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First order iterative functional-differential equation with parameter. (English) Zbl 1212.47039

Summary: We consider the following first order iterative functional-differential equation with parameter \[ \begin{aligned} x'(t)&=f(t,x(t),x(x(t)))+\lambda,\qquad t\in [a,b];\\ x(t)&=\varphi (t),\qquad a_1\leq t\leq a,\\ x(t)&=\psi (t),\qquad b\leq t\leq b_1. \end{aligned} \] Using Schauder’s fixed point theorem, we first establish an existence theorem, then by means of the contraction principle state an existence and uniqueness theorem, and after that a data dependence result. Finally, we give some examples which illustrate our results.

MSC:

47H10 Fixed-point theorems
34K10 Boundary value problems for functional-differential equations
34K20 Stability theory of functional-differential equations
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