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A note on fixed point stability for generalized multivalued contractions
Applied Mathematics Letters 25 (2012) 1708–1710
Contents lists available at SciVerse ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
A note on fixed point stability for generalized multivalued contractions S.L. Singh a , S.N. Mishra a , W. Sinkala b,∗ a
Department of Mathematics, Walter Sisulu University, Nelson Mandela Drive, Private Bag X1, Mthatha 5117, South Africa
b
Department of Applied Mathematics, Walter Sisulu University, Nelson Mandela Drive, Private Bag X1, Mthatha 5117, South Africa
article
abstract
info
Article history: Received 26 November 2010 Received in revised form 27 October 2011 Accepted 30 January 2012
A classical lemma of Lim [T.C. Lim, Fixed point stability for set valued contractive mappings with applications to generalized differential equations, J. Math. Anal. Appl. 110 (1985) 436–441] has been extended in two ways, and one of them is used to obtain a stability result for generalized multivalued contractions. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Fixed point Hausdorff metric Metric space λ-contraction Multivalued map Stability
1. Introduction Let (X , d) be a metric space and CL(X ) the family of all nonempty closed subsets of X . (CL(X ), D) equipped with the Hausdorff metric D defined by
D(A, B) = max sup d(x, A), sup d(x, B) , x∈B
x∈B
where A, B ∈ CL(X ) and d(x, K ) = infz ∈K d(x, z ), is called the generalized hyperspace of (X , d). A multivalued map T : X → CL(X ) is a generalized λ-contraction [1] if 0 < λ < 1 and
D(Tx, Ty) ≤ λ max d(x, y), d(x, Tx), d(y, Ty),
d(x, Ty) + d(y, Tx)
2
(1.1)
for all x, y ∈ X . If the space X is complete then every generalized multivalued λ-contraction T has a fixed point, i.e., a point z ∈ X such that z ∈ Tz. The set of all fixed points of T is denoted by F (T ). A multivalued map T : X → CL(X ) is a λ-contraction (see, for instance, [2,3] and [4, p. 163]), if D(Tx, Ty) ≤ λd(x, y)
(1.2)
for all x, y ∈ X , where 0 ≤ λ ≤ 1. Evidently, (1.2) implies (1.1), that is T satisfying (1.2) also satisfies (1.1). Investigations regarding the stability of fixed points of multivalued contractions were initiated by Nadler [3] and Markin [5]. The stability results for multivalued contractions have been found useful in the area of generalized differential equations and other contexts (see, for instance, [6–8]). In this paper we study the stability problem for generalized multivalued λ-contractions. We also obtain an extension of Lim’s result to a new class of contractions (Theorem 2.4).
∗
Corresponding author. Tel.: +27 47 502 2878; fax: +27 47 502 2725. E-mail addresses: [email protected] (S.L. Singh), [email protected] (S.N. Mishra), [email protected], [email protected] (W. Sinkala).
0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.01.042
S.L. Singh et al. / Applied Mathematics Letters 25 (2012) 1708–1710
1709
2. The main results The following generalizes an important result of Lim [6, Lemma 1] (see also [4, p. 163]). Theorem 2.1. Let X be a complete metric space and multivalued maps T1 , T2 : X → CL(X ) be λ-generalized contractions. Then D(F (T1 ), F (T2 )) ≤
1 sup D(T1 x, T2 x). 1 − λ x∈X
Proof. Write K = supx∈X D(T1 x, T2 x). We may assume that K is finite. Let q > 1. Pick x0 ∈ F (T1 ). Since D(T1 x0 , T2 x0 ) ≤ K , we may choose x1 ∈ T2 x0 such that d(x1 , x0 ) ≤ qK . Now we can choose x2 ∈ T2 x1 such that d(x2 , x1 ) ≤ qD(T2 x1 , T2 x0 ). Similarly, we choose x3 ∈ T2 x2 such that d(x3 , x2 ) ≤ qD(T2 x2 , T2 x1 ). Inductively we choose a sequence {xn } such that xn+1 ∈ T2 xn and d(xn+1 , xn ) ≤ qD(T2 xn , T2 xn−1 ),
n ≥ 1.
