Existence of extremal solutions by approximation to a first-order initial dynamic equation with Carathéodory's conditions and discontinuous non-linearities*

This paper is devoted to the study of the existence of extremal solutions to a first-order initial value problem on an interval of an arbitrary time scale. We prove the existence of extremal solutions for problems satisfying Carathéodory's conditions. Moreover, they are approximated uniformly by a sequence of lower and upper solutions to this problem, respectively.

We also can warrant the existence and approximation of extremal solutions for the problem by relaxing their continuity properties.     

一个非局部方程解的存在性

本文旨在讨论一个线性非局部方程和相应非线性方程解存在的条件，对非局部积分核作一定限制后，得到在积分核变号条件下解的存在唯一性结果。     

On Solvability of Some Initial-Boundary Problems for Mathematical Models of the Motion of Nonlinearly Viscous and Viscoelastic Fluids

In the paper, the settings of initial-boundary and initial value problems arising in a number of models of movement of nonlinearly viscous or viscoelastic incompressible fluid are considered, and existence theorems for these problems are presented. In particular, the settings of initial-boundary value problems appearing in the regularized model of the movement of viscoelastic fluid with Jeffris constitutive relation are described. The theorems for the existence of weak and strong solutions for these problems in bounded domains are given. The initial value problem for a nonlinearly viscous fluid on the whole space is considered. The estimates on the right-hand side and initial conditions under which there exist local and global solutions of this problem are presented. The modification of Litvinov's model for laminar and turbulent flows with a memory is described. The existence theorem for weak solutions of initial-boundary value problem appearing in this model is given.     

The uniqueness of solutions to discrete, vector, two-point boundary value problems

Difference equations which may arise as discrete approximations to two-point boundary value problems for systems of second-order, ordinary differential equations are investigated and conditions are formulated under which solutions to the discrete problem are unique. Some existence, uniqueness implies existence, and convergence theorems for solutions to the discrete problem are also presented.     

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.     

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.     

On the proximal point method for equilibrium problems in Hilbert spaces

We analyse a proximal point method for equilibrium problems in Hilbert spaces, improving upon previously known convergence results. We prove global weak convergence of the generated sequence to a solution of the problem, assuming existence of solutions and rather mild monotonicity properties of the bifunction which defines the equilibrium problem, and we establish existence of solutions of the proximal subproblems. We also present a new reformulation of equilibrium problems as variational inequalities ones.     

Qualitative Properties of Eigenvectors Related to Multivalued Operators and some Existence Results

In this work, we study a class of nonlinear eigenvalue problems related to fully discontinuous operators. In particular, we prove the existence of a critical point for two distinct problems. Connected with this problem, we also study a minimization problem with constraint, and we investigate the existence of solutions for a resonant case near zero. Moreover, we give some estimates and qualitative properties of solutions by using the relative rearrangement theory.     

Strictly feasible solutions and strict complementarity in multiple objective linear optimization

Recently, Luc defined a dual program for a multiple objective linear program. The dual problem is also a multiple objective linear problem and the weak duality and strong duality theorems for these primal and dual problems have been established. Here, we use these results to prove some relationships between multiple objective linear primal and dual problems. We extend the available results on single objective linear primal and dual problems to multiple objective linear primal and dual problems. Complementary slackness conditions for efficient solutions, and conditions for the existence of weakly efficient solution sets and existence of strictly primal and dual feasible points are established. We show that primal-dual (weakly) efficient solutions satisfying strictly complementary conditions exist. Furthermore, we consider Isermann’s and Kolumban’s dual problems and establish conditions for the existence of strictly primal and dual feasible points. We show the existence of primal-dual feasible points satisfying strictly complementary conditions for Isermann’s dual problem. Also, we give an alternative proof to establish necessary conditions for weakly efficient solutions of multiple objective programs, assuming the Kuhn–Tucker (KT) constraint qualification. We also provide a new condition to ensure the KT constraint qualification.     

Existence and Multiplicity of Solutions of Second-Order Difference Boundary Value Problems

This paper concerns the existence and multiplicity of solutions of second-order difference boundary value problems. Under the assumptions which guarantee the existence of at least one nontrivial solution of the homogeneous problem, we obtain the existence of exactly three solutions of the nonhomogeneous problem with some other suitable conditions.     

Spectral Bounds for Tricomi Problems and Application to Semilinear Existence and Existence with Uniqueness Results

For the linear Tricomi problem, it is shown that real eigenvalues corresponding to generalized eigenfunctions must be positive and that the energy integral methods used to prove solvability results can give lower bounds on the spectrum. Exploiting the linear solvability theory and spectral information, standard nonlinear analysis tools are employed to yield results on existence and uniqueness for semilinear problems. In particular, using the Leray-Schauder principle, existence of generalized solutions with sublinear nonlinearities is established. For sublinear or asymptotically linear nonlinearities that satisfy a Lipschitz condition, the contraction mapping principle is employed to give results on existence with uniqueness. The Lipschitz constant depends on lower bounds for the spectrum of the linear problem. For certain superlinear problems, maximum principles for the linear problem are used via the method of upper and lower solutions to give results on existence.     

On the convergence of a regularization method for variational inequalities

For variational inequalities in a finite-dimensional space, the convergence of a regularization method is examined in the case of a nonmonotone basic mapping. It is shown that a fairly general sufficient condition for the existence of solutions to the original problem also guarantees the convergence and existence of solutions to perturbed problems. Examples of applications to problems on order intervals are presented.     

非齐次边界条件下两点边值问题单调解的存在性

研究了一类二阶非线性微分方程在非齐次边界条件下的两点边值问题单调解的存在性.运用锥拉伸与锥压缩不动点定理,分别得到了边值问题单调递增正解和单调递减负解存在的充分条件.     

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.     

Local and nonlocal boundary-value problems for third-order hyperbolic equations

This paper studies the problems of existence of classical solutions to the Goursat and Dirichlet problems and also to some nonlocal boundary-value problems for a linear third-order hyperbolic equation in a rectangular domain. The problems studied are reduced to a uniquely solvable integral equation. Thus, theorems of existence, uniqueness, and stability of solutions of the problem are proved. The effects of the influence of coefficients standing by lower derivatives on the well-posedness of the problems studied are revealed.     

二阶退化双曲型方程的Darboux型问题

本文讨论二阶退化双曲型方程的一些边值问题.文中先给出第一Darboux问题和一般斜微商边值问题的提法和解的表示式,然后使用复分析方法证明了上述问题解的存在唯一性.     

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.     

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.     

A global bifurcation phenomenon for second order singular boundary value problems

We establish an existence result of unbounded continuum of solutions from the trivial branch for a second order singular boundary value problem. As an application, we prove several existence and multiplicity results of positive solutions for certain singular boundary value problems.     

Topologically‐based characterizations of the existence of solutions of optimization‐related problems

Necessary and sufficient conditions for the existence of solutions of optimization‐related problems, defined on general sets, are established by using topologically‐based structures, without the usual linear structure. First, we prove that the existence of solutions to a variational relation problem is equivalent to the existence of either a KKM‐structure or a connectedness structure, satisfying additionally some verifiable conditions. Then, applying these results, we obtain such full (two‐way) characterizations for the existence of invariant points, solutions of quasiequilibrium problems of the Stampacchia‐Minty type, saddle points, and Nash equilibria for noncooperative games. The main advantages of our scheme with the mentioned two structures are that full characterizations for existence are obtained in a unified way, with no convexity assumptions, and also that one has flexibility to choose a suitable structure in investigations. (This flexibility is decisive when the natural underlying structure of the given problem is scarcely employed.)