D. Otrocol Tiberiu Ppoviciu Institute of Numerical Analysis
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D. Otrocol, Ulam stabilities of differential equation with abstract Volterra operator in a Banach space, Nonlinear Functional Analysis and Applications, 15 (2010), no. 4, 613-619.
2010-Otrocol-NFAA-Ulam stabilies of de with abstract Volterra op
ULAM STABILITIES OF DIFFERENTIAL EQUATION WITH ABSTRACT VOLTERRA OPERATOR IN A BANACH SPACE
Diana Otrocol"Tiberiu Popoviciu" Institute of Numerical AnalysisP.O.Box. 68-1, 400110, Cluj-Napoca, Romaniae-mail: dotrocol@ictp.acad.ro
Abstract
The paper is devoted to the study of Ulam-Hyers stability and Ulam-HyersRassias stability for a class of abstract Volterra equations.
1. Introduction
The equations involving abstract Volterra operators have been investigated since 1928 by many authors: L. Tonelli (1928), S. Cinquini (1930), D. Graffi (1930), A.N. Tychonoff (1938). Such operators appear in many areas of investigation: control theory, continuum mechanics, engineering, dynamics of the nuclear reactors. Applications of such operators are contained in [1], [3], [6], [9], [7].
Equation stability is an important subject in the applications. Despite the large amount of works on Volterra integral equations, only the work [5] studies the conditions which ensure Ulam-Hyers-Rassias and Ulam-Hyers stability of a certain type of Volterra integral equations (see [4], [5], [8], [12]).
In the present paper we shall present Ulam-Hyers stability and generalized Ulam-Hyers-Rassias stability for a differential equation with abstract Volterra operator in a Banach space
x^(')(t)=f(t,x(t),V(x)(t)),t in I subR,x^{\prime}(t)=f(t, x(t), V(x)(t)), t \in I \subset \mathbb{R},
where
(i) I=[a,b]I=[a, b] or I=[a,oo[I=[a, \infty[;
(ii) ( B,|*|\mathbb{B},|\cdot| ) is a Banach space;
(iii) f in C([a,b]xxB^(2),B),V in C((C[a,b],B),(C[a,b],B))f \in C\left([a, b] \times \mathbb{B}^{2}, \mathbb{B}\right), V \in C((C[a, b], \mathbb{B}),(C[a, b], \mathbb{B})).
2. Preliminaries
Let (B,|*|)(\mathbb{B},|\cdot|) be a Banach space and V:(C[a,b],B)rarr(C[a,b],B)V:(C[a, b], \mathbb{B}) \rightarrow(C[a, b], \mathbb{B}) an abstract Volterra operator.
For f in C(I xxB^(2),B),epsi > 0f \in C\left(I \times \mathbb{B}^{2}, \mathbb{B}\right), \varepsilon>0 and varphi in C(I,R_(+))\varphi \in C\left(I, \mathbb{R}_{+}\right)we consider the Cauchy problem
{:[(2.1)x^(')(t)=f(t","x(t)","V(x)(t))","t in I],[('")"x(a)=alpha","alpha inR]:}\begin{align*}
& x^{\prime}(t)=f(t, x(t), V(x)(t)), t \in I \tag{2.1}\\
& x(a)=\alpha, \alpha \in \mathbb{R} \tag{$\prime$}
\end{align*}
and the following inequations
{:[(2.2)|y^(')(t)-f(t,y(t),V(y)(t))| <= epsi","t in I],[(2.3)|y^(')(t)-f(t,y(t),V(y)(t))| <= varphi(t)","t in I]:}\begin{gather*}
\left|y^{\prime}(t)-f(t, y(t), V(y)(t))\right| \leq \varepsilon, t \in I \tag{2.2}\\
\left|y^{\prime}(t)-f(t, y(t), V(y)(t))\right| \leq \varphi(t), t \in I \tag{2.3}
\end{gather*}
Following [11] we present some definitions and remarks.
