Some Nonclassical Boundary Value Problems for Linear Mixed-Type Equations

Well-posedness of nonlocal boundary value problems is studied for some class of mixed-type equations which includes the Chaplygin equation and parabolic equations with varying time direction.     

On Nonlocal Boundary Value Problems for Nonlinear Integro-differential Equations of Arbitrary Fractional Order

In this paper, we prove the existence of solutions of a nonlocal boundary value problem for nonlinear integro-differential equations of fractional order given by $$ \begin{array}{ll} ^cD^qx(t) = f(t,x(t),(\phi x)(t),(\psi x)(t)), \quad 0 < t < 1,\\x(0) = \beta x(\eta), x'(0) =0, x''(0) =0, \ldots, x^{(m-2)}(0) =0, x(1)= \alpha x(\eta), \end{array}$$ where $${q \in (m-1, m], m \in \mathbb{N}, m \ge 2}$, $0< \eta <1$$ , and ${\phi x}$ and ${\psi x}$ are integral operators. The existence results are established by means of the contraction mapping principle and Krasnoselskii’s fixed point theorem. An illustrative example is also presented.     

The Singularly Perturbed Nonlinear Nonlocal Initial Boundary Value Problems for Reaction Diffusion Equations

The singularly perturbed nonlinear nonlocal initial boundary value problem for reaction diffusion equations is discussed. Under suitable conditions, the outer solution of the original problem is obtained. By using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed. By using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied, and by educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are considered.     

Results on Nonlocal Boundary Value Problems

In this article, we provide a variational theory for nonlocal problems where nonlocality arises due to the interaction in a given horizon. With this theory, we prove well-posedness results for the weak formulation of nonlocal boundary value problems with Dirichlet, Neumann, and mixed boundary conditions for a class of kernel functions. The motivating application for nonlocal boundary value problems is the scalar stationary peridynamics equation of motion. The well-posedness results support practical kernel functions used in the peridynamics setting.

We also prove a spectral equivalence estimate which leads to a mesh size independent upper bound for the condition number of an underlying discretized operator. This is a fundamental conditioning result that would guide preconditioner construction for nonlocal problems. The estimate is a consequence of a nonlocal Poincaré-type inequality that reveals a horizon size quantification. We provide an example that establishes the sharpness of the upper bound in the spectral equivalence.     

Mixed Boundary Value Problems for some Linear and Non-linear Elliptic Difference Equations

We obtain a priori bounds on difference quotients for some mixedboundary value problems associated with uniformly elliptic differenceequations over rectangular regions. These results are appliedto mildly quasi-linear and mildly non-linear problems to establishexistence and uniqueness criteria. Our estimates are usefulin the numerical computation of these solutions.     

Multiple Positive Solutions of Nonlocal Boundary Value Problems for p-Laplacian Equations with Fractional Derivative

In this paper,we study the multiple positive solutions of integral boundary value problems for a class of p-Laplacian differential equations involving the Caputo fractional derivative.Using a fixed point theorem due to Avery and Peterson,we obtain the existence of at least three positive decreasing solutions of the nonlocal boundary value problems. We give an example to illustrate our results.     

奇摄动高阶非线性椭圆型方程非局部边值问题

利用微分不等式理论研究了一类奇摄动高阶椭圆型微分方程非局部边值问题．得到了其解一致有效的渐近展开式．     

具有分数导数的p-Laplace方程非局部边值问题的多个正解

In this paper,we study the multiple positive solutions of integral boundary value problems for a class of p-Laplacian differential equations involving the Caputo fractional derivative.Using a fixed point theorem due to Avery and Peterson,we obtain the existence of at least three positive decreasing solutions of the nonlocal boundary value problems. We give an example to illustrate our results.     

Triple Positive Solutions for Multipoint Conjugate Boundary Value Problems

For the nth order nonlinear differential equation y ⁽ⁿ⁾(t)=f(y(t)), t [0,1], satisfying the multipoint conjugate boundary conditions, y ^(j)(a_i) = 0,1 i k, 0 j n _i - 1, 0 =a ₁ < a ₂ < < a _k = 1, and _i=1 ^k n _i =n, where f: [0, ) is continuous, growth condtions are imposed on f which yield the existence of at least three solutions that belong to a cone.     

Multipoint Boundary Value Problems for Odes. Part II

We apply the method of quasilinearization to multipoint boundary value problems for ordinary differential equations showing that the corresponding monotone iterations converge to the unique solution of our problem and this convergence is quadratic.     

Well-Posed Boundary Value Problems for Integrable Evolution Equations on a Finite Interval

We consider boundary value problems posed on an interval [0,L] for an arbitrary linear evolution equation in one space dimension with spatial derivatives of order n. We characterize a class of such problems that admit a unique solution and are well posed in this sense. Such well-posed boundary value problems are obtained by prescribing N conditions at x=0 and n–N conditions at x=L, where N depends on n and on the sign of the highest-degree coefficient _n in the dispersion relation of the equation. For the problems in this class, we give a spectrally decomposed integral representation of the solution; moreover, we show that these are the only problems that admit such a representation. These results can be used to establish the well-posedness, at least locally in time, of some physically relevant nonlinear evolution equations in one space dimension.     

Initial/Boundary Value Problems for Simultaneous Evolution Equations in Three Dimensions

A novel procedure recently introduced by the senior author is adapted here for the analysis of initial/boundary value problems for pairs of linear dispersive evolution equations in three dimensions. Such simultaneous equations are shown to arise naturally out of linear representations for 2+1-dimensional nonlinear integrable equations. The method presented emanates from the encoding of such simultaneous equations as the condition that a certain differential 1-form is closed.     

Banach空间中二阶非线性奇异微分方程多点无穷边值问题的正解

利用锥理论和Mōnch不动点定理结合单调迭代技巧,研究了Banach空间中一类二阶非线性奇异微分方程多点无穷边值问题,获得了正解的存在性定理和正解的迭代序列.     

一类高阶非线性椭圆型方程奇摄动非局部边值问题

本文利用微分不等式理论研究了一类高阶椭圆型微分方程非局部边值问题的奇摄动.得到了其解一致有效的渐近展开式.     

Solvability of Nonlocal Problems for Systems of Sobolev-Type Differential Equations with a Multipoint Condition

Russian Mathematics - We consider a nonlocal problem for a system of loaded differential equations of the Sobolev type with a multipoint constraint. By introducing additional unknown functions, we...