On the Solution of a Class of Nonlinear Systems Governed by an 𝑀-Matrix

Abstract

We consider a weakly nonlinear system of the form (𝐼+𝜑(𝑥)𝐴)𝑥=𝑝, where 𝜑(𝑥) is a real function of the unknown vector 𝑥, and (𝐼+𝜑(𝑥)𝐴) is an 𝑀-matrix. We propose to solve it by means of a sequence of linear systems defined by the iteration procedure (𝐼+𝜑(𝑥_𝑟)𝐴)𝑥_𝑟+1=𝑝, 𝑟=0,1,…. The global convergence is proved by considering a related fixed-point problem.

1. Introduction

In recent years [1, 2], we approached the numerical resolution of the following nonstandard integrodifferential problems arising in the study of the kinetic theory of dusty plasmas:𝜈(𝑡)𝑦(𝑡)=^∞₀𝑘(𝑠)𝑦(𝑠)𝑑𝑠𝐷(𝑡)𝑦^′(𝑡)^′+𝑝(𝑡),𝑡>0,𝑦^′(0)=0,𝑦(∞)=0, (1.1) whose discretization, by means of a difference scheme and a quadrature rule [2], leads to a particular kind of nonlinear system of equations. Solving it by means of a fixed-point (FP) iteration process, we noted that such a procedure seems to globally converge, that is, it converges independently of the choice of the starting point. Our aim, here, is to explain the reason of this “nice” behavior. Throughout the paper, the notation 𝑀≥0, with 𝑀∈𝑅^𝑁×𝐾, means that each element of 𝑀 is nonnegative, whereas 𝑀>0 means 𝑀≥0 and 𝑀≠0. The nonlinear system under investigation is𝐼+𝜑(𝑥)𝐴𝑥=𝑝, (1.2) where 𝐼 is the 𝑁-dimensional identity matrix, 𝑥∈𝑅^𝑁 is an unknown vector, and we make the following assumptions which will be sufficient for the global convergence of the iterative process we propose (see Theorem 3.3):(i)𝑝∈𝑅^𝑁with 𝑝>0;(ii)𝐴∈𝑅^𝑁×𝑁 has nonnegative main diagonal and nonpositive off-diagonal entries (𝑍-matrix);(iii)𝐴 is rowwise weakly diagonal dominant;(iv)𝜑∶𝑅^𝑁→𝑅 is a differentiable function such that 𝜑(𝑥)≥0, for all 𝑥≥0 and it is homogeneous of degree 𝛾≤1, that is, for all 𝛼>0𝜑(𝛼𝑥)=𝛼^𝛾𝜑(𝑥);(v)grad(𝜑(𝑥))>0forall𝑥≥0, where grad(𝜑(𝑥)) denotes the (row vector) gradient of 𝜑(𝑥).

Systems of type (1.2) fall into the class of weakly nonlinear systems (see, e.g., [3, 4]) and they also arise from the discretization, by finite-difference methods, of mathematical models less cumbersome than (1.1), as, for instance, the differential problems of the form:𝑦(𝑡)+𝑏𝑦𝑎(𝑡)𝑦^′(𝑡)=𝑝(𝑡),𝑡≥0,𝑦(0)=0. (1.3)

A central rule in our results is played by the matrix (𝐼+𝜑(𝑥)𝐴) appearing in (1.2) which, as it can be immediately seen from the assumption (ii)‒(iv), is an 𝑀-matrix forall𝑥≥0 (see, e.g., [5, page 137, prop. 𝑀₃₅]).

The organization of the paper is the following. In Section 2, we investigate a particular class of one variable functions 𝑓 such that the FP iterations 𝑡_𝑟+1=𝑓(𝑡_𝑟) converge independently of the choice of the starting value 𝑡₀. By exploiting this result jointly with the properties of the matrix 𝐼+𝜑(𝑥)𝐴, we are lead to state that, under the assumptions (i)–(v), the FP iteration procedure (𝐼+𝜑(𝑥_𝑟)𝐴)𝑥_𝑟+1=𝑝 globally converges to a solution of (1.2). This constitutes our main result which is contained in Section 3. Finally, some numerical experiments showing the sharpness of the required conditions and the performance of the iteration scheme are reported in Section 4.

2. The One Variable Case

In order to prove our main theorem, we need the following result on a particular class of numerical sequences.

