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

Banaś, Józef; El-Sayed, Wagdy Gomaa

J. Math. Anal. Appl. 167, No. 1, 133-151 (1992).

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)].

Reviewer: J.Appell (Würzburg)

MSC:

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

Keywords:

solvability; nonlinear Volterra equation; locally bounded variation; Vitali compactness criterion; measure of weak non-compactness

Citations:

Zbl 0365.46015

References:

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