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Search: MSC category 34 ( Ordinary differential equations )

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1. CMB Online first

Alhasanat, Ahmad; Ou, Chunhua
Periodic steady-state solutions of a liquid film model via a classical method
In this paper, periodic steady-state of a liquid film flowing over a periodic uneven wall is investigated via a classical method. Specifically, we analyze a long-wave model that is valid at the near-critical Reynolds number. For the periodic wall surface, we construct an iteration scheme in terms of an integral form of the original steady-state problem. The uniform convergence of the scheme is proved so that we can derive the existence and the uniqueness, as well as the asymptotic formula, of the periodic solutions.

Keywords:film flow, classical methods, asymptotic analysis
Categories:34E05, 34E10, 34E15

2. CMB Online first

Zhang, Guo-Bao; Tian, Ge
Stability of Traveling Wavefronts for a Two-Component Lattice Dynamical System Arising in Competition Models
In this paper, we study a two-component Lotka-Volterra competition system on an one-dimensional spatial lattice. By the method of the comparison principle together with the weighted energy, we prove that the traveling wavefronts with large speed are exponentially asymptotically stable, when the initial perturbation around the traveling wavefronts decays exponentially as $j+ct \rightarrow -\infty$, where $j\in\mathbb{Z}$, $t\gt 0$, but the initial perturbation can be arbitrarily large on other locations. This partially answers an open problem by J.-S. Guo and C.-H. Wu.

Keywords:lattice dynamical system, competition model, traveling wavefront, stability
Categories:34A33, 34K20, 92D25

3. CMB Online first

Bu, Shangquan; Cai, Gang
Hölder continuous solutions of degenerate differential equations with finite delay
Using known operator-valued Fourier multiplier results on vector-valued Hölder continuous function spaces $C^\alpha (\mathbb R; X)$, we completely characterize the $C^\alpha$-well-posedness of the first order degenerate differential equations with finite delay $(Mu)'(t) = Au(t) + Fu_t + f(t)$ for $t\in\mathbb R$ by the boundedness of the $(M, F)$-resolvent of $A$ under suitable assumption on the delay operator $F$, where $A, M$ are closed linear operators on a Banach space $X$ satisfying $D(A)\cap D(M) \not=\{0\}$, the delay operator $F$ is a bounded linear operator from $C([-r, 0]; X)$ to $X$ and $r \gt 0$ is fixed.

Keywords:well-posedness, degenerate differential equation, $\dot{C}^\alpha$-multiplier, Hölder continuous function space
Categories:34N05, 34G10, 47D06, 47A10, 34K30

4. CMB Online first

Huang, Yanhe; Sottile, Frank; Zelenko, Igor
Injectivity of generalized Wronski maps
We study linear projections on Plücker space whose restriction to the Grassmannian is a non-trivial branched cover. When an automorphism of the Grassmannian preserves the fibers, we show that the Grassmannian is necessarily of $m$-dimensional linear subspaces in a symplectic vector space of dimension $2m$, and the linear map is the Lagrangian involution. The Wronski map for a self-adjoint linear differential operator and pole placement map for symmetric linear systems are natural examples.

Keywords:Wronski map, Plücker embedding, curves in Lagrangian Grassmannian, self-adjoint linear differential operator, symmetric linear control system, pole placement map
Categories:14M15, 34A30, 93B55

5. CMB 2017 (vol 60 pp. 381)

Rousseau, C.
The Bifurcation Diagram of Cubic Polynomial Vector Fields on $\mathbb{C}\mathbb{P}^1$
In this paper we give the bifurcation diagram of the family of cubic vector fields $\dot z=z^3+ \epsilon_1z+\epsilon_0$ for $z\in \mathbb{C}\mathbb{P}^1$, depending on the values of $\epsilon_1,\epsilon_0\in\mathbb{C}$. The bifurcation diagram is in $\mathbb{R}^4$, but its conic structure allows describing it for parameter values in $\mathbb{S}^3$. There are two open simply connected regions of structurally stable vector fields separated by surfaces corresponding to bifurcations of homoclinic connections between two separatrices of the pole at infinity. These branch from the codimension 2 curve of double singular points. We also explain the bifurcation of homoclinic connection in terms of the description of Douady and Sentenac of polynomial vector fields.

Keywords:complex polynomial vector field, bifurcation diagram, Douady-Sentenac invariant
Categories:34M45, 32G34

6. CMB 2017 (vol 60 pp. 364)

Preda, Ciprian
On the Roughness of Quasinilpotency Property of One–parameter Semigroups
Let $\mathbf{S}:=\{S(t)\}_{t\geq0}$ be a C$_0$-semigroup of quasinilpotent operators (i.e. $\sigma(S(t))=\{0\}$ for each $t\gt 0$). In the dynamical systems theory the above quasinilpotency property is equivalent to a very strong concept of stability for the solutions of autonomous systems. This concept is frequently called superstability and weakens the classical finite time extinction property (roughly speaking, disappearing solutions). We show that under some assumptions, the quasinilpotency, or equivalently, the superstability property of a C$_0$-semigroup is preserved under the perturbations of its infinitesimal generator.

