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.
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 B. Ahmad A. Alsaedi S.K. Ntouyas J. Tariboon: Hadamard-type fractional differential equations inclusions and inequalities. Springer International Publishing (2017).
 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.
 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.
 R.P. Agarwal D. O’Regan: Infinite Interval Problems for Differential Difference and Integral Equations. Kluwer Academic Dordrecht (2001).
 R.G. Bartle: A modern theory of integration. Amer. Math. Soc. Providence Rhode Island (2001).
 C. Corduneanu: Integral Equations and Stability of Feedback Systems. Academic Press New York (1973).
 S. Das: Functional Fractional Calculus for System Identification and Controls. Springer-Verlag Berlin Heidelberg (2008).
 K. Diethelm: The Analysis of Fractional Differential Equations. Springer Berlin (2010).
 A.A. Kilbas H.H. Srivastava J.J. Trujillo: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V Amsterdam (2006).
 V. Kiryakova: A brief story about the operators of the generalized fractional calculus. Frac. Calc. Appl. Anal. 11 (2) (2008) 203–220.
 V. Kiryakova: Generalized Fractional Calculus and Applications. Longman and John Wiley New York (1994).
 V. Kiryakova Y. Luchko: Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators. Cent. Eur. J. Phys. 11 (10) (2013) 1314–1336.
 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).
 Y. Luchko: Operational rules for a mixed operator of the Erdélyi-Kober type. Fract. Calc. Appl. Anal. 7 (3) (2007) 339–364.
 Y. Luchko J. Trujillo: Caputo-type modification of the Erdélyi-Kober fractional derivative. Fract. Calc. Appl. Anal. 10 (3) (2007) 249–267.
 H. Maagli A. Dhifli: Positive solutions to a nonlinear fractional Dirichlet problem on the half-space. Electron. J. Differ. Equ. 50 (2014) 1–7.
 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.
 I. Podlubny: Fractional Differential Equations Mathematics in Science and Engineering. Academic Press New York (1999).
 J. Sabatier O.P. Agrawal J.A. Tenreiro Machado: Advances in Fractional Calculus Theoretical Developments and Applicationsin Physics and Engineering. Springer (2007).
 S.G. Samko A.A. Kilbas O.I. Marichev: Fractional Integral and Derivatives Theory and Applications. Gordon and Breach Switzerland (1993).
 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.
 I.N. Sneddon: Mixed Boundary Value Problems in Potential Theory. North-Holland Publ. Amsterdam (1966).
 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.
 I.N. Sneddon: The Use of Operators of Fractional Integration in Applied Mathematics. RWN Polish Sci. Publ. Warszawa-Poznan (1979).
 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.
 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.
 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.
 Z. Zhao: Positive solutions of nonlinear second order ordinary differential equations. Proc. Amer. Math. Soc. 121 (2) (1994) 465–469.
 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.
 X. Zhao W. Ge: Unbounded solutions for a fractional boundary value problems on the infinite interval. Acta Appl. Math. 109 (2010) 495–505.