Generalized Krasnoselskii fixed point theorem involving auxiliary ...
Applied Mathematics and Computation 248 (2014) 323–327
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Generalized Krasnoselskii fixed point theorem involving auxiliary functions in bimetric spaces and application to two-point boundary value problem Maher Berzig a,⇑, Sumit Chandok b, Mohammad Saeed Khan c a
Department of Mathematics, Tunis College of Sciences and Techniques, Tunis University, 5 Avenue Taha Hussein, Tunis, Tunisia Department of Mathematics, Khalsa College of Engineering & Technology (Punjab Technical University), Ranjit Avenue, Amritsar 143001, Punjab, India c Department of Mathematics and Statistics, College of Science Sultan Qaboos University, POBox 36, PCode 123 Al-Khod, Muscat, Oman b
a r t i c l e
i n f o
Keywords: Bimetric space Fixed point Coincidence point Two-point boundary value problem
a b s t r a c t In this paper, we introduce a generalized contraction of Krasnoselskii-type using auxiliary functions, and obtained some sufficient conditions for existence and uniqueness of fixed point for such mappings on bimetric spaces. We also establish a result on coincidence point of two mappings, and derive several corollaries of our main theorems. As application, we establish an existence result for a two-point boundary value problem of second order differential equation. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction and preliminaries Fixed point theory is of an intrinsic theoretical interest but it is also a useful tool for studying a wide class of practical problems. In particular, there is an exhaustive variety of results concerning fixed point theory in both Banach spaces and metric spaces which are subject to different types of contractive conditions (see [1–16]). In [13], Jachymski shows the equivalence between eight contractive definitions. The contraction of Krasnoselskii is one among them, therefore a generalization of Krasnoselskii result implies also the generalization of all others results. In this paper, we introduce a generalized contraction of Krasnoselskii-type and we prove fixed and common fixed point results for such contractive mappings in the setting of bimetric spaces. Several interesting corollaries are also obtained. The results presented in this paper generalize and extend several well-known results in the literature. As application, we establish an existence result for a second order differential equation. We recall Maia’s fixed point theorem: Theorem 1.1 [15]. Let ðX; d; dÞ be a bimetric space. Assume that for T : X ! X, the following conditions are satisfied: (i) dðx; yÞ 6 dðx; yÞ for all x; y 2 X; (ii) X is complete with respect to d; (iii) there exists a constant a 2 ½0; 1Þ such that
⇑ Corresponding author. E-mail addresses: [email protected] (M. Berzig), [email protected], [email protected] (S. Chandok), [email protected] (M.S. Khan). http://dx.doi.org/10.1016/j.amc.2014.09.096 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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dðTx; TyÞ 6 adðx; yÞ: Then T has a unique fixed point in X.
2. Main results Denote with H the family of auxiliary functions h : ½0; þ1Þ ! ½0; þ1Þ such that limn!1 hn ðtÞ ¼ 0 for each t > 0, where hn ðtÞ is the nth iterate of h. Furthermore, for each h 2 H; 0 < hðtÞ < t for all t > 0. For example, hðtÞ ¼ at; t 2 ½0; þ1Þ; a 2 ½0; 1Þ, then h 2 H. Definition 2.1. Let ðX; dÞ be a metric space and T : X ! X be a given mapping. We say that T is generalized Krasnoselskii contractive mapping if for any 0 < a < b < 1 there exists ha;b 2 H such that
