Existence and uniqueness of solutions of initial value problems for nonlinear langevin equation involving two fractional orders

The solvability of initial value problems for nonlinear Langevin equation involving two fractional orders are discussed in this paper. An existence result for the solution is obtained using the Leray–Schauder nonlinear alternative. In addition, sufficient conditions for unique solution are established under the Banach contraction principle. The existence results for the initial value problems of nonlinear classical Langevin equation follow as a special case of our results.     

Global solvability of initial boundary-value problems for nonlinear analogs of the Boussinesq equation

The solvability of the natural (first, second, and mixed) initial boundary-value problems for nonlinear analogs of the Boussinesq equation is studied. Uniqueness theorems for regular solutions and global solvability theorems are proved.     

Dirichlet and neumann problems for a fractional ordinary differential equation with constant coefficients

For an ordinary differential equation with constant coefficients, we find a general representation of solutions; by using it, we solve the Dirichlet and Neumann problems, obtain conditions for the unique solvability, and construct closed-form representations of solutions.     

A majorant criterion for the total preservation of global solvability of controlled functional operator equation

We prove the unique solvability of a nonlinear controlled functional operator equation in a Banach ideal space. We also establish sufficient conditions for the global solvability of all controls from a pointwise bounded set, provided that some majorant equation for the given family of these controls is globally solvable. We give examples of controlled boundary value problems reducible to the considered equation.     

一阶非线性椭圆型复方程的间断Riemam边值问题

§1 问题的提法 本文讨论一阶非线性一致椭圆型复方程 的间断非线性Riemann边值问题。以E表示复平面,D~ 表示有界N 1连通区域,其边界由     

Boundary value problem for a mixed-type equation with a difference-differential operator

The Tricomi problem for a mixed-type equation with retarded argument in an unbounded domain is considered. The unique solvability of the problem is proved without restrictions on the delay magnitude. The existence of a solution follows from the solvability of a difference equation. Closed-form integral representations for the solutions are derived.     

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.     

一类四阶非线性奇摄动方程的边值问题

利用匹配渐近展开法,讨论了一类四阶非线性方程的具有两个边界层的奇摄动边值问题．引进伸长变量,根据边界条件与匹配原则,在一定的可解性条件下,给出了外部解和左右边界层附近的内层解,得到了该问题的二阶渐近解,并举例说明了这类非线性问题渐近解的存在性．     

Characterization of the Cone and Applications in Banach Spaces

Abstract

In this article, we first discuss the subduality and orthogonality of the cones and the dual cones when the norm is monotone in Banach spaces. Then, under different assumptions, the necessary and sufficient conditions for the ordering increasing property of the metric projection onto cones and order intervals are studied. Moreover, representations of the metric projection onto cones and order intervals are obtained. As applications, the solvability and approximation results of solutions to nonlinear discontinuous variational inequality and complementarity problems are proved by partial ordering methods.     

Boundary Control Problem for a Nonlinear Convection–Diffusion–Reaction Equation

Computational Mathematics and Mathematical Physics - The solvability of boundary-value and extremum problems for a nonlinear convection–diffusion–reaction equation with mixed boundary...     

On solutions of a quadratic integral equation of Hammerstein type

We study the solvability of a nonlinear quadratic integral equation of Hammerstein type. Using the technique of measures of noncompactness we prove that this equation has solutions on an unbounded interval. Moreover, we also obtain an asymptotic characterization of these solutions. Several special cases of this integral equation are discussed and applications to real world problems are indicated.     

Determination of the Robin coefficient in a nonlinear boundary condition for a steady‐state problem

In this paper, a constant heat transfer coefficient present in a nonlinear Robin‐type boundary condition associated with an elliptic equation is reconstructed uniquely from a single boundary energy measurement. Two types of such boundary energy measurement are considered, and solvability theorems for the solution of the resulting nonlinear inverse problems are provided. Further, one‐dimensional numerical results are presented and discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

Gellerstedt problem with nonclassical matching conditions for the solution gradient on the type change line with data on internal characteristics

We study the solvability of the Gellerstedt problem for the Lavrent’ev–Bitsadze equation. An initial function is posed in the ellipticity domain of the equation on the boundary of the unit half-circle with center the origin. Zero conditions are posed on characteristics in the hyperbolicity domain of the equation. “Frankl-type conditions” are posed on the type change line of the equation. We show that the problem is either conditionally solvable or uniquely solvable. We obtain a closed-form solvability condition in the case of conditional solvability. We derive integral representations of the solution in all cases.     

