Positive solutions of fractional differential equation boundary value problems at resonance

In this article, we study a class of fractional differential equations with resonant boundary value conditions. Some sufficient conditions for the existence of positive solutions are considered by means of the spectral theory of linear operator and the fixed point index theory.     

Positive solutions of second-order differential systems with nonhomogeneous boundary conditions

This paper is to investigate the positive solutions of the systems of second-order ordinary differential equations with nonhomogeneous multi-point boundary conditions. By the lower and upper solutions method, Schauder fixed point theorem and fixed point index theory, under certain conditions, it is proved that there exist appropriate regions of parameters in which the problem has at least two, at least one or no positive solution.     

Nontrivial solutions of nonlocal boundary value problems for nonlinear higher-order singular fractional differential equations

This paper deals with the existence and multiplicity of nontrivial solutions of nonlocal boundary value problems for nonlinear higher-order singular fractional differential equations with sign-changing nonlinear term. The main tool used in the proof is topological degree theory. Some examples explain that our results cannot be obtained by the method of cone theory.     

含积分边界条件的分数阶微分方程边值问题的正解的存在性

研究了含积分边界条件的分数阶微分方程的边值问题,首先给出格林函数及性质,其次将问题转化为一个等价的积分方程,最后应用Krasnoselkii及Leggett-Williams不动点定理得到了一个及多个正解的存在性,推广了以往的结果．     

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 nonnegative solutions of systems with coupled nonlinear boundary conditions

Using the theory of fixed point index, we discuss the existence and multiplicity of nonnegative solutions of a wide class of boundary value problems with coupled nonlinear boundary conditions. Our approach is fairly general and covers a variety of situations. We illustrate in an example that all the constants that occur in our theory can be computed. Copyright © 2013 John Wiley & Sons, Ltd.     

具p-Laplace算子的分数阶微分方程边值问题的正解

考虑具有p-Laplace算子的分数阶泛函微分方程边值问题,利用锥上的不动点定理,得到了其正解及多个正解存在的充分条件,所得结果推广了已有的结论,并举例说明了结论的适用性.     

Existence of positive solutions to a coupled system of fractional differential equations

We consider the following system of fractional differential equations where is the Riemann‐Liouville fractional derivative of order α,f,g : [0,1] × [0, ∞ ) × [0, ∞ ) → [0, ∞ ). Sufficient conditions are provided for the existence of positive solutions to the considered problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Positive solutions for a coupled system of nonlinear fractional differential equations

Using the Krasnoselskii–Zabreiko fixed point theorem, we establish two existence theorems for positive solutions of a coupled system of nonlinear fractional differential equations. Power functions and nonnegative matrices are used to characterize coupling behavior of our nonlinearities, so nonlinearities may grow differently; in fact, one may grow superlinearly, and the other may grow sublinearly. Copyright © 2014 John Wiley & Sons, Ltd.     

Unique existence results and numerical solutions for fourth-order impulsive differential equations with nonlinear boundary conditions

The work is concerned with three kinds of fourth-order impulsive differential equations with nonlinear boundary conditions. We at first focused on studying the existence and uniqueness of positive solutions for these kinds of problems. By converting the problem to an equivalent integral equation, then applying the new class of fixed point theorems for the sum operator on cone, we obtain the sufficient conditions which not only guarantee the existence of a unique positive solution, but also be applied to construct two iterative sequences for approximating it. Further, we present the numerical methods for solving the fourth-order differential equations. At last, some examples are given with numerical verifications to illustrate the main results.     

Positive solutions of singular Caputo fractional differential equations with integral boundary conditions

In this paper, we investigate the existence of positive solutions of singular super-linear (or sub-linear) integral boundary value problems for fractional differential equation involving Caputo fractional derivative. Necessary and sufficient conditions for the existence of C³[0, 1] positive solutions are given by means of the fixed point theorems on cones. Our nonlinearity f(t, x) may be singular at t = 0 and/or t = 1.     

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 where 0≤λ < 1,^CD^α 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: where , 0≤λ < k + 1; Problem 2: where 0≤λ≤α and D^α 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.     

Existence of positive solutions for fractional boundary value problems

In this paper, by introducing a new operator, improving and generating a p-Laplace operator for some $p > 1$, we discuss the existence and multiplicity of positive solutions to the four point boundary value problems of nonlinear fractional differential equations. Our results extend some recent works in the literature.     

Nonzero positive solutions of fractional Laplacian systems with functional terms

We study the existence of nonzero positive solutions of a class of systems of differential equations driven by fractional powers of the Laplacian. Our approach is based on the notion of fixed point index, and allows us to deal with nonlocal functional weights and functional boundary conditions. We present two examples to shed light on the type of functionals and growth conditions that can be considered with our approach.     

Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations

In this paper, we study the three-point boundary value problems for systems of nonlinear second order ordinary differential equations of the form

Under some conditions, we show the existence and multiplicity of positive solutions of the above problem by applying the fixed point index theory in cones.     

Existence and nonexistence of positive solutions of a fractional thermostat model with a parameter

In this paper, we investigate positive solutions for boundary value problem of a fractional thermostat model with a parameter. Under different conditions of the function f, existence and nonexistence results for positive solutions are derived in terms of different values of λ. The results are illustrated with an example. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence of solutions for fractional differential equation three-point boundary value problems

In this paper, by using some fixed point theorems, the existence of unique solution and the existence of at least one solution for a fractional differential equation three-point boundary value problems are established. Finally, some illustrative examples are presented to demonstrate the validity of the main results.     

Existence of positive solutions to a boundary value problem for a delayed singular high order fractional differential equation with sign-changing nonlinearity

In this paper, we discuss the existence of positive solutions to the boundary value problem for a high order fractional differential equation with delay and singularities including changing sign nonlinearity. By using the properties of the Green function, Guo-krasnosel"skii fixed point theorem, Leray-Schauder"s nonlinear alternative theorem, some existence results of positive solutions are obtained, respectively.