Expand all Collapse all | Results 1 - 25 of 33 |
1. CMB Online first
Orbits of Geometric Descent We prove that quasiconvex functions always admit descent trajectories
bypassing all non-minimizing critical points.
Keywords:differential inclusion, quasiconvex function, self-contracted curve, sweeping process Categories:34A60, 49J99 |
2. CMB 2011 (vol 56 pp. 39)
Comparison Theorem for Conjugate Points of a Fourth-order Linear Differential Equation In 1961, J. Barrett showed that if the first conjugate point
$\eta_1(a)$ exists for the differential equation $(r(x)y'')''=
p(x)y,$ where $r(x)\gt 0$ and $p(x)\gt 0$, then so does the first
systems-conjugate point $\widehat\eta_1(a)$. The aim of this note is to
extend this result to the general equation with middle term
$(q(x)y')'$ without further restriction on $q(x)$, other than
continuity.
Keywords:fourth-order linear differential equation, conjugate points, system-conjugate points, subwronskians Categories:47E05, 34B05, 34C10 |
3. CMB 2011 (vol 56 pp. 388)
Application of Measure of Noncompactness to Infinite Systems of Differential Equations In this paper we determine the Hausdorff measure of noncompactness on
the sequence space $n(\phi)$ of W. L. C. Sargent.
Further we apply
the technique of measures of noncompactness to the theory of infinite
systems of differential equations in the Banach sequence spaces
$n(\phi)$ and $m(\phi)$. Our aim is to present some existence results
for infinite systems of differential equations formulated with the help
of measures of noncompactness.
Keywords:sequence spaces, BK spaces, measure of noncompactness, infinite system of differential equations Categories:46B15, 46B45, 46B50, 34A34, 34G20 |
4. CMB 2011 (vol 56 pp. 366)
Multiple Solutions for Nonlinear Periodic Problems We consider a nonlinear periodic problem driven by a
nonlinear nonhomogeneous differential operator and a
CarathÃ©odory reaction term $f(t,x)$ that exhibits a
$(p-1)$-superlinear growth in $x \in \mathbb{R}$
near $\pm\infty$ and near zero.
A special case of the differential operator is the scalar
$p$-Laplacian. Using a combination of variational methods based on
the critical point theory with Morse theory (critical groups), we
show that the problem has three nontrivial solutions, two of which
have constant sign (one positive, the other negative).
Keywords:$C$-condition, mountain pass theorem, critical groups, strong deformation retract, contractible space, homotopy invariance Categories:34B15, 34B18, 34C25, 58E05 |
5. CMB 2011 (vol 56 pp. 102)
Eigenvalue Approach to Even Order System Periodic Boundary Value Problems We study an even order system boundary value problem with
periodic boundary conditions. By establishing
the existence of a positive eigenvalue of an associated linear system
Sturm-Liouville problem, we obtain new conditions for the boundary
value problem to have a positive solution. Our major tools are the
Krein-Rutman theorem for linear spectra and the fixed point index theory
for compact operators.
Keywords:Green's function, high order system boundary value problems, positive solutions, Sturm-Liouville problem Categories:34B18, 34B24 |
6. CMB 2011 (vol 55 pp. 214)
Positive Solutions of Impulsive Dynamic System on Time Scales In this paper, some criteria for the existence of positive solutions of a class
of systems of impulsive dynamic equations on time scales are obtained by
using a fixed point theorem in cones.
Keywords:time scale, positive solution, fixed point, impulsive dynamic equation Categories:39A10, 34B15 |
7. CMB 2011 (vol 55 pp. 285)
Uniqueness Implies Existence and Uniqueness Conditions for a Class of $(k+j)$-Point Boundary Value Problems for $n$-th Order Differential Equations |
Uniqueness Implies Existence and Uniqueness Conditions for a Class of $(k+j)$-Point Boundary Value Problems for $n$-th Order Differential Equations For the $n$-th order nonlinear differential equation, $y^{(n)} = f(x, y, y',
\dots, y^{(n-1)})$, we consider uniqueness implies uniqueness and existence
results for solutions satisfying certain $(k+j)$-point
boundary conditions for $1\le j \le n-1$ and $1\leq k \leq n-j$. We
define $(k;j)$-point unique solvability in analogy to $k$-point
disconjugacy and we show that $(n-j_{0};j_{0})$-point
unique solvability implies $(k;j)$-point unique solvability for $1\le j \le
j_{0}$, and $1\leq k \leq n-j$. This result is
analogous to
$n$-point disconjugacy implies $k$-point disconjugacy for $2\le k\le
n-1$.
