In this paper, by virtue of an inequality and sane analysis techniques, we prove sane convergence theorems cm the iterative process for nonlinear mappings of-pseudo contractive type in named linear spaces, which exten...In this paper, by virtue of an inequality and sane analysis techniques, we prove sane convergence theorems cm the iterative process for nonlinear mappings of-pseudo contractive type in named linear spaces, which extend and improve the corresponding results obtained by others recently.展开更多
In this paper, some new existence and uniqueness of common fixed points for three mappings of Lipschitz type are obtained. The conditions are greatly weaker than the classic conditions in cone metric spaces. These res...In this paper, some new existence and uniqueness of common fixed points for three mappings of Lipschitz type are obtained. The conditions are greatly weaker than the classic conditions in cone metric spaces. These results improve and generalize several wellknown comparable results in the literature. Moreover, our results are supported by some examples.展开更多
This article introduces the concept of commutative semigroups of almost asymptotically nonexpansive-type mappings in a Banach space X which has the Opial property and whose norm is UKK, and establishes the weak conver...This article introduces the concept of commutative semigroups of almost asymptotically nonexpansive-type mappings in a Banach space X which has the Opial property and whose norm is UKK, and establishes the weak convergence theorems for almostorbits of this class of commutative semigroups. The author improves, extends and develops some recent and earlier results.展开更多
The purpose of this paper is to introduce the concept of Φ_pseudo contractive type mapping and to study the convergence problem of Ishikawa and Mann iterative processes with error for this kind of mappings. The resul...The purpose of this paper is to introduce the concept of Φ_pseudo contractive type mapping and to study the convergence problem of Ishikawa and Mann iterative processes with error for this kind of mappings. The results presented in this paper improve and extend many authors'recent results.展开更多
A new class of almost asymptotically nonexpansive type mappings in Banach spaces is introduced,which includes a number of known classes of nonlinear Lipschitzian mappings and non_Lipschitzian mappings in Banach spaces...A new class of almost asymptotically nonexpansive type mappings in Banach spaces is introduced,which includes a number of known classes of nonlinear Lipschitzian mappings and non_Lipschitzian mappings in Banach spaces as special cases; for example,the known classes of nonexpansive mappings,asymptotically nonexpansive mappings and asymptotically nonexpansive type mappings.The convergence problem of modified Ishikawa iterative sequences with errors for approximating fixed points of almost asymptotically nonexpansive type mappings is considered.Not only S.S.Chang's inequality but also H.K.Xu's one for the norms of Banach spaces are applied to make the error estimate between the exact fixed point and the approximate one.Moreover,Zhang Shi_sheng's method(Applied Mathematics and Mechanics(English Edition),2001,22(1):25-34) for making the convergence analysis of modified Ishikawa iterative sequences with errors is extended to the case of almost asymptotically nonexpansive type mappings. The new convergence criteria of modified Ishikawa iterative sequences with errors for finding fixed points of almost asymptotically nonexpansive type mappings in uniformly convex Banach spaces are presented. Also,the new convergence criteria of modified Mann iterative sequences with errors for this class of mappings are immediately obtained from these criteria.The above results unify,improve and generalize Zhang Shi_sheng's ones on approximating fixed points of asymptotically nonexpansive type mappings by the modified Ishikawa and Mann iterative sequences with errors.展开更多
The main purpose of this paper is devoted to generalizing the results of Browder[1,2]This paper consists of four parts. In the first part, we introduce the concepts of multivalued (S) and (S), type mappings and the co...The main purpose of this paper is devoted to generalizing the results of Browder[1,2]This paper consists of four parts. In the first part, we introduce the concepts of multivalued (S) and (S), type mappings and the concepts of the limits of multivalued (S) and (S) + type mappings. These kinds of mappings contain many monotone type mappings, such as maximal monotone mapping, bounded pseudo-monotone mapping and bounded generalized pseudo-monotone mapping, as its special cases. In the second part we define the pseudo-degree for (S) type mapping and the degree for (S)+ type mapping. These two kinds of degrees are all the generalizations of the degree defined by Browder[1,2] As applications, we utilize the degree theory presented in part 2 to study the existence of solutions for the multivalued operator equations (see part 3) and to obtain some new fixed point theorems in part 4.展开更多
