We prove a general principle in Random Fixed Point Theory by introducing a condition named P which was inspired by some of Petryshyn's work, and then we apply our result to prove some random fixed points theorems, including generalizations of some Bharucha-Reid theorems.

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Abstract. The purpose of this paper is to study the convergence of a path that begins at the unique fixed point of a strongly pseudo-contractive operator defined on a closed and convex subset of a reflexive Banach space and converges to a fixed point of a pseudo-contractive mapping. Primarily, it is proven that a convex combination of these two operators is indeed strongly pseudo-contractive under the weakly inward condition. This fact generalizes a result of Barbu for accretive operators.

In 1985, the second author proved a surjective result for m-accretive and φ-expansive mappings for uniformly smooth Banach spaces. However, in this case, we have been able to remove the uniform smoothness of the Banach space, without any additional assumption.