Zeitschrift für Analysis und ihre Anwendungen
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Published online: 2009-12-23
On a Class of Measures of Noncompactness in Banach Algebras and Their Application to Nonlinear Integral EquationsJózef Banaś and Leszek Olszowy (1) Rzeszów University of Technology, Poland
(2) Rzeszów University of Technology, Poland
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.
Keywords: Measure of noncompactness, Banach algebra, product of two operators, ﬁxed point theorem, nonlinear integral equation
Banaś Józef, Olszowy Leszek: On a Class of Measures of Noncompactness in Banach Algebras and Their Application to Nonlinear Integral Equations. Z. Anal. Anwend. 28 (2009), 475-498. doi: 10.4171/ZAA/1394