Fixed points for weakly inward mappings in Banach spaces

AbstractS. Hu and Y. Sun [S. Hu, Y. Sun, Fixed point index for weakly inward mappings, J. Math. Anal. Appl. 172 (1993) 266–273] defined the fixed point index for weakly inward mappings, investigated its properties and studied the fixed points for such mappings. In this paper, following S. Hu and Y. Sun, we continue to investigate boundary conditions, under which the fixed point index for the completely continuous and weakly inward mapping, denoted by i(A,Ω,P), is equal to 1 or 0. Correspondingly, we can obtain some new fixed point theorems of the completely continuous and weakly inward mappings and existence theorems of solutions for the equations Ax=μx, which extend many famous theorems such as Leray–Schauder's theorem, Rothe's two theorems, Krasnoselskii's theorem, Altman's theorem, Petryshyn's theorem, etc., to the case of weakly inward mappings. In addition, our conclusions and methods are different from the ones in many recent works

Fixed Point Theorems of Quasicontractions on Cone Metric Spaces with Banach Algebras

We introduce the concept of quasicontractions on cone metric spaces with Banach algebras, and by a new method of proof, we will prove the existence and uniqueness of fixed points of such mappings. The main result generalizes the well-known theorem of Ćirić (Ćirić 1974)

Coupled Coincidence Points for Mixed Monotone Random Operators in Partially Ordered Metric Spaces

The aim of this work is to prove some coupled random coincidence theorems for a pair of compatible mixed monotone random operators satisfying weak contractive conditions. These results are some random versions and extensions of results of Karapınar et al. (2012). Our results generalize the results of Shatanawi and Mustafa (2012)

Therapeutic Effect of Vagus Nerve Stimulation on Depressive-Like Behavior, Hyperglycemia and Insulin Receptor Expression in Zucker Fatty Rats

Depression and type 2 diabetes (T2D) are common comorbid diseases and highly prevalent in the clinical setting with an unclarified mechanism. Zucker diabetic fatty (ZDF, fa/fa) rats natively develop T2D with hyperglycemia and hyperinsulinemia. Here we studied whether ZDF rats also innately develop depression, what a correlation is between depression and T2D, whether insulin receptor (IR) expression is involved in, and whether transcutaneous auricular vagus nerve stimulation (taVNS) would be beneficial in amelioration of the comorbidity. Six week old male ZDF and Zucker lean (ZL, fa/+) littermates were randomly divided into naïve (ZDF, n = 6; ZL, n = 7) and taVNS (ZDF-taVNS, n = 8; ZL-taVNS, n = 6) groups. Once daily 30 min-taVNS sessions were administrated under anesthesia for 34 consecutive days in taVNS groups. Blood glucose levels were tested weekly, and plasma glycosylated hemoglobin (HbAlc) level and immobility time in forced swimming test were determined on day 35 in all groups. The expression of insulin receptor (IR) in various tissues was also detected by immunostaining and Western blot. We found that naïve ZDF rats developed hyperglycemia steadily. These ZDF rats showed a strong positive correlation between longer immobility time and higher plasma HbAlC level. Long term taVNS treatment simultaneously prevented the development of depression-like behavior and progression of hyperglycemia in ZDF rats. The expression of IR in various tissues of naïve ZDF rats is lower than in naïve ZL and long-term taVNS treated ZDF rats. Collectively, our results indicate that in ZDF rats, i) depression and T2D develop simultaneously, ii) immobility time and HbAlc concentrations are highly and positively correlated, iii) a low expression of IR may be involved in the comorbidity of depression and T2D, and iv) taVNS is antidiabetic and antidepressive possibly through IR expression upregulation

Transcutaneous Auricular Vagus Nerve Stimulation Triggers Melatonin Secretion and Is Antidepressive in Zucker Diabetic Fatty Rats

Decreased circulating melatonin is implicated in depression. We recently found that Zucker diabetic fatty rats (ZDF, fa/fa) develop depression-like behaviors and that transcutaneous auricular vagus nerve stimulation (taVNS) is antidepressive in ZDF rats. Here we studied whether the ZDF rats could be used as a depression rodent model and whether the antidepressive effect of taVNS is mediated through modulation of melatonin secretion. Adult male ZDF and Zucker lean (ZL, fa/+) littermates were used. 30 min-taVNS procedures (2/15 Hz, 2 mA) were administered once daily under anesthesia for 34 consecutive days in pineal intact ZDF (n = 8) and ZL (n = 6) rats, as well as in pinealectomized ZDF rats (n = 8). Forced swimming test (FST) was used to determine depression-like behavior and ELISA to detect plasma melatonin concentration on day 35. We found that naïve ZDF rats had a longer immobility time in FST and that long-term (34 days) taVNS treatment ameliorated the depression-like behavior. In both pineal intact and pinealectomized ZDF rats, taVNS induced acute melatonin secretion, both during and after the taVNS session. A low melatonin level is related to the poor FST performance in ZDF rats (R = −0.544) in contrast to ZL rats (R = 0.247). In conclusion, our results show that ZDF rats are ideal candidates of innate depression and that taVNS is antidepressive through triggering melatonin secretion and increasing its production

Effect of transcutaneous auricular vagus nerve stimulation on impaired glucose tolerance: a pilot randomized study

Background: Impaired glucose tolerance (IGT) is a pre-diabetic state of hyperglycemia that is associated with insulin resistance, increased risk of type II diabetes, and cardiovascular pathology. Recently, investigators hypothesized that decreased vagus nerve activity may be the underlying mechanism of metabolic syndrome including obesity, elevated glucose levels, and high blood pressure. Methods: In this pilot randomized clinical trial, we compared the efficacy of transcutaneous auricular vagus nerve stimulation (taVNS) and sham taVNS on patients with IGT. 72 participants with IGT were single-blinded and were randomly allocated by computer-generated envelope to either taVNS or sham taVNS treatment groups. In addition, 30 IGT adults were recruited as a control population and not assigned treatment so as to monitor the natural fluctuation of glucose tolerance in IGT patients. All treatments were self-administered by the patients at home after training at the hospital. Patients were instructed to fill in a patient diary booklet each day to describe any side effects after each treatment. The treatment period was 12 weeks in duration. Baseline comparison between treatment and control group showed no difference in weight, BMI, or measures of systolic blood pressure, diastolic blood pressure, fasting plasma glucose (FPG), 2-hour plasma glucose (2hPG), or glycosylated hemoglobin (HbAlc). Results: 100 participants completed the study and were included in data analysis. Two female patients (one in the taVNS group, one in the sham taVNS group) dropped out of the study due to stimulation-evoked dizziness. The symptoms were relieved after stopping treatment. Compared with sham taVNS, taVNS significantly reduced the two-hour glucose tolerance (F(2) = 5.79, p = 0.004). In addition, we found that taVNS significantly decreased (F(1) = 4.21, p = 0.044) systolic blood pressure over time compared with sham taVNS. Compared with the no-treatment control group, patients receiving taVNS significantly differed in measures of FPG (F(2) = 10.62, p < 0.001), 2hPG F(2) = 25.18, p < 0.001) and HbAlc (F(1) = 12.79, p = 0.001) over the course of the 12 week treatment period. Conclusions: Our study suggests that taVNS is a promising, simple, and cost-effective treatment for IGT/ pre-diabetes with only slight risk of mild side-effects