MAPPING AND ITS APPLICATIONS. J. Jeyachristy Priskillal 1, P. Thangavelu 2
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1 International Journal of Pure and Applied Mathematics Volume 11 No , ISSN: (printed version); ISSN: (on-line version) url: doi: /ijpam.v11i1.14 PAijpam.eu ψ-contractive TYPE FUZZY MAPPING AND ITS APPLICATIONS J. Jeyachristy Priskillal 1, P. Thangavelu 1 Department of Mathematics Karunya University Coimbatore, Tamil Nadu, , INDIA Ramanujam Centre for Mathematical Sciences Thiruppuvanam, Tamil Nadu, , INDIA Abstract: This manuscript contains a fixed fuzzy point theorems using ψ-contractive fuzzy mapping in a complete metric space and gives applications to fuzzy differential equations. AMS Subject Classification: 47H10, 34A07, 37C5, 54E40, 54E50 Key Words: metric space, fuzzy mapping, fixed fuzzy point, fuzzy differential equations 1. Introduction Vital tool in theory of metric spaces is the Banach fixed point theorem. Many Mathematicians has studied the concept of fixed point theorems and its applications. As an overview of Banach contraction theorem, ψ-contractive type mapping was introduced by Berinde []. Fuzzy mapping was introduced by Heilpern [6]. Further, contractive type fuzzy mapping was introduced by Byung Soo Lee and Sung Jin Cho [3]. Continuing this, many authors had considered the fixed fuzzy point theorem for fuzzy mappings [1], [4], [5], [11]. In 003, Lakshmikantham and Mohapatra [10] applied the Banach contraction theorem Received: November 9, 016 Revised: January 1, 017 Published: January 6, 017 c 017 Academic Publications, Ltd. url: Correspondence author
2 17 J.J. Priskillal, P. Thangavelu to fuzzy differential equations. Progressing this, Hemant Kumar Nashine [7] et al proved a fixed fuzzy point theorem and gave application of fuzzy differential equations [], [9]. This manuscript contains a fixed fuzzy point theorems using ψ-contractive fuzzy mapping in a complete metric space and gives applications to fuzzy differential equations.. Preliminaries According to Berinde [] the function ψ is defined as follows: Let Ψ be the family of nondecreasing functions + n=1 ψn (s) < +, s > 0, where ψ n is the n-th iterate of ψ. Lemma.1. [] If ψ Ψ, then the following are satisfied. 1. ψ(s) < s, s > 0;. ψ(0) = 0; 3. ψ is right continuous at s = 0; Definition.. [] In a metric space (X,d), we say that a mapping F : X X is an ψ-contraction if a function ψ Ψ d(fu,fv) ψ(d(u,v)),for all u,v X. Definition.3. [6] Let A be a fuzzy set in X and α [0,1]. Then the α-level set A α of A is defined as A α = {u : A(u) α}. Definition.4. [6] Let A be a fuzzy set in a metric linear space (X,d) and α [0,1]. A is an approximate quantity iff A α is compact and convex in X and sup u X Au = 1. Definition.5. [6] Let (X,d) be a metric space and W(X) be a collection of approximation quantities. The family W α (X) = {A I X : A α is nonempty, compact and convex}. Let A,B W(X). Then we shall define a distance between two approximate quantities. Define, p α (A,B) = inf d(u,v), u A α,v B α D α (A,B) = H(A α,b α ), D(A,B) = supd α (A,B). α p α is called a α-space, D α is called a α-distance, D is called the distance and H is called the Hausdorff distance.
