Abstract
In this paper, we extend the concept of contraction mappings in b-metric spaces and utilize this concept to prove the existence and uniqueness of fixed point theorems for such mappings in such a space. We also prove the generalized Ulam-Hyers stability and well-posed results for a fixed point equation employing the concept of α-admissibility in b-metric spaces. We shall construct some examples to support our novel results.
MSC:46S40, 47S40, 47H10.
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1 Introduction
The classical Banach contraction principle is a very important tool in solving existence problems in many branches of mathematics. Over the years, it has been generalized in several different directions by several mathematicians (see [1–7]). In 1993, Czerwik [8] introduced and proved the contraction mapping principle in b-metric spaces that generalized the famous Banach contraction principle in such spaces. Subsequently several other authors [9–15] have studied and established the existence of fixed points of a contractive mapping in b-metric spaces.
The study of stability problems for various functional equations play the most important role in mathematical analysis. In the fall of 1940, Ulam [16] discussed a number of important unsolved mathematical problems. Among them, a question concerning the stability of homomorphisms seemed too abstract for anyone to reach any conclusion. In the following year, Hyers [17] gave a first affirmative partial answer to Ulam’s question for Banach spaces, this type of stability is called Ulam-Hyers stability. A large number of papers have been published in connection with various generalizations of Ulam-Hyers stability results in fixed point theory and remarkable result on the stability of certain classes of functional equations via fixed point approach (see [18–29] and references therein).
On the other hand, Samet et al. [30] introduced the concepts of α-ψ-contractive mapping and α-admissible self-mappings. Also, they proved some fixed point results for such mappings in complete metric spaces. Naturally, many authors have started to investigate the existence of a fixed point theorem via α-admissible mappings for single valued and multivalued mappings (see [31–38]). Recently Bota et al. [39] considered the existence and the uniqueness of fixed point theorems and generalized Ulam-Hyers stability results via α-admissible mappings in b-metric spaces.
In this paper, we extend the concept of α-ψ-contractive mapping in b-metric spaces. By using this concept, we establish the existence and uniqueness of fixed point for some new types of contractive mappings in b-metric spaces and give an example to illustrate our main results. Moreover, we study and prove the generalized Ulam-Hyers stability and well-posed results by using fixed point method via α-admissible mappings in b-metric spaces.
2 Preliminaries
Throughout this paper, we shall use the following notation.
Let X be a nonempty set and the functional satisfy:
(b1) if and only if ,
(b2) for all ,
(b3) there exists a real number such that , for all .
Then d is called a b-metric on X and a pair is called a b-metric space with coefficient s.
Remark 2.2 If we take in above definition then b-metric spaces turns into usual metric spaces. Hence, the class of b-metric spaces is larger than the class of usual metric spaces.
Examples of b-metric spaces were given in [8, 40–43].
Example 2.3 The set with , where , together with the functional ,
(where ) is a b-metric spaces with coefficient . Notice that the above result holds for the general case with , where X is a Banach spaces.
Example 2.4 Let X be a set with the cardinal . Suppose that is a partition of X such that . Let be arbitrary. Then the functional defined by
is a b-metric on X with coefficient .
Definition 2.5 ([42])
Let be a b-metric spaces. Then a sequence in X is called
-
(a)
convergent if and only if there exists such that as ;
-
(b)
Cauchy if and only if as .
Lemma 2.6 ([41])
Let be a b-metric spaces and let . Then
Definition 2.7 ([21])
A mapping is called a comparison function if it is increasing and as , for any .
If is a comparison function, then
-
(1)
is also a comparison function, where is nth iterate of ψ;
-
(2)
ψ is continuous at 0;
-
(3)
, for any .
The concept of -comparison function was introduced by Berinde [44] in the following definition.
Definition 2.9 A function is said to be a -comparison function if
-
(1)
ψ is increasing;
-
(2)
there exist , and a convergent series of nonnegative terms such that , for and any .
Here we recall the definitions of the following class of -comparison function as given by Berinde [45] in order to extend some fixed point results to the class of b-metric spaces.
Definition 2.10 ([45])
Let be a real number. A mapping is called a -comparison function if the following conditions are fulfilled:
-
(1)
ψ is increasing;
-
(2)
there exist , , and a convergent series of nonnegative terms such that , for and any .