8x; y 2 X : a 6 dðx; yÞ 6 b ) dðTx; TyÞ 6 ha;b ðdðx; yÞÞ:
ð2:1Þ
Our first main result is the following. Theorem 2.1. Let ðX; d; dÞ be a bimetric space. Assume that for T : X ! X, the following conditions are satisfied: (i) Every d-Cauchy sequence in X is d-Cauchy sequence too; (ii) X is complete with respect to d; (iii) T is generalized Krasnoselskii contractive mapping. Then T has a unique fixed point in X. Proof. Let x0 2 X, and let x1 ¼ Tx0 . If x1 ¼ x0 we get Tx0 ¼ x0 and the coincidence point holds. Suppose that x1 – x0 , then for a large integer n0 , we have 1=n0 6 dðx0 ; x1 Þ 6 n0 , which implies by (2.1) that
dðTx0 ; Tx1 Þ 6 h1=n0 ;n0 ðdðx0 ; x1 ÞÞ < dðx0 ; x1 Þ Again, let x2 2 X such that x2 ¼ Tx1 . If x2 ¼ x1 we get Tx1 ¼ x1 and the fixed point holds. Suppose that x2 – x1 , then for a large integer n1 , we have 1=n1 6 dðx1 ; x2 Þ 6 n1 , which implies by (2.1) that
dðTx1 ; Tx2 Þ 6 h1=n1 ;n1 ðdðx1 ; x2 ÞÞ < dðx1 ; x2 Þ Continuing this process, we can suppose that xnþ1 – xn for all n 0 otherwise xn becomes a fixed point. Therefore, we can construct the sequences fxk g defined by xkþ1 ¼ Txk , for all k 0. Hence, from (2.1), we obtain
dðxkþ1 ; xk Þ ¼ dðTxk ; Txkþ1 Þ 6 h1=nk ;nk ðdðxk ; xk1 ÞÞ < dðxk ; xk1 Þ: Consequently, the nonnegative sequence fdðxn ; xnþ1 Þg is nondecreasing, thus we get
lim dðxn ; xnþ1 Þ ¼ e P 0:
n!1
Now, we shall prove that
e ¼ 0. Thus, by contradiction, suppose that e > 0. Then, for sufficiently large N, we have
e 6 dðxNþk1 ; xNþk Þ 6 e þ 1; for all k ¼ 1; 2; . . . ; which implies by (2.1) that
dðTxNþk1 ; TxNþk Þ 6 he;eþ1 ðdðxNþk1 ; xNþk ÞÞ;
for all k ¼ 1; 2; . . . :
Thus, by induction, we get
dðxNþk ; xNþkþ1 Þ 6 hke;eþ1 ðdðxN ; xNþ1 ÞÞ 6 hke;eþ1 ðe þ 1Þ;
for all k ¼ 1; 2; . . .
Since hke;eþ1 ðe þ 1Þ ! 0 as k ! 1, then dðxNþk ; xNþkþ1 Þ ! 0 as k ! 1 which implies a contradiction. Let e > 0 and choose N sufficiently large such that
dðxN ; xNþ1 Þ 6
e 2
1 he=2;e ðeÞ: 2
ð2:2Þ
Let B be the ball defined by B :¼ fx 2 X : dðx; xN Þ 6 eg. Now, we will prove that TðBÞ # B. Let x 2 B, we distinguish two cases. In first case dðx; xN Þ 6 2e , for a sufficiently large integer L, we have
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325
dðTx; TxN Þ 6 dðTx; TxN1 Þ þ dðTxN1 ; TxN Þ 6 h1=L;e=2 ðdðx; xN ÞÞ þ dðxN ; xNþ1 Þ e e 1 1 þ he=2;e ðeÞ 6 e he=2;e ðeÞ 6 e: 6 h1=L;e=2 2 2 2 2 In the second case for
e 2
6 dðx; xN Þ 6 e, we have
dðTx; TxN Þ 6 dðTx; TxN1 Þ þ dðTxN1 ; TxN Þ 6 he=2;e ðdðx; xN ÞÞ þ dðxN ; xNþ1 Þ 6 he=2;e ðeÞ þ
e 2
1 he=2;e ðeÞ 6 e: 2
Hence, we deduce that TðBÞ # B holds. Next, by (2.2), we have xNþ1 2 B, which implies that xNþ2 2 B and so on, we get xn 2 B for all k > N. Thus we proved that fxn g is a d-Cauchy sequence in X. Now, using (i), we obtain that fxn g is a d-Cauchy sequence too. Further, from (ii), we have that ðX; dÞ is a complete metric space, so there exists x 2 X such that
lim dðxn ; x Þ ¼ 0:
n!1
Next, since from the contraction (2.1) it follows that T is continuous with respect to d, then it is easy to show that x is a fixed point of T. Finally, we shall show that T has a unique fixed point. For this, assume that there exists y 2 X with y – x such that Ty ¼ y and Tx ¼ x . Then, there exists n sufficiently large such that
1=n 6 dðx ; y Þ 6 n; which implies that
dðx ; y Þ ¼ dðTx ; Ty Þ 6 h1=n;n ðdðx ; y ÞÞ < dðx ; y Þ; which is a contradiction. Therefore, dðx ; y Þ ¼ 0, that is, the fixed point is unique.
