Discontinuity, Nonlinearity, and Complexity
Nonlinear Fourpoint Impulsive Fractional Differential Equations with pLaplacian Operator
Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 467486  DOI:10.5890/DNC.2015.11.009
Fatma Tokmak Fen$^{1}$, Ilkay Yaslan Karaca$^{2}$
$^{1}$ Department of Mathematics, Gazi University, 06500 Teknikokullar, Ankara, Turkey
$^{2}$ Department of Mathematics, Ege University, 35100 Bornova, Izmir, Turkey
Download Full Text PDF
Abstract
In this paper, we investigate the existence of solutions for a fourpoint nonlocal boundary value problem of nonlinear impulsive differential equations of fractional order α ∈ (2,3]. By using some well known fixed point theorems, sufficient conditions for the existence of solutions are established. Some illustrative examples are also discussed.
Acknowledgments
We would like to thank the referees for their valuable comments and suggestions.
References

[1]  Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006), Theory and applications of fractional differential equations, Elsevier, Amsterdam. 

[2]  Lakshmikantham, V., Leela, S. and Vasundhara Devi, J. (2009), Theory of Fractional Dynamic Systems, Cambridge Academic Publisher, Cambridge. 

[3]  Agarwal, R.P., Belmekki, M. and Benchohra, M. (2009), A survey on semilinear differential equations and inclusions involving RiemannLiouville fractional derivative, Advances in Difference Equations, 2009, Art. ID 981728. 

[4]  Ahmad, B. and Nieto, J.J. (2009), Existence results for a coupled system of nonlinear fractional differential equations with threepoint boundary conditions, Computers & Mathematics with Applications. An International Journal, 58, 18381843. 

[5]  Balachandran, K. and Trujillo, J.J. (2010), The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 72, 45874593. 

[6]  Belmekki, M., Nieto, J.J. and RodriguezLopez, R. (2009), Existence of periodic solution for a nonlinear fractional differential equation, Boundary Value Problems, Art. ID 324561, 18. 

[7]  Benchohra, M., Hamani, S. and Ntouyas, S.K. (2009), Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 71, 23912396. 

[8]  Darwish, M.A. and Ntouyas, S.K. (2010), On initial and boundary value problems for fractional order mixed type functional differential inclusions, Computers & Mathematics with Applications. An International Journal, 59, 1253 1265. 

[9]  Edelman,M. and Tarasov, V.E. (2009), Fractional standard map, Physics Letters A, 374(2), 279285. 

[10]  Machado,J.A.T. (2013), Fractional generalization of memristor and higher order elements, Communications in Nonlinear Science and Numerical Simulation, 18, 264275. 

[11]  Sabatier, J., Agrawal, O.P. and Machado, J.A.T. (2007), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht. 

[12]  Wang, G., Liu, W. and Ren, C. (2012), Existence of solutions for multipoint nonlinear differential equations of fractional orders with integral boundary conditions, Electronic Journal of Differential Equations, 54, 10. 

[13]  Wang, G., Liu, S., Baleanu, D. and Zhang, L. (2013), Existence results for nonlinear fractional differential equations involving different RiemannLiouville fractional derivatives, Advances in Difference Equations, 2013:280, 7. 

[14]  Zhang, S.Q. (2010), Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Computers & Mathematics with Applications. An International Journal, 59, 13001309. 

[15]  Samoilenko, A.M. and Perestyuk, N.A. (1995), Impulsive Differential Equations,World Scientific, Singapore. 

[16]  Benchohra, M., Henderson, J. and Ntouyas, S. (2006), Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York. 

[17]  Akhmet, M. (2010), Principles of Discontinuous Dynamical Systems, Springer, New York. 

[18]  Karaca, I.Y. (2009), On positive solutions for fourthorder boundary value problem with impulse, Journal of Computational and Applied Mathematics, 225, 356364. 

[19]  Jankowski, T. (2011), Positive solutions to thirdorder impulsive SturmLiouville boundary value problems with deviated arguments and onedimensional pLaplacian, Dynamic Systems and Applications, 20, 575586. 

