Fixed point results for multivalued contractions on graphs and their applications

Nous présentons dans cette thèse des théorèmes de point fixe pour des contractions multivoques définies sur des espaces métriques, et, sur des espaces de jauges munis d’un graphe. Nous illustrons également les applications de ces résultats à des inclusions intégrales et à la théorie des fractales. Cette thèse est composée de quatre articles qui sont présentés dans quatre chapitres. Dans le chapitre 1, nous établissons des résultats de point fixe pour des fonctions multivoques, appelées G-contractions faibles. Celles-ci envoient des points connexes dans des points connexes et contractent la longueur des chemins. Les ensembles de points fixes sont étudiés. La propriété d’invariance homotopique d’existence d’un point fixe est également établie pour une famille de Gcontractions multivoques faibles. Dans le chapitre 2, nous établissons l’existence de solutions pour des systèmes d’inclusions intégrales de Hammerstein sous des conditions de type de monotonie mixte. L’existence de solutions pour des systèmes d’inclusions différentielles avec conditions initiales ou conditions aux limites périodiques est également obtenue. Nos résultats s’appuient sur nos théorèmes de point fixe pour des G-contractions multivoques faibles établis au chapitre 1. Dans le chapitre 3, nous appliquons ces mêmes résultats de point fixe aux systèmes de fonctions itérées assujettis à un graphe orienté. Plus précisément, nous construisons un espace métrique muni d’un graphe G et une G-contraction appropriés. En utilisant les points fixes de cette G-contraction, nous obtenons plus d’information sur les attracteurs de ces systèmes de fonctions itérées. Dans le chapitre 4, nous considérons des contractions multivoques définies sur un espace de jauges muni d’un graphe. Nous prouvons un résultat de point fixe pour des fonctions multivoques qui envoient des points connexes dans des points connexes et qui satisfont une condition de contraction généralisée. Ensuite, nous étudions des systèmes infinis de fonctions itérées assujettis à un graphe orienté (H-IIFS). Nous donnons des conditions assurant l’existence d’un attracteur unique à un H-IIFS. Enfin, nous appliquons notre résultat de point fixe pour des contractions multivoques définies sur un espace de jauges muni d’un graphe pour obtenir plus d’information sur l’attracteur d’un H-IIFS. Plus précisément, nous construisons un espace de jauges muni d’un graphe G et une G-contraction appropriés tels que ses points fixes sont des sous-attracteurs du H-IIFS.

Stochastic delay equations with non-negativity constraints driven by fractional Brownian motion

In this note we prove an existence and uniqueness result for the solution of multidimensional stochastic delay differential equations with normal reflection. The equations are driven by a fractional Brownian motion with Hurst parameter H > 1/2. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann¿Stieltjes integral.

Asymptotically almost periodic solutions for abstract neutral integro-differential equations

We study the existence of asymptotically almost periodic classical solutions for a class of abstract neutral integro-differential equation with unbounded delay. A concrete application to partial neutral integro-differential equations which arise in the study of heat conduction in fading memory material is considered. (C) 2011 Elsevier Inc. All rights reserved.

Controllability results for impulsive mixed type functional integro-differential evolution equations with nonlocal conditions

In this paper, we establish the controllability for a class of abstract impulsive mixed-type functional integro-differential equations with finite delay in a Banach space. Some sufficient conditions for controllability are obtained by using the Mönch fixed point theorem via measures of noncompactness and semigroup theory. Particularly, we do not assume the compactness of the evolution system. An example is given to illustrate the effectiveness of our results.

Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions

In this paper, a new class of generalized backward doubly stochastic differential equations is investigated. This class involves an integral with respect to an adapted continuous increasing process. A probabilistic representation for viscosity solutions of semi-linear stochastic partial differential equations with a Neumann boundary condition is given.

