Functional difference equations and eigenfunctions of a Schrödinger operator with δ ′ −interaction on a circular conical surface

Eigenfunctions and their asymptotic behaviour at large distances for the Laplace operator with singular potential, the support of which is on a circular conical surface in three-dimensional space, are studied. Within the framework of incomplete separation of variables an integral representation of the Kontorovich–Lebedev (KL) type for the eigenfunctions is obtained in terms of solution of an auxiliary functional difference equation with a meromorphic potential. Solutions of the functional difference equation are studied by reducing it to an integral equation with a bounded self-adjoint integral operator. To calculate the leading term of the asymptotics of eigenfunctions, the KL integral representation is transformed to a Sommerfeld-type integral which is well adapted to application of the saddle point technique. Outside a small angular vicinity of the so-called singular directions the asymptotic expression takes on an elementary form of exponent decreasing in distance. However, in an asymptotically small neighbourhood of singular directions, the leading term of the asymptotics also depends on a special function closely related to the function of parabolic cylinder (Weber function).

Oscillation of nonlinear third order perturbed functional difference equations

This paper deals with oscillatory and asymptotic behavior of all solutions of perturbed nonlinear third order functional difference equation\Delta {\left( {{b_n}\Delta ({a_n}(\Delta {x_n}} \right)^\alpha })) + {p_n}f\left( {{x_{\sigma \left( n \right)}}} \right) = g\left( {n,{x_n},{x_{\sigma (n)}},\Delta {x_n}} \right),\,\,\,n \ge {n_0}.By using comparison techniques we present some new sufficient conditions for the oscillation of all solutions of the studied equation. Examples illustrating the main results are included.

Oscillation of a Class of Third Order Generalized Functional Difference Equation

The authors intend to establish new oscillation criteria for a class of generalized third order functional difference equation of the form \begin{equation}{\label{eq01}} \Delta_{\ell}\left(a_2(n)\left[\Delta_{\ell}\left(a_1(n)\left[\Delta_{\ell}z(n)\right]^{\beta_1}\right)\right]^{\beta_2}\right)+q(n)f(x(g(n)))=0, ~~n\geq n_0, \end{equation} where $z(n)=x(n)+p(n)x(\tau(n))$. We also present sufficient conditions for the solutions to converges to zero. Suitable examples are presented to validate our main results.

Some Properties on Complex Functional Difference Equations

We obtain some results on the transcendental meromorphic solutions of complex functional difference equations of the form∑λ∈Iαλ(z)(∏j=0nf(z+cj)λj)=R(z,f∘p)=((a0(z)+a1(z)(f∘p)+ ⋯ +as(z) (f∘p)s)/(b0(z)+b1(z)(f∘p)+ ⋯ +bt(z)(f∘p)t)), whereIis a finite set of multi-indexesλ=(λ0,λ1,…,λn),c0=0,cj∈ℂ∖{0} (j=1,2,…,n)are distinct complex constants,p(z)is a polynomial, andαλ(z) (λ∈I),ai(z) (i=0,1,…,s), andbj(z) (j=0,1,…,t)are small meromorphic functions relative tof(z). We further investigate the above functional difference equation which has special type if its solution has Borel exceptional zero and pole.

A note on the existence of three positive periodic solutions of functional difference equation

It is shown that, under certain assumptions, the functional difference equations have at least three positive periodic solutions. Applications are given to illustrate the main results.