# Positive Solutions (positive + solution)

Distribution by Scientific Domains

## Selected Abstracts

### Positive solutions of non-positone Dirichlet boundary value problems with singularities in the phase variables

MATHEMATISCHE NACHRICHTEN, Issue 5 2008
Ravi P. Agarwal
Abstract The paper presents existence results for positive solutions of the differential equations x , + ,h (x) = 0 and x , + ,f (t, x) = 0 satisfying the Dirichlet boundary conditions. Here , is a positive parameter and h and f are singular functions of non-positone type. Examples are given to illustrate the main results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### Positive solutions and multiple solutions for periodic problems driven by scalar p -Laplacian

MATHEMATISCHE NACHRICHTEN, Issue 12 2006
Shouchuan Hu
Abstract In this paper we study a nonlinear second order periodic problem driven by a scalar p -Laplacian and with a nonsmooth, locally Lipschitz potential function. Using a variational approach based on the nonsmooth critical point theory for locally Lipschitz functions, we first prove the existence of nontrivial positive solutions and then establish the existence of a second distinct solution (multiplicity theorem) by strengthening further the hypotheses. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### Positive solutions of a higher order neutral differential equation

MATHEMATISCHE NACHRICHTEN, Issue 1 2003
John R. Graef
Abstract In this paper, we consider the higher order neutral delay differential equation where p : [0, ,) , (0, ,) is a continuous function, r > 0 and , > 0 are constants, and n > 0 is an odd integer. A positive solution x(t) of Eq. (*) is called a Class,I solution if y(t) > 0 and y,(t) < 0 eventually, where y(t) = x(t) , x(t , r). We divide Class,I solutions of Eq. (*) into four types. We first show that every positive solution of Eq. (*) must be of one of these four types. For three of these types, a necessary and sufficient condition is obtained for the existence of such solutions. A necessary condition for the existence of a solution of the fourth type is also obtained. The results are illustrated with examples. [source]

### Existence, uniqueness, stochastic persistence and global stability of positive solutions of the logistic equation with random perturbation

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 1 2007
Chunyan Ji
Abstract This paper discusses a randomized logistic equation (1) with initial value x(0)=x0>0, where B(t) is a standard one-dimension Brownian motion, and ,,(0, 0.5). We show that the positive solution of the stochastic differential equation does not explode at any finite time under certain conditions. In addition, we study the existence, uniqueness, boundedness, stochastic persistence and global stability of the positive solution. Copyright © 2006 John Wiley & Sons, Ltd. [source]

### Asymptotic behaviour of solutions of quasilinear evolutionary partial differential equations of parabolic type on unbounded spatial intervals

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2006
Poul
Abstract We study the long-time behaviour of solutions to a quasilinear parabolic problem on a half-line. The main result lies in showing the existence of a positive solution that converges to the travelling wave of solution to the stationary problem on the whole line. The main tools used here are the zero number theory and the concentration compactness principle. This result is a generalization of a result know for semilinear parabolic equations. Copyright © 2006 John Wiley & Sons, Ltd. [source]

### Blow-up estimates for a quasi-linear reaction,diffusion system

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2003
Yang Zuodong
In this paper, some sufficient conditions under which the quasilinear elliptic system -div(,,u,p-2,u) = uv, -div(,,u,q-2,u) = uv in ,N(N,3) has no radially symmetric positive solution is derived. Then by using this non-existence result, blow-up estimates for a class of quasilinear reaction,diffusion systems ut = div (,,u,p-2,u)+uv,vt = div(,,v,q-2,v) +uv with the homogeneous Dirichlet boundary value conditions are obtained. Copyright © 2003 John Wiley & Sons, Ltd. [source]

### Positive solutions of a higher order neutral differential equation

MATHEMATISCHE NACHRICHTEN, Issue 1 2003
John R. Graef
Abstract In this paper, we consider the higher order neutral delay differential equation where p : [0, ,) , (0, ,) is a continuous function, r > 0 and , > 0 are constants, and n > 0 is an odd integer. A positive solution x(t) of Eq. (*) is called a Class,I solution if y(t) > 0 and y,(t) < 0 eventually, where y(t) = x(t) , x(t , r). We divide Class,I solutions of Eq. (*) into four types. We first show that every positive solution of Eq. (*) must be of one of these four types. For three of these types, a necessary and sufficient condition is obtained for the existence of such solutions. A necessary condition for the existence of a solution of the fourth type is also obtained. The results are illustrated with examples. [source]

