Monotone Iterative Method Research Articles

Parabolic logistic equation with harvesting involving the fractional Laplacian

This paper deals with a class of parabolic reaction-diffusion equations driven by the fractional Laplacian as the diffusion operator over a bounded domain with zero Dirichlet external condition. Using a comparison principle and monotone iteration method, we establish existence and uniqueness results. We apply the existence result to the logistic growth problems with constant yield harvesting by constructing an ordered pair of positive sub- and supersolution of the corresponding elliptic problem.

Radial Solutions for p-k-Hessian Equations and Systems with Gradient Term

This paper studies the existence of entire radial solutions to the p-k-Hessian equation with nonlinear gradient term σk(λDi|Du|p-2Dj(u)+α|∇u|(p-1)k=a(|x|)fk(u),x∈Rn,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sigma _{k}\\left. (\\lambda \\left( D_{i}\\left( |D u|^{p-2} D_{j}(u)\\right) \\right) +\\alpha |\ abla u|^{(p-1) k}\\right. =a(|x|) f^{k}(u), ~~x \\in \\mathbb {R}^{n}, \\end{aligned}$$\\end{document}and system with nonlinear gradient term σk(λDi|Du|p-2Dj(u)+α|∇u|(p-1)k=a(|x|)fk(v),x∈Rn,σk(λDi|Dv|p-2Dj(v)+β|∇v|(p-1)k=b(|x|)gk(u),x∈Rn.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l} \\sigma _{k}\\left. (\\lambda \\left( D_{i}\\left( |D u|^{p-2} D_{j}(u)\\right) \\right) +\\alpha |\ abla u|^{(p-1) k}\\right. =a(|x|) f^{k}(v), ~~x \\in \\mathbb {R}^{n}, \\\\ \\sigma _{k}\\left. (\\lambda \\left( D_{i}\\left( |D v|^{p-2} D_{j}(v)\\right) \\right) +\\beta |\ abla v|^{(p-1) k}\\right. =b(|x|) g^{k}(u), ~~x \\in \\mathbb {R}^{n}. \\end{array}\\right. \\end{aligned}$$\\end{document}By adopting monotone iteration method, we derive the existence and asymptotic behavior of the radial solutions.

An existence theorem for generalized Abelian Higgs equations and its application

In this note, we construct self-dual vortices and cosmic strings from the generalized Abelian Higgs theory in which the Higgs potential is a polynomial whose degree depends on the number m . When | m | > 0 , we obtain sharp existence theorems for vortices and strings corresponding to the absence and presence of gravity, respectively, over the whole plane. In the absence of gravity, we prove the existence of vortices by using a monotone iteration method. When gravity is taken into account, a regularization method and a fixed-point theorem are used to show that multiple string solutions exist under a sufficient condition imposed only on the total number of strings. In addition, a series of properties with respect to vortices and strings have also been established.

On p-k-convex solutions for the p-k-Hessian system with the gradient term

In this paper, we consider a class of nonlinear p-k-Hessian systems with the gradient term where γ > 0, p > 1, and 1 ≤ k ≤ N is an integer. We show some existence, boundedness and blow-up results of the entire positive p-k-convex radial solutions based on the monotone iterative method. Finally, we present an example to illustrate our result.

Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative

Abstract In this work, we consider a class of fuzzy fractional delay integro-differential equations with the generalized Caputo-type Atangana-Baleanu (ABC) fractional derivative. By using the monotone iterative method, we not only obtain the existence and uniqueness of the solution for the given problem with the initial condition but also give the monotone iteration sequence converging to the unique solution of the problem. Furthermore, we also give the continuous dependence of the unique solution on initial value. Finally, an example is presented to illustrate the main results obtained. The results presented in this study are new and open a new avenue of research for fuzzy fractional delay integro-differential equations with the generalized ABC fractional derivative.

Extreme solution for fractional differential equation with nonlinear boundary condition

In this paper, we investigate a class of fractional equations with nonlinear boundary condition. We establish a new comparison principle related to linear fractional equation and show the existence of extreme solution by using monotone iterative method and lower and upper solutions method.

