In this paper some random fixed point theorems FR´ECHET SPACES have been introduced. Some of them will be the stochastic analogue of the well known fixed point theorems. Also as their applications we see some notable results of fixed points in Banach and linear topological SPACES.

For the Fr echet algebras (A; (pk)) and (B; (qk)) and n 2 N, n  2, a linear map T: A! B is called almost n-multiplicative, with respect to (pk) and (qk), if there exists "  0 such that...

In this paper, we design a data structure for the follow-ing problem. Let  be an x-monotone trajectory with n vertices in the plane and  > 0. We show how to preprocess  and  into a data structure such that for any horizontal query segment Q in the plane, one can quickly determine the minimal continuous fraction of  whose Fr echet and Hausdor distance to the horizon-tal query segment Q is at most some threshold value  . We present a data structure for this query that needs O(n log n) preprocessing time, O(n) space, and O(log n) query time.

In 1989, T. J. Ransford presented a short proof of Johnson's uniqueness of norm theorem. We follow the same method to show that if A and B are Fréchet algebras, B is semisimple and T: A → B is a surjective homomorphism with a certain condition, then T is continuous. In particular, when A is a Banach algebra we conclude that every epimorphism T: A → B is automatically continuous and hence every semisimple Banach algebra has a unique topology as a Fréchet algebra, which is an extension of Johnson's uniqueness of norm theorem.If A and B are Banach algebras, B is semisimple and T: A → B is a dense range homomorphism, then the continuity of T is a long-standing open question. In this work we give a positive answer to this open question with an extra condition on B and then present a partial answer to the well-known Michael's problem. We also obtain similar results for dense range homomorphisms of Fréchet algebras. Finally, we show that if the above question on the continuity of dense range homomorphisms of Banach algebras has a positive answer then the same question has a positive answer for Fréchet algebras.

Let X be a hemicompact k-space and A be a uniform Frechet algebra on X. In this note we first show that if each element of a dense subset of A has square root in A then A=C(X) under certain condition. Then we show that G(C(X)), the group of invertible elements of C(X), is dense in C(X) if and only if dimX, the covering dimension of X, does not exceed 1. Using this result we give a necessary and sufficient condition under which each continuous function on X is the square of another

In this paper, we consider the semi-infinite programming problems with non-differentiable emerging functions. Firstly, we give a counterexample showing that Theorem 3.1 of Ref. [10] is not true. Then, by modifying the assumptions of this theorem, we establish a new necessary Theorem for optimal solution of the problem.

In this paper we apply the technique of measures of noncompactness to the theory of in nite system of integral equations in the Fr echet SPACES. Our aim is to provide a few generalization of Tychono  xed point theorem and prove the existence of solutions for in nite systems of nonlinear integral equations with help of the technique of measures of noncompactness and a generalization of Tychono  xed point theorem. Also, we present an example of nonlinear integral equations to show the e ciency of our results. Our results extend several comparable results obtained in the previous literature.