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We investigate the spatial propagation in a delayed spruce budworm diffusive model ∂w(t,x)∂t=dΔw(t,x)−δw(t,x)−w2(t,x)1+w2(t,x)+pw(t−τ1,x)e−aw(t−τ2,x),$$ \frac{\partial w\left(t,x\right)}{\partial t}=d\Delta w\left(t,x\right)-\delta w\left(t,x\right)-\frac{w^2\left(t,x\right)}{1+{w}^2\left(t,x\right)}+ pw\left(t-{\tau}_1,x\right){e}^{- aw\left(t-{\tau}_2,x\right)}, $$ where τ1$$ {\tau}_1 $$ and τ2$$ {\tau}_2 $$ represent, respectively, the incubation and the maturation delays for the spruce budworm. We find the minimal wave speed c∗$$ {c}^{\ast } $$ to determine the existence of traveling wave fronts of the model. More specifically, the model admits traveling wave fronts when c≥c∗$$ c\ge {c}^{\ast } $$; the model has no traveling wave solutions when c∈(0,c∗)$$ c\in \left(0,{c}^{\ast}\right) $$. The proofs are based on combining the upper and lower solutions with the approach of Wu and Zou's theorems, the limit arguments, and Laplace transform. The obtained results help us to understand the spreading patterns and the spreading speed of spruce budworm population. [ABSTRACT FROM AUTHOR]

This paper presents the dynamics that one microorganism spatially expands with two nutrients in space-time periodic environment. The generalized principal eigenvalue defined by the linear periodic parabolic operator is applied as a threshold to discuss the associated initial value problem. When the generalized principal eigenvalue is nonnegative, the solutions approach to the microorganism-extinction equilibrium uniformly. When the generalized principal eigenvalue is negative, the spreading speeds of the model are established when the initial distribution of microorganism has nonempty compact support, which are determined by a family of periodic parabolic eigenvalue problems. In the homogeneous environment, we show that the solutions locally uniformly converge to the microorganism-existent steady state by constructing upper and lower solutions. Finally, numerical simulations are performed to illustrate that the leftward or rightward spreading speed is approximately a constant under various circumstances. [ABSTRACT FROM AUTHOR]

We present an existence and localization result for periodic solutions of second-order non-linear coupled planar systems, without requiring periodicity for the non-linearities. The arguments for the existence tool are based on a variation of the Nagumo condition and the Topological Degree Theory. The localization tool is based on a technique of orderless upper and lower solutions, that involves functions with translations. We apply our result to a system of two coupled Van der Pol oscillators with a forcing component. [ABSTRACT FROM AUTHOR]

We are concerned with the existence of homoclinic solutions for the nonlinear problems P u ′ ′ + ω u ′ - k u = f (t , u , u ′) , t ∈ R , lim | t | → + ∞ u (t) = 0 , where ω ∈ R , k > 0 are real constants, and f : R 3 → R is an L 1 - Carathéodory function. Under some suitable conditions, the existence of homoclinic solutions for problem (P) and the corresponding coupled systems are provided. The proofs of the main results are based on the method of upper and lower solutions. [ABSTRACT FROM AUTHOR]

In this work, the traveling wave solutions for a modified Leslie-Gower predator–prey model under continuous and discrete diffusion cases are investigated respectively. First, the existence of weak traveling wave solutions is shown by Schauder's fixed point theorem with the help of positive upper and lower solutions, which can remove the singularity of the system with b = 0 at u = 0 . Then we use two methods, a squeeze method and a Lyapunov function method, to prove that, under additional conditions, the weak traveling wave solutions are actually traveling wave solutions. Finally, by showing the nonexistence of traveling wave solutions, the minimal wave speed is obtained. [ABSTRACT FROM AUTHOR]

This paper investigates a nonlocal dispersal epidemic model under the multiple nonlocal distributed delays and nonlinear incidence effects. First, the minimal wave speed c ∗ and the basic reproduction number R 0 are defined, which determine the existence of traveling wave solutions. Second, with the help of the upper and lower solutions, Schauder's fixed point theorem, and limiting techniques, the traveling waves satisfying some asymptotic boundary conditions are discussed. Specifically, when ℛ 0 > 1 , for every speed c > c ∗ there exists a traveling wave solution satisfying the boundary conditions, and there is no such traveling wave solution for any 0 < c < c ∗ when ℛ 0 > 1 or c > 0 when ℛ 0 < 1. Finally, we analyze the effects of nonlocal time delay on the minimum wave speed. [ABSTRACT FROM AUTHOR]

