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In this paper we investigate quasi equilibrium problems in a real Banach space under the assumption of Brezis pseudomonotonicity of the function involved. To establish existence results under weak coercivity conditions we replace the quasi equilibrium problem with a sequence of penalized usual equilibrium problems. To deal with the non compact framework, we apply a regularized version of the penalty method. The particular case of set-valued quasi variational inequalities is also considered.

In this paper we investigate the property of engulfing for H-convex functions defined on the Heisenberg group $${\mathbb {H}^n}$$ H n . Starting from the horizontal sections introduced by Capogna and Maldonado (Proc Am Math Soc 134:3191–3199, 2006) , we consider a new notion of section, called $${\mathbb {H}^n}$$ H n -section, as well as a new condition of engulfing associated to the $${\mathbb {H}^n}$$ H n -sections, for an H-convex function defined in $$\mathbb {H}^n.$$ H n . These sections, that arise as suitable unions of horizontal sections, are dimensionally larger; as a matter of fact, the $${\mathbb {H}^n}$$ H n -sections, with their engulfing property, will lead to the definition of a quasi-distance in $${\mathbb {H}^n}$$ H n in a way similar to Aimar et al. in the Euclidean case (J Fourier Anal Appl 4:377–381, 1998). A key role is played by the property of round H-sections for an H-convex function, and by its connection with the engulfing properties.

In this paper we prove the existence of solutions of regularized set-valued variational inequalities involving Brezis pseudomonotone operators in reflexive and locally uniformly convex Banach spaces. By taking advantage of this result, we approximate a general set-valued variational inequality with suitable regularized set-valued variational inequalities, and we show that their solutions weakly converge to a solution of the original one. Furthermore, by strengthening the coercivity conditions, we can prove the strong convergence of the approximate solutions.

In this note, we provide a simple proof of some properties enjoyed by convex functions having the engulfing property. In particular, making use only of results peculiar to convex analysis, we prove that differentiability and strict convexity are conditions intrinsic to the engulfing property.

Optical detection techniques based on surface enhanced Raman spectroscopy (SERS) and capable of providing relevant information on molecular and protein composition of biological samples, are gaining rising attention in clinical research as alternatives to traditional detection assays. Meanwhile, due to the technological advances in compact instrumentation as well as in nanofabrication processes, SERS probes based on portable guided-wave systems, have been implemented thus providing for easier accessibility into complex environments and enabling for real-time in situ detection of low concentrated target analytes. In the present study, low-cost fabrication processes were successfully combined with surface functionalization strategies for the fabrication of disposable SERS-active substrates, engineered to tightly fit the distal end of portable Raman instruments. Being based on a polymer casting fabrication process, the overall design of the substrates can be easily adapted to the varied geometry of the probes to be fit, thus guaranteeing high design versatility. SERS-functionality was achieved by immobilization of gold nanoparticles whose size and shape directly affect the plasmonic properties of the substrates. Moreover, SERS substrates can be further modified by covalently binding molecules acting as baits to selectively fish target biomarkers within heterogeneous samples thereby increasing the specificity of SERS signals. Finally, these sensors represent a powerful tool potentially implementable for the early diagnosis of widespread pathologies by real-time SERS analysis of liquid biopsy.

The aim of this paper is to show that every representative function of a maximally monotone operator is the Fitzpatrick transform of a bifunction corresponding to the operator. In fact, for each representative function \(\varphi \) of the operator, there is a family of equivalent saddle functions (i.e., bifunctions which are concave in the first and convex in the second argument) each of which has \(\varphi \) as Fitzpatrick transform. In the special case where \(\varphi \) is the Fitzpatrick function of the operator, the family of equivalent saddle functions is explicitly constructed. In this way we exhibit the relation between the recent theory of representative functions, and the much older theory of saddle functions initiated by Rockafellar.

In this paper we study the linear openness of the composition of set-valued maps carried out thanks to applications of Nadler’s fixed point theorem and Lim’s lemma. As a byproduct, we obtain the Lipschitz property of the solution map of a generalized parametric equation and of parametric approximate variational inequalities, as well.

In this paper, we provide sufficient conditions entailing the existence of weak sharp efficient points of a multiobjective optimization problem. The approach uses variational analysis techniques, like regularity and subregularity of the diagonal subdifferential map related to a suitable scalar equilibrium problem naturally associated to the multiobjective optimization problem.

