## Positive solutions for a nonlinear differential equation on a measure chain.(English)Zbl 0963.34020

Summary: The authors are concerned with proving the existence of positive solutions to general two-point boundary value problems for the nonlinear equation $Lx(t):= -[r(t) x^\Delta(t)]^\Delta= f(t, x(t)).$ They use fixed-point theorems concerning cones in a Banach space. Important results concerning Green functions for general two-point boundary value problems for $Lx(t):= -[r(t) x^\Delta(t)]^\Delta= 0$ are given.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems

### Keywords:

positive solutions; two-point boundary value problems
Full Text:

### References:

 [1] Hilger, S, Analysis on measure chains—A unified approach to continuous and discrete calculus, Results in mathematics, 18, 18-56, (1990) · Zbl 0722.39001 [2] Agarwal, R; Bohner, M, Basic calculus on time scales and some of its applications, Results in mathematics, 35, 3-22, (1999) · Zbl 0927.39003 [3] Agarwal, R; Bohner, M, Quadratic functional for second order matrix equations on time scales, Nonlinear analysis, 33, 675-692, (1998) · Zbl 0938.49001 [4] Agarwal, R; Bohner, M; Wong, P, Sturm-Liouville eigenvalue problems on time scales, Applied mathematics and computation, 99, 153-166, (1999) · Zbl 0938.34015 [5] Erbe, L; Hilger, S, Sturmian theory on measure chains, Differential equations and dynamical systems, 1, 223-246, (1993) · Zbl 0868.39007 [6] Erbe, L.H; Peterson, A, Green’s functions and comparison theorems for differential equations on measure chains, Dynamics of continuous, discrete and impulsive systems, 6, 121-137, (1999) · Zbl 0938.34027 [7] Ahlbrandt, C; Peterson, A, Discrete Hamiltonian systems: difference equations, continued fractions, and Riccati equations, (1996), Kluwer Academic Boston · Zbl 0860.39001 [8] Kelley, W; Peterson, A, Difference equations: an introduction with applications, (1991), Academic Press [9] Aulbach, B; Hilger, S, Linear dynamic processes with inhomogeneous time scale, () · Zbl 0719.34088 [10] Deimling, K, Nonlinear functional analysis, (1985), Springer New York · Zbl 0559.47040 [11] Krasnoselskii, M, Positive solutions of operator equations, (1964), Noordhoff Groningen [12] Erbe, L.H; Hu, S; Wang, H, Multiple positive solutions of some boundary value problems, Journal of mathematical analysis and applications, 184, 640-648, (1994) · Zbl 0805.34021 [13] Erbe, L.H; Tang, M, Positive radial solutions to nonlinear boundary value problems for semilinear elliptic problems, (), 45-53 · Zbl 0845.35033 [14] Erbe, L.H; Tang, M, Existence and multiplicity of positive solutions to nonlinear boundary value problems, Differential equations and dynamical systems, 4, 313-320, (1996) · Zbl 0868.35035 [15] Wang, H, On the existence of positive solutions for semilinear elliptic equations in the annulus, Journal of differential equations, 109, 1-7, (1994) · Zbl 0798.34030 [16] Kaymakcalan, B; Laksmikantham, V; Sivasundaram, S, Dynamical systems on measure chains, (1996), Kluwer Academic Boston · Zbl 0869.34039
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