Existence and data dependence for multivalued weakly Ćirić-contractive operators

Transcription

1 Acta Univ. Sapientiae, Mathematica, 1, 2 (2009) Existence and data dependence for multivalued weakly Ćirić-contractive operators Liliana Guran Babeş-Bolyai University, Department of Applied Mathematics, Kogălniceanu 1, , Cluj-Napoca, Romania. gliliana.math@gmail.com Adrian Petruşel Babeş-Bolyai University, Department of Applied Mathematics, Kogălniceanu 1, , Cluj-Napoca, Romania. petrusel@math.ubbcluj.ro Abstract. In this paper we define the concept of weakly Ćirić-contractive operator and give a fixed point result for this type of operators. Then we study the data dependence for the fixed point set. 1 Introduction Let (X, d) be a metric space. A singlevalued operator T from X into itself is called contractive if there exists a real number r [0, 1) such that d(t(x), T(y)) rd(x, y) for every x, y X. It is well known that if X is a complete metric space, then a contractive operator from x into itself has a uniue fixed point in X. In 1996, Japanese mathematicians O. Kada, T. Suzuki and W. Takahashi introduced the w-distance (see [4]) and discussed some properties of this new distance. Later, T. Suzuki and W. Takahashi, starting by the definition above, gave some fixed points result for a new class of operators, weakly contractive operators (see [8]). The purpose of this paper is to give a fixed point theorem for a new class of operators, namely the so-called weakly Ćirić-contractive operators. Then, we present a data dependence result for the fixed point set. AMS 2000 subject classifications: 47H10, 54H25 Key words and phrases: w-distance, weakly Ćirić-contraction, fixed point, multivalued operator 151

2 152 L. Guran, A. Petruşel 2 Preliminaries Let (X, d) be a complete metric space. We will use the following notations: P(X) - the set of all nonempty subsets of X; P(X) = P(X) P cl (X) - the set of all nonempty closed subsets of X; P b (X) - the set of all nonempty bounded subsets of X; P b,cl (X) - the set of all nonempty bounded and closed, subsets of X; For two subsets A, B P b (X), we recall the following functionals: D : P(X) P(X) R +,D(Z, Y) = inf{d(x, y) : x Z, y Y}, Z X the gap functional. δ : P(X) P(X) R +, δ(a, B) := sup{d(a, b) x A, b B} the diameter functional; ρ : P(X) P(X) R +, ρ(a, B) := sup{d(a, B) a A} the excess functional; H : P(X) P(X) R +, H(A, B) := max{sup inf d(a, b), sup inf d(a, b)} a A b B the Pompeiu-Hausdorff functional; Fix F := {x X x F(x)} the set of the fixed points of F; b B a A The concept of w-distance was introduced by O. Kada, T. Suzuki and W. Takahashi (see [4]) as follows: Let (X,d) be a metric space, w : X X [0, ) is called w-distance on X if the following axioms are satisfied : 1. w(x, z) w(x, y) + w(y, z), for any x, y, z X; 2. for any x X : w(x, ) : X [0, ) is lower semicontinuous; 3. for any ε > 0, there exists δ > 0 such that w(z, x) δ and w(z, y) δ implies d(x, y) ε. Let us give some examples of w-distances (see [4]). Example 1 Let (X, d) be a metric space. Then the metric d is a w-distance on X. Example 2 Let X be a normed liniar space with norm. Then the function w : X X [0, ) defined by w(x, y) = x + y for every x, y X is a w- distance.