Keywords:one-parameter semigroups, quasinilpotency, superstability, essential spectrum
Categories:34D05, 34D10, 34E10

7. CMB 2016 (vol 59 pp. 849)

Nah, Kyeongah; Röst, Gergely
Stability Threshold for Scalar Linear Periodic Delay Differential Equations
We prove that for the linear scalar delay differential equation $$ \dot{x}(t) = - a(t)x(t) + b(t)x(t-1) $$ with non-negative periodic coefficients of period $P\gt 0$, the stability threshold for the trivial solution is $r:=\int_{0}^{P} \left(b(t)-a(t) \right)\mathrm{d}t=0,$ assuming that $b(t+1)-a(t)$ does not change its sign. By constructing a class of explicit examples, we show the counter-intuitive result that in general, $r=0$ is not a stability threshold.

Keywords:delay differential equation, stability, periodic system
Categories:34K20, 34K06

8. CMB 2016 (vol 59 pp. 363)

Li, Dan; Ma, Wanbiao
Dynamical Analysis of a Stage-Structured Model for Lyme Disease with Two Delays
In this paper, a nonlinear stage-structured model for Lyme disease is considered. The model is a system of differential equations with two time delays. The basic reproductive rate, $R_0(\tau_1,\tau_2)$, is derived. If $R_0(\tau_1,\tau_2)\lt 1$, then the boundary equilibrium is globally asymptotically stable. If $R_0(\tau_1,\tau_2)\gt 1$, then there exists a unique positive equilibrium whose local asymptotical stability and the existence of Hopf bifurcations are established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions is derived by using the normal form and the center manifold theory. Some numerical simulations are performed to confirm the correctness of theoretical analysis. At last, some conclusions are given.

Keywords:Lyme disease, stage-structure, time delay, Lyapunov functional stability Hopf bifurcation.

9. CMB 2015 (vol 58 pp. 818)

Llibre, Jaume; Zhang, Xiang
On the Limit Cycles of Linear Differential Systems with Homogeneous Nonlinearities
We consider the class of polynomial differential systems of the form $\dot x= \lambda x-y+P_n(x,y)$, $\dot y=x+\lambda y+ Q_n(x,y),$ where $P_n$ and $Q_n$ are homogeneous polynomials of degree $n$. For this class of differential systems we summarize the known results for the existence of limit cycles, and we provide new results for their nonexistence and existence.

Keywords:polynomial differential system, limit cycles, differential equations on the cylinder
Categories:34C35, 34D30

10. CMB 2014 (vol 58 pp. 44)

Daniilidis, A.; Drusvyatskiy, D.; Lewis, A. S.
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

11. CMB 2011 (vol 56 pp. 39)

Ben Amara, Jamel
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

12. CMB 2011 (vol 56 pp. 388)

Mursaleen, M.
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

13. CMB 2011 (vol 56 pp. 366)

Kyritsi, Sophia Th.; Papageorgiou, Nikolaos S.
Multiple Solutions for Nonlinear Periodic Problems
We consider a nonlinear periodic problem driven by a nonlinear nonhomogeneous differential operator and a Carathéodory 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

14. CMB 2011 (vol 56 pp. 102)

Kong, Qingkai; Wang, Min
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

15. CMB 2011 (vol 55 pp. 214)

Wang, Da-Bin
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

16. CMB 2011 (vol 55 pp. 285)

Eloe, Paul W.; Henderson, Johnny; Khan, Rahmat Ali
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

17. CMB 2011 (vol 55 pp. 736)

Hernández, Eduardo; O'Regan, Donal
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

18. CMB 2011 (vol 55 pp. 400)

Sebbar, Abdellah; Sebbar, Ahmed
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

19. CMB 2011 (vol 55 pp. 882)

Xueli, Song; Jigen, Peng
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

20. CMB 2011 (vol 54 pp. 506)

Neamaty, A.; Mosazadeh, S.
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

21. CMB 2011 (vol 55 pp. 3)

Agarwal, Ravi P.; Mustafa, Octavian G.
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

22. CMB 2011 (vol 54 pp. 580)

Baoguo, Jia; Erbe, Lynn; Peterson, Allan
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

23. CMB 2010 (vol 54 pp. 527)

Preda, Ciprian; Sipos, Ciprian
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

24. CMB 2010 (vol 54 pp. 364)

Preda, Ciprian; Preda, Petre
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

25. CMB 2010 (vol 53 pp. 475)

Jankowski, Tadeusz
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
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