scholarly journals Three Homoclinic Solutions for Second-Order -Laplacian Differential System

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Jia Guo ◽  
Bin-Xiang Dai
Keyword(s):  
Differential System ◽  
Critical Points ◽  
Second Order ◽  
Homoclinic Solutions ◽  
Three Critical Points Theorem ◽  
New Criterion

We consider second-order -Laplacian differential system. By using three critical points theorem, we establish the new criterion to guarantee that this -Laplacian differential system has at least three homoclinic solutions. An example is presented to illustrate the main result.

scholarly journals TWO ALMOST HOMOCLINIC SOLUTIONS FOR A CLASS OF PERTURBED HAMILTONIAN SYSTEMS WITHOUT COERCIVE CONDITIONS

2015 ◽  
Vol 20 (1) ◽  
pp. 112-123 ◽  
Author(s):  
Ziheng Zhang ◽  
Rong Yuan
Keyword(s):  
Hamiltonian Systems ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Second Order ◽  
Homoclinic Solutions ◽  
New Criterion

In this paper we consider the existence of almost homoclinic solutions for the following second order perturbed Hamiltonian systems ü- L(t)u + ∇W (t, u) = f (t), (PHS) where is a symmetric and positive definite matrix for all t ∈ R, W ∈ C1(R×Rn, R) and ∇W (t, u) is the gradient of W (t, u) at u, f ∈ C(R, Rn) and belongs to L2(R, Rn). The novelty of this paper is that, assuming L(t) is bounded in the sense that there are two constants 0 < τ1 < τ2 < ∞ such that τ1 ∣u∣2 ≤ (L(t)u, u) ≤ τ2 ∣u∣2 for all (t, u) ∈ R × Rn, W(t, u) satisfies Ambrosetti-Rabinowitz condition and some other reasonable hypotheses, f (t) is sufficiently small in L2(R, Rn), we obtain some new criterion to guarantee that (PHS) has at least two nontrivial almost homoclinic solutions. Recent results in the literature are generalized and significantly improved.

An Asymptotical Stability in Variational Sense for Second Order Systems

2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Dariusz Idczak ◽  
Stanisław Walczak
Keyword(s):  
Sobolev Space ◽  
Differential System ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Second Order ◽  
Functional Parameter ◽  
Asymptotical Stability ◽  
Parameter Control ◽  
Initial Condition ◽  
Second Order Systems

AbstractIn this paper, a new, variational concept of asymptotical stability of zero solution to an ordinary differential system of the second order, considered in Sobolev space, is presented. Sufficient conditions for an asymptotical stability in a variational sense with respect to initial condition and functional parameter (control) are given. Relation to the classical asymptotical stability is illustrated.

scholarly journals Second Approximate Solution of Duffing Equation with Strong Nonlinearity by Homotopy Perturbation Method

1970 ◽  
Vol 30 ◽  
pp. 59-75
Author(s):  
M Alhaz Uddin ◽  
M Abdus Sattar
Keyword(s):  
Approximate Solution ◽  
Nonlinear Systems ◽  
Perturbation Method ◽  
Differential System ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Second Order ◽  
Strong Nonlinearity ◽  
Analytical Technique ◽  
Homotopy Perturbation

 In this paper, the second order approximate solution of a general second order nonlinear ordinary differential system, modeling damped oscillatory process is considered. The new analytical technique based on the work of He’s homotopy perturbation method is developed to find the periodic solution of a second order ordinary nonlinear differential system with damping effects. Usually the second or higher order approximate solutions are able to give better results than the first order approximate solutions. The results show that the analytical approximate solutions obtained by homotopy perturbation method are uniformly valid on the whole solutions domain and they are suitable not only for strongly nonlinear systems, but also for weakly nonlinear systems. Another advantage of this new analytical technique is that it also works for strongly damped, weakly damped and undamped systems. Figures are provided to show the comparison between the analytical and the numerical solutions. Keywords: Homotopy perturbation method; damped oscillation; nonlinear equation; strong nonlinearity. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 59-75  DOI: http://dx.doi.org/10.3329/ganit.v30i0.8504

scholarly journals Multiplicity of Solutions for Nonlocal Elliptic System of(p,q)-Kirchhoff Type

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Bitao Cheng ◽  
Xian Wu ◽  
Jun Liu
Keyword(s):  
Elliptic System ◽  
Critical Points ◽  
Three Solutions ◽  
Multiplicity Of Solutions ◽  
Periodic Condition ◽  
Kirchhoff Type ◽  
Three Critical Points Theorem ◽  
Palais Smale Condition

This paper is concerned with the following nonlocal elliptic system of (p,q)-Kirchhoff type−[M1(∫Ω|∇u|p)]p−1Δpu=λFu(x,u,v), in Ω,−[M2(∫Ω|∇v|q)]q−1Δqv=λFv(x,u,v), in Ω,u=v=0, on∂Ω.Under bounded condition onMand some novel and periodic condition onF, some new results of the existence of two solutions and three solutions of the above mentioned nonlocal elliptic system are obtained by means of Bonanno's multiple critical points theorems without the Palais-Smale condition and Ricceri's three critical points theorem, respectively.

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