Abstract
This article is concerned with a class of singular nonlinear fractional boundary value problems with p‐Laplacian operator, which contains Riemann–Liouville fractional derivative and Caputo fractional derivative. The boundary conditions are made up of two kinds of Riemann–Stieltjes integral boundary conditions and nonlocal infinite‐point boundary conditions, and different fractional orders are involved in the boundary conditions and the nonlinear term, respectively. Based on the method of reducing the order of fractional derivative, some properties of the corresponding Green’s function, and the fixed point theorem of mixed monotone operator, an interesting result on the iterative sequence of the uniqueness of positive solutions is obtained under the assumption that the nonlinear term may be singular in regard to both the time variable and space variables. And we obtain the dependence of solution upon parameter. Furthermore, two numerical examples are presented to illustrate the application of our main results.
Highlights
In the past decades, fractional di erential equations arise in many mathematical disciplines as the analogue modeling of systems and processes in many scienti c elds, such as control theory and engineering
In [15], Xu and Wei investigated the positive solutions of the following fractional di erential equations:
We investigate the following fractional differential equation, which is a generalized form of the problem (2): cDα0+φpDβ0+v(t) + ft, v(t), Dβ0+1 v(t), . . . , Dβ0+n− 2 v(t) 0, 0 < t < 1, (4)
Summary
Fractional di erential equations arise in many mathematical disciplines as the analogue modeling of systems and processes in many scienti c elds, such as control theory and engineering. Liu et al in [16] investigated the existence of positive solution of the following fractional differential equations with p-Laplacian operator:. A nonnegative function v ∈ C[0, 1] is called a positive solution of boundary value problem (4)-(5) E key argument of the problem (4)-(5) and (6)-(5) is the iterative positive solution by applying the method of reducing the order of fractional derivative and the fixed point theory of mixed monotone operator. Different orders of Riemann–Liouville’s fractional derivative are involved in the nonlinearity f, which is solved in a more complex space, in most cases. By using the properties of relevant Green’s function and cone, the theory of mixed monotone operator could be applied on the research of fractional boundary value problems.
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