## On a class of retarded integro-differential equations with nonlocal initial conditions.(English)Zbl 1208.45004

The paper deals with the local existence and uniqueness of a mild solution for the Cauchy problem formed by a fractional integro-differential equation with time-delay and a nonlocal initial condition: $u'(t) - \int_0^t \frac{(t-s)^{\mu -2}}{\Gamma(\mu -1)} Au(s)ds = F(t,u(t),u(\kappa(t))),\quad t\geq 0;$
$u(t) + H_t(u) = \phi(t),\quad-\tau \leq t \leq 0.$ Here, $$1< \mu < 2$$, $$\tau >0$$, $$A: D(A) \subset X \rightarrow X$$ is a generator of a solution operator on a complex Banach space $$X$$, $$\kappa: [0,\,\infty) \rightarrow [-\tau,\,\infty)$$ is a function representing the delay, $$H_t: [-\tau,\,0] \times{\mathcal C}([-\tau,\,0],\,X)\rightarrow X$$ is an operator. The convolution integral in the equation is of the Riemann-Liouville fractional integral type. The existence of a global solution is also proven here. An illustrative example is presented at the end of the paper.

### MSC:

 45J05 Integro-ordinary differential equations 26A33 Fractional derivatives and integrals 45G10 Other nonlinear integral equations
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### References:

 [1] Cuevas, C.; de Souza, Julio César, $$S$$-asymptotically $$\omega$$-periodic solutions of semilinear fractional integro-differential equations, Appl. math. lett., 22, 865-870, (2009) · Zbl 1176.47035 [2] Cuevas, C.; de Souza, Julio César, Existence of $$S$$-asymptotically $$\omega$$-periodic solutions for fractional order functional integro-differential equations with infinite delay, Nonlinear anal., 72, 1683-1689, (2010) · Zbl 1197.47063 [3] E. Bazhlekova, Fractional evolution equations in Banach spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001. [4] Podlubny, I.; Petras˘, I.; Vinagre, B.M.; O’Leary, P.; Dorc˘ak, L., Analogue realizations of fractional-order controllers: fractional order calculus and its applications, Nonlinear dynam., 29, 281-296, (2002) · Zbl 1041.93022 [5] Anh, V.V.; Leonenko, N.N., Spectral analysis of fractional kinetic equations with randomdata, J. stat. phys., 104, 1349-1387, (2001) · Zbl 1034.82044 [6] Anh, V.V.; Mcvinish, R., Fractional differential equations driven by levy noise, J. appl. stoch. anal., 16, 2, 97-119, (2003) · Zbl 1042.60034 [7] Miller, K.S.; Ross, B., An introduction to the fractional calculus and differential equations, (1993), John Wiley New York · Zbl 0789.26002 [8] Agarwal, R.P.; de Andrade, B.; Cuevas, C., On type of periodicity and ergodicity to a class of fractional order differential equations, Adv. differential equations, 2010, 25 pages, (2010), Article ID 179750 · Zbl 1194.34007 [9] R.P. Agarwal, B. de Andrade, C. Cuevas, Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations, Nonlinear Anal. RWA, in press (doi:10.1016/j.nonrwa.2010.01.002). [10] Agarwal, R.P.; Belmekki, M.; Benchohra, M., A survey on semilinear differential equations and inclusions involving riemann – liouville fractional derivative, Adv. difference equ., 2009, 47 pages, (2009), Article ID 981728 · Zbl 1182.34103 [11] Agarwal, R.P.; Benchohra, M.; Hamani, S., A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta appl. math., 109, 973-1033, (2010) · Zbl 1198.26004 [12] Agarwal, R.P.; Lakshmikantham, V.; Nieto, J.J., On the concept of solution for fractional differential equations with uncertainty, Nonlinear anal., 72, 2859-2862, (2010) · Zbl 1188.34005 [13] Cuevas, C.; Lizama, C., Almost automorphic solutions to a class of semilinear fractional differential equations, Appl. math. lett., 21, 1315-1319, (2008) · Zbl 1192.34006 [14] Cuevas, C.; Rabelo, M.; Soto, H., Pseudo almost automorphic solutions to a class of semilinear fractional differential equations, Commun. appl. nonlinear anal., 17, 33-48, (2010) [15] Lakshmikantham, V., Theory of fractional differential equations, Nonlinear anal., 69, 10, 3337-3343, (2008) · Zbl 1162.34344 [16] Lakshmikantham, V.; Vatsala, A., Basic theory of fractional differential equations, Nonlinear anal., 69, 8, 2677-2682, (2008) · Zbl 1161.34001 [17] Podlubny, I., Fractional differential equations, (1999), San Diego Academic Press · Zbl 0918.34010 [18] Hilfer, H., Applications of fractional calculus in physics, (2000), World Scientific Publ. Co. Singapore · Zbl 0998.26002 [19] Henriquez, H.R., Approximation of abstract functional differential equations with unbounded delay, Indian J. pure appl. math., 27, 4, 357-386, (1996) · Zbl 0853.34072 [20] Hino, Y.; Murakami, S.; Naito, T., () [21] Byszewski, L., Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. math. anal. appl., 162, 494-505, (1991) · Zbl 0748.34040 [22] Balachandran, K.; Park, J.Y., Sobolev type integrodifferential equation with nonlocal condition in Banach spaces, Taiwanese J. math., 7, 155-163, (2003) · Zbl 1032.45013 [23] N’Guérékata, G.M., Existence and uniqueness of an integral solution to some Cauchy problem with nonlocal conditions, (), 843-849 · Zbl 1147.35329 [24] N’Guérékata, G.M., A Cauchy problem for some fractional abstract differential equation with nonlocal conditions, Nonlinear anal., 70, 1873-1876, (2009) · Zbl 1166.34320 [25] Liang, J.; Xiao, T.J., Semilinear integrodifferential equations with nonlocal initial conditions, Comput. math. appl., 47, 863-875, (2004) · Zbl 1068.45014 [26] Mophou, G.M.; N’Guérékata, G.M., Existence of mild solution for some fractional differential equations with nonlocal conditions, Semigroup forum, 79, 315-322, (2009) · Zbl 1180.34006 [27] Deng, K., Exponential decay of solutions of semilinear parabolic equations with non-local initial conditions, J. math. anal. appl., 179, 630-637, (1993) · Zbl 0798.35076 [28] Henriquez, H.R.; Hernandez, E.; Akca, H., Global solutions for an abstract Cauchy problem with nonlocal conditions, Internat. J. math. manuscripts, 1, (2007) [29] Liang, J.; Liu, J.H.; Xiao, T.J., Nonlocal Cauchy problems governed by compact operator families, Nonlinear anal., 57, 183-189, (2004) · Zbl 1083.34045 [30] Dubey, Shruti A.; Bahuguna, Dhirendra, Existence and regularity of solutions to nonlocal retarded differential equations, Appl. math. comput., 215, 2413-2424, (2009) · Zbl 1193.34162 [31] Henriquez, H.R.; Pierri, M.; Goncalves, G., Existence results for an impulsive abstract partial differential equation with state-dependent delay, Comput. math. appl., 52, 411-420, (2006) · Zbl 1153.35396 [32] Kolmanovskii, V.; Myshkis, A., Introduction to the theory and applications of functional differential equations, (1999), Kluwer Academic Dordrecht · Zbl 0917.34001 [33] Byszewski, L.; Akca, H., On a mild solution of a semilinear functional-differential evolution nonlocal problem, J. appl. math. stoch. anal., 10, 265-271, (1997) · Zbl 1043.34504 [34] Cuesta, E., Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete contin. dyn. syst., 277-285, (2007) · Zbl 1163.45306 [35] Lizama, C., Regularized solutions for abstract Volterra equations, J. math. anal. appl., 243, 278-292, (2000) · Zbl 0952.45005 [36] Engel, K.J.; Nagel, R., One parameter semigroups for linear evolution equations, (1995), Springer-Verlag [37] Agarwal, S.; Bahuguna, D., Existence of solutions to Sobolev-type partial neutral differential equations, J. appl. math. stoch. anal., 2006, (2006), Article ID 16308 · Zbl 1119.34060 [38] Sadovskii, B.N., On a fixed point principle, Funct. anal. appl., 1, 74-76, (1967) · Zbl 0165.49102
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