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1. CMB Online first

Zhang, Tao; Zhou, Chunqin
 Classification of solutions for harmonic functions with Neumann boundary value In this paper, we classify all solutions of $\left\{ \begin{array}{rcll} -\Delta u &=& 0 \quad &\text{ in }\mathbb{R}^{2}_{+}, \\ \dfrac{\partial u}{\partial t}&=&-c|x|^{\beta}e^{u} \quad &\text{ on }\partial \mathbb{R}^{2}_{+} \backslash \{0\}, \\ \end{array} \right.$ with the finite conditions $\int_{\partial \mathbb{R}^{2}_{+}}|x|^{\beta}e^{u}ds \lt C, \qquad \sup\limits_{\overline{\mathbb{R}^{2}_{+}}}{u(x)}\lt C.$ Here, $c$ is a positive number and $\beta \gt -1$. Keywords:Neumann problem, singular coefficient, classification of solutionsCategories:35A05, 35J65

2. CMB 2017 (vol 60 pp. 422)

Tang, Xianhua
 New Super-quadratic Conditions for Asymptotically Periodic SchrÃ¶dinger Equations This paper is dedicated to studying the semilinear SchrÃ¶dinger equation $$\left\{ \begin{array}{ll} -\triangle u+V(x)u=f(x, u), \ \ \ \ x\in {\mathbf{R}}^{N}, \\ u\in H^{1}({\mathbf{R}}^{N}), \end{array} \right.$$ where $f$ is a superlinear, subcritical nonlinearity. It focuses on the case where $V(x)=V_0(x)+V_1(x)$, $V_0\in C(\mathbf{R}^N)$, $V_0(x)$ is 1-periodic in each of $x_1, x_2, \ldots, x_N$ and $\sup[\sigma(-\triangle +V_0)\cap (-\infty, 0)]\lt 0\lt \inf[\sigma(-\triangle +V_0)\cap (0, \infty)]$, $V_1\in C(\mathbf{R}^N)$ and $\lim_{|x|\to\infty}V_1(x)=0$. A new super-quadratic condition is obtained, which is weaker than some well known results. Keywords:SchrÃ¶dinger equation, superlinear, asymptotically periodic, ground state solutions of Nehari-Pankov typeCategories:35J20, 35J60

3. CMB 2016 (vol 59 pp. 528)

Jahan, Qaiser
 Characterization of Low-pass Filters on Local Fields of Positive Characteristic In this article, we give necessary and sufficient conditions on a function to be a low-pass filter on a local field $K$ of positive characteristic associated to the scaling function for multiresolution analysis of $L^2(K)$. We use probability and martingale methods to provide such a characterization. Keywords:multiresolution analysis, local field, low-pass filter, scaling function, probability, conditional probability and martingalesCategories:42C40, 42C15, 43A70, 11S85

4. CMB 2015 (vol 59 pp. 73)

Gasiński, Leszek; Papageorgiou, Nikolaos S.
 Positive Solutions for the Generalized Nonlinear Logistic Equations We consider a nonlinear parametric elliptic equation driven by a nonhomogeneous differential operator with a logistic reaction of the superdiffusive type. Using variational methods coupled with suitable truncation and comparison techniques, we prove a bifurcation type result describing the set of positive solutions as the parameter varies. Keywords:positive solution, bifurcation type result, strong comparison principle, nonlinear regularity, nonlinear maximum principleCategories:35J25, 35J92

5. CMB 2015 (vol 58 pp. 723)

Castro, Alfonso; Fischer, Emily
 Infinitely Many Rotationally Symmetric Solutions to a Class of Semilinear Laplace-Beltrami Equations on Spheres We show that a class of semilinear Laplace-Beltrami equations on the unit sphere in $\mathbb{R}^n$ has infinitely many rotationally symmetric solutions. The solutions to these equations are the solutions to a two point boundary value problem for a singular ordinary differential equation. We prove the existence of such solutions using energy and phase plane analysis. We derive a Pohozaev-type identity in order to prove that the energy to an associated initial value problem tends to infinity as the energy at the singularity tends to infinity. The nonlinearity is allowed to grow as fast as $|s|^{p-1}s$ for $|s|$ large with $1 \lt p \lt (n+5)/(n-3)$. Keywords:Laplace-Beltrami operator, semilinear equation, rotational solution, superlinear nonlinearity, sub-super critical nonlinearityCategories:58J05, 35A24

