Existence of a positive solution for a singular fractional boundary value problem with fractional boundary conditions using convolution and lower order problems

Existence of a positive solution is shown for two singular two-point fractional boundary value problems with fractional boundary conditions using fixed point theory, lower order problems, and convolution of Green’s functions. A nontrivial example is included.

PDF

___

[1] Agarwal RP, O’Regan D, Stanêk S. Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. Journal of Mathematical Analysis and Applications 2010; 371 (1): 57-68. doi: 10.1016/j.jmaa.2010.04.034
[2] Cui Y. Existence results for singular boundary value problem of nonlinear fractional differential equation. Abstract and Applied Analysis 2011; 2011: 1-9. doi: 10.1155/2011/605614
[3] Diethelm K. The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Berlin, Germany: Springer, 2004.
[4] Eloe PW, Lyons JW, Neugebauer JT. An ordering on Green’s functions for a family of two-point boundary value problems for fractional differential equations. Communications in Applied Analysis 2015; 19 (3-4): 453-462.
[5] Eloe PW, Neugebauer JT. Convolutions and Green’s functions for two families of boundary value problems for fractional differential equations. Electronic Journal of Differential Equations 2016; 2016 (297): 1-13.
[6] Gatica JA, Oliker V, Waltman P. Singular nonlinear boundary value problems for second- order ordinary differential equations. Journal of Differential Equations 1989; 79 (1): 62-78. doi: 10.1016/0022-0396(89)90113-7
[7] Henderson J, Luca R. Existence of positive solutions for a singular fraction boundary value problem. Nonlinear Analysis: Modelling and Control 2017; 22 (1): 99-114.
[8] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. In: Van Mill J (editor). North-Holland Mathematics Studies, Vol. 204. Amsterdam, Netherlands: Elsevier Science B.V., 2006.
[9] Krasnosel’skii MA. Topological Methods in the Theory of Nonlinear Integral Equations. New York, NY, USA: The Macmillan Co., 1964.
[10] Lyons JW, Neugebauer JT. Positive solutions of a singular fractional boundary value problem with a fractional boundary condition. Opuscula Mathematica 2017; 37 (3): 421-434.
[11] Lyons JW, Neugebauer JT. Two point fractional boundary value problems with a fractional boundary condition. Fractional Calculus and Applied Analysis 2018; 21 (2): 442-461. doi: 10.1515/fca-2018-0025
[12] Mâagli H, Mhadhebi N, Zeddini N. Existence and estimates of positive solutions for some singular fractional boundary value problems. Abstract and Applied Analysis 2014; 2014: 1-7. doi: 10.1155/2014/120781
[13] Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York, NY, USA: John Wiley & Sons, Inc., 1993.
[14] Neugebauer JT. Existence of positive solutions of a singular fractional boundary value problem. Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 2018; 25 (4): 257-266.
[15] Podlubny I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Cambridge, MA, USA: Academic Press Inc., 1999.
[16] Stanêk S. The existence of positive solutions of singular fractional boundary value problems. Computers & Mathematics with Applications 2011; 62 (3): 1379-1388. doi: 10.1016/j.camwa.2011.04.048
[17] Xu X, Jiang D, Yuan C. Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation. Nonlinear Analysis 2009; 71 (10): 4676-4688. doi: 10.1016/j.na.2009.03.030
[18] Yuan C, Jiang D, Xu X. Singular positone and semipositone boundary value problems of nonlinear fractional differential equations. Mathematical Problems in Engineering 2009; 2009: 1-17. doi: 10.1155/2009/535209
[19] Zhang X, Mao C, Wu Y, Su H. Positive solutions of a singular nonlocal fractional order differential system via Schauder’s fixed point theorem. Abstract and Applied Analysis 2014; 2014: 1-8. doi: 10.1155/2014/457965