Existence of local fractional integral equation via a measure of non-compactness with monotone property on Banach spaces

In this paper, we discuss fixed point theorems for a new χ -set contraction condition in partially ordered Banach spaces, whose positive cone K is normal, and then proceed to prove some coupled fixed point theorems in partially ordered Banach spaces. We relax the conditions of a proper domain of an underlying operator for partially ordered Banach spaces. Furthermore, we discuss an application to the existence of a local fractional integral equation.