It is well known that the Sturm-Liouville equation has many applications in different areas of science. Thus, it is important to review different versions of the well-known equation. The technique of α-admissible α-ψ-contractions was introduced by Samet et al. in (Nonlinear Anal. 75:2154-2165, 2012). Our aim in this work is to study a fractional hybrid version of the Sturm-Liouville equation by mixing the technique of Samet. In fact, by using the technique of α-admissible α-ψ-contractions, we investigate the existence of solutions for the fractional hybrid Sturm-Liouville equation by using the multi-point boundary value conditions. Also, we review the existence of solutions for a fractional hybrid version of the problem under the integral boundary value conditions. Finally, we provide two examples to illustrate our main results.

In this paper, we investigate some properties of sequences in a gradual normed space. We define some new notions such as gradual convergent sequences, gradual Cauchy sequences, etc., and then we state some theorems about these notions. Finally, in the last section, we bring some illustrative examples.

In this paper, we have investigated some topological properties of sets in a given gradual normed space. We have stated gradual Hausdorff property and then,we have studied the relationship between gradual closed sets and gradual compact sets. Also, we have given a result about having the closure point for an innite set in a gradual normed space. In the end, we have provided some illustrative examples.

We investigate biprojectivity and biflatness of generalized module extension Banach algebra A Z B, in which A and B are Banach algebras and B is an algebraic Banach A-bimodule, with multiplication: (a, b)?(a',b') = (aa', ab' + ba' + bb')

Let A be a Banach algebra and ( A ″, □) be its second dual with first Arens product. The third dual of A can be regarded as dual of A ″ ( A ‴ = ( A ″)′) or as the second dual of A ′ ( A ‴ = ( A ′)″), so there are two ( A ″, □)-bimodule structures on A ‴ that are not always equal. This paper determines the conditions that make these structures equal. As a consequence, there are some relations between weak amenability of A and ( A ″, □).