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Search: All articles in the CMB digital archive with keyword solution

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1. 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

2. 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

3. CMB 2011 (vol 56 pp. 80)

Islam, Muhammad N.
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 theorem
Categories:45D05, 45J05

4. 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

5. 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 iteration
Categories:35B33, 35B40, 35J60

6. 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 function
Categories:32A15, 35F20

7. 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 solutions
Categories:58J05, 35P30

8. 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

9. 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 Itô equation with locally Lipschitz nonlinearites. An example of the equation is given.

Keywords:evolution equation, mild solution, non-Lipschitz drift, Ito integral
Category:60H20

10. 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 equations
Categories:34K45, 34B37

11. 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 solutions
Categories:34B16, 34B18, 34B40, 76D10

12. 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 Theorem
Categories:35J20, 35J60, 35J85

13. CMB 2005 (vol 48 pp. 69)

Fabian, M.; Montesinos, V.; Zizler, V.
Biorthogonal Systems in Weakly Lindelöf 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 sák spaces
Categories:46B03, 46B20., 46B26

14. 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 solutions
Categories:11A55, 11R11, 11D09

15. 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 estimate
Categories:35J60, 58G03

16. CMB 2000 (vol 43 pp. 3)

Adin, Ron; Blanc, David
Resolutions of Associative and Lie Algebras
Certain canonical resolutions are described for free associative and free Lie algebras in the category of non-associative algebras. These resolutions derive in both cases from geometric objects, which in turn reflect the combinatorics of suitable collections of leaf-labeled trees.

Keywords:resolutions, homology, Lie algebras, associative algebras, non-associative algebras, Jacobi identity, leaf-labeled trees, associahedron
Categories:18G10, 05C05, 16S10, 17B01, 17A50, 18G50

17. CMB 1997 (vol 40 pp. 395)

Boudhraa, Zineddine
$D$-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, boundary
Categories:54D20, 54B99, 54D10, 54D30

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