Abstract Volterra operators

Diana Otrocol “T. Popoviciu” Institute of Numerical Analysis
PO Box 68-1, 400110, Cluj-Napoca, Romania dotrocol@ictp.acad.ro

Abstract.

The aim of this paper is to study a differential equation with abstract Volterra operator. Existence and uniqueness, inequalities of Čaplygin type and data dependence (monotony, continuity) results for the solution of the Cauchy problem are obtained using weakly Picard operators theory.

Key words and phrases:

Abstract Volterra operators, weakly Picard operators, data dependence.

2000 Mathematics Subject Classification:

34L05, 47H10.

1. Introduction

The equations involving abstract Volterra operators have been investigated since 1928 by many authors. We mention here the contributions made by L. Tonelli (1928), S. Cinquini (1930), D. Graffi (1930), A.N. Tychonoff (1938). The definition of abstract Volterra operator introduced by A.N. Tychonoff is very easy to grasp:

An operator is abstract Volterra operator if any two functions coinciding on an interval $[a,t]$ have equal images on $[a,t],t\in[a,b]$ .

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 [2], [4], [5].

The Volterra equation with abstract operator are usually written in the form

x^{\prime}(t)=V(x)(t),\ t\in[a,b].

where $V$ stands for such an operator.

This functional equation have been investigated by many authors and results pertaining existence and uniqueness, continuous dependence of solution to the Cauchy’s problem and even more specialized topics can be found in [3], [6], [7], [9]. The fixed point method is certainly one of the most used to study the existence of solution, as well as other basic problems, related to the equation 1, see [1], [4], [7].

This paper has as main objective to provide existence and uniqueness, inequalities of Čaplygin type and data dependence (monotony, continuity) results for the solution of a differential equation with abstract Volterra operator using weakly Picard operators theory. As an illustration of the fact that weakly Picard operators theory can be used with success in case of abstract Volterra operator, we shall give some application of the theory presented below.

We consider the following Cauchy problem

(1)

x^{\prime}(t)=f(t,x(t),V(x)(t)),\ t\in[a,b]

(2)

x(a)=\alpha

where

(C₁)

$\alpha\in\mathbb{R}$ , $f\in C([a,b]\times\mathbb{R}^{2},\mathbb{R}),\ V\in C(C[a,b],C[a,b])$ are given;
(C₂)

there exists $L_{f}>0$ such that

$\left|f(t,u_{1},u_{2})-f(t,v_{1},v_{2})\right|\leq L_{f}\sum_{i=1}^{2}\left|u_{i}-v_{i}\right|,$

$\forall t\in[a,b],u_{i},v_{i}\in\mathbb{R},i=1,2$ ;
(C₃)

there exists $L_{V}>0$ such that

$\left|V(x)(t)-V(y)(t)\right|\leq L_{V}\left|x(t)-y(t)\right|,$

$\forall x,y\in C[a,b],\ t\in[a,b]$ .

In the condition $(C_{1})$ , the problem (1)–(2), is equivalent with the fixed point equation

(3)

x(t)=\alpha+\int_{a}^{t}f(s,x(s),V(x)(s))ds,\ t\in[a,b],\ x\in C[a,b]

and the equation (1) is equivalent with

(4)

x(t)=x(a)+\int_{a}^{t}f(s,x(s),V(x)(s))ds,\ t\in[a,b],x\in C[a,b].

Let us consider the following operators $B_{f},E_{f}:C[a,b]\rightarrow C[a,b]$ defined by $B_{f}(x)(t):=\text{second part of (\ref{ec3.aV1})}$ and $E_{f}(x)(t):=\text{second part of (\ref{ec4.aV1}).}$ For $\alpha\in\mathbb{R},$ we consider $X_{\alpha}:=\{x\in C[a,b]|\ x(a)=\alpha\}.$ We remark that $C[a,b]=\underset{\alpha\in\mathbb{R}}{\cup}X_{\alpha}$ is a partition of $C[a,b].$ We have

Lemma 1.

