On Fractional Neutral Integro-differential Systems with State-dependent Delay in Banach Spaces

• Autorzy: Kailasavalli S. , Suganya S. , Mallika Arjunan M.

Identyfikatory

DOI

10.3233/FI-2017-1482

• Języki publikacji: EN

Abstrakty

EN

Based on concepts for semigroups, fractional calculus, resolvent operator and Banach contraction principle, this manuscript is principally involved with existence results for fractional neutral integro-differential systems with state-dependent delay in Banach spaces. To acquire the main results, our working concepts are that the functions deciding the equation fulfill certain Lipschitz conditions of local type which is similar to the hypotheses [1]. Finally an examples are additionally offered to exhibit the achieved hypotheses.

Słowa kluczowe

EN

fractional order differential equations; state-dependent delay; Banach fixed point theorem; resolvent operators; semigroup theory

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2017
• Tom: Vol. 151, nr 1/4
• Strony: 109--133
• Opis fizyczny: Bibliogr. 40 poz.

Twórcy

autor

Kailasavalli S.

• Department of Mathematics, PSNA College of Engineering and Technology, Dindigul-624622, Tamil Nadu, India

autor

Suganya S.

• Department of Mathematics, C. B. M. College, Kovaipudur, Coimbatore - 641 042, Tamil Nadu, India

autor

Mallika Arjunan M.

• Department of Mathematics, C. B. M. College, Kovaipudur, Coimbatore - 641 042, Tamil Nadu, India

