Some Fixed Point Results for (α, β)-(ψ, φ)-contractive Mappings
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1 Filomat 28:3 214), DOI 12298/FIL143635A Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Some Fixed Point Results for α, β)-ψ, φ)-contractive Mappings Sattar Alizadeh a, Fridoun Moradlou b, Peyman Salimi c a Department of Mathematics, Marand Branch, Islamic Azad University, Marand, Iran b Department of Mathematics, Sahand University of Technology, Tabriz, Iran c Young Researchers and Elite Club, Rasht Branch, Islamic Azad University, Rasht, Iran Abstract In this paper, we introduce the notions of cyclic α, β)-admissible mappings, α, β)-ψ, φ)- contractive and weak α β ψ rational contraction mappings via cyclic α, β)-admissible mappings We prove some new fixed point results for such mappings in the setting of complete metric spaces The obtained results generalize, unify and modify some recent theorems in the literature Some examples and an application to integral equations are given here to illustrate the usability of the obtained results 1 Introduction In the fixed point theory of continuous mappings, a well-known theorem of Banach [3] states that if X, d) is a complete metric space and if f is a self-mapping on X which satisfies the inequality d f x, f y) kdx, y) for some k [, 1) and all x, y X, then f has a unique fixed point z and the sequence of successive approximations { f n x} converges to z for all x X On the other hand, the condition d f x, f y) < dx, y) does not insure that f has a fixed point In the last decades, the Banach s theorem [3] has been extensively studied and generalized on many settings, see for example [1] and [4]-[35] In 28, Dutta and Choudhury [12] proved the following theorem: Theorem 11 Let X, d) be a complete metric space and f : X X be a mapping such that ψd f x, f y)) ψdx, y)) φdx, y)), for all x, y X, where ψ, φ : [, + ) [, + ) are continuous, non-decreasing and ψt) = φt) = if and only if t = Then, f has a unique fixed point x X Notice that above Theorem remains true if the hypothesis on φ is replaced by φ is lower semi-continuous and φt) = if and only if t = see eg [1, 11]) Recently, Samet et al [31] introduced the concept of α-ψ-contractive type mappings and established various fixed point theorems for such mappings in complete metric spaces Very recently, Salimi et al [3] 21 Mathematics Subject Classification Primary 47H1; Secondary 54H25 Keywords α, β)-ψ, φ)-contractive mapping, fixed point, cyclic contractive mapping, cyclic α, β)-admissible mapping, integral equations Received: 1 April 213; Accepted: 27 January 214 Communicated by Vladimir Rakocević Corresponding author: S Alizadeh addresses: salizadeh@marandiauacir, s_alizadeh_s@yahoocom Sattar Alizadeh ), moradlou@sutacir,fridounmoradlou@gmailcom Fridoun Moradlou), salimipeyman@gmailcom Peyman Salimi)
2 Sattar Alizadeh et al / Filomat 28:3 214), modified the concept of α-ψ-contractive type mappings and established some new fixed point theorems for certain contractive conditions Also see [13, 17, 22] and references therein) Motivated by [3], we introduce the notions of α, β)-ψ, φ)-contractive and weak α β ψ- rational contraction mappings via cyclic α, β)-admissible mappings and establish some results of fixed point for this class of mappings in the setting of metric spaces The obtained results generalize, unify and modify some recent theorems in the literature Some examples and an application to integral equations are given here to illustrate the