# Scalar Advanced and Mixed Differential Equations on Semiaxes

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## Abstract

Equations with advanced and mixed (both advanced and delayed) terms occur when the rate of change may depend not only on the state in the past, but also on the future state of the system. Such equations arise, for example, in mathematical economics. For equations with several advanced terms and positive coefficients, sufficient nonoscillation conditions are obtained using the methods similar to the previous chapters (excluding, however, solution representations which are unknown in this case). In addition, results on the asymptotics of nonoscillatory solutions are presented.

For the study of nonoscillation of equations with delayed and advanced terms, the main investigation method is the fixed point theory. Equations with two positive terms, two negative terms, and the terms of different signs are considered. For mixed equations with two coefficients of different signs, the asymptotics of nonoscillatory solutions can be defined, once the relation between the two coefficients is known; it will also depend on the deviation (delay or advanced) of the positive term.

## Keywords

Mixed Differential Equations Nonoscillatory Solution Advanced Term Nonoscillation Conditions Main Investigation Method## References

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