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Open Physics

formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

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Volume 16, Issue 1

Issues

Volume 13 (2015)

A new method for solving nonlinear systems of equations that is based on functional iterations

Joaquín Moreno / Miguel A. López / Raquel Martínez
Published Online: 2018-10-22 | DOI: https://doi.org/10.1515/phys-2018-0079

Abstract

Regarding solving nonlinear equations systems, there is a main problem that is the number and complexity of the linear algebra operations, and the functional evaluations of the applied algorithm. In this paper, an alternative solution will be proposed by means of constructing a converse of the Banach Theorem fixed-point, only to ℝ2 and ℝ3, in the following sense, this being: each root of a non-linear equations system has been considered as a fixed-point. Besides, the compact set and the continuous functions that fulfil the Banach Theorem are built under certain conditions, those that must satisfy the systemfunctions. Thus each iteration only requires the evaluation of two or three functions.

Keywords: Nonlinear systems; fixed point iterations; zeros; algorithms

PACS: 02.60.Cb; 02.60.Pn

1 Introduction

Nonlinear systems arise, either directly or as a part of computing tasks, in many important mathematical areas, such as finite element methods, optimization, with or without constraints, nonlinear least square problems or nonlinear integral equations [1]. On the other side, they are very useful in a large number of fields of the Engineering [2], Physics and Chemistry [3],[4],[5], EGHS, BB, Astronomy [6] and Manufacturing [7].

To give a specific example, in the field of Physics, nonlinear evolution equations and obtaining their traveling wave solutions have a lot of importance due to the applications to nonlinear optics, quantum field theory, solid state physics, plasma physics and fluid mechanics. These problems require the solutions of nonlinear differential equations such as the Tzitzeica-Dodd-Bullough equation, the nonlinear telegraph equation and the generalized Benjamin-Bona-Mahony equation [8]. One of the more efficient ways to solve them is the trial equation method proposed by Liu [9], in which the solution of nonlinear systems of equations plays a very important role in the computing process [8].

In all of these contexts, nonlinear systems are solved using numerical techniques. The most expanded is the classical Newton’s method, whose well known iteration formula is:

xk+1=xk(F(xk))1F(xk)(1)

Newton’s method is also very implemented in the scalar case. See for example [10].

The reader can find a whole development of the theoretical bases of nonlinear systems and nonlinear equations, in general, and the classical Newton’s method, in particular, in [11].

Using Newton’s method can be very expensive, due to the computation of the Jacobian matrix, F(xk), and the resolution of the linear system (1) in each iteration. This fact motivated the development of a great diversity of Newton’s method variant, that we are going to call, in a general sense, Newton-like methods, and which can be represented as a generalization of (1), as follows:

xk+1=xkBk1F(xk)(2)

where Bk is a matrix or a combination of them, that intended to be approximations of F(xk), by using a wide variety of procedures, from the usage of finite differences to the application of the gaussian quadrature integral formulae, for instance. In many methods, the computation of (2) does not involve computing derivatives at all. Moreover, in some particular methods, Bk+11 is obtained from Bk1 , using simple techniques, thanks to which the linear algebra cost involved in (2) is much less than the one involved in (1) [12-17].

In the last decades, lots of root-finding methods have been published, focusing specially on the improving of the order of convergence of mentioned Newton-like methods, most frequently by composing two or more of them (multi-point or multi-step iterative methods). In the recent paper [18], the reader can see an introduction and bibliography of this issue.

Nowadays, there are still three problems left unsolved:

  1. The selection of the initial value of the iterations.

  2. The determination and the computation of all the real roots in a given feasible region.

  3. The number and complexity of the linear algebra operations and functional evaluations of each iteration of the algorithms.

With respect to the first question, as it is well known, given an initial approximation, x0, all the described above methods provide a sequence of iterates xk,k=1,2,⋯ in such a way that, hopefully, the approximation to some solution is progressively improved, but the convergence is not guaranteed in the general case.

In relation to the second issue, it is very frequent in engineering, for example, to search for solutions of nonlinear systems of equations within a certain feasible region, where the problem lies. Nevertheless, the published papers about this theme are referred to find only one root. Nothing is said about other possible ones, herein where needed.

Regarding the third one, it is true that the computational cost and the convergence velocity have been improved, however the number of the linear algebra operations and the functional evaluations is still very high.

In this paper we propose a solution for the third problem, but only for systems of two or three equations and two or three unknowns, respectively. With such a proposition throughout this paper, we develop the following idea: Each root of a nonlinear system of equations can be considered as a fixed point. Banach Theorem ensures its existence for continuous functions, defined in compact sets. We proceed conversely. We suppose the existence of a fixed point (a root of the system) and we are constructing the compact set and the continuous functions that fulfil Banach Theorem, under certain conditions that must satisfy the functions of the system. In this way, each iteration requires only the evaluation of two or three functions.

The following nomenclature has been set:

Throughout this paper, firstly we search for solutions of nonlinear systems of two equations and two unknowns, (x1,x2), written in the form:

F(x¯)=(f1(x¯),f2(x¯))=(0,0);x¯G.(3)

fi(x1,x2),i=1,2, being C1 functions in an open set, G⊂ℝ2. We suppose that system (3) has a root r¯=(r1,r2)∈G. And then we search for solutions of nonlinear systems of three equations and three unknowns, (x1,x2,x3), written in the form:

F(x¯)=(f1(x¯),f2(x¯),f3(x¯))=0¯;x¯G.(4)

fi(x¯),i=1,2,3, being C1 functions in an open set, G⊂ℝ3. We suppose that system (4) has a root r¯=(r1,r2,r3)∈G.

2 Fixed point Banach Theorem converse inℝ2

In this section we prove the converse of the fixed point Banach Theorem for functions from ℝ2 to ℝ2, in the context of the search for solutions of nonlinear systems. We also develop some previous results, we need to prove the converse of the fixed point Banach Theorem for functions from ℝ3 to ℝ3 in the same environment.

Theorem 2.1

The following statements hold:

  1. Let F1 and F2 be two real differentiable functions, defined in an interval [a,b].

  2. F1(x1)>0 and F2(x1)>0,∀x1∈[a,b].

  3. There exists r1∈(a,b), so that there is r2=F1(r1)=F2(r1).

  4. Consider, without loss of generality, that F1(x1)>F2(x1),∀x1∈[a,r1) and F2(x1)>F1(x1),∀x1∈(r1,b].

  5. F1(x1)−F2(x1)<0,∀x1∈[a,b].

  6. Let [c,d] be the image of F1, in other words: [c,d]=F1([a,b]).

Then:

1. |F1(x1)|<|F2(x1)|,∀x1∈[a,b].

2. The functions:

F1F21:[c,d][c,d]F21F1:[a,b][a,b](5)

are well defined.

3. The sequence:

q0[c,d]qn=F1F21(qn1),n1(6)

is convergent to r2. Moreover, if q0<r2 it is increasing and if q0>r2, decreasing.

4. The sequence:

p0[a,b]pn=F21F1(pn1),n1(7)

is convergent to r1. Besides, if p0<r1 it is increasing and if p0>r1, decreasing.

Proof

Notice that, as F1(x1)−F2(x1)<0, then r1 is unique.

First. F1(x1) and F2(x1) are positive, as F1(x1)−F2(x1)<0, then |F2(x1)|>|F1(x1)|,∀x1∈[a,b].

Second. We show that F1([a,b])⊂F2([a,b]):

x[a,b]F2(a)<F1(a)F1(x)F1(b)<F2(b)[c,d]=F1([a,b])F2([a,b])

Therefore, functions (5) are well defined.

Third. For all x2∈[c,d], we have that:

(F1F21)(x2)=F1(F21(x2))(F21)(x2)=F1(F21(x2))F2(F21(x2))>0

Hence, F1F21 is increasing in [c,d].

If q0<r2, then q1=F1F21(q0)<F1F21(r2)=r2(r2)=r2. So, q1<r2.

On the other hand, let u0∈[a,b] so that, q0=F2(u0). Then:

q0<r2u0=F21(q0)<F21(r2)=r1

since F1F21 is increasing. From the fourth hypothesis of the Theorem, we have that:

u0<r1q0=F2(u0)<F1(u0)=v1

Taking into account that F11 is increasing:

q0<v1F21(q0)=u0=F11F1(u0)=F11(v1)>F11(q0)

From here, we deduce that:

F21(q0)>F11(q0)q0=F2F21(q0)>F2F11(q0)=q1

Therefore, we arrive at q0>q1>r2. Consider, now, that qn<r2, then from the same reasons qn<qn+1<r2. So the sequence qn, ∀n≥0, is increasing and bounded by r2. As a consequence, the sequence (6) is convergent. Let q∈[c,d] be its limit. Let p=p=F21(q)(q), so p∈[a,b], Then, by continuity:

q=F1F21(q)F11(q)=F21(q)=pF1(p)=F2(p)=qq=r2 and p=r1

due to the uniqueness of (r1,r2). In a similar manner, if q0>r2, we can infer that the sequence is decreasing and r2 being a lower bound.

We analyze the convergence of sequence (7). Let q0∈[c,d] so that p0=p0=F21(q0)(q0).

p1=F21F1(p0)p1=F21F1F21(q0)F2(p1)=F1F21(q0)=q1

In the same way:

p2=F21F1(p1)p2=F21F1F21(q1)F2(p2)=F1F21(q1)=q2

Suppose now that: F2(pn)=qn, then:

pn+1=F21F1(pn)pn+1=F21F1F21(qn)F2(pn+1)=F1F21(qn)=qn+1

So, the sequence is convergent, by the continuity of the function F21.

If p0<r1, then q0=F2(p0)<F2(r1)=r2. In agreement with the above, the sequence qn is increasing and qn<r2, ∀n≥0, therefore pn=F21(qn)(qn), ∀n≥0 is also increasing and pn=F21(qn)<F21(r2)=r1(qn)<F1F21(r2)=r1, ∀n≥0.

On the contrary, if p0>r1, in a similar way, we can deduce that the sequence is decreasing, r1 being a lower bound. □

Theorem 2.2

Let us introduce F1 and F2 under the same hypotheses 1, 3, 4 and 5 as Theorem 2.1. Besides:

  1. F1(x1)<0 and F2(x1)<0, ∀x1∈[a,b].

  2. Let [c,d] be the image of F2, that is, [c,d]=F2([a,b]).

Then:

1. |F1(x1)|>|F2(x1)|,∀x1∈[a,b].

2. The functions:

F2F11:[c,d][c,d]F11F2:[a,b][a,b]

are well defined.

3. The sequence:

q0[c,d]qn=F2F11(qn1),n1

is convergent to r2. Moreover, if q0<r2 it is increasing and if q0>r2, decreasing.

4. The sequence:

p0[a,b]pn=F11F2(pn1),n1

is convergent to r1. Besides, if p0<r1 it is increasing and if p0>r1, decreasing.

Proof

The proof follows the same techniques as Theorem 2.1. □

Theorem 2.3

Theorem \labeltheo:increasingdecreasing3 Let us introduce F1 and F2 under the same hypotheses 1, 3 and 4 as Theorem 2.1. Besides:

1. F1 and F2 are continuous, with F1(x1)<0, F2(x1)>0 and F1(x1)+F2(x1)<0, ∀x1∈[a,b].

2. Let [c,d] be the image of F2.

Then:

1. |F1(x1)|>|F2(x1)|;∀x1∈[a,b]

2. The functions:

F2F11:[c,d][c,d]F11F2:[a,b][a,b](8)

are well defined.