Keywords:boundary value problem, uniqueness, existence, unique solvability, nonlinear interpolation Categories:34B15, 34B10, 65D05 |
8. CMB 2011 (vol 55 pp. 736)
Existence of Solutions for Abstract Non-Autonomous Neutral Differential Equations In this paper we discuss the existence of mild and classical solutions for a class of abstract non-autonomous
neutral functional differential equations. An application to partial neutral differential equations is considered.
Keywords:neutral equations, mild solutions, classical solutions Categories:35R10, 34K40, 34K30 |
9. CMB 2011 (vol 55 pp. 400)
Eisenstein Series and Modular Differential Equations The purpose of this paper is to solve various differential
equations having Eisenstein series as coefficients using various tools and techniques. The solutions
are given in terms of modular forms, modular functions, and
equivariant forms.
Keywords:differential equations, modular forms, Schwarz derivative, equivariant forms Categories:11F11, 34M05 |
10. CMB 2011 (vol 54 pp. 506)
On the Canonical Solution of the Sturm-Liouville Problem with Singularity and Turning Point of Even Order |
On the Canonical Solution of the Sturm-Liouville Problem with Singularity and Turning Point of Even Order In this paper, we are going to investigate the canonical property of solutions of
systems of differential equations having a singularity and turning
point of even order. First, by a replacement, we transform the system
to the Sturm-Liouville equation with turning point. Using of the
asymptotic estimates provided by Eberhard, Freiling, and Schneider
for a special fundamental system of solutions of the Sturm-Liouville
equation, we study the infinite product representation of solutions of the systems. Then we
transform the Sturm-Liouville equation with
turning point to the
equation with singularity, then we study the asymptotic behavior of its solutions. Such
representations are relevant to the inverse spectral problem.
Keywords:turning point, singularity, Sturm-Liouville, infinite products, Hadamard's theorem, eigenvalues Categories:34B05, 34Lxx, 47E05 |
11. CMB 2011 (vol 55 pp. 882)
Equivalence of $L_p$ Stability and Exponential Stability of Nonlinear Lipschitzian Semigroups
$L_p$ stability and exponential stability are two important concepts
for nonlinear dynamic systems. In this paper, we prove that a
nonlinear exponentially bounded Lipschitzian semigroup is
exponentially stable if and only if the semigroup is $L_p$ stable
for some $p>0$. Based on the equivalence, we derive two sufficient
conditions for exponential stability of the nonlinear semigroup. The
results obtained extend and improve some existing ones.
Keywords:exponentially stable, $L_p$ stable, nonlinear Lipschitzian semigroups Categories:34D05, 47H20 |
12. CMB 2011 (vol 55 pp. 3)
On a Local Theory of Asymptotic Integration for Nonlinear Differential Equations We improve several recent results in the
asymptotic integration theory of nonlinear ordinary differential
equations via a variant of the method devised by J. K. Hale and
N. Onuchic The results
are used for investigating the existence of positive solutions to
certain reaction-diffusion equations.
Keywords:asymptotic integration, Emden-Fowler differential equation, reaction-diffusion equation Categories:34E10, 34C10, 35Q35 |
13. CMB 2011 (vol 54 pp. 580)
Kiguradze-type Oscillation Theorems for Second Order Superlinear Dynamic Equations on Time Scales Consider the second order superlinear dynamic equation
\begin{equation*}
(*)\qquad
x^{\Delta\Delta}(t)+p(t)f(x(\sigma(t)))=0\tag{$*$}
\end{equation*}
where $p\in C(\mathbb{T},\mathbb{R})$, $\mathbb{T}$ is a time scale,
$f\colon\mathbb{R}\rightarrow\mathbb{R}$ is
continuously differentiable and satisfies $f'(x)>0$, and $xf(x)>0$ for
$x\neq 0$. Furthermore, $f(x)$ also satisfies a superlinear condition, which
includes the nonlinear function $f(x)=x^\alpha$ with $\alpha>1$, commonly
known as the Emden--Fowler case. Here the coefficient function $p(t)$ is
allowed to be negative for arbitrarily large values of $t$. In addition to
extending the result of Kiguradze for \eqref{star1} in the real case $\mathbb{T}=\mathbb{R}$, we
obtain analogues in the difference equation and $q$-difference equation cases.
