Using the equation for and the initial **data** (0) = 1 one sees that (0) > 0 so that has a local minimum atτ = 0. Thus, at least for positive values ofτ close to 0 one has that must be increasing. Furthermore, asχ > 0 for r > r , one finds that β 2 ≤ . This last differential inequality can be integrated to yield ≥ cosh(βτ ). One concludes that is increasing for all τ ≥ 0. A similar argument with the function η ≡ − ς satisfying the equation
η − (β 2 +χ)η = 1, η(0) = 0, η (0) = 0,
shows that ≥ ς for allτ ≥ 0.