November  2013, 33(11&12): 5525-5537. doi: 10.3934/dcds.2013.33.5525

Weighted pseudo almost automorphic mild solutions to semilinear integral equations with $S^{p}$-weighted pseudo almost automorphic coefficients

1. 

Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China, China

2. 

Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, M.D. 21251, United States

Received  November 2011 Published  May 2013

In this paper, we consider the existence of weighted pseudo almost automorphic solutions of the semilinear integral equation $x(t)= \int_{-\infty}^{t}a(t-s)[Ax(s) + f(s,x(s))]ds, \ t\in\mathbb{R}$ in a Banach space $\mathbb{X}$, where $a\in L^{1}(\mathbb{R}_{+})$, $A$ is the generator of an integral resolvent family of linear bounded operators defined on the Banach space $\mathbb{X}$, and $f : \mathbb{R}\times\mathbb{X} \rightarrow \mathbb{X}$ is a weighted pseudo almost automorphic function. The main results are proved by using integral resolvent families, suitable composition theorems combined with the theory of fixed points.
Citation: Rui Zhang, Yong-Kui Chang, G. M. N'Guérékata. Weighted pseudo almost automorphic mild solutions to semilinear integral equations with $S^{p}$-weighted pseudo almost automorphic coefficients. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5525-5537. doi: 10.3934/dcds.2013.33.5525
References:
[1]

C. Cuevas and C. Lizama, Almost automorphic solutions to integral equations on the line,, Semigroup Forum, 79 (2009), 461.  doi: 10.1007/s00233-009-9154-0.  Google Scholar

[2]

H. R. Henríquez and C. Lizama, Compact almost automorphic solutions to integral equations with infinite delay,, Nonlinear Anal., 71 (2009), 6029.  doi: 10.1016/j.na.2009.05.042.  Google Scholar

[3]

Z. H. Zhao, Y. K. Chang and G. M. N'Guérékata, Pseudo-almost automorphic mild solutions to semilinear integral equations in a Banach space,, Nonlinear Anal., 74 (2011), 2887.  doi: 10.1016/j.na.2011.01.018.  Google Scholar

[4]

R. Zhang, Y. K. Chang and G. M. N'Guérékata, New composition theorems of Stepanov-like weighted pseudo almost automorphic functions and applications to nonautonomous evolution equations,, Nonlinear Anal. RWA, 13 (2012), 2866.  doi: 10.1016/j.nonrwa.2012.04.016.  Google Scholar

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J. Liang, G. M. N'Guérékata, T. J. Xiao and J. Zhang, Some properties of pseudo almost automorphic functions and applications to abstract differential equations,, Nonlinear Anal., 70 (2009), 2731.  doi: 10.1016/j.na.2008.03.061.  Google Scholar

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T. J. Xiao, X. X. Zhu and J. Liang, Pseudo almost automorphic mild solutions to nonautomous differential equations and applications,, Nonlinear Anal., 70 (2009), 4079.  doi: 10.1016/j.na.2008.08.018.  Google Scholar

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C. Lizama, Regularzed solutions for abstract Volterra equations,, J. Math. Anal. Appl., 243 (2000), 278.  doi: 10.1006/jmaa.1999.6668.  Google Scholar

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J. Prüss, "Evolutionary Integral Equations and Applications,", Monographs Math., 87 (1993).  doi: 10.1007/978-3-0348-8570-6.  Google Scholar

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G. Gripenberg, S. -O.Londen and O. Staffans, Volterra integral and functional equations,, in, 34 (1990).  doi: 10.1017/CBO9780511662805.  Google Scholar

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C. Lizama, On approximation and representation of $k$-regularized resolvent families,, Integral Equations Operator Theory, 41 (2001), 223.  doi: 10.1007/BF01295306.  Google Scholar

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C. Lizama and J. Sánchez, On perturbation of $k$-regularized resolvent families,, Taiwanese J. Math., 7 (2003), 217.   Google Scholar

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S. Y. Shaw and J. C. Chen, Asymptotic behavior of $(a,k)$-regularized families at zero,, Taiwanese J. Math., 10 (2006), 531.   Google Scholar

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G. M. N'Guérékata, "Topics in Almost Automorphy,", Springer, (2005).   Google Scholar

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T. Diagana, Weighted pseudo almost periodic functions and applications,, C. R. Acad. Sci. Paris, 343 (2006), 643.  doi: 10.1016/j.crma.2006.10.008.  Google Scholar

[15]

J. Blot, G. M. Mophou, G. M. N'Guérékata and D. Pennequin, Weighted pseudo almost automorphic functions and applications to abstract differential equations,, Nonlinear Anal., 71 (2009), 903.  doi: 10.1016/j.na.2008.10.113.  Google Scholar

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G. M. Mophou, Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations,, Appl. Math. Comp., 217 (2011), 7579.  doi: 10.1016/j.amc.2011.02.048.  Google Scholar

[17]

T. Diagana, G. M. Mophou and G. M. N'Guérékata, Existence of weighted pseudo almost periodic solutions to some classes of differential equations with $S^p$-weighted pseudo almost periodic coefficients,, Nonlinear Anal., 72 (2010), 430.  doi: 10.1016/j.na.2009.06.077.  Google Scholar

