On the stability of fuzzy set-valued functional equations

Transcription

1 Mursaleen et al., Cogent Mathematics (07, 4: APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE On the stability of fuzzy set-valued functional equations M. Mursaleen,, S.A. Mohiuddine * and Khursheed J. Ansari 3 Received: 3 July 06 Accepted: 04 January 07 First Published: 6 January 07 *Corresponding author: S.A. Mohiuddine, Operator Theory and Applications Research Group, Faculty of Science, Department of Mathematics, King Abdulaziz University, P.O. Box 8003, Jeddah 589, Saudi Arabia mohiuddine@gmail.com Reviewing editor: Peiguang Wang, Hebei University, China Additional information is available at the end of the article Abstract: We introduce some fuzzy set-valued functional equations, i.e. the generalized Cauchy type (in n variables, the Quadratic type, the Quadratic-Jensen type, the Cubic type and the Cubic-Jensen type fuzzy set-valued functional equations and discuss the Hyers-Ulam-Rassias stability of the above said functional equations. These results can be regarded as an important extension of stability results corresponding to single-valued and set-valued functional equations, respectively. Subjects: Applied Mathematics; Mathematics & Statistics; Science Keywords: fuzzy sets; various functional equations; fixed point; Ulam stability 00 Mathematics subject classifications: 46S40; 39B5; 39B8. Introduction The theory of fuzzy sets was introduced by Zadeh (965. After the pioneering work of Zadeh, there has been a great effort to obtain analogues of classical theories. It is a very powerful handset for modeling uncertainty and vagueness in various problems arising in the field of science and ABOUT THE AUTHORS M. Mursaleen is a full Professor & Chairman of Department of Mathematics. As an active researcher, Prof. Mursaleen has authored seven books and five book chapters, in addition to his contributions to more than 00 research papers. His main research interests are Summability Theory, Approximation Theory, Fixed Point Theory and Operator Theory. He has been recipient of the Outstanding Researcher of the Year-04 Award of Aligarh Muslim University. S.A. Mohiuddine is Associate Professor of Mathematics and has visited some foreign reputed universities including Imperial College, London, UK and Yuzuncu Yil University, Van, Turkey. His main research interests in the field Functional Analysis: Summability Theory, Measures of Noncompactness, Ulam Stability Problem and Approximation theory. Mohiuddine has authored one book, two book chapters and published more than hundred research papers to various international journals. Khursheed J. Ansari is working as Assistant Professor in the Department of Mathematics, King Khalid University, Abha, Saudi Arabia. His area of research interests are Approximation Theory and Ulam Stability Problem. PUBLIC INTEREST STATEMENT In 940, S. M. Ulam proposed the Ulam stability problem: When does a linear transformation near an approximately linear transformation exist? Since then, many specialists on this famous Ulam problem, have investigated interesting functional equations. The aim of this paper is to introduce the generalized Cauchy type (in n variables, the Quadratic type, the Quadratic-Jensen type, the Cubic type, the Cubic-Jensen type fuzzy set-valued functional equations and then we determine some stability results concerning these fuzzy set-valued functional equations. 07 The Author(s. This open access article is distributed under a Creative Commons Attribution (CC-BY 4.0 license. Page of

