Fixed point theorems for operator in cone Banach spaces
© Mutlu and Yolcu; licensee Springer 2013
Received: 10 December 2012
Accepted: 25 February 2013
Published: 14 March 2013
In this paper a class of self-mappings on cone Banach spaces which have at least one fixed point is considered. More precisely, for a closed and convex subset C of a cone Banach space with the norm , if there exist a, b, c, r and satisfies the conditions and for all , then T has at least one fixed point.
Keywordscone metric space complete cone metric space fixed point
for all has a unique fixed point. Recently, many results on fixed point theorems have been extended to cone metric spaces in [3, 4]. Karapınar  extended some of well known results in the fixed point theory to cone Banach spaces which were defined and used in  where the existence of fixed points for self-mappings on cone Banach spaces were investigated.
In this study, we prove the fixed point theorems of operator for cone Banach spaces.
Throughout this paper E means Banach algebra, stands for real Banach space. Let always be a closed nonempty subset of E. P is called a cone if for all and nonnegative real numbers a, b where and . Given a cone , we define a partial ordering ≤ with respect to P by if and only if . We will write to indicate that but , while will stand for , where intP denotes the interior of P. The cone P is called normal if there is a number such that implies for all . The least positive number satisfying the above is called the normal constant.
for all and if and only if ,
for all ,
for all .
Then is called a cone metric space (CMS).
Definition 1 (see )
for all ,
if and only if ,
for all ,
for all and for all ,
then is called a cone norm on X, and the pair is called a cone normed space (CNS).
Definition 2 (see )
converges to x whenever for every with , there is a natural number N such that for all . It is denoted by or ;
is a Cauchy sequence whenever for every with , there is a natural number N, such that for all ;
is a complete cone normed space if every Cauchy sequence is convergent.
Complete cone normed spaces will be called cone Banach spaces.
the sequence converges to x if and only if as ,
the sequence is Cauchy if and only if as ,
the sequence converges to x and the sequence converges to y, then .
Proof See  Lemmas 1, 4 and 5. □
Definition 4 (see )
then T is called nonexpansive if it satisfies the condition (2).
2 Main result
From now on, will be a cone Banach space, P will be a normal cone with a normal constant K and T, a self-mapping operator defined on a subset C of X. Let be an increasing, positive and self-mapping operator defined on E, where E is a Banach algebras. In this paper, we give a generalization of Theorem 2.4 in  for operator.
Definition 5 Let E be Banach algebra and be a Banach space. is an increasing and positive mapping, i.e., , where .
If , then is a p-Laplacian operator, i.e., for some .
if , then for all ,
is a continuous bijection and its inverse mapping is also continuous,
for all ,
for all .
for all , where in E. Then T has at least one fixed point.
Since , is a Cauchy sequence in C. Because C is a closed and convex subset of a cone Banach space, thus sequence converges to some . That is, , .
as , then . Thus .
when , . Then . □
for all , where , then T has at least one fixed point.
Thus is a Cauchy sequence in C and thus it converges to some . As in the proof of Theorem 6, we can show . □
The authors did not provide this information
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