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

Rui Zhang; Yong-Kui Chang; G. M. N'Guérékata

doi:10.3934/dcds.2013.33.5525

Article Contents

2013, Volume 33, Issue 11&12: 5525-5537. Doi: 10.3934/dcds.2013.33.5525

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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
2.
Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, M.D. 21251

Received: October 31, 2011

Published: October 2013

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Abstract

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.

Keywords:

Stepanov-like weighted pseudo almost automorphic,
weighted pseudo almost automorphic,
semilinear integral equations,
integral resolvent family,
fixed point.

Mathematics Subject Classification: Primary: 534K14, 60H10, 35B15, 34F05.

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References

[1]	C. Cuevas and C. Lizama, Almost automorphic solutions to integral equations on the line, Semigroup Forum, 79 (2009), 461-472.doi: 10.1007/s00233-009-9154-0.
[2]	H. R. Henríquez and C. Lizama, Compact almost automorphic solutions to integral equations with infinite delay, Nonlinear Anal., 71 (2009), 6029-6037.doi: 10.1016/j.na.2009.05.042.
[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), 2887-2894.doi: 10.1016/j.na.2011.01.018.
[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), 2866-2879.doi: 10.1016/j.nonrwa.2012.04.016.
[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), 2731-2735.doi: 10.1016/j.na.2008.03.061.
[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), 4079-4085.doi: 10.1016/j.na.2008.08.018.
[7]	C. Lizama, Regularzed solutions for abstract Volterra equations, J. Math. Anal. Appl., 243 (2000), 278-292.doi: 10.1006/jmaa.1999.6668.
[8]	J. Prüss, "Evolutionary Integral Equations and Applications," Monographs Math., 87, birkhäuser Verlag, 1993.doi: 10.1007/978-3-0348-8570-6.
[9]	G. Gripenberg, S. -O.Londen and O. Staffans, Volterra integral and functional equations, in "Encyclopedia of Mathematics and Applications," 34, Cambridge University Press, Cambridge, New York, (1990).doi: 10.1017/CBO9780511662805.
[10]	C. Lizama, On approximation and representation of $k$-regularized resolvent families, Integral Equations Operator Theory, 41 (2001), 223-229.doi: 10.1007/BF01295306.
[11]	C. Lizama and J. Sánchez, On perturbation of $k$-regularized resolvent families, Taiwanese J. Math., 7 (2003), 217-227.
[12]	S. Y. Shaw and J. C. Chen, Asymptotic behavior of $(a,k)$-regularized families at zero, Taiwanese J. Math., 10 (2006), 531-542.
[13]	G. M. N'Guérékata, "Topics in Almost Automorphy," Springer, New York, Boston, Dordrecht, London, Moscow, 2005.
[14]	T. Diagana, Weighted pseudo almost periodic functions and applications, C. R. Acad. Sci. Paris, Ser. I, 343 (2006), 643-646.doi: 10.1016/j.crma.2006.10.008.
[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), 903-909.doi: 10.1016/j.na.2008.10.113.
[16]	G. M. Mophou, Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations, Appl. Math. Comp., 217 (2011), 7579-7587.doi: 10.1016/j.amc.2011.02.048.
[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), 430-438.doi: 10.1016/j.na.2009.06.077.
[18]	G. M. N'Guérékata and A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal., 69 (2008), 2658-2667.doi: 10.1016/j.na.2007.02.012.
[19]	H. Lee and H. Alkahby, Stepanov-like almost automorphic solutions of nonautonomous semilinear evolution equations with delay, Nonlinear Anal., 69 (2008), 2158-2166.doi: 10.1016/j.na.2007.07.053.
[20]	A. Granas and J. Dugundji, "Fixed Point Theory," Springer-Velag, New York, 2003.