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Application of the Brouwer and the Kakutani fixedpoint theorems to a discrete equation with a double singular structure
Fixed Point Theory and Applicationsvolume 2018, Article number: 24 (2018)
Abstract
Applying the method consisting of a combination of the Brouwer and the Kakutani fixedpoint theorems to a discrete equation with a double singular structure, that is, to a discrete singular equation of which the denominator contains another discrete singular operator, we prove that the equation has a solution.
Introduction
A large number of discrete models are constructed in natural and social sciences. Many of them are expressed in terms of various kinds of discrete nonlinear equations (DNEs). It is important to study the DNEs. Several DNEs have been studied mathematically (see, e.g., [1–4] and [13]), but many DNEs have not been studied fully (see, e.g., [9, pp. 13–15]). In particular, it is very difficult to study discrete singular equations (DSEs), and there have been only a few studies on DSEs (see, e.g., [11, 12], and [16]).
Fixedpoint theory can play an indispensable role in overcoming the difficulties thus encountered. Moreover, it helps its own progress to apply fixedpoint theory to various DSEs. In light of the close and cooperative interaction between fixedpoint theory and DSEs, we find it beneficial to broaden the application of fixedpoint theory to new DSEs.
On the basis of these considerations, we are concerned with applying fixedpoint theory to the existence of a solution to the following new DSE:
Here \(x=x(i)\) denotes an unknown function contained in L, where L denotes the set of all realvalued functions of \(i\in D\). By D we denote the set of all integers i such that \(1\leq i\leq N\), that is, \(D:=\{ 1,\ldots,N\}\), where N is an arbitrary integer such that \(N\geq2\). We can regard L as an Ndimensional Euclidean space, and each element of L can be regarded as a point of the Euclidean space. However, for convenience, we refer to each element of L not as point but as function of \(i\in D\). We define \(f_{k}(u,v)=f_{k}(u,v)(i)\), \(k=1,2\), as the following operators which act on \((u,v)=(u(i),v(i))\in L\times L\):
where \(K(i,j) \) is a known function of \((i,j)\in D\times D\), \(b_{k}(j)\in L\), \(k=1,2\), are given functions of \(j\in D\), and \(c_{k}\), \(k=1,2\), are known constants. We define \(a(x)=a(x)(i)\) as an operator that has the following form:
where \(a_{i}(r)\) is a known function of \(r\in\mathbb{R}\) that satisfies the following conditions for each \(i\in D\):
It follows from (3)–(7) and (2) with \(k=2\) that \(f_{2}(x,a(x))\) is a discrete singular operator acting on \(x=x(i)\). Moreover, observing operator (2) with \(k=1\), we see that the righthand side of (1) is a discrete singular operator of which the denominator contains the discrete singular operator \(f_{2}(x,a(x))\). Hence, we can say that (1) has a double singular structure.
A large number of DSEs with the same singular structure as (1) have been constructed in spatial economics. Hence DSE (1) is not a special one. However, fixedpoint theory has not been fully applied to such DSEs. Hence, in this paper, applying fixedpoint theory to DSE (1), we prove that DSE (1) has a solution. The main result of this paper is Theorem 1 that is stated in Sect. 3.
Methods
We propose a method consisting of a combination of the Brouwer and the Kakutani fixedpoint theorems. Making use of the method, we prove that DSE (1) has a positive solution. This new method is widely applicable to DSEs with double singular structure (this application is discussed in the Appendix).
In this paper we impose no condition on (3) in addition to (4)–(7). In Sect. 3, we impose conditions on \(K(i,j)\), \(c_{k}\), \(k=1,2\), and \(b_{k}(j)\), \(k=1,2\), and we state and discuss Theorem 1. In Sect. 4 we prove estimates for the discrete singular operator contained in DSE (1). In Sect. 5, making use of the Brouwer fixedpoint theorem, we extend this discrete singular operator to a setvalued operator with no singularity. In Sect. 6, applying the Kakutani fixedpoint theorem to this setvalued operator, we prove Theorem 1. Section 7 is the conclusion section.
In this article, we make use of no advanced knowledge of DSEs and fixedpoint theory. Indeed we use the Brouwer and the Kakutani fixedpoint theorems, but they are ones of the most fundamental fixedpoint theorems (see, e.g., [5] and [8]). In the Appendix we make use of no advanced knowledge of spatial economics. Hence, this article can be easily understood even without having advanced knowledge of DSEs, fixedpoint theory, and spatial economics.
Results and discussion
In what follows, throughout the paper, we assume that
Indeed condition (9) is restrictive, but conditions (8) and (10)–(12) are sufficiently general. By (3) and (9), we can rewrite DSE (1) as follows:
where
Applying (7) to (15), we see that (14) is a discrete singular operator acting on \(x=x(i)\). This discrete singular operator is contained in the denominator of the righthand side of (13). Hence, even if we assume condition (9), then DSE (13) retains the double singular structure.
We define the following norm in L:
We define the following closed subset of L:
We divide this subset into two disjoint subsets as follows:
where
It follows from (8) and (10) that
where
Making use of (17) and (21), we define the following simplex contained in \(L_{0+}\):
Applying (5) to (15), and recalling definition (20), we see that
Applying this result, (10), and (11) to (14), we see that
Hence, we see that
However, recalling (6), we see easily that
Hence, we define that if \(x(i)\in L_{+}\) satisfies (13) for each \(i\in D\), then \(x(i)\) is a solution to (13). The following theorem is the main result of this paper.
Theorem 1

