The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems by George Isac, Monica G. Cojocaru

“…In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. …”
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Effectiveness of transposed inverse sets in faber regions by J. A. Adepoju, M. Nassif

“…The first main result of the present work (Theorem 2.1), on the one hand, provides a lower bound of the class of functions for which the normalized transposed inverse set is effective in the Faber region. …”
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Existence Results for Nonsmooth Vector Quasi-Variational-Like Inequalities by Mohammed Alshahrani, Qamrul Hasan Ansari, Suliman Al-Homidan

“…We establish some existence results for solutions of these inequalities by using Fan-KKM theorem and a maximal element theorem. By using the technique and methodology adopted in Al-Homidan et al. (2012), one can easily derive the relations among these inequalities and a vector quasi-optimization problem. …”
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Differentiable semigroups are Lie groups by John P. Holmes, Mitch Anderson

“…We present here a modern, detailed proof to the following theorem which was introduced by Garrett Birkhoff [1] in 1938. …”
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Some Qualitative Properties of Traveling Wave Fronts of Nonlocal Diffusive Competition-Cooperation Systems of Three Species with Delays by Meiping Yao, Pengzhi Qiao, Yang Wang

“…By modified version of Ikehara’s theorem, we prove that the traveling wave fronts of such system decay exponentially at negative infinity, and one component of such solutions also decays exponentially at positive infinity. …”
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A Study on ψ-Caputo-Type Hybrid Multifractional Differential Equations with Hybrid Boundary Conditions by Fouad Fredj, Hadda Hammouche, Mohammed S. Abdo, Wedad Albalawi, Abdulrazak H. Almaliki

“…Using an advantageous generalization of Krasnoselskii’s fixed point theorem, we establish results of at least one solution, whereas the uniqueness of solution is derived via Banach’s fixed point theorem. …”
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Backward Stochastic Differential Equations Coupled with Value Function and Related Optimal Control Problems by Tao Hao, Juan Li

“…We prove the existence and the uniqueness theorem as well as a comparison theorem for such BSDEs coupled with value function by using the approximation method. …”
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Graph Invariants and Large Cycles: A Survey by Zh. G. Nikoghosyan

“…These simplest kind of relations having no forerunners in the area actually form a source from which nearly all possible hamiltonian results (including well-known Ore's theorem, Posa's theorem, and many other generalizations) can be developed further by various additional new ideas, generalizations, extensions, restrictions, and structural limitations.…”
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Partition detection for large scale ad hoc networks by LI Jian-dong, TIAN Ye, SHENG Min, ZHANG Yan, YAO Jun-liang

“…As the failure of a critical node will directly partition a network,a theorem for critical node identification was proved,which indicates that node degree Ni and elementary loop degree Mi of node i are two decisive factors for the exis-tence of a critical node and shows that Ni-Mi≥ 2 is the necessary and sufficient condition for node i being critical.Based on the theorem,a distributed topology partition detection algorithm（distributed partition detection protocol,DPDP） was presented for large scale networks,which achieves the goal of partition detection efficiently by detecting critical nodes in a local area.Theoretic analysis and experiment results show that DPDP has the advantages of low complexity,high accuracy,low cost as well as good scalability,and is superior to other algorithms.…”
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Asymptotic Behavior of Solutions of Delayed Difference Equations by J. Diblík, I. Hlavičková

“…This is achieved by the application of a more general theorem which deals with systems of first-order difference equations. …”
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Structure of weakly periodic rings with potent extended commutators by Adil Yaqub

“…A well-known theorem of Jacobson (1964, page 217) asserts that a ring R with the property that, for each x in R, there exists an integer n(x)>1 such that xn(x)=x is necessarily commutative. …”
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Fixed points of a certain class of mappings in spaces with uniformly normal structure by Jong Soo Jung, Balwant Singh Thakur, Daya Ram Sahu

“…This result generalizes a fixed point theorem of Górnicki [1].…”
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Nonlinear Langevin Equation of Hadamard-Caputo Type Fractional Derivatives with Nonlocal Fractional Integral Conditions by Jessada Tariboon, Sotiris K. Ntouyas, Chatthai Thaiprayoon

“…A variety of fixed point theorems are used, such as Banach’s fixed point theorem, Krasnoselskii’s fixed point theorem, Leray-Schauder’s nonlinear alternative, and Leray-Schauder’s degree theory. …”
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Dynamic System Analysis of Investment in Both Production and R&D with Time Delay by Debao Gao

“…When both time delays are zero, it is proved that the model is positively invariant, uniformly bounded, and globally asymptotically stable by using the comparison principle and Bendixson–Dulac theorem. When at least one time delay is not zero, according to Hopf bifurcation theorem, the conditions of local asymptotic stability and existence of periodic solution of investment model are obtained. …”
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Rings with a finite set of nonnilpotents by Mohan S. Putcha, Adil Yaqub

“…The ring R is called a θn-ring if the number of elements in R which are not in N is at most n. The following theorem is proved: If R is a θn-ring, then R is nil or R is finite. …”
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Time series aggregation, disaggregation and long memory by Dmitrij Celov, Remigijus Leipus

“…Finally the theorem is illustrated by FARUMA mixture densityÆs example. …”
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An Impulsive Three-Species Model with Square Root Functional Response and Mutual Interference of Predator by Zhen Wang, Yuanfu Shao, Xianjia Fang, Xiangmin Ma

“…By using techniques of impulsive perturbations, comparison theorem, and Floquet theory, the existence and global asymptotic stability of prey-eradication periodic solution are investigated. …”
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On the Convergence of a Family of Chlodowsky Type Bernstein-Stancu-Schurer Operators by Lian-Ta Shu, Guorong Zhou, Qing-Bo Cai

“…We obtain the estimates of moments and central moments of these operators, obtain weighted approximation theorem, establish local approximation theorems by the usual and the second order modulus of continuity, estimate the rate of convergence, give a convergence theorem for the Lipschitz continuous functions, and also obtain a Voronovskaja-type asymptotic formula. …”
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An Analytical and Approximate Solution for Nonlinear Volterra Partial Integro-Differential Equations with a Weakly Singular Kernel Using the Fractional Differential Transform Metho... by Rezvan Ghoochani-Shirvan, Jafar Saberi-Nadjafi, Morteza Gachpazan

“…The result of the theorem will be used to solve a weakly singular Volterra integral equation later on.…”
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Solutions of Second-Order m-Point Boundary Value Problems for Impulsive Dynamic Equations on Time Scales by Xue Xu, Yong Wang

“…The existence and uniqueness of positive solutions are established by using the mixed monotone fixed point theorem on cone and Krasnosel’skii fixed point theorem. …”
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