Theorem 2.1. Let 𝑓 be a one variable scalar function such that(a)𝑓∶𝑡∈𝑋⊆[0,+∞)→(0,𝑀_𝑓), with 𝑀_𝑓>0, 𝑋 closed and 𝑓(𝑋)⊂𝑋;(b)𝑓 continuous;(c)𝑡𝑓(𝑡) strictly increasing.Then, starting from any 𝑡₀∈𝑋, the sequence obtained by the functional iteration process 𝑡_𝑟+1=𝑓𝑡_𝑟,𝑟=0,1,2,…, (2.1) converges to 𝑡^∗, a fixed point of 𝑓.

In this section, we are going to prove this theorem.

Denote by 𝑔 the function 𝑔(𝑡)=𝑡−𝑓(𝑡), then we have𝑔𝑡_𝑟=𝑡_𝑟−𝑓𝑡_𝑟=𝑡_𝑟−𝑡_𝑟+1. (2.2) Of course, if there exists a point 𝑡_𝑟 such that 𝑔(𝑡_𝑟)=0, then 𝑡_𝑟=𝑡_𝑟+𝑛 for all 𝑛=0,1,… and the statement is trivial. Similarly, if the sequence 𝑔(𝑡_𝑟) is ultimately nonnegative (nonpositive), we have that the sequence 𝑡_𝑟 is ultimately decreasing (increasing) and it trivially converges by hypothesis (a). Therefore, from now on, we assume that the sequence 𝑔(𝑡_𝑟) satisfies∀𝑟≥0,∃𝑚>𝑟∶𝑔𝑡_𝑟𝑔𝑡_𝑚<0. (2.3) The following lemma holds.

Lemma 2.2. If 𝑔(𝑡_𝑟)<0(𝑟𝑒𝑠𝑝.>0) and 𝑔(𝑡_𝑟+𝑚)>0(𝑟𝑒𝑠𝑝.<0), 𝑚=1,…,𝑛, then 𝑡_𝑟<𝑡_𝑟+𝑚+2𝑟𝑒𝑠𝑝.𝑡_𝑟>𝑡_𝑟+𝑚+2,𝑚=−1,0,1,…,𝑛. (2.4)

Proof. Let us prove (2.4) by induction on 𝑛≥1. Assume 𝑛=1, that is, we are assuming that 𝑔𝑡_𝑟<0,𝑔𝑡_𝑟+1>0, (2.5) and we want to prove 𝑡_𝑟<𝑡_𝑟+𝑚+2,𝑚=−1,0,1. (2.6) From the first of (2.5) and (2.2), we immediately get (2.6) with 𝑚=−1, that is, we have 𝑡_𝑟<𝑡_𝑟+1, (2.7) and so, from the hypothesis (c), we deduce 𝑡_𝑟𝑓𝑡_𝑟<𝑡_𝑟+1𝑓𝑡_𝑟+1, (2.8) which, recalling that 𝑓(𝑡_𝑟)=𝑡_𝑟+1, implies 𝑡_𝑟𝑡_𝑟+1<𝑡_𝑟+1𝑡_𝑟+2, (2.9) and, therefore, (2.6) holds with 𝑚=0, that is, 𝑡_𝑟<𝑡_𝑟+2. (2.10) By using (c) once again, (2.10) gives 𝑡_𝑟𝑓(𝑡_𝑟)<𝑡_𝑟+2𝑓(𝑡_𝑟+2) or equivalently 𝑡_𝑟𝑡_𝑟+1<𝑡_𝑟+2𝑡_𝑟+3. (2.11) But the second of (2.5) implies 𝑡_𝑟+2<𝑡_𝑟+1, so that from (2.11), we obtain 𝑡_𝑟𝑡_𝑟+2<𝑡_𝑟𝑡_𝑟+1<𝑡_𝑟+2𝑡_𝑟+3, (2.12) which assures that (2.6) holds for 𝑚=1 too.
Now, we assume that the lemma holds for 𝑚=𝑛, that is, suppose that 𝑔𝑡_𝑟<0,𝑔𝑡_𝑟+𝑚>0,𝑚=1,…,𝑛, (2.13) implies 𝑡_𝑟<𝑡_𝑟+𝑚+2,𝑚=−1,0,1,…,𝑛. (2.14) In order to prove the lemma for 𝑚=𝑛+1, assume 𝑔𝑡_𝑟<0,𝑔𝑡_𝑟+𝑚>0,𝑚=1,…,𝑛+1. (2.15) Of course (2.14) holds, and the second of (2.15) implies 𝑡_𝑟+1>𝑡_𝑟+2>⋯>𝑡_𝑟+𝑛+2. (2.16) But from (2.14) with 𝑚=𝑛 and (c), we have 𝑡_𝑟𝑡_𝑟+1<𝑡_𝑟+𝑛+2𝑡_𝑟+𝑛+3, (2.17) which in view of (2.16) leads to 𝑡_𝑟𝑡_𝑟+𝑛+2<𝑡_𝑟+𝑛+2𝑡_𝑟+𝑛+3. (2.18) Hence, 𝑡_𝑟<𝑡_𝑟+𝑛+3, that is, (2.14) holds for 𝑚=𝑛+1 too, and the desired result is proved.