Monotone iterative method for boundary value problem with linear condition of first order delay differential equation

In this paper we discuss the boundary value problem for a first order delay differential equation of the type, \(y'(t) + \lambda y(t) = f(t, y(t - r))\). We prove the existence of solution between weakly coupled lower and upper solution by assuming \(f\) to be a non-decreasing function in the second coordinate. Further, we use this existence result to establish monotone iterative method, where we obtain increasing as well as decreasing sequence of functions whose limits are a solution of the boundary value problem. The sequence of functions obtained are solutions of some defined boundary value problem with linear condition of linear delay differential equation.

AN ALGORITHMIC APPROACH TO LOCAL SOLUTION OF THE NONLINEAR HIGHER ORDER ORDINARY HYBRID DIFFERENTIAL EQUATIONS

In this paper, we introduce a new notion of local or neighborhood solutions and establish a couple of approximation results for local existence and uniqueness of the solution of an IVP of nonlinear higher order ordinary hybrid differential equations by using the Dhage monotone iteration method based on the recent hybrid fixed point theorems of Dhage (2023). An approximation result for Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established. Our main abstract results of this paper are also illustrated with a couple of numerical examples. Finally, we compare our existence and uniqueness results with those existing in the literature via other operator theoretic methods from nonlinear functional analysis. We claim that the method and the results of this paper are new to the literature.

Dynamical behavior of a temporally discrete non-local reaction-diffusion equation on bounded domain

This paper focuses on the study of global dynamics of a class of temporally discrete non-local reaction-diffusion equations on bounded domains. Similar to classical reaction diffusion equations and integro-difference equations, temporally discrete reaction-diffusion equations can also be used to describe the dispersal phenomena in population dynamics. In this paper, we first derived a temporally discrete reaction diffusion equation model with time delay and nonlocal effects to model the evolution of a single species population with age-structured located in a bounded domain. By establishing a new maximum principle and applying the monotone iteration method, the global stabilities of the trivial solution and the positive steady state solution are obtained respectively under some appropriate assumptions.

USING LEARNING ABILITIES OF COMPUTATION IN COUNTING

In this paper, we introduce a new notion of local or neighborhood solutions and establish a couple of approximation results for local existence and uniqueness of the solution of an IVP of nonlinear higher order ordinary hybrid differential equations by using the Dhage monotone iteration method based on the recent hybrid fixed point theorems of Dhage (2023). An approximation result for Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established. Our main abstract results of this paper are also illustrated with a couple of numerical examples. Finally, we compare our existence and uniqueness results with those existing in the literature via other operator theoretic methods from nonlinear functional analysis. We claim that the method and the results of this paper are new to the literature.

A monotone iterative method for second order nonlinear problems with boundary conditions driven by maximal monotone multivalued operators

A monotone iterative method for second order nonlinear problems with boundary conditions driven by maximal monotone multivalued operators

On the Existence of a Solution of a Boundary Value Problem on a Graph for a Nonlinear Equation of the Fourth Order

A fourth-order nonlinear differential equation on a network that is a model of a system of Euler–Bernoulli rods is considered. Based on the monotone iteration method, the existence of a solution of a boundary value problem on a graph for this equation is established using the positiveness of the Green’s function and the maximum principle for the corresponding linear differential equation. An example is given to illustrate the results.

Monotone iterative technique for evolution equations with delay and nonlocal conditions in ordered Banach space

In this paper, based on monotone iterative method in the presence of the lower and upper solutions, we investigate the existence and uniqueness of the S-asymptotically ω-periodic mild solutions to a class of nonlocal problems of evolution equations with delay in ordered Banach spaces. Firstly, we introduce the concept of lower S-asymptotically ω-periodic solution and upper S-asymptotically ω-periodic solution. Secondly, we construct monotone iterative method in the presence of the lower and upper solutions to evolution equations with delay, and obtain the existence of maximal and minimal S-asymptotically ω-periodic mild solutions for the mentioned system under wide monotone conditions and noncompactness measure condition of nonlinear term. Finally, as the application of abstract results, an example is given to illustrate our main results.

