Nonzero solutions of Hammerstein integral equations with discontinuous kernels.

*(English)*Zbl 1008.45004The authors study the nonlinear second-order equation
\[
u''(t)= f(t, u(t))\quad\text{for }0< t< 1,
\]
subject to the three-point boundary conditions \(u(0)= 0\) and \(u(1)=\alpha u'(\eta)\) for \(\eta\in (0,1)\) fixed. The main tool is to pass as usual to an equivalent Hammerstein integral equation and to apply the Leray-Schauder fixed point index; in contrast to classical results, however, the kernel of the corresponding integral operator may be discontinuous and change sign. In this way, the authors obtain existence of multiple nontrivial (but not necessarily positive) solutions in case \(\alpha< 0\) or \(0\leq\alpha< 1-\eta\).

Similar problems for other choices of \(\alpha\) and different boundary conditions have been considered in previous papers by K. Q. Lan [Differ. Equ. Dyn. Syst. 8, No. 2, 175-192 (2000; Zbl 0977.45001) and J. Lond. Math. Soc., II. Ser. 63, No. 3, 690-704 (2001; Zbl 1032.34019)], by K. Lan and J. R. L. Webb [J. Differ. Equations 148, No. 2, 407-421 (1998; Zbl 0909.34013)], and by J. R. L. Webb [Nonlin. Anal. 47, 4319-4332 (2001)].

Similar problems for other choices of \(\alpha\) and different boundary conditions have been considered in previous papers by K. Q. Lan [Differ. Equ. Dyn. Syst. 8, No. 2, 175-192 (2000; Zbl 0977.45001) and J. Lond. Math. Soc., II. Ser. 63, No. 3, 690-704 (2001; Zbl 1032.34019)], by K. Lan and J. R. L. Webb [J. Differ. Equations 148, No. 2, 407-421 (1998; Zbl 0909.34013)], and by J. R. L. Webb [Nonlin. Anal. 47, 4319-4332 (2001)].

Reviewer: Jürgen Appell (Würzburg)

##### MSC:

45G10 | Other nonlinear integral equations |

47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |

34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

47H11 | Degree theory for nonlinear operators |

##### Keywords:

nonzero solutions; discontinuous kernels; Hammerstein integral equation; Leray-Schauder fixed point index
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\textit{G. Infante} and \textit{J. R. L. Webb}, J. Math. Anal. Appl. 272, No. 1, 30--42 (2002; Zbl 1008.45004)

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##### References:

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