Fixed point theorems for φ-contractions
© Samreen et al.; licensee Springer 2014
Received: 19 February 2014
Accepted: 13 June 2014
Published: 22 July 2014
This paper deals with the fixed point theorems for mappings satisfying a contractive condition involving a gauge function φ when the underlying set is endowed with a b-metric. Our results generalize/extend the main results of Proinov and thus we obtain as special cases some results of Mysovskih, Rheinboldt, Gel’man, and Huang. We also furnish an example to substantiate the validity of our results. Subsequently, an existence theorem for the solution of initial value problem has also been established.
1 Introduction and preliminaries
The Banach contraction principle has been extensively used to study the existence of solutions for the nonlinear Volterra integral equations and nonlinear integro-differential equations and to prove the convergence of algorithms in computational mathematics. These applications elicit the significance of fixed point theory. Therefore mathematicians have been propelled to contribute enormously in the field of fixed point theory by finding the fixed point(s) of self-mappings or nonself-mappings defined on several ambient spaces and satisfying a variety of conditions. Among these fixed point theorems only a few have practical importance, i.e., they provide a constructive method for finding fixed point(s). This provides information on the convergence rate along with error estimates. The Banach contraction principle is one of such theorems wherein the proposed iterative scheme converges linearly. Commonly, the iterative procedures serve as constructive methods in fixed point theory. Furthermore, it is also of crucial importance to have prior and posterior estimates for such methods. In this context, Proinov  extended the Banach contraction principle with a higher order of convergence. He proposed an iterative scheme for a mapping satisfying a contractive condition which involves a gauge function of order and obtained error estimates as well. His results include as special cases some results of Mysovskih , Rheinboldt , Gel’man , Huang , and others. In  the authors extended the results of Proinov to the case of multivalued mappings.
For the last few decades fixed point theory has rapidly been evolving, not only in metric structure but also in many different generalized spaces and the b-metric space is one of them. The notion of a b-metric space was initiated in some works of Bourbaki, Bakhtin, Czerwik, and Heinonen. Several papers appeared which deal with the fixed point theory for single valued and multivalued functions in a b-metric space [7–11]etc.
Inspired by the work of Proinov  in this paper we investigate whether the consequences of his results hold when the underlying structure is replaced with a b-metric space. We give an affirmative answer to this question. Our results generalize main results of Proinov  and thus subsume many results of authors [2–5]. We establish an example to substantiate the validity of our results. Consequently, in Section 3 we also obtain an existence theorem for the solution of an initial value problem.
Let X be a nonempty set and be a given real number. A function is said to be a b-metric space if and only if for all the following conditions are satisfied:
(d1) if and only if ;
The pair is called a b-metric space with the coefficient s.
The following example shows that the class of b-metric spaces is essentially larger than the class of metric spaces.
- (2)Let be the space of all real functions , such that . Define as
Then is a b-metric space with coefficient
A sequence in a b-metric space X is convergent if and only if there exists such that as and we write ; it is Cauchy if and only if as . A b-metric space is complete if every Cauchy sequence in X converges.
Let be a b-metric space; then a convergent sequence has a unique limit; every convergent sequence is Cauchy; and in general the b-metric d is not a continuous functional .
Definition 1.3 Let be a b-metric space and A be a nonempty subset of X then closure of A is the set consisting of all points of A and its limit points. Moreover, A is closed if and only if .
In the following the b-metric version of Cantor’s intersection theorem is given, which can easily be established running along the same lines as in the proof of its metric version.
Theorem 1.4 
Let be a complete b-metric space, then every nested sequence of closed balls has a nonempty intersection.
Let and there exist some such that the set . The set is known as an orbit of . We recall that a function G from D into the set of real numbers is said to be f-orbitally lower semi-continuous at if and implies .
Throughout this paper let J always denote an interval in containing 0 i.e., an interval of the form , or ( is a trivial interval). Let denote a polynomial of the form and . Let denote the n th iterate of a function .
Definition 1.5 
for all and ,
for all .
The condition (i) of Definition 1.5 elicits and is nondecreasing on . A gauge function is said to be a Bianchini-Grandolfi gauge function if for all .
where f satisfies (1.1).
2 b-Bianchini-Grandolfi gauge functions
where s is the coefficient of b-metric d.
, is a gauge function of order 1 on ;
(, ) is a gauge function of order r on where .
It is essential to mention here that to establish the fixed point theorem (see Theorem 3.7) we do not necessarily require the gauge functions φ satisfying (2.1), (2.2). But we consider the gauge function such that for all where s is a coefficient of b-metric space.
for all ,
for all and .
Remark 2.3 When d is a simple metric, then . In such case every gauge function satisfying is of the form where ϕ is nonnegative nondecreasing function on J (see ). Thus in such case the condition for all becomes superfluous and directly follows from Lemma 2.2.
The following lemma is fundamental to our main results.
for all ,
for all .
which completes the proof. □
It is easy to see that every b-Bianchini-Grandolfi gauge function is also a Bianchini-Grandolfi  gauge function but the converse may not hold. A b-Bianchini-Grandolfi gauge function having coefficient is also a b-Bianchini-Grandolfi gauge function having coefficient for every .
