The present paper indicate a method of obtaining the Hewitt realcompactification vX of a Tychonoff space X, by considering a distinguished family of maximal regular congruences, viz., those which are real, on the hemiring C+(X) of all the non-negative real valued continuous functions on X.

In this paper we investigate the existence of mild solutions defined on a semiinfinite interval for initial value problems for a differential equation with a nonlocal condition. The results is based on the Schauder-Tychonoff fixed point theorem and rely on a priori bounds on solutions.

In this paper, we prove some fixed point theorems associated with Tychonoff theorem and measure of noncompactness in the Fr?chet spaces. Moreover, as an application our results, analyze existence solutions for infinite system integral equations Volterra together Hammerstein type. Finally, present example to illustrate effectiveness results.