Continuous Dependence of the Solutions of Nonlinear Integral Quadratic Volterra Equation on the Parameter

Abstract

We prove results on the existence and continuous dependence of solutions of a nonlinear quadratic integral Volterra equation on a parameter. This dependence is investigated in terms of Hausdorff distance. The considerations are placed in the Banach space and the Fréchet space.

On quadratic integral equations of Volterra type in Frechet spaces

“…Example 1 cannot be covered by the method used in [4] because the function (x 1 + x 2 ) exp x 1 +x 2 is not bounded. Example 2 cannot be treated by the method in [9] because Ω ⊂ R 2 and f depends on the unknown u.…”

“…Our aim in this work is twofold: first we extend the results obtained in [9,12,14] to the case of unbounded open subsets of R n for a quite general quadratic Volterra type equation of the form (1.1) and secondly we slightly relax the assumptions in [9]. This is the content of Section 3, where the functions f and g satisfy some Lipschitz conditions while h obeys a general growth condition in its third argument.…”

“…Recently, in [9,12], the authors defined a sequence of measures of noncompactness and obtained existence results in the Fréchet space of continuous functions on the real half-axis.…”

“…y, u(y)dy, x ≥ 0, is solved in[9] via a measure of noncompactness and it is clearly covered by Equation (1.1). From Theorem 3.5, the existence of solution is obtained when Λ(x) = (0, x), the nonlinearity f satisfies the Lipschitz condition (H 3 ) in the second argument, and the function ν is variable-separated dominated as in (H 4 ) while it is assumed bounded in the third argument in [9, (H 3 ) , (33)], namely |ν(x, y, u)| ≤ g(x, y).…”

Evolution equations in Fréchet spaces

“…We recall the following definition of the notion of a sequence of measures of noncompactness [10,11].…”