Mild solution for impulsive neutral fractional partial differential inclusions with nonlocal conditions

详细信息    查看全文

• 作者：Alka Chadha ; Dwijendra N. Pandey
• 关键词：Fractional calculus ; Caputo derivative ; Impulsive ; Resolvent operator ; Neutral fractional differential inclusion ; 26A33 ; 34K37 ; 34K40 ; 34K45 ; 35R11 ; 45J05 ; 45K05
• 刊名：Collectanea Mathematica
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：67
• 期：1
• 页码：85-111
• 全文大小：571 KB
• 参考文献：1.Benchohra, M., Henderson, J., Ntouyas, S.K.: Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications, vol. 2. Hindawi Publishing Corporation, New York (2006)CrossRef
  2.Lakshmikantham, V., Baǐnov, D., Simeonov, P.S.: Theory of impulsive differential equations. World Scientific, Singapore-London (1989)
  3.Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 497–505 (1991)MathSciNet CrossRef
  4.Byszewski, L., Lakshmikantham, V.: Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal. 40, 11–19 (1990)MathSciNet CrossRef
  5.Li, F., N’ Gu\(\acute{e}\) kata, G.M.: An existence result for neutral delay integrodifferential equations with fractional order and nonlocal conditions. In: Abstract and Applied Analysis, pp 20 (2011) (article id, 952782)
  6.Chang, Y.-K., Nieto, J.: Existence of solutions for impulsive neutral integro-differential inclusions with nonlocal initial conditions via fractional operators. Numer. Funct. Anal. Optim. 30, 227–244 (2009)MATH MathSciNet CrossRef
  7.Chang, Y.-K., Anguraj, A., Arjunan, M.M.: Existence results for impulsive neutral functional differential equations with infinite delay. Nonlinear Anal. Hybrid Syst. 2, 209218 (2008)MathSciNet CrossRef
  8.Benchohra, M., Ziane, M.: Impulsive evolution inclusions with state-dependent delay and multivalued jumps. Electron. J. Qual. Differ. Equ. 2013(42), 1–21 (2013)CrossRef
  9.Park, J.Y., Jeong, J.U.: Existence results for impulsive neutral stochastic functional integro-differential inclusions with infinite delays. Adv. Differ. Equ. 17 (2014)
  10.Yan, Z., Zhang, H.: Existence of solutions to impulsive fractional partial neutral stochastic integro-differential inclusions with state-dependent delay. Electron. J. Differ. Equ. 81, 1–21 (2013)
  11.Yan, Z., Zhang, H.: Asymptotic stability of fractional impulsive neutral stochastic partial integro-differential equations with state-dependent delay. Electron. J. Differ. Equ. 206, 1–29 (2013)
  12.Mophou, M.G.: Existence and uniqueness of mild solutions to implusive fractional differential equations. Nonlinear Anal. TMA 72, 1604–1615 (2010)MATH MathSciNet CrossRef
  13.Shu, X.B., Lai, Y., Chen, Y.: The existence of mild solutions for impulsive fractional partial differential equations. Nonlinear Anal. Theory Method Appl. 74, 2003–2011 (2011)MATH MathSciNet CrossRef
  14.Zhang, X., Huang, X., Liu, Z.: The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay. Noninear Anal. Hybrid Syst. 4, 775–781 (2010)MATH MathSciNet CrossRef
  15.Balachandran, K., Samuel, F.P.: Existence of mild solutions for quasilinear integrodifferential equations with impulsive conditions. Electron. J. Differ. Equ. 84, 1–9 (2009)
  16.Balachandran, K., Kiruthika, S., Trujillo, J.J.: Existence results for fractional impulsive integrodifferential equations in Banach spaces. Commun. Nonlinear Sci. Numer. Simul. 16, 1970–1977 (2011)MATH MathSciNet CrossRef
  17.Balachandran, K., Kiruthika, S., Trujillo, J.J.: On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces. Comput. Math. Appl. 62, 1157–1165 (2011)MATH MathSciNet CrossRef
  18.Agarwal, R.P., Benchohra, M., Slimani, B.A.: Existence results for differential equations with fractional order and impulses. Mem. Differ. Equ. Math. Phys. 44, 1–21 (2008)MATH MathSciNet CrossRef
  19.Henderson, J., Ouahab, A.: Impulsive differential inclusions with fractional order. Comput. Math. Appl. 59, 1191–1226 (2010)MATH MathSciNet CrossRef
  20.Abbas, M.I.: Existence for fractional order impulsive integrodifferential inclusions with nonlocal initial conditions. Int. J. Math. Anal. 6, 1813–1828 (2012)MATH
  21.Chauhan, A., Dabas, J.: Existence of mild solutions for impulsive fractional order semilinear evolution equations with nonlocal conditions. Electron. J. Differ. Equ. 2011, 1–10 (2011)MathSciNet
  22.Wang, J.R., Fec̆kan, M., Zhou, Y.: On the new concept of solution and existence results for impulsive fractional evolution equations. Dyn. PDE 8, 345–361 (2011)MATH MathSciNet
