# Existence of a common solution for a system of nonlinear integral equations via fixed point methods inb-metric spaces

Open Mathematics (2016)

• Volume: 14, Issue: 1, page 128-145
• ISSN: 2391-5455

top

## Abstract

top
In this paper we introduce a property and use this property to prove some common fixed point theorems in b-metric space. We also give some fixed point results on b-metric spaces endowed with an arbitrary binary relation which can be regarded as consequences of our main results. As applications, we applying our result to prove the existence of a common solution for the following system of integral equations: x (t) =  ∫ a b K 1  (t,r,x(r)) dr, x (t) =  ∫ a b K 2  (t,r,x(r)) dr,       $x\left(t\right)=\underset{a}{\overset{b}{\int }}{K}_{1}\left(t,r,x\left(r\right)\right)dr,x\left(t\right)=\underset{a}{\overset{b}{\int }}{K}_{2}\left(t,r,x\left(r\right)\right)dr,$ where a, b ∈ ℝ with a < b, x ∈ C[a, b] (the set of continuous real functions defined on [a, b] ⊆ ℝ) and K1, K2 : [a, b] × [a, b] × ℝ → ℝ are given mappings. Finally, an example is also given in order to illustrate the effectiveness of such result.

## How to cite

top

Oratai Yamaod, Wutiphol Sintunavarat, and Yeol Je Cho. "Existence of a common solution for a system of nonlinear integral equations via fixed point methods inb-metric spaces." Open Mathematics 14.1 (2016): 128-145. <http://eudml.org/doc/276924>.

@article{OrataiYamaod2016,
abstract = {In this paper we introduce a property and use this property to prove some common fixed point theorems in b-metric space. We also give some fixed point results on b-metric spaces endowed with an arbitrary binary relation which can be regarded as consequences of our main results. As applications, we applying our result to prove the existence of a common solution for the following system of integral equations: x (t) =  ∫ a b K 1  (t,r,x(r)) dr, x (t) =  ∫ a b K 2  (t,r,x(r)) dr,       $\{x (t) = \int \limits \_a^b \{\{K\_1\}\} (t, r, x(r))dr, & & x(t) = \int \limits \_a^b \{\{K\_2\}\}(t, r, x(r))dr,\}$ where a, b ∈ ℝ with a < b, x ∈ C[a, b] (the set of continuous real functions defined on [a, b] ⊆ ℝ) and K1, K2 : [a, b] × [a, b] × ℝ → ℝ are given mappings. Finally, an example is also given in order to illustrate the effectiveness of such result.},
author = {Oratai Yamaod, Wutiphol Sintunavarat, Yeol Je Cho},
journal = {Open Mathematics},
keywords = {b-metric spaces; Coincidence points; Common fixed points; Integral equations; -metric spaces; coincidence points; common fixed points; integral equations},
language = {eng},
number = {1},
pages = {128-145},
title = {Existence of a common solution for a system of nonlinear integral equations via fixed point methods inb-metric spaces},
url = {http://eudml.org/doc/276924},
volume = {14},
year = {2016},
}

TY - JOUR
AU - Oratai Yamaod
AU - Wutiphol Sintunavarat
AU - Yeol Je Cho
TI - Existence of a common solution for a system of nonlinear integral equations via fixed point methods inb-metric spaces
JO - Open Mathematics
PY - 2016
VL - 14
IS - 1
SP - 128
EP - 145
AB - In this paper we introduce a property and use this property to prove some common fixed point theorems in b-metric space. We also give some fixed point results on b-metric spaces endowed with an arbitrary binary relation which can be regarded as consequences of our main results. As applications, we applying our result to prove the existence of a common solution for the following system of integral equations: x (t) =  ∫ a b K 1  (t,r,x(r)) dr, x (t) =  ∫ a b K 2  (t,r,x(r)) dr,       ${x (t) = \int \limits _a^b {{K_1}} (t, r, x(r))dr, & & x(t) = \int \limits _a^b {{K_2}}(t, r, x(r))dr,}$ where a, b ∈ ℝ with a < b, x ∈ C[a, b] (the set of continuous real functions defined on [a, b] ⊆ ℝ) and K1, K2 : [a, b] × [a, b] × ℝ → ℝ are given mappings. Finally, an example is also given in order to illustrate the effectiveness of such result.
LA - eng
KW - b-metric spaces; Coincidence points; Common fixed points; Integral equations; -metric spaces; coincidence points; common fixed points; integral equations
UR - http://eudml.org/doc/276924
ER -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.