The existence of infinitely many solutions all bifurcating from λ = 0

1991 ◽  
Vol 118 (3-4) ◽  
pp. 295-303 ◽  
Author(s):  
Wolfgang Rother
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Critical Point ◽  
Weak Solutions ◽  
Critical Point Theory ◽  
Point Theory ◽  
Infinitely Many Solutions ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

SynopsisWe consider the non-linear differential equationand state conditions for the function q such that (*) has infinitely many distinct pairs of (weak) solutions such that holds for all k ∈ ℕ. The main tools are results from critical point theory developed by A. Ambrosetti and P. H. Rabinowitz [1].

Solutions of a non-linear differential equation. I

1967 ◽  
Vol 63 (3) ◽  
pp. 743-754 ◽  
Author(s):  
C. E. Billigheimer
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

We consider in this paper solutions of the equationwhere the primes indicate differentiation with respect to s, and a, b, c are constants.

19.—Concavity and the Evolutionary Properties of a Class of General Materials

1975 ◽  
Vol 72 (3) ◽  
pp. 239-243 ◽  
Author(s):  
R. N. Hills ◽  
R. J. Knops
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Initial Data ◽  
Linear Differential Equation ◽  
Weak Solutions ◽  
Finite Time ◽  
Escape Time ◽  
Non Linear ◽  
Linear Differential

Concavity arguments have been used by Knops, Levine and Payne [1] to discuss evolutionary properties of weak solutions to an abstract non-linear differential equation in a Hilbert space. These authors demonstrate that provided the non-linearity is suitably restricted and for specified initial data, the norm of the solution becomes unbounded in a finite time. In other words, the solution possesses a finite escape time.

scholarly journals Weak Solutions for ap-Laplacian Antiperiodic Boundary Value Problem with Impulsive Effects

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Keyu Zhang ◽  
Jiafa Xu ◽  
Wei Dong
Keyword(s):  
Differential Equation ◽  
Variational Method ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Critical Point ◽  
Weak Solutions ◽  
Critical Point Theory ◽  
Impulsive Differential Equation ◽  
Point Theory ◽  
Existence Of Weak Solutions

By virtue of variational method and critical point theory, we will investigate the existence of weak solutions for ap-Laplacian impulsive differential equation with antiperiodic boundary conditions.

scholarly journals Existence of limit cycles for a class of autonomous systems

1983 ◽  
Vol 28 (3) ◽  
pp. 331-337
Author(s):  
Anthony Sofo
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Autonomous Systems ◽  
Non Linear ◽  
Stable Periodic ◽  
Linear Differential ◽  
Image Position

A proof is given for the existence of at least one stable periodic limit cycle solution for the polynomial non-linear differential equation of the formin some cases where the Levinson-Smith criteria are not directly applicable.

Infinitely many solutions for non-homogeneous Neumann problems in Orlicz-Sobolev spaces

2018 ◽  
Vol 68 (4) ◽  
pp. 867-880
Author(s):  
Saeid Shokooh ◽  
Ghasem A. Afrouzi ◽  
John R. Graef
Keyword(s):  
Variational Methods ◽  
Critical Point ◽  
Neumann Problem ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Critical Point Theory ◽  
Point Theory ◽  
Infinitely Many Solutions ◽  
Neumann Problems

Abstract By using variational methods and critical point theory in an appropriate Orlicz-Sobolev setting, the authors establish the existence of infinitely many non-negative weak solutions to a non-homogeneous Neumann problem. They also provide some particular cases and an example to illustrate the main results in this paper.

A Generalization of Picard's Method of Successive Approximation

1936 ◽  
Vol 32 (1) ◽  
pp. 86-95 ◽  
Author(s):  
H. O. Hirschfeld
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Boundary Value Problem ◽  
Linear Differential Equation ◽  
Successive Approximation ◽  
Linear Integral Equation ◽  
Non Linear ◽  
Linear Differential ◽  
Linear Integral ◽  
Image Position

It is well known that the boundary value problem for the non-linear differential equationcan be reduced with help of a Green's function K (x, ξ) to a non-linear integral equation of the type

Solutions of a non-linear differential equation. II

1968 ◽  
Vol 64 (1) ◽  
pp. 127-139
Author(s):  
C. E. Billigheimer
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position ◽  
Solution Behaviour

We consider in this paper the solution behaviour as s → 0 of the equationwhere the primes indicate differentiation with respect to s, and a, b, c are constants.

scholarly journals Steady Motion of Ice Sheets

1980 ◽  
Vol 25 (92) ◽  
pp. 229-246 ◽  
Author(s):  
L. W. Morland ◽  
I. R. Johnson
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Power Law ◽  
Perturbation Expansion ◽  
Ice Sheet ◽  
Leading Order ◽  
Plane Flow ◽  
General Law ◽  
Non Linear ◽  
Linear Differential

AbstractSteady plane flow under gravity of a symmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding according to a shear-traction-velocity power law, is treated. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, with illustrations presented for Glen’s power law, the polynomial law of Colbeck and Evans, and a Newtonian fluid. Uniform temperature is assumed so that effects of a realistic temperature distribution on the ice response are not taken into account. In dimensionless variables a small paramter ν occurs, but the ν = 0 solution corresponds to an unbounded sheet of uniform depth. To obtain a bounded sheet, a horizontal coordinate scaling by a small factor ε(ν) is required, so that the aspect ratio ε of a steady ice sheet is determined by the ice properties, accumulation magnitude, and the magnitude of the central thickness. A perturbation expansion in ε gives simple leading-order terms for the stress and velocity components, and generates a first order non-linear differential equation for the free-surface slope, which is then integrated to determine the profile. The non-linear differential equation can be solved explicitly for a linear sliding law in the Newtonian case. For the general law it is shown that the leading-order approximation is valid both at the margin and in the central zone provided that the power and coefficient in the sliding law satisfy certain restrictions.

A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

1986 ◽  
Vol 102 (3-4) ◽  
pp. 253-257 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Series ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
The Real ◽  
Linear Differential ◽  
Image Position

SynopsisIn an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equationhas the asymptotic expansionas |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.We show that if the real valued function q admits the expansionin a neighbourhood of 0, then

Existence of infinitely many solutions for fractional p-Laplacian Schrödinger–Kirchhof-Type equations with general potentials

2021 ◽  
pp. 2250095
Author(s):  
Ghania Benhamida ◽  
Toufik Moussaoui
Keyword(s):  
Critical Point ◽  
Critical Point Theory ◽  
Point Theory ◽  
Infinitely Many Solutions ◽  
Kirchhoff Type ◽  
Laplacian Equations

In this paper, we use the genus properties in critical point theory to prove the existence of infinitely many solutions for fractional [Formula: see text]-Laplacian equations of Schrödinger-Kirchhoff type.

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