New generalization of contraction mapping by new control function
NNew generalization of contraction mapping by new controlfunction
S. M. A. Aleomraninejad
Department of Mathematics, Qom University of Technology, Qom, Iran email: [email protected]
Abstract.
In this paper, we introduce the new generalization of contraction mapping by a newcontrol function and an altering distance . We establish some existence results of fixed point for suchmappings. Our results reproduce several old and new results in the literature.
Keywords:
Altering distance, Contraction mapping, Control function, Fixed point
Mathematics Subject Classification (2010) :
The first important result obtained on fixed points for contractive-type mappings wasthe well-known Banach contraction theorem, published for the first time in 1922 ([2]).In the general setting of complete metric spaces, this theorem runs as follows.
Theorem 1.1.
Let ( X; d ) be a complete metric space, (cid:12) (0 ; and let T : X ! X be a mapping such that for each x; y X;d ( T x; T y ) (cid:20) (cid:12)d ( x; y ) : Then T has a unique fixed point a X such that for each x X , lim n !1 T n x = a . In order to generalize this theorem, several authors have introduced various typesof contraction inequalities. In 2002 Branciari proved the following result (see [4]).
Theorem 1.2.
Let ( X; d ) be a complete metric space, (cid:12) (0 ; and T : X (cid:0)! X amapping such that for each x; y X , ∫ d ( T x;T y ) f ( t ) dt (cid:20) (cid:12) ∫ d ( x;y ) f ( t ) dt; where f : [0 ; ) ! (0 ; ) is a Lebesgue integrable mapping which is summable (i.e.,with finite integral on each compact subset of [0 ; ) ) and for each " > , ∫ " f ( t ) dt > : Then T has a unique fixed point a X such that for each x X , lim n !1 T n x = a . Rhoades [13] and Djoudi et al. [6] extended the result of Branciari and proved thefollowing fixed point theorems. 1 heorem 1.3. [13] Let ( X; d ) be a complete metric space, k [0 ; , T : X ! X amapping satisfying for each x; y X; ∫ d ( T x;T y )0 φ ( t ) dt (cid:20) k ∫ M ( x;y )0 φ ( t ) dt where M ( x; y ) = max f d ( x; y ) ; d ( x; T x ) ; d ( y; T y ) ;
12 [ d ( x; T y ) + d ( y; T x )] g and φ : R + ! R + be as in theorem 1.2. Then T has a unique fixed point x X . Theorem 1.4. [6] Let ( X; d ) be a complete metric space and T : X ! X a mappingsatisfying for each x; y X; ∫ d ( T x;T y )0 φ ( t ) dt (cid:20) h ( ∫ M ( x;y )0 φ ( t ) dt ) where M ( x; y ) = max f d ( x; y ) ; d ( x; T x ) ; d ( y; T y ) ; d ( x; T y ) ; d ( y; T x ) g ;h : R + ! R + is subadditive, nondecreasing and continuous from the right such that h ( t ) < t , for all t > and φ : R + ! R + be as in theorem 1.2. Then T has a uniquefixed point x X . In 1984, M.S. Khan, M. Swalech and S. Sessa [9] expanded the research of themetric fixed point theory to the category (cid:9) by introducing a new function which theycalled an altering distance function. For : R + ! R + we say that (cid:9) if1. ( t ) = 0 if and only if t = 0 ,2. is monotonically non-decreasing,3. is continuous.The following lemma shows that contractive conditions of integral type can be inter-preted as contractive conditions involving an altering distance. Lemma 1.5.
Let φ : R + ! R + be as in Theorem 1.2. Define ( t ) = ∫ t φ ( (cid:28) ) d(cid:28) , for t R + . Then is an altering distance. Khan et al. using this altering distance to extend the Banach Contraction Principleas follows:
Theorem 1.6. [9] Let ( X; d ) be a complete metric space, (cid:12) (0 ; and T : X (cid:0)! X a mapping such that for each x; y X , [ d ( T x; T y )] (cid:20) (cid:12) [ d ( x; y )] where (cid:9) . Then T has a unique fixed point a X such that for each x X , lim n !1 T n x = a .
