Integrable solutions of a functional-integral equation. (English) Zbl 0755.45005

A theorem about the existence of solutions of the functional-integral equation (1) \(x(t)=f\left(t,\int^ 1_ 0k(t,s)g(s,x(s))ds\right)\), \(t\in[0,1]\), is proved. The technique used in the proof depends on an interesting conjunction of the notions of the measure of weak noncompactness and the Schauder fixed point principle.
It is worth while to mention that the existence theorem for (1) is proved under rather general and natural hypotheses.


45G10 Other nonlinear integral equations
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[1] J. Appell and E. De Pascale, Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili , Boll. Un Mat. Ital. Serie VI, II -B, (1984), 497-515. · Zbl 0507.46025
[2] J. Banas, Integrable solutions of Hammerstein and Urysohn integral equations , J. Austral. Math. Soc. 46 (1989), 61-68. · Zbl 0666.45008
[3] J. Banas and Z. Knap, Integrable solutions of a functional-integral equation , Rev. Mat. Univ. Complut. Madrid 2 (1989), 31-38. · Zbl 0679.45003
[4] F.S. De Blasi, On a property of the unit sphere in a Banach space , Bull. Math. Soc. Sci. Math. R.S. Roumanie 21 (1977), 259-262. · Zbl 0365.46015
[5] K. Deimling, Nonlinear functional analysis , Springer Verlag, New York, 1985. · Zbl 0559.47040
[6] G. Emmanuele, About the existence of integrable solutions of a functional integral equation , to appear in Rev. Mat. Univ. Complut. Madrid. · Zbl 0746.45004
[7] ——–, Integrable solutions of Hammerstein integral equations , to appear in Appl. Anal. · Zbl 0795.45004 · doi:10.1080/00036819308840198
[8] M.A. Krasnoselskii, Topological methods in the theory of nonlinear integral equations , Pergamon Press, Elmsford, 1964. · Zbl 0111.30303
[9] R.H. Martin, Nonlinear operators and differential equations in Banach spaces , Wiley and Sons, New York, 1976. · Zbl 0333.47023
[10] G. Scorza Dragoni, Un teorema sulle funzioni continue rispetto ad una e misurabili rispetto ad un’altra variabile , Rend. Sem. Mat. Univ. Padova 17 (1948), 102-106. · Zbl 0032.19702
[11] P.P. Zabrejko, A.I. Koshelev, M.A. Krasnoselskii, S.G. Mikhlin, L.S. Rakovsh-chik and V.J. Stecenko, Integral equations , Noordhoff, Leyden, 1975. · Zbl 0293.45001
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