A class of linear difference equations with variable coefficients is 
considered. Sufficient conditions and necessary conditions for the oscillation of 
the solutions are established. In the special cases where the coefficients are 
constant or periodic the conditions become both necessary and sufficient.

Ch. G. Philos

I. K. Purnaras

Oscillations in linear difference equations with variable
coefficients

In this paper, a generalized dynamical theory of thermoelasticity 
is employed to study disturbances in an infinite elastic solid containing a 
spherical cavity which is subjected to step rise in temperature in its inner 
boundary and an impulsive dynamic pressure on its surface. The 
problem is solved by the use of the Laplace transform on time. The 
short time approximations for the stress, displacement and temperature 
are obtained to examine their discontinuities at the respective 
wavefronts. It is shown that the instantaneous change in pressure and 
temperature at the cavity wall gives rise to elastic and thermal 
disturbances which travel with finite velocities <mml:math alttext="$v_1 $" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>v</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> and <mml:math alttext="$v_2 \left( { &gt; v_1 } \right)$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>v</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>&gt;</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> 
respectively. The stress, displacement and temperature are found to 
experience discontinuities at the respective wavefronts. One of the 
significant findings of the present analysis is that there is no diffusive 
nature of the waves as found in classical theory.

Basudeb Mukhopadhyay

Rasajit Bera

Lokenath Debnath

On generalized thermoelastic disturbances in an elastic solid with a spherical cavity

Differential equations of the form <mml:math alttext="$y' = f\left( {t,y,y'} \right)$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> where <mml:math alttext="$f$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math> is not 
necessarily linear in its arguments represent certain physical phenomena and 
are known for quite some time. The well known Clairut's and Chrystal's 
equations fall into this category. Earlier we established the existence of a 
(unique) solution of the nonstandard initial value problem <mml:math alttext="$y' = f\left( {t,y,y'} \right)$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>, <mml:math alttext="$y\left( {t_0 } \right) = y_0 $" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>y</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> under certain natural hypotheses on <mml:math alttext="$f$" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math>. In this paper, 
we studied the stability of solutions of a nonstandard first order ordinary
differential system.

M. Venkatesulu

P. D. N. Srinivasu

Stability of solutions of a nonstandard ordinary differential
system by Lyapunov's second method

The author studies the queueing process in a single-server bulk queueing system. Upon completion of a previous service, the server can take a group 
of random size from customers that are available. Or, the server can wait until 
the queue attains a desired level.
The author establishes an ergodicity criterion for both the queueing 
process with continuous time parameter and the imbedded process. Under this 
criterion, the author obtains explicit formulas for the stationary distributions 
of both processes by using semi-regenerative techniques.

Jewgeni H. Dshalalow

A single-server queue with random accumulation level

In this paper we consider the questions of existence and 
uniqueness of solutions of certain semilinear and quasilinear evolution 
equations on Banach space. We consider both deterministic and 
stochastic systems. The approach is based on semigroup theory and 
fixed point theorems. Our results allow the nonlinear perturbations in all 
the semilinear problems to be bounded or unbounded with reference to 
the base space, thereby increasing the scope for applications to partial 
differential equations. Further, quasilinear stochastic evolution equations 
seemingly have never been considered in the literature.

N. U. Ahmed

Nonlinear evolution equations on Banach space

Denote by <mml:math alttext="$S_n $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>S</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> the set of all distinct rooted trees with <mml:math alttext="$n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math> labeled 
vertices. A tree is chosen at random in the set <mml:math alttext="$S_n $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>S</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math>, assuming that all the 
possible <mml:math alttext="$n^{n - 1} $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> choices are equally probable. Define <mml:math alttext="$\tau _n \left( m \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#964;</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> as the number 
of vertices in layer <mml:math alttext="$m$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math>, that is, the number of vertices at a distance <mml:math alttext="$m$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math> 
from the root of the tree. The distance of a vertex from the root is the 
number of edges in the path from the vertex to the root. This paper is 
concerned with the distribution and the moments of <mml:math alttext="$\tau _n \left( m \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#964;</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> and their 
asymptotic behavior in the case where <mml:math alttext="$m = \left[ {2\alpha \sqrt n } \right]$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>&#945;</mml:mi><mml:msqrt><mml:mi>n</mml:mi></mml:msqrt></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:math>, <mml:math alttext="$0 &#60; \alpha  &#60; \infty $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo>&#60;</mml:mo><mml:mi>&#945;</mml:mi><mml:mo>&#60;</mml:mo><mml:mi>&#8734;</mml:mi></mml:math> and <mml:math alttext="$n \to \infty $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi><mml:mo>&#8594;</mml:mo><mml:mi>&#8734;</mml:mi></mml:math>. 
In addition, more random trees, branching processes, the Bernoulli 
excursion and the Brownian excursion are also considered.

Lajos  Tak&#225;cs

On the distribution of the number of vertices in layers of random trees

This paper is concerned with the generalization, numerical 
implementation and testing of the non-parameter penalty function algorithm 
which was initially developed for solving <mml:math alttext="$n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-dimensional optimization problems. 
It uses this method to transform a constrained optimal control problem into a 
sequence of unconstrained optimal control problems. It is shown that the 
solutions to the original constrained problem. Convergence results are proved 
both theoretically and numerically.

