^{1}

^{*}

^{1}

In this paper, we study the existence of solutions for the semilinear equation
, where
*A* is a
,
,
and
is a nonlinear continuous function. Assuming that the Moore-Penrose inverse
*A*
^{T}(
*AA*
^{T})
^{-1 }exists (
*A *denotes the transposed matrix of
*A*) which is true whenever the determinant of the
matrix
*AA*
^{T }is different than zero, and the following condition on the nonlinear term
satisfied
. We prove that the semilinear equation has solutions for all
. Moreover, these solutions can be found from the following fixed point relation
.

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

A x + f ( x ) = b , b ∈ I R n , x ∈ I R m (1.1)

where A is a m × n matrix, m ≥ n , b ∈ I R n and f : I R m → I R n is a nonlinear continuous function.

Definition 1.1. The Equation (1.1) is said to be solvable if for all b ∈ I R n there exists x ∈ I R m such that

A x + f ( x ) = b .

Proposition 1.1. The Equation (1.1) is solvable if, and only if, the operator A + f ( ⋅ ) : I R m → I R n is surjective.

The corresponding linear equation A x = b has been studied in [

In this paper, using Moore-Penrose inverse A + and the Rothe’s Fixed Theorem [

Theorem 1.1. If A T ( A A T ) − 1 exists and f is continuous and satisfies the condition

lim ‖ x ‖ → ∞ ‖ f ( x ) ‖ ‖ x ‖ = 0 , (1.2)

then Equation (1.1) is solvable.

Moreover, for each b ∈ I R n there exists x b ∈ I R m such that

A x b + f ( x b ) = b ,

where x b = A T ( A A T ) − 1 ( b − f ( x b ) ) .

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

Theorem 1.2. (Rothe’s Fixed Theorem [

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 K : I R m → I R m by

K ( x ) = A T ( A A T ) − 1 ( b − f ( x ) ) ,

and from condition (1.2) we obtain that

lim ‖ x ‖ → ∞ ‖ K ( x ) ‖ ‖ x ‖ = 0 . (2.3)

Claim. The operator K has a fixed point. In fact, for a fixed 0 < ρ < 1 , there exists R > 0 big enough such that

‖ K ( x ) ‖ ≤ ρ ‖ x ‖ , ‖ x ‖ = R .

Hence, if we denote by B ( 0, R ) the ball of center zero and radius R > 0 , we get that K ( ∂ B ( 0, R ) ) ⊂ B ( 0, R ) . Since K is compact and maps the sphere ∂ B ( 0, R ) into the interior of the ball B ( 0, R ) , we can apply Rothe’s fixed point Theorem 1.2 to ensure the existence of a fixed point x b ∈ B ( 0, R ) ⊂ I R m such that

x b = K ( x b ) . (2.4)

Then,

x b = A T ( A A T ) − 1 ( b − f ( x b ) ) .

Then

A x b = b − f ( x b ) ⇔ A x b + f ( x b ) = 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 : W → W is a Lipschitz function with a Lipschitz constant 0 < h < 1 and consider F ( w ) = w + H w . Then F is an homeomorphism whose inverse is a Lipschitz function with a Lipschitz constant ( 1 − h ) − 1 .

Theorem 2.2. If the Moore-Penrose A T ( A A T ) − 1 exists and the following condition holds

‖ f ( x 2 ) − f ( x 1 ) ‖ ≤ L ‖ x 2 − x 1 ‖ , x 1 , x 2 ∈ I R m , (2.5)

and

‖ A T ( A A T ) − 1 ‖ L < ρ < 1 , (2.6)

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

x b = A T ( A A T ) − 1 ( I + f ∘ Γ ) − 1 ( b ) , (2.7)

where Γ = A T ( A A T ) − 1 .

Proof. Define the operator F = A + f : I R m → I R n . Then F ∘ Γ = I + f ∘ Γ and

‖ ( f ∘ Γ ) ( b 2 ) − ( f ∘ Γ ) ( b 1 ) ‖ ≤ ‖ Γ ‖ L ‖ b 2 − b 1 ‖ , ∀ b 1 , b 2 ∈ I R n ,

and from condition (2.6)

‖ Γ ‖ L < ρ < 1 . (2.8)

Therefore, from Theorem 2.1 and (2.8) we have that F ∘ Γ = I + f ∘ Γ is a homeomorphism Lipschitizian with a Lipschitz constant 1 1 − ρ .

Then,

F ∘ ( Γ ∘ ( I + f ∘ Γ ) − 1 ) = I .

