In this section, the method proposed in [12, 13] will be applied to (1.2). As already mentioned in the introduction, the main idea is to transform (1.2) into an equivalent operator equation in an abstract Banach space and from this to deduce the equivalent difference equation (1.8). This method can be applied only when the ordinary differential equation under consideration is studied in the Banach space defined by (1.9). Moreover, the solution of (1.8), which will eventually give the solution of (1.2), belongs to the Banach space of absolutely summable sequences defined by (1.10).
2.1. Basic Definitions and Propositions
First of all, define the Hilbert space by
where . Denote now by an abstract separable Hilbert space over the real field, with the orthonormal base , Denote by and the inner product and the norm in , respectively. Define also in the shift operator and its adjoint
as well as the diagonal operator
is a one-by-one mapping from onto which preserves the norm, where , , is the complete system in of eigenvectors of and an element of .
The unique element appearing in (2.4) is called theabstract form of in . In general, if is a function from to and is the unique element in for which
then is called the abstract form of in .
Consider now the linear manifold of all which satisfy the condition . Define the norm . Then, this manifold becomes the Banach space defined by (1.9). Denote also by the corresponding by the representation (2.4), abstract Banach space of the elements for which .
The following properties hold [38–40]:
is invariant under the operators , , as well as under every bounded diagonal operator;
the abstract form of is the element , that is, ;
the abstract form of is the element , that is,
the operator is the Frechét differentiable in .
The linear function
is an isomorphism from onto , that is, it is a 1 − 1 mapping from onto which preserves the norm .
The basic Propositions 2.1 and 2.2 were originally proved for complex valued sequences and functions ( also in ), as well as for , defined over the complex field. However, in the present paper a restriction to the real plane is made due to the physical applications of the logistic equation.
2.2. Derivation of (1.8)
In order to apply the method of [12, 13] to the logistic differential equation (1.2), it is considered that , finite and (1.2) is restricted to by using the simple transformation , . Then, (1.2) becomes
Using Proposition 2.1 and what mentioned in Section 2.1, (2.8) is rewritten as
which holds for all , . But is the complete system in of eigenvectors of , which gives the following equivalent operator equation:
By taking the inner product of both parts of (2.10) with and taking into consideration Proposition 2.2 one obtains
where , , which is (1.8), the discrete equivalent logistic equation. It is obvious that in (2.11), it is and that , since and .
Of course, for all the above to hold, one has to assure that and . This is guaranteed by the theorems presented in the next section.
2.3. Existence and Uniqueness Theorems
As mentioned in Section 2.2, conditions must be found so that and . In order to do so, it is helpful to work with the operator equation (2.10), which is equivalent to both (2.8) and (2.11). Equation (2.10) can be rewritten as
where is the bounded operator , or as
due to the definition of , where is a constant which can be defined by taking the inner product of both parts of (2.13) with the element . Indeed, this gives
since . Thus (2.13) becomes
In order to assure the existence of a unique solution of the nonlinear operator equation (2.15) in , some conditions must be imposed on the parameters appearing in the equation. Moreover, since it is a non linear equation, a fixed-point theorem would be useful. Indeed, the following well-known theorems concerning the inversion of linear operators and the existence of a unique fixed point of an equation will be used.
If is a linear bounded operator of a Hilbert space or a Banach space , with , then is invertible with and is defined on all or (see, e.g., [41, pages 70-71] ).
If is holomorphic, that is, its Fréchet derivative exists, and lies strictly inside , then has a unique fixed point in , where is a bounded, connected, and open subset of a Banach space . (By saying that a subset of lies strictly inside , it is meant that there exists an such that for all and ) .
If it is assumed that
then and due to Theorem 2.4, the operator is defined on all and is bounded by . Thus, (2.15) takes the form
from which one finds that
Suppose that . Then, from (2.18) it is obvious that
Define the function , which attains its maximum at the point . Then, for , , it follows that if
then (2.19) gives , which means that Theorem 2.5 is applied to (2.17). Thus, the following has just been proved.
If conditions (2.16) and (2.21) hold, then the abstract operator equation (2.10) has a unique solution in bounded by .
Equivalently, this theorem can be "translated" to the following two.
If conditions (2.16) and (2.21) hold, then the discrete equivalent logistic equation (2.11), has a unique solution in bounded by .
If conditions (2.16) and (2.21) hold, then the logistic differential equation (1.2) has a unique analytic solution of the form bounded by , which together with its first derivative converges absolutely for . (The coefficients are defined of course by (2.11)).
Following the same technique as the one applied for the proof of Theorems 2.6 and 2.7, conditions were given in , so that the difference equation (1.4) is to have a unique solution in or , . Indeed, it was proved that
if and , then (1.4) has a unique solution in and
if and then (1.4) has a unique solution in , .
It is obvious that conditions (2.16) and (2.21) are very similar to the conditions derived in .