Existence and nonexistence of positive solutions via natural Riemann-Liouville convolution kernels for third-order lifted fractional boundary value problems
Britney Hopkins , William Knuth , Jeffrey W. Lyons
Electronic Research Archive ›› 2026, Vol. 34 ›› Issue (9) : 6153 -6176.
We study existence and nonexistence of positive solutions for nonlinear fractional boundary value problems of order $\alpha+3$, where $\alpha \in(m-1, m]$ with $m \geq 3$, governed by the Riemann-Liouville (RL) operator $D_{0+}^{\alpha+3}$. The Green’s function is constructed by pairing the RL semigroup property $D^{3}\left(D_{0+}^{\alpha} u\right)=D_{0+}^{\alpha+3} u$ with a convolution of the RL core $G_{\alpha}$ and a third-order integer bridge, which takes one of two boundary-adapted forms: left-clamped $G_{LC}$ or right-clamped $G_{RC}$. A key structural finding is that left and right convolutions produce fundamentally different boundary condition types, providing selectivity in the Green’s function construction. An explicit closed-form formula for $\mathcal{G}_{L C}=G_{\alpha} * G_{L C}$ is derived via Beta function reduction; $\mathcal{G}_{R C}=G_{\alpha} * G_{R C}$ is evaluated numerically as it consists of incomplete Beta functions. Positivity, monotonicity, and cone estimates for both kernels are established to utilize the Guo-Krasnosel’skii fixed-point theorem to identify parameter intervals for existence and nonexistence. Hybrid pairings and their structural implications are discussed; a worked example illustrates the results.
fractional boundary value problem / RL derivative / Caputo derivative / Green’s function / convolution / natural kernel / existence / nonexistence / fixed-point theorem
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