In this work, we denote by the class of -comparison function . It is evident that the concept of -comparison function reduces to that of -comparison function when .
Lemma 2.11 ([43])
If is a -comparison function, then we have the following:
-
(i)
the series converges for any ;
-
(ii)
the function , defined by , , is increasing and continuous at 0.
Next, we will present the concept of α-admissible mappings introduced by Samet et al. [30].
Definition 2.12 ([30])
Let X be a nonempty set, and . We say that f is an α-admissible mapping if it satisfies the following condition:
Example 2.13 Let . Define and by
and
Then f is α-admissible.
Example 2.14 Let . Define and by
and
Then f is α-admissible.
3 Fixed point theorems for α-admissible mapping in b-metric spaces
In this section, we prove the existence and uniqueness of fixed point theorems in a b-metric space.
Definition 3.1 Let be a b-metric space with coefficient s. A mapping is said to be a generalized α-ψ-contraction in b-metric space if there exist functions and such that the following condition holds:
Theorem 3.2 Let be a complete b-metric space with coefficient s and f be a generalized α-ψ-contraction. Suppose that the following conditions hold:
-
(a)
f is an α-admissible;
-
(b)
there exists such that ;
-
(c)
if is sequence in X such that as and for all , then .
Then f has a unique fixed point in X such that .
Proof Let such that (from condition (b)). We define the sequence in X such that
Since f is an α-admissible and
we deduce that
By continuing this process, we get for all . This implies that
for all . From (3.1), we obtain
for all . By repeating the above process, we get
for all . Next, we show that is a Cauchy sequence in X. For with , we have
Denote for all . This implies that
By Lemma 2.11 we know that the series converges. Therefore, is Cauchy sequence in X. By the completeness of X, there exists such that as . Using condition (c), we get . Also, we have for all . From the assumption (3.1), we have
Letting , it follows that , that is, is a fixed point of f such that .
Next, we prove the uniqueness of the fixed point of f. Let be another fixed point of f such that
Therefore, we get
It follows that
which is a contradiction. Therefore, is the unique fixed point of f such that . This completes the proof. □
In view of Theorem 3.2, we have the following corollary.
Corollary 3.3 Let be a complete b-metric space with coefficient s, , , and be three mappings. Suppose that the following conditions hold:
-
(a)
f is an α-admissible;
-
(b)
there exists such that ;
-
(c)
if is sequence in X such that as and for all , then ;
-
(d)
f satisfies the following condition:
(3.4)
for all .
Then f has a unique fixed point in X such that .
Corollary 3.4 Let be a complete b-metric space with coefficient s, , , and be three mappings. Suppose that the following conditions hold:
-
(a)
f is an α-admissible;
-
(b)
there exists such that ;
-
(c)
if is sequence in X such that as and for all , then ;
-
(d)
f satisfies the following condition:
(3.5)
for all , where .
Then f has a unique fixed point in X such that .
Corollary 3.5 Let be a complete b-metric space with coefficient s, , , and be three mappings. Suppose that the following conditions hold:
-
(a)
f is an α-admissible;
-
(b)
there exists such that ;
-
(c)
if is sequence in X such that as and for all , then ;
-
(d)
f satisfies the following condition:
(3.6)
for all , where .
Then f has a unique fixed point in X such that .
If we set for all in Theorem 3.2, we get the following results.
Corollary 3.6 Let be a complete b-metric space with coefficient s and be a mapping. Suppose that f satisfies
for all , where . Then f has a unique fixed point in X.
If the coefficient in Corollary 3.6, we immediately get the following result.
Corollary 3.7 [46]
Let be a complete metric space and be -comparison function. Suppose that be a mapping satisfies
for all . Then f has a unique fixed point in X.
Remark 3.8 If , where in Corollary 3.7, we get the Banach contraction principle.
Next, we give an example showing that the contractive conditions in our results are independent. Also, our results are real generalizations of the Banach contraction principle in b-metric spaces and several results in literature.
Example 3.9 Let and for all . Then d is a complete b-metric space on X with coefficient . Define by
Also, define and by
and for all .
