## Abstract and Applied Analysis

- Abstr. Appl. Anal.
- Volume 2013 (2013), Article ID 607204, 20 pages.

### Some Bounded Linear Integral Operators and Linear Fredholm Integral Equations in the Spaces ${H}_{\alpha ,\delta ,\gamma}$$((a,b)\times (a,b),X)$ and ${H}_{\alpha ,\delta}$$((a,b),X)$

İsmet Özdemir, Ali M. Akhmedov, and Ö. Faruk Temizer

#### Abstract

The spaces ${H}_{\alpha ,\delta ,\gamma}((a,b)\times (a,b),\mathbb{R})$ and ${H}_{\alpha ,\delta}((a,b),\mathbb{R})$ were defined in ((Hüseynov (1981)), pages 271–277). Some singular integral operators on Banach spaces were examined, (Dostanic (2012)), (Dunford (1988), pages 2419–2426 and (Plamenevskiy (1965)). The solutions of some singular Fredholm integral equations were given in (Babolian (2011), Okayama (2010), and Thomas (1981)) by numerical methods. In this paper, we define the sets ${H}_{\alpha ,\delta ,\gamma}((a,b)\times (a,b),X)$ and ${H}_{\alpha ,\delta}((a,b),X)$ by taking an arbitrary Banach space $X$ instead of $\mathbb{R}$, and we show that these sets which are different from the spaces given in (Dunford (1988)) and (Plamenevskiy (1965)) are Banach spaces with the norms ${\parallel \xb7\parallel}_{\alpha ,\delta ,\gamma}$ and ${\parallel \xb7\parallel}_{\alpha ,\delta}$. Besides, the bounded linear integral operators on the spaces ${H}_{\alpha ,\delta ,\gamma}((a,b)\times (a,b),X)$ and ${H}_{\alpha ,\delta}((a,b),X)$, some of which are singular, are derived, and the solutions of the linear Fredholm integral equations of the form $f(s)=\varphi (s)+\lambda {\int}_{a}^{b}A(s,t)f(t)dt,f(s)=\varphi (s)+\lambda {\int}_{a}^{b}A(t,s)f(t)dt$ and $f(s,t)=\varphi (s,t)+\lambda {\int}_{a}^{b}A(s,t)f(t,s)dt$ are investigated in these spaces by analytical methods.

#### Article information

**Source**

Abstr. Appl. Anal., Volume 2013 (2013), Article ID 607204, 20 pages.

**Dates**

First available in Project Euclid: 27 February 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.aaa/1393511803

**Digital Object Identifier**

doi:10.1155/2013/607204

**Mathematical Reviews number (MathSciNet)**

MR3035391

**Zentralblatt MATH identifier**

1381.45007

#### Citation

Özdemir, İsmet; Akhmedov, Ali M.; Temizer, Ö. Faruk. Some Bounded Linear Integral Operators and Linear Fredholm Integral Equations in the Spaces ${H}_{\alpha ,\delta ,\gamma }$ $\left(\left(a,b\right)×\left(a,b\right),X\right)$ and ${H}_{\alpha ,\delta }$ $\left(\left(a,b\right),X\right)$. Abstr. Appl. Anal. 2013 (2013), Article ID 607204, 20 pages. doi:10.1155/2013/607204. https://projecteuclid.org/euclid.aaa/1393511803