2 Mursaleen et al., Cogent Mathematics (07, 4: engineering. It has also very useful applications in various fields, e.g., population dynamics, chaos control, computer programming, non-linear dynamical systems and control theory etc. Set-valued functional equations in Banach spaces have received a lot of attention in the literature (see Arrow & Debreu, 954; Aumann, 965; Debreu, 966. The pioneering papers by Aumann (965 and Debreu (966 were inspired by problems arising in Control Theory and Mathematical Economics. The concept of stability for functional equations arises when one replaces a functional equation by an inequality which acts a perturbation in the equation. The study of stability problems for functional equations is related to a question posed by Ulam (960 in a conference at Wisconsin University, Madison in 940: Let G be a group and G a metric group with the metric ρ(.,.. Given a number ε>0, does there exists a δ >0 such that if a mapping f :G G satisfies the inequality ρ(f (xy, f (xf (y <ε for all x, y G, then there exist a homomorphism h:g G with ρ(f (x, h(x δ for all x G? If the answer is affirmative the equation f (xy =f (xf (y of the homomorphism is called stable (see Brzdek & Jung, 0; Hyers, Isac, & Rassias, 998. In other words, the equation of homomorphism is stable if every approximat solution can be approximated by a solution of this equation. The first answer to Ulam s problem was given by Hyers (94 for the Cauchy functional equation in Banach spaces. In fact, he proved: Let X, Y be Banach spaces, ε a non-negative number, f :X Y a function f (x + y f (x f (y ε for all x, y X, then there exists a unique additive mapping h:x Y with the property f (x h(x ε Due to the question of Ulam and the result of Hyers this type of stability is called today Hyers-Ulam stability of functional equations. So the Cauchy functional equation f (x + y =f (x+f (y is Hyers-Ulam stable. After Hyers result a large number of literature was devoted to study the Hyers-Ulam stability for various functional equations. A new type of stability for functional equations was introduced by Aoki (950 (for some historical comments regarding the work of Aoki, see Moslehian and Rassias (007 for additive mappings and by Rassias (978 for linear mappings in which the Cauchy difference is allowed to be unbounded by replacing ε with a function depending on x and y in the Hyers theorem. However, the paper of Rassias (978 has provided a lot of influence in the development what we call generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. Until now, the stability problems for different types of functional equations in various spaces have been extensively studied. Gajda (988 determined the stability of Cauchy equation on semigroups. The stability of approximately additive mappings have been studied by Găvruta (994. For more details, the reader can refer to Rassias (989, Rassias (999, Xu, Rassias, and Xu (0, Jung (0 and Jung (998. Among the studies of these problems, it is worth mentioning that Radu (003 proposed a novel method to establish the stability of Cauchy functional equation via fixed point approach in lieu of the direct method which was frequently in used (see also Gordji, Park, & Savadkouhi, 00. Recently, Cieplińsky (0 summarized some applications of several types of fixed point theorems to the Hyers-Ulam stability of functional equations. As of now, this method has been successfully used in the study of stability problems of many types of functional equations in abstract spaces. In Mirmostafaee and Moslehian (008 initiated the study of stability problems of functional equations in fuzzy setting. Specifically, they considered the stability of the cauchy functional equation in fuzzy normed spaces. Since then, the fuzzy stability problems of various types of functional equations have been extensively investigated by different authors Jang, Lee, Park, and Shin (009 and Lee, Jang, Park, and Shin (00. At the same time, the fixed point method has been widely used to prove the fuzzy stability of several types of functional equations (Mohiuddine & Alotaibi, 0; Park, 009. For other results on the Hyers-Ulam stability of functional equations in intuitionistic fuzzy/random normed spaces, one can refer to Al-Fhaid and Mohiuddine (03, Alotaibi and Mohiuddine (0, Mohiuddine (009, Mohiuddine, Alotaibi and Obaid (0, Mohiuddine, Cancan and Şevli (0, Page of

3 Mursaleen et al., Cogent Mathematics (07, 4: Mohiuddine and Şevli (0, Mursaleen and Ansari (03, Mursaleen and Mohiuddine (009 and Mohiuddine and Alghamdi (0. In summary, one can see that the (fuzzy stability for a single-valued functional equation is whether, for a given mapping almost a functional equation (which means that the mapping is close to a solution of the functional equation, there exists an exact solution of the functional equation which can be used to approximate the given mapping. Typically, a metric associated with the corresponding space is chosen to characterize the functional inequality. In Nikodem and Popa (009 considered the general solution of set valued maps linear inclusion relation, which can be regarded as a generalization of the additive single-valued functional equation. By means of the inclusion relation, Lu and Park (0 and Park, Regan, and Saadati (0 investigated the stability of several types set-valued functional equations. However, it should be pointed out that, in their studies, the inclusion relation is applied to characterize the set-valued functional inequality rather than an appropriate metric. Recently, similar to the method is used to deal with the singlevalued functional equations, Kenary, Rezaei, Gheisari, and Park (0 proved the stability of several types of set-valued functional equations via the fixed point approach, in which the Hausdorff metric is adopted to characterize the set-valued functional ineqaulity. Recently, Shen, Lan, and Chen (04 extended the set-valued functional equations and establish some stability results for these fuzzy set-valued functional equations. Notice that the supremum metric, as a generalization of the Hausdorff metric is applied to characterize the fuzzy set-valued functional inequality. They discussed the Hyers-Ulam-Rassias stability of those fuzzy set-valued functional equations which are of additive type. The aim of this paper is to introduce some fuzzy set-valued functional equations, i.e. the generalized Cauchy type (in n variables, the Quadratic type, the Quadratic-Jensen type, the Cubic type, the Cubic-Jensen type fuzzy set-valued functional equations and then discuss the Hyers-Ulam-Rassias stability of the above said functional equations. Interestingly, the corresponding single-valued and set-valued functional equations acted as special cases will be included in our results.. Preliminaries In what follows, we begin with some related concepts and fundamental results, which are mainly derived from Castaing and Valadier (977, Diamond and Kloeden (994 and Inoue (99. Let R, R +, and R n denote the set of all real numbers, the set of all non-negative real numbers and the n-dimensional Euclidean space, respectively. Let Y be a separable Banach space with the norm Y. We denote the set of all nonempty compact subsets and the set of all nonempty compact convex subsets of Y by Y and C (Y, respectively. Let A and B be two nonempty subsets of Y and let λ R. The (Minkowski addition and scalar multiplication can be defined by A + B ={a + b a A, b B}, λa ={λa a A} Notice that the sets Y, C (Y are closed under the operations of addition and scalar multiplication. In fact, these two operations induce a linear structure o Y and C (Y with zero element {0}, respectively. It should be noted that this linear structure is just a cone rather that a vector space because, in general, A + ( A {0}. Moreover, for all λ, μ R, it follows that (. λ(a + B =λa + λb, (λ + μa λa + μa. (. In particular, if A is convex and λμ 0, then (λ + μa = λa + μa. Page 3 of

