Periodic solutions for Liénard type p-Laplacian equation with a deviating argument

In this paper, we use the coincidence degree theory to establish new results on the existence of T-periodic solutions for the Liénard type p-Laplacian equation with a deviating argument of the form:

(_?p(x^′(t)))^′+f(x(t))x^′(t)+g(t,x(t-τ(t)))=e(t).$({?}_p(x^{\prime }(t)){)}^{\prime }+f(x(t))x^{\prime }(t)+g(t,x(t-\tau (t)))=e(t)\text{.}$

On existence of periodic solutions of Rayleigh equation of retarded type

Existence of periodic solutions for a kind of non-autonomous Rayleigh equations of retarded type

x^″(t)+f(t,x^′(t-σ))+g(t,x(t-τ(t)))=p(t)$x^{″}(t)+f(t,x^{\prime }(t-\sigma ))+g(t,x(t-\tau (t)))=p(t)$

Periodic solutions to a second order nonlinear neutral functional differential equation in the critical case

In this paper, the authors study the existence of periodic solutions for a second order neutral functional differential equation

(x(t)-cx(t-τ))^″=f(x(t))x^′(t)+g(t,x(t-μ(t)))+e(t)$(x(t)-\mathit{cx}(t-\tau ){)}^{″}=f(x(t))x^{\prime }(t)+g(t,x(t-\mu (t)))+e(t)$

Invariant tori for a reversible oscillator with a nonlinear damping and periodic forcing term

In this paper, the boundedness of all solutions of the oscillator

x^″+f(x,x^′)+ω²x+?(x)=p(t)$x^{″}+f(x,x^{\prime })+{\omega }^{2}x+?(x)=p(t)$

Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations

Consider in a real Hilbert space H the Cauchy problem (_P0$P_0$): u^′(t)+Au(t)+Bu(t)=f(t)$u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, where −A   is the infinitesimal generator of a _C0$C_0$-semigroup of contractions, B is a nonlinear monotone operator, and f is a given H-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem (_P0$P_0$) the following regularization (_Pε$P_{\epsilon }$): −εu^″(t)+u^′(t)+Au(t)+Bu(t)=f(t)$-\epsilon u^{″}(t)+u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, u^′(T)=_uT$u^{\prime }(T)=u_T$, where ε>0$\epsilon >0$ is a small parameter. We investigate existence, uniqueness and higher regularity for problem (_Pε$P_{\epsilon }$). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (_Pε$P_{\epsilon }$). Problem (_Pε$P_{\epsilon }$) turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of C([0,T];H)$C([0,T];H)$. However, the boundary layer of order one is not visible through the norm of L²(0,T;H)$L^2(0,T;H)$.     

Decay rate estimations for linear quadratic optimal regulators

Let u(t)=−Fx(t)$u(t)=-Fx(t)$ be the optimal control of the open-loop system x^′(t)=Ax(t)+Bu(t)$x^{\prime }(t)=Ax(t)+Bu(t)$ in a linear quadratic optimization problem. By using different complex variable arguments, we give several lower and upper estimates of the exponential decay rate of the closed-loop system x^′(t)=(A−BF)x(t)$x^{\prime }(t)=(A-BF)x(t)$. Main attention is given to the case of a skew-Hermitian matrix A. Given an operator A, for a class of cases, we find a matrix B that provides an almost optimal decay rate.     

Positive solutions for second-order nonlinear differential equations

We prove the existence of positive solutions to the scalar equation y^″(x)+F(x,y,y^′)=0$y^{″}(x)+F(x,y,y^{\prime })=0$. Applications to semilinear elliptic equations in exterior domains are considered.     

Multiplicity of layered solutions for Allen–Cahn systems with symmetric double well potential

We study the existence of solutions u:R³→R²$u:{\mathbb{R}}^3\to {\mathbb{R}}^2$ for the semilinear elliptic systems

equation(0.1)

−Δu(x,y,z)+∇W(u(x,y,z))=0,$-\mathrm{\Delta }u(x,y,z)+\mathrm{\nabla }W(u(x,y,z))=0,$

where W:R²→R$W:{\mathbb{R}}^2\to \mathbb{R}$ is a double well symmetric potential. We use variational methods to show, under generic non-degenerate properties of the set of one dimensional heteroclinic connections between the two minima _a±${\mathbf{a}}_{\pm }$ of W, that (0.1) has infinitely many geometrically distinct solutions u∈C²(R³,R²)$u\in C^2({\mathbb{R}}^3,{\mathbb{R}}^2)$ which satisfy u(x,y,z)→_a±$u(x,y,z)\to {\mathbf{a}}_{\pm }$ as x→±∞$x\to \pm \infty$ uniformly with respect to (y,z)∈R²$(y,z)\in {\mathbb{R}}^2$ and which exhibit dihedral symmetries with respect to the variables y and z  . We also characterize the asymptotic behavior of these solutions as |(y,z)|→+∞$|(y,z)|\to +\infty$.     