Now, let us partition the sequence {𝑡_𝑟}_𝑟 into two subsequences {𝑡_𝑟𝑖}_𝑖 and {𝑡_𝑠𝑗}_𝑗 defined by𝑔𝑡_𝑟𝑖<0,𝑔𝑡_𝑠𝑗>0,𝑖,𝑗=1,2,… (2.19)

In order to state Theorem 2.1, we are going to prove that these subsequences converge to the same limit. First of all, the following result holds.

Lemma 2.3. The sequences {𝑡_𝑟𝑖}_𝑖 and {𝑡_𝑠𝑗}_𝑗 are strictly increasing and decreasing respectively.

Proof. Let us prove that the sequence {𝑡_𝑟𝑖} is strictly increasing, the proof about {𝑡_𝑠𝑗} is analogous, and it is omitted. Consider 𝑟_𝑖 such that 𝑔(𝑡_𝑟𝑖)<0. Then, two cases may occur: (I) 𝑔(𝑡_𝑟𝑖+1)<0or (II) 𝑔(𝑡_𝑟𝑖+1)>0.(I)In this case, it is clear that 𝑡_𝑟𝑖+1 belongs to the same subsequence {𝑡_𝑟𝑖}_𝑖 and we immediately have 𝑡_𝑟𝑖<𝑡_𝑟𝑖+1=𝑡_𝑟𝑖+1. (2.20)(II)From (2.3), there exists 𝑛≥2 such that 𝑡_𝑟𝑖+1=𝑡_{𝑟𝑖+𝑛}, that is, we have𝑔𝑡_𝑟𝑖<0,𝑔𝑡_{𝑟𝑖+𝑚}>0for𝑚=1,…,𝑛−1,𝑔𝑡_{𝑟𝑖+𝑛}<0. (2.21) Then, Lemma 2.2 with 𝑟=𝑟_𝑖 assures that 𝑡_𝑟𝑖<𝑡_{𝑟𝑖+𝑚+2},𝑚=−1,0,1,…,𝑛−1, (2.22) which, for 𝑚=𝑛−2≥0, gives 𝑡_𝑟𝑖<𝑡_{𝑟𝑖+𝑛}=𝑡_𝑟𝑖+1. (2.23) The desired result comes out from (2.20) and (2.23).

Lemma 2.4. The sequences {𝑡_𝑟𝑖}_𝑖 and {𝑡_𝑠𝑗}_𝑗 provide two separate sets, that is, 𝑡_𝑟𝑖≤𝑡_𝑠𝑗,∀𝑖,𝑗∈ℕ. (2.24)

