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Multiple solutions of a third order boundary value problem. (English) Zbl 0755.34017
The aim of the authors is to give a lower estimation for the number of solutions to the two-point boundary-value problem associated with a third order nonlinear differential equation of the form (1) \(x'''=f(t,x,x',x'')\), \(t\in[a,b]\), (2) \(x(a)=A\), \(x'(a)=A_ 1\), \(x(b)=B\). Here \(f\) as well as its first derivatives \(f_ x\), \(f_{x'}\) and \(f_{x''}\) are supposed to be continuous on \([a,b]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\). The result is stated through the solution of a certain linear differential equation which is established by using any one of the solutions to the above-mentioned boundary-value problem, say \(\xi(t)\), namely: (3) \(y'''=f_{x''}(t,\xi,\xi',\xi'')y''+f_{x'}(t,\xi,\xi',\xi'')y'+f_ x(t,\xi,\xi',\xi'')y\), (4) \(y(a)=y'(a)=0\), \(y''(a)=1\). They prove that under certain conditions (1)–(2) has at least \(m+1\) solutions if the unique solution to (3)–(4) has \(m\) zeros in the interval \((a,b)\).

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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References:
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