Common fixed points of biased maps of type (A) and applications

International Journal of Mathematics and Mathematical Sciences, Aug 2018

A generalization of compatible maps of type (A) called “biased maps of type (A)” is introduced and used to prove fixed point theorems for certain contractions of four maps. Extensions of known results are thereby obtained, i.e., the results of Pathak, Prasad, Jungck et al. are improved. Some problems on convergence of self-maps and fixed points are also discussed. Further, we use our main results to show the existence of solutions of nonlinear integral equations.

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Common fixed points of biased maps of type (A) and applications

Internat. J. Math. & Math. Sci. VOL. COMMON FIXED POINTS OF BIASED MAPS OF TYPE (A) AND APPUCATIONS 0 S.M. KANG Department of Mathematics and Research Institute of Natural Science Gyeongsang National University Chinju 660-701 , KOREA 1 Y.J. CHO Department of Mathematics and Research Institute of Natural Science Gyeongsang National University Chinju 660-701 , KOREA 2 H.K. PATHAK Department of Mathematics Kalyan Mahavidyalaya Bhilai Nagar (M. P.) 490 006 , INDIA A generalization of compatible maps of type (A) called "biased maps of type (A)" is introduced and used to prove fixed point theorems for certain contractions of four maps. Extensions of known results are thereby obtained, i.e., the results of Pathak, Prasad, Jungck et al. are improved. Some problems on convergence of self-maps and fixed points are also discussed. Further, we use our main results to show the existence of solutions of nonlinear integral equations. AND PHRASES; Compatible maps and compatible maps of type (A); A-biased and S-biased maps; weakly A-biased and S-biased maps; common fixed point; simultaneous VolterraHammerstein equation - O, d(ASx.,SSx.) 0 whenever {x.} is a sequence in X such that Ax. and Sx. 6 X. Compatible maps of type (A) are, in fact, equivalent to the concept of compatible maps under continuity of maps ([ 12 ]). Recall that self-maps A and S of X are said to be compatible ([ 6 ]) if d(ASx,, SAx, 0 whenever {x. } is a sequence in X such that Ax., Sx. 6 X. It may be remarked that compatible maps were introduced in [ 6 ] as a generalization of commuting maps and weakly commuting maps ([ 21 ]) and have been proved a sharper tool for obtaining more comprehensive fixed point theorems ([ 1 ]-[ 9 ], [ 14 ], [ 15 ], [ 19 ], [ 20 ]) and in the study of periodic points [ 10 ]. We now introduce the concept of biased maps of type (A). Our new concept is an appreciable generalization of compatible maps of type (A) which, as we shall see, proves useful in the "fixed point" arena. Further, we use our main results to show the existence of solutions of nonlinear integral equations. 2. BIASED MAPS OF TYPE (A) In this section, we show that the concepts of biased maps of type (A) is a legitimate generalization of compatible maps of type (A) and give several properties of biased maps of type (A) for our main results. Definition 2.1. Let A and S be self-maps of a metric space (X,d). The pair {A,S} is said to be S-biased and A-biased of type (A), respectively, if, whenever {z } is a sequence in X and Az, Sx --,rEX, ad(SSz, Axe) <_ ad(ASz, Sz), ad(AAx,Sx) <_ ad(SAx,Ax), (Sb) (Ab) respectively, where a lira inf and if a lira sup_. Of course, if the inequality in (Sb) (or (Ab)) holds with a lim. (one has to presuppose that the indicated limit exists), then liminf,_ limsup_ lirn_ and (Sb) (or (Ab)) is satisfied. We shall frequently use this fact in our further discussion. As to notations, we shall use N, R+, Q, L and I to denote the positive integers, non-negative real numbers, the rational numbers, the irrational numbers, and [ 0,1 ], respectively. The following example shows why we could not restrict a to "lim_." if the concept of biased maps of type (A) is to generalize compatibility of type (A). Example 2.2. Let X I and define mappings A, S" X X by Ax=Sx= 0 v [0,1/2] for x eQN (],1] e Example 2.4. Let X );by Let x2 and x2_ for n N. Then Sx 1 as k or, SSx2 O, SSz2_, 1, and, therefore, lim_. d(SSx, Sx) does not exist although lim. d(SSx, SSx) 0. In fact, the pair {S,S} is trivially compatible of type (A) for any function S. Be assured that the concept of "biased maps of type (A)" arises naturally in the context of contractive or relatively nonexpansive maps ([ 8 ]). See also Proposition 2.5 below. Remark 2.3. If the pair {A,S} is compatible of type (A), then it is both S-biased and A-biased of type (A). From d(SSx,Ax) <_ d(SSx,ASz) + d(ASx,Sx) + d(Sx,,Ax), d(AAx,Sx,) <_ d(AAx,SAx) + d(SAx,Ax) + d(Ax,Sx) for n N, it follows that ad(SSx, Axe) <_ 0 + ad(ASx, Sx) + O, . ad(AAx,Sx) <_ O+ ad(,qAx,Ax) + O, respectively, if Axe, Sx X, {A,S} is a compatible pair of type (A) and a is either liminf.. or lira sup.... Therefore, the pair {A, S} is both S-biased and A-biased of type (A). However, the converse of Remark 2.3 is not necessarily true. To this end, consider the following example: [0, x) with the usual metric d(x, y) Ix- Yl. Define mappings A, S" X Sx [1, :), respectively. Then A and S are not continuous at z 1. Now, we assert that the pair {A, S} is not compatible of type (A), but it is S-biased and A-biased of type (A). To show this, we first note that Ax,Sx E X ifft= I andx --0+.Then, ifAx,Sx 1, it follows that Ax l+z 1+ and Sx 1- x 1-. Thus, since 1 + x. > 1 and 1- x. < 1 for all n E N, we have ASx 2SAx 2, SSx x. and AAx 1. Thus 1 alSSx-Ax. <_ alASx.-Sx.] 1 and 0 a[AAx. Sxl < a[SAx Ax.[ 1. Therefore, the pair {A, S} is S-biased and A-biased of type (A), but it is not compatible of type (A). The (...truncated)


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H. K. Pathak, Y. J. Cho, S. M. Kang. Common fixed points of biased maps of type (A) and applications, International Journal of Mathematics and Mathematical Sciences, 21, DOI: 10.1155/S0161171298000945