Sublinear Case Research Articles

A NONOSCILLATION THEOREM FOR SUBLINEAR EMDEN-FOWLER EQUATIONS

Consider the Emden-Fowler equation (E) : y″ + a(x)|y|γ-1y = 0, where γ > 0 and a(x) is a positive continuous function on (0, ∞). I. T. Kiguradze showed in 1962 that if x(γ+3)/2+δa(x) is nonincreasing for any δ > 0, then equation (E) is nonoscillatory when γ > 1. We prove in this paper that the same theorem remains valid in the sublinear case, i.e., equation (E) when 0 < γ < 1.

Classification of Razor Blades to the filtration equation—the sublinear case

We study nonnegative solutions of the filtration equation u t =Δ ϕ( u) in R N(N⩾2) , where ϕ is continuous, increasing and sublinear. More precisely, we study the Razor Blades, i.e., solutions which may be singular at | x|=0 for t>0, and start with zero initial data. We first prove a nonexistence result when ϕ is too sublinear and we show an axial trace result in the other case: there exist a time τ≡ τ( u) and a Radon measure λ on [0, τ) such that u t= Δϕ(u)+δ 0⊗λ(t) in D′( R N×[0,τ)). Then u has a strong singularity on | x|=0 for any t> τ. We then prove existence of such solutions for any λ and τ as above, and give a uniqueness result for those solutions. Finally, we make a complete study of self-similar solutions (in the power case) which classify the possible asymptotic behaviours.

Mathematical modelling of drug transport in tumour multicell spheroids and monolayer cultures

In this paper we adapt an avascular tumour growth model to compare the effects of drug application on multicell spheroids and on monolayer cultures. The model for the tumour is based on nutrient driven growth of a continuum of live cells, whose birth and death generates volume changes described by a velocity field. The drug is modelled as an externally applied, diffusible material capable of killing cells, both linear and Michaelis–Menten kinetics for drug action on cells being studied. Numerical solutions of the resulting system of partial differential equations for the multicell spheroid case are compared with closed form solutions of the monolayer case, particularly with respect to the effects on the cell kill of the drug dosage and of the duration of its application. The results show an enhanced survival rate in multicell spheroids compared to monolayer cultures, consistent with experimental observations, and indicate that the key factor determining this is drug penetration. An analysis of the large time tumour spheroid response to a continuously applied drug at fixed concentration reveals up to three stable large time solutions, namely the trivial solution (i.e. a dead tumour), a travelling wave (continuously growing tumour) and a sublinear growth case in which cells reach a pseudo-steady-state in the core. Each of these possibilities is formulated and studied, with the bifurcations between them being discussed. Numerical solutions reveal that the pseudo-steady-state solutions persist to a significantly higher drug dose than travelling wave solutions.

Positive Solutions and Nonlinear Eigenvalue Problem of One-Dimensional p -Laplacian FDE

In this paper we are concerned with the nonlinear eigenvalue problem consisting of the functional differential equation with p -Laplacian operator, ( { p ( u '))' + u a ( t ) f ( t , u t ) = 0, { p ( s ) = | s | p m 2 s , p > 1, the initial condition, u ( s ) = } ( s ), m h s h 0, and the boundary condition, u (0) = 0 = u (1), where a ( t ) is allowed to be singular at the end points of (0, 1). Our results apply to more than just the sublinear and superlinear cases discussed in other references.

Fourt-order nonlinear oscillations of difference equations

The authors consider the fourth-order difference equation ▪ where f( n, u) may be classified as superlinear, sublinear, strongly superlinear, and strongly sublinear and Σ ∞ n=n 0 ( n/ r n ) < ∞. In superlinear and sublinear cases, necessary and sufficient conditions are obtained for the existence of nonoscillatory solutions of (*) with a specified asymptotic behavior. Further, in strongly superlinear and strongly sublinear cases, necessary and sufficient conditions are given for all solutions of (*) to be oscillatory.