Dynamics of a rotating layer of an ideal electrically conducting incompressible fluid

A system of nonlinear partial differential equations is considered that models perturbations in a layer of an ideal electrically conducting rotating fluid bounded by spatially and temporally varying surfaces with allowance for inertial forces. The system is reduced to a scalar equation. The solvability of initial boundary value problems arising in the theory of waves in conducting rotating fluids can be established by analyzing this equation. Solutions to the scalar equation are constructed that describe small-amplitude wave propagation in an infinite horizontal layer and a long narrow channel.     

Criterion for the solvability of a class of nonlinear two-point boundary value problems on the plane

We study the solvability of a class of nonlinear two-point boundary value problems for systems of ordinary second-order differential equations on the plane. In these boundary value problems, we single out the leading nonlinear terms, which are positively homogeneous mappings. On the basis of properties of the leading nonlinear terms, we prove a criterion for the solvability of boundary value problems under arbitrary perturbations in a given set by using methods for the computation of the winding number of vector fields.     

Some Fundamental Issues of Mathematical Simulation in Biology

Some directions of simulation in biology leading to original formulations of mathematical problems are overviewed. Two of them are discussed in detail: the correct solvability of first-order linear equations with unbounded coefficients and the construction of a reaction–diffusion equation with nonlinear diffusion for a model of genetic wave propagation.     

The Solvability of Nonlinear Split Ordered Variational Inequality Problems in Partially Ordered Vector Spaces

The concept of nonlinear split ordered variational inequality problems on partially ordered Banach spaces extends the concept of the linear split vector variational inequality problems on Banach spaces, while the latter is a natural extension of vector variational inequality problems on Banach spaces. In this article, we prove the solvability of some nonlinear split vector variational inequality problems by using fixed-point theorems on partially ordered Banach spaces. It is important to notice that in the results obtained in this article, the considered mappings are not required to have any type of continuity and they just satisfy some order-monotonic conditions. Consequently, both the solvability of linear split vector variational inequality problems and vector variational inequality problems will be immediately obtained from the solvability of nonlinear split vector variational inequality problems. We will apply these results to solving nonlinear split vector optimization problems. The underlying spaces of the considered variational inequality problems may just be vector spaces which do not have topological structures, the considered mappings are not required to satisfy any continuity conditions, which just satisfy some order-increasing conditions.     

On Some Inverse Problems for First Order Operator-Differential Equations

Siberian Mathematical Journal - We study solvability of the inverse problems of recovering an unknown function on the nonlinear right-hand side of a first order operator-differential equation in...     

A majorant-minorant criterion for the total preservation of global solvability of a functional operator equation

We study a nonlinear controlled functional operator equation in an ideal Banach space. We establish sufficient conditions for the global solvability for all controls from a given set, and obtain a pointwise estimate for solutions. Using upper and lower estimates of the functional component in the right-hand side of the initial equation (with a fixed operator component), we obtain majorant and minorant equations. We prove the stated theorem, assuming the monotonicity of the operator component in the right-hand side and the global solvability of both majorant andminorant equations. We give examples of the reduction of controlled initial boundary value problems to the equation under consideration.     

Solvability conditions for infinite systems of difference equations

The Fredholm property of some linear infinite dimensional difference operators is studied. In the case corresponding to discretization of differential equations on the real axis, the index of the corresponding operators is computed and solvability conditions for the nonhomogeneous problem are established. In the multi-dimensional case, conditions of the normal solvability of the corresponding discrete operators are formulated in terms of limiting problems. The results on the location of the spectrum and the solvability conditions allow various applications to linear and nonlinear problems.