If $(C_{1})$ is satisfied, then

(a)

$B_{f}(C[a,b])\subset X_{\alpha}$ and $E_{f}(X_{\alpha})\subset X_{\alpha},\ \forall\alpha\in\mathbb{R};$
(b)

$B_{f}|_{X_{\alpha}}=E_{f}|_{X_{\alpha}},\ \forall\alpha\in\mathbb{R}.$

For a better understanding of the weakly Picard operator theory, we shall present some notions and results from Rus [10]–[12].

2. Weakly Picard operators

Let $(X,d)$ be a metric space and $A:X\rightarrow X$ an operator. We shall use the following notations:

$F_{A}:=\{x\in X\mid A(x)=x\}$ - the fixed point set of $A$ ;

$I(A):=\{Y\subset X\mid A(Y)\subset Y,Y\neq\emptyset\}$ - the family of the nonempty invariant subsets of $A$ ;

$A^{n+1}:=A\circ A^{n},\;A^{0}=1_{X},\;A^{1}=A,\;n\in\mathbb{N}$ ;

Definition 2.

Let $(X,d)$ be a metric space. An operator $A:X\rightarrow X$ is a Picard operator (PO) if there exists $x^{\ast}\in X$ such that:

(i)

$F_{A}=\{x^{\ast}\};$
(ii)

the sequence $(A^{n}(x_{0}))_{n\in\mathbb{N}}$ converges to $x^{\ast}$ for all $x_{0}\in X$ .

Remark 3.

Accordingly to the definition, the contraction principle insures that, if $A:X\rightarrow X$ is an $\alpha$ -contraction on the complete metric space $X$ , then it is a Picard operator.

Definition 4.

Let $(X,d)$ be a metric space. An operator $A:X\rightarrow X$ is a weakly Picard operator (WPO) if the sequence $(A^{n}(x))_{n\in\mathbb{N}}$ converges for all $x\in X$ , and its limit ( which may depend on $x$ ) is a fixed point of $A$ .

Definition 5.

If $A$ is weakly Picard operator then we consider the operator $A^{\infty}$ defined by

A^{\infty}:X\rightarrow X,\ A^{\infty}(x):=\underset{n\rightarrow\infty}{\lim}A^{n}(x).

Remark 6.

It is clear that $A^{\infty}(X)=F_{A}.$

Definition 7.

Let $A$ be a weakly Picard operator and $c>0.$ The operator $A\$ is $c$ -weakly Picard operator if

d(x,A^{\infty}(x))\leq cd(x,A(x)),\ \forall x\in X.

Lemma 8.

Let $(X,d,\leq)$ be an ordered metric space and $A:X\rightarrow X$ an operator. We suppose that:

(i)

$A$ is WPO;
(ii)

$A$ is increasing.

Then, the operator $A^{\infty}$ is increasing.

Lemma 9.

Let $(X,d,\leq)$ an ordered metric space and $A,B,C:X\rightarrow X$ be such that:

$(i)$ the operator $A,B,C$ are WPOs;

$(ii)$ $A\leq B\leq C$ ;

$(iii)$ the operator $B$ is increasing.

Then $x\leq y\leq z$ implies that $A^{\infty}(x)\leq B^{\infty}(y)\leq C^{\infty}(z)$ .

For some examples of WPOs see [8], [11], [12].

3. Existence

Consider the problem (1)–(2) where $V:C[a,b]\rightarrow[a,b],\ x\rightarrow V(x)$ and $f\in C([a,b]\times J^{2}),J\subset\mathbb{R}$ a compact interval. This problem is equivalent with the integral equation (3).

Let $\overline{B}(a,r)\subset(C[a,b],\left\|\cdot\right\|_{C})$ and suppose that the compatibility condition between the sphere and the compact $J$ hold, i.e.

y\in\overline{B}(a,r)\text{ imply that }y(x)\in J,\forall x\in[a,b].

We note

M=\underset{x\in[a,b]}{\max}\left|f(x,u,v)\right|,\ u,v\in J.

Then we have

Theorem 10.

We suppose:

(i)

$f\in C([a,b]\times J^{2}),J\subset\mathbb{R}$ a compact interval;
(ii)

$V\in C(C[a,b],C[a,b]);$
(iii)

$y\in\overline{B}(a,r)$ imply that $y(x)\in J,\forall x\in[a,b];$
(iv)

$M(b-a)\leq r.$

Then, the problem (1)–(2) has in $\overline{B}(a,r)$ at least one solution.

Proof.

We consider the operator $A:\overline{B}(a,r)\rightarrow C[a,b],$

A(x)(t)=\alpha+\int_{a}^{t}f(s,x(s),V(x)(s))ds,t\in[a,b].

We have $A(x)(t)\leq M(b-a)\leq r$ , so the condition (iv) assures the invariance of the sphere. From this we have $A:\overline{B}(a,r)\rightarrow\overline{B}(a,r)$ .

The set $\overline{B}(a,r)$ is bounded, closed and convex in $C[a,b].$ The operator $A$ is compact and continuous, so $A$ is complete continuous. $A(\overline{B}(a,r))$ is bounded and equicontinuous and from Arzela-Ascoli theorem we have that $A(\overline{B}(a,r))$ is relative compact. We apply Schauder fixed point theorem. ∎

4. Existence and uniqueness

In what follows we consider the problem (1)–(2) in the conditions (C₁)–(C₃).

Theorem 11.

We suppose that the conditions $(C_{1}),(C_{2})$ and $(C_{3})$ are satisfied. Then the problem (1)–(2) has in $C[a,b]$ a unique solution and this solution is the uniform limit of the successive approximations.

Proof.

The problem (1)–(2) is equivalent with the fixed point equation

B_{f}(x)=x,\ x\in C[a,b].

On the other hand we have that

	$\displaystyle\left\|B_{f}(x)(t)-B_{f}(y)(t)\right\|$	$\displaystyle\leq\int_{a}^{t}\left\|f(s,x(s),V(x)(s))-f(s,y(s),V(y)(s))\right\|ds\leq$
		$\displaystyle\leq L_{f}(\int_{a}^{t}\left\|x(s)-y(s)\right\|ds+\int_{a}^{t}\left\|V(x)(t)-V(y)(t)\right\|ds)$
		$\displaystyle\leq L_{f}(\int_{a}^{t}\left\|x(s)-y(s)\right\|e^{-\tau(s-a)}e^{\tau(s-a)}ds+$
		$\displaystyle\quad+\int_{a}^{t}\left\|V(x)(t)-V(y)(t)\right\|e^{-\tau(s-a)}e^{\tau(s-a)}ds)$
		$\displaystyle\leq\frac{e^{\tau t}}{\tau}(L_{f}+L_{f}L_{V})\left\\|x-y\right\\|_{B}.$

So,

\left\|B_{f}(x)-B_{f}(y)\right\|_{B}\leq\tfrac{1}{\tau}L_{f}(1+L_{V})\left\|x-y\right\|_{B},\ \forall x,y\in C[a,b],

i.e., $B_{f}$ is a contraction w.r.t. Bielecki norm on $C[a,b]$ . The proof follows from the contraction principle. ∎

Remark 12.

In the conditions of Theorem 11, the operator $B_{f}$ is PO. But

B_{f}|_{X_{\alpha}}=E_{f}|_{X_{\alpha}},\ \forall\alpha\in\mathbb{R}.

Hence, the operator $E_{f}$ is WPO and $F_{E_{f}}\cap X_{\alpha}=\{x_{\alpha}^{\ast}\},\forall\alpha\in\mathbb{R},$ where $x_{\alpha}^{\ast}$ is the unique solution of the problem (1)–(2).

5. Inequalities of Čaplygin type

In this section we shall study the relation between the solution of the problem (1)–(2) and the subsolution of the same problem. We have

Theorem 13.

We suppose that:

(a)

the conditions $(C_{1}),\ (C_{2})$ and $(C_{3})$ are satisfied;
(b)

$f(x,\cdot,\cdot):\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is increasing;
(c)

$V:C[a,b]\rightarrow C[a,b]$ is increasing.

Let $x$ be a solution of equation (1) and $y$ a solution of the inequality

y^{\prime}(t)\leq f(t,y(t),V(y)(t)),\ t\in[a,b].

Then $y(a)\leq x(a)\text{ implies that }y\leq x$ .

Proof.

