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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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