## Existence and multiplicity of solutions to $$2m$$th-order ordinary differential equations.(English)Zbl 1119.34014

Summary: The existence and multiplicity of solutions are obtained for the $$2m$$th-order ordinary differential equation two-point boundary value problems $(-1)^mu^{(2 m)}(t)+\sum^m_{i=1}(-1)^{m-i}a_i u^{(2(m-i))}(t)=f(t,u(t))\text{ for all }t\in [0,1]$ subject to Dirichlet, Neumann, mixed and periodic boundary value conditions, respectively, where $$f$$ is continuous, $$a_i\in\mathbb{R}$$ for all $$i=1,2,\dots,m$$. Since these four boundary value problems have some common properties and they can be transformed into the integral equation of form $u+ \sum^m_{i=1}a_iT^iu=T^m{\mathbf f}u,$ we firstly deal with this nonlinear integral equation. By using the strongly monotone operator principle and the critical point theory, we establish some conditions on $$f$$ which are able to guarantee that the integral equation has a unique solution, at least one nonzero solution, and infinitely many solutions. Furthermore, we apply the abstract results on the integral equation to the above four $$2m$$th-order two-point boundary problems and successfully resolve the existence and multiplicity of their solutions.

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations
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### References:

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