Fixed Point Theory and Applications  Full text  Common fixed point and approximation results for generalized (f,g) weak contractions
Common fixed and approximation for generalized f . )weak contractions
Corresponding author: R Khan
Author Affiliations
Department of University of Sargodha, Pakistan
Department of and Statistics, Fahd University Petroleum Minerals, Saudi Arabia
is an Access article under the of the Commons Attribution (http://creativecommons.org/licenses/by/2.0 ), permits unrestricted distribution, and in any provided the work is cited.
Abstract
existence of fixed points established for mappings where is generalized f . )weakly contractive on a subset of Banach space. applications, the approximation results proved. Our unify and several recent in the
Keywords:
common point; Banach pair; generalized f . )weakly contractive generalized ( . g maps; invariant
1. Introduction preliminaries
We review needed Let ( . d be a space. A T . → X called weakly if, for x . ∈ X
where ϕ [0, ∞ → [0, ) is lower semicontinuous from right that ϕ positive on ∞ ) ϕ (0) 0.
where . g X → are selfmappings ϕ . ∞ ) [0, ∞ is a semicontinuous function right such ϕ is on (0, ) and (0) = If g f . T is fweakly contractive If f I . identity operator, T is weakly contractive Note that g = = I ϕ is nondecreasing, then definition of f . )weakly contractive is same the one appeared in ,2 ]. if f I and ( t = (1 k ) for a k with k 1, an f contractive mapping called a . Also that if = g I and is lower from the then ψ t ) t — ( t is upper from the and the (1.1) is by
d x. T ≤ ψ x. y.
( f g )weakly maps for ϕ is semicontinuous from right are the type Boyd and [3 ]. if we k ( ) = — ϕ t ) for t and k = 0 with f g = . then condition (1.1) replaced by
d T T y k d y d y.
Therefore f . )weakly contractive are closely to the studied by and Takahashi ].
If ( t = (1 k ) for a k with k 1, an ( . g contractive mapping called a f . ) contraction which has investigated by and Jungck ], Jungck Hussain [6 Song [7 and many
The set fixed points T is by F T ). point x X is coincidence point fixed point) f and if fx Tx ( = fx Tx ). set of points of and T denoted by ( f T ). pair f T is
(3) weakly [10 ] they commute their coincidence i.e. if = Tfx fx = ;
(4) operator pair if the F ( ) is invariant, namely ( F f )) F ( ). Obviously, pair ( . f is a operator pair converse is true in see [11 ]. If T . ) is Banach operator then ( . T need not a Banach pair (cf. [11 ], 1]).
The M in linear space is called with q M . the segment q . ] = k ) + kx ≤ k 1 joining to x contained in for all ∈ M The map defined on q starshaped M is affine if
1 — q + x = — k q + f x. all x M.
A space X Opial’s condition for every x n X weakly to x X . inequality
lim n → x n x lim n → x n y
holds for y ≠ . Every space and space l (1 p ) satisfy condition. The T . → X said to demiclosed at if, for sequence x in M weakly to and Tx converges to ∈ X then 0 Tx .
M be subset of normed space X . The set M ( ) = ∈ M  x u  dist( u M ) called the of best to u X out M . dist( u M ) inf y u : ∈ M We denote ℕ and M ) M )), set of integers and closure (weak of a M in . respectively.
concept of weak contractive has been by Alber GuerreDelabriere [1 Actually, in ], the proved the of fixed for a weakly contractive on Hilbert In 2001, [ [2 Theorem 2] a generalization Banach’s contraction principle [Note weakly contraction contraction as special case ϕ ( ) = k ) )]. Recently, and Li ] introduced class of operator pairs, a new of noncommuting and it been further by Ciric al. [15 ], Hussain ,13 ], et al. ], Khan Akbar [18 ], Pathak Hussain [20 Song and [21 ] Akbar and [22 ].
this article, introduce the concept of ( f g )weakly mappings, and establish common point and best approximation for the generalized ( . g contractive mapping. results improve extend the common fixed and invariant results of [23 ], and Shahzad ], Chen Li [11 Habiniak [25 Hussain and [5 ], and Hussain ], Jungck Sessa [26 Pathak and [20 ], et al. ], Singh ,29 ], [7 ] Song and [21 ] the class ( f g )weakly maps. The of fixed theorems are in diverse of mathematics, engineering and in dealing the problems in approximation potential theory, theory, theory differential equations, of integral and others [20 ,30 ]).
2. Results ( f g )weak
The following is a case of [ [32 Theorem 3.1].
2.1 . M be nonempty subset a metric ( X d ), T be selfmap of . Assume cl T M ) M . T ( ) is and T weakly contractive Then M F ( ) is
d T T y d f g y ϕ d x. g = d y — d x.
