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We investigate the resonance problem and prove the existence of multiple periodic solutions to a second order parameter-dependent equation $ x''+f(t, x) = sp(t) $. We weaken the usual requirement on the sublinearity of perturbations when $ |x| $ becomes large; and develop a more general method to investigate the rotational characterizations of the Landesman-Lazer conditions. Furthermore, $ f $ does not satisfy the common sign condition, and even the global existence of the solution is not guaranteed.

BackgroundSleep disorders are one of the most common non-motor symptoms in PD. It can cause a notable decrease in quality of life and functioning in PD patients, as well as place a huge burden on both patients and caregivers. Currently, there are numerous non-pharmacological interventions available to improve sleep quality in PD, with disagreement as to which intervention is most effective. This network meta-analysis was performed to compare and rank non-pharmacological interventions to explore their efficacy in improving sleep quality in PD and to select the best interventions, with a view to providing references and bases for the development of clinical treatments and care programs.MethodsThe PubMed, Embase, Cochrane Central Register of Controlled Trials (CENTRAL), Web of Science, China National Knowledge Infrastructure (CNKI), and Wanfang databases were searched from inception to December 6, 2023. Two authors independently screened all studies, extracted the data, and evaluated risk of bias of included studies. STATA software version 17.0 was used to conduct the network meta-analysis.ResultsOur network meta-analysis included 29 studies involving 1,477 participants and 16 non-pharmacological interventions. Although most nonpharmacological interventions showed non-significant effects, the surface under the cumulative ranking curve (SUCRA) values indicated that the best non-pharmacological intervention for sleep disorders was massage therapy (97.3%), followed by music therapy (94.2%), and Treadmill training (85.7%).ConclusionMassage therapy can be considered as an effective therapy for improving sleep quality in patients with PD. Due to limited quantity and quality of the included studies, more high quality studies are required to verify the conclusions of this network meta-analysis.Systematic review registrationidentifier CRD42023429339, PROSPERO (york.ac.uk).

Background: Multi-organ metastases represent a substantial life-threatening risk for breast cancer (BC) patients. Nonetheless, the current dearth of assessment tools for patients with multi-organ metastatic BC adversely impacts their evaluation. Methods: We conducted a retrospective analysis of BC patients with multi-organ metastases using data from the SEER database from 2010 to 2019. The patients were randomly allocated into a training cohort and a validation cohort in a 7:3 ratio. Univariate COX regression analysis, the LASSO, and multivariate Cox regression analyses were performed to identify independent prognostic factors in the training set. Based on these factors, a nomogram was constructed to estimate overall survival (OS) probability for BC patients with multi-organ metastases. The performance of the nomogram was evaluated using C-indexes, ROC curves, calibration curves, decision curve analysis (DCA) curves, and the risk classification system for validation. Results: A total of 3626 BC patients with multi-organ metastases were included in the study, with 2538 patients in the training cohort and 1088 patients in the validation cohort. Age, grade, metastasis location, surgery, chemotherapy, and subtype were identified as significant independent prognostic factors for OS in BC patients with multi-organ metastases. A nomogram for predicting 1-year, 3-year, and 5-year OS was constructed. The evaluation metrics, including C-indexes, ROC curves, calibration curves, and DCA curves, demonstrated the excellent predictive performance of the nomogram. Additionally, the risk grouping system effectively stratified BC patients with multi-organ metastases into distinct prognostic categories. Conclusion: The developed nomogram showed high accuracy in predicting the survival probability of BC patients with multi-organ metastases, providing valuable information for patient counseling and treatment decision making. [ABSTRACT FROM AUTHOR]