h
Next, we need the following lemma to prove some common fixed point results. Lemma 2.1 [12]. Let X be a nonempty set and g : X ! X a function. Then there exists a subset Y X such that gðYÞ ¼ gðXÞ and g : Y ! X is one-to-one. Theorem 2.2. Let ðX; d; dÞ be a bimetric space and g : X ! X be a mapping. Assume that for T : X ! X, the following conditions are satisfied: (i)TðXÞ # gðXÞ; (ii)gðXÞ is a complete subset of X with respect to d; (iii)Tand g commute at their coincidence points; (iv)Every d-Cauchy sequence in X is d-Cauchy sequence too; (v)X is complete with respect to d; (vi)for any 0 < a < b < 1, there exists ha;b 2 H such that
8x; y 2 X : a 6 dðgx; gyÞ 6 b ) dðTx; TyÞ 6 ha;b ðdðgx; gyÞÞ:
ð2:3Þ
Then T and g have a unique point of coincidence in X. Proof. By Lemma 2.1, there exists Y # X such that gðYÞ ¼ gðXÞ and g : Y ! X is one-to-one on Y. Hence, by (ii), we can define a mapping S : gðYÞ ! gðYÞ by SðgxÞ ¼ Tx. Thus, the contraction (2.3), will be given by
8x; y 2 X : a 6 dðgx; gyÞ 6 b ) dðSðgxÞ; SðgyÞÞ 6 ha;b ðdðgx; gyÞÞ: Since gðXÞ ¼ gðYÞ is complete with respect to d, by using Theorem 2.1, there exists a unique x 2 X such that Sðgx Þ ¼ gx . Consequently, T and g have a unique point of coincidence. Finally, it is easy to prove that T and g have a unique common fixed point whenever T and g commute at their coincidence points. h The following corollaries can be derived immediately from previous theorems by choosing d ¼ d with/without ha;b ¼ kða; bÞt for kða; bÞ 2 ð0; 1Þ whenever 0 < a < b < 1. Corollary 2.1. Let ðX; dÞ be a complete metric space and T : X ! X be a mapping. Suppose that for any 0 < a < b < 1, there exists ha;b 2 H such that
8x; y 2 X : a 6 dðx; yÞ 6 b ) dðTx; TyÞ 6 ha;b ðdðx; yÞÞ: Then T has a unique fixed point in X.
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Corollary 2.2 [16]. Let ðX; dÞ be a complete metric space and T : X ! X be a mapping. Suppose that for any 0 < a < b < 1, there exists kða; bÞ 2 ð0; 1Þ such that
8x; y 2 X : a 6 dðx; yÞ 6 b ) dðTx; TyÞ 6 kða; bÞdðx; yÞ: Then T has a unique fixed point in X. Corollary 2.3. Let ðX; dÞ be a complete metric space. Assume that for T : X ! X, the following conditions are satisfied: (i) (ii) (iii) (iv)
TðXÞ # gðXÞ; gðXÞ is a complete subset of X with respect to d; T and g commute at their coincidence points; for any 0 < a < b < 1, there exists ha;b 2 H such that
8x; y 2 X : a 6 dðgx; gyÞ 6 b ) dðTx; TyÞ 6 ha;b ðdðgx; gyÞÞ: Then T and g have a unique point of coincidence in X. Corollary 2.4. Let ðX; dÞ be a complete metric space. Assume that for T : X ! X, the following conditions are satisfied: (i) (ii) (iii) (iv)
TðXÞ # gðXÞ; gðXÞ is a complete subset of X with respect to d; T and g commute at their coincidence points; for any 0 < a < b < 1, there exists kða; bÞ 2 ð0; 1Þ such that
8x; y 2 X : a 6 dðgx; gyÞ 6 b ) dðTx; TyÞ 6 kða; bÞðdðgx; gyÞÞ: Then T and g have a unique point of coincidence in X.
3. Application to two-point boundary value problem In this section we apply our main results to obtain a solution of second order differential equation. We now prove the existence of a solution for the following two-point boundary value problem of second order differential equation:
u00 ðtÞ ¼ f ðt; uðtÞÞ; ¼ a;
uðaÞ
t 2 I;
ð3:1Þ
uðbÞ ¼ b;
where I ¼ ½a; b with 0 < a < b < 1, and f : I R ! R. Consider the space CðIÞ, the class of real-valued continuous functions defined on I ¼ ½a; b endowed with the bimetric d and d given by dðx; yÞ ¼ supfjxðtÞ yðtÞj : t 2 Ig and dðx; yÞ ¼ 12 dðx; yÞ for all x; y 2 CðIÞ. Clearly, ðCðIÞ; dÞ is a complete metric space. We consider the following assumptions: (H1) f : I R ! R is continuous; 16 (H2) jf ðt; xÞ f ðt; yÞj 6 pð1þbaÞðbaÞ 2 arctanðjx yjÞ for all x; y 2 R; We have the following result. Theorem 3.1. Suppose that (H1)–(H2) hold. Then (3.1) has at least one solution x 2 C 2 ðIÞ. Proof. The problem (3.1) is equivalent to
