Neumann problems with resonance in the first eigenvalue

The aim of this work is to present results of existence of solutions for a class of superlinear asymmetric elliptic systems with resonance in the first eigenvalue. The asymmetry that we consider has linear behavior on and superlinear on . To obtain these results we apply topological degree theory.     

Duality,area-considerations,and the Kalai–Smorodinsky solution

We introduce a new solution concept for $2$-person bargaining problems, which can be considered as the dual of the Equal-Area solution (EA) (see Anbarc? and Bigelow (1994)). Hence, we call it the Dual Equal-Area solution (DEA). We show that the point selected by the Kalai–Smorodinsky solution (see Kalai and Smorodinsky (1975)) lies in between those that are selected by EA and DEA. We formulate an axiom–area-based fairness–and offer three characterizations of the Kalai–Smorodinsky solution in which this axiom plays a central role.     

On the semicontinuity problem of fibers and global F-regularity

In this article, we discuss the semicontinuity problem of certain properties on fibers for a morphism of schemes. One aspect of this problem is local. Namely, we consider properties of schemes at the level of local rings, in which the main results are established by solving the lifting and localization problems for local rings. In particular, we obtain the localization theorems in the case of seminormal and F-rational rings, respectively. Another aspect of this problem is global, which is often related to the vanishing problem of certain higher direct image sheaves. As a test example, we consider the deformation of the global F-regularity.     

A nonhomogeneous boundary value problem for the Kuramoto–Sivashinsky equation in a quarter plane

We study the initial boundary value problem for the one‐dimensional Kuramoto–Sivashinsky equation posed in a half line with nonhomogeneous boundary conditions. Through the analysis of the boundary integral operator, and applying the known results of the Cauchy problem of the Kuramoto–Sivashinsky equation posed on the whole line , the initial boundary value problem of the Kuramoto–Sivashinsky equation is shown to be globally well‐posed in Sobolev space for any s >?2. Copyright © 2017 John Wiley & Sons, Ltd.     

A modified pseudospectral method for solving trajectory optimization problems with singular arc

This paper presents a direct method based on Legendre–Radau pseudospectral method for efficient and accurate solution of a class of singular optimal control problems. In this scheme, based on a priori knowledge of control, the problem is transformed to a multidomain formulation, in which the switching points appear as unknown parameters. Then, by utilizing Legendre‐Radau pseudospectral method, a nonlinear programming problem is derived which can be solved by the well‐developed parameter optimization algorithms. The main advantages of the present method are its superior accuracy and ability to capture the switching times. Accuracy and performance of the proposed method are examined by means of some numerical experiments. Copyright © 2016 John Wiley & Sons, Ltd.     

Hybrid projection methods for equilibrium problems with non‐Lipschitz type bifunctions

Based on the extended extragradient‐like method and the linesearch technique, we propose three projection methods for finding a common solution of a finite family of equilibrium problems. The linesearch used in the proposed algorithms has allowed to reduce some conditions imposed on equilibrium bifunctions. The strongly convergent theorems are established without the Lipschitz‐type condition of bifunctions. The paper also helps in the design and analysis of practical algorithms and gives us a generalization of some previously known problems. Copyright © 2017 John Wiley & Sons, Ltd.     

A posteriori error estimation for the Stokes–Darcy coupled problem on anisotropic discretization

This paper presents an a posteriori error analysis for the stationary Stokes–Darcy coupled problem approximated by finite element methods on anisotropic meshes in or 3. Korn's inequality for piecewise linear vector fields on anisotropic meshes is established and is applied to non‐conforming finite element method. Then the existence and uniqueness of the approximation solution are deduced for non‐conforming case. With the obtained finite element solutions, the error estimators are constructed and based on the residual of model equations plus the stabilization terms. The lower error bound is proved by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the upper error bound, it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration. To measure this correspondence, a so‐called matching function is defined, and its discussion shows it to be useful tool. With its help, the upper error bound is shown by means of the corresponding anisotropic interpolation estimates and a special Helmholtz decomposition in both media. Copyright © 2016 John Wiley & Sons, Ltd.     

A free boundary problem arising from DCIS mathematical model

Ductal carcinoma in situ – a special cancer – is confined within the breast ductal only. We derive the mathematical ductal carcinoma in situ model in a form of a nonlinear parabolic equation with initial, boundary, and free boundary conditions. Existence, uniqueness, and stability of problem are proved. Algorithm and illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2016 John Wiley & Sons, Ltd.