ABSTRACT

In this paper, we study the existence of solutions for the semilinear equation $Ax+f\left(x\right)=b$ , where $A$ is a $m\times n$ , $m\ge n$ , $b\in I{R}^n$ and $f\mathrm{<mo>\; :\; </mo>}I{R}^m\to I{R}^n$ is a nonlinear continuous function. Assuming that the Moore-Penrose inverse $A^{\text{T}}{\left(A{A}^{\text{T}}\right)}^{-1}$ exists ( $A^{\text{T}}$ denotes the transposed matrix of $A$ ) which is true whenever the determinant of the $n\times n$ matrix $A{A}^{\text{T}}$ is different than zero, and the following condition on the nonlinear term $f$ satisfied $\underset{\Vert x\Vert \to \infty }{\lim }\frac{\Vert f\left(x\right)\Vert }{\Vert x\Vert }=0$ . We prove that the semilinear equation has solutions for all $b\in I{R}^n$ . Moreover, these solutions can be found from the following fixed point relation $x_b=A^{\text{T}}{\left(A{A}^{\text{T}}\right)}^{-1}\left(b-f\left(x_b\right)\right)$ .

Keywords:

Semilinear Equations, Moore-Penrose Inverse, Rothe’s Fixed Point Theorem

1. Introduction

This work is devoted to study the existence of solutions for the following semilinear equation

$Ax+f\left(x\right)=b\mathrm{,}b\in I{R}^n\mathrm{,}x\in I{R}^m$ (1.1)

where $A$ is a $m\times n$ matrix, $m\ge n$ , $b\in I{R}^n$ and $f\mathrm{<mo>\; :\; </mo>}I{R}^m\to I{R}^n$ is a nonlinear continuous function.

Definition 1.1. The Equation (1.1) is said to be solvable if for all $b\in I{R}^n$ there exists $x\in I{R}^m$ such that

$Ax+f\left(x\right)=b$ .

Proposition 1.1. The Equation (1.1) is solvable if, and only if, the operator $A+f(\cdot )\mathrm{<mo>\; :\; </mo>}I{R}^m\to I{R}^n$ is surjective.

The corresponding linear equation $Ax=b$ has been studied in [1] where a generalization of Cramer’s Rule is given applying the Moore-Penrose inverse $A^{+}=A^{\text{T}}{\left(A{A}^{\text{T}}\right)}^{-1}$ that can be used when ${\left(A{A}^{\text{T}}\right)}^{-1}$ 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 $A^{+}$ and the Rothe’s Fixed Theorem [5] [6] [7] , we shall prove the following theorem:

Theorem 1.1. If $A^{\text{T}}{\left(A{A}^{\text{T}}\right)}^{-1}$ exists and $f$ is continuous and satisfies the condition

$\underset{\Vert x\Vert \to \infty }{\lim }\frac{\Vert f\left(x\right)\Vert }{\Vert x\Vert }=0$ , (1.2)

then Equation (1.1) is solvable.

Moreover, for each $b\in I{R}^n$ there exists $x_b\in I{R}^m$ such that

$A{x}_b+f\left(x_b\right)=b$ ,

where $x_b=A^{\text{T}}{\left(A{A}^{\text{T}}\right)}^{-1}\left(b-f\left(x_b\right)\right)$ .

The following theorem will be used to prove our main result.

Theorem 1.2. (Rothe’s Fixed Theorem [4] [5] [6] ) Let $E$ be a Banach space. Let $B\subset E$ be a closed convex subset such that the zero of $E$ is contained in the interior of $B$ . Let $\Phi \mathrm{<mo>\; :\; </mo>}B\to E$ be a continuous mapping with $\Phi \left(B\right)$ relatively compact in $E$ and $\Phi \left(\partial B\right)\subset B$ . Then there is a point $x^{\mathrm{<mo>\; *\; </mo>}}\in B$ such that $\Phi \left(x^{*}\right)=x^{*}$ .

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 $\mathcal{K}\mathrm{<mo>\; :\; </mo>}I{R}^m\to I{R}^m$ by

$\mathcal{K}\left(x\right)=A^{\text{T}}{\left(A{A}^{\text{T}}\right)}^{-1}\left(b-f\left(x\right)\right)$ ,

and from condition (1.2) we obtain that

$\underset{\Vert x\Vert \to \infty }{\lim }\frac{\Vert \mathcal{K}\left(x\right)\Vert }{\Vert x\Vert }=0$ . (2.3)

Claim. The operator $\mathcal{K}$ has a fixed point. In fact, for a fixed $0<\rho <1$ , there exists $R>0$ big enough such that

$\Vert \mathcal{K}\left(x\right)\Vert \le \rho \Vert x\Vert \mathrm{,}\Vert x\Vert =R$ .

Hence, if we denote by $B\left(\mathrm{0,}R\right)$ the ball of center zero and radius $R>0$ , we get that $\mathcal{K}\left(\partial B\left(\mathrm{0,}R\right)\right)\subset B\left(\mathrm{0,}R\right)$ . Since $\mathcal{K}$ is compact and maps the sphere $\partial B\left(\mathrm{0,}R\right)$ into the interior of the ball $B\left(\mathrm{0,}R\right)$ , we can apply Rothe’s fixed point Theorem 1.2 to ensure the existence of a fixed point $x_b\in B\left(\mathrm{0,}R\right)\subset I{R}^m$ such that

$x_b=\mathcal{K}\left(x_b\right)$ . (2.4)

Then,

$x_b=A^{\text{T}}{\left(A{A}^{\text{T}}\right)}^{-1}\left(b-f\left(x_b\right)\right)$ .