uðtÞ ¼ a
bt ta þb þ ba ba
Z
b
Gðt; sÞ½f ðs; uðsÞÞds;
a
for all t 2 I, where Gðt; sÞ is the Green’s function given by
( Gðt; sÞ ¼
ðsaÞðbtÞ ; ba ðtaÞðbsÞ ; ba
a6s6t6b a6t6s6b
Rb bt ta Define F : CðIÞ ! CðIÞ by FuðtÞ ¼ a ba þ b ba þ a Gðt; sÞ½f ðs; uðsÞÞds. Clearly, u 2 C 2 ðIÞ is a solution of given problem if u 2 CðIÞ is a fixed point of F. Now, for each u; v 2 CðIÞ, consider
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jðFuÞðtÞ ðF v ÞðtÞj ¼ j
Z
Z
b
Gðt; sÞ½f ðs; uðsÞÞ f ðs; v ðsÞÞdsj 6
a b
Gðt; sÞds
6 a
Z 6
6
b
Gðt; sÞjf ðs; uðsÞÞ f ðs; v ðsÞÞjds
a b
jf ðs; uðsÞÞ f ðs; v ðsÞÞjds
a
b
Gðt; sÞds
a
6
Z
Z
Z
a
b
16
pð1 þ b aÞðb aÞ2
16 2
pð1 þ b aÞðb aÞ 16
2
pð1 þ b aÞðb aÞ
sup t2I
Z
arctanðjuðsÞ v ðsÞjÞds
1
Gðt; sÞds
Z
0
b
arctanðjuðsÞ v ðsÞjÞds
a
arctanðdðu; v ÞÞ sup t2I
Z 0
1
Gðt; sÞds 6
2
pð1 þ b aÞ
arctanðdðu; v ÞÞ
Rb 2 2 It is easy to verify that for all t 2 I; supt2I a Gðt; sÞds ¼ ðbaÞ . So, we have supt2I jFuðtÞ F v ðtÞj 6 pð1þbaÞ arctanðdðu; v ÞÞ. Put 8 2 ha;b ðtÞ ¼ pð1þbaÞ arctanðtÞ, we obtain dðFu; F v Þ 6 ha;b ðdðu; v ÞÞ, for all u; v 2 CðIÞ and 0 < a < b < 1. By applying Theorem 2.1, F has a fixed point in CðIÞ, that is, there exists x 2 CðIÞ such that x ¼ Fx , and so x is a solution to (3.1). h References [1] R.P. Agarwal, M.A. El-Gebeily, D. O’Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal. 87 (2008) 109–116. [2] V. Berinde, Iterative Approximation of Fixed Points, Springer-Verlag, 2007. [3] M. Berzig, Coincidence and common fixed point results on metric spaces endowed with an arbitrary binary relation and applications, J. Fix. Point Theory Appl. 12 (1–2) (2012) 221–238. [4] M. Berzig and E. Karapnar, Fixed point results for (aw; bu)-contractive mappings for a generalized altering distance, Fixed Point Theory and Appl. vol. 2013, article 205, 2013. [5] M. Berzig, E. Karapnar, and A. Roldán, Discussion on generalized-(aw; bu)-contractive mappings via generalized altering distance function and related fixed point theorems, Abstract and Applied Analysis, vol. 2014, Article ID 259768, 12 pages, 2014. [6] M. Berzig, M.-D. Rus, Fixed point theorems for a-contractive mappings of Meir–Keeler type and applications, Nonlinear Anal. Modell. Control 19 (2) (2014) 178–198. [7] M. Berzig, B. Samet, An extension of coupled fixed points concept in higher dimension and applications, Comput. Math. Appl. 63 (2012) 1319–1334. [8] S. Chandok, On common fixed points for generalized contractive type mappings in ordered metric spaces, Proc. Jangjeon Math. Soc. 16 (2013) 327–333. [9] S. Chandok and S. Dinu, Common fixed points for weak w-contractive mappings in ordered metric spaces with applications, Abs. Appl. Anal. 2013(2013) Article ID 879084. [10] S. Chandok, M.S. Khan, M. Abbas, Common fixed point theorems for nonlinear weakly contractive mappings, Ukr. Math. J. (2014). [11] S. Chandok, B.S. Choudhury, N. Metiya, Some fixed point results in ordered metric spaces for rational type expressions with auxiliary functions, J. Egypt. Math. Soc. (2014). [12] R.H. Haghi, Sh. Rezapour, N. Shahzad, Some fixed point generalizations are not real generalizations, Nonlinear Anal. 74 (5) (2011) 1799–1803. [13] J. Jachymski, Equivalence of some contractivity properties over metrical structures, Proc. Am. Math. Soc. 125 (8) (1997) 2327–2335. [14] M.S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc. 30 (1984) 1–9. [15] M.G. Maia, Un’osservazione sulle contrazioni metriche, Rend. Semin. Mat. Univ. Padova 40 (1968) 139–143. [16] M.A. Krasnoselskii, P.P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, 1984.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Generalized Krasnoselskii fixed point theorem involving auxiliary functions in bimetric spaces and application to two-point boundary value problem Maher Berzig a,⇑, Sumit Chandok b, Mohammad Saeed Khan c a
