## Existence and nonexistence of positive solutions of fourth order nonlinear boundary value problems.(English)Zbl 1031.34025

Summary: The authors consider boundary value problems for fourth-order ordinary differential equations of the form $u''''(t)= \lambda a(t)f(u),\;0<t<1,\tag{E}$ with the boundary conditions $u(0)=u'(1)= u''(0)= u'''(1)=0,\text{ or}\tag $$B_1$$$
$u(0)=u'(1)=u''(1)=u'''(0)=0.\tag $$B_2$$$ They give sufficient conditions for problems (E)–(B$$_1)$$ and (E)-(B$$_2)$$ to have at least one positive solution. They also give sufficient conditions for these problems to have no positive solutions. Examples to illustrate that the results are sharp are also included.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
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