Hyers–Ulam Stability of a Class of Nonlocal Boundary Value Problem Having Triple Solutions

Abstract

This manuscript is concerned with the investigation of three-point boundary value problem (BVP) of nonlinear fractional differential equations (FDEs) as provided by $$\begin{aligned} \begin{aligned} -&\mathbf {D}^\omega w(t)=\varphi (t,w(t));\ t\in (0, 1),\ \omega \in (1, 2],\\ {}&\mathbf {D}^\theta w(0)=0, \ w(1)= \gamma \mathbf {D}^{\theta }w(\eta ),\ \theta \in (0, 1], \end{aligned} \end{aligned}$$ where usual Riemann–Liouville derivative is denoted by $$\mathbf {D}^q$$ with order $$\omega $$ and $$\gamma , \eta \in (0, 1)$$ . Also $$\varphi :[0,1]\times [0,\infty )\rightarrow [0,\infty )$$ is continuous nonlinear function. Thank to classical Leggett–Williams fixed point theorem, appropriate conditions are established for the existence and multiplicity results. Further some useful results about Hyers–Ulam stability for the obtained solutions are also provided. The establish theory is properly justified by providing two examples.