Department of Mathematics, Tunis College of Sciences and Techniques, Tunis University, 5 Avenue Taha Hussein, Tunis, Tunisia Department of Mathematics, Khalsa College of Engineering & Technology (Punjab Technical University), Ranjit Avenue, Amritsar 143001, Punjab, India c Department of Mathematics and Statistics, College of Science Sultan Qaboos University, POBox 36, PCode 123 Al-Khod, Muscat, Oman b
a r t i c l e
i n f o
Keywords: Bimetric space Fixed point Coincidence point Two-point boundary value problem
a b s t r a c t In this paper, we introduce a generalized contraction of Krasnoselskii-type using auxiliary functions, and obtained some sufficient conditions for existence and uniqueness of fixed point for such mappings on bimetric spaces. We also establish a result on coincidence point of two mappings, and derive several corollaries of our main theorems. As application, we establish an existence result for a two-point boundary value problem of second order differential equation. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction and preliminaries Fixed point theory is of an intrinsic theoretical interest but it is also a useful tool for studying a wide class of practical problems. In particular, there is an exhaustive variety of results concerning fixed point theory in both Banach spaces and metric spaces which are subject to different types of contractive conditions (see [1–16]). In [13], Jachymski shows the equivalence between eight contractive definitions. The contraction of Krasnoselskii is one among them, therefore a generalization of Krasnoselskii result implies also the generalization of all others results. In this paper, we introduce a generalized contraction of Krasnoselskii-type and we prove fixed and common fixed point results for such contractive mappings in the setting of bimetric spaces. Several interesting corollaries are also obtained. The results presented in this paper generalize and extend several well-known results in the literature. As application, we establish an existence result for a second order differential equation. We recall Maia’s fixed point theorem: Theorem 1.1 [15]. Let ðX; d; dÞ be a bimetric space. Assume that for T : X ! X, the following conditions are satisfied: (i) dðx; yÞ 6 dðx; yÞ for all x; y 2 X; (ii) X is complete with respect to d; (iii) there exists a constant a 2 ½0; 1Þ such that
⇑ Corresponding author. E-mail addresses: [email protected] (M. Berzig), [email protected], [email protected] (S. Chandok), [email protected] (M.S. Khan). http://dx.doi.org/10.1016/j.amc.2014.09.096 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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M. Berzig et al. / Applied Mathematics and Computation 248 (2014) 323–327
dðTx; TyÞ 6 adðx; yÞ: Then T has a unique fixed point in X.
2. Main results Denote with H the family of auxiliary functions h : ½0; þ1Þ ! ½0; þ1Þ such that limn!1 hn ðtÞ ¼ 0 for each t > 0, where hn ðtÞ is the nth iterate of h. Furthermore, for each h 2 H; 0 < hðtÞ < t for all t > 0. For example, hðtÞ ¼ at; t 2 ½0; þ1Þ; a 2 ½0; 1Þ, then h 2 H. Definition 2.1. Let ðX; dÞ be a metric space and T : X ! X be a given mapping. We say that T is generalized Krasnoselskii contractive mapping if for any 0 < a < b < 1 there exists ha;b 2 H such that
