Avery-Peterson Fixed Point Theorem Research Articles

Multiple positive solutions of fractional differential equations with improper integral boundary conditions on the half-line

This paper investigates the existence of positive solutions for a class of fractional boundary value problems involving an improper integral and the infinite-point on the half-line by making use of properties of the Green function and Avery–Peterson fixed point theorem. In addition, an example is presented to illustrate the applicability of our main result.

Positive Solutions for a System of Hadamard Fractional Boundary Value Problems on an Infinite Interval

Our investigation is devoted to examining the existence, uniqueness, and multiplicity of positive solutions for a system of Hadamard fractional differential equations. This system is defined on an infinite interval and is subject to coupled nonlocal boundary conditions. These boundary conditions encompass both Hadamard fractional derivatives and Riemann–Stieltjes integrals, and the nonlinearities within the system are non-negative functions that may not be bounded. To establish the main results, we rely on the utilization of mathematical theorems such as the Schauder fixed-point theorem, the Banach contraction mapping principle, and the Avery–Peterson fixed-point theorem.

A coupled system of $ p $-Laplacian implicit fractional differential equations depending on boundary conditions of integral type

<abstract><p>The objective of this article is to investigate a coupled implicit Caputo fractional $ p $-Laplacian system, depending on boundary conditions of integral type, by the substitution method. The Avery-Peterson fixed point theorem is utilized for finding at least three solutions of the proposed coupled system. Furthermore, different types of Ulam stability, i.e., Hyers-Ulam stability, generalized Hyers-Ulam stability, Hyers-Ulam-Rassias stability and generalized Hyers-Ulam-Rassias stability, are achieved. Finally, an example is provided to authenticate the theoretical result.</p></abstract>

Existence of positive solutions for singular fractional boundary value problems with p-Laplacian

In this paper, we obtain the existence of positive solutions for the singular fractional boundary value problem with p-Laplacian. The existence of positive solutions is established using the Avery-Peterson fixed point theorem. In addition, we include an example for the demonstration of our main result.

Existence of positive solutions for boundary value problems with p-Laplacian operator

Abstract This paper is concerned with the existence of positive solutions for three point boundary value problems of Riemann-Liouville fractional differential equations with p-Laplacian operator. By means of the properties of the Green’s function and Avery-Peterson fixed point theorem, we establish a condition ensuring the existence of at least three positive solutions for the problem.

Existence of Triple Positive Solutions to a Four-Point Boundary Value Problem of a Fractional Differential Equations

In this paper, the existence result of at least triple positive solutions to a boundary value problem of a fractional differential equations is achieved by means of the Avery-Peterson fixed point theorem and the careful analysis of the associate Green's function in which the derivative of unknown function is involved in the nonlinear term explicitly. An example illustrating our main result is given. Our results complement the previous work in the area of the positive solutions of fractional differential equations.

Existence of multiple positive solutions for a class of infinite-point singular p-Laplacian fractional differential equation with singular source terms

Based on properties of Green’s function and by Avery–Peterson fixed point theorem, the existence of multiple positive solutions are obtained for singular p-Laplacian fractional differential equation with infinite-point boundary conditions, and an example is given to demonstrate the validity of our main results.

EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR CAPUTO FRACTIONAL DIFFERENTIAL EQUATION WITH INFINITE-POINT BOUNDARY VALUE CONDITIONS

<abstract> In this paper, the existence of positive solutions for Caputo fractional differential equations boundary value problem with infinite points contained in the boundary value condition, and based on Green's function and its properties, the existence of multiple positive solutions are obtained by Avery-Peterson fixed point theorem. </abstract>

Existence of positive solutions of Hadamard fractional defferential equations with integral boundary conditions

In this paper, We study the existence of positive solutions for Hadamard fractional differential equations with integral conditions. We employ Avery-Peterson fixed point theorem and properties of Green's function to show the existence of positive solutions of our problem. Furthermore, we present an example to illustrate our main result.

Positive solutions of IBVPs for $ q $-difference equations with $ p $-Laplacian on infinite interval

<abstract> Nowadays, many researches have considerable attention to the nonlinear $ q $-difference equations boundary value problems as important and useful tool for modeling of different phenomena in various research fields. In this work, we investigate a class of $ q $-difference equations boundary value problems with integral boundary conditions with $ p $-Laplacian on infinite intervals. By applying the Avery-Peterson fixed point theorem in a cone, we establish the existence of three positive solutions for the above boundary value problem. Finally, the main results is illustrated with the aid of an example. </abstract>

EXISTENCE OF SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS WITH FRACTIONAL-ORDER DERIVATIVE TERMS

The study in this paper is made on the nonlinear fractional differential equation whose nonlinearity involves the explicit fractional order <i>D</i>_0⁺<i>^β</i> <i>u</i>(<i>t</i>). The corresponding Green's function is derived first, and then the completely continuous operator is proved. Besides, based on the Schauder's fixed point theorem and the Krasnosel'skii's fixed point theorem, the sufficient conditions for at least one or two existence of positive solutions are established. Furthermore, several other sufficient conditions for at least three, <i>n</i> or 2<i>n</i>-1 positive solutions are also obtained by applying the generalized AveryHenderson fixed point theorem and the Avery-Peterson fixed point theorem. Finally, several simulation examples are provided to illustrate the main results of the paper. In particularly, a novel efficient iterative method is employed for simulating the examples mentioned above, that is, the interesting point of this paper is that the approximation graphics for the solutions are given by using the iterative method.

