Int. J. Anal. Appl. (2022), 20:2 An Affirmative Result on Banach Space V. Srinivas1,2, T. Thirupathi2,∗ 1Department of Mathematics, University College of Science, Saifabad,Hyderabad, Telangana, India 2Department of Mathematics, Sreenidhi Institute of Science and Technology, Hyderabad, Telangana, India ∗Corresponding author: thotathirupathi1986gmail.com Abstract. The aim of this paper is to establish a common fixed point theorem on Banach space using occasionally weakly compatible (OWC) mappings. 1. Introduction Fixed point theory is one of the most powerful topics of modern mathematics and might be taken as main subject of analysis.For the past many years, fixed point theory has been evolved as the area of research for many researchers. Banach contraction principle is one such result proposed by Banach to name a few.For the study of discontinuous and noncompatible mappings in fixed point theory we refer the literature like [4] and [5]. Pathak and others [1] proved a fixed point theorem on complete metric space using continuity and weakly compatible mappings.Thereafter Sushil Sharma, Bhavana Deshpande, and Alok Pandey [2] proved some more results on Banach space.Further several theo- rems [3], [6], [7], [8] , [9] and [10]are being generated on Banach space using various conditions.The focus of this work is now on proving a result in Banach space without a continuity constraint using OWC mappings to prove a common fixed point theorem. Before we prove our theorem, we’ll present some definitions and examples. Received: Aug. 28, 2021. 2010 Mathematics Subject Classification. 47H10, 54H25. Key words and phrases. Banach space; weakly compatible mapping; occasionally weakly compatible (OWC) mappings. https://doi.org/10.28924/2291-8639-20-2022-2 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-2 2 Int. J. Anal. Appl. (2022), 20:2 2. Preliminaries A pair (I,J) of a Banach space is said to be Definition 2.1 Weakly commuting iff ‖IJα−JIα‖≤‖Iα−Jα‖ for all α ∈ X. Definition 2.2 Cmpatible iff ‖IJαj −JIαj‖=0 as j→∞ whenever {αj} is a sequence in X such that ‖Iαj −Jαj‖=0 as j →∞ for some η ∈ X. Definition 2.3 Weakly compatible if Iη = Jη for some η ∈ X such that IJη = JIη Definition 2.4 OWC if and only if there exists a point η ∈ X such that Iη = Jη implies IJη = JIη. We now discuss some examples to find the relation among the above definitions. Example 2.5 Let X = [0,1] is a Banach space with ‖u −v‖= |u −v|, u,v ∈ X. Define T and S as T(α)= { 