(i)
Equation (13) has a solution \(x=x(i)\in L_{+}\).

(ii)
If equation (13) has a solution \(x=x(i)\in L_{+}\), then
$$ x=x(i)\in S. $$(28)
Equation (13) is a new DSE that has not been fully studied. Theorem 1(i) implies that (13) has a positive solution. It follows from Theorem 1(ii) that all positive solutions are contained in the simplex (23). Theorem 1 is proved in Sect. 6.
Theorem 1 is widely applicable to many DSEs constructed in spatial economics, since those DSEs have the same double singular structure as (13). For example, applying Theorem 1 to DSEs [6, (5.3)–(5.5)] and [6, (7.1)–(7.8), (7.14)–(7.17), (14.1)–(14.12), (15.1)–(15.4), (15 A.1)–(15 A.10), (16.1)–(16.8)], we can prove that there exist positive solutions to the DSEs. This application is fully discussed in the Appendix.
Indeed we could prove the existence of solutions to DSE [6, (5.3)–(5.5)] more easily than Theorem 1 [14, 15]. However, it is difficult to apply the method developed in [14] and [15] to (13), since this method greatly depends on spatial economic properties of the known functions contained in DSE [6, (5.3)–(5.5)]. In order to prove Theorem 1, we need the method developed in this paper.
Estimates for operators
Replacing an unknown function x by u and v in the righthand side of (13), we define the following operator acting on u and v:
Making use of this operator, we rewrite (13) as follows:
Multiplying both sides of (30) by (15) with \(r=x(i)\), we obtain
where
It follows from (24) that
Hence, we have only to seek a fixed point of \(F_{2}(x,x)\) in \(L_{+}\).
It follows from (8), (17), and (20) that
Hence, observing (29) and (32), in the same way as (26) and (27) we see easily that
but that
Let us obtain estimates for \(F_{k}(u,v)\), \(k=1,2\), when
Lemma 1
If (37) holds, then
Proof
Applying (4), (5), (11), (15), (24), (25), (34), and (37) to (29) and (32), we obtain (38) and (39). □
The following lemma is a key lemma of this paper (see (21)).
Lemma 2
If (37) holds, then
where
Proof
Substituting (29) and (32) in the lefthand side of (40), we see that
Exchange \(\sum_{i\in D}\) and \(\sum_{j\in D}\), and apply (12) to the righthand side. Recalling definition (14) with \(x=v\), we see that the righthand side of the equality thus obtained contains both \(g(v)(j)\) and \(1/g(v)(j)\), which cancel each other out. Hence, we see that
Substituting (21) and (22) in the righthand side, we obtain (40). □
Let us discuss this key lemma. Observing (14) and (29), we see that (32) is expressed in terms of the double summation. Hence, the lefthand side of (40) is expressed in terms of the triple summation with double singular structure. However, the righthand side of (40) is expressed in terms of the single summation (41) with no singularity. By (40) we overcome the difficulty caused by the double singular structure of (13). We make use of (40) to prove Lemma 3(i). We make use of Lemma 3(i) to prove Theorem 1(i). It is difficult to prove that (29) satisfies a useful equality similar to (40). This is the reason why we define (32) instead of (29).
Recalling definition (23), we see easily that
Combining this inclusion relation and (35), we see that
Let us obtain estimates for (32) when
Lemma 3

