Monotone Iterative Technique Research Articles

Existence of positive solutions for discrete delta-nabla fractional boundary value problems with p-Laplacian

In this paper, we consider a discrete delta-nabla boundary value problem for the fractional difference equation with p-Laplacian △v−2β(φp(b∇νx(t)))+λf(t−ν+β+1,x(t−ν+β+1),[b∇εx(t)]t−ν+β+ε+1)=0,x(b)=0,[b∇νx(t)]ν−2=0,x(−1)=∑t=0b−1x(t)A(t),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& {\\triangle_{v-2}^{\\beta}} \\bigl({\\varphi_{p} \\bigl({_{b}\\nabla^{\\nu }}x(t) \\bigr)} \\bigr)+{\\lambda} {f \\bigl(t-\\nu+\\beta+1,x(t-\\nu+\\beta +1), \\bigl[_{b}\\nabla^{\\varepsilon}x(t) \\bigr]_{t-\\nu+\\beta+\\varepsilon+1} \\bigr)}=0, \\\\& x(b)=0,\\quad\\quad \\bigl[_{b}\\nabla^{\\nu}x(t) \\bigr]_{\\nu-2}=0,\\quad\\quad x(-1)=\\sum_{t=0}^{b-1}{x(t)A(t)}, \\end{aligned}$$ \\end{document} where tinmathbb{T}=[nu-beta-1,b+nu-beta-1]_{mathbb{N}_{nu-beta-1}}. {triangle_{nu-2}^{beta}}, {_{b}nabla^{nu}} are left and right fractional difference operators, respectively, and varphi_{p}(s)=|s|^{p-2}s, p>1.By using the method of upper and lower solution and the Schauder fixed point theorem, we obtain the existence of positive solutions for the above boundary value problem; and applying a monotone iterative technique, we establish iterative schemes for approximating the solution.

Positive and negative solutions of a boundary value problem for a fractional q-difference equation

In this work, we study a boundary value problem for a fractional q-difference equation. By using the monotone iterative technique and lower-upper solution method, we get the existence of positive or negative solutions under the nonlinear term is local continuity and local monotonicity. The results show that we can construct two iterative sequences for approximating the solutions.

Existence of positive periodic solutions of some nonlinear fractional differential equations

The paper is devoted to study of existence and uniqueness of periodic solutions for a particular class of nonlinear fractional differential equations admitting its right-hand side with certain singularities. Our approach is based on Krasnosel’skiĭ and Schauder fixed point theorems and monotone iterative technique which enable us to extend some previously known results. The discussed problems are characterized by a Green’s function which has integrable singularities disallowing a direct use of classical techniques known from theory of ordinary differential equations, therefore proper modifications are proposed. Furher, the paper presents simple numerical algorithms directly built on the iterative technique used in theoretical proofs. Illustrative examples conclude the paper.

Positive global solutions of nonlocal boundary value problems for the nonlinear convection reaction-diffusion equations

In this paper, the nonlocal boundary value problems for a class of nonlinear functional convection reaction-diffusion equations with the singular reaction function are studied by using the method of upper and lower solutions and monotone iterative technique. Some of sufficient results on the existence and uniqueness of positive global solutions or positive solutions for the boundary value problems are presented, which are a generalization of some recent results in the area.

Monotone iterative technique via initial time different coupled lower and upper solutions for fractional differential equations

In this paper, we investigate the extremal solutions for a class of nonlinear fractional differential equations with order q 2 (0; 1) by means of monotone iterative technique via initial time different coupled upper and lower solutions.

Symmetric positive solutions for the systems of higher-order boundary value problems on time scales

AbstractIn this paper, we discuss the existence of symmetric positive solutions for the systems of higher-order boundary value problems on time scales. Our results extend some recent work in the literature. The analysis of this paper mainly relies on the monotone iterative technique.

Existence of minimal and maximal solutions to RL fractional integro-differential initial value problems

In this work we investigate integro-differential initial value problems with Riemann Liouville fractional derivatives where the forcing function is a sum of an increasing function and a decreasing function. We will apply the method of lower and upper solutions and develop two monotone iterative techniques by constructing two sequences that converge uniformly and monotonically to minimal and maximal solutions. In the first theorem we will construct two natural sequences and in the second theorem we will construct two intertwined sequences. Finally, we illustrate our results with an example.

Positive solutions for eigenvalue problem of non-linear fractional differential equations

In this paper, by using the fixed point theorem of cone expansion and compression of norm type and monotone iterative technique, we study the eigenvalue problem of non-linear fractional differential equations. The existence and iteration of positive solutions are obtained.