  23.Wang, J.R., Li, X., Wei, W.: On the natural solution of an impulsive fractional differential equation of order q\(\in \) (1,2). Commun. Nonlinear Sci. Numer. Simul. 17, 4384–4394 (2012)MATH MathSciNet CrossRef
  24.Liu, Y., Ahmad, B.: A study of impulsive multiterm fractional differential equations with single and multiple base points and applications. Sci. World J. 28 (2014) (Article ID 194346)
  25.Yan, Z., Jia, X.: Impulsive problems for fractional partial neutral functional integro-differential inclusions with infinite delay and analytic resolvent operators. Mediterr. J. Math. 36 (2013)
  26.Mahto, L., Abbas, S., Favini, A.: Analysis of Caputo impulsive fractional order differential equations with applications. Int. J. Differ. Equ. 2013, 11 (2013)MathSciNet CrossRef
  27.Pazy, A.: Semi-Groups of Linear Operator and Applications of Partial Differential Equations. Springer, New York (1983)CrossRef
  28.Podlubny, I.: Fractional Differential Equations. Acadmic press, New York (1993)
  29.Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)MATH
  30.Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publisher, Yverdon (1993)MATH
  31.Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)MATH
  32.Banas, J., Goebel, K.: Measure of noncompactness in Banach spaces. In: Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, New York (1980)
  33.Deimling, K.: Multivalued Differential Equations. de Gruyter, Berlin (1992)MATH CrossRef
  34.Kamenskii, M., Obukhovskii, V., Zecca, P.: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, vol. 7 of de Gruyter Series in Nonlinear Analysis and Applications. Walter de Gruyte, Berlin (2001)
  35.Akhmerov, R.R., Kamenskiǐ, M.I., Potapov, A.S., Rodkina, A.E., Sadovskiǐ, B.N.: Measures of noncompactness and condensing operators. Birkhäuser, Boston-Basel (1992)
  36.Agarwal, R.P., Santos, J.P.C., Cuevas, C.: Analytic resolvent operator and existence results for fractional order evolutionary integral equations. J. Abstr. Differ. Equ. Appl. 2, 26–47 (2012)MathSciNet
  37.de Andrade, B., Santos, J.P.C.: Existence of solutions for a fractional neutral integro-differential equation with unbounded delay. Electron. J. Differ. Equ. 2012, 1–13 (2012)CrossRef
  38.Araya, D., Lizama, C.: Almost automorphic mild solutions to fractional differential equations. Nonlinear Anal. TMA 69, 3692–3705 (2008)MATH MathSciNet CrossRef
  39.de Andrade, B., Santos, J.P.C.: Existence of solutions for a fractional neutral integro-differential equation with unbounded delay. Electron. J. Differ. Equ. 2012, 1–13 (2012)CrossRef
  40.Li, K., Peng, J., Jia, J.: Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives. J. Funct. Anal. 263, 476–510 (2012)MATH MathSciNet CrossRef
  41.Ezzinbia, K., Xianlong, F.: Existence and regularity of solutions for some neutral partial differential equations with nonlocal conditions. Nonlinear Anal. TMA 57, 1029–1041 (2004)CrossRef
  42.Ezzinbia, K., Xianlong, F., Hilal, K.: Existence and regularity in the \(\alpha \) -norm for some neutral partial differential equations with nonlocal conditions. Nonlinear Anal. TMA 67, 1613–1622 (2007)CrossRef
  43.Yosida, K.: Functional Analysis, 6th edn. Springer, Berlin (1980)MATH CrossRef
  44.Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)MATH CrossRef
  45.Dhage, B.C.: Fixed-point theorems for discontinuous multi-valued operators on ordered spaces with applications. Comput. Math. Appl. 51, 589–604 (2006)MATH MathSciNet CrossRef
  46.Lizama, C.: Regularized solutions for abstract Volterra equations. J. Math. Anal. Appl. 243, 278–292 (2000)MATH MathSciNet CrossRef
  
• 作者单位：Alka Chadha (1)
  Dwijendra N. Pandey (1)
  
  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algebra
  Analysis
  Applications of Mathematics
  Geometry
  
• 出版者：Springer Milan
• ISSN：2038-4815

文摘

In the present paper, we study the existence of a mild solution of a fractional order nonlocal differential inclusion with impulsive condition in a Banach space E. We obtain the sufficient condition for the existence of the mild solution by using a fixed point theorem for multi-valued operators due to Dhage and resolvent semigroup theory with approximate techniques. Keywords Fractional calculus Caputo derivative Impulsive Resolvent operator Neutral fractional differential inclusion