Boundedness in Nonlinear Asymmetric Oscillations

In this paper, the boundedness of all solutions of the nonlinear equation (?_p(x′))′+(p-1)[α?_p(x⁺)−β?_p(x⁻)]+f(x)+g(x)=e(t) is discussed, where e(t)∈C⁷ is 2π_p-periodic, f,g are bounded C⁶ functions, ?_p(u)=∣u∣^p−2u, p?2,α,β are positive constants, x⁺=max{x,0},x⁻=max{−x,0}.     

The behavior of solutions of second order delay differential equations

In this paper, we study the behavior of solutions of second order delay differential equation

y^″(t)=p₁y^′(t)+p₂y^′(t−τ)+q₁y(t)+q₂y(t−τ),     

Existence and uniqueness of periodic solutions for a kind of first order neutral functional differential equations

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T-periodic solutions for the first order neutral functional differential equation of the form

(x(t)+Bx(t−δ)^)′=g₁(t,x(t))+g₂(t,x(t−τ))+p(t).     

Meromorphic Functions That Share Fixed-Points

In this paper, we prove the following result: Let f(z) and g(z) be two nonconstant meromorphic(entire) functions, n ≥ 11(n ≥ 6) a positive integer. If fⁿ(z)f′(z) and gⁿ(z)g′(z) have the same fixed-points, then either f(z) = c₁e^cz2, g(z) = c₂e^− cz2, where c₁, c₂, and c are three constants satisfying 4(c₁c₂)^n + 1c² = −1, or f(z) ≡ tg(z) for a constant t such that t^n + 1 = 1.     

Two-Phase Stefan Problem as the Limit Case of Two-Phase Stefan Problem with Kinetic Condition

Both one-dimensional two-phase Stefan problem with the thermodynamic equilibrium condition u(R(t),t)=0 and with the kinetic rule u_ε(R_ε(t),t)=εR_ε′(t) at the moving boundary are considered. We prove, when ε approaches zero, R_ε(t) converges to R(t) in C^1+δ/2[0,T] for any finite T>0, 0<δ<1.     

The Dichotomy Theorem for evolution bi-families

We prove that the operator G, the closure of the first-order differential operator −d/dt+D(t) on L₂(R,X), is Fredholm if and only if the not well-posed equation u^′(t)=D(t)u(t), t∈R, has exponential dichotomies on R₊ and R₋ and the ranges of the dichotomy projections form a Fredholm pair; moreover, the index of this pair is equal to the Fredholm index of G. Here X is a Hilbert space, D(t)=A+B(t), A is the generator of a bi-semigroup, B(⋅) is a bounded piecewise strongly continuous operator-valued function. Also, we prove some perturbations results and consider various examples of not well-posed problems.     

On oscillatory solutions of certain forced Emden-Fowler like equations

We give a constructive proof of existence to oscillatory solutions for the differential equations x^″(t)+a(t)λ|x(t)|sign[x(t)]=e(t), where t?t₀?1 and λ>1, that decay to 0 when t→+∞ as O(t−μ) for μ>0 as close as desired to the “critical quantity” . For this class of equations, we have limt→+∞E(t)=0, where E(t)<0 and E^″(t)=e(t) throughout [t₀,+∞). We also establish that for any μ>μ^? and any negative-valued E(t)=o(t−μ) as t→+∞ the differential equation has a negative-valued solution decaying to 0 at + ∞ as o(t−μ). In this way, we are not in the reach of any of the developments from the recent paper [C.H. Ou, J.S.W. Wong, Forced oscillation of nth-order functional differential equations, J. Math. Anal. Appl. 262 (2001) 722-732].     

Periodic solutions for a higher order p-Laplacian neutral functional differential equation with a deviating argument

In this paper, a higher order p-Laplacian neutral functional differential equation with a deviating argument:

[φ_p([x(t)−c(t)x(t−σ)]⁽ⁿ⁾)]^(m)+f(x(t))x^′(t)+g(t,x(t−τ(t)))=e(t)     

Existence of three positive solutions for boundary value problems with p-Laplacian

By using fixed point theorem, we study the following equation g(u^′′(t))+a(t)f(u)=0 subject to boundary conditions, where g(v)=|v|^p−2v with p>1; the existence of at least three positive solutions is proved.     

On the instability of differential systems with varying delay

The unstable properties of the null solution of the nonautonomous delay system x′(t)=A(t)x(t)+B(t)x(t−r₁(t))+f(t,x(t),x(t−r₂(t))) are examined; the nonconstant delays r₁, r₂ are assumed to be continuous bounded functions. The case A=constant is reviewed, where a theorem, recalling the Perron instability theorem for ordinary differential equations, is obtained.     