An-Qing  Xing

The non-parameter penalty function method in constrained optimal control problems

A new simple proof of existence and uniqueness of solutions of the 
Volterra integral equation in Lebesque spaces is given. It is shown that the 
weighted norm technique and the Banach contraction mapping principle can 
be applied (as in the case of continuous functions space).

Jaroslaw  Kwapisz

Weighted norms and Volterra integral equations in <svg style="vertical-align:-0.0pt" height="16.012501" version="1.1" viewBox="0 0 18.025 16.012501" width="18.025" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,15.95)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z" /></g><g transform="matrix(.012,-0,0,-.012,9.763,7.8)"><path id="x1D443" d="M619 482q0 -69 -40.5 -119t-96 -72.5t-118.5 -26.5h-44l-70 20l-31 -151q-14 -67 0.5 -83t89.5 -22l-5 -28h-287l8 28q65 7 81.5 22t29.5 83l79 398q12 56 0.5 70.5t-78.5 20.5l7 28h255q108 0 164 -43t56 -125zM524 478q0 141 -146 141q-25 0 -47 -8q-16 -6 -20.5 -13.5&#xA;t-10.5 -39.5l-43 -241q37 -13 83 -13q67 0 125.5 45t58.5 129z" /></g></svg> spaces

Using distributions theory technique we introduce parabolic 
potentials for the heat equation with one time-dependent coefficient 
(not everywhere positive and continuous) at the highest space-derivative, discuss their properties, and apply obtained results to three illustrative problems. Presented technique allows to deal with some equation 
of the degenerate/mixed type.

Igor  Malyshev

On the parabolic potentials in degenerate-type heat equation

In this paper, the existence of periodic solutions of impulsive 
differential systems is considered. Since the solutions of such a system 
are peicewise continuous, it is necessary to introduce piecewise 
continuous Lyapunov functions. By means of such functions, together 
with the comparison principle, some sufficient conditions for the 
existence of periodic solutions of impulsive differential systems are 
established.

L. H. Erbe

Xinzhi  Liu

Existence of periodic solutions of impulsive differential systems

Let <mml:math alttext="$F_n \left( x \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>F</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> and <mml:math alttext="$G_n \left( x \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> be the empirical distribution functions of two 
independent samples, each of size <mml:math alttext="$n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>, in the case where the elements of the 
samples are independent random variables, each having the same continuous 
distribution function <mml:math alttext="$V\left( x \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>V</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> over the interval <mml:math alttext="$\left( {0,1} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>. Define a statistic <mml:math alttext="$\theta _n $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#952;</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> by
<mml:math display="block" alttext="${{\theta _n } \mathord{\left/  {\vphantom {{\theta _n } n}} \right.  \kern-\nulldelimiterspace} n} = \int_0^1 {\left[ {F_n \left( x \right) - G_n \left( x \right)} \right]dV\left( x \right) - \mathop {\min }\limits_{0 \leqslant x \leqslant 1} \left[ {F_n \left( x \right) - G_n \left( x \right)} \right]} $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:msub><mml:mi>&#952;</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mo>&#8747;</mml:mo><mml:mn>0</mml:mn><mml:mn>1</mml:mn></mml:msubsup><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#8722;</mml:mo><mml:msub><mml:mi>G</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mi>V</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#8722;</mml:mo><mml:munder><mml:mrow><mml:mo>min</mml:mo></mml:mrow><mml:mrow><mml:mn>0</mml:mn><mml:mo>&#8804;</mml:mo><mml:mi>x</mml:mi><mml:mo>&#8804;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munder><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#8722;</mml:mo><mml:msub><mml:mi>G</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:math>.
In this paper the limits of <mml:math alttext="$E\left\{ {\left( {{{\theta _n } \mathord{\left/  {\vphantom {{\theta _n } {\sqrt {2n} }}} \right.  \kern-\nulldelimiterspace} {\sqrt {2n} }}} \right)^r } \right\}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>E</mml:mi><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi>&#952;</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msqrt></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>r</mml:mi></mml:msup></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math><mml:math alttext="$\left( {r = 0,1,2, \ldots } \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>&#8230;</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> and <mml:math alttext="$P\left\{ {{{\theta _n } \mathord{\left/  {\vphantom {{\theta _n } {\sqrt {2n} }}} \right.  \kern-\nulldelimiterspace} {\sqrt {2n} }} \leqslant x} \right\}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>P</mml:mi><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi>&#952;</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msqrt></mml:mrow></mml:mrow><mml:mo>&#8804;</mml:mo><mml:mi>x</mml:mi></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math>
are determined for <mml:math alttext="$n \to \infty $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi><mml:mo>&#8594;</mml:mo><mml:mi>&#8734;</mml:mi></mml:math>. The problem of finding the asymptotic behavior of 
the moments and the distribution of <mml:math alttext="$\theta _n $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#952;</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> as <mml:math alttext="$n \to \infty $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi><mml:mo>&#8594;</mml:mo><mml:mi>&#8734;</mml:mi></mml:math> has arisen in a study of the 
fluctuations of the inventory of locomotives in a randomly chosen railway 
depot.

On a probability problem connected with Railway traffic

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International Journal of Stochastic Analysis - Table of contents 1991

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International Journal of Stochastic Analysis

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International Journal of Stochastic Analysis