Ion exchange in silver nitrate is a well-established technique for modifying the chemical-physical properties of the glass materials and is widely used in the fields of Integrated Optics (IO) and, more recently, in Plasmonics due to the possibility to induce the formation of silver nanoparticles in the glass matrix by an ad-hoc thermal post-process. Here we report the application of this technology for the realization of SERS planar substrates in commercial soda-lime glasses. This study is preparatory for the development of a glass optical fibre probe, treated by ion-exchange, for SERS applications. The main results obtained on planar samples as well as the preliminary ones on the glass optical fibre will be presented.

We solve a general vector variational inequality problem in a finite—dimensional setting, where only approximation sequences are known instead of exact values of the cost mapping and feasible set. We establish a new equivalence property, which enables us to replace each vector variational inequality with a scalar set-valued variational inequality. Then, we approximate the scalar set-valued variational inequality with a sequence of penalized problems, and we study the convergence of their solutions to solutions of the original one.

In this paper we prove that maximal H-monotone operators T : H n ⇉ V 1 whose domain is all the Heisenberg group H n are locally bounded. This implies that they are upper semicontinuous. As a consequence, maximal H-monotonicity of an operator on H n can be characterized by a suitable version of Minty’s type theorem.

Keratoconus is an eye disorder that features a reduced stiffness of the cornea and its consequent pathological deformation. Cross-Linking (CXL) treatment has proven useful in hindering the progression of keratoconus, offering a minimally-invasive alternative to corneal surgical transplantation. In this study, the biomechanical characteristics of a human keratoconic cornea were clinically examined in vivo soon before keratoplasty, and the morphological alterations of the collagen scaffold in the same cornea were examined ex vivo by means of Second-Harmonic Generation (SHG) microscopy. A healthy cornea and a CXL-treated keratoconus were compared. In particular, the lamellar organization in the three corneal samples was characterized in different stromal layers by detecting both forward- and backwardscattered SHG signal and then considering the forward/backward (F/B) ratio as parameter. The F/B ratio was used to characterize the morphological organization of collagen lamellae within different stromal layers, finding an increased disorder at the level of Bowman's membrane, opposed to a more regular organization within deeper stromal layers in all the examined samples. The organization of collagen lamellae in CXL-treated keratoconic samples was similar to that one found in healthy corneas, demonstrating that the CXL is able to rearrange the collagen scaffold and partially recover the properties of a healthy condition. The obtained results are in agreement with previous results obtained in studies aimed at monitoring the organization of fibrillar collagen using F/B SHG ratio. In conclusion, the proposed method might be useful for both diagnosing keratoconus as well as for monitoring the effects of the CXL treatment.

In this paper we investigate the Aubin property of the solution map of a parametric equilibrium problem, by providing a connection with a suitable behaviour of the diagonal subdifferential operator associated to the equilibrium bifunction. In particular, we shed some light on the relationship between metric regularity and subregularity of the diagonal subdifferential, on one side, and some properties of the bifunction, on the other side.

SHG microscopy was successfully used to characterize the orientation of sutural lamellae within corneal samples by means of an epi-detection scheme. In particular, the attention was focused on the organization and orientation of corneal collagen lamellae in the first 30 mu m of the stromal layer below Bowman's membrane. In fact, it is thought that the irregular orientation of these specific lamellae, an in particular the existence of so-called sutural lamellae, affects the mechanical properties of the cornea and is responsible of the overall corneal stiffness. The methodology was tested by acquiring image stacks of the central portion of the cornea in ex vivo samples of healthy corneas, keratoconic corneas and keratoconic corneas treated with cross-linking. These samples represent a good benchmark for testing the methodology, considering that keratoconus is an ophthalmic disease in which the cornea loses its stiffness and acquires an abnormal conical shape because of a different organization of sutural lamellae. The samples were first imaged and characterized on the basis of forward/backward SHG ratio, finding a different ratio in keratoconus with respect to both the healthy cornea and cross-linked keratoconus. The inclination of corneal sutural lamellae was then characterized by means of a three-dimensional correlation analysis on SHG images, acquired using a backward detection geometry. Such method provided good discrimination capabilities, demonstrating that this approach can be used not only for diagnosing keratoconus in a very early stage, but also for performing treatment follow-up on cross-linked corneas.