ContinuousMultimodal Global Optimization with Differential Evolution-Based Methods

Metaheuristic methods have become increasingly popular approaches in solving global optimization problems. From a practical viewpoint, it is often desirable to perform multimodal optimization which, enables the search of more than one optimal solution to the task at hand. Population-based metaheuristic methods offer a natural basis for multimodal optimization. The topic has received increasing interest especially in the evolutionary computation community. Several niching approaches have been suggested to allow multimodal optimization using evolutionary algorithms. Most global optimization approaches, including metaheuristics, contain global and local search phases. The requirement to locate several optima sets additional requirements for the design of algorithms to be effective in both respects in the context of multimodal optimization. In this thesis, several different multimodal optimization algorithms are studied in regard to how their implementation in the global and local search phases affect their performance in different problems. The study concentrates especially on variations of the Differential Evolution algorithm and their capabilities in multimodal optimization. To separate the global and local search search phases, three multimodal optimization algorithms are proposed, two of which hybridize the Differential Evolution with a local search method. As the theoretical background behind the operation of metaheuristics is not generally thoroughly understood, the research relies heavily on experimental studies in finding out the properties of different approaches. To achieve reliable experimental information, the experimental environment must be carefully chosen to contain appropriate and adequately varying problems. The available selection of multimodal test problems is, however, rather limited, and no general framework exists. As a part of this thesis, such a framework for generating tunable test functions for evaluating different methods of multimodal optimization experimentally is provided and used for testing the algorithms. The results demonstrate that an efficient local phase is essential for creating efficient multimodal optimization algorithms. Adding a suitable global phase has the potential to boost the performance significantly, but the weak local phase may invalidate the advantages gained from the global phase.

On the limit cycles of a class of piecewise linear differential systems in R-4 with two zones

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Optimal mean-variance control for discrete-time linear systems with Markovian jumps and multiplicative noises

In this paper, we consider the stochastic optimal control problem of discrete-time linear systems subject to Markov jumps and multiplicative noises under two criteria. The first one is an unconstrained mean-variance trade-off performance criterion along the time, and the second one is a minimum variance criterion along the time with constraints on the expected output. We present explicit conditions for the existence of an optimal control strategy for the problems, generalizing previous results in the literature. We conclude the paper by presenting a numerical example of a multi-period portfolio selection problem with regime switching in which it is desired to minimize the sum of the variances of the portfolio along the time under the restriction of keeping the expected value of the portfolio greater than some minimum values specified by the investor. (C) 2011 Elsevier Ltd. All rights reserved.

Results of analyses of fluid inclusions from ODP Site 735B, Leg 118 and 176

Microthermometric and isotopic analyses of fluid inclusions in primitive olivine gabbros, oxide gabbros, and evolved granitic material recovered from Ocean Drilling Program Hole 735B at the Southwest Indian Ridge provide new insights into the evolution of C-O-H-NaCl fluids in the plutonic foundation of the oceanic crust. The variably altered and deformed plutonic rocks span a crustal section of over 1500 m and record a remarkably complex magma-hydrothermal history. Magmatic fluids within this suite followed two chemically distinct paths during cooling through the subsolidus regime: the first path included formation of CO2+CH4+H2O+C fluids with up to 43 mole% CH4; the second path produced hypersaline brines that contain up to 50% NaCl equivalent salinities. Subsequent to devolatilization, respeciation of magmatic CO2, attendant graphite precipitation, and cooling from 800°C to 500°C promoted formation of CH4-enriched fluids. These fluids are characterized by average d13C(CH4) values of -27.1+/-4.3 per mil (N=45) with associated d13C(CO2) compositions ranging from -24.9 per mil to -1.9 per mil (N=39), and average dD values of exsolved vapor of -41+/-12 per mil (N=23). In pods, veins, and lenses of highly fractionated residual material, hypersaline brines formed during condensation and by direct exsolution in the absence of a conjugate vapor phase. Entrapped CO2+CH4+H2O-rich fluids within many oxide-bearing rocks and felsic zones are significantly depleted in 13C (with d13C(CO2) values down to about -25 per mil) and contain CO2 concentrations higher than those predicted by equilibrium devolatilization models. We hypothesize that lower effective pressures in high-temperature shear zones promoted infiltration of highly fractionated melts and compositionally evolved volatiles into focused zones of deformation, significantly weakening the rock strength. In felsic-rich zones, volatile build-up may have driven hydraulic fracturing of gabbroic wall rocks resulting in the formation of magmatic breccias. Comparison of isotopic compositions of fluids in plutonic rocks from 735B, the MARK area of the Mid-Atlantic Ridge, and the Mid-Cayman Rise indicate (1) that the carbon isotope composition of the lower oceanic crust may be far more heterogeneous than previously believed and (2) that carbon-bearing species in the oceanic crust and their distribution at depth are highly variable.