Soil type maps at the scale of 1︰1 000 000 are used extensively to provide soil spatial distribution information for soil erosion assessment and watershed management models in China.However,the soil property maps pro...Soil type maps at the scale of 1︰1 000 000 are used extensively to provide soil spatial distribution information for soil erosion assessment and watershed management models in China.However,the soil property maps produced through conventional direct linking method usually suffer low accuracy as well as the lack of spatial details within a soil type polygon.This paper presents an effective method to produce detailed soil property map based on representative samples which were extracted from each polygon on the 1︰1000 000 soil type map.The representative sample of each polygon is defined as the location that can represent the largest area within the polygon.The representativeness of a candidate sample is determined by calculating the soil-forming environment condition similarities between the sample and other locations.Once the representative sample of each polygon has been chosen,the property values of the existing typical samples are assigned to the corresponding representative samples with the same soil type.Finally,based on these representative samples,the detailed soil property map could be produced by using existing digital soil mapping methods.The case study in XuanCheng City,Anhui Province of China,demonstrated the proposed method could produce soil property map at a higher level of spatial details and accuracy:1)The soil organic matter(SOM)map produced based on the representative samples can not only depict the detailed spatial distribution of SOM within a soil type polygon but also largely reduce the abrupt change of soil property at the boundaries of two adjacent polygons.2)The Root Mean Squared Error(RMSE)of the SOM map based on the representative samples is1.61,and it is 1.37 for the SOM map produced by using conventional direct linking method.Therefore,the proposed method is an effective approach to produce spatial detailed soil property map with higher accuracy for environment simulation models.展开更多
Let C be a closed convex weakly Cauchy subset of a normed space X. Then we define a new {a,b,c} type nonexpansive and {a,b,c} type contraction mapping T from C into C. These types of mappings will be denoted respectiv...Let C be a closed convex weakly Cauchy subset of a normed space X. Then we define a new {a,b,c} type nonexpansive and {a,b,c} type contraction mapping T from C into C. These types of mappings will be denoted respectively by {a,b,c}-ntype and {a,b,c}-ctype. We proved the following: 1. If T is {a,b,c}-ntype mapping, then inf{ || T(x)-x|| :x C C} =0, accordingly T has a unique fixed point. Moreover, any sequence {Xn}n∈NN in C with limn→∞||T(xn) - Xn|| = 0 has a subsequence strongly convergent to the unique fixed point of T. 2. If T is {a,b,c}-ctype mapping, then T has a unique fixed point. Moreover, for any x∈C the sequence of iterates {Tn (x)}n∈N has subsequence strongly convergent to the unique fixed point of T. This paper extends and generalizes some of the results given in [2,4, 7] and [13].展开更多
Let X be a uniformly convex Banach space X such that its dual X^* has the KK property. Let C be a nonempty bounded closed convex subset of X and G be a directed system. Let ={Tt : t ∈ G} be a family of asymptotica...Let X be a uniformly convex Banach space X such that its dual X^* has the KK property. Let C be a nonempty bounded closed convex subset of X and G be a directed system. Let ={Tt : t ∈ G} be a family of asymptotically nonexpansive type mappings on C. In this paper, we investigate the asymptotic behavior of {Ttx0 : t∈ G} and give its weak convergence theorem.展开更多
In this article, first, the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings of type B and order B on the unit ball in complex Ba- nach spaces are given. Second, the sharp estimat...In this article, first, the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings of type B and order B on the unit ball in complex Ba- nach spaces are given. Second, the sharp estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in (in are also established. In particular, the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings (include quasi-convex mappings of type A and quasi-convex mappings of type B) in several complex variables are get accordingly. Our results state that a weak version of the Bieber- bach conjecture for quasi-convex mappings of type B and order a in several complex variables is proved, and the derived conclusions are the generalization of the classical results in one complex variable.展开更多
In this article,some coupled coincidence point results for two mappings g:X→X and G:X×X→X satisfying F-contractive type conditions are obtained,and some coupled fixed point results are derived in partially orde...In this article,some coupled coincidence point results for two mappings g:X→X and G:X×X→X satisfying F-contractive type conditions are obtained,and some coupled fixed point results are derived in partially ordered metric spaces.A sufficient condition for uniqueness of coupled point of coincidence are established for F type contraction,and a coupled common fixed point theorem is obtained.Some examples are given to support our results.展开更多
