Advances in Difference Equations

http://www.advancesindifferenceequations.com/

List of Papers (Total 3,184)

Multiple Positive Solutions in the Sense of Distributions of Singular BVPs on Time Scales and an Application to Emden-Fowler Equations

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.

Existence and Multiple Solutions for Nonlinear Second-Order Discrete Problems with Minimum and Maximum

Consider the multiplicity of solutions to the nonlinear second-order discrete problems with minimum and maximum: , , , , where are fixed numbers satisfying are satisfying , ,.

Eigenvalue Problems for -Laplacian Functional Dynamic Equations on Time Scales

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.

Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations

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 σ(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...

Eigenvalue Problems for p-Laplacian Functional Dynamic Equations on Time Scales

This paper is concerned with the existence and nonexistence of positive solutions of the p-Laplacian functional dynamic equation on a time scale, [ϕp(x▵(t))]∇+λa(t)f(x(t),x(u(t)))=0, t∈(0,T), x0(t)=ψ(t), t∈[−τ,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...

Almost-Periodic Weak Solutions of Second-Order Neutral Delay-Differential Equations with Piecewise Constant Argument

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.

Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations

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...

On the Growth of Nonoscillatory Solutions for Difference Equations with Deviating Argument

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...

On the Solutions of Systems of Difference Equations

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...

Do All Integrable Equations Satisfy Integrability Criteria?

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...

Almost-Periodic Weak Solutions of Second-Order Neutral Delay-Differential Equations with Piecewise Constant Argument

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.

On the Solutions of Systems of Difference Equations

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 .

Do All Integrable Equations Satisfy Integrability Criteria?

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...

Asymptotic Representation of the Solutions of Linear Volterra Difference Equations

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...

Asymptotic Representation of the Solutions of Linear Volterra Difference Equations

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...

Neural Network Adaptive Control for Discrete-Time Nonlinear Nonnegative Dynamical Systems

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...

q-Genocchi Numbers and Polynomials Associated with q-Genocchi-Type l-Functions

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...

Multiple Twisted q-Euler Numbers and Polynomials Associated with p-Adic q-Integrals

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...

Stability of Linear Dynamic Systems on Time Scales

We examine the various types of stability for the solutions of linear dynamic systems on time scales and give two examples.

The Periodic Character of the Difference Equation xn+1=f(xn−l+1,xn−2k+1)

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...

Linearized Riccati Technique and (Non-)Oscillation Criteria for Half-Linear Difference Equations

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...

Multiple Twisted -Euler Numbers and Polynomials Associated with -Adic -Integrals

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...

Positive Solutions of Two-Point Right Focal Eigenvalue Problems on Time Scales

We offer criteria for the existence of positive solutions for two-point right focal eigenvalue problems where are fixed and is a time scale.

Three Solutions to Dirichlet Boundary Value Problems for p-Laplacian Difference Equations

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.

Positive Solutions of Two-Point Right Focal Eigenvalue Problems on Time Scales

We offer criteria for the existence of positive solutions for two-point right focal eigenvalue problems (−1)n−pyΔn(t)=λf(t,y(σn−1(t)),yΔ(σn−2(t)),…,yΔp−1(σn−p(t))), t∈[0,1]∩T,yΔi(0)=0, 0≤i≤p−1,yΔi(σ(1))=0, p≤i≤n−1, where λ>0, n≥2,1≤p≤n...