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Applications of generalized trigonometric functions with two parameters

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Dedicated to Professor Yoshio Yamada on the occasion of his retirement

The work of S. Takeuchi was supported by JSPS KAKENHI Grant Number 17K05336

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  • Generalized trigonometric functions (GTFs) are simple generalization of the classical trigonometric functions. GTFs are deeply related to the $p$-Laplacian, which is known as a typical nonlinear differential operator, and there are a lot of works on GTFs concerning the $p$-Laplacian. However, few applications to differential equations unrelated to the $p$-Laplacian are known. We will apply GTFs with two parameters to nonlinear nonlocal boundary value problems without $p$-Laplacian. Moreover, we will give integral formulas for the functions, e.g. Wallis-type formulas, and apply the formulas to the lemniscate function and the lemniscate constant.

    Mathematics Subject Classification: Primary: 33B10, 34B10.


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  • Figure 1.  Graphs of solutions of (4) with $H = 1$ for $m = 0.5, \ 1.0$ and $10.0$.

    Figure 2.  Graphs of solutions of (13) for $p = 1.1, \ 2.0$ and $5.0$.

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