usability of the obtained results 2 Main Results We denote by Ψ the set of continuous and increasing functions ψ : [, + ) [, + ) and denote by Φ the set of lower semicontinuous functions φ : [, + ) [, + ) such that φt) = iff t = Let X be a nonempty set and let f : X X be an arbitrary mapping We say that x X is a fixed point for f, if x = f x We denote by Fix f ) the set of all fixed points of f Definition 21 Let f : X α, β)-admissible mapping if X be a mapping and α, β : X R + be two functions We say that f is a cyclic i) αx) 1 for some x X implies β f x) 1, ii) βx) 1 for some x X implies α f x) 1 Example 22 Let f : R R be defined by f x = x + x 3 ) Suppose that α, β : R R + are given by αx) = e x for all x R and βy) = e y for all y R Then f is a cyclic α, β)-admissible mapping Indeed, if αx) = e x 1, then x which implies f x Therefore, β f x) = e f x 1 Also, if βy) = e y 1, then y which implies f y So, α f y) = e f y 1 Definition 23 Let X, d) be a metric space and f : X X be a cyclic α, β)-admissible mapping We say that f is a α, β)-ψ, φ)-contractive mapping if αx)βy) 1 = ψd f x, f y)) ψdx, y)) φdx, y)), 1) for x, y X, where ψ Ψ and φ Φ Now we are ready to prove our first theorem Theorem 24 Let X, d) be a complete metric space and f : X X be a α, β)-ψ, φ)-admissible mapping Suppose that the following conditions hold: a) there exists x X such that αx ) 1 and βx ) 1; b) f is continuous, or c) if {x n } is a sequence in X such that x n x and βx n ) 1 for all n, then βx) 1; then f has a fixed point Moreover, if αx) 1 and βy) 1 for all x, y Fix f ), then f has a unique fixed point Proof Define a sequence {x n } by x n = f n x = f x n 1 for all n N Since f is a cyclic α, β)-admissible mapping and αx ) 1 then βx 1 ) = β f x ) 1 which implies αx 2 ) = α f x 1 ) 1 By continuing this process, we get αx 2n ) 1 and βx 2n 1 ) 1 for all n N Again, since f is a cyclic α, β)-admissible mapping and βx ) 1, by the similar method, we have βx 2n ) 1 and αx 2n 1 ) 1 for all n N That is, αx n ) 1 and βx n ) 1 for all n N {} Equivalently, αx n 1 )βx n ) 1 for all n N Therefore by 1), we have ψdx n, x n+1 )) ψdx n 1, x n )) φdx n 1, x n )) ψdx n 1, x n )), 2)
3 and since ψ is increasing, we get dx n, x n+1 ) dx n 1, x n ), Sattar Alizadeh et al / Filomat 28:3 214), for all n N So, {d n := dx n, x n+1 )} is a non-increasing sequence of positive real numbers Then there exist r such that lim n d n = r We shall show that r = By taking the limsup on both sides of 2), we have ψr) ψr) φr) Hence φr) = That is r = Then lim dx n, x n+1 ) = n Now, we want to show that {x n } is a Cauchy sequence Suppose, to the contrary, that {x n } is not a Cauchy sequence Then there are ε > and sequences {mk)} and {nk)} such that for all positive integers k, nk) > mk) > k, dx nk), x mk) ) ε and dx nk), x mk) 1 ) < ε Now for all k N, we have ε dx nk), x mk) ) dx nk), x mk) 1 ) + dx mk) 1, x mk) ) < ε + dx mk) 1, x mk) ) Taking the limit as k + in the above inequality and using 3), we get 3) lim dx nk), x mk) ) = ε k + 4) Since, dx nk), x mk) ) dx mk), x mk)+1 ) + dx mk)+1, x nk)+1 ) + dx nk)+1, x nk) ), and dx nk)+1, x mk)+1 ) dx mk), x mk)+1 ) + dx mk), x nk) ) + dx nk)+1, x nk) ), then by taking the limit as k + in above inequalities and using 3) and 4), we deduce that lim dx nk)+1, x mk)+1 ) = ε k + 5) Now, by 1), we get ψdx nk)+1, x mk)+1 )) ψdx nk), x mk) )) φdx nk), x mk) )), 6) since, αx nk) )βx mk) ) 1 for all k N By taking the limsup on both sides of 6) and applying 4) and 5), we obtain ψε) ψε) φε) That is, ε = which is a contradiction Hence {x n } is a Cauchy sequence Since X, d) is a complete metric space, then there is z X such that x n z as n First, we assume that f is continuous Hence, we deduce f z = lim n f x n = lim n x n+1 = z So z is a fixed point of f Now, assume that c) is held, That is, αx n )βz) 1 From 1) we have
