Existence of solutions for a damped nonlinear impulsive problem with Dirichlet boundary conditions

In this paper, we consider the existence of solutions for second‐order nonlinear damped impulsive differential equations with Dirichlet boundary condition. By critical point theory, we obtain some existence theorems of solutions for the nonlinear problem. We extend and improve some recent results. Copyright © 2013 John Wiley & Sons, Ltd.     

Variational approach to some damped Dirichlet problems with impulses

In this paper, we study the existence of solutions for damped nonlinear impulsive differential equations with Dirichlet boundary conditions. By using critical point theory and variational methods, we give some new criteria to guarantee that the impulsive problems have at least one solution. Some recent results are extended and significantly improved. Finally, some examples are presented to illustrate our main results. Copyright © 2013 John Wiley & Sons, Ltd.     

Variational formulation of the roller contact problem

We formulate the roller contact problem in terms of variational inequality with certain pseudodifferentional operator, and discuss a finite element technique for its numerical solution.     

Variational formulation of the problem of retained viscoplastic oil

A variational formulation is given for the problem of determining the limiting equilibrium of retained viscoplastic oil during its displacement with water from a stratified bed. It is shown that the basic approximation of the formulation admitting of an effective solution by the methods of the plane problem of non-linear filtering /1–5/ follows naturally from the variational formulation proposed, provided that the class of functions in which the solution is sought is restricted. Some estimates of the volume of the bed from which the oil is displaced are obtained on the basis of the variational formulation.     

Variational formulation for a non-local elliptic problem in the theory of varistors

The electric potential in a three-dimensional non-linear conductor of electricity in series with a one-dimensional resistor and a direct current generator is studied with a nonlocal elliptic problem. We give a variational formulation and a theorem of existence and uniqueness.     

Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations

In this paper, we study a linear Sturm-Liouville impulsive problem and a nonlinear Sturm-Liouville impulsive problem. By applying a new approach via critical point theory and variational methods, the existence results of positive solutions are obtained.     

Solvability of a class of impulsive damped vibration problems

In this paper, a class of impulsive damped vibration problems are considered. Existence results are obtained by using variational method and critical point theorem. The obtained results are also valid and new even if the impulsive damped vibration problem is reduced to impulsive Hamiltonian system and Hamiltonian system. Examples are presented to illustrate the feasibility and effectiveness of the results. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimal preconditioner for boundary element formulation of the Dirichlet problem in elasticity

An optimal preconditioning procedure for the numerical solution of two-dimensional Dirichlet problem for Lamé equations by boundary element method is constructed. An efficient algorithm for the above problem is also developed.     

Variational formulation of a modified couple stress theory and its application to a simple shear problem

A variational formulation is provided for the modified couple stress theory of Yang et al. by using the principle of minimum total potential energy. This leads to the simultaneous determination of the equilibrium equations and the boundary conditions, thereby complementing the original work of Yang et al. where the boundary conditions were not derived. Also, the displacement form of the modified couple stress theory, which is desired for solving many problems, is obtained to supplement the existing stress-based formulation. All equations/expressions are presented in tensorial forms that are coordinate-invariant. As a direct application of the newly obtained displacement form of the theory, a simple shear problem is analytically solved. The solution contains a material length scale parameter and can capture the boundary layer effect, which differs from that based on classical elasticity. The numerical results reveal that the length scale parameter (related to material microstructures) can have a significant effect on material responses.      

Variational formulation of a modified couple stress theory and its application to a simple shear problem

A variational formulation is provided for the modified couple stress theory of Yang et al. by using the principle of minimum total potential energy. This leads to the simultaneous determination of the equilibrium equations and the boundary conditions, thereby complementing the original work of Yang et al. where the boundary conditions were not derived. Also, the displacement form of the modified couple stress theory, which is desired for solving many problems, is obtained to supplement the existing stress-based formulation. All equations/expressions are presented in tensorial forms that are coordinate-invariant. As a direct application of the newly obtained displacement form of the theory, a simple shear problem is analytically solved. The solution contains a material length scale parameter and can capture the boundary layer effect, which differs from that based on classical elasticity. The numerical results reveal that the length scale parameter (related to material microstructures) can have a significant effect on material responses.     