4 Mursaleen et al., Cogent Mathematics (07, 4: Furthermore, we can define the Hausdorff separation of B from A by d H (B, A =inf{ε >0 B A + εs }, (.3 where S denotes the closed unit ball in Y; i.e. S ={y Y y Y }. Meantime, the Hausdorff separation of A from B can also be defined in a similar way. Based on these two types of separations the Hausdorff distance between two nonempty subsets A and B is defined by d H (A, B =max{d (A, B, H d H (B, A}. (.4 In general, if A, B (Y or C (Y, then d H (λa, λb = λ d H (A, B for all λ R. In addition, according to some of the properties of Hausdorff distance, if we restrict our attention to the nonempty closed subsets (Y of Y, then it can be verified that ( (Y, d H is a metric space. In fact, it follows from Cădariu and Radu (003 that ( (Y, d H is a complete metric space. Clearly, (Y and C (Y are closed subsets of (Y. Hence, ( (Y, d H and ( C (Y, d H are also complete metric space. In Inoue (99 introduced the concept of Banach space valued fuzzy sets in order to extend the usual fuzzy sets defined on R or R n. In other words, the base space of a fuzzy set is replaced by a more general Banach space. For a given real separable Banach space Y, a fuzzy set defined on Y is a mapping u:y [0, ]. Denote by (Y the set of all fuzzy sets defined on Y. Let K (Y denote the class of fuzzy sets u:y [0, ] with the following properties: (i u is normal, i.e. [u] ={y Y u(y } is nonempty; (ii u is upper semi-continuous; (iii [u] α ={y Y u(y α} is compact for each α (0, ]; (iv [u] 0 = α (0,] [u]α. Notice that the conditions (ii and (iv imply that [u] 0 is also compact. Moreover, we use the notation KC (Y to denote the subspace of (Y whose members also satisfy (v u is fuzzy convex; i.e. [u] α is convex for each α (0, ]. A linear structure can be defined in (Y in a similar way to fuzzy sets in R or R n by (u v(z = sup min{u(x, v(y}, x+y=z (u v(z = (u(x v(y ( x, y, z R + z=x.y (.5 (.6 { (γu(z = u ( z γ if γ 0, I 0 (z if γ = 0, (.7 for u, v (Y and γ R, where I 0 (z =0 if z 0 and I 0 (0 =. Then (Y is closed under these operations and level set-wise [u v] α =[u] α +[v] α, [u v] α =[u] α [v] α, [λu] α = λ[u] α (.8 for each α (0, ] and λ R. Similar to the closeness of C (Y, it is easy to know that KC (Y is closed under these operations. Based on the statement mentioned above, we can easily obtain the following lemma. Page 4 of