3. The sequence:

q0[c,d]qn=F2F11(qn1),n1(9)

is convergent to r2. Moreover, if Q0=q0<r2, the sequence:

Q0[c,d]Qn=F2F11F2F11(Qn1),n1(10)

is increasing and if Q0=q0>r2, decreasing.

4. The sequence:

p0[a,b]pn=F11F2(pn1),n1

is convergent to r1. Besides, if P0=p0<r1, the sequence:

P0[a,b]Pn=F11F2F11F2(Pn1),n1

is increasing and if p0>r1, decreasing.

Proof

Without loss of generality, we can take the interval [a,b] in such a way that F2(a)=F1(b), since if F2(a)<F1(b), we redefine the extreme a, of the interval [a,b], as a1=F21F1(b)F1(b), and then F2(a1)=F1(b). Notice that:

1. F1(b)∈F2([a,b]), due to the fact that:

F2(a)<F1(b)<F1(r1)=F2(r1)<F2(a)

2. a<a1<r1, because of:

a=F21F2(a)<F21F1(b)=a1<F21F1(r1)=r1

and, on the contrary, if F2(a)>F1(b), we redefine the extreme b, of the interval [a,b], as b=F11F2(a)F2(a). In the same way:

1. F2(a)∈F1([a,b]), due to the fact that:

F1(b)<F2(a)<F2(r1)=F1(r1)<F1(a)

2. b>b1>r1, because of:

b=F11F1(b)>F11F2(a)=b1>F11F2(r1)=r1

So, from now, we are supposing that F1(b)=F2(a).

First. F1(x1)+F2(x1)=−|F1(x1)|+|F2(x1)|<0 ∀x1∈[a,b], what implies that:

|F1(x1)|>|F2(x1)|;x1[a,b]

Second. We have that:

0<F2(x1)<F1(x1)ar1F2(x1)dx1<ar1F1(x1)dx1F2(r1)F2(a)<F1(a)F1(r1)r2F2(a)<F1(a)r2(11)

(Notice that the integral is well defined, since F1 and F2 are continuous). On the other hand:

0<F2(x1)<F1(x1)r1bF2(x1)dx1<r1bF1(x1)dx1F2(b)F2(r1)<F1(r1)F1(b)F2(b)r2<r2F1(b)(12)

Therefore, by adding inequalities (11) and (12)

F2(b)F2(a)<F1(a)F1(b)F2(b)<F1(a)

since F1(b)=F2(a), we have that:

F1(b)=F2(a)<F2(b)<F1(a)

Hence, if x1∈[a,b]:

F1(b)=F2(a)F2(x1)F2(b)<F1(a)

and F2([a,b])⊂F1([a,b]), so functions (8) are well defined.

Third. The function: |F2(x1)||F1(x1)| is continuous in [a,b], so it reaches a maximum, (α,δ), with α∈[a,b], hence:

|F2(x1)||F1(x1)||F2(α)||F1(α)|=δ<1,x1[a,b]

We have that:

|(F2F11)(x2)|=|F2(F11(x2))(F11)(x2)|=|F2(F11(x2))||F1(F11(x2))||F2(α)||F1(α)|=δ<1

So F2F11 is contractive and sequence (9) is convergent. Let q∈[c,d] be its limit. Let p=F11(q)[a,b](q)∈[a,b], by continuity:

q=F2F11(q)F21(q)=F11(q)=pF2(p)=F1(p)=q and q=r2; and p=r1

due to the uniqueness of (r1,r2).

Sequence (10) is also convergent to the same limit, since it is a subsequence of (9).

F2F11F2F11F2F11 is increasing because it is the composition of two decreasing functions, since:

(F2F11)(x2)=F2(F11(x2))F1(F11(x2))<0,x2[c,d].

Therefore, if q0<r2, then Q1=F2F11F2F11(q0)<F2F11F2F11(r2)=r2F2F11(q0)<F2F11F2F11(r2)=r2. So, Q1<r2.

On the other hand, let u0∈[a,b] so that, q0=F2(u0). Then:

q0<r2u0=F21(q0)<F21(r2)=r1

since F1F21 is increasing. From the fourth hypothesis of the Theorem 2.1, we have that:

u0<r1q0=F2(u0)<F1(u0)=v1

Taking into account that F11 is decreasing:

q0<v1F21(q0)=u0=F11F1(u0)=F11(v1)<F11(q0)

From here, we deduce that:

F21(q0)<F11(q0)q0=F2F21(q0)<F2F11(q0)

As F2F11 is decreasing:

q0<F2F11(q0)q0<q1 and Q1=q2<q1(13)

We know that:

Q1Q0=q2q0=q2q1+q1q0

From (13) q2q1<0 and q1q0>0. As F2F11 is contractive, then:

|q2q1|<|q1q0| and Q1Q0>0

Therefore, we arrive at q0=Q0<Q1<r2. Consider, now, that Qn<r2, then from the same reasons Qn<Qn+1<r2. So the sequence Qn, ∀n≥0, is increasing and r2 being an upper bound.

Fourth. This part of the Theorem can be proven with the same procedures we have seen in the preceding paragraphs. □Using identical techniques has also proven Theorem 2.4.

Theorem 2.4

Theorem \labeltheo:increasingdecreasing4 Let us introduce F1 and F2 under the same hypotheses 1, 3 and 4 as Theorem 2.1. As in the preceeding theorem, without loss of generality, we can take the interval [a,b] in such a way that F1(a)=F2(b). Besides:

1. F1 and F2 are continuous, with F1(x1)<0, F2(x1)>0 and F1(x1)+F2(x1)>0, ∀x1∈[a,b].

Sequence qn when |F2′(x1)|>|F1′(x1)|;∀x1∈[a,b]$|F_2'(x_1)| \gt |F_1'(x_1)|; \;\forall \; x_1 \in [a,b]$
Figure 1

Sequence qn when |F2(x1)|>|F1(x1)|;x1[a,b]

2. Let [c,d] be the image of F1.

Then:

1. |F1(x1)|<|F2(x1)|;∀x1∈[a,b]

2. The functions:

F1F21:[c,d][c,d]F21F1:[a,b][a,b]

are well defined.

3. The sequence:

q0[c,d]qn=F1F21(qn1),n1

is convergent to r2 (see Figure 1). Moreover, if Q0=q0<r2, the sequence:

Q0[c,d]Qn=F1F21F1F21(Qn1),n1

is increasing and if Q0=q0>r2, decreasing.

4. The sequence:

p0[a,b]pn=F21F1(pn1),n1

is convergent to r1. Besides, if P0=p0<r1, the sequence:

P0=p0[a,b]Pn=F21F1F21F1(pn1),n1

is increasing and if P0=p0>r1, decreasing.

Theorem 2.5

Let f1(x1,x2) and f2(x1,x2) be two functions defined in an open set G∈ℝ2, with continuous partial derivatives. Besides:

  1. ***

  2. ***

Then (r1,r2) is a root of the system if and only if exist a continuous function:

T:[a1,b1]×[c1,d1]⊂G⇒[a1,b1]×[c1,d1]

Proof

If (r1,r2) is a root of the system, by applying the Implicit Function Theorem, there exist neighborhoods of r1: Br11R ∈R and Br12R ∈R, and there are also neighborhoods of r2: Ar21R and Ar22R, where the functions:

F1:Br11Ar21F2:Br12Ar22

are well defined and the equalities f1(x1,F1(x1))=0 and f2(x1,F2(x1))=0 hold.

F1:Br11Ar21F2:Br12Ar22

As by hypothesis:

F1(x1)=f1(x1,x2)x1f1(x1,x2)x2=0F2(x1)=f2(x1,x2)x1f2(x1,x2)x2=0

Then:

f1x1f2x2±f1x2f2x1=0x¯G.

Since, there is a neighborhood of r1, Cr1, in such a way that the function:

H(x1)=f1x1f1x2(x1,F1(x1))±f2x1f2x2(x1,F2(x1))=0

Hence:

F1(x1)±F2(x1)=0;x1Cr1

Let [a1,b1] be a closed interval so that [a1,b1]⊂[a1,b1]Br11Br12Cr1[a1,b1]Br11Br12Cr1Cr1, with r1∈(a1,b1).

Suppose without loss of generality, that F1(x)>F2(x1) ∀x1∈[a1,r1) and F1(x)<F2(x1)∀x1∈(r1,b1]. We have that:

  1. If F1(x1),F2(x1)>0, ∀x1∈[a1,b1], then the hypotheses of Theorem 2.1 are verified.

  2. If F1(x1),F2(x1)<0, ∀x1∈[a1,b1], then the hypotheses of Theorem \reftheo:increasingdecreasing2 are satisfied.

  3. If F1(x1)>0 and F2(x1)<0, with F1(x1)+F2(x1)<0 ∀x1∈[a1,b1], then the hypotheses of Theorem \reftheo:increasingdecreasing3 hold.

  4. If F1(x1)>0 and F2(x1)<0, with F1(x1)+F2(x1)>0 ∀x1∈[a1,b1], then the hypotheses of Theorem 2.4 fulfil.

Therefore, there exists an interval [c1,d1] where either the functions:

F21F1:[a1,b1][a1,b1]F1F21:[c1,d1][c1,d1]

or the functions:

F11F2:[a1,b1][a1,b1]F2F11:[c1,d1][c1,d1]

are well defined. Hence either the function:

T([a1,b1]×[c1,d1])[a1,b1]×[c1,d1]Tx1x2=F11F2(x1)F2F11(x2)

or the function:

T([a1,b1]×[c1,d1])[a1,b1]×[c1,d1]Tx1x2=F21F1(x1)F1F21(x2)

is well defined.

Conversely. If there is such a function T, defined in the compact set [a1,b1]×[c1,d1], then T has a fixed point, that is (r1,r2) in agreement with the cited theorems. □

3 Definition of the first condition of derivation and its consequences

Definition 3.1

The system (4) is said to satisfy the first condition of derivation if it fulfils:

fi(xˉ)xj≠0,i,j=1,2,3;xˉG(14)

The jacobian determinant of system (4):

J=f1x1f2x1f3x1f1x2f2x2f3x2f1x3f2x3f3x3(15)

its nine minors:

m(3,3),m(3,2),m(3,1),m(1,3),m(1,2),m(1,1),m(2,3),m(2,2),m(2,1)(16)

where m(i,j) is the one obtained by removing the row i and the column j of J, and, finally, the determinats:

f1x1f2x1f1x2f2x2;f1x1f3x1f1x2f3x2;f2x1f3x1f2x2f3x2;f1x2f2x2f1x3f2x3;f1x2f3x2f1x3f3x3;f2x2f3x2f2x3f3x3;f1x1f2x1f1x3f2x3;f1x1f3x1f1x3f3x3;f2x1f3x1f2x3f3x3;(17)

are, all of them ((15), (16) and (17)) no equal to zero ∀x¯∈G.

From now, we assume that system (4) verifies the first condition of derivation.

Definition 3.2

We define the functions φji , 1≤i,j≤3, as follows:

From (14), by applying Implicit Function Theorem, there are suitable neighborhoods of (rp,rq) and ri, B(rp,rq)ij and Ariij respectively, with 1≤p<q≤3, i=p̸,q, where the functions:

φji:B(rp,rq)ijXpXqAriijXi with:xi=φji(xp,xq); so that:fj(φji(xp,xq),xp,xq)=0 if i<p<qfj(xp,φji(xp,xq),xq)=0 if p<i<qfj(xp,xq,φji(xp,xq))=0 if p<q<i(18)

are well defined. The superscript i and the subscript j mean that we solve the unknown xifrom the equation fj of the system (4).

Proposition 3.3

Functions \eqrefeq:varphifunctions have partial derivatives different to zero in G.

