Periodic solutions of nonlinear integro-differential equations with an impulse effect

The paper applies a numerical-analytical method for finding periodic solutions of the system of integro-differential equations $$\begin{gathered} \dot x = f(t,x,\mathop \smallint \limits_0^t \varphi (t,s,x(s))ds), t \ne t_i (x), \hfill \\ \Delta x|_{t = t_i (x)} = I_i (x). \hfill \\ \end{gathered} $$ Two theorems for existence of periodic solutions are proved for the cases whent = t _i andt = t _i(x).     

Small prime solutions of some additive equations

Essentially sharp bounds for small prime solutionsp _j ,q _i of the following two different types of equations are obtained.

The lattice point discrepancy of a torus in ℝ3

This article provides an asymptotic formula for the number of integer points in the three-dimensional body $$ \left( \begin{gathered} x \hfill \\ y \hfill \\ z \hfill \\ \end{gathered} \right) = t\left( \begin{gathered} (a + r\cos \alpha )\cos \beta \hfill \\ (a + r\cos \alpha )\sin \beta \hfill \\ r\sin \alpha \hfill \\ \end{gathered} \right),0 \leqq \alpha ,\beta < 2\pi ,0 \leqq r \leqq b, $$ for fixed a > b > 0 and large t.     

Scanning control of a vibrating string

We construct scanning feedback controls {γ _i (t)} for the vibrating string equation $$\begin{gathered} y_{tt} (x,t) = y_{xx} (x,t) + Ry(x,t) + \sum\limits_{i = 1}^N {\phi (x - \gamma _i } (t))y(x,t), \hfill \\ 0< x< 1,y = 0 at x = 0,1. \hfill \\ \end{gathered} $$ so that (y, y _t ) → (0,0) ast → ∞ in the weak topology ofH ₀ ¹ (0,1) ×L ₂ (0,1). In particular we show that ifφ is an even polynomial of degreeN with nonpositive coefficients that forR <π ² we can find such stabilizingγ _i (t), i=1,?,N.     

Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

Necessary conditions for optimality in the identification of elliptic systems with parameter constraints

We consider the problems of dientifying the parametersa _ij (x), b _i (x), c(x) in a 2nd order, linear, uniformly elliptic equation, $$\begin{gathered} - \partial _i (a_{ij} (x)\partial _j u) + b_i (x)\partial _i u + c(x)u = f(x),in\Omega , \hfill \\ \partial _v u|_{\partial \Omega } = \phi (s),s \in \partial \Omega , \hfill \\ \end{gathered} $$ on the basis of measurement data $$u(s) = z(s),s \in B \subset \partial \Omega ,$$ with an equality constraint and inequality constraints on the parameters. The cost functionals are one-sided Gâteaux differentiable with respect to the state variables and the parameters. Using the Duboviskii-Milyutin lemma, we get maximum principles for the identification problems, which are necessary conditions for the existence of optimal parameters.     

On the existence of periodic solutions for the equation\ddot x + f(x)\dot x + g(x) = 0

We establish in this work sufficient conditions for the existence of periodic solutions for the Liénard equation .     

THE BEST APPROXIMATION CONSTANT FOR THE JACKSON''''S TYPE OPERATOR J_(n,3) (f;x)

In this paper we obtain the best approximation constant of function f(x)(∈C_(2π))by theJackson's type operator J_(π3)(f;x),i.e.‖J_(n,3)(f,x)-f(x)‖_c≤(4-6/π)ω(f,1/n),‖J_(n,3)(f,x)-f(x)‖_c≤(8-17/π)ω_2(f,1/n)     

Lower semi-continuous nonconvex perturbation of m-accrative differential inclusions in Banach spaces

In this paper we prove the existence of solutions of the differential inclusions $$\left\{ \begin{gathered} \dot X(t) \in - A_t (X(t)) + F(t,X(t)),,0 \leqslant t \leqslant T_0 \hfill \\ X(0) = x_0 \hfill \\ \end{gathered} \right.$$ whereA _t is a multivaluedm-accretive operator on a Banach spaceE andF is a measurable multifunction defined on the set \(G = \overline {\{ (t,x):A_t (x) \ne 0/\} } \) , lower semicontinuous inx and its values are not necessarily convex inE. This result generalizes some results in [1] and [9].     

A practical method for determining Green's functions using Hadamard's variational formula

This article studies the behavior of the Green's function as a function of the length of the interval. The solution of the boundary-value problem

$$\begin{gathered} \ddot u - q(t)u = - g(t), 0< t< a \hfill \\ u(0) = 0, \dot u(a) = 0 \hfill \\ \end{gathered} $$     

Nonautonomous locally Lipschitz continuous functional differential equations in spaces of continuous functions

A local existence and uniqueness result for the functional differential equation in a Banach space X (FDE) $$\begin{gathered} x\prime (t) \in f(t)x(t) + g(t)x_t ,0 \leqslant t \leqslant T, \hfill \\ x_0 = \phi \in C( - R,0;X) \hfill \\ \end{gathered} $$ is obtained, for the case where the operatorsf(t) satisfy only a local dissipativity condition and the operatorsg(t) are only locally Lipschitz continuous. The conditions include equations defined on cones.     

