Filomat 30:14 (016), 3875 3883 DOI 10.98/FIL1614875C Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Invariant Approximation Results of Generalized Contractive Mappings Ljubomir Ćirić a, Sumit Chandok b, Mujahid Abbas c a Department of Mathematics, University of Belgrade, Serbia. b School of Mathematics, Thapar University, Patiala-147004, Punjab, India. c Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood road, Pretoria 000, South Africa. Abstract. Abbas, Ali and Salvador [Fixed and periodic points of generalized contractions in metric spaces, Fixed Point Theory Appl. 013, 013:43] extended the concept of F contraction mapping introduced in [1], to two mappings. The aim of this paper is to introduce the notion of a generalized F weak contraction mapping and to study sufficient conditions for the existence of common fixed points for such class of mappings. As applications, related invariant approximation results are derived. The results obtained herein unify, generalize and complement various known results in the literature. 1. Introduction and Preliminaries The Banach contraction principle appeared in explicit form in Banach s thesis [3] in 19 where it was used to establish the existence of a solution for an integral equation. Since then, it has become a very popular tool in solving existence problems in many branches of mathematics. Extensions of this principle were obtained either by generalizing the domain of mappings or by extending the contractive condition on the mappings (see, e.g., [1],[],[4]-[] and references mentioned therein). Recently, Wardowski [1] introduced a new type of contractive mapping called F-contraction and proved a fixed point theorem as a generalization of the Banach contraction principle. Abbas et al. ([]) introduced the concept of F contraction with respect to a self mapping and obtained common fixed point results in an ordered metric space. Meinardus [15] was the first who employed fixed point theorem to prove the existence of invariant approximation in Banach spaces. Subsequently, several interesting and valuable results have appeared about invariant approximations ([4], [5], [13] ). Following this research direction, we introduce a notion of generalized F -contraction mappings and obtain common fixed point results employing such contractions. Some results on invariant approximation for such mappings are also obtained which in turn extend and strengthen various known results. In the sequel the letters N, R + and and R will denote the set of all natural numbers, the set of all nonnegative real numbers and the set of all real numbers, respectively. Next we give some definitions which will be used in the sequel. 010 Mathematics Subject Classification. Primary 47H10; Secondary 54H5, 41A50. Keywords. Common fixed point, F-weak contraction mappings, invariant approximations, weakly compatible mappings. Received: 0 November 014; Accepted: 09 July 015 Communicated by Dragan S. Djordjević Email addresses: lciric@rcub.bg.ac.rs (Ljubomir Ćirić), chansok.s@gmail.com (Sumit Chandok), mujahid.abbas@up.ac.za (Mujahid Abbas)
L. Ćirić et al. / Filomat 30:14 (016), 3875 3883 3876 Definition 1.1. ([1]) Let Ϝ be the collection of all mappings F : R + R such that (F1) F is strictly increasing, that is, for all α, β R + such that α < β implies that F(α) < F(β); (F) For any sequence {α n } of positive numbers, lim α n = 0 and lim F(α n ) = are equivalent; (F3) There exists k (0, 1) such that lim αk F(α) = 0. α 0 + For examples of mappings satisfying conditions ( F1 F3 ), we refer to [1] and []. Let F be the collection of all mappings F : R + R which satisfy conditions (F1) and (F3). Definition 1.. ([1]) Let (X, d) be a metric space. A map T : X X is said to be F-contraction on X if there exist F Ϝ and τ > 0 such that the following holds: τ + F(d(Tx, Ty)) F(d(x, y)) for all x, y X with Tx Ty. Abbas et al. ([]) extended the above definition to two mappings. Recently, Wardowski and Dung [] used a contractive inequality introduced and investigated by Ćirić [10], [11] and extended the concept of F contraction mappings to F weak contraction mappings as follows: Definition 1.3. Let (X, d) be a metric space. A map T : X X is said to be F-weak contraction on X if there exist F Ϝ and τ > 0 