Fixed Point Theory and Applications  Full text  Common fixed point and approximation results for generalized (f,g) weak contractions
Common fixed point and approximation results for generalized ( f . g )weak contractions
* Corresponding author: Abdul R Khan arahim@kfupm.edu.sa
Author Affiliations
1 Department of Mathematics, University of Sargodha, Sargodha, Pakistan
2 Department of Mathematics and Statistics, King Fahd University of Petroleum Minerals, Dhahran, Saudi Arabia
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The existence of common fixed points is established for three mappings where T is generalized ( f . g )weakly contractive mapping on a nonempty subset of a Banach space. As applications, the invariant approximation results are proved. Our results unify and improve several recent results in the literature.
Keywords:
common fixed point; Banach operator pair; generalized ( f . g )weakly contractive maps; generalized ( f . g )nonexpansive maps; invariant approximation
1. Introduction and preliminaries
We first review needed definitions. Let ( X . d ) be a metric space. A map T . X → X is called weakly contractive if, for each x . y ∈ X ,
where ϕ . [0, ∞ ) → [0, ∞ ) is a lower semicontinuous function from right such that ϕ is positive on (0, ∞ ) and ϕ (0) = 0.
where f . g . X → X are selfmappings and ϕ . [0, ∞ ) → [0, ∞ ) is a lower semicontinuous function from right such that ϕ is positive on (0, ∞ ) and ϕ (0) = 0. If g = f . then T is called fweakly contractive . If f = I . the identity operator, then T is called weakly contractive . Note that if g = f = I and ϕ is continuous nondecreasing, then the definition of ( f . g )weakly contractive maps is same as the one which appeared in [1 ,2 ]. Further if f = I and ϕ ( t ) = (1 — k ) t for a constant k with 0 k 1, then an f weakly contractive mapping is called a contraction . Also note that if f = g = I and ϕ is lower semicontinuous from the right, then ψ ( t ) = t — ϕ ( t ) is upper semicontinuous from the right and the condition (1.1) is replaced by
d T x. T y ≤ ψ d x. y.
Therefore ( f . g )weakly contractive maps for which ϕ is lower semicontinuous from the right are of the type of Boyd and Wong [3 ]. And if we set k ( t ) = 1 — ϕ ( t ) /t for t 0 and k (0) = 0 together with f = g = I . then the condition (1.1) is replaced by
d T x. T y ≤ k d x. y d x. y.
Therefore ( f . g )weakly contractive maps are closely related to the maps studied by Mizoguchi and Takahashi [4 ].
If ϕ ( t ) = (1 — k ) t for a constant k with 0 k 1, then an ( f . g )weakly contractive mapping is called a ( f . g ) contraction . which has been investigated by Hussain and Jungck [5 ], Jungck and Hussain [6 ], Song [7 ] and many others.
The set of fixed points of T is denoted by F ( T ). A point x ∈ X is a coincidence point (common fixed point) of f and T if fx = Tx ( x = fx = Tx ). The set of coincidence points of f and T is denoted by C ( f . T ). The pair f . T is called;
(3) weakly compatible [10 ] if they commute at their coincidence points, i.e. if fTx = Tfx whenever fx = Tx ;
(4) Banach operator pair . if the set F ( f ) is T invariant, namely T ( F ( f )) ⊆ F ( f ). Obviously, commuting pair ( T . f ) is a Banach operator pair but converse is not true in general; see [11 13 ]. If ( T . f ) is a Banach operator pair, then ( f . T ) need not be a Banach operator pair (cf. [ [11 ], Example 1]).
The set M in a linear space X is called qstarshaped with q ∈ M . if the segment [ q . x ] = — k ) q + kx :0 ≤ k ≤ 1 joining q to x is contained in M for all x ∈ M . The map f defined on a q starshaped set M is called affine if
f 1 — k q + k x = 1 — k f q + k f x. for all x ∈ M.
A Banach space X satisfies Opial’s condition if, for every sequence x n in X weakly convergent to x ∈ X . the inequality
lim inf n → ∞ x n — x lim inf n → ∞ x n — y
holds for all y ≠ x . Every Hilbert space and the space l p (1 p ∞ ) satisfy Opial’s condition. The map T . M → X is said to be demiclosed at 0 if, for every sequence x n in M converging weakly to x and Tx n converges to 0 ∈ X . then 0 = Tx .