2t is easy to see that if ( t ) = t , we obtain the Banach Contraction Principle andby lemma 1.5, we obtain theorem 1.2. Dutta et al. [8], Dori [7], Choudhury et al. [5]and Morals et al. [11] extended the results of Khan and proved the following fixedpoint theorems. Theorem 1.7. [8] Let ( X; d ) be a complete metric space and let T : X ! X be amapping satisfying ( d ( T x; T y )) (cid:20) ( d ( x; y )) (cid:0) h ( d ( x; y )) for each x; y X; where ; h : R + ! R + are continuous and non-decreasing functionsuch that ( t ) = h ( t ) = 0 if and only if t = 0 . Then T has a unique fixed point x X . Theorem 1.8. [7] Let ( X; d ) be a complete metric space and let T : X ! X be amapping satisfying ( d ( T x; T y )) (cid:20) ( M ( x; y )) (cid:0) h ( M ( x; y ))) for each x; y X; where M ( x; y ) = max f d ( x; y ) ; d ( x; T x ) ; d ( y; T y ) ;
12 [ d ( x; T y ) + d ( y; T x )] g ; (cid:9) and h : R + ! R + is a lower semi-continuous function such that h ( t ) = 0 ifand only if t = 0 . Then T has a unique fixed point x X . Theorem 1.9. [5] Let ( X; d ) be a complete metric space and let T : X ! X be amapping satisfying ( d ( T x; T y )) (cid:20) ( M ( x; y )) (cid:0) h (max f d ( x; y ) ; d ( y; T y ) g ) for each x; y X; where M ( x; y ) = max f d ( x; y ) ; d ( x; T x ) ; d ( y; T y ) ;
12 [ d ( x; T y ) + d ( y; T x )] g ; (cid:9) and h : R + ! R + is a continuous function such that h ( t ) = 0 if and only if t = 0 . Then T has a unique fixed point x X . Theorem 1.10. [11] Let ( X; d ) be a complete metric space and T : X (cid:0)! X amapping which satisfies the following condition: [ d ( T x; T y )] (cid:20) a [ d ( x; y )] + b [ m ( x; y )] for all x; y X , a > , b > , a + b < where m ( x; y ) = d ( y; T y ) 1 + d ( x; T x )1 + d ( x; y ) for all x; y X . where (cid:9) . Then T has a unique fixed point a X such that foreach x X , lim n !1 T n x = a .
3n the other hand, in 2008, Suzuki introduced a new method in [14] and then hismethod was extended by some authors (see for example [1], [10], [12]). The aim ofthis paper is to provide a new and more general condition for T which guarantees theexistence of its fixed point. Our results generalize several old and new results in theliterature. In this way, consider (cid:8) the set of all control function ϕ : [0 ; ) k (cid:0)! [0 ; ) satisfying(i) ϕ (0 ; ; :::;
0) = 0 ,(ii) lim n !1 ϕ ( t n ; t n ; :::; t kn ) (cid:20) ϕ ( t ; t ; :::; t k ) whenever ( t n ; t n ; :::; t kn ) ! ( t ; t ; :::; t k ) ,and R the set of all continuous function g : [0 ; ) (cid:0)! [0 ; ) satisfying the followingconditions:(i) g (1 ; ; ; ; ; g (1 ; ; ; ; (0 ; ; (ii) g is subhomogeneous, i.e. g ( (cid:11)x ; (cid:11)x ; (cid:11)x ; (cid:11)x ; (cid:11)x ) (cid:20) (cid:11)g ( x ; x ; x ; x ; x ) for all (cid:11) (cid:21) : (iii) if x i ; y i [0 ; ) ; x i (cid:20) y i for i = 1 ; :::; we have g ( x ; x ; x ; x ; x ) (cid:20) g ( y ; y ; y ; y ; x ) Example 1.1.
Define g ( x ; x ; x ; x ; x ) = max f x i g i =1 . It is obvious that g R . Example 1.2.
Define g ( x ; x ; x ; x ; x ) = max f x ; x ; x ; x + x g . It is obviousthat g R . Proposition 1.11. If g R and u; v [0 ; ) are such that u < max f g ( v; v; u; v; u ) ; g ( v; u; v; v + u; g ; then u < v: Proof.
Without loss of generality, we can suppose u < g ( v; u; v; v + u; . If v (cid:20) u ,then u < g ( v; u; v; v + u; (cid:20) g ( u; u; u; u; (cid:20) ug (1 ; ; ; ; (cid:20) u which is a contradiction. Thus u < v . Lemma 1.12.