Definition 2.1. The equation (2.1) is Ulam-Hyers stable if there exists a real number c > 0c>0 such that for each epsi > 0\varepsilon>0 and for each solution y inC^(1)(I,B)y \in C^{1}(I, \mathbb{B}) of (2.2) there exists a solution x inC^(1)(I,B)x \in C^{1}(I, \mathbb{B}) of (2.1) with
|y(t)-x(t)| <= c epsi,AA t in I|y(t)-x(t)| \leq c \varepsilon, \forall t \in I
Definition 2.2. The equation (2.1) is generalized Ulam-Hyers-Rassias stable with respect to varphi\varphi, if there exists c_(varphi) > 0c_{\varphi}>0, such that for each solution y inC^(1)(I,B)y \in C^{1}(I, \mathbb{B}) of the inequation (2.3) there exists a solution x inC^(1)(I,B)x \in C^{1}(I, \mathbb{B}) of (2.1) with
|y(t)-x(t)| <= c_(varphi)varphi(t),quad AA t in I|y(t)-x(t)| \leq c_{\varphi} \varphi(t), \quad \forall t \in I
Remark 2.3. A function y inC^(1)(I,B)y \in C^{1}(I, \mathbb{B}) is a solution of (2.2) if and only if there exists a function g in C(I,B)g \in C(I, \mathbb{B}) (which depend on yy ) such that
(i) |g(t)| <= epsi,AA t in I|g(t)| \leq \varepsilon, \forall t \in I;
(ii) y^(')(t)=f(t,y(t),V(y)(t))+g(t),AA t in Iy^{\prime}(t)=f(t, y(t), V(y)(t))+g(t), \forall t \in I.
Remark 2.4. A function y inC^(1)(I,B)y \in C^{1}(I, \mathbb{B}) is a solution of (2.3) if and only if there exists a function widetilde(g)in C(I,B)\widetilde{g} \in C(I, \mathbb{B}) (which depend on yy ) such that
(i) | widetilde(g)(t)| <= varphi(t),AA t in I;|\widetilde{g}(t)| \leq \varphi(t), \forall t \in I ;
(ii) y^(')(t)=f(t,y(t),V(y)(t))+ widetilde(g)(t),AA t in Iy^{\prime}(t)=f(t, y(t), V(y)(t))+\widetilde{g}(t), \forall t \in I.
Remark 2.5. If y inC^(1)(I,B)y \in C^{1}(I, \mathbb{B}) is a solution of the inequation (2.2), then yy is a solution of the following integral equation
|y(t)-y(a)-int_(a)^(t)f(s,y(s),V(y)(s))ds| <= (t-a)epsi,AA t in I\left|y(t)-y(a)-\int_{a}^{t} f(s, y(s), V(y)(s)) d s\right| \leq(t-a) \varepsilon, \forall t \in I
Remark 2.6. If y inC^(1)(I,B)y \in C^{1}(I, \mathbb{B}) is a solution of the inequation (2.3), then yy is a solution of the following integral equation
|y(t)-y(a)-int_(a)^(t)f(s,y(s),V(y)(s))ds| <= int_(a)^(t)varphi(s)ds,AA t in I\left|y(t)-y(a)-\int_{a}^{t} f(s, y(s), V(y)(s)) d s\right| \leq \int_{a}^{t} \varphi(s) d s, \forall t \in I
3. Ulam-Hyers stability on a compact interval I=[a,b]I=[a, b]
This section is totaly devoted to find out conditions under which the Volterra equation (2.1) admits the Ulam-Hyers stability on a compact interval I=[a,b]I=[a, b]. This is assembled in the next theorem.
Theorem 3.1. We suppose that
(a) f in C([a,b]xxB^(2),B),V in C((C[a,b],B),(C[a,b],B));f \in C\left([a, b] \times \mathbb{B}^{2}, \mathbb{B}\right), V \in C((C[a, b], \mathbb{B}),(C[a, b], \mathbb{B})) ;
(b) there exists L_(f) > 0L_{f}>0 such that
|V(x)(t)-V(y)(t)| <= L_(V)|x(t)-y(t)|,AA x,y in C[a,b],t in[a,b]|V(x)(t)-V(y)(t)| \leq L_{V}|x(t)-y(t)|, \forall x, y \in C[a, b], t \in[a, b]
Then
(i) the problem (2.1)-(2.1') has a unique solution in C([a,b],B)C([a, b], \mathbb{B});
(ii) the equation (2.1) is Ulam-Hyers stable.
Proof. Let y inC^(1)([a,b],B)y \in C^{1}([a, b], \mathbb{B}) be a solution of the inequation (2.2).
