Sience De Matier

In this paper, we consider a class of stochastic control problems for stochastic differential equations with random coefficients. The control domain need not to be convex but the control process is not allowed to enter in diffusion term. Moreover, the terminal cost involves a non linear term of the expected value of terminal state. Our purpose is to derive a new version of the Pontryagin's stochastic maximum principle by adopting an idea inspired from the work of Peng [S. Peng, Maximum Principle for Stochastic Optimal Control with Nonconvex Control Domain, Lecture Notes in Control & Information Sciences, 114, (1990), pp. 724-732]. More specifically, we show that if we combine the spike perturbation of the optimal control combined with the stochastic Feynman-Kac representation of linear backward stochastic partial differential equations (BSPDE, for short), a new version of the stochastic maximum principle can be derived. We also investigate sufficient conditions of optimality. In the last part of this paper, motivated by our version of SMP, an interesting class of forward backward stochastic partial differential equations is naturally introduced and the solvability of such kind of equations is briefly presented.

To study circadian BP patterns in patients with congenital central hypoventilation syndrome (CCHS). Design: Case-control study. Setting: Teaching hospital in Paris, France. Patients: Eleven patients with CCHS (median age, 13 years; range, 6 to 18 years) and 11 sex-and height-matched control subjects. Intervention: None. Methods: Each subject underwent 24-h ambulatory BP monitoring. Oxygen saturation and end-tidal PCO 2 were monitored noninvasively. Polysomnography was performed to determine sleep times. All patients with CCHS received mechanical ventilation during sleep. Mean values for systolic BP (SBP) and diastolic BP (DBP) during wakefulness and sleep were analyzed. Nocturnal BP "dipping" was defined as the difference in mean SBP (and/or DBP) between wakefulness and sleep, divided by individual waking mean values. BP "dippers" were defined as subjects showing at least 10% nocturnal dipping. Results: Patients with CCHS had BPs in the low normal range of normative data. As compared to control subjects, patients with CCHS had lower BP during wakefulness (p ‫؍‬ 0.003 and p ‫؍‬ 0.016 for SBP and DBP, respectively), and higher BP during sleep (p ‫؍‬ 0.016 and p ‫؍‬ 0.002). Nocturnal BP dipping was abnormally reduced in patients with CCHS (p ‫؍‬ 0.000). Ten of the 11 patients with CCHS were BP nondippers, compared to none of the control subjects. Conclusion: The abnormal circadian BP pattern observed in children and adolescents with CCHS may be related to autonomic nervous dysfunction. Lifelong cardiovascular follow-up is recommended for patients with CCHS.

In this work, we study the existence and uniqueness of a pseudo-almost automorphic solution for some partial functional differential equations of neutral type. We assume that the undelayed part is not necessarily densely defined and satisfies the Hille-Yosida condition. The delay part is assumed to be pseudo-almost automrophic in time. We prove if the homogeneous linear equation has an exponential dichotomy, then the nonhomogeneous linear equation has a unique pseudo-almost automorphic solution. An application is given for some nonlinear equations. We present also a new concept of generalized pseudo-almost automorphy.

In this work, we study the existence of mild solutions for some partial functional integrodifferential equations with finite delay in a Fréchet spaces. We assume that the linear part has a resolvent operator in the sense given by Grimmer (Trans Am Math Soc 273: 333-349, 1982). The nonlinear part is a sum of a Lipschitzian function and another satisfies the Carathéodory's conditions. Our approach is based on a nonlinear alternative of Avramescu type and the resolvent operators theory. An application is provided to a reaction-diffusion equation with delay. Keywords Mild solution • Partial functional integrodifferential equations with delay • Resolvent operator • Nonlinear alternative of Avramescu • C 0-semigroup Mathematics Subject Classification 35D30 Weak solutions • 45K05 Integro-partial differential equations • 47H08 Measures of non-compactness and condensing mappings, K-set contractions • 47H10 Fixed-point theorems • 92D25 Population dynamics (general) 1 Introduction Differential equations play an important role in describing many real-world processes. For many years the models are successfully used to study a number of physical, biological. A

The aim of this work is to establish several results on the existence of mild solutionse nondensely nonautonomous partial functional differential equations with infinite delay in Banach spaces. We assume that the linear part is not necessarily densely defined and generates an evolution family. We show the local existence of the mild solutions which may blow up at the finite time. Secondly, we get the existence of global mild solutions. For illustration, we propose applications for distributed delay and discrete delay.