Monotone Iterative Technique for a Class of Slanted Cantilever Beam Equations

In this paper, we deal with the existence and uniqueness of the solutions of two-point boundary value problem of fourth-order ordinary differential equation: u4(t)=f(t,u(t),u′(t)), t∈[0,1], u(0)=u′(0)=u′′(1)=u′′′(1)=0, where f:[0,1]×R2→R is a continuous function. The problem describes the static deformation of an elastic beam whose left end-point is fixed and right is freed, which is called slanted cantilever beam. Under some weaker assumptions, we establish a new maximum principle by the perturbation of positive operator and construct the monotone iterative sequence of the lower and upper solutions, and, based on this, we obtain the existence and uniqueness results for the slanted cantilever beam.

New Fixed Point Theorems for Mixed Monotone Operators with Perturbation and Applications

By using the properties of cone and the fixed point theorem for mixed monotone operators in ordered Banach spaces, we investigate the mixed monotone operators of a new type with perturbation. We establish some sufficient conditions for such operators to have a new existence and uniqueness fixed point and provide monotone iterative techniques which give sequences convergent to the fixed point. Finally, as applications, we apple the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.

Existence of solutions for a class of nonlinear higher-order fractional differential equation with fractional nonlocal boundary condition

In this paper, we study the existence of solutions for a class of nonlinear higher-order fractional differential equation with fractional nonlocal boundary condition by using the monotone iterative technique based on the method of upper and lower solutions and give a specific iterative equation about its solutions.

Explicit iteration to Hadamard fractional integro-differential equations on infinite domain

This paper investigates the existence of the unique solution for a Hadamard fractional integral boundary value problem of a Hadamard fractional integro-differential equation with the monotone iterative technique. Two examples to illustrate our result are given.

Monotone iterative technique for the initial value problem for differential equations with non-instantaneous impulses

An algorithm for constructing two monotone sequences of upper and lower solutions of the initial value problem for a scalar nonnlinear differential equation with non-instantaneous impulses is given. The impulses start abruptly at some points and their action continue on given finite intervals. We prove that the functional sequences are convergent and their limits are minimal and maximal solutions of the considered problem. An example is given to illustrate the results.

The method of lower and upper solutions for mixed fractional four-point boundary value problem with p-Laplacian operator

Based on the monotone iterative technique, a new method of lower and upper solutions which is used to study the multi-point boundary value problem of nonlinear fractional differential equations with mixed fractional derivatives and p -Laplacian operator is proposed. Some new results on the existence of positive solutions for the four-point boundary value problem are established. Finally, an example is presented to illustrate the wide range of potential applications of our main results.

Positive solutions of boundary value problems for fractional order differential equations by monotonic iteration method

In this article, we discuss a class of boundary value problems for nonlinear singular fractional order differential equations. By applying monotone iterative techniques, we not only obtain the existence of positive solutions but also establish an iterative scheme for approximating the solutions. An example is also included to illustrate the main result.

Mixed monotone operators and their application to integral equations

In this paper, we investigate the mixed monotone operators of a new type in a Cartesian product of a Banach space with an order determined by a normal cone. We establish sufficient conditions for such operators to have a unique fixed point and provide monotone iterative techniques which give sequences convergent to the fixed point. We also apply the obtained results to prove the existence and uniqueness of a solution of a nonlinear integral equation.

Double Perturbations for Impulsive Differential Equations in Banach Spaces

In this article, we are concerned with the existence of extremal solutions to the initial value problem of impulsive differential equations in ordered Banach spaces. The existence and uniqueness theorem for the solution of the associated linear impulsive differential equation is established. With the aid of this theorem, the existence of minimal and maximal solutions for the initial value problem of nonlinear impulsive differential equations is obtained under the situation that the nonlinear term and impulsive functions are not monotone increasing by using perturbation methods and monotone iterative technique. The results obtained in this paper improve and extend some related results in abstract differential equations. An example is also given to illustrate the feasibility of our abstract results.

Monotone iterative technique for a nonlinear fractional q-difference equation of Caputo type

By establishing a comparison theorem and applying the monotone iterative technique combined with the method of lower and upper solutions, we investigate the existence of extremal solutions of the initial value problem for fractional q-difference equation involving Caputo derivative. An example is presented to illustrate the main result.

Monotonic iteration positive symmetric solutions to a boundary value problem with p-Laplacian and Stieltjes integral boundary conditions

This paper investigates the existence of monotonic iteration positive symmetric solutions to a boundary value problem with p-Laplacian and Stieltjes integral boundary conditions. The main tool is a monotone iterative technique. Meanwhile, an example is worked out to demonstrate the main results.

Extremal Solutions by Monotone Iterative Technique for Hybrid Fractional Differential Equations

This paper highlights the mathematical model of biological experiments, that have an effect on our lives. We suggest a mathematical model involving fractional differential operator, kind of hybrid iterative fractional differential equations. Our technique is based on monotonous iterative in the nonlinear analysis. The monotonous sequences described extremal solutions converging for hybrid monotonous fractional iterative differential equations. We apply the monotonous iterative method under appropriate conditions to prove the existence of extreme solutions. The tool relies on the Dhage fixed point Theorem. This theorem is required in biological studies in which increasing or decreasing know freshly split bacterial and could control.