We study the autowave front propagation in a medium with discontinuous characteristics and the conditions for its stabilization to a stationary solution with a large gradient at the interface between media in the one-dimensional case. The asymptotic method of differential inequalities, based on constructing an asymptotic approximation of the solution, is the main method of study. We develop an algorithm for constructing such an approximation for the solution of the moving front form in a medium with discontinuous characteristics. The application of such an algorithm requires a detailed analysis of the behavior of the solution in neighborhoods of two singular points: the front localization point and the medium discontinuity point. As a result, we obtain a system of equations for the front propagation speed; this distinguishes this paper from the previously published ones. The developed algorithm can be used to describe autowave propagation in layered media. The results can also be extended to the multidimensional case. [ABSTRACT FROM AUTHOR]

In this paper, we provide existence results to the following boundary value problem {−u' = q(t)f(t, u(t), u' (t)), t ∈ (0, +∞), u(0) = 0, u' (∞)=0, where q ∈ L¹((0, +∞), R) and the nonlinearity f: [0, +∞[×R × R → R is continuous. [ABSTRACT FROM AUTHOR]

This study focuses on constructing solutions for a specific type of second order fractional differential equation involving nonlocal boundary conditions. We additionally provide exemplifications that demonstrate the application of our results. [ABSTRACT FROM AUTHOR]

We present some existence and localization results for periodic solutions of impulsive first‐order coupled nonlinear systems of two equations, without requiring periodicity for the nonlinearities. The arguments are based on Schauder's fixed point theorem together with the upper and lower solution methods, where the upper and lower solutions are not necessarily well ordered. In addition, results on equi‐regulated functions are required for the impulsive analysis. An application to a Wilson‐Cowan system of two strongly coupled neurons illustrates one of the main results. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Zhejiang University (Science Edition) is the property of Journal of Zhejiang University (Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In the study the existence of solutions of a class of fractional integro-differential equations with boundary conditions was considered. The main tool, we employ, is the conventional monotone iterative technique, which is highly effective method to examine the quantitative and qualitative characteristics of various nonlinear problems. This technique produces monotone sequences whose iterations are unique solutions of the certain linear problems. These bounds converge uniformly to the maximal solutions of the given problems. Some types of coupled solutions are considered to obtain the claim of the main results under suitable conditions.

This paper is devoted to the study of the existence and stability of traveling wavefronts for a two dimensional discrete diffusion system with nonlocal delay effects. The existence result can be derived by applying the technique of monotone iteration with the help of a pair of explicit upper and lower solutions. Then, using the weighted energy method together with the comparison principle, we prove that the traveling wavefronts with large speed are exponentially stable, when the initial perturbation around the traveling wavefronts decays exponentially as $ x\rightarrow -\infty $, but can be arbitrarily large in other locations. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the existence of positive solutions for a singular Hessian equation with a negative augmented term. By finding more appropriate upper and lower solutions, we not only overcome the difficulty due to the negative augmented term but also remove a critical condition required in the existing work and establish new results for the existence of positive solutions of the equations under study. Our results improve and complement many existing works. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of nonlinear second‐order functional differential equations with piecewise constant arguments with applications to a thermostat that is controlled by the introduction of functional terms in the temperature and the speed of change of the temperature at some fixed instants. We first prove some comparison results for boundary value problems associated to linear delay differential equations that allow to give a priori bounds for the derivative of the solutions, so that we can control not only the values of the solutions but also their rate of change. Then, we develop the method of upper and lower solutions and the monotone iterative technique in order to deduce the existence of solutions in a certain region (and find their approximations) for a class of boundary value problems, which include the periodic case. In the approximation process, since the sequences of the derivatives for the approximate solutions are, in general, not monotonic, we also give some estimates for these derivatives. We complete the paper with some examples and conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with traveling waves in a class of reaction–diffusion equations with nonlocal delays. By introducing new variables, we transform equations with nonlocal delays to non‐delayed system; the existence of traveling waves is obtained by means of Routh–Hurwitz criterion, and by considering the regularity of the Cauchy problem and using spectral theory, we investigate the global stability of traveling waves and then the uniqueness of wave speeds with the help of upper and lower solution method. Finally, our results are applied to two nonlocal delayed reaction–diffusion equations. [ABSTRACT FROM AUTHOR]

