** Authors:**
NGUYEN DAT THUC, LE THI PHUONG NGOC, NGUYEN THANH LONG

** Abstract: **
This paper is devoted to the study of the following nonlinear functional integral equation
\begin{equation}
f(x)=\sum\limits_{i=1}^{q}\alpha _{i}(x)f(\tau_{i}(x))+\int_{0}^{\sigma_{1}(x)}\Psi \left( x,t,f(\sigma _{2}(t)),\int_{0}^{\sigma_{3}(t)}f(s)ds\right) dt+g(x),\text{ }\forall x\in \lbrack 0,1], \tag{E}
\label{E}
\end{equation}
where $\tau _{i},$ $\sigma _{1},$ $\sigma _{2},$ $\sigma
_{3}:[0,1]\rightarrow \lbrack 0,1];$ $\alpha _{i},$ $g:[0,1]\rightarrow \mathbb{R};$ $\Psi :[0,1]\times \lbrack 0,1]\times \mathbb{R}^{2}\rightarrow \mathbb{R}$ are the given continuous functions and $f:[0,1]\,\rightarrow \mathbb{R}$ is an unknown function. First, two sufficient conditions for the existence and some properties of solutions of Eq. (E) are proved. By using Banach's fixed point theorem, we have the first sufficient condition yielding existence, uniqueness and stability of the solution. By applying Schauder's fixed point theorem, we have the second sufficient condition for the existence and compactness of the solution set. An example is also given in order to illustrate the results obtained here.
Next, in the case of $\Psi \in C^{2}([0,1]\times \lbrack 0,1]\times \mathbb{R}^{2}; \mathbb{R}),$ we investigate the quadratic convergence for the solution of Eq. (E). Finally, the smoothness of the solution depending on data is established.

** Keywords: **
Nonlinear functional integral equation, quadratic convergence, stability and smoothness of solutions, compactness of the solution set

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