Local behaviour of the solutions of a class of nonlinear elliptic systems

Here we study the behaviour near a punctual singularity of the positive solutions of semilinear elliptic systems in ${\mathbb R}^{N}(N\geq 3)$ given by \[ \left\{ \begin{array}{c} \Delta u+\left| x\right| ^{a}u^{s}v^{p}=0, \\ \Delta v+\left| x\right| ^{b}u^{q}v^{t}=0, \end{array} \right. \] (where $a,b,p,q,s,t\in $ ${\mathbb R}$ , $p,q>0,s,t\geq 0$). We describe the first undercritical case, and the sublinear and linear cases. The proofs do not use any variational methods, but lie essentially upon comparison properties between the two solutions $u$ and $v$, and the properties of the subsolutions and supersolutions of the scalar equation \[ \Delta f+\left| x\right| ^{\sigma }f^{\eta }=0 \] ($\sigma ,\eta \in $ ${\mathbb R}$ , $\eta >0$). This extends the classical study of the scalar equation when $0 <\eta <\max $ $(N,(N+\sigma ))/(N-2)$.

Existence and multiplicity results for quasilinear elliptic differential systems

We use a nonsmooth critical point theory to prove existence results for a variational system of quasilinear elliptic equations in both the sublinear and superlinear cases. We extend a technique of Bartsch to obtain multiplicity results when the system is invariant under the action of a compact Lie group. The problem is rather different from its scalar version, because a suitable condition on the coefficients of the system seems to be necessary in order to prove the convergence of the Palais-Smale sequences. Such condition is in some sense a restriction to the "distance" between the quasilinear operator and a semilinear one.

On the Existence of Positive Solutions of Quasilinear Elliptic Boundary Value Problems

We establish the existence of positive solutions to a class of quasilinear anisotropic problems which have either sublinear or superlinear nonlinearity. With a, b nonnegative constants and α, β positive constants, one example is If b−a<1 (sublinear case), then for each λ∈[0,∞), (1) has a solution. On the other hand, if b−a>1 (superlinear case), then there exists a λ*>0 such that for 0⩽λ<λ*, (1) has at least one solution, and for λ>λ* no solution exists.

Bifurcation for an elliptic system coupled in the linear part

We study linearly coupled elliptic systems without any assumption on the signs of the coefficients of the linear part. We derive first the Leray-Schauder degree of the associated linear system. Then we obtain sufficient conditions on the nonlinear terms for having bifurcation to coexistence states. Our results include the sublinear case. Keys words : Linearly coupled elliptic systems, Bifurcation to positive solutions, Coexistence states, Leray-Schauder degree. ∗email: jfleck@cict.fr †rpardo@sunma4.mat.ucm.es

On a nonlinear boundary value problem for fourth order equations

In this paper, we consider the fourth order ordinary differential equation together with the boundary conditions Some existence results for the problem (E)—(B) are obtained. The results here apply to more than just the sublinear and superlinear cases discussed by most other authors.

Nonoscillation theorems for second order nonlinear differential equations

We prove nonoscillation theorems for the second order Emden-Fowler equation (E): y + a ( x ) | y | γ − 1 y = 0 y+a(x)|y|^{\gamma -1}y=0 , γ &gt; 0 \gamma &gt;0 , where a ( x ) ∈ C ( 0 , ∞ ) a(x)\in C(0,\infty ) and γ ≠ 1 \gamma \not =1 . It is shown that when x ( γ + 3 ) / 2 + δ a ( x ) x^{(\gamma +3)/2+\delta }a(x) is nondecreasing for any δ &gt; 0 \delta &gt;0 and is bounded above, then (E) is nonoscillatory. This improves a well-known result of Belohorec in the sublinear case, i.e. when 0 &gt; γ &gt; 1 0&gt;\gamma &gt;1 and 0 &gt; δ &gt; ( 1 − γ ) / 2 0&gt;\delta &gt;(1-\gamma )/2 .

Necessary and Sufficient Conditions for the Oscillation of Forced Nonlinear Second Order Differential Equations with Delayed Argument

For a special class of the external forceg(t) and nonnegative potentiala(t), we give necessary and sufficient conditions for the oscillation of all solutions of a nonlinear second order forced differential equation with delayed argument of Emden–Fowler type in the formx″(t)+a(t)|x(φ(t))|νsgnx(φ(t))=g(t) in the superlinear (ν>1) and the sublinear cases (ν<1).

Oscillatory and asymptotic behaviour of fourth order nonlinear difference equations

This paper considers a class of fourth order nonlinear difference equations Δ2(rn Δ2(yn) + Δ2(rn,f(n,nn)=0,n ∈N(n0) wheref(n, y) may be classified as superlinear, sublinear, strongly superlinear and strongly sublinear. In superlinear and sublinear cases, necessary and sufficient conditions are obtained for the difference equation to admit the existence of nonoscillatory solutions with special asymptotic properties. In strongly superlinear and strongly sublinear cases, sufficient conditions are given for all solutions to be oscillatory.