5 Mursaleen et al., Cogent Mathematics (07, 4: Lemma. For any u, v KC (Y and λ, μ R, the following equalities hold: (i λ(u v =λu λv; (ii λ(μu =(λμu; (iii (λ + μu = λu μu for any λ, μ 0. Integer powers of a fuzzy number can be defined as follows (Kaufmann & Gupta, 984: Let u be a fuzzy number in R: u n = u u u (.9 ( First case: u R + ; here α [0, ]; therefore from [u] α =[u α, uα + ], we have ([u]α n = [(u α n, (u α + n ]. ( Second and general case: u R. This is a much more complicated case and requires the examination of the parity of n (the evenness or oddness. For n N: (i u u + 0: [u, u + ] n =[u n, un + ], if n is odd, =[u n, + un ], if n is even. (ii u 0 u +, u u + : [u, u + ] n =[u u n +, un + ]. (iii u 0 u +, u u + : [u, u + ] n =[u n, un u + ], if n is odd, =[u n + un ], if n is even. (iv 0 u u + : [u, u + ] n =[u n, un + ]. Remark. The Lemma. shows that KC (Y is just a cone defined on Y rather than a vector space. As a generalization of the Hausdorff metric d H in (Y, we will define the supremum metric in C (Y. For u, v C (Y, the supremum metric is defined by (u, v = sup d H ([u] α, [v] α α (0,] (.0 Remark.3 Every ordinary crisp subset A of Y can be identified with with the fuzzy set on Y by its characteristic function χ A :Y {0, }, i.e. with χ A (y = if y A and χ A (y =0 if y A. Therefore, if A (Y(or A C (Y, then χ A C (Y(or KC (Y, and vice versa. From Remark.3, for any A, B (Y(or KC (Y, it follows that ( (χ a, χ B = sup d H [χa ] α, [χ B ] α = d H (A, B. α (0,] (. In particular, if A and B degenerate into two singleton sets {a} and {b}, then we can infer from equality (. that (χ a, χ B =d(a, b, where d denotes the usual metric between a and b. Page 5 of

6 Mursaleen et al., Cogent Mathematics (07, 4: In view of the property of the Hausdorff metric, it is easy to see that (λu, λv =λ (u, v for any λ 0. Restricting attention to the set KC (Y, we can prove that ( KC (Y, is a complete metric space by the method analogous to that used in Diamond and Kloeden (994 (see Proposition Finally, we quote a fundamental result in fixed point theory. Theorem.4 (Diaz & Margolis, 968 Let (X, d be a complete generalized metric space, i.e. one for which d may assume infinite values. Suppose that J:X X is a strictly contractive mapping with Lipschitz constant L <. Then, for each given element x X, either d ( J n x, J n+ x = for all n 0 or there exists an n 0 N such that (i d ( J n x, J n+ x < for all n n 0 ; (ii the sequence {J n x} converges to a fixed point y of J; (iii y is the unique fixed point of J in the set Y ={y X d ( J n 0 x, y < }; (iv d(y, y ( ( Ld(y, Jy for all y Y. 3. Stability of the generalized Cauchy type additive fuzzy set-valued functional equation in n variables In this section, we will establish the Hyers-Ulam-Rassias stability of the generalized Cauchy type additive fuzzy set-valued functional equation by using fixed point alternative approach. Definition 3. Let X be a cone with the vertex 0 and let f :X KC (Y be a fuzzy set-valued mapping. The generalized Cauchy type additive fuzzy set-valued functional equation in n variables is defined by ( f x i = k f (x i (3. for all x, y X and for some > 0 with k. Especially, if =, for each i {,,, k}, then (3. is called the standard Cauchy type additive fuzzy set-valued functional equation in n variables. Every solution of (3. is called a generalized Cauchy type additive fuzzy set-valued mapping. Example 3. Let X = R + and Y = R. Suppose that f :R + KC (R is a triangular fuzzy set-valued mapping, i.e. for every t R +, f(t is a triangular fuzzy number in R, which is defined by f (t =(t at, t, t + bt, t R +, (3. where a and b are two non-negative real numbers. By the definition of α-level set, we can obtain that [f (t] α =[t at( α, t + bt( α] (3.3 for every t R +. Then, for every α [0, ], it is easy to verify that [ ( ] α f x i = [f (x i ] α for all x, x R + and > 0 with. That is, f is a solution of (3. in R +. (3.4 Remark 3.3 A triangular fuzzy number u KC (R is characterized by an ordered triple (x l, x c, x r R 3 with x l x c x r such that the support set [u] 0 =[x l, x r ] and -level set [u] ={x c }. Page 6 of