(i)
If (44) holds, then
$$ F_{2}(u,v)\in S\cap L_{+}. $$(45) 
(ii)
If (44) holds and \(d_{k}\subseteq D\), \(k=1,2\), are nonempty, then
$$ 0< F_{2}(u,v) (i)\leq\biggl(\frac{c_{1}}{c_{2}}\biggr) \biggl( \frac{\mathbf{b}+\mathbf {b}_{1}}{ \vert d_{2} \vert }\biggr) \biggl(\frac{\overline{K}}{\underline{K}}\biggr) \biggl( \frac {A_{i}(v(i))}{\underline{A}(d_{2},v)}\biggr)\quad \textit{for all }i\in d_{1}, $$(46)
where
and we denote the number of points of \(d\subseteq D\) by
Proof
Applying (10) and definition (23) to (41), we deduce that
Substituting this equality in (40), recalling definition (23), and making use of (38) with \(k=2\), we obtain (45).
Let us prove (ii). It follows from (11) and (47) that
It follows from (44) that (24) holds. Applying (10), (24), and (51) to (14), we see that each term of the righthand side of (14) with \(x=v\) is positive. Hence, replacing \(\sum_{j\in D}\) and \(K(i,j)\) with \(\sum_{j\in d_{2}}\) and \(\underline{K}\), respectively, in the righthand side of (14) with \(x=v\), we obtain
Applying (16), (24), (48), and (49) with \(d=d_{2}\) to this inequality, we deduce that
Making use of (10), (34), (51), and (52), we see easily that each term of the righthand side of (29) is positive. Applying this result, (24), and (51) to (29) and (32), we obtain
Applying (22), (23), and (53) to this inequality when \(i\in d_{1}\subseteq D\), we obtain (46). Note that the righthand side of (46) is independent of \(u\in S\). □
Setvalued operators
The purpose of this section is to extend the discrete operator (32), which has the double singular structure, to a setvalued operator with no singularity.
Let \(v\in S\cap L_{+}\) be fixed. By (43) we regard \(F_{2}(u,v)\) as an operator acting on \(u\in S\) for each fixed \(v\in S\cap L_{+}\). Making use of (45), we see that this operator is an operator from S to \(S\cap L_{+}\). We refer to the operator thus defined as the partially fixed operator. We denote it by the same symbol \(F_{2}(u,v)\). No confusion should arise. By (43) and (45), we define \(\mathbf{F}(v)\) as a setvalued operator that maps \(v\in S\cap L_{+}\) to the set of all fixed points of the partially fixed operator \(F_{2}(u,v)\) as follows:
Definition 1
In the next section, we apply the Brouwer fixedpoint theorem to the partially fixed operator \(F_{2}(u,v)\) for each \(v\in S\cap L_{+}\) in order to prove that \(\mathbf{F}(v)\) is nonempty for every \(v\in S\cap L_{+}\).
Recalling (36) with \(k=2\), we find it difficult to define (54) for each \(v\in S\cap L_{0}\). In order to overcome such a difficulty, we define \(\mathbf{F}(v)\) for each \(v\in S\cap L_{0}\) as follows:
Definition 2
where
Recalling (18), we see that \(\mathbf{F}(v)\) defined in (54) and (55) is a setvalued operator from S to \(2^{S}\).
Conditions of the Kakutani fixedpoint theorem
Making use of the following lemma, we apply the Kakutani fixedpoint theorem to the setvalued operator \(\mathbf{F}(v)\).
Lemma 4

