Solvability of an integro-differential equation with partial derivatives on the half-axis

The paper considers the following integro-differential equation on the semi-axis: $ \frac{{\partial f(t,x)}} {{\partial t}} + \frac{{\partial f(t,x)}} {{\partial x}} + qf(t,x) = \int_0^\infty {k(x - x')f(t,x')dx'} , $ where 0 ≤ k(x) ? L ₁(?∞,∞), q = const and f(t, x) is the unknown function. This equation has significant applications in different fields of natural sciences, particularly in econometrics (see [1]), where the unknown function is considered as the density of the national income distribution function, q characterizes the mean savings etc., k(x) is the function of income rearrangement. A structural theorem of the solution is proved, whose asymptotic at infinity is found.     

The method of lower and upper solutions for third order singular four point boundary value problems

We mainly study the existence of positive solutions for the following third order singular four point boundary value problem $$\begin{cases}x^{(3)}(t)+f(t,x,x',-x'')=0,\quad 0<t<1,\\x(0)-\alpha x(\xi)=0,\quad x'(1)-\beta x'(\eta)=0,\quad x''(0)=0.\end{cases}$$ where 0≤α<1, 0≤β<1, 0<ξ<1,0<η<1. And we obtain some necessary and sufficient conditions for the existence of C ²[0,1] positive solutions by means of the lower and upper solution method. Our nonlinearity f(t,x,y,z) may be singular at x,y,z,t=0 and/or t=1.     

Existence results for second order boundary value problems using the barrier strip technique

Using barrier strip type arguments we investigate the existence of solutions of the boundary value problem ${x''=f(t,x),\;t\in(0,1),\;x(0)=A,\;x'(1)=0,}Using barrier strip type arguments we investigate the existence of solutions of the boundary value problem x"=f(t,x),  t ? (0,1),  x(0)=A,  x￠(1)=0,{x'=f(t,x),\;t\in(0,1),\;x(0)=A,\;x'(1)=0,} where the scalar function f(t, x) may be singular at x = A.     

Eventually positive solutions of first order nonlinear differential equations with a deviating argument

The following first order nonlinear differential equation with a deviating argument $ x'(t) + p(t)[x(\tau (t))]^\alpha = 0 $ is considered, where α > 0, α ≠ 1, p ∈ C[t ₀; ∞), p(t) > 0 for t ≧ t ₀, τ ∈ C[t ₀; ∞), lim _t→∞ τ(t) = ∞, τ(t) < t for t ≧ t ₀. Every eventually positive solution x(t) satisfying lim _t→∞ x(t) ≧ 0. The structure of solutions x(t) satisfying lim _t→∞ x(t) > 0 is well known. In this paper we study the existence, nonexistence and asymptotic behavior of eventually positive solutions x(t) satisfying lim _t→∞ x(t) = 0.     

Oscillating solutions of scalar delay-differential equations with state dependence

This paper is concerned with the oscillatory behavior of the delay-differential equation X'(t)=F(t,x_t) including the equations x'(t)=-a(t)x(t-r(t,x(t))), [display math001] as special cases.We give conditions for the existence of a nonoscillatory solution of (1) and criteria for the oscillation of all solutions of (1), aiming at extending or generalizing to (1) some of the recent oscillation and nonoscillation results for delay equations of the form x'(t)=-a(t)x(t-p)).     

Oscillation of second order quasilinear neutral delay differential equations

In this paper, by employing Riccati transformation technique, some new sufficient conditions for the oscillation criteria are given for the second order quasilinear neutral delay differential equations with delayed argument in the form $$\bigl(r(t)\bigl|z'(t)\bigr|^{\alpha-1}z'(t)\bigr)'+q(t)f\bigl(x\bigl(\sigma(t)\bigr)\bigr)=0,\quad t\geq t_0,$$ where z(t)=x(t)?p(t)x(??(t)), 0??p(t)??p<1, lim _t???? p(t)=p ₁<1, q(t)>0, ??>0. Two examples are considered to illustrate the main results.     

