ABSTRACT
In view of the above restriction on the initial conditions of Eq(2.1), the equilibrium points of Eq(2.1) are the nonnegative solutions of the equation
x = α+ (β + γ)x A+ (B + C)x
(2.3)
or equivalently (B + C)x2 − (β + γ − A)x− α = 0. (2.4)
Zero is an equilibrium point of Eq(2.1) if and only if
α = 0 and A > 0. (2.5)
When (2.5) holds, in addition to the zero equilibrium , Eq(2.1) has a positive equilibrium if and only if
is given by
x = β + γ − A B + C
. (2.6)
When α = 0 and A = 0
the only equilibrium point of Eq(2.1) is positive and is given by
x = β + γ B + C
. (2.7)
Note that in view of Condition (2.2), when α = 0, the quantity β + γ is positive. Finally when
α > 0
the only equilibrium point of Eq(2.1) is the positive solution
x = β + γ − A+√(β + γ − A)2 + 4α(B + C)
2(B + C) (2.8)
of the quadratic equation (2.4). If we had allowed negative initial conditions then the negative solution of Eq(2.4)
would also be an equilibrium worth investigating. The problem of investigating Eq(2.1) when the initial conditions and the coefficients of the equation are real numbers is of great mathematical importance and, except for our presentation in Section 1.6 about the Riccati equation
xn+1 = α+ βxn A+Bxn
, n = 0, 1, . . . ,
there is very little known. (See [13], [18], [32], and [69] for some results in this regard.) In summary, it is interesting to observe that when Eq(2.1) has a positive equilibrium
x, then x is unique, it satisfies Eqs(2.3) and (2.4), and it is given by (2.8). This observation simplifies the investigation of the local stability of the positive equilibrium of Eq(2.1).