(i)
S is a nonempty, compact, and convex subset of the Ndimensional Euclidean space L.

(ii)
\(\mathbf{F}(v)\) is nonempty for every \(v\in S\cap L_{+}\).

(iii)
\(\mathbf{F}(v)\subseteq S\cap L_{+}\) for all \(v\in S\cap L_{+}\).

(iv)
\(\mathbf{F}(v)\) is nonempty for every \(v\in S\cap L_{0}\).

(v)
\(\mathbf{F}(v)\subseteq S\cap L_{0}\) for all \(v\in S\cap L_{0}\).

(vi)
\(\mathbf{F}(v)\) is a convex subset of S for every \(v\in S\).

(vii)
\(\mathbf{F}=\mathbf{F}(v)\) has a closed graph.

(viii)
\(\mathbf{F}=\mathbf{F}(v)\) has no fixed point in \(S\cap L_{0}\).
Proof of Lemma 4(i)–(vi)
Applying (17) and (21) to (23), we see that S is a simplex contained in \(L_{0+}\). Hence, we obtain (i).
Let \(v\in S\cap L_{+}\) be fixed. Making use of (42), (45), and (39) with \(V=v\), we see that the partially fixed operator \(F_{2}(u,v)\) is a continuous operator from S to \(S\cap L_{+}\subset S\) for each fixed \(v\in S\cap L_{+}\). Hence, making use of (i) of this lemma, we apply the Brouwer fixedpoint theorem to the partially fixed operator \(F_{2}(u,v)\) for each fixed \(v\in S\cap L_{+}\). Hence, for each \(v\in S\cap L_{+}\), there exists \(u_{+}\in S\) such that \(u_{+}=F_{2}(u_{+},v)\). Recalling (54), we obtain (ii).
Assume that \(u\in S\) and \(v\in S\cap L_{+}\) satisfy that \(u\in\mathbf {F}(v)\). Applying this assumption to (54) and applying (45) to the righthand side of the equality mentioned in (54), we see that \(u\in S\cap L_{+}\). Hence, we obtain (iii).
Considering definitions (19), (21), (23), (56), and (57), we see easily that
Making use of this result and recalling (49), we define the following function for each \(v\in S\cap L_{0}\):
Recalling (23), (56), and (57), we see easily that \(u_{0}\in S\) and \(D_{0}(u_{0})=D_{+}(v)\). Applying (55) to this result, we see that \(u_{0}\in\mathbf{F}(v)\). Hence, we obtain (iv).
Assume that \(u\in S\) and \(v\in S\cap L_{0}\) satisfy that \(u\in\mathbf {F}(v)\). Applying (58) to (55), we see that the righthand side of the inclusion relation mentioned in (55) is nonempty. Hence, the lefthand side \(D_{0}(u) \) is nonempty. Recalling definitions (19) and (56), we see that \(u\in S\cap L_{0}\). Hence, we obtain (v).
Let \(v\in S\cap L_{+}\) be fixed. Observing (29) and (32), we see easily that the partially fixed operator \(F_{2}(u,v)\) is linear with respect to \(u(j)+b_{1}(j)\). Applying this result to the equality mentioned in (54), we easily obtain (vi) when \(v\in S\cap L_{+}\). Assume that \(v\in S\cap L_{0}\). Considering definition (56) and the inclusion relation mentioned in (55), we easily obtain (vi) when \(v\in S\cap L_{0}\). □
Proof of Lemma 4(vii)
Assume that \(\{(u^{n},v^{n})\}\) is a convergent sequence such that
Applying (i) of this lemma to this sequence, we see that
where
Applying definition (57) to (62), we see that there exists \(n_{0}>0\) such that if \(n\geq n_{0}\), then
We have only to prove that
Let us prove (65) when
Applying definitions (20) and (57) to (66), we see that \(D_{+}(v^{\infty})=D\). Substituting this equality in (64), we deduce that \(D_{+}(v^{n})=D\) for all \(n\geq n_{0}\), that is, that \(v^{n}\in L_{+}\) for all \(n\geq n_{0}\). Combining this result and (59), we see that
Applying this result, (iii) of this lemma, and (54) to (60), we deduce that if \(n\geq n_{0}\), then
Making use of (42), (61), (62), (66), and (69), we can substitute
