Category: SCIENCE BLOG

Fixed point theory occupies a prominent place in the study of metric spaces. One of the important questions that may arise in this connection is whether metric spaces really provide enough space for this theory or not. Recently, in Huang and Zhang (2007), the authors rather implied that the answer is no. Actually, they introduced the notion of cone metric space, and gave an example of a function which is a contraction in the category of cone metric spaces but not a contraction if considered over metric spaces and hence, by proving a fixed point theorem in cone metric spaces, ensured that this map must have a unique fixed point. After that a series of articles about cone metric spaces started to appear. Some of those articles dealt with fixed point theorems in those spaces, specially in complete ones, and others dealt with the structure of the spaces themselves.
On the other hand Cone metric spaces were introduced in Huang and Zhang (2007). The authors there described convergence in cone metric spaces, and introduced completeness. Then they proved some fixed point theorems of contractive mappings on cone metric spaces, where they also provided an example of a contractive mapping on a cone metric space with E = R2 and P = (x, y) ∈ R2: x, y ≥ 0. However this mapping is not contractive in the Euclidean metric space. Then recently, in (Ili and Rako, 2008; Abbas, and Jungck, 2008) some common fixed point theorems were proved for maps on cone metric spaces. In this paper we introduce some basic topological concepts and definitions in cone metric spaces. Actually, we prove that every cone metric space is a topological space. Then we generalize the concepts diametrical contractivity and asymptotical diametrical contractivity to cone metric spaces. Finally we obtain some fixed point theorems for such mappings. By this we generalize some results from Xu(2004) to cone metric spaces.

Herbs are those plants that are used either as a food or for treating diseases. There are several herbs that we use on daily basis in our kitchen which are very helpful to maintain a good health. Let me inform you some of them.

Ferula Asafoetida: It is the member of Apiaceae, commonly known as Hing, have very unpleasant smell but this oleo-resin gum is carminative, antispasmodic, stimulant, digestive, sedative, expectorant and diuretic. It is helpful in treating flatulent colic, dyspepsia, asthama, hysteria, and cholera.

Curcuma Longa: The member of family Zingiberaceae, commonly known as Haldi or turmeric. Its rhizome is aromatic, stimulant, anti-periodic, tonic, and carminative. It is given in diarrhoea, dropsy, jaundice, liver disorder, and urinary troubles. It is commonly used for cleaning foul ulcers. Fresh juice is used for many skin infections.

Zingiber Officinale: Member of family Zingiberaceae, commonly known as Adrak or Ginger. Its rhizome is used as stimulant, carminative, and flavouring agent. It is given in dyspepsia and flatulent colic. It makes valuable drugs for disorders of digestive systems, rheumatism, piles, pulmonary disease, dropsy etc.

Cinnamomum Zeylanicum: Member of family Lauraceae, commonly known as Dalchini or Cinnamon. Its bark is given in diarrhoea, nausea, gastric irritation, flatulence, vomiting, spasmodic affection of bowels, tooth ache, paralysis of tongue, and labour caused by defective utring contraction.

Cuminum Cyminum: Member of family Umbelliferae, commonly known as Zeera or Cumin. Its fruit is a good source of thymol. It is prescribed in dyspepsia, diarrhoea, and hoarseness of voice. The fruit is given with lime juice to pregnant women for checking nausea and for promoting secretion of milk, cumin seeds are given after child birth.

Foeniculum Vulgare: Member of family Umbelliferae, commonly known as Saunf or Fennel. Its fruit is sweet, laxative, stomachic, stimulant, aromatic, corrective of flatulence, appetizer, and carminative. It is used in curing burning sensation, fever, excessive thirst, dysentery, chest diseases, and headache.