Then

$A{x}_b=b-f\left(x_b\right)\iff A{x}_b+f\left(x_b\right)=b$ .

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 $W$ be a Hilbert space and $H\mathrm{<mo>\; :\; </mo>}W\to W$ is a Lipschitz function with a Lipschitz constant $0<h<1$ and consider $F\left(w\right)=w+Hw$ . Then $F$ is an homeomorphism whose inverse is a Lipschitz function with a Lipschitz constant ${\left(1-h\right)}^{-1}$ .

Theorem 2.2. If the Moore-Penrose $A^{\text{T}}{\left(A{A}^{\text{T}}\right)}^{-1}$ exists and the following condition holds

$\Vert f\left(x_2\right)-f\left(x_1\right)\Vert \le L\Vert x_2-x_1\Vert \mathrm{,}{x}_1\mathrm{,}{x}_2\in I{R}^m$ , (2.5)

and

$\Vert A^{\text{T}}{\left(A{A}^{\text{T}}\right)}^{-1}\Vert L<\rho <1$ , (2.6)

then the Equation (1.1) is solvable and a solution of it is given by

$x_b=A^{\text{T}}{\left(A{A}^{\text{T}}\right)}^{-1}{\left(I+f\circ \Gamma \right)}^{-1}\left(b\right)$ , (2.7)

where $\Gamma =A^{\text{T}}{\left(A{A}^{\text{T}}\right)}^{-1}$ .

Proof. Define the operator $F\mathrm{<mo>\; =\; </mo>}A+f\mathrm{<mo>\; :\; </mo>}I{R}^m\to I{R}^n$ . Then $F\circ \Gamma =I+f\circ \Gamma$ and

$\Vert \left(f\circ \Gamma \right)\left(b_2\right)-\left(f\circ \Gamma \right)\left(b_1\right)\Vert \le \Vert \Gamma \Vert L\Vert b_2-b_1\Vert \mathrm{,}\forall b_1\mathrm{,}{b}_2\in IR{}^n$ ,

and from condition (2.6)

$\Vert \Gamma \Vert L<\rho <1$ . (2.8)

Therefore, from Theorem 2.1 and (2.8) we have that $F\circ \Gamma =I+f\circ \Gamma$ is a homeomorphism Lipschitizian with a Lipschitz constant $\frac{1}{1-\rho }$ .

Then,

$F\circ \left(\Gamma \circ {\left(I+f\circ \Gamma \right)}^{-1}\right)=I$ .

Hence, $x_b=\left(\Gamma \circ {\left(I+f\circ \Gamma \right)}^{-1}\right)\left(b\right)$ is a solution of (1). In fact,

$F\left(x_b\right)=b\iff A{x}_b+f\left(x_b\right)=b$ ,

and this complete the proof. □

3. Practical Example

Now, we shall apply Theorem 1.1 to find one solution of the following semilinear system

$\{\begin{array}{l}{x}_1+x_2+\sin \left(x_1{x}_2\right)=1\\ -x_1+x_2+x_3+\cos \left(x_2{x}_3\right)=1\end{array}$ (3.9)

In this case, the vector of unknown $x$ , the operators $A$ , $f\left(x\right)$ and the system second member $b$ are:

$x=\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right),A=\left(\begin{array}{ccc}1& 1& 0\\ -1& 1& 1\end{array}\right)\text{and}f\left(x\right)=\left(\begin{array}{c}\sin \left(x_1{x}_2\right)\\ \cos \left(x_2{x}_3\right)\end{array}\right),b=(\; 1\; 1\; )$

Therefore, (3.9) can be written in the form of (1.1).

$Ax+f\left(x\right)=b$ (3.10)

Applying Theorem 1.1 a solution of (3.10) can be obtained as a solution of the fixed-point problem:

$x=A^{\text{T}}{\left(A{A}^{\text{T}}\right)}^{-1}\left(b-f\left(x\right)\right)$ (3.11)

In this particular example, one has:

$A^{\text{T}}{\left(A{A}^{\text{T}}\right)}^{-1}=\left(\begin{array}{cc}\frac{1}{2}& -\frac{1}{2}\\ \frac{1}{2}& \frac{1}{2}\\ 0& \frac{1}{2}\end{array}\right)$ (3.12)

To solve this problem numerically, one uses fixed-point iterations directly, i.e. one uses the following fixed point method:

$\{\begin{array}{l}l{x}_{n+1}=A^{\text{T}}{\left(A{A}^{\text{T}}\right)}^{-1}\left(b-f\left(x_n\right)\right)\\ x_0=\left(\begin{array}{r}\hfill 20\\ \hfill 10\\ \hfill -1\end{array}\right)\end{array}$ (3.13)

and an error tolerance of ${10}^{-10}$ , where the error is defined for each iteration as

$\text{Error}\left(n\right)=\Vert x_n-x_{n-1}\Vert \mathrm{,}\text{for}n=\mathrm{1,2},\mathrm{...}$ (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 $x_0$ , 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.

The approximated value obtained for $x$ solution of (3.13) is:

$\left(\begin{array}{r}\hfill 414.511990290326e-003\\ \hfill 414.511990290326e-003\\ \hfill 0.00000000000000e+000\end{array}\right)$

Here in, one presents the value Table 1 of fixed-point iteration.

Supported

This work has been supported by Yachay Tech University.

Cite this paper

Leiva, H. and Manzanilla, R. (2018) Moore-Penrose Inverse and Semilinear Equations. Advances in Linear Algebra & Matrix Theory, 8, 11-17. https://doi.org/10.4236/alamt.2018.81002

References