Almost periodic homogeneous linear difference systems without almost periodic solutions

Almost periodic homogeneous linear difference systems are considered. It is supposed that the coefficient matrices belong to a group. The aim was to find such groups that the systems having no non-trivial almost periodic solution form a dense subset of the set of all considered systems. A closer examination of the used methods reveals that the problem can be treated in such a generality that the entries of coefficient matrices are allowed to belong to any complete metric field. The concepts of transformable and strongly transformable groups of matrices are introduced, and these concepts enable us to derive efficient conditions for determining what matrix groups have the required property.     

Analytic representation of periodic solutions of linear differential systems

We study the problem of periodic solutions of linear differential systems with small parameter. We establish new conditions for the existence and uniqueness of periodic solutions of these systems, which can be efficiently verified. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 731–735, May, 1997.     

Multiple periodic solutions for asymptotically linear Duffing equations with resonance

We investigate multiple periodic solutions of asymptotically linear Duffing equation with resonance using index theory and Morse theory and obtain a new result.     

Almost-everywhere regularity results for solutions of non linear elliptic systems

We prove almost-everywhere regularity of weak solutions of non linear elliptic systems of arbitrary order.Dedicated to Hans Lewy and Charles B. Morrey, Jr.     

Rotating periodic solutions for asymptotically linear second‐order Hamiltonian systems with resonance at infinity

In this paper, we consider a class of asymptotically linear second‐order Hamiltonian system with resonance at infinity. We will use Morse theory combined with the technique of penalized functionals to obtain the existence of rotating periodic solutions.     

Cauchy matrix for linear almost periodic systems and some consequences

A novel method to construct the fundamental matrix for a linear almost periodic system is proposed, provided that the diagonal terms satisfy an average separation condition and the off-diagonal coefficients are L^∞-small. The idea is to transform the system in a set of Riccati type equations and use exponential dichotomy and its consequences. It is shown that the method yields easy computation procedures with simple and direct conditions depending on the coefficients. Finally, our result enables us to obtain: (i) explicit almost periodic matrices Q(t), Q⁻¹(t) and Q^′(t), which diagonalize the original system and (ii) sufficient conditions for the stability. Two illustrative examples are shown.     

Unboundedness of the large solutions of some asymmetric oscillators at resonance

In this paper, we consider the unboundedness of solutions of the following differential equation (φ_p(x′))′ + (p ? 1)[αφ_p(x⁺) ? βφ_p(x^?)] = f(x)x′ + g(x) + h(x) + e(t) where φ_p(u) = |u|^{p? 2} u, p > 1, x^± = max {±x, 0}, α and β are positive constants satisfying with m, n ∈ N and (m, n) = 1, f and g are continuous and bounded functions such that lim_x→±∞g(x) ? g(±∞) exists and h has a sublinear primitive, e(t) is 2π_p‐periodic and continuous. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On periodic solutions of countable systems of linear and quasilinear difference equations with periodic coefficients

We present conditions for the existence of periodic solutions of linear difference equations with periodic coefficients in spaces of bounded number sequences. In the case where the generating linear equation has a unique periodic solution, we indicate sufficient conditions for the existence of a periodic solution of a quasilinear difference equation.     

Computations of critical groups and periodic solutions for asymptotically linear Hamiltonian systems

The purpose of this paper is two-fold. Firstly, we will give some parabolic-like conditions which improve the well-known angle conditions and allow further computations of the critical groups both at degenerate critical points and at infinity. As an application, we then consider the second-order Hamiltonian systems

     

Bounds for the singular set of solutions to non linear elliptic systems

We give new estimates for the Hausdorff dimension of the singular set of solutions to elliptic systems If the vector fields a and b are Hölder continuous with respect to the variables (x,u) with exponent , then, under suitable assumptions, the Hausdorff dimension of the singular set of any weak solution is at most . We consider natural growth assumptions on a(x,u,Du) with respect to u and critical ones on the right hand side b(x,u,Du), with respect to Du.Accepted: 12 March 2003, Published online: 16 May 2003     

Periodic solutions of asymptotically linear autonomous Newtonian systems with resonance

In this paper, we study the existence, continuation and bifurcation from infinity of2π$2\pi$-periodic solutions of autonomous Newtonian systems. We underline that the resonant case is considered. To prove the results, we apply the degree for S¹$S^1$-equivariant gradient maps defined by Rybicki (1994) in [15] and the angle condition introduced by Bartsch and Li (1997) in [16].     

Existence of periodic solutions for second-order Hamiltonian systems with asymptotically linear conditions

We consider a class of asymptotically linear nonautonomous second-order Hamiltonian systems. Using the Saddle Point Theorem, we obtain the existence result, which extends some previously known results.     

Branching of periodic solutions of quasilinear autonomous systems in the resonance case

Sufficient conditions for the existence of periodic solutions of quasilinear autonomous systems are obtained, using the theory of branching of nonlinear equations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 6, pp. 760–770, June, 1991.