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Bounds for positive solutions for a focal boundary value problem. (English) Zbl 0933.39040
Summary: We are concerned with the focal boundary value problem \[ (-1)^n \Delta^n\bigl[p(t) \Delta^n y(t-n)\bigr]= f\bigl(h,t,y(t), \dots, \Delta^{n-1}y(t)\bigr), \] \[ \Delta^iy(0)= \Delta^{n+i} y(b+1)=0,\;0\leq i\leq n-1. \] Using cone theory in a Banach space, we show that under certain assumptions on \(f\), this focal boundary value problem has two positive solutions. In the special case \[ -\Delta^2y(t-1)=h^2 \bigl[y^\alpha (t)+ y^\beta (t)\bigr],\;y(0)=\Delta y(b+1)=0, \] where \(0<\alpha <1<\beta\), we are able to exhibit upper and lower bounds for these two positive solutions.

MSC:
39A12 Discrete version of topics in analysis
47B60 Linear operators on ordered spaces
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