In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form $\left( {{P_{a,b}}} \right)\left\{ {_{u\left( 0 \right) = u\left( 1 \right) = 0,{D^{\alpha - 3}}u\left( 0 \right) = a,u'\left( 1 \right) = - b}^{{D^\alpha }u\left( x \right) + f\left( {x,u\left( x \right)} \right) = 0,x \in \left( {0,1} \right)}} \right.$, where 3 < α ≤ 4, D α is the standard Riemann-Liouville fractional derivative and a, b are nonnegative constants. First the authors suppose that f(x, t) = −p(x)t ^σ, with σ ∈ (−1, 1) and p being a nonnegative continuous function that may be singular at x = 0 or x = 1 and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Schäuder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem (P _0,0). Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a + b > 0 and f(x, t) = t φ(x, t), with φ(x, t) being a nonnegative continuous function in (0, 1)×[0,∞) that is required to satisfy some suitable integrability condition. Using estimates on the Green’s function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem (P _a,b), which behaves like the unique solution of the homogeneous problem corresponding to (P _a,b). Some examples are given to illustrate the existence results.

In this article, the authors establish the local null controllability property for semilinear parabolic systems in a domain whose boundary moves in time by a single control force acting on a prescribed subdomain. The proof is based on Kakutani’s fixed point theorem combined with observability estimates for the associated linearized system.

In this paper, the authors prove the existence of solutions for degenerate elliptic equations of the form −div(a(x)∇ _p u(x)) = g(λ, x, |u| ^p−2u) in ℝ ^N, where ∇ _p u = |∇u|^p−2∇ _u and a(x) is a degenerate nonnegative weight. The authors also investigate a related nonlinear eigenvalue problem obtaining an existence result which contains information about the location and multiplicity of eigensolutions. The proofs of the main results are obtained by using the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality and by using a specific minimax method, but without making use of the Palais-Smale condition.

In this paper, the continuous-time independent edge-Markovian random graph process model is constructed. The authors also define the interval isolated nodes of the random graph process, study the distribution sequence of the number of isolated nodes and the probability of having no isolated nodes when the initial distribution of the random graph process is stationary distribution, derive the lower limit of the probability in which two arbitrary nodes are connected and the random graph is also connected, and prove that the random graph is almost everywhere connected when the number of nodes is sufficiently large.

Quillen proved that if a Hermitian bihomogeneous polynomial is strictly positive on the unit sphere, then repeated multiplication of the standard sesquilinear form to this polynomial eventually results in a sum of Hermitian squares. Catlin-D’Angelo and Varolin deduced this positivstellensatz of Quillen from the eventual positive-definiteness of an associated integral operator. Their arguments involve asymptotic expansions of the Bergman kernel. The goal of this article is to give an elementary proof of the positive-definiteness of this integral operator.

There are various adjunctions between model (co)slice categories. The author gives a proposition to characterize when these adjunctions are Quillen equivalences. As an application, a triangle equivalence between the stable category of a Frobenius category and the homotopy category of a non-pointed model category is given.

The authors show that linear simple groups L ₂(q) with q ∈ {17, 27, 29} can be uniquely determined by nse(L ₂(q)), which is the set of numbers of elements with the same order.

This paper deals with an attraction-repulsion chemotaxis model (ARC) in multi-dimensions. By Duhamel’s principle, the implicit expression of the solution to (ARC) is given. With the method of Green’s function, the authors obtain the pointwise estimates of solutions to the Cauchy problem (ARC) for small initial data, which yield the W ^s,p (1 ≤ p ≤ ∞) decay properties of solutions.

In this paper, the spatial Hill lunar problem is investigated, and the existence of invariant tori of hyperbolic type in a neighborhood of its equilibrium is shown. Moreover, the author checks the non-degenerate condition analytically and obtains two-dimensional elliptic invariant tori on its central manifold as well.

By using the moving frame method, the authors obtain a kind of asymmetric kinematic formulas for the total mean curvatures of hypersurfaces in the n-dimensional Euclidean space.

For a class of cubic systems, the authors give a representation of the nth order Liapunov constant through a chain of pseudo-divisions. As an application, the center problem and the isochronous center problem of a particular system are considered. They show that the system has a center at the origin if and only if the first seven Liapunov constants vanish, and cannot have an isochronous center at the origin.