In this paper we extend some classical results of Convex Analysis to the sub-Riemannian setting of the Heisenberg group. In particular, we provide a horizontal version of Minty’s theorem concerning maximal H-monotone operators defined in the Heisenberg group with values in the first layer of its Lie algebra.

Over recent years, gold nanorods (GNRs) have emerged as a promising material in biomedical optics and have been proposed as contrast agents for the photothermal therapy and the photoacoustic imaging of tumors. A pioneering approach to target tumors is the use of cellular vehicles, i.e. cells of the immune system that exhibit an innate tropism to tumors and that can be serve as Trojan horses. This strategy relies on cell types, such as tumor-associated macrophages or T cells, that are recruited by or naturally traffic to the microenvironment of tumors and that can be isolated from a patient and loaded with plasmonic particles in vitro. In this work, GNRs were synthesized and designed to combine high optical and photo-stability and the ability to accumulate into cells of the immuno system. Particles were silanized, PEGylated and conjugated with cationic moieties. Different cationic compounds were tested and the cell viability and uptake of the particles were studied on complementary cell types. The cytotoxicity test was based on a colorimetric WST-8 assay while the intracellular amount of gold and the optical absorbance of the cells were quantified by spectrophotometry. Moreover, we investigated the effect of GNRs on the cell migration and the production of cytokines in the presence of pro-inflammatory stimuli, which provide a functional overview on the feasibility of this approach to target.

Keratoconus is an ophthalmic disease in which the cornea acquires an abnormal conical shape that prevents the correct focusing on the retina, causing visual impairment. The late diagnosis of keratoconus is among the principal causes of corneal transplantation surgery. In this study, we characterize the morphology of keratoconic corneas by means of the correlation of SHG images, finding that keratoconus can be diagnosed by analyzing the inclination of lamellae below Bowman’s membrane. In addition, imaging performed with both sagittal and “en face” geometry demonstrated that this morphological features can be highlighted both ex vivo and in vivo.

In this paper we introduce some notions of well-posedness for scalar equilibrium problems in complete metric spaces or in Banach spaces. As equilibrium problem is a common extension of optimization, saddle point and variational inequality problems, our definitions originates from the well-posedness concepts already introduced for these problems.We give sufficient conditions for two different kinds of well-posedness and show by means of counterexamples that these have no relationship in the general case. However, together with some additional assumptions, we show via Ekeland's principle for bifunctions a link between them.Finally we discuss a parametric form of the equilibrium problem and introduce a well-posedness concept for it, which unifies the two different notions of well-posedness introduced in the first part. (C) 2009 Elsevier Ltd. All rights reserved.

We introduce and study two notions of well-posedness for vector equilibrium problems in topological vector spaces; they arise from the well-posedness concepts previously introduced by the same authors in the scalar case, and provide an extension of similar definitions for vector optimization problems. The first notion is linked to the behaviour of suitable maximizing sequences, while the second one is defined in terms of Hausdorff convergence of the map of approximate solutions. In this paper we compare them, and, in a concave setting, we give sufficient conditions on the data in order to guarantee well-posedness. Our results extend similar results established for vector optimization problems known in the literature. © 2008 Springer-Verlag.

In this paper, the authors deal with bifunctions defined on complete metric spaces and with values in locally convex spaces ordered by closed convex cones. The aim is to provide a vector version of Ekeland's theorem related to equilibrium problems. To prove this principle, a weak notion of continuity of a vector-valued function is considered, and some of its properties are presented. Via the vector Ekeland's principle, existence results for vector equilibria are proved in both compact and noncompact domains. (c) 2006 Elsevier Ltd. All rights reserved.

