EXISTENCE OF SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS WITH FRACTIONAL-ORDER DERIVATIVE TERMS

Ai Sun; Youhui Su; Qingchun Yuan; Tongxiang Li

doi:10.11948/20200072

2021 Volume 11 Issue 1

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Ai Sun, Youhui Su, Qingchun Yuan, Tongxiang Li. EXISTENCE OF SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS WITH FRACTIONAL-ORDER DERIVATIVE TERMS[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 486-520. doi: 10.11948/20200072

Citation:

Ai Sun, Youhui Su, Qingchun Yuan, Tongxiang Li. EXISTENCE OF SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS WITH FRACTIONAL-ORDER DERIVATIVE TERMS[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 486-520. doi: 10.11948/20200072

EXISTENCE OF SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS WITH FRACTIONAL-ORDER DERIVATIVE TERMS

1.
College of Science, Shenyang University of Technology, Shenliao Road, Shenyang 110870, Liaoning, China
2.
School of Mathematics and Statistics, Xuzhou University of Technology, Lishui Road, Xuzhou 221018, Jiangsu, China

Corresponding author: Email address:suyh02@163.com(Y. Su)
Fund Project: The authors were supported by the Foundation of XZIT (XKY2020102)

Abstract

Abstract

The study in this paper is made on the nonlinear fractional differential equation whose nonlinearity involves the explicit fractional order D_0⁺^β u(t). The corresponding Green's function is derived first, and then the completely continuous operator is proved. Besides, based on the Schauder's fixed point theorem and the Krasnosel'skii's fixed point theorem, the sufficient conditions for at least one or two existence of positive solutions are established. Furthermore, several other sufficient conditions for at least three, n or 2n-1 positive solutions are also obtained by applying the generalized AveryHenderson fixed point theorem and the Avery-Peterson fixed point theorem. Finally, several simulation examples are provided to illustrate the main results of the paper. In particularly, a novel efficient iterative method is employed for simulating the examples mentioned above, that is, the interesting point of this paper is that the approximation graphics for the solutions are given by using the iterative method.
- Fractional differential equation /
- Green's function /
- the fixed point theorems /
- numerical simulation
MSC: 34A08, 34B18, 34K37

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References

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