Existence Criteria and HyersUlam Theorem for a Coupled PLaplacian System of Fractional Differential Equations
Vol. 5, Issue 5, PP. 7686, May 2018
Keywords: Fractional differential equations (FDEs), HyerUlam stability (HUS), topological degree theory, existence and uniqueness of solutions (EUS)
Download PDF

Dealing with high order coupled systems of FDEs through nonlinear pLaplacian operator. We analyze existence, uniqueness & HyerUlam stability (HUS) of the solutions by means of topological degree method. For this purpose, we transform the supposed problem into an integral system via Green�s function(s) and assume certain operator equivalent to the integral form of the problem. Then after, the results are proved with some necessary assumptions.
 Kiran Tabassum: College of Science, Hohai University, 210098, Nanjing, P. R. China.email: kirantabassam@gmail.com
 Liu Xiangyang: College of Science, Hohai University, 210098, Nanjing, P. R. China. email: liuxy@hhu.edu.cn
 Syed Furqan Rafique: Department of Electrical Engineering, North China Electric Power University, Beijing, P.R.China. email: syedfurqan@ncepu.edu.cn
 Irfan Jamil: College of Energy and Electrical Engineering, Hohai University, 210098, Nanjing, P.R.China. email: I.jamil@hhu.edu.cn
Kiran Tabassum Liu Xiangyang Syed Furqan Rafique Irfan Jamil
 D. Baleanua, O.G. Mustafa and R.P. Agarwal, An existence result for a superlinear fractional differential equation, Appl. Math. Lett., Vol. 23(9) (2010), 1129  1132.
 P. Kumam, A. Ali, K. Shah, R. A. Khan, Existence results and HyersUlam stability to a class of nonlinear arbitrary order differential equations, J. Nonlinear Sci. Appl., 10 (2017), 29862997.
 D. Baleanu, R. P. Agarwal, H. Mohammadi, S. Rezapour, Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces, Bound. Value Probl., 2013 (2013), 8 pages.
 D. Baleanu, O. G. Mustafa, R. P. Agarwal, On the solution set for a class of sequential fractional differential equations,J.Phys. A, 43 (2010), 7 pages.
 N. I. Mahmudov, S. Unul, Existence of solutions of order fractional threepoint boundary value problems with integral conditions, Abstr. Appl. Anal., 2014 (2014), 12 pages.
 N. I. Mahmudov, S. Unul, Existence of solutions of fractional boundary value problems with pLaplacian operator, Bound.ValueProbl., 2015 (2015), 16 pages. 1.
 N. I. Mahmudov, S. Unul, On existence of BVP�s for impulsive fractional differential equations, Adv. Difference Equ.,2017 (2017), 16 pages.
 L. Hu, S. Zhang,On existence results for nonlinear fractional differential equations involving thepLaplacian at resonance. Mediterr. J. Math.13, 955966 (2016).
 A. Ali, B. Samet, K. Shah, R. A. Khan, Existence and stability of solution to a toppled systems of differential equations of noninteger order. Bound.Value Probl.2017, 16 (2017).
 J. Mawhin,Topological Degree Methods in Nonlinear Boundary Value Problems CMBS Regional Conference Series in Mathematics, vol. 40. Am. Math. Soc., Providence (1979).
 F. Isaia, On a nonlinear integral equation without compactness. Acta Math.Univ. Comen.75, 233240 (2006).
 J. Wang, Y. Zhou, W. Wei, Study in fractional differential equations by means of topological degree methods. Numer.Funct. Anal. Optim.33(2), 216238 (2012).
 K. Shah, R. A. Khan, Existence and uniqueness results in a coupled system of fractional order boundary value problems by topological degree theory. Numer.Funct.Anal.Optim.37, 887899 (2016).
 K. Shah, A. Ali, R. A. Khan, Degree theory and existence of positive solutions to coupled systems of multipoint boundary value problems. Bound. Value Probl.2016(1), 1 (2016).
