Stepanov-like pseudo-almost periodicity and its applications to some nonautonomous differential equations. (English) Zbl 1169.34330
Let \(X\) be a Banach space, and \(p\geq 1.\) The author proves the existence and uniqueness of a pseudo-almost periodic mild solution to the nonautonomous differential equation
\[ u'(t)=A(t)u(t)+f(t),\quad t\in\mathbb R, \]
where \(A(t):D(A(t))\subset X\rightarrow X\) is a family of densely defined closed linear operators with a common domain \(D=D(A(t)),\) independent of \(t,\) whose associated evolution family of operators is asymptotically stable, and \(f:R\rightarrow X\) is a continuous and Stepanov-like pseudo-almost periodic (or \(S^{p}\)-pseudo-almost periodic) function. This notion of \(S^{p}\)-pseudo-almost periodicity was introduced by the author [Commun. Math. Anal. 3, No. 1, 9–18 (2007; Zbl 1286.44007)], and generalizes the classical notion of pseudo-almost periodicity due to C. Y. Zhang [J. Math. Anal. Appl. 181, No. 1, 62–76 (1994; Zbl 0796.34029) and 192, No. 2, 543–561 (1995; Zbl 0826.34040)].
\[ u'(t)=A(t)u(t)+f(t),\quad t\in\mathbb R, \]
where \(A(t):D(A(t))\subset X\rightarrow X\) is a family of densely defined closed linear operators with a common domain \(D=D(A(t)),\) independent of \(t,\) whose associated evolution family of operators is asymptotically stable, and \(f:R\rightarrow X\) is a continuous and Stepanov-like pseudo-almost periodic (or \(S^{p}\)-pseudo-almost periodic) function. This notion of \(S^{p}\)-pseudo-almost periodicity was introduced by the author [Commun. Math. Anal. 3, No. 1, 9–18 (2007; Zbl 1286.44007)], and generalizes the classical notion of pseudo-almost periodicity due to C. Y. Zhang [J. Math. Anal. Appl. 181, No. 1, 62–76 (1994; Zbl 0796.34029) and 192, No. 2, 543–561 (1995; Zbl 0826.34040)].
Reviewer: Behzad Djafari-Rouhani (El-Paso)
MSC:
| 34G20 | Nonlinear differential equations in abstract spaces |
| 34C27 | Almost and pseudo-almost periodic solutions to ordinary differential equations |
| 42A75 | Classical almost periodic functions, mean periodic functions |
Keywords:
almost periodic; pseudo-almost periodic; Stepanov-like pseudo-almost periodicity; evolution family of operators; mild solution; Bochner transformReferences:
| [1] | Amir, B.; Maniar, L., Composition of pseudo-almost periodic functions and Cauchy problems with operator of nondense domain, Ann. Math. Blaise Pascal, 6, 1, 1-11 (1999) · Zbl 0941.34059 |
| [2] | Cuevas, C.; Pinto, M., Existence and uniqueness of pseudo almost periodic solutions of semilinear Cauchy problems with non-dense domain, Nonlinear Anal. (Ser. A: Theory Methods), 45, 1, 73-83 (2001) · Zbl 0985.34052 |
| [3] | Casarino, V., Characterization of almost automorphic groups and semigroups, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 5, 24, 219-235 (2000) |
| [4] | Diagana, T.; N’Guérékata, G. M., Stepanov-like almost automorphic functions and applications to some semilinear equations, Appl. Anal., 86, 6, 723-733 (2007) · Zbl 1128.43006 |
| [5] | Diagana, T., Stepanov-like pseudo almost periodic functions and their applications to differential equations, Commun. Math. Anal., 3, 1, 9-18 (2007) · Zbl 1286.44007 |
| [6] | Diagana, T., Pseudo almost periodic solutions to some differential equations, Nonlinear Anal., 60, 7, 1277-1286 (2005) · Zbl 1061.34040 |
| [7] | Diagana, T., Existence and uniqueness of pseudo almost periodic solutions to some classes of partial evolution equations, Nonlinear Anal., 66, 2, 384-395 (2007) · Zbl 1105.35304 |
| [8] | Diagana, T., Weighted pseudo almost periodic functions and applications, C. R. Acad. Sci. Paris, Ser. I, 343, 10, 643-646 (2006) · Zbl 1112.43005 |
| [9] | Diagana, T.; Hernàndez, E., Existence and uniqueness of pseudo almost periodic solutions to some abstract partial neutral functional-differential equations and applications, J. Math. Anal. Appl., 327, 2, 776-791 (2007) · Zbl 1123.34060 |
| [10] | Diagana, T.; Hernández, E.; Rabello, M., Pseudo almost periodic solutions to some non-autonomous neutral functional differential equations with unbounded delay, Math. Comput. Modelling, 45, 1241-1252 (2007) · Zbl 1133.34042 |
| [11] | Li, H.-X.; Huang, F.-L.; Li, J.-Y., Composition of pseudo almost-periodic functions and semilinear differential equations, J. Math. Anal. Appl., 255, 2, 436-446 (2001) · Zbl 1047.47030 |
| [12] | Lunardi, A., Analytic semigroups and optimal regularity in parabolic problems, (PNLDE, vol. 16 (1995), Birkhäuser Verlag: Birkhäuser Verlag Basel) · Zbl 1261.35001 |
| [13] | G.M. N’Guérékata, A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal., in press (doi:10.1016/j.na.2007.02.012; G.M. N’Guérékata, A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal., in press (doi:10.1016/j.na.2007.02.012 |
| [14] | Pankov, A., Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations (1990), Kluwer: Kluwer Dordrecht · Zbl 0712.34001 |
| [15] | Zhang, C. Y., Pseudo almost periodic solutions of some differential equations, J. Math. Anal. Appl., 151, 62-76 (1994) · Zbl 0796.34029 |
| [16] | Zhang, C. Y., Pseudo almost periodic solutions of some differential equations II, J. Math. Anal. Appl., 192, 543-561 (1995) · Zbl 0826.34040 |
| [17] | Zhang, C. Y., Integration of vector-valued pseudo almost periodic functions, Proc. Amer. Math. Soc., 121, 167-174 (1994) · Zbl 0818.42003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
