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

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.