## Multiple positive solutions for nonlinear dynamical systems on a measure chain.(English)Zbl 1045.39007

Authors’ summary: We consider the following dynamical system on a measure chain: $u_1^{\Delta\Delta}(t)+f_1(t,u_1(\sigma(t)),u_2(\sigma(t)))=0, \quad t\in [a,b],$
$u_2^{\Delta\Delta}(t)+f_2(t,u_1(\sigma(t)),u_2(\sigma(t)))=0, \quad t\in [a,b],$ with the Sturm-Liouville boundary value conditions $\alpha u_i(a)-\beta u_i^{\Delta}(a)=0, \gamma u_i(\sigma(b))+\delta u_i^\Delta(\sigma(t))=0, \;\text{ for}\;i=1,2.$ Some results are obtained for the existence of three positive solutions of the above problem by using Leggett-Williams fixed point theorem.

### MSC:

 39A11 Stability of difference equations (MSC2000) 39A12 Discrete version of topics in analysis 34B24 Sturm-Liouville theory
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### References:

 [1] Agarwal, R.; Bohner, M., Quadratic functionals for second order matrix equations on time scales, Nonlinear anal. TMA, 33, 675-692, (1998) · Zbl 0938.49001 [2] Agarwal, R.; Bohner, M., Basic calculus on time scales and some of its applications, Results math., 35, 3-22, (1999) · Zbl 0927.39003 [3] Agarwal, R.; Bohner, M.; Wong, P., Sturm – liouville eigenvalue problems on time scales, Appl. math. comput., 99, 153-166, (1999) · Zbl 0938.34015 [4] Ahlbrandt, C.; Peterson, A., Discrete Hamiltonian systems: difference equations, continued fractions, and Riccati equations, (1996), Kluwer Academic Publishers Boston · Zbl 0860.39001 [5] B. Aulbach, S. Hilger, Linear dynamic processes with inhomogeneous time scale, in: Nonlinear Dynamics and Quantum Dynamical Systems, Akademie Verlag, Berlin, 1990. · Zbl 0719.34088 [6] Bohner, M.; Peterson, A., Dynamic equations on time scales: an introduction with applications, (2001), Birkhauser Boston, MA · Zbl 0978.39001 [7] Bohner, M.; Peterson, A., Advances in dynamic equations on time scales, (2003), Birkhauser Boston, MA · Zbl 1025.34001 [8] Chyan, C.J.; Henderson, J., Eigenvalue problems for nonlinear differential equations on a measure chain, J. math. anal. appl., 245, 547-559, (2000) · Zbl 0953.34068 [9] Deimling, K., Nonlinear functional analysis, (1985), Springer New York · Zbl 0559.47040 [10] Erbe, L.; Hilger, S., Sturmian theory on measure chains, Differential equations dynamical systems, 1, 223-246, (1993) · Zbl 0868.39007 [11] 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 [12] Erbe, L.; Peterson, A., Green’s functions and comparison theorems for differential equations on measure chains, dynamics continuous, Discrete impulsive systems, 6, 121-137, (1999) · Zbl 0938.34027 [13] Erbe, L.; Peterson, A., Positive solutions for a nonlinear differential equation on a measure chain, Math. comput. modelling, 32, 5-6, 571-585, (2000) · Zbl 0963.34020 [14] Erbe, L.H.; Tang, M., Existence and multiplicity of positive solutions to nonlinear boundary value problems, Differential equations dynamical systems, 4, 313-320, (1996) · Zbl 0868.35035 [15] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press San Diego · Zbl 0661.47045 [16] Hilger, S., Analysis on measure chains—A unified approach to continuous and discrete calculus, Results math., 18, 18-56, (1990) · Zbl 0722.39001 [17] Kelley, W.; Peterson, A., Difference equations: an introduction with applications, (1991), Academic Press New York [18] Krasnoselskii, M., Positive solutions of operator equations, (1964), Noordhoff Groningen [19] Lakshmikantham, V.; Sivasundaram, S.; Kaymakcalan, B., Dynamic systems on a measure chain, (1996), Kluwer Academic Publishers Boston · Zbl 0869.34039 [20] Leggett, R.W.; Williams, L.R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana univ. math. J., 28, 673-688, (1979) · Zbl 0421.47033 [21] Shi, Y.; Chen, S., Spectral theory of second order vector difference equations, J. math. anal. appl., 239, 195-212, (1999) · Zbl 0934.39002 [22] J.P. Sun, W.T. Li, H.R. Sun, Positive solutions for Sturm-Liouville boundary value problems on a measure chain, submitted. [23] Wang, H., On the existence of positive solutions for semilinear elliptic equations in the annulus, J. differential equations, 109, 1-7, (1994) · Zbl 0798.34030
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