Compactness of support of solutions to nonlinear second-order elliptic and parabolic equations in a half-cylinder

We study equations of the form $$\begin{gathered} u_{tt} + Lu + b(x,t)u_t = a(x,t)\left| u \right|^{\sigma - 1} u, \hfill \\ - u_t + Lu = a(x,t)\left| u \right|^{\sigma - 1} u \hfill \\ \end{gathered}$$ , whereL is a uniformly elliptic operator and 0<σ<1. In the half-cylinder II_0,∞={(x, t):x= (x ₁,...,x _n )∈ ω,t>0}, where ? ? ? _n is a bounded domain, we consider solutions satisfying the homogeneous Neumann condition forx∈?ω andt>0. We find conditions under which these solutions have compact support and prove statements of the following type: ifu(x, t)=o(t ^γ) ast→∞, then there exists aT such thatu(x, t)≡0 fort>T. In this case γ depends on the coefficients of the equation and on the exponent σ.     

Boundary value problems for complete quasi-parabolic differential equations of odd order with variable domains of operators

We prove the well-posed solvability in the strong sense of the boundary value Problems

$$\begin{gathered} ( - 1)\frac{{_m d^{2m + 1} u}}{{dt^{2m + 1} }} + \sum\limits_{k = 0}^{m - 1} {\frac{{d^{k + 1} }}{{dt^{k + 1} }}} A_{2k + 1} (t)\frac{{d^k u}}{{dt^k }} + \sum\limits_{k = 1}^m {\frac{{d^k }}{{dt^k }}} A_{2k} (t)\frac{{d^k u}}{{dt^k }} + \lambda _m A_0 (t)u = f, \hfill \\ t \in ]0,t[,\lambda _m \geqslant 1, \hfill \\ {{d^i u} \mathord{\left/ {\vphantom {{d^i u} {dt^i }}} \right. \kern-\nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} \mathord{\left/ {\vphantom {{d^j u} {dt^j }}} \right. \kern-\nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., \hfill \\ \end{gathered} $$

where the unbounded operators A _s (t), s > 0, in a Hilbert space H have domains D(A _s (t)) depending on t, are subordinate to the powers A ^1?(s?1)/2m (t) of some self-adjoint operators A(t) ≥ 0 in H, are [(s+1)/2] times differentiable with respect to t, and satisfy some inequalities. In the space H, the maximally accretive operators A ₀(t) and the symmetric operators A _s (t), s > 0, are approximated by smooth maximally dissipative operators B(t) in such a way that

$$\begin{gathered} \mathop {lim}\limits_{\varepsilon \to 0} Re(A_0 (t)B_\varepsilon ^{ - 1} (t)(B_\varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H \geqslant c(A(t)u,u)_H \hfill \\ \forall u \in D(A_0 (t)),c > 0, \hfill \\ \end{gathered} $$

, where the smoothing operators are defined by

$$B_\varepsilon ^{ - 1} (t) = (I - \varepsilon B(t))^{ - 1} ,(B_\varepsilon ^{ - 1} (t)) * = (I - \varepsilon B^ * (t))^{ - 1} ,\varepsilon > 0.$$

.

     

Some further stability and boundedness results of some differential equations of the fourth order

In this paper sufficient conditions (Theorem 1 and Corollary 1) for the asymptotic stability (in the large) of the trivial solution x=0 of the differential equations $$D_1 (x) = x^{(4)} + f_1 (\ddot x)\dddot x + f_2 (\dot x,\ddot x) + g(\dot x) + h(x,\dot x) = 0$$ , and $$D_2 (x) = x^{(4)} + F_1 (\ddot x)\dddot x + F_2 (\dot x,\ddot x)\ddot x + G(\dot x)\dot x + H(x,\dot x)x = 0$$ are given. A result (Theorem 2) on the boundedness of the solutions of the differential equations D₁(x)=p₁(t) and D₂(x)=p₂(t) is also established. Further, the results which we obtain reduce to results which are more general than those obtained by Ezeilo [1] for the differential equation $$x^{(4)} + f_1 (\ddot x)\dddot x + a_2 \ddot x + g(\dot x) + a_4 x = p(t)$$ .     

Existence and iteration of positive solution for a three-point boundary value problem with ap-Laplacian operator

In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for BVPs $$\left\{ \begin{gathered} (\phi _p (u\prime ))\prime + q(t)f(t, u) = 0,0< t< 1, \hfill \\ u(0) - B(u\prime (\eta )) = 0, u\prime (1) = 0 \hfill \\ \end{gathered} \right.$$ and $$\left\{ \begin{gathered} (\phi _p (u\prime ))\prime + q(t)f(t, u) = 0,0< t< 1, \hfill \\ u\prime (0) = 0, u(1) + B(u\prime (\eta )) = 0 \hfill \\ \end{gathered} \right.$$ The main tool is the monotone iterative technique. Here, the coefficientq(t) may be singular att = 0,1.     