In the terms of the operator $E_{f},$ we have

x=E_{f}(x)\text{ and }y\leq E_{f}(y),

and $y(a)\leq x(a).$ From the conditions $(C_{1}),\ (C_{2})$ and $(C_{3})$ we have that the operator $E_{f}$ is WPO. From the condition (b), $E_{f}^{\infty}$ is increasing (see [11]). If $\alpha\in\mathbb{R},$ then we denote by $\widetilde{\alpha}$ the following function

\widetilde{\alpha}:[a,b]\rightarrow\mathbb{R},\ \widetilde{\alpha}(t)=\alpha,\ \forall t\in[a,b].

We have

y\leq E_{f}(y)\leq\ldots\leq E_{f}^{\infty}(y)=E_{f}^{\infty}(\widetilde{y}(a))\leq E_{f}^{\infty}(\widetilde{x}(a))=x.

∎

6. Data dependence: monotony

In this section we study the monotony of the system (1)–(2) with respect to $f$ . For this we use Lemma 9.

Theorem 14.

We suppose that $f_{i}\in C([a,b]\times\mathbb{R}^{2},\mathbb{R}),i=1,2,$ satisfy the conditions $(C_{1}),(C_{2})$ and $(C_{3})$ . Furthermore, we suppose that:

(i)

$f_{1}\leq f_{2}\leq f_{3};$
(ii)

$f_{2}(t,\cdot,\cdot):\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is increasing;
(iii)

$V:C[a,b]\rightarrow C[a,b]$ is increasing.

Let $x_{i}\in C^{1}[a,b]$ be a solution of the equation

x_{i}^{\prime}(t)=f_{i}(t,x(t),V(x)(t)),\ t\in[a,b]\text{ and }i=1,2,3.

If, $x_{1}(a)\leq x_{2}(a)\leq x_{3}(a)$ , then $x_{1}\leq x_{2}\leq x_{3}.$

Proof.

From Theorem 11 we have that the operator $E_{f_{i}},i=1,2,3,\$ are WPOs. From the condition (ii) the operator $E_{f_{2}}$ is monotone increasing. From the condition (i) it follows that $E_{f_{1}}\leq E_{f_{2}}\leq E_{f_{3}}$ .

Let $\widetilde{x}_{i}(a)\in C[a,b]$ be defined by $\widetilde{x}_{i}(a)(t)=x_{i}(a),\ \forall t\in[a,b]$ . It is clear that

\widetilde{x}_{1}(a)(t)\leq\widetilde{x}_{2}(a)(t)\leq\widetilde{x}_{3}(a)(t),\ \forall t\in[a,b].

From Lemma 9 we have that

E_{f_{1}}^{\infty}(\widetilde{x}_{1}(a))\leq E_{f_{2}}^{\infty}(\widetilde{x}_{2}(a))\leq E_{f_{3}}^{\infty}(\widetilde{x}_{3}(a)).

But, $x_{i}=E_{fi}^{\infty}(\widetilde{x}_{i}(a)),$ and $x_{1}\leq x_{2}\leq x_{3}.$ ∎

7. Data dependence: continuity

Consider the Cauchy problem (1)–(2) and suppose the conditions of the Theorem 11 are satisfied. Denote by $x^{\ast}(\cdot;\alpha,f),\$ the solution of this problem. We shall use the data dependence theorem (see [11]).

Theorem 15.

We suppose that $\alpha_{i},f_{i},i=1,2$ satisfy the conditions $(C_{1}),(C_{2})$ and $(C_{3})$ . Furthermore, we suppose that there exists $\eta_{i}>0,i=1,2$ such that

(i)

$\left|\alpha_{1}(t)-\alpha_{2}(t)\right|\leq\eta_{1},\forall t\in[a,b];$
(ii)

$\left|f_{1}(t,u_{1},u_{2})-f_{2}(t,u_{1},u_{2})\right|\leq\eta_{2},\forall t\in[a,b],u_{i}\in\mathbb{R},i=1,2.$

Then

\left\|x_{1}^{\ast}(t;\alpha_{1},f_{1})-x_{2}^{\ast}(t;\alpha_{2},f_{2})\right\|\leq\frac{\eta_{1}+(b-a)\eta_{2}}{1-\tfrac{1}{\tau}L_{f}(1+L_{V})(b-a)},

where $x_{i}^{\ast}(t;\alpha_{i},f_{i}),i=1,2$ are the solution of the problem (1)–(2) with respect to $\alpha_{i},f_{i}$ and $L_{f}=\max(L_{f_{1}},L_{f_{2}}).$

Proof.

