Uniqueness of continuous solution to $q$-Hilfer fractional hybrid integro-difference equation of variable order

In this paper, the authors introduced a novel definition based on Hilfer fractional derivative, which name $q$-Hilfer fractional derivative of variable order. And the uniqueness of solution to $q$-Hilfer fractional hybrid integro-difference equation of variable order of the form \eqref{eq:varorderfrac} with $0 < \alpha(t) < 1$, $0 \leq \beta \leq 1$, and $0 < q < 1$ is studied. Moreover, an example is provided to demonstrate the result.

Oscillation theory of q-difference equation

In this research article, the authors present the oscillation theory of the q-difference equation K(t)y(qt)+k(t/q)y(t/q)= r(t)y(t) where r(t) = k(t)+k(t/q)-q(t) In particular we prove that this q-difference equation is oscillatory or non-oscillatory for different conditions.

Closed-Form Solution of a Rational Difference Equation

In this paper, we study the solution of the difference equation Ω m + 1 = Ω m − 7 q + 6 / 1 + ∏ t = 0 5 Ω m − q + 1 t − q , where the initials are positive real numbers.

The Neimark–Sacker Bifurcation and Global Stability of Perturbation of Sigmoid Beverton–Holt Difference Equation

We present the bifurcation results for the difference equation x n + 1 = x n 2 / a x n 2 + x n − 1 2 + f where a and f are positive numbers and the initial conditions x − 1 and x 0 are nonnegative numbers. This difference equation is one of the perturbations of the sigmoid Beverton–Holt difference equation, which is a major mathematical model in population dynamics. We will show that this difference equation exhibits transcritical and Neimark–Sacker bifurcations but not flip (period-doubling) bifurcation since this difference equation cannot have period-two solutions. Furthermore, we give the asymptotic approximation of the invariant manifolds, stable, unstable, and center manifolds of the equilibrium solutions. We give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium.

Positive solutions for eigenvalue problems of fractional q-difference equation with ϕ-Laplacian

The aim of this paper is to investigate the boundary value problem of a fractional q-difference equation with ϕ-Laplacian, where ϕ-Laplacian is a generalized p-Laplacian operator. We obtain the existence and nonexistence of positive solutions in terms of different eigenvalue intervals for this problem by means of the Green function and Guo–Krasnoselskii fixed point theorem on cones. Finally, we give some examples to illustrate the use of our results.

Algorithm design and analysis of fire rescue problems

With the rapid economic development, the urban space environment is becoming more and more complex, various accidents and disasters occur frequently, and safety risks are increasing. The rescue tasks involved in the fire brigade are showing a trend of diversification and complexity. The fire rescue team always puts the people first and insists on serving the people wholeheartedly. It is the guardian of maintaining social stability in our country and safeguarding the health and safety of people’s lives and property and various disaster affairs. The society needs the participation in the fire rescue team. Aiming at the fire rescue problem, this paper uses the fire rescue call data onto 2016 to 2019 to predict the number of fire rescues / rescued calls based on the difference equation to improve the rescue efficiency of the fire brigade. Taking into account the impact on the domestic epidemiced in 2020 on people’s lives, the adjustment value was introduced to adjust part of the alarm data onto 2020 to ensure the accuracy and reliability of the data. Finally, the second-order difference equation is used to predict the alarm data onto 2021 through the least square method, which verifies the accuracy of the model.

Local and Global Stability of Certain Mixed Monotone Fractional Second Order Difference Equation with Quadratic Terms

This paper investigates the local and global character of the unique positive equilibrium of a mixed monotone fractional second-order difference equation with quadratic terms. The corresponding associated map of the equation decreases in the first variable, and it can be either decreasing or increasing in the second variable depending on the corresponding parametric values. We use the theory of monotone maps to study global dynamics. For local stability, we use the center manifold theory in the case of the non-hyperbolic equilibrium point. We show that the observed equation exhibits three types of global behavior characterized by the existence of the unique positive equilibrium, which can be locally stable, non-hyperbolic when there also exist infinitely many non-hyperbolic and stable minimal period-two solutions, and a saddle. Numerical simulations are carried out to better illustrate the results.

Universal strategies for the two-alternative big data processing

We consider the two-alternative processing of big data in the framework of the two-armed bandit problem. We assume that there are two processing methods with different, fixed but a priori unknown efficiencies which are due to different reasons including those caused by legislation. Results of data processing are interpreted as random incomes. During control process, one has to determine the most efficient method and to provide its primary usage. The difficulty of the problem is caused by the fact that its solution essentially depends on distributions of one-step incomes corresponding to results of data processing. However, in case of big data we show that there are universal processing strategies for a wide class of distributions of one-step incomes. To this end, we consider Gaussian two-armed bandit which naturally arises when batch data processing is analyzed. Minimax risk and minimax strategy are searched for as Bayesian ones corresponding to the worst-case prior distribution. We present recursive integro-difference equation for computing Bayesian risk and Bayesian strategy with respect to the worst-case prior distribution and a second order partial differential equation into which integro-difference equation turns in the limiting case as the control horizon goes to infinity. We also show that, in case of big data, processing of data one-by-one is not more efficient than optimal batch data processing for some types of distributions of one-step incomes, e.g. for Bernoulli and Poissonian distributions. Numerical experiments are presented and show that proposed universal strategies provide high performance of two-alternative big data processing.