Abstract
The aim of this study is to investigate the existence and other properties of solution of nonlinear fractional integro–differential equations with constant coefficient. Also with the help of Pachpatte’s inequality, we prove the continuous dependence of the solutions.
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Abbas, S., Benchohra, M., N’Guérékata, G.M.: Topics in Fractional Differential Equations, vol. 27. Springer, New York (2012). https://doi.org/10.1007/978-1-4614-4036-9
Aghajani, A., Pourhadi, E., Trujillo, J.: Application of measure of noncompactness to a Cauchy problem for fractional differential equations in Banach spaces. Fract. Calc. Appl. Anal. 16(4), 962–977 (2013). https://doi.org/10.2478/s13540-013-0059-y
Anastassiou, G.: Advances on Fractional Inequalities. Springer, New York (2011)
Balachandran, K., Park, J.Y.: Nonlocal Cauchy problem for abstract fractional semilinear evolution equations. Nonlinear Anal. 71(10), 4471–4475 (2009). https://doi.org/10.1016/j.na.2009.03.005
Baleanu, D., Güvenç, Z., Machado, J.: New Trends in Nanotechnology and Fractional Calculus Applications. Springer, New York (2000). https://doi.org/10.1007/978-90-481-3293-5
Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus. Models and Numerical Methods. World Scientific, New York (2012)
Cabrera, I., Harjani, J., Sadarangani, K.: Existence and uniqueness of solutions for a boundary value problem of fractional type with nonlocal integral boundary conditions in Hölder spaces. Mediterr. J. Math. 15, 1–15 (2018). https://doi.org/10.1007/s00009-018-1142-8
Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, New York (2010)
Diethelm, K., Ford, N.J.: Analysis of Fractional Differential Equations. J. Math. Anal. Appl. 265(2), 229–248 (2002). https://doi.org/10.1006/jmaa.2000.7194
Dong, X., Wang, J., Zhou, Y.: On nonlocal problems for fractional differential equations in Banach spaces. Opuscula Math. 31(3), 341–357 (2011)
Furati, K.M., Tatar, N.: Long time behavior for a nonlinear fractional model. J. Math. Anal. Appl. 332(1), 441–454 (2007). https://doi.org/10.1016/j.jmaa.2006.10.027
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, New York (2000)
Jagtap, T.B., Kharat, V.V.: On existence of solution to nonlinear fractional integrodifferential system. J. Trajectory 22(1), 40–46 (2014)
Kharat, V.V.: On existence and uniqueness of fractional integrodifferential equations with an integral fractional boundary condition. Malaya J. Mat. 6(3), 485–491 (2018)
Kendre, S.D., Jagtap, T.B., Kharat, V.V.: On nonlinear fractional integrodifferential equations with nonlocal condition in Banach spaces. Nonlinear Anal. Differ. Equat. 1(3), 129–141 (2013)
Kendre, S.D., Kharat, V.V., Jagtap, T.B.: On abstract nonlinear fractional integrodifferential equations with integral boundary condition. Comm. Appl. Nonlinear Anal. 22(3), 93–108 (2015)
Kendre, S.D., Kharat, V.V., Jagtap, T.B.: On fractional integrodifferential equations with fractional non-separated boundary conditions. Int. J. Appl. Math. Sci. 13(3), 169–181 (2013)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, 204th edn. Elsevier, Amsterdam (2006)
Lakshmikantham, V., Leela, S., Vasundhara, D.J.: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge (2009)
Liang, J., Liu, Z., Wang, X.: Solvability for a couple system of nonlinear fractional differential equations in a Banach space. Fract. Calc. Appl. Anal. 16(1), 51–63 (2013). https://doi.org/10.2478/s13540-013-0004-0
Luchko, Y.U.R.I.I., Gorenflo, R.: An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math. Vietnam. 24(2), 207–233 (1999)
N’Guérékata, G.M.: A Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Anal. 70(5), 1873–1876 (2009). https://doi.org/10.1016/j.na.2008.02.087
N’Guérékata, G.M.: Corrigendum: A Cauchy Problem for some Fractional Differential Equations. Commun. Math. Anal. 7(1), 11 (2009). http://math-res-pub.org/cma/7
Nieto, J., Ouahab, A., Venktesh, V.: Implicit fractional differential equations via the Liouville–Caputo derivative. Mathematics 3(2), 398–411 (2015). https://doi.org/10.3390/math3020398
Pachpatte, B.: Inequalities for Differential and Integral Equations. Academic Press, New York (1998)
Pierri, M., O’Regan, D.: On non-autonomous abstract nonlinear fractional differential equations. Appl. Anal. 94(5), 879–890 (2015). https://doi.org/10.1080/00036811.2014.905679
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
Ren, Y., Qin, Y., Sakthivel, R.: Existence results for fractional order semilinear integro-differential evolution equations with infinite delay. Integr. Equ. Oper. Theory 16(1), 33–49 (2010). https://doi.org/10.1007/s00020-010-1767-x
Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives. Gordon and Breach, Yverdon (1993)
Tarasov, V.E.: Fractional dynamics: applications of fractional calculus to dynamics of particles. Higher Education Press, Heidelberg (2010)
Tate, S., Dinde, H.T.: Some theorems on Cauchy problem for nonlinear fractional differential equations with positive constant coefficient. Mediterr. J. Math. 14(2), 1–17 (2017). https://doi.org/10.1007/s00009-017-0886-x
Tidke, H.L.: Some theorems on fractional semilinear evolution equations. J. Appl. Anal. 18(2), 209–224 (2012). https://doi.org/10.1515/jaa-2012-0014
Wang, J., Li, X.: A uniform method to UlamHyers stability for some linear fractional equations. Mediterr. J. Math. 13(2), 625–635 (2016). https://doi.org/10.1007/s00009-015-0523-5
Zhou, Y., Shen, X.H., Zhang, L.: Cauchy problem for fractional evolution equations with Caputo derivative. Eur. Phys. J. Spec. Top. 222(8), 1749–1765 (2013). https://doi.org/10.1140/epjst/e2013-01961-5
Zhou, Y., Jiao, F., Pečarić, J.: Abstract Cauchy problem for fractional functional differential equations. Topol Methods Nonlinear anal. 42(1), 119–136 (2013)
Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. Real World Appl. 11(5), 4465–4475 (2010). https://doi.org/10.1016/j.nonrwa.2010.05.029
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Tate, S., Kharat, V.V. & Dinde, H.T. On Nonlinear Fractional Integro–Differential Equations with Positive Constant Coefficient. Mediterr. J. Math. 16, 41 (2019). https://doi.org/10.1007/s00009-019-1325-y
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DOI: https://doi.org/10.1007/s00009-019-1325-y
Keywords
- Fractional integro–differential equation
- existence of solution
- continuous dependence
- fixed point theorem
- Pachpatte’s inequality