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Multiple solutions of ordinary differential systems with minmax terms and applications to the fuzzy differential equations
Advances in Difference Equations volume 2015, Article number: 379 (2015)
Abstract
In this paper, we investigate the existence of multiple solutions for a class of ordinary differential systems with minmax 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.
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 minmax terms.
One of the earliest suggestions to define the concept of differentiability for fuzzy functions and, in consequence, to study fuzzy differential equations is the Hukuhara derivative [2, 15]. Naturally, the solution of fuzzy differential equations interpreted by the Hukuhara derivative became fuzzier as time went by [16]. Recently, there were two extensions of the concept of Hukuhara derivatives with different differences [1, 17]. The different definition of the Hukuhara derivative will lead to a different levelcut system. The new generalization of the Hukuhara difference, due to Stefanini [17], raising the fuzzy differential equations associated with the levelcut system involving the minmax terms, is as follows:
There are more results on the Hukuhara derivative in the references, for example in [18–23].
In this paper, we consider the differential system (1) with minmax 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.
Existence results
Theorem 2.1
Let \(F, G :I \times R\times R\rightarrow R\) be two continuous functions, F and G being Lipschitz continuous relative to their last two arguments, i.e. there exist real numbers \(L_{1}, L_{2}, L_{3}, L_{4}> 0\) such that, for \((t, u_{1}, v_{1}), (t, u_{2}, v_{2})\in I\times R \times R \),
Moreover, if \(f,g\in C(I)\) and \(f(t)\leq g(t)\) for all \(t\in I\) implies that \(F(t,f(t), g(t))\leq G(t,f(t), g(t))\), then the problem (1) has at least one solution on I provided \(u_{0}\leq v_{0}\).
Proof
Let \(u(t)=\int_{0}^{t}f(s)\,ds+u_{0}\) and \(v(t)=\int _{0}^{t}g(s)\,ds+v_{0}\), where \(f,g\in C(I)\). It is easy to see that the pair \((u(t),v(t))\) is a solution of (1) if and only if the pair \((f(t),g(t))\) is a solution of following group integral equations with minmax terms:
Define the successive iterations \(\{f_{n}(t)\}\) and \(\{g_{n}(t)\}\) by
Since \(u_{0}\leq v_{0}\), we have \(F(t,u_{0},v_{0})\leq G(t,u_{0},v_{0})\). Thus \(f_{1}(t)\leq g_{1}(t)\) for all \(t\in I\). Also, for \(t\in I\), by \(\int_{0}^{t}f_{1}(s)\,ds+u_{0}\leq\int_{0}^{t}g_{1}(s)\,ds+v_{0}\) and the assumptions, we see that \(f_{2}(t)\leq g_{2}(t)\) for all \(t\in I\). By mathematical induction, we conclude that \(f_{n}(t)\leq g_{n}(t)\) for all \(t\in I\) and \(n\in\mathbb{N}\).
Next, we would prove that the successive iterations \(\{f_{n}(t)\}\) and \(\{g_{n}(t)\}\) are uniformly convergent on I. To this end, we choose a constant \(c>0\) satisfying \(\frac{L_{1}+L_{2}+L_{3}+L_{4}}{c}<1\) and introduce the norm \(\\cdot\\) in \(C(I)\) by
Then \(x(t)\leq e^{ct} \x\\) for all \(t\in I\).
On the other hand, from the successive iterations, we see that
Thus
This implies that
Similarly, we have
From the above two inequalities, we see that
Let \(\gamma=\frac{L_{1}+L_{3}+L_{2}+L_{4}}{c}\), then \(\gamma<1\) and
Then, for any \(k\in\mathbb{N}\), we have
This means that both successive iterations \(\{f_{n}(t)\}\) and \(\{g_{n}(t)\} \) are uniformly convergent on I. Let \(\lim_{n\rightarrow\infty }f_{n}(t)=\bar{f}(t)\), \(\lim_{n\rightarrow\infty}g_{n}(t)=\bar{g}(t)\), then \(\bar{f}(t)\leq\bar{g}(t)\) for \(t\in I\). Also, we have
Then \(u(t)=\int_{0}^{t}\bar{f}(s)\,ds+u_{0}\), \(v(t)=\int_{0}^{t}\bar{g}(s)\,ds+v_{0}\) is a solution of problem (1) on I. The proof of Theorem 2.1 is complete. □
Define the successive iteration
then we have the following theorem.