On forced nonlinear oscillations

The existence of a 1-periodic solution of the generalized Liénard equation x″ + g(x)x′ + f(t, x) = e(t), where g(x) is continuous, e(t) is continuous, periodic of period 1 and with mean value 0 and f is continuous, periodic of period 1 in t, is proved under one of the following conditions: (i) there exists M ? 0 such that $\text{f(t, x)x ? 0}\text{for}\text{¦x¦? M}$ and

$\text{lim sup}\text{|x|?+∞}\text{|f(t,x)|}\text{| x |}\text{<}\text{4π}{}^2\text{2π + 1}$

(ii) there exists M ? 0 such that $\text{f(t, x)x ? 0}\text{for}\text{¦x¦? M}$. Earlier results of A. C. Lazer, J. Mawhin and R. Reissig are obtained as particular cases.     

Periodic solutions of non-smooth friction oscillators

We study the differential equation x"+g(x￠)+m(x) sgn x￠+f(x)=j(t)x''+g(x')+\mu(x)\,{\rm sgn}\, x'+f(x)=\varphi(t) with T-periodic right-hand side, which models e.g. a mechanical system with one degree of freedom subjected to dry friction and periodic external force. If, in particular, the damping term g is present and acts, up to a bounded difference, like a linear damping, we get existence of a T-periodic solution.¶In the more difficult case g = 0, we concentrate on the model equation x"+m(x) sgn x￠+x=j(t)x''+\mu(x)\,{\rm sgn}\,x'+x=\varphi(t) and obtain sufficient conditions for the existence of a T-periodic solution by application of Brouwer's fixed point theorem. For this purpose we show that a certain associated autonomous differential equation admits a periodic orbit such that the surrounded set (minus some neighborhood of the equilibria) is forward invariant for the equation above. Under additional assumptions on 7 we prove boundedness of all solutions.¶Finally, we provide a principle of linearized stability for periodic solutions without deadzones, where the "linearized" differential equation is an impulsive Hill equation.     

Solvability of a Kind of Sturm–Liouville Boundary Value Problems with Impulses via Variational Methods

The purpose of this paper is to use a three critical point theorem due to Ricceri to obtain the existence of at least three solutions for the following Sturm–Liouville boundary value problem with impulses

$\begin{cases}(\phi_{p}(x'(t)))'=(a(t)\phi_{p}(x)+\lambda f(t,x)+\mu h(x))g(x'(t)),\quad \mbox{a.e. }t\in[0,1],\\\Delta G(x'(t_{i}))=I_{i}(x(t_{i})),\quad i=1,2,\ldots,k,\\\alpha_{1}x(0)-\alpha_{2}x'(0)=0,\\\beta_{1}x(1)+\beta_{2}x'(1)=0,\end{cases}$

where p>1, φ _p (x)=|x|^p?2 x, λ, μ are positive parameters, \(G(x)=\int_{0}^{x}\frac{(p-1)|s|^{p-2}}{g(s)}\,ds\). The interest is that the nonlinear term includes x′. We exhibit the existence of at least three solutions and h(x) can be an arbitrary C ¹ functional with compact derivative. As an application, an example is given to illustrate the results.

     

Global Dynamics of Froude-Type Oscillators with Superlinear Damping Terms

This paper deals with the damped superlinear oscillator $$x'' + a(t)\phi_p\bigl(x' \bigr) + b(t)\phi_q\bigl(x'\bigr)+ \omega^2x = 0, $$ where a(t) and b(t) are continuous and nonnegative for t≥0; p and q are real numbers greater than or equal to 2; ? _r (x′)=|x′| ^r?2 x′. This equation is a generalization of nonlinear ship rolling motion with Froude’s expression, which is very familiar in marine engineering, ocean engineering and so on. Our concern is to establish a necessary and sufficient condition for the equilibrium to be globally asymptotically stable. In particular, the effect of the damping coefficients a(t), b(t) and the nonlinear functions ? _p (x′), ? _q (x′) on the global asymptotic stability is discussed. The obtained criterion is judged by whether the integral of a particular solution of the first-order nonlinear differential equation $$u' + \omega^{p-2}a(t)\phi_p(u) + \omega^{q-2}b(t)\phi_q(u) + 1 = 0 $$ is divergent or convergent. In addition, explicit sufficient conditions and explicit necessary conditions are given for the equilibrium of the damped superlinear oscillator to be globally attractive. Moreover, some examples are included to illustrate our results. Finally, our results are extended to be applied to a more complicated model.     

