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Abstract. This paper is devoted to the study of the following perturbed system of nonlinear functional equationsx P Ω " r´b, bs, i " 1, . . . , n, where ε is a small parameter, a ijk , b ijk are the given real constants, R ijk , S ijk , X ijk : Ω Ñ Ω, gi : Ω Ñ R, Ψ : ΩˆR 2 Ñ R are the given continuous functions and fi : Ω Ñ R are unknown functions. First, by using the Banach fixed point theorem, we find sufficient conditions for the unique existence and stability of a solution of (E). Next, in the case of Ψ P C 2 pΩˆR 2 ; Rq, we investigate the quadratic convergence of (E). Finally, in the case of Ψ P C N pΩˆR 2 ; Rq and ε sufficiently small, we establish an asymptotic expansion of the solution of (E) up to order N`1 in ε. In order to illustrate the results obtained, some examples are also given.

This work is concerned with a multi-dimensional viscoelastic pseudo-parabolic equation with critical Sobolev exponent. First, with some suitable conditions, we prove that the weak solution exists globally. Next, we show that the stability of the system holds for a much larger class of kernels than the ones considered in previous literature. More precisely, we consider the kernel g : [ 0 , ∞ ) ⟶ ( 0 , ∞ ) g:{[}0,\infty )\hspace{0.33em}\longrightarrow \hspace{0.33em}(0,\infty ) satisfying g ′ ( t ) ⩽ − ξ ( t ) G ( g ( t ) ) {g}^{^{\prime} }(t)\leqslant -\xi (t)G(g(t)) , where ξ \xi and G G are functions satisfying some specific properties.

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In this paper, we study the existence and the compactness of the set of solutions for a nonlinear integrodifferential equation in N variables. The main tools are the fixed point theorem of Krasnosel'skii with the definition of a suitable Banach space and a sufficient condition for relative compactness of subsets in this space. An illustrative example is given.