From [7], the problem (2.1)-(2.1') has a unique solution in C^(1)([a,b],B)C^{1}([a, b], \mathbb{B}). We denote by x inC^(1)([a,b],B)x \in C^{1}([a, b], \mathbb{B}) the unique solution of the Cauchy problem
4. Generalized Ulam-Hyers-Rassias stability on I=[a,oo[I=[a, \infty[
This section is devoted to the analysis of the generalized Ulam-HyersRassias stability of the Volterra equation (2.1) but when considering infinite intervals. Such stability is here obtained for this case under the conditions of the next result.
Theorem 4.1. We suppose that
(a) f in C([a,oo[xxB^(2),B),V in C((C[a,b],B),(C[a,b],B));:}f \in C\left(\left[a, \infty\left[\times \mathbb{B}^{2}, \mathbb{B}\right), V \in C((C[a, b], \mathbb{B}),(C[a, b], \mathbb{B})) ;\right.\right.
(b) there exists l_(f)inL^(1)([a,oo[,R_(+)):}l_{f} \in L^{1}\left(\left[a, \infty\left[, \mathbb{R}_{+}\right)\right.\right.such that |f(t,u_(1),u_(2))-f(t,v_(1),v_(2))| <= l_(f)(t)(|u_(1)-v_(1)|+|u_(2)-v_(2)|),AA t in[a,oo[,u_(i),v_(i)inB;:}\left|f\left(t, u_{1}, u_{2}\right)-f\left(t, v_{1}, v_{2}\right)\right| \leq l_{f}(t)\left(\left|u_{1}-v_{1}\right|+\left|u_{2}-v_{2}\right|\right), \forall t \in\left[a, \infty\left[, u_{i}, v_{i} \in \mathbb{B} ;\right.\right.
(c) there exists l_(V)inL^(1)([a,oo[,R_(+)):}l_{V} \in L^{1}\left(\left[a, \infty\left[, \mathbb{R}_{+}\right)\right.\right.such that |V(x)(t)-V(y)(t)| <= l_(V)(t)|x(t)-y(t)|,quad AA x,y in C[a,oo[,t in[a,oo[;|V(x)(t)-V(y)(t)| \leq l_{V}(t)|x(t)-y(t)|, \quad \forall x, y \in C[a, \infty[, t \in[a, \infty[;
(d) the function varphi in C[a,oo[\varphi \in C[a, \infty[ is increasing;
(e) there exists lambda > 0\lambda>0 such that
int_(a)^(t)varphi(s)ds <= lambda varphi(t),t in[a,oo[\int_{a}^{t} \varphi(s) d s \leq \lambda \varphi(t), t \in[a, \infty[
Then
(i) the problem (2.1)-(2.1') has a unique solution in C([a,oo[,B)C([a, \infty[, \mathbb{B});
(ii) the equation (2.1) is generalized Ulam-Hyers-Rassias stable with respect to varphi\varphi.
Proof. Let y inC^(1)([a,oo[,B)y \in C^{1}([a, \infty[, \mathbb{B}) be a solution of the inequation (2.3).
From [7], the problem (2.1)-(2.1') has a unique solution in C^(1)([a,oo[,B)C^{1}([a, \infty[, \mathbb{B}). We denote by x inC^(1)([a,oo[,B)x \in C^{1}([a, \infty[, \mathbb{B}) the unique solution of the Cauchy problem
x(t)=y(a)+int_(a)^(t)f(s,x(s),V(x)(s))ds,t in[a,oo[x(t)=y(a)+\int_{a}^{t} f(s, x(s), V(x)(s)) d s, t \in[a, \infty[
From (2.3) we have
{:[|y(t)-y(a)-int_(a)^(t)f(s,y(s),V(y)(s))ds| <= ],[ <= int_(a)^(t)varphi(s)ds <= lambda varphi(t)","t in[a","oo[]:}\begin{aligned}
& \left|y(t)-y(a)-\int_{a}^{t} f(s, y(s), V(y)(s)) d s\right| \leq \\
& \leq \int_{a}^{t} \varphi(s) d s \leq \lambda \varphi(t), t \in[a, \infty[
\end{aligned}
From the above relations it follows
