×

Stepanov-like weighted pseudo almost automorphic solutions to fractional order abstract integro-differential equations. (English) Zbl 1327.35391

The aim of this work is to study the existence of Stepanov-like weighted pseudo almost automorphic solutions for some fractional-order abstract integro-differential equations. The fractional derivative is taken in the Riemann-Liouville sense. The linear part is assumed to be sectorial in a Banach space. In the first part of this work, the authors assume that the nonlinear part is continuous and Lipschitz with respect to the second argument and then they use the strict contraction principle to prove the existence and uniqueness of a Stepanov-like weighted almost automorphic solution. In the second part, where the nonlinear part is not Lipshitz, the authors prove the existence of at least one Stepanov-like weighted almost automorphic solution, by using Schauder’s fixed point theorem.
To illustrate these results, an example is provided for a relaxation-oscillation partial differential equation of fractional order.

MSC:

35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abbas S, Existence of solutions to fractional order ordinary and delay differential equations and applications, Electron. J. Diff. Equ.2011(9) (2011) 1-11 · Zbl 1211.34096
[2] Abbas S, Weighted pseudo almost automorphic solutions of fractional functional differential equations, Cubo16(1) (2014) 21-35 · Zbl 1301.34102
[3] Abbas S, Chang Y K and Hafayed M, Stepanov type weighted pseudo almost automorphic sequences and their applications to difference equations, Nonlinear Studies21(1) (2014) 99-111 · Zbl 1297.39017
[4] Agarwal R P, Benchohra M and Hamani S, Boundary value problems for fractional differential equations, Georgian Math. J.16(3) (2009) 401-411 · Zbl 1179.26011
[5] Agarwal R P, Andradec B and Cuevas C, Weighted pseudo almost periodic solutions of a class of semilinear fractional differential equations, Nonlinear Anal., RWA11 (2010) 3532-3554 · Zbl 1248.34004
[6] Blot J, Mophu G M, N’Guerekata G M and Pennequin D, Weighted pseudo almost automorphic functions and applications to abstract differential equations, Nonlinear Anal.71 (2009) 903-909 · Zbl 1177.34077
[7] Bochner S, A new approach to almost periodicity, Proc. Natl. Acad. Sci. USA48 (1962) 2039-2043 · Zbl 0112.31401
[8] Bochner S and Neumann J V, On compact solutions of operational-differential equations.I, Ann. Math.36(1) (1935) 255-291 · JFM 61.0442.01
[9] Casarino V, Characterization of almost automorphic groups and semigroups, Rend. Accad. Naz. Sci. XL Mem. Appl.5 (2000) 219-235
[10] Cuesta E, Asymptotic bahaviour of the solutions of fractional integrodifferential equations and some time discretizations, Discrete Continuum Dynamics Systems (Supplement) (2007), 277-285. · Zbl 1163.45306
[11] Cuevas C and Henriquez H, Solutions of second order abstract retarded functional differential equations on the line, Journal of Nonlinear and Convex Analysis12(2) (2011) 225-240 · Zbl 1231.34135
[12] Cuevas C and Lizama C, Almost automorphic solutions to a class of semilinear fractional differential equations, Appl. Math. Lett.21(12) (2008) 1315-1319 · Zbl 1192.34006
[13] Cuevas C, Pierri M and Sepulveda A, Weighted S-asymptotically ω-periodic solutions of a class of fractional differential equations, Adv. Differ. Equ.2011 (2011) Article ID 584874, 13 pages. doi: 10.1155/2011/584874 · Zbl 1210.34006
[14] Cuevas C, Sepulveda A and Soto H, Almost periodic pseudo-almost periodic solutions to fractional differential and integro-differential equations, Appl. Math. Comput.218 (2011) 1735-1745 · Zbl 1246.45012
[15] Cuevas C, N’Guerekata G M and Sepulveda A, Pseudo almost automorphic solutions to fractional differential and integro-differential equations, Commun. Appl. Anal.16(1) (2012) 131-152 · Zbl 1264.43005
[16] Diagana T, Weighted pseudo alomst periodic functions and applications, Compt. Rendus Math.343(103) (2006) 643-646 · Zbl 1112.43005
[17] Diagana T, Stepanov like pseudo alomst periodic functions and their applications to differential equations, Comm. Math. Anal.3(1) (2007) 9-18 · Zbl 1286.44007
[18] Diagana T, Stepanov like pseudo almost periodic functions and their applications to nonautonomous differential equations, Nonlinear Anal., TMA69(12) (2008) 4227-4285 · Zbl 1169.34330
