A Generalized Nonlinear Gronwall-Bellman Inequality with Maxima in Two Variables

This paper deals with a generalized form of nonlinear retarded Gronwall-Bellman type integral inequality in which the maximum of the unknown function of two variables is involved. This form includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals. Requiring neither monotonicity nor separability of given functions, we apply a technique of monotonization to estimate the unknown function. Our result can be used to weaken conditions for some known results. We apply our result to a boundary value problem of a partial differential equation with maxima for uniqueness

Calculus of Variations on Time Scales and Discrete Fractional Calculus

We study problems of the calculus of variations and optimal control within the framework of time scales. Specifically, we obtain Euler-Lagrange type equations for both Lagrangians depending on higher order delta derivatives and isoperimetric problems. We also develop some direct methods to solve certain classes of variational problems via dynamic inequalities. In the last chapter we introduce fractional difference operators and propose a new discrete-time fractional calculus of variations. Corresponding Euler-Lagrange and Legendre necessary optimality conditions are derived and some illustrative examples provided.Comment: PhD thesis, University of Aveiro, 2010. Supervisor: Delfim F. M. Torres; co-supervisor: Martin Bohner. Defended 26/July/201

Qualitative analysis of some models of delay differential equations

This thesis concerns the study of the global dynamics of delay differential equations of the so-called production and destruction type, which find applications to the modelling of several phenomena in areas such as population growth dynamics, economics, cell production, etc. For instance, by applying tools coming from discrete dynamics, we provide sufficient conditions for the existence of globally attracting equilibria for families of scalar or multidimensional equations. Moreover, we extend some known results in the scalar non-autonomous case by the use of integral inequalities. Finally, the existence of periodic solutions is analysed in the general context of infinite delay, impulses and periodic coefficients

Convergence and Convergence Rate of Stochastic Gradient Search in the Case of Multiple and Non-Isolated Extrema

The asymptotic behavior of stochastic gradient algorithms is studied. Relying on results from differential geometry (Lojasiewicz gradient inequality), the single limit-point convergence of the algorithm iterates is demonstrated and relatively tight bounds on the convergence rate are derived. In sharp contrast to the existing asymptotic results, the new results presented here allow the objective function to have multiple and non-isolated minima. The new results also offer new insights into the asymptotic properties of several classes of recursive algorithms which are routinely used in engineering, statistics, machine learning and operations research

Convergence Rate of Stochastic Gradient Search in the Case of Multiple and Non-Isolated Minima

The convergence rate of stochastic gradient search is analyzed in this paper. Using arguments based on differential geometry and Lojasiewicz inequalities, tight bounds on the convergence rate of general stochastic gradient algorithms are derived. As opposed to the existing results, the results presented in this paper allow the objective function to have multiple, non-isolated minima, impose no restriction on the values of the Hessian (of the objective function) and do not require the algorithm estimates to have a single limit point. Applying these new results, the convergence rate of recursive prediction error identification algorithms is studied. The convergence rate of supervised and temporal-difference learning algorithms is also analyzed using the results derived in the paper

Hunting French Ducks in a Noisy Environment

We consider the effect of Gaussian white noise on fast-slow dynamical systems with one fast and two slow variables, containing a folded-node singularity. In the absence of noise, these systems are known to display mixed-mode oscillations, consisting of alternating large- and small-amplitude oscillations. We quantify the effect of noise and obtain critical noise intensities above which the small-amplitude oscillations become hidden by fluctuations. Furthermore we prove that the noise can cause sample paths to jump away from so-called canard solutions with high probability before deterministic orbits do. This early-jump mechanism can drastically influence the local and global dynamics of the system by changing the mixed-mode patterns.Comment: 60 pages, 9 figure

Unique solvability of boundary value problem for functional differential equations with involution

In this paper we consider a boundary value problem for systems of Fredholm type integral-differential equations with involutive transformation, containing derivative of the required function on the right-hand side under the integral sign. Applying properties of an involutive transformation, original boundary value problem is reduced to a boundary value problem for systems of integral-differential equations, containing derivative of the required function on the right side under the integral sign. Assuming existence of resolvent of the integral equation with respect to the kernel K˜2(t, s) (this is the kernel of the integral equation that contains the derivative of the desired function) and using properties of the resolvent, integral-differential equation with a derivative on the right-hand side is reduced to a Fredholm type integral-differential equation, in which there is no derivative of the desired function on the right side of the equation. Further, the obtained boundary value problem is solved by the parametrization method created by Professor D. Dzhumabaev. Based on this method, the problem is reduced to solving a special Cauchy problem with respect to the introduced new functions and to solving systems of linear algebraic equations with respect to the introduced parameters. An algorithm to find a solution is proposed. As is known, in contrast to the Cauchy problem for ordinary differential equations, the special Cauchy problem for systems of integral-differential equations is not always solvable. Necessary conditions for unique solvability of the special Cauchy problem were established. By using results obtained by Professor D. Dzhumabaev, necessary and sufficient conditions for the unique solvability of the original problem were established