# A Note on Positivity of One-Dimensional Elliptic Differential Operators

## Authors

• Allaberen Ashyralyev Department of Elementary Mathematics Education, Fatih University, 34500 Buyukcekmece, Istanbul, Turkey, and Department of Mathematics, ITTU, 74400 Gerogly Street,Ashgabat, Turkmenistan
• Sema Akturk Department of Mathematics, Fatih University, 34500 Buyukcekmece, Istanbul, Turkey

## Keywords:

Positive operator, fractional spaces, Green's function, H\"{o}lder spaces

## Abstract

We consider a structure of fractional spaces $E_{\alpha }(C\left( \mathbb{R}_{+}\right) ,A)$ generated by the positive differential operator $A$ definedby the formula $Au(t)=-u_{tt}(t)+u(t)$ with domain \\ $D(A)=\{u:u_{tt},u\in C\left( \mathbb{R}_{+}\right) ,u(0)=0,u(\infty )=0\},$ where $\mathbb{R}_{+}=\left[ 0,\infty \right) .$ It is established that for any $0<\alpha <1/2,$the norms in the spaces $E_{\alpha }(C\left( \mathbb{R}_{+}\right) ,A)$ and $C^{2\alpha }\left( \mathbb{R}_{+}\right)$ are equivalent. The positivity of the differential operator $A$ in $C^{2\alpha }\left( \mathbb{R}_{+}\right)$is established.

2016-04-30

## Section

Differential Equations

## How to Cite

A Note on Positivity of One-Dimensional Elliptic Differential Operators. (2016). European Journal of Pure and Applied Mathematics, 9(2), 165-174. https://www.ejpam.com/index.php/ejpam/article/view/2610

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