Boundary value problems via vector field. An alternative approach (Q2730963)

The existence of positive solutions is obtained for boundary value problems for the equation NEWLINE\[NEWLINEx''+\varepsilon \cdot f(t,x)=0,\quad t\in [0,1],\;\varepsilon= \pm 1,NEWLINE\]NEWLINE with \(f\in C([0,1] \times[0,\infty), [0,\infty))\). The boundary conditions are either NEWLINE\[NEWLINE\alpha x(0)-\beta x'(0)=0,\;\gamma x(1)-\delta x'(1)=0, \quad\text{when }\varepsilon=1,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\alpha x(0) \pm\beta x'(0)= 0,\;\gamma x(1)\pm \delta x'(1)=0, \quad\text{when }\varepsilon=-1,NEWLINE\]NEWLINE with \(\alpha,\beta, \gamma,\delta >0\) and \(\varepsilon(\beta \gamma+ \varepsilon \alpha\gamma -\alpha\delta) >0\), or NEWLINE\[NEWLINEg\bigl(x(0), x'(0)\bigr)=0,\quad h\bigl(x (1), x'(1)\bigr) =0,NEWLINE\]NEWLINE where the functions \(g\) and \(h\) are such that the graph of \(g(x,y)=0\) and \(h(x,y)=0\) can be parametrized. The following two cases are considered: NEWLINE\[NEWLINE\lim_{x\to 0^+} \max_{0\leq t\leq 1}{f(t,x)\over x}=0,\quad \lim_{x \to+\infty} \min_{0\leq t\leq 1}{f(t,x) \over x}=+ \inftyNEWLINE\]NEWLINE and NEWLINE\[NEWLINE\lim_{x\to 0^+} \min_{0\leq t\leq 1}{f(t,x) \over x}=+ \infty,\quad \lim_{x\to +\infty} \max_{0\leq t\leq 1}{f(t,x) \over x}=0.NEWLINE\]NEWLINE The results rely on an analysis of the corresponding vector field in the \((x,x')\) phase plane and Kneser's property of the cross-sections of the solutions found.