Nonlinear Fractional Differential Equation Boundary Research Articles

On singular φ−Laplacian BVPs of nonlinear fractional differential equation

This paper investigates the existence of multiple positive solutions for a class of φ−Laplacian boundary value problem with a nonlinear fractional differential equation and fractional boundary conditions. Multiple solutions are proved under slight conditions on a possibly degenerating source term. Approximation techniques together with the fixed-point index theory a on cone of a Banach space are employed. Some illustrating examples of are also supplied. Mathematics Subject Classification (2010): 34A08, 34B15, 34B18, 47H10. Received 27 July; Accepted 13 December 2021

Eigenvalue Criteria for Existence of Positive Solutions to Fractional Boundary Value Problem

The existence and multiplicity of positive solutions for the nonlinear fractional differential equation boundary value problem (BVP) DC0+αyx+fx,yx=0, 0<x<1, y0=y′1=y″0=0 is established, where 2<α≤3, CD0+α is the Caputo fractional derivative, and f:0,1×0,∞⟶0,∞ is a continuous function. The conclusion relies on the fixed-point index theory and the Leray-Schauder degree theory. The growth conditions of the nonlinearity with respect to the first eigenvalue of the related linear operator is given to guarantee the existence and multiplicity.

Fixed points of mixed monotone operators for existence and uniqueness of nonlinear fractional differential equations

In this paper we study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems by using new fixed point results of mixed monotone operators on cones.

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.

The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions

In this paper, first, we consider the existence of a positive solution for the nonlinear fractional differential equation boundary value problem urn:x-wiley:mma:media:mma4105:mma4105-math-0002 where 0≤λ < 1,CDα is the Caputo's differential operator of order α, and f:[0,1] × [0,∞)→[0,∞) is a continuous function. Using some cone theoretic techniques, we deduce a general existence theorem for this problem. Then, we consider two following more general problems for arbitrary α, 1≤n < α≤n + 1:Problem 1: urn:x-wiley:mma:media:mma4105:mma4105-math-0003 where , 0≤λ < k + 1;Problem 2: urn:x-wiley:mma:media:mma4105:mma4105-math-0004 where 0≤λ≤α and Dα is the Riemann–Liouville fractional derivative of order α.For these problems, we give existence results, which improve recent results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.

Singular boundary value problems of fractional differential equations with changing sign nonlinearity and parameter

In this paper, we consider singular boundary value problems for the following nonlinear fractional differential equations with delay: $$\left \{ \textstyle\begin{array}{l@{\quad}l} D^{\alpha}x(t)+\lambda f (t,x(t-\tau) )=0, &t\in(0,1)\backslash \{\tau\}, x(t)=\eta(t), &t\in[-\tau,0], x'(1)=x'(0)=0, \end{array}\displaystyle \right . $$ where $2<\alpha\leq3$ , $D^{\alpha}$ denotes the Riemann-Liouville fractional derivative, λ is a positive constant, $f(t,x)$ may change sign and be singular at $t=0$ , $t=1$ , and $x=0$ . By means of the Guo-Krasnoselskii fixed point theorem, the eigenvalue intervals of the nonlinear fractional functional differential equation boundary value problem are considered, and some positive solutions are obtained, respectively.

The solution for a class of sum operator equation and its application to fractional differential equation boundary value problems

In this paper we study a class of sum operator equation $Ax+Bx+C(x,x)=x$ on ordered Banach spaces, where A is an increasing operator, B is a decreasing operator, and C is a mixed monotone operator. The existence and uniqueness of its positive solution are obtained by using the properties of cone and fixed point theorems for mixed monotone operators. As an application, we utilize the obtained results to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.

Positive solutions of multi-point boundary value problem of fractional differential equation

By means of two fixed-point theorems on a cone in Banach spaces, some existence and multiplicity results of positive solutions of a nonlinear fractional differential equation boundary value problem are obtained. The proofs are based upon some properties of Green’s function, which are also the key of the paper.

Solvability of boundary value problems of nonlinear fractional differential equations

In this paper, we study the existence of multiple positive solutions for the nonlinear fractional differential equation boundary value problem $D^{\alpha}_{0^{+}}u(t)+f(t,u(t))=0 $ , $0< t<1$ , $u(0)=u(1)=u'(0)=0$ , where $2<\alpha\leq3$ is a real number, $D^{\alpha}_{0^{+}}$ is the Riemann-Liouville fractional derivative. By the properties of the Green’s function, the lower and upper solution method and the Leggett-Williams fixed point theorem, some new existence criteria are established. As applications, examples are presented to illustrate the main results.

Results for nonlinear fractional differential equations on pseudo-metric spaces with three boundary val

Abstract. In this paper, we investigate the existence of solutions in Gauge spaces for the nonlinear fractional differential equation boundary value problem With different boundary value and and different order 1 < a 2 and 0 < a < 1. where 0 < /31 < /32 < T and µ [O, T]and + is the standard Riemann-Lowville differentiation with f(t, u(t)) C([O, T] x [O, ), R). By using some fixed-point results on gauge spaces, some existence results of positive solutions are obtained.

