Study on the effects of a perturbation parameter on the existence of positive solutions for Hadamard type fractional differential equation

Abstract

We consider a class of Hadamard type fractional differential equations with disturbance parameter and present the number $\varrho^*$ that sharply demarcates the existence and nonexistence of positive solutions for the corresponding boundary value problem across all non-negative parameter values by using fixed point theory. Specifically, we establish that no positive solutions exist if the parameter $\varrho> \varrho^*$, while at least one positive solution emerges when $\varrho$ lies in $[0,\varrho^*]$. Notably, within the subcritical regime $(0,\varrho^*)$, the problem admits at least two distinct positive solutions. Therefore, in this paper, the influence of parameter on solution is clearly demonstrated, with no ambiguities or indistinct regions. Some existing results failed to prove the existence of this type of point. The present study not only demonstrates the existence of one such point but also determines its imprecise range, thus complementing and enhancing previous findings. Finally, an example is presented and analyzed to illustrate the practical implications of our theoretical results.