such that for any x, y X with d(tx, Ty) > 0, the following hold: τ + F(d(Tx, Ty)) F(max{d(x, y), d(x, Tx), d(y, Ty), d(x, Ty) + d(y, Tx) }). Every F contraction mapping is F weak contraction but converse does not hold in general ([]). To begin with, we recall the following result of []. Theorem 1.4. Let (X, d) be a complete metric space and T : X X be F-weak contraction. If T or F is continuous, then we have (1) T has a unique fixed point x 1 X. () For all x X, the sequence {T n x} is convergent to x 1. Now, we give the following definition. Definition 1.5. Let (X, d) be a metric space and T, : X X. A mapping T is called F -weak contraction on X if there exist F F and τ > 0 such that for any x, y X with d(tx, Ty) > 0, the following holds: τ + F(d(Tx, Ty)) F(max{d( x, y), d( x, Tx), d( y, Ty), d( x, Ty) + d( y, Tx) }). Definition 1.6. Let (X, d) be a metric space and T, : X X. A mapping T is called F -contraction on (X, d) if there exist F F and τ > 0 such that, for all x, y X satisfying d(tx, Ty) > 0, the following holds: τ + F(d(Tx, Ty)) F(d( x, y)). The following example shows that a mapping T is F -weak contraction but not F -contraction. Example 1.7. Let X = [0, 1], and d(x, y) = x y. Define T, : X X by T(x) = 1 3, x [0, 1 3 ) 1, x [ 1 3, 1) 0, x = 1
L. Ćirić et al. / Filomat 30:14 (016), 3875 3883 3877 and (x) = 8 10, x [0, 1 3 ) 9 10, x [ 1 3, 1]. Note that T is F -weak contraction when F(t) = ln(t), t (0, + ) and τ = ln 1.045 but not F -contraction if x = 1 and y = 0. A subset M of a normed space E is said to be (1) convex if M includes every line segment joining any two of its points () z starshaped or starshaped with respect to z M if for any x M and q [0, 1], qz + (1 q)x M. In this case, z is called a star-centre of M. A selfmapping f on M is said to be demiclosed at y if for any sequence {x n } in M with {x n } converges weakly to x M and { f x n } converges to y strongly, then f x = y. Definition 1.8. Let (X, d) be a metric space and f, : X X. The pair ( f, ) is called (a) commuting if f x = f x for all x X (b) compatible if for any {x n } in X with lim f x n = lim x n = t for some t in X, we have lim d( f x n, f x n ) = 0 (c) weakly compatible if they commute at their coincidence points, that is, if f x = f x whenever x = f x. Let f and be two self maps on X. A point x X is called (1) a fixed point of f if f (x) = x; () a common fixed point of and f if x = x = f x; (3) coincidence point of and f if x = f x. The set of all common fixed points ( coincidence points) of and f is denoted by F(, f ) ( C(, T), respectively). Let M be a subset of metric space (X, d). For any x X, define P M (x) = {y M : d(x, y) = d(x, M)}, where d(x, M) = inf{d(x, y) : y M}. The set P M (x) is the set of best approximations of x from M. If for each x X, P M (x) is nonempty, then M is called proximinal. Observe that if M is closed, then P M (x) is also closed. The reader interested in the interplay of fixed points and approximation theory in normed spaces is referred to the pioneer work of Park [18] and Singh ([0]).. Main Results Theorem.1. Let D be a subset of a metric space (X, d) and and T selfmappings of D with T(D) (D). If T(D) is complete, F F is a continuous mapping, T is F -weak contraction mapping on D, then T and have a unique coincidence point. Moreover if (, T) is weakly compatible, then F(T) F( ) is a singleton. Proof. Let x 0 in D be arbitrary. Choose a point x 1 D such that x 1 = Tx 0. This can be done because T(D) (D). As x 1 (D), there exists x D such that x = Tx 1. Continuing this process, having chosen x n in D with (x n ) D, we obtain x n+1 in D such that x n+1 = Tx n. Thus we obtain a sequence {x n } in D such that x n+1 = Tx n for n 0. If there exists some n 0 in N {0} such that Tx n0 = Tx n0 +1, then Tx n0 = x n0 +1 implies that x n0 +1 = Tx n0 +1 and x n0 +1 is coincidence point of and T. We assume that Tx n+1 Tx n for all n in N {0}. Consider F(d(Tx n+1, Tx n )) F(max{d( x n+1, x n ), d( x n+1, Tx n+1 ), d( x n, Tx n ) d( x n+1, Tx n ) + d( x n, Tx n+1 ) = F(max{d(Tx n, Tx n 1 ), d(tx n, Tx n+1 ), d(tx n 1, Tx n ) d(tx n, Tx n ) + d(tx n 1, Tx n+1 ) F(max{d(Tx n, Tx n 1 ), d(tx n, Tx n+1 ), d(tx n 1, Tx n ), d(tx n 1, Tx n ) + d(tx n, Tx n+1 ) = F(max{d(Tx n, Tx n 1 ), d(tx n, Tx n+1 ).