Consider the operators $B_{\alpha_{i},f_{i}},i=1,2.$ From Theorem 11 these operators are contractions. Additionally

\left\|B_{\alpha_{1},f_{1}}(x)-B_{\alpha_{2},f_{2}}(x)\!\right\|\leq\eta_{1}+(b-a)\eta_{2},\forall x\in C[a,b]

Now the proof follows from the the data dependence theorem (see [11]). ∎

In what follows we shall use the $c$ -WPOs techniques to give some data dependence results.

Theorem 16.

[12] Let $(X,d)$ be a metric space and $A_{i}:X\rightarrow X,\ i=1,2.$ Suppose that

(i)

the operator $A_{i}$ is $c_{i}$ -weakly Picard operator, $i\!=\!1,2;$
(ii)

there exists $\eta>0$ such that

$d(A_{1}(x),A_{2}(x))\leq\eta,\ \forall x\in X.$

Then $H(F_{A_{1}},F_{A_{2}})\leq\eta\max(c_{1},c_{2}).$

We have

Theorem 17.

We suppose that $f_{1}$ and $f_{2}$ satisfy the conditions $(C_{1}),(C_{2})$ and $(C_{3})$ . Let $S_{E_{f_{1}}},S_{E_{f_{2}}}$ be the solution set of system (1) corresponding to $f_{1}$ and $f_{2}$ . Suppose that there exists $\eta>0,$ such that

(5)

\left|f_{1}(t,u_{1},u_{2})-f_{2}(t,u_{1},u_{2})\right|\leq\eta

for all $t\in[a,b],u_{i}\in\mathbb{R},i=1,2.$ Then

H_{\left\|\cdot\right\|_{C}}(S_{E_{f_{1}}},S_{E_{f_{2}}})\leq\frac{(b-a)\eta}{1-\tfrac{1}{\tau}L_{f}(1+L_{V})(b-a)},

where $L_{f}=\max(L_{f_{1}},L_{f_{2}})$ and $H_{\left\|\cdot\right\|_{C}}$ denotes the Pompeiu-Housdorff functional with respect to $\left\|\cdot\right\|_{C}$ on $C[a,b].$

Proof.

In the condition of Theorem 11, the operators $E_{f_{1}}$ and $E_{f_{2}}$ are $c_{i}$ -weakly Picard operators, $i=1,2$ . Let $X_{\alpha}:=\{x\in C[a,b]|\ x(a)=\alpha\}$ .

It is clear that $E_{f_{1}}|_{X_{\alpha}}=B_{f_{1}},\ E_{f_{2}}|_{X_{\alpha}}=B_{f_{2}}.$ Therefore,

\left|E_{f_{1}}^{2}(x)-E_{f_{1}}(x)\right|\leq\tfrac{1}{\tau}L_{f_{1}}(1+L_{V})(b-a)\left|E_{f_{1}}(x)-x\right|,

\left|E_{f_{2}}^{2}(x)-E_{f_{2}}(x)\right|\leq\tfrac{1}{\tau}L_{f_{2}}(1+L_{V})(b-a)\left|E_{f_{2}}(x)-x\right|,

for all $x\in C[a,b].$ Now, choosing $\alpha_{1}=\tfrac{1}{\tau}L_{f_{1}}(1+L_{V})(b-a)\ \text{and }\alpha_{2}=\tfrac{1}{\tau}L_{f_{2}}(1+L_{V})(b-a)$ , we get that $E_{f_{1}}$ and $E_{f_{2}}$ are $c_{i}$ -weakly Picard operators, $i=1,2$ with $c_{1}=(1-\alpha_{1})^{-1}\$ and $\ c_{2}=(1-\alpha_{2})^{-1}$ . From (5) we obtain that

\left\|E_{f_{1}}(x)-E_{f_{2}}(x)\right\|_{C}\leq(b-a)\eta,

$\forall x\in C[a,b].$ Applying Theorem 16 we have that

H_{\left\|\cdot\right\|_{C}}(S_{E_{f_{1}}},S_{E_{f_{2}}})\leq\frac{(b-a)\eta}{1-\tfrac{1}{\tau}L_{f}(1+L_{V})(b-a)},

where $L_{f}=\max(L_{f_{1}},L_{f_{2}})$ and $H_{\left\|\cdot\right\|_{C}}$ is the Pompeiu-Housdorff functional with respect to $\left\|\cdot\right\|_{C}$ on $C[a,b].$ ∎

