In this paper we summarize an abstract approach to inertial manifolds for nonautonomous dynamical systems. Our result on the existence of inertial manifolds requires only two geometrical assumptions, called cone invariance and squeezing property, and some additional technical assumptions like boundedness or smoothing properties. We apply this result to processes (two-parameter semiflows) generated by nonautonomous semilinear parabolic evolution equations.

The paper presents fractional-order semilinear, continuous, finite-dimensional dynamical systems with multiple delays both in controls and nonlinear function $f$. The constrained relative controllability of the presented semilinear system and corresponding linear one are discussed. New criteria of constrained relative controllability for the fractional semilinear systems with delays under assumptions put on the control values are established and proved. The conical type constraints are considered....

MSC 2010: 34A08, 34A37, 49N70Here we investigate a problem of approaching terminal (target) set by a system of impulse differential equations of fractional order in the sense of Caputo. The system is under control of two players pursuing opposite goals. The first player tries to bring the trajectory of the system to the terminal set in the shortest time, whereas the second player tries to maximally put off the instant when the trajectory hits the set, or even avoid this meeting at all. We derive...

Conjugacy and disconjugacy criteria are established for the equation $$u^{\text{'}\text{'}}+p\left(t\right)u=0,$$ where $p:]-\infty ,+\infty [\to ]-\infty ,+\infty [$ is a locally summable function.

Oscillation properties of the self-adjoint, two term, differential equation $${(-1)}^n{\left(p\left(x\right)y^{\left(n\right)}\right)}^{\left(n\right)}+q\left(x\right)y=0(*)$$ are investigated. Using the variational method and the concept of the principal system of solutions it is proved that (*) is conjugate on $R=(-\infty ,\infty )$ if there exist an integer $m\in \{0,1,\cdots ,n-1\}$ and $c_0,\cdots ,c_m\in R$ such that $${\int }_{\infty }^0{x}^{2(n-m-1)}{p}^{-1}\left(x\right)dx=\infty ={\int }_0^{\infty }{x}^{2(n-m-1)}{p}^{-1}\left(x\right)dx$$ and $$\underset{x_1\downarrow -\infty ,x_2\uparrow \infty }{lim\; sup}{\int }_{x_1}^{x_2}q\left(x\right){(c_0+c_{1}x+\cdots +c_m{x}^m)}^{2}dx\le 0,q\left(x\right)\neg \equiv 0.$$ Some extensions of this criterion are suggested.

Sufficient conditions on the function $c\left(t\right)$ ensuring that the half-linear second order differential equation $$\left(\right|u^{\text{'}}{|}^{p-2}{u}^{\text{'}}{)}^{\text{'}}+c\left(t\right){\left|u\left(t\right)\right|}^{p-2}u\left(t\right)=0,p>1$$ possesses a nontrivial solution having at least two zeros in a given interval are obtained. These conditions extend some recently proved conjugacy criteria for linear equations which correspond to the case $p=2$.

We give sufficient conditions for a diffeomorphism in the plane to be analytically conjugate to a shift in a complex neighborhood of a segment of an invariant curve. For a family of functions close to the identity uniform estimates are established. As a consequence an exponential upper estimate for splitting of separatrices is established for diffeomorphisms of the plane close to the identity. The constant in the exponent is related to the width of the analyticity domain of the limit flow separatrix....

In a recent paper [9] we presented a Galerkin-type Conley index theory for certain classes of infinite-dimensional ODEs without the uniqueness property of the Cauchy problem. In this paper we show how to apply this theory to strongly indefinite elliptic systems. More specifically, we study the elliptic system $-\Delta u={\partial }_{v}H(u,v,x)$ in Ω, $-\Delta v={\partial }_{u}H(u,v,x)$ in Ω, u = 0, v = 0 in ∂Ω, (A1) on a smooth bounded domain Ω in ${ℝ}^N$ for "-"-type Hamiltonians H of class C² satisfying subcritical growth assumptions on their first order derivatives....

This paper is based mainly on the joint paper with W. Kryszewski [Dzedzej, Z., Kryszewski, W.: Conley type index applied to Hamiltonian inclusions. J. Math. Anal. Appl. 347 (2008), 96–112.], where cohomological Conley type index for multivalued flows has been applied to prove the existence of nontrivial periodic solutions for asymptotically linear Hamiltonian inclusions. Some proofs and additional remarks concerning definition of the index and special cases are given.

We prove an existence theorem for connected branches of solutions to nonlinear operator equations in Banach spaces. This abstract result is applied to the asymptotically equivalent solutions to nonlinear ordinary differential equations.