7 Mursaleen et al., Cogent Mathematics (07, 4: Remark 3.4 More generally, if X = R +, by Lemma., it is easy to see that f (t =tu 0 is a solution of (3. for any t R + and any fixed u 0 KC (Y. Theorem 3.5 Let φ:x k [0, + be a function such that there exists a positive constant L < (( ( ( L φ(x, x ( φ x, x,, x k (3.5 for all x i X and for some > 0 with k a i, i k. Suppose that f :X KC (Y is a mapping ( k (f x i, f (x i φ(x, x (3.6 for all x i X, i k. Then ( A(x =lim n x f ( n (3.7 exists for each x X and defines a unique generalized Cauchy type additive fuzzy set-valued mapping A:X KC (Y such that ( L f (x, A(x ( φ(x, x ( L (3.8 for all x. Proof Replacing x, x by x in (3.6, by Lemma., we get (( (f x, f (x φ(x, x,, x (3.9 Thus we can obtain f (x, x f ( x y L φ (, ( ( φ(x, x,, x (3.0 Consider the set E ={g g:x KC (Y, g(0 =I 0 } and introduce the generalized metric D on E, which is defined by Page 7 of

8 Mursaleen et al., Cogent Mathematics (07, 4: D(g, h =inf{μ (0, (g(x, h(x μφ(x, x,, x, x X} (3. where, as usual, inf =. It can be easily verified that (E, D is a complete generalized metric space (see Radu, 003. Now, we consider the linear mapping J:E E such that Jg(x = x g ( (3. Let g, h be given such that D(g, h ε. Then ( g(x, h(x εφ(x, x,, x (3.3 Hence, we have ( x Jg(x, Jh(x = d g (, = ε x h ( = Lεφ(x, x,, x So D(g, h ε implies that D ( Jg, Jh Lε. This mans that x x g (, h ( x x φ (, ( ( L ( ε D ( Jg, Jh LD(g, h (3.5 for all g, h E. Evidently, J is a strictly contractive self-mapping on E with the Lipschitz constant L <. ( Moreover, it follows from (3.0 that D(f, Jf L. According to Theorem.4, there exists a mapping A:X KC (Y the following. (i A is a fixed point of J: i.e. x ( A(x =A ( (3.6 Page 8 of

9 Mursaleen et al., Cogent Mathematics (07, 4: The mapping A is the unique fixed point of J in the set M ={g E D(f, g < }, (3.7 which implies that A is the unique mapping (3.6 such that there exists an r (0, ( f (x, A(x rφ(x, x,, x (3.8 (ii D ( J n f, A 0 as n. This implies the equality lim ( n x f ( n = A(x (3.9 (iii D(f, A ( ( LD ( f, Jf, which implies the inequality D(f, A L ( = (. ( L (3.0 By (3.5, we can obtain that ( n ( ( x n f k x i ( n, f i ( n ( n x φ x ( n, x ( n,, k ( n ( n L n. ( n φ(x, x,, x k = L n φ(x, x (3. which tends to zero as n for all x, x. Thus, ( A x i = k A(x i (3. for all x, x X and therefore the mapping A:X KC (Y is a generalized Cauchy type additive fuzzy set-valued mapping as desired. Page 9 of

10 Mursaleen et al., Cogent Mathematics (07, 4: Corollary 3.6 Let X be a cone with the vertex 0 contained in a real normed space and let p, θ be positive real numbers with p > (resp. 0 < p <. Suppose that f :X KC (Y is a mapping ( (f x i, ( k f (x i θ x i p (3.3 for all x, x X and for some > 0 with k > (resp. <. Then ( A(x =lim n x f ( n (3.4 exists for each x X and defines a unique generalized Cauchy type additive fuzzy set-valued mapping A:X KC (Y such that ( θk x p f (x, A(x ( p ( (3.5 ( ( k p Proof In Theorem 3.5, let φ(x, x =θ x i p. Then we can choose L = and we get the desired result. Corollary 3.7 Let X be a cone with the vertex 0 contained in a real normed space and let p, θ be positive real numbers with p > (resp. p <. Suppose that f :X KC (Y is a mapping ( (f x i, k k f (x i θ x i p (3.6 for all x, x X and for some > 0 with k > (resp. <. Then A(x =lim ( n x f ( n (3.7 exists for each x X and defines a unique generalized Cauchy type additive fuzzy set-valued mapping A:X KC (Y such that ( θ x kp f (x, A(x ( p ( (3.8 ( p k Proof In Theorem 3.5, let φ(x, x =θ x i p. Then we can choose L = and we get the desired result. Corollary 3.7 Let X be a cone with the vertex 0 contained in a real normed space and let p i ( i k, θ be positive real numbers with p = k p i > (resp. p = k p i <. Suppose that f :X KC (Y is a mapping Page 0 of

