We prove some fixed point theorems in partially ordered sets, providing an extension of the Banach contractive mapping theorem. Having studied previously the nondecreasing case, we consider in this paper nonincreasing...We prove some fixed point theorems in partially ordered sets, providing an extension of the Banach contractive mapping theorem. Having studied previously the nondecreasing case, we consider in this paper nonincreasing mappings as well as non monotone mappings. We also present some applications to first-order ordinary differential equations with periodic boundary conditions, proving the existence of a unique solution admitting the existence of a lower solution.展开更多
This paper provides data envelopment analysis methods based on partially ordered set theory.These methods reveal the special relationships between two decision making units from the perspective of mathematical theory ...This paper provides data envelopment analysis methods based on partially ordered set theory.These methods reveal the special relationships between two decision making units from the perspective of mathematical theory and offer the classification,projection and improvement methods of decision making units.It is proved that an efficient decision making unit must be a maximal element of the related poset,and the maximal element may not be efficient.For this,we introduce the concepts of minimum envelope and efficiency envelope which further reveal the special relationship among efficient and inefficient decision making units.Compared with the previous methods,this method not only reveals theoretically the complex relationship among decision making units and the causes of the ineffectiveness,but also gives a new importance and competitiveness measurement method to each decision making unit.Finally,related algorithm and examples are given for the application of these methods to complex decision making problems.展开更多
By using the partial ordering method,a more general type,of Ekeland’s ariational principle and a set-valued Caristi’s coincidence theorem in probabilistic metric spaces are obtained.In addition,a direct simple proof...By using the partial ordering method,a more general type,of Ekeland’s ariational principle and a set-valued Caristi’s coincidence theorem in probabilistic metric spaces are obtained.In addition,a direct simple proof of the equivalence between these two theorems in probabilistic metric spaces is given.展开更多
Some new properties of lattice filters are presented based on the order-preserving mapping and lattice homomorphism, and two necessary and sufficient conditions for lattice filters under the chain type are given. Then...Some new properties of lattice filters are presented based on the order-preserving mapping and lattice homomorphism, and two necessary and sufficient conditions for lattice filters under the chain type are given. Then, the relations between lattice filter and lattice implication algebras (LIAs), i. e., the relations between lattice filter and LIA-filters, and the related properties are investigated. In addition, three necessary and sufficient conditions for LIA-filters are discussed. The obtained results may serve as some theoretical supports to lattice-valued logical system.展开更多
Monotonic regression (MR) is a least distance problem with monotonicity constraints induced by a partiaily ordered data set of observations. In our recent publication [In Ser. Nonconvex Optimization and Its Applicat...Monotonic regression (MR) is a least distance problem with monotonicity constraints induced by a partiaily ordered data set of observations. In our recent publication [In Ser. Nonconvex Optimization and Its Applications, Springer-Verlag, (2006) 83, pp. 25-33], the Pool-Adjazent-Violators algorithm (PAV) was generalized from completely to partially ordered data sets (posets). The new algorithm, called CPAV, is characterized by the very low computational complexity, which is of second order in the number of observations. It treats the observations in a consecutive order, and it can follow any arbitrarily chosen topological order of the poset of observations. The CPAV algorithm produces a sufficiently accurate solution to the MR problem, but the accuracy depends on the chosen topological order. Here we prove that there exists a topological order for which the resulted CPAV solution is optimal. Furthermore, we present results of extensive numerical experiments, from which we draw conclusions about the most and the least preferable topological orders.展开更多
基金Ministerio de Educacióny Ciencia and FEDER,Project MTM2004-06652-C03-01Xunta de Galicia and FEDER,Projects PGIDIT02PXIC20703PN and PGIDIT05PXIC20702PN
文摘We prove some fixed point theorems in partially ordered sets, providing an extension of the Banach contractive mapping theorem. Having studied previously the nondecreasing case, we consider in this paper nonincreasing mappings as well as non monotone mappings. We also present some applications to first-order ordinary differential equations with periodic boundary conditions, proving the existence of a unique solution admitting the existence of a lower solution.
基金supported by the National Natural Science Foundation of China under Grant No.71961026the National Natural Science Foundation of Inner Mongolia under Grant No.2019MS07001.
文摘This paper provides data envelopment analysis methods based on partially ordered set theory.These methods reveal the special relationships between two decision making units from the perspective of mathematical theory and offer the classification,projection and improvement methods of decision making units.It is proved that an efficient decision making unit must be a maximal element of the related poset,and the maximal element may not be efficient.For this,we introduce the concepts of minimum envelope and efficiency envelope which further reveal the special relationship among efficient and inefficient decision making units.Compared with the previous methods,this method not only reveals theoretically the complex relationship among decision making units and the causes of the ineffectiveness,but also gives a new importance and competitiveness measurement method to each decision making unit.Finally,related algorithm and examples are given for the application of these methods to complex decision making problems.
文摘By using the partial ordering method,a more general type,of Ekeland’s ariational principle and a set-valued Caristi’s coincidence theorem in probabilistic metric spaces are obtained.In addition,a direct simple proof of the equivalence between these two theorems in probabilistic metric spaces is given.
基金The National Natural Science Founda-tion of China (No.60474022)the Specialized Research Fund for the Doctoral Program of Higher Education of China(No.20060613007)
文摘Some new properties of lattice filters are presented based on the order-preserving mapping and lattice homomorphism, and two necessary and sufficient conditions for lattice filters under the chain type are given. Then, the relations between lattice filter and lattice implication algebras (LIAs), i. e., the relations between lattice filter and LIA-filters, and the related properties are investigated. In addition, three necessary and sufficient conditions for LIA-filters are discussed. The obtained results may serve as some theoretical supports to lattice-valued logical system.
文摘Monotonic regression (MR) is a least distance problem with monotonicity constraints induced by a partiaily ordered data set of observations. In our recent publication [In Ser. Nonconvex Optimization and Its Applications, Springer-Verlag, (2006) 83, pp. 25-33], the Pool-Adjazent-Violators algorithm (PAV) was generalized from completely to partially ordered data sets (posets). The new algorithm, called CPAV, is characterized by the very low computational complexity, which is of second order in the number of observations. It treats the observations in a consecutive order, and it can follow any arbitrarily chosen topological order of the poset of observations. The CPAV algorithm produces a sufficiently accurate solution to the MR problem, but the accuracy depends on the chosen topological order. Here we prove that there exists a topological order for which the resulted CPAV solution is optimal. Furthermore, we present results of extensive numerical experiments, from which we draw conclusions about the most and the least preferable topological orders.