zbMATH — the first resource for mathematics

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.

39A12 Discrete version of topics in analysis
47B60 Linear operators on ordered spaces
Full Text: DOI
[1] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press San Diego, CA · Zbl 0661.47045
[2] Krasnosel’skii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen · Zbl 0121.10604
[3] Erbe, L.H.; Hu, S.; Wang, H., Multiple positive solutions of some boundary value problems, J. math. anal. appl., 184, 640-648, (1994) · Zbl 0805.34021
[4] Erbe, L.H.; Wang, H., On the existence of positive solutions of ordinary differential equations, (), 743-748 · Zbl 0802.34018
[5] Hankerson, D.; Peterson, A.C., Comparison of eigenvalues for focal point problems for nth order difference equations, Differential and integral equations, 3, 363-380, (1990) · Zbl 0733.39002
[6] Merdivenci, F., Two positive solutions of a boundary value problem for difference equations, Journal of difference equations and applications, 1, 262-270, (1995) · Zbl 0854.39001
[7] Merdivenci, F., Green’s matrices and positive solutions of a discrete boundary value problem, Panamerican mathematical journal, 5, 25-42, (1995) · Zbl 0839.39002
[8] Merdivenci, F., Positive solutions for focal point problems for 2nth order difference equations, Panamerican mathematical journal, 5, 71-82, (1995) · Zbl 0839.39003
[9] Peterson, A.C., Boundary value problems for an nth order difference equation, J. math. anal., 15, 124-132, (1984) · Zbl 0532.39001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.