The existence of solutions and generalized Lyapunov-type inequalities to boundary value problems of differential equations of variable order

Shuqin Zhang; Lei Hu; Shuqin Zhang; Lei Hu

doi:10.3934/math.2020189

AIMS Mathematics

2020, Volume 5, Issue 4: 2923-2943. doi: 10.3934/math.2020189

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Research article

The existence of solutions and generalized Lyapunov-type inequalities to boundary value problems of differential equations of variable order

Shuqin Zhang ^{1
,
,},
Lei Hu ²

1.
School of Science, China University of Mining and Technology-Beijing, Ding No.11 Xueyuan Road, Haidian District, Beijing, China
2.
School of Science, Shandong Jiaotong University, 5001 Haitang Road, Changqing District, Jinan, Shandong Province, China

Received: 04 November 2019 Accepted: 10 February 2020 Published: 19 March 2020
MSC : 26A33, 34B15

In this paper, we discuss the existence of solutions to a boundary value problem of differential equations of variable order, which is a piecewise constant function. Our results are based on the Schauder fixed point theorem. Then, under some assumptions on the nonlinear term, we obtain a generalized Lyapunov-type inequality to the two-point boundary value problem considered. To the best of our knowledge, there is no paper dealing with Lyapunov-type inequalities for boundary value problems in term of variable order. In addition, some examples of the obtained inequalities are given.
Citation: Shuqin Zhang, Lei Hu. The existence of solutions and generalized Lyapunov-type inequalities to boundary value problems of differential equations of variable order[J]. AIMS Mathematics, 2020, 5(4): 2923-2943. doi: 10.3934/math.2020189

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Abstract

In this paper, we discuss the existence of solutions to a boundary value problem of differential equations of variable order, which is a piecewise constant function. Our results are based on the Schauder fixed point theorem. Then, under some assumptions on the nonlinear term, we obtain a generalized Lyapunov-type inequality to the two-point boundary value problem considered. To the best of our knowledge, there is no paper dealing with Lyapunov-type inequalities for boundary value problems in term of variable order. In addition, some examples of the obtained inequalities are given.

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