New Integral Inequalities Connected with Retarded Terms and Their Applications to Differential Equations

In this paper, we establish several retarded integral inequalities of Gronwall–Bihari–Pachpatte type for nonnegative functions involving a delayed argument α(t)≤t. The proposed results cover both linear and nonlinear settings and provide explicit upper bounds through an auxiliary integral function and its inverse. In particular, we consider inequalities containing delayed nonlinear terms, kernels depending on both t and s, weighted contributions, and iterated integrals. The considered structures are motivated, in part, by energy estimates arising in nonlinear delay differential systems, where the nonlinear control function is naturally determined by the growth of the nonlinearities. The obtained inequalities extend several classical Gronwall-, Bihari-, and Pachpatte-type estimates by retaining the effect of the variable time delay in the resulting bounds. As applications, suitable a priori estimates are derived for generalized delayed Liénard and Rayleigh systems and are used to establish the global existence of their solutions. The developed framework therefore provides useful tools for the qualitative analysis of nonlinear differential and integral equations with retarded arguments.