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Existence of positive solution for some class of nonlinear fractional differential equations. (English) Zbl 1026.34008
The author considers nonlinear differential equations involving a generalised fractional differential operator corresponding to a generalised Riemann-Liouville fractional integral. He focuses on questions of existence and uniqueness of positive solutions to the equations. In the first two sections of the paper, the author introduces the definitions used, and gives basic theory. In Section 3, he gives a sequence of existence and uniqueness results followed by several lemmas and theorems that guarantee, under suitable conditions, the existence of multiple (infinite sequences of) positive solutions.
MSC:
34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)
26A33Fractional derivatives and integrals (real functions)
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
34A34Nonlinear ODE and systems, general