Bibliografia

• [1] Andrade F, Cuevas C, and Henriquez HR. Periodic solutions of abstract functional differential equations with state-dependent delay. Mathematical Methods in the Applied Sciences. 2016. doi: 10.1002/mma.3837.
• [2] Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. vol. 44, Springer-Verlag. New York. 1983. doi: 10.1007/978-1-4612-5561-1.
• [3] Baleanu D, Machado JAT, and Luo ACJ. Fractional Dynamics and Control. Springer. New York. USA. 2012. doi: 10.1007/978-1-4614-0457-6.
• [4] Kilbas AA, Srivastava HM, and Trujillo JJ. Theory and Applications of Fractional Differential Equations. Elsevier. Amsterdam. 2006. ISBN: 0444518320.
• [5] Bonanno G, Rodriguez-Lopez R, and Tersian S. Existence of solutions to boundary value problem for impulsive fractional differential equations. Fractional Calculus and Applied Analysis. 2014; 17 (3): 717- 744. doi: 10.2478/s13540-014-0196-y.
• [6] Rodriguez-Lopez R, and Tersian S. Multiple solutions to boundary value problem for impulsive differential equations. Fractional Calculus and Applied Analysis. 2014; 17 (4): 1016-1038. doi: 10.2478/s13540-014-0212-2.
• [7] Agarwal RP. Lupulescu V, Regan D’O, and Rahman G. Fractional calculus and fractional differential equations in nonreflexive Banach spaces. Communications in Nonlinear Science and Numerical Simulation. 2015; 20 ( 1): 59-73. doi: 10.1016/j.cnsns.2013.10.010.
• [8] Keshavarz E. Ordokhani Y, and Razzaghi M. Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Applied Mathematical Modelling. 2014; 38: 6038-6051. doi: 10.1016/j.apm.2014.04.064.
• [9] Lv Z, and Chen B. Existence and uniqueness of positive solutions for a fractional switched system. Abstract and Applied Analysis. Volume 2014 (2014), Article ID 828721, 7 pages. URL http://dx.doi.org/10.1155/2014/828721.
• [10] Wang Y, Liu L, and Wu Y. Positive solutions for a class of higher-order singular semipositone fractional differential systems with coupled integral boundary conditions and parameters. Advances in Difference Equations. 2014; 268: 1-24. doi: 10.1186/1687-1847-2014-268.
• [11] Ahmad B, Ntouyas SK and Alsaed A. Existence of solutions for fractional q-integro-difference inclusions with factional q-integral boundary conditions. Advances in Difference Equations. 2014: 257: 1-18. doi: 10.1186/1687-1847-2014-257.
• [12] Zhang Y, Yang X, Cattani C, Rao RV, Wang S and Phillips P. Tea Category Identification Using a Novel Fractional Fourier Entropy and Jaya Algorithm. Entropy. 2016; 18 (3): 77. doi: 10.3390/e18030077.
• [13] Wang S, Zhang Y, Yang X, Sun P, Dong Z, Liu A and Yuan TF. Pathological Brain Detection by a Novel Image Feature-Fractional Fourier Entropy. Entropy. 2015; 17 (12): 8278-8296. doi: 10.3390/e17127877.
• [14] Agarwal RP, Andrade B and Siracusa G. On fractional integro-differential equations with state-dependent delay. Computers and Mathematics with Applications. 2011; 62: 1143-1149. URL http://dx.doi.org/ 10.1016/j.camwa.2011.02.033.
• [15] Benchohra M and Berhoun F. Impulsive fractional differential equations with state-dependent delay. Communications in Applied Analysis. 2010; 14 (2): 213-224. URL http://www.acadsol.eu/en/articles/14/2/7.pdf.
• [16] Aissani K and Benchohra M. Fractional integro-differential equations with state-dependent delay. Advances in Dynamical Systems and Applications. 2014; 9 (1): 17-30. URL https://www.ripublication.com/adsav2/v9n1p2.pdf.
• [17] Dabas J and Gautam GR. Impulsive neutral fractional integro-differential equation with state-dependent delay and integral boundary condition. Electronic Journal of Differential Equations. 2013; 2013 (273): 1-13. URL http://ejde.math.txstate.edu/Volumes/2013/273/dabas.pdf.
• [18] Suganya S, Mallika Arjunan M and Trujillo JJ. Existence results for an impulsive fractional integro-differential equation with state-dependent delay. Applied Mathematics and Computation. 2015; 266: 54-69. URL http://dx.doi.org/10.1016/j.amc.2015.05.031.
• [19] Dos Santos JPC, Cuevas C and de Andrade B. Existence results for a fractional equations with state dependent delay. Advances in Difference Equations. 2011; 2011: 1-15. doi: 10.1155/2011/642013.
• [20] Dos Santos JPC, Mallika Arjunan M and Cuevas C. Existence results for fractional neutral integro-differential equations with state-dependent delay. Computers and Mathematics with Applications. 2011; 62: 1275-1283. URL http://dx.doi.org/10.1016/j.camwa.2011.03.048.
• [21] Yan Z. Approximate controllability of fractional neutral integro-differential inclusions with state-dependent delay in Hilbert spaces. IMA Journal of Mathematical Control and Information. 2013; 30: 443-462. doi: 10.1093/imamci/dns033.
