This work is devoted to study the existence of solutions for the following semilinear equation
where is a matrix, , and is a nonlinear continuous function.
Definition 1.1. The Equation (1.1) is said to be solvable if for all there exists such that
Proposition 1.1. The Equation (1.1) is solvable if, and only if, the operator is surjective.
The corresponding linear equation has been studied in  where a generalization of Cramer’s Rule is given applying the Moore-Penrose inverse that can be used when exists, and a result from  . More information about the Moore-Penrose inverse can be found in  and  .
In this paper, using Moore-Penrose inverse and the Rothe’s Fixed Theorem    , we shall prove the following theorem:
Theorem 1.1. If exists and is continuous and satisfies the condition
then Equation (1.1) is solvable.
Moreover, for each there exists such that
The following theorem will be used to prove our main result.
Theorem 1.2. (Rothe’s Fixed Theorem    ) Let be a Banach space. Let be a closed convex subset such that the zero of is contained in the interior of . Let be a continuous mapping with relatively compact in and . Then there is a point such that .
2. Proof of the Main Theorems
In this section we shall prove the main results of this paper, Theorem 1.1, formulated in the introduction of this paper, which concern with the solvability of the semilinear Equation (1.1).
Proof of Theorem 1.1. Using the Moore-Penrose inverse we define the operator by
and from condition (1.2) we obtain that
Claim. The operator has a fixed point. In fact, for a fixed , there exists big enough such that
Hence, if we denote by the ball of center zero and radius , we get that . Since is compact and maps the sphere into the interior of the ball , we can apply Rothe’s fixed point Theorem 1.2 to ensure the existence of a fixed point such that
This complete the proof. □
From Banach Fixed Point Theorem it is easy to prove the following theorem that we will use to prove the next result of this paper.
Theorem 2.1. Let be a Hilbert space and is a Lipschitz function with a Lipschitz constant and consider . Then is an homeomorphism whose inverse is a Lipschitz function with a Lipschitz constant .
Theorem 2.2. If the Moore-Penrose exists and the following condition holds
then the Equation (1.1) is solvable and a solution of it is given by
Proof. Define the operator . Then and
and from condition (2.6)
Therefore, from Theorem 2.1 and (2.8) we have that is a homeomorphism Lipschitizian with a Lipschitz constant .
Hence, is a solution of (1). In fact,
and this complete the proof. □
3. Practical Example
Now, we shall apply Theorem 1.1 to find one solution of the following semilinear system
In this case, the vector of unknown , the operators , and the system second member are:
Therefore, (3.9) can be written in the form of (1.1).
Applying Theorem 1.1 a solution of (3.10) can be obtained as a solution of the fixed-point problem:
In this particular example, one has:
To solve this problem numerically, one uses fixed-point iterations directly, i.e. one uses the following fixed point method:
and an error tolerance of , where the error is defined for each iteration as
In the following figures one shows the convergence process to obtain the approximate solution. Thus, Figure 1 shows the fixed-point iterations (3.13) for different groups of iterations, i.e. in the subfigure “Iteration from 0 to 7” it being showed the seven first fixed-point iteration values and the initial condition , thus in the figure “Iteration from 8 to 15” it being showed the next eight the fixed-point iteration values and so on for the other subfigures. By changing the scale in the subfigures, one observes the accumulation of the point-fixed iteration values in a specific place of space and that is an indicative of fixed-point iterations convergence.
As in the previous figure, Figure 2 shows the convergence error (3.14) of the fixed-point iterations for different groups of iterations. Herein, one can appreciate error convergences to zero quickly.
Figure 1. Convergence of fixed-point iterations.
Figure 2. Error for each iteration.
The approximated value obtained for solution of (3.13) is:
Here in, one presents the value Table 1 of fixed-point iteration.
Table 1. Fixed-point iteration values.
This work has been supported by Yachay Tech University.