Mixed Monotone Operators Research Articles

Existence of positive solutions for a class of p-Laplacian fractional differential equations with nonlocal boundary conditions

This article is devoted to proving the uniqueness of positive solutions for p-Laplacian equations with Caputo and Riemann-Liouville fractional derivative. The uniqueness result and the dependence of the solution on a parameter are established based on the fixed point point theorem of mixed monotone operators. In the end, a numerical simulation is given to verify the main results.

Coupled Fixed Point Theory in Subordinate Semimetric Spaces

The aim of this paper is to study the coupled fixed point of a class of mixed monotone operators in the setting of a subordinate semimetric space. Using the symmetry between the subordinate semimetric space and a JS-space, we generalize the results of Senapati and Dey on JS-spaces. In this paper, we obtain some coupled fixed point results and support them with some examples.

Positive solutions for the Riemann–Liouville-type fractional differential equation system with infinite-point boundary conditions on infinite intervals

In this paper, we study the existence and uniqueness of positive solutions for a class of a fractional differential equation system of Riemann–Liouville type on infinite intervals with infinite-point boundary conditions. First, the higher-order equation is reduced to the lower-order equation, and then it is transformed into the equivalent integral equation. Secondly, we obtain the existence and uniqueness of positive solutions for each fixed parameter λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda >0$\\end{document} by using the mixed monotone operators fixed-point theorem. The results obtained in this paper show that the unique positive solution has good properties: continuity, monotonicity, iteration, and approximation. Finally, an example is given to demonstrate the application of our main results.

A study of the multipoint boundary value problem for nonlinear differential equations of arbitrary fractional order based on the fixed point theorem for composite nonlinear operators

Abstract The existence and distinctiveness of solutions, as well as the iterative approximations of the distinctive answer to the initial boundary value issue of nonlinear differential equations, may be solved with the help of the theory of nonlinear operators, which offers a strong theoretical guarantee and a fundamental tool. In this work, we examine the fixed point theory for composites nonlinear operators, which may offer a fresh approach to solving linear differential equations having arbitrary order fractional multipoint boundary value problems. The existing result of the boundary value issue’s solution is confirmed by the fixed point theorem for composites nonlinear operators, and the four-point boundary value problem for fractional-order linear equations of the Riemann-Liouville type is examined. A group of fractional-order nonlinear differential equations with deviating quantities are examined to examine the presence of an unusual positive solution for this boundary value problem. The existence of this unique positive solution is determined by the use of both the mixed monotone operator and the combined nonlinear operator fixed point theorem. Ultimately, the nonlinear Bagley-Torvik equation with a fractional order variable coefficient four-point boundary value problem is converted into a Fredholm-Hammerstein integral the formula of the second type with weakly singular or continuous kernels, and the fixed point principle is used to demonstrate the uniqueness of the solution in the space of continuous functions. Approximate solutions are provided for the second class of nonlinear Fredholm-Hammerstein integral equations with weakly singular kernels. The outcomes demonstrate the applicability and validity of the fixed point theorem to composite nonlinear operators, offering a fresh approach to solving the multipoint boundary value issue for nonlinear differential equations of any fractional order.

On the solvability of a cantilever-type boundary value problem by using the mixed monotone operator

In the present paper, by using the mixed monotone operator method we prove the existence and uniqueness of positive solution to the following cantilever-type boundary value problem u(4)(t)=f(t,u(t),u(αt))+g(t,u(t)),0<t<1,α∈(0,1),u(0)=u′(0)=u′′(1)=u′′′(1)=0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\displaystyle \\left\\{ \\begin{array}{l} u^{(4)}(t)=f(t,u(t),u(\\alpha t))+g(t,u(t)),\\quad 0<t<1, \\quad \\alpha \\in (0,1),\\\\ u(0)=u'(0)=u''(1)=u'''(1)=0. \\end{array}\\right. \\end{aligned}$$\\end{document}Moreover, in order to illustrate the results we present an example.

New results on the existence and uniqueness of positive almost periodic solution for the generalized Mackey–Glass hematopoietic model

This paper is concerned with the solution of the generalized hematopoietic model with multiple variable delays and multiple exponents. By using the fixed point theorem for mixed monotone operators, we prove that the generalized Mackey–Glass hematopoietic model has a unique positive almost periodic solution, and derive various new existence and uniqueness results under weaker conditions. Then, an iterative sequence is established and shown to converge to the unique solution under various conditions. Finally, four examples are given to illustrate the application of our theoretical results.

Existence and Uniqueness of Non-Negative Solution to a Coupled Fractional q-Difference System with Mixed q-Derivative via Mixed Monotone Operator Method

In this paper, we study a nonlinear Riemann-Liouville fractional a q-difference system with multi-strip and multi-point mixed boundary conditions under the Caputo fractional q-derivative, where the nonlinear terms contain two coupled unknown functions and their fractional derivatives. Using the fixed point theorem for mixed monotone operators, we constructe iteration functions for arbitrary initial value and acquire the existence and uniqueness of extremal solutions. Moreover, a related example is given to illustrate our research results.

Use of Mixed Operator Method to a fractional Hadamard Dirichlet boundary value problem

The purpose of this paper is to deal with the following nonlinear Hadamard fractional boundary value problem HDα1+ u(t) + f(t, u(t), u(t)) + g(t, u(t)) = 0,1 &lt; t &lt; e, 1 &lt; α ≤ 2,u(1) = u(e) = 0, where HDα 1+ is the Hadamard fractional derivative operator. Using the mixed monotone operator method, we prove an existence and uniqueness result for this mixed fractional Hadamard boundary value problem. As an application of this result, we give one example to establish an existence and uniqueness of a positive solution.

