Measure functional differential equations and functional dynamic equations on time scales

We study measure functional differential equations and clarify their relation to generalized ordinary differential equations. We show that functional dynamic equations on time scales represent a special case of measure functional differential equations. For both types of equations, we obtain results on the existence and uniqueness of solutions, continuous dependence, and periodic averaging.

Boundedness of solutions of retarded functional differential equations with variable impulses via generalized ordinary differential equations

In this paper, we give sufficient conditions for the uniform boundedness and uniform ultimate boundedness of solutions of a class of retarded functional differential equations with impulse effects acting on variable times. We employ the theory of generalized ordinary differential equations to obtain our results. As an example, we investigate the boundedness of the solution of a circulating fuel nuclear reactor model.

Periodic solutions of abstract neutral functional differential equations

We characterize the existence of periodic solutions of some abstract neutral functional differential equations with finite and infinite delay when the underlying space is a UMD space. (C) 2011 Elsevier Inc. All rights reserved.

Existence of Solutions for Abstract Non-Autonomous Neutral Differential Equations

In this paper we discuss the existence of mild and classical solutions for a class of abstract non-autonomous neutral functional differential equations. An application to partial neutral differential equations is considered.

On abstract differential equations with nonlocal conditions involving the temporal derivative of the solution

In this paper we discuss the existence of solutions for a class of abstract differential equations with nonlocal conditions for which the nonlocal term involves the temporal derivative of the solution. Some concrete applications to parabolic differential equations with nonlocal conditions are considered. (C) 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

Periodic averaging theorems for various types of equations

We prove a periodic averaging theorem for generalized ordinary differential equations and show that averaging theorems for ordinary differential equations with impulses and for dynamic equations on time scales follow easily from this general theorem. We also present a periodic averaging theorem for a large class of retarded equations.

EXISTENCE OF SOLUTIONS FOR A FRACTIONAL NEUTRAL INTEGRO-DIFFERENTIAL EQUATION WITH UNBOUNDED DELAY

In this article, we study the existence of mild solutions for fractional neutral integro-differential equations with infinite delay.

Semilinear functional difference equations with infinite delay

We obtain boundedness and asymptotic behavior of solutions for semilinear functional difference equations with infinite delay. Applications to Volterra difference equations with infinite delay are shown. (C) 2011 Elsevier Ltd. All rights reserved.

Differential cellular FGF-2 upregulation in the rat facial nucleus following axotomy, functional electrical stimulation and corticosterone: a possible therapeutic target to Bell's palsy

Abstract Background The etiology of Bell's palsy can vary but anterograde axonal degeneration may delay spontaneous functional recovery leading the necessity of therapeutic interventions. Corticotherapy and/or complementary rehabilitation interventions have been employed. Thus the natural history of the disease reports to a neurotrophic resistance of adult facial motoneurons leading a favorable evolution however the related molecular mechanisms that might be therapeutically addressed in the resistant cases are not known. Fibroblast growth factor-2 (FGF-2) pathway signaling is a potential candidate for therapeutic development because its role on wound repair and autocrine/paracrine trophic mechanisms in the lesioned nervous system. Methods Adult rats received unilateral facial nerve crush, transection with amputation of nerve branches, or sham operation. Other group of unlesioned rats received a daily functional electrical stimulation in the levator labii superioris muscle (1 mA, 30 Hz, square wave) or systemic corticosterone (10 mgkg-1). Animals were sacrificed seven days later. Results Crush and transection lesions promoted no changes in the number of neurons but increased the neurofilament in the neuronal neuropil of axotomized facial nuclei. Axotomy also elevated the number of GFAP astrocytes (143% after crush; 277% after transection) and nuclear FGF-2 (57% after transection) in astrocytes (confirmed by two-color immunoperoxidase) in the ipsilateral facial nucleus. Image analysis reveled that a seven days functional electrical stimulation or corticosterone led to elevations of FGF-2 in the cytoplasm of neurons and in the nucleus of reactive astrocytes, respectively, without astrocytic reaction. Conclusion FGF-2 may exert paracrine/autocrine trophic actions in the facial nucleus and may be relevant as a therapeutic target to Bell's palsy.

Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.

Differential calculus and integration of generalized functions over membranes

In this paper we continue the development of the differential calculus started in Aragona et al. (Monatsh. Math. 144: 13-29, 2005). Guided by the so-called sharp topology and the interpretation of Colombeau generalized functions as point functions on generalized point sets, we introduce the notion of membranes and extend the definition of integrals, given in Aragona et al. (Monatsh. Math. 144: 13-29, 2005), to integrals defined on membranes. We use this to prove a generalized version of the Cauchy formula and to obtain the Goursat Theorem for generalized holomorphic functions. A number of results from classical differential and integral calculus, like the inverse and implicit function theorems and Green's theorem, are transferred to the generalized setting. Further, we indicate that solution formulas for transport and wave equations with generalized initial data can be obtained as well.

