1. Introduction
This work is devoted to study the existence of solutions for the following semilinear equation
(1.1)
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 [1] where a generalization of Cramer’s Rule is given applying the Moore-Penrose inverse
that can be used when
exists, and a result from [2] . More information about the Moore-Penrose inverse can be found in [3] and [4] .
In this paper, using Moore-Penrose inverse
and the Rothe’s Fixed Theorem [5] [6] [7] , we shall prove the following theorem:
Theorem 1.1. If
exists and
is continuous and satisfies the condition
, (1.2)
then Equation (1.1) is solvable.
Moreover, for each
there exists
such that
,
where
.
The following theorem will be used to prove our main result.
Theorem 1.2. (Rothe’s Fixed Theorem [4] [5] [6] ) 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
. (2.3)
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
. (2.4)
Then,
.
Then
.
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
, (2.5)
and
, (2.6)
then the Equation (1.1) is solvable and a solution of it is given by
, (2.7)
where
.
Proof. Define the operator
. Then
and
,
and from condition (2.6)
. (2.8)
Therefore, from Theorem 2.1 and (2.8) we have that
is a homeomorphism Lipschitizian with a Lipschitz constant
.
Then,
.
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
(3.9)
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).
(3.10)
Applying Theorem 1.1 a solution of (3.10) can be obtained as a solution of the fixed-point problem:
(3.11)
In this particular example, one has:
(3.12)
To solve this problem numerically, one uses fixed-point iterations directly, i.e. one uses the following fixed point method:
(3.13)
and an error tolerance of
, where the error is defined for each iteration as
(3.14)
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.
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.
Supported
This work has been supported by Yachay Tech University.