In this paper, we will study the existence of solutions in the sense of distributions for the quasilinear $p(x)$-elliptic problem, $$ Au + g(x,u,\nabla u) = f,$$ where $A$ is a Leray-Lions operator from $W_{0}^{1,p(\cdot)}(\Omega)$ into its dual, the nonlinear term $g(x,s,\xi)$ has a growth condition with respect to $\xi$ and the sign condition with respect to $s.$ The datum $\>f\>$ is assumed in the dual space $\>W^{-1,p'(\cdot)}(\Omega),\>$ and then in $\>L^{1}(\Omega).$

If W is a subset of a metric space $(X,d)$ then for a given $\varepsilon>0$, an element $y_0\in W$ is called a good approximation or $\varepsilon-$approximation for $x\in X$ if $d(x,y_0)\leq d(x,W)+\varepsilon.$ We denote by $P_{W,\varepsilon}(x)$ the set of all such $y_0\in W$ i.e. $P_{W,\; \varepsilon}(x)=\{y\in W:d(x,y)\leq d(x,W)+\varepsilon\}$. In particular, for $\varepsilon=0$ we get the set of all best approximations to $x$ in W. Given a subset M of W, what are the necessary and sufficient conditions in order that every element $y_0\in M$ is an element of good approximation to $x$ by the elements of W? The paper mainly deals with this problem of simultaneous characterization of elements of good approximation in metric spaces. The proved results extend and generalize several known results on the subject.

In this paper, we generalize the Cartan's fundamental formula on lightlike hypersurfaces, then we use it to show that a screen conformal lightlike hypersurface of a Lorentzian Euclidean space is locally a lightlike triple product manifold.

In this paper a Beddington-DeAngelis predator-prey model with time lag for predator is proposed and analyzed. Mathematical analysis regard to boundedness of solutions, nature of equilibria, uniform persistence, and stability are analyzed. We show that if the positive equilibrium is unstable, an orbitally asymptotically stable periodic solution exists.

In this paper, we are concerned in a non-classical boundary value problem for heat equation. More precisely, we study a linear heat equation without initial condition but with a homogeneous Dirichlet condition on the whole boundary and a nonhomogeneous Neumann condition on a part of the boundary. Under sufficient conditions on the data, we prove that the problem has a unique solution. The proof combines optimal control and controllability theories.

In this paper, we study the topological structure of solution sets for the following first-order impulsive evolution inclusion with initial conditions: $$ \left\{ \begin{array}{rlll} y'(t)-Ay(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash \{t_{1},\ldots,t_{m}\},\\ y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1,\ldots,m\\ y(0)&=&a\in E, \end{array} \right. $$where $J:=[0,b]$ and $0 = t_0 < t_1 < \,... \,< t_m < b$, $A$ is the infinitesimal generator of a $C_0-$semigroup of linear operator $T(t)$ on a separable Banach space $E$ and $F$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1, \ldots ,m$). The continuous selection of the solution set is also investigated.