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For a singularly perturbed reaction–diffusion equation, we study the structure of the internal transition layer in the case of a balanced reaction with a weak discontinuity. The existence of solutions with an internal transition layer (contrast structures) is proved, the question of their stability is investigated, and asymptotic approximations to solutions of this type are obtained. It is shown that in the case of reaction balance, the presence of even a weak (asymptotically small) reaction discontinuity can lead to the formation of contrast structures of finite size, both stable and unstable. [ABSTRACT FROM AUTHOR]

We study integrally bounded solutions of the differential equation , where is a linear differential operator of order defined on functions () and is a finite-dimensional Euclidean space). The right-hand side is an integrally bounded function on ranging in and satisfying the inequality , , . Conditions are given on the operator and the function that guarantee an inverse inequality of the following form for the solutions under consideration: where the constant is independent of the choice of a real number and function . [ABSTRACT FROM AUTHOR]

We study a boundary value problem for a quasilinear reaction–diffusion–advection ordinary differential equation with a KPZ-nonlinearity containing the squared gradient of the unknown function. The noncritical and critical cases of existence of an internal transition layer are considered. An asymptotic approximation to the solution is constructed, and the asymptotics of the transition layer point is determined. Existence theorems are proved using the asymptotic method of differential inequalities, the Lyapunov asymptotic stability of solutions is proved by the narrowing barrier method, and instability theorems are proved with the use of unordered upper and lower solutions. [ABSTRACT FROM AUTHOR]

For differential inequalities with the -Laplacian in the principal part, we find conditions for the absence of solutions in unbounded domains. Examples are given that demonstrate the accuracy of these conditions. [ABSTRACT FROM AUTHOR]

We establish estimates (interior and up to the boundary) of solutions of the inequality , where is a linear second-order differential operator with impulse singularities. The results in particular apply to the Neumann boundary value problem. The relationship between this inequality and the class of concave functions of one variable is studied. [ABSTRACT FROM AUTHOR]

We prove the existence and study the stability of time-periodic boundary layer solutions for a two-dimensional reaction–diffusion problem with a small parameter multiplying the parabolic operator for the case of singularly perturbed boundary conditions of the third kind. An asymptotic approximation to such solutions with respect to the small parameter is constructed. Conditions under which these solutions are Lyapunov asymptotically stable, as well as conditions under which such solutions are unstable, are obtained. For the proof, we used results based on applying the asymptotic method of differential inequalities and the Krein–Rutman theorem. [ABSTRACT FROM AUTHOR]

We study boundary value and control problems using methods based on results on operator equations in partially ordered spaces. Sufficient conditions are obtained for the existence of a coincidence point for two mappings acting from a partially ordered space into an arbitrary set, an estimate for such a point is found, and corollaries about a fixed point for a mapping that acts in a partially ordered space and is not monotone are derived. The established results are applied to the study of functional and differential equations. For the Nemytskii operator in the space of measurable vector functions, sufficient conditions for the existence of a fixed point are obtained and it is shown that these conditions do not follow from the classical fixed point theorems. Assertions on the existence and estimates of the solution of the Cauchy problem are proved, and the solutions are given to a periodic boundary value problem and a control problem for systems of ordinary differential equations of the first order unsolved for the derivative of the desired vector function. [ABSTRACT FROM AUTHOR]

For the solutions of the linear differential inequality , where is a linear differential operator of order defined on functions of one variable, we establish estimates of the form , where , , , is the Sobolev space of times differentiable functions, is the Lebesgue space of integrable functions, and the constant is independent of the choice of the function . [ABSTRACT FROM AUTHOR]

We consider a multidimensional singularly perturbed stationary diffusion model with a cubic nonlinearity. For models of this type, a modified asymptotic method of boundary functions, which extends the classical asymptotic analysis methods to the case of multidimensional problems, and the asymptotic method of differential inequalities, which is based on the comparison principle, are used to study the existence of asymptotically Lyapunov stable solutions with internal layers as stationary solutions of the corresponding parabolic problems. Sufficient conditions are established for the existence of such solutions in the form of some conditions on the coefficients of the equation, an asymptotic approximation to the solution of an arbitrary accuracy order with coefficients is constructed in closed form, and the formal constructions are justified. This result can be used for creating efficient numerical algorithms for direct and coefficient inverse problems for stationary equations of the reaction–diffusion–advection type as well as for constructing test examples. Heat and mass transfer problems occurring in chemical industry are pointed out as possible applications of our results. [ABSTRACT FROM AUTHOR]