4 Sattar Alizadeh et al / Filomat 28:3 214), ψdx n+1, f z)) ψdx n, z)) φdx n, z)) 7) By taking the limsup on both sides of 7), we get ψdz, f z)) = Then dz, f z) = ie, z = f z To prove the uniqueness of fixed point, suppose that z and z are two fixed points of f Since αz)βz ) 1, it follows from 1) that ψdz, z )) ψdz, z )) φdz, z )) So φdz, z )) = and hence dz, z ) = ie, z = z Example 25 Let X = R endowed with the usual metric dx, y) = x y for all x, y X and f : X X be defined by 1 x, if x [ 1, 1] 2 f x =, 2x, if x R\[ 1, 1] and α, β : X [, + ) be given by 1, if x [ 1, ] αx) =, otherwise Also define ψ, φ : [, ) [, ) by 1, if x [, 1] & βx) =, otherwise ψt) = t and φt) = 1 4 t Now, we prove that f is a cyclic α, β)-admissible mapping and the hypotheses a) and c) of Theorem 24 are satisfied by f and hence f has a fixed point Moreover, the result of Dutta and Choudhury [12] can not be applied to f Let αx) 1 for some x X Then x [ 1, ] and so f x [, 1] Therefore, β f x) 1 Similarly, if βx) 1 then α f x) 1 Then f is a cyclic α, β)-admissible mapping Now, if {x n } is a sequence in X such that βx n ) 1 and x n x as n Therefore, x n [, 1] Hence x [, 1], ie, βx) 1 Let αx)βy) 1 Then x [ 1, ] and y [, 1] and so we have ψd f x, f y)) = f x f y = 1 2 x y 3 x y 4 = x y 1 x y = ψdx, y)) φdx, y)) 4 So the hypotheses of Theorem 24 hold and therefore, f has a fixed point But, if x = 2 and y = 3 then ψd f 2, f 3)) = 2 > 3 = ψd2, 3)) φd2, 3)) 4 That is, Theorem 11 can not be applied to f Corollary 26 Let X, d) be a complete metric space and f : X X be a cyclic α, β)-admissible mapping such that αx)βy)ψd f x, f y)) ψdx, y)) φdx, y)), 8) for all x, y X where ψ Ψ and φ Φ Suppose that the following assertions hold:
5 a) there exists x X such that αx ) 1 and βx ) 1; b) f is continuous, or Sattar Alizadeh et al / Filomat 28:3 214), c) if {x n } is a sequence in X such that x n x and βx n ) 1 for all n, then βx) 1; then f has a fixed point Moreover, if αx) 1 and βy) 1 for all x, y Fix f ), then f has a unique fixed point Proof Let αx)βy) 1 for x, y X Then by 8), we get ψd f x, f y)) ψdx, y)) φdx, y)) This implies that the inequality 1) holds Therefore, the proof follows from Theorem 24 Corollary 27 Let X, d) be a complete metric space and f : X X be a cyclic α, β)-admissible mapping such that )ψd f x, f y)) αx)βy) ψdx, y)) φdx, y)), for all x, y X where ψ Ψ and φ Φ Suppose that the following assertions hold: a) there exists x X such that αx ) 1 and βx ) 1; b) f is continuous, or c) if {x n } is a sequence in X such that x n x and βx n ) 1 for all n, then βx) 1; then f has a fixed point Moreover, if αx) 1 and βy) 1 for all x, y Fix f ), then f has a unique fixed point Example 28 Let X = R endowed with the usual metric dx, y) = x y for all x, y X and f : X X be defined by 1 6 x + x3 ), if x [ 1, 1] f x =, ln x, if x R\[ 1, 1] and α, β : X [, + ) be given by 1, if x [ 1, ] αx) =, otherwise Also, define ψ, φ : [, ) [, ) by ψt) = t and φt) = 1 2 t & βx) = 1, if x [, 1], otherwise Now, we prove that the hypotheses a) and c) of Corollary 27 are satisfied by f and hence f has a fixed point Proceeding as in the Example 25, we deduce that f is a cyclic α, β)-admissible mapping and that the hypotheses a) and c) of Corollary 27 hold Now, for all x [ 1, ] and all y [, 1], we get ) ψd f x, f y)) f x f y αx)βy) + 1 = 2 = x y 1+x2 +xy+y x y 1+x2 +y x y = 2 x y 1 2 x y = 2 ψdx,y)) φdx,y))
