First Sentence:
In the study of finite difference equations one has often to deal with certain finite difference inequalities which provide explicit bounds on the unknown functions.
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Key Phrases - Statistically Improbable Phrases (SIPs):
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finite difference inequalities, miscellaneous inequalities, other available inequalities, finite difference inequality, new discrete inequalities, submultiplicative function, stochastic difference system, nonlinear finite difference equations, certain new applications, certain new classes, yields the required inequality, perturbed difference equations, following useful inequality, following theorem deals, discrete generalization, nonnegative functions, pth mean, nondecreasing continuous function, given initial boundary conditions, constants formulae, continuous nondecreasing function, integral inequalities, desired inequality, nonlinear variation, integral inequality
Key Phrases - Capitalized Phrases (CAPs):
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Proof Define, Proof Let, Proof First, Proof Rewrite, Proof Assume
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