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A generalized nonlinear sum-difference inequality of product form. (English) Zbl 1397.39001

Summary: We establish a generalized nonlinear discrete inequality of product form, which includes both nonconstant terms outside the sums and composite functions of nonlinear function and unknown function without assumption of monotonicity. Upper bound estimations of unknown functions are given by technique of change of variable, amplification method, difference and summation, inverse function, and the dialectical relationship between constants and variables. Using our result we can solve both the discrete inequality in [B. G. Pachpatte, J. Math. Anal. Appl. 195, No. 3, 638–644 (1995; Zbl 0842.92022); ibid. 189, No. 1, 128–144 (1995; Zbl 0824.26010)]. Our result can be used as tools in the study of difference equations of product form.

MSC:

39A10 Additive difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
26D15 Inequalities for sums, series and integrals
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