Proof

It is due to the fact that:

φji(xp,xq)xp=fj(x¯)xpfj(x¯)xi=0φji(xp,xq)xq=fj(x¯)xqfj(x¯)xi=0

C1={x¯R3;f1(x¯)=0;f3(x¯)=0}C2={x¯R3;f2(x¯)=0;f3(x¯)=0}C3={x¯R3;f1(x¯)=0;f2(x¯)=0}

Definition 3.4

We define the curves C1, C2 and C3, given by:

C1={xˉR3;f1(xˉ)=0;f3(xˉ)=0}C2={xˉR3;f2(xˉ)=0;f3(xˉ)=0}C3={xˉR3;f1(xˉ)=0;f2(xˉ)=0}

Definition 3.5

We define the functions G21 in the open set B(r1,r3)21B(r1,r3)22B(r1,r3)23 as follows:

G21(x1,x3)=φ32(x1,x3)φ12(x1,x3)G22(x1,x3)=φ32(x1,x3)φ22(x1,x3)G23(x1,x3)=φ22(x1,x3)φ12(x1,x3)(19)

Proposition 3.6

There is a neighborhood of (r1,r3), that, for the sake of clarity and without loss of generality, we can take as B(r1,r3)21B(r1,r3)22B(r1,r3)23 (in other case, we could always make the intersection) where the partial derivatives of functions \eqrefeq:differencefunctions are different to zero.

Proof

From (16), the minors m(1,2) valued at the root (r1,r2,r3)=(r1,φ32(r1,r3),r3)=(r1,φ12(r1,r3),r3) is different to zero. Therefore:

f1x2f3x2f1x3f3x3=f1x2f3x3f1x3f3x2=/0f3x3f3x2+f1x3f1x2=φ32x3φ12x3=/0G21(r1,r3)x3=/0

On the other hand:

f1x1f3x1f1x2f3x2=f1x1f3x2f1x2f3x1=/0f3x1f3x2+f1x1f1x2=φ32x1φ12x1=/0G21(r1,r3)x1=/0

Taking into account that the minor m(3,2) is also different to zero at the root and that (r1, r2, r3)(r1,r2,r3)=(r1,φ32(r1,r3),r3)=(r1,φ12(r1,r3),r3) we arrive at:

f2x2f3x2f2x3f3x3=f2x2f3x3f2x3f3x2=/0f3x3f3x2+f2x3f2x2=φ32x3φ22x3=/0G22(r1,r3)x3=/0(20)

And:

f2x1f3x1f2x2f3x2=f2x1f3x2f2x2f3x1=/0f3x1f3x2+f2x1f2x2=φ32x1φ22x1=/0G22(r1,r3)x1=/0

Reasoning equivalently:

f1x2f2x2f1x3f2x3=f1x2f2x3f2x2f1x3=/0f2x3f2x2+f1x3f1x2=φ22x3φ12x3=/0G23(r1,r3)x3=/0(21)

And:

f1x1f2x1f1x2f2x2=f1x1f2x2f2x1f1x2=/0f2x1f2x2+f1x1f1x2=φ22x1φ12x1=/0G23(r1,r3)x1=/0

By continuity, there is a neighborhood of (r1,r3), that we take as B(r1,r3)21B(r1,r3)22B(r1,r3)23 , in which all the derivatives are not equal to zero. □

Proposition 3.7

There is a neighborhood of (r1,r3), that, for the sake of clarity and without loss of generality, we can take as B(r1,r3)21B(r1,r3)22B(r1,r3)23, where the partial derivatives of the functions:

H21(x1,x3)=φ32(x1,x3)+φ12(x1,x3)H22(x1,x3)=φ32(x1,x3)+φ22(x1,x3)H23(x1,x3)=φ22(x1,x3)+φ12(x1,x3)

are different to zero.

Proof

From (17), the following determinant, valued at the root (r1,r2,r3)=(r1,φ32(r1,r3),r3)=(r1,φ12(r1,r3),r3) is different to zero. Hence:

f1x2f3x2f1x3f3x3=f1x2f3x3+f1x3f3x20f3x3f3x2f1x3f1x2=φ32x3+φ12x30H21(r1,r3)x30(22)

The rest of the proof uses identical techniques to Proposition 3.6. □

Definition 3.8

We define the functions λ1, λ2 and λ3 as follows:

λ1:Cr11R3λ1(x1)=(x1,φ32(x1,P1(x1)),P1(x1))(23)

in such a way that:

f1(x1,φ32(x1,P1(x1)),P1(x1))=0f3(x1,φ32(x1,P1(x1)),P1(x1))=0(24)

with:

P1:Cr11X1Dr31X3x3=P1(x1), so that G21(x1,P1(x1))=0

Cr11 and Dr31 being neighborhoods of r1 and r3, respectively.

Function λ1 is well defined, since G21(r1,r3)=0 and from (22), we can apply Implicit Function Theorem to the function G21.

With the same foundation, from (20) and (21), we introduce:

λ2:Cr12R3λ2(x1)=(x1,φ32(x1,P2(x1)),P2(x1))

in such a way that:

f2(x1,φ32(x1,P2(x1)),P2(x1))=0f3(x1,φ32(x1,P2(x1)),P2(x1))=0(25)

with:

P2:Cr12X1Dr32X3x3=P2(x1), so that G22(x1,P2(x1))=0

where Cr12 and Dr32 are neighborhoods of r1 and r3, respectively.

And:

λ3:Cr13R3λ3(x1)=(x1,φ12(x1,P3(x1)),P3(x1))

in such a way that:

f1(R1(x2),x2,φ33(R1(x2),x2))=0f3(R1(x2),x2,φ33(R1(x2),x2))=0(26)

with:

P3:Cr13X1Dr33X3x3=P3(x1), so that G23(x1,P3(x1))=0

where Cr13 and Dr33 are neighborhoods of r1 and r3, respectively.

Remark

Observe that functions λ1, λ2 and λ3 are parameterizations of the curves C1, C2and C3 with respect to the parameter x1.

Definition 3.9

We define the functions G31 , G32 and G33 in the open set B(r1,r3)21B(r1,r3)22B(r1,r3)23 as follows:

G31(x1,x2)=φ33(x1,x2)φ13(x1,x2)G32(x1,x2)=φ33(x1,x2)φ23(x1,x2)G33(x1,x2)=φ23(x1,x2)φ13(x1,x2)

Proposition 3.10

There is a neighborhood of (r1,r2), that, for the sake of clarity and without loss of generality, we can take as B(r1,r3)21B(r1,r3)22B(r1,r3)23, where the partial derivatives of the functions G33(x1,x2) are different to zero.

proof

It follows from (16), following the same reasoning as in Proposition 3.6. □

Proposition 3.11

There is a neighborhood of (r1,r2), that, for the sake of clarity and without loss of generality, we can take as B(r1,r3)21B(r1,r3)22B(r1,r3)23, where the partial derivatives of the functions:

H31(x1,x2)=φ33(x1,x2)+φ13(x1,x2)H32(x1,x2)=φ33(x1,x2)+φ23(x1,x2)H33(x1,x2)=φ23(x1,x2)+φ13(x1,x2)

are different to zero.

proof

The same as Proposition 3.7. □

Definition 3.12

With the same criteria as in Definition \refdef:lambdafunctions, let us introduce the functions γ1, γ2 and γ3 in the following manner:

γ1:Cr21R3γ1(x2)=(R1(x2),x2,φ33(R1(x2),x2))

in such a way that:

f1(R1(x2),x2,φ33(R1(x2),x2))=0f3(R1(x2),x2,φ33(R1(x2),x2))=0

with:

R1:Cr21X2Dr11X1x1=R1(x2), so that G31(R1(x2),x2)=0

where Cr21 and Dr11 are neighborhoods of r2 and r1, respectively.

γ2:Cr22R3γ2(x2)=(R2(x2),x2,φ33(R2(x2),x2))

in such a way that:

f2(R2(x2),x2,φ33(R2(x2),x2))=0f3(R2(x2),x2,φ33(R2(x2),x2))=0(27)

with:

R2:Cr22X2Dr12X1x1=R2(x2), so that G32(R2(x2),x2)=0

where Cr22 and Dr12 are neighborhoods of r2 and r1, respectively.

γ3:Cr23R3γ3(x2)=(R3(x2),x2,φ23(R3(x2),x2))

in such a way that:

f1(R3(x2),x2,φ23(R3(x2),x2))=0f2(R3(x2),x2,φ23(R3(x2),x2))=0

with:

R3:Cr23X2Dr13X1x1=R3(x2), with G33(R3(x2),x2)=0

where Cr23 and Dr13 are neighborhoods of r2 and r1, respectively.

Remark

Observe that functions γ1, γ2 and γ3 are parameterizations of the curves C1, C2and C3 with respect to the parameter x2.

Definition 3.13

We define the functions G11 , G12 and G13 in B(r2,r3)11B(r2,r3)12B(r2,r3)13 as follows:

G11(x2,x3)=φ31(x2,x3)φ11(x2,x3)G12(x2,x3)=φ21(x2,x3)φ31(x2,x3)G13(x2,x3)=φ11(x2,x3)φ21(x2,x3)

Proposition 3.14

There is a neighborhood of (r2,r3), that, for the sake of clarity and without loss of generality, we can take as B(r2,r3)11B(r2,r3)12B(r2,r3)13, where the partial derivatives of the functions G13(x2,x3) are different to zero.

proof

It follows from (16), following the same reasoning as in Proposition 3.6. □

Proposition 3.15

There is a neighborhood of (r2,r3), that, for the sake of clarity and without loss of generality, we can take as B(r2,r3)11B(r2,r3)12B(r2,r3)13, where the partial derivatives of the functions:

H11(x2,x3)=φ31(x2,x3)+φ11(x2,x3)H12(x2,x3)=φ31(x2,x3)+φ21(x2,x3)H13(x2,x3)=φ21(x2,x3)+φ11(x2,x3)

are different to zero.

proof

The same as Proposition 3.7. □

Definition 3.16

With the same criteria as in Definition \refdef:lambdafunctions, let us introduce, next, the functions β1, β2 and β2:

β1:Cr31R3β1(x3)=(φ31(Q1(x3),x3),Q1(x3),x3)

in such a way that:

f1(φ31(Q1(x3),x3),Q1(x3),x3)=0f3(φ31(Q1(x3),x3),Q1(x3),x3)=0(28)

with:

Q1:Cr31X3Dr21X2x2=Q1(x3), so that G11(Q1(x3),x3)=0

where Cr31 and Dr21 are neighborhoods of r3 and r2, respectively.

β2:Cr32R3β2(x3)=(φ31(Q2(x3),x3),Q2(x3),x3)

in such a way that:

f2(φ31(Q2(x3),x3),Q2(x3),x3)f3(φ31(Q2(x3),x3),Q2(x3),x3)(2)

with:

Q2:Cr32X3Dr22X2x2=Q2(x3), so that G12(Q2(x3),x3)=0

where Cr32 and Dr22 are neighborhoods of r3 and r2, respectively.

β3:Cr33R3β3(x3)=(φ11(Q3(x3),x3),Q3(x3),x3)

in such a way that:

f1(φ11(Q3(x3),x3),Q3(x3),x3)=0f2(φ11(Q3(x3),x3),Q3(x3),x3)=0(30)

with:

Q3:Cr33X3Dr23X2x2=Q3(x3), with G13(Q3(x3),x3)=0

where Cr33 and Dr23 are neighborhoods of r3 and r2, respectively.

