In [3] the authors introduced a {\em relative category} for a map that differ from the {\em sectional category} by just one. The relative category has specific properties (for instance a homotopy pushout does not increase it) which make it a convenient tool to study the sectional category. The question to know when secat equals relcat arises. We give here some sufficient conditions. Applications are given to the {\em topological complexity}, which is nothing but the sectional category of the diagonal.

In this paper we prove that if $k\geq3$ and $d$ are positive integers and the set $\{k-2,k+2,4k^3-4k,d\}$ has the property that the product of any two of its distinct elements increased by $4$ is a perfect square, then $d=4k$ or $d=4k^5-12k^3+8k$.

In this paper, we introduce the new notions of $\phi$-biflatness, $\phi$-biprojectivity, $\phi$-Johnson amenability and $\phi$-Johnson contractibility for Banach algebras, where $\phi$ is a non-zero homomorphism from a Banach algebra $A$ into $\mathbb{C}$. We show that a Banach algebra $A$ is $\phi$-Johnson amenable if and only if it is $\phi$-inner amenable and $\phi$-biflat. Also we show that $\phi$-Johnson amenability is equivalent with the existence of left and right $\phi$-means for $A$. We give some examples to show differences between these new notions and the classical ones. Finally, we show that ${L^{1}(G)}$ is $\phi$-biflat if and only if $G$ is an amenable group and $A(G)$ is $\phi$-biprojective if and only if $G$ is a discrete group.

We first define a generalized form of a Hurwitz-Lerch zeta type function and then use it in constructing certain classes of analytic functions in the unit disk. In our investigation, we obtain various results for the classes introduced, thereby, exhibiting their useful properties and characteristics by adopting the techniques of differential subordination. Several consequences of the main results are considered and relevant connections with some of the known results are also pointed out.

Let $H$ be a real Hilbert space and let $A:D(A)\subset H\rightarrow H$ be a (possibly multivalued) maximal monotone operator. We are concerned with the difference equation \begin{equation*} \Delta u_{n} + c_{n} A u_{n+1}\ni f_{n},\qquad n=0,1, ... , \end{equation*} where $(c_{n})\subset (0,+\infty)$, $(f_{n})\subset H$ are $p$-periodic sequences for a positive integer $p$. We investigate the existence of periodic solutions to this equation as well as the weak or strong convergence of solutions to $p$-periodic solutions. The first result of this paper (Theorem 1) is a discrete analogue of the 1977 result by Baillon and Haraux (on the periodic forcing problem for the continuous counterpart of the above equation) and was essentially stated by Djafari Rouhani and Khatibzadeh in a recent paper. Here we provide a simpler proof of this result that is based on old existing results due to Browder and Petryshyn and Opial. A strong convergence result is also given and some examples are discussed to illustrate the theoretical results.

A new notion of injectivity is introduced. It is shown that approximate Connes-amenability and approximate injectivity are the same properties. As a consequence, approximate Connes-amenability of the direct sum of dual Banach algebras is discussed. A characterization is given for approximate Connes-amenability of dual Banach algebras in terms of the approximate splitting of certain short exact sequence.

We study the Bousfield-Kan spectral sequence associated to the Munson-Volić cosimplicial model for the space of long links with $\ell$ strings in $\mathbb R^N$. We compute explicitly some Euler-Poincaré series associated to the second page of that spectral sequence and deduce exponential growth of their Betti numbers.

Classical solutions of initial problems for nonlinear functional differential equations of Hamilton--Jacobi type are approximated by solutions of associated differential difference systems. A method of quasilinearization is adopted. Sufficient conditions for the convergence of the method of lines and error estimates for approximate solutions are given. Nonlinear estimates of the Perron type with respect to functional variables for given operators are assumed. The proof of the stability of differential difference problems is based on a comparison technique. The results obtained here can be applied to differential integral problems and differential equations with deviated variables.

The extinction phenomenon of solutions for the the initial-boundary value problem of the $p$-Laplacian parabolic equation $$u_t=\mbox{div} ( |\nabla u|^{p-2}\nabla u)+\lambda|u|^{m-1}u-\beta u$$ is studied. Sufficient conditions about the extinction and decay estimates of solutions are obtained by using $L^p$-integral model estimate methods and two crucial lemmas on differential inequality. Non-extinction results are obtained by super and sub-solution method.

In this paper, we consider the radially symmetric solutions for $p$-Laplacian with nonlocal boundary condition in a perforated-like domain. We obtain the existence, the uniqueness and some other properties of the radially symmetric solution. The nonexistence of solution is also studied.

A relationship between curved differential algebras and corings is established and explored. In particular it is shown that the category of semi-free curved differential graded algebras is equivalent to the category of corings with surjective counits. Under this equivalence, comodules over a coring correspond to integrable connections or quasi-cohesive curved modules, while contramodules over a coring correspond to a specific class of curved modules introduced and termed $\mathbb Z$-divergences in here.

In this paper we analyze some applications of the category of exterior spaces to the study of dynamical systems (flows). We study the notion of an absorbing open subset of a dynamical system; i.e., an open subset that contains the ``future part'' of all the trajectories. The family of all absorbing open subsets is a quasi-filter which gives the structure of an exterior space to the flow. The limit space and end space of an exterior space are used to construct the limit spaces and end spaces of a dynamical system. On the one hand, for a dynamical system two limits spaces $L^{\r}(X)$ and $\bar L^{\r}(X)$ are constructed and their relations with the subflows of periodic, Poisson stable points and $\Omega $-limits of $X$ are analyzed. On the other hand, different end spaces are also associated to a dynamical system having the property that any positive semi-trajectory has an end point in these end spaces. This type of construction permits us to consider the subflow containing all trajectories finishing at an end point $a$. When $a$ runs over the set of all end points, we have an induced decomposition of a dynamical system as a disjoint union of stable (at infinity) subflows.