in (39). Applying the equality thus obtained and (62) to (68), we deduce that
Applying this equality, (61), and (66) to (54), we obtain (65) when (66) holds.
Let us prove (65) when
Making use of (18) and (iii), (v) of this lemma, we can divide the convergent sequence (59) into two disjoint subsequences as follows:
where
At least one of these subsequences is a convergent infinite sequence.
Assume that (73) is a convergent infinite sequence. Applying (55) to (73), we see that
Taking the complements of both sides of (63), we deduce that if \(n\geq n_{0}\), then
Recalling that (73) is a subsequence of (59), we replace \((u^{n},v^{n})\) with \((u_{0}^{n},v_{0}^{n})\) in (64) and (76). Combining the inclusion relations thus obtained and (75), we see that
Applying (55), (61), and (71) to this inclusion relation, we obtain (65).
Assume that (74) is a convergent infinite sequence. Applying (54) to (74), we obtain
We need to obtain (65) from (78) when (71) holds. Noting that (37) is imposed on (39), we see that \((U,V,u,v)=(u_{+}^{n},v_{+}^{n},u^{\infty},v^{\infty})\) cannot be substituted in (39). Hence, we find it difficult to perform the same calculations as done when obtaining (70) from (68). In order to overcome this difficulty, we make use of (46) instead of (39).
Making use of (71), (74), and (58) with \(v=v^{\infty}\), we can substitute
in (44) and (46). Recall that (74) is a convergent infinite subsequence of (59). Considering definition (56) with \(v=v^{\infty}\), and applying (7) and (62) to (15) with \(r=v_{+}^{n}(i)\), where \(i\in D_{0}(v^{\infty })\), we see that
Applying this result to (48) with (79), we see easily that
Considering definition (57) with \(v=v^{\infty}\) and making use of (4), (5), (62), and (64), we deduce that
Applying this result and (62) to (15) with \(r=v_{+}^{n}(i)\), where \(i\in D_{+}(v^{\infty})\), we deduce that
Applying this result and (80) to (46) with (79), we see that
Applying this result and (62) to (78), we see that \(u^{\infty}(i)=0\) for all \(i\in D_{+}(v^{\infty})\), that is, that (77) holds. Applying (55), (61), and (71) to (77), we obtain (65). Therefore we obtain (vii). □
Proof of Lemma 4(viii)
Assume that there exists \(v_{0}\in S\cap L_{0}\) such that \(v_{0}\in \mathbf{F}(v_{0})\). Applying this assumption to (55), we see that \(D_{0}(v_{0})\supseteq D_{+}(v_{0})\). Applying (58) with \(v=v_{0}\) to this inclusion relation, we deduce that both sides of this inclusion relation are nonempty. Recalling definitions (56) and (57), we see that this inclusion relation leads us to a contradiction. Hence, we obtain (viii). □
Proof of Theorem 1
Making use of Lemma 4(i), (ii), (iv), (vi), (vii) and (18), we apply the Kakutani fixedpoint theorem to \(\mathbf{F}=\mathbf{F}(v)\). Hence, we see that there exists \(x=x(i)\in S\) such that \(x\in\mathbf{F}(x)\). Applying (18), (54), and Lemma 4(viii) to this result, we deduce that \(x=x(i)\) is contained in \(S\cap L_{+}\) and satisfies (31). Recalling (33), we obtain (i).
Note that not (44) but (37) is imposed on (40). Assume that \(x\in L_{+}\) satisfies (31). Substituting (31) in the lefthand side of (40) with \(u=v=x\), we see that
Substituting (41) with \(u=x\) in the righthand side of this equality, making use of (10), and recalling definition (23), we obtain (28). □
Conclusions