Maternal hypotension during spinal anaesthesia for Caesarean delivery is a common event, with potential detrimental consequences. We led a prospective, randomized study to compare the effects of two strategies on neonatal and maternal wellbeing.Parturients scheduled for elective Caesarean section in spinal anaesthesia were preoperatively studied with a supine stress test. Those with a positive test were enrolled in the study and received a solution of 0.5% hyperbaric bupivacaine 12.5 mg and 0.02% morphine 200 µg intrathecally. Patients received a 37.5 mg/h preventive intravenous (IV) infusion of ephedrine (Pharmacologic Group), or a 15° left lateral tilt (Non-Pharmacologic Group). In Pharmacolgic Group hypotension was treated for 20% drops in systolic blood pressure; in Non-Pharmacolgic Group only severe hypotension - defined as a 40% drop in systolic blood pressure - was treated.Thirty-six patients were studied. Study groups were statistically similar in terms of demographic variables and intraoperative times. No statistical differences were found in terms of umbilical arterial blood base excess [-1.4 (-3.7 to -0.3) mEq/l Pharmacologic Group vs. -1.7 (-2.7 to -1.0) mEq/l Non-Pharmacologic Group; p=0.815] and other umbilical blood gas values. Apgar scores were statistically similar between study groups. Treatment for hypotension was required by 13 (72.2%) patients in Pharmacologic Group and 9 (50%) patients in Non-Pharmacologic Group (p=0.171). No differences were found at the analysis of serial changes in vital signs.Both studied strategies guaranteed a comparable safe outcome in terms of maternal and neonatal wellbeing. (www.actabiomedica.it).

Several attempts were performed to use efficient organic dyes for DSSCs as building integrated (BIPV) windows, that could substitute the more expensive Ru complexes. For this particular application, transparency in the 500 - 600 nm region is a stringent demand that in most cases implied the use a co-sensitizing NIR dye in addition to a dye sensitizer in the visible region. We employed also gold nanorods (GNRs) to enhance the absorption of the two dyes, with the aim of improving also cell efficiency, which was expected to be lower with respect to those obtained in panchromatic applications.

In this paper, we present the notions of openness and metric regularity for a set-valued map with respect to two fixed sets, proving their equivalence. By using different approaches, we show the stability, with respect to the sum of maps, of the openness property, both in the setting of Banach spaces and of metric spaces. Finally, we infer the regularity of the map solving a generalized parametric equation defined via a parametric map that is, in its turn, perturbed by the sum with another map.

In the literature, when dealing with equilibrium problems and the existence of their solutions, the most used assumptions are the convexity of the domain and the generalized convexity and monotonicity, together with some weak continuity assumptions, of the function. In this paper, we focus on conditions that do not involve any convexity concept, neither for the domain nor for the function involved. Starting from the well-known Ekeland's theorem for minimization problems, we find a suitable set of conditions on the function f that lead to an Ekeland's variational principle for equilibrium problems. Via the existence of c-solutions, we are able to show existence of equilibria on general closed sets for equilibrium problems and systems of equilibrium problems. (c) 2004 Elsevier Inc. All rights reserved.

The study of the existence of solutions of equilibrium problems on unbounded domains involves usually the same sufficient assumptions as for bounded domains together with a coercivity condition. We focus on two different conditions: the first is obtained assuming the existence of a bounded set such that no elements outside is a candidate for a solution; the second allows the solution set to be unbounded. Our results exploit the generalized monotonicity properties of the function f defining the equilibrium problem. It turns out that, in both the pseudomonotone and the quasimonotone setting, an equivalence can be stated between the nonemptyness and boundedness of the solution set and these coercivity conditions. In the pseudomonotone case, we compare our coercivity conditions with various coercivity conditions that appeared in the literature.

In the recent literature, the connection between maximal monotone operators and the Fitzpatrick function is investigated. Subsequently, this relation has been extended to maximal monotone bifunctions and their Fitzpatrick transform. In this paper we generalize some of these results to maximal $$n$$ n -cyclically monotone and maximal cyclically monotone bifunctions, by introducing and studying the Fitzpatrick transforms of order $$n$$ n or infinite order for bifunctions.