Let (cid:9) and ϕ (cid:8) such that for every t i R + , ϕ ( t ; t ; ::; t k ) < ( max i =1 ;:::;k t i ) : If for t; s i R + we have ( t ) (cid:20) ϕ ( s ; s ; :::; s k ) ; then t < max i =1 ;:::;k s i : Proof.
Let S = max i =1 ;:::;k s i . Suppose that t (cid:21) S . Then ( t ) (cid:21) ( S ) > ϕ ( s ; s ; :::; s k ) ; which is a contradiction. 4 emma 1.13. Suppose that f s n g be a sequence of non-negative real numbers suchthat s n +1 (cid:20) s n . Then s n is convergent. Lemma 1.14. [2] Let ( X; d ) be a metric space and f x n g be a sequence in X suchthat lim n !1 d ( x n ; x n +1 ) = 0 : If f x n g is not a Cauchy sequence in X , then there exist an " > and sequences ofpositive integers m k and n k with m k > n k > k such that d ( x m k ; x n k ) (cid:21) " ; d ( x m k (cid:0) ; x n k ) < " and(i) lim k !1 d ( x m k (cid:0) ; x n k +1 ) = " ; (ii) lim k !1 d ( x m k ; x n k ) = " ; (iii) lim k !1 d ( x m k (cid:0) ; x n k ) = " : The following theorem is the main result of this paper.
Theorem 2.1.
Let ( X; d ) be a complete metric space, T : X (cid:0)! X a mapping, (cid:11) (0 ; ] , (cid:9) and ϕ (cid:8) such that for every t i R + with ( t ; t ; :::; t k ) ̸ = (0 ; ; :::; , ϕ ( t ; t ; ::; t k ) < ( max i =1 ;:::;k t i ) : Suppose that f g i g ki =1 be a sequence in R and (cid:11)d ( x; T x ) (cid:20) d ( x; y ) implies [ d ( T x; T y )] (cid:20) ϕ ( g ( M xy ) ; g ( M xy ) ; :::; g k ( M xy )) for all x; y X , where M xy = ( d ( x; y ) ; d ( y; T y ) ; d ( x; T x ) ; d ( x; T y ) ; d ( y; T x )) for all x; y X . Then T has a unique fixed point in X .Proof. Fix arbitrary x X and let x = T x . We have (cid:11)d ( x ; T x ) < d ( x ; x ) .Hence, [ d ( T x ; T x )] (cid:20) ϕ ( g ( M x x ) ; g ( M x x ) ; :::; g k ( M x x )) : Then by lemma 1.12 we have d ( x ; T x ) < max i =1 ;:::;k g i ( M x x )= max i =1 ;:::;k g i ( d ( x ; x ) ; d ( x ; T x ) ; d ( x ; T x ) ; d ( x ; T x ) ; d ( x ; T x )) max i =1 ;:::;k g i ( d ( x ; x ) ; d ( x ; T x ) ; d ( x ; x ) ; d ( x ; T x ) ; (cid:20) max i =1 ;:::;k g i ( d ( x ; x ) ; d ( x ; T x ) ; d ( x ; x ) ; d ( x ; x ) + d ( x ; T x ) ; : By proposition 1.11, we obtain d ( x ; T x ) < d ( x ; x ) . Now let x = T x . Since (cid:11)d ( x ; T x ) < d ( x ; x ) , by using the assumption we have [ d ( T x ; T x )] (cid:20) ϕ ( g ( M x x ) ; g ( M x x ) ; :::; g k ( M x x )) : Then by lemma 1.12 we have d ( x ; T x ) < max i =1 ;:::;k g i ( M x x )= max i =1 ;:::;k g i ( d ( x ; x ) ; d ( x ; T x ) ; d ( x ; T x ) ; d ( x ; T x ) ; d ( x ; T x ))= max i =1 ;:::;k g i ( d ( x ; x ) ; d ( x ; T x ) ; d ( x ; x ) ; d ( x ; T x ) ; (cid:20) max i =1 ;:::;k g i ( d ( x ; x ) ; d ( x ; T x ) ; d ( x ; x ) ; d ( x ; x ) + d ( x ; T x ) ; : By proposition 1.11, we obtain d ( x ; T x ) < d ( x ; x ) . Now by continuing this pro-cess, we obtain a sequence f x n g n (cid:21) in X such that x n +1 = T x n and d ( x n ; x n +1 )
