Geodesic
Remarks on existence of positive solutions of some integral equations
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 44 (2005) no. 1, pp. 71-82.

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We study the existence of positive solutions of the integral equation \[ x(t) = \mu \int _0^1 k(t, s) f(s, x(s), x^{\prime }(s), \ldots , x^{(n-1)} (s))\, ds, \quad n \ge 2 \] in both $ C^{n-1} [0, 1] $ and $ W^{n-1, p} [0, 1] $ spaces, where $ p \ge 1 $ and $ \mu > 0 $. Throughout this paper $k$ is nonnegative but the nonlinearity $f$ may take negative values. The Krasnosielski fixed point theorem on cone is used.
Classification : 34B10, 34B15, 34G20, 34K10, 45B05, 45G10, 45M20
Keywords: Positive solutions; Fredholm integral equations; cone; boundary value problems; fixed point theorem. 
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Ligęza, Jan. Remarks on existence of positive solutions of some integral equations. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 44 (2005) no. 1, pp. 71-82. http://geodesic.mathdoc.fr/item/AUPO_2005__44_1_a7/