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EXISTENCE OF THREE SOLUTIONS FOR IMPULSIVE FRACTIONAL DIFFERENTIAL SYSTEMS THROUGH VARIATIONAL METHODS

Yıl 2019, Cilt: 9 Sayı: 3, 646 - 657, 01.09.2019

Öz

This paper is devoted to the study of the multiplicity results of existence of solutions for a class of impulsive fractional di erential systems. Indeed, we will use variational methods for smooth functionals, de ned on the re exive Banach spaces in order to achieve the existence of at least three solutions for these systems. In particular, in the scalar case, we will prove that the impulsive fractional di erential problem has three non-negative solutions. Finally, by presenting two examples, we will ensure the applicability of our results.

Kaynakça

  • Benchohra, M., Henderson, J., Ntouyas, S., (2006), Theory of Impulsive Differential Equations, Con- temporary Mathematics and Its Applications, 2. Hindawi Publishing Corporation, New York.
  • Bonanno, G., Rodr´ıguez-L´opez, R., Tersian, S., (2014), Existence of solutions to boundary-value problem for impulsive fractional differential equations, Fract. Calc. Appl. Anal., 17, pp. 717-744.
  • Carter, T. E., (2000), Necessary and sufficient conditions for optimal impulsive rendezvous with linear equations of motion, Dyn. Control, 10, pp. 219-227.
  • Chen, H., He, Z., (2012), New results for perturbed Hamiltonian systems with impulses, Appl. Math. Comput., 218, pp. 9489-9497.
  • Diethelm, K., (2010), The Analysis of Fractional Differential Equation, Springer, Heidelberg.
  • Galewski, M., Molica Bisci, G., (2016), Existence results for one-dimensional fractional equations, Math. Meth. Appl. Sci., 39, pp. 1480–1492.
  • Heidarkhani, S., Ferrara, M., Caristi, G., Salari, A., (2017), Existence of three solutions for impulsive nonlinear fractional boundary value problems, Opuscula Math., 37, pp. 281-301.
  • Heidarkhani, S., Salari, A., (2017), Existence of three solutions for impulsive perturbed elastic beam fourth-order equations of Kirchhoff-type, Studia Sci. Math. Hungar., 54, pp. 119-140.
  • Heidarkhani, S., Salari, A., (2016), Nontrivial Solutions for impulsive fractional differential systems through variational methods, Comput. Math. Appl., http://dx.doi.org/10.1016/j.camwa.2016.04.016.
  • Heidarkhani, S., Zhou, Y., Caristi, G., Afrouzi, G. A., Moradi, S., (2016), Existence results for fractional differential systems through a local minimization principle, Comput. Math. Appl., http://dx.doi.org/10.1016/j.camwa.2016.04.012.
  • Hilfer, R., (2000), Applications of Fractional Calculus in Physics, World Scientific, Singapore.
  • Jiao, F., Zhou, Y., (2011), Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl., 62, pp. 1181-1199.
  • Kilbas, A., Srivastava, H. M., Trujillo, J. J., (2006), Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, pp. 1-523.
  • Kong, L., (2013), Existence of solutions to boundary value problems arising from the fractional ad- vection dispersion equation, Electron. J. Diff. Equ., 2013, pp. 1-15.
  • Ricceri, B., (2009), A further three critical points theorem, Nonlinear Anal. TMA, 71, pp. 4151-4157. [16] Sun, J., Chen, H., Nieto, J. J., Otero-Novoa, M., (2010), The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive efects, Nonlinear Anal. TMA, 72, pp. 4575-4586.
  • Wang, J., Fe˘ckan, M., Zhou, Y., (2011), On the new concept of solutions and existence results for impulsive fractional evolution equations, Dyn. Partial Differ. Equ., 8, pp. 345-361.
  • Zhao, Y., Chen, H., Qin, B., (2015), Multiple solutions for a coupled system of nonlinear fractional differential equations via variational methods, Appl. Math. Comput., 257, pp. 417-427.
  • Zhao, Y., Chen, H., Zhang, Q., (2016), Infinitely many solutions for fractional differential system via variational method, J. Appl. Math. Comput., 50, pp. 589-609.
  • Zhao, Y., Chen, H., Zhang, Q., Multiplicity of solutions for perturbed nonlinear fractional differential system via variational method, preprint.
  • Shapour Heidarkhani is an Associate Professor of Mathematics in the Department of Mathematics at Razi University, Kermanshah/Iran.
Yıl 2019, Cilt: 9 Sayı: 3, 646 - 657, 01.09.2019