Remark

Observe that the functions β1, β2 and β3 are parameterizations of the curves C1, C2 and C3 with respect to the parameter x3.

Theorem 3.17

The determinants:

G2i(x1,x3)x1G2i(x1,x3)x3G2j(x1,x3)x1G2j(x1,x3)x3;1i<j3;G3i(x1,x2)x1G3i(x1,x2)x2G3j(x1,x2)x1G3j(x1,x2)x2;1i<j3;G1i(x2,x3)x2G1i(x2,x3)x3G1j(x2,x3)x2G1j(x2,x3)x3;1i<j3;(31)

are different to zero in a neighborhood of r¯, Lr, that we can take as LrLrLr3.

proof

We show only the case of G21 and G22 . Suppose the determinant is zero in (r1,r3), what means that:

φ32x1φ12x1φ32x3φ12x3φ32x1φ22x1φ32x3φ22x3=0

in the point (r1,r3). This expression, written according to the functions of system (4) leads to:

f3x1f3x2+f1x1f1x2f3x3f3x2+f1x3f1x2f3x1f3x2+f2x1f2x2f3x3f3x2+f2x3f2x2=0

In the point (r1,φ32(r1,r3),r3)=(r1,r2,r3) in the case of f3, (r1,φ12(r1,r3),r3)=(r1,r2,r3) in the case of f1 and (r1,φ12(r1,r3),r3)=(r1,r2,r3) in the case of f2, the determinant we arrive at:

f1x1f3x2f1x2f3x1f1x3f3x2f1x2f3x3f2x1f3x2f2x2f3x1f2x3f3x2f2x2f3x3=0

If we write the columns of the determinant in matrix form, then there exists a λ0=/0, in such a manner that:

f1x1f2x1f3x2f1x2f2x2f3x1=f1x3f2x3λ0f3x2f1x2f2x2λ0f3x3

What is equivalent to:

f1x2f2x2f3x1+λ0f3x3+f1x1f2x1f3x2f1x3f2x3λ0f3x2=00(32)

From (16) and Rouché-Fröbenius Theorem, there is only an α and a β, so that:

f1x2f2x2αf1x1f2x1βf1x3f2x3=00(33)

Taking into account \eqrefeq:jacobian1:

αf3x1+f3x3λ0=f3x2βf3x1+f3x3λ0=λ0f3x2(34)

Therefore:

λ0=βα

Substituting λ0 in the first equation of (34), we obtain:

f3x2=αf3x1+βf3x3

And in concordance with \eqrefeq:jacobian3, we get:

f1x2f2x2f3x2=αf1x1f2x1f3x1+βf1x3f2x3f3x3

So, the jacobian of system (4) is equal to zero at the point (r1,r2,r3), in contradiction with (15). Therefore by continuity, there is a neighborhood of r¯, that we name as Lr=LrLrLr3, where the determinant is different to zero. □

4 Construction of the projection functions and their properties

In this section we introduce the projections functions of the curves C1, C2 and C3, that will play a very important role in the construction of the fixed point functions. About them we will talk in the following section.

Definition 4.1

Let us introduce the neighborhood of r1, r2 and r3, Cr1, Cr2 and Cr3 in this way:

1. Cr1=Cr11Cr12Cr13 (see Definition 3.8)

2. Cr2=Cr21Cr22Cr23 , (see Definition 3.12)

3. Cr3=Cr31Cr32Cr33 , (see Definition 3.16)

respectively. Then:

C2X1X3 are said to be the projections of the curves C1, C2 and C3 on CrX1.

C2X1X2 are said to be the projections of the curves C1, C2 and C3 on CrX2.

C3X2X3 are said to be the projections of the curves C1, C2 and C3 on CrX3.

Definition 4.2

We define the projection functions F1X1X2 , F2X1X2 and F3X1X2 as follows:

F1X1X2,F2X1X2,F3X1X2:Cr1X2F1X1X2(x1)=φ32(x1,P1(x1))F2X1X2(x1)=φ32(x1,P2(x1))F3X1X2(x1)=φ12(x1,P3(x1))

where φ32 and φ12 were introduced in (18), and P1, P2 and P3, in Definition 3.8 (see Figure 2).

Projection functions in the plane X1X2 of Example 1
Figure 2

Projection functions in the plane X1X2 of Example 1

Proposition 4.3

The following assertions hold:

  1. C1X1X2={(x1,x2);x1Cr1,x2=F1X1X2(x1)}

  2. C2X1X2={(x1,x2);x1Cr1;x2=F2X1X2(x1)}

  3. C3X1X2={(x1,x2);x1Cr1;x2=F3X1X2(x1)}

proof

It is a consequence of (23), (25) and (26), since the projection π3 from ℝ3 to ℝ2, given by π3(x1,x2,x3)=(x1,x2), applied to the curves C1, C2 and C3, respectively, are:

  1. π3(x1,φ32(x1,P1(x1)),P1(x1))=(x1,φ32(x1,P1(x1))

  2. π3(x1,φ32(x1,P2(x1)),P2(x1))=(x1,φ32(x1,P2(x1))

  3. π3(x1,φ12(x1,P3(x1)),P3(x1))=(x1,φ12(x1,P3(x1))

□In an analogous manner, we next define the projection functions in the planes X2X3 and X1X3, respectively.

Definition 4.4

We define the projection functions F1X2X3 , F2X2X3 and F3X2X3 as follows:

F1X2X3,F2X2X3,F3X2X3:Cr2X3F1X2X3(x2)=φ33(R1(x2),x2)F2X2X3(x2)=φ33(R2(x2),x2)F3X2X3(x2)=φ23(R3(x2),x2)

where φ33 and φ32 were introduced in (18), and R1, R2 and R3, in Definition 3.12.

Proposition 4.5

The following assertions hold:

  1. C1X2X3={(x2,x3);x2Cr2,x3=F1X2X3(x2)}

  2. C2X2X3={(x2,x3);x2Cr2;x3=F2X2X3(x2)}

  3. C3X2X3={(x2,x3);x2Cr2;x3=F3X2X3(x2)}

Definition 4.6

We define the projection functions F1X1X3 , F2X1X3 and F3X1X3 as follows:

F1X1X3,F2X1X3,F3X1X3:Cr3X1F1X1X3(x3)=φ31(Q1(x3),x3)F2X1X3(x3)=φ31(Q2(x3),x3)F3X1X3(x3)=φ11(Q3(x3),x3)

where φ11 and φ31 were introduced in (18), and Q1, Q2 and Q3, in Definition 3.16.

Proposition 4.7

The following assertions hold:

  1. C1X1X3={(x1,x3);x3Cr3,x1=F1X1X3(x3)}

  2. C2X1X3={(x1,x3);x3Cr3;x1=F2X1X3(x3)}

  3. C3X1X3={(x1,x3);x3Cr3;x1=F3X1X3(x3)}

Definition 4.8

System (4) is said to satisfy the second condition of derivation if:

  1. (FiX1X2)(x1)+(FjX1X2)(x1)=0

  2. 1i<j3

  3. (FiX1X3)(x3)+(FjX1X3)(x3)=0

for x1Pr1, x2Rr2, x3Qr3, Pr1, Rr2, Qr3 being neighborhoods of r1, r2and r3, respectively.

From now, we shall assume that system (4) satisfies the second condition of derivation.

Proposition 4.9 provides a necessary and sufficient condition for second condition of derivation to be fulfilled, focusing, for the sake of clarity, on the case of the functions F1X1X2 and F2X1X2 . The reasoning for the remaining ones is identical.

Proposition 4.9

There exists a neighborhood of r1, Pr112 , where:

(F1X1X2)(x1)+(F2X1X2)(x1)=0,

if and only if there exists a neighborhood of r1, Pr112, where:

m(2,2)(x1,x2,x3)m(1,1)(x1,x2,x3)+m(2,1)(x1,x2,x3)m(1,2)(x1,x2,x3)=0

m(2,2), m(1,1), m(2,1) and m(1,2) were introduced in (16), with x2=F1X1X2(x1) , x3=P1(x1) and with x2=x2=F2X1X2(x1) , x3=P2(x1).

proof

In agreement with (24) we have that:

f1x1+f1x2(F1X1X2)(x1)+f1x3P1(x1)=0f3x1+f3x2(F1X1X2)(x1)+f3x3P1(x1)=0

So, by Cramer’s rule:

(F1X1X2)(x1)=m(2,2)(x1,x2,x3)m(1,2)(x1,x2,x3)

with x2=F1X1X2(x1) , x3=P1(x1). Therefore:

(F1X1X2)(r1)=m(2,2)(r1,r2,r3)m(1,2)(r1,r2,r3)

In the same way, from (25):

f2x1+f2x2(F2X1X2)(x1)+f2x3P2(x1)=0f3x1+f3x2(F2X1X2)(x1)+f3x3P2(x1)=0

Hence:

(F2X1X2)(x1)=m(2,1)(x1,x2,x3)m(1,1)(x1,x2,x3)

with x2=F1X1X2(x1), x3=P2(x1). Accordingly:

(F2X1X2)(r1)=m(2,1)(r1,r2,r3)m(1,1)(r1,r2,r3)

What leads to:

(F1X1X2)(r1)+(F2X1X2)(r1)=m(2,2)(r1,r2,r3)m(1,2)(r1,r2,r3)+m(2,1)(r1,r2,r3)m(1,1)(r1,r2,r3)

that is to say:

m(2,2)(r1,r2,r3)m(1,1)(r1,r2,r3)+m(2,1)(r1,r2,r3)m(1,2)(r1,r2,r3)=0Pr112 with m(2,2)(x1,x2,x3)m(1,1)(x1,x2,x3)+m(2,1)(x1,x2,x3)m(1,2)(x1,x2,x3)=0(F1X1X2)(x1)+(F2X1X2)(x1)=0

Remark

Observe that by applying Proposition 4.9 to the functions F1X1X2 and F2X1X2 we have obtained Pr112. In the same way, if we apply Proposition 4.9 to the functions F2X1X2 and F3X1X2, we get Pr123 ; and to the functionsF1X1X2 and F3X1X2, P13r1. Then the neighborhood Pr1, introduced in Definition 4.8, is given by:

Pr1=Pr112Pr123Pr113

and, similarly:

  1. Rr2=R12r2∩R13r2∩R23r2

  2. Qr3=Q12r3∩Q13r3∩Q23r3

The following propositions are also true, on the one side, for the projections functions on the plane X2X3, and on the other side, for the projections functions on the plane X1X3.

Proposition 4.10

There is a neighborhood of r1, Lr1introduced in Theorem 3.17, where (FiX1X2)(x1)(FjX1X2)(x1)=0 0, 1≤i<j≤3.

proof

We only prove the case forF1X1X2 and F2X1X2.

(F1X1X2)(x1)=φ32(x1,x3)x1+φ32(x1,x3)x3P1(x1)

with x3=P1(x1).

(F2X1X2)(x1)=φ32(x1,x3)x1+φ32(x1,x3)x3P2(x1)

with x3=P2(x1). Therefore:

(F1X1X2)(r1)(F2X1X2)(r1)=φ32(r1,r3)x3(P1(r1)P2(r1))

taking in consideration that P1(r1)=P2(r1)=r3. On the other hand:

P1(x1)=G21(x1,P1(x1))x1G21(x1,P1(x1))x3

and:

P2(x1)=G22(x1,P2(x1))x1G22(x1,P2(x1))x3

Therefore:

P1(r1)P2(r1)=G21(r1,P(r1))x1G21(r1,P1(r1))x3G22(r1,P2(r1))x1G22(r1,P2(r1))x3=0

in agreement with (31). Hence, in consonance with Proposition 3.3,

(F1X1X2)(x1)(F2X1X2)(x1)=0x1Lr1

Remark

Proposition 4.10 can also be applied to the pairs of functions FjX2X3 , 1≤i<j≤3, if we change Lr1 by Lr2; and, finally, Proposition 4.10 is also valid for the functions FiX1X3 , 1≤i<j≤3, if we change Lr1 by Lr3.

