Discontinuity, Nonlinearity, and Complexity
Boundary Value Problems for Impulsive Fractional Evolution Integrodifferential Equations with Gronwall’s Inequality in Banach Spaces
Discontinuity, Nonlinearity, and Complexity 3(1) (2014) 3348  DOI:10.5890/DNC.2014.03.003
Dimplekumar N. Chalishajar$^{1}$; K. Karthikeyan$^{2}$
1Department of Applied Mathematics, Virginia Military Institute (VMI), 431 Mallory Hall, Lexington, VA24450, USA
$^{2}$ Department of Mathematics, KSR College of Technology, Tiruchengode637 215, Tamilnadu, India
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Abstract
In this paper, we study boundary value problems for impulsive fractional evolution integrodifferential equations with Caputo derivative in Banach spaces. A generalized singular type Gronwall inequality is given to obtain an important priori bounds. Some sufficient conditions for the existence solutions are established by virtue of fractional calculus and fixed point method under some mild conditions. An example is given to illustrate the results.
Acknowledgments
1. First author expresses his thanks to JacksonHope grant of VMI (JH 2014) for kind support..
2. Authors express their gratitude to the anonymous referee for valuable comments and suggestions which are helpful to modify the manuscript.
References

[1]  Caputo, M. (1969), Elasticita e Dissipazione, Bolonga. 

[2]  Benchohra, M. and Slimani, B.A. (2009), Existence and uniqueness of solutions to impulsive fractional differential equations, Electronic Journal of Differential Equations, 2009 (10), 111. 

[3]  Mophou, G.M. (2010), Existence and uniqueness of mild solutions to impulsive fractional differential equations, Nonlinear Analysis, 72, 16041615. 

[4]  Ahmad, B. and Sivasundaram, S. (2009), Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Analysis: Hybrid Systems, 3(3), 251258. 

[5]  Agarwal, R.P., Benchohra, M., and Hamanani, S. (2010), A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Applicandae Mathematicae , 109, 9731033. 

[6]  Chalishajar, D.N. and Karthikeyan, K. (2013), Existence and uniqueness results for boundary value problems of higher order fractional integrodifferential equations involving Grownwall's inequality in Banach spaces, Acta Mathematica Scientia (Science Direct), 33B(3), 758772. 

[7]  Chalishajar, D.N. and Karthikeyan,K. (2012), Existence of Mild Solutions for Fractional Impulsive Semilinear Integrodifferential Equations in Banach Spaces, Communications on Applied Nonlinear Analysis, 19 (4), 4556. 

[8]  Agarwal, R.P., Benchohra, M., and Slimani, B.A. (2008), Existence results for differential equations with fractional order and impulses, Memors on Differential Equations and Mathematical Physics, 44, 121. 

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

[10]  Podlubny, I. (1993), Fractional Differential Equations, Academic Press, New York. 

[11]  Hilfer, R., Metzler, R., Blumen, A., and Klafter, J. (2002), Strange kinetics, Chemical Physics, 284 (12), 399, 12. 

[12]  Hilfer, R. (2003), On fractional relaxation, Fractals, 11, 251257. 

[13]  Klafter, J., Lin, S.C., and Metzler, R. (2011), Fractional Dynamics, Recent Advances, World scientific Publishing Comapany, Singapore. 

[14]  Sandev, T., Metzler, R., and Tomovski, Z. (2011), Fractional diffusion equation with a generalized RiemannLiouville time fractional derivative, Journal of Physics A: Mathematical and Theoretical, 44, 255203. 

[15]  Tomovski, Z., Sandev, T., Metzler, R., and Dubbeldam, J. (2012), Generalized spacetime fraction diffusion equation with composite fractional time derivative, Physica A, Statistical Mechanics and its Applications, 391, 25272542. 

[16]  Metzler, R. and Klafter, J. (2000), The random walk's guide to anomalous difusion: a fractional dynamics approach, Physics Reports , 339 (1) , 177. 

[17]  Metzler, R., Barkie, E., and Klafter, J. (1999), Anomalous diffusion and relaxation close to thermal equilibrium: A fractional FokkerPlanck equation approach, Physical Review Letters, 82 (18), 3563. 