Some necessary and sufficient conditions for existence of positive solutions for third order singular super-linear multi-point boundary value problems

We mainly study the existence of positive solutions for the following third order singular super-linear multi-point boundary value problem $$ \left \{ \begin{array}{l} x^{(3)}(t)+ f(t, x(t), x'(t))=0,\quad0 where \(0\leq\alpha_{i}\leq\sum_{i=1}^{m_{1}}\alpha_{i}<1\) , i=1,2,…,m ₁, \(0<\xi_{1}< \xi_{2}< \cdots<\xi_{m_{1}}<1\) , \(0\leq\beta_{j}\leq\sum_{i=1}^{m_{2}}\beta_{i}<1\) , j=1,2,…,m ₂, \(0<\eta_{1}< \eta_{2}< \cdots<\eta_{m_{2}}<1\) . And we obtain some necessary and sufficient conditions for the existence of C ¹[0,1] and C ²[0,1] positive solutions by means of the fixed point theorems on a special cone. Our nonlinearity f(t,x,y) may be singular at t=0 and t=1.     

Potential theory associated with Uhlenbeck-Ornstein process

Some parallel results of Gross' paper (Potential theory on Hilbert space, J. Functional Analysis1 (1967), 123–181) are obtained for Uhlenbeck-Ornstein process U(t) in an abstract Wiener space (H, B, i). Generalized number operator $\text{N}$ is defined by $\text{N}$f(x) = ?lim_∈←0{E[f($\text{U}$(τ^∈_ξ))] ? f(x)}/E[τ^∈_ξ, where τ_x^? is the first exit time of U(t) starting at x from the ball of radius ? with center x. It is shown that $\text{N}$f(x) = ?trace D²f(x)+〈Df(x),x〉 for a large class of functions f. Let r_t(x, dy) be the transition probabilities of U(t). The λ-potential G_λf, λ > 0, and normalized potential Rf of f are defined by G_λf(X) = ∫₀^∞e^?λtr_tf(x) dt and Rf(x) = ∫₀^∞ [r_tf(x) ? r_tf(0)] dt. It is shown that if f is a bounded Lip-1 function then trace D²G_λf(x) ? 〈DG_λf(x), x〉 = ?f(x) + λG_λf(x) and trace D²Rf(x) ? 〈DRf(x), x〉 = ?f(x) + ∫_Bf(y)p₁(dy), where p₁ is the Wiener measure in B with parameter 1. Some approximation theorems are also proved.     

Weak and generic bang-bang properties for continuous evolution inclusions and Baire’s method

We consider evolution inclusions, in a separable and reflexive Banach space ${\mathbb{E}}$ , of the form ${(\ast) x'(t) \in Ax(t) + F(t, x(t)), x(t_0) = c}$ and ${(**) x'(t) \in Ax(t) + {\rm ext} F(t,x(t)), x(t_0) = c}$ , where A is the infinitesimal generator of a C ₀-semigroup, F is a continuous and bounded multifunction defined on ${[t_0, t_1] \times \mathbb{E}}$ with values F(t, x) in the space of all closed convex and bounded subsets of ${\mathbb{E}}$ with nonempty interior, and ext F(t, x(t)) denotes the set of the extreme points of F(t, x(t)). For (*) and (**) we prove a weak form of the bang-bang property, namely, the closure of the set of the mild solutions of (**) contains the set of all internal solutions of (*). The proof is based on the Baire category method. This result is used to prove the following generic bang-bang property, that is, if A is the infinitesimal generator of a compact C ₀-semigroup then for most (in the sense of the Baire categories) continuous and bounded multifunctions, with closed convex and bounded values ${F(t, x) \subset \mathbb{E}}$ , the bang-bang property is actually valid, that is, the closure of the the set of the mild solutions of (**) is equal to the set of the mild solutions of (*).     