Since T2 is a generalized multivalued λ-contraction, we have d(xn+1 , xn ) ≤ qλ max {d(xn , xn−1 ), d(xn , xn+1 )} . This yields d(xn+1 , xn ) ≤ qλd(xn , xn−1 ).
(2.1)
This is true for n ≥ 1. Evidently we may choose q > 1 such that qλ < 1. Hence {xn } is a Cauchy sequence. Since the space X is complete, the sequence {xn } has a limit in X . Call it z. Therefore d(xn+1 , T2 z ) ≤ D(T2 xn , T2 z )
d(xn , T2 z ) + d(z , T2 xn ) . ≤ λ max d(xn , z ), d(xn , T2 xn ), d(z , T2 z ), 2
Making n → ∞, we obtain d(z , T2 z ) ≤ λd(z , T2 z ). Since λ < 1, it follows that d(z , T2 z ) = 0 proving that z ∈ F (T2 ). Furthermore, using (2.1), d(x0 , z ) ≤
∞
d(xn , xn+1 )
n=0
≤
1 1 − qλ
d(x0 , x1 ) ≤
q 1 − qλ
K.
Reversing the roles of T1 and T2 , we also conclude that for each y0 ∈ F (T2 ), there exists y1 ∈ T1 y0 and ω ∈ F (T1 ) such that d(y0 , ω) ≤
q 1 − qλ
K.
Hence D(F (T1 ), F (T2 )) ≤
q 1 − qλ
K.
This completes the proof when we take q → 1.
(2.2)
The following theorem generalizes Lim’s stability results [6] for a sequence of multivalued λ-contractions. Theorem 2.2. Let X be a complete metric space and Tn : X → CL(X ) a sequence of λ-generalized contractions, n = 0, 1, 2, . . . . If the sequence {Tn }∞ n=1 converges to T0 uniformly on X , then lim D(F (Tn ), F (T0 )) = 0.
n→∞
Proof. Since {Tn }∞ n=1 converges to T0 uniformly on X , for a given ε > 0, we can find a positive integer n0 such that sup D(Tn x, T0 x) < (1 − λ)ε x∈X
for all n ≥ n0 . Then D(F (Tn ), F (T0 )) < ε for all n ≥ n0 by Theorem 2.1.
1710
S.L. Singh et al. / Applied Mathematics Letters 25 (2012) 1708–1710
Extending the notion of multivalued λ-contractions, Kikkawa and Suzuki [9] recently obtained the following result. Theorem 2.3. Let (X , d) be a complete metric space and T a multivalued map from X into CB(X ), the class of nonempty closed bounded subsets of X . Assume that there exists λ ∈ [0, 1) such that 1 1+λ
d(x, Tx) ≤ d(x, y)
(2.3)
implies the condition (1.2). Then T has a fixed point. Theorem 2.3 has recently been extended in [10], by considering the following condition instead of (1.2). A map T : X → CL(X ) is a Suzuki–Zamfirescu λ-contraction (or simply an S–Z λ-contraction) if and only if there exists λ ∈ [0, 1) such that (2.3) implies
D(Tx, Ty) ≤ λ max d(x, y),
d(x, Tx) + d(y, Ty) d(x, Ty) + d(y, Tx) 2
,
2
(2.4)
for all x, y ∈ X . We remark that the condition (2.4) includes generalizations of Kikkawa and Suzuki’s contraction condition (cf. Theorem 2.3) studied by Mot and Petrusel [11] (see also [12,13]). Theorem 2.4. Let X be a complete metric space and let multivalued maps T1 , T2 : X → CL(X ) be S–Z λ-contractions. Then the conclusion of Theorem 2.1 is true. Proof. By [10, Corollary 3.2], T1 and T2 have fixed points. Now, the proof may be completed following Mot and Petrusel [11, Theorem 3.2] and the proof of Theorem 2.1. Remark 2.1. A result akin to Theorem 2.4 has recently been obtained in [11, Theorem 3.2] for when T1 and T2 are Kikkawa–Suzuki multivalued contractions. Acknowledgments The authors thank the Directorate of Research Development of Walter Sisulu University for financial support. A special word of thanks is due to an anonymous referee whose constructive comments were useful in improving the first draft of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Lj. B. Cirić, Fixed points for generalized multivalued contractions, Mat. Vesnik 9 (24) (1972) 265–272. S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (2) (1969) 475–488. S.B. Nadler Jr., Hyperspaces of Sets, Dekker, New York, Basel, 1978. K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, 