6 Otherwise, αx)βy) = and so we have αx)βy) + 1) ψd f x, f y)) = 1 2 ψdx,y)) φdx,y)) Therefore, Corollary 27 implies that f has a fixed point Sattar Alizadeh et al / Filomat 28:3 214), Corollary 29 Let X, d) be a complete metric space and f : X X be a cyclic α, β)-admissible mapping Assume that there exists l > 1 such that )αx)βy) ψd f x, f y)) + l ψdx, y)) φdx, y)) + l, for all x, y X where ψ Ψ and φ Φ Suppose that the following assertions hold: a) there exists x X such that αx ) 1 and βx ) 1; b) f is continuous, or c) if {x n } is a sequence in X such that x n x and βx n ) 1 for all n, then βx) 1; then f has a fixed point Moreover, if αx) 1 and βy) 1 for all x, y Fix f ), then f has a unique fixed point Example 21 Let X = [, + ) endowed with the usual metric dx, y) = x y for all x, y X and f : X X be defined by 1 8 x + x2 ), if x [, 1] f x =, 3x, if x 1, + ) and α, β : X [, + ) be given by 1, if x [, 1] αx) = βx) =, otherwise Also, define ψ, φ : [, ) [, ) by ψt) = 1 2 t and φt) = 1 8 t Now, we prove that the Corollary 29 can be applied to f At first, proceeding as in the proof of Example 25, we deduce that f is a cyclic α, β)-admissible mapping and that the conditions a) and c) of Corollary 29 hold Now, suppose that l > 1 Let x, y [, 1] Then, )αx)βy) ψd f x, f y)) + l = 1 2 f x f y + l = 1 x y 1 + x + y + l 16 3 x y + l 8 = 1 2 x y 1 8 x y + l = ψdx, y)) φdx, y)) + l
7 Otherwise, αx)βy) = and hence we get Sattar Alizadeh et al / Filomat 28:3 214), )αx)βy) ψd f x, f y)) + l = 1 ψdx, y)) φdx, y)) + l So, it follows from Corollary 29 that f has a fixed point Definition 211 Let X, d) be a metric space and f : X X be a cyclic α, β)-admissible mapping We say that f is a weak α β ψ rational contraction if αx)βy) 1 for some x, y X implies d f x, f y) Mx, y) ψ Mx, y) ), where ψ Ψ and { Mx, y) = max dx, y), [1 + dx, Tx)]dy, Ty) } dx, y) + 1 Theorem 212 Let X, d) be a complete metric space and f : X X be a weak α β ψ- rational contraction Assume that the following assertions hold: a) there exists x X such that αx ) 1 and βx ) 1; b) f is continuous, or c) if {x n } is a sequence in X such that x n x and βx n ) 1 for all n, then βx) 1; then f has a fixed point Moreover, if αx) 1 and βy) 1 for all x, y Fix f ), then f has a unique fixed point Proof Similar to the proof of Theorem 24, we define a sequence {x n } by x n = f n x and see that, αx n 1 )βx n ) 1 for all n N Also we assume that x n 1 x n for all n N Since f is a weak α β ψ rational contraction, we obtain dx n, x n+1 ) Mx n 1, x n ) ψ Mx n 1, x n ) ), 9) where Mx n 1, x n ) = { max dx n 1, x n ), [dx } n 1, x n ) + 1]dx n, x n+1 ) dx n 1, x n ) + 1 = max{dx n 1, x n ), dx n, x n+1 )} Now, suppose that there exists n N such that dx n, x n +1) > dx n 1, x n ) Therefore Mx n 1, x n, x n, x n +1) = dx n, x n +1) and so from 9), we get dx n, x n +1) dx n, x n +1) ψ dx n, x n +1) ) This implies that ψ dx n, x n +1) ) =, ie, dx n, x n +1) =, which is a contradiction Hence, dx n, x n+1 ) dx n 1, x n ) for all n N Therefore the sequence {d n := dx n, x n+1 )} is decreasing and so there exists d R + such that d n d as n Taking the limit as n in 9), we have d d ψd) This yields that ψd) = Therefore, the property of ψ implies that d = That is, lim dx n+1, x n ) = n We next prove that {x n } is a Cauchy sequence Assume toward a contradiction that {x n } is not a Cauchy sequence Then there are ε > and sequences {mk)} and {nk)} such that for all positive integers k, nk) > mk) > k, dx nk), x mk) ) ε and dx nk), x mk) 1 ) < ε 1)
