In this article, the equivalence and symmetries of underdetermined differential equations and differential equations with deviations of the first order are considered with respect to the pseudogroup of transformations $\overline{x}=\varphi \left(x\right),$$\overline{y}...$

We consider a Strang-type splitting method for an abstract semilinear evolution equation $${\partial }_{t}u=Au+F\left(u\right).$$ Roughly speaking, the splitting method is a time-discretization approximation based on the decomposition of the operators $A$ and $F.$ Particularly, the Strang method is a popular splitting method and is known to be convergent at a second order rate for some particular ODEs and PDEs. Moreover, such estimates usually address the case of splitting the operator into two parts. In this paper, we consider the splitting...

2000 Mathematics Subject Classification: Primary 34C07, secondary 34C08.We obtain an upper bound for the number of zeros of the Abelian integral.The work was partially supported by contract No 15/09.05.2002 with the Shoumen University “K. Preslavski”, Shoumen, Bulgaria.

We study asymptotic properties of solutions for a system of second differential equations with $p$-Laplacian. The main purpose is to investigate lower estimates of singular solutions of second order differential equations with $p$-Laplacian ${\left(A\left(t\right){\Phi }_p\left(y^{\text{'}}\right)\right)}^{\text{'}}+B\left(t\right)g\left(y^{\text{'}}\right)+R\left(t\right)f\left(y\right)=e\left(t\right)$. Furthermore, we obtain results for a scalar equation.

By using Mawhin’s continuation theorem, the existence of even solutions with minimum positive period for a class of higher order nonlinear Duffing differential equations is studied.

The work characterizes when is the equation $y^{\left(n\right)}+\mu p\left(x\right)y=0$ eventually disconjugate for every value of $\mu$ and gives an explicit necessary and sufficient integral criterion for it. For suitable integers $q$, the eventually disconjugate (and disfocal) equation has 2-dimensional subspaces of solutions $y$ such that $y^{\left(i\right)}>0$, $i=0,...,q-1$, ${(-1)}^{i-q}{y}^{\left(i\right)}>0$, $i=q,...,n$. We characterize the “smallest” of such solutions and conjecture the shape of the “largest” one. Examples demonstrate that the estimates are sharp.

We study the exact multiplicity and bifurcation curves of positive solutions of generalized logistic problems$$\left\{\begin{array}{cc}-{\left[\phi \left(u^{\text{'}}\right)\right]}^{\text{'}}=\lambda u^p\left(1-{\displaystyle \frac{u}{N}}\right)\hfill & \text{in}(-L,L),\hfill \\ u(-L)=u\left(L\right)=0,\hfill \end{array}\right.$$ where $p>1$, $N>0$, $\lambda >0$ is a bifurcation parameter, $L>0$ is an evolution parameter, and $\phi \left(u\right)$ is either $\phi \left(u\right)=u$ or $\phi \left(u\right)=u/\sqrt{1-u^2}$. We prove that the corresponding bifurcation curve is $\subset$-shape. Thus, the exact multiplicity of positive solutions can be obtained.