Bounds on the critical times for the Fisher-KPP equation

The Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation is one of the prototypical reaction–diffusion equations and is encountered in many areas, primarily in population dynamics. An important consideration for the phenomena modelled by diffusion equations is the length of the diffusive process. In this paper, three definitions of the critical time are given, and bounds are obtained by a careful construction of the upper and lower solutions. The comparison functions satisfy the nonlinear, but linearizable, partial differential equations of Fisher–KPP type. Results of the numerical simulations are displayed. Extensions to some classes of reaction–diffusion systems and an application to a spatially heterogeneous harvesting model are also presented. doi:10.1017/S1446181121000365

Method for Predicting Failure Rate of Airborne Equipment Based on Optimal Combination Model

Accurate prediction of airborne equipment failure rate can provide correct repair and maintenance decisions and effectively establish a health management mechanism. This plays an important role in ensuring the safe use of the aircraft and flight safety. This paper proposes an optimal combination forecasting model, which mixes five single models (Multiple Linear Regression model (MLR), Gray model GM (1, N), Partial Least Squares model (PLS), Artificial Neural Network model (BP), and Support Vector Machine model (SVM)). The combined model and its single model are compared with the other three algorithms. Seven classic comparison functions are used for predictive performance evaluation indicators. The research results show that the combined model is superior to other models in terms of prediction accuracy. This paper provides a practical and effective method for predicting the airborne equipment failure rate.

BOUNDS ON THE CRITICAL TIMES FOR THE GENERAL FISHER–KPP EQUATION

The Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation is one of the prototypical reaction–diffusion equations and is encountered in many areas, primarily in population dynamics. An important consideration for the phenomena modelled by diffusion equations is the length of the diffusive process. In this paper, three definitions of the critical time are given, and bounds are obtained by a careful construction of the upper and lower solutions. The comparison functions satisfy the nonlinear, but linearizable, partial differential equations of Fisher–KPP type. Results of the numerical simulations are displayed. Extensions to some classes of reaction–diffusion systems and an application to a spatially heterogeneous harvesting model are also presented.

A GENERALIZATION OF AN EXTENDED b-METRIC SPACE AND SOME FIXED POINT THEOREMS

In this paper, we present fixed point theorems for contraction mappings in a generalization of an extended $b$-metric space where the product of the Lipschitz constant and functions of the underlying space in the limit are bounded by one for sequences in an orbit. Futhermore, we prove fixed point results in which the contraction involves $b$-comparison functions.

A Relation-Theoretic Matkowski-Type Theorem in Symmetric Spaces

In this paper, we present a fixed-point theorem in R-complete regular symmetric spaces endowed with a locally T-transitive binary relation R using comparison functions that generalizes several relevant existing results. In addition, we adopt an example to substantiate the genuineness of our newly proved result. Finally, as an application of our main result, we establish the existence and uniqueness of a solution of a periodic boundary value problem.

On stochastic inverse problem of construction of stable program motion

One of the inverse problems of dynamics in the presence of random perturbations is considered. This is the problem of the simultaneous construction of a set of first-order Ito stochastic differential equations with a given integral manifold, and a set of comparison functions. The given manifold is stable in probability with respect to these comparison functions.

A Common Fixed Point Theorem for Nonlinear Quasi-Contractions on b -Metric Spaces with Application in Integral Equations

In this paper, we present a common fixed point result for a pair of mappings defined on a b-metric space, which satisfies quasi-contractive inequality with nonlinear comparison functions. An application in solving a class of integral equations will support our results.

On the solutions of a class of nonlinear integral equations in cone $b$-metric spaces over Banach algebras

In this paper, we study the existence of the solutions of a class of functional integral equations by using some fixed point results in cone $b$-metric spaces over Banach algebras. In order to obtain these results we introduced and proved some properties of generalized weak $\varphi$-contractions, in which the $\varphi$ are nonlinear weak comparison functions. The obtained results are generalizations of results of Van Dung N., Le Hang V. T., Huang H., Radenovic S. and Deng G. Also, some suitable examples are given to illustrate obtained results.

On (Λ,Υ,ℜ)-Contractions and Applications to Nonlinear Matrix Equations

In this paper, we study the behavior of Λ , Υ , ℜ -contraction mappings under the effect of comparison functions and an arbitrary binary relation. We establish related common fixed point theorems. We explain the significance of our main theorem through examples and an application to a solution for the following nonlinear matrix equations: X = D + ∑ i = 1 n A i ∗ X A i − ∑ i = 1 n B i ∗ X B i X = D + ∑ i = 1 n A i ∗ γ X A i , where D is an Hermitian positive definite matrix, A i , B i are arbitrary p × p matrices and γ : H ( p ) → P ( p ) is an order preserving continuous map such that γ ( 0 ) = 0 . A numerical example is also presented to illustrate the theoretical findings.