Keywords:Oscillation, Emden-Fowler equation, superlinear Categories:34K11, 39A10, 39A99 |
14. CMB 2010 (vol 54 pp. 527)
On the Dichotomy of the Evolution Families: A Discrete-Argument Approach
We establish a discrete-time criteria guaranteeing the existence of an
exponential dichotomy in the continuous-time
behavior of an abstract evolution family. We prove that an evolution
family ${\cal U}=\{U(t,s)\}_{t
\geq s\geq 0}$ acting on a Banach space $X$ is uniformly
exponentially dichotomic (with respect to its continuous-time
behavior) if and only if the
corresponding difference equation with the inhomogeneous term from
a vector-valued Orlicz sequence space $l^\Phi(\mathbb{N}, X)$
admits
a solution in the same $l^\Phi(\mathbb{N},X)$. The technique of
proof effectively eliminates the continuity hypothesis on the
evolution family (\emph{i.e.,} we do not assume that $U(\,\cdot\,,s)x$
or $U(t,\,\cdot\,)x$ is continuous on $[s,\infty)$, and respectively
$[0,t]$). Thus, some known results given by
Coffman and Schaffer, Perron, and Ta Li are extended.
Keywords:evolution families, exponential dichotomy, Orlicz sequence spaces, admissibility Categories:34D05, 47D06, 93D20 |
15. CMB 2010 (vol 54 pp. 364)
Lyapunov Theorems for the Asymptotic Behavior of Evolution Families on the Half-Line
Two theorems regarding the
asymptotic behavior of evolution families are established in
terms of the solutions of a certain Lyapunov operator equation.
Keywords:evolution families, exponential instability, Lyapunov equation Categories:34D05, 47D06 |
16. CMB 2010 (vol 53 pp. 475)
Nonlinear Multipoint Boundary Value Problems for Second Order Differential Equations In this paper we shall discuss nonlinear multipoint boundary value problems for second order differential equations when deviating arguments depend on the unknown solution. Sufficient conditions under which such problems have extremal and quasi-solutions are given. The problem of when a unique solution exists is also investigated. To obtain existence results, a monotone iterative technique is used. Two examples are added to verify theoretical results.
Keywords:second order differential equations, deviated arguments, nonlinear boundary conditions, extremal solutions, quasi-solutions, unique solution Categories:34A45, 34K10 |
17. CMB 2010 (vol 53 pp. 367)
Almost Periodicity and Lyapunov's Functions for Impulsive Functional Differential Equations with Infinite Delays |
Almost Periodicity and Lyapunov's Functions for Impulsive Functional Differential Equations with Infinite Delays This paper studies the existence and uniqueness of almost periodic solutions of nonlinear impulsive functional differential equations with infinite delay. The results obtained are based on the Lyapunov--Razumikhin method and on differential inequalities for piecewise continuous functions.
Keywords:almost periodic solutions, impulsive functional differential equations Categories:34K45, 34B37 |
18. CMB 2009 (vol 53 pp. 347)
Multiple Nontrivial Solutions for Doubly Resonant Periodic Problems We consider semilinear periodic problems with the right-hand side nonlinearity satisfying a double resonance condition between two successive eigenvalues. Using a combination of variational and degree theoretic methods, we prove the existence of at least two nontrivial solutions.
Keywords:double resonance, generalized LL-condition, Leray-Schauder degree, Cerami condition Category:34B15 |
19. CMB 2009 (vol 53 pp. 193)
On the Oscillation of a Second Order Strictly Sublinear Differential Equation We establish a flexible oscillation criterion based on an averaging technique that improves upon a result due to C.~G. Philos.
Keywords:oscillation theory, averaging method Categories:34C10, 34C15, 34C29 |
20. CMB 2009 (vol 52 pp. 315)
Generic Quasi-Convergence for Essentially Strongly Order-Preserving Semiflows By employing the limit set
dichotomy for essentially strongly order-preserving semiflows and
the assumption that limit sets have infima and suprema in the
state space, we prove a generic quasi-convergence principle
implying the existence of an open and dense set of stable
quasi-convergent points. We also apply this generic quasi-convergence principle
to a model for biochemical feedback in protein
synthesis and obtain some results about the model which are of theoretical
and realistic significance.
