## The existence and uniqueness of solutions for a class of nonlinear fractional differential equations with infinite delay.(English)Zbl 1277.34110

Summary: We prove the existence and uniqueness of solutions for two classes of infinite delay nonlinear fractional-order differential equations involving Riemann-Liouville fractional derivatives. The analysis is based on the alternative of the Leray-Schauder fixed-point theorem, the Banach fixed-point theorem, and the Arzela-Ascoli theorem in $$\Omega = \{y : (-\infty, b] \to \mathbb R : y \mid_{(-\infty, 0]} \in \mathcal B\}$$ such that $$y \mid_{[0, b]}$$ is continuous and $$\mathcal B$$ is a phase space.

### MSC:

 34K37 Functional-differential equations with fractional derivatives 47N20 Applications of operator theory to differential and integral equations
Full Text:

### References:

 [1] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, xvi+523, (2006), Amsterdam, The Netherlands: Elsevier Science B.V., Amsterdam, The Netherlands · Zbl 1092.45003 [2] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations. An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, xvi+366, (1993), New York, NY, SUA: John Wiley & Sons, New York, NY, SUA · Zbl 0789.26002 [3] Oldham, K. B.; Spanier, J., The Fractional Calculus, xiii+234, (1974), New York, NY, USA: Academic Press, New York, NY, USA [4] Podlubny, I., Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering, 198, xxiv+340, (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0924.34008 [5] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives, xxxvi+976, (1993), Yverdon, Switzerland: Gordon and Breach Science, Yverdon, Switzerland · Zbl 0818.26003 [6] Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J., Fractional Calculus. Fractional Calculus, Series on Complexity, Nonlinearity and Chaos, 3, xxiv+400, (2012), Hackensack, NJ, USA: World Scientific, Hackensack, NJ, USA · Zbl 1248.26011 [7] Glockle, W. G.; Nonnenmacher, T. F., A fractional calculus approach of self-similar protein dynamics, Biophysical Journal, 68, 46-53, (1995) [8] Baleanu, D.; Günvenc, Z. B. G.; Machado, J. A. T., New Trends in Nanotechnology and Fractional Calculus Applications, xii+531, (2010), New York, NY, USA: Springer, New York, NY, USA · Zbl 1196.65021 [9] Diethelm, K.; Freed, A. D.; Keil, F.; Mackens, W.; Voss, H.; Werther, J., On the solution of fractional order differential equations used in the modeling of viscoplasticity, Scientific Computing in Chemical Engineering II-Computional Fluid Dynamics, Reaction Engineering and Molecular Properties, 217-224, (1999), Heidelberg, Germany: Springer, Heidelberg, Germany [10] Babakhani, A.; Daftardar-Gejji, V., Existence of positive solutions of nonlinear fractional differential equations, Journal of Mathematical Analysis and Applications, 278, 2, 434-442, (2003) · Zbl 1027.34003 [11] Yu, C.; Gao, G., Existence of fractional differential equations, Journal of Mathematical Analysis and Applications, 310, 1, 26-29, (2005) · Zbl 1088.34501 [12] Babakhani, A.; Daftardar-Gejji, V., Existence of positive solutions for N-term non-autonomous fractional differential equations, Positivity, 9, 2, 193-206, (2005) · Zbl 1111.34006 [13] Băleanu, D.; Mustafa, O. G.; Agarwal, R. P., Asymptotically linear solutions for some linear fractional differential equations, Abstract and Applied Analysis, 2010, (2010) · Zbl 1210.34005 [14] Cabada, A.; Wang, G., Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, Journal of Mathematical Analysis and Applications, 389, 1, 403-411, (2012) · Zbl 1232.34010 [15] Ahmad, B.; Nieto, J. J.; Alsaedi, A.; El-Shahed, M., A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear Analysis: Real World Applications, 13, 2, 599-606, (2012) · Zbl 1238.34008 [16] Bhalekar, S.; Daftardar-Gejji, V.; Baleanu, D.; Magin, R., Fractional Bloch equation with delay, Computers & Mathematics with Applications, 61, 5, 1355-1365, (2011) · Zbl 1217.34123 [17] Agarwal, R. P.; Ahmad, B., Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions, Computers & Mathematics with Applications, 62, 3, 1200-1214, (2011) · Zbl 1228.34009 [18] Ahmad, B.; Nieto, J. J.; Alsaedi, A., A nonlocal three-point inclusion problem of Langevin equation with two different fractional orders, Advances in Difference Equations, 2012, article 54, (2012) · Zbl 1291.34004 [19] Baleanu, D.; Trujillo, J. J., On exact solutions of a class of fractional Euler-Lagrange equations, Nonlinear Dynamics, 52, 4, 331-335, (2008) · Zbl 1170.70328 [20] Diethelm, K.; Ford, N. J., Analysis of fractional differential equations, Journal of Mathematical Analysis and Applications, 265, 2, 229-248, (2002) · Zbl 1014.34003 [21] Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, Electronic Transactions on Numerical Analysis, 5, 1-6, (1997) · Zbl 0890.65071 [22] Odibat, Z.; Momani, S., Numerical methods for nonlinear partial differential equations of fractional order, Applied Mathematical Modelling, 32, 28-39, (2008) · Zbl 1133.65116 [23] Diethelm, K.; Walz, G., Numerical solution of fractional order differential equations by extrapolation, Numerical Algorithms, 16, 3-4, 231-253, (1997) · Zbl 0926.65070 [24] Maraaba, T.; Baleanu, D.; Jarad, F., Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, Journal of Mathematical Physics, 49, 8, (2008) · Zbl 1152.81550 [25] Ye, H.; Ding, Y.; Gao, J., The existence of a positive solution of a fractional differential equation with delay, Positivity, 11, 341-350, (2007) [26] Hale, J. K.; Kato, J., Phase space for retarded equations with infinite delay, Funkcialaj Ekvacioj, 21, 1, 11-41, (1978) · Zbl 0383.34055 [27] Granas, A.; Dugundji, J., Fixed Point Theory. Fixed Point Theory, Springer Monographs in Mathematics, xvi+690, (2003), New York, NY, USA: Springer, New York, NY, USA · Zbl 1025.47002 [28] Golberg, R. R., Methods of Real Analysis, (1970), New Delhi, India: Oxford and IBH Publishing Company, New Delhi, India [29] Henry, D., Geometric Theory of Semilinear Parabolic Partial Differential Equations, (1989), Berlin, Germany: Springer, Berlin, Germany
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.