Measure of noncompactness for an infinite system of fractional Langevin equation in a sequence space

A new sequence space related to the space $\ell _{p}$ ℓ p , $1\leq p<\infty $ 1 ≤ p < ∞ (the space of all absolutely p-summable sequences) is established in the present paper. It turns out that it is Banach and a BK space with Schauder basis. The Hausdorff measure of noncompactness of this space is presented and proven. This formula with the aid the Darbo’s fixed point theorem is used to investigate the existence results for an infinite system of Langevin equations involving generalized derivative of two distinct fractional orders with three-point boundary condition.

Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions

In this paper, we present the concepts of generalized derivative, directional generalized derivative, subdifferential and conjugate for n-dimensional fuzzy-number-valued functions and discuss the characterizations of generalized derivative and directional generalized derivative by, respectively, using the derivative and directional derivative of crisp functions that are determined by the fuzzy mapping. Furthermore, the relations among generalized derivative, directional generalized derivative, subdifferential and convexity for n-dimensional fuzzy-number-valued functions are investigated. Finally, under two kinds of partial orderings defined on the set of all n-dimensional fuzzy numbers, the duality theorems and saddle point optimality criteria in fuzzy optimization problems with constraints are discussed.