8x; y 2 X : a 6 dðx; yÞ 6 b ) dðTx; TyÞ 6 ha;b ðdðx; yÞÞ:
ð2:1Þ
Our first main result is the following. Theorem 2.1. Let ðX; d; dÞ be a bimetric space. Assume that for T : X ! X, the following conditions are satisfied: (i) Every d-Cauchy sequence in X is d-Cauchy sequence too; (ii) X is complete with respect to d; (iii) T is generalized Krasnoselskii contractive mapping. Then T has a unique fixed point in X. Proof. Let x0 2 X, and let x1 ¼ Tx0 . If x1 ¼ x0 we get Tx0 ¼ x0 and the coincidence point holds. Suppose that x1 – x0 , then for a large integer n0 , we have 1=n0 6 dðx0 ; x1 Þ 6 n0 , which implies by (2.1) that
dðTx0 ; Tx1 Þ 6 h1=n0 ;n0 ðdðx0 ; x1 ÞÞ < dðx0 ; x1 Þ Again, let x2 2 X such that x2 ¼ Tx1 . If x2 ¼ x1 we get Tx1 ¼ x1 and the fixed point holds. Suppose that x2 – x1 , then for a large integer n1 , we have 1=n1 6 dðx1 ; x2 Þ 6 n1 , which implies by (2.1) that
dðTx1 ; Tx2 Þ 6 h1=n1 ;n1 ðdðx1 ; x2 ÞÞ < dðx1 ; x2 Þ Continuing this process, we can suppose that xnþ1 – xn for all n 0 otherwise xn becomes a fixed point. Therefore, we can construct the sequences fxk g defined by xkþ1 ¼ Txk , for all k 0. Hence, from (2.1), we obtain
dðxkþ1 ; xk Þ ¼ dðTxk ; Txkþ1 Þ 6 h1=nk ;nk ðdðxk ; xk1 ÞÞ < dðxk ; xk1 Þ: Consequently, the nonnegative sequence fdðxn ; xnþ1 Þg is nondecreasing, thus we get
lim dðxn ; xnþ1 Þ ¼ e P 0:
n!1
Now, we shall prove that
e ¼ 0. Thus, by contradiction, suppose that e > 0. Then, for sufficiently large N, we have
e 6 dðxNþk1 ; xNþk Þ 6 e þ 1; for all k ¼ 1; 2; . . . ; which implies by (2.1) that
dðTxNþk1 ; TxNþk Þ 6 he;eþ1 ðdðxNþk1 ; xNþk ÞÞ;
for all k ¼ 1; 2; . . . :
Thus, by induction, we get
dðxNþk ; xNþkþ1 Þ 6 hke;eþ1 ðdðxN ; xNþ1 ÞÞ 6 hke;eþ1 ðe þ 1Þ;
for all k ¼ 1; 2; . . .
Since hke;eþ1 ðe þ 1Þ ! 0 as k ! 1, then dðxNþk ; xNþkþ1 Þ ! 0 as k ! 1 which implies a contradiction. Let e > 0 and choose N sufficiently large such that
dðxN ; xNþ1 Þ 6
e 2
1 he=2;e ðeÞ: 2
ð2:2Þ
Let B be the ball defined by B :¼ fx 2 X : dðx; xN Þ 6 eg. Now, we will prove that TðBÞ # B. Let x 2 B, we distinguish two cases. In first case dðx; xN Þ 6 2e , for a sufficiently large integer L, we have
M. Berzig et al. / Applied Mathematics and Computation 248 (2014) 323–327
325
dðTx; TxN Þ 6 dðTx; TxN1 Þ þ dðTxN1 ; TxN Þ 6 h1=L;e=2 ðdðx; xN ÞÞ þ dðxN ; xNþ1 Þ e e 1 1 þ he=2;e ðeÞ 6 e he=2;e ðeÞ 6 e: 6 h1=L;e=2 2 2 2 2 In the second case for
e 2
6 dðx; xN Þ 6 e, we have
dðTx; TxN Þ 6 dðTx; TxN1 Þ þ dðTxN1 ; TxN Þ 6 he=2;e ðdðx; xN ÞÞ þ dðxN ; xNþ1 Þ 6 he=2;e ðeÞ þ
e 2
1 he=2;e ðeÞ 6 e: 2
Hence, we deduce that TðBÞ # B holds. Next, by (2.2), we have xNþ1 2 B, which implies that xNþ2 2 B and so on, we get xn 2 B for all k > N. Thus we proved that fxn g is a d-Cauchy sequence in X. Now, using (i), we obtain that fxn g is a d-Cauchy sequence too. Further, from (ii), we have that ðX; dÞ is a complete metric space, so there exists x 2 X such that
lim dðxn ; x Þ ¼ 0:
n!1
Next, since from the contraction (2.1) it follows that T is continuous with respect to d, then it is easy to show that x is a fixed point of T. Finally, we shall show that T has a unique fixed point. For this, assume that there exists y 2 X with y – x such that Ty ¼ y and Tx ¼ x . Then, there exists n sufficiently large such that
1=n 6 dðx ; y Þ 6 n; which implies that
dðx ; y Þ ¼ dðTx ; Ty Þ 6 h1=n;n ðdðx ; y ÞÞ < dðx ; y Þ; which is a contradiction. Therefore, dðx ; y Þ ¼ 0, that is, the fixed point is unique.