8 Sattar Alizadeh et al / Filomat 28:3 214), Now for all k N, we have ε dx nk), x mk) ) dx nk), x mk) 1 ) + dx mk) 1, x mk) ) < ε + dx mk) 1, x mk) ) Taking the limit as k + in the above inequality and using 1) we get lim dx nk), x mk) ) = ε k + 11) On the other hand, dx nk), x mk) ) dx mk), x mk)+1 ) + dx mk)+1, x nk)+1 ) + dx nk)+1, x nk) ), and dx nk)+1, x mk)+1 ) dx mk), x mk)+1 ) + dx mk), x nk) ) + dx nk)+1, x nk) ) Taking the limit as k + in above two inequalities, by using 1) and 11), we deduce that lim dx nk)+1, x mk)+1 ) = ε k + 12) Again since f is a weak α β ψ rational contraction and αx nk) )αx mk) ) 1, then dx nk)+1, x mk)+1 ) Mx nk), x mk) ) ψ Mx nk), x mk) ) ), 13) where { Mx nk), x mk) ) = max dx nk), x mk) ), [1 + dx nk), x nk)+1 )]dx mk), x mk)+1 ) dx nk), x mk) ) + 1 Letting k in 13) and using 1), 11) and 12), we get ε ε ψε) So ε =, which is a contradiction Hence, {x n } is a Cauchy sequence Since X is complete, then there exists z X such that x n z Suppose that c) is held That is, αx n )βz) 1 Since f is a weak α β ψ rational contraction, then dx n+1, f z) Mx n, z) ψ Mx n, z) ), 14) where { Mx n, z) = max dx n, z), [1 + dx n, x n+1 )]dz, Tz) dx n, z) + 1 } Taking the limit as n in 14), we get z = f z To prove the uniqueness of fixed point, suppose that z and z are two fixed points of f Since αz)βz ) 1, it follows that ψdz, z )) ψmz, z )) φmz, z )), where { Mz, z ) = max dz, y), [1 + dz, Tz)]dz, Tz ) dz, z ) + 1 So φdz, z )) = and hence, dz, z ) = ie, z = z } }
9 Sattar Alizadeh et al / Filomat 28:3 214), Also we can obtain the following corollaries from Theorem 212 Corollary 213 Let X, d) be a complete metric space and f : X X be a cyclic α, β)-admissible mapping such that αx)βy)d f x, f y) Mx, y) ψmx, y)), 15) for all x, y X where ψ Ψ Suppose that the following assertions hold: a) there exists x X such that αx ) 1 and βx ) 1; b) f is continuous, or c) if {x n } is a sequence in X such that x n x and βx n ) 1 for all n, then βx) 1; then f has a fixed point Moreover, if αx) 1 and βy) 1 for all x, y Fix f ), then f has a unique fixed point Corollary 214 Let X, d) be a complete metric space and f : X X be a cyclic α, β)-admissible mapping such that )d f x, f y) αx)βy) Mx, y) ψmx, y)), for all x, y X where ψ Ψ Suppose that the following assertions hold: a) there exists x X such that αx ) 1 and βx ) 1; b) f is continuous, or c) if {x n } is a sequence in X such that x n x and βx n ) 1 for all n, then βx) 1; then f has a fixed point Moreover, if αx) 1 and βy) 1 for all x, y Fix f ), then f has a unique fixed point Corollary 215 Let X, d) be a complete metric space and f : X X be a cyclic α, β)-admissible mapping Assume that there exists l > 1 such that )αx)βy) d f x, f y) + l Mx, y) ψmx, y)) + l, for all x, y X where ψ Ψ and l > Suppose that the following assertions hold: a) there exists x X such that αx ) 1 and βx ) 1; b) f is continuous, or c) if {x n } is a sequence in X such that x n x and βx n ) 1 for all n, then βx) 1; then f has a fixed point Moreover, if αx) 1 and βy) 1 for all x, y Fix f ), then f has a unique fixed point 3 Some cyclic contraction via cyclic α, β)-admissible mapping In this section, in a natural way, we apply the Theorem 24 for proving a fixed point theorem involving a cyclic mapping For more results see [19, 25, 26, 28, 29, 32] Theorem 31 Let A and B be two closed subsets of complete metric space X, d) such that A B and f : A B A B be a mapping such that f A B and f B A Assume that ψd f x, f y)) ψdx, y)) φdx, y)), 16) for all x A and y B where ψ Ψ and φ Φ Then f has a unique fixed point in A B
