Let $K$ be a nonempty closed convex subset of a real Hilbert space $H$ such that $K\pm K\subset K$, $TK\to H$ a $k$-strict pseudo-contraction for some $0\le k<1$ such that $F\left(T\right)=\{x\in Kx=Tx\}\ne \varnothing$. Consider the following iterative algorithm given by $$\forall x_1\in K,x_{n+1}={\alpha }_n\gamma f\left(x_n\right)+{\beta }_n{x}_n+((1-{\beta }_n)I-{\alpha }_{n}A)P_{K}S{x}_n,n\ge 1,$$ where $SK\to H$ is defined by $Sx=kx+(1-k)Tx$, $P_K$ is the metric projection of $H$ onto $K$, $A$ is a strongly positive linear bounded self-adjoint operator, $f$ is a contraction. It is proved that the sequence $\left\{x_n\right\}$ generated by the above iterative algorithm converges strongly to a fixed point of $T$, which solves a variational inequality related...

This paper deals with feedback stabilization of second order equations of the form ytt + A0y + u (t) B0y (t) = 0, t ∈ [0, +∞[, where A0 is a densely defined positive selfadjoint linear operator on a real Hilbert space H, with compact inverse and B0 is a linear map in diagonal form. It is proved here that the classical sufficient ad-condition of Jurdjevic-Quinn and Ball-Slemrod with the feedback control u = ⟨yt, B0y⟩H implies the strong stabilization. This result is derived from a general compactness...

This paper deals with feedback stabilization of second order equations of the form ytt + A0y + u (t) B0y (t) = 0, t ∈ [0, +∞[, where A0 is a densely defined positive selfadjoint linear operator on a real Hilbert space H, with compact inverse and B0 is a linear map in diagonal form. It is proved here that the classical sufficient ad-condition of Jurdjevic-Quinn and Ball-Slemrod with the feedback control u = ⟨yt, B0y⟩H implies the strong stabilization. This result is derived from a general compactness theorem...

In this paper, we use a modification of Krasnoselskii’s fixed point theorem introduced by Burton (see [Burton, T. A.: Liapunov functionals, fixed points and stability by Krasnoseskii’s theorem. Nonlinear Stud., 9 (2002), 181–190.] Theorem 3) to obtain stability results of the zero solution of the totally nonlinear neutral differential equation with variable delay $$x^{\text{'}}\left(t\right)=-a\left(t\right)h\left(x\left(t\right)\right)+\frac{d}{dt}Q\left(t,x\left(t-\tau \left(t\right)\right)\right)+G\left(t,x\left(t\right),x\left(t-\tau \left(t\right)\right)\right).$$ The stability of the zero solution of this eqution provided that $h\left(0\right)=Q\left(t,0\right)=G\left(t,0,0\right)=0$. The Caratheodory condition is used for the functions $Q$ and $G$.

We consider the summation equation, for $t\in {[\mu -2,\mu +b]}_{{\mathbb{N}}_{\mu -2}}$, $$\begin{array}{ccc}\hfill y\left(t\right)={\gamma }_1\left(t\right)H_1\left(\sum _{i=1}^n{a}_{i}y\left({\xi }_i\right)\right)& +{\gamma }_2\left(t\right)H_2\left(\sum _{i=1}^m{b}_{i}y\left({\zeta }_i\right)\right)\hfill & \hfill +\lambda \sum _{s=0}^{b}G(t,s)f(s+\mu -1,y(s+\mu -1))\end{array}$$ in the case where the map $(t,s)\mapsto G(t,s)$ may change sign; here $\mu \in (1,2]$ is a parameter, which may be understood as the order of an associated discrete fractional boundary value problem. In spite of the fact that $G$ is allowed to change sign, by introducing a new cone we are able to establish the existence of at least one positive solution to this problem by imposing some growth conditions on the functions $H_1$ and $H_2$. Finally, as an application of the abstract existence result,...