### Convergence analysis of the Newton,Shamanskii method for a nonsymmetric algebraic Riccati equation

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 6 2008
Yiqin Lin
Abstract In this paper, we consider the nonsymmetric algebraic Riccati equation arising in transport theory. An important feature of this equation is that its minimal positive solution can be obtained via computing the minimal positive solution of a vector equation. We apply the Newton,Shamanskii method to solve the vector equation. Convergence analysis shows that the sequence of vectors generated by the Newton,Shamanskii method is monotonically increasing and converges to the minimal positive solution of the vector equation. Numerical experiments show that the Newton,Shamanskii method is feasible and effective, and outperforms the Newton method. Copyright © 2008 John Wiley & Sons, Ltd. [source]

### Existence, uniqueness, stochastic persistence and global stability of positive solutions of the logistic equation with random perturbation

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 1 2007
Chunyan Ji
Abstract This paper discusses a randomized logistic equation (1) with initial value x(0)=x0>0, where B(t) is a standard one-dimension Brownian motion, and ,,(0, 0.5). We show that the positive solution of the stochastic differential equation does not explode at any finite time under certain conditions. In addition, we study the existence, uniqueness, boundedness, stochastic persistence and global stability of the positive solution. Copyright © 2006 John Wiley & Sons, Ltd. [source]

### Existence of multiple positive solutions for ,,u,,(u/,x,2)=u2*,1+,f(x)

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2002
Ting Cheng
Abstract In this paper, we consider the semilinear elliptic problem where ,,,N (N,3) is a bounded smooth domain such that 0,,, ,>0 is a real parameter, and f(x) is some given function in L,(,) such that f(x),0, f(x),,0 in ,. Some existence results of multiple solutions have been obtained by implicit function theorem, monotone iteration method and Mountain Pass Lemma. Copyright © 2002 John Wiley & Sons, Ltd. [source]

### A critical exponent in a degenerate parabolic equation

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2002
Michael Winkler
We consider positive solutions of the Cauchy problem in for the equation $$u_t=u^p\,\Delta u+u^q,\quad p\geq1,\; q\geq 1$$\nopagenumbers\end and show that concerning global solvability, the number q = p + 1 appears as a critical growth exponent. Copyright © 2002 John Wiley & Sons, Ltd. [source]

### On the numerical computation of blowing-up solutions for semilinear parabolic equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2001
Theoretical aspects related to the approximation of the semilinear parabolic equation: $u_t=\Delta u+f(u)$\nopagenumbers\end, with a finite unknown ,blow-up' time Tb have been studied in a previous work. Specifically, for , a small positive number, we have considered coupled systems of semilinear parabolic equations, with positive solutions and ,mass control' property, such that: \def\ve{^\varepsilon}$$u_t\ve=\Delta u\ve+f(u\ve)v\ve\qquad v_t\ve=\Delta v\ve-\varepsilon f(u\ve)v\ve$$\nopagenumbers\end The solution \def\ve{^\varepsilon}$$\{u\ve,v\ve\}$$\nopagenumbers\end of such systems is known to be global. It is shown that $$\|(u^\varepsilon-u)(\, .\, ,t)\|_\infty\leq C(M_T)\varepsilon$$\nopagenumbers\end, \def\lt{\char'74}$t\leq T \lt T_b$\nopagenumbers\end where $M_T=\|u(\, .\, ,T)\|_\infty$\nopagenumbers\end and $C(M_T)$\nopagenumbers\end is given by (6). In this paper, we suggest a numerical procedure for approaching the value of the blow-up time Tb and the blow-up solution u. For this purpose, we construct a sequence $\{M_\eta\}$\nopagenumbers\end, with $\lim_{\eta\rightarrow 0}M_\eta=\infty$\nopagenumbers\end. Correspondingly, for $\varepsilon\leq1/2C(M_\eta+1)=\eta^\alpha$\nopagenumbers\end and \def\lt{\char'74}$0\lt\alpha\lt\,\!1$\nopagenumbers\end, we associate a specific sequence of times $\{T_\varepsilon\}$\nopagenumbers\end, defined by $\|u^\varepsilon(\, .\, ,T_\varepsilon)\|_\infty=M_\eta$\nopagenumbers\end. In particular, when $\varepsilon=\eta\leq\eta^\alpha$\nopagenumbers\end, the resulting sequence $\{T_\varepsilon\equiv T_\eta\}$\nopagenumbers\end, verifies, $\|(u-u^\eta)(\, .\, ,t)\|_\infty\leq{1\over2}(\eta)^{1-\alpha}$\nopagenumbers\end, \def\lt{\char'74}$0\leq t\leq T_\eta\lt T_{\rm b}$\nopagenumbers\end with $\lim_{\eta\rightarrow 0}T_\eta=T_{\rm b}$\nopagenumbers\end. The two special cases of a single-point blow-up where $f(u)=\lambda{\rm e}^u$\nopagenumbers\end and $f(u)=u^p$\nopagenumbers\end are then studied, yielding respectively sequences $\{M_\eta\}$\nopagenumbers\end of order $O(\ln|\ln(\eta)|)$\nopagenumbers\end and $O(\{|\ln(\eta)|\}^{1/p-1})$\nopagenumbers\end. The estimate $|T_\eta-T_{\rm b}|/T_{\rm b}=O(1/|\ln(\eta)|)$\nopagenumbers\end is proven to be valid in both cases. We conduct numerical simulations that confirm our theoretical results. Copyright © 2001 John Wiley & Sons, Ltd. [source]

