Existence and asymptotic behavior of solutions to a singular parabolic equation

In this paper, we study the initial-boundary value problem for a class of singular parabolic equations. Under some conditions, we obtain the existence and asymptotic behavior of solutions to the problem by parabolic regularization method and the sub-super solutions method. As a byproduct, we prove the existence of solutions to some problems with gradient terms, which blow up on the boundary.     

Higher order boundary value problems on time scales

In this paper we study the existence of solutions for Lidstone boundary value problems on time scale. Firstly, by using Schauder fixed point theorem in a cone, we obtain the existence of solutions to a Lidstone boundary value problem (LBVP). Secondly, existence result for this problem is also given by the monotone method. Finally, by using Krasnosel'skii fixed point theorem, it is proved that the LBVP has a positive solution.     

Quasi-variational relation problems and generalized Ekeland's variational principle with applications

In this article, we study the existence of solutions for a quasivariational relation problem and then give applications to the existence of solutions for set-valued Ekeland's principle, generalized vector Ekeland's variational principle and generalized equilibrium problems. Our results and techniques of proof are different from any existence result in the literature.     

Existence and multiplicity of weak solutions for a singular semilinear elliptic equation

In this paper we study existence and multiplicity of weak solutions of the homogenous Dirichlet problem for a singular semilinear elliptic equation with a quadratic gradient term. The proofs for the main results are based on a priori estimates of solutions of approximate problems.     

An inverse pollution problem in porous media

In this paper, we deal with an inverse source problem of integro-differential parabolic equations, which comes from nonlinear pollution problems in porous media. We study the existence and uniqueness of solutions for the direct problem as well as the existence of quasisolutions of the inverse source problem in an appropriate class of admissible source functions.     

Nonlinear eigenvalue Neumann problems with discontinuities

In this paper we study nonlinear eigenvalue problems with Neumann boundary conditions and discontinuous terms. First we consider a nonlinear problem involving the p-Laplacian and we prove the existence of a solution for the multivalued approximation of it, then we pass to semilinear problems and we prove the existence of multiple solutions. The approach is based on the critical point theory for nonsmooth locally Lipschitz functionals.     

Nonlinear elliptic differential equations with multivalued nonlinearities

In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all ℝ. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of ℝ. This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).     

Principal frequency and existence of solutions for quasilinear elliptic obstacle problems

In this paper, we are interested in the first eigenvalue of p-Laplacian and the relation between the first eigenvalue and the existence (or nonexistence) of nontrivial (positive) solution for quasilinear elliptic obstacle problems. Utilizing the fact that obstacle problem have consanguineous relations with corresponding equation, we get a simple approach to study the properties of solutions of obstacle problems, such as existence and nonexistence, regularity and stability, etc. In this paper we are mainly concerned with the existence and nonexistence.     

Existence of Solutions for Nonlinear Implicit Differential Equations: An Equilibrium Problem Approach

In this article, we study the existence of solutions for nonlinear implicit differential equations associated to a time-dependent pseudomonotone (respectively, quasimonotone) operator by using a new approach based on the theory of equilibrium \hboxproblems. More precisely, we first establish some existence results of solutions for mixed equilibrium problems where the \hboxbifunctions are, respectively, maximal monotone and \hboxpseudomonotone (quasimonotone) in the topological sense. Then, we write the nonlinear implicit differential equations in the form of a mixed equilibrium problem. By using the existence results for mixed equilibrium problems, we establish the existence of solutions of nonlinear implicit differential equations. This new approach provides some new and nice results which improve and unify most of the recent results obtained in this direction.     

Optimal control problems for the reaction–diffusion–convection equation with variable coefficients

The solvability of optimal control problems is proved on both weak and strong solutions of a boundary value problem for the nonlinear reaction–diffusion–convection equation with variable coefficients. In the second case, the requirements for smoothness of the multiplicative control are reduced. The study of extremal problems is based on the proof of the solvability of the corresponding boundary value problems and on the qualitative analysis of their solutions properties. The large data existence results for weak solutions, the maximum principle as well as the local existence and uniqueness of a strong solution are established. Moreover, an optimal feedback control problem is considered. Using methods of the theory of topological degree for set-valued perturbations (with aspheric image sets) of generalized monotone operators, we obtain sufficient conditions for the solvability of this problem in the class of weak solutions.     

Strong solutions of periodic parabolic problems with discontinuous nonlinearities

We study the problem of finding time-periodic solutions of a parabolic equation with the homogeneous Dirichlet boundary condition and with a discontinuous nonlinearity. We assume that the nonlinearity is equal to the difference of two superpositionally measurable functions nondecreasing with respect to the state variable. For such a problem, we prove the principle of lower and upper solutions for the existence of strong solutions without additional constraints on the “jumping-up” discontinuities in the nonlinearity. We obtain existence theorems for strong solutions of this class of problems, including theorems on the existence of two nontrivial solutions.     

