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In this paper, we discuss the fractional boundary value problem containing left and right fractional derivative operators and p-Laplacian. By using critical point theory we obtain some results on the existence of weak solutions of such a fractional boundary value problem.MSC: 26A33; 34B15; 58E05

The objective of this study was to investigate the treatment and prognosis of patients with spinal tuberculosis in Guizhou province. A total of 863 patients with spinal tuberculosis admitted to our hospital from 2006 to 2017 were included in this study. All patients underwent standardized quadruple antituberculosis treatment. Eighty patients were lost to follow-up due to a change of their contact information or noncompliance. A total of 783 patients completed the follow-up. The average follow-up period was 20.33 ± 8.77 months (range: 6 to 38 months). Among these patients, 145 patients underwent conservative treatment, while 638 patients underwent surgical treatment. All patients in the surgery group were treated with lesion removal, bone graft fusion, and internal fixation. Preoperative and postoperative standard quadruple antituberculosis treatment was administered. The clinical efficacy was evaluated according to erythrocyte sedimentation rate (ESR), c-reactive protein (CRP), visual analogue scale (VAS), Cobb angle correction, neurological functional recovery, and interbody fusion with bone graft and tuberculosis outcome. A total of 608 patients achieved clinical cure. The symptoms, physical signs, blood tests and imaging findings were improved in 143 patients. Twenty patients showed refractory clinical symptoms, and 12 patients had local tuberculosis recurrence. Conservative and surgical treatments are the mainstream treatments for spinal tuberculosis. According to the patients' individual conditions, individualized treatments should be used to achieve good efficacy. Standardized antituberculosis treatment should be applied over the course of spinal tuberculosis.

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Background Improper occipitocervical alignment after occipitocervical fusion (OCF) may lead to devastating complications, such as dysphagia and/or dyspnea. The occipital to C2 angle (O-C2a), occipital and external acoustic meatus to axis angle (O-EAa) have been used to evaluate occipitospinal alignment. However, it may be difficult to identify the inferior endplate of the C2 vertebra in patients with C2–3 Klippel-Feil syndrome (KFS). The purpose of this study aimed to compare four different parameters for predicting dysphagia after OCF in patients with C2–3 KFS. Methods There were 40 patients with C2–3 KFS undergoing OCF between 2010 and 2019. Radiographs of these patients were collected to measure the occipital to C3 angle (O-C3a), O-C2a, occipito-odontoid angle (O-Da), occipital to axial angle (Oc-Axa), and narrowest oropharyngeal airway space (nPAS). The presence of dysphagia was defined as the patient complaining of difficulty or excess endeavor to swallow. Patients were divided into two groups according to whether they had postoperative dysphagia. We evaluated the relationship between each of the angle parameters and nPAS and analyzed their influence to the postoperative dysphagia. Results The incidence of dysphagia after OCF was 25% in patients with C2–3 KFS. The Oc-Axa, and nPAS were smaller in the dysphagia group compared to non-dysphagia group at the final follow-up (p < 0.05). Receiver-operating characteristic (ROC) curves showed that dO-C3a had the highest accuracy as a predictor of the dysphagia with an area under the curve (AUC) of 0.868. The differences in O-C3a, O-C2a, O-Da, and Oc-Axa were all linearly correlated with nPAS scores preoperatively and at the final follow-up within C2–3 KFS patients, while there was a higher R2 value between the dO-C3a and dnPAS. Multiple linear regression analysis showed that the difference of O-C3a was the only significant predictor for dnPAS (β = 0.670, p < 0.001). Conclusions The change of O-C3a (dO-C3a) is the most reliable indicator for evaluating occipitocervical alignment and predicting postoperative dysphagia in C2–3 KFS patients. Moreover, dO-C3a should be more than − 2° during OCF to reduce the occurrence of postoperative dysphagia.

In this article, we consider the new results for the Kirchhoff-type p-Laplacian Dirichlet problem containing the Riemann-Liouville fractional derivative operators. By using the mountain pass theorem and the genus properties in the critical point theory, we get some new results on the existence and multiplicity of nontrivial weak solutions for such Dirichlet problem.

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