Global existence of solutions for intervalvalued integrodifferential equations under generalized Hdifferentiability
 Vinh An Truong^{1},
 Van Hoa Ngo^{2}Email author and
 Dinh Phu Nguyen^{3}
https://doi.org/10.1186/168718472013217
© Truong et al.; licensee Springer 2013
Received: 10 January 2013
Accepted: 8 July 2013
Published: 22 July 2013
Abstract
In this study, we consider the intervalvalued integrodifferential equations (IIDEs) under generalized Hdifferentiability
The global existence of solutions for intervalvalued integrodifferential equations with initial conditions under generalized Hdifferentiability is studied. Theorems for global existence of solutions are given and proved on $[{t}_{0},\mathrm{\infty})$. Some examples are given to illustrate these results.
MSC:34K05, 34K30, 47G20.
Keywords
1 Introduction
The setvalued differential and integral equations are an important part of the theory of setvalued analysis, and they play an important role in the theory and application of control theory; and they were first studied in 1969 by De Blasi and Iervolino [1]. Recently, setvalued differential equations have been studied by many scientists due to their applications in many areas. For the basic theory on setvalued differential and integral equations, the readers can be referred to the following books and papers [2–13] and references therein. Integrodifferential equations are encountered in many areas of science, where it is necessary to take into account aftereffect or delay (for example, in control theory, biology, ecology, medicine, etc. [14–16]). Especially, one always describes a model which possesses hereditary properties by integrodifferential equations in practice.
The intervalvalued analysis and interval differential equations (IDEs) are the special cases of the setvalued analysis and setvalued differential equations, respectively. In many situations, when modeling realworld phenomena, information about the behavior of a dynamic system is uncertain and one has to consider these uncertainties to gain better meaning of full models. Intervalvalued differential equation is a natural way to model dynamic systems subject to uncertainties. Recently, many works have been done by several authors in the theory of intervalvalued differential equations (see, e.g., [17–20]). There are several approaches to the study of interval differential equations. One popular approach is based on Hdifferentiability. The approach based on Hderivative has the disadvantage that it leads to solutions which have an increasing length of their support. Recently, Stefanini and Bede [17] solved the above mentioned approach under strongly generalized differentiability of intervalvalued functions. In this case, the derivative exists and the solution of an intervalvalued differential equation may have decreasing length of the support, but the uniqueness is lost. The paper of Stefanini and Bede was the starting point for the topic of intervalvalued differential equations (see [19, 20]) and later also for fuzzy differential equations. Also, a very important generalization and development related to the subject of the present paper is in the field of fuzzy sets, i.e., fuzzy calculus and fuzzy differential equations under the generalized Hukuhara derivative. Recently, several works, e.g., [7, 10, 16, 21–39], have been done on setvalued differential equations, fuzzy differential equations and random fuzzy differential equations.
where ′ denotes two kinds of derivatives, namely the classical Hukuhara derivative and the second type Hukuhara derivative (generalized Hukuhara differentiability). The existence and uniqueness of a Cauchy problem is then obtained under an assumption that the coefficients satisfy a condition with the Lipschitz constant (see [17]). The proof is based on the application of the Banach fixed point theorem. In [20], under the generalized Lipschitz condition, Malinowski obtained the existence and uniqueness of solutions to both kinds of IDEs.
In this paper, we study two kinds of solutions to IIDEs. The different types of solutions to IIDEs are generated by the usage of two different concepts of intervalvalued derivative. This direction of research is motivated by the results of Stefanini and Bede [17], Malinowski [19, 20] concerning deterministic IDEs with generalized intervalvalued derivative.
This paper is organized as follows. In Section 2, we recall some basic concepts and notations about interval analysis and intervalvalued differential equations. In Section 3, we present the global existence of solutions to the intervalvalued integrodifferential equations under two kinds of the Hukuhara derivative. Finally, we give some examples for IIDEs in Section 4.
2 Preliminaries

If $A\ominus B$, $A\ominus C$ exist, then $H(A\ominus B,A\ominus C)=H(B,C)$;

If $A\ominus B$, $C\ominus D$ exist, then $H(A\ominus B,C\ominus D)=H(A+D,B+C)$;

If $A\ominus B$, $A\ominus (B+C)$ exist, then there exist $(A\ominus B)\ominus C$ and $(A\ominus B)\ominus C=A\ominus (B+C)$;

If $A\ominus B$, $A\ominus C$, $C\ominus B$ exist, then there exist $(A\ominus B)\ominus (A\ominus C)$ and $(A\ominus B)\ominus (A\ominus C)=C\ominus B$.
