Some Integral Inequalities On Time Scales And Qualitative Analysis For Solutions For Dynamic Equations

For many differential equations, difference equations, and dynamic equations on timescales, if it is unavailable to obtain their exact solutions, it is important to fulfill qualitativeanalysis for their solutions such as boundedness, uniqueness, as well as continuous dependenceon initial data and parameters. Gronwall-Bellman type inequalities play important roles in theresearch of boundedness, uniqueness, as well as continuous dependence on initial data andparameters. Many researching results have been established for such inequalities so far in theliterature. But we notice that little attention have been paid to Gronwall-Bellman typeinequalities on time scales as well as Gronwall-Bellman type inequalities with discontinuousfunctions.Concerning the research on oscillatory and asymptotic properties for solutions fordifferential equations, there have been many results so far. Yet we notice that little attention havebeen paid on oscillatory and asymptotic properties for solutions for fractional differentialequations and third order dynamic equations with damping on time scales.On the other hand, for some differential equations with certain forms, it is feasible to obtaintheir exact solutions. There have been a lot of effective methods for obtaining exact solutions fordifferential equations such as the Exp function method，Jacobi elliptic function method,homogeneous balancing method and so on, while relatively less attention have been paid toobtain exact solutions for differential-difference equations, which is worthy to be furtherresearched.Based on the analysis above, in this thesis, we fulfill the following research. In Chapter I,we present a summary of the background of the research as well as some important definitionsand theorems on the theory of time scales. In Chapter II, based on the theory of time scales, weresearch and establish some Gronwall-Bellman-Volterra-Fredholm type inequalities on timescales, nonlinear Gronwall-Bellman type delay inequalities on time scales and nonlinearPachpatte type delay inequalities on time scales. New bounds are obtained for unknownfunctions concerned based on these inequalities. We also research qualitative properties forsolutions for some certain dynamic equations on time scales. The established results on one handgeneralize many existing Gronwall-Bellman type continuous and discrete inequalities, on theother hand unify continuous and discrete analysis. In Chapter III, based on the theory of time scales, generalized Riccati technique, inequality and integration average technique, we researchoscillatory and asymptotic properties for solutions for a class of third order dynamic equationswith damping on time scales and a class of third order delay functional dynamic equations withdamping on time scales, and present some sufficient conditions for judging oscillation andasymptotic behavior. Some examples are also presented. In Chapter IV, based on generalizedRiccati technique, inequality and integration average technique, we research oscillatoryproperties for solutions for several fractional differential equations with damping as well aswithout damping, and obtain some sufficient conditions for oscillation. The fractional derivativeis defined in the sense of right-hand side Liouville derivative. Some examples are also presented.In chapter V, we establish some Gronwall-Bellman type integral inequalities with discontinuousfunctions, and apply them to analysis of boundedness for solutions for some certain differentialand integral equations. The established inequalities extend some known results in the literature.In Chapter VI, we extend the known Riccati sub-equation method to obtain exact solutions fordifferential-difference equations. By use of this method, and with the aid of mathematicalsoftware Maple, abundant exact solutions including hyperbolic function solutions, trigonometricfunction solutions and rational function solutions for the Hybrid lattice equation are obtained. Atthe same time, many variable coefficient exact solutions for a class of (2+1) dimensional Todalattice equation are derived.