We consider the heat equation on a polygonal domain $\Omega$ of the plane in weighted $L^p$-Sobolev spaces \begin{equation} \label{ab1} \begin{array}{cl} \partial_t u -\Delta u = h, & \mbox{in } \Omega \times {]0,T[}, \\ u=0,& \mbox{on } \partial\Omega \times {[0,T]}, \\ u(\cdot, 0)=0,& \mbox{in } \Omega. \end{array} \end{equation} Here $h$ belongs to $L^p(0,T;L^p_\mu(\Omega))$, where $L^p_\mu(\Omega)=\{v \in L^p_{loc}(\Omega): r^\mu v\in L^p(\Omega)\},$ with a real parameter $\mu$ and $r(x)$ the distance from $x$ to the set of corners of $\Omega$. We give sufficient conditions on $\mu$, $p$ and $\Omega$ that guarantee that problem (\ref{ab1}) has a unique solution $u\in L^p(0,T;L^p_\mu(\Omega))$ that admits a decomposition into a regular part in weighted $L^p$-Sobolev spaces and an explicit singular part.

In this paper, we investigate the uniform exponential stability of solutions to abstract integro-differential equations in Hilbert spaces by the theory of operator semigroups and Laplace transforms of vector-valued functions. New criterions are given based on the growth property of associated vector-valued functions on the right half plane. Examples are presented to illustrate our results.

We establish new and general existence theorems of Lipschitz continuous solutions to integral equations with infinite delay in Banach spaces, as well as of strong and classical solutions to the Cauchy problem for some evolution equations with infinite delay. An example is given to illustrate the abstract result.

We apply the quasilinearization method to boundary value problems for first order functional differential equations with delayed arguments. We formulate sufficient conditions for semi-quadratic or quadratic convergence of corresponding monotone sequences to a unique solution.

In this paper we generalize and extend Brosowski-Meinardus type results on invariant points from the set of best approximation to the set of ε-simultaneous approximation. As a consequence some results on ε-approximation and best approximation are also deduced. The results proved extend and generalize some of the results of R.N. Mukherjee and V. Verma [Bull. Cal. Math. Soc. 81(1989) 191-196; Publ. de l'Inst. Math. 49(1991) 111-116], T.D. Narang and S. Chandok [Mat. Vesnik 61(2009) 165-171; Selçuk J. Appl. Math. 10(2009) 75-80; Indian J. Math. 51(2009) 293-303], G.S. Rao and S.A. Mariadoss [Serdica-Bulgaricae Math. Publ. 9(1983) 244-248] and of few others.

We investigate the existence of entire solutions of non-linear differential equations of type $f^{2}(z)+cf'(z)=h(z),$ where $h(z)$ is a given entire function, whose zeros form an $A-$set. As a by-product of the studies, we give a negative answer to an open question raised in [4].

We study an eigenvalue problem involving a degenerate and singular elliptic operator on the whole space $\mathbb R^N$. We prove the existence of an unbounded and increasing sequence of eigenvalues. Our study generalizes to the case of degenerate and singular operators a result of A. Szulkin and M. Willem.

It is well known after R. Swan that $\tilde K_0(R[X,Y,Z]/(X^2+Y^2+Z^2-1))$ is isomorphic to the integers $\mathbb Z$, whenever $R$ is a field of characteristic not two which contains the squared root of $-1$. \par First, we give explicit idempotent matrices $\gamma^p$ of order two, corresponding to the integer $p,$ in the isomorphism above, if $R$ is a field of characteristic zero. Then, we use the algebraic de Rham cohomology of Kähler differentials to define Brouwer degree for polynomial homomorphisms of $R[X,Y,Z]/(X^2+Y^2+Z^2-1)$ to itself, and relate the problem of finding hermitian representatives for $R=K(i),$ $K$ a field not containing $i,$ to some unsolved problems of representing Brouwer degrees by polynomial maps.

In this paper we deal with non$-$smooth vector fields on the plane. We prove that the analysis of their local behavior around certain typical singularities can be treated via singular perturbation theory. In fact, after a regularization of a such system and a blow$-$up we are able to bring out some results that bridge the space between non$-$smooth dynamical systems presenting typical singularities and singularly perturbed smooth systems.

With the aid of weighted sharing method we study the uniqueness of entire functions concerning general nonlinear differential polynomials sharing fixed points. The results of the paper improve and generalize some results due to Zhang and Qi-Dou.

We show the existence of periodic and symmetric solutions of parametric weakly nonlinear ODE possessing time-reversal symmetries. Local asymptotic behaviours of these solutions are established as well. Concrete examples are presented to illustrate the general theory.

We classify rational homotopy types of elliptic spaces with homotopy Euler characteristic zero for dim < 8.

By making use of the principle of subordination between analytic functions, we introduce a certain subclass of multivalent analytic functions defined by the Liu-Owa operator. Results such as subordination and superordination properties, convolution properties, distortion theorems and inequality properties, are proved.