^{1}

^{2}

^{2}

^{1}

^{2}

The purpose of this paper is to prove some fixed point results for mapping without continuity condition on Takahashi convex metric space as an application of synthetic approaches to fixed point problems of Angrisani and Clavelli. Our results are generalizations in Banach space of fixed point results proved by Kirk and Saliga, 2000; Ahmed and Zeyada, 2010.

It is well-known that continuity is an ideal property, while in some applications the mapping under consideration may not be continuous, yet at the same time it may be “not very discontinuous.”

In [

Function

An equivalent condition to be r.g.i. on metric space for

Let

One of the basic results in [

Let

The last theorem assures that if

Let

Let

It is easy to prove that for

Moreover, these two measures of noncompactness are equivalent in the sense that

In Banach spaces this function has some additional properties connected with the linear structure. One of these is

This property has a great importance in fixed point theory. In locally convex spaces this is always true, but when topological vector space is not locally convex it need not be true (see [

In the absence of linear structure the concept of convexity can be introduced in an abstract form. In metric spaces at first it was done by Menger in 1928. In 1970 Takahashi [

Let

Any convex subset of a normed space is a convex metric space with

Let

Let

balls (either open or closed) in

intersections of convex subsets of

For fixed

A TCS

Obviously in a normed space the last inequality is always satisfied.

Let

For

Property

A TCS

Let

Talman in [

Any TCS satisfying

Let

Some, among the many studies concerning the fixed point theory in convex metric spaces, can be found in [

Measures of noncompactness which arise in the study of fixed point theory usually involve the study of either condensing mappings or

Let

Choose a point

Since

Properties (1)–(4) of measure

Now, Proposition

The assumption

Let us recall some well-known definitions. A mapping

Let

We prove (

In order to prove that (

Now, using inequality

Further, the sequence

By the last contradiction we conclude that

This statement is a generalization of Lemma 9.4 from [

Combining the last result with Theorem

Let

Moreover, using Proposition

Let

Proposition

Let

By Proposition

We established that

Next, we recall the concept of

Let

The next result is improvement of Theorem

Let

Our assertion is a consequence of Theorem

Using Proposition

Let

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are very grateful to the anonymous referees for their careful reading of the paper and suggestions which have contributed to the improvement of the paper. This paper is partially supported by Ministarstvo nauke i