Complete Existence Theory for Singular Second-Order Boundary Value Problems: A Fixed-Point Approach with Precision Analysis
DOI:
https://doi.org/10.61952/jlabw.v1i3.196Keywords:
Existence Theory, Boundary Value problems, Fixed-Point Theorem, Cubic-Root Singularity, Green’s Function AnalysisAbstract
This work establishes the first complete existence theory for a class of singular second-order boundary value problems characterized by competing cubic-root singularities. By unifying an extended Leggett-Williams fixed point theorem, weighted cone theory, and precision Green's function analysis, we not only overcome longstanding analytical challenges but also validate our results with error-controlled numerical techniques. The framework developed herein is applicable to various interdisciplinary models, ranging from non-Newtonian fluid dynamics to plasma physics.
References
Agarwal, R., O'Regan, D., & Staněk, S. (2023). Singular Dirichlet problems for mixed-type differential equations. Journal of Differential Equations, 367, 1–31.
Bai, C., & Zhang, D. (2023). Multiple solutions for fractional boundary value problems with singular nonlinearities. Applied Mathematics Letters, 142, 108621.
Benslimane, S., Georgiev, S. G., & Mebarki, K. (2023). Expansion-compression fixed point theorem of Leggett-Williams type for the sum of two operators and applications for some classes of BVPs. Studia Universitatis Babeș-Bolyai Mathematica, 68 (2), 345–360.
Cabada, A., & Infante, G. (2022). Fixed point theorems for sum operators with applications to nonlinear systems. Nonlinear Analysis: Real World Applications, 68, 103701.
Chen, Y. (2021). Recent advances in singular BVPs. Journal of Nonlinear Science, 33, 45–67.
Deimling, K. (1985). Nonlinear functional analysis. Springer.
Deng, J., & Li, Y. (2023). A generalized Leggett–Williams theorem and applications to fractional BVPs. Communications in Nonlinear Science and Numerical Simulation, 119, 107105.
Georgiev, S. G., & Zennir, K. (2023). New fixed point theorems in cones and applications to reaction-diffusion models. Mathematics, 11 (5), 1216.
Graef, J. R., & Kong, L. (2022). Green’s function techniques for singular BVPs with nonlocal conditions. Boundary Value Problems, 2022 (1), 1–25.
Guo, D., & Lakshmikantham, V. (2004). Nonlinear problems in abstract cones. Academic Press.
Henderson, J., & Luca, R. (2023). Positive solutions for singular second-order differential equations with integral boundary conditions. Electronic Journal of Qualitative Theory of Differential Equations, 2023 (25), 1–18.
Jleli, M., & Samet, B. (2022). On a class of singular fractional equations with p-Laplacian operator. Journal of Computational and Applied Mathematics, 413, 114384.
Khan, H., & Tunç, C. (2023). Existence results for singular fractional q-difference equations via topological methods. AIMS Mathematics, 8 (4), 9365–9380.
Li, Y., & Wang, Y. (2023). Precision analysis for nonlinear BVPs with double singularities using cone compression theory. Journal of Mathematical Analysis and Applications, 525 (2), 127154.
Mavridis, K. G. (2021). A Leggett-Williams fixed point theorem type for sums of operators and applications in PDEs. Differential Equations & Applications, 13 (3), 321–344.
Mebarki, K., & Benslimane, S. (2023). Hybrid fixed point theorems for multi-term singular BVPs in weighted spaces. Mediterranean Journal of Mathematics, 20 (4), 207.
Ntouyas, S. K., & Samei, M. E. (2022). Singular fractional integro-differential equations with Hilfer derivatives. Fractal and Fractional, 6 (12), 713.
Sousa, J. V. C., & Kucche, K. D. (2023). On the Hilfer fractional differential equations with non-instantaneous impulses. Quaestiones Mathematicae, 46 (5), 861–880.
Wang, G., & Pei, K. (2023). Positive solutions for singular third-order boundary value problems via Leggett–Williams theorem. Journal of Applied Analysis & Computation, 13 (3), 1127–1141.
Altwaty, A., & Eloe, P. W. (2013). Concavity of solutions of a 2n-th order problem with symmetry. Opuscula Mathematica, 33 (4), 603–615.
