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Nonlinear boundary value problems on time scales. (English) Zbl 0995.34016
This paper contains several results on the existence of one or two nonnegative solutions to the nonlinear second-order dynamic equation on a measure chain (time scale) ${\bold T}$, i.e., $$y^{\Delta\Delta}(t)+f(t,y(\sigma(t)))=0, \quad t\in[a,b]\cap {\bold T},$$ subject to the boundary conditions $$y(a)=0, \quad y^\Delta(\sigma(b))=0.$$ The theory of dynamic equations on measure chains unifies and extends the differential (${\bold T}={\bbfR}$) and difference (${\bold T}={\bbfZ}$) equations theories. The results extend the ones by {\it L. Erbe} and {\it A. Peterson} [Math. Comput. Modelling 32, No. 5-6, 571---585 (2000; Zbl 0963.34020)], and are also closely related to results by {\it C. J. Chyan, J. Henderson} and {\it H. C. Lo} [Tamkang J. Math. 30, No. 3, 231-240 (1999; Zbl 0995.34017)]. The proofs are based on an application of a nonlinear alternative of Leray-Schauder-type or Krasnoselskii fixed-point theorems. The paper will be useful for researchers interested in nonnegative solutions to differential or difference equations, and/or in dynamic equations on measure chains (time scales).

##### MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B45 Boundary value problems for ODE on graphs and networks 39A99 Difference equations
Full Text:
##### References:
 [1] Agarwal, R. P.; Bohner, M.: Basic calculus on time scales and some of its applications. Results math. 35, 3-22 (1999) · Zbl 0927.39003 [2] R.P. Agarwal, D. O’Regan, P.J.Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, 1999. [3] B. Aulbach, S. Hilgar, Linear dynamic processes with inhomogeneous time scale, Nonlinear Dynamics and Quantum Dynamical Systems, Akademie Verlag, Berlin, 1990. · Zbl 0719.34088 [4] Erbe, L. H.; Peterson, A.: Green’s functions and comparison theorems for differential equations on measure chains, dynamics continuous. Discrete impulsive systems 6, 121-138 (1999) · Zbl 0938.34027 [5] L.H. Erbe, A. Peterson, Positive solutions for a nonlinear differential equation on a measure chain, Mathematical and Computer Modelling. · Zbl 0963.34020 [6] Hilgar, S.: Analysis on measure chains -- a unified approach to continuous and discrete calculus. Results math. 18, 18-56 (1990) [7] B. Kaymakcalan, V. Lakshmikantham, S. Sivasundaram, Dynamical Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996. · Zbl 0869.34039 [8] J.W. Lee, D. O’Regan, Existence principles for differential equations and systems of equations, in: A. Granas, M. Frigon (Eds.), Topological Methods in Differential Equations and Inclusions, NATO ASI Series C, Vol. 472, Kluwer Academic Publishers, Dordrecht, 1995, pp. 239--289. · Zbl 0834.34074 [9] J. Munkres, Topology, Prentice-Hall, Englewood Cliffs, 1975.