Journal of The Korean Mathematical Society 

Abstract: The paper is devoted to the study of fractional integra- tion and dierentiation on a finite interval ( a;b) of the real axis in the frame of Hadamard setting The constructions under consider- ation generalize the modified integration R x a (t=x) " f(t)dt=t and the modified dierentiation - +" (- = xD; D = d=dx) with real ", be- ing taken n times Conditions are given for such a Hadamard-type fractional integration operator to be bounded in the space X p c(a;b) of Lebesgue measurable functions f on R+ = (0;1) such that Z b a jt c f(t)j p dt t < 1 (1 • p < 1); ess supatb(u c jf(t)j) < 1 (p = 1); for c 2 R = (i1;1), in particular in the space L p (0;1) (1 • p • 1) The existence almost everywhere is established for the corresponding Hadamard-type fractional derivative for a function g(x) such that x " g(x) have - derivatives up to order n i 1 on (a;b) and - ni1 (x " g(x)) is absolutely continuous on (a;b) Semigroup and reciprocal properties for the above operators are proved

Abstract: We prove that a non-empty separable metrizable space X admits a totally bounded metric for which the metric dimension of X in Assouad's sense equals the topological dimension of X, which leads to a characterization for the latter. We also give a characterization based on this Assouad dimension for the demension (embedding dimension) of a compact set in a Euclidean space. We discuss Assouad dimension and these results in connection with porous sets and measures with the doubling property. The elementary properties of Assouad dimension are proved in an appendix.

Abstract: A demi-closed theorem and some new weak conver- gence theorems of iterative sequences for asymptotically nonexpan- sive and nonexpansive mappings in Banach spaces are obtained. The results presented in this paper improve and extend the corre- sponding results of (1), (8)-(10), (12), (13), (15), (16), and (18).

Abstract: In this paper, we will prove the Hyers-Ulam stability of the quadratic functional equation of Pexider type, f1(x + y) + f2(x y) = f3(x) + f4(y).