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(D_{0+}^\alpha u) = 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}_{LC} = G_\alpha* G_{LC} $ is derived via Beta function reduction; $ \mathcal{G}_{RC} = G_\alpha* G_{RC} $ 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.
Citation: Britney Hopkins, William Knuth, Jeffrey W. Lyons. Existence and nonexistence of positive solutions via natural Riemann-Liouville convolution kernels for third-order lifted fractional boundary value problems[J]. Electronic Research Archive, 2026, 34(9): 6153-6176. doi: 10.3934/era.2026271
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(D_{0+}^\alpha u) = 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}_{LC} = G_\alpha* G_{LC} $ is derived via Beta function reduction; $ \mathcal{G}_{RC} = G_\alpha* G_{RC} $ 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.
| [1] | P. W. Eloe, J. T. Neugebauer, Convolutions and Green's functions for two families of boundary value problems for fractional differential equations, Electron. J. Differ. Equations, 2016 (2016), 1–13. |
| [2] |
J. W. Lyons, J. T. Neugebauer, Two point fractional boundary value problems with a fractional boundary condition, Fract. Calc. Appl. Anal., 21 (2018), 442–461. https://doi.org/10.1515/fca-2018-0025 doi: 10.1515/fca-2018-0025
|
| [3] |
J. T. Neugebauer, A. G. Wingo, Positive solutions for a fractional boundary value problem with Lidstone-like boundary conditions, Kragujevac J. Math., 48 (2024), 309–322. https://doi.org/10.46793/kgjmat2402.309n doi: 10.46793/kgjmat2402.309n
|
| [4] |
J. W. Lyons, J. T. Neugebauer, A. G. Wingo, Existence and nonexistence of positive solutions for fractional boundary value problems with Lidstone-inspired fractional conditions, Mathematics, 13 (2025), 1336. https://doi.org/10.3390/math13081336 doi: 10.3390/math13081336
|
| [5] |
J. W. Lyons, Solutions of higher-order fractional boundary value problems with fractional left-focal like conditions, Math. Syst. Sci., 3 (2025), 3577. https://doi.org/10.54517/mss3577 doi: 10.54517/mss3577
|
| [6] |
J. W. Lyons, Positive solutions for fractional boundary value problems with fractional conditions of right-focal type utilizing lower-order problems, Open J. Math. Anal., 9 (2025), 26–40. https://doi.org/10.30538/psrp-oma2025.0163 doi: 10.30538/psrp-oma2025.0163
|
| [7] | J. W. Lyons, J. T. Neugebauer, A. Wingo, An application of the Leray-Schauder nonlinear alternative to a Riemann-Liouville fractional boundary value problem with Lidstone-inspired fractional conditions, Fract. Differ. Calc., (in press). |
| [8] |
J. R. Graef, C. Qian, B. Yang, A three point boundary value problem for nonlinear fourth order differential equations, J. Math. Anal. Appl., 287 (2003), 217–233. https://doi.org/10.1016/S0022-247X(03)00545-6 doi: 10.1016/S0022-247X(03)00545-6
|
| [9] |
J. R. Graef, B. Yang, Positive solutions to a multi-point higher order boundary-value problem, J. Math. Anal. Appl., 316 (2006), 409–421. https://doi.org/10.1016/j.jmaa.2005.04.038 doi: 10.1016/j.jmaa.2005.04.038
|
| [10] | J. R. Graef, J. Henderson, B. Yang, Positive solutions of a nonlinear higher order boundary-value problem, Electron. J. Differ. Equations, 2007 (2007), 1–10. |
| [11] | I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. |
| [12] | K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, 2010. https://doi.org/10.1007/978-3-642-14574-2 |
| [13] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006. |
| [14] | K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. |
| [15] | D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988. |
| [16] | M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York, 1964. |
| [17] | P. Eloe, J. W. Lyons, J. T. Neugebauer, An ordering on Green's functions for a family of two-point boundary value problems for fractional differential equations, Commun. Appl. Anal., 19 (2015), 453–462. |
| [18] |
R. P. Agarwal, D. O'Regan, S. Staněk, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. Math. Anal. Appl., 371 (2010), 57–68. https://doi.org/10.1016/j.jmaa.2010.04.034 doi: 10.1016/j.jmaa.2010.04.034
|
Figures(7) / Tables(1)
Britney Hopkins, William Knuth, Jeffrey W. Lyons. Existence and nonexistence of positive solutions via natural Riemann-Liouville convolution kernels for third-order lifted fractional boundary value problems[J]. Electronic Research Archive, 2026, 34(9): 6153-6176. doi: 10.3934/era.2026271
DownLoad: