Solvability of infinite systems of singular integral equations in Fréchet space of continuous functions

AbstractThe aim of this paper is to show how some measures of noncompactness in the Fréchet space of continuous functions defined on an unbounded interval can be applied to an infinite system of singular integral equations. The results obtained generalize and improve several ones

On existence of solutions of a neutral differential equation with deviation argument

We establish theorems on the existence and asymptotic characterization of solutions of a differential equation of neutral type with deviated argument on neutral type. The mentioned differential equation admits both delayed and advanced arguments. In our considerations we use techniques linking measures of noncompactness with the Tikhonov fixed point principle in suitable Frechet space. This approach admits us to improve and extend some result

On existence of solutions of a quadratic Urysohn integral equation on an unbounded interval

We show that $\omega_0 (X) = \lim_{T\to\infty} \lim_{\varepsilon\to 0} \omega^T (X, \varepsilon)$ is a measure of noncompactness defined on some subsets of the space $C(\mathbb{R}^+) = \{x\colon \mathbb{R}^+ \to \mathbb{R},\ x\ \text{continuous}\}$ furnished with the distance defined by the family of seminorms $|x|_n$. Moreover, using a technique associated with the measures of noncompactness, we prove the existence of solutions of a quadratic Urysohn integral equation on an unbounded interval. This measure allows to obtain theorems on the existence of solutions of a integral equations on an unbounded interval under a weaker assumptions then the assumptions of theorems obtained by applying two-component measures of noncompactness

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

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