Proof. Suppose, for example, 𝑟_𝑖<𝑠_𝑗 and precisely let 𝑠_𝑗=𝑟_𝑖+𝑘. If 𝑘=1, we get (2.24) directly from the assumption 0>𝑔(𝑡_𝑟𝑖)=𝑡_𝑟𝑖−𝑡_𝑟𝑖+1. If 𝑘≥2, observe that the numerical sequence {𝑔(𝑡_𝑟𝑖),𝑔(𝑡_𝑟𝑖+1),…,𝑔(𝑡_{𝑟𝑖+𝑘})} presents at least one change of sign, in view of (2.19). Let us examine the case of an unique change of sign, by supposing that there exists an integer 𝑙, with 0≤𝑙≤𝑘−1, such that 𝑔𝑡_{𝑟𝑖+𝑚}<0for𝑚=0,1,…,𝑙,𝑔𝑡_{𝑟𝑖+𝑙+𝑚}>0for𝑚=1,…,𝑘−𝑙, (2.25) the other cases being analogous.
By Lemma 2.3, we get 𝑡_𝑟𝑖≤𝑡_𝑟𝑖+1≤⋯≤𝑡_{𝑟𝑖+𝑙},𝑡_{𝑟𝑖+𝑘}≤𝑡_{𝑟𝑖+𝑘−1}≤⋯≤𝑡_{𝑟𝑖+𝑙+1}, (2.26) and by applying Lemma 2.2 with 𝑟=𝑟_𝑖+𝑙, we deduce 𝑡_{𝑟𝑖+𝑙}<𝑡_{𝑟𝑖+𝑙+𝑚+2},𝑚=−1,0,1,…,𝑘−𝑙. (2.27) Hence, by taking 𝑚=𝑘−𝑙−2≥−1, we conclude 𝑡_𝑟𝑖<𝑡_𝑟𝑖+1<⋯<𝑡_{𝑟𝑖+𝑙}<𝑡_{𝑟𝑖+𝑘}<𝑡_{𝑟𝑖+𝑘−1}<⋯<𝑡_{𝑟𝑖+𝑙+1}. (2.28)

By the previous lemmas, we get the next result which completes the proof of the convergence of the FP iteration process.

Theorem 2.5. For any choice of 𝑡₀∈𝑋, the sequence {𝑡_𝑟}_𝑟 defined in (2.1) converges.

Proof. It is sufficient to prove that the subsequences {𝑡_𝑟𝑖} and {𝑡_𝑠𝑗} defined in (2.19) converge to the same limit, satisfying lim_𝑖𝑡_𝑟𝑖=lim_𝑗𝑡_𝑠𝑗=lim_𝑟𝑡_𝑟. (2.29) By the previous lemmas, we partitioned the sequence {𝑡_𝑟} into two subsequences, {𝑡_𝑟𝑖} and {𝑡_𝑠𝑗}, which are separate, strictly monotone, and bounded sequences. Hence, they converge and 0≤sup_𝑖𝑡_𝑟𝑖=lim_𝑖𝑡_𝑟𝑖=∶𝑙₁≤𝑙₂∶=lim_𝑗𝑡_𝑠𝑗=inf_𝑗𝑡_𝑠𝑗≤𝑀_𝑓. (2.30) Assume ab absurdo that 𝑙₁<𝑙₂. (2.31) Let us consider the subsequence {𝑡_𝜌𝑖} of {𝑡_𝑟𝑖} and the subsequence {𝑡_𝜎𝑗} of {𝑡_𝑠𝑗}, such that 𝑔𝑡_𝜌𝑖−1>0,𝑔𝑡_𝜌𝑖<0,𝑔𝑡_𝜎𝑗−1<0,𝑔𝑡_𝜎𝑗>0. (2.32) In other words, 𝑡_𝜌𝑖 and 𝑡_𝜎𝑗 are those elements of {𝑡_𝑟𝑖} and {𝑡_𝑠𝑗}, respectively, whose previous elements, in the main sequence {𝑡_𝑟}, belong to the other subsequence {𝑡_𝑠𝑗} and {𝑡_𝑟𝑖} respectively, that is, we have that ∃𝑘∈ℕsuchthat𝑡_𝜌𝑖−1=𝑡_𝜎𝑘,∃ℎ∈ℕsuchthat𝑡_𝜎𝑗−1=𝑡_𝜌ℎ. (2.33) Of course, as {𝑡_𝜌𝑖} is a subsequence of the convergent sequence {𝑡_𝑟𝑖}, there results lim_𝑖𝑡_𝜌𝑖=𝑙₁, (2.34) and the same is true for {𝑡_𝜎𝑗}, that is, lim_𝑗𝑡_𝜎𝑗=𝑙₂, (2.35) where 𝑙₁ and 𝑙₂ are given in (2.30). On the other hand, recalling that 𝑡_𝜌𝑖=𝑓(𝑡_𝜌𝑖−1), from (b) and (2.33), we have 𝑙₁=lim_𝑖𝑡_𝜌𝑖=lim_𝑖𝑓𝑡_𝜌𝑖−1=lim_𝑘𝑓𝑡_𝜎𝑘=𝑓𝑙₂,𝑙₂=lim_𝑗𝑡_𝜎𝑗=lim_𝑗𝑓𝑡_𝜎𝑗−1=lim_ℎ𝑓𝑡_𝜌ℎ=𝑓𝑙₁. (2.36) But by the hypothesis (c), the assumption (2.31) implies 𝑙₁𝑓(𝑙₁)<𝑙₂𝑓(𝑙₂), that is, in view of (2.36), we have 𝑙₁𝑙₂<𝑙₂𝑙₁, which is absurd. Thus, the result 𝑙₁=𝑙₂=lim_𝑟𝑡_𝑟 is achieved.