Hence, x b = ( Γ ∘ ( I + f ∘ Γ ) − 1 ) ( b ) is a solution of (1). In fact,

F ( x b ) = b ⇔ A x b + f ( x b ) = b ,

and this complete the proof. □

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

{ x 1 + x 2 + sin ( x 1 x 2 ) = 1 − x 1 + x 2 + x 3 + cos ( x 2 x 3 ) = 1 (3.9)

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

x = ( x 1 x 2 x 3 ) , A = ( 1 1 0 − 1 1 1 ) and f ( x ) = ( sin ( x 1 x 2 ) cos ( x 2 x 3 ) ) , b = ( 1 1 )

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

A x + f ( x ) = 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 T ( A A T ) − 1 ( b − f ( x ) ) (3.11)

In this particular example, one has:

A T ( A A T ) − 1 = ( 1 2 − 1 2 1 2 1 2 0 1 2 ) (3.12)

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

{ l x n + 1 = A T ( A A T ) − 1 ( b − f ( x n ) ) x 0 = ( 20 10 − 1 ) (3.13)

and an error tolerance of 10 − 10 , where the error is defined for each iteration as

Error ( n ) = ‖ x n − x n − 1 ‖ , for n = 1,2 , ... (3.14)

In the following figures one shows the convergence process to obtain the approximate solution. Thus,

As in the previous figure,

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

( 414.511990290326 e − 003 414.511990290326 e − 003 0.00000000000000 e + 000 )

Here in, one presents the value

i | W 1 | W 2 | W 3 |
---|---|---|---|

0 | 20.0000000000000e+000 | 10.0000000000000e+000 | −1.00000000000000e+000 |

1 | 17.1128840687711e−003 | 1.85618441314522e+000 | 919.535764538226e−003 |

2 | −83.6859033311723e−003 | 1.05192657611689e+000 | 567.806239724032e−003 |

3 | 457.390133251325e−003 | 630.527636166526e−003 | 86.5687514576005e−003 |

4 | 357.047377906710e−003 | 358.536714067085e−003 | 744.668080187583e−006 |

5 | 436.167364616208e−003 | 436.167400258264e−003 | 17.8210279311308e−009 |

6 | 405.451739883687e−003 | 405.451739883687e−003 | 0.00000000000000e+000 |

7 | 418.174158181806e−003 | 418.174158181806e−003 | 0.00000000000000e+000 |

8 | 413.010123030581e−003 | 413.010123030581e−003 | 0.00000000000000e+000 |

9 | 415.124320120457e−003 | 415.124320120457e−003 | 0.00000000000000e+000 |

10 | 414.261735970446e−003 | 414.261735970446e−003 | 0.00000000000000e+000 |

11 | 414.614167222791e−003 | 414.614167222791e−003 | 0.00000000000000e+000 |

12 | 414.470255524096e−003 | 414.470255524096e−003 | 0.00000000000000e+000 |

13 | 414.529034297305e−003 | 414.529034297305e−003 | 0.00000000000000e+000 |

14 | 414.505029226087e−003 | 414.505029226087e−003 | 0.00000000000000e+000 |

15 | 414.514833210466e−003 | 414.514833210466e−003 | 0.00000000000000e+000 |

16 | 414.510829199791e−003 | 414.510829199791e−003 | 0.00000000000000e+000 |

17 | 414.512464474425e−003 | 414.512464474425e−003 | 0.00000000000000e+000 |

18 | 414.511796615081e−003 | 414.511796615081e−003 | 0.00000000000000e+000 |

19 | 414.512069374520e−003 | 414.512069374520e−003 | 0.00000000000000e+000 |

20 | 414.511957977294e−003 | 414.511957977294e−003 | 0.00000000000000e+000 |

21 | 414.512003472857e−003 | 414.512003472857e−003 | 0.00000000000000e+000 |

22 | 414.511984892088e−003 | 414.511984892088e−003 | 0.00000000000000e+000 |

23 | 414.511992480630e−003 | 414.511992480630e−003 | 0.00000000000000e+000 |

24 | 414.511989381406e−003 | 414.511989381406e−003 | 0.00000000000000e+000 |

25 | 414.511990647155e−003 | 414.511990647155e−003 | 0.00000000000000e+000 |

26 | 414.511990130213e−003 | 414.511990130213e−003 | 0.00000000000000e+000 |

27 | 414.511990341336e−003 | 414.511990341336e−003 | 0.00000000000000e+000 |

28 | 414.511990255112e−003 | 414.511990255112e−003 | 0.00000000000000e+000 |

29 | 414.511990290326e−003 | 414.511990290326e−003 | 0.00000000000000e+000 |

This work has been supported by Yachay Tech University.

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