6. CMB 2015 (vol 58 pp. 651)

Tang, Xianhua
 Ground State Solutions of Nehari-Pankov Type for a Superlinear Hamiltonian Elliptic System on ${\mathbb{R}}^{N}$ This paper is concerned with the following elliptic system of Hamiltonian type $\left\{ \begin{array}{ll} -\triangle u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N}, \\ -\triangle v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N}, \\ u, v\in H^{1}({\mathbb{R}}^{N}), \end{array} \right.$ where the potential $V$ is periodic and $0$ lies in a gap of the spectrum of $-\Delta+V$, $W(x, s, t)$ is periodic in $x$ and superlinear in $s$ and $t$ at infinity. We develop a direct approach to find ground state solutions of Nehari-Pankov type for the above system. Especially, our method is applicable for the case when $W(x, u, v)=\sum_{i=1}^{k}\int_{0}^{|\alpha_iu+\beta_iv|}g_i(x, t)t\mathrm{d}t +\sum_{j=1}^{l}\int_{0}^{\sqrt{u^2+2b_juv+a_jv^2}}h_j(x, t)t\mathrm{d}t,$ where $\alpha_i, \beta_i, a_j, b_j\in \mathbb{R}$ with $\alpha_i^2+\beta_i^2\ne 0$ and $a_j\gt b_j^2$, $g_i(x, t)$ and $h_j(x, t)$ are nondecreasing in $t\in \mathbb{R}^{+}$ for every $x\in \mathbb{R}^N$ and $g_i(x, 0)=h_j(x, 0)=0$. Keywords:Hamiltonian elliptic system, superlinear, ground state solutions of Nehari-Pankov type, strongly indefinite functionalsCategories:35J50, 35J55

7. CMB 2014 (vol 58 pp. 174)

Raffoul, Youssef N.
 Periodic Solutions of Almost Linear Volterra Integro-dynamic Equation on Periodic Time Scales Using Krasnoselskii's fixed point theorem, we deduce the existence of periodic solutions of nonlinear system of integro-dynamic equations on periodic time scales. These equations are studied under a set of assumptions on the functions involved in the equations. The equations will be called almost linear when these assumptions hold. The results of this papers are new for the continuous and discrete time scales. Keywords:Volterra integro-dynamic equation, time scales, Krasnoselsii's fixed point theorem, periodic solutionCategories:45J05, 45D05

8. 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

9. 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

10. CMB 2011 (vol 56 pp. 80)

 Three Fixed Point Theorems: Periodic Solutions of a Volterra Type Integral Equation with Infinite Heredity In this paper we study the existence of periodic solutions of a Volterra type integral equation with infinite heredity. Banach fixed point theorem, Krasnosel'skii's fixed point theorem, and a combination of Krasnosel'skii's and Schaefer's fixed point theorems are employed in the analysis. The combination theorem of Krasnosel'skii and Schaefer requires an a priori bound on all solutions. We employ Liapunov's direct method to obtain such an a priori bound. In the process, we compare these theorems in terms of assumptions and outcomes. Keywords:Volterra integral equation, periodic solutions, Liapunov's method, Krasnosel'skii's fixed point theorem, Schaefer's fixed point theoremCategories:45D05, 45J05

11. 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

12. CMB 2011 (vol 55 pp. 537)

Kang, Dongsheng
 Asymptotic Properties of Solutions to Semilinear Equations Involving Multiple Critical Exponents In this paper, we investigate a semilinear elliptic equation that involves multiple Hardy-type terms and critical Hardy-Sobolev exponents. By the Moser iteration method and analytic techniques, the asymptotic properties of its nontrivial solutions at the singular points are investigated. Keywords:elliptic problem, solution, Hardy-Sobolev inequality, singularity, Moser iterationCategories:35B33, 35B40, 35J60

13. CMB 2011 (vol 55 pp. 249)

Chang, Der-Chen; Li, Bao Qin
 Description of Entire Solutions of Eiconal Type Equations The paper describes entire solutions to the eiconal type non-linear partial differential equations, which include the eiconal equations $(X_1(u))^2+(X_2(u))^2=1$ as special cases, where $X_1=p_1{\partial}/{\partial z_1}+p_2{\partial}/{\partial z_2}$, $X_2=p_3{\partial}/{\partial z_1}+p_4{\partial}/{\partial z_2}$ are linearly independent operators with $p_j$ being arbitrary polynomials in $\mathbf{C}^2$. Keywords:entire solution, eiconal equation, polynomial, transcendental functionCategories:32A15, 35F20

