## 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

### Keywords:

resonance condition; coincidence degree theory
Full Text:

### References:

 [1] Bicadze, AV; Samarskiĭ, 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
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