A Solution of the System of Integral Equations in Product Spaces via Concept of Measures of Noncompactness

Hemant Kumar Nashine (), Reza Arab () and Rabha W. Ibrahim ()
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Hemant Kumar Nashine: School of Advanced Sciences Vellore Institute of Technology, Department of Mathematics
Reza Arab: Sari Branch Islamic Azad University, Department of Mathematics
Rabha W. Ibrahim: Informetrics Research Group, Ton Duc Thang University

Chapter Chapter 7 in Advances in Metric Fixed Point Theory and Applications, 2021, pp 133-155 from Springer

Abstract: Abstract In this chapter, we present the role of measures of noncompactness and related fixed point results to study the existence of solutions for the system of integral equations of the form $$\begin{aligned} \begin{aligned} x_{i}(t)&=a_{i}(t)+f_{i}(t,x_{1}(t),x_{2}(t),\ldots ,x_{n}(t))\\&\quad +g_{i}(t,x_{1}(t),x_{2}(t),\ldots ,x_{n}(t))\int _0^ {\alpha (t)} k_{i}(t,s,x_{1}(s),x_{2}(s),\ldots ,x_{n}(s))) ds, \end{aligned} \end{aligned}$$ x i ( t ) = a i ( t ) + f i ( t , x 1 ( t ) , x 2 ( t ) , … , x n ( t ) ) + g i ( t , x 1 ( t ) , x 2 ( t ) , … , x n ( t ) ) ∫ 0 α ( t ) k i ( t , s , x 1 ( s ) , x 2 ( s ) , … , x n ( s ) ) ) d s , for all $$t\in \mathbb {R_{+}},\, x_{1},x_{2},\ldots ,x_{n}\in E=BC(\mathbb {R_{+}})$$ t ∈ R + , x 1 , x 2 , … , x n ∈ E = B C ( R + ) and $$1\le i\le n$$ 1 ≤ i ≤ n . We mainly focus on introducing new notion of $$\mu -(F,\varphi ,\psi )-$$ μ - ( F , φ , ψ ) - set contractive operator and establishing some new generalization of Darbo fixed point theorem and Krasnoselskii fixed point result associated with measures of noncompactness. Moreover, we deal with a system of fractional integral equations when $$k_{i}$$ k i is defined in a fractal space.

Keywords: Measures of noncompactness; Set contractive map; Integral equations; Darbo fixed point; Krasnoselskii fixed point (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-981-33-6647-3_7

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DOI: 10.1007/978-981-33-6647-3_7

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