• [22] Yan Z and Jia X. Approximate controllability of partial fractional neutral stochastic functional integro-differential inclusions with state-dependent delay. Collectanea Mathematica. 2015; 66: 93-124. doi: 10.1007/s13348-014-0109-8.
• [23] Vijayakumar V, Ravichandran C and Murugesu R. Approximate controllability for a class of fractional neutral integro-differential inclusions with state-dependent delay. Nonlinear Studies. 2013; 20 (4): 513-532. URL http://nonlinearstudies.com/index.php/nonlinear/article/view/847.
• [24] Agarwal RP, Dos Santos JPC and Cuevas C. Analytic resolvent operator and existence results for fractional integro-differential equations. Journal of Abstract Differential Equations and Applications. 2012; 2 (2): 26-47. URL http://math-res-pub.org/jadea/2/2.
• [25] Vijayakumar V, Selvakumar A and Murugesu R. Controllability for a class of fractional neutral integro-differential equations with unbounded delay. Applied Mathematics and Computation. 2014; 232: 303-312. URL http://dx.doi.org/10.1016/j.amc.2014.01.029.
• [26] de Andrade B and Dos Santos JPC. Existence of solutions for a fractional neutral integro-differential equation with unbounded delay. Electronic Journal of Differential Equations. 2012; 90: 1-13. URL http://ejde.math.txstate.edu/Volumes/2012/90/andrade.pdf.
• [27] Murugesu R, Vijayakumar V and Dos Santos JPC. Existence of mild solutions for nonlocal Cauchy problem for fractional neutral integro-differential equation with unbounded delay. Communications in Mathematical Analysis. 2013; 14 (1): 59-71. URL http://projecteuclid.org/euclic.cma/1364216230.
• [28] Shu X and Wang Q. The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order 1 < α < 2. Computers and Mathematics with Applications. 2012; 64: 21002110. URL http://dx.doi.org/10.1016/j.camwa.2012.04.006.
• [29] Yan Z and Jia X. Impulsive problems for fractional partial neutral functional integro-differential inclusions with infinite delay and analytic resolvent operators. Mediterranean Journal of Mathematics. 2014; 11:393-428. doi: 10.1007/s00009-013-0349-y.
• [30] Yan Z and Lu F. On approximate controllability of fractional stochastic neutral integro-differential inclusions with infinite delay. Applicable Analysis. 2014: 12351258. URL http://dx.doi.org/10.1080/00036811.2014.924214.
• [31] Kexue L, Jigen P and Jinghuai G. Controllability of nonlocal fractional differential systems of order α ϵ (1,2] in Banach spaces. Reports in Mathematical Physics. 2013; 71: 33-43. URL http://dx.doi.org/10.1016/S0034-4877(13)60020-8.
• [32] Sakthivel R, Ganesh R, Ren Y and Marshal Anthoni S. Approximate controllability of nonlinear fractional systems. Communications in Nonlinear Science and Numerical Simulation. 2013; 18: 3498-3508. URL http://dx.doi.org/10.1016/j.cnsns.2013.05.015.
• [33] Rajivganthi C, Muthukumar P and Ganesh Priya G. Approximate controllability of fractional stochastic integro-differential equations with infinite delay of order 1 < α < 2. IMA Journal of Mathematical Control and Information. 2015: 1-15, Available Online, doi: 10.1093/imamci/dnv005.
• [34] Balasubramaniam P, Kumaresan N, Ratnavelu K and Tamilalagan P. Local and global existence of mild solution for impulsive fractional stochastic differential equations. Bulletin of the Malaysian Mathematical Science Society. 2015; 38: 867-884. doi: 10.1007/s40840-014-0054-4.
• [35] Balasubramaniam P and Tamilalagan P. Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardis function. Applied Mathematics and Computation. 2015; 256: 232-246. URL http://dx.doi.org/10.1016/j.amc.2015.01.035.
• [36] Dabas J and Chauhan A. Existence and uniqueness of mild solution for an impulsive fractional integro-differential equation with infinite delay. Mathematical and Computer Modelling. 2013; 57 (3-4): 754-763. URL http://dx.doi.org/10.1016/j.mcm.2012.09.001.
• [37] Hale J and Kato J. Phase space for retarded equations with infinite delay. Funkcialaj Ekvacioj. 1978; 21: 11-41. URL http://fe.math.kobe-u.ac.jp/FE/Free/vol21/fe21-1-2.pdf.
• [38] Hino Y, Murakami S and Naito T. Functional Differential Equations with Infinite Delay. Springer-Verlag. Berlin. 1991. ISBN: 978-3-540-47388-6, URL http://www.springer.com/us/book/9783540540847.
• [39] Fu X and Huang R. Existence of solutions for neutral integro-differential equations with state-dependent delay. Applied Mathematics and Computation. 2013; 224: 743-759. URL http://dx.doi.org/10.1016/j.amc.2013.09.010.
• [40] Lakshmikantham V, Bainov DD and Simeonov PS. Theory of Impulsive Differential Equations. World Scientific. Singapore. Teaneck. London. 1989. ISBN: 978-9971-5-0970-5, URL http://www.worldscientific.com/worldscibooks/10.1142/0906.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).