Variants of Meir-Keeler Fixed Point Theorem And Applications of Soft Set-Valued Maps

In this paper, we prove a Meir-Keeler type common fixed point theorem for two mappings for which the range set of the first one is a family of soft sets, called soft set-valued map and the second is a point-to-point mapping. In addition, it is also shown that under some suitable conditions, a soft set-valued map admits a selection having a unique fixed point. In support of the obtained result, nontrivial examples are provided. The novelty of the presented idea herein is that it extends the Meir-Keeler fixed point theorem and the theory of selections for multivalued mappings from the case of crisp mappings to the frame of soft set-valued maps. Finally, an application of soft setvalued maps in decision making problems is considered

Existence results of delay and fractional differential equations via fuzzy weakly contraction mapping principle

[EN] The purpose of this article is to extend the results derived through former articles with respect to the notion of weak contraction into intuitionistic fuzzy weak contraction in the context of (T,N,∝) -cut set of an intuitionistic fuzzy set. We intend to prove common fixed point theorem for a pair of intuitionistic fuzzy mappings satisfying weakly contractive condition in a complete metric space which generalizes many results existing in the literature. Moreover, concrete results on existence of the solution of a delay differential equation and a system of Riemann-Liouville Cauchy type problems have been derived. In addition, we also present illustrative examples to substantiate the usability of our main result.Tabassum, R.; Azam, A.; Mohammed, SS. (2019). Existence results of delay and fractional differential equations via fuzzy weakly contraction mapping principle. Applied General Topology. 20(2):449-469. https://doi.org/10.4995/agt.2019.11683SWORD449469202H. M. Abu-Donia, Common fixed points theorems for fuzzy mappings in metric space under $varphi$-contraction condition, Chaos Solitons & Fractals 34 (2007), 538-543. https://doi.org/10.1016/j.chaos.2005.03.055A. Z. Al-Abedeen, Existence theorem on differential equation of generalized order, Al-Rafidain J. Sci. Mosul University, Iraq, 1 (1976), 95-104.Y. I. Alber and S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert spaces, in: New Results in Operator Theory and Its Applications, Birkhäuser, Basel (1997), 7-22. https://doi.org/10.1007/978-3-0348-8910-0_2H. L. Arora and J. G. Alshamani, Stability of differential equations of noninteger order through fixed point in the large, Indian J. Pure Appl. Math. 11, no. 3 (1980), 307-313.K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and Systems 20, no. 1 (1986), 87-96. https://doi.org/10.1016/S0165-0114(86)80034-3A. Azam, M. Arshad and P. Vetro, On a pair of fuzzy $varphi$-contractive mappings, Mathematical and Computer Modelling 52, no. 1 (2010), 207-214. https://doi.org/10.1016/j.mcm.2010.02.010A. Azam and M. Rashid, A fuzzy coincidence theorem with applications in a function space, Journal of Intelligent and Fuzzy Systems 27, no. 4 (2014), 1775-1781.A. Azam, R. Tabassum and M. Rashid, Coincidence and fixed point theorems of intuitionistic fuzzy mappings with applications, Journal of Mathematical Analysis 8, no. 4 (2017), 56-77.A. Azam and R. Tabassum, Existence of common coincidence point of intuitionistic fuzzy maps, Journal of Intelligent and Fuzzy Systems 35 (2018), 4795-4805. https://doi.org/10.3233/JIFS-18411J. S. Bae, Fixed point theorems for weakly contractive multivalued maps, Journal of Mathematical Analysis and Applications 284, no. 2 (2003), 690-697. https://doi.org/10.1016/S0022-247X(03)00387-1I. Beg and M. Abbas, Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition, Fixed Point Theory and Applications 2006 (2006), 1-7. https://doi.org/10.1155/FPTA/2006/74503M. A. Al-Bassam, Some existence theorems on differential equations of generalized order, J. Reine Angew. Math. 218, no. 1 (1965), 70-78. https://doi.org/10.1515/crll.1965.218.70V. Berinde, Approximating fixed points of weak contractions, Fixed Point Theory 4 (2003), 131-142.S. M. Ciupe, B. L. de Bivort, D. M. Bortz and P. W. Nelson, Estimates of kinetic parameters from HIV patient data during primary infection through the eyes of three different models, Math. Biosci., to appear.K. Cooke, Y. Kuang and B. Li, Analyses of an antiviral immune response model with time delays, Canad. Appl. Math. Quart. 6, no. 4 (1998), 321-354.K. L. Cooke, P. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol. 39 (1999), 332-352. https://doi.org/10.1007/s002850050194P. Z. Daffer and H. Kaneko, Fixed points of generalized contractive multi-valued mappings, Journal of Mathematical Analysis and Applications 192, no. 2 (1995), 655-666. https://doi.org/10.1006/jmaa.1995.1194S. K. De, R. Biswas and A. R. Roy, An application of intuitionistic fuzzy sets in medical diagnosis, Fuzzy Sets and Systems 117, no. 2 (2001), 209-213. https://doi.org/10.1016/S0165-0114(98)00235-8D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Appl. 204, no. 2 (1996), 609-625. https://doi.org/10.1006/jmaa.1996.0456S. Heilpern, Fuzzy mappings and fixed point theorems, Journal of Mathematical Analysis and Applications 83, no. 2 (1981), 566-569. https://doi.org/10.1016/0022-247X(81)90141-4Z. Jia, L. Amselang and P. Gros, Content-based image retrieval from a large image database, Pattern Recognition 11, no. 5 (2008), 1479-1495. https://doi.org/10.1016/j.patcog.2007.06.034A. Kharal, Homeopathic drug selection using intuitionistic fuzzy sets, Homeopathy 98, no. 1 (2009), 35-39. https://doi.org/10.1016/j.homp.2008.10.003A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science Limited, 2006.S. Konjik, L. Oparnica and D. Zorica, Waves in viscoelastic media described by a linear fractional model, Integral Transforms Spec. Funct. 22 (2011), 283-291. https://doi.org/10.1080/10652469.2010.541039A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis, Nauka, Moscow, 1968.D. F. Li, Multiattribute decision making models and methods using intuitionistic fuzzy sets, J. Comput. Syst. Sci. 70 (2005), 73-85. https://doi.org/10.1016/j.jcss.2004.06.002D. Martinetti, V. Janis and S. Montes, Cuts of intuitionistic fuzzy sets respecting fuzzy connectives, Information Sciences 232 (2013), 267-275. https://doi.org/10.1016/j.ins.2012.12.026S. B. Nadler Jr, Multi-valued contraction mappings, Pacific Journal of Mathematics 30, no. 2 (1969), 475-488. https://doi.org/10.2140/pjm.1969.30.475P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includesan intracellular delay. Math. Biosci. 163 (2000), 201-215. https://doi.org/10.1016/S0025-5564(99)00055-3B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Analysis 4, no. 47 (2001), 2683-2693. https://doi.org/10.1016/S0362-546X(01)00388-1A. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives : Theory and Applcations, Gordon and Breach Science Publishers, Switzerland, 1993.A. M. A. El-Sayed and A. G. Ibrahim, Multivalued fractional differential equations, Appl. Math. Comp. 68, no. 1 (1995), 15-25. https://doi.org/10.1016/0096-3003(94)00080-NA. A. Kilbas and J. J. Trujillo, Differential equations of fractional order: methods, results and problems, I. Appl. Anal. 78, no. 1-2 (2001), 153-192. https://doi.org/10.1080/00036810108840931P. Turchin and A. D. Taylor, Complex dynamics in ecological time series, Ecology 73 (1992), 289-305. https://doi.org/10.2307/1938740D. Valerjo, D. Machadoa and J. T. Kryakova, Some pioneers of the applications of fractional calculus, Fract. Calc. Appl. Anal. 17 (2014), 552-578. https://doi.org/10.2478/s13540-014-0185-1B. Vielle and G. Chauvet, Delay equation analysis of human respiratory stability, Math. Biosci. 152, no. 2 (1998), 105-122. https://doi.org/10.1016/S0025-5564(98)10028-7M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth, J. Math. Biol. 47, no. 3 (2003), 270-294. https://doi.org/10.1007/s00285-003-0211-0Y. H. Shen, F. X. Wang and W. Chen, A note on intuitionistic fuzzy mappings, Iranian Journal of Fuzzy Systems 9, no. 5 (2012), 63-76.L. A. Zadeh, Fuzzy sets, Information and Control 8, no. 3 (1965), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-

Combined DNA extraction and antibody elution from filter papers for the assessment of malaria transmission intensity in epidemiological studies.

