Preferred Language
Details
Publication Date
Thu Sep 30 2021
Journal Name
Iraqi Journal Of Science
Volume
62
Issue Number
9
DOI
10.24996/ijs.2021.62.9.25
Choose Citation Style
Statistics
Abstract Views
174
Galley Views
101
Statistics
Scopus (1)
Articles
/
ijs-3525
On Invariant Approximations in Modular Spaces
Banach operator pair
best approximation
fixed point
modular space
non-expansive mappings
Nadia Jasim
Salwa Salman
...Show More Authors

     This article is devoted to presenting results on invariant approximations over a non-star-shsped weakly compact subset of a complete modular space by introduced a new notion called S-star-shaped with center f:  if   be a mapping and , . Then the existence of common invariant best approximation is proved for Banach operator pair of mappings by combined the hypotheses with Opial’s condition or demi-closeness condition

Scopus Crossref
View Publication Preview PDF
Quick Preview PDF
Related publications
Publication Date
Sat Sep 01 2018
Journal Name
Polyhedron
Novel dichloro (bis {2-[1-(4-methylphenyl)-1H-1, 2, 3-triazol-4-yl-κN3] pyridine-κN}) metal (II) coordination compounds of seven transition metals (Mn, Fe, Co, Ni, Cu, Zn and Cd)
The 1
3-dipolar cycloaddition “click” reaction between an azide and an alkyne to give a 1
2
3-triazole was reported by Huisgen in 1961 [1]. In 2002 the Copper(I)-catalyzed Azide-Alkyne Cycloaddition (CuAAC) to prepare a 1
2
3-triazole was reported [2]
[3]. The existence of relatively basic nitrogen atoms in the 1
2
3-triazole rings
and the possibility of introducing additional donor groups in the substituents (Fig. 1)
made the CuAAC “click” reaction an attractive method to prepare differently substituted 1
2
3-triazoles. These compounds have been used as ligands to coordinate to various metal ions that display a range of applications such as in electrochemical and photochemical studies
in supramolecular chemistry
magnetism
metal-ion sensing and catalysis [4]. The reasons for the success of the “click” reaction
is that it is easy to carry out and is widely applicable. It is not affected by a variety of functional groups
and can be carried out with a variety of Cu(I) catalysts and solvents
including aqueous conditions. The Cu(I) catalysts overcome the high activation energy barrier of the non-catalyzed Huisgen reaction by changing the mechanism of the reaction. A large variety of copper catalysts can be used for the CuAAC reaction
on condition that the maximum concentration of Cu(I) species is generated during the reaction. The pre-catalyst can be a Cu(II) salt (usually CuSO4) together with a reducing agent (often sodium ascorbate) or a Cu(I) compound in the presence of a base or amine ligand and a reducing agent to prevent oxidation to Cu(II). Some strong oxidising cupric salts or complexes such as Cu(OAc)2 also work. The solvent is very flexible from organic to aqueous
with the most commonly used combination water + an alcohol (t-BuOH
MeOH or EtOH). The key role of the solvent or solvent mixture is to solubilize the substrates and Cu(I) catalyst in order to ensure rapid reactions. Such aqueous conditions are very useful for biochemical conjugations
as well as for organic syntheses. We recently reported on the synthesis of a series of differently substituted 1
2
3-triazole chromophores
the substituted 2-(1-phenyl-1H-1
2
3-triazol-4-yl)pyridine ligands [5]
see Fig. 2 middle
with substituents R = H (L1)
CH3 (L2)
OCH3 (L3)
COOH
F
Cl
CN
CF3
O(CH2)3CH3 and N(CH3)2. These versatile ligands were found to coordinate to various first row transition metals
such as manganese
cobalt and nickel [6]. Here we extend the series to include more first row transition metal(II) coordination compounds
iron
copper and zinc
as well as a second row transition metal(II) coordination compound
cadmium
containing the 2-(1-(4-methyl-phenyl)-1H-1
2
3-triazol-1-yl)pyridine chromophore (Fig. 2 right with R = CH3). This series of seven novel coordination compounds is the first series of pyridyl-triazole based transition metal coordination compounds where seven different transition metals are coordinated to the same 1
2
3-triazole chromophore
namely 2-(1-(4-methyl-phenyl)-1H-1
2
3-triazol-1-yl)pyridine.
Kinaan M
...Show More Authors

Preview PDF