[8] **viXra:1810.0441 [pdf]**
*replaced on 2018-11-06 03:17:06*

**Authors:** Johan Aspegren

**Comments:** 5 Pages.

In this article we will prove the Kakeya set conjecture. In addition we will prove that in the usual approach to the Kakeya maximal function conjecture we can assume that the tube-sets are maximal. Third, we build a direct connection between line incidence theorems and Kakeya type conjectures.

**Category:** Functions and Analysis

[7] **viXra:1810.0313 [pdf]**
*submitted on 2018-10-19 06:28:06*

**Authors:** Fayowole David Ayadi

**Comments:** 3 Pages.

Abstract:I can still remember my expression and feeling when we were asked to show that sup(A + B) = sup(A) + sup(B). It was an herculean task because the concept was too difficult to grasp with the use of approximation property until I discovered an easy route. In a bid to restrict my papers to just few pages, I will focus more on examples than theorems.

**Category:** Functions and Analysis

[6] **viXra:1810.0312 [pdf]**
*submitted on 2018-10-19 06:32:53*

**Authors:** Fayowole David Ayadi, Olabiyi Tobi David, Oluwajoba Godsfavour Favour, Oluwusi Faith Tolu, Isaleye Dorcas, Olorunisola Femi Stephen

**Comments:** 13 Pages.

Throughout these discussions the numbers epsilon > 0 and delta > 0 should be thought of as very small numbers. The aim of this part is to provide a working definition for the integral of a bounded function f(x) on the interval [a, b]. We will see that the real number "f(x)dx" is really the limit of sums of areas of rectangles.

**Category:** Functions and Analysis

[5] **viXra:1810.0308 [pdf]**
*submitted on 2018-10-19 12:22:45*

**Authors:** Zaid Laadjal

**Comments:** 4 Pages.

In this work, we apply the fixed point theorems, we study the existence and uniqueness of solutions for Langevin differential equations involving two ractional orders with multi-point boundary conditions on the half-line.

**Category:** Functions and Analysis

[4] **viXra:1810.0303 [pdf]**
*replaced on 2019-02-07 19:36:18*

**Authors:** Johan Aspegren

**Comments:** 2 Pages.

In this preprint we will prove the isotropic constant conjecture.

**Category:** Functions and Analysis

[3] **viXra:1810.0170 [pdf]**
*submitted on 2018-10-10 15:15:29*

**Authors:** Zaid Laadjal

**Comments:** 4 Pages.

In this paper, we study the existence and uniqueness of solutions for Langevin differential equations of Riemman-Liouville fractional derivative with boundary value conditions on the half-line. By a classical fixed point theorems, several new existence results of solutions are obtained.

**Category:** Functions and Analysis

[2] **viXra:1810.0169 [pdf]**
*submitted on 2018-10-10 15:21:57*

**Authors:** Zaid Laadjal

**Comments:** 5 Pages.

In this paper, we investigate the existence and uniqueness of solutions for the following fractional Langevin equations with boundary conditions $$\left\{\begin{array}{l}D^{\alpha}( D^{\beta}+\lambda)u(t)=f(t,u(t)),\text{ \ \ \ }t\in(0,+\infty),\\ \\u(0)=D^{\beta}u(0)=0,\\ \\ \underset{t\rightarrow+\infty}{\lim}D^{\alpha-1}u(t)=\underset{t\rightarrow+\infty}{\lim}D^{\alpha +\beta-1}u(t)=au(\xi),\end{array}\right.$$ where $1<\alpha \leq2$ and$\ 0<\beta \leq1,$ such that $1<\alpha +\beta \leq2,$ with $\ a,b\in\mathbb{R},$ $\xi \in\mathbb{R}^{+},$\ and $D^{\alpha}$, $D^{\beta }$ are the Riemman-Liouville fractional derivative. Some new results are obtained by applying standard fixed point theorems.

**Category:** Functions and Analysis

[1] **viXra:1810.0168 [pdf]**
*submitted on 2018-10-10 15:28:53*

**Authors:** Zaid Laadjal

**Comments:** 6 Pages.

In this work, we use the fixed point theorems, we investigate the existence and uniqueness of solutions for a class of fractional Langevin equations with boundary value conditions on an infinite interval.

**Category:** Functions and Analysis