Ordinary Differential Inequalities Research Articles

On p-harmonic maps and convex functions

We prove that, in general, given a $p$-harmonic map $F:M\to N$ and a convex function $H:N\to\mathbb{R}$, the composition $H\circ F$ is not $p$-subharmonic. By assuming some rotational symmetry on manifolds and functions, we reduce the problem to an ordinary differential inequality. The key of the proof is an asymptotic estimate for the $p$-harmonic map under suitable assumptions on the manifolds.

The critical exponent for an ordinary fractional differential problem

We consider the Cauchy problem for an ordinary fractional differential inequality with a polynomial nonlinearity with variable coefficient. A nonexistence result is proved and the critical exponent separating existence from nonexistence is found. This is proved in the absence of any regularity assumptions.

On differential inequalities with point singularities on the boundary

We prove the nonexistence of solutions for some nonlinear ordinary differential equations and inequalities, for quasilinear partial differential equations and inequalities in bounded domains with singular points on the boundary, and for systems of such equations and inequalities. The proofs are based on the method of nonlinear capacity. We also give examples demonstrating that the conditions obtained are sharp in the class of problems under consideration.

The initial boundary value problem for quasi-linear wave equation with viscous damping

In this paper, the existence and uniqueness of the local generalized solution and the local classical solution for the initial boundary value problem of the quasi-linear wave equation with viscous damping are proved. The nonexistence of the global solution for this problem is discussed by an ordinary differential inequality. Finally, an example is given.

Hessian equations with infinite Dirichlet boundary

In this paper, we study the Hessian equation with infinite Dirichlet (blow-up) boundary value conditions. Using radial functions and techniques of ordinary differential inequality, we construct various barrier functions (super-solution and sub-solution). Existence and non-existence theorems are proved by those barriers, maximum principle and theory of viscous solutions. Furthermore, generic boundary blow-up rates for the solutions are derived.

Necessary Conditions for the Existence of Global Solutions of Systems of Higher-Order Nonlinear Ordinary Differential Equations and Inequalities

Necessary Conditions for the Existence of Global Solutions of Systems of Higher-Order Nonlinear Ordinary Differential Equations and Inequalities

On the asymptotic behaviour of solutions of some nonlinear elliptic and parabolic equations

In this paper, we investigate the spatial asymptotic behaviour of some nonlinear elliptic and parabolic equations related with the process of diffusion–convection. We obtain results of the kind of the Phragmén–Lindelof alternative. The main idea of the paper is to obtain a nonlinear ordinary differential inequality which is satisfied by a certain measure of the solutions. This ordinary inequality was studied by Horgan and Payne in a recent paper. We obtain our results with the help of estimates known for this inequality. When the solutions tend to zero, we also obtain an estimate for the amplitude term.

On Necessary Conditions for the Existence of Local Solutions to Singular Nonlinear Ordinary Differential Equations and Inequalities

We obtain conditions for the nonexistence of global solutions and estimates of existence time for local solutions to the problem $$\frac{{d^k y}}{{dt^k }} \geqslant a_1 (t)\left| y \right|^{q_1 } + a_2 (t)\left| y \right|^{q_2 } + \cdots + a_n (t)\left| y \right|^{q_n } .$$ The proofs are based on the method of trial functions developed by Mitidieri and Pokhozhaev.

Necessary Conditions for the Existence of Global Solutions of Higher-Order Nonlinear Ordinary Differential Inequalities

Necessary Conditions for the Existence of Global Solutions of Higher-Order Nonlinear Ordinary Differential Inequalities

Nonoscillation of Ordinary Differential Equations and Inequalities on Spatial Networks

Nonoscillation of Ordinary Differential Equations and Inequalities on Spatial Networks

Convergence and stability analysis of system of partial differential equations under Markovian structural perturbations–ii:vextor Lyapunov–like functionals

In this paper we employ the theory of dererministic ordinary differential inequalities together with the concept of vector Lyapunov–like functional to develop basic comparison theorems for system of partial differential equations of parabolic type under Markovian structural perturbations.These results will be utilized to give sufficient conditions for the convergence and stability of the solution process of the system.We also characterize the effects of the random structural perturbations on the qualitative properties of such system. Moreover,the Lyapunov–like functional approach provides a mechanism to characterize the diffusion effects on the qualitative properties of the system.

