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D. Otrocol
Tiberiu Popoviciu 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.
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Nonlinear Functional Analysis and Applications
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ULAM STABILITIES OF DIFFERENTIAL EQUATION WITH ABSTRACT VOLTERRA OPERATOR IN A BANACH SPACE
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})).
(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
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.
(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.
(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
(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
|f(t,u_(1),u_(2))-f(t,v_(1),v_(2))| <= L_(f)sum_(i=1)^(2)|u_(i)-v_(i)|,AA t in[a,b],u_(i),v_(i)inR,i=1,2\left|f\left(t, u_{1}, u_{2}\right)-f\left(t, v_{1}, v_{2}\right)\right| \leq L_{f} \sum_{i=1}^{2}\left|u_{i}-v_{i}\right|, \forall t \in[a, b], u_{i}, v_{i} \in \mathbb{R}, i=1,2
(c) there exits L_(V) > 0L_{V}>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.
(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
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
{:[x^(')(t)=f(t","x(t)","V(x)(t))","t in[a","b]],[x(a)=y(a)]:}\begin{aligned}
x^{\prime}(t) & =f(t, x(t), V(x)(t)), t \in[a, b] \\
x(a) & =y(a)
\end{aligned}
From condition (a) we have
x(t)=y(a)+int_(a)^(t)f(s,x(s),V(x)(s))ds,t in[a,b]x(t)=y(a)+\int_{a}^{t} f(s, x(s), V(x)(s)) d s, t \in[a, b]
From Remark 2.5 we have
|y(t)-y(a)-int_(a)^(t)f(s,y(s),V(y)(s))ds| <= (t-a)epsi,t in[a,b]\left|y(t)-y(a)-\int_{a}^{t} f(s, y(s), V(y)(s)) d s\right| \leq(t-a) \varepsilon, t \in[a, b]
From above relations we have
{:[|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],[ <= (t-a)epsi+L_(f)(int_(a)^(t)|y(s)-x(s)|ds+int_(a)^(t)|V(y)(s)-V(x)(s)|ds)],[ <= (t-a)epsi+L_(f)(1+L_(V))int_(a)^(t)|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 & (t-a) \varepsilon+L_{f}\left(\int_{a}^{t}|y(s)-x(s)| d s+\int_{a}^{t}|V(y)(s)-V(x)(s)| d s\right) \\
\leq & (t-a) \varepsilon+L_{f}\left(1+L_{V}\right) \int_{a}^{t}|y(s)-x(s)| d s
\end{aligned}
From the Gronwall Lemma (see [10], Example 6.2) we have that
|y(t)-x(t)| <= (t-a)epsie^(L_(f)(1+L_(V))(b-a))=c epsi,t in[a,b]|y(t)-x(t)| \leq(t-a) \varepsilon e^{L_{f}\left(1+L_{V}\right)(b-a)}=c \varepsilon, t \in[a, b]
i.e. the equation (2.1) is Ulam-Hyers stable.
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
(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.
(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
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)=f(t","x(t)","V(x)(t))","t in[a","oo[],[x(a)=y(a)]:}\begin{aligned}
x^{\prime}(t) & =f(t, x(t), V(x)(t)), t \in[a, \infty[ \\
x(a) & =y(a)
\end{aligned}
We have that
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[.
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].
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
(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
|f(t,u_(1),u_(2))-f(t,v_(1),v_(2))| <= L_(f)sum_(i=1)^(2)|u_(i)-v_(i)|,AA t in[a,b],u_(i),v_(i)inR,i=1,2\left|f\left(t, u_{1}, u_{2}\right)-f\left(t, v_{1}, v_{2}\right)\right| \leq L_{f} \sum_{i=1}^{2}\left|u_{i}-v_{i}\right|, \forall t \in[a, b], u_{i}, v_{i} \in \mathbb{R}, i=1,2
(c) there exists L_(k) > 0L_{k}>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[.
References
[1] C. Corduneanu, Existence of solutions for neutral functional differential equations with causal operators, Journal of Differential Equations 168 (2000), 93-101.
[2] C. Corduneanu, Abstract Volterra equations (a survey), Math. and Comp. Modelling 32 (2000), 1503-1528.
[3] D. Guo, V. Lakshmikantham, X. Liu, Nonlinear Integral Equations in Abstract Spaces, Kuwer Academic Publishers, Dordrecht, Boston, London, 1996.
[4] D. H. Hyers, G. Isac, Th. M. Rassias, Stability of functional equations in several variables, Progress in Nonlinear Differential Equations and their Applications, 34, Birkhäuser, Boston, 1998.
[5] S. M. Jung, A fixed point approach to the stability of a Volterra integral equation, Fixed Point Theory Appl. 2007, Article ID 57064, 9p.
[6] V. Kolmanovskii, A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer, 1992.
[7] D. Otrocol, Abstract Volterra operators, Carpathian Journal of Mathematics 24 (2008), No. 3, 370-377.
[8] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.
[9] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), No. 1, 91-96.
[10] I. A. Rus, Gronwall lemmas; ten open problems, Scientiae Mathematicae Japonicae 70 (2009), No. 2, 221-228, :e2009, 257-264.
[11] I. A. Rus, Ulam stability of ordinary differential equations, Studia Univ. "BabesBolyai" Mathematica 54 (2009), in press.
[12] S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, 8, Interscience Publishers, New York, 1960.
[2] C. Corduneanu, Abstract Volterra equations (a survey), Math. and Comp. Modelling 32 (2000), 1503-1528.
[3] D. Guo, V. Lakshmikantham, X. Liu, Nonlinear Integral Equations in Abstract Spaces, Kuwer Academic Publishers, Dordrecht, Boston, London, 1996.
[4] D. H. Hyers, G. Isac, Th. M. Rassias, Stability of functional equations in several variables, Progress in Nonlinear Differential Equations and their Applications, 34, Birkhäuser, Boston, 1998.
[5] S. M. Jung, A fixed point approach to the stability of a Volterra integral equation, Fixed Point Theory Appl. 2007, Article ID 57064, 9p.
[6] V. Kolmanovskii, A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer, 1992.
[7] D. Otrocol, Abstract Volterra operators, Carpathian Journal of Mathematics 24 (2008), No. 3, 370-377.
[8] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.
[9] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), No. 1, 91-96.
[10] I. A. Rus, Gronwall lemmas; ten open problems, Scientiae Mathematicae Japonicae 70 (2009), No. 2, 221-228, :e2009, 257-264.
[11] I. A. Rus, Ulam stability of ordinary differential equations, Studia Univ. "BabesBolyai" Mathematica 54 (2009), in press.
[12] S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, 8, Interscience Publishers, New York, 1960.
- ^(0){ }^{0} Received . Revised .
^(0)2000{ }^{0} 2000 Mathematics Subject Classification: 34L05, 34K20, 47H10.
^(0){ }^{0} Keywords: Ulam-Hyers stability, Ulam-Hyers-Rassias stability, abstract Volterra operators.