3 ψ-contractive TYPE FUZZY Definition.6. [6] Let α (0,1]. The fuzzy point u α of X is the fuzzy set of X defined by u α (u) = α and u α (w) = 0 if w u. Definition.7. [6] Let X be an arbitrary set, Y be any metric linear space and I = [0,1]. A fuzzy set of X is an element of I X. T is a fuzzy mapping iff T is a mapping from the set X into a family W(X) I X, that is T(u) W(X) u X. Definition.. [6] Let (X,d) be a metric space. Let A,B W(X) and u,v X. 1. u α A if p α (u,a) = 0,. p α (u,a) d(u,v)+p α (v,a), 3. If u α A, then p α (u,a) D α (A,B). Definition.9. [6] A fuzzy point u α in X is called a fixed fuzzy point of a fuzzy mapping T if (u) α Tu, that is, Tu(u) α or u (Tu) α. If (u) 1 Tu, then u is a fixed point of fuzzy mapping T. 3. Main Results Definition 3.1. In a metric space (X,d), the fuzzy mapping T : X W α (X)isanψ-contractive fuzzymappingif afunctionψ Ψ D α (T(u),T(v)) ψ(d(u,v)), u,v X. Theorem 3.. Let (X,d) be a complete metric linear space and T : X W α (X) be an ψ-contractive fuzzy mapping. Then there exists a fixed fuzzy point. Proof. Let u 0 X and T : X W α (X) be a fuzzy mapping. Suppose u 1 (T(u 0 )) α. Since T(u 1 )) α is nonempty compact subset of X, then u (T(u 1 )) α by lemma.(3) and by our hypothesis, d(u 1,u ) = p α (u 1,T(u 1 )) D α (T(u 0 ),T(u 1 )) ψ(d(u 0,u 1 )). Proceeding like this, we assemble a sequence {u n } in X u n (T(u n 1 )) α,
4 10 J.J. Priskillal, P. Thangavelu by lemma.(3) and by our hypothesis, d(u n,u n+1 ) = p α (u n,t(u n )) Continuing this process, we can get,. D α (T(u n 1 ),T(u n )) ψ(d(u n 1,u n )) = ψ(p α (u n 1,T(u n 1 ))) ψ(d α (T(u n ),T(u n 1 ))) ψ(ψ(d(u n,u n 1 ))) ψ n (d(u 0,u 1 )). d(u n,u n+m ) d(u n,u n+1 )+...+d(u n+m 1,u n+m ) ψ n (d(u 0,u 1 ))+...+ψ n+m 1 (d(u 0,u 1 )) = n+m 1 i=n ψ i (d(u 0,u 1 )) Since n=1 ψn (s) <,{u n } is a Cauchy sequence in X. Since (X,d) is a complete metric space, {u n } converges to some u X, that is, (d(u n,u)) 0. Now by lemma.(,3) and ψ is continuous at the origin, p α (u,t(u)) d(u,u n )+p α (u n,t(u)) d(u,u n )+D α (T(u n 1 ),T(u)) d(u,u n )+ψ(d(u n 1,u)) 0+0 = 0. Therefore, p α (u,t(u)) = 0 and by lemma.(1),u α T(u). Example 3.3. [1] Let X = [0,1],d : X X X bethe Euclidean metric and α (0, 1 ). The fuzzy mapping T : X IX is defined by 1 when u = 0, 1 when u = 0, T(0)(u) = α when u (0, 1 ], T(1)(u) = α when u (0, 1 α when u ( 1,1], ], α when u ( 1,1],
5 ψ-contractive TYPE FUZZY and for w (0,1), 1 when u = 0, T(w)(u) = α when u (0, 1 ], 0 when u ( 1,1], Then T(0) 1 = T(w) 1 = T(1) 1 = {0},T(0) α = T(w) α = T(1) α = [0, 1 ] and T(0)α = T(1) α = [0,1],T(w) α = [0, 1 ]. Now, D 1 (T(u),T(v)) = H(T(u) 1,T(v) 1 ) = 0, u,v X, D α (T(u),T(v)) = H(T(u) α,t(v) α ) = 0, u,v X, Dα (T(u),T(v)) = H(T(u) α,t(v) α) = 0, u,v {0,1} and u,v (0,1), Dα (T(u),T(v)) = H(T(u) α,t(v) α ) = 1, u {0,1} and v (0,1). Define ψ : [0, ) [0, ) by ψ(s) = s s+1. Clearly, n=1 ψn (s) < where s (0, ). We obtain D α (T(u),T(v)) = 0 ψ(d(u,v)), u,v X. Then 0 is the fixed fuzzy point. The theorem is justified. Theorem 3.4. Let α (0,1],(X,d) beacomplete metric space, T 1 and T be two fuzzy mappings from X onto W α (X) and ψ Ψ D α (T 1 (u),t (v)) ψ(d(u,v)), u,v X. Then u α is a common fixed fuzzy point of T 1 and T. Proof. Let u 0 X and T 1 : X W α (X) and T 1 : X W α (X) be two fuzzymappings. Since(T 1 (u 0 )) α isnonemptysubsetofx, then u 1 (T 1 (u 0 )) α,also since (T 1 (u 1 )) α is nonempty subset of X, then u (T (u 1 )) α such that by lemma.(3) and by our hypothesis, d(u 1,u ) = p α (u 1,T (u 1 )) D α (T 1 (u 0 ),T (u 1 )) ψ(d(u 0,u 1 )). By induction we construct a sequence {u n } in X u n+1 (T 1 (u n )) α and
6 1 J.J. Priskillal, P. Thangavelu u n+ (T (u n+1 )) α, by lemma.(3) and by our hypothesis, d(u n+1,u n+ ) = p α (u n+1,t (u n+1 )) D α (T (u n ),T (x n+1 )) ψ(d(u n,u n+1 )) = ψ(p α (u n,t 1 (u n ))) ψ(d α (T (u n 1 ),T 1 (u n ))) ψ(ψ(d(u n 1,u n ))). ψ n+1 (d(u 0,u 1 )). That is, d(u n+1,u n+ ) ψ n+1 (d(u 0,u 1 )). Similarly we can prove, Continuing this process, we can get, d(u n,u n+1 ) ψ n (d(u 0,u 1 )). d(u n,u n+m ) d(u n,u n+1 )+...+d(u n+m 1,u n+m ) ψ n (d(u 0,u 1 ))+...+ψ n+m 1 (d(u 0,u 1 )) = n+m 1 i=n ψ i (d(u 0,u 1 )) Since n=1 ψn (t) <,{u n } is a Cauchy sequence in X. Since (X,d) is a complete metric space, {u n } converges to some x X, that is, (d(u n,u)) 0. Suppose n is odd, by lemma.(,3) and ψ is continuous at the origin, p α (u,t 1 (u)) d(u,u n )+p α (u n,t 1 (u)) d(u,u n )+D α (T 1 (u n 1 ),T 1 (u)) d(u,u n )+ψ(d(u n 1,u)) 0+0 = 0. Therefore, p α (u,t 1 (u)) = 0 and by lemma.(1),u α T 1 (u). Similarly, Suppose n is even, we can prove u α T (u). Therefore, u α is a common fixed fuzzy point of T 1 and T.
