On a probabilistic version of Meir-Keeler type fixed point theorem for a family of discontinuous operators

[EN] A Meir-Keeler type fixed point theorem for a family of mappings is proved in Mengerprobabilistic metric space (Menger PM-space). We establish that completeness of the space isequivalent to fixed point property for a larger class of mappings that includes continuous as wellas discontinuous mappings. In addition to it, a probabilistic fixed point theorem satisfying (ϵ - δ)type non-expansive mappings is established.Bisht, RK.; Rakocević, V. (2021). On a probabilistic version of Meir-Keeler type fixed point theorem for a family of discontinuous operators. Applied General Topology. 22(2):435-446. https://doi.org/10.4995/agt.2021.15561OJS435446222R. K. Bisht and R. P. Pant, A remark on discontinuity at fixed point, J. Math. Anal. Appl. 445 (2017), 1239-1242. https://doi.org/10.1016/j.jmaa.2016.02.053R. K. Bisht, A probabilistic Meir-Keeler type fixed point theorem which characterizes metric completeness, Carpathain J. Math. 36, no. 2 (2020), 215-222. https://doi.org/10.37193/CJM.2020.02.05R. K. Bisht and V. Rakočević, Generalized Meir-Keeler type contractions and discontinuity at fixed point, Fixed Point Theory 19, no. 1 (2018), 57-64. https://doi.org/10.24193/fpt-ro.2018.1.06R. K. Bisht and V. Rakočević, Discontinuity at fixed point and metric completeness, Appl. Gen. Topol. 21, no. 2 (2020), 349-362. https://doi.org/10.4995/agt.2020.13943Lj. B. Ćirić, On contraction type mappings, Math. Balkanica 1 (1971), 52-57.T. Hicks and B. E. Rhoades, Fixed points and continuity for multivalued mappings, International J. Math. Math. Sci. 15 (1992), 15-30. https://doi.org/10.1155/S0161171292000024D. S. Jaggi, Fixed point theorems for orbitally continuous functions, Indian J. Math. 19, no. 2 (1977), 113-119.G. F. Jungck, Generalizations of continuity in the context of proper orbits and fixed pont theory, Topol. Proc. 37 (2011), 1-15.A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969), 326-329. https://doi.org/10.1016/0022-247X(69)90031-6K. Menger, Statistical metric, Proc. Nat. Acad. Sci. USA 28 (1942), 535-537. https://doi.org/10.1073/pnas.28.12.535A. Pant and R. P. Pant, Fixed points and continuity of contractive maps, Filomat 31, no. 11 (2017), 3501-3506. https://doi.org/10.2298/FIL1711501PA. Pant, R. P. Pant and M. C. Joshi, Caristi type and Meir-Keeler type fixed point theorems, Filomat 33, no. 12 (2019), 3711-3721. https://doi.org/10.2298/FIL1912711PA. Pant, R. P. Pant and W. Sintunavarat, Analytical Meir-Keeler type contraction mappings and equivalent characterizations, RACSAM 37 (2021), 115. https://doi.org/10.1007/s13398-020-00939-8R. P. Pant, Discontinuity and fixed points, J. Math. Anal. Appl. 240 (1999), 284-289. https://doi.org/10.1006/jmaa.1999.6560R. P. Pant, N. Y. Özgür and N. Tac s, On discontinuity problem at fixed point, Bull. Malays. Math. Sci. Soc. 43, no. 1 (2020), 499-517. https://doi.org/10.1007/s40840-018-0698-6R. P. Pant, A. Pant, R. M. Nikolić and S. N. Ješić, A characterization of completeness of Menger PM-spaces, J. Fixed Point Theory Appl. 21, (2019) 90. https://doi.org/10.1007/s11784-019-0732-9R. P. Pant, N. Y. Özgür and N. Taş, Discontinuity at fixed points with applications, Bulletin of the Belgian Mathematical Society-Simon Stevin 25, no. 4 (2019), 571-589. https://doi.org/10.36045/bbms/1576206358O. Popescu, A new type of contractions that characterize metric completeness, Carpathian J. Math. 31, no. 3 (2015), 381-387. https://doi.org/10.37193/CJM.2015.03.15B. E. Rhoades, Contractive definitions and continuity, Contemporary Mathematics 72 (1988), 233-245. https://doi.org/10.1090/conm/072/956495S. Romaguera, w-distances on fuzzy metric spaces and fixed points, Mathematics 8, no. 11 (2020), 1909. https://doi.org/10.3390/math8111909B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), 415-417. https://doi.org/10.2140/pjm.1960.10.313B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland, New York, Elsevier 1983.V. M. Sehgal and A. T. Bharucha-Reid, Fixed points of contraction mappings in PM-spaces, Math. System Theory 6 (1972), 97-102. https://doi.org/10.1007/BF01706080P. V. Subrahmanyam, Completeness and fixed points, Monatsh. Math. 80 (1975), 325-330. https://doi.org/10.1007/BF01472580T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136, no. 5 (2008), 1861-1869. https://doi.org/10.1090/S0002-9939-07-09055-7N. Taş and N. Y. Özgür, A new contribution to discontinuity at fixed point, Fixed Point Theory 20, no. 2 (2019), 715-728. https://doi.org/10.24193/fpt-ro.2019.2.4