L. Ćirić et al. / Filomat 30:14 (016), 3875 3883 3878 If for some n 1, max{d(tx n, Tx n 1 ), d(tx n, Tx n+1 )} = d(tx n, Tx n+1 ) then we have F(d(Tx n+1, Tx n )) F(d(Tx n+1, Tx n )) τ < F(d(Tx n+1, Tx n )) a contradiction. Hence max{d(tx n, Tx n 1 ), d(tx n, Tx n+1 )} = d(tx n, Tx n 1 ) for all n 1. Thus, we have F(d(Tx n+1, Tx n )) F(d(Tx n, Tx n 1 )) τ for all n 1. This further implies that F(d(Tx n+1, Tx n )) F(d(Tx 1, Tx 0 )) nτ for all n 1. On taking limit as n, we have lim F(d(Tx n+1, Tx n )) =. We now show that lim d(tx n+1, Tx n ) = 0. Assume on contrary that lim d(tx n+1, Tx n ) = α 0. Using Archmidean property of real numbers, there exists some α 1 R + such that 0 < α 1 < α. By nondecreasing property of F, we obtain that F(α 1 ) <, a contradiction. Hence lim d(tx n+1, Tx n ) = 0. By (F3), there exists k (0, 1) such that lim (d(tx n+1, Tx n )) k F(d(Tx n+1, Tx n )) = 0. As F(d(Tx n+1, Tx n )) F(d(Tx 1, Tx 0 )) nτ, so and hence (d(tx n+1, Tx n )) k [F(d(Tx n+1, Tx n )) F(d(Tx 1, Tx 0 ))] (d(tx n+1, Tx n )) k nτ 0, lim (d(tx n+1, Tx n )) k n = 0. Thus, there exists n 1 N such that (d(tx n+1, Tx n )) k n 1 for all n n 1. That is, d(tx n+1, Tx n ) 1 n n 1. Therefore for all m > n 1, we have d(tx m, Tx n ) d(tx m, Tx m 1 ) +... + d(tx n+1, Tx n ) < d(tx i+1, Tx i ) i=n i=n 1 i 1/k < n 1 k for all This proves that {Tx n } is a Cauchy sequence in g(x). As x n = Tx n 1, { x n } is also a Cauchy sequence in T(D). By the completeness of T(D), there is some w T(D) (D) such that lim x n+1 = lim Tx n = w. Let z D be such that (z) = w, that is, lim x n+1 = lim Tx n = z. (1) Let us assume that d(tz, Tx n ) > 0 and d( z, Tz) > 0. As T is a weak F-contraction, we have F(d(Tz, Tx n )) F(max{d( z, x n ), d( z, Tz), d( x n, Tx n ), d( z, Tx n) + d( x n, Tz). ()
L. Ćirić et al. / Filomat 30:14 (016), 3875 3883 3879 Then from (1) and () there is a sufficiently large n 0, such that F(d(Tz, Tx n )) F(d(Tz, z)) τ, (3) for all n n 0. Note that if lim d(tz, Tx n ) = d(tz, z) > 0, then by (F), lim F(d(Tz, Tx n )). Now, taking the limit as n in the above inequality (3), we get, as lim F(d(Tz, Tx n)) = F(d(Tz, z)), F(d(Tz, z)) F(d(Tz, z)) τ. Hence we have τ 0, a contradiction with the assumption τ > 0. Therefore, our supposition d( z, Tz) > 0 was wrong. Thus d( z, Tz) = 0. Hence Tz = z. So we proved that z is a coincidence point of T and. Now suppose that T and are weakly compatible. Since w = T(z) = (z), then T(w) = T( (z)) = (T(z)) = (w). Now we show that (w) = w. If T(z) T(w), then we have d( z, Tw) + d( w, Tz) F(d(T(z), T(w))) F(max{d( z, w), d( z, Tz), d( w, Tw), d(tz, Tw) + d(tw, Tz) = F(max{d(Tz, Tw), = F(d(Tz, Tw)) τ, a contradiction. Hence T(z) = T(w) = (w) = w. If z 1, z are two common fixed points of T and such that z 1 z, then we have F(d(z 1, z )) = F(d(T(z 1 ), T(z ))) a contradiction. Hence z 1 = z. F(max{d( z 1, z ), d( z 1, Tz 1 ), d( z, Tz ), d( z 1, Tz ) + d( z, Tz 1 ) = F(max{d(z 1, z ), d(z 1, z 1 ), d(z, z ), d(z 1, z ) + d(z, z 1 ) = F(d(z 1, z )) τ. Corollary.. Let D be a subset of a metric space (X, d) and and T self-mappings on D with T(D) (D). If T(D) is complete, F F is a