3. The Solution of the Nonlinear System

In this section, we come back to the nonlinear system (1.2). First of all, we recall that, under the hypotheses (ii)–(iv), for any 𝑥≥0, the matrix (𝐼+𝜑(𝑥)𝐴)) appearing in (1.2) is an M-matrix and it satisfies (see, e.g., [5, prop. 𝑁₃₈])𝐼+𝜑(𝑥)𝐴⁻¹>0,∀𝑥≥0. (3.1) Now, we define the function 𝐹 as𝐹∶𝑞∈[0,+∞)⟶𝜑𝐼+𝑞𝐴⁻¹𝑝, (3.2) and we are going to prove that it satisfies the properties stated in the following theorem, where 𝛾 is the homogeneity degree of 𝜑 appearing in (iv) and the classical notation 𝑒=(1,…,1)^𝑇∈𝑅^𝑁 is used.

Theorem 3.1. Assume that (i)–(v) hold. Then, 𝐹 is a continuous, differentiable function, and for any 𝑞≥0, it satisfies 𝐹𝑞∈0,𝑀_𝐹𝑤𝑖𝑡ℎ𝑀_𝐹=𝑝^𝛾_∞𝜑(𝑒),𝐹^′𝑞=−grad𝜑𝐼+𝑞𝐴⁻¹𝑝𝐼+𝑞𝐴⁻¹𝐴𝐼+𝑞𝐴⁻¹𝑝,𝑠𝑞∶=𝑞^𝛾𝐹𝑞𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔. (3.3)(3.4)(3.5)

Proof. As the inverse of (𝐼+𝑞𝐴) is defined for all 𝑞≥0, then 𝐹 is clearly a continuous function. Moreover, it can be easily seen that 𝑑𝑑𝑞𝐼+𝑞𝐴⁻¹=−𝐼+𝑞𝐴⁻¹𝐴𝐼+𝑞𝐴⁻¹, (3.6) hence, (3.4) is immediately true. In view of Theorem A.1 in [6], we observe that (ii) and (iii) assures ‖𝐼+𝑞𝐴⁻¹‖_∞≤1,∀𝑞≥0. (3.7) Therefore, for any 𝑞≥0, the vector 𝑥(𝑞)∶=(𝐼+𝑞𝐴)⁻¹𝑝 satisfies ‖𝑥(𝑞)‖_∞≤‖𝑝‖_∞ and because of (v), we get 𝐹(𝑞)=𝜑(𝑥(𝑞))≤𝜑(‖𝑝‖_∞𝑒). Consequently, assertion (3.3) holds taking into account (iv). Finally, again in view of (iv), we have 𝑠𝑞∶=𝑞^𝛾𝐹𝑞=𝑞^𝛾𝜑𝑥𝑞=𝜑𝑞𝑥𝑞⟹𝑠^′𝑞=grad𝜑𝑞𝑥𝑞𝑥𝑞+𝑞𝑥^′𝑞, (3.8) where, recalling (3.6), it results 𝑥𝑞+𝑞𝑥^′𝑞=𝐼+𝑞𝐴⁻¹−𝐼+𝑞𝐴⁻¹𝑞𝐴𝐼+𝑞𝐴⁻¹𝑝=𝐼+𝑞𝐴⁻²𝑝. (3.9) Thus, by (v) and (3.1), we deduce (3.5) from 𝑠^′𝑞=grad𝜑𝑞𝑥𝑞𝐼+𝑞𝐴⁻²𝑝>0. (3.10)

Using this theorem, we can easily claim that the function 𝐹 in (3.2) has the same characteristics as the function 𝑓 introduced in the previous section.

