Infinite-point Boundary Value Research Articles

Positive solutions for a class of fractional differential equations with infinite-point boundary conditions on infinite intervals

In this paper, the existence of the multiple positive solutions for a class of higher-order fractional differential equations on infinite intervals with infinite-point boundary value conditions is mainly studied. First, we construct the Green function and analyze its properties, and then by using the Leggett–Williams fixed point theorem, some new results on the existence of positive solutions for the boundary value problem are obtained. Finally, we illustrate the application of our conclusion by an example.

Existence of Monotone Positive Solutions for Caputo–Hadamard Nonlinear Fractional Differential Equation with Infinite-Point Boundary Value Conditions

Numerical solutions and approximate solutions of fractional differential equations have been studied by mathematicians recently and approximate solutions and exact solutions of fractional differential equations are obtained in many kinds of ways, such as Lie symmetry, variational method, the optimal ADM method, and so on. In this paper, we obtain the positive solutions by iterative methods for sum operators. Green’s function and the properties of Green’s function are deduced, then based on the properties of Green’s function, the existence of iterative positive solutions for a nonlinear Caputo–Hadamard infinite-point fractional differential equation are obtained by iterative methods for sum operators; an example is proved to illustrate the main result.

Existence and uniqueness of positive solutions for a class of singular fractional differential equation with infinite-point boundary value conditions

Existence and uniqueness of positive solutions for a class of singular fractional differential equation with infinite-point boundary value conditions

POSITIVE SOLUTIONS IN THE SPACE OF LIPSCHITZ FUNCTIONS TO A CLASS OF FRACTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE-POINT BOUNDARY VALUE CONDITIONS

<abstract> The purpose of this paper is to study the existence of positive solutions to a class of fractional differential equations with infinite-point boundary value conditions. Our solutions are placed in the space of Lipschitz functions and the main tools used in the proof of the results are a sufficient condition about the relative compactness in Holder spaces and the classical Schauder fixed point theorem. </abstract>

Positive Solution for the Integral and Infinite Point Boundary Value Problem for Fractional-Order Differential Equation Involving a Generalized ϕ-Laplacian Operator

In this paper, we establish the existence of nontrivial positive solution to the following integral-infinite point boundary-value problem involving ϕ-Laplacian operator D0+αϕx,D0+βux+fx,ux=0,x∈0,1,D0+σu0=D0+βu0=0,u1=∫01 gtutdt+∑n=1n=+∞ αnuηn, where ϕ:0,1×R→R is a continuous function and D0+p is the Riemann-Liouville derivative for p∈α,β,σ. By using some properties of fixed point index, we obtain the existence results and give an example at last.

Existence and uniqueness of monotone positive solutions for fractional higher-order BVPs

We discuss the existence and uniqueness of monotone positive solutions for a class of higher-order nonlinear fractional differential equations infinite-point boundary value problems (for short, BVPs) on cones. The existence and uniqueness of solutions are obtained via applying the properties of the Green function and a fixed point theorem. Our analysis is based on the operator equation T ω + S ω = ω on an ordered Banach space. Finally, a example is given to illustrate our results.

Positive solutions to p-Laplacian fractional differential equations with infinite-point boundary value conditions

This paper is devoted to an investigation of the multiple positive solutions to a class of infinite-point boundary value problems of nonlinear fractional differential equations coupled with the p-Laplacian operators and infinite-point boundary value conditions. By means of the properties of Green’s function and fixed point theorems, we establish the suitable criteria to guarantee the existence of positive solutions. Finally, an example is given in order to illustrate the main results.

Positive solutions for a class of fractional infinite-point boundary value problems

In this paper, we consider a class of fractional differential equations with infinite-point boundary value conditions. Under some conditions concerning the spectral radius with respect to the relevant linear operator, both the existence of uniqueness and the nonexistence of positive solution are obtained by means of the iterative technique.

Sum operator methods for the existence and uniqueness of solution to infinite-point boundary value problems for fractional differential equations

In this paper, we study infinite-point boundary value problems for a class of higher-order nonlinear factional differential equations involving the Riemann-Liouville derivative. By using sum operator methods, the existence and uniqueness of solution to this kind of problems is obtained and iterative sequence of the positive solution is structured. Two examples are provided for our new results.

Sum operator methods for the existence and uniqueness of solution to infinite-point boundary value problems for fractional differential equations

In this paper, we study infinite-point boundary value problems for a class of higher-order nonlinear factional differential equations involving the Riemann-Liouville derivative. By using sum operator methods, the existence and uniqueness of solution to this kind of problems is obtained and iterative sequence of the positive solution is structured. Two examples are provided for our new results.

Solvability to infinite-point boundary value problems for singular fractional differential equations on the half-line

In this paper, we investigate the solvability of infinite-point boundary value problems for a class of higher-order factional differential equations on the half-line involving Riemann–Liouville derivatives. Under a new compactness criterion, some new results are obtained by means of the properties of the Green’s function and some appropriate fixed point theorems on cone.

The positive solutions of infinite-point boundary value problem of fractional differential equations on the infinite interval

In this paper, we consider a class of infinite-point boundary value problems of fractional differential equations on the infinite interval [0,+infty) with a disturbance parameter. By using the method of upper and lower solutions, fixed point index theory and some fixed point theorems, the existence, multiplicity and nonexistence for the positive solution of the boundary value problem are obtained, respectively. The impact of the disturbance parameters on the existence of positive solutions is also given. Finally, some examples are presented to illustrate the wide range of potential applications of our main results.

Existence of positive solutions of singular fractional differential equations with infinite-point boundary conditions

Using height functions of the nonlinear term on some bounded sets and considering integrations of these height functions, we obtain the existence of positive solutions for a class of singular fractional differential equations with infinite-point boundary value conditions.

Existence of solutions to a coupled system of fractional differential equations with infinite-point boundary value conditions at resonance

By means of coincidence degree theory, we present an existence result for the solution of a higher-order coupled system of nonlinear fractional differential equations with infinite-point boundary conditions at resonance.

Existence of solutions for fractional Langevin equation with infinite-point boundary conditions

In this paper, we investigate a class of nonlinear Langevin equation involving two fractional orders \(\alpha \in (0,1]\) and \(\gamma \in (1,2]\) with infinite-point boundary value conditions. By Leray–Schauder’s nonlinear alternative and Leray–Schauder degree theory, several new existence results of solutions are obtained. An example is given to show the applicability of our main results.

Positive solutions for a class of singular fractional differential equation with infinite-point boundary value conditions

We consider the following nonlinear fractional differential equation with infinite-point boundary value conditions { D 0 + α u ( t ) + q ( t ) f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0 , u ( i ) ( 1 ) = ∑ j = 1 ∞ α j u ( ξ j ) , where α > 2 , n − 1 < α ≤ n , i ∈ [ 1 , n − 2 ] is a fixed integer, α j ≥ 0 , 0 < ξ 1 < ξ 2 < ⋯ < ξ j − 1 < ξ j < ⋯ < 1 ( j = 1 , 2 , … ) , Δ − ∑ j = 1 ∞ α j ξ j α − 1 > 0 , Δ = ( α − 1 ) ( α − 2 ) ⋯ ( α − i ) . The nonlinear term f permits singularities with respect to both the time and space variables. By introducing height functions of the nonlinear term on some bounded sets and considering integrations of these height functions, several local existence and multiplicity of positive solutions theorems are obtained.