×

Existence of solutions to a nonlocal boundary value problem with nonlinear growth. (English) Zbl 1213.34029

Summary: This paper deals with the existence of solutions for the following differential equation:
\[ x''(t)=f(t,x(t),x'(t)),\quad t\in (0,1), \]
subject to the boundary conditions:
\[ x(0)=\alpha x(\xi),\;x'(1)=\int^1_0x'(s)\,dg(s), \]
where \(\alpha\geq 0\), \(0<\xi<1\), \(f:[0,1]\times \mathbb R^2\to\mathbb R\) is a continuous function, \(g:[0,1]\to[0,\infty)\) is a nondecreasing function with \(g(0)=0\). Under the resonance condition \(g(1)=1\), some existence results are given for the boundary value problems. Our method is based upon the coincidence degree theory of Mawhin. We also give an example to illustrate our results.

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Bicadze, AV; Samarskiń≠, AA, Some elementary generalizations of linear elliptic boundary value problems, Doklady Akademii Nauk SSSR, 185, 739-740, (1969)
[2] Il’pin, VA; Moiseev, EI, Nonlocal boundary value problems of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects, Differential Equations, 23, 803-810, (1987) · Zbl 0668.34025
[3] Il’cprimein, VA; Moiseev, EI, Nonlocal boundary value problems of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects, Differential Equations, 23, 979-987, (1987) · Zbl 0668.34024
[4] Karakostas, GL; Tsamatos, PCh, Sufficient conditions for the existence of nonnegative solutions of a nonlocal boundary value problem, Applied Mathematics Letters, 15, 401-407, (2002) · Zbl 1028.34023
[5] Du, Z; Lin, X; Ge, W, On a third-order multi-point boundary value problem at resonance, Journal of Mathematical Analysis and Applications, 302, 217-229, (2005) · Zbl 1072.34012
[6] Du, Z; Lin, X; Ge, W, Some higher-order multi-point boundary value problem at resonance, Journal of Computational and Applied Mathematics, 177, 55-65, (2005) · Zbl 1059.34010
[7] Feng, W; Webb, JRL, Solvability of three point boundary value problems at resonance, Nonlinear Analysis, 30, 3227-3238, (1997) · Zbl 0891.34019
[8] Liu, B, Solvability of multi-point boundary value problem at resonance. II, Applied Mathematics and Computation, 136, 353-377, (2003) · Zbl 1053.34016
[9] Gupta, CP, A second order \(m\)-point boundary value problem at resonance, Nonlinear Analysis, 24, 1483-1489, (1995) · Zbl 0824.34023
[10] Zhang, X; Feng, M; Ge, W, Existence result of second-order differential equations with integral boundary conditions at resonance, Journal of Mathematical Analysis and Applications, 353, 311-319, (2009) · Zbl 1180.34016
[11] Du, B; Hu, X, A new continuation theorem for the existence of solutions to \(p\)-Laplacian BVP at resonance, Applied Mathematics and Computation, 208, 172-176, (2009) · Zbl 1169.34307
[12] Mawhin, J; Fitzpertrick, PM (ed.); Martelli, M (ed.); Mawhin, J (ed.); Nussbaum, R (ed.), Opological degree and boundary value problems for nonlinear differential equations, No. 1537, (1991), New York, NY, USA
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.