1990. J.T. Markin, Continuous dependence of fixed point sets, Proc. Amer. Math. Soc. 38 (3) (1973) 545–547. T.C. Lim, Fixed point stability for set valued contractive mappings with applications to generalized differential equations, J. Math. Anal. Appl. 110 (1985) 436–441. B. Ricceri, Smooth extensions of lipschitzian real functions, Proc. Amer. Math. Soc. 104 (2) (1988) 641–642. A. Tolstonogov, Differential Inclusions in a Banach Space, Kluwer Academic Publishers, 2000. M. Kikkawa, T. Suzuki, Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal. 69 (9) (2008) 2942–2949. S.L. Singh, S.N. Mishra, Coincidence theorems for certain classes of hybrid contractions, Fixed Point Theory Appl. 2010 (2010) Article ID 898109, 14 pages. G. Mot, A. Petrusel, Fixed point theory for a new type of contractive multivalued operators, Nonlinear Anal. 70 (9) (2009) 3371–3377. S. Dhompongsa, H. Yingtaweesittikul, Fixed points for multivalued mappings and the metric completeness, Fixed Point Theory Appl. 2009 (2009) Article ID 972395, 15 pages. S.L. Singh, S.N. Mishra, Remarks on recent fixed point theorems, Fixed Point Theory Appl. 2010 (2010) Article ID 452905, 18 pages.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
A note on fixed point stability for generalized multivalued contractions S.L. Singh a , S.N. Mishra a , W. Sinkala b,∗ a
Department of Mathematics, Walter Sisulu University, Nelson Mandela Drive, Private Bag X1, Mthatha 5117, South Africa
b
Department of Applied Mathematics, Walter Sisulu University, Nelson Mandela Drive, Private Bag X1, Mthatha 5117, South Africa
article
abstract
info
Article history: Received 26 November 2010 Received in revised form 27 October 2011 Accepted 30 January 2012
A classical lemma of Lim [T.C. Lim, Fixed point stability for set valued contractive mappings with applications to generalized differential equations, J. Math. Anal. Appl. 110 (1985) 436–441] has been extended in two ways, and one of them is used to obtain a stability result for generalized multivalued contractions. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Fixed point Hausdorff metric Metric space λ-contraction Multivalued map Stability
1. Introduction Let (X , d) be a metric space and CL(X ) the family of all nonempty closed subsets of X . (CL(X ), D) equipped with the Hausdorff metric D defined by
D(A, B) = max sup d(x, A), sup d(x, B) , x∈B
x∈B
where A, B ∈ CL(X ) and d(x, K ) = infz ∈K d(x, z ), is called the generalized hyperspace of (X , d). A multivalued map T : X → CL(X ) is a generalized λ-contraction [1] if 0 < λ < 1 and
D(Tx, Ty) ≤ λ max d(x, y), d(x, Tx), d(y, Ty),
d(x, Ty) + d(y, Tx)
2
(1.1)
for all x, y ∈ X . If the space X is complete then every generalized multivalued λ-contraction T has a fixed point, i.e., a point z ∈ X such that z ∈ Tz. The set of all fixed points of T is denoted by F (T ). A multivalued map T : X → CL(X ) is a λ-contraction (see, for instance, [2,3] and [4, p. 163]), if D(Tx, Ty) ≤ λd(x, y)
(1.2)
for all x, y ∈ X , where 0 ≤ λ ≤ 1. Evidently, (1.2) implies (1.1), that is T satisfying (1.2) also satisfies (1.1). Investigations regarding the stability of fixed points of multivalued contractions were initiated by Nadler [3] and Markin [5]. The stability results for multivalued contractions have been found useful in the area of generalized differential equations and other contexts (see, for instance, [6–8]). In this paper we study the stability problem for generalized multivalued λ-contractions. We also obtain an extension of Lim’s result to a new class of contractions (Theorem 2.4).