Applying the method consisting of the combination of the Brouwer and the Kakutani fixedpoint theorems to the discrete equation with double singular structure, we prove that the equation has a positive solution and that all positive solutions to the equation are contained in the simplex (23) (Theorem 1).
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Acknowledgements
The authors would like to express their deepest gratitude to Professor Yoshitsugu Kabeya for his valuable suggestions.
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Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study. Materials sharing is not applicable to this article as no materials were generated or analyzed during the current study.
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Minoru Tabata and Nobuoki Eshima are supported in part by Grantinaid for Scientific Research of Japan (15K05005, 26330045).
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Appendix
Appendix
Spatial economics is an interdisciplinary area between economics and geography. In about 1990, Krugman began seminal research in this area. His research has since grown into one of the major branches of spatial economics, and now it is known as the New Economic Geography (NEG) (see, e.g., [6] and [7]). In 2008, Krugman was awarded the Nobel Memorial Prize in Economic Sciences for his great contribution to spatial economics [10].
Many discrete models have been constructed in the NEG. Among those models, the Krugman’s coreperiphery model (KCP model) is one of the most important models. In the KCP model, economic activities are conducted at each point of a finite set of points, and each point represents a region. The economy consists of agriculture and manufacturing, and the population consists of farmers and workers. The KCP model is described by a DSE that is referred to as the wage equation [6, pp. 61–95].
Let us apply the result of this paper to the wage equation. We assume that the finite set of points is equal to D defined in Sect. 1, that is, that each integer contained in D represents a region. Substitute
in (13)–(15), where \(w(i)\) denotes an unknown function contained in \(L_{+}\), \(\lambda(i)\), \(\phi(i)\), and \(T(i,j)\) are known functions of \(i,j\in D\), and σ and μ are known constants. In spatial economics, \(w(i)\) denotes the distribution of nominal wages, and σ and μ denote the elasticity of substitution and the manufacturing expenditure, respectively. Moreover, \(\lambda(i)\), \(\phi(i)\), and \(T(i,j)\) denote the distribution of workers, the distribution of farmers, and the transportcost function, respectively. Then we obtain the following DSE, which is the wage equation [6, (5.3)–(5.5)]:
where
We assume that
These assumptions are the most general conditions in spatial economics [6, pp. 46–49]. It follows from these conditions that (81)–(83) satisfy (4)–(12). Hence, applying Theorem 1(i) to the wage equation (84), we see that (84) has a positive solution \(w(i)\in L_{+}\).
Performing calculations similar to those done above, we can prove that there exist positive solutions to DSEs [6, (7.1)–(7.8), (7.14)–(7.17), (14.1)–(14.12), (15.1)–(15.4), (15 A.1)–(15 A.10), (16.1)–(16.8)], since these DSEs are constructed as extensions of the KCP model.
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Keywords
 Positive solution
 Setvalued operator
 Discrete singular equation
 The Brouwer fixedpoint theorem
 The Kakutani fixedpoint theorem