We present some models of Cahn-Hilliard equations
based on a microforce balance proposed by M. Gurtin. We then study the existence and uniqueness of solutions.

Alain  Miranville

Generalized Cahn-Hilliard equations based on a microforce balance

With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of a delayed ratio-dependent predator-prey system with stage structure for predator. The approach involves some new technique of priori estimate. For the system without delay, by constructing a suitable Lyapunov function, some sufficient conditions which guarantee the existence of a unique global attractive positive periodic solution are obtained. Those results have further applications in population dynamics.

Fengde Chen

Periodicity in a ratio-dependent predator-prey system with stage structure for predator

Graceful labeling is one of the best known labeling methods of
graphs. Despite the large number of papers published on the
subject of graph labeling, there are few particular techniques to
 be used by researchers to gracefully label graphs. In this paper,
first a new approach based on the mathematical programming
technique is presented to model the graceful labeling problem.
Then a &#x93;branching method&#x94; is developed to solve the problem for
special classes of graphs. Computational results show the
efficiency of the proposed algorithm for different classes of
graphs. One of the interesting results of our model is in the
class of trees. The largest tree known to be graceful has at most
27 vertices but our model can easily solve the graceful labeling
for trees with 40 vertices.

Kourosh Eshghi

Parham Azimi

Applications of mathematical programming in graceful labeling of graphs

We consider a mathematical model which describes the contact
between a deformable body and an obstacle, the so-called
foundation. The body is assumed to have a viscoelastic behavior
that we model with the Kelvin-Voigt constitutive law. The contact
is frictionless and  is modeled with the well-known Signorini
condition in a form with a zero gap function. We present
two alternative yet equivalent weak formulations of the problem
and establish existence and uniqueness results for both
formulations. The proofs are based on a general result on
evolution equations with maximal monotone operators. We then
study a semi-discrete numerical scheme for the problem, in terms
of displacements. The numerical scheme has a unique solution. We
show the convergence of the scheme under the basic solution
regularity. Under appropriate regularity assumptions on the
solution, we also provide optimal order error estimates.

Mikäel Barboteu

Weimin Han

Mircea Sofonea

A frictionless contact problem for viscoelastic materials

We present a result on the averaging for functional differential equations on finite time intervals. The result is formulated in both classical mathematics and nonstandard analysis; its proof uses some methods of nonstandard analysis.

Mustapha  Lakrib

Time averaging for functional differential equations

We prove existence results for solutions of periodic boundary value problems concerning the <mml:math alttext="$n$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>th-order differential equation
with <mml:math alttext="$p$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-Laplacian <mml:math alttext="$[\phi (x^{(n-1)}(t))]'  =  f(t, x(t), x'(t),  \dotsc,\linebreak x^{(n-1)}(t))$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mi>&#966;</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow><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>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>'</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mn>...</mml:mn><mml:mo>,</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> and the boundary value conditions <mml:math alttext="$x^{(i)}(0)\!=\! x^{(i)}(T)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>i</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>i</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>T</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>, <mml:math alttext="$i= 0, \dotsc, n-1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>...</mml:mn><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:math>. Our method is based upon the coincidence degree theory of Mawhin. It is interesting that <mml:math alttext="$f$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi></mml:math> may be a polynomial and the degree of some variables among <mml:math alttext="$x_0, x_1, \dotsc, x_{n-1}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mn>...</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> in the function <mml:math alttext="$f(t, x_0, x_1, \dotsc, x_{n-1})$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mn>...</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> is allowed to be greater than <mml:math alttext="$1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn></mml:math>.

Yuji  Liu

Weigao  Ge

Periodic boundary value problems for <svg style="vertical-align:-0.1638pt" height="7.9499998" version="1.1" viewBox="0 0 8.6625004 7.9499998" width="8.6625004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g></svg>th-order ordinary differential equations with <svg style="vertical-align:-3.50804pt" height="13.3" version="1.1" viewBox="0 0 10.2 13.3" width="10.2" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,8.863)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36&#xA;t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g></svg>-laplacian

When the phase space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>P</mml:mi></mml:math>
of a Hamiltonian 
				<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math>-system <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>P</mml:mi><mml:mo>,</mml:mo><mml:mi>&#969;</mml:mi><mml:mo>,</mml:mo><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>J</mml:mi><mml:mo>,</mml:mo><mml:mi>H</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>
has an almost K&#228;hler structure a preferred
connection, called <em>abstract mechanical connection</em>, can
be defined by declaring horizontal spaces at each point to be
metric orthogonal to the tangent to the group orbit. Explicit
formulas for the corresponding connection one-form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#119964;</mml:mi></mml:math> are derived in terms of the momentum map,
symplectic and complex structures. Such connection can play the
role of the <em>reconstruction connection</em> (due to the work of A.
Blaom), thus
significantly simplifying computations of the corresponding
dynamic and geometric phases for an Abelian group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math>. These
ideas are illustrated using the example of the resonant
three-wave interaction. Explicit formulas for the connection
one-form and the phases are given together with some new results
on the symmetry reduction of the Poisson structure.

Sergey  Pekarsky

Jerrold E. Marsden

Abstract mechanical connection and abelian reconstruction for
almost K&#228;hler manifolds

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<p><em>Journal of Applied Mathematics</em> publishes original research papers and review articles in all areas of applied, computational, and industrial mathematics.</p>


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<p>Chief Editor, Professor Theodore E. Simos, is based at Ulyanovsk State Technical University, Russia. His main research interest is the numerical analysis of differential equations.</p>


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Journal of Applied Mathematics publishes original research papers and review articles in all areas of applied, computational, and industrial mathematics.

Journal of Applied Mathematics