Advances in Analysis

6

Authors: Yaé Ulrich Gaba

In this article, we solve an open problem initially suggested in [2], namely: Let (X,d) be a Hausdorff left K-complete T0-quasi-pseudometric space, φ : X → R be a bounded from above function and  the preorder induced by φ. Let F : X × X → X and Gi : X → X;i = 1, 2, · · · , N for N > 2 be N + 1 d-sequentially continuous mapping on X such that the pairs {F; Gi};i = 1, 2, · · · , N are weakly left-related. Problem: 1. Can we prove that F, G1, · · · , GN have a common coupled fixed point in X? 2. Alternatively, what could be a correct formulation of the statement, using the induced preorder and the weakly left-related property that guarantees a positive answer? We answer this question by the affirmative. In fact we prove that a more general result holds when F : X n → X for a natural number n > 2.

8

Authors: Yang Li, Yongshun Liang

In the present paper, upper bound estimation of upper Box dimension of RiemannLiouville fractional integral of order v of any continuous functions on a closed interval has been proved to be no more than 2 − v when 0 < v < 1. If a continuous function which satisfies α-Hölder condition on a closed interval, upper Box dimension of its Riemann-Liouville fractional integral is no more than 2 − α when 0 < α < 1. Upper bound of upper Box dimension of Riemann-Liouivlle fractional integral of certain type of fractal functions has been proved to be no more than Box dimension of functions themselves.

13

Authors: R.Sathya, K.Balachandran

In this paper we study the controllability of damped second order neutral impulsive stochastic functional differential system with infinite delay in Hilbert spaces. Sufficient conditions for controllability results are obtained by using the theory of cosine families of bounded linear operators and fixed point technique. An example is provided to illustrate the theory.

17

Authors: Chenkuan Li, Changpin Li, Bailey Kacsmar, Roque Lacroix, Kyle Tilbury

This paper is to study the following generalized Abel’s integral equation g(x) = 1 Γ(1 − α) Z x 0 f(ζ) (x − ζ) α dζ, where α ∈ R and its variant in the distributional (Schwartz) sense based on fractional calculus of distributions. We obtain a number of interesting and new results which are not achievable in the classical sense.