In this paper, the existence and uniqueness of mild solution for a class of multi-term time-fractional delay differential system have been discussed in ordered Banach space by enforcing monotone iterative technique. The generalized semigroup theory, fractional calculus and measure of noncompactness have been implemented to obtain the required results. A new set of sufficient conditions with the coefficients in the equations satisfying some monotonic properties has been obtained. Finally, an application is given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of spreading properties and traveling wave solutions for a vector-host disease system, which models the invasion of vectors and hosts to a new habitat. Combining the uniform persistence idea from dynamical systems with the properties of the corresponding entire solutions, we investigate the propagation phenomena in two different cases: (1) fast susceptible vector; (2) slow susceptible vector when the disease spreads. It turns out that in the former case, the susceptible vector may spread faster than the infected vector and host under appropriate conditions, which leads to multi-front spreading with different speeds; while in the latter case, the infected vector and host always catch up with the susceptible vector, and they spread at the same speed. We further obtain the existence and nonexistence of traveling wave solutions connecting zero to the endemic equilibrium. We also conduct numerical simulations to illustrate our analytic results. [ABSTRACT FROM AUTHOR]

This article is devoted to enhancing a class of generalized Suzuki-type nonlinear contractions following Pant to a class of Suzuki–Ćirić-type nonlinear contractions via comparison functions via a locally ζ-transitive relation and implemented the same to ascertain certain fixed-point results. The outcomes presented herewith unify and generalize a few existing findings. An illustrative examples is offered to explain our findings. Our outcomes assist us in figuring out the unique solution to a boundary value problem.

We present some existence and localization results for periodic solutions of first-order coupled nonlinear systems of two equations, without requiring periodicity for the nonlinearities. The arguments are based on Schauder's fixed point theorem together with not necessarily well-ordered upper and lower solutions. A real-case scenario shows the applicability of our results to some population dynamics models, describing the interaction between a criminal and a non-criminal population with a law enforcement component.

In this paper, we construct a new generalized result to study the existence of solutions of nonlinear fractional boundary value problems (FBVPs). The proposed results unify the existence criteria of certain FBVPs including periodic and antiperiodic as special cases that have been previously studied separately in the literature. The method we employ is topological in its nature and manifests themselves in the forms of differential inequalities (lower and upper solutions, and coupled lower and upper solutions (CLUSs)). Two examples are given to demonstrate the applicability of the developed theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some existence results for the integral boundary value problems of nonlinear fractional q-difference equations. The differential operator is taken in the Riemann-Liouville sense. [ABSTRACT FROM AUTHOR]

This manuscript is concerned about the study of the existence of solutions for the class of nonlinear neutral Caputo-Hadamard fractional differential equations including integral terms. In order to establish the necessary conditions of solvability for the proposed problem, we apply the semi-group property of Hadamard fractional integral operator. Also, under the appropriate conditions, we demonstrate that the solution set for the proposed problem is non-empty by using Arzelá-Ascoli theorem and the method of upper and lower solutions. In contrast to the fundamental results graphical example is also presented in order to validate the findings. [ABSTRACT FROM AUTHOR]

An algorithm is presented for the construction of an asymptotic approximation of a stable stationary solution to a diffusion equation system in a two-dimensional domain with a smooth boundary and a source function that is discontinuous along some smooth curve lying entirely inside the domain. Each of the equations contains a small parameter as a factor in front of the Laplace operator, and as a result, the system is singularly perturbed. In the vicinity of the curve, the solution of the system has a large gradient. Such a problem statement is used in the model of urban development in metropolitan areas. The discontinuity curves in this model are the boundaries of urban biocenoses or large water pools, which prevent the spread of urban development. The small parameter is the ratio of the city's outskirts linear size to the whole metropolis linear size. The algorithm includes the construction of an asymptotic approximation to a solution with a large gradient at the media interface as well as the steps for obtaining the existence conditions. To prove the existence and stability theorems, we use the upper and lower solutions, which are constructed as modifications of the asymptotic approximation to the solution. The latter is constructed using the Vasil'yeva algorithm as an expansion of a small parameter exponent. [ABSTRACT FROM AUTHOR]

Steel corrosion plays a central role in different technological fields. In this article, we consider a simple case of a corrosion phenomenon which describes a pure iron dissolution in sodium chloride. This article is devoted to prove rigorously that under rather general hypotheses on the initial data, the solution of this iron dissolution model converges to a self-similar profile as $t\rightarrow +\infty$. We will do so for an equivalent formulation as presented in the book of Avner Friedman about parabolic equations (Friedman (1964) Partial Differential Equations of Parabolic Type , Prentice-Hall, Inc., Englewood Cliffs, NJ.). In order to prove the convergence result, we apply a comparison principle together with suitable upper and lower solutions. [ABSTRACT FROM AUTHOR]

This paper studies the initial value problems and traveling wave solutions in an SIRS model with general incidence functions. Linearizing the infected equation at the disease free steady state, we can define a threshold if the corresponding basic reproduction ratio in kinetic system is larger than the unit. When the initial condition for the infected is compactly supported, we prove that the threshold is the spreading speed for three unknown functions. At the same time, this threshold is the minimal wave speed for traveling wave solutions modeling the disease spreading process. If the corresponding basic reproduction ratio in kinetic system is smaller than the unit, then we confirm the extinction of the infected and the nonexistence of nonconstant traveling waves.