BACKGROUND: Informing and evaluating malaria control efforts relies on knowledge of local transmission dynamics. Serological and molecular tools have demonstrated great sensitivity to quantify transmission intensity in low endemic settings where the sensitivity of traditional methods is limited. Filter paper blood spots are commonly used a source of both DNA and antibodies. To enhance the operational practicability of malaria surveys, a method is presented for combined DNA extraction and antibody elution. METHODS: Filter paper blood spots were collected as part of a large cross-sectional survey in the Kenyan highlands. DNA was extracted using a saponin/chelex method. The eluate of the first wash during the DNA extraction process was used for antibody detection and compared with previously validated antibody elution procedures. Antibody elution efficiency was assessed by total IgG ELISA for malaria antigens apical membrane antigen-1 (AMA-1) and merozoite-surface protein-1 (MSP-142). The sensitivity of nested 18S rRNA and cytochrome b PCR assays and the impact of doubling filter paper material for PCR sensitivity were determined. The distribution of cell material and antibodies throughout filter paper blood spots were examined using luminescent and fluorescent reporter assays. RESULTS: Antibody levels measured after the combined antibody/DNA extraction technique were strongly correlated to those measured after standard antibody elution (p < 0.0001). Antibody levels for both AMA-1 and MSP-142 were generally slightly lower (11.3-21.4%) but age-seroprevalence patterns were indistinguishable. The proportion of parasite positive samples ranged from 12.9% to 19.2% in the different PCR assays. Despite strong agreement between outcomes of different PCR assays, none of the assays detected all parasite-positive individuals. For all assays doubling filter paper material for DNA extraction increased sensitivity. The concentration of cell and antibody material was not homogenously distributed throughout blood spots. CONCLUSION: Combined DNA extraction and antibody elution is an operationally attractive approach for high throughput assessment of cumulative malaria exposure and current infection prevalence in endemic settings. Estimates of antibody prevalence are unaffected by the combined extraction and elution procedure. The choice of target gene and the amount and source of filter paper material for DNA extraction can have a marked impact on PCR sensitivity

Factors associated with high heterogeneity of malaria at fine spatial scale in the Western Kenyan highlands.

BACKGROUND: The East African highlands are fringe regions between stable and unstable malaria transmission. What factors contribute to the heterogeneity of malaria exposure on different spatial scales within larger foci has not been extensively studied. In a comprehensive, community-based cross-sectional survey an attempt was made to identify factors that drive the macro- and micro epidemiology of malaria in a fringe region using parasitological and serological outcomes. METHODS: A large cross-sectional survey including 17,503 individuals was conducted across all age groups in a 100 km(2) area in the Western Kenyan highlands of Rachuonyo South district. Households were geo-located and prevalence of malaria parasites and malaria-specific antibodies were determined by PCR and ELISA. Household and individual risk-factors were recorded. Geographical characteristics of the study area were digitally derived using high-resolution satellite images. RESULTS: Malaria antibody prevalence strongly related to altitude (1350-1600 m, p < 0.001). A strong negative association with increasing altitude and PCR parasite prevalence was found. Parasite carriage was detected at all altitudes and in all age groups; 93.2 % (2481/2663) of malaria infections were apparently asymptomatic. Malaria parasite prevalence was associated with age, bed net use, house construction features, altitude and topographical wetness index. Antibody prevalence was associated with all these factors and distance to the nearest water body. CONCLUSION: Altitude was a major driver of malaria transmission in this study area, even across narrow altitude bands. The large proportion of asymptomatic parasite carriers at all altitudes and the age-dependent acquisition of malaria antibodies indicate stable malaria transmission; the strong correlation between current parasite carriage and serological markers of malaria exposure indicate temporal stability of spatially heterogeneous transmission

The Impact of Hotspot-Targeted Interventions on Malaria Transmission in Rachuonyo South District in the Western Kenyan Highlands: A Cluster-Randomized Controlled Trial.

