AIMS Mathematics, 2020, 5(4): 2923-2943. doi: 10.3934/math.2020189.

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

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

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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Keywords derivatives and integrals of variable order; differential equations of variable order; piecewise constant functions; existence; generalized Lyapunov-type inequality

Citation: Shuqin Zhang, Lei Hu. The existence of solutions and generalized Lyapunov-type inequalities to boundary value problems of differential equations of variable order. AIMS Mathematics, 2020, 5(4): 2923-2943. doi: 10.3934/math.2020189


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