In this work, we introduced new notions of a new contraction named $S$ -weakly contraction; after that, we obtained the p-common best proximity point results for different types of contractions in the setting of complete metric spaces by using weak $P_p$ -property and proved the uniqueness of these points. Also, we presented some examples to prove the validity of our results.

In this paper, we introduce a new concept of random $\alpha$ -proximal admissible and random $\alpha$ - $\mathcal{Z}$ -contraction. Then we establish random best proximity point theorems for such mapping in complete separable metric spaces.

In this paper, we prove some random fixed point theorems for generalized random $\alpha-\psi-$contractive mappings in a Polish space and, as some applications, we show the existence of random solutions of second order random differential equation.

In this paper, we introduced many new notions and new contraction named as S-weakly contraction after that we obtained the p-common best proximity point theorems for different types of contractions in the setting of complete metric spaces by using weak Pp-property and proved the uniqueness of these points. Also we presented some examples to prove the validity of our results.

The main purpose in this paper is to define the modification form of random $\alpha$ -admissible and random $\alpha$ - $\psi$ -contractive maps. We establish new random fixed point theorems in complete separable metric spaces. The interpretation of our results provide the main theorems of Tchier and Vetro (2017) as directed corollaries. In addition, some applications to second order random differential equations are presenred here to interpret the usability of the obtained results.