The composition of the protosolar disk and the formation conditions for comets

Conditions in the protosolar nebula have left their mark in the composition of cometary volatiles, thought to be some of the most pristine material in the solar system. Cometary compositions represent the end point of processing that began in the parent molecular cloud core and continued through the collapse of that core to form the protosun and the solar nebula, and finally during the evolution of the solar nebula itself as the cometary bodies were accreting. Disentangling the effects of the various epochs on the final composition of a comet is complicated. But comets are not the only source of information about the solar nebula. Protostellar disks around young stars similar to the protosun provide a way of investigating the evolution of disks similar to the solar nebula while they are in the process of evolving to form their own solar systems. In this way we can learn about the physical and chemical conditions under which comets formed, and about the types of dynamical processing that shaped the solar system we see today. This paper summarizes some recent contributions to our understanding of both cometary volatiles and the composition, structure and evolution of protostellar disks.Comment: To appear in Space Science Reviews. The final publication is available at Springer via http://dx.doi.org/10.1007/s11214-015-0167-

Strong convergence of implicit iteration processes for nonexpansive semigroups in Banach spaces

Let $C$ be a convex compact subset of a uniformly convex Banach space. Let $\{T_t\}_{t \geq0}$ be a strongly-continuous nonexpansive semigroup on $C$. Consider the iterative process defined by the sequence of equations $$x_{k+1} =c_k T_{t_{k+1}}(x_{k+1})+(1-c_k)x_k.$$ We prove that, under certain conditions on $\{c_k\}$ and $\{t_k\}$, the sequence $\{x_k\}_{n=1}^\infty$ converges strongly to a common fixed point of the semigroup $\{T_t\}_{t \geq0}$. There are known results on convergence of such iterative processes for nonexpansive semigroups in Hilbert spaces and Banach spaces with the Opial property, and also weak convergence results in Banach spaces that are simultaneously uniformly convex and uniformly smooth. In this paper, we do not assume the Opial property or uniform smoothness of the norm

On the construction of common fixed points for semigroups of nonlinear mappings in uniformly convex and uniformly smooth Banach spaces

Let $C$ be a bounded, closed, convex subset of a uniformly convex and uniformly smooth Banach space $X$. We investigate the weak convergence of the generalized Krasnosel'skii-Mann and Ishikawa iteration processes to common fixed points of semigroups of nonlinear mappings $T_t\colon C \to C$. Each of $T_t$ is assumed to be pointwise Lipschitzian, that is, there exists a family of functions $\alpha_t\colon C \to [0, \infty)$ such that $\|T_t(x) - T_t (y)\| \leq\alpha_t (x)\|x -y\|$ for $x, y \in C$. The paper demonstrates how the weak compactness of $C$ plays an essential role in proving the weak convergence of these processes to common fixed points

On Nonlinear Differential Equations in Generalized Musielak-Orlicz Spaces

We consider ordinary differential equations $u'(t)+(I-T)u(t)=0$, where an unknown function takes its values in a given modular function space being a generalization of Musielak-Orlicz spaces, and $T$ is nonlinear mapping which is nonexpansive in the modular sense. We demonstrate that under certain natural assumptions the Cauchy problem related to this equation can be solved. We also show a process for the construction of such a solution. This result is then linked to the recent results of the fixed point theory in modular function spaces