Suppose f is a spirallike function of type β and order α on the unit disk D. Let Ωn,p1,p2,···,pn = {z = (z1,z2,··· ,zn) ∈ Cn : n: n j=1 |zj|pj < 1}, where 1 ≤ p1 ≤ 2,pj ≥1,j = 2...Suppose f is a spirallike function of type β and order α on the unit disk D. Let Ωn,p1,p2,···,pn = {z = (z1,z2,··· ,zn) ∈ Cn : n: n j=1 |zj|pj < 1}, where 1 ≤ p1 ≤ 2,pj ≥1,j = 2,··· ,n, are real numbers. In this paper, we will prove that Φn,β2,γ2,···,βn,γn(f)(z) =(f(z1),(f(zz11 ))β2(f (z1))γ2z2,··· ,(f(zz11 ))βn(f (z1))γnzn) preserves spirallikeness of type βand order α on Ωn,p1,p2,···,pn.展开更多
A sequence of periodic attractors has been observed in a two-dimensional discontinuous map, which canbe considered as a model of impact oscillator. The so-called 'transfer number', which is defined as the mean...A sequence of periodic attractors has been observed in a two-dimensional discontinuous map, which canbe considered as a model of impact oscillator. The so-called 'transfer number', which is defined as the mean numberof transfer from non-impact state to impact state per iteration, is locked onto a lot of rational values to form a curveconsisting of many steps. Our numerical investigation confirms that every step is confined by conditions created by thecollision between the periodic orbit and the discontinuous boundary of the system. After the last collision the systemshows a chaotic motion with intermittent characteristics. Therefore the staircase can be addressed as a 'prelude staircaseto type V intermittency'. The similar phenomenon has also been observed in a model of electric circuit. These resultsof our study suggest that this kind of staircases is common in two (or even higher) dimensional discontinuous maps.展开更多
In terms of 34-year monthly mean temperature series in 1946-1979,the multi-level maPPing model of neural netWork BP type was applied to calculate the system's fractual dimension Do=2'8,leading tO a three-level...In terms of 34-year monthly mean temperature series in 1946-1979,the multi-level maPPing model of neural netWork BP type was applied to calculate the system's fractual dimension Do=2'8,leading tO a three-level model of this type with ixj=3x2,k=l,and the 1980 monthly mean temperture predichon on a long-t6rm basis were prepared by steadily modifying the weighting coefficient,making for the correlation coefficient of 97% with the measurements.Furthermore,the weighhng parameter was modified for each month of 1980 by means of observations,therefore constrcuhng monthly mean temperature forecasts from January to December of the year,reaching the correlation of 99.9% with the measurements.Likewise,the resulting 1981 monthly predictions on a long-range basis with 1946-1980 corresponding records yielded the correlahon of 98% and the month-tO month forecasts of 99.4%.展开更多
The purpose of this paper is to study a new two-step iterative scheme with mean errors of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and prove str...The purpose of this paper is to study a new two-step iterative scheme with mean errors of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and prove strong convergence theorems for the new two-step iterative scheme in uniformly convex Banach spaces.展开更多
The invariance of strong and almost spirallike mappings of type β and order α is discussed in this paper. From the maximum modulus principle of holomorphic functions, we obtain that the generalized Roper-Suffridge o...The invariance of strong and almost spirallike mappings of type β and order α is discussed in this paper. From the maximum modulus principle of holomorphic functions, we obtain that the generalized Roper-Suffridge operators preserve strong and almost spirallike-hess of type β and order α on the unit ball B^n in C^n and on bounded and complete Reinhardt domains. Therefore we obtain that the generalized Roper-Suffridge operators preserve strong spirllikeness of type β, strong and almost starlikeness of order α, strong starlikeness on the corresponding domains.Thus we can construct more subclasses of spirallike mappings in several complex variables.展开更多
In this paper, we first establish several sharp inequalities of homogeneous expansion for biholomorphic quasi-convex mappings of type B and order α on the unit ball E in a complex Banach space X by applying the metho...In this paper, we first establish several sharp inequalities of homogeneous expansion for biholomorphic quasi-convex mappings of type B and order α on the unit ball E in a complex Banach space X by applying the method and technique of complex analysis. Then, as the application of these sharp inequalities, we derive the sharp estimate of third homogeneous expansions for the above mappings defined on the unit polydisk U^n in C^n.展开更多
We establish a common fixed-point theorem for six self maps under the compatible mappings of type (C) with a contractive condition [1], which is independent of earlier contractive conditions.
