Du Dt Research Articles

Permanence and existence of periodic solutions for a generalized system with feedback control

Sufficient conditions are obtained for the permanence and the existence of positive periodic solutions of the delay differential system with feedback control (0.1) dx dt = x ( t ) [ F ( t , x ( t - τ 1 ( t ) ) , … , x ( t - τ n ( t ) ) ) - c ( t ) u ( t - δ ( t ) ) ] , du dt = - η ( t ) u ( t ) + a ( t ) x ( t - σ ( t ) ) . The method involves the application of estimation for uniform upper and lower bounds of solutions. When these results are applied to some special population models with multiple delays, some new results are obtained and some known results are generalized. Especially, our conclusions generalize and complement the results in Chen et al. [F.D. Chen, J.H. Yang, L.J. Chen, X.D. Xie, On a mutualism model with feedback controls, Appl. Math. Comput. 214 (2009) 581–587] and Huo and Li [H.F. Huo, W.T. Li, Positive periodic solutions of a class of delay differential system with feedback control, Appl. Math. Comput. 148 (2004) 35–46].

A note on a nonlinear functional differential system with feedback control

In this note we apply Avery-Peterson multiple fixed point theorem to investigate the existence of multiple positive periodic solutions to the following non- linear non-autonomous functional differential system with feedback control ( dx dt = r(t)x(t) +F(t,xt,u(t �(t))), du dt = h(t)u(t) +g(t)x(t �(t)). We prove the system above admits at least three positive periodic solutions under certain growth conditions imposed on F.

The stabilizing contribution of thymine in duplexes of (dA)24 with (dU)24, (dT)24, (dU12–dT12), (dU–dT)12, (dU2–dT2)6, or (dU3–dT3)4: Nearest neighbor and next‐nearest neighbor effects

Ultraviolet melting curves are used to determine the effect of the pyrimidine 5-methyl group on the stability of duplexes of (dA)(24) with (dU)(24), (dT)(24), (dU(12)-dT(12)), (dU-dT)(12), (dU(2)-dT(2))(6), and (dU(3)-dT(3))(4). Substitution of a T for a U results in an increase in stability, which is attributed to an increase in strength of dipole-induced dipole and dispersion (van der Waals) interactions. Significant additional enhancement occurs when two T residues are adjacent. A further increase in the number of adjacent T's has a relatively slight effect on T(m). The sequence effect appears to be largely attributable to an increment in dispersion forces. The CD spectra of the duplexes are all closely similar except in the region between 260 and 290 nm. A band near 272 nm associated with the presence of U in the spectrum of (dA)(24).(dU)(24) decreases in intensity when T's are incorporated in the pyrimidine strand. The band is completely replaced in the spectrum of (dA)(24).(dT)(24) with a new maximum at 282 nm and a minimum at 268 nm, both of lower magnitude. The emergence of the two new bands is correlated with the presence of adjacent T's once more, and only two adjacent T's appear necessary for a major part of the change to occur. The degree of cation release on thermal dissociation of the oligomer dimers ranges from Deltai = 0.14 to 0.16, about the same or slightly less than values reported for polynucleotide duplexes and less than predicted from theoretical calculations.

Multiple positive periodic solutions of nonlinear functional differential system with feedback control

In this paper, we employ Avery–Henderson fixed point theorem to study the existence of positive periodic solutions to the following nonlinear nonautonomous functional differential system with feedback control: dx dt =−r(t)x(t)+F(t,x t,u(t−δ(t))), du dt =−h(t)u(t)+g(t)x(t−σ(t)). We show that the system above has at least two positive periodic solutions under certain growth condition imposed on F.

Stability of θ-methods for delay integro-differential equations

Stability of θ-methods for delay integro-differential equations (DIDEs) is studied on the basis of the linear equation du dt =λu(t)+μu(t−τ)+κ ∫ t−τ t u(σ) dσ, where λ, μ, κ are complex numbers and τ is a constant delay. It is shown that every A-stable θ-method possesses a similar stability property to P-stability, i.e., the method preserves the delay-independent stability of the exact solution under the condition that κ is real and τ/ h is an integer, where h is a step-size. It is also shown that the method does not possess the same property if τ/ h is not an integer. As a result, no θ-method can possess a similar stability property to GP-stability with respect to DIDEs.

