Search

In a recent paper [I. Altun and A. Tasdemir, On best proximity points of interpolative proximal contractions, Quaestiones Math. (2020), DOI: 10.2989/16073606.2020.1785576] the authors proved two best proximity point theorems for a new class of non-self mappings, called interpolative Reich- Rus-Ciric type proximal contractions of the rst and second kinds. Our purpose is to show that their results follow from a xed point theorem for interpolative Reich-Rus-Ciric type contraction mappings [E. Karapinar, R. Agarwal and H. Aydi, Interpolative Reich-Rus-Ciric contractions on partial metric spaces, Mathematics 6 (2018), 7 pages].

In a recently published theorem on the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings, Tang et al. (J. Inequal. Appl. 2015:305, 2015) studied a uniformly convex and 2-uniformly smooth real Banach space with the Opial property and best smoothness constant κ satisfying the condition $0 0 < κ < 1 2 , as a real Banach space more general than Hilbert spaces. A well-known example of a uniformly convex and 2-uniformly smooth real Banach space with the Opial property is $E=l_{p}$ E = l p , $2\leq p 2 ≤ p < ∞ . It is shown in this paper that, if κ is the best smoothness constant of E and satisfies the condition $0 0 < κ ≤ 1 2 , then E is necessarily $l_{2}$ l 2 , a real Hilbert space. Furthermore, some important remarks concerning the proof of this theorem are presented.

Fuzzy Cognitive Maps (FCM) is a tool merging the properties of Artificial Neural Networks (NN) and fuzzy systems. FCMs are bipolar directed fuzzy graphs consisting of Concepts (nodes) assigned fuzzy membership degrees expressing the intensity or level of the property associated with the given concept; and of directed influence values (edges) which have a sign (expressing whether the influence is positive or negative) and a fuzzy membership degree of the intensity of the influence. E.g., if C1 and C2 are two concepts, $\mu 1$ is the degree of C1 and $\mu 2$ of C2, the directed edges e12 and e21 have ±$\mu 12$ and ±$\mu 21$, respectively, denoting that C1 influences C2 either in the positive sense (increasing its level) with + $\mu 12$, or in the negative sense (decreasing C2) with - $\mu 2.$ FCMs are suitable to model multicomponent systems with multiple mutual influences, such as, e.g., a management system. Because of the NN-type behaviour of FCMs, they may be used for simulating a system that changes in time, and possibly converges to a steady state (called fixed point attractor). It is interesting to fit the FCM to a series of historical data and calculate the expected development of the system in the future ± a way for time series prediction. Such fitting may happen by various machine learnning techniques, such as evolutionary and memetic algorithms, among others. In this case the FCM model will be determined by theis fitting, including all membership degrees at the concepts and at the edges. On the other hand, often experts determine these values (approximately) in advance, and thus the FCM model is ready for prediction - starting from a known initial state. In our past reserch we found real life applications (such as waste management, bank management, etc.) where one or both of the starting data were available. Especially interesting is the case when experts determine the membership degrees and then the FCM is matched to the historical data. In some case we found contradiction in the contents and this fact started a deeper methodological research about how to fit the proper FCM to a given problem, and how to transform the size and complexity of such models, so that theyt reemain intuitive, and tyransparent, while the fineness and precision of the model is satisfactory from the point of view of the application field. This talk will give some formal definitions in connection with FCMs, and then show the iterative (NN-like) behaviour. The next question to be discussed will be the existence and number of fixed point attractors (sustainable state of the system to be modelled), a topic that has not yet been satisfactorily clarified in the literature. In this respect we have presented some entirely new results in the form of mathematical statements. In the next a novel extension, the Fuzzy Grey Cognitive Map will be introduced, and it will be shown that they are more adequate in modelling real life systems, as human experts often hesitate in giving concrete degrees, and subjectivity and uncertainty of the experts prevent any clear and deterministic membership values at the concepts and edges. The next real application motivated extension is the application of fuzzy numbers for determining the memberships. Some results about the existense and uniqueness of a fixed point attractor are also stated here. The other main stream of the research reported here focuses on the transformation of FCMs (as far in the classic sense), especially by reducing the size of the model, namely, the number of concepts, which of course also results in the drastic reduction of the number of edges in the new model. A novel reduction technique, based on a very special FCM compatible metrics (a distance measure between two concepts where all the influence degrees to other concepts and the way around are known and compared) is proposed, illustrated by a real life problem, where the obvious contradiction between the expert assessment and the historical data points towrds the necessity of the refinement of the mmodel, and then, towards a transformation that leads to the emerging of new concepts that have to be interpreted in the context of the application problem. An extensive simulation, applying a large number of real life and randomly generated examples for large size FCMs and respective reductions allows the discussion of to what extent such a transformation preserves the properties of the original FCM, with a special stress on the existence, number and parameters of the fixed points. The last topic of the talk is about the sensitivity of the FCMs, where it may be assumed that expert assessments are subjective and uncertain, and the uncertainty or error in the membership values may cause instability, even, a total change of behaviour of the FCM. One more real life example will be presented and evaluated.

