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Solution curves of \(2m\)-th order boundary-value problems. (English) Zbl 1060.34011

The author considers the \(2m\)th-order nonlinear boundary value problem \[ \begin{gathered} (-1)^m u^{(2m)}(x)+ \sum^{m- 1}_{i=0} (- 1)^i p_i u^{(2i)}(x)= \lambda f(u(x)),\quad x\in (-1,1),\\ u^{(i)}(- 1)= u^{(i)}(1)= 0,\quad i= 0,\dots, m- 1,\end{gathered} \] where \(m\geq 2\), \(p_i\geq 0\), \(i= 0,\dots, m-1\), are constants; \(\lambda\in \mathbb{R}_+:= [0,\infty)\); the function \(f: \mathbb{R}\to \mathbb{R}\) is \(C^2\) and satisfies \(f(\xi)> 0\), \(\xi\in \mathbb{R}\).
Under various convexity- or concavity-type assumptions on \(f\), the author shows that this problem has a smooth curve \(S_0\) of solutions \((\lambda, u)\) emanating from \((\lambda_0, u)= (0,0)\), and describes the shape and asymptotics of \(S_0\). All solutions on \(S_0\) are positive and all solutions for which \(u\) is stable lie on \(S_0\).

MSC:

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