Ma, Ruyun Existence results of a \(m\)-point boundary value problem at resonance. (English) Zbl 1070.34028 J. Math. Anal. Appl. 294, No. 1, 147-157 (2004). Let \(0<\xi_1<\xi_2<\cdots<\xi_{m-2}<1\), and \(a_i>0\), \(i=1,\,2,\,\cdots,\,m-2\), be given constants with \(\sum_{i=1}^{m-2}a_i=1\). This paper studies the existence of a solution to the second-order multipoint boundary value problem \[ x''=f(t,\,x,\,x')=0,\qquad t\in (0,\,1),\quad x'(0)=0,\;x(1)=\sum_{i=1}^{m-2}a_i u(\xi_i), \] where \(f:[0,\,1]\times \mathbb{R}^2\to \mathbb{R}\) is continuous. The author improves some results of his paper [J. Math. Anal. Appl. 211, 545–555 (1997; Zbl 0884.34024)], where the essential condition, \(x\,f(t,\,x,\,0)\geq \delta\) for \(| x| >M\), where \(M\) and \(\delta\) are positive constants, is replaced by the weaker sign conditions that \(f(t,\,-M,\,0)\leq 0\) and \(f(t,\,M,\,0)\geq 0\). The arguments are based on the nonlinear alternative of Leray-Schauder. Reviewer: Yongxiang Li (Lanzhou) Cited in 32 Documents MSC: 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:multipoint boundary value problem; existence; nonlinear alternative; sign conditions Citations:Zbl 0884.34024 PDFBibTeX XMLCite \textit{R. Ma}, J. Math. Anal. Appl. 294, No. 1, 147--157 (2004; Zbl 1070.34028) Full Text: DOI References: [1] Feng, W., On an \(m\)-point boundary value problems, Nonlinear Anal., 30, 5369-5374 (1997) · Zbl 0895.34014 [2] Feng, W.; Webb, J. R.L., Solvability of three-point boundary value problems at resonance, Nonlinear Anal., 30, 3227-3238 (1997) · Zbl 0891.34019 [3] Gupta, C. P., Existence theorems for a second order \(m\)-point boundary value problem at resonance, Internat. J. Math. Math. Sci., 18, 705-710 (1995) · Zbl 0839.34027 [4] Kelevedjiev, P., Existence of solutions for two-point boundary value problems, Nonlinear Anal., 22, 217-224 (1994) · Zbl 0797.34019 [5] Ma, R., Existence of positive solutions for superlinear semipositone \(m\)-point boundary-value problems, Proc. Edinburgh Math. Soc., 46, 279-292 (2003) · Zbl 1069.34036 [6] Ma, R., Existence theorems for a second order \(m\)-point boundary value problem, J. Math. Anal. Appl., 211, 545-555 (1997) · Zbl 0884.34024 [7] O’Regan, D., Boundary value problems for second and higher order differential equations, Proc. Amer. Math. Soc., 113, 761-775 (1991) · Zbl 0742.34023 [8] Rachunkova, I., On four-point boundary value problem without growth conditions, Czechoslovak Math. J., 49, 241-248 (1999) · Zbl 0955.34008 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.