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Asymptotics of the Solution of a Singularly Perturbed Second-Order Delay Differential Equation
Differential Equations, 2020The authors analyze the existence and asymptotic behavior of a solution with an internal transition or boundary layer for a singularly perturbed second-order delay differential equation \[ \mu^2 y''=F(y(t),y(t-\sigma),t,\mu),\; 0 < t < T, \] satisfying the boundary conditions \[ y(t)=\varphi(t), \ -\sigma\leq t\leq 0,\ y(T)=y^T, \] where \(\mu>0\) is a
Ni, M. K. +2 more
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Multiplicity of periodic solutions for a class of second-order perturbed Hamiltonian systems
Journal of Mathematical Analysis and Applications, 2020The authors consider the following second-order Hamiltonian systems with non-autonomous perturbed term \[\left\{\begin{array}{l} \ddot{u}(t)+\nabla F(u) = \nabla_u G(t,u), ~~t \in \mathbb R,\\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,~~ T>0, \end{array}\right.\] where \(F(u)=-K(u)+W(u),~K,~W \in C^2(\mathbb R^N,\mathbb R),~G \in C^2(\mathbb R \times \mathbb R^
Liu, Yan, Guo, Fei
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NONTRIVIAL SOLUTION FOR A SECOND-ORDER BOUNDARY VALUE PROBLEM WITH p-LAPLACIAN AND PERTURBATION
Advances in Differential Equations and Control Processes, 2018Summary: This paper concerns with the existence of nontrivial solutions for a second-order boundary value problem with a \(p\)-Laplacian and a perturbation. By mountain-pass theorem, some sufficient conditions for the existence of solutions for a second-order boundary value problem are obtained which show that a nontrivial solution is generated by the ...
Liu, Xi-Lan, Liu, Nan-Nan, Wu, Shan
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Mediterranean Journal of Mathematics, 2016
The existence of homoclinic orbits is studied for the class of differential equations \[ -\ddot{u}(t)+L(t)u(t)=W_u(t,u(t))+G_u(t,u(t)). \] The existence of infinitely many homoclinic solutions is proven by using the theory of Bolle's perturbation method in critical point. The paper reports some generalizations of known results.
Zhang, Liang, Tang, Xianhua, Chen, Yi
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The existence of homoclinic orbits is studied for the class of differential equations \[ -\ddot{u}(t)+L(t)u(t)=W_u(t,u(t))+G_u(t,u(t)). \] The existence of infinitely many homoclinic solutions is proven by using the theory of Bolle's perturbation method in critical point. The paper reports some generalizations of known results.
Zhang, Liang, Tang, Xianhua, Chen, Yi
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Solutions of a class of singular second-order IVPs by homotopy-perturbation method
Physics Letters A, 2007In this Letter, solutions of a class of singular initial value problems (IVPs) in the second-order ordinary differential equations (ODEs) by homotopy-perturbation method (HPM) are presented. HPM yields solutions in convergent series forms with easily computable terms, and in some cases, yields exact solutions in one iteration.
M.S.H. Chowdhury, I. Hashim
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Acta Applicandae Mathematicae, 2014
The authors investigate the periodic boundary value problem: \[ \begin{aligned} &u''(t) + V_u(t,u(t)) = 0, \quad t \in (0,T) \setminus \{s_1,\ldots,s_m\},\\ &\triangle u'(s_k) = \lambda f_k(u(s_k)) + \mu g_k(u(s_k)),\\ &u(0) - u(T) = u'(0) - u'(T) = 0, \end{aligned} \] where \(0 < s_1 < s_2 < \ldots < s_m < T\), \(f_k = \nabla F_k\), \(g_k = \nabla G_k\
Heidarkhani, Shapour +2 more
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The authors investigate the periodic boundary value problem: \[ \begin{aligned} &u''(t) + V_u(t,u(t)) = 0, \quad t \in (0,T) \setminus \{s_1,\ldots,s_m\},\\ &\triangle u'(s_k) = \lambda f_k(u(s_k)) + \mu g_k(u(s_k)),\\ &u(0) - u(T) = u'(0) - u'(T) = 0, \end{aligned} \] where \(0 < s_1 < s_2 < \ldots < s_m < T\), \(f_k = \nabla F_k\), \(g_k = \nabla G_k\
Heidarkhani, Shapour +2 more
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Acta Applicandae Mathematicae, 2009
The authors are concerned with the oscillation of a perturbed nonlinear differential equation \[ \left( a\left( t\right) \psi\left( x\left( t\right) \right) x^{\prime }\left( t\right) \right) ^{\prime}+Q\left( t,x\left( t\right) \right) =P\left( t,x\left( t\right) ,x^{\prime}\left( t\right) \right) ,\tag{1} \] where \(a\) and \(\psi\) are positive ...
Zhang, Quanxin, Wang, Lei
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The authors are concerned with the oscillation of a perturbed nonlinear differential equation \[ \left( a\left( t\right) \psi\left( x\left( t\right) \right) x^{\prime }\left( t\right) \right) ^{\prime}+Q\left( t,x\left( t\right) \right) =P\left( t,x\left( t\right) ,x^{\prime}\left( t\right) \right) ,\tag{1} \] where \(a\) and \(\psi\) are positive ...
Zhang, Quanxin, Wang, Lei
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Journal of Elasticity, 1998
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BATRA R. C. +2 more
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BATRA R. C. +2 more
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