1+2α 2 if 0≤ α ≤ 2 3 ; 5α+2 8 if 2 3 < α ≤ 1. S(α)= { 1−2α if 0≤ α ≤ 2 3 ; α if 2 3 ≤ α ≤ 1. Take a sequence αk as αk = 1 6 − 1 k T(αk)= T( 1 6 − 1 k )= 1+2(1 6 −1 k ) 2 =2 3 S(αk)= S( 1 6 − 1 k )=1−2(1 6 − 1 k )=2 3 Tαk=Sαk= 2 3 ask →∞. Now TS(αk)= GJ( 1 6 − 1 k )=G(1−2(1 6 − 1 k )) =G(2 3 + 2 k )= 3(2 3 )+2 k +1 5 =3 5 as k →∞. STαk = S[T( 1 6 − 1 k )]=S[1−2(1 6 − 1 k )] =S(2 3 + 2 k )=2 3 + 2 k =2 3 as k →∞. limk →∞‖(TSαk −STαk)‖ 6=0 Therefore the pair (T,S) is not compatible. But T(1 6 )=S(1 6 )=(2 3 ). TS(1 6 )=T[1−2(1 6 )]=G(2 3 )=2 3 and ST(1 6 )= J(G(1 6 ))= 2(2 3 )+1 2 = 2 3 . Hence the pair (T,S) weakly compatiable. Example 2.6 Let X = [0,1] is a Banach space with ‖u −v‖= |u −v|, ∀u,v ∈ X. Define four maps G,J,H and I as follows G(α)= I(α)= { α+2 8 if 0≤ α < 1 2 ; 1−α if α ≥ 1 2 . Int. J. Anal. Appl. (2022), 20:2 3 J(α)= H(α)= { α+1 4 if 0≤ α < 1 2 ; 4α−1 2 if α ≥ 1 2 . Here G(0)= J(0)= 1 4 and G(1 2 )= J(1 2 )= 1 2 . Clearly α =0 and α = 1 2 are two coincidence points. If α =0 then GJ(0)= G(1 4 )= (1 4 +2) 8 = 9 32 and JG(0)= J(1 4 )= (1 4 )+1 4 = 5 16 . Therefore GJ(0) 6= JG(0). Now JG(1 2 )= J(1− 1 2 )= J(1 2 )= 4(1 2 )−1 2 = 1 2 GJ(1 2 )= G(1 2 )=1− 1 2 = 1 2 Therefore GJ(1 2 )= JG(1 2 ). Thus G and J are OWC, but not weakly compatible. The following Theorem was proved in metric space [1]. 3. Theorem Let X be a comlete metic space and self mappings G,H,I and J are satisfying (b1) G(X)⊆ H(X) and I(X)⊆ J(X) (b2) d(Gu,Iv)2p ≤ [aφ0(d(Ju,Hv)2p)+(1−a)max{φ1(d(Ju,Hv)2p), φ2(d(Ju,Gv) qd(Hv,Iv)q ′ ),φ3(d(Ju,Iv) rd(Hv,Gv)r ′ ), φ4(d(Ju,Gv) sd(Hv,Gu)s ′ ),φ5(d(Ju,Iv) ld(Hv,Iv)l ′ )}] for all u,v∈X where φj ∈ φ,j =0,1,2,3,4,5and a takes values from0 to1 inclusive and p,q,q ′ , r, r ′ .s,s ′ , l, l ′ takes from 0 exclusive to 1 inclusive such that 2p = q +q′ = r + r ′ = s + s′ = l + l′. (b3) The mapping G or I is a continuous (b4) the pairs (G,J) and (I,H) are weakly compatible mappings. Then G,H,I and J have a unique fixed point which is common. Now we’ll present a list of key lemmas that are useful to our main result. Lemma 3.1 [10] If φj ∈ φ and j ∈ {0,1,2,3,4,5} ,φ is upper semicontinuous and also contractive modulus such that max{φj(t)}≤ φ(t) for all t > 0 and also φ(t) < t for t > 0. Lemma 3.2 [1] Let φj ∈ φ and βj be a non-negative real sequence.If βj+1 ≤ φ(βj) for j ∈ N, then the sequence converges to 0. Now we genealize the existence of the above theorem by extending it on to Banach Space under the following modified conditions. 