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Multiple solutions for fourth-order boundary value problem. (English) Zbl 1094.34012

A twopoint fourth-order nonlinear boundary value problem is studied of the form \[ x^{(4)}(t)=f(x(t),-x''(t)),\quad x(0)=x(1)=x''(0)= x''(1)=0. \]
The function \(f\) is continuous, satisfies \(f(0,0)=0\), and such that \(f(u,v)\geq 0\) for \(u>0\), \(v>0\), while \(f(u,v)=0\) for \(u<0\), \(v<0\). Sufficient conditions are given for the existence of at least six nontrivial solutions: two positive, two negative, and two sign-changing. In the case of an odd function \(f(-u,-v)\equiv -f(u,v)\), there exist at least eight nontrivial solutions. The proof employ the theory of fixed-point index in a cone and Leray-Schauder degree. Some corollaries are given, in particular, for the case if the right-hand side is independent of the second-order derivative.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47H11 Degree theory for nonlinear operators
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