3 Existence and data dependence 153 Example 3 Let (X,d) be a metric space and let g : X X a continuous mapping. Then the function w : X Y [0, ) defined by: for every x, y X is a w-distance. w(x, y) = max{d(g(x), y), d(g(x), g(y))} For the proof of the main results we need the following crucial result for w-distance (see [8]). Lemma 1 Let (X, d) be a metric space, and let w be a w-distance on X. Let (x n ) and (y n ) be two seuences in X, let (α n ), (β n ) be seuences in [0,+ [ converging to zero and let x, y, z X. Then the following holds: 1. If w(x n, y) α n and w(x n, z) β n for any n N, then y = z. 2. If w(x n, y n ) α n and w(x n, z) β n for any n N, then (y n ) converges to z. 3. If w(x n, x m ) α n for any n, m N with m > n, then (x n ) is a Cauchy seuence. 4. If w(y, x n ) α n for any n N, then (x n ) is a Cauchy seuence. 3 Existence of fixed points for multivalued weakly Ćirić-contractive operators At the beginning of this section let us define the notion of multivalued weakly Ćirić-contractive operators. Definition 1 Let (X, d) be a metric space and T : X P(X) a multivalued operator. Then T is called weakly Ćirić-contractive if there exists a w-distance on X such that for every x, y X and u T(x) there is v T(y) with w(u, v) max{w(x, y), D w (x, T(x), D w (y, T(y)), 1 2 D w(x, T(y))}, for every [0, 1). Let (X, d) be a metric space, w be a w-distance on X x 0 X and r > 0. Let us define: B w (x 0 ; r) := {x X w(x 0, x) < r} the open ball centered at x 0 with radius r with respect to w;

4 154 L. Guran, A. Petruşel B w (x 0 ; r) := {x X w(x 0, x) r} the closed ball centered at x 0 with radius r with respect to w; d B w (x0 ; r)- the closure in (X, d) of the set B w (x 0 ; r). One of the main results is the following fixed point theorem for weakly Ćirić-contractive operators. Theorem 1 Let (X, d) be a complete metric space, x 0 X, r > 0 and T : B w (x 0 ; r) P cl (X) a multivalued operator such that: (i) T is weakly Ćirić-contractive operator; (ii) D w (x 0, T(x 0 )) (1 )r. Then there exists x X such that x T(x ). Proof. Since D w (x 0, T(x 0 )) (1 )r, then for every x 0 X there exists x 1 T(x 0 ) such that D w (x 0, T(x 0 )) w(x 0, x 1 ) (1 )r < r. Hence x 1 B w (x 0 ; r). For x 1 B w (x 0 ; r), there exists x 2 T(x 1 ) such that: i. w(x 1, x 2 ) w(x 0, x 1 ) ii. w(x 1, x 2 ) D w (x 0, T(x 0 )) w(x 0, x 1 ) iii. w(x 1, x 2 ) D w (x 1, T(x 1 )) w(x 1, x 2 ) iv. w(x 1, x 2 ) 2 D w(x 0, T(x 1 )) 2 w(x 0, x 2 ) w(x 1, x 2 ) 2 [w(x 0, x 1 ) + w(x 1, x 2 )] (1 2 )w(x 1, x 2 ) 2 w(x 0, x 1 ) w(x 1, x 2 ) 2 w(x 0, x 1 ). Then w(x 1, x 2 ) max {, 2 }w(x 0, x 1 ) Since > 2 for every [0, 1), then w(x 1, x 2 ) w(x 0, x 1 ) (1 )r. Then w(x 0, x 2 ) w(x 0, x 1 )+w(x 1, x 2 ) < (1 )r+(1 )r = (1 2 )r < r. Hence x 2 B w (x 0 ; r). For x 1 B w (x 0 ; r) and x 2 T(x 1 ), there exists x 3 T(x 2 ) such that i. w(x 2, x 3 ) w(x 1, x 2 ) ii. w(x 2, x 3 ) D w (x 1, T(x 1 )) w(x 1, x 2 ) iii. w(x 2, x 3 ) D w (x 2, T(x 2 )) w(x 2, x 3 ) iv. w(x 2, x 3 ) 2 D w(x 1, T(x 2 )) 2 w(x 1, x 3 ) w(x 2, x 3 ) 2 [w(x 1, x 2 ) + w(x 2, x 3 )] (1 2 )w(x 2, x 3 ) 2 w(x 1, x 2 ) w(x 2, x 3 ) 2 w(x 1, x 2 ). Then w(x 2, x 3 ) max {, 2 }w(x 1, x 2 ). Since > 2 for every [0, 1), then w(x 2, x 3 ) w(x 1, x 2 ) 2 (x 0, x 1 ) 2 (1 )r.