Continuous dependence for integrodifferential equations with infinite delay

Continuous dependence for integrodifferential equation with infinite delay $$\begin{gathered} \dot x = h(t,x) + \int_{ \sim \infty }^t {q(t,s,x(s))ds} + F(t,x(t),Sx(t))t \geqslant 0 \hfill \\ x(t) = \Phi (t) \hfill \\ \end{gathered} $$ where \(Sx(t) = \int_{ \sim \infty }^t {k(t,s,x(s))} ds\) is studied under the assumption of existence of unique solution.     

On the null-controllability in function space of nonlinear systems of neutral functional differential equations with limited controls

Consider the following functional equations of neutral type: $$\begin{gathered} (i) (d/dt)D(t,x_t ) = L(t,x_t ), \hfill \\ (ii) (d/dt)D(t,x_t ) = L(t,x_t ) + B(t)u(t), \hfill \\ (iii) (d/dt)D(t,x_t ) = L(t,x_t ) + B(t)u(t) + f(t,x(t),u(t)), \hfill \\ \end{gathered} $$ whereD, L are bounded linear operators fromC([?h, 0],E ⁿ) intoE ⁿ for eacht?(σ, ∞) =J, B is ann ×m continuous matrix function,u:J →C ^m is square integrable with values in the unitm-dimensional cubeC ^m, andf(t, 0, 0)=0. We prove that, if the system (i) is uniformly asymptotically stable and if the controlled system (ii) is controllable, then the system (iii) is null-controllable with constraints, provided that $$f = f_1 + f_2 $$ , where $$\begin{gathered} |f_1 (t,\phi ,0)| \leqslant \varepsilon \parallel \phi \parallel , |f_2 (t,\phi ,0)| \leqslant \pi (t)\parallel \phi \parallel , t \geqslant \sigma , \hfill \\ \Pi = \int_0^\infty {\pi (t)dt< \infty .} \hfill \\ \end{gathered} $$      

Mixed boundary-value problems for singular second-order ordinary differential equations

It is proved that the boundary-value problem

Oscillation of solutions of nonlinear partial differential equations of neutral type

In this paper, we deal with the oscillatory behavior of solutions of the neutral partial differential equation of the form $$\begin{gathered} \frac{\partial }{{\partial t}}\left[ {p\left( t \right)\frac{\partial }{{\partial t}}(u\left( {x,t} \right) + \sum\limits_{i = 1}^t {p_i \left( t \right)u\left( {x,t - \tau _i } \right)} )} \right] + q\left( {x,t} \right)f_j (u(x,\sigma _j (t))) \hfill \\ = a\left( t \right)\Delta u\left( {x,t} \right) + \sum\limits_{k = 1}^n {a_k \left( t \right)} \Delta u\left( {x,\rho _k \left( t \right)} \right), \left( {x,t} \right) \in \Omega \times R_ + \equiv G \hfill \\ \end{gathered} $$ where Δ is the Laplacian in EuclideanN-spaceR ^N, R₊=(0, ∞) and Ω is a bounded domain inR ^N with a piecewise smooth boundary δΩ.     

An optimization algorithm for the pile driver problem

In this paper we present the analysis of an algorithm of Uzawa type to compute solutions of the quasi variational inequality $$\begin{gathered} (QVI)\left( {\frac{{\partial ^2 u}}{{\partial t^2 }},\upsilon - \frac{{\partial u}}{{\partial t}}} \right) + \left( {\frac{{\partial u}}{{\partial x}},\frac{{\partial \upsilon }}{{\partial x}} - \frac{{\partial ^2 u}}{{\partial x\partial t}}} \right) + \left( {\frac{{\partial ^2 u}}{{\partial x\partial t}},\frac{{\partial \upsilon }}{{\partial x}} - \frac{{\partial ^2 u}}{{\partial x\partial t}}} \right) + \hfill \\ + \left[ {u(1,t) + \frac{{\partial u}}{{\partial t}}(1,t)} \right]\left[ {\upsilon (1) - \frac{{\partial u}}{{\partial t}}(1,t)} \right] + J(u;\upsilon ) - J\left( {u;\frac{{\partial u}}{{\partial t}}} \right) \geqslant \hfill \\ \geqslant \left( {f,\upsilon - \frac{{\partial u}}{{\partial t}}} \right) + F(t)\left[ {\upsilon (0) - \frac{{\partial u}}{{\partial t}}(0,t)} \right],t > 0,\forall \upsilon \in H^1 (0,1), \hfill \\ \end{gathered} $$ which is a model for the dynamics of a pile driven into the ground under the action of a pile hammer. In (QVI) (...) is the scalar product inL ²(0, 1) andJ(u;.) is a convex functional onH ¹(0, 1), for eachu, describing the soil-pile friction effect.