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.     

Symmetry breaking of an elliptic equation in expanding annuli

We obtain non-radial bifurcation from radial solutions of a semilinear elliptic equation in expanding annuli of \(\mathbb {R}^N\). To obtain the main results, we use a blow-up argument via the Morse index of the regular entire solutions of the equation

$$\begin{aligned} -\Delta u=\lambda u^p \quad \text {in}\quad \mathbb {R}^N. \end{aligned}$$

(0.1)

The main results of this paper can be seen as applications of the properties of regular entire solutions of (0.1).

     

Asymptotic behavior of certain delay differential equations with forcing term

We study the asymptotic behavior of solutions of the following forced delay differential equation:

Some sufficient conditions that guarantee every solution of the equation to converge to zero are obtained. The results obtained are applied to some well-known delay differential ecological equations with forcing term.     

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:

     

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.

     

Optimization of the control by elastic boundary forces at two ends of a string in an arbitrarily large time interval

We further develop the method, devised earlier by the authors, which permits finding closed-form expressions for the optimal controls by elastic boundary forces applied at two ends, x = 0 and x = l, of a string. In a sufficiently large time T, the controls should take the string vibration process, described by a generalized solution u(x, t) of the wave equation

$$u_{tt} (x,t) - u_{tt} (x,t) = 0,$$

from an arbitrary initial state

$$\{ u(x,0) = \varphi (x), u_t (x,0) = \psi (x)$$

to an arbitrary terminal state

$$\{ u(x,T) = \hat \varphi (x), u_t (x,T) = \hat \psi (x).$$

     

THE EXISTENCE OF GLOBAL SOLUTION AND THE BLOWUP PROBLEM FOR SOME p-LAPLACE HEAT EQUATIONS

This article consider, for the following heat equation ut/|x|s-△pu=uq,(x,t)∈Ω×(0,T), u(x,t)=0,(x,t)∈(?)Ω×(0,T), u(x,0)=u0(x),u0(x)≥0,u0(x)(?)0 the existence of global solution under some conditions and give two sufficient conditions for the blow up of local solution in finite time, whereΩis a smooth bounded domain in RN(N>p),0∈Ω,△pu=div(|▽u|p-2▽u),0≤s≤2,p≥2,p-1相似文献   

The Vortex Filament Equation as a Pseudorandom Generator

In this paper, we consider the evolution of the so-called vortex filament equation (VFE),

$$\mathbf{X}_t = \mathbf{X}_s\wedge\mathbf{X}_{ss}, $$

taking a planar regular polygon of M sides as initial datum. We study VFE from a completely novel point of view: that of an evolution equation which yields a very good generator of pseudorandom numbers in a completely natural way. This essential randomness of VFE is in agreement with the randomness of the physical phenomena upon which it is based.

     

Operational Research Projects in Health and Welfare Services in the United Kingdom and Ireland

The paper presents a comparison of Health and Welfare O.R. in the U.K. and Ireland as represented by two registers of current projects, compiled in 1972 and 1977 respectively. The survey shows that

1. i)
   There appear to be very few O.R. studies of social services;
    
2. ii)
   The major focus has apparently moved away from hospital-based tactical studies, towards more broadly-based strategic studies;
    
3. iii)
   Related to this, the number of projects labelled as concerned with "planning" has increased;
    
4. iv)
   The number of projects involving "optimisation" has decreased, whilst those involving simulation have shown some increase; and
    
5. v)
   The majority of projects in both years were described as involving "applied common sense", rather than any specific technique.
    

     

Stability for non-hyperbolic fixed points of scalar difference equations

We give some new criteria to determine the stability of a non-hyperbolic fixed point of the scalar difference equation

where and f is a sufficient smooth function. Our results are based on higher order derivative at a fixed point of .     

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→∞.     

The coexistence of quasi-periodic and blow-up solutions in a superlinear Duffing equation

In this paper, we construct a continuous positive periodic function p(t) such that the corresponding superlinear Duffing equation x′′+ a(x)~(x2n+1)+p(t)x~(2m+1)= 0, n + 2≤2 m+12n+1 possesses a solution which escapes to infinity in some finite time, and also has infinitely many subharmonic and quasi-periodic solutions, where the coefficient a(x) is an arbitrary positive smooth periodic function defined in the whole real axis.     

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.     

On a class of equations stemming from various quadrature rules

We deal with the functional equation

$F(y)-F(x)=(y-x)\sum_{i=1}^n a_if(\lambda_ix+(1-\lambda_i)y),\quad x,y\in \mathbb {R}$

motivated by quadrature rules of approximate integration. In previous results the solutions of this equation were found only in some particular cases. For example the coefficients λ _i were supposed to be rational or the equation in question was solved only for n=2. In the current paper we do not assume any particular form of coefficients occurring in this equation and we allow n to be any positive integer. Moreover, we obtain a solution of our equation without any regularity assumptions concerning the functions f and F.

     

Boundedness character of a fourth order nonlinear difference equation

In this paper, among others, we give a complete picture of the boundedness character of the positive solutions of the following nonlinear difference equation

where the parameters A and p are positive real numbers. This is a natural extension of a difference equation equivalent to an equation arising in automatic control theory. We present some new methods for investigating the boundedness character of nonlinear difference equations.     

Superstability of the Cauchy equation with squares in finite-dimensional normed algebras

Our purpose is to provide an affirmative answer to Moszner’s problem [cf. Moszner (Ann Univ Paed Crac Stud Math XI:69–94, 2012), p. 93] concerning the superstability of the Cauchy equation with squares

$$f(x + y)^2 = (f(x) + f(y))^2$$

in the class of functions mapping an Abelian semigroup into a finite-dimensional normed algebra without divisors of zero.

     

A Production Planning and Stock Control Problem in the Textile Industry

This paper describes the techniques used in an enquiry into production planning and stock holding problems encountered by one of our member firms.The investigation of the problem falls into three parts:

1. i)
   The analysis of demand.
    
2. ii)
   The determination of optimum stock levels.
    
3. iii)
   The solution of the allocation problem which arises if the capacity available for the production of stock lines is not equal to the amount required to bring stocks back to the optimum levels. This problem has not been fully investigated, but a proposed method of solution is given in an appendix.
    

     

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.