Roshdi Khalil

Best Simultaneous Approximation in Lp(I,E)

Let G be a reflexive subspace of the Banach space E and let Lp(I,E) denote the space of all p-Bochner integrable functions on the interval I=0,1] with values in E, 1?p?∞. Given any norm N(?,?) on R2, N nondecreasing in each coordinate on the set R2+, we prove that Lp(I,G) is N-simultaneously proximinal in Lp(I,E). Other results are also obtained.

REMOTALITY OF CLOSED BOUNDED CONVEX SETS IN REFLEXIVE SPACES

Let X be a Banach space and E be a closed bounded subset of X. For x ∈ X, we define D(x, E) = sup{‖ x − e‖:e ∈ E}. The set E is said to be remotal (in X) if, for every x ∈ X, there exists e ∈ E such that D(x, E) = ‖x − e‖. The object of this paper is to characterize those reflexive Banach spaces in which every closed bounded convex set is remotal. Such a result enabled us to produce a convex closed and bounded set in a uniformly convex Banach space that is not remotal. Further, we characterize Banach spaces in which every bounded closed set is remotal.

Total fractional differentials with applications to exact fractional differential equations

ABSTRACT In this paper we introduce and study exact fractional differential equations, where we use the conformable fractional derivative. This forces us to introduce the fractional differential function.

Some norms estimates of hermite type interpolation operators

The object of this paper is to find some estimates for norms of certain interpolation operators on the space of continuously differentiable functions on [−1,1]. Further, we improve known estimates.