56 research outputs found
Sturm-Picone type theorems for second order nonlinear impulsive differential equations
No full textWe obtain Sturm Picone type comparison theorems for nonlinear impulsive differential equations. Our results cover the previous results existing in the literature and useful in investigating qualitative behaviour of solutions of such equations
Goodman and Kruskal’s gamma coefficient for ordinalized bivariate distributions
No full textWe consider a bivariate normal distribution with linear correlation ρ whose random components are discretized according to two assigned sets of thresholds. On the resulting bivariate ordinal random variable, one can compute Goodman and Kruskal’s gamma coefficient, γ, which is a common measure of ordinal association. Given the known analytical monotonic relationship between Pearson’s ρ and Kendall’s rank correlation τ for the bivariate normal distribution, and since in the continuous case, Kendall’s τ coincides with Goodman and Kruskal’s γ, the change of this association measure before and after discretization is worth studying. We consider several experimental settings obtained by varying the two sets of thresholds, or, equivalently, the marginal distributions of the final ordinal variables. This study, confirming previous findings, shows how the gamma coefficient is always larger in absolute value than Kendall’s rank correlation; this discrepancy lessens when the number of categories increases or, given the same number of categories, when using equally probable categories. Based on these results, a proposal is suggested to build a bivariate ordinal variable with assigned margins and Goodman and Kruskal’s γ by ordinalizing a bivariate normal distribution. Illustrative examples employing artificial and real data are provided
An existence and uniqueness result for linear fractional impulsive boundary value problems as an application of Lyapunov type inequality
No full textA new and different approach to the investigation of the existence and uniqueness of solution of nonhomogenous impulsive boundary value problems involving the Caputo fractional derivative of order a (1 < 2) is brought by using Lyapunov type inequality. To express and to analyze the unique solution, Green's function and its bounds are established, respectively. As far as we know, this approach based on the link between fractional boundary value problems and Lyapunov type inequality, has not been revealed even in the absence of impulse effect. Besides, the novel Lyapunov type inequality generalizes the related ones in the literature
Lyapunov type inequalities and their applications for quasilinear impulsive systems
No full textA novel Lyapunov-type inequality for Dirichlet problem associated with the quasilinear impulsive system involving the (p(j), q(j))-Laplacian operator for j = 1,2 is obtained. Then utility of this new inequality is exemplified in finding disconjugacy criterion, obtaining lower bounds for associated eigenvalue problems and investigating boundedness and asymptotic behaviour of oscillatory solutions. The effectiveness of the obtained disconjugacy criterion is illustrated via an example. Our results not only improve the recent related results but also generalize them to the impulsive case
Existence and Uniqueness Results For Linear Impulsive Fractional Boundary Value Problems (BVPs) via Lyapunov Type Inequality
No full textKesirli Mertebeden Lineer İmpalsif Diferansiyel Denklemler İçin Homojen Olmayan Sınır Değer Problemleri (SDP)
No full textİmpals etkisi altındaki lineer ve lineer olmayan sistemler için lyapunov tipi eşitsizlikler ve uygulamaları.
No full textIn this thesis, Lyapunov type inequalities and their applications for impulsive systems of linear/nonlinear differential equations are studied. Since systems under impulse effect are one of the fundamental problems in most branches of applied mathematics, science and technology, investigation of their theory has developed rapidly in the last three decades. In addition, Lyapunov type inequalities have become a popular research area in recent years due to the fact that they provide not only better understanding of the qualitative nature of the solutions of ordinary and impulsive systems, for instance oscillation, disconjugacy, stability and asymptotic behavior of solutions, but also deeper analysis for boundary and eigenvalue problems. This thesis consists of 7 chapters. Chapter 1 is introductory and contains detailed literature review, and brief information about the linear systems of impulsive differential equations and Hamiltonian systems. The main contributions of the thesis, which are presented in the second and third chapters, are to derive Lyapunov type inequalities for the linear 2nx 2n Hamiltonian system with impulsive perturbations and to prove the existence and uniqueness criteria for the solutions of inhomogenous boundary value problems to such systems, respectively. Since changing the impulsive perturbation or assuming different conditions on the impulses leads to different inequalities, presence of the impulse effect provides various Lyapunov type inequalities. This shows that the systems of impulsive equations is richer and more fruitful in the applications than the systems of ordinary differential equations and that is why we are interested in these systems. Besides, the obtained inequalities are new even in the nonimpulsive case and therefore they improve and generalize the previous ones existing in the literature. In Chapter 3, the connection, which has not been noticed even for the nonimpulsive case, between Lyapunov type inequalities and boundary value problems has been revealed for the first time and two existence and uniqueness criteria for the solutions of inhomogenous BVPs are proved by using the Lyapunov type inequalities obtained in the previous chapter. Furthermore, the unique solution of inhomogenous BVPs is expressed in terms of Green’s function (pair) and the properties of Green’s function (pair) are listed. Chapter 4 is devoted to the stability theory, which is the application of Lyapunov type inequalities, for the linear planar Hamiltonian systems with impulsive perturbations. Two pairs of stability criteria are obtained, one of which is the generalization of the results obtained for systems of ordinary differential equations to the impulsive case and the latter is new and alternative to the former. In Chapter 5 and 6, we establish several Lyapunov type inequalites, some of which are generalizations of the nonimpulsive case while the others are new for nonlinear and quasilinear impulsive systems, respectively. As an application of Lyapunov type inequalities, we investigate disconjugacy intervals and study the asymptotic behaviour of oscillatory solutions for the systems under considerations and find a lower bound for the eigenvalues of the associated eigenvalue problems. The last chapter serves as a conclusion and is a summary of our findings
AN EXISTENCE AND UNIQUENESS RESULT FOR LINEAR SEQUENTIAL FRACTIONAL BOUNDARY VALUE PROBLEMS (BVPS) VIA LYAPUNOV TYPE INEQUALITY
No full textA sufficient condition for the existence and uniqueness of solution of nonhomogenous fractional boundary value problem involving sequential fractional derivative of Riemann Liouville type is established by using a new Lyapunov type inequality and disconjugacy criterion. Green's function and some of its properties are also presented. Our approach is quite new and to the best of our knowledge, the uniqueness of solution of nonhomogenous fractional boundary value problems is proved by employing Lyapunov type inequality for the first time and this Lyapunov type inequality improves and generalizes the previous ones
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