# Mathematical Methods in the Applied Sciences

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_{0}. We can find a critical wave speed

*c*

^{∗}if ℜ

_{0}> 1, by employing the Schauder's...

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*p*,

*q*)‐Laplacian‐type impulsive fractional differential equation for 1 <

*p*≤

*q*<

*∞*, with a nonlinearity

*f*containing fractional derivatives ${}_{0}{D}_{t}^{\alpha}u$ and ${}_{0}{D}_{t}^{\beta}u$ simultaneously and being imposed on mild assumptions contrasting with previous works. The underlying idea is based on the Mountain pass theorem and the iterative technique to obtain the existence...

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*m*< 1 along a tangential curve and estimate the capacitary dimension of the divergence set. We extend the results of Cho and Shiraki (2021) for the case

*m*> 1 to the case 0 <

*m*< 1, which is sharp up to the endpoint.