[20]  Ozen, O.B., Karaca, I.Y. and Tokmak, F. (2013), Existence results for pLaplacian boundary value problems of impulsive dynamic equations on time scales, Advances in Difference Equations, 2013:334, 14. 

[21]  Abbas, S. and Benchohra, M. (2010), Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order, Nonlinear Analysis. Hybrid Systems. An International Multidisciplinary Journal, 4, 406413. 

[22]  Agarwal, R.P. and Ahmad, B. (2011), Existence of solutions for impulsive antiperiodic boundary value problems of fractional semilinear evolution equations,Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis, 18, 457470. 

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

[24]  Ahmad, B. and Sivasundaram, S. (2010), Existence of solutions for impulsive integral boundary value problems of fractional order, Nonlinear Analysis. Hybrid Systems. An International Multidisciplinary Journal, 4, 134141. 

[25]  Debbouche, A. and Baleanu, D. (2011), Controllability of fractional evolution nonlocal impulsive quasilinear delay integrodifferential systems, Computers & Mathematics with Applications. An International Journal, 62, 14421450. 

[26]  Mophou, G.M. (2010), Existence and uniqueness of mild solutions to impulsive fractional differential equation, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methodse, 72, 16041615. 

[27]  Tian, Y. and Bai, Z. (2010), Existence results for the threepoint impulsive boundary value problem involving fractional differential equations, Computers & Mathematics with Applications. An International Journal, 59, 26012609. 

[28]  Wang, G., Ahmad, B. and Zhang, L. (2011), Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions, Computers & Mathematics with Applications. An International Journal, 62, 13891397. 

[29]  Wang, G., Ahmad, B. and Zhang,L. (2013), New existence results for nonlinear impulsive integrodifferential equations of fractional order with nonlocal boundary conditions, Nonlinear Studies, 20, 119130. 

[30]  Chai, G.Q. (2012), Positive solutions for boundary value problem of fractional differential equation with pLaplacian operator, Boundary Value Problems, 2012:18, 20. 

[31]  Chen, T. and Liu, W.B. (2012), An antiperiodic boundary value problem for the fractional differential equation with a pLaplacian operator, Applied Mathematics Letters. An International Journal of Rapid Publication, 25, 16711675. 

[32]  Chen, T. and Liu, W.B. (2013), Solvability of some boundary value problems for fractional pLaplacian equation, Abstract and Applied Analysis, Art. ID 432509, 6. 

[33]  He, J. and Song, X. (2014), The uniqueness of solution for a class of fractional order nonlinear systems with p Laplacian operator, Abstract and Applied Analysis, Art. ID 921209, 8. 

[34]  Liu, X.P., Jia, M. and Xiang, X.F. (2012), Nucleic acid content of microscope, Computers & Mathematics with Applications. An International Journal, 64, 32673272. 

[35]  Prasad, K.R. and Krushna, B.M.B. (2014), Multiple positive solutions for a coupled system of pLaplacian fractional order twopoint boundary value problems, International Journal of Differential Equations, Art. ID 485647, 10. 

[36]  Wang, J.H., Xiang, H.J. and Liu, Z.G. (2010), Existence of concave positive solutions for boundary value problem of nonlinear fractional differential equation with pLaplacian operator, International Journal of Mathematics and Mathematical Sciences, Art. ID 495138, 17. 

[37]  Liu, Z., Lu, L. and Szántó, I. (2013), Existence of solutions for fractional impulsive differential equations with pLaplacian operator, Acta Mathematica Hungarica, 141, 203219. 

[38]  Karaca, I.Y. and Tokmak, F. (2014), Existence of solutions for nonlinear impulsive fractional differential equations with pLaplacian operator,Mathematical Problems in Engineering, Art. ID 692703, 11. 

[39]  Ahmad, B. and Wang, G. (2011), A study of an impulsive fourpoint nonlocal boundary value problem of nonlinear fractional differential equations, Computers & Mathematics with Applications. An International Journal, 62, 1341 1349. 

[40]  Sun, J.X. (2008), Nonlinear Functional Analysis and its Application, Science Press, Beijing. 