The goal of this paper is to prove some proposed integro-differential inequalities equations as control Gronwall formulation. Thus, we have been established and studied their analytic inequalities with properties of differential control equations to give the related estimator of the derivative for the solutions of considered problem. Also, the inequalities classes is presented with distinct input control functions and delay functions in the first time, in which introduce and proved new analytic results for proposed such class. Finally, some applications is presented and illustrated in different situations of this problem.

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