In this paper, we obtain unique common fixed point theorems for two mappings satisfying the variable coefficient linear contraction of integral type and the implicit contraction of integral type respectively in metric...In this paper, we obtain unique common fixed point theorems for two mappings satisfying the variable coefficient linear contraction of integral type and the implicit contraction of integral type respectively in metric spaces.展开更多
文摘In this paper, by virtue of an inequality and sane analysis techniques, we prove sane convergence theorems cm the iterative process for nonlinear mappings of-pseudo contractive type in named linear spaces, which extend and improve the corresponding results obtained by others recently.
基金Supported by the Foundation of Education Ministry of Hubei Province(D20102502)
文摘In this paper, some new existence and uniqueness of common fixed points for three mappings of Lipschitz type are obtained. The conditions are greatly weaker than the classic conditions in cone metric spaces. These results improve and generalize several wellknown comparable results in the literature. Moreover, our results are supported by some examples.
基金Project supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE,P.R.C., by the Dawn Program Foundation in Shanghai, and by Shanghai Leading Academic Discipline Project Fund (T0401).
文摘This article introduces the concept of commutative semigroups of almost asymptotically nonexpansive-type mappings in a Banach space X which has the Opial property and whose norm is UKK, and establishes the weak convergence theorems for almostorbits of this class of commutative semigroups. The author improves, extends and develops some recent and earlier results.
文摘The purpose of this paper is to introduce the concept of Φ_pseudo contractive type mapping and to study the convergence problem of Ishikawa and Mann iterative processes with error for this kind of mappings. The results presented in this paper improve and extend many authors'recent results.
文摘A new class of almost asymptotically nonexpansive type mappings in Banach spaces is introduced,which includes a number of known classes of nonlinear Lipschitzian mappings and non_Lipschitzian mappings in Banach spaces as special cases; for example,the known classes of nonexpansive mappings,asymptotically nonexpansive mappings and asymptotically nonexpansive type mappings.The convergence problem of modified Ishikawa iterative sequences with errors for approximating fixed points of almost asymptotically nonexpansive type mappings is considered.Not only S.S.Chang's inequality but also H.K.Xu's one for the norms of Banach spaces are applied to make the error estimate between the exact fixed point and the approximate one.Moreover,Zhang Shi_sheng's method(Applied Mathematics and Mechanics(English Edition),2001,22(1):25-34) for making the convergence analysis of modified Ishikawa iterative sequences with errors is extended to the case of almost asymptotically nonexpansive type mappings. The new convergence criteria of modified Ishikawa iterative sequences with errors for finding fixed points of almost asymptotically nonexpansive type mappings in uniformly convex Banach spaces are presented. Also,the new convergence criteria of modified Mann iterative sequences with errors for this class of mappings are immediately obtained from these criteria.The above results unify,improve and generalize Zhang Shi_sheng's ones on approximating fixed points of asymptotically nonexpansive type mappings by the modified Ishikawa and Mann iterative sequences with errors.
文摘The main purpose of this paper is devoted to generalizing the results of Browder[1,2]This paper consists of four parts. In the first part, we introduce the concepts of multivalued (S) and (S), type mappings and the concepts of the limits of multivalued (S) and (S) + type mappings. These kinds of mappings contain many monotone type mappings, such as maximal monotone mapping, bounded pseudo-monotone mapping and bounded generalized pseudo-monotone mapping, as its special cases. In the second part we define the pseudo-degree for (S) type mapping and the degree for (S)+ type mapping. These two kinds of degrees are all the generalizations of the degree defined by Browder[1,2] As applications, we utilize the degree theory presented in part 2 to study the existence of solutions for the multivalued operator equations (see part 3) and to obtain some new fixed point theorems in part 4.
