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Integrable solutions of Hammerstein and Urysohn integral equations

Published online by Cambridge University Press:  09 April 2009

Józef Banaś
Józef Banaś
Affiliation:
Department of Mathematics, I. Lukasiewicz Technical University, 35-084 Rzeszów, Poznańska 2, Poland
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Abstract

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In this paper we prove theorems on the existence of integrable and monotonic solutions of Hammserstein and Urysohn integral equations. The basic tool used in the proof is the fixed point principle for contractions with respect to the so-called measure of weak noncompactness.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 46 , Issue 1 , February 1989 , pp. 61 - 68
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Appell, J., ‘Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator’, J. Math. Anal. Appl. 83 (1981), 251–263.CrossRefGoogle Scholar
[2]Appell, J., ‘On the solvability of nonlinear noncompact problems in function spaces with applications to integral and differential equations’, Boll. Un. Mat. Ital. B (6) 1 (1982), 1161–1177.Google Scholar
[3]Appell, J. and De Pascale, E., ‘Su alcuni parametri connesi con la misura di non compattezza di Hausdorff in spazi di functioni misurabili’, Boll. Un. Mat. Ital. B (6) 3 (1984), 497–515.Google Scholar
[4]Banaá, J., ‘On the superposition operator and integrable solutions of some functional equations’, Nonlinear Anal. TMA (preprint).Google Scholar
[5]De Blasi, F. S., ‘On a property of the unit sphere in a Banach space’, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 21 (1977), 259–262.Google Scholar
[6]Dunford, N. and Schwartz, J., Linear operators (Interscience, Leyden, 1963).Google Scholar
[7]Emmanuele, G., ‘Measure of weak noncompactness and fixed point theorems’, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 25 (1981), 353–358.Google Scholar
[8]Juberg, R. K., ‘The measure of noncompactness in Lp for a class of integral operators’, Indiana Univ. Math. J. 23 (1974), 925–936.CrossRefGoogle Scholar
[9]Krasnosel'skii, M. A., ‘On the continuity of the operator Fu(x) = f(x, u(x))’, Dokl. Akad. Nauk SSSR 77 (1951), 185–188 (in Russian).Google Scholar
[10]Krasnosel'skii, M. A., Zabrejko, P. P., Pustyl'nik, J. I. and Sobolevskii, P. J., Integral operators in spaces of summable functions (Nauka, Moscow, 1966) [English translation: Noordhoff, Leyden, 1976].Google Scholar
[11]Krasnosel'skii, M. A., and Zabrejko, P. P., Geometric methods of nonlinear analysis (Moscow 1975; in Russian).Google Scholar
[12]Shragin, V. I., ‘On the weak continuity of the Nemytskii operator’, Uchen. Zap. Moskov. O.P.I. 57 (1957), 73–79 (in Russian).Google Scholar
[13]Stuart, C. A., ‘The measure of noncompactness of some linear integral operators’, Proc. Roy. Soc. Edinburgh 71 (1973), 167–179.Google Scholar
[14]Zabrejko, P. P., Koshelev, A. I., Krasnosel'skii, M. A., Mikhlin, S. C., Rakovshchik, L. S. and Stecenko, V. J., Integral equations (Nauka, Moscow, 1968; in Russian).Google Scholar
[15]Zabrejko, P. P. and Pustyl'nik, J. I., ‘On the continuity and complete continuity of nonlinear integral operators in Lp spaces’, Uspehi Mat. Nauk 19 (1964), 204–205 (in Russian).Google Scholar