Now, we show that f is a generalized α-ψ-contraction mapping. For with
we get . Then we have
It is easy to see that f is an α-admissible mapping. There exists such that
Also, we can easily to prove that condition (c) in Theorem 3.2 holds. Therefore, all of conditions in Theorem 3.2 hold. In this example, we have 1 is a unique fixed point of f and .
Remark 3.10 We observe that the contractive condition in Corollary 3.4 cannot be applied to this example. Indeed, for and , we obtain
where and . Therefore, Corollary 3.4 cannot be applied to this case. Also, by a similar method, we can show that Corollary 3.5 cannot be applied to this case.
Also, we can see that the fixed point result for Banach contraction principle in b-metric spaces cannot be applied to this case. Indeed, for and , we get
4 The generalized Ulam-Hyers stability in b-metric spaces
In this section, we prove the generalized Ulam-Hyers stability in b-metric spaces for which Theorem 3.2 holds.
Let be a b-metric spaces with coefficient s and be an operator. Let us consider the fixed point equation
and the inequality
Theorem 4.1 Let be a complete b-metric space with coefficient s. Suppose that all the hypotheses of Theorem 3.2 hold and also that the function defined by is strictly increasing and onto. If for all which is an ε-solution, then the fixed point equation (4.1) is generalized Ulam-Hyers stable.
Proof By Theorem 3.2, we have , that is, is a solution of the fixed point equation (4.1). Let and is an ε-solution, that is,
Since are ε-solution, we have
Also, we have
Now, we obtain
It follows that
Since , we have
It implies that
Notice that exists, is increasing, continuous at 0 and . Therefore, the fixed point equation (4.1) is generalized Ulam-Hyers stable. □
Corollary 4.2 Let be a complete b-metric space with coefficient s. Suppose that all the hypotheses of Corollary 3.3 hold and also that the function defined by is strictly increasing and onto. If for all which is an ε-solution, then the fixed point equation (4.1) is generalized Ulam-Hyers stable.
Corollary 4.3 Let be a complete b-metric space with coefficient s. Suppose that all the hypotheses of Corollary 3.4 hold and also that the function defined by is strictly increasing and onto. If for all which is an ε-solution, then the fixed point equation (4.1) is generalized Ulam-Hyers stable.
Corollary 4.4 Let be a complete b-metric space with coefficient s. Suppose that all the hypotheses of Corollary 3.5 hold and also that the function defined by is strictly increasing and onto. If for all which is an ε-solution, then the fixed point equation (4.1) is generalized Ulam-Hyers stable.
5 Well-posedness of a function with respect to α-admissibility in b-metric spaces
In this section, we present and prove well-posedness of a function with respect to an α-admissible mapping in b-metric spaces.
Definition 5.1 Let be a complete b-metric spaces with coefficient s and , . The fixed point problem of f is said to be well-posed with respect to α if
-
(i)
f has a unique fixed point in X such that ;
-
(ii)
for sequence in X such that , as , then , as .
In the following next theorems, we add a new condition to assure the well-posedness via α-admissibility.
-
(S)
If is sequence in X such that , as , then for all .
Theorem 5.2 Let be a complete b-metric space with coefficient s, , , and . Suppose that all the hypotheses of Theorem 3.2 and condition (S) hold. Then the fixed point equation (4.1) is well-posed with respect to α.
Proof By Theorem 3.2, there unique exists such that and . Let be sequence in X such that , as . By condition (S), we get
Also, we get
Now, we have
ψ is continuous at 0 and as . It implies that as , that is, , as . Therefore, the fixed point equation (4.1) is well-posed with respect to α. □
Corollary 5.3 Let be a complete b-metric space with coefficient s, , , and . Suppose that all the hypotheses of Corollary 3.3 and condition (S) hold. Then the fixed point equation (4.1) is well-posed with respect to α.
Corollary 5.4 Let be a complete b-metric space with coefficient s, , , and . Suppose that all the hypotheses of Corollary 3.4 and condition (S) hold. Then the fixed point equation (4.1) is well-posed with respect to α.
Corollary 5.5 Let be a complete b-metric space with coefficient s, , , and . Suppose that all the hypotheses of Corollary 3.5 and condition (S) hold. Then the fixed point equation (4.1) is well-posed with respect to α.
