Boundary value problems for second order delay differential equations (Q5891675)

The author obtains sufficient conditions for the existence of unique solutions of a class of boundary value problems for functional differential equations of second order of the form NEWLINE\[NEWLINE x^{\prime\prime}(t)=f(t, x_{t}),\quad t\in J=[0,T],\;T>0 NEWLINE\]NEWLINE NEWLINE\[NEWLINE x_{0}=\varphi,\;x^\prime(T)=\beta x^\prime(0),\quad \beta>1, NEWLINE\]NEWLINE where \( f:J\times C([-\tau,0],\mathbb R)\rightarrow\mathbb R \) is a given function and \( \varphi\in C([-\tau, 0],\mathbb R)\), \(\tau> 0\). A quasilinearization technique is employed to construct two monotone sequences which converge to the unique solution of problem (1), (2). The main result of the paper is the following one.NEWLINENEWLINE {Theorem}. Suppose that \( f\in C(J\times C([-\tau, 0],\mathbb R),\mathbb R)\) and there exists \( m\in L^{1}([0, T],\mathbb R_{+}) \) such that NEWLINE\[NEWLINE |f(t,u_{1})-f(t,u_{2})|\leq m(t) \parallel u_{1}-u_{2}\parallel_{0} NEWLINE\]NEWLINE for all \( t\in [0, T]\), \(u_{1}, u_{2}\in C([-\tau, 0],\mathbb R) \) and \( M(T)< \frac{\ln \beta}{T} \), where \( M(t)=\int _{0}^{t} m(s)\, ds\) and \( \parallel\varphi\parallel_{0}= \sup _{\theta \in [-\tau , 0]}\mid \varphi( \theta )\mid \) for \( \varphi \in C([-\tau , 0]\mathbb R) \). Then problem (1), (2) admits a unique solution \( x\in C^{*} \), where NEWLINE\[NEWLINE C^{*}= C([-\tau, T],\mathbb R)\cap C^{2}( [0, T],\mathbb R). NEWLINE\]NEWLINE Examples are given to illustrate the obtained results.