The purpose of this paper is to bring a new light on the state-dependent Hamilton–Jacobi equation and its connection with the Hopf–Lax formula in the framework of a Carnot group \((\mathbf G ,\circ ).\) The equation we shall consider is of the form $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} u_{t}+ \Psi (X_{1}u, \ldots , X_{m}u)=0\qquad &{}(x,t)\in \mathbf G \times (0,\infty ) \\ {u}(x,0)=g(x)&{}x\in \mathbf G , \end{array} \right. \end{aligned}$$ where \(X_{1},\ldots , X_{m}\) are a basis of the first layer of the Lie algebra of the group \(\mathbf G ,\) and \(\Psi : \mathbb{R }^{m} \rightarrow \mathbb{R }\) is a superlinear, convex function. The main result shows that the unique viscosity solution of the Hamilton–Jacobi equation can be given by the Hopf–Lax formula $$\begin{aligned} u(x,t) = \inf _{y\in \mathbf G }\left\{ t \Psi ^\mathbf{G }\left( \delta _{\frac{1}{t}}(y^{-1}\circ x)\right) + g(y) \right\} , \end{aligned}$$ where \(\Psi ^\mathbf{G }:\mathbf G \rightarrow \mathbb{R }\) is the \(\mathbf G \)-Legendre–Fenchel transform of \(\Psi ,\) defined by a control theoretical approach. We recover, as special cases, some known results like the classical Hopf–Lax formula in the Euclidean spaces \(\mathbb{R }^n,\) showing that \(\Psi ^{\mathbb{R }^n}\) is the Legendre–Fenchel transform \(\Psi ^*\) of \(\Psi ;\) moreover, we recover the result by Manfredi and Stroffolini in the Heisenberg group for particular Hamiltonian function \(\Psi .\) In this paper we follow an optimal control problem approach and we obtain several properties for the value functions \(u\) and \(\Psi ^\mathbf G .\)

In this paper we extend a (scalarized) generalized type-I invexity into a vector invexity (V-type I). A number of sufficiency results are established using Lagrange multiplier conditions and under various types of generalized V-type I requirements. Weak, strong, and converse duality theorems are proved in the generalized V-invexity type I setting. © 2001 Academic Press.

In this paper, we consider some well–known equilibrium problems and their duals in a topological Hausdorff vector space X for a bifunction F defined on \kk,where K is a convex subset of X. Some necessary conditions are investigated, proving different results depending on the behaviour of F on the diagonal set. The concept of proper quasimonotonicity for bifunctions is defined, and the relationship with generalized monotonicity is investigated. The main result proves that the condition of proper quasimonotonicity is sharp in order to solve the dual equilibrium problem on every convex set.

General definitions of monotonocity, quasimonotonocity, pseudomonotonocity and strong monotonocity of a vector valued function defined on a subset of an n-dimensional Euclidean space are given. Some properties and characterizations are studied in the genera] setting. A relationship between generalized monotonocity and (Φ1, Φ2)– convexity introduced earlier by the authors, is described under certain conditions. Applications of the general definitions are studied in terms of invexity and B-vexity.

Convexity of a function and of a set are generalized. The new class being introduced, include many well known classes as its subclasses. The defining functions involved are required to satisfy certain regularity conditions. Some properties are studied with or without differentiability; in the differentiable case, first and second order conditions are stated.

This work deals with non-global inverse map results for the sum of two maps. We prove two theorems which shed some new light on this aspect. Some implications in terms of sensitivity of parametric generalized equations are investigated. Finally, a class of well-conditioned operators is identified.

We present a laser-assisted system for dispensing liquid micro-droplets by near infrared illumination of a pyroelectric crystal functionalized with gold nanorods embedded into polyvinyl alcohol. The non-invasive near infrared source resonates with the plasmon oscillations of the gold nanorods, providing a controlled thermal stimulus able to generate the pyroelectric effect. The resulting electric field interacts electro-hydrodynamically with a liquid reservoir leading to precise drawing of micro-litre droplets. This laser-assisted electro-hydrodynamic technique may open the way to the development of more compact and non-invasive nano-dispensing devices. (C) 2013 AIP Publishing LLC.

Definition of convexity of a function is weakened to extend some separation theory and theorems of the alternative. As an application, necessary conditions are given for a multiobjective minimization (in the Pareto sense) problem.

