More than 30 years ago, Hedetniemi made a conjecture which says that the categorical product of two n-chromatic graphs is still n-chromatic. The conjecture is still open, despite many dierent approaches from dierent point of views. This article surveys methods and partial results; and discuss problems related to or motivated by this conjecture.

Several new fixed point theorems in H-space are first proved. Next, by applying the fixed point theorems, some minimax inequalities and existence theorems of maximal elements for $\cal{L}$$_F$ correspondences and $\cal{L}$$_F$-majorized correspondences in H-spaces are obtained. Finally, using the existence theorems of maximal elements, some equilibrium existence theorems for one-person games, qualitative games and noncompact abstract economies with $\cal{L}$$_F$-majorized correspondences in H-spaces are obtained. Our theorems improve and generalize most known results due to Border, Borglin-Keiding, Ding-Kim-Tan, Ding-Tan, Ding-Tarafdar, Mehta-Tarafdar, Shafer-Sonnenschein, Tan-Yuan, Tarafdar, Toussaint, Tulcea, Yannelis and Yannelis-Prabhakar etc.

A class $S_s^\ast (\alpha ,\beta )$ of functions $f$, regular and univalent in $D = \{z: | z | \lt 1\}$ given by $f(z) = z+\sum\limits^\infty _{n=2} a_n z^n$ and satisfying the condition $$ \left |\displaystyle\frac{zf'(z)}{f(z)-f(-z)}-1\right |\lt \beta\left|\displaystyle\frac{\alpha zf'(z)}{f(z)-f(-z)}+1\right |, $$ $z\in D, 0 \leq \alpha \leq 1, 0 \lt \beta \leq 1$ is introduced and studied. An analogous class $S_c^\ast (\alpha ,\beta )$ is also examined.

We introduce the Wiener space and then consider wavelets which are not necessarily compactly supported but have a decay condition at innity. Under the Wiener condition, a wavelet and its bi-orthogonal wavelet have the same rate of decay at innity and the same moment. Under a certain sucient condition, we prove the existence of a pre-wavelet basis in the multi-dimensional case. Furthermore, pre-wavelets and their bi-orthogonal pre-wavelets have the same rate of decay at innity and the same number of vanishing moments.

In the present paper, we shall establish one of our earlier conjectures by proving that on compact subsets of a $*$-foundation semigroup $S$ with identity and with a locally bounded Borel measurable weight function $w$, the pointwise convergence and the uniform convergence of a sequence of $w$-bounded positive definite functions on $S$ which are also continuous at the identity are equivalent..

The purpose of this paper is to prove xed point theorems for certain type of asymptotically nonexpansive mappings in product of two locally convex spaces. These results generalize the theorems of Kirk and Yanez [7] and the author [9].

A certain differential operator $D^n$ is introduced for functions of the form $$ f(z) = \displaystyle \frac{1}{z^p}+\displaystyle\sum^\infty _{k=0} a_kz^k $$ which are analytic in $E^{\ast } = \{z : 0 \lt |z| \lt 1\}$. The object of the present paper is to give an application of the above operator $D^n$ to the differential inequalities.

Let $E$ be a vector space in which some notion of boundedness is defined. Then $T:E\to E$ is said to have the boundedness stability property (BSP) if for each $x\in E$, the sequence $(T^n_x)^\infty_{n=1}$ is bounded whenever a subsequence $(T^{n_i}x)^\infty_{i=1}$ is bounded. It is shown that (1) every affine operator on a finite-dimensional Banach space has the (BSP); (2) every affine operator on an infinite-dimensional vector space has the functional (BSP); (3) when $E$ is an infinite-dimensional Banach space, an affine operator $T$ on $E$ has the (BSP) if its linear part $A_T=T-T(0)$ is a compact perturbation of a bounded linear operator with spectral radius less than one and (4) when $E$ is a Hilbert space, every normal or subnormal bounded linear operator has the (BSP). Some results on affine operators on a Hilbert space whose linear parts are normal or subnormal are also obtained. Finally, some problems are posed.