8. Applications

Let us now discuss some applications of the general result. Consider the first order integro-differential equation of Volterra type (see [3], [4], [6])

(6)

x^{\prime}(t)=f(t,x(t),\int\nolimits_{a}^{t}k(t,s,x(s)ds),t\in[a,b],

with the initial condition

(7)

x(0)=\alpha.

For this equations the conditions (C₁)-(C₃) becomes

(C₁)

$\alpha\in\mathbb{R}$ , $f\in C([a,b]\times\mathbb{R}^{2},\mathbb{R}),\ k\in C([a,b]\times[a,b]\times\mathbb{R},\mathbb{R})$ are given;

(C₂)

there exists $L_{f}>0$ such that

\left|f(t,u_{1},u_{2})-f(t,v_{1},v_{2})\right|\leq L_{f}\sum_{i=1}^{2}\left|u_{i}-v_{i}\right|,t\in[a,b],u_{i},v_{i}\in\mathbb{R},i=1,2.

(C₃)

there exists $L_{k}>0$ such that

$\left|k(t,s,u)-k(t,s,v)\right|\leq L_{k}\left|u-v\right|,\forall t,s\in[a,b],\ u,v\in\mathbb{R}.$

From this conditions and the above results we have

Theorem 18.

We suppose that the conditions $(C_{1}),(C_{2})$ and $(C_{3})$ are satisfied. Then the problem (6)–(7) has in $C[a,b]$ a unique solution and this solution is the uniform limit of the successive approximations.

Theorem 19.

We suppose that:

(a)

the conditions $(C_{1}),\ (C_{2})$ and $(C_{3})$ are satisfied;
(b)

$f(t,\cdot,\cdot):\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is increasing;
(c)

$k:C([a,b]\times[a,b]\times\mathbb{R),\mathbb{R}}\rightarrow\mathbb{R}$ is increasing.

Let $x$ be a solution of equation (1) and $y$ a solution of the inequality

y^{\prime}(t)\leq f(t,y(t),\int\nolimits_{a}^{t}k(t,s,y(s)ds),\ t\in[a,b].

Then $y(a)\leq x(a)\text{ implies that }y\leq x$ .

Theorem 20.

We suppose that $f_{i}\in C([a,b]\times\mathbb{R}^{2},\mathbb{R}),i=1,2,$ satisfy the conditions $(C_{1}),(C_{2})$ and $(C_{3})$ . Furthermore, we suppose that:

(i)

$f_{1}\leq f_{2}\leq f_{3};$
(ii)

$f_{2}(t,\cdot,\cdot):\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is increasing;
(iii)

$k:C([a,b]\times[a,b]\times\mathbb{R,\mathbb{R})}\rightarrow\mathbb{R}$ is increasing.

Let $x_{i}\in C^{1}[a,b]$ be a solution of the equation

x_{i}^{\prime}(t)=f_{i}(t,x(t),\int\nolimits_{a}^{t}k(t,s,y(s)ds),\ t\in[a,b]\text{ and }i=1,2,3.

If, $x_{1}(a)\leq x_{2}(a)\leq x_{3}(a)$ , then $x_{1}\leq x_{2}\leq x_{3}.$

Theorem 21.

We suppose that $\alpha_{i},f_{i},i=1,2$ satisfy the conditions $(C_{1}),(C_{2})$ and $(C_{3})$ . Furthermore, we suppose that there exists $\eta_{i}>0,i=1,2$ such that

(i)

$\left|\alpha_{1}(t)-\alpha_{2}(t)\right|\leq\eta_{1},\forall t\in[a,b];$
(ii)

$\left|f_{1}(t,u_{1},u_{2})-f_{2}(t,u_{1},u_{2})\right|\leq\eta_{2},\forall t\in[a,b],u_{i}\in\mathbb{R},i=1,2.$

Then

\left\|x_{1}^{\ast}(t;\alpha_{1},f_{1})-x_{2}^{\ast}(t;\alpha_{2},f_{2})\right\|\leq\frac{\eta_{1}+(b-a)\eta_{2}}{1-\tfrac{1}{\tau}L_{f}(1+L_{k})(b-a)},

where $x_{i}^{\ast}(t;\alpha_{i},f_{i}),i=1,2$ are the solution of the problem (6)–(7) with respect to $\alpha_{i},f_{i}$ and $L_{f}=\max(L_{f_{1}},L_{f_{2}}).$

Acknowledgement 22.