基金Under the auspices of Program of International Science & Technology Cooperation,Ministry of Science and Technology of China(No.2010DFB24140)National Natural Science Foundation of China(No.41023010,41001298)National High Technology Research and Development Program of China(No.2011AA120305)
文摘Soil type maps at the scale of 1︰1 000 000 are used extensively to provide soil spatial distribution information for soil erosion assessment and watershed management models in China.However,the soil property maps produced through conventional direct linking method usually suffer low accuracy as well as the lack of spatial details within a soil type polygon.This paper presents an effective method to produce detailed soil property map based on representative samples which were extracted from each polygon on the 1︰1000 000 soil type map.The representative sample of each polygon is defined as the location that can represent the largest area within the polygon.The representativeness of a candidate sample is determined by calculating the soil-forming environment condition similarities between the sample and other locations.Once the representative sample of each polygon has been chosen,the property values of the existing typical samples are assigned to the corresponding representative samples with the same soil type.Finally,based on these representative samples,the detailed soil property map could be produced by using existing digital soil mapping methods.The case study in XuanCheng City,Anhui Province of China,demonstrated the proposed method could produce soil property map at a higher level of spatial details and accuracy:1)The soil organic matter(SOM)map produced based on the representative samples can not only depict the detailed spatial distribution of SOM within a soil type polygon but also largely reduce the abrupt change of soil property at the boundaries of two adjacent polygons.2)The Root Mean Squared Error(RMSE)of the SOM map based on the representative samples is1.61,and it is 1.37 for the SOM map produced by using conventional direct linking method.Therefore,the proposed method is an effective approach to produce spatial detailed soil property map with higher accuracy for environment simulation models.
文摘Let C be a closed convex weakly Cauchy subset of a normed space X. Then we define a new {a,b,c} type nonexpansive and {a,b,c} type contraction mapping T from C into C. These types of mappings will be denoted respectively by {a,b,c}-ntype and {a,b,c}-ctype. We proved the following: 1. If T is {a,b,c}-ntype mapping, then inf{ || T(x)-x|| :x C C} =0, accordingly T has a unique fixed point. Moreover, any sequence {Xn}n∈NN in C with limn→∞||T(xn) - Xn|| = 0 has a subsequence strongly convergent to the unique fixed point of T. 2. If T is {a,b,c}-ctype mapping, then T has a unique fixed point. Moreover, for any x∈C the sequence of iterates {Tn (x)}n∈N has subsequence strongly convergent to the unique fixed point of T. This paper extends and generalizes some of the results given in [2,4, 7] and [13].
基金Foundation item: the National Natural Science Foundation of China (No. 10571150) the Natural Science Foundation of Jiangsu Education Committee of China (No. 07KJB110131) and the Natural Science Foundation of Yangzhou University (No. FK0513101).
文摘Let X be a uniformly convex Banach space X such that its dual X^* has the KK property. Let C be a nonempty bounded closed convex subset of X and G be a directed system. Let ={Tt : t ∈ G} be a family of asymptotically nonexpansive type mappings on C. In this paper, we investigate the asymptotic behavior of {Ttx0 : t∈ G} and give its weak convergence theorem.
基金Supported by National Natural Science Foundation of China(11471111)Guangdong Natural Science Foundation(2014A030307016)
文摘In this article, first, the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings of type B and order B on the unit ball in complex Ba- nach spaces are given. Second, the sharp estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in (in are also established. In particular, the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings (include quasi-convex mappings of type A and quasi-convex mappings of type B) in several complex variables are get accordingly. Our results state that a weak version of the Bieber- bach conjecture for quasi-convex mappings of type B and order a in several complex variables is proved, and the derived conclusions are the generalization of the classical results in one complex variable.
基金Supported by the National Natural Science Foundation of China(11701245)the Scientific Research Fund of Sichuan Provincial Education Department(18ZB0272)
文摘In this article,some coupled coincidence point results for two mappings g:X→X and G:X×X→X satisfying F-contractive type conditions are obtained,and some coupled fixed point results are derived in partially ordered metric spaces.A sufficient condition for uniqueness of coupled point of coincidence are established for F type contraction,and a coupled common fixed point theorem is obtained.Some examples are given to support our results.