continuous mapping, T is F -contraction mapping on D, then T and have a unique coincidence point in D. Moreover if (, T) is weakly compatible, then F(T) F( ) is a singleton. If =identity mapping, then we have the following results. Corollary.3. Let D be a nonempty subset of a metric space (X, d) and T : D D. If T(D) D, T(D) is complete, F F is a continuous mapping, and T is an F-weak contraction, then F(T) is singleton. Corollary.4. Let D be a nonempty subset of a metric space (X, d) and T : D D. If T(D) D, T(D) is complete, F F is a continuous mapping, and T is an F-contraction mapping, then F(T) is a singleton. As an application of Corollary.3, we obtain the following general result. Theorem.5. Let D be a nonempty subset of a metric space (X, d) and, T : D D. If F( ) is nonempty, T(F( )) F( ), T(D) is complete, F F is a continuous mapping, and T is F -weak contraction, then F(, T) is a singleton.
Proof. Let x, y F( ). Note that L. Ćirić et al. / Filomat 30:14 (016), 3875 3883 3880 d( x, Ty) + d( y, Tx) F(d(Tx, Ty)) F(max{d( x, y), d( x, Tx), d( y, Ty), d(x, Ty) + d(y, Tx) = F(max{d(x, y), d(x, Tx), d(y, Ty),. Hence T is an F weak contraction on F( ). The completeness of T(D) implies the completeness of T(F( )). It follows from Corollary.3 that F(, T) = F( ) F(T) is singleton. In the sequel, we assume that D is a q-starshaped subset of a normed space E and, T : D D. Define δ( x, Ty) := inf{ x T k y : 0 k 1} for all x, y D where T k x := ktx + (1 k)q. For a subset D of a normed space E, D w denotes weak closure of D in E Definition.6. Let X be a normed linear space and T, : X X. A mapping T is called generalized F -nonexpansive on X if there exist F F and τ > 0 such that for any x, y X with d(tx, Ty) > 0, we have F( Tx Ty ) F(max{ x y, δ( x, Tx), δ( y, Ty), δ( x, Ty), δ( y, Tx). Theorem.7. Let D be a subset of a normed space E and, T : D D. If F( ) is q-starshaped, T(F( )) F( ), T(D) is compact, T is continuous and generalized F -nonexpansive on D, F F is a continuous mapping, then F(, T) is nonempty. Moreover, if we replace T(F( )) F( ) with T(F( )) w F( ), compactness of T(D) with weak compactness of T(D) and continuity of T with demiclosedness of T at zero, then the same conclusion holds provided that E is a Banach space. Proof. Let {λ n } be a sequence in (0, 1) such that λ n 1 as n. For each n 1, define T n x = λ n Tx + (1 λ n )q for all x F( ). As F( ) is q-starshaped and T(F( )) F( ), so we have T n (F( )) F( ) for each n 1. Note that F( T n x T n y ) = F( λ n Tx + (1 λ n )q λ n Ty (1 λ n )q ) = F( λ n (Tx Ty) ). Since F is strictly increasing and λ n < 1 for each n 1, therefore F( T n x T n y ) < F( (Tx Ty) ) < F(max{ x y, δ( x, Tx), δ( y, Ty), δ( x, Ty), δ( y, Tx) < F(max{ x y, x T n x, y T n y, x T n y, y T n x holds for all x, y F( ). As T(D) is compact, so T n (D) is compact for each n 1. Thus T n (D) is complete for each n 1. It follows from Theorem.5 that for each n 1, F(, T n ) = F( ) F(T n ) = {x n } for some x n F( ). Since T(D) is compact, there exists a subsequence {Tx nk } of {Tx n } such that Tx nk z T(D). Since {Tx n } is a sequence in T(F( )), then z T(F( )) F( ). Moreover, x nk = T nk x nk = λ nk Tx nk + (1 λ nk )q z. As T is continuous on D, we have Tz = z. Next, assume that T(D) w is weakly compact and T is demiclosed at zero. Thus there exists a subsequence {Tx nk } of {Tx n } such that Tx nk w z T(F( )) w. Since {Tx n } is a sequence in T(F( )), then
