In this manuscript, we investigate RANDOM FIXED POINTS by examining RANDOM mappings that satisfy RANDOM Sehgal-type inequalities within the framework of separable complete metric spaces.  The principal contributions of this study both encompass and significantly broaden a range of existing results in the pertinent literature.  There is a novel approach to finding the RANDOM FIXED POINTS as the proposed results in the RANDOM setting. To demonstrate the effectiveness and strength of our results, we also provide illustrative examples. Moreover, the results are also applied to Ulam Hyers Stability (UHS) and the Limit Shadowing Properties (LSP) of RANDOM FIXED point problems. Subsequently, we utilize the derived results to systematically consolidate and affirm the existence and uniqueness of RANDOM solutions for nonlinear RANDOM Fredholm integral equations.

Let (W, S, m,) be a complete probability measure space, E be a real separable Banach space, K a nonempty closed convex subset of E. Let T W×K ® K, such that {Ti}Ni =1, be N-uniformly Li-Lipschitzian asymptotically hemicontractive RANDOM maps of K with F = ÇNi=1 F(Ti)¹f F(Ti) 6= ;. We construct an explicit iteration scheme and prove neccessary and sufficient conditions for approximating common FIXED POINTS of finite family of asymptotically hemicontractive RANDOM maps.

Let B(X) and Mn (F) be the algebra of all bounded linear operators on a complex Banach space X with dimX³3 and the algebra of all n×n matrices over a field F with F¹2, respectively. Also let F(A)be the space of all FIXED POINTS of an operator AÎB(X). In this paper, we characterize the forms of linear maps f:B(X)®B(X) which satisfy F(A)=0ÛF(f(A))=0 and linear maps f: Mn(F)® Mn(F) which preserve the FIXED POINTS of matrices.

Here, the existence of FIXED POINTS for weakly compatible maps is studied. The results are new generalization of the results of [5]. Finally, we study the new common FIXED point theorems.