Browder Fixed Point Theorem Research Articles

Strict Vector Equilibrium Problems of Multi-Product Supply–Demand Networks with Capacity Constraints and Uncertain Demands

This paper considers a multi-product, multi-criteria supply–demand network equilibrium model with capacity constraints and uncertain demands. Strict network equilibrium principles are proposed both in the case of a single criterion and multi-criteria, respectively. Based on a single criterion, it proves that strict network equilibrium flows are equivalent to vector variational inequalities, and the existence of strict network equilibrium flows is derived by virtue of the Fan–Browder fixed point theorem. Based on multi-criteria, the scalarization of strict network equilibrium flows is given by using Gerstewitz’s function without any convexity assumptions. Meanwhile, the necessary and sufficient conditions of strict network equilibrium flows are derived in terms of vector variational inequalities. Finally, an example is given to illustrate the application of the derived theoretical results.

A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation

Abstract In this article, a numerical solution for the Rosenau-Kawahara equation is considered. A new conservative numerical approximation scheme is presented to solve the initial boundary value problem of the Rosenau-Kawahara equation, which preserves the original conservative properties. The proposed scheme is based on the finite difference method. The existence of the numerical solutions for the scheme has been shown by Browder fixed point theorem. The priori bound and error estimates, as well as the conservation of discrete mass and discrete energy for the finite difference solutions, are discussed. The discrepancies of discrete mass and energy are computed and shown by the curves of these quantities over time. Unconditional stability, second-order convergence, and uniqueness of the scheme are proved based on the discrete energy method. Numerical examples are given to show the effectiveness of the proposed scheme and confirm the theoretical analysis.

Existence of solutions of a generalized bifunction-set optimization problem

<p style='text-indent:20px;'>An existence result is established for a generalized bifunction-set optimization problem, under some verifiable assumptions. Its proof is based on a Browder fixed point theorem and the KKMFan-Lemma. As applications of the main result of this paper, we obtain new existence results for some generalized versions of Kuroiwa set optimization problems and strong vector quasiequilibrium problems. Examples are provided. One of them is a non-cooperative generalized game with utility multifunctions.</p>

Convergence analysis of the Galerkin finite element method for the fourth-order Rosenau equation

In this paper, the Galerkin finite element method (FEM) for solving the fourth-order Rosenau equation is proposed with the bicubic Hermite element. The existence and uniqueness of the approximate solution is demonstrated through the Browder fixed point theorem and the convergent result of order O(h2) in H2-norm is derived for the semi-discrete scheme. The linearized fully-discrete scheme is constructed and its error estimation of order O(h2+τ2) in H2-norm is deduced. Finally, some numerical results are provided to confirm our theoretical analysis. Here and later h and τ denote the mesh size and the time step, respectively.

A two-grid mixed finite volume element method for nonlinear time fractional reaction-diffusion equations

<abstract><p>In this paper, a two-grid mixed finite volume element (MFVE) algorithm is presented for the nonlinear time fractional reaction-diffusion equations, where the Caputo fractional derivative is approximated by the classical $ L1 $-formula. The coarse and fine grids (containing the primal and dual grids) are constructed for the space domain, then a nonlinear MFVE scheme on the coarse grid and a linearized MFVE scheme on the fine grid are given. By using the Browder fixed point theorem and the matrix theory, the existence and uniqueness for the nonlinear and linearized MFVE schemes are obtained, respectively. Furthermore, the stability results and optimal error estimates are derived in detailed. Finally, some numerical results are given to verify the feasibility and effectiveness of the proposed algorithm.</p></abstract>

Pointwise error estimate and stability analysis of fourth-order compact difference scheme for time-fractional Burgers’ equation

A novel implicit high-order compact difference scheme is established for the time-fractional Burgers’ equation based on a newly developed nonlinear compact operator and the reduction order technique under the uniform mesh and graded mesh. Uniqueness, boundedness, convergence in the pointwise sense and stability in L2-norm of the discrete solutions are proved by the Browder fixed point theorem and energy argument. Numerical examples with the smooth solution and nonsmooth solution are provided to confirm numerical theoretical results and demonstrate the efficiency of the high-order compact difference scheme.

Vector Variational Inequalities In G-Convex Spaces

Inthispaper,westudysomeexistencetheoremsofsolutionsforvectorvariational inequality by using the generalized KKM theorem. Also, we investigate the properties of so- lution set of the Minty vector variational inequality in G–convex spaces. Finally, we prove the equivalence between a Browder fixed point theorem type and the vector variational in- equality in G-convex spaces.

Browder type fixed point theorems and Nash equilibria in generalized games

In this paper, we present two generalizations of the well-known Browder fixed point theorem, one of which is equivalent to the well-known Fan–Knaster–Kuratowski–Mazurkiewicz theorem. As applications, we apply these fixed point theorems to derive existence theorems for Nash equilibria in generalized games which generalize some existing existence theorems in the literature, including the well-known equilibrium existence theorem by Arrow and Debreu (Econometrica 22:265–290, 1954) and the existence theorem by Cubiotti (Int J Game Theory 26:267–273, 1997).