L. Ćirić et al. / Filomat 30:14 (016), 3875 3883 3881 z T(F( )) w F( ). From the boundedness of {Tx nk }, we conclude that {x xnk } is bounded too. As λ nk 1 and x nk Tx nk = x nk Tx nk = (1 λ nk )q + λ nk Tx nk Tx nk (1 λ nk ) q Tx nk (1 λ nk ) ( q + Tx nk ), for each k 1. We have x nk Tx nk w 0. It follows from the demiclosedness of T at zero that z Tz = ( T)z = 0 and hence Tz = z. The result follows. Let M be a nonempty subset of a metric space (X, d). Suppose that C = P M (u) C f M (u), where C M (u) = {x M : x P M (u)}. Corollary.8. Let E be a normed linear space and T, self mappings of E. If u E, D G, G = D F( ) is q- starshaped, T(G) G, T(D) is compact, and T is continuous generalized F -nonexpansive on D, F F is a continuous mapping, then P M (u) F(, T) is nonempty. Moreover, if we replace T(G) G with T(G) w G, compactness of T(D) with weak compactness of T(D) and continuity of T with demiclosedness of T at zero, then the same conclusion holds provided that E is a Banach space. Corollary.9. Let E be a normed linear space and T, f self mappings of X. If u E, D P M (u), G = D F( f ) is q- starshaped, T(G) G, T(D) is compact and T is continuous generalized F -nonexpansive on D, F F is a continuous mapping, then P M (u) F( f, T) is nonempty. Moreover, if we replace T(G) G with T(G) w G, compactness of T(D) with weak compactness of T(D) and continuity of T with demiclosedness of f T at zero, then the same conclusion holds provided that E is a Banach space. Let G 0 denotes the class of closed convex subsets of a normed space X containing 0. For M G 0 and p X, let M p = {x M : x p }. Then P M (p) M p G 0 (see [4], [17]). Theorem.10. Let E be a normed linear space and T, self mappings on X. If p E and M G 0 such that T(M p ) M, T(M p ) is compact, and Tx p x p for all x M p, then P M (p) is nonempty, closed and convex with T(P M (p)) P M (p). If, in addition, D is a subset of P M (p), G = D F( ) is q-starshaped, T(G) G and T is continuous generalized F -nonexpansive on D, F F is a continuous mapping, then P M (p) F(, T) is nonempty. Moreover if we replace compactness of T(M p ) with weak compactness of T(M p ) w, T(G) G with T(G) w G and continuity of T with demiclosedness of T at 0, then the same conclusion holds provided that E is a Banach space. Proof. If p M then the result follows. Assume that p M. If x M\M p, then x > p and hence p x x p > p d(p, M). Thus α = d(p, M) p. As T(M p ) is compact so by the continuity of norm, there exists z T(M p ) such that β = d(p, cl T(M p )) = z p. On the other hand, if T(M p ) w is weakly compact, then by Lemma 5.5 in ([0]), there exists z T(M p ) w such that β = d(p, T(M p ) w ) = z p. Thus in both cases, we have α = d(p, M) d(p, T(M p )) = β = d(p, T(M p )) Tx p x p
