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Application of the Brouwer and the Kakutani fixed-point theorems to a discrete equation with a double singular structure
- Minoru Tabata^{1}Email authorView ORCID ID profile and
- Nobuoki Eshima^{2}
https://doi.org/10.1186/s13663-018-0649-9
© The Author(s) 2018
- Received: 3 July 2018
- Accepted: 9 October 2018
- Published: 1 November 2018
Abstract
Applying the method consisting of a combination of the Brouwer and the Kakutani fixed-point 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.
Keywords
- Positive solution
- Set-valued operator
- Discrete singular equation
- The Brouwer fixed-point theorem
- The Kakutani fixed-point theorem
1 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]).
Fixed-point theory can play an indispensable role in overcoming the difficulties thus encountered. Moreover, it helps its own progress to apply fixed-point theory to various DSEs. In light of the close and cooperative interaction between fixed-point theory and DSEs, we find it beneficial to broaden the application of fixed-point theory to new DSEs.
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 right-hand 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, fixed-point theory has not been fully applied to such DSEs. Hence, in this paper, applying fixed-point 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.
2 Methods
We propose a method consisting of a combination of the Brouwer and the Kakutani fixed-point 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 fixed-point theorem, we extend this discrete singular operator to a set-valued operator with no singularity. In Sect. 6, applying the Kakutani fixed-point theorem to this set-valued operator, we prove Theorem 1. Section 7 is the conclusion section.
In this article, we make use of no advanced knowledge of DSEs and fixed-point theory. Indeed we use the Brouwer and the Kakutani fixed-point theorems, but they are ones of the most fundamental fixed-point 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, fixed-point theory, and spatial economics.
3 Results and discussion
Theorem 1
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.
4 Estimates for operators
Lemma 1
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
Proof
Let us discuss this key lemma. Observing (14) and (29), we see that (32) is expressed in terms of the double summation. Hence, the left-hand side of (40) is expressed in terms of the triple summation with double singular structure. However, the right-hand 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).
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)
Proof
5 Set-valued operators
The purpose of this section is to extend the discrete operator (32), which has the double singular structure, to a set-valued 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 set-valued 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 fixed-point 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
Recalling (18), we see that \(\mathbf{F}(v)\) defined in (54) and (55) is a set-valued operator from S to \(2^{S}\).
6 Conditions of the Kakutani fixed-point theorem
Making use of the following lemma, we apply the Kakutani fixed-point theorem to the set-valued operator \(\mathbf{F}(v)\).
Lemma 4
- (i)
S is a nonempty, compact, and convex subset of the N-dimensional 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 fixed-point 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 right-hand side of the equality mentioned in (54), we see that \(u\in S\cap L_{+}\). Hence, we obtain (iii).
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 right-hand side of the inclusion relation mentioned in (55) is nonempty. Hence, the left-hand 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)
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 fixed-point 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).
7 Conclusions
Applying the method consisting of the combination of the Brouwer and the Kakutani fixed-point 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).
Declarations
Acknowledgements
The authors would like to express their deepest gratitude to Professor Yoshitsugu Kabeya for his valuable suggestions.
Availability of data and materials
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.
Funding
Minoru Tabata and Nobuoki Eshima are supported in part by Grant-in-aid for Scientific Research of Japan (15K05005, 26330045).
Authors’ contributions
Each author equally contributed to this article, read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Anastassiou, G.A., Kester, M.: Discrete Approximation Theory. Vol. 20. World Scientific, Singapore (2016) MATHGoogle Scholar
- Askhabov, S.N., Karapetyants, N.K.: Discrete equations of convolution type with monotone nonlinearity. Differ. Uravn. 25(10), 1777–1784 (1989) MathSciNetMATHGoogle Scholar
- Askhabov, S.N.: Approximate solution of nonlinear discrete equations of convolution type. J. Math. Sci. 201(5), 566–580 (2014) MathSciNetView ArticleGoogle Scholar
- Askhabov, S.N., Karapetyants, N.K.: Discrete equations of convolution type with a monotone nonlinearity in complex spaces. Dokl. Math. 45(1), 206–210 (1992) MathSciNetMATHGoogle Scholar
- Farmakis, I., Moskowitz, M.: Fixed Point Theorems and Their Applications. World Scientific, Singapore (2013) View ArticleGoogle Scholar
- Fujita, M., Krugman, P.R., Venables, A.: The Spatial Economy: Cities, Regions, and International Trade. MIT Press, Cambridge (2001) MATHGoogle Scholar
- Fujita, M., Thisse, J.-F.: Economics of Agglomeration: Cities, Industrial Location, and Globalization. Cambridge University Press, Cambridge (2013) View ArticleGoogle Scholar
- Granas, A., Dugundji, J.: Fixed Point Theory. Springer, Berlin (2013) MATHGoogle Scholar
- Hritonenko, N., Yatsenko, Y.: Mathematical Modeling in Economics, Ecology and the Environment. Kluwer Academic, Dordrecht (1999) View ArticleGoogle Scholar
- Krugman, P.: The Official Homepage of the Nobel Prize in Economic Sciences (2008) http://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/ Google Scholar
- Lewis, F.L., Mertzios, B.G.: On the analysis of discrete linear time-invariant singular systems. IEEE Trans. Autom. Control 35(4), 506–511 (1990) MathSciNetView ArticleGoogle Scholar
- Mendez, V., Fedotov, S., Horsthemke, W.: Reaction-Transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities. Springer, Berlin (2010) View ArticleGoogle Scholar
- Moadab, M.H.: Existence of bounded solutions to nonlinear discrete equations. Libertas Math. 9, 127–132 (1989) MathSciNetMATHGoogle Scholar
- Tabata, M., et al.: The existence and uniqueness of short-run equilibrium of the Dixit–Stiglitz–Krugman model in an urban–rural setting. IMA J. Appl. Math. 80(2), 474–493 (2015) MathSciNetView ArticleGoogle Scholar
- Tabata, M., Nobuoki, E.: Existence of a short-run equilibrium of the Dixit–Stiglitz–Krugman model. Discrete Dyn. Nat. Soc. 2018, Article ID 2193070 (2018) MathSciNetView ArticleGoogle Scholar
- Vasil’ev, A.V., Vasil’ev, V.B.: On the solvability of certain discrete equations and related estimates of discrete operators. Dokl. Math. 92(2), 585–589 (2015) MathSciNetView ArticleGoogle Scholar