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**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.

\[ x''(t)=f(t,x(t),x'(t)),\quad t\in (0,1), \]

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 |

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\textit{X. Lin}, Bound. Value Probl. 2011, Article ID 416416, 15 p.. (2011; Zbl 1213.34029)

### References:

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