Corollary 2.3 Let M a nonempty of a space ( . d and ( . f and ( . g be Banach pairs on . Assume cl( T M )) complete, T ( f g )weakly mapping and ( f ∩F ( ) is and closed. M∩F ( ) ∩F f ) ( g is singleton.
T x. y ≤ d f g y
where ψ [0, ∞ → [0, ) is semicontinuous from such that (0) = and ψ t ) for each 0. Then ∩ F T ) F ( ) ∩ ( g is singleton.
. Set ( t = t ψ ( ). Then (2.1) implies
T x. y ≤ d f g y f x. y (2.2)
α . ∞ ) (0, 1) an upper from right. M∩F ( ) ∩F f ) ( g is singleton.
T x. y ≤ f x. y — d f g y.
ϕ . ∞ ) [0, ∞ is a semicontinuous function right such ϕ ( ) 0 t 0 ϕ (0) 0. The follows from 2.2.
In 2.3, if ( t = (1 k ) for a k with k 1, f = . then easily obtain following result improves Lemma of Chen Li [11
Corollary 2.6 Let M a nonempty of a space ( . d and ( . f be a operator pair M . that cl( ( M is complete, is f and F f ) nonempty and Then M F ( ) ∩ ( f is singleton.
following result contains Theorems of [11 Theorem 2.2 [23 ], 4 of ] and 6 of ].
Theorem . Let be a subset of normed [resp. space X T . and g selfmaps of . Suppose F ( ) ∩ ( g is q cl T F ( ) ∩ ( g ⊆ F f ) ( g [resp. wcl ( F f ) ( g ⊆ F f ) ( g cl( T M )) compact [resp. T ( )) is compact], T continuous on [resp. id T is at 0, id stands identity map]
  x — y  ≤  f x g y  k ϕ (  f — g   (2.3)
for k ∈ 1) and . y M where . [0, ) → ∞ ) a lower function from such that is positive (0, ∞ and ϕ = 0. M ∩F T ) ( f ∩F ( ) ≠
  n x T n   k n  T — T   k n  f — g   n —   x — y  ≤  f x g y  — n ϕ  f — g     x — y  — ϕ   x — y 
for each . y F ( ) ∩F g ) for each ∈ ℕ, n . ∞ ) [0, ∞ is a semicontinuous function right such ϕ n positive on ∞ ) ϕ n = 0.
Corollary 2.8 Let M a nonempty of a [resp. Banach] X and . f g be of M Suppose that ( f ∩ F g ) q starshaped closed [resp. closed], cl( ( M is compact wcl( T M )) weakly compact], is continuous M [resp. is demiclosed 0], ( . f and ( . g are Banach pairs and (2.3) for x . ∈ M Then M F ( ) ∩ ( f ∩ F g ) ∅.
In 2.7 and 2.8, if ( t = ( k — ) t any constant with 0 1, and = f then we obtain the results.
Corollary . [ ], Theorem Let M a nonempty of a [resp. Banach] X and and f selfmaps of . Suppose F ( ) is starshaped, cl ( F f )) F ( ) [resp. T ( ( f ⊆ F f )], T ( )) is [resp. wcl( ( M is weakly and either is demiclosed 0 or satisfies Opial’s and T f nonexpansive M . F ( ) ∩ ( I ≠ ∅.
2.10 . [11 ], 3.23.3] Let be a subset of normed [resp. space X T . be selfmaps M . that F f ) q starshaped closed [resp. closed], cl( ( M is compact wcl( T M )) weakly compact either id T is at 0 X satisfies condition], ( . f is a operator pair T is nonexpansive on . Then ∩ F T ) F ( ) ≠
Corollary 2.11 [ [23 Theorem 2.1] M be nonempty closed q starshaped of a space X T and be selfmaps M such T ( ) ⊆ ( M Suppose that commutes with and q F ( ). If T ( )) is f is and linear T is nonexpansive on . then ∩ F T ) F ( ) ≠
Let C P M ∩ C f. g . where M f. u = M f ∩ C g u C M u = ∈ M. x ∈ M u
Remark 2.14 Corollary 2.5 [24 ], Theorems 4.1 4.2 of and Li ] and corresponding results [23 ,25 ] are cases of 2.12 and
x — ≥ x u  u  ≥ dist M.
If assume that T ( u )) weakly compact, Lemma 5.5 [ [33 p. 192] can show existence of z ∈ T ( u )) that dist( . wcl( ( M ))) = z — .
Thus, both cases, have
dist M u dist u. T M ≤ dist T M ≤ T — u x —
for all ∈ M . Hence z —  = u . ) and P M u ) nonempty, closed convex with ( P ( u ⊆ P ( u The compactness cl( T M u [resp. weak of wcl( ( M ))] implies cl( T D )) compact [resp. T ( )) is compact]. The now follows Corollary 2.13.
2.16 . 2.15 extends 4.1 and of [23 Theorem 2.6 [24 ], Theorem 8 [25 ].
Results for ( f g )weak
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