Multiple solutions of ordinary differential systems with min-max terms and applications to the fuzzy differential equations
- Yicheng Liu^{1}Email author and
- Jun Wu^{2}
https://doi.org/10.1186/s13662-015-0708-z
© Liu and Wu 2015
Received: 17 July 2015
Accepted: 23 November 2015
Published: 18 December 2015
Abstract
In this paper, we investigate the existence of multiple solutions for a class of ordinary differential systems with min-max terms. We present two fundamental results for the existence of solutions. An illustrative example shows that the uniqueness of solution does not hold although the Lipschitz condition is added. Finally, there are some nontrivial applications of the considered theory to fuzzy differential equations with the generalized Hukuhara derivative.
Keywords
MSC
1 Introduction
Crisp differential equations are popular models to approach various phenomena in the real world when the conditions/hypotheses are clear. If the occurrence of the phenomenon or the initial data is not precisely known, fuzzy differential equations [1–3] and stochastic fuzzy differential equations [4–8] appear to be a natural way to model the aleatory and epistemic uncertainty. For example these equations are used to model cell growth and the dynamics of populations [4, 6–9], dry friction [10], tumor growth [11], and the phenomenon of nuclear disintegration [12] and the transition from HIV to AIDS [13] under uncertainty. It would have better application prospect to investigate the foundational theory of fuzzy differential equation deeply. Especially, there are many approaches to interpret a solution for various fuzzy differential equations. As for most differential equations, the Lipschitz condition is a popular assumption as regards the uniqueness of the local solution. Also, the relationship between crisp differential equations defined in terms of the Hukuhara derivative and set differential equations was investigated by Lakshmikantham et al. [14]. As we observe, the uniqueness of the solution does not hold although the Lipschitz condition is added to the ordinary differential equation with min-max terms.
In this paper, we consider the differential system (1) with min-max terms subject to the initial value conditions \(u(0)=u_{0}\), \(v(0)=v_{0}\), where \(I=[0,1]\) is an interval, \(F,G \in C(I \times R\times R, R)\). We say the pair \((u(t),v(t))\) is a solution of system (1) if \(u,v \in C(I)\) and \(u(t)\), \(v(t)\) satisfy equations (1) and the initial value conditions.
2 Existence results
Theorem 2.1
Proof
Theorem 2.2
Proof
Remark 2.1
The popular examples for the functions F and G in Theorem 2.2 are \(F(t,x,y)=\min\{f_{1}(t,x,y), \ldots,f_{k}(t,x,y)\}\) and \(G(t,x,y)=\max\{g_{1}(t,x,y),\ldots,g_{k}(t,x,y)\}\), where \(f_{i}\) and \(g_{i}\) are Lipschitz continuous relative to their last two arguments. Following the Appendix, we see that both min function and max function are also Lipschitz continuous.
Next, we show an example to illustrate that the initial value problem (1) admits a solution on I although some assumptions in Theorem 2.1 and Theorem 2.2 do not hold.
Example 2.1
For equations (5) and (6), it is sufficient for the pair \((x(t), y(t))\) to be a solution of (4) that \(a x(t)+3\leq a y(t)+1\) for all \(t\in I\). Thus, if \(a(x_{0}-y_{0})+2\leq0\) then (5) is a solution of problem (4) in \([0,1]\). Similarly, if \(a(x_{0}-y_{0})\leq0\) then (6) is a solution of problem (4) in \([0,1]\). Thus for the problem (4), we can conclude the following.
Conclusion 2.1
If \(a(x_{0}-y_{0})\leq-2\), then the initial value problem (4) has two solutions on \([0,1]\), which are given by (5) and (6), respectively. If \(-2< a(x_{0}-y_{0})\leq0\), then the initial value problem (4) has a unique solution on \([0,1]\), which is given by (6). If \(a(x_{0}-y_{0})> 0\), then the initial value problem (4) has no solution on \([0,1]\).
3 Application to the fuzzy differential equation
- (i)
u is normal, i.e. there exists \(s_{0}\in R\) such that \(u(s_{0}) = 1\),
- (ii)
u is a convex fuzzy set (i.e. \(u(ts + (1-t)r ) \geq \min\{u(s), u(r )\}\), for \(t\in[0, 1]\), \(s, r\in R\)),
- (iii)
u is upper semicontinuous on R,
- (iv)
\(\operatorname{cl}\{s \in R|u(s) > 0\}\) is compact where cl denotes the closure of a subset.
Then \(\mathbb{R}_{F}\) is called the space of fuzzy numbers. For \(x, y \in \mathbb{R}_{F} \), if there exists a fuzzy number \(z \in\mathbb{R}_{F}\) such that \(y + z = x\), then z is called the H-difference of x, y and is denoted by \(x \ominus y \) (see e.g. [26]). If there exists a fuzzy number \(z \in\mathbb{R}_{F}\) such that \(y + z = x\) or \(y=x+(-1)z\), then z is called the gH-difference of x and y and is denoted by \(x \ominus_{g} y \) (see e.g. [17]).
Lemma 3.1
Definition 3.1
[17]
Example 3.1
Case 1. \(a>0\): If \(\underline{u}_{0\alpha}-\bar{u}_{0\alpha }\leq\frac{2\alpha-2}{a}(e^{a}-1)\), then both (8) and (9) are solutions of problem (7).
If \(\frac{2\alpha-2}{a}(e^{a}-1)<\underline{u}_{0\alpha}-\bar {u}_{0\alpha }\leq0 \), then there is a unique solution (8) on \([0,1]\).
Case 2. \(a<0\): If \(2(\alpha-1)(e^{a}-1)\leq a(\underline {u}_{0\alpha}-\bar{u}_{0\alpha})\leq2(1-\alpha)\), by direct computation, we see that there is a unique solution of problem (7) on \([0,1]\), which is given by (9).
Example 3.2
4 Two counterexamples
Theorem 4.1
([27], Theorem 1)
Theorem 4.2
([27], Theorem 2)
Example 4.1
Let \(a=0\), \(J=[0,2]\), \(k(t,s,u)\equiv0\), \(f(t,u(t))=3(t-2)^{2}\mu_{1}\), then all the assumptions in Theorem 4.1 hold. Unfortunately, there are no the proper solutions of the problem (15) on J. More details now follow.
Theorem 4.3
There is a unique mixed solution and no proper solution for (15) on J.
Proof
Example 4.2
5 Conclusions
As we know, the crisp differential equations are popular models to approach the various phenomena in the real world when the conditions/hypotheses are clear. Also, the stochastic fuzzy differential equation is a candidate to describe the occurrence of the phenomenon or the unknown initial data. In this paper, we build a relationship between differential system with min-max terms and fuzzy differential equations, and we investigate the existence and multiple solutions for a class of first-order differential system with min-max terms. As applications, the existence results for some linear fuzzy differential equations are obtained. Work in progress uses fixed point theorems for nonlinear operators to study the existence and multiple solutions of interval and fuzzy differential equations (see e.g. [28, 29]).
Declarations
Acknowledgements
This work was partially supported by National Natural Science Foundation of China (11201481 and 11301039).
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
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