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We study a boundary value problem for a system of the third order semi-linear partial differential equations with nonlocal boundary conditions. We establish sufficient conditions of existence, uniqueness, regularity and sign-preserving property of solutions of the studied problem and construct an iterative method for its approximation., Mathematical Physics

This thesis provides a firm mathematical foundation for nonlinear systems of singular, second order boundary value problems (BVPs). In particular, we shall advance the current state of knowledge by answering the question: “When do nonlinear systems of singular BVPs have solutions?”. The significance of these singular problems lies in their application to science, engineering, medicine and technology, where traditional non–singular problems are unsuitable.The thesis is divided into two parts, each of which treats the different nonlinear forms: Singular BVPs with derivative independence in the right hand side, Singular BVPs with derivative dependence in the right hand side.Discussing the above cases separately enables the content herein to evolve naturally. The methods include topological techniques and fixed point theory to obtain existence of solutions. Underpinning these ideas are determining a priori bounds on possible solutions to the BVPs. These bounds are achieved via the application of new differential inequalities, which, in turn, enable novel existence results.In the derivative independent case, there are two distinct inequalities that are investigated that both deal with problems without growth restrictions. The first inequality stems from Lyapunov functions and uses the equivalent integral representation of the BVP to prove that every possible solution to the BVP satisfies a particular a priori bound. The latter inequality achieves the same conclusion, however uses the technique of maximum principles instead. Under the assumptions of these bounds, topological techniques are applied to obtain the existence of solutions for systems of singular, second order BVP.Next, for the derivative dependent case; the same ideas apply. The pronounced difference between the forms is the dependence of the derivative term in the right hand side of the problem. This naturally gives rise to a particular discussion on finding a priori bounds for the derivative of possibl

Motivation by the question of testing for a nonlinear parameter in a regression model with independent, homoscedastic normal residuals, Hotelling (1939) was led to consider the geometric problem of computing the volume of a tube of given radius around a curve in s to the n-1 power the unit sphere in ir to the nth power. The answer involves only the arc length of the curve and not its curvature, providing the radius of the tube is sufficiently small that there is no self overlap in the tube. Starting from a somewhat different statistical setting Naiman (1986) arrived at the same geometric problem and showed that Hotelling's result (properly interpreted) is an upper bound for the volume of a tube of arbitrary radius. The purpose of this note is to give two new derivations of the Hotelling Naiman results. The first involves differential inequalities. The second is probabilistic, using the concept of upcrossing borrowed from the theory of Gaussian processes. In the context of Gaussian processes Knowles (1987) has observed that approximations obtained from Hotelling's result and bounds derived via upcrossings are related.

The maximum principle and the general theory of differential inequalities are used to obtain elementary estimates for solutions of second order ordinary differential equations. These estimates are applied to obtain results on the limiting behavior of solutions of singular perturbation problems. Some of the results presented here are new and others shed light on known results. (Author)

A classical theorem on systems of ordinary differential inequalities states that (vector notation) v(0) = or < w(0), v' - f(t, v) = or < w' - f(t, w) in J = (0, T) implies v = or > w in J, if the function f(t, x) = (f sub 1, ..., f sub n) is quasimonotone increasing in x, i.e., if (f sub i)(t, x sub 1, ..., x sub n) is increasing in x sub j for i not equal to j. An appropriate concept of quasimonotonicity is formulated for arbitrary ordered Banach spaces, and the above theorem is generalized to the case of differential inequalities in ordered Banach spaces. (Author)

The paper proves that the feedback controls synthesizing the equilibrium solutions to linear quadratic differential games possess certain stability characteristics. (Author)

It is shown that all solutions of certain recursively defined, infinite systems of differential inequalities converge to zero. Convergence is global for initial value inequalities, but only local for boundary value inequalities. Applications to the stability of differential games are sketched. (Author), Prepared in cooperation with Maryland Univ., College Park, under Grant NSF-GP-7844, Jagiellonian Univ., Krakow (Poland), and Karlsruhe Univ. (West Germany).

The report describes activity during the reporting period in the following areas: Controllability of systems with time delay; Structure and realization theory for linear systems; Finite time stability.