10 Sattar Alizadeh et al / Filomat 28:3 214), Proof Define α, β : X [, + ) by 1, if x A 1, if x B αx) = and βx) =, otherwise, otherwise Let αx)βy) 1 Then x A and y B Hence, by 16) we have ψd f x, f y)) ψdx, y)) φdx, y)), for all x, y A B Let αx) 1 for some x X, then x A Hence, f x B and so β f x) 1 Now, let βx) 1 for some x X, so x B Hence, f x A and then α f x) 1 Therefore f is a cyclic α, β)-admissible mapping Since A B is nonempty, then there exists x A B such that αx ) 1 and βx ) 1 Now, Let {x n } be a sequence in X such that βx n ) 1 for all n N and x n x as n, then x n B for all n N Therefore x B This implies that βx) 1 So the conditions a) and c) of Theorem 24 hold So f has a fixed point in A B, say z Since z A, then z = f z B and since z B, then z = f z A Therefore z A B The uniqueness of the fixed point follows easily from 16) Example 32 Let X = R endowed with the usual metric dx, y) = x y for all x, y X and f : A B A B be defined by f x = x/5 where A = [ 1, ] and B = [, 1] Also define ψ, φ : [, ) [, ) by ψt) = t and φt) = 4 5t Then f has a unique fixed point here, x = is a unique fixed point of f ) Indeed, for all x A and all y B, we have ψd f x, f y)) = f x f y = 1 5 x y 1 x y = ψdx, y)) φdx, y)) 5 Therefore, the conditions of Theorem 31 hold and f has a unique fixed point Similarly, we can prove the following Theorem Theorem 33 Let A and B be two closed subsets of complete metric space X, d) such that A B and f : A B A B be a mapping such that f A B and f B A Assume that d f x, f y) Mx, y) ψmx, y)) 17) for all x A and all y B where ψ Ψ Then f has a unique fixed point in A B 4 Application to the existence of solutions of integral equations Let X = C[, T], R) be the set of real continuous functions defined on [, T] and d : X X [, + ) be defined by dx, y) = x y, for all x, y X Then X, d) is a complete metric space Consider the integral equation xt) = pt) + and the mapping F : X X defined by where Fxt) = pt) + St, s) f s, xs))ds, 18) St, s) f s, xs))ds, 19)
11 A) f : [, T] R R is continuous, B) p : [, T] R is continuous, C) S : [, T] [, T] [, + ) is continuous, Sattar Alizadeh et al / Filomat 28:3 214), D) there exist θ, ϑ, π : X R such that if θx) and ϑy) for some x, y X, then for every s [, T] we have f s, xs)) f s, ys)) πy) xs) ys), F) St, s) πy) ds < 1, G) there exists x X such that θx ) and ϑx ), H) and θx) for some x X implies ϑfx), ϑx) for some x X implies θfx), I) if {x n } is a sequence in X such that θx n ) for all n N {} and x n x as n +, then θx) Theorem 41 Under the assumptions A)-I), the integral equation 18) has a solution in X = C[, T], R) Proof Let F : X X be defined by 19) and let x, y X be such that θx) and ϑy) By the condition D), we deduce that Fxt) Fyt) = St, s)[ f s, xs)) f s, ys))]ds Then Fx Fy St, s) f s, xs)) f s, ys)) ds St, s) πys)) xs) ys) ds St, s) πys)) x y ds ) = x y St, s) πys)) ds St, s) πy) ds x y Now, define α, β : X [, + ) by 1, if θx) 1, if ϑy) αx) = and βy) =, otherwise, otherwise