for all x M p. Therefore α = β = d(p, M), that is, d(p, M) = d(p, T(M p )) = p z L. Ćirić et al. / Filomat 30:14 (016), 3875 3883 388 and z P M (P). Hence P M (p) is nonempty. The closedness and convexity of P M (p) follows from that of M. Now we prove that T(P M (p)) P M (p). If y T(P M (p)). Then y = Tx for some x P M (p). Since p y = p Tx p x = d(p, M). Therefore y P M (p) M p. Now T(P M (p)) M gives that y M. The compactness of T(M p ) (weak compactness of T(M p ) w, respectively) implies that T(D) compact (respectively, T(D) w is weakly compact). Hence the result follows from Corollary.9. Example.11. Let M = [0, 1]. Define T, :M M by T(x) = and (x) = x. Note that 0, x [0, 1 ) 1 8, x [ 1, 1] τ + F(d(Tx, Ty)) F(max{d( x, y), d( x, Tx), d( y, Ty), d( x, Ty) + d( y, Tx) }) holds with F(t) = ln(t), t (0, + ) and τ = ln 1.5. Moreover 0 is common fixed point of T and. Remarks.11.1. 1) It must be noted that the assumption of linearity or affinity for I is necessary in almost all known results dealing with common fixed points of maps T, and I when T is I-nonexpansive under the conditions of commuting, weakly commuting, R-subweakly commuting, compatibility or Banach operator pairs (see [4]-[9], [14] and the literature cited therein), but our results in this paper are independent of the linearity or affinity condition on I. ) In [], it was shown that from different types of F-weak contractions, one can obtain the variety of well-known contractions. Using the concept the of Wardowski [1], we introduce the F -weak contraction mappings by omitting condition (F) which was assumed in [1, ]. Also, in Example 1.7, we showed that F -contraction is F -weak contraction but not conversely. Therefore, the obtained results generalize and extend many of the well-known results existing in the literature to F -weak contraction mappings. Acknowledgement: The authors thank editor and reviewers for their valuable suggestions that helped us to improve the final version of the paper. References [1] M. Abbas, B. Ali and S. Romaguera, Generalized contraction and invariant approximation results on nonconvex subsets of normed spaces, Abstract and Applied Analysis, 014(014), Article 39195, 5 pages. [] M. Abbas, B. Ali and S. Romaguera, Fixed and periodic points of generalized contractions in metric spaces, Fixed Point Theory Appl. 013, 013:43. [3] S. Banach, Sur les op erations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math., 3(19), 133 181. [4] M. A. Al-Thagafi, Common fixed points and best approximation, J. Approx. Theory 85(1996), 318-33. [5] M. A. Al-Thagafi and N. Shahzad, Noncommuting self-maps and invariant approximations, Nonlinear Anal. 64(006), 778-786. [6] M. A. Al-Thagafi and N. Shahzad, Banach operator pairs, common fixed points, invariant approximations, and -nonexpansive multimaps, Nonlinear Analysis 69 (008), 733-739. [7] S. Chandok, Some common fixed point theorems for generalized nonlinear contractive mappings, Computers Math. Appl. 6 (10)(011), 369-3699. [8] S. Chandok, Common fixed points, invariant approximation and generalized weak contractions, Internat. J. Math. Math. Sci. Vol. 01 (01) Article ID 10980.
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