Integro-differential equations have been discussed in many applied fields, such as biological, physical and engineering problems. Particulary, the Fredholm-Volterra equations are interesting because they put together two types of integrals that may have different approaches from the solution point of view.

The purpose of the paper is to study existence, data dependence and comparison results for the solutions of the functional integral equations of the form

The new idea of this paper compared to [15] and [16] is the Fredholm term and the delay argument $h(t)$.

The paper is organized as follows. In section 2, we recall some definitions and facts concerning the weakly Picard operator theory. In section 3, we prove first the existence and uniqueness theorem and then we give some properties of the solution regarding the order preserving property and the continuity of the solution of the problem (1.1). In the last section some applications are given.

For the basic theory of functional integral equations see [1]-[3], [6] and [7]. Fredholm-Volterra integro-differential equations were studied in [15] and [16].

In this paper we use the terminologies and notations from [11]-[12]. For the convenience of the reader we recall some of them.

Let $(X,d)$ be a metric space and $A:X\rightarrow X$ an operator.

We denote by$\;A^{0}=1_{X},\;A^{1}=A,\;A^{n+1}:=A\circ A^{n},n\in\mathbb{N}$ the iterates of the operator $A$;

We also use the following notations:

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

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

The following results are very useful in the sequel.

Another important notion is

We note that most (in the sense of Baire category) operators in the space of nonexpansive operators are (weakly) Picard operators. See, for example, the two papers by S. Reich and A. J. Zaslavski [13] and [14]. For some examples of WPOs see [4], [5] and [8]-[10].

We consider now

• (C₁)

  $g\in C([a,b]\times\mathbb{R}^{2};\mathbb{R}),\ f\in C([a,b]\times\mathbb{R},\mathbb{R)}$, $K\in C([a,b]\times\mathbb{R},\mathbb{R)}$, $h(t)\leq t$, $h\in C([a,b],\mathbb{[}a_{1},b\mathbb{])},a_{1}\leq a$;

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• (C${}_{1}^{\prime}$)

  $x(t)=\varphi(t),t\in[a_{1},a],\varphi\in C([a_{1},a],[a,b])$;

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• (C₂)

  there exists $l_{g}>0$ such that $g$ is Lipschitz with respect to the last two arguments, i.e.,

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  $$\left|g(t,u_{1},u_{2})-g(t,v_{1},v_{2})\right|\leq l_{g}\left(\left|u_{1}-v_{1}\right|+\left|u_{2}-v_{2}\right|\right),$$

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

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• (C₃)

  there exists $l_{f}>0$ such that $f$ is Lipschitz with respect to the second argument, i.e.,

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  $$\left|f(t,u)-f(t,v)\right|\leq l_{f}\left|u-v\right|,\ \forall t\in[a,b],u,v\in\mathbb{R};$$

• (C₄)

  there exists $l_{K}>0$ such that $K$ is Lipschitz with respect to the second argument, i.e.,

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  $$\left|K(t,u)-K(t,v)\right|\leq l_{K}\left|u-v\right|,\ \forall t\in[a,b],u,v\in\mathbb{R};$$

• (C₅)

  $\varphi(a)=g(a,x(a),x(h(a)))+\int\nolimits_{a}^{b}K(s,x(h(s)))ds;$

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• (C₆)

  $2l_{g}+(b-a)(l_{f}+l_{K})<1.$

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The numbers $m_{g},M_{g},M_{f},M_{k}>0$ are such that

• (C₇)

  $M_{g}+(M_{f}+M_{K})(b-a)\leq b;$

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• (C₈)

  $a\leq M_{g}-(M_{f}+M_{K})(b-a).$

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With respect to the equation (1.1) we consider the equation (in $\beta\in\mathbb{R}$)

Let $S_{g,K}$ be the solution set of the equation (3.1). By a solution of equation (1.1) we understand a function $x\in C([a_{1},b],\mathbb{R})$. So we consider the Banach space $X:=\left(C([a_{1},b];\mathbb{R}),\left\|\cdot\right\|_{C}\right)$ where $\left\|\cdot\right\|_{C}$ is the Chebyshev norm defined by $\left\|x\right\|_{C}=\underset{t\in[a,b]}{\sup}(\left|x(t)\right|).$

Let the operator $B:X\rightarrow X$ be defined by

where $\varphi(a)=\beta,\ \beta\in S_{g,K}$. It is obvious that the solution set of the equation (1.1) is $F_{B}.$

Let $L>0$ and

It is easy to see that $C_{L}$ is convex and compact in $X$, thus it is a complete metric space with respect to $d_{\left\|\cdot\right\|_{C}}.$

Let $X_{\varphi}:=\{x\in X|\ x(t)=\varphi(t),t\in[a_{1},a]\}.$ Notice that $X=\underset{\varphi\in C(\mathbb{[}a_{1},b\mathbb{]},[a,b]\mathbb{)}}{\cup}X_{\varphi}$ is a partition of $X$ and we have the following lemma (see [15]).

We denote by $Y:=\underset{\varphi\in C\mathbb{[}a_{1},a\mathbb{]}}{\cup}(X_{\varphi}\cap C_{L}).$

Our first main result is the following one:

Next we shall study the relation between the solution of the problem (1.1) with the initial condition

and the subsolution of the same problem. We have

Let $x$ be a solution of equation (1.1) and $y$ a solution of the inequality

Then $y(t)\leq x(t)\text{ with }t\in[a_{1},a]\text{\ implies that }y\leq x$ on $[a,b]$.

In the following part of this section we study the order preserving property of the problem (1.1)–(4.1) with respect to $K.$ For this we use Lemma 2.8.

In the last part of this section we consider the Cauchy problem (1.1)–(4.1) and we suppose that the conditions of the Theorem 4.1 are satisfied. We denote by $x^{\ast}(\cdot;g,f,K),\$the solution of this problem. The last result is a data dependence result for the solutions of two similar problems with different parameters.

Let us now discuss some applications for the general result.