In this paper, we present two new fixed point theorems in Fréchet algebras and Fréchet spaces. Our fixed point results are expressed with the help of family of measures of noncompactness and generalizes Darbo theorem. As an application, we establish some existence results for various types of nonlinear integral equations.

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.

We study the solvability of some nonlinear functional integral equations in the Banach algebra of real functions defined, continuous, and bounded on the real half axis. We apply the technique of measures of noncompactness in order to obtain existence results for equations in question. Additionally, that technique allows us to obtain some characterization of considered integral equations. An example illustrating the obtained results is also included.

In this paper, we formulate necessary and sufficient conditions for relative compactness in the space $$BG({\mathbb {R}}_+,E)$$ B G ( R + , E ) of regulated and bounded functions defined on $${\mathbb R}_+$$ R + with values in the Banach space E. Moreover, we construct four new measures of noncompactness in the space $$BG({\mathbb {R}}_+,E)$$ B G ( R + , E ) . We investigate their properties and we describe relations between these measures. We provide necessary and sufficient conditions so that the superposition operator (Niemytskii) maps $$BG({\mathbb {R}}_+,E)$$ B G ( R + , E ) into $$BG({\mathbb {R}}_+,E)$$ B G ( R + , E ) and, additionally, be compact.

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