Asymptotic behavior of solutions of nonlinear impulsive delay differential equations with positive and negative coefficients

This paper is concerned with the nonlinear impulsive delay differential equations with positive and negative coefficients

(*)

Sufficient conditions are obtained for every solution of equation (*) tending to a constant as t→∞.     

On the existence of positive periodic solutions for neutral functional differential equation with multiple deviating arguments

By means of the abstract continuation theory for k-contractions, some criteria are established for the existence and nonexistence of positive periodic solutions of the following neutral functional differential equation:

     

Numerical Hopf bifurcation of Runge–Kutta methods for a class of delay differential equations

In this paper, we consider the discretization of parameter-dependent delay differential equation of the form

It is shown that if the delay differential equation undergoes a Hopf bifurcation at τ=τ^*, then the discrete scheme undergoes a Hopf bifurcation at τ(h)=τ^*+O(h^p) for sufficiently small step size h, where p1 is the order of the Runge–Kutta method applied. The direction of numerical Hopf bifurcation and stability of bifurcating invariant curve are the same as that of delay differential equation.     

Oscillation criteria of second order half linear delay dynamic equations on time scales

In this paper, we establish some new oscillation criteria for a non autonomous second order delay dynamic equation

$${\left( {r\left( t \right)g\left( {{x^\Delta }\left( t \right)} \right)} \right)^\Delta } + p\left( t \right)f\left( {x\left( {\tau \left( t \right)} \right)} \right) = 0,$$

on a time scale T. Oscillation behavior of this equation is not studied before. Our results not only apply on differential equations when T=?, difference equations when T=? but can be applied on different types of time scales such as when T=q^? for q > 1 and also improve most previous results. Finally, we give some examples to illustrate our main results.

     

Oscillation of third order differential equation with damping term

We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term

$$x'''(t) + q(t)x'(t) + r(t)\left| x \right|^\lambda (t)\operatorname{sgn} x(t) = 0,{\text{ }}t \geqslant 0.$$

We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case λ ? 1 and if the corresponding second order differential equation h″ + q(t)h = 0 is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions.

     

On the optimal control of systems described by implicit evolution equations

We obtain conditions for the existence and uniqueness of an optimal control for the linear nonstationary operator-differential equation

$\frac{d}{{dt}}[A(t)y(t)] + B(t)y(t) = K(t)u(t) + f(t)$

with a quadratic performance criterion. The operators A(t) and B(t) are closed and may have nontrivial kernels. The results are applied to differential-algebraic equations and to partial differential equations that do not belong to the Cauchy-Kowalewskaya type.

     

Oscillation Criteria for Certain Fourth-Order Nonlinear Delay Differential Equations

In this article, we establish some new criteria for the oscillation of fourth-order nonlinear delay differential equations of the form

$$(r_2(t)(r_1(t)(y''(t))^\alpha)')' + p(t)(y''(t))^\alpha + q(t)f(y(g(t))) = 0$$

provided that the second-order equation

$$(r_2(t)z'(t))') + \frac{p(t)}{r_1(t)}z(t) = 0$$

is nonoscillatory or oscillatory.

     

Existence of a positive solution of a fourth-order boundary value problem

In this paper, the existence of a positive solution of the boundary value problem of the following fourth-order nonlinear differential equation:

is discussed.     

The modulus semigroup for linear delay equations III

In this paper, we describe the modulus semigroup of the C₀-semigroup associated with the linear differential equation with delay

in the Banach lattice X×L_p(-h,0;X), where X is a Banach lattice with order continuous norm. The progress with respect to previous papers is that A may be an unbounded generator of a C₀-semigroup possessing a modulus semigroup.     

On an approach to the analysis of asymptotic properties of solutions of first-order ordinary delay differential equations

For the first-order ordinary delay differential equation

$$u'(t) + p(t)u(r(t)) = 0,$$

where p ∈ L _loc(?₊; ?₊), τ ∈ C(?₊; ?₊), τ(t) ≤ t for t ∈ ?₊, lim_t→+∞ τ(t) = +∞, and ?₊:= [0, ∞), we obtain new criteria for the existence of sign-definite and oscillating solutions, thus generalizing some earlier-known results.

     

Existence of positive homoclinic solutions for damped differential equations

This paper is concerned with the existence of positive homoclinic solutions for the second-order differential equation

$$\begin{aligned} u^{\prime \prime }+cu^{\prime }-a(t)u+f(t,u)=0, \end{aligned}$$

where \(c\ge 0\) is a constant and the functions a and f are continuous and not necessarily periodic in t. Under other suitable assumptions on a and f, we obtain the existence of positive homoclinic solutions in both cases sub-quadratic and super-quadratic by using critical point theorems.

     

An approximating algorithm for the solution of an integral equation from epidemics

The following delay integral equation

$ x(t)=\int\limits_{t-\tau}^{t}f(s,x(s))ds,\quad t\in \mathbb{R}, $

has been proposed by Cooke and Kaplan to describe the spread of certain infectious diseases with periodic contact rate that varies seasonally. This mathematical model can also be interpreted as an evolution equation of a single species population. The purpose of this paper is to present an approximating algorithm for the continuous positive solution of this integral equation from the theory of epidemics. This algorithm is obtained by applying the successive approximations method and the rectangle formula, used for the calculation of the approximate value of integrals which appear in the right-hand-side of the terms of the sequence of successive approximations. In order to establish this approximating algorithm, we will suppose that this integral equation has a unique solution. The main result contains also the error of approximation of the solution obtained by applying this approximating algorithm.

