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Abstract

• Abstract

  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.

  Keywords:
  • Generalized trigonometric functions,
  • $p$-Laplacian,
  • Gaussian hypergeometric functions,
  • Wallis-type formulas.
  Mathematics Subject Classification:Primary: 33B10, 34B10.

  Citation:

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

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

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