Definition 2.1 [20]
We say that the intervalvalued mapping $X:[a,b]\subset {R}^{+}\to {K}_{C}(\mathbb{R})$ is continuous at the point $t\in [a,b]$ if for every $\u03f5>0$ there exists $\delta =\delta (t,\u03f5)>0$ such that, for all $s\in [a,b]$ such that $ts<\delta $, one has $H(X(t),X(s))\le \u03f5$.
The strongly generalized differentiability was introduced in [17] and studied in [19, 36–38].
 (i)for all $h>0$ sufficiently small, $\mathrm{\exists}X(t+h)\ominus X(t)$, $\mathrm{\exists}X(t)\ominus X(th)$ and the limits$\underset{h\searrow 0}{lim}H(\frac{X(t+h)\ominus X(t)}{h},{D}_{H}^{g}X(t))=0,\phantom{\rule{2em}{0ex}}\underset{h\searrow 0}{lim}H(\frac{X(t)\ominus X(th)}{h},{D}_{H}^{g}X(t))=0,$
 (ii)for all $h>0$ sufficiently small, $\mathrm{\exists}X(t)\ominus X(t+h)$, $\mathrm{\exists}X(th)\ominus X(t)$ and the limits$\underset{h\searrow 0}{lim}H(\frac{X(t)\ominus X(t+h)}{h},{D}_{H}^{g}X(t))=0,\phantom{\rule{2em}{0ex}}\underset{h\searrow 0}{lim}H(\frac{X(th)\ominus X(t)}{h},{D}_{H}^{g}X(t))=0,$
 (iii)for all $h>0$ sufficiently small, $\mathrm{\exists}X(t+h)\ominus X(t)$, $\mathrm{\exists}X(th)\ominus X(t)$ and the limits$\underset{h\searrow 0}{lim}H(\frac{X(t+h)\ominus X(t)}{h},{D}_{H}^{g}X(t))=0,\phantom{\rule{2em}{0ex}}\underset{h\searrow 0}{lim}H(\frac{X(th)\ominus X(t)}{h},{D}_{H}^{g}X(t))=0,$
 (iv)for all $h>0$ sufficiently small, $\mathrm{\exists}X(t)\ominus X(t+h)$, $\mathrm{\exists}X(t)\ominus X(th)$ and the limits$\underset{h\searrow 0}{lim}H(\frac{X(t)\ominus X(t+h)}{h},{D}_{H}^{g}X(t))=0,\phantom{\rule{2em}{0ex}}\underset{h\searrow 0}{lim}H(\frac{X(t)\ominus X(th)}{h},{D}_{H}^{g}X(t))=0$
(h at denominators means $\frac{1}{h}$). In this definition, case (i) ((i)differentiability for short) corresponds to the classical Hderivative, so this differentiability concept is a generalization of the Hukuhara derivative. In [17], Stefanini and Bede considered four cases for the derivative. In this paper, we consider only the two first items of Definition 2.2. In other cases, the derivative is trivial because it is reduced to a crisp element.
If for intervals $X,Y,Z\in {K}_{C}(\mathbb{R})$ there exist Hukuhara differences $X\ominus Y$, $X\ominus Z$, then $H(X\ominus Y,\{0\})=H(X,Y)$ and $H(X\ominus Y,X\ominus Z)=H(Y,Z)$.
 (i)
If X is (i)differentiable, then it is continuous.
 (ii)
If X, Y are (i)differentiable and $\lambda \in \mathbb{R}$, then ${D}_{H}^{g}(X+Y)(t)={D}_{H}^{g}X(t)+{D}_{H}^{g}Y(t)$, ${D}_{H}^{g}(\lambda X)(t)=\lambda {D}_{H}^{g}X(t)$.
 (iii)
Let X be (i)differentiable and assume that ${D}_{H}^{g}X$ is integrable over $[a,b]$. Then we have $X(t)=X(a)+{\int}_{a}^{t}{D}_{H}^{g}X(s)\phantom{\rule{0.2em}{0ex}}ds$.
 (iv)
If X is (i)differentiable on $[a,b]$, then the real function $t\to len(X(t))$ is nondecreasing on $[a,b]$.
 (v)
Let X be (ii)differentiable and assume that ${D}_{H}^{g}X$ is integrable over $[a,b]$. Then we have $X(a)=X(t)+(1){\int}_{a}^{t}{D}_{H}^{g}X(s)\phantom{\rule{0.2em}{0ex}}ds$.
