zbMATH — the first resource for mathematics

Nonzero solutions of Hammerstein integral equations with discontinuous kernels. (English) Zbl 1008.45004
The 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)].

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
Full Text: DOI
[1] Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM rev., 18, 620-709, (1976) · Zbl 0345.47044
[2] Gupta, C.P.; Ntouyas, S.K.; Tsamatos, P.Ch., On an m-point boundary value problem for second order ordinary differential equations, Nonlinear anal., 23, 1427-1436, (1994) · Zbl 0815.34012
[3] Gupta, C.P.; Trofimchuk, S.I., A sharper condition for the solvability of a three-point second order boundary value problem, J. math. anal. appl., 205, 586-597, (1997) · Zbl 0874.34014
[4] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press · Zbl 0661.47045
[5] G. Infante, J.R.L. Webb, Three point boundary value problems with solutions that change sign, submitted · Zbl 1055.34023
[6] Lan, K.Q., Multiple positive solutions of semilinear differential equations with singularities, J. London math. soc., 63, 690-704, (2001) · Zbl 1032.34019
[7] Lan, K.Q., Multiple positive solutions of Hammerstein integral equations with singularities, Differential equations dynam. systems, 8, 175-195, (2000) · Zbl 0977.45001
[8] Lan, K.Q.; Webb, J.R.L., Positive solutions of semilinear differential equations with singularities, J. differential equations, 148, 407-421, (1998) · Zbl 0909.34013
[9] Ma, R., Positive solutions of a nonlinear three-point boundary-value problem, Electron. J. differential equations, 34, 1-8, (1999)
[10] Martin, R.H., Nonlinear operators & differential equations in Banach spaces, (1976), Wiley New York
[11] Webb, J.R.L., Positive solutions of some three point boundary value problems via fixed point index theory, Nonlinear anal., 47, 4319-4332, (2001) · Zbl 1042.34527
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.