∗
Corresponding author. Tel.: +27 47 502 2878; fax: +27 47 502 2725. E-mail addresses: [email protected] (S.L. Singh), [email protected] (S.N. Mishra), [email protected], [email protected] (W. Sinkala).
0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.01.042
S.L. Singh et al. / Applied Mathematics Letters 25 (2012) 1708–1710
1709
2. The main results The following generalizes an important result of Lim [6, Lemma 1] (see also [4, p. 163]). Theorem 2.1. Let X be a complete metric space and multivalued maps T1 , T2 : X → CL(X ) be λ-generalized contractions. Then D(F (T1 ), F (T2 )) ≤
1 sup D(T1 x, T2 x). 1 − λ x∈X
Proof. Write K = supx∈X D(T1 x, T2 x). We may assume that K is finite. Let q > 1. Pick x0 ∈ F (T1 ). Since D(T1 x0 , T2 x0 ) ≤ K , we may choose x1 ∈ T2 x0 such that d(x1 , x0 ) ≤ qK . Now we can choose x2 ∈ T2 x1 such that d(x2 , x1 ) ≤ qD(T2 x1 , T2 x0 ). Similarly, we choose x3 ∈ T2 x2 such that d(x3 , x2 ) ≤ qD(T2 x2 , T2 x1 ). Inductively we choose a sequence {xn } such that xn+1 ∈ T2 xn and d(xn+1 , xn ) ≤ qD(T2 xn , T2 xn−1 ),
n ≥ 1.
Since T2 is a generalized multivalued λ-contraction, we have d(xn+1 , xn ) ≤ qλ max {d(xn , xn−1 ), d(xn , xn+1 )} . This yields d(xn+1 , xn ) ≤ qλd(xn , xn−1 ).
(2.1)
This is true for n ≥ 1. Evidently we may choose q > 1 such that qλ < 1. Hence {xn } is a Cauchy sequence. Since the space X is complete, the sequence {xn } has a limit in X . Call it z. Therefore d(xn+1 , T2 z ) ≤ D(T2 xn , T2 z )
d(xn , T2 z ) + d(z , T2 xn ) . ≤ λ max d(xn , z ), d(xn , T2 xn ), d(z , T2 z ), 2
Making n → ∞, we obtain d(z , T2 z ) ≤ λd(z , T2 z ). Since λ < 1, it follows that d(z , T2 z ) = 0 proving that z ∈ F (T2 ). Furthermore, using (2.1), d(x0 , z ) ≤
∞
d(xn , xn+1 )
n=0
≤
1 1 − qλ
d(x0 , x1 ) ≤
q 1 − qλ
K.
Reversing the roles of T1 and T2 , we also conclude that for each y0 ∈ F (T2 ), there exists y1 ∈ T1 y0 and ω ∈ F (T1 ) such that d(y0 , ω) ≤
q 1 − qλ
K.
Hence D(F (T1 ), F (T2 )) ≤
q 1 − qλ
K.
This completes the proof when we take q → 1.
(2.2)
The following theorem generalizes Lim’s stability results [6] for a sequence of multivalued λ-contractions. Theorem 2.2. Let X be a complete metric space and Tn : X → CL(X ) a sequence of λ-generalized contractions, n = 0, 1, 2, . . . . If the sequence {Tn }∞ n=1 converges to T0 uniformly on X , then lim D(F (Tn ), F (T0 )) = 0.
n→∞
Proof. Since {Tn }∞ n=1 converges to T0 uniformly on X , for a given ε > 0, we can find a positive integer n0 such that sup D(Tn x, T0 x) < (1 − λ)ε x∈X
for all n ≥ n0 . Then D(F (Tn ), F (T0 )) < ε for all n ≥ n0 by Theorem 2.1.