This work has been supported by grant 2-CEx06-11-96/19.09.2006.

References

[1] Azbelev, N.V. (ed), Functional-differential equations (Russian), Perm. Politekh. Inst., Perm, 1985
[2] Corduneanu, C., Existence of solutions for neutral functional differential equations with causal operators, Journal of Differential Equations, 168 (2000), 93-101
[3] Corduneanu, C., Abstract Volterra equations (a survey), Math. and Comp. Modelling, 32 (2000), 1503-1528
[4] Guo, D., Lakshmikantham, V. and Liu, X., Nonlinear Integral Equations in Abstract Spaces, Kuwer Academic Publishers, Dordrecht, Boston, London, 1996
[5] Kolmanovskii, V. and Myshkis, A., Applied Theory of Functional Differential Equations, Kluwer, 1992
[6] Kwapisz, M., An extension of Bielecki’s method of proving global existence and uniqueness results for functional equations, Ann. Univ. Mariae Curie-Sklodowska, Sect. A, 38 (1984), 59-68
[7] Lupulescu, V., Causal functional differential equations in Banach spaces, Nonlinear Analysis, (2007), doi:10.1016/j.na.2007.11.028
[8] Otrocol, D., Lotka-Volterra systems with retarded argument (in Romanian), Cluj University Press, 2007
[9] O’Regan, D., A note on the topological structure of the solution set of abstract Volterra equations, Math. Proc. of the Royal Irish Academy, 99A (1) (1999), 67-74
[10] Rus, I.A., Picard operators and applications, Scientiae Mathematicae Japonicae, 58 (2003), No. 1, 191-219
[11] Rus, I.A., Generalized contractions, Cluj University Press, 2001
[12] Rus, I.A., Weakly Picard operators and applications, Seminar on Fixed Point Theory, Cluj-Napoca, 2 (2001), 41-58

	$\displaystyle\left\|B_{f}(x)(t)-B_{f}(y)(t)\right\|$	$\displaystyle\leq\int_{a}^{t}\left\|f(s,x(s),V(x)(s))-f(s,y(s),V(y)(s))\right\|ds\leq$
		$\displaystyle\leq L_{f}(\int_{a}^{t}\left\|x(s)-y(s)\right\|ds+\int_{a}^{t}\left\|V(x)(t)-V(y)(t)\right\|ds)$
		$\displaystyle\leq L_{f}(\int_{a}^{t}\left\|x(s)-y(s)\right\|e^{-\tau(s-a)}e^{\tau(s-a)}ds+$
		$\displaystyle\quad+\int_{a}^{t}\left\|V(x)(t)-V(y)(t)\right\|e^{-\tau(s-a)}e^{\tau(s-a)}ds)$
		$\displaystyle\leq\frac{e^{\tau t}}{\tau}(L_{f}+L_{f}L_{V})\left\\|x-y\right\\|_{B}.$

Abstract Volterra operators

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Abstract Volterra operators

Abstract.

Key words and phrases:

2000 Mathematics Subject Classification:

1. Introduction

Lemma 1.

2. Weakly Picard operators

Definition 2.

Remark 3.

Definition 4.

Definition 5.

Remark 6.

Definition 7.

Lemma 8.

Lemma 9.

3. Existence

Theorem 10.

Proof.

4. Existence and uniqueness

Theorem 11.

Proof.

Remark 12.

5. Inequalities of Čaplygin type

Theorem 13.

Proof.

6. Data dependence: monotony

Theorem 14.

Proof.

7. Data dependence: continuity

Theorem 15.

Proof.

Theorem 16.

Theorem 17.

Proof.

8. Applications

Theorem 18.

Theorem 19.

Theorem 20.

Theorem 21.

Acknowledgement 22.

References

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