We introduce a class of measures of noncompactness in Banach algebras satisfying certain condition and we prove a ﬁxed point theorem for the product of two operators being contractions with respect to such measures of noncompactess. We also indicate measures of noncompactness in some Banach function algebras which satisfy the mentioned condition. The obtained results are applied to prove a few theorems on the existence of solutions of nonlinear integral equations in Banach algebras. Some characterizations of solutions of considered integral equations are also derived.
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Józef Banaś, Leszek Olszowy, On a Class of Measures of Noncompactness in Banach Algebras and Their Application to Nonlinear Integral Equations. Z. Anal. Anwend. 28 (2009), no. 4, pp. 475–498DOI 10.4171/ZAA/1394