We prove some new results on existence of solutions to first-order ordinary differential equations with deviated arguments.
It is easy to see that differential integral equations and equations with deviated variables are particular cases of (28).
For some recent work on equations of fractional order with deviated arguments, see [46 48] and the references therein.
This seems to be a peculiar feature of equations with deviated arguments, see [1] for an example with a second-order equation.
As far as we know, fractional order differential equations with deviated arguments have not been much studied and many aspects of these equations are yet to be explored.
For the general theory and applications of integer order differential equations with deviated arguments, we refer the reader to the references [39 45].
It is clear that more complicated partial differential systems with deviated variables and partial differential integral systems can be obtained from (3) and (6) by a suitable definition of f and g.
For higher-order functional differential equations and, particularly, for nonautonomous differential equations with deviated arguments, new sufficient conditions for the existence and uniqueness of a periodic solution are established.
This implies, in particular, that we can study equations with deviated arguments of the form u ″ ( t ) = f ( t, u ( t ), u ( τ ( t ) ) ), where τ : t ∈ ( a, b ) → [ a − r −, b + r + ] is measurable.
Differential equations with deviated arguments are found to be important mathematical tools for the better understanding of several real world problems in physics, mechanics, engineering, economics, etc. [37, 38].