     

Periodic Solutions of a Second-Order Functional Differential Equation with State-Dependent Argument

In this paper, we use Schauder and Banach fixed point theorem to study the existence, uniqueness and stability of periodic solutions of a class of iterative differential equation

$$\begin{aligned} c_0x''(t)+c_1x'(t)+c_2x(t)=x(p(t)+bx(t))+h(t). \end{aligned}$$

     

Existence of positive solutions for a general nonlinear eigenvalue problem

Let Ω R n be a bounded domain, H = L 2 (Ω), L : D(L) H → H be an unbounded linear operator, f ∈ C(■× R, R) and λ∈ R. The paper is concerned with the existence of positive solutions for the following nonlinear eigenvalue problem Lu = λf (x, u), u ∈ D(L), which is the general form of nonlinear eigenvalue problems for differential equations. We obtain the global structure of positive solutions, then we apply the results to some nonlinear eigenvalue problems for a second-order ordinary differential equation and a fourth-order beam equation, respectively. The discussion is based on the fixed point index theory in cones.     

Uniqueness of a positive radially symmetric Solution of the Dirichlet problem for certain nonlinear differential equation of the second order in a ball

We consider the Dirichlet problem

$u_\Gamma = 0$

for the nonlinear differential equation

$\Delta u + \left| x \right|^m \left| u \right|^p = 0, x \in S,$

with constant m ≥ 0 and p > 1 in the unit ball S = {x ∈ R ⁿ : |x| < 1}(n ≥ 3) with the boundary Γ. We prove that with p ≤ m+n/n?2 this problem has a unique positive radially symmetric solution.

     

On the Asymptotic Behavior of Nonoscillatory Solutions of Certain Fractional Differential Equations

We present the conditions under which every nonoscillator solution x(t) of the forced fractional differential equation

$$\begin{aligned} ^{\mathrm{C}}D_{\mathrm{c}}^{\alpha } y ( t ) = e ( t ) +f ( {t, x ( t )} ), c > 1,\alpha \in ( {0,1} ), \quad \mathrm{{and}} \,\, \delta \ge 1, \end{aligned}$$

where \(y(t)= ( {a(t) ( {{x}'(t)} )^{\delta }})^{\prime },c_0 =\frac{y(c)}{\Gamma (1)}= y(c)\), is a real constant which satisfies

$$\begin{aligned} |x(t)|=O\left( {t^{1/\delta }e^{t}\int _{\mathrm{c}}^{t} {a^{-1/\delta }} (s)\mathrm{d}s} \right) , \quad t \rightarrow \infty \end{aligned}$$

It is shown that the technique can be applied to some related fractional differential equations. Examples are inserted to illustrate the relevance of the obtained results.

     

On the stability of a quadratic Jensen type functional equation

In this paper we obtain the general solution of the quadratic Jensen type functional equation

and prove the stability of this equation in the spirit of Hyers, Ulam, Rassias, and G vruta.     

On the behavior of solutions to the Cauchy problem for a degenerate parabolic equation with inhomogeneous density and a source

The Cauchy problem for a degenerate parabolic equation with a source and inhomogeneous density of the form

$\rho (x)\frac{{\partial u}}{{\partial t}} = div(u^{m - 1} \left| {Du} \right|^{\lambda - 1} Du) + \rho (x)u^p $

is studied. Time global existence and nonexistence conditions are found for a solution to the Cauchy problem. Exact estimates of the solution are obtained in the case of global solvability.

     

Multiple solutions for a quasilinear second order differential equation depending on a parameter

The purpose of this paper is to use a very recent three critical points theorem due to Bonanno and Marano to establish the existence of at least three solutions for the quasilinear second order differential equation on a compact interval [a, b] ? R

$$\left( {\begin{array}{*{20}{c}}{t - 1} \\3 \end{array}} \right) {u\left( a \right)}$$

under appropriate hypotheses. We exhibit the existence of at least three (weak) solutions and, and the results are illustrated by examples.

     

Existence theorems for nonlinear differential equations having trichotomy in banach spaces

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation

$$\dot x\left( t \right) = \mathcal{L}\left( t \right)x\left( t \right) + f\left( {t,x\left( t \right)} \right),t \in \mathbb{R}$$

(P)

where {?(t): t ∈ R} is a family of linear operators from a Banach space E into itself and f: R × E → E. By L(E) we denote the space of linear operators from E into itself. Furthermore, for a < b and d > 0, we let C([?d, 0],E) be the Banach space of continuous functions from [?d, 0] into E and f ^d : [a, b] × C([?d, 0],E) → E. Let \(\hat {\mathcal{L}}:[a,b] \to L(E)\) be a strongly measurable and Bochner integrable operator on [a, b] and for t ∈ [a, b] define τ _t x(s) = x(t + s) for each s ∈ [?d, 0]. We prove that, under certain conditions, the differential equation with delay

$$\dot x\left( t \right) = \hat {\mathcal{L}}\left( t \right)x\left( t \right) + {f^d}\left( {t,{\tau _t}x} \right),ift \in \left[ {a,b} \right],$$

(Q)

has at least one weak solution and, under suitable assumptions, the differential equation (Q) has a solution. Next, under a generalization of the compactness assumptions, we show that the problem (Q) has a solution too.