families and abstract nonlinear second order differential equations. Acta Math. Acad. Sci. Hungar....
  • Zhu, B.,Liu, L.,Wu, Y.. (2017). Local and global existence of mild solutions for a class of semilinear fractional integro-differential equations....