12 Also, define ψ, φ : [, + ) [, + ) by ψt) = t and φt) = 1 St, s) πy) ds Consequently, for all x, y X we have αx)βy)dfx), Fy)) ψdx, y)) φdx, y)) Sattar Alizadeh et al / Filomat 28:3 214), ) It easily shows that all the hypotheses of Corollary 26 are satisfied and hence the mapping F has a fixed point which is a solution of the integral equation 18) in X = C[, T], R) t Acknowledgements The authors would like to thank Marand Branch, Islamic Azad University for the Financial support of this research, which is based on a research project contract References [1] M Abbas and D Dorić, Common fixed point theorem for four mappings satisfying generalized weak contractive condition, Filomat 24 2) 21) 1-1 [2] MA Al-Thafai and N Shahzad, Convergence and existence results for best proximity points, Nonlinear Anal 7 29) [3] S Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund Math ) [4] V Berinde and F Vetro, Common fixed points of mappings satisfying implicit contractive conditions, Fixed Point Theory Appl 212, 212:15 [5] B S Choudhury and A Kundu, ψ, α, β)-weak contractions in partially ordered metric spaces, Appl Math Lett ) 6-1 [6] D W Boyd and J S Wong, On nonlinear contractions, Proc Amer Math Soc ) [7] Lj B Ćirić, A Generalization of Banach s contraction principle, Proc Amer Math Soc ) [8] B Damjanovic, B Samet and C Vetro, Common fixed point theorems for multi-valued maps, Acta Math Sci Ser B Engl Ed ) [9] C Di Bari, M Milojević, S Radenović and P Vetro, Common fixed points for self-mappings on partial metric spaces, Fixed Point Theory Appl, 212:14 212) [1] C Di Bari, Z Kadelburg, H Nashine and S Radenović, Common fixed points of g-quasicontractions and related mappings in -complete partial metric spaces, Fixed Point Theory Appl, 212: ) [11] D Dorić, Common fixed point for generalized ψ, φ)-weak contractions, Appl Math Lett 22 29) [12] P N Dutta and B S Choudhury, A generalization of contraction principle in metric spaces, Fixed Point Theory Appl Volume 28 Article ID [13] S Gülyaz, E Karapınar, V Rakocević and P Salimi, Existence of a solution of integral equations via fixed point theorem, Journal of Inequalities and Applications 213, 213:529 [14] J Harjani and K Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal 72 21) [15] R Kannan, Some results on fixed points, Bull Cal Math Soc ) [16] E Karapınar and P Salimi, Fixed point theorems via auxiliary functions, J Appl Math, 212 Articled Id: [17] E Karapınar, P Kumam and P Salimi, On α-ψ-meir-keeler contractive mappings, Fixed Point Theory Appl, 213, 213:94 [18] M Kikkawa and T Suzuki, Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal, 69 28), [19] W A Kirk, P S Srinavasan and P Veeramani, Fixed points for mapping satisfying cyclical contractive conditions, Fixed Point Theory 4 23) [2] S G Matthews, Partial metric topology, in: Proc 8th Summer Conference on General Topology and Applications, in: Ann New York Acad Sci ) [21] F Moradlou, P Salimi and P Vetro, Some new extensions of Edelstein-Suzuki-type fixed point theorem to G-metric and G-cone metric spaces, Acta Math Sci Ser B Engl Ed 213, 33 B 4): [22] F Moradlou, P Salimi and P Vetro, Fixed point results for r η, ξ, ψ)-contractive mappings of type I), II) and III), Filomat ), [23] S B Nadler Jr, Multi-valued contraction mappings, Pacific J Math ) [24] D Paesano and P Vetro, Suzuki s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces, Topology Appl, ), [25] M Păcurar and IA Rus, Fixed point theory for cyclic ϕ-contractions, Nonlinear Anal ) 21) [26] G Petruşel, Cyclic representations and periodic points, Studia Univ Babes-Bolyai Math 5 25) [27] S Reich, Kannan s fixed point theorem, Boll Un Mat Ital ) 1 11
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