Abstract and Applied Analysis

Positive Solutions for the Initial Value Problem of Fractional Evolution Equations

He Yang and Yue Liang

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Abstract

By using the fixed point theorems and the theory of analytic semigroup, we investigate the existence of positive mild solutions to the Cauchy problem of Caputo fractional evolution equations in Banach spaces. Some existence theorems are obtained under the case that the analytic semigroup is compact and noncompact, respectively. As an example, we study the partial differential equation of the parabolic type of fractional order.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 428793, 7 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511818

Digital Object Identifier
doi:10.1155/2013/428793

Mathematical Reviews number (MathSciNet)
MR3039148

Zentralblatt MATH identifier
1291.35430

Citation

Yang, He; Liang, Yue. Positive Solutions for the Initial Value Problem of Fractional Evolution Equations. Abstr. Appl. Anal. 2013 (2013), Article ID 428793, 7 pages. doi:10.1155/2013/428793. https://projecteuclid.org/euclid.aaa/1393511818


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References

  • K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
  • V. Lakshmikantham, S. Leela, and J. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge, UK, 2009.
  • K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010.
  • R. P. Agarwal, M. Belmekki, and M. Benchohra, “A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative,” Advances in Difference Equations, vol. 2009, Article ID 981728, 47 pages, 2009.
  • R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010.
  • R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, “On the concept of solution for fractional differential equations with uncertainty,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 6, pp. 2859–2862, 2010.
  • M. M. El-Borai, “Some probability densities and fundamental solutions of fractional evolution equations,” Chaos, Solitons and Fractals, vol. 14, no. 3, pp. 433–440, 2002.
  • M. M. El-Borai, “The fundamental solutions for fractional evolution equations of parabolic type,” Journal of Applied Mathematics and Stochastic Analysis, no. 3, pp. 197–211, 2004.
  • M. M. El-Borai, “Semigroups and some nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 149, no. 3, pp. 823–831, 2004.
  • M. El-Borai, K. El-Nadi, and E. El-Akabawy, “Fractional evolution equations with nonlocal conditions,” International Journal of Applied Mathematics and Mechanics, vol. 4, no. 6, pp. 1–12, 2008.
  • Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations,” Nonlinear Analysis. Real World Applications, vol. 11, no. 5, pp. 4465–4475, 2010.
  • J. Wang and Y. Zhou, “A class of fractional evolution equations and optimal controls,” Nonlinear Analysis. Real World Applications, vol. 12, no. 1, pp. 262–272, 2011.
  • Y. Zhou and F. Jiao, “Existence of mild solutions for fractional neutral evolution equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1063–1077, 2010.
  • J. Wang, Y. Zhou, and W. Wei, “A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 10, pp. 4049–4059, 2011.
  • Z. Tai and X. Wang, “Controllability of fractional-order impulsive neutral functional infinite delay integrodifferential systems in Banach spaces,” Applied Mathematics Letters, vol. 22, no. 11, pp. 1760–1765, 2009.
  • A. Debbouche and D. Baleanu, “Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1442–1450, 2011.
  • G. M. Mophou, “Existence and uniqueness of mild solutions to impulsive fractional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1604–1615, 2010.
  • R.-N. Wang, T.-J. Xiao, and J. Liang, “A note on the fractional Cauchy problems with nonlocal initial conditions,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1435–1442, 2011.
  • P. Sobolevskii, “Equations of parabolic type in a Banach space,” American Mathematics Society Translations. Series 2, vol. 49, pp. 1–62, 1966.
  • H. Amann, “Periodic solutions of semilinear parabolic equations,” in Nonlinear Analysis, pp. 1–29, Academic Press, New York, NY, USA, 1978.
  • A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983.
  • H. Liu and J.-C. Chang, “Existence for a class of partial differential equations with nonlocal conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 9, pp. 3076–3083, 2009.
  • K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
  • D. Guo, V. Lakshmikantham, and X. Liu, Nonlinear Integral Equations in Abstract Spaces, vol. 373 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
  • J. Liang and T.-J. Xiao, “Solvability of the Cauchy problem for infinite delay equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 58, no. 3-4, pp. 271–297, 2004.
  • H. Amann, “Nonlinear operators in ordered Banach spaces and some applications to nonlinear boundary value problems,” in Nonlinear Operators and the Calculus of Variations, vol. 543 of Lecture Notes in Mathematics, pp. 1–55, Springer, Berlin, Germany, 1976.
  • Y. Li, “Existence and asymptotic stability of periodic solution for evolution equations with delays,” Journal of Functional Analysis, vol. 261, no. 5, pp. 1309–1324, 2011.