## Pseudo asymptotic solutions of fractional order semilinear equations.(English)Zbl 1275.47092

In this paper, the authors study the fractional order differential equation $D^{\alpha+1}_tu(t)+\mu D^\beta_tu(t)-Au(t)=f(t,u(t))\,, \qquad t>0,$ with prescribed initial conditions $$u(0)$$ and $$u'(0)$$. Here, $$A:D(A)\subset X \to X$$ is a sectorial operator, $$f$$ is a vector-valued function, $$\alpha$$ and $$\beta$$ are two parameters such that $$0<\alpha\leq \beta\leq 1$$, $$\mu\geq 0$$, and $$D^\gamma_t$$ denotes the Caputo fractional derivative of order $$\gamma$$.
In the main results of the paper, the authors prove the existence and uniqueness of solutions for this problem, using the contraction mapping theorem.

### MSC:

 47D06 One-parameter semigroups and linear evolution equations 34A08 Fractional ordinary differential equations 35R11 Fractional partial differential equations 45N05 Abstract integral equations, integral equations in abstract spaces
Full Text:

### References:

 [1] D. Araya and C. Lizama, Almost automorphic mild solutions to fractional differential equations , Nonlinear Anal. 69 (2008), 3692-3705. · Zbl 1166.34033 [2] W. Arendt, C. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems , Monographs in Mathematics, 96 . Birkhäuser, Basel, 2001. · Zbl 0978.34001 [3] E. Bazhlekova, Fractional Evolution Equations in Banach Spaces , Ph.D. Thesis, Eindhoven University of Technology, 2001. · Zbl 0989.34002 [4] S. Bochner, Continuous mappings of almost automorphic and almost periodic functions , Proc. Nat. Acad. Sci. USA 52 (1964), 907-910. · Zbl 0134.30102 [5] S. Bochner, Uniform convergence of monotone sequences of functions , Proc. Nat. Acad. Sci. USA 47 (1961), 582-585. · Zbl 0103.05304 [6] S. Bochner, A new approach in almost-periodicity , Proc. Nat. Acad. Sci. USA 48 (1962), 2039-2043. · Zbl 0112.31401 [7] S. Bochner and J. von Neumann, On compact solutions of operational-differential equations I , Ann. Math. 36 (1935), 255-290. · Zbl 0011.02002 [8] B. De Andrade, C. Cuevas and E. Henriquez, Almost automorphic solutions of hyperbolic evolution equations , Banach J. Math. Anal. 6 (2012), no. 1, 90-100. · Zbl 1253.34054 [9] R. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order , CIMS Lecture Notes, http://arxiv.org/0805.3823. · Zbl 0934.35008 [10] N. Heymans and I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives , Rheologica Acta 45 (2006), no. 5, 765-771. [11] R. Hilfer, Applications of Fractional Calculus in Physics , World Scientific Publ. Co., Singapore, 2000. · Zbl 0998.26002 [12] V. Keyantuo, C. Lizama and M. Warma, Asymptotic behavior of fractional order semilinear evolution equations , · Zbl 1299.35309 [13] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations , Elsevier, Amsterdam, 2006. · Zbl 1092.45003 [14] J. Liang, J. Zhang and T.J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions , J. Math. Anal. Appl. 340 (2008), no. 2, 1493-1499. · Zbl 1134.43001 [15] J.H. Liu and X.Q. Song, Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations , J. Funct. Anal. 258 (2010), 196-207. · Zbl 1194.47047 [16] C. Lizama, Regularized solutions for abstract Volterra equations , J. Math. Anal. Appl. 243 (2000), 278-292. · Zbl 0952.45005 [17] C. Lizama, An operator theoretical approach to a class of fractional order differential equations , Appl. Math. Lett. 24 (2011), no. 2, 184-190. · Zbl 1226.47048 [18] C. Lizama and G.M. N’Guérékata, Bounded mild solutions for semilinear integro-differential equations in Banach spaces , Integral Equations and Operator Theory 68 (2010), 207-227. · Zbl 1209.45007 [19] G. M. N’Guérékata, Almost automorphic and almost periodic functions in abstract spaces. Kluwer Academic/Plenum Publishers, New York, 2001. [20] I. Podlubny, Fractional Differential Equations , Academic Press, San Diego, 1999 · Zbl 0924.34008 [21] J. Prüss, Evolutionary Integral Equations and Applications . Monographs Math. 87 , Birkhäuser Verlag, 1993. · Zbl 0784.45006 [22] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications , Gordon and Breach, New York (1993) [Translation from the Russian edition, Nauka i Tekhnika, Minsk (1987)]. · Zbl 0818.26003 [23] M. Stojanović and R. Gorenflo, Nonlinear two-term time fractional diffusion-wave problem , Nonlinear Anal. Real World Appl. 11 (2010), no. 5, 3512-3523. · Zbl 1203.35289 [24] T.J. Xiao, J. Liang and J. Zhang, Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces , Semigroup Forum 76 (2008), no. 3, 518-524. · Zbl 1154.46023 [25] C.Y. Zhang, Almost Periodic Type Functions and Ergodicity , Science Press, Kluwer Academic Publishers, New York, 2003. · Zbl 1068.34001 [26] C.Y. Zhang, Pseudo almost periodic solutions of some differential equations , J. Math. Anal. Appl. 151 (1994), 62-76. · Zbl 0796.34029 [27] C.Y. Zhang, Pseudo almost periodic solutions of some differential equations II , J. Math. Anal. Appl. 192 (1995), 543-561. · Zbl 0826.34040
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.