11 Mursaleen et al., Cogent Mathematics (07, 4: ( (f x i, ( k k f (x i θ x i p i + x i p (3.9 for all x, x X and for some > 0 with k > (resp. <. Then A(x =lim ( n x f ( n (3.30 exists for each x X and defines a unique generalized Cauchy type additive fuzzy set-valued mapping A:X KC (Y such that ( θ(k + x p f (x, A(x ( p ( (3.3 ( k Proof In Theorem 3.5, let φ(x, x =θ x i p i + x i p. Then we can choose ( p L = and we get the desired result. 4. Stability of the quadratic type fuzzy set-valued functional equation In this section, we will establish the Hyers-Ulam-Rassias stability of the quadratic type fuzzy setvalued functional equation by applying the same method as used in the previous Section 3. Definition 4. Let X be a cone with the vertex 0 and let f :X KC (Y be a fuzzy set valued mapping. The quadratic type fuzzy set-valued functional equation is defined by f (x + y f (x y =f (x f (y for all x, y X. Every solution of (4. is called a quadratic type fuzzy set-valued mapping. (4. Remark 4. Obviously, it can be checked that for the triangular fuzzy set-valued mapping f as defined in Example 3., f (t =tu 0 is a solution of (4. for any t R+ and any fixed u 0 KC (Y. Theorem 4.3 Let j {, } be fixed and let φ:x X [0, + be a function such that there exists a positive constant L < φ(x, x 4 j Lφ( j x, j x Moreover, assume that φ satisfies lim 4 jn φ( jn x, jn y=0 (4. (4.3 for all x, y X. If a mapping f :X KC (Y satisfies f (0 =I 0 and the inequality (f (x + y f (x y, f (x f (y φ(x, y (4.4 for all x, y X, then A(x =lim 4 jn f ( jn x (4.5 exists for each x X and defines a unique quadratic type fuzzy set-valued mapping A:X KC (Y such that Page of

12 Mursaleen et al., Cogent Mathematics (07, 4: (f (x, A(x Proof L 4( L 4( L Letting y = x in (4.4. Since u I 0 = u for any u KC (Y, we get ( f (x, 4 f (x φ(x, x 4 φ(x, x if j = φ(x, x if j = (4.6 (4.7 Furthermore, it follows from (4. that ( f (x,4f ( x φ(x, x (4.8 Consider the set E ={g g:x KC (Y, g(0 =I 0 } and introduce the generalized metric D on E, which is defined by D(g, h =inf{μ (0, ( g(x, h(x μφ(x, x, x X} (4.9 where, as usual, inf =. It can be easily verified that (E, D is a complete generalized metric space (see Radu, 003. Now, we consider the linear mapping J:E E such that Jg(x =4 j g( j x (4.0 Moreover, we can infer from (4.7 and (4.8 that D ( f, Jf { L 4 4 if j = if j =. (4. The rest of the proof is similar to the proof of Theorem 3.5. Corollary 4.4 Let j {, } be fixed and let p, θ be positive real numbers with p, and let X be a cone with the vertex 0 contained in a real normed space. Suppose that f :X KC (Y is a mapping (f (x + y f (x y,f (x f (y θ( x p + y p (4. for all x, y X. Then A(x =lim 4 jn f ( jn x (4.3 exists for each x X and defines a unique quadratic type fuzzy set-valued mapping Q:X KC (Y such that (f (x, Q(x { θ x p p, if j =, p > θ x p p, if j =, 0 < p < (4.4 Page of

