Some results in the new direction of multiplicative metric space

Inspired by the recent work of Stojan Radenovi\'c and Bessem Samet \cite{stojan new direction}, we prove some fixed point theorems in multiplicative metric space which are not equivalent to the corresponding theorems in standard metric space using the function \(\varphi : [1, \infty) \to [0, \infty)\) that satisfies the condition: $ \varphi(s) \geq b (\ln s)^c, s > 1, $ where \(b\) and \(c\) are positive constants. We give an application in boundary value problem for the second-order differential equation to validate our results.