Anticipated Generalized Backward Doubly Stochastic Differential Equations

In this paper, we explore a new class of stochastic differential equations called anticipated generalized backward doubly stochastic differential equations (AGBDSDEs), which not only involve two symmetric integrals related to two independent Brownian motions and an integral driven by a continuous increasing process but also include generators depending on the anticipated terms of the solution (Y, Z). Firstly, we prove the existence and uniqueness theorem for AGBDSDEs. Further, two comparison theorems are obtained after finding a new comparison theorem for GBDSDEs.

A class of backward stochastic Bellman–Bihari’s inequality and its applications

In this paper, we propose and prove several different forms of backward stochastic Bellman–Bihari’s inequality. Then, as two applications, two different types of the comparison theorems for backward stochastic differential equation with stochastic non-Lipschitz condition are presented.

Upper bounds for Steklov eigenvalues of subgraphs of polynomial growth Cayley graphs

We study the Steklov problem on a subgraph with boundary $$(\Omega ,B)$$ ( Ω , B ) of a polynomial growth Cayley graph $$\Gamma$$ Γ . For $$(\Omega _l, B_l)_{l=1}^\infty$$ ( Ω l , B l ) l = 1 ∞ a sequence of subgraphs of $$\Gamma$$ Γ such that $$|\Omega _l| \longrightarrow \infty$$ | Ω l | ⟶ ∞ , we prove that for each $$k \in {\mathbb {N}}$$ k ∈ N , the kth eigenvalue tends to 0 proportionally to $$1/|B|^{\frac{1}{d-1}}$$ 1 / | B | 1 d - 1 , where d represents the growth rate of $$\Gamma$$ Γ . The method consists in associating a manifold M to $$\Gamma$$ Γ and a bounded domain $$N \subset M$$ N ⊂ M to a subgraph $$(\Omega , B)$$ ( Ω , B ) of $$\Gamma$$ Γ . We find upper bounds for the Steklov spectrum of N and transfer these bounds to $$(\Omega , B)$$ ( Ω , B ) by discretizing N and using comparison theorems.

Moderately Discontinuous Homotopy

We introduce a metric homotopy theory, which we call moderately discontinuous homotopy, designed to capture Lipschitz properties of metric singular subanalytic germs. It matches with the moderately discontinuous homology theory recently developed by the authors and E. Sampaio. The $k$-th MD homotopy group is a group $MDH^b_{\bullet }$ for any $b\in [1,\infty ]$ together with homomorphisms $MD\pi ^b\to MD\pi ^{b^{\prime}}$ for any $b\geq b^{\prime}$. We develop all its basic properties including finite presentation of the groups, long homotopy sequences of pairs, metric homotopy invariance, Seifert van Kampen Theorem, and the Hurewicz Isomorphism Theorem. We prove comparison theorems that allow to relate the metric homotopy groups with topological homotopy groups of associated spaces. For $b=1$, it recovers the homotopy groups of the tangent cone for the outer metric and of the Gromov tangent cone for the inner one. In general, for $b=\infty $, the $MD$-homotopy recovers the homotopy of the punctured germ. Hence, our invariant can be seen as an algebraic invariant interpolating the homotopy from the germ to its tangent cone. We end the paper with a full computation of our invariant for any normal surface singularity for the inner metric. We also provide a full computation of the MD-homology in the same case.

Note on the conformable boundary value problems: Sturm’s theorems and Green’s function

Recently, the conformable derivative and its properties have been introduced. In this paper, we propose and prove some new results on conformable Boundary Value Problems. First, we introduce a conformable version of classical Sturm´s separation, and comparison theorems. For a conformable Sturm-Liouville problem, Green's function is constructed, and its properties are also studied. In addition, we propose the applicability of the Green´s Function in solving conformable inhomogeneous linear differential equations with homogeneous boundary conditions, whose associated homogeneous boundary value problem has only trivial solution. Finally, we prove the generalized Hyers-Ulam stability of the conformable inhomogeneous boundary value problem.