In this paper, we show that the beautiful theory developed by M. Schechter and K. Tintarev in [9] can be applied to the eigenvalue problem $$ \begin{cases} -\Delta u = \lambda f(u) & {\rm in} \,\,\, \Omega \\ u = 0 & {\rm on\,\, \partial} \Omega \end{cases} $$ when $$ \limsup_{|\xi|\to +\infty}{{\int_0^{\xi}f(t)dt}\over {\xi^2}}\lt +\infty$$ and, for each $\lambda$ in a suitable interval, the problem has a unique positive solution.

We prove a multiple combinatorial Stokes’ theorem and a multiple Sperner’s lemma and formulate their matroid versions. The combinatorial properties of pseudomanifolds with matroid structures are discussed.

It is famous that Hilbert proved that, for any positive integer n, there exists an entire function fn(·, ·, ·) of three complex variables which cannot be represented as any n-time nested superposition constructed from several entire fuctions of two complex variables. In this paper, a finer classification of the 13th problem formulated by Hilbert is given. This classification is applied to the theorem showing that there exists an entire function f(·, ·, ·) of three complex variables which cannot be represented as any finite-time nested superposition constructed from several entire functions of two complex variables. The original result proved by Hilbert can be derived from this theorem.

We construct Mond-Weir and Wolfe types of nondifferentiable multiobjective second order symmetric dual problems with cone constraints over arbitrary closed convex cones. Weak, strong, and converse duality theorems for weakly efficient solutions are established under the assumptions of second order invex and pseudo-invex functions. Several known results are obtained as special cases of our symmetric duality.

In this paper, KKM theorems or coincidence theorems on abstract convex spaces are applied to obtain the Fan-Browder type fixed point theorems, existence of maximal elements, existence of economic equilibria and some related results. Consequently, we obtain generalizations or improvements of a number of known equilibria results, especially, in a recent work of Ding and Wang [3] on the so-called FC-spaces.

In this paper we study a discrete time version of optimization problems of Bolza type. The functionals are assumed to be merely lower semicontinuous. We obtain optimality conditions which are always necessary and which are also sufficient in the convex case whenever the given problem satisfies a qualification condition.

In this paper, we introduce a new iterative scheme based on the hybrid method and the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of a nonexpansive mapping and the set of the variational inequality for a monotone, Lipschitz-continuous mapping. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces. Based on this result, we also get some new and interesting results. The results in this paper generalize, extend and unify some well-known strong convergence theorems in the literature.

We prove, under appropriate assumptions, the existence of solutions for a second order evolution inclusion with delay in a separable reflexive Banach space. Several applications are investigated via discretization techniques and variational convergence results.

We prove that an approximately convex function on an open subset of an Asplund space is generically Fr´echet differentiable, as are genuine convex functions. Thus, we give a positive answer to a question raised by S. Rolewicz. We also prove a more general result of that type for regular functions on an open subset of an Asplund space.

In this paper we use the penalty approach in order to study two constrained minimization problems in Banach spaces. A penalty function is said to have the generalized exact penalty property if there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. In this paper we show that the generalized exact penalty property holds and is stable under perturbations of objective functions, constraint functions and the right-hand side of constraints.

We study the influence of errors on the convergence of orbits of nonexpansive mappings in Banach and metric spaces.

This paper outlines some of my recent joint works with Q. Takahashi on fixed point properties or ergodic properties for semigroup of nonexpansive mappings on closed convex subsets of a Banach space and their relationship with existence of left invariant mean on certain subspaces of bounded realvalued functions on the semigroup.

et $\{T(t);t\ge 0\}$ be a uniformly bounded $(C_0)$-semigroup of operators on a Banach space $X$ with generator $A$ such that all orbits are relatively weakly compact. Let $\{\phi_\alpha\}$ and $\{\psi_\alpha\}$ be two nets of continuous linear functionals on the space $C_b[0,\infty)$ of all bounded continuous functions on $[0,\infty)$. $\{\phi_\alpha\}$ and $\{\psi_\alpha\}$ determine two nets $\{A_\alpha\},\ \{B_\alpha\}$ of operators satisfying $\langle A_\alpha x,x^*\rangle=\phi_\alpha(\langle T(\cdot)x,x^*\rangle)$ and $\langle B_\alpha x,x^*\rangle=\psi_\alpha(\langle T(\cdot)x,x^*\rangle)$ for all $x\in X$ and $x^*\in X^*$. Under suitable conditions on $\{\phi_\alpha\}$ and $\{\psi_\alpha\}$, this paper discusses: 1) the convergence of $\{A_\alpha\}$ and $\{B_\alpha\}$ in operator norm; 2) rates of convergence of $\{A_\alpha x\}$ and $\{A_\alpha y\}$ for each $x\in X$ and $y\in R(A)$.

In our previous paper we introduced the sequence of Extended Common Multiples (abr. ECM) and the ECM condition for a pair of rationally independent irrational numbers to estimate the recurrent dimensions of quasiperiodic orbits. In this paper first we define the quasi-periodic properties for the functions, which have recurrent properties related with some ECM sequences. Next we study a second order nonlinear evolution equation with a quasi-periodic perturbation term, which gives an ECM sequence, and we show the existence of solutions, which have the quasi-periodic properties induced by this ECM sequence. Furthermore, we estimate the recurrent dimensions of the discrete orbits given by this solution for the case where the irrational frequencies of the q.p. pertubations satisfy the ECM condition.

We propose a new nonlinear method for finding a coalition array in a cooperative game, a new coherent criterion for concluding whether its array corresponds to the nucleolus or not, and an adjusting process for obtaining the nucleolus strictly from its nonlinear approximation.

The author proposed a distance on $R^n$ in \cite{hK08} which measures improvement of social welfare. Continuity of the distance with respect to the Euclidean metric of $R^n$ is proved in this note, and conversely, the Euclidean metric is continuous with respect to our distance on some subset of $R^n$.