Corollary 3.2. Assume that (i)–(v) hold. Then, the function 𝐹 verifies all the hypotheses (a)–(c).

Proof. Assertions (a) and (b) directly follow from Theorem 3.1. Moreover, in view of (3.5), (c) is immediately obtained by 𝑞𝐹(𝑞)=𝑠(𝑞)𝑞^1−𝛾, recalling that 𝛾<1 implies 𝑞^1−𝛾 strictly increasing.

Now, we are ready to state our main result about the convergence of the sequence {𝑥_𝑟} defined as𝑥_𝑟+1=𝐼+𝜑𝑥_𝑟𝐴⁻¹𝑝,𝑟=0,1,…,𝑥₀≥0. (3.11)

Theorem 3.3. Assume that the hypotheses (i)–(v) hold. Then, the sequence {𝑥_𝑟} converges to a solution 𝑥^∗ of (1.2).

Proof. Let us start from an arbitrary 𝑥₀≥0 and put 𝑞_𝑟=𝜑𝑥_𝑟,𝑟=0,1,… (3.12) From (3.11), we have that (3.12) can also be written as 𝑞_𝑟+1=𝐹(𝑞_𝑟),𝑟=0,1,… Therefore, Corollary 3.2 assures that the sequence {𝑞_𝑟} is convergent. Denoted by 𝑞^∗ its limit, we set 𝑥^∗∶=(𝐼+𝑞^∗𝐴)⁻¹𝑝, and we obtain the statement by observing that lim_𝑟𝑥_𝑟+1=lim_𝑟𝐼+𝜑𝑥_𝑟𝐴⁻¹𝑝=lim_𝑟𝐼+𝑞_𝑟𝐴⁻¹𝑝=𝐼+𝑞^∗𝐴⁻¹𝑝=𝑥^∗. (3.13)

Remark 3.4. Of course, in view of (3.7), any solution of (1.2) satisfies ‖𝑥^∗‖_∞≤‖𝑝‖_∞, but nothing seems to imply its uniqueness. Anyway, we conjecture it, because in a large variety of experimental tests (including also high-dimensional systems), we never found more than one solution. Of course, a sufficient condition for the uniqueness is ‖grad(𝜑(𝑥))‖₁‖𝐴‖_∞‖𝑝‖_∞<1, for any 𝑥≥0, which assures |𝐹^′(𝑞)|<1, for any q≥0, by virtue of (3.4) and (3.7). Nevertheless, as it can be observed in the next section, it seems that the solution is unique even if such a condition is largely not satisfied.

4. Numerical Experiments

In order to verify our theoretical results and to check the performances of the FP iteration scheme on problem (1.2), we carried out a large variety of numerical experiments. Here, the most significant are reported. From the previous section, it appears clear that the iterative scheme (3.11), which furnishes a vector at each iteration, requires the same computational effort as (3.12), which instead provides scalar values. Hence, for the sake of simplicity, all the numerical tests reported here, are referred to (3.12), that is, we test the iterative procedure:𝑞_𝑟+1=𝜑𝐼+𝑞_𝑟𝐴⁻¹𝑝=𝐹𝑞_𝑟. (4.1) In particular, in all the section, we assume that the convergence is reached whenever |𝑞_𝑟+1−𝑞_𝑟|<𝑡𝑜𝑙 and we put 𝑡𝑜𝑙=10⁻⁶.

Let us consider the following two problems:𝑁=3,𝜑(𝑥)=‖𝑥‖₂5,𝐴=11−10−.51−.5−.5−12,𝑝=10,𝛾=1;𝑁=3,𝜑(𝑥)=√𝑥₁𝑥exp¹.1𝑥₁+100𝑥₂+𝑥₃,𝑥=𝑥₁,𝑥₂,𝑥₃^𝑇,21𝐴=10−100−1015−.50−12,𝑝=.5100,𝛾=2 (4.2)(4.3) which, as it can be easily checked, satisfy the hypotheses of Theorem 3.3. In Tables 1 and 2, we report the number of iterations we performed in order to compute the fixed point 𝑞^∗ of problems (4.2) and (4.3), respectively, for different choices of the starting value 𝑞₀. Moreover, we also report the number of iterations performed by the Matlab routine fzero. This is given by the sum of the number of iterations to find an interval containing the zero (first addendum in Table 1) and the number of zero-finding iterations (second addendum in Table 1).