Long-time dynamics and semi-wave of a delayed nonlocal epidemic model with free boundaries

This paper is concerned with a nonlocal reaction–diffusion system with double free boundaries and two time delays. The free boundary problem describes the evolution of faecally–orally transmitted diseases. We first show the well-posedness of global solution, and then establish the monotonicity and asymptotic property of basic reproduction number for the epidemic model without delays, which is defined by spectral radius of the next infection operator. By introducing the generalized principal eigenvalue defined in general domain, we obtain an upper bound of the limit value of basic reproduction number. We discuss the spreading and vanishing phenomena in terms of the basic production number. By employing the perturbed approximation method and monotone iteration method, we establish the existence, uniqueness and monotonicity of solution to semi-wave problem. When spreading occurs, we determine the asymptotic spreading speeds of free boundaries by constructing suitable upper and lower solutions from the semi-wave solutions. Moreover, spreading speeds for partially degenerate diffusion case are provided in a similar way.

Monotone traveling waves in a general discrete model for populations with long term memory

In this paper we consider the existence of monotone traveling waves for a class of general integral difference models for populations that are dependent on the previous state term and also on long term memory. This allows us to consider multiple past states. For this model we will have to deal with the non-compactness of the evolution operator when we prove the existence of a fixed point. This difficulty will be overcome by using the Monotone Iteration Method and Dini’s Theorem to show uniform convergence of an iterative evolution operator to a continuous wave function.

Approximating local solution of IVPs of nonlinear first order ordinary hybrid integrodifferential equations

In this paper, we prove a couple of approximation results for local existence and uniqueness of the solution of a IVP of nonlinear first order ordinary hybrid integrodifferential equations by using the Dhage monotone iteration method based on a hybrid fixed point theorem of Dhage (2022) and Dhage {et al.} (2022). An approximation result for the Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established. Finally, our main abstract results are also illustrated with a couple of numerical examples.

Approximation results for PBVPs of nonlinear first order ordinary functional differential equations in a closed subset of the Banach space

In this paper we prove the approximation results for existence and uniqueness of the solution of PBVPs of nonlinear first order ordinary functional differential equations in a closed subset of the Banach space. We employ the Dhage monotone iteration method based on a recent hybrid fixed point theorem of Dhage (2022) and Dhage {\em et al.} (2022) for the main results of this paper. Finally an example is indicated to illustrate the abstract ideas involed in the approximation results.

Existence and stability of traveling waves for $p$-Laplacian equations with time delay

We consider the existence and stability of traveling wave solutions for Nicholson's blowflies equation with degenerate $p$-Laplacian diffusion and time delay. Existence results are proved by the monotone iteration method. Furthermore, we prove the existence and regularity of original solution for the time-delayed degenerate Nicholson's blowflies equation via compactness analysis. Finally, we prove the stability and exponential convergence rate of traveling waves by an approximated weighted energy method.

Existence and stability of traveling wavefronts for a three‐species Lotka–Volterra competitive‐cooperative system with nonlocal dispersal

The purpose of this work is to investigate the existence and stability of traveling wavefronts for a three‐species competitive‐cooperative system with nonlocal dispersal. By applying monotone iteration method combining with a pair of suitable super‐ and sub‐solutions, we establish the existence of traveling wavefronts. The nonexistence of traveling wavefronts is obtained by a contradiction argument. Finally, by using the weighted energy method together with the comparison principle, we prove that the traveling wavefronts with relatively large speeds are exponentially stable as perturbation in some exponentially weighted spaces, when the difference between initial data and traveling wavefronts decays exponentially at negative infinity, but in other locations, the initial data can be very large.

Nonlocal Problems for Hilfer Fractional q-Difference Equations

In the paper, we investigate a kind of Hilfer fractional q-difference equations with nonlocal condition. Firstly, the existence and uniqueness results of solutions are obtained by using topological degree theory and Banach fixed point theorem. Subsequently, the existence of extremal solutions in an ordered Banach space is discussed by monotone iterative method. In that following, we consider the Ulam stability results for equations. Finally, two examples are given to illustrate the effectiveness of theory results.