Theorem 2.2
Suppose that \(F, G :I \times R\times R\rightarrow R\) are two continuous functions, Lipschitz continuous relative to their last two arguments. Moreover, if \(F(t,f(t), g(t))\leq G(t,f(t), g(t))\) for all \(f,g\in C(I)\) and \(t\in I\), then both the successive iteration pairs \((P_{n} (t), Q_{n}(t))\) and \((U_{n}(t),V_{n}(t))\) converge to the solutions of problem (1) on I, where
Proof
It follows from the proof of Theorem 2.1 that the successive iteration pair \(\{(P_{n}(t), Q_{n}(t))\}\) converges to the solution of problem (1) on I. Thus it is sufficient to prove that the pair \(\{(U_{n}(t),V_{n}(t))\}\) converges to the solution of problem (1) on I. Since \(F(t,f(t), g(t))\leq G(t,f(t), g(t))\) for all \(f,g\in C(I)\) and \(t\in I\), we have
With similar arguments to the proof of Theorem 2.1, we see that \((u_{n}(t),v_{n}(t))\) uniformly converge to the continuous functions \(u(t)\) and \(v(t)\) on I. Thus the pair \((U_{n}(t),V_{n}(t))\) converges to the pair \((\int_{0}^{t}u(s)\,ds+u_{0},\int_{0}^{t}v(s)\,ds+v_{0})\), which is a solution of problem (1) on I. The proof is complete. □
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
Consider the following system with minmax terms:
where \(a\neq0\) is a constant.
By direct computations, the corresponding solutions of the subsystems
and
are given as follows:
and
respectively.
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 \(a(x_{0}y_{0})\leq2\), 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 \([0,1]\).
Application to the fuzzy differential equation
Let us denote by \(\mathbb{R}_{F}\) the class of fuzzy subsets of the real axis (i.e. \(u : R \rightarrow[0, 1]\)) satisfying the following properties [16, 24, 25]:

(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. \(u(ts + (1t)r ) \geq \min\{u(s), u(r )\}\), for \(t\in[0, 1]\), \(s, r\in R\)),

(iii)
u is upper semicontinuous on R,

(iv)
\(\operatorname{cl}\{s \in Ru(s) > 0\}\) 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 Hdifference 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 gHdifference of x and y and is denoted by \(x \ominus_{g} y \) (see e.g. [17]).
If \(v\in\mathbb{R}_{f}\) then the αlevel set
are closed bounded intervals, which is denoted \([v]^{\alpha}= [\underline {v}_{\alpha},\bar{v}_{\alpha}]\).
Lemma 3.1
For \(x, y\in\mathbb{R}_{f}\) and k be scalar. For \(\alpha \in(0,1]\),
Definition 3.1
[17]
For fuzzy value functions \(f:[a,b]\rightarrow\mathbb{R}_{f}\). If \(x\in(a,b)\) and \(x+h\in(a, b)\), the gHderivative of f at x is defined as the limit
provided that the gHdifferences exist for small h.
Also, the fuzzy derivative, in terms of levelcuts, is
Example 3.1
Consider the following fuzzy differential equation:
where \(a\in R\), \(u_{0}, \gamma\in\mathbb{R}_{F}\) are given by \([\gamma ]^{\alpha}=[\alpha1,1\alpha]\) and \([u_{0}]^{\alpha}=[\underline {u}_{0\alpha },\bar{u}_{0\alpha}]\) and \(\underline{u}_{0\alpha}<\bar {u}_{0\alpha}\) for some \(\alpha\in[0,1]\), ′ denotes the gHderivative.
By the definition of the gHderivative, the levelcut of a fuzzy derivative for a fuzzyvalued function \(u(t)\) should be given by
Thus the αlevelcut system of (7) should be
Noting the claims in Conclusion 2.1, we have
and
Obviously, if both \(a\underline{u}_{\alpha}(t)+\alpha1\leq a\bar {u}_{\alpha}(t)+1\alpha\) and \(\underline{u}_{\alpha}(t)\leq\bar {u}_{\alpha}(t)\) hold on \([0,1]\), then the pair \((\underline{u}_{\alpha}(t), \bar {u}_{\alpha}(t))\) would be a solution of (7) on \([0,1]\). Thus, for (8), if
for all \(t\in[0,1]\), \(\alpha\in[0,1]\), then it is a solution of problem (7). Also, if
for all \(t\in[0,1]\), \(\alpha\in[0,1]\), then (9) is a solution of problem (7). Next we would discuss the solutions of problems (7) by three cases: \(a>0\), \(a<0\), and \(a=0\).
Case 1. \(a>0\): If \(\underline{u}_{0\alpha}\bar{u}_{0\alpha }\leq\frac{2\alpha2}{a}(e^{a}1)\), then both (8) and (9) are solutions of problem (7).
If \(\frac{2\alpha2}{a}(e^{a}1)<\underline{u}_{0\alpha}\bar {u}_{0\alpha }\leq0 \), then there is a unique solution (8) on \([0,1]\).
