Boundary value problems for differential equations with deviating arguments

Aequationes mathematicae -     

Nonoscillatory solutions of linear differential equations with deviating arguments

Summary The equation to be considered is of the form (1) x(ⁿ)(t)+p(t)x(g(t))=0 (t>a), where =±1, p(t) > 0 for ta and g(t) as t. It is well- known that a nonoscillatory solution x(t) of (1) satisfies (2) x(t)x(ⁱ)(t)>0 (0il), (–1)^i–lx(t)x(ⁱ)(t)>0 (lin) for some integer l, 0ln, (–1)^n–l–1=1. In this paper, for a given l such that 0n–l–1=1, necessary conditions and sufficient conditions are found for (1) to have a solution x(t) which satisfies (2), and a necessary and sufficient condition is established in order that for every >0 the equation x(ⁿ)(t)+p(t)x(g(t))=0 (t>a) has a solution x(t) which satisfies (2). Related results are also contained.     

FORCED OSCILLATIONS OF HYPERBOLIC DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

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Oscillatory properties of strongly superlinear differential equations with deviating arguments

We regard a graph G as a set {1,…, v} together with a nonempty set E of two-element subsets of {1,…, v}. Let p = (p₁,…, p_v) be an element of $\text{R}$^nv representing v points in $\text{R}$ⁿ and consider the realization G(p) of G in $\text{R}$ⁿ consisting of the line segments [p_i, p_j] in $\text{R}$ⁿ for {i, j} ?E. The figure G(p) is said to be rigid in $\text{R}$ⁿ if every continuous path in $\text{R}$^nv, beginning at p and preserving the edge lengths of G(p), terminates at a point q ? $\text{R}$^nv which is the image (Tp₁,…, Tp_v) of p under an isometry T of $\text{R}$ⁿ. We here study the rigidity and infinitesimal rigidity of graphs, surfaces, and more general structures. A graph theoretic method for determining the rigidity of graphs in $\text{R}$² is discussed, followed by an examination of the rigidity of convex polyhedral surfaces in $\text{R}$³.     

An oscillation criterion for functional differential equations with deviating arguments

An oscillation criterion is given for the differential equation

On second order functional differential equations and inequalities with deviating arguments

A necessary and sufficient condition is established in order that (i) the retarded differential equation $$y''(t) = p_0 y(t) + f(y(t - \tau _1 ),...,y(t - \tau _N ))$$ has no bounded nonoscillatory solution and (ii) the advanced differential equation $$y''(t) = p_0 y(t) + f(y(t + \tau _1 ),...,y(t + \tau _N ))$$ has no unbounded nonoscillatory solution, wherep ₀≥0 and τ _j > 0,1 ?i ?N, are constants. Differential inequalities related to (^*) and (^**) are also studied. Finally, an oscillation criterion is given for a class of differential equations containing both retarded and advanced arguments.     

Oscillation theory of first order functional differential equations with deviating arguments

Summary New oscillation criteria are established for the first order functional differential equation (*) y'(t)+p(t)y(g(t))=0and its nonlinear analogue. The results are presented so that a remarkable duality existing between the case where (*) is retarded (g(t)t) is apparent. Possible extension of the results for (*) to equations with several deviating arguments is attempted. Finally, it is shown that there exists a class of autonomous equations for which the oscillation situation can be completely characterized.     

Oscillation theorems for nth-order differential equations with deviating arguments

This paper is concerned with the study of the oscillatory behavior of solutions of the nth-order nonlinear differential equation (a(t)x^{(n ? v)}(t))^(v) + q(t)f(x[g(t)]) = 0, where n is even and 1 ? v ? n ? 1. A systematic study is attempted which extends and correlates a number of existing results.     

Oscillation of even order partial differential equations with distributed deviating arguments

Some Kamenev-type oscillation criteria are established for a class of boundary value problems associated with even-order partial differential equations with distributed deviating arguments. Our approach is to reduce the high-dimensional oscillation problem to a one-dimensional oscillation one, and the general means developed by Philos and Wong is used as the main tool. The results obtained here extend and improve some known results in the literature.     

Weighted Cauchy problem for nonlinear singular differential equations with deviating arguments

For higher-order nonlinear differential equations with deviating arguments and with nonintegrable singularities with respect to the time variable, we establish sharp sufficient conditions for the Cauchy problem to be solvable and well-posed.     

具连续偏差变元的中立型向量抛物偏微分方程的H-振动性

讨论一类具连续偏差变元的中立型向量抛物偏泛函微分方程的H-振动性,利用内积降维的方法和Green公式,得到了该类方程在Robin边值条件下所有解Hm-振动的若干充分判据,这里H是Rm中的单位向量.     

Kamenev-type oscillation criteria for even order neutral differential equations with deviating arguments

Sufficient conditions are established for the oscillation of even order neutral differential equations of the form

where n  2 is even integer.