[19] Diagana T, Existence of pseudo-almost automorphic solutions to some abstract differential equations with Sppseudo-almost automorphic coefficients, Nonlinear Anal.70 (2009) 3781-3790 · Zbl 1178.43004
[20] Diagana T, Double-weighted pseudo almost periodic functions, Afr. Diaspora J. Math.12 (2011) 121-136 · Zbl 1247.42008
[21] Diagana T, Evolution equations in generalized Stepanov-like pseudo almost automorphic spaces, Electron. J. Differ. Equ.2012(49) (2012) 1-19 · Zbl 1238.58006
[22] Diagana T and N’Guerekata G M, Stepanov-like almost automorphic functions and applications to some semilinar equations, Appl. Anal.86 (2007) 723-733 · Zbl 1128.43006
[23] Ding H, Liang J and Xiao T, Almost automorphic solutions to nonautonomous semilinear evolution equations in Banach spaces, Nonlinear Anal.73 (2010) 1426-1438 · Zbl 1192.43005
[24] Ding H, Liang J and Xiao T, Almost automorphic solutions to abstract fractional differential equations, Advances in Difference Equations, doi: 10.1155/2010/508374 · Zbl 1197.34105
[25] El-Sayed A M A, On the fractional differential equations, Appl. Math. Comput.49(2-3) (1992) 205-213 · Zbl 0757.34005
[26] El-Sayed A M A, Nonlinear functional-differential equations of arbitrary orders, Nonlinear Anal.33(2) (1998) 181-186 · Zbl 0934.34055
[27] Giona M, Cerbelli S and Roman H E, Fractional diffusion equation and relaxation in complex viscoelastic materials, Physica A191 (1992) 449-453
[28] Granas A and Dugundji J, Fixed Point Theory (2003) (New York: Springer-Verlog)
[29] Liang J, Zhang J and Xiao Ti-Jun, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl.340(2) (2008) 1493-1499 · Zbl 1134.43001
[30] Mahto L, Abbas S and Favini A, Analysis of Caputo impulsive fractional order differential equations with applications, International Journal of Differential Equations2013 (2013) Article ID 704547, 11 pages. doi: 10.1155/2013/704547 · Zbl 1275.34010
[31] Mainardi F, Fractional Calculus: Some basic problems in continuum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics (eds) A Carpinteri and F Mainardi (1997) (New York: Springer) · Zbl 0917.73004
[32] N’Guerekata G M, Almost Automrphic Functions and Almost Periodic Functions in Abstract Spaces (2001) (New York, Berlin, Moscow: Kluwer Academic/Plenum Publishers) · Zbl 1001.43001
[33] N’Guerekata G M, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal.68 (2008) 2658-2667 · Zbl 1140.34399
[34] N’Guerekata G M and Pankov A, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal.68(9) (2008) 2658-2667 · Zbl 1140.34399
[35] Pazy A, Semigroups of Linear Operators and Applications to Partial Differential Equations, in: Applied Mathematical Sciences 44 (1983) (New York, Berlin, Heidelberg, Tokyo: Springer-Verlag) · Zbl 0516.47023
[36] Podlubny I, Fractional Differential Equations (1999) (London: Academic Press) · Zbl 0924.34008
[37] Stepanov V, Uber einige Verallgemeinerungen der fastperiodischen Funktionen, Math. Ann.95 (1926) 473-498 · JFM 52.0262.01
[38] Tarallo M, A Stepanov version for Farvard theory, Arch. Math.87 (2008) 53-59 · Zbl 1154.34332
[39] Xia Z, Weighted pseudo almost automorphic solutions of hyperbolic semilinear integro-differential equations, Nonlinear Anal.95 (2014) 50-65 · Zbl 1283.65121
[40] Xia Z N and Fan M, Weighted Stepanov-like pseudo almost automorphy and applications, Nonlinear Anal.75 (2012) 2378-2397 · Zbl 1306.35140
[41] Xiao T and Liang J, Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup Forum76 (2008) 518-524 · Zbl 1154.46023
[42] Zhang C Y, Pseudo almost periodic solutions of some differential equations, J. Math. Anal. Appl.181(1)(90) (1994) 62-76 · Zbl 0796.34029
[43] Zhang L and Li H, Weighted pseudo-almost periodic solutions for some abstract differential equations with uniform continuity, Bull. Aust. Math. Soc.82 (2010) 424-436 · Zbl 1213.34064
[44] Zhang R, Chang Y K and N’Guerekata G M, New composition theorems of Stepanov-like weighted pseudo almost automorphic functions and applications to nonautonomous evolution equations, Nonlinear Anal.: Real World Appl.13 (2012) 2866-2879 · Zbl 1316.34064
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.