Existence of nonzero nonnegative solutions of Sturm-Liouville boundary value problems and applications
Bibliographic record
Abstract
Sufficient conditions for the boundary value problems (BVPs) of linear Sturm-Liouville (S-L) homogeneous equations subject to the separated boundary conditions (BCs) to have only zero solution are provided in this paper for the first time. Some previous papers and classical books used the assertion that the BVPs have only zero solution as a hypothesis and did not provide any sufficient conditions to ensure that the assertion holds. The sufficient conditions obtained in this paper are a key toward obtaining both the Green's functions to such BVPs and uniqueness of solutions for the linear S-L nonhomogeneous BVPs including the one-dimensional elliptic BVPs. New results on the existence of nonzero nonnegative or strictly positive solutions for the BVPs of nonlinear S-L equations with the separated BCs are obtained by using the fixed point index theory for nowhere normal-outward maps in Banach spaces . The new results allow the nonlinearities in the S-L BVPs to take negative values and have no lower bounds and are applied to deal with the logistic type population models which contain such nonlinearities.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".