We study the Lyapunov asymptotic stability of the stationary solution of the spatially one-dimensional initial–boundary value problem for a nonlinear singularly perturbed differential equation of the reaction–diffusion–advection (RDA) type in the case where the advection and reaction terms undergo a discontinuity of the first kind at some interior point of an interval. Sufficient conditions are derived for the existence of a stable stationary solution with a large gradient near the point of discontinuity. An asymptotic method of differential inequalities is used to prove the existence and stability theorems. The resulting stability conditions can be employed to create mathematical models and develop numerical methods for solving "stiff" problems arising in various applications, for example, when simulating combustion processes and in nonlinear wave theory. [ABSTRACT FROM AUTHOR]

We consider a singularly perturbed boundary value problem for a second-order ordinary differential equation with nonlinear right-hand side containing functions of delayed argument. We prove the existence of a solution with a transition layer that has a more sophisticated structure than the ones studied before and construct a uniform asymptotic approximation to this solution with respect to a small parameter. Vasil'eva's method is used when constructing the asymptotic approximation, while the existence theorem is proved by combining the matching method and the asymptotic differential inequality method. Conditions for the existence of a solution with monotone internal transition and boundary layers are stated. An example illustrating the class of problems studied here is given. [ABSTRACT FROM AUTHOR]

We study a nonlocal boundary value problem for a parabolic equation in the multidimensional case. A locally one-dimensional difference scheme is constructed to solve this problem numerically. A priori estimates are derived by the method of energy inequalities in the differential and difference settings. The uniform convergence of the locally one-dimensional scheme is proved. [ABSTRACT FROM AUTHOR]

We prove the existence and the Lyapunov asymptotic stability of a stationary boundary layer solution of the initial–boundary value problem for a two-dimensional singularly perturbed reaction–diffusion–advection equation. We construct an asymptotic approximation to this solution using the boundary function method. The proofs are based on the applicability of the asymptotic method of differential inequalities. [ABSTRACT FROM AUTHOR]

We consider a reaction-diffusion-type equation in a two-dimensional domain containing the interface between media with distinct characteristics along which the reactive term has a discontinuity of the first kind. We assume that the interface between the media, as well as the functions describing the reactions, periodically varies in time. We study the existence of a stable periodic solution of a problem with an internal layer. To prove the existence, stability, and local uniqueness of the solution, we use the asymptotic method of differential inequalities, which we generalized to a new class of problems with discontinuous nonlinearities. [ABSTRACT FROM AUTHOR]

We define the set of ordered covering of a mapping that acts in partially ordered spaces; we suggest a method for finding the set of ordered covering of vector functions of several variables and the Nemytskii operator acting in Lebesgue spaces. We prove assertions on operator inequalities in arbitrary partially ordered spaces. We obtain conditions that use a set of ordered covering of the corresponding mapping and ensure that the existence of an element u such that f( u) ≥ y implies the solvability of the equation f( x) = y and the estimate x ≤ u for its solution. We study the problem on the existence of the minimal and least solutions. These results are used for the analysis of an implicit differential equation. For the Cauchy problem, we prove a theorem on an inequality of the Chaplygin type. [ABSTRACT FROM AUTHOR]

For a linear system of functional-differential equations of delay type, by the Lyapunov direct method, we prove a sufficient criterion for exponential dichotomy in the H-topology in terms of operator inequalities. The original problem is reduced to the same problem for a difference equation of the form u = Λ u in the state space H(0, 1) with compact operator Λ. Subspaces realizing the dichotomy are constructed. The constructions are carried out in a basis taking into account the specific properties of the space H and coordinated with the metric chosen in it. An illustrative example is considered. [ABSTRACT FROM AUTHOR]

We consider a singularly perturbed initial-boundary value problem for a system of parabolic equations with distinct powers of a small parameter. We construct and justify the asymptotics of a solution with an internal transition layer. The proof is carried out with the use of the asymptotic method of differential inequalities. [ABSTRACT FROM AUTHOR]

Some results due to Fang and Peterson on generalized variational inequalities in the space ℝ are extended to infinite-dimensional spaces. Theorems on the existence of solutions of such inequalities under generalized coercivity conditions are obtained. [ABSTRACT FROM AUTHOR]

For higher-order nonlinear differential equations with deviating arguments and with nonintegrable singularities with respect to the time variable, we establish sharp sufficient conditions for the Cauchy problem to be solvable and well-posed. [ABSTRACT FROM AUTHOR]

On the basis of the nonlinear capacity method, we obtain exact solvability conditions for some classes of stationary and evolution nonlinear differential inequalities with singular coefficients. [ABSTRACT FROM AUTHOR]