Functional evolution equations

Of concern is the functional evolution equation du dt ∈Bu+F(t,u t), t>0,u 0=φ(s)∈C 1 [−r,0];X , where u t ( s)= u( t+ s), r>0, s∈[− r,0], and X is a real Banach space. It is shown by the method of lines combined with the Crandall–Pazy theorem, that for the initial data u 0 in a generalized domain, this equation has a limit solution, which is Lipschitz continuous in t, and that this limit solution is a unique strong one if further assumptions on B are imposed, and that the zero solution is asymptotically stable. An application is given to a class of partial functional differential boundary value problems.

Energy bounds for some nonstandard problems in partial differential equations

We consider problems of the form d 2u dt 2 +Au=F, αu(0)+u(T)=g, β du dt (0)+ du dt (T)=h, for t∈(0, T), where A is a densely defined, linear, time independent, positive definite symmetric operator and α and β are constants. Although most of our results would hold for more general operators A, we restrict attention to the case in which A is a differential operator and determine ranges of values of α and β for which it is possible to obtain energy bounds, uniqueness results, and, in a special case, pointwise bounds. Some extensions which include a damping term or a term which arises in a generalization of the Kirchhoff string model are also discussed.

Stability of Runge–Kutta methods for delay integro-differential equations

We study stability of Runge–Kutta (RK) methods for delay integro-differential equations with a constant delay on the basis of the linear equation du dt =Lu(t)+Mu(t−τ)+K ∫ t−τ t u(θ) dθ, where L, M, K are constant complex matrices. In particular, we show that the same result as in the case K=0 (Koto, BIT 34 (1994) 262–267) holds for this test equation, i.e., every A-stable RK method preserves the delay-independent stability of the exact solution whenever a step-size of the form h= τ/ m is used, where m is a positive integer.

Viability for Semilinear Differential Inclusions via the Weak Sequential Tangency Condition

Let X be a reflexive and separable Banach space, A: D( A) ⊂ X → X the generator of a C 0-semigroup S( t) : X → X, t ≥ 0, D a locally weakly sequentially closed subset in X, and F: D → 2 X a nonempty, closed, convex, and bounded valued mapping which is weakly-weakly upper semi-continuous. The main result of the paper is: Theorem . Under the general assumptions above a necessary and sufficient condition in order that for each ξ ∈ D there exists at least one mild solution u of du dt t ∈ Au t + F u t satisfying u(0) = ξ is the so-called “weak sequential tangency condition” below. ( W S T C ) For each ξ ∈ D there exists y ∈ F(ξ) and two sequences ( t n ) n ∈ N in R + and ( p n ) n ∈ N in X such that t n → 0, p n → 0 and satisfying S( t n )ξ + t n ( y + p n ) ∈ D.

A new approach to evolution

Let B be a Banach space, and ϕ 0,ϕ 1:B→ ]−∞,+∞] be two given functions. A generalized solution of the equation ∇ϕ 1( du dt )+∇ϕ 0(u)=0 in B′ is introduced via the search for a minimal element of a certain partial order relation on the trajectories. This is based on the use of the evolution potential [Φ(u)](t):= ∫ 0 t ϕ 1( du dτ (τ)) dτ+ϕ 0(u(t)) . This approach is extended to higher order ODEs and PDEs.

A modification of Adomian's solution for nonlinear oscillatory systems

In this paper, we consider nonlinear oscillatory equations of the form d 2u dt 2 + ϖ 2u = εf(u, du dt ) , whose solutions can be obtained by the decomposition method. But these solutions do not exhibit periodicity, which is characteristic of oscillatory systems. We use an alternative technique by which the solution obtained by the decomposition method is made periodic. The method is described and illustrated with examples. There is no smallness assumption on the parameter ε occuring in the equation.

Explicit energy conservative difference schemes for nonlinear dynamical systems with at most quartic potentials

We report a conservative difference scheme for nonlinear dynamical systems with energy of the form H = ( du dt ) 2 2+V(u) . If the potential V(u) is a polynomial of degree four or less, then the scheme is explicit and identical with McMillan's integrable mapping of the plane. We extend the method to systems of several variables. We also discuss whether or not the schemes are symplectic.

Dirichlet operators: A priori estimates and the uniqueness problem

We study the equation du dt + Hu = 0 , where H is the operator associated with the Dirichlet form of quasi-invariant measure on Hilbert space. A priori estimates of the first and second order derivatives of the solutions of the equation are obtained. Then these estimates are applied to the problem of essential self-adjointness of the operator H on domains which consist of smooth functions.