Table 1 

Table 2 

From these two tests, we observe that, according to Theorem 3.3, the FP iteration method converges for any 𝑞₀ in both the cases. Of course, when we know the interval where a zero lies, fzero does converge and it requires much less iterations, but in many cases this information is unavailable and, see Table 2, fzero fails to provide a value. In fact, at the third line of Table 2, the symbol AB stays for “aborted” and means that the MATLAB routine does not succeed in finding an interval containing a sign change. The last line of Table 2 shows that the best results can be obtained by a combined solution strategy, which starts with a certain number of FP iterations (first addendum) and takes the final steps using fzero iterations (second addendum). At the fifth column, we take the right balance between FP and fzero iterations, while in the remaining columns, we displayed the minimum number of FP iterations which are necessary to get a value from fzero. Thus, we can fruitfully exploit the global convergence of our FP iteration method in order to create more efficient combined strategy with fzero. Of course, this could be done also by means of other well-known methods like Newton’s one.

In order to verify whether the convergence of the FP procedure could be ascribed to the Banach contraction theorem, we plotted the absolute value of the derivative of the function 𝐹 in (4.1) (not reported) and we noted that the starting value in bold, that is, 𝑞₀=.2, 𝑞₀=.4 in Table 1 and 𝑞₀=.05, 𝑞₀=.1 in Table 2, belongs to intervals where |𝐹^′(𝑞)|>1. Hence we can assert that, at least in these cases, the convergence of the FP method is not “helped” by the contraction principle.

As we already wrote in Remark 3.4, from these numerical experiments and many others, starting from very different values of 𝑞₀, the method seems to converge to the same fixed point. For example, in problems (4.2) and (4.3), we obtain 𝑞^∗≈4.7954 and 𝑞^∗≈2.6085, respectively, for a very large number of initial guesses (much larger than the ones reported in Tables 1 and 2). This numerical evidence let us conjecture that the assumptions in Theorem 3.3 are also sufficient to assure the uniqueness of the solution of problem (1.2).

In Figures 1 and 2, we plotted the first 100 elements of the error sequence {|𝑞_𝑟−𝑞^∗|}, where 𝑞_𝑟 is computed by applying the FP iterations to the following two problems:𝑁=3,𝜑(𝑥)=𝑥₁sin²20𝑥₁.1𝑥₁+𝑥₂+𝑥₃,𝑞^∗2=1.1210,𝐴=10−100−1015−.50−12,𝑝=.5100,𝛾=1;𝑁=1,𝜑(𝑥)=𝑥^3/2,𝑞^∗3=0.0898,𝐴=100,𝑝=2,𝛾=2. (4.4)(4.5) It can be easily seen that both the problems do not satisfy the hypotheses of Theorem 3.3. To be more precise, (4.4) does not fulfill (v), whereas (4.5) does not verify (iv), having 𝛾>1. The figures clearly show that the method fails to converge, suggesting us that the assumptions of our main theorem are not too restrictive.

Figure 1 

The error sequence {|𝑞𝑟−𝑞∗|}𝑟=1,…,100 for problem (4.4).

Figure 2 

The error sequence {|𝑞𝑟−𝑞∗|}𝑟=1,…,100 for problem (4.5).

Moreover, we point out that, in our experience, some problems arising in the the kinetic theory of dusty plasmas, are more cumbersome than (1.1), and therefore, the application of Newton or other available iterative procedures is not always simple, convenient, or possible. A comparison between FP and other iteration processes on problem of type (1.1) will be the subject of future investigations. Finally, in order to test the performance of our method for larger systems arising from the applications, we discretized a problem of type (1.1) with stepsize ℎ on a sufficiently large interval (0,𝑇), with 𝑇=𝑁ℎ. In this way, we get a nonlinear system of type (1.2), which has dimension 𝑁. By applying FP iteration method to such a system for different values of 𝑁, with 10²<𝑁<10⁵, we observe that the number of iterations does not vary with 𝑁 and so with ℎ, but only with the choice of the starting point 𝑞₀. Moreover, we underline that in this case the linear systems arising from the FP iterations are tridiagonal and then very easy to be solved.