We prove some new bounds on exponential sums for nonlinear recurring sequences over residue rings. In addition, we also show similar novel results when the modulus is almost squarefree, thereby improving the results in El-Mahassni, Shparlinski, and Winterhof [11] and El-Mahassni and Winterhof [13] This is done by using a technique employed by Niederreiter and Winterhof [26] and through the generalisation of a Lemma found in [11] and [13]. Lastly, applications to the distribution of nonlinear congruential pseudorandom numbers are also given.

In this paper, we consider and study a new class of mixed variational inequality, which is called the extended general mixed variational inequality. We use the auxiliary principle technique to study the existence of a solution of the extended general mixed variational inequality. Several special cases are also discussed.

We prove the existence of common fixed points of a generalized contraction / Zamfirescu pair of maps in a complete cone metric space. Our results generalize the results of Huang anf Zhang [L-G. Huang, X. Zhang: Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476] and extend the results of Rezapour and Hamlbarani [Sh. Rezapour, R. Hamlbarani: Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”, J. Math. Anal. Appl. 345 (2008) 719–724].