Chemical composition of glass inclusions of volcanic rocks from the Bonin Arc

The chemical composition of glass inclusions in phenocrystic plagioclase and pyroxene from Sites 792 and 793, drilled during Ocean Drilling Program Leg 126 in the Bonin Arc, is examined. Immiscible liquid, which is preserved as glass inclusions with unmixed textures in plagioclase, is observed in a high-magnesian andesite, which suggests an important role of liquid immiscibility in the fractionation of high-magnesian andesite. In other andesitic rocks (SiO2 = 57-60 wt%), such unmixed textures of glass inclusions in calcic plagioclase with a similar percentage of An (around 80%) is not found. The degree of fractionation and mixing of liquid are inferred from the glass composition in pyroxene.

Chemical composition of glass inclusions from ODP Holes 157-953C and 157-956B

We report S concentrations and relative proportions of (SO4)2- and S2- in OL- and CPX-hosted glass inclusions and in host glassy lapilli from Miocene basaltic hyaloclastites drilled north and south of Gran Canaria during ODP Leg 157. Compositions of glass inclusions and lapilli resemble those of subaerial Miocene shield basalts on Gran Canaria and comprise mafic to more evolved tholeiitic to alkali basalt and basanite (10.3-3.7 wt.% MgO, 44.5-56.9 wt.% SiO2). Glass inclusions fall into three groups based on their S concentrations: a high-sulfur group (1050 to 5810 ppm S), an intermediate-sulfur group (510 to 1740 ppm S), and a low-sulfur group (<500 ppm S). The most S-rich inclusions have the highest and nearly constant proportion of sulfur dissolved as sulfate determined by electron microprobe measurements of SKa peak shift. Their average S6+/S_total value is 0.75+/-0.09, unusually high for ocean island basalt magmas. The low-sulfur group inclusions have low S6+/S_total ratios (0.08+/-0.05), whereas intermediate sulfur group inclusions show a wide range of S6+/S_total (0.05-0.83). Glassy lapilli and their crystal-hosted glass inclusions with S concentrations of 50 to 1140 ppm S have very similar S6+/S_total ratios of 0.36+/-0.06 implying that sulfur degassing does not affect the proportion of (SO4)2- and S2- in the magma. The oxygen fugacities estimated from S6+/S_total ratios and from Fe3+/Fe2+ ratios in spinel inclusions range from NNO-1.1 to NNO+1.8. The origin of S-rich magmas is unclear. We discuss (1) partial melting of a mantle source at relatively oxidized fO2 conditions, and (2) magma contamination by seawater either directly or through magma interaction with seawater-altered Jurassic oceanic crust. The intermediate sulfur group inclusions represent undegassed or slightly degassed magmas similar to submarine OIB glasses, whereas the low-sulfur group inclusions are likely to have formed from magmas significantly degassed in near-surface reservoirs. Mixing of these degassed magmas with stored volatile-rich ones or volatile-rich magma replenishing the chamber filled by partially degassed magmas may produce hybrid melts with strongly varying S concentrations and S6+/S_total ratios.