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The existence of solutions for mixed fractional resonant boundary value problem with p(t)-Laplacian operator

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Abstract

In this paper, we consider the following mixed fractional resonant boundary value problem with p(t)-Laplacian operator

$$\begin{aligned} \left\{ \begin{array}{lll} ^{C} D^{\beta }_{0^{+}}\varphi _{p(t)}(D^{\alpha }_{0^{+}}u(t))=f(t, u(t), D^{\alpha }_{0^{+}}u(t)),~~t\in [0, T],\\ t^{1-\alpha }u(t)\mid _{t=0}=0,~~D^{\alpha }_{0^{+}}u(0)=D^{\alpha }_{0^{+}}u(T), \end{array}\right. \end{aligned}$$

where \(^{C} D^{\beta }_{0^{+}}\) is Caputo fractional derivative, \(D^{\alpha }_{0^{+}}\) is Riemann–Liouville fractional derivative, \(\varphi _{p(t)}\) is p(t)-Laplacian operator, \(p(t)>1\), \(p(t)\in C^{1}[0, T]\) with \(p(0)=p(T)\). Under the appropriate conditions of the nonlinear term, the existence of solutions for the above mixed fractional resonant boundary value problem is obtained by using the continuation theorem of coincidence degree theory, which enrich the existing literatures. In addition, an example is included to demonstrate the main result.

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Acknowledgements

The authors thank the anonymous reviewers for their valuable comments and suggestions on improving the presentation of this paper. The work is supported by the Natural Science Foundation of China (No. 11761038, No. 11761039), the Science and Technology Project of Department of Education of Jiangxi Province (No. GJJ180583) and Natural Science Foundation of Jiangxi Province of China (No. 20171BAB202010).

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  1. School of Mathematics and Physics, Jinggangshan University, Ji’an, 343009, China

    Xiaosong Tang, Xinchang Wang, Zhiwei Wang & Peichang Ouyang

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  1. Xiaosong Tang
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Tang, X., Wang, X., Wang, Z. et al. The existence of solutions for mixed fractional resonant boundary value problem with p(t)-Laplacian operator. J. Appl. Math. Comput. 61, 559–572 (2019). https://doi.org/10.1007/s12190-019-01264-z

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