Generalization of the Banach Fixed Point Theorem for Mappings in (R, ϕ)-spaces

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1 International Mathematical Forum, Vol. 10, 2015, no. 12, HIKARI Ltd, Generalization of the Banach Fixed Point Theorem for Mappings in (R, ϕ)-spaces Andrzej Mach Faculty of Engineering and Economics Department of Informatics The State Higher School of Vocat. Education Narutowicza 9, Ciechanów, Poland Copyright c 2015 Andrzej Mach. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The paper includes theorem giving the sufficient condition to the existence of a fixed point for mappings in arbitrary set equipped with the family of binary reflexive and symmetric relations satisfying some conditions. Mathematics Subject Classification: Primary 47H10; Secondary 54H25, 55M20 Keywords: fixed point, binary relation, ϕ-transitivity, contraction, R- contraction, Banach fixed point theorem, (R, ϕ)-space Introduction If (X, d) is a complete metric space, then by contraction ([1], [2]) we understand a mapping F : X X for which there exists L ]0, 1[ such that d(f x, F y) L d(x, y), for all x, y X. The well known Banach Fixed Point Theorem formulated for (X, d) ([1], [2]) reads as follows. If F : X X is a contraction, then there exists exactly one fixed point, i.e. the solution of F x = x. The Banach Fixed Point Theorem is an important tool in mathematical analysis and has been investigated under various conditions and developed in different directions ([3], [4], [5], [6], [7]). In the presented paper we obtain a fixed point

2 580 Andrzej Mach theorem for arbitrary set X with a family of binary reflexive and symmetric relations in X forming a chain satisfying some conditions. 1 Notations, definitions, lemma Let X, T be the arbitrary sets and ϕ : T T be a fixed bijection. Moreover, let R = {R t } t T be a family of binary reflexive and symmetric relations in X forming a chain such that t T R t = X X, t T R t = I, where I = {(x, x) : x X}. Additionally we suppose that the family R satisfies the following condition of ϕ-transitivity ] t T x, y, z X : [(x, y) R t, (y, z) R t (x, z) R ϕ(t). The examples of such families are given in the next part of the paper. Definition 1.1. A set X with the family of relations described above will be called in the sequel by (R, ϕ)-space. Below is defined the notion of convergent sequence in (R, ϕ)-space X. Definition 1.2. We say that a sequence (x n ) n N of elements of (R, ϕ)-space X is R-convergent to x 0 X in X, which is denoted by lim n x R n = x 0 (or R x 0 ), if and only if x n t T N(t) N n N(t) : (x n, x 0 ) R t. In this case x 0 will be called R-limit. Lemma 1.3. Each sequence in (R, ϕ)-space X has at most one R-limit. Proof. Let us suppose ad absurdum that a sequence (x n ) n N has two distinct R-limits x 0 x 0. Therefore there exists t T for which (x 0, x 0) / R t. Indeed, in the opposite case, by supposition t T R t = I we would get (x 0, x 0) I, so the contradiction x 0 = x 0. Let us take t 0 T such that ϕ(t 0 ) = t. By R-convergence of the sequence (x n ) n N to x 0 and to x 0 we have (x n, x 0 ) R t0 and (x n, x 0) R t0, for sufficiently large n N. Hence, by supposition of ϕ-transitivity, we get the contradiction (x 0, x 0) R ϕ(t0 ) = R t. Definition 1.4. A sequence (x n ) n N of elements of (R, ϕ)-space X is R- Cauchy sequence if and only if t T N(t) N n, m N(t) : (x n, x m ) R t.

3 Generalization of the Banach fixed point theorem Definition 1.5. An (R, ϕ)-space X is called complete if and only if every R-Cauchy sequence is R-convergent in X. The next definition gives a generalization of the notion of classical contraction. Definition 1.6. For (R, ϕ)-space X, a mapping f : X X is called R- contraction if the following condition is satisfied for all x, y X [ )] t T : (x, y) R t ( t T : (f(x), f(y)) R t and R t R t. Definition 1.7. An (R, ϕ)-space X is called strong if and only if the intersection of every sequence (R tn ) n N of shrinking relations equals I and t 1, t 2 T : R t1 R t2 R ϕ(t1 ) R ϕ(t2 ) (1.1) and for every n N, n 3, for all t 1, t 2,..., t n 1 T if R tn 1 R tn 2... R t1 then (x 1, x 2 ) R t1, (x 2, x 3 ) R t2,..., (x n 1, x n ) R tn 1 (x 1, x n ) R ϕ(t1 ) (1.2) for all x 1,..., x n X. 2 Main theorem Theorem 2.1. Let X be strong and complete (R, ϕ)-space and let the mapping f : X X be an R-contraction. The above suppositions imply the existing of exactly one solution of f(x) = x. Then f has exactly one fixed point. Proof. Choose x 0 X. Define the iterative sequence x n = f(x n 1 ), for n N, so x 1 = f(x 0 ), x 2 = f(x 1 ) = f 2 (x 0 ),..., x n 1 = f(x n 2 ) = f n 1 (x 0 ), x n = f(x n 1 ) = f n (x 0 ),.... Let n, m N, n < m. We have (x n, x m ) = (f n (x 0 ), f m (x 0 )). Let us assume that (x 0, x 1 ) R t0. Since the mapping f is R-contraction then we have (x 1, x 2 ) = (f(x 0 ), f(x 1 )) R t1 and R t1 R t0, Continuing (x 2, x 3 ) = (f(x 1 ), f(x 2 )) R t2 and R t2 R t1. (x n 1, x n ) = (f(x n 2 ), f(x n 1 )) R tn 1 and R tn 1 R tn 2,