Global Existence of Solutions to Nonlinear Wave Equations

T. Global in time solutions of the equations Ou = H(u, u') on Minkowski space-time are considered. Results available so far involve complicated decay and energy estimates and also careful choice of Banach spaces and associated ordinary differential inequalities. This work tries to simplify some of the existing arguments and to develop a new technique for other nonlinear evolution equations. The method is motivated by the work of Christodoulou and Baez, Segal, and Zhou, on nonlinear wave equations. The key idea is to use the Penrose conformal compactification that transforms the equations from Minkowski space to the Einstein universe in order to change the global existence question to the local one.

Ordinary funcitonal differential equations and inequalities in the sense of carathéodory

We consider the linear IVP (i) and the corresponding nonlinear IVP (ii) where A solution u belongs to the Operator P: L is linear, positive and of Volterra type. Here The essential results are (a) A Positivity Theorem for the linear equation (i) (b) A Comparison Theorem for the nonlinear equation (ii) (c) A precise characterization of inequalities like u(t)≥0 or v(t) ≤w(t) with respect to strict inequalities of components (d) A far—reaching generalization of M.Hirsch's theorem on strongly monotone flows e) Existence and Uniqueness Theorems under arathéodory hypotheses.

Ordinary differential inequalities and quasimonotonicity in ordered topological vector spaces

A well known comparison theorem on ordinary differential inequalities with quasimonotone right-hand side f ( t , x ) f(t,x) was carried over by Volkmann (1972) to (pre)ordered topological vector spaces. We prove that the quasimonotonicity of f f is a necessary condition here if f f is continuous. Then it is shown that quasimonotonicity can be verified by considering only a few positive continuous linear functionals in the definition (for instance in ℓ ∞ \ell _{\infty } by taking coordinate functionals).

Hopf bifurcation and ordinary differential inequalities

Hopf bifurcation and ordinary differential inequalities

A Boltzmann-like model for a problem of outgassing and contamination with velocity dependent boundary conditions

Abstract A mathematical model for the description of processes of outgassing and contamination is proposed. The source kinetics, the mass transfer across the cavity and the contamination process are assumed to obey Boltzmann-like equations, with the appropriate boundary conditions. The theory of semigroups is used to show that a unique positive strict solution exists. A system of ordinary differential inequalities for the total number of contaminant particles is also derived.

Qualitative analysis of dynamical systems determined by differential inequalities with applications to robust stability

In this paper we develop a Lyapunov stability theory for finite dimensional continuous-time dynamical systems described by a system of first-order ordinary differential inequalities. We utilize this theory to establish sufficient robust stability criteria for a large class of finite dimensional, continuous-time dynamical systems described by systems of ordinary differential equations. We demonstrate the applicability of the methodology advanced herein by means of a specific example that has been considered in the literature. In terms of computational complexity and conservatism of stability criteria, the present results frequently offer improvements over existing results. >

Lyapunov Stability of Dynamical Systems determinined by Differential Inequalities with Applications to Robust Stability

We develop a Lyapunov stability theory for finite dimensional continuous-time dynamical systems described by a system of first order ordinary differential inequalities. We utilize this theory to establish sufficient robust stability criteria for a large class of finite dimensional, continuous-time dynamical systems described by systems of ordinary differential equations. We demonstrate the applicability of the methodology advanced herein by means of a specific example which has been considered in the literature.In terms of computational complexity and conservatism of stability criteria, the present results frequently offer improvements over existing results.

Maximum principles for ordinary differential inequalities of fourth and sixth order

The well-known maximum principle for second order ordinary differen- tial inequalities has been extended to some cases of fourth order inequalities. In [l], the following result was proved. If U(X) is a real valued function satisfying d4) 2 0, x E (4 b), u’(a) 3 0, u’(b) < 0, and attains its minimum value at a point c E (a, b), then u is identically constant on (a, b). The previous result has been extended in [2] to the differential inequality d4) + f(x) u”’ + h(x) u” 3 0, where f(x) and h(x) are bounded on every closed subinterval of (a, b), h(x) d 0. In the present paper we obtain similar results for a more general class of differential inequalities of fourth order and for a wide class of inequalities of sixth order. The maximum principle has applications to the question of uniqueness

On the zeros of solutions of beam equations

We consider extensible beam equations, Timoshenko beam equations and the system of coupled beam equations. We show that, under suitable conditions, there are bounded domains in which every solution satisfying certain end conditions has a zero. End conditions to be considered are hinged ends and hinged-sliding ends. The results are based on the conditions for the nonexistence of positive solutions of ordinary differential inequalities.