We present some existence and localization results for periodic solutions of first-order coupled nonlinear systems of two equations, without requiring periodicity for the nonlinearities. The arguments are based on Schauder's fixed point theorem together with not necessarily well-ordered upper and lower solutions. A real-case scenario shows the applicability of our results to some population dynamics models, describing the interaction between a criminal and a non-criminal population with a law enforcement component. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of traveling wave solutions of a higher dimensional lattice delayed cooperation system with nonlocal diffusion. For sufficiently small intraspecific cooperative delays, we construct upper and lower solutions under two different parameters conditions. And then, by using the monotone iterative and Schauder's fixed point theorem, we obtain the existence of traveling wave solutions. The lower bound of the wave speed is in accordance with the properties of linear determined. [ABSTRACT FROM AUTHOR]

We use asymptotic analysis to study the existence of solutions of a one-dimensional nonlinear system of ordinary differential equations with different powers of a small parameter at higher derivatives. A specific feature of the problem is the presence of a discontinuity of the first kind in the right-hand side of the equation in the unknown variable at the level , while the right-hand side of the second equation is assumed to be smooth in all variables. We define a generalized solution of the problem is in terms of differential inclusions. Conditions under which generalized solutions turn into strong ones are proposed, and the possibility that the -component of the solution intersects zero only once is studied. The existence theorems are proved by using the asymptotic method of differential inequalities. [ABSTRACT FROM AUTHOR]

A new method is developed for studying unstable contrast structures (solutions with an internal transition layer), based on the construction of sufficiently accurate unordered upper and lower solutions and the application of a corollary of the Krein–Rutman theorem. Conditions are formulated for the existence of Lyapunov-unstable one-dimensional step-type contrast structures as stationary solutions of singularly perturbed parabolic reaction–diffusion equations with a discontinuous right-hand side. It is shown that the results obtained can be extended to other singularly perturbed one-dimensional reaction–diffusion–advection problems with discontinuous nonlinearities. [ABSTRACT FROM AUTHOR]

We study boundary value problems for systems of second-order ordinary differential equations with quasimonotonicity conditions typical of problems of activator–inhibitor type and with solutions containing domains with large gradients. We obtain sufficient conditions for the existence of a stable stationary solution. Using the asymptotic method of differential inequalities, we prove the existence and stability theorems. [ABSTRACT FROM AUTHOR]

Our main concern in this study is to present the generalized results to investigate the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. The framework of the presented results relies on the lower and upper solutions approach which allows us to ensure the existence of solutions in a sector defined by well-ordered coupled lower and upper solutions. It is worth mentioning that the presented results unify the existence criteria of certain problems which were treated on a case-by-case basis in the literature. Two examples are supplied to support the results.

In this paper, we investigate a new class of nonlinear fractional integrodifferential systems that includes the Ψ -Riemann–Liouville fractional integral term. Using the technique of upper and lower solutions, the solvability of the system is examined. We add two examples to demonstrate and validate the main result. The main results highlight crucial contributions to the general theory of fractional differential equations. [ABSTRACT FROM AUTHOR]

We consider the initial-boundary value problem of reaction-diffusion-advection that has a solution of a front form. The statement comes from the theory of wave physics. We study the question of the solution stabilizing to the stationary one. Proof of the stabilization theorem is based on the concepts of upper and lower solutions and corollaries from comparison theorems. The upper and lower solutions with large gradients are constructed as modifications of the formal moving front asymptotic approximation in a small parameter. The main idea of the proof is to show that the upper and lower solutions of the initial-boundary value problem get into the attraction domain of the asymptotically stable stationary solution on a sufficiently large time interval. The study conducted in this work gives an answer about the non-local attraction domain of the stationary solution and can give some stationing criteria. The results are illustrated by computational examples. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence, uniqueness and asymptotic stability of traveling wave solutions in a delayed competitive system with spatial diffusion. When the interspecific competition is strong, the corresponding functional differential system has two stable and two unstable steady states. Firstly, we establish the existence of bistable traveling wave solutions by the theory of monotone semiflows. With the help of squeezing technique based on comparison principle, the asymptotic stability of traveling wave solutions is proved, which further implies the uniqueness of wave speed and wave profile in the sense of phase shift. [ABSTRACT FROM AUTHOR]