 T. Shen, W. Liu and X. Shen, Existence and uniqueness of solutions for several BVPs of fractional differential equations with pLaplacian operator, Mediterr. J. Math. (2016) DOI 10.1007/s0000901607669.
 I. Area, J. Losada, and J. J. Nieto, A note on fractional logistic equation. Physica A444, 182187 (2016).
 K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, pp. 209217. Wiley,New York (1993).
 R. Hilfer, Application of Fractional Calculus in Physics. World Scientific, Singapore (2000).
 A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations.NorthHolland Mathematics Studies, vol. 24.Amsterdam (2006).
 R. Agarwal, S. Hristova, and D. O�Regan, Stability of solutions to impulsive Caputo fractional differential equations. Electron. J. Differ. Equ.2016, Article ID 58 (2016).
 D. Anderson, R. Avery, Fractionalorder boundary value problem with SturmLiouville boundary conditions. Electron.J. Differ. Equ.2015, Article ID 29 (2015).
 I. Bachar, H. Maagli, and V. Radulescu, Fractional Navier boundary value problems. Bound.Value Probl.2016, Article ID 79 (2016).
 N. D. Cong, T. S. Doan, S. Siegmund, and H. T. Tuan, Linearized asymptotic stability for fractional differential equations.Electron. J. Qual. Theory Differ. Equ.2016, Article ID 39 (2016).
 M. Ghergu, V. Radulescu, Nonlinear PDEs. Mathematical Models in Biology, Chemistry and Population Genetics.Springer Monographs in Mathematics. Springer, Heidelberg (2012).
 S. Kumar, D. Kumar, J. Singh, Fractional modeling arising in unidirectional propagation of long waves in dispersive media. Adv. Nonlinear Anal.5(4), 383394 (2016).
 S. Peng, J. R. Wang, Existence and UlamHyers stability of ODEs involving two Caputo fractional derivatives.Electron.J.Qual. Theory Differ. Equ.2015, Article ID 52 (2015).
 H. Jafari, D. Baleanu, H. Khan, R. A. Khan, and A. Khan, Existence criterion for the solutions of fractional orderpLaplacian boundary value problems.Bound.Value Probl.2015, 164 (2015).
 L. Diening, P. Lindqvist, B. Kawohl, MiniWorkshop: thepLaplacian operator and applications. OberwolfachRep.10(1), 433482 (2013).
 X. Han, X. Yang, Existence andmultiplicity of positive solutions for a systemof fractional differential equation with parameters.Bound.Value Probl.2017, 78 (2017).
 E. Zhi, X. Liu, and F. Li, Non local boundary value problems of fractional differential equations withpLaplacian.Math.Methods Appl. Sci.37, 26512662 (2014).
 L. Zhang, W. Zhang, X. Liu, and M. Jia, Existence of positive solutions for integral boundary value problems of fractional differential equations with pLaplacian. Adv. Differ. Equ.2017, 36 (2017).
 R. A. Khan, A. Khan, A. Samad, H. Khan, On existence of solutions for fractional differential equations withpLaplacian operator. J. Fract. Calc. Appl.5(2), 2837 (2014).
 E. Cetin, F. S. Topal, Existence of solutions for fractional four point boundary value problems withpLaplacian operator. J. Comput. Anal.Appl.19(1), 892903 (2015).
 S. Liang, J. Zhang, Existence and uniqueness of positive solutions for integral boundary problems of nonlinear fractional differential equations with pLaplacian operator. Rocky Mt. J. Math.44(1), 953974 (2014).
 H. Khan, Y. Li, W. Chen, D. Baleanu, and A. Khan, Existence theorems and HyersUlam stability for a coupled system of fractional differential equations with pLaplacian operator. Boundary Value Problems (2017) 2017:157. DOI 10.1186/s1366101708786.
 H. Khan, Y. Li, H. Sun, and A. Khan, Existence of solution and HyersUlam stability for a coupled system of fractional differential equations with pLaplacian operator. J.Nonlinear Sci. Appl., 10 (2017), 52195229.