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This paper investigates first-order variable metric backward forward dynamical systems associated with monotone inclusion and convex minimization problems in real Hilbert space. The operators are chosen so that the backward-forward dynamical system is closely related to the forward-backward dynamical system and has the same computational complexity. We show existence, uniqueness, and weak asymptotic convergence of the generated trajectories and strong convergence if one of the operators is uniformly monotone. We also establish that an equilibrium point of the trajectory is globally exponentially stable and monotone attractor. As a particular case, we explore similar perspectives of the trajectories generated by a dynamical system related to the minimization of the sum of a nonsmooth convex and a smooth convex function. Numerical examples are given to illustrate the convergence of trajectories., 3 figures

276 Background: NET are malignant tumors arising from neuroendocrine cells throughout the body. Everolimus (EVE), a mammalian target of rapamycin inhibitor, is approved for the treatment of advanced, well-differentiated pancreatic NET. There is an unmet medical need in GI and lung NET; targeted therapies, such as everolimus, are of particular interest. Methods: Patients with advanced nonfunctional NET of GI or lung origin with progressive disease (PD) within the past 6 months were randomized (2:1) to EVE 10 mg/d or placebo, both with best supportive care. Concomitant use of somatostatin analogue (SSA) was not allowed during the study, except for control of emergent carcinoid symptoms not manageable by standard therapy. Patients were stratified based on tumor sites, prior SSA exposure, and WHO performance status (PS) at baseline. Primary endpoint was progression-free survival (PFS) as assessed by central radiology review using modified RECIST 1.0 criteria. Primary analysis is planned after ~176 PFS events. Crossover to open label EVE after progression would not be allowed prior to the primary analysis. Overall survival was the key secondary endpoint. Results: Recruitment is completed. Of 388 patients screened, 302 were randomized (planned, 285). Median age was 63 years, 53% were females, and majority of them (76.2%) were white. The most common tumor sites were lung (29.8%), ileum (23.5%), and rectum (13.2%). WHO PS was 0 in 219 (72.5%) patients and 1 in 82 (27.2%) patients; 52% had received SSA prior to study entry. As of Sep 16, 2013, 173 (57.3%) patients remain on treatment, 127 (42.1%) discontinued treatment and 2 (0.7%) were not treated. PD (24.2%) and adverse events (10.6%) were the most common reasons for treatment discontinuation. Results of primary analysis are expected by early 2015. Conclusions: RADIANT-4 is the first phase III study to assess the efficacy and safety of EVE in patients with nonfunctional NET of GI or lung origin. Non-crossover design and prospective stratification of the population based on known prognostic factors should minimize confounding in the estimation of the treatment effect. Clinical trial information: NCT01524783.

We consider the following generalized quasi-variational inequality problem introduced in [7]: given a real normed space $X$ with topological dual $X^*$, two sets $C,D \subseteq X$ and two multifunctions $S: C \to 2^D$ and $T: C \to 2^{X^*}$, find $(\hat x,\hat\varphi) \in C \times X^*$ such that $$\hat x \in S(\hat x), \quad\hat\varphi \in T(\hat x) \quad{\rm and} \quad \langle\hat\varphi,\hat x-y\rangle \le 0 \quad{\rm for\ all}\quad y\in S(\hat x).$$ We prove an existence theorem where $T$ is not assumed to have any continuity or monotonicity property, improving some aspects of the main result of [7]. In particular, the coercivity assumption is meaningfully weakened. As an application, we prove a theorem of the alternative for the fixed points of a Hausdorff lower semicontinuous multifunction. In particular, we obtain sufficient conditions for the existence of a fixed point which belongs to the relative boundary of the corresponding value.

This paper is mainly devoted to applications of modern tools of variational analysis and generalized differentiation to the metric regularity of parametric generalized inequality systems in infinite-dimensional spaces. The basic tools of our analysis involve the Mordukhovich normal coderivatives of set-valued mappings, the limiting subgradient estimate for the marginal functions, and the Ekeland variational principle. Using these tools, we establish new sufficient conditions for the metric regularity of parametric generalized inequality systems. Our results extend the corresponding results in [23] which established some pointbased sufficient conditions for the metric regularity in the Robinson’s sense of implicit multifunctions in finite-dimensional setting.

The purpose of this paper is to introduce and study two modified hybrid proximal-point algorithms for finding a common element of the solution set EP of a generalized equilibrium problem and the set for two maximal monotone operators and defined on a Banach space . Strong and weak convergence theorems for these two modified hybrid proximal-point algorithms are established.

We study the controllability problem for a system governed by a degenerate semilinear functional differential inclusion in a Banach space with infinite delay. Notice that we are not assuming that the generalized semigroup generated by the linear part of inclusion is compact. Instead we suppose that the multivalued nonlinearity satisfies the regularity condition expressed in terms of the Hausdorff measure of noncompactness. It allows to obtain the general controllability principle in the terms of the topological degree theory for condensing multivalued operators. Two realizations of this principle are considered.