We consider a nonlinear boundary value problem for an equation of reaction-diffusion-advection type whose solutions have internal transition layers (contrast structures). We prove the existence of such solutions, construct their asymptotics, and estimate its accuracy. The analysis is carried out with the use of the asymptotic method of differential inequalities and permits one to prove the Lyapunov stability of the contrast structures. [ABSTRACT FROM AUTHOR]

We consider initial-boundary value problems for a class of singularly perturbed nonlinear integro-differential equations. In applications, they are referred to as nonlocal reactionadvection-diffusion equations, and their solutions have moving interior transition layers (fronts). We construct the asymptotics of such solutions with respect to a small parameter and estimate the accuracy of the asymptotics. To justify the asymptotics, we use the asymptotic differential inequality method. [ABSTRACT FROM AUTHOR]

For a triple { V, H, V*} of Hilbert spaces, we consider an evolution inclusion of the form u′( t)+ A( t) u( t)+ δϕ( t, u( t)) ∋ f( t), u(0) = u0, t ∈ (0, T ], where A( t) and ϕ( t, ·), t ∈ [0, T], are a family of nonlinear operators from V to V * and a family of convex lower semicontinuous functionals with common effective domain D( ϕ) ⊂ V. We indicate conditions on the data under which there exists a unique solution of the problem in the space H(0, T; V)∩ W (0, T;H) and the implicit Euler method has first-order accuracy in the energy norm. [ABSTRACT FROM AUTHOR]

We obtain a generalization of the complete Perron effect whereby the characteristic exponents of all solutions change their sign from negative for the linear approximation system to positive for a nonlinear system with perturbations of higher-order smallness [ Differ. Uravn., 2010, vol. 46, no. 10, pp. 1388-1402]. Namely, for arbitrary parameters λ ≤ λ < 0 and m > 1 and for arbitrary intervals [ b, d) ⊂ [ λ,+∞), i = 1, 2, with boundaries d ≤ b, we prove the existence of (i) a two-dimensional linear differential system with bounded coefficient matrix A( t) infinitely differentiable on the half-line t ≥ 1 and with characteristic exponents λ( A) = λ ≤ λ( A) = λ < 0; (ii) a perturbation f( t, y) of smallness order m > 1 infinitely differentiable with respect to time t > 1 and continuously differentiable with respect to y and y, y = ( y, y) ∈ R such that all nontrivial solutions y( t, c), c ∈ R, of the nonlinear system .y = A( t) y + f( t, y), y ∈ R, t ≥ 1, are infinitely extendible to the right and have characteristic exponents λ[ y] ∈ [ b, d) for c = 0 and λ[ y] ∈ [ b, d) for c ≠ 0. [ABSTRACT FROM AUTHOR]

We study the stability property in a game-theoretic problem on the approach of a conflict control system to an objective set at a fixed time. [ABSTRACT FROM AUTHOR]

We consider a scalar singularly perturbed nonlinear delay differential-difference equation modeling an individual neuron. We study the existence, asymptotics, and stability of its relaxation cycle. [ABSTRACT FROM AUTHOR]

We consider the behavior of solutions in unbounded domains as | x| → ∞ and in a neighborhood of an isolated singular point for a class of nonlinear elliptic equations and inequalities degenerating with respect to the coordinate variable, the solution, and the solution gradient. [ABSTRACT FROM AUTHOR]

For the wave equation with variable coefficients and boundary conditions of the first kind, we consider mutually dual problems with irregular zonal controls and regular zonal observations. Constructive estimates of well-posed solvability are obtained for the observation problem with strong generalized solutions on sufficiently large time intervals. These estimates contain information necessary for the construction of stable approximations to solutions of both problems with the use of the earlier suggested variational method. [ABSTRACT FROM AUTHOR]

We consider some differential inequalities of Sobolev type in the entire space ℝ N and find the Fujita critical exponents. Pokhozhaev’s nonlinear capacity method is substantially used. [ABSTRACT FROM AUTHOR]

A class of parabolic variational inequalities with zero obstacle inside the domain is considered. For this class, exact penalty operators are defined, which are then used to construct and study regularization methods. The following estimates are obtained for the closeness of the original and regularized problem: [ABSTRACT FROM AUTHOR]

The article discusses the role of the Laplace operator in the equation presented where L is assumed as an arbitrary linear uniformly elliptic in divergence form. It states that the fundamental solution are subjected to restrictive additional conditions, however, its complexity is removed in which the function occurring on the right-hand side of the equation satisfies the inequality.

The article discusses the classical boundary value problems for systems and the subdifferential boundary value problems for hydrodynamic equations and Maxwell equations. It examines the existence and uniqueness of the solution of subdifferential inclusion in terms of the development of the theory of abstract evolution equations and Navier-Stokes inequalities.