In this paper, we study periodic solutions in distribution of mean-field stochastic differential equations. We introduce the notion of upper and lower solutions of mean-field stochastic differential equations. With the help of the comparison principle, we prove the existence of periodic solutions in distribution. [ABSTRACT FROM AUTHOR]

In this paper, we consider stochastic lattice differential equations (SLDEs). Firstly, we discuss the well-posedness of solutions for SLDEs. Then, via upper and lower solutions, we obtain a pair of monotone sequences starting at them respectively, and we prove the existence of periodic solutions in distribution. [ABSTRACT FROM AUTHOR]

The study considers a directed dynamics reaction-diffusion competition model to study the density of evolution for a single species population with harvesting effect in a heterogeneous environment, where all functions are spatially distributed in time series. The dispersal dynamics describe the growth of the species, which is distributed according to the resource function with no-flux boundary conditions. The analysis investigates the existence, positivity, persistence, and stability of solutions for both time-periodic and spatial functions. The carrying capacity and the distribution function are either arbitrary or proportional. It is observed that if harvesting exceeds the growth rate, then eventually, the population drops down to extinction. Several numerical examples are considered to support the theoretical results. [ABSTRACT FROM AUTHOR]

This paper deals with some differential inequalities for generalized fractional integro-differential equations by using the technique of upper and lower solutions. The fractional differential operator is taken in Caputo’s sense and the nonlinear term divided into two parts depends on the fractional integrals of an unknown function with two different fractional orders. The results are studied by employing a variety of coupled upper and lower solutions. These theorems have some potential for extending the iterative techniques to fractional order integro-differential equations and to coupled systems of integro-differential fractional equations to obtain the existence of solutions as well as approximate solutions for the considered problem.

Copyright of Journal of Zhejiang University (Science Edition) is the property of Journal of Zhejiang University (Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Our main concern in this study is to present the generalized results to investigate the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. The framework of the presented results relies on the lower and upper solutions approach which allows us to ensure the existence of solutions in a sector defined by well-ordered coupled lower and upper solutions. It is worth mentioning that the presented results unify the existence criteria of certain problems which were treated on a case-by-case basis in the literature. Two examples are supplied to support the results. [ABSTRACT FROM AUTHOR]

Traveling wave propagation is a significant phenomenon observed in population biology. Due to the occurrence of nonlocal effect in recursive systems, a deep understanding of the wavefront in the propagation direction (speed sign) is challenging. In this paper we study the sign of wave speed for bistable traveling waves to a two-species competitive system that biologically models the dynamics of two species in competition for a common resource. By a proper choice of the kernel functions, we transfer our model into a coupled localized differential system and shed a new light on how to determine the wave speed sign. Suffcient conditions with symmetry are obtained on the propagation directions of the wavefronts. This symmetry is further verified in the final analysis and numerical simulations are provided to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

This paper studies the convergence and minimal wave speed of traveling wave solutions in a delayed competitive model. Due to the effect of time delays, this system may not satisfy the classical comparison principle. When the domain is bounded and equipped with homogeneous Neumann boundary condition, we obtain the persistence and stability of any positive mild solutions by constructing proper contracting rectangles. When the domain is whole space R , we study the minimal wave speed of traveling wave solutions by presenting the existence or nonexistence of nontrivial traveling wave solutions with any wave speeds, which models the coinvasion-coexistence of these species. Our main recipe on minimal wave speed is the generalized upper and lower solutions, contracting rectangles and spreading speeds. [ABSTRACT FROM AUTHOR]

This article is devoted to a numerical analysis of two types of fractional nonlinear reaction-diffusion problems with periodic conditions or with initial conditions. We also consider two types of derivative in time, the first one is that of Caputo and the second is the conformable derivative; we discretize the problem via a finite difference method. We construct two iterative schemes by the upper and lower solutions method which converges monotonically towards a maximal solution or a minimal solution of the considered problem when the mesh decreases to zero, depending on whether the initial iteration is an upper solution or a lower solution. A comparison lemma for the different monotone sequences is also proved. The presented iterative scheme is used to show that the finite difference system converges to continuous solutions of the fractional reaction-diffusion problem. Finally to validate the theoretical results some examples with numerical simulations of reaction-diffusion problem are also presented and discussed in detail. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)