INVESTIGATING A COUPLED SYSTEM WITH RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE BY MODIFIED MONOTONE ITERATIVE TECHNIQUE
Bibliographic record
Abstract
In recent time, the area of arbitrary order differential equations (AODEs)has been considered very well. Different aspects have been investigated for the saidarea. One of the important and most warm area is devoted to study multiplicityresults along with existence and uniqueness of solutions for the said equations. In thisregard various techniques have been utilized to investigate the said area. Monotoneiterative technique (MIT) coupled with the method of extremal solutions has beenused recently to investigate multiplicity of solutions to some AODEs. In this researchwork, we deal a coupled system of nonlinear AODEs under boundary conditions (BCs)involving Riemann-Liouville fractional derivative by using fixed point theorems dueto Perov’s and Schuader’s to study existence and uniqueness results. Using Perove’sfixed point theorem ensures uniqueness of solution to systems of equations, whileexistence of at least one solution is achieved by Schauder’s fixed point theorem.Then we come across the multiplicity of solutions and establish some criteria for theiterative solutions via using updated type MIT together with the method of upper andlower solutions for the considered system of AODEs. Corresponding to multiplicityresults of solutions, we first establish two sequences of extremal solutions. One of thesequence is monotonically decreasing and converging to lower solution. On the otherhand, the other sequence is monotonically increasing and converging to the uppersolution. In last we give suitable examples to illustrate the main results.
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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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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".