In this chapter we present basic theory of symplectic difference systems. We show that these systems incorporate as special cases many important equations or systems, such as the Sturm-Liouville difference equations, symmetric three-term recurrence equations, Jacobi difference equations, linear Hamiltonian difference systems, or trigonometric and hyperbolic systems.

In this chapter we introduce the comparative index as a main mathematical tool for the results in the subsequent chapters of this book.

In this chapter we investigate eigenvalue problems associated with symplectic system ( SDS ), where the coefficient matrix depends on a spectral parameter.

In this chapter we will present some additional topics from the theory of symplectic difference systems ( SDS ), which are closely related to their oscillation or spectral theory.

We discuss the concept of relative oscillation for linear Hamiltonian differential systems. We prove a statement which enables to study this oscillation using the criteria of the classical oscillation theory. We also discuss some other aspects of the problem.

We establish an asymptotic formula for a pair of linearly independent solutions of the subcritical Riemann–Weber type half-linear differential equation. We also complement the results of the author and M. Ünal, Acta Math. Hungar. 120 (2008), 147–163, where the equation was considered in the critical case.

We investigate oscillatory properties of perturbed half-linear Euler differential equation. We give an alternative proof (simpler and more straightforward) of the main result of [O. Doslý, H. Funkova, Abstr. Appl. Anal. 2012, Art. ID 738472] and we prove the extended version of a conjecture formulated in [O. Doslý, J. Math. Anal. Appl. 323(2006), 426–440].

In this paper, we introduce a new modification of the half-linear Prüfer angle. Applying this modification, we investigate the conditional oscillation of the half-linear second order differential equation (∗)tα−1r(t)Φ(x′)′+tα−1−ps(t)Φ(x)=0,Φ(x)=|x|p−1sgnx,where p>1, α≠p, and r,s are continuous functions such that r(t)>0 for large t. We present conditions on the functions r,s which guarantee that Eq. (∗) behaves like the Euler type equation [tα−1Φ(x′)]′+λtα−1−pΦ(x)=0, which is conditionally oscillatory with the oscillation constant λ0=|p−α|p∕pp.

The classical Bohl transformation [4] from 1906 concerns the second order linear differential equations and states, roughly speaking, that a pair of linearly independent solutions of a second order differential equation can be expressed via the sine and cosine functions. Since that time, this transformation has been extended in various directions and became e.g. the theoretical basis for the deeply developed transformation theory of second order linear differential equations [8]. In our paper we discuss this transformation for linear Hamiltonian differential systems and discrete symplectic systems. We provide an alternative proofs to some know results and these new proofs enable to give a new insight into the topics. We also formulate some open problems associated with the discrete Bohl transformation.

We investigate oscillatory properties of even-order half-linear differential equations and conditions for negativity of the associated energy functional. First, using the relationship between positivity of the functional and nonoscillation of the investigated equation, we prove Hille-Nehari type nonoscillation criteria which extend criteria known in the linear case. In the second part of the paper, we present conditions which guarantee that the energy functional attains a negative value, i.e., it is unbounded below.

We establish nonoscillation criteria for even order half-linear differential equations. The principal tool we use is the Wirtinger type inequality combined with various perturbation techniques. Our results extend nonoscillation criteria known for linear higher order differential equations.

We investigate the relationship between oscillatory properties of half-linear even order difference equations and nonpositivity of the associated energy functionals. We convert the investigated difference equation into a Hamiltonian type difference system and using this transformation we establish our main result which says that the existence of two (or more) generalized zeros of a solution of the investigated difference equation implies that the corresponding energy functional attains a nonpositive value.

In this paper we generalize oscillation theorems for discrete Hamiltonian eigenvalue problems with nonlinear dependence on the spectral parameter λ. In our version of the discrete oscillation theorems, we incorporate the case when the block Bk(λ) of the discrete Hamiltonian Hk(λ) has nonconstant rank with respect to λ. We introduce a new notion of weighted focal points for conjoined bases of the Hamiltonian difference systems and we show that the number of weighted focal points plays the role of the classical number of focal points in the discrete oscillation theorems for the Hamiltonian spectral problems with the nonconstant rank of Bk(λ).

In this paper, we investigate oscillation properties of discrete trigonometric systems whose coefficients matrices are simultaneously symplectic and orthogonal. The main result generalizes a necessary and sufficient condition of non-oscillation of trigonometric systems proved by M. Bohner and O. Doly (J. Differential Equations 163 (2000), pp. 113-129) in the case when the block in the upper right corner of the coefficient matrix is symmetric and positive definite. Now, we present this oscillation criterion for an arbitrary trigonometric system. The obtained results are applied to formulate a necessary and sufficient condition for non-oscillation of even-order Sturm-Liouville difference equations.

In this paper we consider the second-order nonlinear differential equation (∗)(tα−1Φ(x′))′+tα−1−pf(x)=0,Φ(x)=|x|p−2x,p>1,α∈R, with f satisfying xf(x)>0, x≠0. We analyze the difference between the cases αp, and α=p. In each case we give a condition on the function f which guarantees that solutions of Eq. (∗) are (non)oscillatory. The principal methods used in this paper are the Riccati technique and its modifications. The results of our paper complement and extend several previously obtained results on the subject.

These proceedings of the 20th International Conference on Difference Equations and Applications cover the areas of difference equations, discrete dynamical systems, fractal geometry, difference equati

We investigate oscillatory properties of second order Euler type half-linear differential equations whose coefficients are given by periodic functions and functions having mean values. We prove the conditional oscillation of these equations. In addition, we prove that the known oscillation constants for the corresponding equations with only periodic coefficients do not change in the studied more general case. The presented results are new for linear equations as well.

We present new relations connecting the number of focal points of conjoined bases of eventually disconjugate symplectic difference systems with different coefficient matrices which obey a majorant condition at $\infty.$ For the case of controllable (near $\infty$) symplectic systems we investigate connections between the number of focal points of their recessive solutions. The consideration is based on the concept of minimal and maximal solutions of the associated Riccati matrix difference equations and the comparative index theory for discrete symplectic systems.