References
Radenović S, Kadelburg Z: Quasi-contractions on symmetric and cone symmetric spaces. Banach J. Math. Anal. 2011, 5: 38–50. 10.15352/bjma/1313362978
Van An T, Van Dung N, Kadelburg Z, Radenović S: Various generalizations of metric spaces and fixed point theorems. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 2014. 10.1007/s13398-014-0173-7
Agarwal RP, Kadelburg Z, Radenović S: On coupled fixed point results in asymmetric G -metric spaces. J. Inequal. Appl. 2013., 2013: Article ID 528
Xu S, Radenović S: Fixed point theorems of generalized Lipschitz mappings on cone metric spaces with Banach algebra without assumption of normality. Fixed Point Theory Appl. 2014., 2014: Article ID 102
Jovanović M, Kadelburg Z, Radenović S: Common fixed point results in metric-type spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 978121 10.1155/2010/978121
Ćojbašić V, Radenović S, Chauhan S: Common fixed point of generalized weakly contractive maps in 0-complete partial metric spaces. Acta Math. Sci. Ser. B 2014,34(4):1345–1356. 10.1016/S0252-9602(14)60088-6
Shah MH, Simić S, Hussain N, Sretenović A, Radenović S: Common fixed points theorems for occasionally weakly compatible pairs on cone metric type spaces. J. Comput. Anal. Appl. 2012,14(2):290–297.
Czerwik S: Contraction mappings in b -metric spaces. Acta Math. Inform. Univ. Ostrav. 1993, 1: 5–11.
Roshan JR, Shobkolaei N, Sedghi S, Parvaneh V, Radenović S:Common fixed point theorems for three maps in discontinuous -metric spaces. Acta Math. Sci. Ser. B 2014,34(5):1–12.
Azam A, Mehmood N, Ahmad J, Radenović S: Multivalued fixed point theorems in cone b -metric spaces. J. Inequal. Appl. 2013., 2013: Article ID 582
Parvaneh V, Roshan JR, Radenović S: Existence of tripled coincidence point in ordered b -metric spaces and applications to a system of integral equations. Fixed Point Theory Appl. 2013., 2013: Article ID 130
Hussain N, Dorić D, Kadelburg Z, Radenovć S: Suzuki-type fixed point results in metric type spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 126
Popović B, Radenović S, Shukla S: Fixed point results to tvs-cone b -metric spaces. Gulf J. Math. 2013, 1: 51–64.
George, R, Radenović, S, Reshma, KP, Shukla, S: Rectangular b-metric spaces and contraction principle. J. Nonlinear Sci. Appl. (2014, in press)
Sintunavarat W, Plubtieng S, Katchang P: Fixed point result and applications on b -metric space endowed with an arbitrary binary relation. Fixed Point Theory Appl. 2013., 2013: Article ID 296
Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1964.
Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941,27(4):222–224. 10.1073/pnas.27.4.222
Bota-Boriceanu MF, Petruşel A: Ulam-Hyers stability for operatorial equations. An. Ştiinţ. Univ. ‘Al.I. Cuza’ Iaşi, Mat. 2011, 57: 65–74.
Lazăr VL: Ulam-Hyers stability for partial differential inclusions. Electron. J. Qual. Theory Differ. Equ. 2012., 2012: Article ID 21
Rus IA: The theory of a metrical fixed point theorem: theoretical and applicative relevances. Fixed Point Theory 2008,9(2):541–559.
Rus IA: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca; 2001.
Rus IA: Remarks on Ulam stability of the operatorial equations. Fixed Point Theory 2009,10(2):305–320.
Tişe FA, Tişe IC: Ulam-Hyers-Rassias stability for set integral equations. Fixed Point Theory 2012,13(2):659–668.
