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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 paper, we are concerned with a class of system of nonlinear singular fractional differential equations with integral boundary conditions. More precisely, we establish sufficient conditions for existence, multiplicity and nonexistence of positive solutions. The results are derived in terms of different values of the parameters. Our approach relies on the Krasnoselskii?s fixed point theorem. 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, the authors aim at proving two existence results of fractional differential boundary value problems of the form $$\left( {{P_{a,b}}} \right)\left\{ {_{u\left( 0 \right) = u\left( 1 \right) = 0,{D^{\alpha - 3}}u\left( 0 \right) = a,u'\left( 1 \right) = - b}^{{D^\alpha }u\left( x \right) + f\left( {x,u\left( x \right)} \right) = 0,x \in \left( {0,1} \right)}} \right.$$ , where 3 < α ≤ 4, D α is the standard Riemann-Liouville fractional derivative and a, b are nonnegative constants. First the authors suppose that f(x, t) = −p(x)t σ , with σ ∈ (−1, 1) and p being a nonnegative continuous function that may be singular at x = 0 or x = 1 and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Schauder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem (P 0,0). Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a + b > 0 and f(x, t) = t φ(x, t), with φ(x, t) being a nonnegative continuous function in (0, 1)×[0,∞) that is required to satisfy some suitable integrability condition. Using estimates on the Green’s function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem (P a,b ), which behaves like the unique solution of the homogeneous problem corresponding to (P a,b ). Some examples are given to illustrate the existence 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

We present existence results for the polyharmonic nonlinear elliptic boundary-value problem $$displaylines{ (-Delta )^m u=f(cdot,u) quad hbox{in }B cr (frac{partial }{partial u })^j u=0quad hbox{on }partial B, quad 0leq jleq m-1. }$$ (in the sense of distributions), where $B$ is the unit ball in $mathbb{R}^n$ and $ngeq 2$. The nonlinearity $f(x,t)$ satisfies appropriate conditions related to a Kato class of functions $K_{m,n}$. Our approach is based on estimates for the polyharmonic Green function with zero Dirichlet boundary conditions and on the Schauder fixed point theorem.