scholarly journals Analysis of the effect of certain forcings on the nonoscillatory solutions of even order equations

1977 ◽  
Vol 24 (2) ◽  
pp. 234-244 ◽  
Author(s):  
Athanassios G. Kartsatos
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Asymptotic Properties ◽  
The Other ◽  
Negative Solution ◽  
Nonoscillatory Solutions ◽  
Existence Of Positive Solutions ◽  
Even Order ◽  
Image Position

AbstractProperties of solutions of, are studied in the causeuH(t, u)>0 foru≠0. It is shown that two inequalities may always be associated with (I) in such a way that if one of these inequalities has a small positive solution and the other inequality has a small negative solution, then (I) is oscillatory. Further asymptotic properties of (I) are studied under assumptions involving intermediate antiderivativesP(i)(t), withP(n)=Q. Several results of this type ensure the non-existence of positive solutions of (I).

scholarly journals Eigenvalue characterization for (n · p) boundary-value problems

1998 ◽  
Vol 39 (3) ◽  
pp. 386-407 ◽  
Author(s):  
Patricia J. Y. Wong ◽  
Ravi P. Agarwal
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Existence Of Positive Solutions ◽  
Image Position ◽  
Special Case

AbstractWe consider the (n, p) boundary value problemwhere λ > 0 and 0 ≤ p ≤ n - l is fixed. We characterize the values of λ such that the boundary value problem has a positive solution. For the special case λ = l, we also offer sufficient conditions for the existence of positive solutions of the boundary value problem.

Bifurcation analysis of a singular nonlinear Sturm–Liouville equation

2014 ◽  
Vol 16 (05) ◽  
pp. 1450012 ◽  
Author(s):  
Hernán Castro
Keyword(s):  
Boundary Condition ◽  
Lower Bound ◽  
Positive Solution ◽  
Positive Solutions ◽  
Liouville Equation ◽  
The Other ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Existence Of Positive Solutions ◽  
Sturm Liouville

In this paper we study existence of positive solutions to the following singular nonlinear Sturm–Liouville equation [Formula: see text] where α > 0, p > 1 and λ are real constants. We prove that when 0 < α ≤ ½ and p > 1 or when ½ < α < 1 and [Formula: see text], there exists a branch of continuous positive solutions bifurcating to the left of the first eigenvalue of the operator ℒαu = -(x2αu′)′ under the boundary condition limx→0x2αu′(x) = 0. The projection of this branch onto its λ component is unbounded in two cases: when 0 < α ≤ ½ and p > 1, and when ½ < α < 1 and [Formula: see text]. On the other hand, when ½ < α < 1 and [Formula: see text], the projection of the branch has a positive lower bound below which no positive solution exists. When 0 < α < ½ and p > 1, we show that a second branch of continuous positive solution can be found to the left of the first eigenvalue of the operator ℒαunder the boundary condition limx→0u(x) = 0. Finally, when α ≥ 1, the operator ℒαhas no eigenvalues under its canonical boundary condition at the origin, and we prove that in fact there are no positive solutions to the equation, regardless of λ ∈ ℝ and p > 1.

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

scholarly journals ON EXISTENCE OF POSITIVE SOLUTIONS OF NEUTRAL DIFFERENCE EQUATIONS

1994 ◽  
Vol 25 (3) ◽  
pp. 257-265
Author(s):  
J. H. SHEN ◽  
Z. C. WANG ◽  
X. Z. QIAN
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Difference Equations ◽  
Real Numbers ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Existence Of Positive Solutions

Consider the neutral difference equation \[\Delta(x_n- cx_{n-m})+p_nx_{n-k}=0, n\ge N\qquad (*) \] where $c$ and $p_n$ are real numbers, $k$ and $N$ are nonnegative integers, and $m$ is positive integer. We show that if \[\sum_{n=N}^\infty |p_n|<\infty \qquad (**) \] then Eq.(*) has a positive solution when $c \neq 1$. However, an interesting example is also given which shows that (**) does not imply that (*) has a positive solution when $c =1$.

scholarly journals Positive solutions and eigenvalues of conjugate boundary value problems

1999 ◽  
Vol 42 (2) ◽  
pp. 349-374 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Martin Bohner ◽  
Patricia J. Y. Wong
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Lower Bounds ◽  
Boundary Value ◽  
Upper And Lower Bounds ◽  
Special Cases ◽  
Image Position

We consider the following boundary value problemwhere λ > 0 and 1 ≤ p ≤ n – 1 is fixed. The values of λ are characterized so that the boundary value problem has a positive solution. Further, for the case λ = 1 we offer criteria for the existence of two positive solutions of the boundary value problem. Upper and lower bounds for these positive solutions are also established for special cases. Several examples are included to dwell upon the importance of the results obtained.

