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De la Vallée-Poussin theorem on the differential inequality for equations with an aftereffect. (English. Russian original) Zbl 0957.34064
Proc. Steklov Inst. Math. 211, 28-34 (1995); translation from Tr. Mat. Inst. Steklova 211, 32-39 (1995).
In this paper, the authors investigate the boundary value problem (BVP) $${\cal L}x=f,\ x(a)=\alpha \text{ and } x(b)= \beta,$$ where ${\cal L}={\cal L}_0-T, {\cal L}_0:W^2[a,b]\to L[a,b]$ and $T:C[a,b]\to L[a,b]$ are linear, bounded Volterra-type operators and $(Tx_1)(t)\le (Tx _2)(t)$ for $t\in [a,b]$, whenever $x_1(t)\ge x_2(t)$ on $[a,b]$. Defining the mapping $H:C[a,b]\to C[a,b],\ (Hx)(t)=\int_a^bG_0(t,s) (Tx) (s)ds$, $t\in[a,b]$, $x\in C([a,b])$, where $G_0$ is the Green function of (BVP), the authors prove that the following statements are equivalent: (i) There exists $\nu\in W^2[a,b]$, such that $\nu(t)\ge 0$, $({\cal L}\nu) (t)\le 0$ $(a\le t\le b)$, and $\nu(a)+ \nu(b)-\int_a^b({\cal L}\nu) (s)ds>0$; (ii) The spectral radius of operator $H$ is less than one; (iii) (BVP) has exactly one solution for any $f\in L[a, b]$ and any $\alpha,\beta$, and its Green operator is antitone; (iv) The fundamental system of ${\cal L}x=0$ is not oscillatory. For the entire collection see [Zbl 0863.00015].

##### MSC:
 34K10 Boundary value problems for functional-differential equations 34A40 Differential inequalities (ODE)