基金Supported by the National Natural Science Foundation of China(10626015, 10571044) Supported by Guangdong Natural Science Foundation(06301315) Supported by the doctoral foundation of Zhanjiang Normal University(Z0420)
文摘Suppose f is a spirallike function of type β and order α on the unit disk D. Let Ωn,p1,p2,···,pn = {z = (z1,z2,··· ,zn) ∈ Cn : n: n j=1 |zj|pj < 1}, where 1 ≤ p1 ≤ 2,pj ≥1,j = 2,··· ,n, are real numbers. In this paper, we will prove that Φn,β2,γ2,···,βn,γn(f)(z) =(f(z1),(f(zz11 ))β2(f (z1))γ2z2,··· ,(f(zz11 ))βn(f (z1))γnzn) preserves spirallikeness of type βand order α on Ωn,p1,p2,···,pn.
文摘A sequence of periodic attractors has been observed in a two-dimensional discontinuous map, which canbe considered as a model of impact oscillator. The so-called 'transfer number', which is defined as the mean numberof transfer from non-impact state to impact state per iteration, is locked onto a lot of rational values to form a curveconsisting of many steps. Our numerical investigation confirms that every step is confined by conditions created by thecollision between the periodic orbit and the discontinuous boundary of the system. After the last collision the systemshows a chaotic motion with intermittent characteristics. Therefore the staircase can be addressed as a 'prelude staircaseto type V intermittency'. The similar phenomenon has also been observed in a model of electric circuit. These resultsof our study suggest that this kind of staircases is common in two (or even higher) dimensional discontinuous maps.
文摘In terms of 34-year monthly mean temperature series in 1946-1979,the multi-level maPPing model of neural netWork BP type was applied to calculate the system's fractual dimension Do=2'8,leading tO a three-level model of this type with ixj=3x2,k=l,and the 1980 monthly mean temperture predichon on a long-t6rm basis were prepared by steadily modifying the weighting coefficient,making for the correlation coefficient of 97% with the measurements.Furthermore,the weighhng parameter was modified for each month of 1980 by means of observations,therefore constrcuhng monthly mean temperature forecasts from January to December of the year,reaching the correlation of 99.9% with the measurements.Likewise,the resulting 1981 monthly predictions on a long-range basis with 1946-1980 corresponding records yielded the correlahon of 98% and the month-tO month forecasts of 99.4%.
文摘The purpose of this paper is to study a new two-step iterative scheme with mean errors of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and prove strong convergence theorems for the new two-step iterative scheme in uniformly convex Banach spaces.
基金supported by NSF of China(11271359U1204618)+1 种基金Science and Technology Research Projects of Henan Provincial Education Department(14B11001514B110016)
文摘The invariance of strong and almost spirallike mappings of type β and order α is discussed in this paper. From the maximum modulus principle of holomorphic functions, we obtain that the generalized Roper-Suffridge operators preserve strong and almost spirallike-hess of type β and order α on the unit ball B^n in C^n and on bounded and complete Reinhardt domains. Therefore we obtain that the generalized Roper-Suffridge operators preserve strong spirllikeness of type β, strong and almost starlikeness of order α, strong starlikeness on the corresponding domains.Thus we can construct more subclasses of spirallike mappings in several complex variables.
基金supported by Guangdong Natural Science Foundation(2018A030313508)Science and Technology Program of Guangzhou,China(201804010171)
文摘In this paper, we first establish several sharp inequalities of homogeneous expansion for biholomorphic quasi-convex mappings of type B and order α on the unit ball E in a complex Banach space X by applying the method and technique of complex analysis. Then, as the application of these sharp inequalities, we derive the sharp estimate of third homogeneous expansions for the above mappings defined on the unit polydisk U^n in C^n.
文摘We establish a common fixed-point theorem for six self maps under the compatible mappings of type (C) with a contractive condition [1], which is independent of earlier contractive conditions.
文摘In this paper, we obtain unique common fixed point theorems for two mappings satisfying the variable coefficient linear contraction of integral type and the implicit contraction of integral type respectively in metric spaces.