Existence of Positive Solutions to Boundary Value Problem of Caputo Fractional Differential Equation

We consider the existence of positive solutions to the nonlinear fractional differential equation boundary value problemD0+αCut+fut,u't=0, t∈0,1, u0=u1=u″0=0, wheref:0,+∞×R→0,+∞is continuous,α∈2,3, andD0+αCis the standard Caputo differentiation. By using fixed point theorems on cone, we give some existence results concerning positive solutions. Here the solutions especially are the interior points of cone.

The existence and nonexistence of positive solutions to a discrete fractional boundary value problem with a parameter

In this paper, we study the existence and nonexistence of positive solutions for the boundary value problem with a parameter { − Δ ν y ( t ) = λ f ( t + ν − 1 , y ( t + ν − 1 ) ) , y ( ν − 2 ) = y ( ν + b + 1 ) = 0 , where t ∈ [ 0 , b + 1 ] N , 1 < ν ≤ 2 is a real number, f : [ ν − 1 , ν + b ] N ν − 1 × R → ( 0 , + ∞ ) is a continuous function, b ≥ 2 is an integer, λ is a parameter. The eigenvalue intervals of the nonlinear fractional differential equation boundary value problem are considered by the properties of the Green function and Guo–Krasnosel’skii fixed point theorem on cones, some sufficient conditions of the nonexistence of positive solutions for the boundary value problem are established. As applications, we give some examples to illustrate the main results.

Existence of multiple positive solutions for singular boundary value problems of nonlinear fractional differential equations

In this paper, we consider the properties of the Green’s function for the nonlinear fractional differential equation boundary value problem , , , where is a real number, and is the standard Riemann-Liouville differentiation. As an application of the Green’s function, we give some multiple positive solutions for singular boundary value problems, and we also give the uniqueness of solution for a singular problem by means of the Leray-Schauder nonlinear alternative, a fixed-point theorem on cones, and a mixed monotone method.

Positive properties of Green’s function for focal-type BVPs of singular nonlinear fractional differential equations and its application

In this paper, we consider the properties of Green’s function for the singular nonlinear fractional differential equation boundary value problem

Uniqueness and existence of positive solutions for a multi-point boundary value problem of singular fractional differential equations

In this paper, we study the uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem , , , , where is a real number, is the standard Riemann-Liouville differentiation and , with . Our analysis relies on a fixed-point theorem in partially ordered set. As an application, an example is presented to illustrate the main result. MSC:26A33, 34B15, 34K37.

Eigenvalue problem for a class of nonlinear fractional differential equations

‎In this paper, we study eigenvalue problem for a class of nonlinear fractional differential equations $$D^\alpha_{0^+}u(t)=\lambda f(u(t)),\quad 0 \lt t \lt 1,$$ $$u(0)=u(1)=u'(0)=u'(1)=0,$$ where $3 \lt \alpha\leq4$ is a real number, $D^\alpha_{0^+}$ is the Riemann-Liouville fractional derivative, $\lambda$ is a positive parameter and $f:(0,+\infty)\to(0,+\infty)$ is continuous. By the properties of the Green function and Guo-Krasnosel'skii fixed point theorem on cones, the eigenvalue intervals of the nonlinear fractional differential equation boundary value problem are considered, some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established. As an application, some examples are presented to illustrate the main results.

The Positive Properties of Green’s Function for Fractional Differential Equations and Its Applications

We consider the properties of Green’s function for the nonlinear fractional differential equation boundary value problem:D0+αu(t)+f(t,u(t))+e(t)=0,0&lt;t&lt;1,u(0)=u'(0)=⋯=u(n-2)(0)=0,u(1)=βu(η), wheren-1&lt;α≤n,n≥3,0&lt;β≤1,0≤η≤1,D0+αis the standard Riemann-Liouville derivative. Here our nonlinearityfmay be singular atu=0. As applications of Green’s function, we give some multiple positive solutions for singular boundary value problems by means of Schauder fixed-point theorem.

On positive solutions for a class of singular nonlinear fractional differential equations

We study the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem

Positive properties of Green's function for three-point boundary value problems of nonlinear fractional differential equations and its applications

In this article, we consider the properties of Green's function for the nonlinear fractional differential equation boundary value problem where 1 < α < 2, 0 < β, η < 1, is the standard Riemann–Liouville derivative. Here our nonlinearity f may be singular at u = 0. As an application of Green's function, we establish some multiple positive solutions for singular positone and semipositone boundary value problems by means of the Leray–Schauder nonlinear alternative, a fixed-point theorem on cones, and also we give uniqueness of a solution for a singular problem by a mixed monotone method.

Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems

The purpose of this paper is to present some new fixed point theorems for mixed monotone operators with perturbation by using the properties of cones and a fixed point theorem for mixed monotone operators. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.