Definition 4.11

We define the compact set Wr¯ as follows:

Wr¯=Wr1×Wr2×Wr3

with Wr1=[ax,bx], Wr2=[cy,dy] and Wr3=[ez,fz], so that:

  1. [ax,bx]⊂Cr1Pr1∩Lr1

  2. [cy,dy]⊂Cr2Qr2∩Lr2

  3. [ez,fz]⊂Cr3Rr3∩Lr3

  4. [ax,bx]×[cy,dy]B(r1,r2)31B(r1,r2)32B(r1,r2)33

  5. [ax,bx]×[ez,fz]B(r1,r3)21B(r1,r3)22B(r1,r3)23

  6. [cy,dy]×[ez,fz]B(r2,r3)11B(r2,r3)12B(r2,r3)13

with r1∈(ax,bx), r2∈(cy,dy) and r3∈(ez,fz), in such a way that all the functions, so far introduced, are well defined in [ax,bx] or [cy,dy] or [ez,fz] or [ax,bx]×[cy,dy] or [ax,bx]×[ez,fz] or [cy,dy]×[ez,fz], respectively.

Now, we introduce some more results about the projections functions, defined in the plane X1X2,F1X1X2, F2X1X2 and F3X1X2, which can be generalized smoothly to the projection functions on the plane X2X3, F1X2X3 , F2X2X3 and F3X2X3 ; and to the projection functions on the plane X1X3, F1X1X3 , F2X1X3 and F3X1X3 .

Proposition 4.12

The following equalities fulfil:

r2=F1X1X2(r1)=F2X1X2(r1)=F3X1X2(r1)

proof

From the definitions of the functions φ32 , φ12 , P2 and P3, we have that:

1. r2=φ32(r1,P1(r1))

2. r2=φ32(r1,P2(r1))

3. r2=φ12(r1,P3(r1))

Proposition 4.13

(F1X1X2)(x1)=0 (x1)=/0, (F2X1X2)(x1)=/0, (F3X1X2)(x1)=/0 for all x1Wr1.

proof

We know that:

F1X1X2(x1)=φ32(x1,P1(x1))=φ12(x1,P1(x1))(35)

Suppose there is a point a0Wr1 so that (F1X1X2)(x1)=0(a0)=0. Then for x1=a0 and x3=P1(a0), we can say that:

dφ32(x1,P1(x1))dx1=φ32(x1,x3)x1+φ32(x1,x3)x3P1(x1)=0

So:

P1(x1)=φ32(x1,x3)x1φ32(x1,x3)x3;x3=P1(x1)(36)

At the point x1=a0, in the same way:

P1(x1)=φ12(x1,x3)x1φ12(x1,x3)x3;x3=P1(x1)(37)

From (36) and (37) we can conclude the proportionality of the quotients of the derivatives of φ32 and φ12 . Hence:

φ32x3φ12x3φ32x1φ12x1=0(38)

Expressing the derivatives of φ32 and φ12 at the point x1=a; x3=P1(a) from f1 and f3, at the point (a0,φ12(a0,P1(a0)),P1(a0)) with respect to f1 and (a0,φ32(a0,P1(a0)),P1(a0))with respect to f3 of the system (4), with(a0,φ12(a0,P1(a0)),P1(a0))=(a0,φ32(a0,P1(a0)),P1(a0)) in agrement with (35), (38) can be written as:

f3x3f3x2f1x3f1x2f3x1f3x2f1x1f1x2=f3x3f1x3f3x1f1x11f3x21f1x2=0

Therefore there exist a point in G, where the minor m(2,2)=0 in contradiction with (16).

The fact that (F2X1X2)(x1)=/0 and (F3X1X2)(x1)=/0 can be proven in the same way.

Proposition 4.14

The following inequalities are true ∀x1Wr1:

|(Fi1X1X2)(x1)|<|(Fi2X1X2)(x1)|<|(Fi3X1X2)(x1)|(39)

for some permutation: i1, i2 and i3 of the set {1,2,3}.

proof

The pairs of functionsF1X1X2 and F2X1X2, F2X1X2 and F3X1X2,F1X1X2 and F3X1X2, defined in [ax,bx], from Propositions 4.10, 4.12 and 4.13, satisfy Theorem 2.1 or 2.2, or 2.3 or 2.4, according to their features of increase or decrease. So, we can derive that:

|(Fi1X1X2)(x1)|<|(Fi2X1X2)(x1)| or |(Fi2X1X2)(x1)|<|(Fi1X1X2)(x1)|;|(Fi1X1X2)(x1)|<|(Fi3X1X2)(x1)| or |(Fi3X1X2)(x1)|<|(Fi1X1X2)(x1)|;|(Fi2X1X2)(x1)|<|(Fi3X1X2)(x1)| or |(Fi3X1X2)(x1)|<|(Fi2X1X2)(x1)|(40)

If (39) is false, then:

  1. Or the two inequalities are false.

  2. Or only the first one is false.

  3. Or only the second one is false.

In the first case, from (39)a is true:

|(Fi1X1X2)(x1)|>|(Fi2X1X2)(x1)|>|(Fi3X1X2)(x1)|

In the second case, from (39)a is true:

|(Fi1X1X2)(x1)|>|(Fi2X1X2)(x1)|

and then, if |(Fi1X1X2)(x1)|<|(Fi3X1X2)(x1)|, , we have that:

|(Fi2X1X2)(x1)|<|(Fi1X1X2)(x1)|<|(Fi3X1X2)(x1)|

On the contrary:

|(Fi1X1X2)(x1)|>|(Fi2X1X2)(x1)||(Fi1X1X2)(x1)|<|(Fi3X1X2)(x1)|

and then:

|(Fi3X1X2)(x1)|>|(Fi1X1X2)(x1)|>|(Fi2X1X2)(x1)|

The proof of the third case is similar to the second one. □

Proposition 4.15

Let (a1,a2,a3) and (b1,b2,b3) be two arbitrary points in C1Wr¯ or two arbitrary points in C2Wr¯ or two arbitrary points in C3Wr¯. Let us introduce the subscripts i=1, j=2, k=3; or, i=2, j=1, k=3; or, i=3, j=1, k=2. Then for m=1,2,3, we have that:

ai<bi;(FmXiXj)<0;(FmXiXk)<0aj>bj;ak>bk

where FmX1X2 is considered equal to FmX1X3 , equal to FmX3X1 ; and FmX2X3 , equal to FmX3X2 .

proof

We show the result only for m=1, since for m=2 and m=3 the reasoning is the same.

Assume, firstly, the subscripts values i=1, j=2 and k=3. Let (a1,a2,a3),(b1,b2,b3)∈C1Wr¯. Their projections on the plane X1X2 are (a1,a2) and (b1,b2), respectively, with a2=FX1X21(a1) and b2=FX1X21(b1). Their projections on the plane X1X3 are (a1,a3) and (b1,b3), respectively, with a1=FX1X31(a3) and b1=FX1X31(b3).

If (F1X1X2)(x1)>0 and a1<b1a2<b2.If (F1X1X3)(x3)>0 and a1<b1a3<b3.

So, a1<b1a2<b2 and a3<b3.

If (F1X1X2)(x1)>0 and a1<b1a2<b2.If (F1X1X3)(x3)<0 and a1<b1a3>b3.

So, a1<b1a2<b2 and a3>b3.

If (F1X1X2)(x1)<0 and a1<b1a2>b2.If (F1X1X3)(x3)>0 and a1<b1a3<b3.

So, a1<b1a2>b2 and a3<b3.

If (F1X1X2)(x1)<0 and a1<b1a2>b2.If (F1X1X3)(x3)<0 and a1<b1a3>b3.

So, a1<b1a2>b2 and a3>b3.

Assume, secondly, the subscripts values i=2, j=1 and k=3. Let (a1,a2,a3),(b1,b2,b3)∈C1Wr¯. Their projections on the plane X1X2 are (a1,a2) and (b1,b2), respectively, with a2=FX1X21(a1) and b2=FX1X21(b1). Their projections on the plane X2X3 are (a2,a3) and (b2,b3), respectively, with a3=FX2X31(a2) and b3=FX2X31(b2).

If (F1X1X2)(x1)>0 and a2<b2a1<b1.If (F1X2X3)(x2)>0 and a2<b2a3<b3.

So, a2<b2a1<b1 and a3<b3.

If (F1X1X2)(x1)>0 and a2<b2a1<b1.If (F1X2X3)(x2)<0 and a2<b2a3>b3.

So, a2<b2a1<b1 and a3>b3.

If (F1X1X2)(x1)<0 and a2<b2a1>b1.If (F1X2X3)(x2)>0 and a2<b2a3<b3.

So, a2<b2a1>b1 and a3<b3.

If (F1X1X2)(x1)<0 and a2<b2a1>b1.If (F1X2X3)(x2)<0 and a2<b2a3>b3.

So, a2<b2a1>b1 and a3>b3.

Assume, thirdly, the subscripts values i=3, j=1 and k=2. Let (a1,a2,a3),(b1,b2,b3)∈C1Wr¯. Their projections on the plane X1X3 are (a1,a3) and (b1,b3), respectively, with a1=FX1X31(a3) and b1=FX1X31(b3). Their projections on the plane X2X3 are (a2,a3) and (b2,b3), respectively, with a3=FX2X31(a2) and b3=FX2X31(b2).

If (F1X1X3)(x3)>0 and a3<b3a1<b1.If (F1X2X3)(x2)>0 and a3<b3a2<b2.

So, a3<b3a1<b1 and a2<b2.

If (F1X1X3)(x3)>0 and a3<b3a1<b1.If (F1X2X3)(x2)<0 and a3<b3a2>b2.

So, a3<b3a1<b1 and a2>b2.

If (F1X1X3)(x3)<0 and a3<b3a1>b1.If (F1X2X3)(x2)>0 and a3<b3a2<b2.

So, a3<b3a1>b1 and a2<b2.

If (F1X1X3)(x3)<0 and a3<b3a1>b1.If (F1X2X3)(x2)<0 and a3<b3a2>b2.

So, a3<b3a1>b1 and a2>b2. □

Corollary 4.16

Let (a1,a2,a3) and (b1,b2,b3) be two arbitrary points in C1Wr¯ or two arbitrary points in C2Wr¯ or two arbitrary points in C3Wr¯. Let us introduce the subscripts i=1, j=2, k=3; or, i=2, j=1, k=3; or, i=3, j=1, k=2. Then for m=1,2,3, we have that:

ai=biaj=bj;ak=bk

Definition 4.17

System (4) is said to satisfy the third condition of derivation if there exists a sequence:

Fj1X1X2,Fj2X1X3,Fj3X2X3(41)

where j1, j2 and j3 are a permutation of the set {1,2,3} where |(FX1X2j1)(x1)| is the greater absolute value in agreement with inequalities (39); |(FX1X3j2)(x3)| is the greater absolute value in agreement with inequalities (39), applied to the projection functions on the plane X1X3 and, |(FX2X3j3)(x2)| is the greater absolute value related to the projection functions on the plane X2X3.

From now, we are supposing that system (4) fulfils the third condition of derivation.