4 Int. J. Anal. Appl. (2022), 20:2 Theorem 3.3 Let (X,‖.‖) be a Banach Space and self mappings G,H,I and J are satisfying (b1) G(X)⊆ H(X) and I(X)⊆ J(X) (b2) ‖(Gu − Iv)‖2p ≤ [aφ0(‖(Ju −Hv)‖2p)+(1−a)max{φ1(‖Ju −Hv‖2p), φ2(‖Ju −Gu‖q‖Hv − Iv‖q ′ ),φ3(‖Ju − Iv‖r‖Hv −Gu‖r ′ ), φ4( 1 2 ‖Ju −Gu‖s‖Hv − Iv‖s ′ ),φ5( 1 2 ‖Ju − Iv‖l‖Hv − Iv‖l ′ )}] for all u,v∈X where φj ∈ φ,j =0,1,2,3,4,5and a takes values from0 to1 inclusive and p,q,q′, r, r ′,s,s′, l, l′ takes from 0 exclusive to 1 inclusive such that 2p = q +q′ = r + r ′ = s + s′ = l + l′. (b3) Two pairs (G,J) and (I,H) have a coincidence point (b4) the pairs maps (G,J) and (I,H) are OWC. Then G,H,I and J have a unique common fixed point. Proof Using the condition (b1), there is a point u0 ∈ X such that Gu0 = Hu1.For this point u1 ∈ X there exists a point u2 in X such that Iu1 = Ju2 and so on. Continuing this process it is possible to construct a sequence {vj} for j = 1,2,3......in X such that v2j = Gu2j = Hu2j+1,v2j+1 = Iu2j+1 = Ju2j+2 for j ≥ 0. We now prove {vj} is a cauchy sequence. Putting u = u2j and v = v2j+1 in (b2), we get ‖v2j −v2j+1‖2p ≤ [aφ0(‖v2j−1 −v2j‖2p)+(1−a)max{φ1(‖v2j−1 −v2j‖2p), φ2(‖v2j−1 −v2j‖q‖v2j −v2j+1‖q ′ ),φ3(‖v2j−1 −v2j+1‖r‖v2j −v2j‖r ′ ), φ4( 1 2 ‖v2j−1 −v2j‖s‖v2j −v2j‖s ′ ),φ5( 1 2 ‖v2j−1 − Iv2j+1‖l‖v2j −v2j+1‖l ′ )}]. Denote ρj = ‖vj −vj+1‖ (ρ2j) 2p ≤ [aφ0(ρ2j−1)2p)+(1−a)max{φ1(ρ2j−1)2p,φ2((ρ2j−1)q(ρ2j)q ′ ),φ3(0), φ4(0),φ5( 1 2 [(ρ2j−1) l +(ρ2j) l′)](ρ2j) l)}]. (ρ2j) 2p ≤ [aφ0(ρ2j−1)2p)+(1−a)max{φ1(ρ2j−1)2p,φ2((ρ2j−1)q(ρ2j)q ′ ),φ3(0), φ4(0),φ5( 1 2 [(ρ2j−1) l(ρ2j) l +(ρl ′ 2j)(ρ2j) l)])}]. If ρ2j > ρ2j−1 then we have (ρ2j) 2p ≤ [aφ0(ρ2j)2p)+(1−a)max{φ1(ρ2j)2p,φ2(ρ2j)q+q ′ ,φ3(0),φ4(0), φ5( 1 2 [(ρ2j) l+l′(ρ2j) l+l′ +(ρl+l ′ 2j )])}(ρ2j)2p)]. ≤ [aφ0(ρ2j)2p)+(1−a)max{φ1(ρ2j)2p,φ2(ρ2j)2p,φ3(0),φ4(0),φ5(ρ2j)2p)]. Using Lemma (3.2) (ρ2j) 2p ≤ φ(ρ2j)2p < (ρ2j)2p. a contradiction. Int. J. Anal. Appl. (2022), 20:2 5 This implies ρ2j ≤ ρ2j−1 then using this inequality the condition (b2) yields ρ2j ≤ φ(ρ2j−1). (3.1) Similarly taking u = u2j+2 and v = u2j+1 in (b2), we get ‖v2j+1 −v2j+2‖2p ≤ [aφ0(‖v2j −v2j+1‖2p)+(1−a)max{φ1(‖v2j −v2j+1‖2p), φ2(‖v2j+1 −v2j+2‖q‖v2j −v2j+1‖q ′ ),φ3(‖v2j+1 −v2j+1‖r‖v2j −v2j+1‖r ′ ), φ4( 1 2 ‖v2j+11 −v2j+2‖s‖v2j −v2j+2‖s ′ ),φ5( 1 2 ‖v2j+2 − Iv2j+1‖l‖v2j −v2j+1‖l ′ )}]. (ρ2j+1) 2p ≤ [aφ0(ρ2j)2p)+(1−a)max{φ1(ρ2j)2p,φ2((ρ2j+1)q(ρ2j)q ′ ),φ3(0), φ4( 1 2 [(ρ2j+1) s(ρ2j) s′ +(ρ2j+1) s′],φ5(0)}]. (ρ2j+1) 2p ≤ [aφ0(ρ2j)2p)+(1−a)max{φ1(ρ2j)2p,φ2((ρ2j+1)q(ρ2j)q ′ ),φ3(0), φ4( 1 2 [(ρ2j+1) s(ρ2j) s′ +(ρ2j+1) s′(ρ2j+1) s′]),φ5(0)}]. If ρ2j+1 > ρ2j,then we have (ρ2j+1) 2p ≤ [aφ0(ρ2j+1)2p)+(1−a)max{φ1(ρ2j+1)2p,φ2((ρ2j+1)q+q ′ ),φ3(0), φ4(ρ2j+1),φ5(0)}]. Using Lemma(3.2) (ρ2j+1) 2p ≤ φ(ρ2j+1)2p < (ρ2j+1)2p which is a contradiction. Thus we must have ρ2j+1 ≤ ρ2j. Again applying (b2) to the above inequality,we obtain ρ2j+1 ≤ φ(ρ2j). (3.2) From (2.1) and (2.2), in general ρj+1 ≤ φ(ρj), for j=0,1,2,3.... by Lemma 3.3 we get ρj → 0 as j →∞. This shows that ρj = ‖vj −vj+1‖→ 0 as j →∞. Hence {vj} is a cauchy sequence. Since X Banach space, ∃ a point η ∈ X such that vj → η as j →∞. Consequently, the subsequences Gα2j,Jα2j, Iα2j+1 and Hα2j of {vj} also converge to the same point η ∈ X. Since the pair (G,J) is OWC ,there exists u ∈ C(G,J) such that Gu = Ju = η(say) and GJu = JGu = η ′ (say). Hence we have Gη = Jη = η ′ (say) (3.3) Since the pair (I,H) is OWC ,there exists v ∈ C(I,H) such that Iv = Hv = δ(say) and IHv = HIv = δ′(say). 6 Int. J. Anal. Appl. (2022), 20:2 Hence we have Iδ = Hδ = δ ′ (say) (3.4) Now we claim that η ′ = δ′ substitute u = η,v = δ in (b2) ‖(Gη − Iδ)‖2p ≤ [φ0(‖(Jη −Hδ)‖2p)+(1−a)max{φ1(‖Jη −Hδ‖2p), φ2(‖Jη −Gη‖q‖Hδ − Iδ‖q ′ ),φ3(‖Jη − Iδ‖r‖Hδ −Gη‖r ′ ), φ4( 1 2 ‖Jη −Gη‖s‖Hδ − Iδ‖s ′ ),φ5( 1 2 ‖Jη − Iδ‖l‖Hδ − Iδ‖l ′ )}]. Using (3) and (4), we get ‖η′ −δ′‖2p ≤ [φ0(‖η′ −δ′‖2p)+(1−a)max{φ1(‖η′ −δ′‖2p), φ2(‖η′ −η′‖q‖δ′ −δ‖q ′ ),φ3(‖η′ −δ′‖r‖δ′ −η′‖r ′ ), φ4( 1 2 ‖η′ −η′‖s‖δ′ −η′‖s ′ ),φ5( 1 2 ‖η′ −δ′‖l‖δ′ −δ′‖l ′ )}]. ‖η′ −δ′‖2p ≤ [φ0(‖η′ −δ′‖2p)+(1−a)max{φ1(‖η′ −δ′‖2p),φ2(0),φ3(‖η′ −δ′‖2p),φ4(0),φ5(0)}]. Since by Lemma (3.2) ‖η′ −δ′‖2p ≤ φ(‖η′ −δ′‖)2p < ‖η′ −δ′‖2p a contradiction. Therefore η′ = δ′. Hence from (2.3) we get Gη = Jη = δ ′ . (3.5) Next claim that η = δ′ substitute u = u and v = δ in (b2) ‖(Gu − Iδ)‖2p ≤ [φ0(‖(Ju −Hδ)‖2p)+(1−a)max{φ1(‖Ju −Hδ‖2p), φ2(‖Ju −Gu‖q‖Hδ − Iδ‖q ′ ),φ3(‖Ju − Iδ‖r‖Hδ −Gu‖r ′ ), φ4( 1 2 ‖Ju −Gu‖s‖Hδ − Iδ‖s ′ ),φ5( 1 2 ‖Ju − Iδ‖l‖Hδ − Iδ‖l ′ )}]. Using Gu = Ju = η and Iδ = Hδ = δ ′ , we get ‖η−δ′‖2p ≤ [φ0(‖η−δ′‖2p)+(1−a)max{φ1(‖η−δ′‖2p),φ2(‖η−η‖q‖δ−δ‖q ′ ),φ3(‖η−δ′‖r‖δ′−η‖r ′ ), φ4( 1 2 ‖η −η‖s‖δ′ −η‖s ′ ),φ5( 1 2 ‖η −δ′‖l‖δ′ −δ′‖l ′ )}]. ‖η −δ′‖2p ≤ [φ0(‖η −δ′‖2p)+(1−a)max{φ1(‖η −δ′‖2p),φ2(0),φ3(‖η −δ′‖2p),φ4(0),φ5(0)}]. By Lemma(3.2) ‖η −δ′‖2p ≤ φ(‖η −δ′‖)2p < ‖η −δ′‖2p which is a contradiction, and hence η = δ′. From (2.5),we get Gη = Jη = η (3.6) and Iδ = Hδ = η. (3.7) Now we claim that η = δ. Again in (b2) putting u = η and v = v ‖Gη − Iv‖2p ≤ [φ0(‖(Jη −Hv)‖2p)+(1−a)max{φ1(‖Jη −Hv‖2p), Int. J. Anal. Appl. (2022), 20:2 7 φ2(‖Jη −Gη‖q‖Hv − Iv‖q ′ ),φ3(‖Jη − Iv‖r‖Hv −Gη‖r ′ ), φ4( 1 2 ‖Jη −Gη‖s‖Hv − Iv‖s ′ ),φ5( 1 2 ‖Ju − Iv‖l‖Hv − Iv‖l ′ )}]. Using Gη = Jη = η and Iv = Hv = δ ‖η−δ‖2p ≤ [φ0(‖η−δ‖2p)+(1−a)max{φ1(‖η−δ‖2p),φ2(‖η−η‖q‖δ−δ‖q ′ ),φ3(‖η−δ‖r‖η−δ‖r ′ ), φ4( 1 2 ‖η −η‖s‖δ −η‖s ′ ),φ5( 1 2 ‖η −δ‖l‖δ −δ‖l ′ )}]. ‖η −δ‖2p ≤ [φ0(‖η −δ‖2p)+(1−a)max{φ1(‖η −δ‖2p),φ2(0),φ3(‖η −δ‖2p),φ4(0),φ5(0)}]. By Lemma(3.2) ‖η −δ‖2p ≤ φ(‖η −δ‖)2p < ‖η −δ‖2p which is a contradiction. Therefore η = δ. From (2.7), we get Iη = Hη = η. (3.8) From (2.6 and (2.8), we get Gη = Jη = Iη = Hη = η. Hence this gives that η is a common fixed point for G, H, I and J. For Uniqueness: Suppose η and η∗ (η 6= η∗) are two common fixed points. Then put u = η and v = η∗ in the inequality (b2) ‖Gη − Iη∗‖2p ≤ [aφ0(‖Jη −Hη∗‖2p)+(1−a)max{φ1(‖Jη −Hη∗‖2p), φ2(‖Jη −Gη‖q‖Hη∗ − Iη∗‖q ′ ),φ3(‖Jη − Iη∗‖r‖Hη∗ −Gη‖r ′ ), φ4( 1 2 ‖Jη −Gη‖s‖Hη∗ −Gη‖s ′ ),φ5( 1 2 ‖Jη − Iη∗‖l‖Hη∗ − Iη∗‖l ′ )}] ‖η−η∗‖2p ≤ [aφ0(‖η−η∗‖2p)+(1−a)max{φ1(‖η−η∗‖2p),φ2(0),φ3(‖η−η∗‖2p),φ4(0),φ5(0)}] ‖η −η∗‖2p ≤ [φ(‖η −η∗‖2p) < ‖η −η∗‖2p a contradiction. Which gives η = η∗, this proves the uniqueness. Example Now we continue to discuss the Example (2.6) to justyfy our Theorem(3.3). Now G(X)=I(X)=[1 4 , 5 16 )∪ (1 2 ) while J(X)=H(X)= [1 4 , 3 8 )∪ (1 2 ) clearly G(X)⊆ H(X), I(X)⊆ J(X) therefore (b1) is satisfied. Now we verify the condition (b2) Case(i). If u,v ∈ [0, 1 2 ),then we have ‖(Gu − Iv)‖= |Gα− Iv| put u = 1 3 ,v = 1 5 . Then the inequality (b2) implies ‖G(1 3 )− I(1 5 )‖2p ≤ [aφ0(‖J(13)−H( 1 5 ))‖2p)+(1−a)max{φ1(‖J(13)−H( 1 5 )‖2p), 8 Int. J. Anal. Appl. (2022), 20:2 φ2(‖J(13)−G( 1 3 )‖q‖H(1 5 )− I(1 5 )‖q ′ ),φ3(‖J(13)− I( 1 3 )‖r‖H(1 5 )−G(1 3 )‖r ′ ), φ4( 1 2 ‖J(1 3 )−G(1 3 )‖s‖H(1 5 )− I(1 5 )‖s ′ ),φ5( 1 2 ‖J(1 3 )− I(1 5 )‖l‖H(1 5 )− I(1 5 )‖l ′ )}] for a = 1 2 and p = q = q′ = r = r ′ = s = s′ = l = l′ = 1 2 ‖0.0166‖≤ [1 2 φ0‖(0.033)‖+(1− 12)max{φ1‖(0.01665)‖, φ2‖(0.0322)‖,φ3‖(0.01870)‖,φ4‖(0.0161)‖,φ5‖(0.019)}] |0.0166| < |0.0327|. Hence the inequality (b2) holds. Also the verification in the remaining intervals is also simple. Here it is evident that 1 2 is the unique common fixed point for the four self mappings. 4. Conclusion In this paper we proved a common fixed point theorem on Banach space using OWC mappings.It is also clear from the example proved that the mappings are neither compatible nor weakly compatible but OWC mappings.Moreover the continuity condition is dropped.Hence we conclude that our Theo- rem stands as an improvement of Theorem (3). Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] H.K. Pathak, M.S. Khan, R. Tiwari, A common fixed point theorem and its application to nonlinear integral equations, Computers Math. Appl. 53 (2007), 961–971. https://doi.org/10.1016/j.camwa.2006.08.046. [2] S. Sharma, B. Deshpande, A. Pandey, Common fixed point point theorem for a pair of weakly compatible mappings on Banach spaces, East Asian Math. J. 27 (2011), 573–583. https://doi.org/10.7858/eamj.2011.27.5.573. [3] A. Djoudi, L. Nisse, Greguš type fixed points for weakly compatible maps, Bull. Belg. Math. Soc. Simon Stevin. 10 (2003), 369–378. https://doi.org/10.36045/bbms/1063372343. [4] S. Sharma, B. Deshpande, Common fixed point theorems for finite number of mappings without continuity and compatibility on intuitionistic fuzzy metric spaces, Chaos Solitons Fractals. 40 (2009), 2242–2256. https://doi. org/10.1016/j.chaos.2007.10.011. [5] R.A. Rashwan, A common fixed point theorem in uniformly convex Banach spaces. Ital. J. Pure Appl. Math. 3 (1998), 117–126. [6] N. Shahzad, S. Sahar, Fixed points of biased mappings in complete metric spaces, Radovi Math. 11 (2002), 249-261. [7] S. Sharma, P. Tilwankar, Some fixed point theorems in intuitionistic fuzzy metric spaces, Tamkang J. Math. 42 (2011), 405–414. https://doi.org/10.5556/j.tkjm.42.2011.683. [8] V. Srinivas, A result on Banach space using property E.A, Indian J. Sci. Technol. 13 (2020), 4490–4499. https: //doi.org/10.17485/IJST/v13i44.1909. [9] V. Srinivas, T. Thirupathi, A result on Banach space using E.A like property, Malaya J. Mat. 8 (2020), 903–908. https://doi.org/10.26637/MJM0803/0029. [10] H.K. Pathak, S.N. Mishra, A.K. Kalinde, Common fixed point theorems with applications to nonlinear integral equations, Demonstr. Math. 32 (1999), 547-564. https://doi.org/10.1515/dema-1999-0310. https://doi.org/10.1016/j.camwa.2006.08.046 https://doi.org/10.7858/eamj.2011.27.5.573 https://doi.org/10.36045/bbms/1063372343 https://doi.org/10.1016/j.chaos.2007.10.011 https://doi.org/10.1016/j.chaos.2007.10.011 https://doi.org/10.5556/j.tkjm.42.2011.683 https://doi.org/10.17485/IJST/v13i44.1909 https://doi.org/10.17485/IJST/v13i44.1909 https://doi.org/10.26637/MJM0803/0029 https://doi.org/10.1515/dema-1999-0310 1. Introduction 2. Preliminaries 3. Theorem 4. Conclusion References