Let M be a subset of a normed space ( X . ·). The set P M ( u ) = x ∈ M .  x — u  = dist( u . M ) is called the set of best approximants to u ∈ X out of M . where dist( u . M ) = inf y — u : y ∈ M . We denote by ℕ and cl( M ) (wcl( M )), the set of positive integers and the closure (weak closure) of a set M in X . respectively.
The concept of the weak contractive mapping has been defined by Alber and GuerreDelabriere [1 ]. Actually, in [1 ], the authors proved the existence of fixed points for a singlevalued weakly contractive mapping on Hilbert spaces. In 2001, Rhoades [ [2 ], Theorem 2] obtained a generalization of Banach’s contraction mapping principle [Note the weakly contraction contains contraction as the special case ( ϕ ( t ) = (1 k ) t )]. Recently, Chen and Li [11 ] introduced the class of Banach operator pairs, as a new class of noncommuting maps and it has been further studied by Ciric et al. [15 ,16 ], Hussain [12 ,13 ], Hussain et al. [17 ], Khan and Akbar [18 ,19 ], Pathak and Hussain [20 ], Song and Xu [21 ] and Akbar and Khan [22 ].
In this article, we introduce the new concept of generalized ( f . g )weakly contractive mappings, and consequently establish common fixed point and invariant best approximation results for the noncommuting generalized ( f . g )weakly contractive mapping. Our results improve and extend the recent common fixed point and invariant approximation results of AlThagafi [23 ], AlThagafi and Shahzad [24 ], Chen and Li [11 ], Habiniak [25 ], Hussain and Jungck [5 ], Jungck and Hussain [6 ], Jungck and Sessa [26 ], Pathak and Hussain [20 ], Sahab et al. [27 ], Singh [28 ,29 ], Song [7 ] and Song and Xu [21 ] to the class of ( f . g )weakly contractive maps. The applications of fixed point theorems are remarkable in diverse disciplines of mathematics, statistics, engineering and economics in dealing with the problems arising in approximation theory, potential theory, game theory, theory of differential equations, theory of integral equations and others (see [20 ,30 ,31 ]).
2. Results for ( f . g )weak contractions
The following result is a particular case of Song [ [32 ], Theorem 3.1].
Lemma 2.1 . Let M be a nonempty subset of a metric space ( X . d ), and T be a selfmap of M . Assume that cl T ( M ) ⊂ M . cl T ( M ) is complete, and T is weakly contractive mapping. Then M ∩ F ( T ) is singleton.
d T x. T y ≤ d f x. g y — ϕ d f x. g y = d x. y — ϕ d x. y
Corollary 2.3 . Let M be a nonempty subset of a metric space ( X . d ), and ( T . f ) and ( T . g ) be Banach operator pairs on M . Assume that cl( T ( M )) is complete, T is ( f . g )weakly contractive mapping and F ( f ) ∩F ( g ) is nonempty and closed. Then M∩F ( T ) ∩F ( f ) ∩F ( g ) is singleton.
d T x. T y ≤ ψ d f x. g y (2.1)
where ψ . [0, ∞ ) → [0, ∞ ) is upper semicontinuous from right such that ψ (0) = 0 and ψ ( t ) t for each t 0. Then M ∩ F ( T ) ∩ F ( f ) ∩ F ( g ) is singleton.
Proof . Set ϕ ( t ) = t — ψ ( t ). Then inequality (2.1) implies
d T x. T y ≤ α d f x. g y d f x. g y (2.2)
where α . [0, ∞ ) → (0, 1) is an upper semicontinuous from right. Then M∩F ( T ) ∩F ( f ) ∩F ( g ) is singleton.
d T x. T y ≤ d f x. g y — ϕ d f x. g y.
where ϕ . [0, ∞ ) → [0, ∞ ) is a lower semicontinuous function from right such that ϕ ( t ) 0 for t 0 and ϕ (0) = 0. The result follows from Theorem 2.2.
In Corollary 2.3, if ϕ ( t ) = (1 — k ) t for a constant k with 0 k 1, and f = g . then we easily obtain the following result which improves Lemma 3.1 of Chen and Li [11 ].
Corollary 2.6 . Let M be a nonempty subset of a metric space ( X . d ), and ( T . f ) be a Banach operator pair on M . Assume that cl( T ( M )) is complete, T is f contraction and F ( f ) is nonempty and closed. Then M ∩ F ( T ) ∩ F ( f ) is singleton.
The following result properly contains Theorems 3.23.3 of [11 ], Theorem 2.2 of [23 ], Theorem 4 of [25 ] and Theorem 6 of [26 ].
Theorem 2.7 . Let M be a nonempty subset of a normed [resp. Banach] space X and T . f and g be selfmaps of M . Suppose that F ( f ) ∩ F ( g ) is q starshaped, cl T ( F ( f ) ∩ F ( g )) ⊆ F ( f ) ∩F ( g ) [resp. wcl T ( F ( f ) ∩F ( g )) ⊆ F ( f ) ∩F ( g )], cl( T ( M )) is compact [resp. wcl( T ( M )) is weakly compact], T is continuous on M [resp. id — T is demiclosed at 0, where id stands for identity map] and
  T x — T y   ≤   f x — g y   k — ϕ (   f x — g y   ). (2.3)