On a new class of abstract integral equations and applications

In this paper we introduce a new class of abstract integral equations which enables us to study in a unified manner several different types of differential equations. (C) 2012 Elsevier Inc. All rights reserved.

Differential regulation of PGC-1α expression in rat liver and skeletal muscle in response to voluntary running

Abstract Background The beneficial actions of exercise training on lipid, glucose and energy metabolism and insulin sensitivity appear to be in part mediated by PGC-1α. Previous studies have shown that spontaneously exercised rats show at rest enhanced responsiveness to exogenous insulin, lower plasma insulin levels and increased skeletal muscle insulin sensitivity. This study was initiated to examine the functional interaction between exercise-induced modulation of skeletal muscle and liver PGC-1α protein expression, whole body insulin sensitivity, and circulating FFA levels as a measure of whole body fatty acid (lipid) metabolism. Methods Two groups of male Wistar rats (2 Mo of age, 188.82 ± 2.77 g BW) were used in this study. One group consisted of control rats placed in standard laboratory cages. Exercising rats were housed individually in cages equipped with running wheels and allowed to run at their own pace for 5 weeks. At the end of exercise training, insulin sensitivity was evaluated by comparing steady-state plasma glucose (SSPG) concentrations at constant plasma insulin levels attained during the continuous infusion of glucose and insulin to each experimental group. Subsequently, soleus and plantaris muscle and liver samples were collected and quantified for PGC-1α protein expression by Western blotting. Collected blood samples were analyzed for glucose, insulin and FFA concentrations. Results Rats housed in the exercise wheel cages demonstrated almost linear increases in running activity with advancing time reaching to maximum value around 4 weeks. On an average, the rats ran a mean (Mean ± SE) of 4.102 ± 0.747 km/day and consumed significantly more food as compared to sedentary controls (P < 0.001) in order to meet their increased caloric requirement. Mean plasma insulin (P < 0.001) and FFA (P < 0.006) concentrations were lower in the exercise-trained rats as compared to sedentary controls. Mean steady state plasma insulin (SSPI) and glucose (SSPG) concentrations were not significantly different in sedentary control rats as compared to exercise-trained animals. Plantaris PGC-1α protein expression increased significantly from a 1.11 ± 0.12 in the sedentary rats to 1.74 ± 0.09 in exercising rats (P < 0.001). However, exercise had no effect on PGC-1α protein content in either soleus muscle or liver tissue. These results indicate that exercise training selectively up regulates the PGC-1α protein expression in high-oxidative fast skeletal muscle type such as plantaris muscle. Conclusion These data suggest that PGC-1α most likely plays a restricted role in exercise-mediated improvements in insulin resistance (sensitivity) and lowering of circulating FFA levels.

Multivariate gene expression analysis reveals functional connectivity changes between normal/tumoral prostates

Abstract Background Prostate cancer is a leading cause of death in the male population, therefore, a comprehensive study about the genes and the molecular networks involved in the tumoral prostate process becomes necessary. In order to understand the biological process behind potential biomarkers, we have analyzed a set of 57 cDNA microarrays containing ~25,000 genes. Results Principal Component Analysis (PCA) combined with the Maximum-entropy Linear Discriminant Analysis (MLDA) were applied in order to identify genes with the most discriminative information between normal and tumoral prostatic tissues. Data analysis was carried out using three different approaches, namely: (i) differences in gene expression levels between normal and tumoral conditions from an univariate point of view; (ii) in a multivariate fashion using MLDA; and (iii) with a dependence network approach. Our results show that malignant transformation in the prostatic tissue is more related to functional connectivity changes in their dependence networks than to differential gene expression. The MYLK, KLK2, KLK3, HAN11, LTF, CSRP1 and TGM4 genes presented significant changes in their functional connectivity between normal and tumoral conditions and were also classified as the top seven most informative genes for the prostate cancer genesis process by our discriminant analysis. Moreover, among the identified genes we found classically known biomarkers and genes which are closely related to tumoral prostate, such as KLK3 and KLK2 and several other potential ones. Conclusion We have demonstrated that changes in functional connectivity may be implicit in the biological process which renders some genes more informative to discriminate between normal and tumoral conditions. Using the proposed method, namely, MLDA, in order to analyze the multivariate characteristic of genes, it was possible to capture the changes in dependence networks which are related to cell transformation.