BACKGROUND: Malaria transmission is highly heterogeneous, generating malaria hotspots that can fuel malaria transmission across a wider area. Targeting hotspots may represent an efficacious strategy for reducing malaria transmission. We determined the impact of interventions targeted to serologically defined malaria hotspots on malaria transmission both inside hotspots and in surrounding communities. METHODS AND FINDINGS: Twenty-seven serologically defined malaria hotspots were detected in a survey conducted from 24 June to 31 July 2011 that included 17,503 individuals from 3,213 compounds in a 100-km2 area in Rachuonyo South District, Kenya. In a cluster-randomized trial from 22 March to 15 April 2012, we randomly allocated five clusters to hotspot-targeted interventions with larviciding, distribution of long-lasting insecticide-treated nets, indoor residual spraying, and focal mass drug administration (2,082 individuals in 432 compounds); five control clusters received malaria control following Kenyan national policy (2,468 individuals in 512 compounds). Our primary outcome measure was parasite prevalence in evaluation zones up to 500 m outside hotspots, determined by nested PCR (nPCR) at baseline and 8 wk (16 June-6 July 2012) and 16 wk (21 August-10 September 2012) post-intervention by technicians blinded to the intervention arm. Secondary outcome measures were parasite prevalence inside hotpots, parasite prevalence in the evaluation zone as a function of distance from the hotspot boundary, Anopheles mosquito density, mosquito breeding site productivity, malaria incidence by passive case detection, and the safety and acceptability of the interventions. Intervention coverage exceeded 87% for all interventions. Hotspot-targeted interventions did not result in a change in nPCR parasite prevalence outside hotspot boundaries (p ≥ 0.187). We observed an average reduction in nPCR parasite prevalence of 10.2% (95% CI -1.3 to 21.7%) inside hotspots 8 wk post-intervention that was statistically significant after adjustment for covariates (p = 0.024), but not 16 wk post-intervention (p = 0.265). We observed no statistically significant trend in the effect of the intervention on nPCR parasite prevalence in the evaluation zone in relation to distance from the hotspot boundary 8 wk (p = 0.27) or 16 wk post-intervention (p = 0.75). Thirty-six patients with clinical malaria confirmed by rapid diagnostic test could be located to intervention or control clusters, with no apparent difference between the study arms. In intervention clusters we caught an average of 1.14 female anophelines inside hotspots and 0.47 in evaluation zones; in control clusters we caught an average of 0.90 female anophelines inside hotspots and 0.50 in evaluation zones, with no apparent difference between study arms. Our trial was not powered to detect subtle effects of hotspot-targeted interventions nor designed to detect effects of interventions over multiple transmission seasons. CONCLUSIONS: Despite high coverage, the impact of interventions targeting malaria vectors and human infections on nPCR parasite prevalence was modest, transient, and restricted to the targeted hotspot areas. Our findings suggest that transmission may not primarily occur from hotspots to the surrounding areas and that areas with highly heterogeneous but widespread malaria transmission may currently benefit most from an untargeted community-wide approach. Hotspot-targeted approaches may have more validity in settings where human settlement is more nuclear. TRIAL REGISTRATION: ClinicalTrials.gov NCT01575613

FIXED POINT RESULTS FOR FUZZY SET-VALUED MAPS ON METRIC SPACE

Among various developments in fuzzy mathematics, progressive efforts have been in process to examine new fuzzy versions of the classical fixed point results and their various applications. Following this trend in this paper, some new fixed point results for fuzzy set-valued maps are established in the framework of metric space. From application viewpoint, a few corresponding fixed point theorems in ordered metric spaces as well as crisp multi-valued and single-valued mappings are pointed out and discussed. A nontrivial example is constructed to support the assertions of our obtained results. Consequently, we note that the ideas presented herein complement, unify and generalize several recently announced results in the related literature of both fuzzy and classical mathematics

Understanding the contribution of malaria parasites to continued residual malaria in a pre-elimination setting in South Africa : implications for elimination strategies

