Isaac Scientific Publishing

Advances in Analysis

Existence of Solutions for Implicit Fractional Differential Systems with Coupled Nonlocal Conditions

Download PDF (488.9 KB) PP. 1 - 9 Pub. Date: January 15, 2017

DOI: 10.22606/aan.2017.11001

Author(s)

  • Mengna Zhang*
    School of Mechanical Engineering, Shandong University, Jinan, 250061, P. R. China
  • Yansheng Liu*
    Department of Mathematics, Shandong Normal University, Jinan, 250014, P. R. China

Abstract

This article concerns the existence of solutions for implicit fractional differential systems with coupled nonlocal conditions of functional type. We use a vectorial version of Krasnoselskii’s fixed point theorem in generalized metric space to overcome the lack of complete continuity of the associated integral operators. Moreover, the sufficient conditions for the existence will be weakened on the subinterval in which our nonlocal conditions act. An example is presented to illustrate the theory.

Keywords

Fractional differential systems, implicit differential equation, coupled nonlocal conditions, fixed point, vector-valued norm, spectral radius of a matrix.

References

[1] S. Abbes, M. Benchohra, G.M. N’Guérékata, Topics in Fractional Differential Equations, Springer-Verlag, New York, 2012.

[2] G. A. Anastassiou, Advances on Fractional Inequalities, Springer, New York, 2011.

[3] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, 2010.

[4] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Science B.V. vol. 204, pp. 2453-2461, 2006.

[5] A.M.A. El-Sayed, Fractional differential equations, Bull.Fac.Sci.Assiut Univ.A, vol. 16, pp. 271-275, 1987.

[6] V.E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of particles, Fields and Media, Springer, Heidelberg; Higher Education Press, Beijing, 2010.

[7] J.J Nieto, A. Ouahab, V. Venktesh, Implicit fractional differential equations via the Liouville-Caputo derivative, Mathematics, vol. 3, pp. 398-411, 2015.

[8] M. Benchohra, J.R. Graef, S. Hamani, “Existence results for boundary value problems with nonlinear fractional differential equations,” Appl. Anal. vol. 87, pp. 851-863, 2008.

[9] M. Benchohra, J.E. Lazreg, Nonlinear fractional implicit differential equations, Commun. Appl. Anal. vol. 17, pp. 471-482, 2013.

[10] M. Benchohra, J.E. Lazreg, “Existence and uniqueness results for nonlinear implicit fractional differential equations with boundary conditions,” Rom. J. Math. Comput. Sci. vol. 4, pp. 60-72, 2014.

[11] A.N. Vityuk, A.V. Mikhailenko, “The Darboux problem for an implicit differential equation of fractional order,” Ukr. Mat. Visn. vol. 7, pp. 439-452, 585, 2010.

[12] A.M.E. El-Sayed, E.M. Hamdallah, K.W. Elkadeky, “Internal nonlocal and integral condition problems of the differential equation x' = f(t, x, x'),” Journal of Nonlinear Sciences & Its Applications, vol. 3, pp. 193-199, 2011.

[13] O. Bolojan, “Implicit first order differential systems with nonlocal conditions,” Electronic Journal of Qualitative Theory of Differential Equations, 2015, 2014.

[14] D. Trif, “Multiple positive solutions of nonlocal initial value problems for first order differential systems,” Nonlinear Analysis Theory Methods & Applications, vol. 75, pp. 5961-5970, 2012.

[15] O. Nica, “Initial-value problems for first-order differential systems with general nonlocal conditions,” Electronic Journal of Differential Equations, vol. 2, pp. 1394-1396, 2012.

[16] O. Nica, “Nonlocal initial value problems for first order differential systems,” Fixed Point Theory, vol. 13, pp. 603-612, 2012.

[17] O. Nica, “On the nonlocal initial value problem for first order differential systems,” International Journal on Fixed Point Theory Computation & Applications, vol. 56, pp. 113-125, 2011.

[18] O. Nica, G. Infante, R. Precup, “Existence results for systems with coupled nonlocal initial conditions,” Nonlinear Analysis Theory Methods & Applications, vol. 94, pp. 231-242, 2014.

[19] R. Precup, Methods in Nonlinear Integral Equations, Kluwer, Dordrecht, 2002.

[20] R. Precup, “The role of matrices that are convergent to zero in the study of semilinear operator systems,” Mathematical & Computer Modelling, vol. 49, pp. 703-708, 2009.

[21] A. Viorel, Contributions to the study of nonlinear evolution equations, Ph.D thesis, Babes-Bolyai University of Cluj-Napoca, 2011.

[22] G. Infante, F.M. Minhlos, P. Pietramala, “Non-negative solutions of systems of ODEs with coupled boundary conditions,” Commun. Nonlinear Sci. Numer. Simul. vol. 17, pp. 4952-4960, 2012.

[23] C. Yuan, D. Jiang, D. OarRegan, R.P. Agarwal, “Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 13, pp. 1-13, 2012.