In this paper, we prove fixed point theorems for weakly compatible self &nbsp;&nbsp;mappings satisfying certain contractive conditions of integral type in G-metric spaces.&nbsp

Athaley, Rucha

Gupta, Rashmi

Jatav, Hotam Singh

Rajput, Anil

English

Scitech Research Journals

                                                                      Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 4 available at www.scitecresearch.com/journals/index.php/jprm                                                 1109|  SCITECH                                                                            Volume 7, Issue 4  RESEARCH ORGANISATION                     Published online: May 18, 2016| Journal of Progressive Research in Mathematics www.scitecresearch.com/journals        Fixed Point Theorems in Ordered Generalized Metric Spaces with Integral Type Anil Rajput1, Rashmi Gupta2, Rucha Athaley3 and Hotam Singh Jatav4 1Chandra Shekhar Azad P. G. College, Sehore, Madhya Pradesh, India. 2Oriental College of Technology, Bhopal, India 3Sardar Ajeet Singh Memorial  Girls College, Bhopal, India 4Science College, Gwalior, MP, India.  Abstract. In this paper, we prove fixed point theorems for weakly compatible self mappings satisfying certain contractive conditions of integral type in 𝐺 - metric spaces.  Keywords: Weakly compatible mapping; 𝐺 −metric space; common fixed point.  1. Introduction and  Preliminaries Fixed points of mappings in ordered metric space are of great use in many mathematical problems in    applied and pure mathematics. The first result in this direction was obtained by Ran and Reurings [9], in this study the authors presented some applications of their obtained results to matrix equations. In [6, 7], Nieto and Lopez extended the result of Ran and Reurings [9] for non-decreasing mappings and applied their result to get a unique solution for a first order differential equation. While Agarwal et al. [1] and O’Regan and Petrutel [8] studied some results for generalized contractions in ordered metric spaces. In 2002, Branciari [2] established a fixed point theorem for a single-valued mapping satisfying a contractive inequality of integral type. Recently, Liu et al. [3] (see also [4,5] obtained fixed point theorems for general classes of contractive mappings of integral type in complete metric spaces. In this paper, using auxiliary functions, we establish some fixed point theorems for self-mappings satisfying a certain contractive inequality of integral type.      