A Simple Observation Concerning Contraction Mappings

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1 Revista Colombiana de Matemáticas Volumen 46202)2, páginas A Simple Observation Concerning Contraction Mappings Una simple observación acerca de las contracciones German Lozada-Cruz a Universidade Estadual Paulista, São José do Rio Preto, Brasil Abstract. In this short note we show that the results obtained by Walter in 4] remain valid if we change the metric σ by another metric. Furthermore, if we use the norm T,ɛ given in 3], Theorem B in4] remains valid. Key words and phrases. Contraction, contraction principle, fixed point. 200 Mathematics Subject Classification. 47H09, 47H0. Resumen. En esta breve nota se muestra que los resultados obtenidos por Walter en 4] siguen siendo válidos si se cambia la métrica σ por otra. Además, si se utiliza la norma T,ɛ usada en 3], el Teorema B en 4] sigue siendo válido. Palabras y frases clave. Contracción, principio de la contracción, punto fijo.. Introduction The main motivation of this note was the paper by W. Walter 4]. Thus, we consider X, ϱ) a metric space and T : X X a nonlinear map. We say that T is Lipschitz continuous if there exists α 0 such that ϱt x, T y) αϱx, y), x, y X, and if in addition 0 α <, the map T is called a contraction. The aim of this short note is to prove the following propositions and make some remarks about them. a Partially supported by FAPESP, Grant: 09/ , Brazil. 229

2 230 GERMAN LOZADA-CRUZ Proposition. Let X, ϱ) be a metric space and T : X X a map such that, for a fixed n N, T n satisfies ϱ T n x, T n y ) α n ϱx, y) for x, y X. ) Then the function ζ defined by ζx, y) := ϱ 2 x, y) + α 2 ϱ2 T x, T y) + + α ϱ2 T n x, T n y ) ] /2 2) 2n ) is a metric on X, and T satisfies ζt x, T y) αζx, y) for x, y X. 3) Moreover, there exist positive constants a, b such that aϱx, y) ζx, y) bϱx, y) 4) if and only if T is Lipschitz continuous with respect to ϱ. Proof. It is not difficult to see that ζ is a metric on X and ϱx, y) ζx, y) for all x, y X. Now, using the definition of ζ we get ζt x, T y) = ϱ 2 T x, T y) + α 2 ϱ2 T T x), T T y) ) + = + α ϱ2 T n T x), T n T y) ) ] /2 2n ) ϱ 2 T x, T y) + α 2 ϱ2 T 2 x, T 2 y) + + α ϱ2 T n x, T n y ) 2n 2) ] /2 + α ϱ2 T n x, T n y ) 2n ) ϱ 2 T x, T y) + α 2 ϱ2 T 2 x, T 2 y) + + α ϱ2 T n x, T n y ) 2n 2) + α2n α 2n ) ϱ2 x, y) α 2 ϱ 2 x, y) + α 2 ϱ2 T x, T y) + + ] /2 α ϱ2 T n x, T n y ) )] /2 2n ) = αζx, y), x, y X, where in the last inequality we have used ). Hence 3) is proved. Also, if ζx, y) bϱx, y), it is not difficult to show that T is Lipschitz continuous with respect to ϱ. In fact, ϱt x, T y) ζt x, T y) αζx, y) αbϱx, y), for all x, y X. Volumen 46, Número 2, Año 202

3 A SIMPLE OBSERVATION CONCERNING CONTRACTION MAPPINGS 23 Conversely, if T is Lipschitz continuous, then the powers of T are also Lipschitz continuous. then If we assume that ϱt k x, T k y) a k ϱx, y), x, y X, k =, 2,..., n, 5) ϱx, y) ζx, y) bϱx, y), for x, y X 6) where b = + a α + + a n α n. To get the last inequality we use the right side of 2) and 5). Proposition 2. Let X, ) be a Banach space and A LX) such that A m = α m. Then the formula x ζ := x 2 + ) /2 α 2 Ax α 2n ) An x 2 defines a norm on X equivalent to the original norm, and for the norm of A, A ζ, we have the inequality A ζ α. Proof. It is not difficult to see that ζ is a norm on X and x x ζ b x for all x X, i.e., the norms and ζ are equivalent. On the other hand, Ax ζ = Ax 2 + α 2 A2 x Ax 2 + α 2 A2 x α 2 x + α 2 A2 x = α x ζ. ) /2 α 2n ) An x 2 α 2n 2) An x 2 + α 2n ) An x 2 α ]) /2 ) /2 α2n x 2 2n ) This proves that A ζ α. 2. Some Remarks Remark 3. Proposition is the same as Proposition A in 4], where we change the metric σ by the metric ζ. Also, we can see that ζx, y) σx, y) for all x, y X. 7) The same applications given in 4] such as Contraction principle, Continuous dependence and Approximate iteration can also be obtained changing the metric σ by ζ. As an example, it is well known that if X, ϱ) is a complete metric space and T : X X is a contraction then there exists an unique x X Revista Colombiana de Matemáticas

4 232 GERMAN LOZADA-CRUZ such that T x = x. This is called the contraction principle or the Banach fixed point theorem. For details on contraction principle see, p.20]. One way to find the fixed point x is: given x 0 X arbitrary, the sequence {x n } X given by { x 0 X, x n = T n 8) x 0, n = 0,, 2,... converges to x. The recursion formula given in 8) is known as the sucessive approximations method to find the fixed point x. Moreover, we have a priori error estimate ϱx n, x) and a posteriori error estimate ϱx n+, x) and, we have the rate of convergence αn α ϱx 0, x ), n = 0,, 2,..., 9) α α ϱx n, x n+ ), n = 0,, 2,..., 0) ϱx n+, x) αϱx n, x), n = 0,, 2,.... ) Now, if T is a map such that, for some n N, T n is a contraction with constant α n < and T satisfies the hypothesis of Proposition then from 3), we have that T is a contraction with respect to ζ with constant α. Thus, the inequalities 9), 0) and ) remain valid if we change the metric ϱ by the metric ζ. For numerical implementation it is important to know the number of iterations, N, to get a good approximation of the fixed point. Setting d = ϱx, T x) and using the a priori error estimate 9), we have a lower bound for N given by lnɛ) + ln α) ln d N >, ln K thus we have ϱx n, x) < ɛ, ɛ > 0. For more details see 2]. Remark 4. Proposition 2 is the same as Proposition B in 4] where we change the norm by the norm ζ. Also, we can easily see that x ζ x for all x X. Remark 5. The norm ζ is the same norm T,ɛ given in 3, p. 32]. If we use the norm given in 4] which is equivalent to the norm ζ, the main result Theorem ) in 3] is still valid. Volumen 46, Número 2, Año 202

5 A SIMPLE OBSERVATION CONCERNING CONTRACTION MAPPINGS 233 References ] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Society for Industrial and Applied Mathematics SIAM), Philadelphia, PA, 2000, Reprint of the 970 original. Classics in Applied Mathematics, 30. 2] R. S. Palais, A Simple Proof of the Banach Contraction Principle, J. Fixed Point Theory Appl ), no. 2, ] H. M. Rodrigues and J. Solà-Morales, A Note on the Relationship Between Spectral Radius and Norms of Bounded Linear Operators, Mat. Contemp ), ] W. Walter, A Note on Contraction, SIAM Rev ), no., 07. Recibido en julio de 202. Aceptado en octubre de 202) Departamento de Matemática IBILCE - Instituto de Biociências, Letras e Ciências Exatas UNESP- Universidade Estadual Paulista , São José do Rio Preto, SP, Brazil german@ibilce.unesp.br Revista Colombiana de Matemáticas