We consider a three-term nonlinear recurrence relation involving a nonlinear filtering function with a positive threshold λ. We work out a complete asymptotic analysis for all solutions of this equation when the threshold varies from 0+ to +∞. It is found that all solutions either tends to 0, a limit 1-cycle, or a limit 2-cycle, depending on whether the parameter λ is smaller...
The existence of asymptotically almost periodic mild solutions for a class of abstract partial neutral integro-differential equations with unbounded delay is studied.
The asymptotic behavior of the solutions of the first-order differential equation ẏ(t)=∑i=1nβi(t)[y(t-δi)-y(t-τi)] containing delays is studied with βi:[t0-τ,∞)→[0,∞), τ=max{τ1,…,τn}, ∑i=1nβi(t)>0, τi>δi>0. The attention is focused on an analysis of the asymptotical convergence of solutions. A criterion for the asymptotical convergence of all solutions, characterized by the...
It is found that every solution of a system of linear delay difference equations has finite limit at infinity, if some conditions are satisfied. These are much weaker than the known sufficient conditions for asymptotic constancy of the solutions. When we impose some positivity assumptions on the coefficient matrices, our conditions are also necessary. The novelty of our results...
This paper describes asymptotic properties of solutions of some linear difference systems. First we consider system of a general form and estimate its solutions by use of a solution of an auxiliary scalar difference inequality assuming that this solution admits certain properties. Then applying this result to linear difference systems of a variable order with constant (or bounded...
In this paper, we investigate the singular Sturm-Liouville problem u′′=λg(u), u′(0)=0, βu′(1)+αu(1)=A, where λ is a nonnegative parameter, β≥0, α>0, and A>0. We discuss the existence of multiple positive solutions and show that for certain values of λ, there also exist solutions that vanish on a subinterval [0,ρ]⊂[0,1), the so-called dead core solutions. The theoretical findings...
A new and general existence and uniqueness theorem of almost automorphic solutions is obtained for the semilinear fractional differential equation Dtαu(t)=Au(t)+Dtα−1f(t,u(t)) (1<α<2), in complex Banach spaces, with Stepanov-like almost automorphic coefficients. Moreover, an application to a fractional relaxation-oscillation equation is given.
This paper summarizes a series of results on the oscillation of impulsive ordinary differential equations. We consider linear, half-linear, super-half-linear, and nonlinear equations. Several oscillation criteria are given. The Sturmian comparison theory for linear and half linear equations is also included.
We investigate several arithmetic properties of (h,q)-Genocchi polynomials and numbers of higher order.
We deal in this paper with the mild solution for the semilinear fractional differential equation of neutral type with infinite delay: Dαx(t)+Ax(t)=f(t,xt), t∈[0,T], x(t)=ϕ(t), t∈]−∞,0], with T>0 and 0<α<1. We prove the existence (and uniqueness) of solutions, assuming that −A is a linear closed operator which generates an analytic semigroup (T(t))t≥0 on a Banach space 𝕏 by means...
We define the twisted q-Bernoulli polynomials and the twisted generalized q-Bernoulli polynomials attached to χ of higher order and investigate some symmetric properties of them. Furthermore, using these symmetric properties of them, we can obtain some relationships between twisted q-Bernoulli numbers and polynomials and between twisted generalized q-Bernoulli numbers and...
This paper is concerned with the nonlinear system x′(t)=f(t,x(t)). We give a converse Lyapunov theorem and prove robustness of uniform exponential dissipativity with respect to unbounded external perturbations, without assuming f being globally Lipschitz in x.
By using a multiple fixed point theorem (Avery-Peterson fixed point theorem) for cones, some criteria are established for the existence of three positive periodic solutions for a class of higher-dimensional functional differential equations with impulses on time scales of the following form: , , , where is a nonsingular matrix with continuous real-valued functions as its elements...
This paper deals with variational optimal-control problems on time scales in the presence of delay in the state variables. The problem is considered on a time scale unifying the discrete, the continuous, and the quantum cases. Two examples in the discrete and quantum cases are analyzed to illustrate our results.
Szekeley observed that the dynamic pattern of the locomotion of salamanders can be explained by periodic vector sequences generated by logical neural networks. Such sequences can mathematically be described by "doubly periodic traveling waves" and therefore it is of interest to propose dynamic models that may produce such waves. One such dynamic network model is built here based...
We systematically explore the periodicity of Liénard type -Laplacian equations on time scales. Sufficient criteria are established for the existence of periodic solutions for such equations, which generalize many known results for differential equations when the time scale is chosen as the set of the real numbers. The main method is based on the Mawhin's continuation theorem.
The existence and uniqueness of solutions and a representation of solution formulas are studied for the following initial value problem: , , , , . Such problems are obtained by transforming second-order delay differential equations to first-order differential equations.
We will investigate the superstability of the (hyperbolic) trigonometric functional equation from the following functional equations: , , , , which can be considered the mixed functional equations of the sine function and cosine function, of the hyperbolic sine function and hyperbolic cosine function, and of the exponential functions, respectively.
We study the asymptotic behavior of the solutions for the following nonlinear difference equation where the initial conditions are arbitrary nonnegative real numbers, are nonnegative integers, , and are positive constants. Moreover, some numerical simulations to the equation are given to illustrate our results.
This paper is concerned with the existence of multiple positive solutions for the third-order -Laplacian dynamic equation with the multipoint boundary conditions , where with . Using the fixed point theorem due to Avery and Peterson, we establish the existence criteria of at least three positive solutions to the problem. As an application, an example is given to illustrate the...
By using variational methods directly, we establish the existence of periodic solutions for a class of nonautonomous differential delay equations which are superlinear both at zero and at infinity.
By using the critical point theory, we establish various sets of sufficient conditions on the nonexistence and existence of solutions for the boundary value problems of a class of fourth-order difference equations.
The variational method and critical point theory are employed to investigate the existence of solutions for 2th-order difference equation for with boundary value condition by constructing a functional, which transforms the existence of solutions of the boundary value problem (BVP) to the existence of critical points for the functional. Some criteria for the existence of at least...
We mainly study the global behavior of the nonlinear difference equation in the title, that is, the global asymptotical stability of zero equilibrium, the existence of unbounded solutions, the existence of period two solutions, the existence of oscillatory solutions, the existence, and asymptotic behavior of non-oscillatory solutions of the equation. Our results extend and...