Classic frameworks of rock-paper-scissors game have been assumed in a closed community that a density of each group is only affected by internal factors such as competition interplay among groups and reproduction itself. In real systems in ecological and social sciences, however, the survival and a change of a density of a group can be also affected by various external factors. One of common features in real population systems in ecological and social sciences is population flow that is characterized by population inflow and outflow in a group or a society, which has been usually overlooked in previous works on models of rock-paper-scissors game. In this paper, we suggest the rock-paper-scissors system by implementing population flow and investigate its effect on biodiversity. For two scenarios of either balanced or imbalanced population flow, we found that the population flow can strongly affect group diversity by exhibiting rich phenomena. In particular, while the balanced flow can only lead the persistent coexistence of all groups which accompanies a phase transition through supercritical Hopf bifurcation on different carrying simplices, the imbalanced flow strongly facilitates rich dynamics such as alternative stable survival states by exhibiting various group survival states and multistability of sole group survivals by showing not fully covered but spirally entangled basins of initial densities due to local stabilities of associated fixed points. In addition, we found that, the system can exhibit oscillatory dynamics for coexistence by relativistic interplay of population flows which can capture the robustness of the coexistence state. Applying population flow in the rock-paper-scissors system can ultimately change a community paradigm from closed to open one, and our foundation can eventually reveal that population flow can be also a significant factor on a group density which is independent to fundamental interactions among groups.

In this paper we consider, discuss and slightly complement recent fixed point results for mappings in b-metric spaces established by Sintunavarat (J Fixed Point Theory Appl 18:397–416, 2016). Thus, all our results are with shorter proofs. In addition, an application to integral equations is given to illustrate the usability of our approach.

In several recent papers, attempts have been made to apply Wardowski's method of $F$-contractions in order to obtain fixed point results for single and multivalued mappings in $b$-metric spaces. In this article, it is shown that in most cases the conditions imposed on respective mappings are too strong and that the results can be obtained directly, i.e., without using most of the properties of auxiliary function $F$.

Savonius rotor is a very popular vertical shaft power generation impeller, with the advantage of small starting torque and low cost. In order to improve the maximum power coefficient of the impeller, this paper presents a method using B spline optimization to get the best solution. The new design of the blades' shape is controlled by three variable points and two threefold fixed points. With the decrease of vertical height, the best power coefficient shows the trend of increasing first and then decreasing. With the increase of the horizontal offset, the optimal power coefficient shows a tendency of increasing first and then decreasing. By comparing all simulation data, the case H1 is the best blade curve. Its maximum power coefficient of 0.288 is 3.2% greater than the original blade.

Results of experiments on the dynamics and kinetic roughening of one-dimensional slow-combustion fronts in three grades of paper are reported. Extensive averaging of the data allows a detailed analysis of the spatial and temporal development of the interface fluctuations. The asymptotic scaling properties, on long length and time scales, are well described by the Kardar-Parisi-Zhang (KPZ) equation with short-range, uncorrelated noise. To obtain a more detailed picture of the strong-coupling fixed point, characteristic of the KPZ universality class, universal amplitude ratios, and the universal coupling constant are computed from the data and found to be in good agreement with theory. Below the spatial and temporal scales at which a cross-over takes place to the standard KPZ behavior, the fronts display higher apparent exponents and apparent multiscaling. In this regime the interface velocities are spatially and temporally correlated, and the distribution of the magnitudes of the effective noise has a power-law tail. The relation of the observed short-range behavior and the noise as determined from the local velocity fluctuations is discussed., RevTeX v3.1, 13 pages, 12 Postscript figures (uses epsf.sty), 3 tables; submitted to Phys. Rev. E

In this paper we correct an error made in our paper [Blouhi, T.; Caraballo, T.; Ouahab, A. Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion. Stoch. Anal. Appl. 34 (2016), no. 5, 792-834]. In fact, in this corrigendum we present the correct hypotheses and results, and highlight that the results can be proved using the same method used in the original work. The main feature is that we used a result which has been proved only when the diffusion term does not depend on the unknown.

