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1. 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 cylinderCategories:34C35, 34D30

2. CMB Online first

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.Category:34D20

3. 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 processCategories:34A60, 49J99

4. 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, subwronskiansCategories:47E05, 34B05, 34C10

5. 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 equationsCategories:46B15, 46B45, 46B50, 34A34, 34G20

6. 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 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 invarianceCategories:34B15, 34B18, 34C25, 58E05

7. 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 problemCategories:34B18, 34B24

8. 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 equationCategories:39A10, 34B15

9. 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 interpolationCategories:34B15, 34B10, 65D05

10. 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 solutionsCategories:35R10, 34K40, 34K30

11. 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 formsCategories:11F11, 34M05

12. 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 semigroupsCategories:34D05, 47H20

13. CMB 2011 (vol 54 pp. 506)

 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, eigenvaluesCategories:34B05, 34Lxx, 47E05

14. 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 equationCategories:34E10, 34C10, 35Q35

15. 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, superlinearCategories:34K11, 39A10, 39A99

16. 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, admissibilityCategories:34D05, 47D06, 93D20

17. 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 equationCategories:34D05, 47D06

18. 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 solutionCategories:34A45, 34K10

19. CMB 2010 (vol 53 pp. 367)

Stamov, Gani Tr.
 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 equationsCategories:34K45, 34B37

20. CMB 2009 (vol 53 pp. 347)

Papageorgiou, Nikolaos S.; Staicu, Vasile
 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 conditionCategory:34B15

21. CMB 2009 (vol 53 pp. 193)

Agarwal, Ravi P.; Avramescu, Cezar; Mustafa, Octavian G.
 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 methodCategories:34C10, 34C15, 34C29

22. CMB 2009 (vol 52 pp. 315)

Yi, Taishan; Zou, Xingfu
 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-convergenceCategories:34C12, 34K25

23. CMB 2008 (vol 51 pp. 386)

Lan, K. Q.; Yang, G. C.
 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 solutionsCategories:34B16, 34B18, 34B40, 76D10

24. CMB 2008 (vol 51 pp. 217)

Filippakis, Michael E.; Papageorgiou, Nikolaos S.
 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 mapCategory:34A60

25. CMB 2007 (vol 50 pp. 377)

Gutierrez, C.; Jarque, X.; Llibre, J.; Teixeira, M. A.
 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
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