The properties of invex functions on manifolds with respect to particular maps are investigated to throw a new light on the relationship between invexity and convexity along curves: geodesic convexity on Rieman-nian manifolds is obtained as a particular case

Generalized monotone maps are studied under affine variable transformations. The results enable us to generate generalized monotone matrices of any size. Various necessary conditions and sufficient conditions for generalized monotone matrices are derived. Furthermore, admissible translations of generalized monotone linear maps are studied. Finally, the maximal domain of generalized monotonicity is characterized

Gold nanorods (GNRs) show enhanced absorption in the near infrared (NIR) region and can be conveniently employed for selective conversion of light into heat. We have succeeded in fabricating hybrid GNRs-polysaccharides composites in the form of semi-solid formulations, films or hydrogels, with good biocompatibility, enhanced stabilization in physiological environment and under laser irradiation and high versatility. Thanks to these properties, these materials are good candidates for applications in many biomedical methodologies including laser-mediated tissue repair and drug delivery. The present results are encouraging toward the development of a novel minimally-invasive technology based on the introduction of bio-inspired nanoplasmonic materials into photothermal applications.

Thermal modifications induced in the corneal stroma were investigated by means of second harmonic generation (SHG) imaging. Whole fresh cornea samples were heated in a water bath at temperatures in the 35-80 °C range for a 4-min time. SHG images of the structural modifications induced at each temperature were acquired from different areas of cross-sectioned corneal stroma by using an 880 nm linearly- and circularly-polarized excitation light emitted by a mode-locked Ti:Sapphire laser. The SHG images were then analyzed by means of both an empirical approach and a 2D-theoretical model. The proposed analyses provide a detailed description of the changes occurring in the structural architecture of the cornea during the thermal treatment. Our results allow us to depict a temperature-dependent biochemical model for the progressive destructuration occurring to collagen fibrils and nonfibrillar components of the stroma.

The aim of this work is to analyze lexicographic equilibrium problems on a topological Hausdorff vector space X, and their relationship with some other vector equilibrium problems. Existence results for the tangled lexicographic problem are proved via the study of a related sequential problem. This approach was already followed by the same authors in the case of variational inequalities. © 2009 Springer Science+Business Media, LLC.

In questo lavoro viene affrontato un problema di decisioni sui margini finanziari che puo essere inquadrato nell'ambito dei problemi di controllo outimo governati da equazioni integrali di Volterra. Dopo una descrizione del problema economico e delle operazioni finanziarie coinvolte, si schematizza il modello nell'ambito del controllo ottimo regolato da equazioni di tipo integrale che ammettono controlli discontinui di tipo bang-bang. Si danno condizioni necessarie per l'esistenza di soluzioni ottime e, infine, si esamina un caso particolare per evidenziare come il metodo considerato consenta di analyzzare e di individuare le strategie adeguate qualora si specifichino le funzioni in questione.

Thermal modifications induced in corneal stroma were investigated with the use of fluorescence microscopy. Tissue samples were heated in a water bath at temperatures in the 35-90 degrees C range. Fluorescence images of the structural modifications induced were acquired after staining with Indocyanine Green (ICG). Discrete Fourier Transform (DFT) and entropy analyses of each image made it possible to characterize the thermally-induced phase transitions in the stroma, and to indicate a threshold value for high thermal damage. The procedure could be proposed as the basis for a real-time controlling system for surgical techniques based on induced thermal effects.

This paper establishes a heat semigroup version of Bernstein's theorem, applicable to any unimodular Lie group. The result has an intrinsic geometric content, involving estimates for the norms of the heat kernels for small time and large time. The theorem is stated in terms of certain Lipschitz spaces whose definition incorporates these two geometric features of the group in question. The geometric content is further underlined by showing that, in a certain sence, the theorem is best-possible. © 1990 Springer-Verlag.

We adapt the Dagum survival function to become a function of space and time and study its theoretical properties as a covariance in the isotropic case. The resulting Dagum class is proved to have certain interesting mathematical properties and shows smooth behaviour at the origin, which has considerable applicability. A simple extension to the spatio-temporal case is provided and interesting points of comparison arise with other models appearing in literature.

In this article, we consider equivalence properties between various kinds of lexicographic variational inequalities. By employing various concepts of monotonicity, we show that the usual sequential variational inequality is equivalent to the direct lexicographic variational inequality or to the dual lexicographic variational inequality. We establish several existence results for lexicographic variational inequalities. Also, we introduce the lexicographic complementarity problem and establish its equivalence with the lexicographic variational inequality. We illustrate our approach by several examples of applications to vector transportation and vector spatial equilibrium problems.