Abstract and Applied Analysis

Existence of Some Semilinear Nonlocal Functional Differential Equations of Neutral Type

Hsiang Liu, Cheng-Wen Liao, and Chin-Tzong Pang

Full-text: Open access

Abstract

This paper is concerned with the existence of mild and strong solutions on the interval [ 0 , T ] for some neutral partial differential equations with nonlocal conditions. The linear part of the equations is assumed to generate a compact analytic semigroup of bounded linear operators, whereas the nonlinear part satisfies the Carathëodory condition and is bounded by some suitable functions. We first employ the Schauder fixed-point theorem to prove the existence of solution on the interval [ δ , T ] for δ > 0 that is small enough, and, then, by letting δ 0 and using a diagonal argument, we have the existence results on the interval [ 0 , T ] . This approach allows one to drop the compactness assumption on a nonlocal condition, which generalizes recent conclusions on this topic. The obtained results will be applied to a class of functional partial differential equations with nonlocal conditions.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 503656, 12 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393449678

Digital Object Identifier
doi:10.1155/2013/503656

Mathematical Reviews number (MathSciNet)
MR3134151

Zentralblatt MATH identifier
1298.34148

Citation

Liu, Hsiang; Liao, Cheng-Wen; Pang, Chin-Tzong. Existence of Some Semilinear Nonlocal Functional Differential Equations of Neutral Type. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 503656, 12 pages. doi:10.1155/2013/503656. https://projecteuclid.org/euclid.aaa/1393449678


Export citation

References

  • L. Byszewski, “Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem,” Journal of Mathematical Analysis and Applications, vol. 162, no. 2, pp. 494–505, 1991.
  • L. Byszewski, “Uniqueness of solutions of parabolic semilinear nonlocal-boundary problems,” Journal of Mathematical Analysis and Applications, vol. 165, no. 2, pp. 472–478, 1992.
  • L. Byszewski and V. Lakshmikantham, “Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space,” Applicable Analysis, vol. 40, no. 1, pp. 11–19, 1991.
  • K. Deng, “Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions,” Journal of Mathematical Analysis and Applications, vol. 179, no. 2, pp. 630–637, 1993.
  • J. Liang, J. van Casteren, and T.-J. Xiao, “Nonlocal Cauchy problems for semilinear evolution equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 50, no. 2, pp. 173–189, 2002.
  • Y. P. Lin and J. H. Liu, “Semilinear integrodifferential equations with nonlocal Cauchy problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 26, no. 5, pp. 1023–1033, 1996.
  • T.-J. Xiao and J. Liang, “Existence of classical solutions to nonautonomous nonlocal parabolic problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5-7, pp. e225–e232, 2005.
  • K. Balachandran and R. Sakthivel, “Existence of solutions of neutral functional integrodifferential equation in Banach spaces,” Proceedings of the Indian Academy of Sciences–-Mathematical Sciences, vol. 109, no. 3, pp. 325–332, 1999.
  • J.-C. Chang and H. Liu, “Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the $\alpha $-norm,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 9, pp. 3759–3768, 2009.
  • J. P. Dauer and K. Balachandran, “Existence of solutions of nonlinear neutral integrodifferential equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 93–105, 2000.
  • Q. Dong, Z. Fan, and G. Li, “Existence of solutions to nonlocal neutral functional differential and integrodifferential equations,” International Journal of Nonlinear Science, vol. 5, no. 2, pp. 140–151, 2008.
  • K. Ezzinbi and X. Fu, “Existence and regularity of solutions for some neutral partial differential equations with nonlocal conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 57, no. 7-8, pp. 1029–1041, 2004.
  • K. Ezzinbi, X. Fu, and K. Hilal, “Existence and regularity in the $\alpha $-norm for some neutral partial differential equations with nonlocal conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 5, pp. 1613–1622, 2007.
  • X. Fu and K. Ezzinbi, “Existence of solutions for neutral functional differential evolution equations with nonlocal conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 54, no. 2, pp. 215–227, 2003.
  • J. Liang, J. Liu, and T.-J. Xiao, “Nonlocal Cauchy problems governed by compact operator families,” Nonlinear Analysis: Theory, Methods & Applications, vol. 57, no. 2, pp. 183–189, 2004.
  • Q. Liu and R. Yuan, “Existence of mild solutions for semilinear evolution equations with non-local initial conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 9, pp. 4177–4184, 2009.
  • A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983.
  • B. N. Sadovskiĭ, “On a fixed point principle,” Functional Analysis and Its Applications, vol. 1, no. 2, pp. 74–76, 1967.
  • Y. Kōmura, “Differentiability of nonlinear semigroups,” Journal of the Mathematical Society of Japan, vol. 21, pp. 375–402, 1969.