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We obtain asymptotic upper bounds for the number of natural solutions of thefollowing diagonal Diophantine equations in a hypercube with side - $N$ in thepaper:  $x_1 = x_2^k+...+x_s^k$,  $x_1^k = x_2^k+...+x_s^k$,  $x_1 = \sum_{j=2}^s {x_j}^{k_j}$, where $k,s,k_j$ are the natural numbers.

Determination of the asymptotic behavior of the number of natural solutions for certain types of diagonal Diophantine equations