Investigation of the Oldroyd model as a generalized incompressible Navier–Stokes equation via the interpolating stabilized element free Galerkin technique

The main objective of the current paper is to propose a meshless weak form to simulate the Oldroyd equation. At the first stage, the Crank-Nicolson method has been utilized to discreet the temporal direction. At the second stage, the meshless element free Galerkin technique has been employed to approximate the spatial direction. The main equation is a non-local model as there is an integral term in the right hand side of the model. The mentioned integral term is approximated by using a difference scheme. The uniqueness and existence of the proposed numerical plan have been proved by using the Browder fixed point theorem. Furthermore, the error estimation of the developed numerical technique based on the stability and convergence order has been studied. Finally, some examples have been investigated to verify the theoretical results.

Analysis of new conservative difference scheme for two-dimensional Rosenau-RLW equation

In this article, numerical solution for the Rosenau-RLW equation in 2D is considered and a conservative Crank–Nicolson finite difference scheme is proposed. Existence of the numerical solutions for the difference scheme has been shown by Browder fixed point theorem. A priori bound and uniqueness as well as conservation of discrete mass and discrete energy for the finite difference solutions are discussed. Unconditional stability and a second-order accuracy on both space and time of the difference scheme are proved. Numerical experiments are given to support our theoretical results.

Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation

Coupled with the Richardson extrapolation, a new conservative Crank–Nicolson finite difference scheme, which has the accuracy of O(τ2+h4) without refined mesh for the symmetric regularized long wave equation is proposed. The corresponding conservative quantities are discussed, and the existence of numerical solution is proved by the Browder fixed point theorem. The convergence, unconditional stability and uniqueness of the scheme are also derived using the energy method. Numerical results are given to verify the accuracy and the efficiency of proposed algorithm.

High-order conservative Crank-Nicolson scheme for regularized long wave equation

Numerical solution for the regularized long wave equation is studied by a new conservative Crank-Nicolson finite difference scheme. By the Richardson extrapolation technique, the scheme has the accuracy of O(τ 2 + h 4 ) without refined mesh. Conservations of discrete mass and discrete energy are discussed, and existence of the numerical solution is proved by the Browder fixed point theorem. Convergence, unconditional stability as well as uniqueness of the solution are also derived using energy method. Numerical examples are carried out to verify the correction of the theory analysis. MSC: 65M06; 65N30

The existence of solutions for general variational inequality and applications in FC spaces

In this article, we prove the existence of solutions for the general variational inequality ϕ(x, y) ≥ f(x) - f(y) and Minty type theorem by using the generalized KKM theorem in topological spaces without linear structure. Some properties of solutions set for the general variational inequality are studied by Minty type theorem. As applications, equivalence between Browder fixed point theorem and Ky Fan's minimax inequality are studied in topological spaces.

New generalizations of basic theorems in the KKM theory

In the present paper, we obtain a new KKM type theorem for intersectionally closed-valued KKM maps and some useful new basic consequences. Typical examples of them are abstract forms of Fan’s matching theorem, Fan’s geometric lemma, the Fan–Browder fixed point theorem, maximal element theorems, Fan’s minimax inequality, variational inequalities, and others.

Browder's Fixed Point Theorem and Some Interesting Results in Intuitionistic Fuzzy Normed Spaces

We define and study Browder's fixed point theorem and relation between an intuitionistic fuzzy convex normed space and a strong intuitionistic fuzzy uniformly convex normed space. Also, we give an example to show that uniformly convex normed space does not imply strongly intuitionistic fuzzy uniformly convex.

On Some Generalized Ky Fan Minimax Inequalities

Some generalized Ky Fan minimax inequalities for vector-valued mappings are established by applying the classical Browder fixed point theorem and the Kakutani-Fan-Glicksberg fixed point theorem.

A new KKM theorem in L-convex spaces and some applications

In this paper, a new KKM theorem is established in L-convex spaces. As applications, a Ky Fan matching theorem for compactly open covers, a Fan–Browder coincidence theorem, a Fan–Browder fixed point theorem and a maximal element theorem are obtained in L-convex spaces. These theorems unify, improve and generalize some recent results in the references therein. Finally, the equilibrium existence theorems for abstract economies and qualitative games in L-convex spaces are yielded.

Fixed points, coincidence points and maximal elements with applications to generalized equilibrium problems and minimax theory

In this paper, using a generalization of the Fan–Browder fixed point theorem, we obtain a new fixed point theorem for multivalued maps in generalized convex spaces from which we derive several coincidence theorems and existence theorems for maximal elements. Applications of these results to generalized equilibrium problems and minimax theory will be given in the last sections of the paper.

Generalized vector variational-type inequalities

Solvabilities of generalized vector variational-type inequalities (GVVTI) are discussed in this work. First, the solvability of GVVTI without monotone assumption for mappings is considered in (reflexive) Banach spaces by using Brouwer fixed point theorem (Browder fixed point theorem). Second, the solvability of GVVTI with monotone assumption for mappings is considered in reflexive Banach spaces by using Ky Fan lemma.

The applications of the Fan–Browder fixed point theorem in topological ordered spaces

In this work, for topological ordered spaces, by using the Fan–Browder fixed point theorem, we obtain existence results for solutions for some generalized quasi-Ky Fan inequalities and Nash equilibrium points for a game system.