 (vi)
If X is (ii)differentiable on $[a,b]$, then the real function $t\to len(X(t))$ is nonincreasing on $[a,b]$.
Corollary 2.1 (see, e.g., [17, 19])
 (i)
If the mapping X is (i)differentiable (i.e., classical Hukuhara differentiability) at $t\in [{t}_{0},T]$, then the realvalued functions ${X}^{}$, ${X}^{+}$ are differentiable at t and ${D}_{H}^{g}X(t)=[{({X}^{})}^{\prime}(t),{({X}^{+})}^{\prime}(t)]$.
 (ii)
If the mapping X is (ii)differentiable at $t\in [{t}_{0},T]$, then the realvalued functions ${X}^{}$, ${X}^{+}$ are differentiable at t and ${D}_{H}^{g}X(t)=[{({X}^{+})}^{\prime}(t),{({X}^{})}^{\prime}(t)]$.
on the interval $[{t}_{0},T]\in \mathbb{R}$, under the strong differentiability condition, (i) or (ii), respectively. We notice that the equivalence between two equations in this lemma means that any solution is a solution for the other one.
for all $t\in [{t}_{0},T]$, where $F:I=[{t}_{0},T]\times {K}_{C}(\mathbb{R})\to {K}_{C}(\mathbb{R})$ and $K:\mathcal{D}\times {K}_{C}(\mathbb{R})\to {K}_{C}(\mathbb{R})$ are continuous intervalvalued mappings on I, with $\mathcal{D}=\{(t,s)\in I\times I:{t}_{0}\le s\le t<T\}$.
Definition 2.3 A mapping $X:[{t}_{0},T]\to {K}_{C}(\mathbb{R})$ is called a solution to problem (2.2) on I if and only if X is a continuous mapping on I and it satisfies one of the following intervalvalued integral equations:
(S1) $X(t)={X}_{0}+({\int}_{{t}_{0}}^{t}F(s,X(s))\phantom{\rule{0.2em}{0ex}}ds+{\int}_{{t}_{0}}^{t}{\int}_{{t}_{0}}^{s}K(s,u,X(u))\phantom{\rule{0.2em}{0ex}}du\phantom{\rule{0.2em}{0ex}}ds)$, $t\in I$, if X is (i)differentiable or (iii)differentiable.
(S2) $X(t)={X}_{0}\ominus (1)({\int}_{{t}_{0}}^{t}F(s,X(s))\phantom{\rule{0.2em}{0ex}}ds+{\int}_{{t}_{0}}^{t}{\int}_{{t}_{0}}^{s}K(s,u,X(u))\phantom{\rule{0.2em}{0ex}}du\phantom{\rule{0.2em}{0ex}}ds)$, $t\in I$, if X is (ii)differentiable or (iv)differentiable.
Definition 2.4 Let $X:[{t}_{0},T]\to {K}_{C}(\mathbb{R})$ be an intervalvalued function which is (i)differentiable. If X and its derivative satisfy problem (2.2), we say X is a (i)solution of problem (2.2).
Definition 2.5 Let $X:[{t}_{0},T]\to {K}_{C}(\mathbb{R})$ be an intervalvalued function which is (ii)differentiable. If X and its derivative satisfy problem (2.2), we say X is a (ii)solution of problem (2.2).
Definition 2.6 A solution $X:[{t}_{0},T]\to {K}_{C}(\mathbb{R})$ is unique if ${sup}_{t\in [{t}_{0},T]}H(X(t),Y(t))=0$ for any mapping $Y:[{t}_{0},T]\to {K}_{C}(\mathbb{R})$ that is a solution to (2.2) on $[{t}_{0},T]$.
Theorem 2.1 (see [42])
for all $t,s\in I$, ${X}_{1},{X}_{2}\in {K}_{C}(\mathbb{R})$. Then there exists the only local solution X to IIDE (2.2) on some intervals $[{t}_{0},\mathbb{T}]$ ($\mathbb{T}\le T{t}_{0}$) for each case ((i)solution and (ii)solution).
3 Main results
for all $t\in J=[{t}_{0},\mathrm{\infty})$, where $F:J\times {K}_{C}(\mathbb{R})\to {K}_{C}(\mathbb{R})$ and $K:\mathcal{D}\times {K}_{C}(\mathbb{R})\to {K}_{C}(\mathbb{R})$ are continuous intervalvalued mappings on J, with $\mathcal{D}=\{(t,s)\in J\times J:{t}_{0}\le s\le t<\mathrm{\infty}\}$.