7 ψ-contractive TYPE FUZZY Example 3.5. [7] Let X = {0,1,},d : X X X be the Euclidean metric and α (0, 1 3 ). Define, two fuzzy mappings T 1 : X W α (X) and T : X W α (X) as follows: α if u = 0, (T 1 0)(u) = (T 1)(u) = α if u = 1, 0 if u =, α if u = 0, (T 1 1)(u) = (T 0)(u) = 0 if u = 1, α 3 if u =, α 3 if u = 0, (T 1 )(u) = (T )(u) = α if u = 1, α 4 if u =. Note that (T 1 0) α = (T 1 1) α = (T 0) α = (T 1) α = {0},(T 1 ) α = (T ) α = {1}. Now, D α (T 1 (u),t (v)) = H(T 1 (u) α,t (v) α ) = 0, u = v and,u,v {0,1}, D α (T 1 (u),t (v)) = H(T 1 (u) α,t (v) α ) = 1, u,v {0,} and,u,v {1,}. Define ψ : [0, ) [0, ) by ψ(t) = t. Clearly, n=1 ψn (t) = n=1 (1 )n t < where t (0, ). For all u,v X,D α (T 1 (u),t (v)) ψ(d(u,v)). The hypotheses in the above theorem are satisfied. Then0is the fixed fuzzy point. The theorem is justified. Remark 3.6. If T 1 = T in Theorem 3.4, we can get Theorem Applications to Fuzzy Differential Equations From the book of Lakshmikantham et al. [10], we have the follwing problem and some lemmas to apply our main results. Let E n be the space of all fuzzy subsets x of R n where x : R n I = [0,1]. Consider the boundary value problem { x (t) = f(t,x(t),x (t)),t J = [a,b], (1) x( ) = x 1,x(t ) = x,,t J,
8 14 J.J. Priskillal, P. Thangavelu wheref : J E n E n E n isacontinuousfunction. Thisproblemisequivalent to the integral equation x(t) = G(t,s)[f(s,x(s),x (s))]ds+β(t), where Green s function G is given by { (t t)(s ) G(t,s) = t s t t, (t s)(t ) t t s t. and β(t) satisfies β = 0,β( ) = x 1,β(t ) = x. Let us recall some properties of G(t, s), namely, G(t,s) ds (t ), and G t (t,s) ds t Now, we shall prove the existence of the result for the above boundary value problem by using our theorem 3. and 3.4. Theorem 4.1. Let f : J E n E n E n. Suppose 0 < γ < 1,0 < δ < 1 with γ δ x,y C 1 (J,E n ), f(t,x,x ) f(t,y,y ) γ x y +δx y, Then the integral equation x(t) = has a solution in C 1 [[,t ],E n ]. G(t,s)[f(s,x(s),x (s))]ds+β(t),t J Proof. Consider C = C 1 [[,t ],E n ] with the metric D(x,y) = max [γ x(t) y(t) +δx (t) y (t) ]. t t The space (C,D) is a complete metric space. Define the operator T : C C by Tx(t) = G(t,s)[f(s,x(s),x (s))]ds+β(t)..