h
Next, we need the following lemma to prove some common fixed point results. Lemma 2.1 [12]. Let X be a nonempty set and g : X ! X a function. Then there exists a subset Y X such that gðYÞ ¼ gðXÞ and g : Y ! X is one-to-one. Theorem 2.2. Let ðX; d; dÞ be a bimetric space and g : X ! X be a mapping. Assume that for T : X ! X, the following conditions are satisfied: (i)TðXÞ # gðXÞ; (ii)gðXÞ is a complete subset of X with respect to d; (iii)Tand g commute at their coincidence points; (iv)Every d-Cauchy sequence in X is d-Cauchy sequence too; (v)X is complete with respect to d; (vi)for any 0 < a < b < 1, there exists ha;b 2 H such that
8x; y 2 X : a 6 dðgx; gyÞ 6 b ) dðTx; TyÞ 6 ha;b ðdðgx; gyÞÞ:
ð2:3Þ
Then T and g have a unique point of coincidence in X. Proof. By Lemma 2.1, there exists Y # X such that gðYÞ ¼ gðXÞ and g : Y ! X is one-to-one on Y. Hence, by (ii), we can define a mapping S : gðYÞ ! gðYÞ by SðgxÞ ¼ Tx. Thus, the contraction (2.3), will be given by
8x; y 2 X : a 6 dðgx; gyÞ 6 b ) dðSðgxÞ; SðgyÞÞ 6 ha;b ðdðgx; gyÞÞ: Since gðXÞ ¼ gðYÞ is complete with respect to d, by using Theorem 2.1, there exists a unique x 2 X such that Sðgx Þ ¼ gx . Consequently, T and g have a unique point of coincidence. Finally, it is easy to prove that T and g have a unique common fixed point whenever T and g commute at their coincidence points. h The following corollaries can be derived immediately from previous theorems by choosing d ¼ d with/without ha;b ¼ kða; bÞt for kða; bÞ 2 ð0; 1Þ whenever 0 < a < b < 1. Corollary 2.1. Let ðX; dÞ be a complete metric space and T : X ! X be a mapping. Suppose that for any 0 < a < b < 1, there exists ha;b 2 H such that
8x; y 2 X : a 6 dðx; yÞ 6 b ) dðTx; TyÞ 6 ha;b ðdðx; yÞÞ: Then T has a unique fixed point in X.
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Corollary 2.2 [16]. Let ðX; dÞ be a complete metric space and T : X ! X be a mapping. Suppose that for any 0 < a < b < 1, there exists kða; bÞ 2 ð0; 1Þ such that
8x; y 2 X : a 6 dðx; yÞ 6 b ) dðTx; TyÞ 6 kða; bÞdðx; yÞ: Then T has a unique fixed point in X. Corollary 2.3. Let ðX; dÞ be a complete metric space. Assume that for T : X ! X, the following conditions are satisfied: (i) (ii) (iii) (iv)
TðXÞ # gðXÞ; gðXÞ is a complete subset of X with respect to d; T and g commute at their coincidence points; for any 0 < a < b < 1, there exists ha;b 2 H such that
8x; y 2 X : a 6 dðgx; gyÞ 6 b ) dðTx; TyÞ 6 ha;b ðdðgx; gyÞÞ: Then T and g have a unique point of coincidence in X. Corollary 2.4. Let ðX; dÞ be a complete metric space. Assume that for T : X ! X, the following conditions are satisfied: (i) (ii) (iii) (iv)
TðXÞ # gðXÞ; gðXÞ is a complete subset of X with respect to d; T and g commute at their coincidence points; for any 0 < a < b < 1, there exists kða; bÞ 2 ð0; 1Þ such that
8x; y 2 X : a 6 dðgx; gyÞ 6 b ) dðTx; TyÞ 6 kða; bÞðdðgx; gyÞÞ: Then T and g have a unique point of coincidence in X.
3. Application to two-point boundary value problem In this section we apply our main results to obtain a solution of second order differential equation. We now prove the existence of a solution for the following two-point boundary value problem of second order differential equation:
u00 ðtÞ ¼ f ðt; uðtÞÞ; ¼ a;
uðaÞ
t 2 I;
ð3:1Þ
uðbÞ ¼ b;
where I ¼ ½a; b with 0 < a < b < 1, and f : I R ! R. Consider the space CðIÞ, the class of real-valued continuous functions defined on I ¼ ½a; b endowed with the bimetric d and d given by dðx; yÞ ¼ supfjxðtÞ yðtÞj : t 2 Ig and dðx; yÞ ¼ 12 dðx; yÞ for all x; y 2 CðIÞ. Clearly, ðCðIÞ; dÞ is a complete metric space. We consider the following assumptions: (H1) f : I R ! R is continuous; 16 (H2) jf ðt; xÞ f ðt; yÞj 6 pð1þbaÞðbaÞ 2 arctanðjx yjÞ for all x; y 2 R; We have the following result. Theorem 3.1. Suppose that (H1)–(H2) hold. Then (3.1) has at least one solution x 2 C 2 ðIÞ. Proof. The problem (3.1) is equivalent to
