Abstract
The purpose of this paper is to study the existence and asymptotic behavior of solutions to a class of second-order nonlinear dynamic equations on unbounded time scales. Four different results are obtained by using the Banach fixed point theorem, the Boyd and Wong fixed point theorem, the Leray-Schauder nonlinear alternative, and the Schauder fixed point theorem. For each theorem, an illustrative example is presented. The results provide unification and some extensions in the time scale setup of the theory of asymptotic integration of nonlinear equations both in the continuous and discrete cases.
Highlights
This work is devoted to the study of the existence and asymptotic behavior of solutions to the nonlinear dynamic equation uΔΔ f t, u 0, t ∈ T, 1.1 where the function f : T × R → R is continuous and T is a time scale i.e., a nonempty closed subset of the real numbers; see 1, 2 and Section 2 below that has a minimal element t0 > 0 and is unbounded above, that is, nl→im∞tn ∞ for some set tn : n ∈ N ⊂ T
Inspired and motivated by the results obtained both for difference and differential equations, our aim in this paper is to extend some of these results to nonlinear dynamic equations on time scales
Specific results regarding the asymptotic behavior of the nonlinear dynamic equation 1.1 have been obtained, extending some known results in the theories of difference and differential equations, for example to q-difference equations see Remark 2.6 and to other cases of arbitrary time scales
Summary
This work is devoted to the study of the existence and asymptotic behavior of solutions to the nonlinear dynamic equation uΔΔ f t, u 0, t ∈ T, 1.1 where the function f : T × R → R is continuous and T is a time scale i.e., a nonempty closed subset of the real numbers; see 1, 2 and Section 2 below that has a minimal element t0 > 0 and is unbounded above, that is, nl→im∞tn ∞ for some set tn : n ∈ N ⊂ T. Inspired and motivated by the results obtained both for difference and differential equations, our aim in this paper is to extend some of these results to nonlinear dynamic equations on time scales. These are four distinct results, each of which guarantees the existence of asymptotically linear solutions according to 1.3.
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