The purpose of this paper is threefold. Firstly, recognizing the concept of Piri and Kumam (Fixed Point Theor Appl 210, 2014), we define generalized *F*-contractive mappings in the framework of *G*-metric spaces and by employing this, some fixed point theorems in the structure of *G*-metric spaces are established that can not be obtained from the existing results in the context of allied metric spaces and do not meet the remarks of Samet et al. (Int J Anal. Article ID 917158, 2013) and Jleli et al. (Fixed Point Theor Appl 210, 2012). Infact, we utilize the pattern, mentioned in Karapinar and Agrawal (Fixed Point Theor Appl 154, 2013), a counter paper to remarks of Samet et al. (Int J Anal. Article ID 917158, 2013) and Jleli et al. (Fixed Point Theor Appl 210, 2012). Secondly, in the setting of *G*-metric spaces, certain fixed point results for integral inequalities under generalized *F*-contraction are presented. Finally, as an application, our results are utilized to establish the existence and uniqueness of solution the equations arising in Oscillation of a spring. In the sequel, another application is given to set-up the existence and uniqueness of solution of functional equations occurring in dynamic programming. Our investigations are also authenticated with the aid of some appropriate and innovative examples.