5 Existence and data dependence 155 Then w(x 0, x 3 ) w(x 0, x 2 ) + w(x 2, x 3 ) (1 2 )r + 2 (1 )r = = (1 )( )r = (1 3 )r < r. Hence x 3 B w (x 0 ; r). By this procedure we get a seuence (x n ) n N X of successive applications for T starting from arbitrary x 0 X and x 1 T(x 0 ), such that (1) x n+1 T(x n ), for every n N; (2) w(x n, x n+1 ) n w(x 0, x 1 ) n (1 )r, for every n N. For every m, n N, with m > n, we have w(x n, x m ) w(x n, x n+1 ) + w(x n+1, x n+2 ) w(x m 1, x m ) n w(x 0, x 1 ) + n+1 w(x 0, x 1 ) m 1 w(x 0, x 1 ) n 1 w(x 0, x 1 ) n r. By Lemma 1(3) we have that the seuence (x n ) n N B w (x 0 ; r) is a Cauchy seuence in (X, d). Since (X, d) is a complete metric space, then there exists x B d w(x 0 ; r) such that x n d x. Fix n N. Since (x m ) m N converge to x and w(x n, ) is lower semicontinuous, we have w(x n, x ) lim m inf w(x n, x m ) n 1 w(x 0, x 1 ) n r. For x B d w(x 0 ; r) and x n T(x n 1 ), there exists u n T(x ) such that i. w(x n, u n ) w(x n 1, x ) n 1 w(x 0, x 1 ) ii. w(x n, u n ) D w (x n 1, T(x n 1 )) w(x n 1, x n )... n w(x 0, x 1 ) iii. w(x n, u n ) D w (x, T(x )) w(x, u n ) n 1 w(x 0, x 1 ) iv. w(x n, u n ) 2 D w(x n 1, T(x )) 2 w(x n 1, u n ) 2 n 1 1 w(x 0, x 1 ) = n 2(1 ) w(x 0, x 1 ). Then w(x n, u n ) max{ n 1, n, n 2(1 ) }w(x 0, x 1 ). Since for [0, 1) we have true n 1 > n and n 1 > we get that w(x n, u n ) n 1 w(x 0, x 1 ) n r. So, for every n N we have: w(x n, x ) n r w(x n, u n ) n r. n 2(1 ) d Then, from 1(2), we obtain that u n x. As u n T(x ) and using the closure of T result that x T(x ). A global result for previous theorem is the following fixed point result for multivalued weakly Ćirić-contractive operators.

6 156 L. Guran, A. Petruşel Theorem 2 Let (X, d) be a complete metric space, T : X P cl (X) a multivalued weakly Ćirić-contractive operator. Then there exists x X such that x T(x ). 4 Data dependence for weakly Ćirić-contractive multivalued operators The main result of this section is the following data dependence theorem with respect to the above global theorem 2. Theorem 3 Let (X, d) be a complete metric space, T 1, T 2 : X P cl (X) be two multivalued weakly Ćirić-contractive operators with i [0, 1) with i = {1, 2}. Then the following are true: 1. FixT 1 FixT 2 ; 2. We suppose that there exists η > 0 such that for every u T 1 (x) there exists v T 2 (x) such that w(u, v) η, (respectively for every v T 2 (x) there exists u T 1 (x) such that w(v, u) η). Then for every u FixT 1, there exists v FixT 2 such that w(u, v ) η 1, where = i for i = {1, 2}; (respectively for every v FixT 2 there exists u FixT 1 such that w(v, u ) η 1, where = i for i = {1, 2}). Proof. From the above theorem we have that FixT 1 FixT 2. Let u 0 FixT 1, then u 0 T 1 (u 0 ). Using the hypothesis (2) we have that there exists u 1 T 2 (u 0 ) such that w(u 0, u 1 ) η. Since T 1, T 2 are weakly Ćririć-contractive with i [0, 1) and i = {1, 2} we have that for every u 0, u 1 X with u 1 T 2 (u 0 ) there exists u 2 T 2 (u 1 ) such that i. w(u 1, u 2 ) w(u 0, u 1 ) ii. w(u 1, u 2 ) D w (u 0, T 2 (u 0 )) w(u 0, u 1 ) iii. w(u 1, u 2 ) D w (u 1, T 2 (u 1 )) w(u 1, u 2 ) iv. w(u 1, u 2 ) 2 D w(u 0, T 2 (u 1 )) 2 w(u 0, u 2 ) w(u 1, u 2 ) 2 [w(u 0, u 1 ) + w(u 1, u 2 )] w(u 1, u 2 ) 2 w(u 0, u 1 ).