Remark

The proofs of the results below, whose differences consist of the distinct properties of increase or decrease of the projection functions, are very similar, so it is not worth to repeat them. Hence, from now on, we suppose, without loss of generality, the following:

1. The functionsF1X1X2, F2X1X2 and F3X1X2 are decreasing in Wr1.

2. The functions F1X1X3, F2X1X3 and F3X1X3 are increasing in Wr3.

3. The functions F1X2X3, F2X2X3 and F3X2X3 are decreasing in Wr2.

4. The following inequalities:

|(F1X1X2)(x1)|>|(F2X1X2)(x1)|>|(F3X1X2)(x1)||(F3X1X3)(x3)|>|(F2X1X3)(x3)|>|(F1X1X3)(x3)||(F2X2X3)(x2)|>|(F1X2X3)(x2)|>|(F3X2X3)(x2)|(42)

hold.

5. Therefore, sequence \eqrefeq:sequenceprojections is:

F1X1X2;F3X1X3;F2X2X3(43)

Proposition 4.18

Let (a1,a2,x30)C1Wr¯C1Wr¯ , (b1,b2,x30)C2Wr¯C2Wr¯ and (c1,c2,x30)C3Wr¯C3Wr¯, with x30<r3 <r3, then r2<b2<a2<c2.

proof

The projections of the points (a1,a2,x30<r3), (b1,b2,x30<r3) and (c1,c2,x30<r3) in the plane X2X3 are (a2,x30<r3), (b2,x30<r3) and (c2,x30<r3), respectively, with a2=(F1X2X3)−1(x30<r3), b2=(F2X2X3)−1(x30<r3) and c2=(F3X2X3)−1(x30<r3).

As x30<r3<r3, b2=(F2X2X3)−1(x30<r3)>(F2X2X3)−1(r3)=r2. On the other hand, the functions F1X2X3, F2X2X3 and F3X2X3 verify (42), therefore:

r2b2(F2X2X3)(x2)dx2>r2b2(F1X2X3)(x2)dx2b2r2(F2X2X3)(x2)dx2>b2r2(F1X2X3)(x2)dx2F2X2X3(r2)F2X2X3(b2)>F1X2X3(r2)F1X2X3(b2)F2X2X3(b2)<F1X2X3(b2)

What leads to:

x30<F1X2X3(b2)(F1X2X3)1(x30)>(F1X2X3)1F1X2X3(b2)a2>b2

The fact that a2<c2 can be proven in the same way. Therefore, we conclude that r2<b2<a2<c2. □The proof of the Propositions 4.19 and 4.20 are performed in the same way as in Proposition 4.18.

Proposition 4.19

Let (a1,x20>r2 ,a3)∈C1Wr¯, (b1,x20>r2,b3)∈C2Wr¯ and (c1,x20>r2,c3)∈C3Wr¯, with x20>r2>r2, then c1<b1<a1<r1.

Proposition 4.20

Let (x10>r1 ,a2,a3)∈C1Wr¯, (x10>r1,b2,b3)∈C2Wr¯ and (x10>r1,c2,c3)∈C3Wr¯, with x10>r1>r1, then a3<b3<c3<r3.

5 The fixed point iteration functions

Let us introduce in this section the fixed point iteration functions. They are built from functions (18). Among all of them we choose two to be used in the plane X1X2, two to be used in the plane X1X3 and two to be used in the plane X2X3. The selection criterion has to do with sequence \eqrefeq:projectionsequence, in agreement with the following Definition.

Definition 5.1

Given sequence (43):

F1X1X2;F3X1X3;F2X2X3

we select, for the plane X1X2, the curve related to the third projection function of the sequence, in this case, C2, that is the intersection between the functions f2 and f3 of the system (4), in agreement with Definition 3.4. From f2 and f3 we choose φ22 (x1,φ13) and φ32(x1,φ13), in other words, the ones that solve the unknown x2.

For the plane X1φ13, we take the curve related to the first projection function of the sequence, in this case, C1, that is the intersection between the functions f1 and f3 of the system (4), in agreement with Definition 3.4. From f1 and f3 we choose φ11(x2,φ13) and φ13(x2,φ13), in other words, the ones that solve the unknown x1.

Finally for the plane X2φ13, we are left with the curve related to the second projection function of the sequence, in this case, C3, that is the intersection between the functions f1and f2 of the system, in agreement with Definition 3.4. From f1 and f2 we choose φ31(x1,x2) and φ32(x1,x2), in other words, the ones that solve the unknown φ13.

This process that given any sequence (43), provides a selection of functions (18) is said to be the selection criterion of the iteration functions.

Definition 5.2

In consonance with Definition 5.1, we define the fixed point iteration functions related to the plane X1X2 as follows (see Figure 3):

For each fixed x30<r3Wr3, we define the functions:

f22x30,f23x30:Wr1X2x2=f22x30(x1)=φ22(x1,x30)x2=f23x30(x1)=φ32(x1,x30)(44)

We define the fixed point iteration functions related to the plane X2φ13 as follows:

001
Figure 3

001

For each fixed x10>r1Wr1:

f32x10,f31x10:Wr2X3x3=f32x10(x2)=φ23(x10,x2)x3=f31x10(x2)=φ13(x10,x2)(45)

We define the fixed point iteration functions related to the plane X1φ13 as follows:

For each fixed x20>r2Wr2:

f11x20,f13x20:Wr3X1x1=f11x20(x3)=φ11(x20,x3)x1=f13x20(x3)=φ31(x20,x3)(46)

Proposition 5.3

Given an arbitrary point, (x10>r1,x20>r2,x30<r3)∈C2, with x10>r1Wr1, consider the set:

Sx30={(x1,x2,x30);x1Wr1;x2R}

Then, in Sx30, the functions F2X1X2, f23x30 and f22x30 satisfy that:

x20=f23x30(x10)=f22x30(x10)x20=F2X1X2(x10)(47)

in the other words: the three pass through the same point.

proof

The point (x10>r1,x20>r2,x30<r3)∈C2, with x10>r1Wr1 therefore:

f2(x10,x20,x30)=0f2(x10,φ22(x10,x30),x30)=0x20=f22x30(x10)

And:

f3(x10,x20,x30)=0f3(x10,φ32(x10,x30),x30)=0x20=f23x30(x10)

On the other hand the projection of the point (x10>r1,x20>r2,x30<r3)∈C2 on the plane X1X2 is(x10>r1,x20>r2), so x20>r2=F2X1X2(x10>r1) (see Figure 3).

□The two following propositions are proved in the same manner.

Proposition 5.4

Given an arbitrary point, ((x11,x21,x31)C1)∈C1, with x31Wr3x31Wr3, consider the set:

Sx21={(x1,x21,x3);x3Wr3;x1R}

Then, in Sx21, the functions F1X1X3, f13x21 and f11x21, satisfy that:

x11=f13x21(x31)=f11x21(x31)x11=F1X1X3(x31)(48)

Proposition 5.5

Given an arbitrary point ((x12,x22,x32)C3)∈C3, with x22Wr2x22Wr2, consider the set:

Sx12={(x12,x2,x3);x2Wr2;x3R}

Then, in Sx12, the functions F3X2X3, f31x12 and f32x12, satisfy that:

x32=f31x12(x22)=f32x12(x22)x32=F3X2X3(x22)(49)

Theorem 5.6

The functions (44), (45) and (46), satisfy the hypothesis of:

  1. Theorem 2.1

  2. or Theorem 2.2

  3. or Theorem 2.3

  4. or Theorem 2.4

in agrement with their features of increase or decrease.

proof

We show the theorem only for the functions f22x30(x1) and f23x30(x1).

  1. From functions (18) they are real functions well defined in [ax,bx] with x30<r3∈[ez,fz].

  2. From Proposition 3.3 (f22x30)(x1)=0 and (f23x30)(x1)=0x1[ax,bx].

  3. From (47) they have a root in [ax,bx].

  4. From Propositions 3.6 and 3.7

(f22x30)(x1)±(f23x30)(x1)=0,x1[ax,bx]

Proposition 5.7

The inverse functions of:

  1. x2=fx0322(x1) is x1=fx0312(x2)

  2. x2=fx0323(x1) is x1=fx0313(x2)

  3. φ13=fx0132(x2) is x2=fx0122(φ13)

  4. φ13=fx0131(x2) is x2=fx0121(φ13)

  5. x1=fx0211(φ13) is φ13=fx0231(x1)

  6. x1=fx0213(φ13) is φ13=fx0233(x1)

proof

It is a direct consequence of their definition. □

Remark

The proofs of the results below, whose differences consist of the distinct properties of increase and decrease of the fixed point functions, are very similar, so it is not worth to repeat them. Hence, from now on, we suppose, without loss of generality, the following:

(f22x30)(x1)<0;(f23x30)(x1)>0;x1Wr1|(f22x30)(x1)|<|(f23x30)(x1)|;x1Wr1(50)

(f31x10)(x2)<0;(f32x10)(x2)<0;x2Wr2|(f31x10)(x2)|<|(f32x10)(x2)|;x2Wr2(51)

(f13x20)(x3)>0;(f11x20)(x3)<0;x3Wr3|(f13x20)(x3)|<|(f11x20)(x3)|;x3Wr3(52)

Theorem 5.8

If system (4) fulfils the first, second and third condition or derivation verifying \eqrefeq:superfacedefinition2, \eqrefeq:superfacedefinition4 and \eqrefeq:superfacedefinition5 then there exists an initial value (x0,y0,z0)Wr¯Wr¯, for which the sequence:

Yn+1=f22Zn(f23Zn)1(Yn)Xn+1=f13Yn+1(f11Yn+1)1(Xn)Zn+1=f31Xn+1(f32Xn+1)1(Zn)(53)

n0 is convergent to the root (r1,r2,r3) ∈ Wr¯ of the system. Where |(f23z0)|>|(f22z0)|

and |(f32x0)|>|(f31x0)|

Besides, sequence \eqrefeq:finalsequence can be substituted by:

Yn+1=f22Znf13Zn(Yn)Xn+1=f13Yn+1f31Yn+1(Xn)Zn+1=f31Xn+1f22Xn+1(Zn)

in agreement with Proposition \refpro:inversefunctionsfab.

proof

In the plane X1X2.

Let a0<r3 be close enough to r3 in order to exist the points (a11,a21,a0)C1Wr¯, (b11,b21,a0)C2Wr¯ and (c11,c21,a0)C3Wr¯, what is possible due to the continuity of parameterizations \eqrefeq:parameterC1x3, \eqrefeq:parameterC2x3 and \eqrefeq:parameterC3x3.

From Proposition 4.18, in agreement with the assumptios of previous Remark, we have that:

r2<b21<a21<c21

Because of b21 is the closest value to r2, we select the point (b11,b21,a0)C2.

We know, from (47), that:

b21=f23a0(b11)=f22a0(b11), with b11Wr1(54)

As (r1,r2,r3)C1C2C3Wr¯ and a0<r3, from Proposition \refpro:comparisonpoints we have that:

a11,b11,c11<r1a21,b21,c21>r2a31=b31=c31=a0<r3

We look for y0, from the functions f23a0 and f22a0, so that:

r2<b21<y0<a21<c21

in the following manner:

In consonance with assumptions \eqrefeq:superfacedefinition2: f23a0(x1) is increasing, f22a0(x1) is decreasing and |(f22a0)(x1)|<|(f23a0)(x1)|. Therefore, from Theorem 2.3 or 2.4, according to the sum of their derivatives is lower or greater than zero, we have that f22a0([ax,bx])f23a0([ax,bx]) and the sequence:

q0=f22a0(ax)qn+1=(f22a0)(f23a0)1(f22a0)(f23a0)1(qn)(55)

n0 is decreasing. Besides it converges to b21, in agreement with \eqrefeq:rootfabtheorem1. So, there exists n0, so that:

b21<qn<a21;nn0.