The article presents information related to the approximation to the Kuramoto-Tsuzki equation by periodic solution of a finite-difference. It is stated that the Kuramoto-Tsuzuki equation is the simplest finite-dimensional analog. Andronov-Hopf cycle has also provided a solution to the problem on the existence and stability of self-similar periodic solutions of equation. It is also told that numerical methods have replaced analytic methods to investigate the inequalities.

The article presents the behavior of solutions of variational inequalities for the biharmonic operator in a certain domain with one-sided constraints on epsilon-periodically arranged subsets of a certain domain. Theorems and proofs of the homogenizational variation inequalities are presented. One theorem discusses that in a case of inequality to be satisfied, then the sequence of solution of variational inequality convergies to the solution of the problem.

The article studies the method of differential inequalities for step-like contrast structures in singularly perturbed integro-differential equations in the spatially two-dimensional case. The asymptotic expansions were justified with the use of the asymptotic method of differential inequalities, extended to a new class of problems.

The article discusses a parabolic variational inequality in unbounded domains. The authors discuss Tikhonov's classical result on the Cauchy problem for the heat equation with a continuous initial function. The linear parabolic variational inequalities in unbounded domains are discussed. The authors analyze a parabolic variational inequality with a semilinear fourth-order elliptic differential operator in unbounded domains.

The article studies the resonance elliptic variational inequalities with discontinuous nonlinearities. The article starts with a statement of the problem and the formulation of the main results. The authors suggest a statement equivalent to a variational inequality in the form of an elliptic boundary value problem with a discontinuous nonlinearity.

Discusses the application of the method of differential inequalities to equations of parabolic type whose right-hand side grows faster than quadratically with respect to the space gradient. Modification of the Amann Theorem on the existence of a solution of parabolic equation; A priori estimates for the derivative of the solution; Construction of the asymptotics of a solution with two boundary layers.

Considers a two-point boundary value problem for a linear singularity perturbed reaction-diffusion equation. Utilization of a classical three-point difference scheme on an arbitrary nonuniform grid; Estimates of the Green function; Presentation of integral theorems.

Discusses a method for singling out solutions bounded at a singular point for some differential-algebraic systems. Difficulties encountered in the generalization of any results obtained for systems of ordinary differential equations; Transformation of the problem; Illustration of mathematical computations.

Discusses the convergence of variable step iterative methods for variational inequalities of the second kind. Suggestion of an iterative regularization method for inequalities with pseudomonotone potential operators; Consideration of variational inequalities with strongly inverse-monotone operators; Presentation of integral theorems.

Focuses on nonlinear boundary value problems for two-dimensional differential systems. Determination of functions that satisfies the local Carathéodory conditions; Class of boundary conditions; Validity of inequalities.

Develops the upper and lower solutions through the Nagumo method. Application of a modified method of differential equations to boundary value problems of the first kind for singularly perturbed second-order differential equations ; Assertions; Proofs.

Examines positive solutions of some semilinear differential inequalities and systems. Features of the modified version of the test function method used in the study; Characteristics of scalar inequalities; Conditions for the instantaneous or finite-time blow-up of nonnegative solutions.

Studies the unique solvability of problem for the case in which p is a nonnegative operator. Determination of a sufficient conditions for the unique solvability of problem which generalize some earlier known results and are in a sense sharp; Derivation of an equation which implies strict inequalities; Formulation of a theory which implies the assertion of the corollary.

Gives a simple proof of Hardy inequalities on the Heisenberg group. Generalization of the some inequalities similar on the Heisenberg group; Statement of some basic results about the Heisenberg groups; Derivation of some specific Hardy type inequalities from inequality.

Presents the main points of the generalized method of differential inequalities. Analysis of the behavior of the solution of problem (1)-(4); Achievement of the conditions on the initial functions guaranteeing the existence of a nonstationary contrast step-like structure; Specification of initial data for which a solution with an interior layer in a neighborhood of point x is formed in the problem.

Studies the homogenization problem for a variational inequality in a periodically perforated domain with two cavities of different scale in the periodicity cell. Equivalence of finding a generalized soution of problem to solving the minimization problem; Use of the Cauchy-Schwarz and Friedrichs inequalities.

Describes a decomposition method for variational inequalities of the second kind with strongly inverse-monotone operators. Feature of the decomposition method; Convergence analysis of an iterative method for solving variational inequalities of the second kind with strongly inverse-monotone potential coercive operators and convex nondifferentiable functionals in Hilbert spaces.