 (i)
$F(t,X)$, $G(t,s,X)$ are locally Lipschitzian for all $t,s\in J$, $X\in {K}_{C}(\mathbb{R})$;
 (ii)$f\in C[J\times [0,\mathrm{\infty}),[0,\mathrm{\infty})]$ and $k\in C[\mathcal{D}\times [0,\mathrm{\infty}),[0,\mathrm{\infty})]$ are nondecreasing in $x\ge 0$, and the maximal solution $r(t,{t}_{0},{x}_{0})$ of the scalar integrodifferential equation${x}^{\prime}(t)=f(t,x(t))+{\int}_{{t}_{0}}^{t}k(t,s,x(s))\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}x({t}_{0})={x}_{0}\ge 0,$(3.2)
 (iii)
$H(F(t,X),\{0\})\le f(t,H(X,\{0\}))$, $H(K(t,s,X),\{0\})\le k(t,s,H(X,\{0\}))$ for all $t,s\in J$, $X\in {K}_{C}(\mathbb{R})$;
 (iv)
$H(X(t,{t}_{0},{X}_{0}),\{0\})\le r(t,{t}_{0},{x}_{0})$, $H({X}_{0},\{0\})\le {x}_{0}$.
Then the largest interval of the existence of any solution $X(t,{t}_{0},{X}_{0})$ of (3.1) for each case ((i)solution and (ii)solution) such that $H({X}_{0},\{0\})\le {x}_{0}$ is J. In addition, if $r(t,{t}_{0},{x}_{0})$ is bounded on J, then ${lim}_{t\to \mathrm{\infty}}X(t,{t}_{0},{X}_{0})$ exists in $({K}_{C}(\mathbb{R}),H)$.
By the assumption (i) again, it follows that $X(t)$ can be extended beyond α, which contradicts our assumption. So, any (i)solution of problem (3.1) exists on $J=[{t}_{0},\mathrm{\infty})$, and so $\alpha =\mathrm{\infty}$. □
on $[{t}_{0},\mathrm{\infty})$. Moreover, we see that $H(a(t)X(t),\{0\})\le a(t)H(X(t),\{0\})=f(t,H(X(t),\{0\}))$ and $H(b(t)X(t),\{0\})\le b(t)H(X(t),\{0\})=k(t,s,H(X(t),\{0\}))$. Therefore, the solutions of problem (3.4) are on $[{t}_{0},\mathrm{\infty})$.
Employing the comparison Theorem 3.1, we shall prove the following global existence result.
 (i)
$F\in C[J\times {K}_{C}(\mathbb{R}),{K}_{C}(\mathbb{R})]$, $K\in C[\mathcal{D}\times {K}_{C}(\mathbb{R}),{K}_{C}(\mathbb{R})]$, F and K are bounded on bounded sets, and there exists a local (i)solution of (3.1) for every $({t}_{0},{X}_{0})$, ${t}_{0}\ge 0$ and ${X}_{0}\in {K}_{C}(\mathbb{R})$;
 (ii)$V\in C[J\times {K}_{C}(\mathbb{R}),[0,\mathrm{\infty})]$; $V(t,A)V(t,B)\le LH(A,B)$, where L is the local Lipschitz constant, for $A,B\in {K}_{C}(\mathbb{R})$, $t\in J$, $V(t,A)\to \mathrm{\infty}$ as $H(A,\{0\})\to \mathrm{\infty}$ uniformly for $[{t}_{0},T]$, for every $T>{t}_{0}$ and for $t\in J$, $A\in {K}_{C}(\mathbb{R})$,$\begin{array}{r}\underset{h\to {0}^{+}}{lim\hspace{0.17em}sup}\frac{1}{h}[V(t+h,A+h\{F(t,A)+{\int}_{{t}_{0}}^{t}K(t,s,A)\phantom{\rule{0.2em}{0ex}}ds\})V(t,A)]\\ \phantom{\rule{1em}{0ex}}\le f(t,V(t,A))+{\int}_{{t}_{0}}^{t}k(t,s,V(s,A))\phantom{\rule{0.2em}{0ex}}ds,\end{array}$
 (iii)The maximal solution $r(t)=r(t,{t}_{0},{x}_{0})$ of the scalar integrodifferential equation${x}^{\prime}(t)=f(t,x(t))+{\int}_{{t}_{0}}^{t}k(t,s,x(s))\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}x({t}_{0})={x}_{0}\ge 0$(3.5)
exists on J and is positive whenever ${x}_{0}>0$.