Single point blow-up for a semilinear initial value problem

The initial value problem on [?R, R] is considered: u_t(t, x) = u_xx(t, x) + u(t, x)^γu(t, ±R) = 0u(0, x) = ?(x), where ? ? 0 and γ is a fixed large number. It is known that for some initial values ? the solution u(t, x) exists only up to some finite time T, and that ∥u(t, ·)∥_∞ → ∞ as t → T. For the specific initial value ? = kψ, where ψ ? 0, ψ_xx + ψ^γ = 0, ψ(±R) = 0, k is sufficiently large, it is shown that if x ≠ 0, then lim_{t → T}u(t, x) and lim_{t → T}u_x(t, x) exist and are finite. In other words, blow-up occurs only at the point x = 0.     

A completely monotonic function involving the tri- and tetra-gamma functions

The di-gamma function ψ(x) is defined on (0,∞) by $\psi (x) = \frac{{\Gamma '(x)}} {{\Gamma (x)}} $ and ψ ⁽ⁱ⁾(x) for i ∈ ? denote the polygamma functions, where Γ(x) is the classical Euler’s gamma function. In this paper we prove that a function involving the difference between [ψ′(x)]² + ψ″(x) and a proper fraction of x is completely monotonic on (0,∞).     

Oscillation theorems for second order nonlinear differential equations

Some oscillation criteria are established for certain second order nonlinear differential equations of the form (a(t)ψ(x(t)) x^. (t))^. + p(t) x^. (t) + q(t)f(x(t)) = 0. These criteria improve upon some of the known results by Kura, Kamenev and Philos.     

Tranchage et cône tangent des courants localement normaux

Let T be a locally normal current on an open set Ω of ℝ″ = ℝ x ℝ″⁻¹ and let π: ℝⁿ → ℝ denote the projection π(x₁, x″) = x₁. We define the current 〈T, π, 0〉 (called slice of T at 0 by π) as the limit, as ɛ → 0, of the family ɛ⁻¹TΛπ ^* (ψ(x¹ /ɛ)dx¹), where ψ is a C^∞ function on ℝ with compact support such that ∝ψ(x¹)dx¹ = 1, provided the limit exists and doesn't depend on the choice of ψ. We first prove that the limit lim_R→+∞(h_R)_#T exists, where h_R(x₁,x″) = (Rx¹,x″). We apply this result to the study of the existence of the tangent cone at 0 associated to a locally normal current, and especially associated to a subanalytic chain. We finally give a necessary and sufficient condition relative to T for the existence of the slice 〈T, π, 0〉.     

Oscillation theorems for second-order superlinear neutral differential equations

The purpose of this paper is to investigate the oscillation of the second-order neutral differential equations of the form (E) $$ (r(t)|z'(t)|^{\alpha - 1} z'(t))' + q(t)|x(\sigma (t))|^{\alpha - 1} x(\sigma (t)) = 0, $$ where z(t) = x(t) + p(t)x(τ(t)). The obtained comparison principles essentially simplify the examination of the studied equations. Further, our results extend and improve the results in the literature.     

Two linear equations in formal power series

The equations [gradφ(x)]^TF(x)=h(x) and F(ψ(x))–ψ(x) are considered. They arise in the stability theory of differential and difference equations. The scalar function h(x) is a given, and the function ψ(x) an unknown, formal power series in the n indeterminates x=(x₁,…,x_n)^T, and h(0)=ψ=0; the elements of the n×n matrix F(x) are also formal power series in x, F(0)=0. It is shown that the solvability of both equations depends on the eigenvalues of the Jacobian F_x(0).