### Positive solutions of non-positone Dirichlet boundary value problems with singularities in the phase variables

MATHEMATISCHE NACHRICHTEN, Issue 5 2008
Ravi P. Agarwal
Abstract The paper presents existence results for positive solutions of the differential equations x , + ,h (x) = 0 and x , + ,f (t, x) = 0 satisfying the Dirichlet boundary conditions. Here , is a positive parameter and h and f are singular functions of non-positone type. Examples are given to illustrate the main results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### On a semilinear elliptic equation with singular term and Hardy,Sobolev critical growth

MATHEMATISCHE NACHRICHTEN, Issue 8 2007
Jianqing ChenArticle first published online: 8 MAY 200
Abstract In a previous work [6], we got an exact local behavior to the positive solutions of an elliptic equation. With the help of this exact local behavior, we obtain in this paper the existence of solutions of an equation with Hardy,Sobolev critical growth and singular term by using variational methods. The result obtained here, even in a particular case, relates with a partial (positive) answer to an open problem proposed in: A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations 177, 494,522 (2001). (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### Positive solutions and multiple solutions for periodic problems driven by scalar p -Laplacian

MATHEMATISCHE NACHRICHTEN, Issue 12 2006
Shouchuan Hu
Abstract In this paper we study a nonlinear second order periodic problem driven by a scalar p -Laplacian and with a nonsmooth, locally Lipschitz potential function. Using a variational approach based on the nonsmooth critical point theory for locally Lipschitz functions, we first prove the existence of nontrivial positive solutions and then establish the existence of a second distinct solution (multiplicity theorem) by strengthening further the hypotheses. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### Best Sobolev constants and quasi-linear elliptic equations with critical growth on spheres

MATHEMATISCHE NACHRICHTEN, Issue 12-13 2005
C. Bandle
Abstract Sharp existence and nonexistence results for positive solutions of quasilinear elliptic equations with critical growth in geodesic balls on spheres are established. The arguments are based on Pohozaev type identities and asymptotic estimates for Emden,Fowler type equations. By means of spherical symmetrization and the concentration-compactness principle existence and nonexistence results for general domains on spheres are obtained. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### Numerical blow-up for the porous medium equation with a source

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2004
Raúl Ferreira
Abstract We study numerical approximations of positive solutions of the porous medium equation with a nonlinear source, where m > 1, p > 0 and L > 0 are parameters. We describe in terms of p, m, and L when solutions of a semidiscretization in space exist globally in time and when they blow up in a finite time. We also find the blow-up rates and the blow-up sets, proving that there is no regional blow-up for the numerical scheme. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004 [source]