New class of boundary value problem for nonlinear fractional differential equations involving Erdélyi-Kober derivative


In this paper, we introduce a new class of boundary value problem for nonlinear fractional differential equations involving the Erdélyi-Kober differential operator on an infinite interval. Existence and uniqueness results for a positive solution of the given problem are obtained by using the Banach contraction principle, the Leray-Schauder nonlinear alternative, and Guo-Krasnosel’skii fixed point theorem in a special Banach space. To that end, some examples are presented to illustrate the usefulness of our main results.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] B. Ahmad, A. Alsaedi, S.K. Ntouyas, J. Tariboon: Hadamard-type fractional differential equations, inclusions and inequalities. Springer International Publishing (2017).

  • [2] B. Ahmad, S.K. Ntouyas, J. Tariboonc, A. Alsaedi: A Study of Nonlinear Fractional-Order Boundary Value Problem with Nonlocal Erdélyi-Kober and Generalized Riemann-Liouville Type Integral Boundary Conditions. Math. Model. Anal. 22 (2) (2017) 121–139.

  • [3] B. Ahmad, S.K. Ntouyas, J. Tariboonc, A. Alsaedi: Caputo Type Fractional Differential Equations with Nonlocal Riemann-Liouville and Erdélyi-Kober Type Integral Boundary Conditions. Filomat 31 (14) (2017) 4515–4529.

  • [4] R.P. Agarwal, D. O’Regan: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht (2001).

  • [5] R.G. Bartle: A modern theory of integration. Amer. Math. Soc., Providence, Rhode Island (2001).

  • [6] C. Corduneanu: Integral Equations and Stability of Feedback Systems. Academic Press, New York (1973).

  • [7] S. Das: Functional Fractional Calculus for System Identification and Controls. Springer-Verlag Berlin Heidelberg (2008).

  • [8] K. Diethelm: The Analysis of Fractional Differential Equations. Springer, Berlin (2010).

  • [9] A.A. Kilbas, H.H. Srivastava, J.J. Trujillo: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V, Amsterdam (2006).

  • [10] V. Kiryakova: A brief story about the operators of the generalized fractional calculus. Frac. Calc. Appl. Anal. 11 (2) (2008) 203–220.

  • [11] V. Kiryakova: Generalized Fractional Calculus and Applications. Longman and John Wiley, New York (1994).

  • [12] V. Kiryakova, Y. Luchko: Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators. Cent. Eur. J. Phys. 11 (10) (2013) 1314–1336.

  • [13] X. Liu, M. Jia: Multiple solutions of nonlocal boundary value problems for fractional differential equations on half-line. Electron. J. Qual. Theory Differ. Equ. 56 (1-14).

  • [14] Y. Luchko: Operational rules for a mixed operator of the Erdélyi-Kober type. Fract. Calc. Appl. Anal. 7 (3) (2007) 339–364.

  • [15] Y. Luchko, J. Trujillo: Caputo-type modification of the Erdélyi-Kober fractional derivative. Fract. Calc. Appl. Anal. 10 (3) (2007) 249–267.

  • [16] H. Maagli, A. Dhifli: Positive solutions to a nonlinear fractional Dirichlet problem on the half-space. Electron. J. Differ. Equ. 50 (2014) 1–7.

  • [17] A.M. Mathai, H.J. Haubold: Erdélyi-Kober Fractional Calculus. Springer Nature, Singapore Pte Ltd (2018).

  • [18] S.K. Ntouyas: Boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions. Opuscula Math. 33 (1) (2013) 117–138.

  • [19] G. Pagnini: Erdélyi-Kober fractional diffusion. Fract. Calc. Appl. Anal. 15 (1) (2012) 117–127.

  • [20] I. Podlubny: Fractional Differential Equations, Mathematics in Science and Engineering. Academic Press, New York (1999).

  • [21] J. Sabatier, O.P. Agrawal, J.A. Tenreiro Machado: Advances in Fractional Calculus Theoretical Developments and Applicationsin Physics and Engineering. Springer (2007).

  • [22] S.G. Samko, A.A. Kilbas, O.I. Marichev: Fractional Integral and Derivatives Theory and Applications. Gordon and Breach, Switzerland (1993).

  • [23] B. A1-Saqabi, V.S. Kiryakova: Explicit solutions of fractional integral and differential equations involving Erdé1yi-Kober operators. Appl. Math. Comput. 95 (1998) 1–13.

  • [24] I.N. Sneddon: Mixed Boundary Value Problems in Potential Theory. North-Holland Publ., Amsterdam (1966).

  • [25] I.N. Sneddon: The use in mathematical analysis of the Erdélyi-Kober operators and some of their applications. In: Lect Notes Math. Springer-Verlag, New York (1975) 37–79.

  • [26] I.N. Sneddon: The Use of Operators of Fractional Integration in Applied Mathematics. RWN Polish Sci. Publ., Warszawa-Poznan (1979).

  • [27] Q. Sun, S. Meng, Y. Cu: Existence results for fractional order differential equation with nonlocal Erdélyi-Kober and generalized Riemann-Liouville type integral boundary conditions at resonance. Adv. Difference Equ. (2018) 243.

  • [28] B. Yan, Y. Liu: Unbounded solutions of the singular boundary value problems for second order differential equations on the half-line. Appl. Math. Comput. 147 (3) (2004) 629–644.

  • [29] B. Yan, D. O’Regan, and R.P. Agarwal: Unbounded solutions for singular boundary value problems on the semi-infinite interval Upper and lower solutions and multiplicity. Int. J. Comput. Appl. Math. 197 (2) (2006) 365–386.

  • [30] Z. Zhao: Positive solutions of nonlinear second order ordinary differential equations. Proc. Amer. Math. Soc. 121 (2) (1994) 465–469.

  • [31] X. Zhao, W. Ge: Existence of at least three positive solutions for multi-point boundary value problem on infinite intervals with p-Laplacian operator. J. Appl. Math. Comput. 28 (1) (2008) 391–403.

  • [32] X. Zhao, W. Ge: Unbounded solutions for a fractional boundary value problems on the infinite interval. Acta Appl. Math. 109 (2010) 495–505.


Journal + Issues