Keywords:Essentially strongly order-preserving semiflow, compactness, quasi-convergence Categories:34C12, 34K25 |
21. CMB 2008 (vol 51 pp. 386)
Positive Solutions of the Falkner--Skan Equation Arising in the Boundary Layer Theory The well-known Falkner--Skan equation is one of the most important
equations in laminar boundary layer theory and is used to describe
the steady two-dimensional flow of a slightly viscous
incompressible fluid past wedge shaped bodies of angles related to
$\lambda\pi/2$, where $\lambda\in \mathbb R$ is a parameter
involved in the equation. It is known that there exists
$\lambda^{*}<0$ such that the equation with suitable boundary
conditions has at least one positive solution for each $\lambda\ge
\lambda^{*}$ and has no positive solutions for
$\lambda<\lambda^{*}$. The known numerical result shows
$\lambda^{*}=-0.1988$. In this paper, $\lambda^{*}\in
[-0.4,-0.12]$ is proved analytically by establishing a singular
integral equation which is equivalent to the Falkner--Skan
equation. The equivalence result
provides new techniques to study properties and existence of solutions of
the Falkner--Skan equation.
Keywords:Falkner-Skan equation, boundary layer problems, singular integral equation, positive solutions Categories:34B16, 34B18, 34B40, 76D10 |
22. CMB 2008 (vol 51 pp. 217)
A Multivalued Nonlinear System with the Vector $p$-Laplacian on the Semi-Infinity Interval We study a second order nonlinear system driven by the vector
$p$-Laplacian, with a multivalued nonlinearity and defined on
the positive time semi-axis $\mathbb{R}_+.$ Using degree
theoretic techniques we solve an auxiliary mixed boundary value
problem defined on the finite interval $[0,n]$ and then via a
diagonalization method we produce a solution for the original
infinite time-horizon system.
Keywords:semi-infinity interval, vector $p$-Laplacian, multivalued nonlinear, fixed point index, Hartman condition, completely continuous map Category:34A60 |
23. CMB 2007 (vol 50 pp. 377)
Global Injectivity of $C^1$ Maps of the Real Plane, Inseparable Leaves and the Palais--Smale Condition |
Global Injectivity of $C^1$ Maps of the Real Plane, Inseparable Leaves and the Palais--Smale Condition We study two sufficient conditions that imply global injectivity
for a $C^1$ map $X\colon \R^2\to \R^2$ such that its Jacobian at any
point of $\R^2$ is not zero. One is based on the notion of
half-Reeb component and the other on the Palais--Smale condition.
We improve the first condition using the notion of inseparable
leaves. We provide a new proof of the sufficiency of the second
condition. We prove that both conditions are not equivalent, more
precisely we show that the Palais--Smale condition implies the
nonexistence of inseparable leaves, but the converse is not true.
Finally, we show that the Palais--Smale condition it is not a
necessary condition for the global injectivity of the map $X$.
Categories:34C35, 34H05 |
24. CMB 2006 (vol 49 pp. 170)
The Geometry of $ d^{2}y^{1}/dt^{2} = f(y, \dot{y},t) \; \text{and} \; d^{2}y^{2}/dt^{2} = g(y,\dot{y},t)$, and Euclidean Spaces |
The Geometry of $ d^{2}y^{1}/dt^{2} = f(y, \dot{y},t) \; \text{and} \; d^{2}y^{2}/dt^{2} = g(y,\dot{y},t)$, and Euclidean Spaces This paper investigates the relationship between a system of
differential equations and the underlying geometry associated with
it. The geometry of a surface determines shortest paths, or
geodesics connecting nearby points, which are defined as the
solutions to a pair of second-order differential equations: the
Euler--Lagrange equations of the metric. We ask when the converse
holds, that is, when solutions to a system of differential
equations reveals an underlying geometry. Specifically, when may
the solutions to a given pair of second order ordinary
differential equations $d^{2}y^{1}/dt^{2} = f(y,\dot{y},t)$ and
$d^{2}y^{2}/dt^{2} = g(y,\dot{y},t)$ be reparameterized by
$t\rightarrow T(y,t)$ so as to give locally the geodesics of a
Euclidean space? Our approach is based upon Cartan's method of
equivalence. In the second part of the paper, the equivalence
problem is solved for a generic pair of second order ordinary
differential equations of the above form revealing the existence
of 24 invariant functions.
Category:34A26 |
25. CMB 2005 (vol 48 pp. 405)
Liouville's Theorem in the Radially Symmetric Case We present a very short proof of Liouville's theorem for solutions
to a non-uniformly elliptic radially symmetric equation. The proof uses
the Ricatti equation satisfied by the Dirichlet to Neumann map.
Categories:35B05, 34A30 |