14. CMB 2010 (vol 53 pp. 674)

Kristály, Alexandru; Papageorgiou, Nikolaos S.; Varga, Csaba
 Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary We study a semilinear elliptic problem on a compact Riemannian manifold with boundary, subject to an inhomogeneous Neumann boundary condition. Under various hypotheses on the nonlinear terms, depending on their behaviour in the origin and infinity, we prove multiplicity of solutions by using variational arguments. Keywords:Riemannian manifold with boundary, Neumann problem, sublinearity at infinity, multiple solutionsCategories:58J05, 35P30

15. CMB 2010 (vol 53 pp. 526)

Milian, Anna
 On Some Stochastic Perturbations of Semilinear Evolution Equations We consider semilinear evolution equations with some locally Lipschitz nonlinearities, perturbed by Banach space valued, continuous, and adapted stochastic process. We show that under some assumptions there exists a solution to the equation. Using the result we show that there exists a mild, continuous, global solution to a semilinear ItÃ´ equation with locally Lipschitz nonlinearites. An example of the equation is given. Keywords:evolution equation, mild solution, non-Lipschitz drift, Ito integralCategory:60H20

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 solutionCategories:34A45, 34K10

17. 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

18. 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

19. CMB 2007 (vol 50 pp. 356)

Filippakis, Michael E.; Papageorgiou, Nikolaos S.
 Existence of Positive Solutions for Nonlinear Noncoercive Hemivariational Inequalities In this paper we investigate the existence of positive solutions for nonlinear elliptic problems driven by the $p$-Laplacian with a nonsmooth potential (hemivariational inequality). Under asymptotic conditions that make the Euler functional indefinite and incorporate in our framework the asymptotically linear problems, using a variational approach based on nonsmooth critical point theory, we obtain positive smooth solutions. Our analysis also leads naturally to multiplicity results. Keywords:$p$-Laplacian, locally Lipschitz potential, nonsmooth critical point theory, principal eigenvalue, positive solutions, nonsmooth Mountain Pass TheoremCategories:35J20, 35J60, 35J85

20. CMB 2005 (vol 48 pp. 69)

Fabian, M.; Montesinos, V.; Zizler, V.
 Biorthogonal Systems in Weakly LindelÃ¶f Spaces We study countable splitting of Markushevich bases in weakly Lindel\"of Banach spaces in connection with the geometry of these spaces. Keywords:Weak compactness, projectional resolutions,, Markushevich bases, Eberlein compacts, Va\v sÃ¡k spacesCategories:46B03, 46B20., 46B26

21. CMB 2002 (vol 45 pp. 428)

Mollin, R. A.
 Criteria for Simultaneous Solutions of $X^2 - DY^2 = c$ and $x^2 - Dy^2 = -c$ The purpose of this article is to provide criteria for the simultaneous solvability of the Diophantine equations $X^2 - DY^2 = c$ and $x^2 - Dy^2 = -c$ when $c \in \mathbb{Z}$, and $D \in \mathbb{N}$ is not a perfect square. This continues work in \cite{me}--\cite{alfnme}. Keywords:continued fractions, Diophantine equations, fundamental units, simultaneous solutionsCategories:11A55, 11R11, 11D09

22. CMB 2001 (vol 44 pp. 210)

Leung, Man Chun
 Growth Estimates on Positive Solutions of the Equation $\Delta u+K u^{\frac{n+2}{n-2}}=0$ in $\R^n$ We construct unbounded positive $C^2$-solutions of the equation $\Delta u + K u^{(n + 2)/(n - 2)} = 0$ in $\R^n$ (equipped with Euclidean metric $g_o$) such that $K$ is bounded between two positive numbers in $\R^n$, the conformal metric $g=u^{4/(n-2)}g_o$ is complete, and the volume growth of $g$ can be arbitrarily fast or reasonably slow according to the constructions. By imposing natural conditions on $u$, we obtain growth estimate on the $L^{2n/(n-2)}$-norm of the solution and show that it has slow decay. Keywords:positive solution, conformal scalar curvature equation, growth estimateCategories:35J60, 58G03

23. CMB 2000 (vol 43 pp. 3)

 $D$-spaces and resolution A space $X$ is a $D$-space if, for every neighborhood assignment $f$ there is a closed discrete set $D$ such that $\bigcup{f(D)}=X$. In this paper we give some necessary conditions and some sufficient conditions for a resolution of a topological space to be a $D$-space. In particular, if a space $X$ is resolved at each $x\in X$ into a $D$-space $Y_x$ by continuous mappings $f_x\colon X-\{{x}\} \rightarrow Y_x$, then the resolution is a $D$-space if and only if $\bigcup{\{{x}\}}\times \Bd(Y_x)$ is a $D$-space. Keywords:$D$-space, neighborhood assignment, resolution, boundaryCategories:54D20, 54B99, 54D10, 54D30