Journal of Applied Nonlinear Dynamics
Existence Result for a Neutral Fractional IntegroDifferential Equation with State Dependent Delay
Journal of Environmental Accounting and Management 7(4) (2018) 371381  DOI:10.5890/JAND.2018.12.005
K. Jothimani$^{1}$, N. Valliammal$^{1}$,$^{2}$, C. Ravichandran$^{3}$
$^{1}$ Department of Mathematics, Sri Eshwar College of Engineering, Coimbatore  641 202, Tamil Nadu, India
$^{2}$ Research and Development Centre, Bharathiar University, Coimbatore  641 046, Tamilnadu, India
$^{3}$ PG & Research Department of Mathematics, Kongunadu Arts & Science College, Coimbatore  641029, Tamil Nadu, India
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Abstract
In this article, we establish the existence of mild solutions for a class of fractional neutral integrodifferential equation with state dependent in Banach space. The results are obtained by Banach contraction principle with resolvent operator technique. An example is offered to explain the theory.
References

[1]  Agarwal, R.P., Meehan, M., and O'Regan, D. (2001), Fixed Point Theory and Applications, Cambridge Tracts in Mathematics, 141, Cambridge University Press, Cambridge. 

[2]  Baleanu, D., Machado, J.A.T., and Luo, A.C.J. (2012), Fractional Dynamics and Control, Springer New York, USA. 

[3]  Hilfer, R. (2000), Applications of Fractional Calculus in Physics, World Scientific, Singapore. 

[4]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and applications of fractional differential equations, In: NorthHolland Mathematics Studies, 204, Elsevier Science, Amsterdam. 

[5]  Lakshmikantham, V., Leela, S., and Vasundhara Devi, J. (2009), Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers. 

[6]  Miller, K.S. and Ross, B. (1993), An Introduction to the Fractional Calculus and Differential Equations, Wiley, New York. 

[7]  Pazy, A. (1983), Semigroups of Linear Operators and Applications to Partial Differential Equations, New York, Springerverlag. 

[8]  Podlubny, I. (1999), Fractional Differential Equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, San Diego. 

[9]  Zhou, Y. (2014), Basic Theory of Fractional Differential Equations, World Scientific, Singapore. 

[10]  Agarwal, R.P., Lupulescu, V., O'Regan, D. and Rahman, G. (2015), Fractional calculus and fractional differential equations in nonreflexive Banach spaces, Commun. Nonlinear. Sci. Numer. Simul., 20(1), 5973. 

[11]  Byszewski, L. (1991), Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal cauchy problem, J. Math. Anal. Appl., 162, 494505. 

[12]  Banano, G.R., and RodriguezLopez, S. (2014), Tersian Existence of solutions to boundary value problem for impulsive fractional differential equations, Frac. calc. Appl.Anal., 17(3), 717744. 

[13]  Chang, Y.K. and Nieto, J.J. (2009), Existence of solutions for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators, Numer. Funct. Anal. Optim., 30(34), 227244. 

[14]  Lv, Z. and Chen, B. (2014), Existence and uniquenes of positive solutions for a fractional swithced system, Abstr. Appl. Anal., 7, 2014, Article ID 828721. 

[15]  Ravichandran, C. and Baleanu, D. (2013), Existence results for fractional neutral functional integrodifferential evolution equations with infinite delay in Banach spaces, J. Adva. Diff. Equ., 1, 215227. 

[16]  Suganya, S., Mallika Arjunan, M., and Trujillo, J.J. (2015), Existence results for an impulsive fractional integrodifferential equation with statedependent delay, App. Math. comput., 266, 5469. 

[17]  Valliammal, N., Ravichandran, C., and Park, J.H. (2017), On the controllability of fractional neutral integrodifferential delay equations with nonlocal conditions, Math. Methods Appl. Sci., 40, 50445055. 

[18]  Vijayakumar, V., Ravichandran, C., and Murugesu, R. (2014), Existence of mild solutions for nonlocal cauchy problem for fractional neutral evolution equations with infinite delay, Surv. Math. Appl., 9, 117129. 

[19]  Vijayakumar, V., Ravichandran, C., and Murugesu, R. (2013), Approximate controllability for a class of fractional neutral integrodifferential inclusions with statedependent delay, Nonlinear Stud., 20(4), 511530. 

[20]  Wang, Y., Liu, L., and Wu, Y. (2014), Positive solutions for a class of higher order singular semipositone fractional differential systems with coupled integral boundary conditions and parameters, Adv. Difference. Equ., 268, 124. 

[21]  Zhou, Y., Zhang, L., and Shen, X.H. (2014), Existence of mild solutions for fractional evolution equations, J. Integral Equations Appl., 25, 557586. 

[22]  Zhou, Y. and Jiao, F. (2010), Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59, 10631077. 

[23]  Dos Santos, J.P.C., Mallika Arjunan, M., and Cuevas, C. (2011), Existence results for fractional neutral integrodifferential equations with statedependent delay, Comput. Math. Appl., 62, 12751283. 

[24]  Dos Santos, J.P.C., Cuevas, C., and de Andrade, B. (2011), Existence results for a fractional equation with statedependent delay, Adv. Difference Equ., 115, DOI: 10.1155/2011/642013. 

[25]  Kailasavalli, S., Suganya, S., and Mallika Arjunan, M. (2017), On fractional neutral integrodifferential systems with statedependent delay in Banach spaces, Fundamenta Informaticae, 151, 109133. 

[26]  Suganya, S., Kalamani, P., and Arjunan, M.M. (2016), Existence of a class of fractional neutral integrodifferential systems with statedependent delay in Banach spaces, Comput. Math. Appl., http://dx.doi.org/10.1016/j. camwa. 

[27]  Suganya, S., Baleanu, D., Selvarasu, S., and Mallika Arjunan, M. (2016), About the existence results of fractional neutral integrodifferential inclusions with statedependent delay in frechet spaces, J. Funct. Spaces, 2016, Article ID 6165804, 19. 

[28]  Andrade, B.D. and Dos Santos, J.P.C. (2012), Existence of solutions for a fractional neutral integrodifferential equation with unbounded delay, Electron. J. Differential Equations, 90, 113. 

[29]  Dos Santos, J.P.C., Vijayakumar, V., and Murugesu, R. (2011), Existence of mild solutions for nonlocal Cauchy problem for fractional neutral integrodifferential equation with unbounded delay, Commun. Math. Anal., X, 113. 

[30]  Fu, X. and Ezzinbi, K. (2003), Existence of solutions for neutral functional differential evolution equations with nonlocal conditions, Nonlinear Anal., 54, 215227. 

[31]  Agarwal, R.P., Dos Santos, J.P.C., and Cuevas, C. (2012), Analytic resolvent operator and existence results for fractional integrodifferential equations, J. Abstr. Differ. Equ. Appl., 2, 2647. 