This paper is devoted to using perturbation and variational techniques to derive some sufficient conditions for the existence of multiple positive solutions in the sense of distributions to a singular second-order dynamic equation with homogeneous Dirichlet boundary conditions, which includes those problems related to the negative exponent Emden-Fowler equation.
Consider the multiplicity of solutions to the nonlinear second-order discrete problems with minimum and maximum: , , , , where are fixed numbers satisfying are satisfying , ,.
This paper is concerned with the existence and nonexistence of positive solutions of the -Laplacian functional dynamic equation on a time scale, , , , , , . We show that there exists a such that the above boundary value problem has at least two, one, and no positive solutions for and , respectively.
It is supposed that the fractional difference equation xn+1=(μ+∑j=0kajxn−j)/(λ+∑j=0kbjxn−j), n=0,1,…, has an equilibrium point x^ and is exposed to additive stochastic perturbations type of Ã(xn−x^)ξn+1 that are directly proportional to the deviation of the system state xn from the equilibrium point x^. It is shown that known results in...
This paper is concerned with the existence and nonexistence of positive solutions of the p-Laplacian functional dynamic equation on a time scale, [Õp(x▵(t))]∇+λa(t)f(x(t),x(u(t)))=0, t∈(0,T), x0(t)=È(t), t∈[−Ä,0], x(0)−B0(x▵(0))=0, x▵(T)=0. We show that there exists a λ∗>0 such that the above boundary value problem has...
We investigate the existence of almost-periodic weak solutions of second-order neutral delay-differential equations with piecewise constant argument of the form , where denotes the greatest integer function, is a real nonzero constant, and is almost periodic.
It is supposed that the fractional difference equation , has an equilibrium point and is exposed to additive stochastic perturbations type of that are directly proportional to the deviation of the system state from the equilibrium point . It is shown that known results in the theory of stability of stochastic difference equations that were obtained via V. Kolmanovskii and L...
The half-linear difference equations with the deviating argument , are considered. We study the role of the deviating argument , especially as regards the growth of the nonoscillatory solutions and the oscillation. Moreover, the problem of the existence of the intermediate solutions is completely resolved for the classical half-linear equation ( = 1). Some analogies or...
We show that every solution of the following system of difference equations xn+1(1)=xn(2)/(xn(2)−1), xn+1(2)=xn(3)/(xn(3)−1),…,xn+1(k)=xn(1)/(xn(1)−1) as well as of the system xn+1(1)=xn(k)/(xn(k)−1), xn+1(2)=xn(1)/(xn(1)−1),…,xn+1(k)=xn(k−1)/(xn(k−1)−1) is periodic with period 2k if k  ≠  0 (mod2), and...
At the price of sacrificing all suspense, we can already announce that the answer to the question of the title is “no.†It is indeed our belief that one may find counterexamples to all integrability conjectures, unless one constrains the definition of integrability to the point that the integrability criterion becomes tautological. This review is devoted to a critical...
We investigate the existence of almost-periodic weak solutions of second-order neutral delay-differential equations with piecewise constant argument of the form (x(t)+x(t−1))′′=qx(2[(t+1)/2])+f(t), where [⋅] denotes the greatest integer function, q is a real nonzero constant, and f(t) is almost periodic.
We show that every solution of the following system of difference equations , as well as of the system , is periodic with period 2 if (2), and with period if (2) where the initial values are nonzero real numbers for .
At the price of sacrificing all suspense, we can already announce that the answer to the question of the title is "no." It is indeed our belief that one may find counterexamples to all integrability conjectures, unless one constrains the definition of integrability to the point that the integrability criterion becomes tautological. This review is devoted to a critical analysis of...
This article analyses the asymptotic behaviour of solutions of linear Volterra difference equations. Some sufficient conditions are presented under which the solutions to a general linear equation converge to limits, which are given by a limit formula. This result is then used to obtain the exact asymptotic representation of the solutions of a class of convolution scalar...
This article analyses the asymptotic behaviour of solutions of linear Volterra difference equations. Some sufficient conditions are presented under which the solutions to a general linear equation converge to limits, which are given by a limit formula. This result is then used to obtain the exact asymptotic representation of the solutions of a class of convolution scalar...
Nonnegative and compartmental dynamical system models are derived from mass and energy balance considerations that involve dynamic states whose values are nonnegative. These models are widespread in engineering and life sciences, and they typically involve the exchange of nonnegative quantities between subsystems or compartments, wherein each compartment is assumed to be...
The main purpose of this paper is to study on generating functions of the q-Genocchi numbers and polynomials. We prove new relation for the generalized q-Genocchi numbers which is related to the q-Genocchi numbers and q-Bernoulli numbers. By applying Mellin transformation and derivative operator to the generating functions, we define q-Genocchi zeta and l-functions, which are...
By using p-adic q-integrals on ℤp, we define multiple twisted q-Euler numbers and polynomials. We also find Witt's type formula for multiple twisted q-Euler numbers and discuss some characterizations of multiple twisted q-Euler Zeta functions. In particular, we construct multiple twisted Barnes' type q-Euler polynomials and multiple twisted Barnes' type q-Euler Zeta...
We examine the various types of stability for the solutions of linear dynamic systems on time scales and give two examples.
In this paper, we consider the nonlinear difference equation xn+1=f(xn−l+1,xn−2k+1), n=0,1,…, where k,l∈{1,2,…} with 2k≠l and gcd(2k,l)=1 and the initial values x−α,x−α+1,…,x0∈(0,+∞) with α=max{l−1,2k−1}. We give sufficient conditions under which every positive solution of this equation converges to a ( not...
We consider the half-linear second-order difference equation Δ(rkΦ(Δxk))+ckΦ(xk+1)=0, Φ(x):=|x|p−2x, p>1, where r, c are real-valued sequences. We associate with the above-mentioned equation a linear second-order difference equation and we show that oscillatory properties of the above-mentioned one can be investigated using properties of this associated linear...
By using -adic -integrals on , we define multiple twisted -Euler numbers and polynomials. We also find Witt's type formula for multiple twisted -Euler numbers and discuss some characterizations of multiple twisted -Euler Zeta functions. In particular, we construct multiple twisted Barnes' type -Euler polynomials and multiple twisted Barnes' type -Euler Zeta functions. Finally, we...
We offer criteria for the existence of positive solutions for two-point right focal eigenvalue problems where are fixed and is a time scale.
We deal with Dirichlet boundary value problems for p-Laplacian difference equations depending on a parameter λ. Under some assumptions, we verify the existence of at least three solutions when λ lies in two exactly determined open intervals respectively. Moreover, the norms of these solutions are uniformly bounded in respect to λ belonging to one of the two open intervals.
We offer criteria for the existence of positive solutions for two-point right focal eigenvalue problems (−1)n−pyΔn(t)=λf(t,y(Ãn−1(t)),yΔ(Ãn−2(t)),…,yΔp−1(Ãn−p(t))), t∈[0,1]∩T,yΔi(0)=0, 0≤i≤p−1,yΔi(Ã(1))=0, p≤i≤n−1, where λ>0, n≥2,1≤p≤n...