Rakhimova/Kudabayeva, Saltanat (2011) The oscillation properties of half-linear second order and higher order differential equations. [Tesi di dottorato]
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The Thesis is dedicated to the investigation of the oscillation properties of half-linear second order and higher order differential equations. Chapter 1 includes a short history of the problem, the statement of the problems and the main results. In this chapter, we present some well-known auxiliary facts and necessary notation on half-linear second order differential equations and weighted Hardy inequalities. In Chapter 2, we consider weighted Hardy inequalities on the set of smooth functions with compact support. We obtain new results, which generalize the known results concerning this theme. In Chapter 3, we investigate the problems of disfocality and disconjugacy of half-linear second order differential equations. We obtain new sufficient conditions and necessary conditions of disfocality and disconjugacy on a given interval. Also, we consider the behavior of some of the solutions on a given interval. In Chapter 4, we apply the results of Chapter 3 in order to obtain sufficient conditions and necessary conditions of nonoscillation of half-linear second order differential equations with nonnegative coefficients. We also obtain new oscillation and nonoscillation conditions of half-linear differential equations by applying the known results of the theory of weighted Hardy type inequalities. Also, Chapter 4 includes the proofs of general statements establishing the link between results of weighted Hardy type inequalities and results ensuring oscillation and nonoscillation of the solutions of half-linear second order differential equation with nonnegative coefficients. We also obtain new results on conjugacy and oscillation in the case in which the second coefficient changes sing, by applying the variational method. In Chapter 5, we investigate the problem of oscillation and nonoscillation of the solutions of two term linear and half-linear equations of higher order with nonnegative coefficients. It seems that our results concerning general conditions on the coefficients for nonoscillatory solutions of half-linear equations of higher order and for oscillatory and nonoscillatory solutions for linear equations of higher order are new. We establish necessary and sufficient conditions of strong oscillation and strong nonoscillation of the solutions for linear equations.
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Questa Tesi consiste di cinque capitoli. La Tesi è dedicata allo studio delle proprietà di oscillazione di equazioni differenziali semi-lineari del secondo ordine e di ordine superiore. Il Capitolo 1 contiene una breve storia del problema, la formulazione dei problemi e i risultati principali. In questo capitolo, presentiamo alcuni risultati ausiliari noti e alcune
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