Hence, we take y0=qn0.

In de plane X1φ13.

We consider the section Sy0.

Let (a12,y0,a32), (b12,y0,b32) and (c12,y0,c32) be points of the curves C1, C2 and C3, respectively. From Proposition \refpro:comparisonpoints, they belong to Wr¯ since, by comparing in C1 the point (a12,y0,a32) with (a11,a21,a0) and (r1,r2,r3), respectively, we arrive at:

r2<y0<a21a11<a12<r1r2<y0<a21a31=a0<a32<r3

And, by comparing in C3 the point (c12,y0,c32) with (c11,c21,a0) and (r1,r2,r3), respectively, we get:

r2<y0<c21c11<c12<r1r2<y0<c21a0<c32<r3(56)

And, by comparing in C2 the point (b12,y0,b32) with (b11,b21,a0) and (r1,r2,r3), respectively, we get

y0>b21>r2b12<b11<r1y0>b21b32<b31=a0<r3

Finally, by the continuity of parameterization \eqrefeq:parameterC2x2 and Corollary \refcorollary, we can take, if needed, y0=qn1, with n1>n0, so close to b21 that a0b32<a0ez and b11b12<b11ax, since when qn, of the sequence (47), goes to b21, then b12 goes to b11 and b32 goes to a0. Therefore, b32>ez and b12>ax.

From Proposition 4.19, We conclude that:

c12<b12<a12<r1

Because of a12 is the closest value to r1, we select the point (a12,y0,a32)C1.

We know, from \eqrefeq:rootsfab1, that:

a12=f13y0(a32)=f11y0(a32), with a32Wr3(57)

We look for x0, from the functions f13y0 and f11y0, so that:

c12<b12<x0<a12<r1

as follows:

In agreement with assumptions \eqrefeq:superfacedefinition5: f13y0(x3) is increasing and f11y0(x3) is decreasing. |(f13y0)(x3)|<|(f11y0)(x3)|, x3Wr3. Therefore, from Theorem 2.3 or 2.4, f13y0([ez,fz])f11y0([ez,fz]) and the sequence:

q0=f11y0(fz)qn+1=(f13y0)(f11y0)1(f13y0)(f11y0)1(qn)(58)

n0 is increasing. Besides it converges to a12, in agreement with \eqrefeq:rootfabtheorem2. So, there exists n0, so that:

b12<qn<a12;nn0

Hence, we take x0=qn0.

In the plane X2φ13.

We consider the section Sx0.

Let (x0,a23,a33), (x0,b23,b33) and (x0,c23,c33) be points of C1, C2 and C3, respectively. From Proposition \refpro:comparisonpoints, they belong to Wr¯ since, by comparing in C2 the point (x0,b23,b33) with (b12,y0,b32) and (r1,r2,r3), we obtain:

b12<x0<r1b22=y0>b23>r2b12<x0<r1b32<b33<r3

And, by comparing in C3 the point (x0,c23,c33) with (c12,y0,c32) and (r1,r2,r3),

c12<x0<r1c22>c23>r2c12<x0<r1c32<c33<r3(59)

And, by comparing in C1 the point (x0,a23,a33) with (a12,y0,a32) and (r1,r2,r3),

r1>a12>x0a23>a22>r2r1>a12>x0a33<a32<r3

Finally, by the continuity of (23) and Corollary \refcorollary, we can select x0 in the way that, a23>cy and a23>cy.

From Proposition 4.20, we have that:

a33<b33<c33<r3

Because of c33 is the closest value to r3, we select the point (x0,c23,c33)C3.

We know, from \eqrefeq:rootsfab3, that:

c33=f32x0(c23)=f31x0(c23), with c23Wr2(60)

From \eqrefeq:a0¡c23 and \eqrefeq:c23¡c33, we deduce that a0<c33. Again, as the above, we look for z0, from the functions f31x0 and f32x0, so that:

a0<z0<c33<r3

in the following way:

In agreement with assumptions \eqrefeq:superfacedefinition4: f31x0(x2) is decreasing and f32x0(x2) is decreasing. |(f31x0)(x2)|<|(f32x0)(x2)|, x2Wr2. Therefore, from Theorem 2.2, f31x0([cy,dy])f32x0([cy,dy]) and the sequence:

q0=f31x0(dy)qn+1=(f31x0)(f32x0)1(qn)(61)

n0 is increasing. Besides it converges to c33, in agreement with \eqrefeq:rootfabtheorem3. So, there exists n0, so that:

a0<qn<c33;nn0.

Hence, we take z0=qn0.

In the plane X1X2.

We consider the section Sz0.

We compare in C2 the point (b11,b21,a0) with (b14,b24,z0) and (r1,r2,r3), respectively. Then:

a0<z0<r3b21>b24>r2a0<z0<r3b11<b14<r1

So, (b14,b24,z0) is also in Wr¯.

We know, from (47), that:

b24=f22z0(b14)=f23z0(b14), with b14Wr1(62)

As in \eqrefeq:sequenceqn1, from Theorem 2.3 or 2.4, we have that the sequence:

q0=y0qn+1=(f22z0)(f23z0)1(f22z0)(f23z0)1(qn)

n0 is decreasing. Besides it converges to b42, in agreement with \eqrefeq:rootfabtheorem4. Therefore, if we take y1 as:

y1=f22z0(f23z0)1f22z0(f23z0)1(y0)

then, we have that:

y0>y1>b24>r2

We compare in C1 the point (a14,y1,a34) with (a12,y0,a32) and (r1,r2,r3), respectively. Then:

r2<y1<y0a12<a14<r1r2<y1<y0a32<a34<r3

So, (a14,y1,a34) is also in Wr¯.

In the plane X1φ13.

We consider the section Sy1.

We know, from \eqrefeq:rootsfab1, that:

a14=f11y1(a34)=f13y1(a34), with a34Wr3(63)

As in \eqrefeq:sequenceqn2, from Theorem 2.3 or 2.4, the sequence:

q0=x0qn+1=(f13y1)(f11y1)1(f13y1)(f11y1)1(qn)

n0 is increasing. Besides it converges to a41, in agreement with \eqrefeq:rootfabtheorem5. Therefore, if we take x1 as:

x1=f13y1(f11y1)1f13y1(f11y1)1(x0)

then:

x0<x1<a14<r1

We compare in C3 the point (x1,c24,c34) with (x0,c23,c33) and (r1,r2,r3), respectively. Then:

x0<x1<r1c24>c23>r2r2<y1<y0c33<c34<r3

So, (x1,c24,c34) also is in Wr¯.

In the plane X2φ13.

We consider the section Sx1.

We know, from \eqrefeq:rootsfab3, that:

c34=f31x1(c24)=f32x1(c24), with c24Wr2

As in \eqrefeq:sequenceqn3, from Theorem 2.2, the sequence:

q0=z0qn+1=f31x1(f32x1)1(qn)

n0 is increasing. Besides it converges to c43.

Therefore, if we take z1 as:

z1=f31x1(f32x1)1(z0)

then:

z0<z1<c34<r3

We compare in C2 the point (b15,b25,z1) with (b14,b24,z0) and (r1,r2,r3), respectively. Then:

z0<z1<r3b14<b15<r1z0<z1<r3b24>b25>r2

So, ((b15,b25,z1) also is in Wr¯.

So far, we have got the points x0, y0, z0 and x1, y1, z1 that fulfil:

y0>y1>r2x0<x1<r1z0<z1<r3

With their corresponding points (b15,b25,z1), (a14,y1,a34) and (x1,c24,c34), placed inC2Wr¯, C1Wr¯, and C3Wr¯, respectively, and that verify:

  1. b51, a41 ¡ r1

  2. b52, c42 ¿ r2

  3. a43, c43 ¡ r3

By induction hypothesis, suppose, now, we have, xp, yp and zp, p>1, so that:

y0>y1>....>yp>r2x0<x1<....<xp<r1z0<z1<....<zp<r3

and besides, the point (b1p+4,b2p+4,zp) is placed in C2Wr¯, the point (a1p+3,yp,a3p+3) is placed in C1Wr¯ and the point (xp,c2p+3,c3p+3) is placed in C3Wr¯, with:

  1. bp+41, ap+31 ¡ r1

  2. bp+42, cp+32 ¿ r2

  3. ap+33, cp+33 ¡ r3

In the plane X1X2.

We consider the section Szp.

We know, from (47), that:

b2p+4=f22zp(b1p+4)=f23zp(b1p+4)(64)

with b1p+4Wr1. As in \eqrefeq:sequenceqn1, from Theorem 2.3 or 2.4, we have that the sequence:

q0=ypqn+1=(f22zp)(f23zp)1(f22zp)(f23zp)1(qn)

n0 is decreasing. Besides it converges to bp+42 in agreement with \eqrefeq:rootfabtheorem7. Therefore, if we take yp+1 as:

yp+1=f22zp(f23zp)1f22zp(f23zp)1(yp)

then, we have that:

yp>yp+1>b2p+4>r2

We compare in C1 the point (a1p+4,yp+1,a3p+4) with (a1p+3,yp,a3p+3) and (r1,r2,r3), respectively. One gets:

yp>yp+1>r2a1p+3<a1p+4<r1yp>yp+1>r2a3p+3<a3p+4<r3

So, we can conclude that (a1p+4,yp+1,a3p+4)C1Wr¯.

In the plane X1φ13.

We consider the section Syp+1.

We know, from \eqrefeq:rootsfab1, that:

a1p+4=f11yp+1(a3p+4)=f13yp+1(a3p+4)(65)

with a3p+4Wr3. As in \eqrefeq:sequenceqn2, from Theorem 2.3 or 2.4, we have that the sequence:

q0=xpqn+1=f11yp+1(f13yp+1)1f11yp+1(f13yp+1)1(xp)

n0 is increasing. Besides it converges to ap+41, in agreement with \eqrefeq:rootfabtheorem8.

If we take xp+1 as:

xp+1=f11yp+1(f13yp+1)1f11yp+1(f13yp+1)1(xp)

that satisfies:

xp<xp+1<a1p+4<r1

We compare in C3 the point (xp+1,c2p+4,c3p+4) with ((xp,c2p+3,c3p+3) and (r1,r2,r3), respectively. One arrives at:

xp<xp+1<r1c2p+3>c2p+4>r2xp<xp+1<r1c3p+4<c3p+3<r3

So, we can conclude that (xp+1,c2p+4,c3p+4)C3Wr¯.

In the plane X2φ13.

We consider the section Sxp+1.

We know, from \eqrefeq:rootsfab3, that:

c3p+4=f31xp+1(c2p+4)=f32xp+1(c2p+4)(66)

with c2p+4Wr2. As in \eqrefeq:sequenceqn3, from Theorem 2.2, we have that the sequence:

q0=zpqn+1=f31xp+1(f32xp+1)1(zp)

n0 is increasing. Besides it converges to c3p+4, in agreement with \eqrefeq:rootfabtheorem9.

If we take zp+1 as:

zp+1=f31xp+1(f32xp+1)1(zp)

then, zp+1 verifies:

zp<zp+1<c3p+4<r3

We compare in C2 the point (b1p+5,b2p+5,zp+1) with (b1p+4,b2p+4,zp) and (r1,r2,r3), respectively. We obtain:

zp<zp+1<r3b1p+4<b1p+5<r1zp<zp+1<r3b2p+5>b2p+4>r2

So, we can conclude that (b1p+5,b2p+5,zp+1)C2Wr¯.