6e claim that for any y X , one of the flowing relations is held: (cid:11)d ( x n ; T x n ) (cid:20) d ( x n ; y ) or (cid:11)d ( x n +1 ; T x n +1 ) (cid:20) d ( x n +1 ; y ) : (1)Otherwise, if (cid:11)d ( x n ; T x n ) > d ( x n ; y ) and (cid:11)d ( x n +1 ; T x n +1 ) > d ( x n +1 ; y ) , we have d ( x n ; x n +1 ) (cid:20) d ( x n ; y ) + d ( x n +1 ; y ) < (cid:11)d ( x n ; T x n ) + (cid:11)d ( x n +1 ; T x n +1 )= (cid:11)d ( x n ; x n +1 ) + (cid:11)d ( x n +1 ; x n +2 ) (cid:20) (cid:11)d ( x n ; x n +1 ) (cid:20) d ( x n ; x n +1 ) which is a contradiction. Now by using the assumption and relation 1, for each n (cid:21) one of the following cases holds:(i) There exists an infinite subset I (cid:26) N such that [ d ( x m ( k )+1 ; x n ( k )+1 )] (cid:20) ϕ ( g ( M x m ( k ) x n ( k ) ) ; g ( M x m ( k ) x n ( k ) ) ; :::; g k ( M x m ( k ) x n ( k ) )) : (ii)There exists an infinite subset J (cid:26) N such that [ d ( x m ( k )+2 ; x n ( k )+1 )] (cid:20) ϕ ( g ( M x m ( k )+1 x n ( k ) ) ; g ( M x m ( k )+1 x n ( k ) ) ; :::; g k ( M x m ( k )+1 x n ( k ) )) : Since M x m ( k ) x n ( k ) = ( d ( x m ( k ) ; x n ( k ) ) ; d ( x n ( k ) ; T x n ( k ) ) ; d ( x m ( k ) ; T x m ( k ) ) ; d ( x m ( k ) ; T x n ( k ) ) ; d ( x n ( k ) ; T x m ( k ) ))= ( d ( x m ( k ) ; x n ( k ) ) ; d ( x n ( k ) ; x n ( k )+1 ) ; d ( x m ( k ) ; x m ( k )+1 ) ; d ( x m ( k ) ; x n ( k )+1 ) ; d ( x n ( k ) ; x m ( k )+1 )) (cid:20) ( d ( x m ( k ) ; x n ( k ) ) ; d ( x n ( k ) ; x n ( k )+1 ) ; d ( x m ( k ) ; x m ( k )+1 ) ;d ( x m ( k ) ; x n ( k ) ) + d ( x n ( k ) ; x n ( k )+1 ) ; d ( x n ( k ) ; x m ( k ) ) + d ( x m ( k ) ; x m ( k )+1 )) ; we have lim n !1 M x m ( k ) x n ( k ) = ( " ; ; ; A; B ) where A (cid:20) " and B (cid:20) " . Then incase (i), we obtain ( " ) (cid:20) ϕ ( g ( " ; ; ; A; B ) ; g ( " ; ; ; A; B ) ; :::; g k ( " ; ; ; A; B )) and then by lemma 1.12 we have " < max i =1 ;:::;k g i ( " ; ; ; A; B ) (cid:20) max i =1 ;:::;k g i ( " ; ; ; " ; " ) (cid:20) " ; which is a contradiction.In case (ii), similar to cas(i), we obtain " < " ; which is a contradiction. This proves our claim that f x n g n (cid:21) is a Cauchy sequencein ( X; d ) . Let lim n !1 x n = x . By relation 1, for each n (cid:21) and y X , eithera) [ d ( T x n ; T y )] (cid:20) ϕ ( g ( M x n x ) ; g ( M x n x ) ; :::; g k ( M x n x )) [ d ( T x n +1 ; T y )] (cid:20) ϕ ( g ( M x n +1 x ) ; g ( M x n +1 x ) ; :::; g k ( M x n +1 x )) In case (a), by using of lemma 1.12 we obtain d ( x; T x ) (cid:20) d ( x; T x n ) + d ( T x n ; T x ) < d ( x; T x n ) + max i =1 ;:::;k g i ( M x n x )= d ( x; T x n ) + max i =1 ;:::;k g i ( d ( x n ; x ) ; d ( x n ; T x n ) ; d ( x; T x ) ; d ( x; T x n ) ; d ( x n ; T x )) : Hence d ( x; T x ) (cid:20) max i =1 ;:::;k g i (0 ; ; d ( x; T x ) ; ; d ( x; T x )) : Now by using Proposition 1.11, we have d ( x; T x ) = 0 and so x = T x .In case (b), by using lemma 1.12, we obtain d ( x; T x ) (cid:20) d ( x; T x n +1 ) + d ( T x n +1 ; T x ) < d ( x; T x n ) + max i =1 ;:::;k g i ( M x n +1 x ) (cid:20) d ( x; T x n +1 )+ max i =1 ;:::;k g i ( d ( x n +1 ; x ) ; d ( x n +1 ; T x n +1 ) ; d ( x; T x ) ; d ( x; T x n +1 ) ; d ( x n +1 ; T x )) : Hence d ( x; T x ) (cid:20) g (0 ; ; d ( x; T x ) ; ; d ( x; T x )) ; and then by using Proposition 1.11, we have d ( x; T x ) = 0 . So x = T x . We claim thatthis fixed point is unique. Suppose that there are two distinct points a; b X suchthat T a = a and T b = b . Since d ( a; b ) > (cid:11)d ( a; T a ) , we have the contradiction < [ d ( a; b )] = [ d ( T a; T b )] (cid:20) ϕ ( g ( M ab ) ; g ( M ab ) ; :::; g k ( M ab )) : Now by lemma 1.12, we obtain d ( a; b ) < max i =1 ;:::;k g i ( d ( a; b ) ; d ( a; T a ) ; d ( b; T b ) ; d ( a; T b ) ; d ( b; T a ))= max i =1 ;:::;k g i ( d ( a; b ) ; ; ; d ( a; b ) ; d ( b; a )) (cid:20) d ( a; b ) : So d ( a; b ) = 0 . Corollary 2.2.
Let ( X; d ) be a complete metric space and T : X ! X be a mappingsatisfying ( d ( T x; T y )) (cid:20) h ( ( M ( x; y ))) for each x; y X; where M ( x; y ) = max f d ( x; y ) ; d ( x; T x ) ; d ( y; T y ) ; d ( x; T y ) ; d ( y; T x ) g ; (cid:9) and h : R + ! R + is a continuous function such that h ( t ) < t for all t > .Then T has a unique fixed point x X . roof. Let g ( t ; t ; t ; t ; t ) = max f t ; t ; t ; t ; t g and define ϕ by ϕ ( t ) = h ( ( t )) .It is easy to see that ϕ (cid:8) and for every t > , ϕ ( t ) < ( t ) : Now by using Theorem2.1, T has a fixed point. Remark 2.1.
By lemma 1.5, we see that theorems 1.2, 1.3 and 1.4 are special casesof theorem 2.1.
Remark 2.2.
Theorem 1.7 is a special case of theorem 2.1.Proof.
Let g = g ( t ; t ; t ; t ; t ) = t and define ϕ by ϕ ( t ; t ) = ϕ ( t ) (cid:0) h ( t ) . Nowby using Theorem 2.1, T has a fixed point. Remark 2.3.
Theorem 1.8 is a special case of theorem 2.1.Proof.
Let g ( t ; t ; t ; t ; t ) = max f t ; t ; t ; ( t + t ) g and define ϕ ( t ) = ( t ) (cid:0) h ( t ) .Now by using Theorem 2.1, T has a fixed point. Remark 2.4.
Theorem 1.9 is a special case of theorem 2.1.Proof.
Let g ( t ; t ; t ; t ; t ) = max f t ; t ; t ; ( t + t ) g , g ( t ; t ; t ; t ; t ) = max f t ; t g and define ϕ ( t ; t ) = ( t ) (cid:0) h ( t ) . Now by using theorem 2.1, T has a fixed point. Remark 2.5.
