Positive solutions of a boundary value problem for a nonlinear fractional differential equation

In this paper we give sufficient conditions for the existence of at least one and at least three positive solutions to the nonlinear fractional boundary value problem \begin{eqnarray*} &&D^{\alpha}u + a(t) f(u) = 0, \quad 0<t<1, 1<\alpha\leq2,\\ &&u(0) = 0 ,u'(1)= 0, \end{eqnarray*} where $D^{\alpha}$ is the Riemann-Liouville differential operator of order $\alpha$, $f: [0,\infty)\rightarrow [0,\infty)$ is a given continuous function and $a$ is a positive and continuous function on $[0,1]$

Positive solutions for nonlinear semipositone nth-order boundary value problems

In this paper, we investigate the existence of positive solutions for a class of nonlinear semipositone $n$th-order boundary value problems. Our approach relies on the Krasnosel'skii fixed point theorem. The result of this paper complement and extend previously known result

Positive solutions of 2mth-order boundary value problems

AbstractWe study the existence of positive solutions of the differential equation (−1)my(2m) (t) = f(t, y(t), y″(t),…, y(2(m−1)) (t)) with the boundary condition y(2i)(0) = 0 = y(2i)(1), 0 ≤ i ≤ m − 1, and y(2i)(0) = 0 = y(2i+1)(1), 0 ≤ i ≤ m − 1. We show the existence of at least one positive solution if f is either superlinear or sublinear by an application of a fixed-point theorem in a cone

Positive solutions of higher order fractional integral boundary value problem with a parameter

In this paper, we study a higher-order fractional differential equation with integral boundary conditions and a parameter. Under different conditions of nonlinearity, existence and nonexistence results for positive solutions are derived in terms of different intervals of parameter. Our approach relies on the Guo–Krasnoselskii fixed point theorem on cones

Unique Solution of a Coupled Fractional Differential System Involving Integral Boundary Conditions from Economic Model

We study the existence and uniqueness of the positive solution for the fractional differential system involving the Riemann-Stieltjes integral boundary conditions , , , , , and , where , , and and are the standard Riemann-Liouville derivatives, and are functions of bounded variation, and and denote the Riemann-Stieltjes integral. Our results are based on a generalized fixed point theorem for weakly contractive mappings in partially ordered sets

Multiple positive solutions of singular fractional differential system involving Stieltjes integral conditions

In this paper, the existence and multiplicity of positive solutions to singular fractional differential system is investigated. Sufficient conditions which guarantee the existence of positive solutions are obtained, by using a well known fixed point theorem. An example is added to illustrate the results

Positive Solutions for (n - 1,1) -Type Singular Fractional Differential System with Coupled Integral Boundary Conditions

We study the positive solutions of the (n - 1,1)-type fractional differential system with coupled integral boundary conditions. The conditions for the existence of positive solutions to the system are established. In addition, we derive explicit formulae for the estimation of the positive solutions and obtain the unique positive solution when certain additional conditions hold. An example is then given to demonstrate the validity of our main results

Study of fractional semipositone problems on RN\mathbb{R}^N

Let $s \in (0,1)$ and $N \geq 2$. In this paper, we consider the following class of nonlocal semipositone problems: \begin{align*} (-\Delta)^s u= g(x)f_a(u) \text { in } \mathbb{R}^N, \; u > 0 \text{ in } \mathbb{R}^N, \end{align*} where the weight $g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)$ is positive, $a>0$ is a parameter, and $f_a \in C(\mathbb{R})$ is negative on $\mathbb{R}^{-}$. For $f_a$ having subcritical growth and weaker Ambrosetti-Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution $u_a$, provided `$a$' is near the origin. To obtain the positivity of $u_a$, we establish a Brezis-Kato type uniform estimate of $(u_a)$ in $L^{\infty}(\mathbb{R}^N)$.Comment: 17 page