A generalization of binomial queues

We give a generalization of binomial queues involving an arbitrary sequence ( m k ) k = 0,1,2,… of integers greater than one. Different sequences lead to different worst case bounds for the priority queue operations, allowing the user to adapt the data structure to the needs of a specific application. Examples include the first priority queue to combine a sub-logarithmic worst case bound for Meld with a sub-linear worst case bound for Delete_min.

Uniqueness theorems for some fourth-order elliptic equations

The purpose of this paper is to prove the uniqueness of positive solutions of some particular biharmonic boundary value problems. We also give some existence results in the sublinear case.

Uniqueness Theorems for Some Fourth-Order Elliptic Equations

The purpose of this paper is to prove the uniqueness of positive solutions of some particular biharmonic boundary value problems. We also give some existence results in the sublinear case.

The Existence of Homoclinic Solutions for Hyperbolic Equations

Studying homoclinic solutions of equations is one of the steps to go deeper in the understanding of dynamics. As it is known to the authors there are no papers studying homoclinic solutions of hyperbolic systems. In the paper we present a new variational method general enough to treat the problem of the existence of homoclinic solutions for the following semi-linear wave equation: xtt(t, y) – xyy(t, y) + g(t, y, x(t, y)) = 0 for 0 < y < Y, t ∈ R, x(t, 0) = 0, x(t, Y) = 0 for t ∈ R. Our approach covers both sublinear and super linear cases.

Oscillation criteria for second order nonlinear differential equations with integrable coefficients

Consider the second order nonlinear differential equation y + a ( t ) f ( y ) = 0 y + a\left ( t \right )f\left ( y \right ) = 0 where a ( t ) ∈ C [ 0 , ∞ ) , f ( y ) ∈ C 1 ( − ∞ , ∞ ) , f ′ ( y ) ≥ 0 a\left ( t \right ) \in C[0,\infty ),f\left ( y \right ) \in {C^1}\left ( { - \infty ,\infty } \right ),f’\left ( y \right ) \geq 0 , and y f ( y ) &gt; 0 yf\left ( y \right ) &gt; 0 for y ≠ 0 {\text {y}} \ne {\text {0}} . Furthermore, f ( y ) f\left ( y \right ) also satisfies either a superlinear or a sublinear condition, which covers the prototype nonlinear function f ( y ) = | y | γ sgn ⁡ y f\left ( y \right ) = |y{|^\gamma }\operatorname {sgn} y with γ &gt; 1 \gamma {\text { &gt; 1}} and 0 &gt; γ &gt; 1 0 &gt; \gamma &gt; 1 respectively. The coefficient a ( t ) a\left ( t \right ) is allowed to be negative for arbitrarily large values of t t and is integrable in the sense that the improper interval ∫ t ∞ a ( s ) d s = A ( t ) \int _t^\infty {a\left ( s \right )ds} = A\left ( t \right ) exists for each t ≥ 0 t \geq 0 . Oscillation criteria involving integrals of A ( t ) A\left ( t \right ) due to Coles and Butler for the superlinear and sublinear cases are shown to remain valid without the additional hypothesis that A ( t ) ≥ 0 A\left ( t \right ) \geq 0 .

Nonoscillatory solutions of second order differential equations with integrable coefficients

The asymptotic behavior of nonoscillatory solutions of the equation x + a ( t ) | x | γ sgn ⁡ x = 0 , γ &gt; 0 x + a\left ( t \right ){\left | x \right |^\gamma }\operatorname {sgn} x = 0,\gamma &gt; 0 , is discussed under the condition that A ( t ) = li m T → ∞ ∫ t T a ( s ) d s A\left ( t \right ) = {\text {li}}{{\text {m}}_{T \to \infty }}\int _t^T {a\left ( s \right )ds} exists and A ( t ) ≥ 0 A\left ( t \right ) \geq 0 for all t t . For the sublinear case of 0 &gt; γ &gt; 1 0 &gt; \gamma &gt; 1 , the existence of at least one nonoscillatory solution is completely characterized.

Diffusion in inhomogeneous media: Localization and positivity

We study nonnegative solutions of the Emden-Fowler equation on R+ in the sub-linear case. The results are used to investigate localization and positivity properties for a class of degenerate parabolic problems.