Kong, Qingkai, 1946-
Ph.D. (Doctor of Philosophy)
Department of Mathematical Sciences
Mathematics||Mathematics--Study and teaching
In this dissertation, we derive Lyapunov-type inequalities for integer and fractional order differential equations and use them to study the nonexistence, uniqueness, and existence-uniqueness criteria for several classes of boundary value problems.||First, we consider third-order half-linear differential equations of the form ([phi][sub [alpha]2] (([phi][sub [alpha]1](x'))'))' + q(t) [phi][sub [alpha]1[alpha]2](x) = 0, where [phi][sub p](x) = |x|^[p-1]x, and [alpha]1, [alpha]2 > 0. We obtain Lyapunov-type inequalities which utilize integrals of both q+(t) and q-(t) rather than those of |q(t)| as in most papers in the literature. Furthermore, by combining these inequalities with the ``uniqueness implies existence'' theorems by many authors, we establish the uniqueness and hence existence-uniqueness for several classes of boundary value problems for third-order linear equations. This is the first time for Lyapunov-type inequalities to be used to deal with the existence-uniqueness of boundary value problems. These inequalities are further extended to higher order half-linear differential equations. Our results cover and improve many results in the literature when the equations become linear. For the third-order linear differential equation x''' + q(t)x = 0, using the Green's function method in a subtle way, we obtain the sharpest Lyapunov-type inequalities in the literature. We further extend these inequalities to more general third-order and higher order linear differential equations. We also discuss their applications to the existence-uniqueness of boundary value problems. Then we investigate boundary value problems for Riemann-Liouville fractional differential equations with certain fractional integral boundary conditions. Such boundary conditions are different from the widely considered pointwise conditions in the sense that they allow solutions to have singularities. We derive Lyapunov-type inequalities for linear fractional differential equations with order [alpha] [epsilon] (1,2] and [alpha] [epsilon] (2,3], respectively. Our results are good in the sense that they are consistent with the existing ones for the second-order and third-order problems when [alpha]=2,3. Finally, we establish some Lyapunov-type inequalities for Riemann-Liouville fractional differential equations with order [alpha] [epsilon] (2,3] and certain pointwise or mixed boundary conditions. Results are first given for univariate case, and then extended to multivariate case. All the results are new and one of them extends and improves substantially the one in the literature for third-order multivariate boundary value problems.
Dhar, Sougata, "Lyapunov-type inequalities and applications to boundary value problems" (2017). Graduate Research Theses & Dissertations. 3843.
vi, 146 pages
Northern Illinois University
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