Measures of weak noncompactness and nonlinear integral equations of convolution type. (English) Zbl 0699.45002

The authors prove that the equation \(x(t)=f[t,\int^{\infty}_{0}k(t- s)x(\phi (s))ds]\) has a monotone solution \(x\in L^ 1(0,\infty)\) if suitable conditions are imposed on the functions f, k, and \(\phi\). The proof builds on measures of weak noncompactness [F. S. De Blasi, Bull. Math. Soc. Sci. Math. R. S. R. n. Ser. 21(69), 259-262 (1977; Zbl 0365.46015)], a fixed point principle of G. Emmanuele [Bull. Math. Soc. Sci. Math. Repub. Soc. Roum. Nouv. Ser. 25, 353-358 (1981; Zbl 0482.47027)], and certain properties of nonlinear superposition operators [P. P. Zdrejko and the referee, J. Aust. Math. Soc. Ser. A 47, 186- 210 (1989; Zbl 0683.47045)].
Reviewer: J.Appell


45G05 Singular nonlinear integral equations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
Full Text: DOI


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