## Existence and approximation of solutions for nonlinear functional differential equations with periodic boundary value conditions.(English)Zbl 0958.34055

Consider the periodic boundary value problem $du(t)/dt =g(t,u(t), u(\Theta(t))), \;u (0)=u(T), \;0 \leq t \leq T,\tag{*}$ where $$g:[0,T] \times \mathbb{R}^2 \rightarrow \mathbb{R}$$ and $$\Theta: [0,T] \rightarrow \mathbb{R}$$ are continuous, $$0 \leq \Theta (t) \leq t.$$ First, the authors study the linear problem $$du(t)/dt+M u(t) +Nu(\Theta (t)) = \sigma (t)$$ with $$\sigma \in C([0,T], \mathbb{R})$$ and prove maximum principles for different boundary inequality conditions. Using these principles, the authors define upper and lower solutions to $$(*)$$ and derive conditions such that the existence of ordered lower and upper solutions imply the existence of a solution to $$(*)$$. Finally, a result for monotone iteration is proved.

### MSC:

 34K13 Periodic solutions to functional-differential equations 34K07 Theoretical approximation of solutions to functional-differential equations 34K05 General theory of functional-differential equations
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### References:

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