In 2013, Liu et al. [3] introduced the following three contractive mappings of integral type:                    𝜓   𝜑 𝑡 𝑑 𝑓𝑥 ,𝑓𝑦 0𝑑𝑡 ≤ 𝜓   𝜑(𝑡)𝑑(𝑥 ,𝑦)0𝑑𝑡 − 𝜙   𝜑(𝑡)𝑑(𝑥 ,𝑦)0𝑑𝑡                                  (1)           Where (𝜑,𝜙,𝜓) ∈ 𝜙1 × 𝜙2 × 𝜙3                                                                       Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 4 available at www.scitecresearch.com/journals/index.php/jprm                                                 1110|               𝜓  𝜑 𝑡 𝑑 𝑓𝑥 ,𝑓𝑦 0𝑑𝑡 ≤ 𝛼 𝑑(𝑥,𝑦) 𝜓   𝜑 𝑡 𝑑𝑡𝑑(𝑥 ,𝑦)0   ∀𝑥,𝑦 ∈ 𝑋                                                 (2) Where (𝜑,𝜓,𝛼) ∈ 𝜙1 × 𝜙3 × 𝜙5  and  𝜓  𝜑 𝑡 𝑑 𝑓𝑥 ,𝑓𝑦 0𝑑𝑡 ≤ 𝛼 𝑑(𝑥,𝑦) 𝜙   𝜑 𝑡 𝑑𝑡𝑑(𝑥 ,𝑦)0 + 𝛽(𝑑(𝑥,𝑦)𝜓  𝜑 𝑡 𝑑𝑡𝑑(𝑥 ,𝑦)0  ∀𝑥, 𝑦 ∈ 𝑋 (3)                                                 Definition 1.1: Let 𝑓 and 𝑔 on a 𝐺 −metric space be two self mappings(𝑋,𝐺). The mappings  𝑓 𝑎𝑛𝑑 𝑔   are said to be compatible if   lim𝑛→∞𝐺 𝑓𝑔𝑥𝑛 ,𝑔𝑓𝑥𝑛 ,𝑔𝑓𝑥𝑛 = 0       whenever  𝑥𝑛  is a sequence in 𝑋 such that           lim𝑛→∞ 𝑓𝑥𝑛 = lim𝑛→∞ 𝑔𝑥𝑛 = 𝑧 for some 𝑧 ∈ 𝑋. Definition 1.2 Let A and S be mappings from a G-metric space (X,G) into itself. Then the mappings are said to be weakly compatible if they commute at their coincidence point, that is,   𝐴𝑥 = 𝑆𝑥 implies than 𝐴𝑆𝑥 = 𝑆𝐴𝑥. Definition 1.3 Let (𝑋,𝐺) be a 𝐺 −metric space, for 𝐴,𝐵,𝐶 ⊂ 𝑋 define 𝛿𝐺 𝐴,𝐵,𝐶 = sup⁡{𝐺 𝑎, 𝑏, 𝑐 ;𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵, 𝑐 ∈ 𝐶} If 𝐴 consists of a single point 𝑎,𝑤𝑒 𝑤𝑟𝑖𝑡𝑒 𝛿𝐺 𝐴,𝐵,𝐶 = 𝛿𝐺 𝑎,𝐵,𝐶  If 𝐵 and 𝐶 also consist of a single point 𝑏 𝑎𝑛𝑑 𝑐,𝑤𝑒 𝑤𝑟𝑖𝑡𝑒 𝛿𝐺 𝐴,𝐵,𝐶 = 𝐺 𝑎, 𝑏, 𝑐  In particular for 𝐴 = 𝐵 = 𝐶 ∈ 𝑋, 𝑎𝑛 = 𝛿𝐺 𝐴 = sup⁡{𝐺 𝑎, 𝑏, 𝑐 : 𝑎, 𝑏, 𝑐 ∈ 𝐴} It follows from the definition that (i) If 𝐴 ⊆ 𝐵 then 𝛿𝐺(𝐴) ≤ 𝛿𝐺(𝐵) For a sequence 𝐴𝑛 = {𝑥𝑛 ,𝑥𝑛+1 ,𝑥𝑛+2 ……… } in G-metric space(𝑋,𝐺), let         𝑎𝑛 = 𝛿𝐺 𝐴𝑛 , for all 𝑛 ∈ 𝑁, then (a) Since 𝐴𝑛 ⊇ 𝐴𝑛+1 ,then 𝑎𝑛+1 ≤ 𝑎𝑛  (b) 𝐺 𝑥𝑙 ,𝑥𝑚 , 𝑥𝑘 ≤ 𝛿𝐺 𝐴𝑛 = 𝑎𝑛   for every 𝑙,𝑚,𝑘 ≥ 𝑛 (c) 0 ≤ 𝛿𝐺 𝐴𝑛 = 𝑎𝑛    and  𝑎𝑛+1 ≤ 𝑎𝑛  for every 𝑛 ≥ 𝑙 Therefore 𝑎𝑛  is decreasing and bounded for all 𝑛 ∈ 𝑁 and so there exist an 0 ≤ 𝑎 such that lim𝑛→∞ 𝑎𝑛 = 𝑎.                                                                       Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 4 available at www.scitecresearch.com/journals/index.php/jprm                                                 1111| Lemma 1.1 Let (𝑋,𝐺) be a 𝐺 −metric space. If Lim𝑛→∞ 𝑎𝑛 = 0 then sequence {𝑥𝑛} is a Cauchy sequence. Theorem 1.1: Let 𝑆,𝑅,𝑇,𝑈 be self  mapping of a complete 𝐺 − metric  space (𝑋,𝐺) satisfying            (i)   𝑆𝑅 ⊆ 𝑇𝑈,𝑇𝑈 is closed subset of 𝑋.            (ii) The pair (𝑆𝑅,𝑇𝑈) is weakly compatible,            (iii) 𝐺(𝑆𝑅𝑥, 𝑆𝑅𝑦, 𝑆𝑅𝑧) ≤ 𝜙 𝐺 𝑇𝑈𝑥,𝑇𝑈𝑦,𝑇𝑈𝑧   for  every 𝑥, 𝑦, 𝑧 ∈ 𝑋, where ∅:  0,∞ →                      0,∞  is  non-decreasing  continuous function with Φ t < 𝑡 for every 𝑡 > 0            (iv)   𝑆,𝑅 , (𝑇,𝑈) are commutative, then S,R,T,U have  a common fixed point in X. 2. Main Results   Theorem 2.1: Let (𝑋,𝐺) be a complete   𝐺-metric space and S, R, T,U be self mappings on                                              (𝑋,𝐺) satisfying (i) 𝑆𝑅 ⊆ 𝑇𝑈,𝑇𝑈 is closed subset of 𝑋.                 (ii)    The pair (𝑆𝑅,𝑇𝑈) is weakly compatible,                 (iii)     𝜑 𝑡 𝑑𝑡 ≤ Φ   𝜑 𝑡 𝑑𝑡𝐺 𝑇𝑈𝑥 ,𝑇𝑈𝑦 ,𝑇𝑈𝑧 0 𝐺 𝑆𝑅𝑥 ,𝑆𝑅𝑦 ,𝑆𝑅𝑧 0 for every 𝑥,𝑦, 𝑧 ∈ 𝑋, where                               ∅:  0,∞ →    0,∞  is non-decreasing  continuous function with Φ t < 𝑡                                     for every 𝑡 > 0                (iv)       𝑆,𝑅 , (𝑇,𝑈) are commutative, then S,R,T,U have  a common fixed point in X. Proof: Let 𝑥0 be an arbitrary point in X, by (i) we can choose a point 𝑥1 in 𝑋 such that 𝑦0 = 𝑆𝑅𝑥0 = 𝑇𝑈𝑥1 and 𝑦1 = 𝑆𝑅𝑥1=TU𝑥2 . In general, there exist a sequence  𝑦𝑛  such that 𝑦𝑛 =𝑆𝑅𝑥𝑛 = 𝑇𝑈𝑥𝑛+1 ,  for 𝑛 = 0,1,2,3…. We prove that the sequence  𝑦𝑛  is a Cauchy sequence. Let 𝐴𝑛 = {𝑦𝑛 ,𝑦𝑛+1 ,𝑦𝑛+2 ,… . . } and 𝑎𝑛 = 𝛿𝐺 𝐴𝑛 ,𝑛 ∈ 𝑁. Then we know that lim𝑛→∞ 𝑎𝑛 = 𝑎 for some 𝑎 ≥ 0. Taking 𝑥 = 𝑥𝑛+𝑘 ,𝑦 = 𝑦𝑚+𝑘  𝑎𝑛𝑑 𝑧 = 𝑧𝑙+𝑘  in (iii) for 𝑘 ≥ 1 𝑎𝑛𝑑 𝑚,𝑛, 𝑙 ≥ 0, we have  𝜑 𝑡 𝑑𝑡𝐺(𝑦𝑛+𝑘 ,𝑦𝑚+𝑘 ,𝑦𝑙+𝑘)0=  𝜑 𝑡 𝑑𝑡𝐺(𝑆𝑅𝑥𝑛+𝑘 ,𝑆𝑅𝑥𝑚+𝑘 ,𝑆𝑅𝑥 𝑙+𝑘)0≤ Φ  𝜑 𝑡 𝑑𝑡𝐺 𝑇𝑈𝑥𝑛+𝑘 ,𝑇𝑈𝑥𝑚+𝑘 ,𝑇𝑈𝑥𝑙+𝑘 0  =  Φ 𝜑 𝑡 𝑑𝑡    G(yn +k−1 ,ym +k−1 ,y l+k−1)0                                                                                                    (1.1) Now we claim that  𝜑 𝑡 𝑑𝑡    G(yn +k−1 ,ym +k−1 ,y l+k−1)0≤  𝜑 𝑡 𝑑𝑡𝑎𝑘−10 for every 𝑛,𝑚, 𝑙 ≥ 0.                                                                       Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 4 available at www.scitecresearch.com/journals/index.php/jprm                                                 1112| Since 𝐴𝑘−1 = {𝑦𝑘−1 ,𝑦𝑘 ,𝑦𝑘+1 …… . . }, 𝑎𝑘 = sup{𝐺 𝑎, 𝑏, 𝑐 ,𝑎, 𝑏. 𝑐 ∈ 𝐴𝑘−1} Also 𝑦𝑘+𝑛−1 ,𝑦𝑘+𝑚−1 ,𝑦𝑘+𝑙−1 ∈ 𝐴𝑘−1 , 𝑖𝑚𝑝𝑙𝑖𝑒𝑠  𝜑 𝑡 𝑑𝑡    G(yn +k−1 ,ym +k−1 ,y l+k−1)0≤  𝜑 𝑡 𝑑𝑡𝑎𝑘−10 Also Φ is increasing in 𝑡,  From (1.1) we get  𝜑 𝑡 𝑑𝑡sup 𝐺(𝑦𝑘+𝑛−1 ,𝑦𝑘+𝑚−1 ,𝑦𝑘+𝑙−1)0  ≤ Φ( 𝜑 𝑡 𝑑𝑡𝑎𝑘−10) Therefore we have  𝜑 𝑡 𝑑𝑡𝑎𝑘0  ≤ Φ( 𝜑 𝑡 𝑑𝑡𝑎𝑘−10). Letting 𝑘 → ∞, we get   𝜑 𝑡 𝑑𝑡𝑎0  ≤ Φ( 𝜑 𝑡 𝑑𝑡𝑎0). If 𝑎 ≠ 0, then  A 𝜑 𝑡 𝑑𝑡𝑎0  ≤ Φ  𝜑 𝑡 𝑑𝑡𝑎0 <  𝜑 𝑡 𝑑𝑡𝑎0    which is a contradiction. Thus  𝑎 = 0. Hence lim𝑛→∞ 𝑎𝑛 = 0. Thus by lemma {𝑦𝑛} is a Cauchy sequence in  𝑋. By completeness of 𝑋, there exist 𝑦1 ∈ 𝑋 such that lim𝑛→∞ 𝑦𝑛 = lim𝑛→∞𝑆𝑅𝑥𝑛 = lim𝑛→∞𝑇𝑈𝑥𝑛+1= 𝑦1. Also 𝑇𝑈(𝑋) is closed, there exist 𝑧 ∈ 𝑋 such that 𝑇𝑈𝑧 = 𝑦1. Now we show that 𝑆𝑅𝑧 = 𝑦1 . For this  set 𝑥𝑛 ,𝑥𝑛 , 𝑧 replacing 𝑥,𝑦, 𝑧 respectively in equation (iii), we get                                 𝜑 𝑡 𝑑𝑡 ≤ Φ𝐺(𝑆𝑅𝑥𝑛 ,𝑆𝑅𝑥𝑛 ,𝑆𝑅𝑧)0( 𝜑 𝑡 𝑑𝑡)𝐺(𝑇𝑈𝑥𝑛 ,𝑇𝑈𝑥𝑛 ,,𝑇𝑈𝑧)0 Taking 𝑛 → ∞, we get                                 𝜑 𝑡 𝑑𝑡 ≤ Φ𝐺 𝑦1.,𝑦1.,𝑆𝑅𝑧 0( 𝜑 𝑡 𝑑𝑡) = 0𝐺 𝑦1.,𝑦1.,𝑦1. 0         Implies 𝑆𝑅𝑧 = 𝑦1..  Since the pair  𝑆𝑅,𝑇𝑈  is weakly compatible (𝑆𝑅)(𝑇𝑈) = (𝑇𝑈)(𝑆𝑅) 𝑇𝑕𝑢s 𝑆𝑅𝑦1. = 𝑇𝑈𝑦1..  Now we prove that 𝑆𝑅𝑦1 = 𝑦1 . If we substitute 𝑥,𝑦, 𝑧 in (iii) by 𝑥𝑛 ,𝑥𝑛 ,𝑦1 respectively                           𝜑 𝑡 𝑑𝑡 ≤𝐺(𝑆𝑅𝑥𝑛 ,𝑆𝑅𝑥𝑛 ,𝑆𝑅𝑦1)0 Φ( 𝜑 𝑡 𝑑𝑡),𝐺(𝑇𝑈𝑥𝑛 ,𝑇𝑈𝑥𝑛 ,,𝑇𝑈𝑦1)0 Taking 𝑛 → ∞, we get  𝜑 𝑡 𝑑𝑡 ≤ Φ𝐺 𝑦1.,𝑦1.,𝑆𝑅𝑦1 0( 𝜑 𝑡 𝑑𝑡) = Φ(𝐺 𝑦1.,𝑦1.,𝑇𝑈𝑦1. 0 𝜑 𝑡 𝑑𝑡𝐺 𝑦1.,𝑦1.,𝑆𝑅𝑦1 0 If 𝑆𝑅𝑦1 ≠ 𝑦1, then 𝐺 𝑦1.,𝑦1., 𝑆𝑅𝑦1 < 𝐺 𝑦1.,𝑦1., 𝑆𝑅𝑦1  is a contradiction. Therefore 𝑆𝑅𝑦1 = 𝑇𝑈𝑦1 = 𝑦1. For uniqueness let 𝑦1 and 𝑦2 b e two fixed points of 𝑆𝑅,𝑇𝑈, Taking 𝑥 = 𝑦 = 𝑦1 and 𝑧 = 𝑦2in (iii) we have                                                                       Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 4 available at www.scitecresearch.com/journals/index.php/jprm                                                 1113|  𝜑 𝑡 𝑑𝑡 =  𝜑 𝑡 𝑑𝑡 ≤𝐺(𝑆𝑅𝑦1 ,𝑆𝑅𝑦1,𝑆𝑅𝑦2)0𝐺(𝑦1 ,𝑦1 ,𝑦2)0 Φ( 𝜑 𝑡 𝑑𝑡)G(TU y1,TU y1 ,TU y2)0 = Φ( 𝜑 𝑡 𝑑𝑡)G 𝑦1 ,𝑦1 ,𝑦1 0<   𝜑 𝑡 𝑑𝑡)𝐺(𝑦1 ,𝑦1 ,𝑦2)0  ,a contradiction. Thus we have 𝑦1 = 𝑦2. Now by (iv) 𝑆,𝑅 , (𝑇,𝑈) are mutually commutative pair of mapping. Consider 𝑆𝑦1 = 𝑆 