9 ψ-contractive TYPE FUZZY By the properties of G(t,s) and by our hypothesis, and Tx(t) Ty(t) (Tx) (t) (Ty) (t) Now, we have G(t,s) f(s,x(s),x (s)) f(s,y(s),y (s)) ds D(x,y) G(t,s) ds (t ) D(x, y) D(x,y). G t (t,s) f(s,x(s),x (s)) f(s,y(s),y (s)) ds D(x,y) G t (t,s) ds (t ) D(x, y) D(x,y). D[Tx,Ty] γ D(x,y) ( 5 δ)d(x,y) = ψ(d(x,y)). +δ D(x,y) and n=1 ψn (s) = n=1 (5 δ)n s <,ψ(0) = 0 and ψ is continuous at the origin. We obtain D(Tx,Ty) ψ(d(x,y)). Therefore, Theorem 3. applies to T which has a fixed point x C, that is x is a solution of the boundary value problem. Theorem 4.. Let f 1,f : J E n E n E n. Suppose there exists 0 < γ < 1,0 < δ < 1 with γ δ such that for all u,v C 1 (J,E n ), f 1 (t,u,u ) f (t,v,v ) γ u v +δu v,
10 16 J.J. Priskillal, P. Thangavelu Then the integral equation u(t) = G(t,s)[f i (s,u(s),u (s))]ds+β(t),t J,i {1,} has a common solution in C 1 [[,t ],E n ]. Proof. Consider C = [[,t ],E n ] with the metric D(u,v) = max [γ u(t) v(t) +δu (t) v (t) ]. t t The space (C,D) is a complete metric space. Define the operator F i : C C by F i (u)(t) = G(t,s)[f i (s,u(s),u (s))]ds+ β(t), t J,i {1,}, where f 1,f C(J E n E n,e n ),u C 1 (J,E n ) and β C(J,E n ). By the properties of G(t, s) and by our hypothesis, and F 1 (u)(t) F (v)(t) G(t,s) f 1 (s,u(s),u (s)) f (s,v(s),v (s)) ds D(u,v) (F 1 (u)) (t) (F (v)) (t) G(t,s) ds (t ) D(u, v) D(u,v). G t (t,s) f 1 (s,u(s),u (s)) f (s,v(s),v (s)) ds D(u,v) G t (t,s) ds (t ) D(u, v) D(u,v).
11 ψ-contractive TYPE FUZZY Now, we have D[F 1 u,f v] γ D(u,v) ( 5 δ)d(u,v) = ψ(d(u,v)). +δ D(u,v) and n=1 ψn (s) = n=1 (5 δ)n s <,ψ(0) = 0 and ψ is continuous at the origin. We obtain D(F 1 u,f v) ψ(d(u,v)). Therefore, Theorem 3.4 applies to F 1 and F have a common fixed point u C, that is u is a solution of the boundary value problem. References [1] M. Abbas, B. Damjanivic, and R. Lazovic, Fuzzy common fixed point theorems for generalized contractive mappings, Applied Mathematics Letter, 3, No. 11 (010), , doi: /j.aml [] V. Berinde, Iterative Approximation of Fixed Points, Lecture Notes in Mathematics, Romania, 007, doi: / [3] Byung Soo Lee and Sung Jin Cho, A fixed point theorem for contractive-type fuzzy mappings, Fuzzy Sets and Systems, 61 (1994), , doi: / (94) [4] L. Ciric, M. Abbas, B. Damjanovic, R. Saadati, Common fuzzy fixed point theorems in ordered metric spaces, Mathematical and Computer Modelling, 53 (011), , doi: /j.mcm [5] V.D. Estruch, A. Vidal, A note on fixed fuzzy points for fuzzy mappings, Rend. Ist. Mat. Univ. Trieste., 3 (001), [6] S. Heilpern, Fuzzy mappings and fixed point theorem, Journal of Mathematical Analalysis and Applications, 3 (191), , doi: /00-47X(1) [7] Hemant Kumar Nashine, Calogero Vetro, Wiyada Kumam and Poom Kumam, Fixed point theorems for fuzzy mappings and applications to ordinary fuzzy differential equations, Advances in Difference Equations, 3, No. 1 (014), 1-14, doi: / [] Jong Yeoul Park, Hyo Keun Han, Fuzzy differential equations, Fuzzy Sets and Systems, 110 (000), 69-77, doi: /S (9)00150-X. [9] Jong Yeoul Park, S.Y. Lee, H.M. Kee, The existence of solution for fuzzy differential equations with infinite delays, Indian Journal of Pure and Applied Mathematics,31, No. 10 (000), [10] V. Lakshmikantham, R. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor and Francis, London, 003, doi: /
12 1 J.J. Priskillal, P. Thangavelu [11] R.A. Rashwan, M.A. Ahmed, Common fixed point theorems for fuzzy mappings, Arch. Math., 3 (00), [1] D. Turkoglu, B.E. Rhoades, A fixed fuzzy point for fuzzy mapping in complete metric spaces, Mathematical Communications, 10 (005),
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