8.—On Second-order Differential Inequalities

1974 ◽  
Vol 72 (2) ◽  
pp. 109-127 ◽  
Author(s):  
F. V. Atkinson
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Second Order ◽  
Differential Inequalities ◽  
Existence Of Positive Solutions ◽  
Image Position

SynopsisThis paper is devoted to a study of differential equations and inequalities of the formandThe results are mainly concerned with the existence of positive solutions, their uniqueness in the case of (*), and bounds for these solutions.

Existence of Positive Global Solutions of Mixed Sublinear-Superlinear Problems

1988 ◽  
Vol 40 (5) ◽  
pp. 1222-1242
Author(s):  
W. Allegretto ◽  
Y. X. Huang
Keyword(s):  
Positive Solutions ◽  
Special Interest ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Linear Type ◽  
Existence Of Positive Solutions ◽  
Image Position ◽  
Quasilinear Problem

Consider the elliptic quasilinear problem:1in Rn, n ≧ 3, whereWe are interested in establishing sufficient conditions on f for the existence of positive solutions u(x) with specified behaviour at ∞. Of special interest to us are criteria which guarantee that u(x) decays at least as fast as |x|−α for some α ≧ 0, given below, in the case f(x, u, ∇u) contains terms of typeThat is: f is of mixed sublinear-super linear type. Our main result is Theorem 3 below which explicitly states sufficient conditions for the existence of such solutions.

scholarly journals The existence of positive solutions for an elliptic boundary value problem

2005 ◽  
Vol 2005 (13) ◽  
pp. 2005-2010
Author(s):  
G. A. Afrouzi
Keyword(s):  
Boundary Conditions ◽  
Positive Solution ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Elliptic Boundary ◽  
Dirichlet Boundary Conditions ◽  
Mountain Pass Lemma ◽  
First Eigenvalue ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

By using the mountain pass lemma, we study the existence of positive solutions for the equation−Δu(x)=λ(u|u|+u)(x)forx∈Ωtogether with Dirichlet boundary conditions and show that for everyλ<λ1, whereλ1is the first eigenvalue of−Δu=λuinΩwith the Dirichlet boundary conditions, the equation has a positive solution while no positive solution exists forλ≥λ1.

scholarly journals Existence of Nonoscillatory Solutions of First-Order Neutral Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Božena Dorociaková ◽  
Anna Najmanová ◽  
Rudolf Olach
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Nonoscillatory Solutions ◽  
Neutral Differential Equations ◽  
First Order ◽  
Existence Of Positive Solutions ◽  
Positive Functions ◽  
Bounded Below

This paper contains some sufficient conditions for the existence of positive solutions which are bounded below and above by positive functions for the first-order nonlinear neutral differential equations. These equations can also support the existence of positive solutions approaching zero at infinity

scholarly journals POSITIVE SOLUTIONS FOR A CLASS OF p(x)-LAPLACIAN PROBLEMS

2009 ◽  
Vol 51 (3) ◽  
pp. 571-578 ◽  
Author(s):  
G. A. AFROUZI ◽  
H. GHORBANI
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Symmetric Functions ◽  
Symmetric Domain ◽  
Sign Conditions ◽  
Image Position

AbstractWe consider the system where p(x), q(x) ∈ C1(RN) are radial symmetric functions such that sup|∇ p(x)| < ∞, sup|∇ q(x)| < ∞ and 1 < inf p(x) ≤ sup p(x) < ∞, 1 < inf q(x) ≤ sup q(x) < ∞, where −Δp(x)u = −div(|∇u|p(x)−2∇u), −Δq(x)v = −div(|∇v|q(x)−2∇v), respectively are called p(x)-Laplacian and q(x)-Laplacian, λ1, λ2, μ1 and μ2 are positive parameters and Ω = B(0, R) ⊂ RN is a bounded radial symmetric domain, where R is sufficiently large. We prove the existence of a positive solution when for every M > 0, $\lim_{u \rightarrow +\infty} \frac{h(u)}{u^{p^--1}} = 0$ and $\lim_{u \rightarrow +\infty} \frac{\gamma(u)}{u^{q^--1}} = 0$. In particular, we do not assume any sign conditions on f(0), g(0), h(0) or γ(0).

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