Some Nonstrandard Result of Continuous, Monotonic Functions

Some Nonstandard Treatment of the Singularity in the Differential Equation

“…Now, if β ∈ ℝ\ℕ st , then the equation ( 7) has a unique analytical solution ϕ(x) such that ϕ(0) = 0. In case ϕ(0) ≅ 0 then by Corollary 2.3 in [19] we obtain that equation (7) has an approximation solution (shadow solution). Let the power series about the origin for y be given as follows…”

“…So, the differential coefficients are convergent in the m((0,0)) and (0,0) is a singular point for (19), where ω k for k=1, 2, 3, 4 is unlimited with ∏ where ω 5 =(max{ω 1 , ω 3 } +max{ω 2 , ω 4 }) − 1, ω 6 = max{ω 2 , ω 4 } and ω 7 = (ω 3 + ω 4 ) − 1. We know that the geometric series 1 + g(x, y) + (g(x, y)) 2 + ⋯ converges to…”