Measures of noncompactness and solvability of an integral equation in the class of functions of locally bounded variation. (English) Zbl 0754.45008

The authors study the solvability of the nonlinear Volterra equation
\(x(t)=u(t)+\int^ t_ 0k(t,s)f[s,x(m(s))]ds\) \((0\leq t<\infty)\) in the space of the functions \(x\in L^ 1(0,\infty)\) of locally bounded variation.
The main tool is the Vitali compactness criterion in \(L^ 1\), combined with a measure of weak non-compactness \(\gamma\) which is equivalent to F. S. De Blasi’s measure \(\beta\) [Bull. Math. Soc. Sci. Math. R. S. R. n. Ser. 21(69), 259-262 (1977; Zbl 0365.46015)].


45G10 Other nonlinear integral equations
45D05 Volterra integral equations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.


Zbl 0365.46015
Full Text: DOI


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