This work deals with the existence of multiple positive solutions for a third order boundary value problem with a $\phi$-Laplacian operator on the halfline. The existence results are obtained both for the regular and the singular cases using the fixed point index theory on a suitable cone of a Banach space. The singularity is treated by an approximation technique and sequential arguments. Examples of applications are included to illustrate the existence results.

For $n\geq 1,$ let $ \mathbf{P}(n) = \mathbb{F}_2[x_1,\ldots,x_n]$ be the polynomial algebra in $n$ variables $x_i,$ of degree one, over the field $\mathbb{F}_2$ of two elements. The mod2 Steenrod algebra ${\cal A}$ acts on $\mathbf{P}(n)$ according to well known rules. Let ${\cal A}^+\mathbf{P}(n)$ denote the image of the action of the positively graded part of ${\cal A}.$ A major problem is that of determining a basis for the quotient vector space $\mathbf{Q}(n) = \mathbf{P}(n)/{\cal A}^+\mathbf{P}(n).$ Both ${\mathbf{P} }(n) = \oplus_{d\geq0}\mathbf{P}^{d}(n)$ and $\mathbf{Q}(n)$ are graded where $\mathbf{P}^{d}(n)$ denotes the set of homogeneous polynomials of degree $d.$ In this paper we show that if $n \geq 2,$ and $d \geq 1$ can be expressed in the form $d = \sum_{i=1}^{n1} (2^{\lambda_i}1)$ with ${\lambda_1}> {\lambda_2} > \ldots >{\lambda_{n2}} \geq {\lambda_{n1}}\geq 1,$ then $${\rm {dim}}(\mathbf{Q}^{d}(n)) \geq \left (\sum_{q=1}^{{\rm min}\{ {\lambda}_{n1},n\}} {{n}\choose {q}}\right ) ({\rm {dim}}(\mathbf{Q}^{d'}(n1)) )$$ where $ d'= \sum_{i=1}^{n1} (2^{\lambda_i \lambda_{n1}}1).$

Let $X$ be a simply connected homogeneous space of which $ \pi_*(X) \otimes \mathbb{Q} $ is finite dimensional. We consider the homology of the free loop space $ {\rm map}(S^1, X) $ with the bracket defined by Chas and Sullivan. We show that the Lie algebra $ s\mathbb{H}_*({\rm map}(S^1, X), \mathbb{Q}) $ is not nilpotent.

In this paper, we characterize the solution of a nonlinear reflected Backward Stochastic Differential Equations (BSDE) as the unique solution of a Stochastic Variational Inequality (SVI). This approach leads to a priori estimate for the increment of the predictable component of the solution of the reflected BSDE.

Solutions of the complex Monge-Ampère Equation are obtained in the Sobolev topology on complex manifolds and through the Delta-Delta-Bar Lemma, in case the manifold is compact Kähler, a simple proof is given of the Aubin-Calabi-Yau Theorem.

We give an explicit and purely algebraic proof for the existence of noetherian differential operators for primary ideals of polynomial algebras. The proof of this important result in [1] uses complicated algebraic and analytic techniques. Later U.Oberst gave an elementary and constructive proof in [6]. In this paper we propose a different proof from the one in [6].

In this paper, we prove a new composition theorem for asymptotically antiperiodic and weighted pseudo antiperiodic functions. We also give some sufficient conditions to ensure invertibility of convolution operators in the space of antiperiodic functions. Then we prove the existence and uniqueness of asymptotically antiperiodic mild solutions to some fractional functional integro-differential equations in a Banach space using the Banach's fixed point theorem.

The cancellation of the singularities of the heat equation in a polygonal domain with cracks is analyzed. Using a density result, a bi-orthogonality property of a family of finite eigenfunctions of the Laplacian and Holmgren's theorem, we obtain a regular solution of the heat equation by an internal control.