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We examine the existence and multiplicity of positive solutions for a class of nonlinear semipositone fractional differential equations involving integral boundary conditions. The results are obtained in terms of different intervals of the parameters by means of the Leray-Schauder and Guo-Krasnoselskii fixed point theorems. Examples are included to verify our main results.

In this paper, under suitable conditions we employ the nonlinear alternative of Leray-Schauder type and the Guo-Krasnosel?skii fixed point theorem to show the existence of positive solutions for a system of nonlinear singular Riemann-Liouville fractional differential equations with sign-changing nonlinearities, subject to integral boundary conditions. Some examples are given to illustrate our main results.

In this work, we consider the following nonlinear fractional differential equation $$\begin{aligned} {\left\{ \begin{array}{ll} -D^{\nu } u(t)=\lambda f(t,u(t))+e(t) \ \ \ in \ \ (0,1),\\ u^{(j)}(0)=0, \ \ 0\le j \le n-2, \ \ [D^{\alpha } u(t)]_{t=1}=0,\\ \end{array}\right. } \end{aligned}$$where $$\lambda >0$$ is a parameter, $$n \ge 3$$, $$n-1< \nu < n$$, $$1 \le \alpha \le n-2$$ and $$D^{\nu }$$ stands for the standard Reimann–Liouville derivative, $$f: [0,1]\times [0,+\infty ) \longrightarrow {\mathbb {R}}$$ is sign-changing continuous function (that is, we have a so-called equation of semipositone problems). The perturbed term $$e: (0,1) \rightarrow {\mathbb {R}}$$ is measurable function and verifies some appropriate conditions. We derive some intervals of $$\lambda $$ such that the problem has positive solutions. Our study relies on Guo-Krasnoselskii fixed point theorem.

This paper provides sufficient conditions to guarantee the existence and multiplicity of positive solutions for a singular semipositone fractional equation, subject to integral boundary value condition and parametric dependence $\mu $. Our approach relies on Krasnoselskii's fixed point theorem. Some examples are presented to illustrate our results.

In this paper, we deal with the following nonlinear fractional differential problem in the half-line $${\mathbb{R}^{+}=(0,+ \infty)}$$ $$\left\{\begin{array}{l}D^{\alpha }u(x)+f(x,u(x),D^{p}u(x))=0,\quad x \in \mathbb{R}^{+},\\ u(0)=u^{\prime } \left( 0\right) = \cdots =u^{\left( m-2\right) }(0)=0,\end{array}\right.$$ where $${m\in \mathbb{N}, m \geq 2, m-1 < \alpha \leq m, 0 < p \leq \alpha -1}$$ , the differential operator is taken in the Riemann–Liouville sense and f is a Borel measurable function in $${\mathbb{R}^{+} \times \mathbb{R}^{+} \times \mathbb{R} ^{+}}$$ satisfying certain conditions. More precisely, we show the existence of multiple unbounded positive solutions, by means of Schauder fixed point theorem. Some examples illustrating our main result are also given.

In this work, we deal with the existence of extremal quasisolutions for the following finite system of nonlinear fractional differential equation \(^{C}D^{q}u\left (t\right ) + f\left (t,u\left (t\right )\right ) = 0\;\text{ in }\;\left (0,1\right ),u\left (0\right ) -\alpha u^{{\prime}}\left (0\right ) =\lambda,u\left (1\right ) +\beta u^{{\prime}}\left (1\right ) =\mu,\) where \(1