The Limit Circle Criteria For Certain Dynamic Equations On Time Scales With Applications

With the further development of natural science, we are getting to know that the changing process in the nature can be attributable to continuous or discrete variables. Neither differential equations nor difference equations are able to explain continuously and discretely changing phenomena at the same time. However, all the phenomena can be perfectly modeled by the dynamic equations on time scales which is originate with Sefan Higer's dissertation in 1988. The theory of dynamic equations not only unify the existing results for the conventional continuous system and discrete system, but also generalize those results to a more extensive class of systems on time scales, especially for the study of biomathematics. The research in the dynamic equations on time scales will enable us to analyze the changing process in the nature comprehensively and completely.In the paper, using the known theories of time scale, we discuss the classification of limit case or limit point case for dynamic equation Ly = -[p(t)yâ–³]^â–³+ q(t)y^σ=λy^σ, on a suitable(?)²(T), where p(t)∈C'_rd,q(t)∈C_rd, q(t) > 0,λ∈C₀ , the result generalize and unify the corresponding results of both differential equations and difference equations.The thesis is divided into three sections according to contents.In the first chapter, we introduce the main contents of this paper.In the second chapter, a new space (?)²(T) on time scales T is defined in this paper. With the aid of theory of Weyl circle, we investigate the classification of limit circle and limit point for the following second-order dynamic equations on time scales Ly=-[p(t)y^â–³]^â–³+ q(t)y^σ=λy^σ, where p(t) G C'_rd, q(t)∈C_rd, q(t) > 0,λ∈C₀ . The problem of limit circle with bounded perturbation for the dynamic equations Ly =λy^σis also involved, and the invariable criteria under the bounded perturbation was obtained.In the third chapter, based on the time scales T (inf T = 0, sup T = co), we investigate the following second-order dynamic equation [p(t)y^â–³]^â–³+ q(t)yσ= 0. With the condition of p(t) = (?), q(t) =α(t)(?)(t)-(?)^â–³(t), a(t) is regressive,(?)(t)≠0 and(?)(t)∈C_rd(T), we considered the boundedness of the solution and the limit circle criteria of the last second-order dynamic equation. Thus a sufficient and necessary condition of the limit circle case about the symmetric differential expression L(y) = -[p(t)y^â–³]^â–³+ q(t)yσis established. Furthermore, we obtain the limit circle criteria of dynamic equations with damping term on time scales.