Öz

Kaynakça

  • Benchohra, M., Henderson, J., Ntouyas, S., (2006), Theory of Impulsive Differential Equations, Con- temporary Mathematics and Its Applications, 2. Hindawi Publishing Corporation, New York.
  • Bonanno, G., Rodr´ıguez-L´opez, R., Tersian, S., (2014), Existence of solutions to boundary-value problem for impulsive fractional differential equations, Fract. Calc. Appl. Anal., 17, pp. 717-744.
  • Carter, T. E., (2000), Necessary and sufficient conditions for optimal impulsive rendezvous with linear equations of motion, Dyn. Control, 10, pp. 219-227.
  • Chen, H., He, Z., (2012), New results for perturbed Hamiltonian systems with impulses, Appl. Math. Comput., 218, pp. 9489-9497.
  • Diethelm, K., (2010), The Analysis of Fractional Differential Equation, Springer, Heidelberg.
  • Galewski, M., Molica Bisci, G., (2016), Existence results for one-dimensional fractional equations, Math. Meth. Appl. Sci., 39, pp. 1480–1492.
  • Heidarkhani, S., Ferrara, M., Caristi, G., Salari, A., (2017), Existence of three solutions for impulsive nonlinear fractional boundary value problems, Opuscula Math., 37, pp. 281-301.
  • Heidarkhani, S., Salari, A., (2017), Existence of three solutions for impulsive perturbed elastic beam fourth-order equations of Kirchhoff-type, Studia Sci. Math. Hungar., 54, pp. 119-140.
  • Heidarkhani, S., Salari, A., (2016), Nontrivial Solutions for impulsive fractional differential systems through variational methods, Comput. Math. Appl., http://dx.doi.org/10.1016/j.camwa.2016.04.016.
  • Heidarkhani, S., Zhou, Y., Caristi, G., Afrouzi, G. A., Moradi, S., (2016), Existence results for fractional differential systems through a local minimization principle, Comput. Math. Appl., http://dx.doi.org/10.1016/j.camwa.2016.04.012.
  • Hilfer, R., (2000), Applications of Fractional Calculus in Physics, World Scientific, Singapore.
  • Jiao, F., Zhou, Y., (2011), Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl., 62, pp. 1181-1199.
  • Kilbas, A., Srivastava, H. M., Trujillo, J. J., (2006), Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, pp. 1-523.
  • Kong, L., (2013), Existence of solutions to boundary value problems arising from the fractional ad- vection dispersion equation, Electron. J. Diff. Equ., 2013, pp. 1-15.
  • Ricceri, B., (2009), A further three critical points theorem, Nonlinear Anal. TMA, 71, pp. 4151-4157. [16] Sun, J., Chen, H., Nieto, J. J., Otero-Novoa, M., (2010), The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive efects, Nonlinear Anal. TMA, 72, pp. 4575-4586.
  • Wang, J., Fe˘ckan, M., Zhou, Y., (2011), On the new concept of solutions and existence results for impulsive fractional evolution equations, Dyn. Partial Differ. Equ., 8, pp. 345-361.
  • Zhao, Y., Chen, H., Qin, B., (2015), Multiple solutions for a coupled system of nonlinear fractional differential equations via variational methods, Appl. Math. Comput., 257, pp. 417-427.
  • Zhao, Y., Chen, H., Zhang, Q., (2016), Infinitely many solutions for fractional differential system via variational method, J. Appl. Math. Comput., 50, pp. 589-609.
  • Zhao, Y., Chen, H., Zhang, Q., Multiplicity of solutions for perturbed nonlinear fractional differential system via variational method, preprint.
  • Shapour Heidarkhani is an Associate Professor of Mathematics in the Department of Mathematics at Razi University, Kermanshah/Iran.
Toplam 20 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Research Article
Yazarlar

S. Heidarkhani Bu kişi benim

A. Salari Bu kişi benim

Yayımlanma Tarihi 1 Eylül 2019
Yayımlandığı Sayı Yıl 2019 Cilt: 9 Sayı: 3

Kaynak Göster