𝑆𝑅𝑦1 = 𝑆 𝑅𝑆𝑦1 = 𝑆𝑅 𝑆𝑦1 , 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 S𝑦1 is the unique fixed point of SR, but 𝑦1 is the unique fixed point of 𝑆𝑅 hence S𝑦1 = 𝑦1 . Also 𝑅𝑦1 = 𝑅 𝑆𝑅𝑦1 =  𝑅𝑆  𝑅𝑦1 = 𝑆𝑅(𝑅𝑦1) implies 𝑅𝑦1 is the fixed point of 𝑆𝑅 but 𝑦1 is the unique fixed point of 𝑆𝑅. Hence 𝑅𝑦1 = 𝑦1 . Thus 𝑆𝑦1 = 𝑅𝑦1 = 𝑦1 . In the same way we have 𝑇𝑦1 = 𝑈𝑦1 = 𝑦1 . Hence the result. References [1] Agarwal, RP, El-Gebeily, MA: O’Regan D: Generalized contractions in partially ordered metric spaces.  Appl Anal. 87, 1–8(2008). doi:10.1080/00036810701714164 [2] Branciari, A: A fixed point theorem for mappings satisfying a general contractive condition of integral  type,  Int.J. Math.  Math. Sci. 29(9), 531-536 (2002) [3] Liu, Z, Li, J, Kang, SM: Fixed point theorems of contractive mappings of integral type.                                 Fixed Point Theory Appl. 2013, Article ID 300 (2013). [4] Liu, Z, Wu, H, Ume, JS, Kang, SM: Some fixed point theorems for mappings satisfying contractive conditions of integral type. Fixed Point Theory Appl. 2014, Article ID 69 (2014) [5] Liu, Z, Han, Y, Kang, SM, Ume, JS: Common fixed point theorems for weakly compatible mappings satisfying contractive conditions of integral type. Fixed Point Theory Appl. 2014, Article ID 132 (2014) [6] Nieto, JJ, López, RR: Contractive mapping theorems in partially ordered sets and pplications to ordinary differential equations. Order. 22, 223–239 (2005). doi: 10.1007/s11083-005-9018-5 [7] Nieto, JJ, López, RR: Existence and uniqueness of fixed point in partially ordered sets and applications to    ordinary differential equations. Acta Math Sinica Engl Ser. 23(12):2205–2212 (2007). doi:10.1007/s10114-005-0769-0 [8] O’Regan, D, Petrutel, A: Fixed point theorems for generalized contractions in ordered metric spaces. J Math Anal Appl.341, 241–1252 (2008) [9] Ran, ACM, Reuring, MCB: A fixed point theorem in partially ordered sets and some applications to  matrix equations, Proc. Am MathSoc132,1435-1443(2004),doi:10,1090/S002-9939-03—07220-4 

Fixed Point Theorems in Ordered Generalized Metric Spaces with Integral Type

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Fixed Point Theorems in Ordered Generalized Metric Spaces with Integral Type

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