Case 2. \(a<0\): If \(2(\alpha1)(e^{a}1)\leq a(\underline {u}_{0\alpha}\bar{u}_{0\alpha})\leq2(1\alpha)\), by direct computation, we see that there is a unique solution of problem (7) on \([0,1]\), which is given by (9).
Case 3. \(a=0\): In this case, it is easy to see that, when \(2(1\alpha)\leq\bar{u}_{0\alpha}\underline{u}_{0\alpha}\) for all \(\alpha\in[0,1]\), both
and
are solutions of problems (7) on \([0,1]\).
When \(2(1\alpha)> \bar{u}_{0\alpha}\underline{u}_{0\alpha}\) for all \(\alpha\in[0,1]\), then
is the unique solution of problems (7) on \([0,1]\).
Example 3.2
Consider the following fuzzy differential equation:
where \(J\subset R\) is an interval, \(a, u_{0}, \gamma\in\mathbb{R}_{F}\) are given by \([a]^{\alpha}=[\underline{a}_{\alpha},\bar{a}_{\alpha}]\) with \(\underline{a}_{\alpha}>0\), \([\gamma]^{\alpha}=[\alpha1,1\alpha]\) and \([u_{0}]^{\alpha}=[\underline{u}_{0\alpha},\bar{u}_{0\alpha}]\) and \(\underline{u}_{0\alpha}<\bar{u}_{0\alpha}\) for some \(\alpha\in[0,1]\), ′ denotes the gHderivative, ⊙ denotes the fuzzy product of two fuzzy number.
First of all, by the definitions of gHderivative and fuzzy product, we can translate the fuzzy equation (10) into the αlevelcuts systems as follows:
Since \(\underline{a}_{\alpha}>0\), we have \(\underline{a}_{\alpha}\underline {u}_{\alpha}(t)\leq\underline{a}_{\alpha}\bar{u}_{\alpha}(t)\) and \(\bar {a}_{\alpha}\underline{u}_{\alpha}(t)\leq\bar{a}_{\alpha}\bar {u}_{\alpha}(t)\) for all \(t\in J\). Thus
and
Thus the αlevelcuts systems can be rewriten as
It follows from Theorem 2.2 that both of the pairs \((P_{n} (t), Q_{n}(t))\) and \((U_{n}(t),V_{n}(t))\) converge to the solutions of the problem (11).
In fact, if \(\underline{a}_{\alpha}\underline{u}_{\alpha}(t)\leq\bar {a}_{\alpha}\underline{u}_{\alpha}(t)\) then \(\underline{u}_{\alpha}(t)\geq 0\). Thus \(\max\{\underline{a}_{\alpha}\bar{u}_{\alpha}(t), \bar{a}_{\alpha}\bar{u}_{\alpha}(t)\}=\bar{a}_{\alpha}\bar{u}_{\alpha}(t)\). Similarly, if \(\underline{a}_{\alpha}\bar{u}_{\alpha}(t)\geq \bar{a}_{\alpha}\bar{u}_{\alpha}(t)\) then \(\bar{u}_{\alpha}(t)\leq0\). Thus \(\max\{\underline{a}_{\alpha}\bar{u}_{\alpha}(t), \bar{a}_{\alpha}\bar{u}_{\alpha}(t)\}=\underline{a}_{\alpha}\bar {u}_{\alpha}(t)\). Then the αcuts system (11) can be reduced to the following subsystem:
or
Following similar arguments to Example 3.1, we see that there would be two solutions for the problems (12) and (13) subject to a certain additional restriction. The details can be found in Figure 1 and Figure 2.
Two counterexamples
Recently, Alikhani et al. [27] considered the following initial value problem for the firstorder fuzzy integrodifferential equation of Volterra type:
where \(J\subset R\) is an interval, \(f \in C(J \times\mathbb{R}_{F}, \mathbb{R}_{F} )\) and \(k\in C(J \times J\times\mathbb{R}_{F} , \mathbb {R}_{F})\). They established the following two results. Unfortunately, both of them are wrong. Two counterexamples are presented in this section.
Theorem 4.1
([27], Theorem 1)
Let \(f : J \times\mathbb{R}_{F} \rightarrow\mathbb{R}_{F}\) and \(k: J \times J \times\mathbb{R}_{F} \rightarrow\mathbb{R}_{F}\) be bonded continuous functions. Then the problem (14) has at least a proper solution which is (i)differentiable on J. Moreover, if f and k are Lipschitz continuous relative to their last argument, i.e. there exist real numbers \(L_{1}, L_{2} >0\) such that
Then the proper solution of the problem (14) is unique on J.