Brzdek J, Chudziak J, Pales Z: A fixed point approach to stability of functional equations. Nonlinear Anal. TMA 2011, 74: 6728–6732. 10.1016/j.na.2011.06.052
Brzdek J, Cieplinski K: A fixed point approach to the stability of functional equations in non-Archimedean metric spaces. Nonlinear Anal. TMA 2011, 74: 6861–6867. 10.1016/j.na.2011.06.050
Brzdek J, Cieplinski K: A fixed point theorem and the Hyers-Ulam stability in non-Archimedean spaces. J. Math. Anal. Appl. 2013, 400: 68–75. 10.1016/j.jmaa.2012.11.011
Cadariu L, Gavruta L, Gavruta P: Fixed points and generalized Hyers-Ulam stability. Abstr. Appl. Anal. 2012., 2012: Article ID 712743
Sintunavarat W: Generalized Ulam-Hyers stability, well-posedness and limit shadowing of fixed point problems for α - β -contraction mapping in metric spaces. Sci. World J. 2014., 2014: Article ID 569174
Kutbi MA, Sintunavarat W: Ulam-Hyers stability and well-posedness of fixed point problems for α - λ -contraction mapping in metric spaces. Abstr. Appl. Anal. 2014., 2014: Article ID 268230
Samet B, Vetro C, Vetro P: Fixed point theorems for α - ψ -contractive type mappings. Nonlinear Anal. 2012, 75: 2154–2165. 10.1016/j.na.2011.10.014
Asl JH, Rezapour S, Shahzad N: On fixed points of α - ψ -contractive multifunctions. Fixed Point Theory Appl. 2012., 2012: Article ID 212
Mohammadi B, Rezapour S, Shahzad N: Some results on fixed points of α - ψ -Ciric generalized multifunctions. Fixed Point Theory Appl. 2013., 2013: Article ID 24
Karapınar E, Samet B: Generalized α - ψ -contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal. 2012., 2012: Article ID 793486
Agarwal RP, Sintunavarat W, Kumam P:PPF dependent fixed point theorems for an -admissible non-self mapping in the Razumikhin class. Fixed Point Theory Appl. 2013., 2013: Article ID 280
Long W, Khaleghizadeh S, Salimi P, Radenović S, Shukla S: Some new fixed point results in partial ordered metric spaces via admissible mappings. Fixed Point Theory Appl. 2014., 2014: Article ID 117
Latif A, Mongkolkeha C, Sintunavarat W: Fixed point theorems for generalized α - β -weakly contraction mappings in metric spaces and applications. Sci. World J. 2014., 2014: Article ID 784207
Kutbi MA, Sintunavarat W: Fixed point theorems for generalized -contraction multivalued mappings in α -complete metric spaces. Fixed Point Theory Appl. 2014., 2014: Article ID 139
Kutbi MA, Sintunavarat W: Fixed point analysis for multi-valued operators with graph approach by the generalized Hausdorff distance. Fixed Point Theory Appl. 2014., 2014: Article ID 142
Bota M, Karapınar E, Mleşniţe O: Ulam-Hyers stability results for fixed point problems via α - ψ -contractive mapping in -metric space. Abstr. Appl. Anal. 2013., 2013: Article ID 825293
Bakhtin IA: The contraction mapping principle in quasimetric spaces. 30. In Functional Analysis. Ulyanowsk Gos. Ped. Inst., Ulyanowsk; 1989:26–37.
Czerwik S: Nonlinear set-valued contraction mappings in b -metric spaces. Atti Semin. Mat. Fis. Univ. Modena 1998, 46: 263–276.
Boriceanu M, Bota M, Petru A: Multivalued fractals in b -metric spaces. Cent. Eur. J. Math. 2010,8(2):367–377. 10.2478/s11533-010-0009-4
Berinde V: Generalized contractions in quasimetric spaces. 3. Seminar on Fixed Point Theory 1993, 3–9. Preprint
Berinde V: Contracţii generalizate şi aplicaţii. Editura Club Press 22, Baia Mare; 1997.
Berinde V: Sequences of operators and fixed points in quasimetric spaces. Stud. Univ. Babeş-Bolyai, Math. 1996,16(4):23–27.
Berinde V: Iterative Approximation of Fixed Points. Editura Efemeride, Baia Mare; 2002.
Acknowledgements
The authors were supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU57000621). Moreover, the authors are grateful Dr. Wutiphol Sintunavarat and the reviewers for careful reading of the paper and for the suggestion, which improved the quality of this work.
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Phiangsungnoen, S., Kumam, P. Generalized Ulam-Hyers stability and well-posedness for fixed point equation via α-admissibility. J Inequal Appl 2014, 418 (2014). https://doi.org/10.1186/1029-242X-2014-418
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DOI: https://doi.org/10.1186/1029-242X-2014-418