Simple geometry facilitates iterative solution of a nonlinear equation via a special transformation to accelerate convergence to third order

Direct substitution _xk+1=g(_xk)$x_{k+1}=g(x_k)$ generally represents iterative techniques for locating a root z   of a nonlinear equation f(x)$f(x)$. At the solution, f(z)=0$f(z)=0$ and g(z)=z$g(z)=z$. Efforts continue worldwide both to improve old iterators and create new ones. This is a study of convergence acceleration by generating secondary solvers through the transformation _gm(x)=(g(x)-m(x)x)/(1-m(x))$g_m(x)=(g(x)-m(x)x)/(1-m(x))$ or, equivalently, through partial substitution _gmps(x)=x+G(x)(g-x)$g_{m\mathrm{ps}}(x)=x+G(x)(g-x)$, G(x)=1/(1-m(x))$G(x)=1/(1-m(x))$. As a matter of fact, _gm(x)≡_gmps(x)$g_m(x)\equiv g_{m\mathrm{ps}}(x)$ is the point of intersection of a linearised g   with the g=x$g=x$ line. Aitken's and Wegstein's accelerators are special cases of _gm$g_m$. Simple geometry suggests that m(x)=(g^′(x)+g^′(z))/2$m(x)=(g^{\prime }(x)+g^{\prime }(z))/2$ is a good approximation for the ideal slope of the linearised g  . Indeed, this renders a third-order _gm$g_m$. The pertinent asymptotic error constant has been determined. The theoretical background covers a critical review of several partial substitution variants of the well-known Newton's method, including third-order Halley's and Chebyshev's solvers. The new technique is illustrated using first-, second-, and third-order primaries. A flexible algorithm is added to facilitate applications to any solver. The transformed Newton's method is identical to Halley's. The use of m(x)=(g^′(x)+g^′(z))/2$m(x)=(g^{\prime }(x)+g^{\prime }(z))/2$ thus obviates the requirement for the second derivative of f(x)$f(x)$. Comparison and combination with Halley's and Chebyshev's solvers are provided. Numerical results are from the square root and cube root examples.     

On super-linear Emden–Fowler type differential equations

We study the second order Emden–Fowler type differential equation

(a(t)|x^′|^αsgnx^′)^′+b(t)|x|^βsgnx=0${(a(t){\left|x^{\prime }\right|}^{\alpha }\mathrm{sgn}{x}^{\prime })}^{\prime }+b(t){|x|}^{\beta }\mathrm{sgn}x=0$

in the super-linear case α<β$\alpha <\beta$. Using a Hölder-type inequality, we resolve the open problem on the possible coexistence on three possible types of nononscillatory solutions (subdominant, intermediate, and dominant solutions). Jointly with this, sufficient conditions for the existence of globally positive intermediate solutions are established. Some of our results are new also for the Emden–Fowler equation.     

Inequalities with exponential weights

Let R=(-∞,∞)$\mathbb{R}=(-\infty ,\infty )$ and let Q∈C²:R→R⁺=[0,∞)$Q\in C^2:\mathbb{R}\to {\mathbb{R}}^{+}=[0,\infty )$ be an even function. Then in this paper we consider the infinite–finite range inequality, an estimate for the Christoffel function, and the Markov–Bernstein inequality with the exponential weights _wρ(x)=|x|^ρe^-Q(x),x∈R$w_{\rho }(x)=|x{|}^{\rho }{\mathrm{e}}^{-Q(x)},x\in \mathbb{R}$.     

Simultaneous bifurcation of limit cycles from a linear center with extra singular points

The period annuli of the planar vector field x^′=−yF(x,y)$x^{\prime }=-yF(x,y)$, y^′=xF(x,y)$y^{\prime }=xF(x,y)$, where the set {F(x,y)=0}$\{F(x,y)=0\}$ consists of k   different isolated points, is defined by k+1$k+1$ concentric annuli. In this paper we perturb it with polynomials of degree n and we study how many limit cycles bifurcate, up to a first order analysis, from all the period annuli simultaneously in terms of k and n  . Additionally, we prove that the associated Abelian integral is piecewise rational and, when k=1$k=1$, the provided upper bound is reached. Finally, the case k=2$k=2$ is also treated.