7 Existence and data dependence 157 Then w(u 1, u 2 ) max{, 2 }w(u 0, u 1 ). Since for [0, 1) we have true > 2, then we have w(u 1, u 2 ) w(u 0, u 1 ). For u 1 X and u 2 T 2 (u 1 ), there exists u 3 T 2 (u 2 ) such that i. w(u 2, u 3 ) w(u 1, u 2 ) ii. w(u 2, u 3 ) D w (u 1, T 2 (u 1 )) w(u 1, u 2 ) iii. w(u 2, u 3 ) D w (u 2, T 2 (u 2 )) w(u 2, u 3 ) iv. w(u 2, u 3 ) 2 D w(u 1, T 2 (u 2 )) 2 w(u 1, u 3 ) w(u 2, u 3 ) 2 [w(u 1, u 2 ) + w(u 2, u 3 )] w(u 2, u 3 ) 2 w(u 1, u 2 ) Then w(u 2, u 3 ) max{, 2 }w(u 1, u 2 ). Since for [0, 1) we have true > 2, then we have w(u 2, u 3 ) w(u 1, u 2 ) 2 w(u 0, u 1 ). By induction we obtain a seuence (u n ) n N X such that (1) u n+1 T 2 (u n ), for every n N; (2) w(u n, u n+1 ) n w(u 0, u 1 ). For n, m N, with m > n we have the ineuality w(u n, u m ) w(u n, u n+1 ) + w(u n+1, u n+2 ) + + w(u m 1, u m ) < n w(u 0, u 1 ) + n+1 w(u 0, u 1 ) + + m 1 w(u 0, u 1 ) n 1 w(u 0, u 1 ) By Lemma 1(3) we have that the seuence (u n ) n N is a Cauchy seuence. Since (X, d) is a complete metric space, we have that there exists v X such d that u n v. By the lower semicontinuity of w(x, ) : X [0, ) we have w(u n, v ) lim inf w(u n, u m ) n m 1 w(u 0, u 1 ). For u n 1, v X and u n T 2 (u n 1 ) there exists z n T 2 (v ) such that we have i. w(u n, z n ) w(u n 1, v ) n 1 w(u 0, u 1 ) ii. w(u n, z n ) D w (u n 1, T 2 (u n 1 )) w(u n 1, u n )... n w(u 0, u 1 ) iii. w(u n, z n ) D w (v, T 2 (v )) w(v, z n ) n 1 w(u 0, u 1 ) iv. w(u n, z n ) 2 D w(u n 1, T 2 (v )) 2 w(u n 1, z n ) n 2(1 ) w(u 0, u 1 ).

8 158 L. Guran, A. Petruşel Then w(u n, z n ) max{ n 1, n, 2(1 ) }w(u 0, u 1 ). Since n 1 > n and n 1 > n 2(1 ) n w(u n, z n ) for every [0, 1) we have that n 1 w(u 0, u 1 ). So, we have: w(u n, v ) n 1 w(u 0, u 1 ) w(u n, z n ) n 1 w(u 0, u 1 ). Applying Lemma 1(2), from the above relations we have that z n d v. Then, we know that z n T 2 (v d ) and z n v. In this case, by the closure of T 2, it results that v T 2 (v ). Then, by w(u n, v ) n 1 w(u 0, u 1 ), with n N, for n = 0, we obtain w(u 0, v ) 1 1 w(u 0, u 1 ) η 1, which completes the proof. References [1] Lj. B. Ćirić, A generalization of Banach s contraction principle, Proc. Amer. Math. Soc., 45 (1974), [2] Lj. B. Ćirić, Fixed points for generalized multi-valued contractions, Mat. Vesnik, 9(24) (1972), [3] A. Granas, J. Dugundji, Fixed Point Theory, Berlin, Springer-Verlag, [4] O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japonica, 44 (1996), [5] N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl., 141 (1989), [6] I. A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, 2001.

9 Existence and data dependence 159 [7] I.A. Rus, A. Petruşel, G. Petruşel, Fixed Point Theory, Cluj University Press, [8] T. Suzuki, W. Takahashi, Fixed points theorems and characterizations of metric completeness, Topological Methods in Nonlinear Analysis, Journal of Juliusz Schauder Center, 8 (1996), [9] J. S. Ume, Fixed point theorems related to Ćirić contraction principle, J. Math. Anal. Appl., 255 (1998), Received: May 18, 2009

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