4 582 Andrzej Mach (x n, x n+1 ) = (f(x n 1 ), f(x n )) R tn and R tn R tn 1,... (x m 1, x m ) = (f(x m 2 ), f(x m 1 )) R tm 1 and R tm 1 R tm 2, From the above by (1.2) (x n, x m ) R ϕ(tn), which means -by (1.1)- that (x n ) n N is an R-Cauchy sequence. Let x be its limit. We have t T : (x n, x ) = (f(x n 1 ), x ) R t, for all sufficiently large n N. Similarly, t T : (x n 1, x ) R t, for all sufficiently large n N and since f is a R-contraction then we have also t T : (f(x n 1 ), f(x )) R t, for all sufficiently large n N. From the above, by ϕ-transitivity t T : (x, f(x )) R ϕ(t), and considering that t T R ϕ(t) = I we get f(x ) = x. To prove that this is the only fixed point remark that for two fixed point we have (x 1, x 2) = (f(x 1), f(x 2)) = (f 2 (x 1), f 2 (x 2)) = (f 3 (x 1), f 3 (x 2)) =... (f 2, f 3,... denote iteration of f) and by Definition 1.6 we observe easily that (x 1, x 2) I, so x 1 = x 2. 3 Examples, Remark Example 3.1. Let (X, d) be a complete metric space. Define T := R + and ϕ : T T, ϕ(t) := 2t. t T : R t := {(x, y) X 2 : d(x, y) t}. The set X forms the complete (R, ϕ)-space. d(x, y) = x y we have Particularly, if X := R and t T : R t := {(x, y) R 2 : x y t}. (3.1)

5 Generalization of the Banach fixed point theorem Example 3.2. Let X := R, T := {..., 2 n,..., 64, 32, 16, 8, 4, 2, 1, 1, 1, 1,..., 1,...} n We put ϕ : T T as ϕ(t) := 2t. Let us define the family R = {R t } t T of reflexive and symmetric binary relations in X as in (3.1). The set X forms the complete and strong (R, ϕ)-space. Example 3.3. For (R, ϕ)-space presented in example 3.2 we define the mapping f : X X as follows f(x) = 1 2 x. One can observe easily that the mapping f is R-contraction. The sequence x 0 := 1; x n = f(x n 1 ), for n N, so 1, 1 2, 1 4, 1 8,... is R-convergent in X and its limit equals 0. We have f(0) = 0. This conclusion is perhaps trivial by classical Banach Fixed Point Theorem but it is confirmed here without using of metric. Example 3.4. Let X := {f : R R} be the set of continuous functions such that f(x) = 0 for x U, where U is a neighborhood of zero. Let T := {..., 2 n,..., 64, 32, 16, 8, 4, 2, 1, 1 2, 1 4, 1 8,..., 1 2 n,...}. We put ϕ : T T as ϕ(t) := t. Let us define the family R = {R t } t T binary reflexive and symmetric relations in X as follows of t T : R t := {(f, g) X 2 : x [ t, t] f(x) = g(x)}. The set X forms the strong (R, ϕ)-space. One can observe that the mapping Φ : X X defined as follows ( 1 ) Φ(f)(x) := f 2 x is R-contraction. Let us consider the following sequence 0, for x [ 1, 1], f 0 (x) = x 1, for x > 1, x 1, for x < 1, 0, for x [ 2, 2], 1 f 1 (x) = Φ(f 0 )(x) = x 1, for x > 2, 2 x 1, for x < 2, 1 2

6 584 Andrzej Mach 0, for x [ 4, 4], 1 f 2 (x) = Φ(f 1 )(x) = x 1, for x > 4, 4 1 x 1, for x < 4, , for x [ 2 n, 2 n ], 1 f n (x) = Φ(f n 1 )(x) = x 1, for x > 2 n, 2 n 1 x 1, for x < 2 n, 2 n... By theorem 2.1 the R-limit of this sequence is f 0 and Φ(f ) = f. Remark 3.5. It seems to be interesting to consider fixed point theorem in spaces with only relations and without metric. Maybe it will be possible using of such result not only in mathematics, physics, but also, for example, in medicine, sociology, economics, psychology and others. Let us imagine a set of objects. The nature of these objects can be arbitrary, may be humans, bacteria, molecules,... Suppose that every such object is characterized by a set of features, genes, properties,... There are relations (between these objects) determined by its features. Perhaps, it will be possible to conclude using proved above result that there exists an object which is fixed with respect to a transformation. Let me suggest to the readers as the problem - find examples of (R, ϕ)-spaces and applications of the proved theorem. References [1] S. Banach, Sur les opération dans l ensembles abstraits et leur application aux équations intégrales, Fundam. Math., 3 (1922), [2] J. Dugundji and A. Granas, Fixed Point Theory, Monografie Matematyczne, Tom 61 vol. I, PWN-Polish Scientific Publishers, [3] S. K. Chatterjea, Fixed-points theorems, C. R. Acad. Bulgare Sci., 25 (1972), [4] J. Jachymski, Common fixed point theorems for some families of mappings, Indian J. Pure Appl. Math., 25 (1994), [5] K. Jha, R. P. Pant, S. L. Singh, Common fixed points for comapatible mappings in metric spaces, Radovi Matematički, 12 (2003), [6] R. Kannan, Some results on fixed points II, Am. Math. Mon., 76 (1969), no. 4,

7 Generalization of the Banach fixed point theorem [7] R. P. Pant, P. C. Joshi, V. Gupta, A Meir-Keeler type fixed point theorem, Indian J. Pure Appl. Math., 32 (2001), no. 6, Received: August 21, 2015; Published: October 23, 2015