[18]

G. M. N'Guérékata and A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations,, Nonlinear Anal., 69 (2008), 2658.  doi: 10.1016/j.na.2007.02.012.  Google Scholar

[19]

H. Lee and H. Alkahby, Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay,, Nonlinear Anal., 69 (2008), 2158.  doi: 10.1016/j.na.2007.07.053.  Google Scholar

[20]

A. Granas and J. Dugundji, "Fixed Point Theory,", Springer-Velag, (2003).   Google Scholar

show all references

References:
[1]

C. Cuevas and C. Lizama, Almost automorphic solutions to integral equations on the line,, Semigroup Forum, 79 (2009), 461.  doi: 10.1007/s00233-009-9154-0.  Google Scholar

[2]

H. R. Henríquez and C. Lizama, Compact almost automorphic solutions to integral equations with infinite delay,, Nonlinear Anal., 71 (2009), 6029.  doi: 10.1016/j.na.2009.05.042.  Google Scholar

[3]

Z. H. Zhao, Y. K. Chang and G. M. N'Guérékata, Pseudo-almost automorphic mild solutions to semilinear integral equations in a Banach space,, Nonlinear Anal., 74 (2011), 2887.  doi: 10.1016/j.na.2011.01.018.  Google Scholar

[4]

R. Zhang, Y. K. Chang and G. M. N'Guérékata, New composition theorems of Stepanov-like weighted pseudo almost automorphic functions and applications to nonautonomous evolution equations,, Nonlinear Anal. RWA, 13 (2012), 2866.  doi: 10.1016/j.nonrwa.2012.04.016.  Google Scholar

[5]

J. Liang, G. M. N'Guérékata, T. J. Xiao and J. Zhang, Some properties of pseudo almost automorphic functions and applications to abstract differential equations,, Nonlinear Anal., 70 (2009), 2731.  doi: 10.1016/j.na.2008.03.061.  Google Scholar

[6]

T. J. Xiao, X. X. Zhu and J. Liang, Pseudo almost automorphic mild solutions to nonautomous differential equations and applications,, Nonlinear Anal., 70 (2009), 4079.  doi: 10.1016/j.na.2008.08.018.  Google Scholar

[7]

C. Lizama, Regularzed solutions for abstract Volterra equations,, J. Math. Anal. Appl., 243 (2000), 278.  doi: 10.1006/jmaa.1999.6668.  Google Scholar

[8]

J. Prüss, "Evolutionary Integral Equations and Applications,", Monographs Math., 87 (1993).  doi: 10.1007/978-3-0348-8570-6.  Google Scholar

[9]

G. Gripenberg, S. -O.Londen and O. Staffans, Volterra integral and functional equations,, in, 34 (1990).  doi: 10.1017/CBO9780511662805.  Google Scholar

[10]

C. Lizama, On approximation and representation of $k$-regularized resolvent families,, Integral Equations Operator Theory, 41 (2001), 223.  doi: 10.1007/BF01295306.  Google Scholar

[11]

C. Lizama and J. Sánchez, On perturbation of $k$-regularized resolvent families,, Taiwanese J. Math., 7 (2003), 217.   Google Scholar

[12]

S. Y. Shaw and J. C. Chen, Asymptotic behavior of $(a,k)$-regularized families at zero,, Taiwanese J. Math., 10 (2006), 531.   Google Scholar

[13]

G. M. N'Guérékata, "Topics in Almost Automorphy,", Springer, (2005).   Google Scholar

[14]

T. Diagana, Weighted pseudo almost periodic functions and applications,, C. R. Acad. Sci. Paris, 343 (2006), 643.  doi: 10.1016/j.crma.2006.10.008.  Google Scholar

[15]

J. Blot, G. M. Mophou, G. M. N'Guérékata and D. Pennequin, Weighted pseudo almost automorphic functions and applications to abstract differential equations,, Nonlinear Anal., 71 (2009), 903.  doi: 10.1016/j.na.2008.10.113.  Google Scholar

[16]

G. M. Mophou, Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations,, Appl. Math. Comp., 217 (2011), 7579.  doi: 10.1016/j.amc.2011.02.048.  Google Scholar

[17]

T. Diagana, G. M. Mophou and G. M. N'Guérékata, Existence of weighted pseudo almost periodic solutions to some classes of differential equations with $S^p$-weighted pseudo almost periodic coefficients,, Nonlinear Anal., 72 (2010), 430.  doi: 10.1016/j.na.2009.06.077.  Google Scholar

[18]

G. M. N'Guérékata and A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations,, Nonlinear Anal., 69 (2008), 2658.  doi: 10.1016/j.na.2007.02.012.  Google Scholar

[19]

H. Lee and H. Alkahby, Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay,, Nonlinear Anal., 69 (2008), 2158.  doi: 10.1016/j.na.2007.07.053.  Google Scholar

[20]

A. Granas and J. Dugundji, "Fixed Point Theory,", Springer-Velag, (2003).   Google Scholar

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