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Keywords:
Fractional differential equations; Boundary value problems; Erdélyi-Kober derivative; Fixed point theorems; Existence; Uniqueness.
Fractional differential equations; Boundary value problems; Erdélyi-Kober derivative; Fixed point theorems; Existence; Uniqueness.
Summary:
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.
References:
[1] Ahmad, B., Alsaedi, A., Ntouyas, S.K., Tariboon, J.: Hadamard-type fractional differential equations, inclusions and inequalities. 2017, Springer International Publishing, MR 3616285
[2] Ahmad, B., Ntouyas, S.K., Tariboonc, J., Alsaedi, A.: 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, DOI 10.3846/13926292.2017.1274920 | MR 3625429
[3] Ahmad, B., Ntouyas, S.K., Tariboonc, J., Alsaedi, A.: Caputo Type Fractional Differential Equations with Nonlocal Riemann-Liouville and Erdélyi-Kober Type Integral Boundary Conditions. Filomat, 31, 14, 2017, 4515-4529, DOI 10.2298/FIL1714515A | MR 3730375
[4] Agarwal, R.P., O'Regan, D.: Infinite Interval Problems for Differential, Difference and Integral Equations. 2001, Kluwer Academic, Dordrecht, MR 1845855 | Zbl 0988.34002
[5] Bartle, R.G.: A modern theory of integration. 32, 2001, Amer. Math. Soc., Providence, Rhode Island, MR 1817647
[6] Corduneanu, C.: Integral Equations and Stability of Feedback Systems. 1973, Academic Press, New York, MR 0358245 | Zbl 0273.45001
[7] Das, S.: Functional Fractional Calculus for System Identification and Controls. 2008, Springer-Verlag Berlin Heidelberg, MR 2414740
[8] Diethelm, K.: The Analysis of Fractional Differential Equations. 2010, Springer, Berlin, MR 2680847 | Zbl 1215.34001
[9] Kilbas, A.A., Srivastava, H.H., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. 2006, Elsevier Science B.V, Amsterdam, MR 2218073 | Zbl 1092.45003
[10] Kiryakova, V.: A brief story about the operators of the generalized fractional calculus. Frac. Calc. Appl. Anal., 11, 2, 2008, 203-220, MR 2401328
[11] Kiryakova, V.: Generalized Fractional Calculus and Applications. 1994, Longman and John Wiley, New York, MR 1265940
[12] Kiryakova, V., Luchko, Y.: Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators. Cent. Eur. J. Phys., 11, 10, 2013, 1314-1336,
[13] Liu, X., Jia, M.: Multiple solutions of nonlocal boundary value problems for fractional differential equations on half-line. Electron. J. Qual. Theory Differ. Equ., 56, 1-14. MR 2825141
[14] Luchko, Y.: Operational rules for a mixed operator of the Erdélyi-Kober type. Fract. Calc. Appl. Anal., 7, 3, 2007, 339-364, MR 2252570
[15] Luchko, Y., Trujillo, J.: Caputo-type modification of the Erdélyi-Kober fractional derivative. Fract. Calc. Appl. Anal., 10, 3, 2007, 249-267, MR 2382781
[16] Maagli, H., Dhifli, A.: Positive solutions to a nonlinear fractional Dirichlet problem on the half-space. Electron. J. Differ. Equ., 50, 2014, 1-7, MR 3177559
[17] Mathai, A.M., Haubold, H.J.: Erdélyi-Kober Fractional Calculus. 2018, Springer Nature, Singapore Pte Ltd, MR 3838388
[18] Ntouyas, S.K.: Boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions. Opuscula Math., 33, 1, 2013, 117-138, DOI 10.7494/OpMath.2013.33.1.117 | MR 3008027
[19] Pagnini, G.: Erdélyi-Kober fractional diffusion. Fract. Calc. Appl. Anal., 15, 1, 2012, 117-127, DOI 10.2478/s13540-012-0008-1 | MR 2872114
[20] Podlubny, I.: Fractional Differential Equations, Mathematics in Science and Engineering. 1999, Academic Press, New York, MR 1658022
[21] Sabatier, J., Agrawal, O.P., Machado, J.A. Tenreiro: Advances in Fractional Calculus Theoretical Developments and Applicationsin Physics and Engineering. 2007, Springer, MR 2432163
[22] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integral and Derivatives Theory and Applications. 1993, Gordon and Breach, Switzerland, MR 1347689
[23] A1-Saqabi, B., Kiryakova, V.S.: Explicit solutions of fractional integral and differential equations involving Erdé1yi-Kober operators. Appl. Math. Comput., 95, 1998, 1-13, MR 1630272
[24] Sneddon, I.N.: Mixed Boundary Value Problems in Potential Theory. 1966, North-Holland Publ., Amsterdam, MR 0216018
[25] Sneddon, I.N.: The use in mathematical analysis of the Erdélyi-Kober operators and some of their applications. Lect Notes Math, 457, 1975, 37-79, Springer-Verlag, New York, DOI 10.1007/BFb0067097 | MR 0487301
[26] Sneddon, I.N.: The Use of Operators of Fractional Integration in Applied Mathematics. 1979, RWN Polish Sci. Publ., Warszawa-Poznan, MR 0604924
[27] Sun, Q., Meng, S., Cu, Y.: 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, MR 3829286
[28] Yan, B., Liu, Y.: 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, MR 2011077 | Zbl 1045.34009
[29] Yan, B., O'Regan, D., Agarwal, and R.P.: 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, MR 2260412
[30] Zhao, Z.: Positive solutions of nonlinear second order ordinary differential equations. Proc. Amer. Math. Soc., 121, 2, 1994, 465-469, DOI 10.1090/S0002-9939-1994-1185276-5 | MR 1185276
[31] Zhao, X., Ge, W.: 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, DOI 10.1007/s12190-008-0113-9 | MR 2430946
[32] Zhao, X., Ge, W.: Unbounded solutions for a fractional boundary value problems on the infinite interval. Acta Appl. Math., 109, 2010, 495-505, DOI 10.1007/s10440-008-9329-9 | MR 2585801
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