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Upper and lower solutions method for a class of singular fractional boundary value problems with \(p\)-Laplacian operator. (English) Zbl 1209.34005

The authors establish the existence of at least one positive solution for a class of singular Riemann-Liouville fractional order boundary value problems with \(p\)-Laplacian operator satisfying four point boundary conditions. Using the upper and lower solutions method, properties of Green’s function, and, applying Schauder’s fixed point theorem, the existence of at least one positive solution of the four point fractional order boundary value problem is established. Here, the singularity of the nonlinear function of the boundary value problem is taken for three different cases. The established results are verified by an example at the end.
Reviewer’s remark: The concept of upper and lower solutions is not used properly. The remaining part of the paper is good.

MSC:

34A08 Fractional ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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References:

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