Theorem 4.2
([27], Theorem 2)
Let \(f : J \times\mathbb{R}_{F} \rightarrow\mathbb{R}_{F}\) and \(k: J \times J \times\mathbb{R}_{F} \rightarrow\mathbb{R}_{F}\) be bounded continuous and Lipschitz continuous functions as mentioned in Theorem 2.1. Let the sequence \(u_{n} : J \rightarrow \mathbb{R}_{F}\) given by
be defined for any \(n \in\mathbb{N}\). Then the problem (14) has a unique proper solution which is (ii)differentiable on J. Furthermore, the successive iteration
converges to this solution.
Example 4.1
Consider the following fuzzy differential equation:
where \(\mu_{0}\), \(\mu_{1}\) are given by
and
Let \(a=0\), \(J=[0,2]\), \(k(t,s,u)\equiv0\), \(f(t,u(t))=3(t2)^{2}\mu_{1}\), 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
Since \([\mu_{0}]^{\alpha}=[1\sqrt{1\alpha}, 1+\sqrt {1\alpha}]\) and \([\mu_{1}]^{\alpha}=[1\sqrt{1\alpha},1+\sqrt {1\alpha}]\), if u is (i)differentiable, then we transform equation (15) into the αlevel system
subject to the initial conditions
Then
Noting that \(\underline{u}^{\alpha}(t)\leq\bar{u}^{\alpha}(t)\), we see that \(t\in[1,2]\). This implies that the problem (15) has an (i)solution on \([1,2]\). Similarly, if u is (ii)differentiable, then we transform equation (15) into the αlevel system
subject to the same initial conditions (16). Then for \(t\in[0,1]\),
Then the problem (15) has a (ii)solution on \([0,1]\). Thus there is a unique mixed solution and no proper solution for (15) on J. □
Example 4.2
Let us consider the linear firstorder fuzzy differential equation
where \([\gamma]^{\alpha}=[\alpha1,1\alpha]\). By direct computations, all the assumptions in Theorem 4.1 and Theorem 4.2 hold for the problem (15); then there would have to be a unique proper solution. But there are two proper solutions of problem (15) ([3], Example 4.1): (i) the solution \(y(t)=e^{t}\gamma\) and (ii) the solution \(y(t)=\cosh(t)\gamma\ominus(1)\sinh(t)\gamma\). Thus the results in both Theorem 4.1 and Theorem 4.2 are invalid.
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 minmax terms and fuzzy differential equations, and we investigate the existence and multiple solutions for a class of firstorder differential system with minmax 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]).
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Acknowledgements
This work was partially supported by National Natural Science Foundation of China (11201481 and 11301039).
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All results in this paper have been obtained by all authors’ in mutual discussion. The first author’s main contributions are the deduction of the theorems and examples; the second author’s main contributions are writing and simulation. Both authors read and approved the final manuscript.
Appendix
Appendix
Lemma A.1
If \(f_{i}: R\times R\rightarrow R\) (\(i=1,2,\ldots,m\)) are Lipschitz continuous functions relative to their two arguments, then both the minimum function \(\min\{f_{i}:i=1,2,\ldots,m\}\) and the maximum function \(\max\{f_{i}:i=1,2,\ldots,m\}\) are also Lipschitz continuous functions.
Proof
Assume that the Lipschitz constants of \(f_{i}\) are \(L_{i1}\) and \(L_{i2}\), that is, for all \((x,y),(u,v)\in R^{2}\), \(f_{i}(x,y)f_{i}(u,v)\leq L_{i1}xu+L_{i2}yv\). Let \(f_{i_{0}}(x,y)=\min\{f_{i}(x,y):i=1,2,\ldots,m\}\) and \(f_{j_{0}}(u,v)=\min \{f_{i}(u,v):i=1,2,\ldots,m\}\), where \(1\leq i_{0},j_{0}\leq m\). Noting that \(f_{i_{0}}(u,v)f_{j_{0}}(u,v)\geq0\) and \(f_{i_{0}}(x,y)f_{j_{0}}(x,y)\leq0\), we have
Thus
This implies that
where \(L_{j}=\max\{L_{ij}: i=1,2,\ldots,m\}\) (\(j=1,2\)). Thus, the minimum function \(\min\{f_{i}:i=1,2,\ldots,m\}\) is a Lipschitz continuous function. Similarly, the maximum function \(\max\{ f_{i}:i=1,2,\ldots,m\}\) is also a Lipschitz continuous function. The proof is complete. □
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Liu, Y., Wu, J. Multiple solutions of ordinary differential systems with minmax terms and applications to the fuzzy differential equations. Adv Differ Equ 2015, 379 (2015). https://doi.org/10.1186/s136620150708z
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MSC
 34A07
 34K36
 26E50
Keywords
 nonlinear fuzzy differential equations
 generalized Hukuhara derivative
 multiple solutions