A Theorem involving the convergence of random functions is presented. The Theorem is
based upon the Arzela-Ascoli Theorem of Real Analysis. The development here may be useful in
transient or asymptotic analysis of random functions and linear systems theory.

Joseph P. Noonan

Henry M. Polchlopek

An Arzela-Ascoli type theorem for random functions

The <mml:math alttext="${\mathrm{EM}}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>EM</mml:mtext></mml:math> fields of charged particles moving with velocity <mml:math alttext="${v}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>v</mml:mi></mml:math> in a physical vacuum with
wave carrier (substratum) are determined by means of the generalized, Galilei covariant Maxwell
equations for inertial frames <mml:math alttext="$\sum $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#8721;</mml:mo></mml:math>. with substratum flow <mml:math alttext="${w}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>w</mml:mi></mml:math>. In this Galilean approach, all velocities
have absolute meaning relative to the substratum rest frame <mml:math alttext="$\sum ^o $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mo>&#8721;</mml:mo><mml:mi>o</mml:mi></mml:msup></mml:math>, and the relative velocity of material
particles is given by the linear Galilean relation <mml:math alttext="${v}_{G} = {v}_{1} - {v}_{2} $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>v</mml:mi><mml:mi>G</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mi>1</mml:mi></mml:msub><mml:mo>&#8722;</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mi>2</mml:mi></mml:msub></mml:math>, permitting in principle superluminal
relative velocities <mml:math alttext="$\left| {{v}_{G} } \right| &#62; c_o $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>G</mml:mi></mml:msub></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>&#62;</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mi>o</mml:mi></mml:msub></mml:math>. Inter alias, the possibility of <mml:math alttext="${\mathrm{EM}}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>EM</mml:mtext></mml:math> shock waves and Cerenkov radiation
in the vacuum substratum is discussed. Experiments are proposed to test the theoretical
predictions.

H. E. Wilhelm

Shock waves and Cerenkov radiation in electromagnetic substratum

In the present paper it is proved that the ordinary dichotomy is preserved under
perturbations of the coefficient matrix of the linear impulsive differential equation.

N. V. Milev

D. D. Bainov

Ordinary dichotomy and perturbations of the coefficient matrix of the linear impulsive differential equation

The Stepanov almost periodic solution of a certain second-order differential equation in a reflexive Banach space is shown to be almost periodic.

Aribindi Satyanarayan Rao

On the stepanov almost periodic solution of a second-order infinitesimal generator differential equation

Conditions are given for the continuity and differentiability of solutions of initial value problems
and boundary value problems for the <mml:math alttext="$n^{th} $" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math> order finite difference equation, <mml:math alttext="$u\left( {m + n} \right) = f\left( {m,u\left( m \right),u\left( {m + 1} \right), \ldots ,u\left( {m + n - 1} \right)} \right),m \in \mathbb{Z}$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo><mml:mo>,</mml:mo><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mi>n</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo>&#x2208;</mml:mo><mml:mi>&#x2124;</mml:mi></mml:math>.

Johnny Henderson

Linda Lee

Continuous dependence and differentiation of solutions of finite difference equations

Let <mml:math alttext="$P\left[ {A,B} \right]$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>P</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>B</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:math>, <mml:math alttext="$ - 1 \leqslant B &#x3C; A \leqslant 1$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo>&#x2264;</mml:mo><mml:mi>B</mml:mi><mml:mo>&#x3C;</mml:mo><mml:mi>A</mml:mi><mml:mo>&#x2264;</mml:mo><mml:mn>1</mml:mn></mml:math>, be the class of functions <mml:math alttext="$p$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> such that <mml:math alttext="$p\left( z \right)$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> is subordinate to
<mml:math alttext="$\tfrac{{1 + Az}} {{1 + Bz}}$" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mstyle scriptlevel="1"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>A</mml:mi><mml:mi>z</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>B</mml:mi><mml:mi>z</mml:mi></mml:mrow></mml:mfrac></mml:mstyle></mml:math>. Let <mml:math alttext="$P\left( {\alpha _1 } \right)$" id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>P</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>&#x3B1;</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> be the class of functions with positive real part greater than <mml:math alttext="$\alpha _1 $" id="E7" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#x3B1;</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math>, <mml:math alttext="$0 \leqslant \alpha _1 \leqslant 1$" id="E8" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo>&#x2264;</mml:mo><mml:msub><mml:mi>&#x3B1;</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#x2264;</mml:mo><mml:mn>1</mml:mn></mml:math>. It is clear that 
<mml:math alttext="$p\left[ {A,B} \right] \subset P\left( {\tfrac{{1 - A}} {{1 - B}}} \right) \subset P\left[ {1, - 1} \right]$" id="E9" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>B</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>&#x2282;</mml:mo><mml:mi>P</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle scriptlevel="1"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>B</mml:mi></mml:mrow></mml:mfrac></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x2282;</mml:mo><mml:mi>P</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:math>. The principal results in this paper are the determination of the radius of
<mml:math alttext="$\beta $" id="E10" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x3B2;</mml:mi></mml:math>-starlikeness and <mml:math alttext="$\beta $" id="E11" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x3B2;</mml:mi></mml:math>-convexity of <mml:math alttext="$f\left( z \right)$" id="E12" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> with <mml:math alttext="$\beta = \tfrac{{1 - A}} {{1 - B}}$" id="E13" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x3B2;</mml:mi><mml:mo>=</mml:mo><mml:mstyle scriptlevel="1"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>B</mml:mi></mml:mrow></mml:mfrac></mml:mstyle></mml:math>, when <mml:math alttext="$f\left( z \right)$" id="E14" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> is restricted to certain classes of univalent
and analytic functions related vith <mml:math alttext="$P\left[ {A,B} \right]$" id="E15" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>P</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>B</mml:mi></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:math>.