for all k ∈ (0, 1) and x . y ∈ M where ϕ . [0, ∞ ) → [0, ∞ ) is a lower semicontinuous function from right such that ϕ is positive on (0, ∞ ) and ϕ (0) = 0. Then M ∩F ( T ) ∩F ( f ) ∩F ( g ) ≠ ∅.
  T n x — T n y   = k n   T x — T y   ≤ k n   f x — g y   k n — ϕ   f x — g y   ≤   f x — g y   — k n ϕ   f x — g y   =   f x — g y   — ϕ n   f x — g y  .
for each x . y ∈ F ( f ) ∩F ( g ) and for each n ∈ ℕ, ϕ n . [0, ∞ ) → [0, ∞ ) is a lower semicontinuous function from right such that ϕ n is positive on (0, ∞ ) and ϕ n (0) = 0.
Corollary 2.8 . Let M be a nonempty subset of a normed [resp. Banach] space X and T . f and g be selfmaps of M . Suppose that F ( f ) ∩ F ( g ) is q starshaped and closed [resp. weakly closed], cl( T ( M )) is compact [resp. wcl( T ( M )) is weakly compact], T is continuous on M [resp. idT is demiclosed at 0], ( T . f ) and ( T . g ) are Banach operator pairs and satisfy (2.3) for all x . y ∈ M . Then M ∩ F ( T ) ∩ F ( f ) ∩ F ( g ) ≠ ∅.
In Theorem 2.7 and Corollary 2.8, if ϕ ( t ) = ( 1 k — 1 ) t for any constant k with 0 k 1, and g = f . then we easily obtain the following results.
Corollary 2.9 . [ [24 ], Theorem 2.4] Let M be a nonempty subset of a normed [resp. Banach] space X and T and f be selfmaps of M . Suppose that F ( f ) is q starshaped, cl T ( F ( f )) ⊆ F ( f ) [resp. wcl T ( F ( f )) ⊆ F ( f )], cl( T ( M )) is compact [resp. wcl( T ( M )) is weakly compact and either idT is demiclosed at 0 or X satisfies Opial’s condition] and T is f nonexpansive on M . Then F ( T ) ∩ F ( I ) ≠ ∅.
Corollary 2.10 . [ [11 ], Theorems 3.23.3] Let M be a nonempty subset of a normed [resp. Banach] space X and T . f be selfmaps of M . Suppose that F ( f ) is q starshaped and closed [resp. weakly closed], cl( T ( M )) is compact [resp. wcl( T ( M )) is weakly compact and either id — T is demiclosed at 0 or X satisfies Opial’s condition], ( T . f ) is a Banach operator pair and T is f nonexpansive on M . Then M ∩ F ( T ) ∩ F ( f ) ≠ ∅.
Corollary 2.11 . [ [23 ], Theorem 2.1] Let M be a nonempty closed and q starshaped subset of a normed space X and T and f be selfmaps of M such that T ( M ) ⊆ f ( M ). Suppose that T commutes with f and q ∈ F ( f ). If cl( T ( M )) is compact, f is continuous and linear and T is f nonexpansive on M . then M ∩ F ( T ) ∩ F ( f ) ≠ ∅.
Let C = P M u ∩ C M f. g u . where C M f. g u = C M f u ∩ C M g u and C M f u = x ∈ M. f x ∈ P M u .
Remark 2.14 . Corollary 2.5 of [24 ], and Theorems 4.1 and 4.2 of Chen and Li [11 ] and the corresponding results in [23 ,25 29 ] are particular cases of Corollaries 2.12 and 2.13.
x — u ≥ x — u   u   ≥ dist u. M.
If we assume that wcl( T ( M u )) is weakly compact, using Lemma 5.5 of [ [33 ], p. 192] we can show the existence of a z ∈ wcl( T ( M u )) such that dist( u . wcl( T ( M u ))) =  z — u .
Thus, in both cases, we have
dist u. M u ≤ dist u. cl T M u ≤ dist u. T M u ≤ T x — u ≤ x — u.
for all x ∈ M u . Hence  z — u  = dist( u . M ) and so P M ( u ) is nonempty, closed and convex with T ( P M ( u )) ⊆ P M ( u ). The compactness of cl( T ( M u )) [resp. weak compactness of wcl( T ( M u ))] implies that cl( T ( D )) is compact [resp. wcl( T ( D )) is weakly compact]. The result now follows from Corollary 2.13.
Remark 2.16 . Theorem 2.15 extends Theorems 4.1 and 4.2 of [23 ], Theorem 2.6 of [24 ], and Theorem 8 of [25 ].
3. Results for generalized ( f . g )weak contractions
Interesting Articles
 Journal of Inequalities and Applications  Full text  Applications of Caristi’s fixed point results
 COMMON COUPLED FIXED POINTS OF GENERALIZED CONTRACTIVE MAPPINGS — ResearchGate
 Ice Chips 06′07′: Transactions
 Ice Chips 06′07′:
 Common fixed point theorems in symmetric spaces employing a new implicit function and common property (E.A). — Bulletin of the Belgian Mathematical Society — Simon Stevin  HighBeam Research
Share this article:
Other articles of the category "Sessa":

Flying bridge 

Estrogens Increase Transcription of the Human Endo...

Explorehoward

Journal of Inequalities and Applications  Full te...

36 Express Cabo Yacht  36X CABO Oyster Harbors Ma...

Key Largo Vacation Accommodation  Islamorada Vaca...

ForumNautico.it Sessa Dorado 36 fly

Press Article — 2012 Tullett Prebon London B...