Therefore ${Z}_{1}(t)$ is a (i)solution of problem (3.1) on $[{t}_{0},{c}_{Z}+\delta )$, and, by repeating the arguments that were used to obtain (3.7), we get $V(t,{Z}_{1}(t))\le r(t)$, $t\in [{t}_{0},{c}_{Z}+\delta )$. This contradicts the maximality of Z, and hence ${c}_{Z}=+\mathrm{\infty}$. The proof is complete. □
 (i)
$F\in C[J\times {K}_{C}(\mathbb{R}),{K}_{C}(\mathbb{R})]$, $K\in C[\mathcal{D}\times {K}_{C}(\mathbb{R}),{K}_{C}(\mathbb{R})]$, F and K are bounded on bounded sets, and there exists a local (ii)solution of (3.1) for every $({t}_{0},{X}_{0})$, ${t}_{0}\ge 0$ and ${X}_{0}\in {K}_{C}(\mathbb{R})$;
 (ii)$V\in C[J\times {K}_{C}(\mathbb{R}),[0,\mathrm{\infty})]$; $V(t,A)V(t,B)\le LH(A,B)$, where L is the local Lipschitz constant, for $A,B\in {K}_{C}(\mathbb{R})$, $t\in J$, $V(t,A)\to \mathrm{\infty}$ as $H(A,\{0\})\to \mathrm{\infty}$ uniformly for $[{t}_{0},T]$, for every $T>{t}_{0}$ and for $t\in J$, $A\in {K}_{C}(\mathbb{R})$,$\begin{array}{r}\underset{h\to {0}^{+}}{lim\hspace{0.17em}sup}\frac{1}{h}[V(t+h,A\ominus (1)h\{F(t,A)+{\int}_{{t}_{0}}^{t}K(t,s,A)\phantom{\rule{0.2em}{0ex}}ds\})V(t,A)]\\ \phantom{\rule{1em}{0ex}}\le f(t,V(t,A))+{\int}_{{t}_{0}}^{t}k(t,s,V(s,A))\phantom{\rule{0.2em}{0ex}}ds,\end{array}$
 (iii)The maximal solution $r(t)=r(t,{t}_{0},{x}_{0})$ of the scalar integrodifferential equation${x}^{\prime}(t)=f(t,x(t))+{\int}_{{t}_{0}}^{t}k(t,s,x(s))\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}x({t}_{0})={x}_{0}\ge 0,$(3.8)
exists on J and is positive whenever ${x}_{0}>0$.
Proof One can obtain this result easily by using the methods as in the proof of Theorem 3.3. □
4 Some examples
In this section, we present some examples being simple illustrations of the theory of IIDEs. We will consider IIDEs (3.1) with (i) and (ii) derivatives, respectively. For convenience, from now on, we denote the solution of IIDE (3.1) with (i) derivative by ${X}_{1}$ and the solution with (ii) derivative by ${X}_{2}$.
where $F:[{t}_{0},T]\to {K}_{C}(\mathbb{R})$ is an intervalvalued function (i.e., $F(t)=[{F}^{}(t),{F}^{+}(t)]$), $k(t,s)$ is a real known function, and ${X}_{0}\in {K}_{C}(\mathbb{R})$. In equation (4.1), we shall solve it by two types of the Hukuhara derivative, which are defined in Definition 2.2. Consequently, based on the type of differentiability, we have the following two cases.
As we see in figures, the first type and second type Hukuhara differentiable intervalvalued solutions X behave in various ways, i.e., one can say that $len({X}_{1}(t))$ in examples is nondecreasing in time (see Figures 1 and 3) and $len({X}_{2}(t))$ in examples is nonincreasing in time (see Figures 2 and 4).
5 Conclusions and further work
From Example 4.1 to Example 4.2, we notice that the solutions under the classical Hukuhara derivative ((i)differentiable) have increasing length of their values. Indeed, we can see this in Figures 1 and 3. However, if we consider the second type Hukuhara derivative ((ii)differentiable), the length of solutions changes. Under the second type Hukuhara derivative, differentiable solutions have nonincreasing length of its values (see Figures 2 and 4). In [17, 18], authors introduced and studied new generalized differentiability concepts for intervalvalued functions. Our point is that the generalization of this concept can be of great help in the dynamic study of intervalvalued differential equations and intervalvalued integrodifferential equations.
Declarations
Acknowledgements
The authors would like to express their gratitude to the anonymous referees for their helpful comments and suggestions, which have greatly improved the paper. The first named author would like to thank the University of Technical Education, Ho Chi Minh City, Vietnam.
Authors’ Affiliations
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