Perron Theorem and Monotone Iteration Method in the Study of Traveling Waves

The existence of a unique solution to equations of the form x'(t) = Ax(t)+f(t) is ofter referred to as Perron Theorem. The positivity of the solution, given f(t) is positive or negative, is an interesting subject to be studied as it is related to the Monotone Iteration Method in proving the existence of traveling waves to reaction-diffusion equation u'_t = d u''_xx +f(u). The reader can find out how this idea can be extended to very complex situation when time delays appear in diffusion term in different models.

•  Existence of Traveling Waves of Lotka Volterra Type Models with Delayed Diffusion Term and Partial Quasimonotonicity

  (William Barker). Preprint: https://arxiv.org/abs/2303.11145

  
  

• Traveling Wave for a diffusive SIR model with delay in diffusion term

  (William Barker). Preprint: https://arxiv.org/abs/2310.13799
  

• Traveling waves in reaction-diffusion equations with delay in both diffusion and reaction terms. (W. Barker, Nguyen Van Minh). https://arxiv.org/abs/2301.11504 
  

• Existence of traveling waves in a Nicholson blowflies model with delayed diffusion term (William Kyle Barker). Proceedings of the A.M.S.

  DOI: https://doi.org/10.1090/proc/17233
  

• Traveling waves for nonlocal Fisher–KPP equations with diffusive delay (Nguyen Truong Thanh, W. Barker, Nguyen Van Minh.

  Journal of Evolution Equations. 03 April 2025 Article: 38 
  

• Perron theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations (Boumenir, Nguyen, Van Minh)

  J. Differential Equations 244 (2008), no. 7, 1551–1570.