[18]  Minardi, F. (1996), The fundamental solutions for the fractional diffusionwave equation, AppliedMathematics Letters, 9(6), 2328. 

[19]  Minardi, F. (1995), The time fractional diffusionwave equation, Radiophysics and Quantum Electronics, 38 (12), 1324. 

[20]  Minardi, F., Luchko, Yu., and Pagnini, G. (2001), The fundamental solution of the spacetime fractional diffusion equation, Fractional Calculus and Applied Analysis, 4 (2), 153192. 

[21]  Sandev, T., Metzler, R., and Tomovski, Z. (2012), Velocity and displacement correlation functions for fractional generalized Langevin equations, Fractional Calculus and Applied Analysis, 15, 426450. 

[22]  Eab, C.H. and Lim, S.C. (2010), Fractional generalized Langevin equation approach to singlefile diffusion, Physica A, 389, 25102521. 

[23]  Uchaikin, V. and Sibatov, R. (2013), Fractional Kinetics in Solids: Anomalous Charge Transport in Semiconductors, Dielectics and Nanosystems, World Scientific Publishing Comapany, Singapore. 

[24]  Zhang, X., Huang, X., and Liu, Z. (2010), The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay, Nonlinear Analysis. Hybrid Systems, 4, 775781. 

[25]  Diethelm, K., and Freed, A.D. (1999), On the solution of nonlinear fractional order differential equation used in the modeling of viscoplasticity, in: F. keil, W. Maskens, H. Voss(Eds), Scientific computing in chemical Engineering IIComputational Fluid Dynamics and Molecular Properties, SpringerVerlag, Heideberg, 217224. 

[26]  Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993), Fractional Intrgrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon. 

[27]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam. 

[28]  Hernández, E., O'Regan, D., and Balachandran, K. (2010), On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Analysis, 73, 34623471. 

[29]  Keskin, Damla, (2009), Caputo Fractional Derivatives and its Applications, Graduatation Project I, Dept. of Math and Comp. Sc., Cankaya University. 

[30]  Crompton, B. (2011), An Introduction to Fractional Calculus and the Fractional DiffusionWave Equation, Honors Program thesis, University of Massachusetts Lowell. 

[31]  Malinowska, A. and Torres, D. (2011), Fractional calculus for a combined Caputo derivative, Fractional Calculus and Applied Analysis, 14 (4), 523537. 

[32]  Ahmed, N.U. (1991), Semigroup Theory with Applications to systems and control, Pitman Research Notes in Mathematics Series, 246 Longman Scientific and Technical, Harlow JohnWiley and Sons, Inc., New York. 

[33]  Engel, K.J. and Nagel, R. (2000),Oneparameter Semigroups for Linear Evolution Equations, SpringerVerlag, New York. 

[34]  Pazy, A. (1983), Semigroups of linear operators and applications to partial differential equations, AppliedMathematical Sciences, 44, SpringerVerlag, New YorkBerlin. 

[35]  Mainardi, F. and Gorenflo, R. (2000), On MittagLefflertype functions in fractional evolution processes, Computational and Applied Mathematics, 118, (12), 283299. 

[36]  Granas, A., and Dugundji, J. (2003), Fixed Point Theory, Springer, New York. 

[37]  Corduneanu, C. (1971), Principles of Differential and Integral Equations, Allyn and Bacon, Boston. 

[38]  Ntouyas, S.K. and O'Regan, D. (2009), Some remarks on controllability of evolution equations in Banach spaces, Electronic Journal of Differential Equations, 2009, 79, 16. 

[39]  Wang, J.R., Xiang, X., Wei, W., and Chen, Q. (2008), The Generalized Grownwall inequality and its applications to periodic solutions of inetegro differential impulsive periodic system on Banach space, Journal of Inequalities and Applications, 2008, 122. 

[40]  Haiping, Y., Jianming, G., Yongsheng, D. (2007), A generalised Grownwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications, 328, 10751081. 

[41]  Lakshmikantham,V. (2008), Theory of fractional functional differential equations, Nonlinear Analysis, 69, 33373343. 