Remarkable progress has been made in averting malaria cases and reducing malaria mortality globally. However, an incomplete understanding of the factors driving the continued transmission in pre-elimination settings challenges further successes in malaria control. For the evidence-based deployment of effective strategies targeting the parasite reservoir in humans, a better understanding of the contribution of asymptomatic parasite carriage to continued transmission is necessary. Primaquine (PQ) is a drug that targets and clears mature gametocytes, the transmissible forms of the malaria parasite. However, the implementation of PQ treatment strategies to reduce Plasmodium falciparum transmission is rather slow. The perceived safety issue surrounding PQ use in glucose-6-phosphate dehydrogenase (G6PD)-deficient individuals and local drug regulations are partly responsible for this slow uptake. When exposed to high doses of PQ, G6PD-deficient individuals risk haemolysis that may be life-threatening. In addition, the human cytochrome P-450 enzymes modulate PQ efficacy. This thesis evaluated locally relevant G6PD-deficiency genotypes and CYP2D6 variants in a pre-elimination setting in South Africa. The low prevalence of these variants supports PQ deployment as a transmission-blocking strategy towards malaria elimination in the country. To better understand which populations should be targeted, the thesis subsequently investigated the contribution of asymptomatic parasite carriage to continued transmission in South Africa. We focussed on genetically characterising the currently circulating parasites in the Vhembe District in Limpopo Province, an area that continues to experience malaria transmission. While the parasites are homogenous based on genotypes, clear distinctions are shown for “local” parasites in contrast to those “imported” from neighbouring countries. When we analysed drug resistance markers, we observed high levels of resistance to sulphadoxine-pyrimethamine and no evidence for artemisinin resistance in the parasite population. This supports the need for continued surveillance to maintain and prolong the efficacy of the existing drugs. Importantly, this thesis uncovers the presence of apparently chronic low-density P. falciparum parasite infections, only detectable by sensitive telomere-associated repetitive element 2 qPCR. Elimination strategies need to consider these low-density infections that may play a role in ongoing transmission. The transmission potential of (low density) infection depends on the production of mature male and female gametocytes. Mosquitoes need to ingest at least one gametocyte of both sexes to become infected with malaria. The thesis describes the development of a novel multiplex qPCR for detecting P. falciparum male and female gametocytes. The P. falciparum male gametocyte enriched transcript (pfmget, PF3D7_1469900) and female specific (ccp4, PF3D7_0903800) mRNA targets were used in qPCR assays on mosquito blood meal samples. This approach fuelled a pilot study to explore the possibility to use mosquito blood meals as a source of parasite material for xenodiagnoses of malaria, either by qPCR or by rapid diagnostic tests. Taken together, this thesis addresses several relevant gaps in our understanding of the role of asymptomatic parasite carriers to ongoing transmission in South African settings. The novel data presented in this thesis may contribute to the rational implementation of malaria interventions in South Africa with the ultimate aim of achieving countrywide malaria elimination and ultimately global malaria eradication.Thesis (PhD)--University of Pretoria, 2018.BiochemistryPhDUnrestricte

On General Class of Z-Contractions with Applications to Spring Mass Problem

One of the latest techniques in metric fixed point theory is the interpolation approach. This notion has so far been examined using standard functional equations. A hybrid form of this concept is yet to be uncovered by observing the available literature. With this background information, and based on the symmetry and rectangular properties of generalized metric spaces, this paper introduces a novel and unified hybrid concept under the name interpolative Y-Hardy&ndash;Rogers&ndash;Suzuki-type Z-contraction and establishes sufficient conditions for the existence of fixed points for such contractions. As an application, one of the obtained results was employed to examine new criteria for the existence of a solution to a boundary valued problem arising in the oscillation of a spring. The ideas proposed herein advance some recently announced important results in the corresponding literature. A comparative example was constructed to justify the abstractions and pre-eminence of our obtained results

On General Class of Z-Contractions with Applications to Spring Mass Problem

One of the latest techniques in metric fixed point theory is the interpolation approach. This notion has so far been examined using standard functional equations. A hybrid form of this concept is yet to be uncovered by observing the available literature. With this background information, and based on the symmetry and rectangular properties of generalized metric spaces, this paper introduces a novel and unified hybrid concept under the name interpolative Y-Hardy–Rogers–Suzuki-type Z-contraction and establishes sufficient conditions for the existence of fixed points for such contractions. As an application, one of the obtained results was employed to examine new criteria for the existence of a solution to a boundary valued problem arising in the oscillation of a spring. The ideas proposed herein advance some recently announced important results in the corresponding literature. A comparative example was constructed to justify the abstractions and pre-eminence of our obtained results