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Optimal mild solutions and weighted pseudo-almost periodic classical solutions of fractional integro-differential equations. (English) Zbl 1213.34089
The aim of this paper is to study the existence of classical solutions to the following fractional integro-differential equation $$\frac{d^\alpha x(t)}{dt^\alpha}= Ax(t)+f(t,x(t),G x(t)),\tag1$$ where $\frac{d^\alpha x(t)}{dt^\alpha}$ stands for the Riemann-Liouville derivative of order $\alpha$, $0<\alpha<1$, $A$ is the infinitesimal generator of an analytic semigroup $\{Q(t)\}_{t\geq 0}$ in a Banach space $\mathbb{X},$ $f:\mathbb{R}\times \mathbb{X}_q\times \mathbb{X}_q\to \mathbb{X}$ is a suitable function, and $\mathbb{X}_q$ a Banach space. $Gx(t)$, which may be interpreted as a control of the system, is defined by $Gx(t)=\int_{t_0}^t k(t,s,x(s))\,ds$ and $k:D\times\mathbb{X}_q\to \mathbb{X}_q$, $D=\{(t,s):t_0\leq s\leq t\leq T\}$. By means of the contraction mapping, the authors prove the existence and uniqueness of a classical solution of the initial value problem associated to (1). Then, they show the existence and uniqueness of an optimal mild solution among all the solutions of (1) which are bounded over $\mathbb{R}$. Note that the notion of optimal solution was introduced by {\it G. M. N’Guérékata} in [Riv. Mat. Univ. Parma, IV. Ser. 9, 145--151 (1983; Zbl 0547.34049)]. Finally, they study sufficient conditions for the existence and uniqueness of a weighted pseudo-almost periodic classic solution. An example is also given to illustrate the abstract results.

##### MSC:
 34K30 Functional-differential equations in abstract spaces 34A08 Fractional differential equations 34C27 Almost and pseudo-almost periodic solutions of ODE
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##### References:
 [1] Zhang, C.: Pseudo almost periodic solutions of some differential equations, J. math. Anal. appl. 151, 62-76 (1994) · Zbl 0796.34029 · doi:10.1006/jmaa.1994.1005 [2] Zhang, C.: Integration of vector-valued pseudo almost periodic functions, Proc. amer. Math. soc. 121, 167-174 (1994) · Zbl 0818.42003 · doi:10.2307/2160378 [3] Zhang, C.: Pseudo almost periodic solutions of some differential equations II, J. math. Anal. appl. 192, 543-561 (1995) · Zbl 0826.34040 · doi:10.1006/jmaa.1995.1189 [4] Dads, E.; Ezzinbi, K.; Arino, O.: Pseudo-almost periodic solutions for some differential equations in a Banach space, Nonlinear anal. TMA 28, 1141-1155 (1997) · Zbl 0874.34041 · doi:10.1016/S0362-546X(97)82865-9 [5] Cuevas, C.; Pinto, M.: Existence and uniqueness of pseudo almost periodic solutions of semilinar Cauchy problems with non dense domain, Nonlinear anal. TMA 45, 73-83 (2001) · Zbl 0985.34052 · doi:10.1016/S0362-546X(99)00330-2 [6] Diagana, T.: Pseudo almost periodic functions in Banach spaces, (2007) · Zbl 1234.43002 [7] Cuevas, C.; Hernádez, H.: Pseudo almost periodic solutions for abstract partial functional differential equations, Appl. math. Lett. 22, 534-538 (2009) · Zbl 1170.35551 · doi:10.1016/j.aml.2008.06.026 [8] Diagana, T.: Weighted pseudo almost periodic functions and applications, C. R. Acad. sci. Paris, ser. I 343, 643-646 (2006) · Zbl 1112.43005 · doi:10.1016/j.crma.2006.10.008 [9] Ahn, V.; Mcvinisch, R.: Fractional differential equations driven by Lévy noise, J. appl. Math. stoch. Anal. 16, 97-119 (2003) · Zbl 1042.60034 · doi:10.1155/S1048953303000078 [10] D. Benson, The fractional advection--dispersion equation, Ph.D. Thesis, University of Nevada, Reno, NV, 1998. [11] Schumer, R.; Benson, D.: Eulerian derivative of the fractional advection--dispersion equation, J. contam. Hydrol. 48, 69-88 (2001) [12] Borai, M.: Some probability densities and fundamental solutions of fractional evolutions equations, Chaos solitons fractals 14, 433-440 (2002) · Zbl 1005.34051 · doi:10.1016/S0960-0779(01)00208-9 [13] Sayed, A.: Nonlinear functional--differential equations of arbitrary orders, Nonlinear anal. TMA 33, 181-186 (1998) · Zbl 0934.34055 · doi:10.1016/S0362-546X(97)00525-7 [14] Mophou, G.; N’guérékata, G.: Existence of mild solution for some fractional differential equations with nonlocal conditions, Semigroup forum 79, 315-322 (2009) · Zbl 1180.34006 · doi:10.1007/s00233-008-9117-x [15] N’guérékata, G.: Cauchy problem for some fractional abstract differential equation with nonlocal conditions, Nonlinear anal. TMA 70, 1873-1876 (2009) · Zbl 1166.34320 · doi:10.1016/j.na.2008.02.087 [16] Lahshmikantham, V.; Devi, J.: Theory of fractional differential equations in Banach spaces, Eur. J. Pure appl. Math. 1, 38-45 (2008) · Zbl 1146.34042 · http://www.ejpam.com/ejpam/index.php/ejpam/article/view/84 [17] Glockle, W.; Nonnemacher, T.: A fractional calculus approach of self-similar protein dynamics, Biophys. J. 68, 46-53 (1995) [18] Metzler, F.; Schick, W.; Kilian, H.; Nonnemacher, T.: Relaxation in filled polymers: a fractional calculus approach, J. chem. Phys. 103, 7180-7186 (1995) [19] Ahmad, B.; Sivasundaram, S.: Some existence results for fractional integrodifferential equations with nonlinear conditions, Commun. math. Anal. 12, 107-112 (2008) · Zbl 1179.45009 [20] Bhaskar, T.; Lakshmikantham, V.; Leela, S.: Fractional differential equations with Krasnoselskii--Krein-type condition, Nonlinear anal. Hybrid syst. 3, 734-737 (2009) · Zbl 1181.34008 · doi:10.1016/j.nahs.2009.06.010 [21] Lakshmikantham, V.; Leela, S.: Nagumo-type uniqueness result for fractional differential equations, Nonlinear anal. TMA 71, 2886-2889 (2009) · Zbl 1177.34003 · doi:10.1016/j.na.2009.01.169 [22] Anguraj, A.; Karthikeyan, P.; N’guérékata, G.: Nonlocal Cauchy problem for some fractional abstract integro--differential equations in Banach spaces, Commun. math. Anal. 6, 31-35 (2009) · Zbl 1167.34387 [23] Bahuguna, D.; Srvastavai, S.: Semilinear integro--differential equations with compact semigroup, J. appl. Math. stoch. Anal. 11, 507-517 (1998) · Zbl 0919.34055 · doi:10.1155/S1048953398000410 [24] Dieudonne, J.: Foundation of modern analysis, (1960) · Zbl 0100.04201 [25] Yorke, J.: A continuous differential equation in Hilbert space without existence, Funkcial. ekvac. 12, 19-21 (1970) · Zbl 0248.34061 [26] Rashid, M.; El-Qaderi, Y.: Semilinear fractional integro--differential equations with compact semigroup, Nonlinear anal. TMA 71, 6276-6282 (2009) · Zbl 1184.45007 · doi:10.1016/j.na.2009.06.035 [27] Balachandran, K.; Park, J.: Nonlocal Cauchy problem for abstract fractional semilinear evolution equations, Nonlinear anal. TMA 71, 4471-4475 (2009) · Zbl 1213.34008 · doi:10.1016/j.na.2009.03.005 [28] Mophou, G.; N’guérékata, G.: Mild solutions for semilinear fractional differential equations, Electron. J. Differential equations 21, 1-9 (2009) · Zbl 1179.34002 · emis:journals/EJDE/Volumes/2009/21/abstr.html [29] Changa, Y.; Kavitha, V.; Arjunan, M.: Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order, Nonlinear anal. TMA 71, 5551-5559 (2009) · Zbl 1179.45010 · doi:10.1016/j.na.2009.04.058 [30] Heard, M.; Rankin, S.: A semilinear parabolic integro--differential equation, J. differential equations 71, 201-233 (1988) · Zbl 0642.45006 · doi:10.1016/0022-0396(88)90023-X [31] Mophou, G.; N’guérékata, G.: A note on a semilinear fractional differential equation of a neutral type with infinite delay, Adv. difference equ. 2010 (2010) · Zbl 1194.34148 [32] Li, F.; N’guérékata, G.: Existence and uniqueness of mild solution for fractional integrodifferential equations, Adv. difference equ. 2010 (2010) · Zbl 1198.45015 · doi:10.1155/2010/158789 [33] I. Podlubny, Fractional Differential Equations, Math. in Science and Eng., Technical University of Kosice, Slovak Republic, 1999. · Zbl 0924.34008 [34] Pazy, A.: Semigroups of linear operators and applications to partial differential equations, (1983) · Zbl 0516.47023 [35] Diagana, T.: Existence of weighted pseudo almost periodic solutions to some classes of hyperbolic evolution equations, J. math. Anal. appl. 350, 18-28 (2009) · Zbl 1167.34023 · doi:10.1016/j.jmaa.2008.09.041 [36] Agarwal, R.; Diagana, T.; Hernández, E.: Weighted pseudo almost periodic solutions to some partial neutral functional differential equations, J. nonlinear convex anal. 8, 397-415 (2007) · Zbl 1155.35104 [37] Diagana, T.: Weighted pseudo almost periodic solutions to some differential equations, Nonlinear anal. TMA 68, 2250-2260 (2008) · Zbl 1131.42006 · doi:10.1016/j.na.2007.01.054 [38] Feller, W.: An introduction to probability theory and its applications, An introduction to probability theory and its applications (1971) · Zbl 0219.60003 [39] N’guérékata, G.: Almost automorphic and almost periodic functions in abstract spaces, (2001) · Zbl 1001.43001 [40] Larsen, R.: Functional analysis, (1973) · Zbl 0261.46001 [41] Yoshizawa, T.: Stability theory and the existence of periodic solutions and almost periodic solutions, (1975) · Zbl 0304.34051 [42] Zaidman, S.: Abstract differential equations, (1979) · Zbl 0465.34002