On Cyclic Generalized Weakly C-Contractions on Partial Metric Spaces

We give new results of a cyclic generalized weakly C-contraction in partial metric space. The results of this paper extend, generalize, and improve some fixed point theorems in the literature

Central ghrelin treatment stimulates ACTH cells in normal-fed, food-restricted and high-fed rats: An immunohistomorphometric and hormonal study

Changes in feeding regime represent serious stress, while ghrelin is considered a key player in energy balance. We investigated the effects of intracerebroventricular (ICV) ghrelin application on pituitary adrenocorticotropic (ACTH) cells in rats fed diets differing in energy content. Before the ICV treatment, male Wistar rats were subjected to three different feeding regimes for 4 weeks: normal-fed (NF), food-restricted (FR) or high-fed (HF) (n = 3 x 14). At the age of 8 weeks, rats from each group were divided into two subgroups and given ICV, either ghrelin (G; 1 mu g ghrelin/5 mu l PBS, n = 7) or solvent alone (5 mu l PBS, n = 7) every 24 h for 5 days. The immunohistochemical appearance and quantitative morphology of pituitary ACTH cells were evaluated, as well as peripheral ACTH and corticosterone levels. Central ghrelin administration increased (p < 0.05) ACTH cell volumes in GNF, GFR and GHF rats by 8.1%, 11.8% and 9.1%, respectively, compared to the controls, while significant increases in ACTH cell volume density were observed in GNF and GHF rats. Circulating ACTH and corticosterone levels were elevated (p < 0.05) in GNF and GFR rats by 72.8% and 80.8%, respectively, when compared to the corresponding controls. Thus, central ghrelin administration stimulated the pituitary-adrenal axis under preserved and negative energy balance states. (C) 2013 Elsevier GmbH. All rights reserved.Ministry of Education, Science and Technological Development of the Republic of Serbia [ON173009, III41025

Central ghrelin treatment stimulates ACTH cells in normal-fed, food-restricted and high-fed rats: An immunohistomorphometric and hormonal study

Changes in feeding regime represent serious stress, while ghrelin is considered a key player in energy balance. We investigated the effects of intracerebroventricular (ICV) ghrelin application on pituitary adrenocorticotropic (ACTH) cells in rats fed diets differing in energy content. Before the ICV treatment, male Wistar rats were subjected to three different feeding regimes for 4 weeks: normal-fed (NF), food-restricted (FR) or high-fed (HF) (n = 3 x 14). At the age of 8 weeks, rats from each group were divided into two subgroups and given ICV, either ghrelin (G; 1 mu g ghrelin/5 mu l PBS, n = 7) or solvent alone (5 mu l PBS, n = 7) every 24 h for 5 days. The immunohistochemical appearance and quantitative morphology of pituitary ACTH cells were evaluated, as well as peripheral ACTH and corticosterone levels. Central ghrelin administration increased (p < 0.05) ACTH cell volumes in GNF, GFR and GHF rats by 8.1%, 11.8% and 9.1%, respectively, compared to the controls, while significant increases in ACTH cell volume density were observed in GNF and GHF rats. Circulating ACTH and corticosterone levels were elevated (p < 0.05) in GNF and GFR rats by 72.8% and 80.8%, respectively, when compared to the corresponding controls. Thus, central ghrelin administration stimulated the pituitary-adrenal axis under preserved and negative energy balance states. (C) 2013 Elsevier GmbH. All rights reserved.Ministry of Education, Science and Technological Development of the Republic of Serbia [ON173009, III41025