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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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