Existence and stability analysis of solutions for a new kind of boundary value problems of nonlinear fractional differential equations

This research work is dedicated to an investigation for a new kind of boundary value problem of nonlinear fractional differential equation supplemented with general boundary condition. A full analysis of existence and uniqueness of positive solutions is respectively proved by Leray–Schauder nonlinear alternative theorem and Boyd–Wong’s contraction principles. Furthermore, we prove the Hyers–Ulam (HU) stability and Hyers–Ulam–Rassias (HUR) stability of solutions. An example illustrating the validity of the existence result is also discussed

Direct shock wave loading of Stishovite to 235 GPa: Implications for perovskite stability relative to an oxide assemblage at lower mantle conditions

Pure stishovite and coesite samples with zero porosity and dimensions appropriate for planar shock wave experiments have been synthesized with multi-anvil high-pressure techniques. The equation of state of stishovite is obtained by direct shock wave loading up to 235 GPa: K_(0T) = 306 ± 5 GPa and K'_(0T) = 5.0 ± 0.2 where K_(0T) and K'_(0T) are ambient bulk modulus and its pressure derivative, respectively. The Hugoniots (shock equations of state) for stishovite, coesite and quartz achieve widely differing internal energy states at equal volume and therefore allow us to determine the Gruneisen parameter of stishovite. On the basis of the resulting P-V-T equation of state for stishovite and previous studies on other phases on the MgO-SiO_2 binary, the breakdown reaction of MgSiO_3-perovskite to MgO and SiO_2 was calculated. Our calculations show that perovskite is thermodynamically stable relative to the stishovite and periclase assemblage at lower mantle conditions. We obtain similar results for a range of models, despite the appreciable differences among these experiment-based thermodynamic parameters

The Existence of Einstein Static Universes and their Stability in Fourth order Theories of Gravity

We investigate whether or not an Einstein Static universe is a solution to the cosmological equations in $f(R)$ gravity. It is found that only one class of $f(R)$ theories admits an Einstein Static model, and that this class is neutrally stable with respect to vector and tensor perturbations for all equations of state on all scales. Scalar perturbations are only stable on all scales if the matter fluid equation of state satisfies $c_s^2>\frac{\sqrt{5}-1}{6}\approx 0.21$. This result is remarkably similar to the GR case, where it was found that the Einstein Static model is stable for $c_s^2>{1/5}$.Comment: Minor changes, To appear in PR

Stability of the Einstein static universe in f(R) gravity

We analyze the stability of the Einstein static universe by considering homogeneous scalar perturbations in the context of f(R) modified theories of gravity. By considering specific forms of f(R), the stability regions of the solutions are parameterized by a linear equation of state parameter w=p/rho. Contrary to classical general relativity, it is found that in f(R) gravity a stable Einstein cosmos with a positive cosmological constant does indeed exist. Thus, we are lead to conclude that, in principle, modifications in f(R) gravity stabilize solutions which are unstable in general relativity.Comment: 7 pages, 2 figures, 2 tables; references adde

Horizon thermodynamics in f(R)f(R) theory

We investigate whether the new horizon first law proposed recently still work in $f(R)$ theory. We identify the entropy and the energy of black hole as quantities proportional to the corresponding value of integration, supported by the fact that the new horizon first law holds true as a consequence of equations of motion in $f(R)$ theories. The formulas for the entropy and energy of black hole found here are in agreement with the results obtained in literatures. For applications, some nontrivial black hole solutions in $f(R)$ theories have been considered, the entropies and the energies of black holes in these models are firstly computed, which may be useful for future researches.Comment: 8 pages, no figur

Hedging of Defaultable Contingent Claims using BSDE with uncertain time horizon

This article focuses on the mathematical problem of existence and uniqueness of BSDE with a random terminal time which is a general random variable but not a stopping time, as it has been usually the case in the previous literature of BSDE with random terminal time. The main motivation of this work is a financial or actuarial problem of hedging of defaultable contingent claims or life insurance contracts, for which the terminal time is a default time or a death time, which are not stopping times. We have to use progressive enlargement of the Brownian filtration, and to solve the obtained BSDE under this enlarged filtration. This work gives a solution to the mathematical problem and proves the existence and uniqueness of solutions of such BSDE under certain general conditions. This approach is applied to the financial problem of hedging of defaultable contingent claims, and an expression of the hedging strategy is given for a defaultable contingent claim or a life insurance contract