Khalida Inayat Noor

On radii of convexity and starlikeness of some classes of analytic functions

We have used the idea of &#8216;quasi inner product&#8217; introduced by L. R.
Bragg in 1986 to consider generating series
<mml:math display="block" alttext="\[ \sum\limits_{n = 0}^\infty {\frac{{H_n^2 \left( x \right)H_n^2 \left( y \right)t^n }} {{2^{2n} \left( {n!} \right)^2 }}} \]" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:munderover><mml:mo>&#8721;</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>&#8734;</mml:mi></mml:munderover><mml:mrow><mml:mfrac><mml:mrow><mml:msubsup><mml:mi>H</mml:mi><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msubsup><mml:mi>H</mml:mi><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>y</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mi>t</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>!</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:math>
studied by L. Carlitz in 1963. The pecularity of the series is that there is <mml:math alttext="$\left( {n!} \right)^2 $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>!</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math>
in the denominator, which has a striking deviation from the usuaI generating series
containing <mml:math alttext="$n!$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi><mml:mo>!</mml:mo></mml:math> in the denominator. Our generating function for the said generating
series is quite different from that of Carlitz, but somewhat analogous to generating
integrals derived by G. N. Watson (Higher Transcendental function Vol.III, P 271-272
for the case of Legendre, Gegenbauer and Jacobi polynomials.

S. K. Chatterjea

S. M. Eaqub Ali

Some formulas of L. Carlitz on Hermite polynomials

The system <mml:math alttext="$x' = Ax + f\left( x \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>x</mml:mi><mml:mo>&#8242;</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mi>A</mml:mi><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> of nonlinear vector differential equations, where
the nonlinear term <mml:math alttext="$f\left( x \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> is quadratic with orthogonality property <mml:math alttext="$x^T f\left( x \right) = 0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>x</mml:mi><mml:mi>T</mml:mi></mml:msup><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math> for all <mml:math alttext="$x$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>x</mml:mi></mml:math>, is
point-dissipative if <mml:math alttext="$u^T Au &#60; 0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>u</mml:mi><mml:mi>T</mml:mi></mml:msup><mml:mi>A</mml:mi><mml:mi>u</mml:mi><mml:mo>&#60;</mml:mo><mml:mn>0</mml:mn></mml:math> for all nontrivial zeros <mml:math alttext="$u$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>u</mml:mi></mml:math> of <mml:math alttext="$f\left( x \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>.

Anile K. Bose

Alan S. Cover

James A. Reneke

On point-dissipative systems of differential equations with quadratic nonlinearity

This paper provides a survey of iteration procedures that have been used to
obtain fixed points for maps satisfying a variety of contractive conditions. The author does
not claim to provide complete coverage of the literature, and admits to certain biases in the
theorems that are cited herein. In spite of these shortcomings, however, this paper should be
a useful reference for those persons wishing to become better acquainted with the area.

B. E. Rhoades

Some fixed point iteration procedures

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International Journal of Mathematics and Mathematical Sciences publishes research across all fields of mathematics and mathematical sciences, such as pure and applied mathematics, mathematical physics, probability and mathematical statistics.

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International Journal of Mathematics and Mathematical Sciences