An analytical solution for a nonlinear differential equation with logarithmic decay

The problem is the ordinary differential equation du dt = − p exp( − q u ) with u(0) = α . We prove that the general three-parameter problem may be reduced through a group invariance to computing a single, parameter-free function w( t). This solution may be most compactly expressed in the form w = 1 ln (t∗ ln 2 (t∗)B(τ)) , where τ ≡ ln( ln(t∗)) and t∗ = t + exp(1) . We compute a Chebyshev series for B( τ) for τϵ [0,6] and show that B( τ) ~ 1 + (4 τ −2) exp(− τ) + 4 τ 2exp(-2 τ) to within 1 part in 10 4 for τ > 6. The problem was motivated by the similar logarithmic decay of certain classes of quasi-solitary waves, such as the “breather” of the φ 4 field theory.

A simple almost-periodicity criterion and applications

We introduce a new almost-periodicity criterion for functions: R → X with X a complete metric space. This result is used to establish asymptotic almost periodicity of precompact positive trajectories to some differential equations of the form du dt + A(t) u(t) ∋ 0, where A(t) is periodic with respect to t ϵ R.

Asymptotic behavior for two-dimensional, quasi-autonomous, almost-periodic evolution equations

Let A be a maximal monotone operator in R 2 and f ( t ): R → R 2 any S 1 -almostperiodic function. If u is a solution of the evolution equation du dt + Au(t) ∋ f(t) which is bounded on R + , then u ( t ): R + → R 2 is asymptotically almost periodic as t → + ∞.

Bifurcation at multiple eigenvalues and stability of bifurcating solutions

If K is a bounded linear operator from the real Banach space U into the real Banach space V and ƒ:U× R → V has the value zero at (0, 0), the existence and linear stability of the equilibrium solutions of the dynamical system K du dt = ƒ(u, α) which are close to the origin in U× R are studied. It is assumed that ƒ u(0, 0): U → V is a Freholm operator of index zero. The only restriction on the dimension of the null space of ƒ u(0, 0) and the order of vanishing, at (0, 0), of ƒ restricted to the null space of Dƒ(0,0): U× R → V, is that they both be finite positive integers. The main result gives conditions under which the equation, which determines the equilibrium solutions in a neighborhood of the origin, also determines the stability of these equilibrium solutions.

Asymptotic behavior of trajectories for some nonautonomous, almost periodic processes

In this paper, asymptotics are studied for some almost periodic processes on a complete metric space ( X, d): (1) It is shown that any precompact positive trajectory of a contractive periodic process is asymptotically almost periodic as t → +∞. This property does not hold for general almost periodic contractive processes. (2) A compactness result is obtained for weakly almost periodic complete trajectories of some (possibly nonlinear) processes in a uniformly convex Banach space. (3) The existence of almost periodic trajectories is studied for “affine” processes in a uniformly convex Banach space. These results are applicable to some evolution equations of the form du dt + A (t) u(t) ϵ f(t) , where ƒ(t) is almost periodic: R → V uniformly convex Banach space and A( t) is a periodic, time-dependent, m-accretive operator in V.

Generalized almost periodic solutions and ergodic properties of quasi-autonomous dissipative systems

Let H be a real Hilbert space, A a maximal monotone operator in H and ƒ R → H a measurable function which is S 1-almost periodic. Assuming that the positive trajectories of the equation du dt + Au(t)∋ f(t) are bounded (in H) for t ⩾ 0, we construct a kind of generalized almost periodic “solution” of the equation, and we show how to deduce information of ergodic type on the asymptotic behavior of the trajectories as t → +∞.

Regularizing effects for ut + Aϑ( u) = 0 in L1

Various initial-boundary value problems and Cauchy problems can be written in the form du dt + Aϑ(u) = 0 , where ϑ: R→ R is nondecreasing and A is the linear generator of strongly continuous nonexpansive semigroup e − tA in an L 1 space. For example, if A = − Δ (subject, perhaps, to suitable boundary conditions) we obtain equations arising in flow in a porous medium or plasma physics (depending on the choice of ϑ) while if A = ∂ ∂x acting in L 1( R ) we have a scalar conservation law. In this paper we show that if M, m > 0 and mϑ′ 2 ⩽ νϑϑ′' ⩽ Mϑ′ 2, where ν ϵ {1,−1}, then (roughly speaking), the norm of t du dt may be estimated in terms of the initial data u 0 in L 1. Such estimates give information about the regularity of solutions, asymptotic behaviour, etc., in applications. Side issues, such as the introduction of sufficiently regular approximate problems on which estimates can be made and the assignment of a precise meaning to the operator Aϑ, are also dealt with. These considerations are of independent interest.