Pseudo almost automorphic and weighted pseudo almost automorphic mild solutions to a partial functional differential equation in Banach spaces

Volume 5, Issue 1, pp 14--26 http://dx.doi.org/10.22436/jnsa.005.01.02

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1709 Downloads
• 2649 Views

Authors

Yong-Kui Chang - Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, Ginsu, China. Zhi-Han Zhao - Institute of Mathematics and Information Engineering, Sanming University, Sanming,365004, Fujian, China. Juan J. Nieto - Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, Santiago de Compostela, 15782, Spain. Zhi-Wei Liu - Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China.

Abstract

In this paper we prove some existence and uniqueness results for pseudo almost automorphic and weighted pseudo almost automorphic mild solutions to a class of partial functional differential equation in Banach spaces. The main technique is based upon some appropriate composition theorems combined with the Banach contraction mapping principle and fractional powers of operators.

Share and Cite

Keywords

• Pseudo almost automorphic
• Weighted pseudo almost automorphic
• Partial functional differential equations
• Fractional powers of operators.

MSC

•  44A35
•  42A85
•  42A75

References

• [1] T. J. Xiao, J. Liang, J. Zhang, Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup Forum , 76 (2008), 518-524.


• [2] S. Bochner, Uniform convergence of monotone sequences of functions, Proc. Natl. Acad. Sci. USA , 47 (1961), 582-585.


• [3] S. Bochner , Continuous mappings of almost automorphic and almost periodic functions, Proc. Natl. Acad. Sci. USA, 52 (1964), 907-910.


• [4] K. Ezzinbi, S. Fatajou, G. M. N'Gurkata, Pseudo almost automorphic solutions to some neutral partial functional defferential equations in Banach spaces, Nonlinear Anal. , 70 (2009), 1641-1647.


• [5] J. Liang, G. M. N'Gurkata, T. J. Xiao, J. Zhang, Some properties of pseudo almost automorphic functions and applications to abstract differential equations, Nonlinear Anal. , 71 (2009), 248-257.


• [6] J. Liang, J. Zhang, T. J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl. , 340 (2008), 1493-1499.


• [7] G. M. N'Gurkata, Almost automorphic functions and almost periodic functions in abstract spaces, Kluwer Academic Plenum Publishers, New York, London, Moscow (2001)


• [8] G. M. N'Gurkata, Topics in Almost Automorphy , Springer, New York, Boston, Dordrecht, London Moscow (2005)


• [9] W. A. Veech, Almost automorphic functions , Proc. Nat. Acad. Sci. USA, 49 (1963), 462-464.


• [10] J. Blot, G. M. Mophou, G. M. N'Gurkata, D. Pennequin, Weighted pseudo almost automorphic functions and applications to abstract differential equations, Nonlinear Anal., 71 (2009), 903-909.


• [11] R. P. Agarwal, T. Diagana, E. Hernndez, Weighted pseudo almost periodic solutions to some partial neutral functional differential equations, J. Nonlinear Convex Anal. , 8(3) (2007), 397-415.


• [12] R. P. Agarwal, B. de Andrade, C. Cuevas, Weighted pseudo almost periodic solutions of a class of semilinear fractional differential equations, Nonlinear Anal. RWA , 11(5) (2010), 3532-3554.


• [13] T. Diagana , Weighted pseudo almost periodic functions and applications, C. R. Acad. Sci. Paris Ser. I , 343 (2006), 643-646.


• [14] T. Diagana, Weighted pseudo-almost periodic solutions to a neutral delay integral equation of advanced type, Nonlinear Anal. , 70 (2009), 298-304.


• [15] Y. Yu, J. J. Nieto, A. Torres, K. Wang, A Viral Infection Model with a Nonlinear Infection Rate, Boundary Value Problems , 2009 (2009), 19 pages.


• [16] P. Cieutat, G. M. N'Gurkata, Bounded and almost automorphic solutions of some nonlinear differential equations in Banach spaces, Nonlinear Anal., 71 (2009), 674-684.


• [17] T. Diagana , Existence of pseudo-almost automorphic solutions to some abstract differential equations with pseudo-almost automorphic coefficients, Nonlinear Anal. , 70 (2009), 3781-3790.


• [18] T. Diagana, G. M. N'Gurkata, Almost automorphic solutions to some classes of partial evolution equations, Appl. Math. Lett. , 20(4) (2007), 462-466.


• [19] T. Diagana, G. M. N'Gurkata, N. V. Minh, Almost automorphic solutions of evolution equations, Proc. Amer. Math. Soc. , 132(11) (2004), 3289-3298.


• [20] H. S. Ding, W. Long, G. M. N'Gurkata, Almost automorphic solutions of nonautonomous evolution equations , Nonlinear Anal., 70 (2009), 2731-2735.


• [21] K. Ezzinbi, G. M. N'Gurkata, Massera type theorem for almost automorphic solutions of functional differential equations of neutral type, J. Math. Anal. Appl. , 316 (2006), 707-721.


• [22] K. Ezzinbi, G. M. N'Gurkata, Almost automorphic solutions for some partial functional differential equations, J. Math. Anal. Appl. , 328(1) (2007), 344-358.


• [23] J. H. Liu, X. Q. Song, Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations, J. Funct. Anal. , 258 (2010), 196-207.


• [24] Z. H. Zhao, Y. K. Chang, J. J. Nieto, Almost automorphic and pseudo almost automorphic mild solutions to an abstract differential equations in Banach spaces, Nonlinear Anal. TMA , 72 (2010), 1886-1894.


• [25] E. Hernndez, M. Rabello, H. R. Henrquez, Existence of solutions for impulsive partial neutral functional differential equations , J. Math. Anal. Appl., 331 (2007), 1135-1158.


• [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Equations, in: Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York (1983)


• [27] T. J. Xiao, X. X. Zhu, J. Liang, Pseudo almost automorphic mild solutions to nonautonomous differential equations and applications, Nonlinear Anal. , 70 (2009), 4079-4085.


• [28] G. M. Mophou, Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations, Appl. Math. Comput. , 217 (2011), 7579-7587.


• [29] Y. K. Chang, Z. H. Zhao, J. J. Nieto, Pseudo almost automorphic and weighted pseudo almost automorphic mild solutions to semi-linear differential equations in Hilbert spaces, Rev. Mat. Complut., 24 (2011), 421-438.