Local existence\u2013uniqueness and monotone iterative approximation of positive solutions for p-Laplacian differential equations involving tempered fractional derivatives

In this paper, we are concerned with a kind of tempered fractional differential equation Riemann–Stieltjes integral boundary value problems with p-Laplacian operators. By means of the sum-type mixed monotone operators fixed point theorem based on the cone P_{h}, we obtain not only the local existence with a unique positive solution, but also construct two successively monotone iterative sequences for approximating the unique positive solution. Finally, we present an example to illustrate our main results.

Existence of the Unique Nontrivial Solution for Mixed Fractional Differential Equations

In this paper, we consider the differential equations with right-sided Caputo and left-sided Riemann-Liouville fractional derivatives. Furthermore, the expression of Green’s function is derived, and its properties are investigated. By the fixed-point theorem for both φ − h , e -concave operators and mixed monotone operators, we get the existence and uniqueness of the solution, respectively. As applications, some examples are provided to illustrate our main results.

Existence and Uniqueness of Positive Solutions for a Class of Nonlinear Fractional Differential Equations with Singular Boundary Value Conditions

This paper focuses on a singular boundary value (SBV) problem of nonlinear fractional differential (NFD) equation defined as follows: D 0 + β υ τ + f τ , υ τ = 0 , τ ∈ 0,1 , υ 0 = υ ′ 0 = υ ″ 0 = υ ″ 1 = 0 , where 3 &lt; β ≤ 4 , D 0 + β is the standard Riemann–Liouville fractional (RLF) derivative. The nonlinear function f τ , υ τ might be singular on the spatial and temporal variables. This paper proves that a positive solution to the SBV problem exists and is unique, taking advantage of Green’s function through a fixed-point (FP) theory on cones and mixed monotone operators.

Existence of positive solutions for a class of fractional differential equations with the derivative term via a new fixed point theorem

In this paper, we firstly establish the existence and uniqueness of solutions of the operator equation A(x,x)+ B(x,x)+C(x)+e = x, where A and B are two mixed monotone operators, C is a decreasing operator, and ein P with theta leq e leq h. Then, using our abstract theorem, we prove a class of fractional boundary value problems with the derivative term to have a unique solution and construct the corresponding iterative sequences to approximate the unique solution.

Positive solutions for a fractional boundary value problem via a mixed monotone operator

Positive solutions for a fractional boundary value problem via a mixed monotone operator

Existence and Uniqueness Results for Two-Term Nonlinear Fractional Differential Equations via a Fixed Point Technique

The work addressed in this paper is to ensure the existence and uniqueness of positive solutions for initial value problems for nonlinear fractional differential equations with two terms of fractional orders. By virtue of recent fixed point theorems on mixed monotone operators, we get some new straightforward results with a wide range of applications.

Probabilistic $ (\omega, \gamma, \phi) $-contractions and coupled coincidence point results

&lt;abstract&gt;&lt;p&gt;In this paper, we introduce the notion of probabilistic $ (\omega, \gamma, \phi) $-contraction and establish the existence coupled coincidence points for mixed monotone operators subjected to the introduced contraction in the framework of ordered Menger $ PM $-spaces with Had${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over z} }}$ić type $ t $-norm. As an application, a corresponding result in the setup of fuzzy metric space is also obtained.&lt;/p&gt;&lt;/abstract&gt;

New fixed point theorems for the sum of two mixed monotone operators of Meir–Keeler type and their applications to nonlinear elastic beam equations

In this paper, we present a new fixed point theorem for the sum of two mixed monotone operators of Meir–Keeler type on ordered Banach spaces through projective metric, which extends the existing corresponding results. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear singular fourth-order elastic beam equations with nonlinear boundary conditions.

Existence-uniqueness of positive solutions to nonlinear impulsive fractional differential systems and optimal control

In this thesis, we investigate a kind of impulsive fractional order differential systems involving control terms. By using a class of φ-concave-convex mixed monotone operator fixed point theorem, we obtain a theorem on the existence and uniqueness of positive solutions for the impulsive fractional differential equation, and the optimal control problem of positive solutions is also studied. As applications, an example is offered to illustrate our main results.

Positive solutions to integral boundary value problems from geophysical fluid flows

The mathematical model of the Antarctic Circumpolar Current with integral boundary conditions is established and the explicit expression of green’s function is obtained. The existence and uniqueness of solutions are proved by using the mixed monotone operator theory. The sufficient conditions for the existence of positive solutions of the model are given and the existence of positive solutions with integral boundary is proved by using the fixed point technique in cone.

Uniqueness of iterative positive solutions for the singular infinite-point p-Laplacian fractional differential system via sequential technique

By sequential techniques and mixed monotone operator, the uniqueness of positive solution for singular p-Laplacian fractional differential system with infinite-point boundary conditions is obtained. Green's function is derived, and some useful properties of Green' function are obtained. Based on these new properties, the existence of unique positive solutions is established, moreover, an iterative sequence and a convergence rate are given, which are important for practical application, and an example is given to demonstrate the validity of our main results.

Existence and Uniqueness of Positive Solutions for Singular Nonlinear Fractional Differential Equation via Mixed Monotone Operator Method

In this article, we discuss the existence and uniqueness of positive solution for a class of singular fractional differential equations, where the nonlinear term contains fractional derivative and an operator. By applying the fixed point theorem in cone, we get the existence and uniqueness of positive solutions for the fractional differential equation. Moreover, we give an example to demonstrate our main result.