Existence of multiple positive solutions for nonlinear three-point problem for Riemann-Liouville fractional differential equation

In this paper, the existence of multiple positive solutions is considered for nonlinear three-point problem for Riemann-Liouville fractional differential equation. We use the Avery-Peterson fixed point theorem to acquire the existence of multiple positive solutions for the boundary value problem. Two examples are also presented to illustrate the effectiveness of the main result.

Existence of multiple positive solutions for nonlinear three-point problem for Riemann-Liouville fractional differential equation

In this paper, the existence of multiple positive solutions is considered for nonlinear three-point problem for Riemann-Liouville fractional differential equation. We use the Avery-Peterson fixed point theorem to acquire the existence of multiple positive solutions for the boundary value problem. Two examples are also presented to illustrate the effectiveness of the main result.

Existence of multiple positive solutions for p-Laplacian fractional q-difference equations with integral boundary conditions

This article studies a boundary value problem of fractional q-difference equations with integral boundary conditions. Some new existence results are obtained by using Avery-Peterson fixed point theorem. An example is also presented to illustrate the effectiveness of the main result.

Multiple Solutions for an Equation of Kirchhoff Type: Theoretical and Numerical Aspects

A nonlinear boundary value problem related to an equation of Kirchhoff type is considered. The existence of multiple positive solutions is proved through Avery-Peterson Fixed Point Theorem. A numerical method based on Levenberg-Marquadt algorithm combined with a heuristic process is present in order to align numerical and theoretical aspects.

Multiple Solutions for an Equation of Kirchhoff Type: Theoretical and Numerical Aspects

A nonlinear boundary value problem related to an equation of Kirchhoff type is considered. The existence of multiple positive solutions is proved through Avery-Peterson Fixed Point Theorem. A numerical method based on Levenberg-Marquadt algorithm combined with a heuristic process is present in order to align numerical and theoretical aspects.

Multiple Positive Solutions for a Boundary Value Problem with Nonlinear Nonlocal Riemann-Stieltjes Integral Boundary Conditions

Abstract In this paper, we study the existence of positive solutions to the fractional boundary value problem D 0 + α x ( t ) + q ( t ) f ( t , x ( t ) ) = 0 , 0 &lt; t &lt; 1 , $$\begin{array}{} \displaystyle D^{\alpha }_{0+}x(t)+q(t)f(t,x(t))=0, \,\, 0\lt t \lt1, \end{array}$$ together with the boundary conditions x ( 0 ) = x ′ ( 0 ) = ⋯ = x ( n − 2 ) ( 0 ) = 0 , D 0 + β x ( 1 ) = ∫ 0 1 h ( s , x ( s ) ) d A ( s ) , $$\begin{array}{} \displaystyle x(0)=x^{\prime}(0)= \cdots = x^{(n-2)}(0)=0, D_{0+}^{\beta }x(1)= \int^{1}_{0}h(s,x(s))\,dA(s), \end{array}$$ where n &gt; 2, n – 1 &lt; α ≤ n, β ∈ [1,α – 1], and D 0 + α $\begin{array}{} \displaystyle D^{\alpha }_{0+} \end{array}$ and D 0 + β $\begin{array}{} \displaystyle D^{\beta }_{0+} \end{array}$ are the standard Riemann-Liouville fractional derivatives of order α and β, respectively. We consider two different cases: f, h : [0, 1] × R → R, and f, h : [0, 1] × [0, ∞) → [0, ∞). In the first case, we prove the existence and uniqueness of the solutions of the above problem, and in the second case, we obtain sufficient conditions for the existence of positive solutions of the above problem. We apply a number of different techniques to obtain our results including Schauder’s fixed point theorem, the Leray-Schauder alternative, Krasnosel’skii’s cone expansion and compression theorem, and the Avery-Peterson fixed point theorem. The generality of the Riemann-Stieltjes boundary condition includes many problems studied in the literature. Examples are included to illustrate our findings.

THREE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM FOR DIFFERENTIAL EQUATION WITH RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE

In this paper, by using the Avery-Peterson fixed point theorem, we establish the existence result of at least three positive solutions of boundary value problem of nonlinear differential equation with Riemann-Liouville's fractional order derivative. An example illustrating our main result is given. Our results complements and extends previous work in the area of boundary value problems of nonlinear fractional differential equations.

Multiple positive solutions for p-Laplacian equations with integral boundary conditions

In this paper, we consider a class of boundary value problems for p-Laplacian equation (Φp(u′))′+h(t)f(t,u(t),u′(t))=0 with integral boundary conditions u(0)−αu′(0)=∫01g1(s)u(s)ds,u(1)+βu′(1)=∫01g2(s)u(s)ds. By using the Avery–Peterson fixed point theorem, we obtain the existence of at least three positive solutions.

Multiplicity of positive radial solutions of p-Laplacian problems with nonlinear gradient term

In the present paper, we prove the existence of at least three radial solutions of the p-Laplacian problem with nonlinear gradient term {Δpv+f(|x|,v,|∇v|)=0in Ω,v=0on ∂Ω,\\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}$$\\textstyle\\begin{cases} \\Delta_{p}v+f(\\vert x\\vert ,v,\\vert \\nabla v\\vert )=0\\quad \\mbox{in }\\Omega, \\\\ v=0\\quad \\mbox{on }\\partial\\Omega, \\end{cases} $$\\end{document} and the corresponding one-parameter problem. Here Ω is a unit ball in mathbb{R}^{N}. Our approach relies on the Avery-Peterson fixed point theorem. In contrast with the usual hypotheses, no asymptotic behavior is assumed on the nonlinearity f with respect to phi_{p}(cdot).