uðtÞ ¼ a
bt ta þb þ ba ba
Z
b
Gðt; sÞ½f ðs; uðsÞÞds;
a
for all t 2 I, where Gðt; sÞ is the Green’s function given by
( Gðt; sÞ ¼
ðsaÞðbtÞ ; ba ðtaÞðbsÞ ; ba
a6s6t6b a6t6s6b
Rb bt ta Define F : CðIÞ ! CðIÞ by FuðtÞ ¼ a ba þ b ba þ a Gðt; sÞ½f ðs; uðsÞÞds. Clearly, u 2 C 2 ðIÞ is a solution of given problem if u 2 CðIÞ is a fixed point of F. Now, for each u; v 2 CðIÞ, consider
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jðFuÞðtÞ ðF v ÞðtÞj ¼ j
Z
Z
b
Gðt; sÞ½f ðs; uðsÞÞ f ðs; v ðsÞÞdsj 6
a b
Gðt; sÞds
6 a
Z 6
6
b
Gðt; sÞjf ðs; uðsÞÞ f ðs; v ðsÞÞjds
a b
jf ðs; uðsÞÞ f ðs; v ðsÞÞjds
a
b
Gðt; sÞds
a
6
Z
Z
Z
a
b
16
pð1 þ b aÞðb aÞ2
16 2
pð1 þ b aÞðb aÞ 16
2
pð1 þ b aÞðb aÞ
sup t2I
Z
arctanðjuðsÞ v ðsÞjÞds
1
Gðt; sÞds
Z
0
b
arctanðjuðsÞ v ðsÞjÞds
a
arctanðdðu; v ÞÞ sup t2I
Z 0
1
Gðt; sÞds 6
2
pð1 þ b aÞ
arctanðdðu; v ÞÞ
Rb 2 2 It is easy to verify that for all t 2 I; supt2I a Gðt; sÞds ¼ ðbaÞ . So, we have supt2I jFuðtÞ F v ðtÞj 6 pð1þbaÞ arctanðdðu; v ÞÞ. Put 8 2 ha;b ðtÞ ¼ pð1þbaÞ arctanðtÞ, we obtain dðFu; F v Þ 6 ha;b ðdðu; v ÞÞ, for all u; v 2 CðIÞ and 0 < a < b < 1. By applying Theorem 2.1, F has a fixed point in CðIÞ, that is, there exists x 2 CðIÞ such that x ¼ Fx , and so x is a solution to (3.1). h References [1] R.P. Agarwal, M.A. El-Gebeily, D. O’Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal. 87 (2008) 109–116. [2] V. Berinde, Iterative Approximation of Fixed Points, Springer-Verlag, 2007. [3] M. Berzig, Coincidence and common fixed point results on metric spaces endowed with an arbitrary binary relation and applications, J. Fix. Point Theory Appl. 12 (1–2) (2012) 221–238. [4] M. Berzig and E. Karapnar, Fixed point results for (aw; bu)-contractive mappings for a generalized altering distance, Fixed Point Theory and Appl. vol. 2013, article 205, 2013. [5] M. Berzig, E. Karapnar, and A. Roldán, Discussion on generalized-(aw; bu)-contractive mappings via generalized altering distance function and related fixed point theorems, Abstract and Applied Analysis, vol. 2014, Article ID 259768, 12 pages, 2014. [6] M. Berzig, M.-D. Rus, Fixed point theorems for a-contractive mappings of Meir–Keeler type and applications, Nonlinear Anal. Modell. Control 19 (2) (2014) 178–198. [7] M. Berzig, B. Samet, An extension of coupled fixed points concept in higher dimension and applications, Comput. Math. Appl. 63 (2012) 1319–1334. [8] S. Chandok, On common fixed points for generalized contractive type mappings in ordered metric spaces, Proc. Jangjeon Math. Soc. 16 (2013) 327–333. [9] S. Chandok and S. Dinu, Common fixed points for weak w-contractive mappings in ordered metric spaces with applications, Abs. Appl. Anal. 2013(2013) Article ID 879084. [10] S. Chandok, M.S. Khan, M. Abbas, Common fixed point theorems for nonlinear weakly contractive mappings, Ukr. Math. J. (2014). [11] S. Chandok, B.S. Choudhury, N. Metiya, Some fixed point results in ordered metric spaces for rational type expressions with auxiliary functions, J. Egypt. Math. Soc. (2014). [12] R.H. Haghi, Sh. Rezapour, N. Shahzad, Some fixed point generalizations are not real generalizations, Nonlinear Anal. 74 (5) (2011) 1799–1803. [13] J. Jachymski, Equivalence of some contractivity properties over metrical structures, Proc. Am. Math. Soc. 125 (8) (1997) 2327–2335. [14] M.S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc. 30 (1984) 1–9. [15] M.G. Maia, Un’osservazione sulle contrazioni metriche, Rend. Semin. Mat. Univ. Padova 40 (1968) 139–143. [16] M.A. Krasnoselskii, P.P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, 1984.