First kind symmetric periodic solutions and their stability for the Kepler problem and anisotropic Kepler problem plus generalized anisotropic perturbation

Motivated by some problems in Celestial Mechanics that combines quasihomogeneous potential in the anisotropic space, we investigate the existence of several families of first kind symmetric periodic solutions for a family of planar perturbed Kepler problem. In addition, we give sufficient conditions for the existence of first kind periodic solutions and also we characterize its type of stability. As an application of this general situation, we discuss the existence of symmetric periodic solutions for the anisotropic Kepler problem plus a generalized anisotropic perturbation, (shortly, p-AKPQ problem) and for the Kepler problem plus a generalized anisotropic perturbation (shortly, p-KPQ problem), as continuation of circular orbits of the two-dimensional Kepler problem. To get this objective, we consider different types of perturbations and then we apply our main result.     

Existence of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions

In this paper, we consider the existence of solutions for a class of nonlinear impulsive problems with Dirichlet boundary conditions. We obtain some new existence theorems of solutions for the nonlinear impulsive problem by using critical point theory. We extend and improve some recent results.     

带任意质量的平面2N＋1体问题的周期解

本文研究平面2N+1-体问题的周期解.利用Hermite矩阵和循环矩阵的性质,给出了几何中心带一个质量的平面正多边形套问题存在周期解的必要和充分条件,证明了该周期解的存在唯一性,推广了文献[9]的结论.     

随机二阶多值周期问题解的存在性

本文研究了随机二阶多值周期问题解的存在性,在非常一般的假定下建立了凸和非凸问题解的存在性定理,并给出了在随机周期反馈控制系统中的应用．     

Optimization by the Homogenization Method for Nonlinear Elliptic Dirichlet Systems

We study the existence of solutions of control problems relative to a nonlinear elliptic system with Dirichlet boundary conditions. In this problem, the control variables are the coefficients of the equations and the open set where they are posed. It is known that this class of problems has no solution in general, but using homogenization results about elliptic systems we show the existence of solutions when the controls are searched in a bigger set. These results are related to the selection of optimal materials and shapes.     

Existence of solutions of boundary value problems for singular fractional <Emphasis Type="Italic">q</Emphasis>-difference equations

In this paper, we study the existence and uniqueness of solutions for a class of singular three-point boundary value problems of fractional q-difference equations invovling fractional q-derivative of Riemann–Liouville type. Based on the generalization of Banach contraction principle, we obtain a sufficient condition for existence and uniqueness of solutions of the problem. By applying the Krasnoselskii’s fixed point theorem, we establish a sufficient condition for the existence of at least one solution of the problem. As applications, two examples are presented to illustrate our main results.     

On a class of elliptic problems with indefinite nonlinearities

In this paper we apply variational and sub-supersolution methods to study the existence and multiplicity of nonnegative solutions for a class of indefinite semilinear elliptic problems that depend on a parameter. The results on the existence of solutions do not impose any growth condition at infinity on the term which depends on the parameter. To derive such results, first we find a positive supersolution by solving an auxiliary problem. Then we use a truncation argument and a global minimization method. The main hypothesis for the existence of two nonzero solutions is that the indefinite term is the product of a weight function, having a thick zero set, and a nonlinear function which satisfies the Ambrosetti–Rabinowitz superlinear condition. Results for some corresponding indefinite problems are also established.     

Efficiency in vector quasi-equilibrium problems and applications

In this paper, we give sufficient conditions for the existence of efficient solutions of a generalized vector quasi-equilibrium problem in topological vector spaces. The motivations for introducing this problem come from practical problems in traffic networks and the optimal control theory for discrete-time dynamical systems. The main results of the paper are proven with the help of a strongly monotonic function which can be constructed from the data of the problem under consideration. Some notions of cone-semicontinuity of set-valued maps, weaker than the usual concepts of semicontinuity, are also used in our study. As applications, we obtain existence results in vector quasi-optimization problems, Stampacchia set-valued vector quasi-variational inequality problems and Pareto vector quasi-saddle point problems. All these results are different from the corresponding ones in the literature.     

Existence of Global Solutions for a Class of Abstract Second-Order Nonlocal Cauchy Problem with Impulsive Conditions in Banach Spaces

In this paper, we are concerned with a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. First, we study the existence of mild solutions for a class of second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces on an interval [0,a]. Later, we study a couple of cases where we can establish the existence of global solutions for a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. We apply our theory to study the existence of solutions for impulsive partial differential equations.