From now on, we always assume that the coefficient of the b-Bianchini-Grandolfi gauge function is at least as large as the coefficient of the b-metric space.
Lemma 2.6 Every gauge function of order defined by (2.1) and (2.2) is a b-Bianchini-Grandolfi gauge function with coefficient .
Proof It is immediately follows from the first part of Lemma 2.4 and using the fact that for and . □
3 Fixed point theorems
Lemma 3.1 Suppose is such that . Assume that ; then for all .
Hence, . Similarly, iterating successively we get for all . □
Definition 3.2 Suppose is such that and . Then for every iterate , we define the closed ball with center at and radius , where is defined by (2.4).
Lemma 3.3 Suppose is such that and . Assume that for some ; then and .
Hence, . □
Definition 3.4 (Initial orbital point)
We say that a point is an initial orbital point of f if and .
The following lemma is obvious.
where and ϕ is nonnegative nondecreasing on J satisfying (2.1) and (2.2).
where , and .
- (1)Proof From definition of we have(3.2)
- (2)From (3.2) we have
- (3)By making use of first part of Lemma 3.5 above we have
- (4)Now by making use of Lemma 2.4 we have
- (5)From (4) we have
Now we proceed to formulate the following fixed point theorems.
If and the function on D is f-orbitally lower semi-continuous at ξ, then ξ is a fixed point of f.
Remark 3.8 Theorem 3.7 gives a generalization of [, Theorem 4.1] and extends it to the case of b-metric spaces. It reduces to [, Theorem 4.1] when . Hence Theorem 3.7 not only extends the result of Proinov  but in turn it also includes results of Bianchini and Grandolfi  and Hicks  as special cases.
Corollary 3.9 [, Theorem 4.1]
where φ is a Bianchini-Grandolfi gauge function on an interval J. Then starting from an initial orbital point of f the iterative sequence remains in and converges to a point ξ which belongs to each of the closed balls , where and . Moreover, if and f is continuous at ξ, then ξ is a fixed point of f.
This implies f-orbital lower semi-continuity of at point ξ. Hence the conclusion follows from Theorem 3.7. □
- (1)The iterative sequence (1.2) remains in and converges with rate of convergence at least to a point ξ which belongs to each of the closed balls , , and(3.7)
- (2)For all the following prior estimate holds:(3.8)
- (3)For all the following posterior estimate holds:(3.9)
- (4)We have(3.10)
If and the function on D is f-orbitally lower semi-continuous at ξ, then ξ is a fixed point of f.
- (2)For ,
- (3)From (3.11) we have for ,
- (4)We have
Its proof runs along the same lines as the proof of Theorem 3.7. □
Remark 3.11 For , Theorem 3.10 reduces to [, Theorem 4.2]. It also generalizes (taking and , ) results of Ortega and Rheinboldt [, Section 12.3.2], Kornstaedt [, Satz 4.1], Hicks and Rhoades , and Park [, Theorem 2]. The first two conclusions of Theorem 3.10 are due to Gel’man [, Theorem 3] (taking and , , ). It also yields some results of Hicks [, Theorem 3].
The iterative sequence (1.2) converges to a fixed point ξ of f.
The operator f has a unique fixed point in .
The estimates (3.7)-(3.10) are valid.
which gives the continuity of f in b-metric space . Thus conclusions (i) and (iii) follow immediately from Theorem 3.10. Let be another fixed point of f in S; then . It follows from (3.16) that , which yields . □
Remark 3.13 For when the b-metric space under consideration is a simple metric space, the above corollary coincides with [, Corollary 4.4]. Thus the conclusions of Corollary 3.12 are consequences of the results of Matkowski .
4 Application and illustrative example
The following example illuminates the degree of generality of our result.
Setting on then φ is a b-Bianchini-Grandolfi gauge function with coefficient having order 2. Moreover, it is easily seen that all conditions of Theorem 3.7 are satisfied.
which contradicts the definition of φ. Hence, one cannot invoke the main results of Proinov [, Theorems 4.1, 4.2, Corollary 4.4].
f is continuous;
- (ii)f satisfies the condition(4.2)
- (iii)f is bounded on R, i.e.,(4.3)
where , and .
Then the initial value problem (4.1) has a unique solution on the interval .
Thus from (4.7) we obtain for all and . Further, for any it is easily seen that , which yields for . Therefore, all the conditions of Corollary 3.12 are satisfied. Hence the iterative sequence ; converges to the unique fixed point t of T at a rate of convergence . On the other hand, Picard’s iterations converge to the solution linearly. □
In Section 3 we have established two convergence theorems in the setting of a b-metric such that the self-mapping satisfies a contraction condition involving a gauge function of order . The gauge function φ has to satisfy the condition where is the coefficient of the underlying b-metric space. An example has been furnished to assess the degree of generality of our results. In Section 4 we established an existence theorem for the solution of an initial value problem, which not only gives the unique solution but also locates the domain for the solution.
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