We show that the main results in the recent paper [4] by Sadiq Basha (2012) are not real generalizations but particular cases of existing fixed point theorems in the literature.

In this paper, we discuss the validity of the result of Khuangsatung and Kangtunyakarn [Existence and convergence theorem for fixed point problem of various nonlinear mappings and variational inequality problems without some assumptions, Filomat 32(1) (2018) 305–309].

In this paper we correct an error made in our paper [Blouhi, T.; Caraballo, T.; Ouahab, A. Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion. Stoch. Anal. Appl. 34 (2016), no. 5, 792-834]. In fact, in this corrigendum we present the correct hypotheses and results, and highlight that the results can be proved using the same method used in the original work. The main feature is that we used a result which has been proved only when the diffusion term does not depend on the unknown.

In this paper we analyze the population dynamics of bacteria competing by anti-bacterial toxins (bacteriocins). Three types of bacteria involved in these dynamics can be distinguished: toxin producers, resistant bacteria and sensitive bacteria. Their interplay can be regarded as a R ock- S cissors- P aper - game (RSP). Here, this is modeled by a reasonable three-dimensional Lotka- Volterra ($L$V) type differential equation system. In contrast to earlier approaches to modeling the RSP game such as replicator equations, all interaction terms have negative signs because the interaction between the three different types of bacteria is purely competitive, either by toxification or by competition for nutrients. The model allows one to choose asymmetric parameter values. Depending on parameter values, our model gives rise to a stable steady state, a stable limit cycle or a heteroclinic orbit with three fixed points, each fixed point corresponding to the existence of only one bacteria type. An alternative model, the May - Leonard model, leads to coexistence only under very restricted conditions. We carry out a comprehensive analysis of the generic stability conditions of our model, using, among other tools, the Volterra-Lyapunov method. We also give biological interpretations of our theoretical results, in particular, of the predicted dynamics and of the ranges for parameter values where different dynamic behavior occurs. For example, one result is that the intrinsic growth rate of the producer is lower than that of the resistant while its growth yield is higher. This is in agreement with experimental results for the bacterium Listeria monocytogenes.

When playing games in groups, it is an advantage for individuals to have accurate statistical information on the strategies of their opponents. Such information may be obtained by remembering previous interactions. We consider a rock-paper-scissors game in which agents are able to recall their last m interactions, used to estimate the behavior of their opponents. At critical memory length, a Hopf bifurcation leads to the formation of stable limit cycles. In a mixed population, agents with longer memories have an advantage, provided the system has a stable fixed point, and there is some asymmetry in the payoffs of the pure strategies. However, at a critical concentration of long memory agents, the appearance of limit cycles destroys their advantage. By introducing population dynamics that favors successful agents, we show that the system evolves toward the bifurcation point.

Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476]. We shall prove that there are no normal cones with normal constant M 1 and for each k > 1 there are cones with normal constant M > k . Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476], we obtain generalizations of the results.

In this note, a small gap is corrected in the assumption of main theorem of T.M. Tuyen (Theorem 3.1, Regularization proximal point algorithm for common fixed points of nonexpansive mappings in Banach spaces, J. Optim. Theory Appl., 152:351–365, 2012). We give another assumption, which allows us to obtain the strong convergence of regularization algorithms.

We study the oscillatory dynamics in the generic three-species rock-paper-scissors games with mutations. In the mean-field limit, different behaviors are found: (a) for high mutation rate, there is a stable interior fixed point with coexistence of all species; (b) for low mutation rates, there is a region of the parameter space characterized by a limit cycle resulting from a Hopf bifurcation; (c) in the absence of mutations, there is a region where heteroclinic cycles yield oscillations of large amplitude (not robust against noise). After a discussion on the main properties of the mean-field dynamics, we investigate the stochastic version of the model within an individual-based formulation. Demographic fluctuations are therefore naturally accounted and their effects are studied using a diffusion theory complemented by numerical simulations. It is thus shown that persistent erratic oscillations (quasi-cycles) of large amplitude emerge from a noise-induced resonance phenomenon. We also analytically and numerically compute the average escape time necessary to reach a (quasi-)cycle on which the system oscillates at a given amplitude., Comment: 25 pages, 9 figures. To appear in the Journal of Theoretical Biology

In this paper, we show that Theorem 2.1 [1] (resp. Theorem 2.2 [1]) is a consequence of Corollary 2.1 [1] ( resp. Corollary 2.2 [1]).