The induction hypothesis is true and we have got the sequences:

x0,y0,z0yn+1=f22zn(f23zn)1f22zn(f23zn)1(yn)xn+1=f11yn+1(f13yn+1)1f11yn+1(f13yn+1)1(xn)zn+1=f31xn+1(f32xn+1)1(zn)

n0, which are monotonic and bounded, so convergent.

Either from Theorem 2.3 or from Theorem 2.4 the sequences:

X0=x0,Y0=y0,Z0=z0Yn+1=f22Zn(f23Zn)1(Yn)Xn+1=f11Yn+1(f13Yn+1)1(Xn)Zn+1=f31Xn+1(f32Xn+1)1(Zn)(67)

n0, are convergent too. Obviously to the same limit.

Let B, A and C be such limits, respectively. Taking limits:

B=f22C(f23C)1(B)A=f11B(f13B)1(A)C=f31A(f32A)1(C)(68)

We have the following equivalences:

A=(f23C)1(B)B=(f32A)1(C)C=(f13B)1(A).(69)

Indeed, taking into account \eqrefeq:secondconvergentfunctions2:

A=(f23C)1(B)B=f23C(A);B=f22C(A)C=f32A(B)B=(f32A)1(C)

B=(f32A)1(C)C=f32A(B)C=f31A(B);A=f11B(C)C=(f11B)1(A)=(f13B)1(A)

This last equality is due to the second equation of \eqrefeq:secondconvergentfunctions2.

C=(f13B)1(A)A=f13B(C)C=(f13B)1(A)

Finally if:

C=(f13B)1(A)A=f13B(C)B=f23A(C)C=(f23A)1(B)B=f23A(C)B=f23C(A)A=(f23C)1(B)

This last equation close the loop of the reasoning.

From \eqrefe:secondconvergentfunctions if A=(f23C)1(B), then (f32A)1(C) and C=(f13B)1(A). Then:

A=f11B(f13B)1(A)A=f13B(C);A=f11B(C)f3(A,B,C)=f3(φ31(B,C),B,C)=0and f1(A,B,C)=f1(φ11(B,C),B,C)=0

and:

C=f31A(f32A)1(C)C=f31A(B);C=f32A(B)f1(A,B,C)=f1(A,B,φ13(A,B))=0and f2(A,B,C)=f2(A,B,φ23(A,B))=0

Therefore (A,B,C)=(r1,r2,r3).

On the contrary, if A1=(f23C)1(B)=, then B=f23C(A1);B=f22C(A1) hold (this last equatily is deduced from the firts equation of \eqrefeq:secondconvergentfunctions2), and (A1,B) is the only cut point of f23C and f22C.

From \eqrefeq:equivalences, it is also true that A=f13B(C1);A=f11B(C1) (this last equation is deduced from the second equation of \eqrefeq:secondconvergentfunctions2), C1=C̸, what means that the only cut point of f11B and f13B is (A,C1).

Finally, C=f32A(B1);C=f31A(B1) also hold, what implies that the only cut point of f31A and f32A.

In this case, the sequence has fallen in the loop:

(A1,B,C)(A,B,C1)(A,B1,C)(A1,B,C)

where (A1,B,C) is a root of f2 and f3, (A,B,C1) is a root of f1 and f3 and (A,B1,C) is a root of f1 and f2.

If that happens, we can restart the sequence making x0=(A+A1)/2; y0=(B+B1)/2; z0=(C+C1)/2.

Finally, in agreement with Proposition \refpro:inversefunctionsfab, \eqrefe:secondconvergentfunctions becomes:

X0=x0,Y0=y0,Z0=z0Yn+1=f22Znf13Zn(Yn)Xn+1=f11Yn+1f33Yn+1(Xn)Zn+1=f31Xn+1f22Xn+1(Zn)

And the result follows. □

6 Example 1

We want to find the root (6.26397, 0.945931, 3.02692) of the system:

f1(x,y,z)=6y2+20y+2x+44z170=0f2(x,y,z)=3y243y7x6z+100=0f3(x,y,z)=z279z+6x210y+4=0(70)

We check if that satisfies the three conditions of derivation in the compact set:

Wr¯=[5.5,6.5]×[0.5,1.5]×[2.5,3.15]

with the sequence introduced in Definition \refeq:fifthcondition:

F1X1X2,F3X1X3,F2X2X3

This allows us to select the following functions (18), in agreement with Definition 5.1:

In X1X2:φ22(x,z)=16(43649+84x+72z)φ32(x,z)=110(4+6x279z+z2)In X1X3:φ31(y,z)=4+10y+79zz26φ11(y,z)=8510y3y222zIn X2X3φ13(x,y)=122(85x10y3y2)φ23(x,y)=16(1007x43y+3y2)

What leads us to the construction of the fixed point iteration functions, introduced in Definition \refdef:curvesplanessystem, given by:

In X1X2:f22x30(x)=φ22(x,x30);x30[2.5,3.15]f23x30(x)=φ32(x,x30);x30[2.5,3.15]In X1X3:f13x20(z)=φ31(x20,z);x20[0.5,1.5]f11x20(z)=φ11(x20,z);x20[0.5,1.5]In X2X3:f31x10(y)=φ13(x10,y);x10[5.5,6.5]f32x10(y)=φ23(x10,y);x10[5.5,6.5]

Taking into account the inverse functions, defined in Proposition \refpro:inversefunctionsfab, then the sequence:

x0=5.5;y0=0.5;z0=2.5zn+1=f31xnf22xn(zn)=φ13(xn,φ22(xn,zn))yn+1=f22zn+1f13zn+1(yn)=φ22(φ31(yn,zn+1),zn+1)xn+1=f13yn+1f31yn+1(xn)=φ31(yn+1,φ13(xn,yn+1))

converges to the root (6.26394, 0.945931, 3.02692), whose results can be seen in Table 1.

Table 1

Results of the system (70)

7 Example 2

In the next example, we illustrate the idea of how to solve nonlinear systems, using solvers of nonlinear equations.

Find the root (0.495104, 0.849685, 1.38957) of the system: \labeleq:system3

f1(x,y,z)=cos(y)zsin(x)f2(x,y,z)=zx1yf3(x,y,z)=exyz2(71)

using numerical solvers of nonlinear equations.

We check that the system satisfies the three conditions of derivation in the compact set:

Wr¯=[0.3,0.5]×[0.78,0.95]×[1,1.5]

with the sequence:

F1X1X2,F2X1X3,F3X2X3

This allows us to select the following functions (18), in agreement with Definition 5.1:

In X1X2:φ12(x,z) and φ32(x,z)In X1X3:φ21(y,z) and φ31(y,z)In X2X3:φ13(x,y) and φ23(x,y)

What leads us to the construction of the fixed point iteration functions, introduced in Definition \refdef:curvesplanessystem, given by:

In X1X2:y=f21x30(x)=φ12(x,x30);x30[1,1.5]y=f23x30(x)=φ32(x,x30);x30[1,1.5]In X1X3:x=f12x20(z)=φ21(x20,z);x20[0.78,0.95]x=f13x20(z)=φ31(x20,z);x20[0.78,0.95]In X2X3:z=f31x10(y)=φ13(x10,y);x10[0.3,0.5]z=f32x10(y)=φ23(x10,y);x10[0.3,0.5]

and the sequence:

x0=0.3;y0=0.78;z0=1.5yn+1=f22znf11zn(yn)=φ22(φ11(yn,zn),zn)xn+1=f11yn+1f33yn+1(xn)=φ11(yn+1,φ33(xn,yn+1))zn+1=f33xn+1f22xn+1(zn)=φ33(xn+1,φ22(xn+1,zn))(72)

converges to the root (0.495104, 0.849685, 1.38957).

In this case we do not use sequence \eqrefeq:sequencesecondex. Instead of it, we use numerical solvers of nonlinear equations in the following manner:

Given a single equation f(x)=0, in [19] an infinity family of numerical solvers are provided. These solvers can be expressed as xn+1=Fm(xn), with 1≤m<∞, integer, in such a way that the velocity of convergence of such iterations increases more and more as mgoes to infinity. For example, therein, F2 is defined as follows:

F2(x)=xf(x)f(x)+f2(x)f(x)2(f)3(x)

and in the cited paper it is also proven that the sequence:

xn+1=F2(xn)

converges to the root of the equation f(x)=0, if the initial value, x0, satisfies that:

f(x0)f′′(x0)(f)2(x0)<1(73)

Besides, it converges to the closest root to x0.

Given the initial value of sequence \eqrefeq:sequencesecondex, x0=0.3; y0=0.78; z0=1.5, then the second equation of system \eqrefeq:system4, f2(x0,y,z0)=0, is a nonlinear equation in the unknown y. If, for each fixed (x0,z0), we consider the function in the variable y:

F22(x0,y,z0)=yf2(x0,y,z0)f2(x0,y,z0)+f22(x0,y,z0)f2(x0,y,z0)2(f2)3(x0,y,z0)

that satisfies the convergence condition \eqrefeq:Convergencecondition for y0=0.78, then the sequence:

y0=0.78;yn+1=F22(x0,yn,z0);n0

converges to φ22(x0,z0), that is the root of f2(x0,y,z0)=0. In the same way, taking into account the first equation of the system \eqrefeq:system4, f1(x,y0,z0)=0 and, for each fixed (y0,z0), the function:

F11(x,y0,z0)=xf1(x,y0,z0)f1(x,y0,z0)+f12(x,y0,z0)f1(x,y0,z0)2(f1)3(x,y0,z0)

that satisfies the convergence condition \eqrefeq:Convergencecondition for x0=0.3, then the sequence:

x0=0.3;xn+1=F11(xn,y0,z0);n0

converges to φ11(y0,z0), that is the root of f1(x,y0,z0)=0.

And, finally, focusing on the third equation of the system, f3(x0,y0,z)=0, and on the function:

F33(x0,y0,z)=zf3(x0,y0,z)f3(x0,y0,z)+f32(x0,y0,z)f3(x0,y0,z)2(f3)3(x0,y0,z)

then, the sequence:

z0=1.5;zn+1=F33(x0,y0,zn)

converges to φ33(x0,y0).

Then, using the sequence:

x0=0.3;y0=0.78;z0=1.5yn+1=F22(F11(xn,yn,zn),yn,zn)xn+1=F11(xn,yn+1,F33(xn,yn+1,zn))zn+1=F33(xn+1,F22(xn+1,yn+1,zn),zn)

Where we have held the same structure as sequence \eqrefeq:sequencesecondex and substituted the exact functions φ11, φ22 and φ33 by their approximate functions F11, F22 and F33 respectively.

We have obtained the required root. See the results in Table 2.

Table 2

Results of the system (71)

8 Conclusions

A new method has been developed to solve nonlinear equations system. This fact reduces significantly the complexity and number of lineal algebra operations and functional evaluations. As a way of illustration, Example 1 as a whole needs thirty-six functional evaluations where, in addition, linear algebra operations have not been required.

Otherwise, there are other equations that cannot be solved in an exact manner, whose functions (18) cannot be computed. In Example 2, this fact is solved by means of combining the proposed method with solvers of nonlinear simple equations.

Besides, this method also provides a collection of very interesting ideas to construct a forthcoming proof of the Banach Theorem fixed-point converse.

Acknowledgement

This work has been partially supported by Fundaciόn S\'eneca de la Regiόn de Murcia grant number 19219/PI/14.

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About the article

Received: 2018-07-09

Accepted: 2018-08-13

Published Online: 2018-10-22


Citation Information: Open Physics, Volume 16, Issue 1, Pages 605–630, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0079.

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© 2018 Joaquín Moreno et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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