{:[|y(t)-x(t)| <= |y(t)-y(a)-int_(a)^(t)f(s,y(s),V(y)(s))ds|+],[+int_(a)^(t)|f(s","y(s)","V(y)(s))-f(s","x(s)","V(x)(s))|ds],[ <= lambda varphi(t)+int_(a)^(t)l_(f)(s)(1+l_(V)(s))|y(s)-x(s)|ds]:}\begin{aligned}
|y(t)-x(t)| \leq & \left|y(t)-y(a)-\int_{a}^{t} f(s, y(s), V(y)(s)) d s\right|+ \\
& +\int_{a}^{t}|f(s, y(s), V(y)(s))-f(s, x(s), V(x)(s))| d s \\
\leq & \lambda \varphi(t)+\int_{a}^{t} l_{f}(s)\left(1+l_{V}(s)\right)|y(s)-x(s)| d s
\end{aligned}
From the Gronwall Lemma (see [10], Example 6.2) we have that
{:[|y(t)-x(t)| <= lambda varphi(t)e^(int_(a)^(t)l_(f)(s)(1+l_(V)(s))ds)],[=[lambdae^(int_(a)^(t)l_(f)(s)(1+l_(V)(s))ds)]varphi(t)=c_(varphi)varphi(t)","t in[a","oo[]:}\begin{aligned}
|y(t)-x(t)| & \leq \lambda \varphi(t) e^{\int_{a}^{t} l_{f}(s)\left(1+l_{V}(s)\right) d s} \\
& =\left[\lambda e^{\int_{a}^{t} l_{f}(s)\left(1+l_{V}(s)\right) d s}\right] \varphi(t)=c_{\varphi} \varphi(t), t \in[a, \infty[
\end{aligned}
i.e the equation (2.1) is generalized Ulam-Hyers-Rassias stable.
5. Applications
Example 5.1.
{:(5.1)x^(')(t)=int_(a)^(t)x(t)dt","t in I:}\begin{equation*}
x^{\prime}(t)=\int_{a}^{t} x(t) d t, t \in I \tag{5.1}
\end{equation*}
For this example we have B=R\mathbb{B}=\mathbb{R} and V(x)(t)=int_(a)^(t)x(t)dtV(x)(t)=\int_{a}^{t} x(t) d t, see [2].
So, the equation (5.1) is Ulam-Hyers stable on I=[a,b]I=[a, b] and is generalized Ulam-Hyers-Rassias stable on I=[a,oo[I=[a, \infty[.
Example 5.2.
{:(5.2)x^(')(t)=f(t","x(t))","t in I:}\begin{equation*}
x^{\prime}(t)=f(t, x(t)), t \in I \tag{5.2}
\end{equation*}
For this example we have B=R\mathbb{B}=\mathbb{R} and V(x)(t)=0V(x)(t)=0.
The equation (5.2) is Ulam-Hyers stable on I=[a,b]I=[a, b] and is generalized Ulam-Hyers-Rassias stable on I=[a,oo[I=[a, \infty[, see [11].
Example 5.3.
{:(5.3)x^(')(t)=f(t,x(t),int_(a)^(t)k(t,s,x(s))ds,t in I:}:}\begin{equation*}
x^{\prime}(t)=f\left(t, x(t), \int_{a}^{t} k(t, s, x(s)) d s, t \in I\right. \tag{5.3}
\end{equation*}
For this example we have B=R,V(x)(t)=int_(a)^(t)k(t,s,x(s))ds\mathbb{B}=\mathbb{R}, V(x)(t)=\int_{a}^{t} k(t, s, x(s)) d s and conditions (a)-(c) from Theorem 3.1 become:
(a) f in C([a,b]xxR^(2),R),k in C([a,b]xx[a,b]xxR,R)f \in C\left([a, b] \times \mathbb{R}^{2}, \mathbb{R}\right), k \in C([a, b] \times[a, b] \times \mathbb{R}, \mathbb{R}) are given;
(b) there exists L_(f) > 0L_{f}>0 such that
|k(t,s,u)-k(t,s,v)| <= L_(k)|u-v|,AA t,s in[a,b],u,v inR|k(t, s, u)-k(t, s, v)| \leq L_{k}|u-v|, \forall t, s \in[a, b], u, v \in \mathbb{R}
In this case, the equation (5.2) has a unique solution in C([a,b],R)C([a, b], \mathbb{R}), see [7][7], is Ulam-Hyers stable on I=[a,b]I=[a, b] and is generalized Ulam-Hyers-Rassias stable on I=[a,oo[I=[a, \infty[.
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