Let g ( t ; t ; t ; t ; t ) = t , g ( t ; t ; t ; t ; t ) = t t t and define ϕ ( t ; t ) = a ( t ) + b ( t ) . Then we obtain theorem 1.10 of theorem 2.1. References [1] S. M. A. Aleomraninejad, Sh. Rezapour, N. Shahzad, On fixedpoint generalizations of Suzuki‘s method, Applied Mathematics Letters,24(2011),1037-1040.[2] G. U. R. Babu, P.P. Sailaja, A fixed point theorem of generalized weaklycontractive maps in orbitally complete metric space, Thai Journal of Math.9 1 (2011) 110.[3] S. Banach, Sur les operations dans les ensembles abstraits et leur applica-tion aux equations integrales, Fund. Math. 3 (1922), 133181 (French).[4] A. Branciari, A fixed point theorem for mappings satisfying a general con-tractive condition of integral type, Hindawi Publishing Corpration, IJMMS29.9 (2002) 531-536.[5] B. S. Choudhury, P. Konar, B. E. Rhoades, N. Metiya, Fixed point the-orems for generalized weakly contractive mappings, Nonlinear Anal. 74(2011) 2116-2126. doi:10.1016/j.na.2010.11.017[6] A. Djoudi, F. Merghadi, Common fixed point theorems for maps undercontractive condition of integral type, J. Math. Anal. Appl. 341(2008) 953-960.
7] D. Dori, Common fixed point for generalized ( , ϕ )-weak contractions.Appl Math Lett. 22 (2009) 18961900. doi:10.1016/j.aml.2009.08.001[8] P. N. Dutta, B. S. Choudhury, A generalisation of contraction principle inmetric spaces, Fixed Point Theory Appl. (2008).[9] MS. Khan, M. Swaleh, S. Sessa, Fixed point theorems by alteringdistances between the points, Bull Austral Math Soc. 30 (1984) 19.doi:10.1017/S0004972700001659[10] M. Kikkawa, T. Suzuki, Three fixed point theorems for generalized con-tractions with constants in complete metric spaces, Nonlinear Analysis. 69(2008) 2942-2949. doi:10.1016/j.na.2007.08.064[11] J. R. Morals, E. M Rojas, Altering distance function and fixed point theo-rem through rational expression, Math F.A. 25 ( 2012).[12] G. Mot, A. Petrusel, Fixed point theory for a new type of contrac-tive multivalued operators, Nonlinear Analysis. 70 (2009) 33713377.doi:10.1016/j.na.2008.05.005[13] BE. Rhoades, Two fixed-point theorems for mappings satisfying a generalcontractive condition of integral type, Int. J. Math. Math. Sci. 63 (2003)4007-4013. doi:10.1155/S0161171203208024.[14] T. Suzuki, A new type of fixed point theorem in metric spaces, NonlinearAnalysis. 71 (2009) 5313-5317. doi:10.1016/j.na.2009.04.017)-weak contractions.Appl Math Lett. 22 (2009) 18961900. doi:10.1016/j.aml.2009.08.001[8] P. N. Dutta, B. S. Choudhury, A generalisation of contraction principle inmetric spaces, Fixed Point Theory Appl. (2008).[9] MS. Khan, M. Swaleh, S. Sessa, Fixed point theorems by alteringdistances between the points, Bull Austral Math Soc. 30 (1984) 19.doi:10.1017/S0004972700001659[10] M. Kikkawa, T. Suzuki, Three fixed point theorems for generalized con-tractions with constants in complete metric spaces, Nonlinear Analysis. 69(2008) 2942-2949. doi:10.1016/j.na.2007.08.064[11] J. R. Morals, E. M Rojas, Altering distance function and fixed point theo-rem through rational expression, Math F.A. 25 ( 2012).[12] G. Mot, A. Petrusel, Fixed point theory for a new type of contrac-tive multivalued operators, Nonlinear Analysis. 70 (2009) 33713377.doi:10.1016/j.na.2008.05.005[13] BE. Rhoades, Two fixed-point theorems for mappings satisfying a generalcontractive condition of integral type, Int. J. Math. Math. Sci. 63 (2003)4007-4013. doi:10.1155/S0161171203208024.[14] T. Suzuki, A new type of fixed point theorem in metric spaces, NonlinearAnalysis. 71 (2009) 5313-5317. doi:10.1016/j.na.2009.04.017