Variational formulation for nonlinear impulsive fractional differential equations with (p,q)-Laplacian operator

In this paper, we focus on a class of generalized (p, q)-Laplacian-type impulsive fractional differential equation for 1 < p≤q < ∞, with a nonlinearity f containing fractional derivatives ${}_0{D}_t^{\alpha }u$ and ${}_0{D}_t^{\beta }u$ simultaneously and being imposed on mild assumptions contrasting with previous works. The underlying idea is based on the Mountain pass theorem and the iterative technique to obtain the existence of nontrivial solutions for our equation. Our work extends and supplements some existing results.     

Variational formulation for a general dynamic financial equilibrium problem: Balance law and liability formula

An evolutionary model for a multi-sector, multi-instrument financial equilibrium problem, with a general utility function, different prices for assets and liabilities and including the expenses for the management of the financial institutions is presented. In this general case, we give the evolutionary financial equilibrium conditions and prove their equivalence with an evolutionary variational inequality, from which existence results of the financial equilibrium follow. Moreover, making use of new nonlinear analysis results, we develop a Lagrange theory which allows one to study the behaviour of the financial equilibrium by means of the Lagrange multipliers. As a product of this analysis, we prove that for the financial model an equilibrium law together with a liability formula must be fulfilled, from which the reorganization of the existing financial disequilibrium depends.     

Variational approach to impulsive evolution equations

This article uses variational method for studying existence and uniqueness of solutions for impulsive evolution equations. The main techniques include Hilbert triple, Sobolev embedding theorem, Galerkin approximation and weak convergence for passing to the limit.     

Positive solutions for a Dirichlet problem

1. IntroductionSince the work of Ambrosetti and Rabinowitz[l], the problems similar to{;t2:<::l">, (l.l)have been studied extensively But it is well known that, for applying the Moulltain PassTheOrem, we atway8 assum that g(x, 8) is suPerlineax in s at indnity; moeove) a strongercondition like (AR) (see later on) is required. If these conditions are not satisfied, can wealso get solutions for problem (1.1) by a Mountain Pass Theorem? So, ill this paPer, westudy the following Dirichlet pr…     

Dirichlet duality and the nonlinear Dirichlet problem

We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form F (Hess u) = 0 on a smoothly bounded domain Ω ? ?ⁿ. In our approach the equation is replaced by a subset F ? Sym²(?ⁿ) of the symmetric n × n matrices with ?F ? { F = 0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric “F‐convexity” assumption on the boundary ?Ω. We also study the topological structure of F‐convex domains and prove a theorem of Andreotti‐Frankel type. Two key ingredients in the analysis are the use of “subaffine functions” and “Dirichlet duality.” Associated to F is a Dirichlet dual set F? that gives a dual Dirichlet problem. This pairing is a true duality in that the dual of F? is F, and in the analysis the roles of F and F? are interchangeable. The duality also clarifies many features of the problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: all branches of the homogeneous Monge‐Ampère equation over ?, ?, and ?; equations appearing naturally in calibrated geometry, Lagrangian geometry, and p‐convex Riemannian geometry; and all branches of the special Lagrangian potential equation. © 2008 Wiley Periodicals, Inc.     

Multiple solutions to damped Hamiltonian systems with impulsive effects

In this paper, we aim to prove the existence of solutions for impulsive Hamiltonian systems with first derivative. Two theorems of the existence of triple solutions are obtained via variational methods and three-critical-points theorems.     

Dirichlet problem for a nonlocal wave equation

We study the Dirichlet problem for a nonlocal wave equation in a rectangular domain. We prove the existence and uniqueness of a solution of the problem and show that determining whether the solution is unique can be reduced to determining whether a function of Mittag-Leffler type has real zeros. The obtained uniqueness condition turns into the uniqueness condition for the solution of the Dirichlet problem for the wave equation as the order of the fractional derivative in the equation tends to 2.