Sign-changing solutions on a kind of fourth-order Neumann boundary value problem

In this paper, we consider the fourth-order Neumann boundary value problem u(4)(t)−2u^″(t)+u(t)=f(t,u(t)) for all t∈[0,1] and subject to u^′(0)=u^′(1)=u^?(0)=u^?(1)=0. Using the fixed point index and the critical group, we establish the existence theorem of solutions that guarantees the problem has at least one positive solution and two sign-changing solutions under certain conditions.     

A Galerkin Approximation for the Initial-Value Problem for Linear Second-Order Differential Equations

In this paper, a Galerkin type algorithm is given for the numerical solution of L(x)=(r(t)x'(t))'-p(t)x(t)=g(t); x(a)=x_a, x'(a)=x'_a, where r (t)>f0, and Spline hat functions form the approximating basis. Using the related quadratic form, a two-step difference equation is derived for the numerical solutions. A discrete Gronwall type lemma is then used to show that the error at the node points satisfies e_k=0(h²). If e(t) is the error function on a?t?b; it is also shown (in a variety of norms) that e(t)?Ch² and e'(t)?C₁h. Test case runs are also included. A (one step) Richardson or Rhomberg type procedure is used to show that e^R_k=0(h⁴). Thus our results are comparable to Runge-Kutta with half the function evaluations.     

Periodic solutions for p-Laplacian neutral functional differential equation with deviating arguments

By using the theory of coincidence degree, we study a kind of periodic solutions to p-Laplacian neutral functional differential equation with deviating arguments such as (φp(x^′(t)−cx^′(t−σ)))+g(t,x(t−τ(t)))=p(t), a result on the existence of periodic solutions is obtained.     

Existence of positive periodic solutions of nonlinear first-order delayed differential equations

We consider the existence of positive ω-periodic solutions for the equation

u^′(t)=a(t)g(u(t))u(t)−λb(t)f(u(t−τ(t))),     

Almost automorphic solutions for differential equations with piecewise constant argument in a Banach space

Using spectral theory we obtain sufficient conditions for the almost automorphy of bounded solutions to differential equations with piecewise constant argument of the form x^′(t)=A(t)x([t])+f(t),t∈R, where A(t) is an almost automorphy operator, f(t) is an X-valued almost automorphic function and X is a finite dimensional Banach space.     

Paired bondage in trees

Let G=(V,E) be a graph with δ(G)≥1. A set D⊆V is a paired dominating set if D is dominating, and the induced subgraph 〈D〉 contains a perfect matching. The paired domination number of G, denoted by γ_p(G), is the minimum cardinality of a paired dominating set of G. The paired bondage number, denoted by b_p(G), is the minimum cardinality among all sets of edges E^′⊆E such that δ(G−E^′)≥1 and γ_p(G−E^′)>γ_p(G). We say that G is a γ_p-strongly stable graph if, for all E^′⊆E, either γ_p(G−E^′)=γ_p(G) or δ(G−E^′)=0. We discuss the basic properties of paired bondage and give a constructive characterization of γ_p-strongly stable trees.     

The numerical solution of forward-backward differential equations: Decomposition and related issues

This paper focuses on the decomposition, by numerical methods, of solutions to mixed-type functional differential equations (MFDEs) into sums of “forward” solutions and “backward” solutions. We consider equations of the form x^′(t)=ax(t)+bx(t−1)+cx(t+1) and develop a numerical approach, using a central difference approximation, which leads to the desired decomposition and propagation of the solution. We include illustrative examples to demonstrate the success of our method, along with an indication of its current limitations.     

Existence of three positive pseudo-symmetric solutions for a one dimensional p-Laplacian

We apply the Five Functionals Fixed Point Theorem to verify the existence of at least three positive pseudo-symmetric solutions for the three point boundary value problem, (g(u′))′+a(t)f(u)=0, u(0)=0, and u(ν)=u(1), where g(v)=|v|^p−2v, with p>1 and ν∈(0,1).     

Stabilization of unbounded bilinear control systems in Hilbert space

We consider bilinear control systems of the form y^′(t)=Ay(t)+u(t)By(t) where A generates a strongly continuous semigroup of contraction (etA)_t?0 on an infinite-dimensional Hilbert space Y whose scalar product is denoted by 〈.,.〉. We suppose that this system is unbounded in the sense that the linear operator B is unbounded from the state Y into itself. Tacking into account eventual control saturation, we study the problem of stabilization by (possibly nonquadratic) feedback of the form u(t)=−f(〈By(t),y(t)〉). Applications to the heat equation is considered.