1710
S.L. Singh et al. / Applied Mathematics Letters 25 (2012) 1708–1710
Extending the notion of multivalued λ-contractions, Kikkawa and Suzuki [9] recently obtained the following result. Theorem 2.3. Let (X , d) be a complete metric space and T a multivalued map from X into CB(X ), the class of nonempty closed bounded subsets of X . Assume that there exists λ ∈ [0, 1) such that 1 1+λ
d(x, Tx) ≤ d(x, y)
(2.3)
implies the condition (1.2). Then T has a fixed point. Theorem 2.3 has recently been extended in [10], by considering the following condition instead of (1.2). A map T : X → CL(X ) is a Suzuki–Zamfirescu λ-contraction (or simply an S–Z λ-contraction) if and only if there exists λ ∈ [0, 1) such that (2.3) implies
D(Tx, Ty) ≤ λ max d(x, y),
d(x, Tx) + d(y, Ty) d(x, Ty) + d(y, Tx) 2
,
2
(2.4)
for all x, y ∈ X . We remark that the condition (2.4) includes generalizations of Kikkawa and Suzuki’s contraction condition (cf. Theorem 2.3) studied by Mot and Petrusel [11] (see also [12,13]). Theorem 2.4. Let X be a complete metric space and let multivalued maps T1 , T2 : X → CL(X ) be S–Z λ-contractions. Then the conclusion of Theorem 2.1 is true. Proof. By [10, Corollary 3.2], T1 and T2 have fixed points. Now, the proof may be completed following Mot and Petrusel [11, Theorem 3.2] and the proof of Theorem 2.1. Remark 2.1. A result akin to Theorem 2.4 has recently been obtained in [11, Theorem 3.2] for when T1 and T2 are Kikkawa–Suzuki multivalued contractions. Acknowledgments The authors thank the Directorate of Research Development of Walter Sisulu University for financial support. A special word of thanks is due to an anonymous referee whose constructive comments were useful in improving the first draft of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
Lj. B. Cirić, Fixed points for generalized multivalued contractions, Mat. Vesnik 9 (24) (1972) 265–272. S.B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (2) (1969) 475–488. S.B. Nadler Jr., Hyperspaces of Sets, Dekker, New York, Basel, 1978. K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, 1990. J.T. Markin, Continuous dependence of fixed point sets, Proc. Amer. Math. Soc. 38 (3) (1973) 545–547. T.C. Lim, Fixed point stability for set valued contractive mappings with applications to generalized differential equations, J. Math. Anal. Appl. 110 (1985) 436–441. B. Ricceri, Smooth extensions of lipschitzian real functions, Proc. Amer. Math. Soc. 104 (2) (1988) 641–642. A. Tolstonogov, Differential Inclusions in a Banach Space, Kluwer Academic Publishers, 2000. M. Kikkawa, T. Suzuki, Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal. 69 (9) (2008) 2942–2949. S.L. Singh, S.N. Mishra, Coincidence theorems for certain classes of hybrid contractions, Fixed Point Theory Appl. 2010 (2010) Article ID 898109, 14 pages. G. Mot, A. Petrusel, Fixed point theory for a new type of contractive multivalued operators, Nonlinear Anal. 70 (9) (2009) 3371–3377. S. Dhompongsa, H. Yingtaweesittikul, Fixed points for multivalued mappings and the metric completeness, Fixed Point Theory Appl. 2009 (2009) Article ID 972395, 15 pages. S.L. Singh, S.N. Mishra, Remarks on recent fixed point theorems, Fixed Point Theory Appl. 2010 (2010) Article ID 452905, 18 pages.