13 Mursaleen et al., Cogent Mathematics (07, 4: Proof In Theorem 4.3, let φ(x, y =θ( x p + y p. Then we can choose L = j(p and we get the desired result. Corollary 4.5 Let j {, } be fixed and let p, θ be positive real numbers with p, and let X be a cone with the vertex 0 contained in a real normed space. Suppose that f :X KC (Y is a mapping (f (x + y f (x y,f (x f (y θ x p y p (4.5 for all x, y X. Then A(x =lim 4 jn f ( jn x (4.6 exists for each x X and defines a unique quadratic type fuzzy set-valued mapping Q:X KC (Y such that (f (x, Q(x θ x p 4(4 p θ x p 4( 4 p, if j =, p >, if j =, 0 < p < (4.7 Proof result. In Theorem 4.3, let φ(x, y =θ x p y p. then we can choose L = j(p and we get the desired Corollary 4.6 Let j {, } be fixed and let p, θ be positive real numbers with p + q, and let X be a cone with the vertex 0 contained in a real normed space. Suppose that f :X KC (Y is a mapping (f (x + y f (x y, f (x f (y θ ( x p y q + x p+q + y p+q (4.8 for all x, y X. Then A(x =lim 4 jn f ( jn x (4.9 exists for each x X and defines a unique quadratic type fuzzy set-valued mapping Q:X KC (Y such that (f (x, Q(x { 3θ x p+q, p+q 4 if j =, p + q > 3θ x p+q, 4 p+q if j =, 0 < p + q < (4.0 Proof In Theorem 4.3, let φ(x, y =θ ( x p y q + x p+q + y p+q. Then we can choose L = j(p+q and we get the desired result. 5. Stability of the quadratic-jensen type fuzzy set-valued functional equation In this section, we will prove the Hyers-Ulam-Rassias stability of the quadratic-jensen type fuzzy set-valued functional equation by using the same method as applied in Section 4. Page 3 of

14 Mursaleen et al., Cogent Mathematics (07, 4: Definition 5. Let X be a cone with the vertex 0 and let f :X KC (Y be a fuzzy set valued mapping. The quadratic-jensen type fuzzy set-valued functional equation is defined by ( x + y f f ( x y = f (x f (y for all x, y X. Every solution of (5. is called a Jensen type quadratic fuzzy set-valued mapping. Remark 5. Obviously, it can be checked that for the triangular fuzzy set-valued mapping f as defined in Example 3., f (t =tu 0 is a solution of (5. for any t R+ and any fixed u 0 KC (Y. (5. Theorem 5.3 Let j {, } be fixed and let φ:x X [0, + be a function such that there exists a positive constant L < φ(x,0 4 j Lφ( j x,0 Moreover, assume that φ satisfies lim 4jn φ( jn x, jn y=0 for all x, y X. If a mapping f :X KC Y satisfies f (0 =I 0 and the inequality ( x + y (f f ( x y, f (x f (y φ(x, y (5. (5.3 (5.4 for all x, y X, then A(x =lim 4 jn f ( jn x (5.5 exists for each x X and defines a unique quadratic-jensen type fuzzy set-valued mapping A:X KC (Y such that (f (x, A(x L ( L ( L φ(x,0 if j = φ(x,0 if j = (5.6 Proof Letting y = 0 in (5.4. Since u I 0 = u for any u KC (Y, we get ( f (x,4f ( x φ(x,0 Furthermore, it follows from (5.7 that ( f (x, 4 f (x Lφ(x,0 (5.7 (5.8 Consider the set E ={g g:x KC (Y, g(0 =I 0 } and introduce the generalized metric D on E, which is defined by D(g, h =inf{μ (0, ( g(x, h(x μφ(x,0, x X} (5.9 where, as usual, inf =. It can be easily verified that (E, D is a complete generalized metric space (see Radu, 003. Page 4 of

15 Mursaleen et al., Cogent Mathematics (07, 4: Now, we consider the linear mapping J:E E such that Jg(x =4 j g( j x (5.0 Moreover, we can infer from (5.7 and (5.8 that D ( f, Jf { L if j = if j =. (5. The rest of the proof is similar to the proof of Theorem 3.5. Corollary 5.4 Let j {, } be fixed and let p, θ be positive real numbers with p, and let X be a cone with the vertex 0 contained in a real normed space. Suppose that f :X KC (Y is a mapping ( x + y (f f for all x, y X. Then ( x y, f (x f (y θ( x p + y p (5. A(x =lim 4 jn f ( jn x (5.3 exists for each x X and defines a unique quadratic type fuzzy set-valued mapping Q:X KC (Y such that (f (x, Q(x { θ x p, p if j =, p > θ x p, if j =, 0 < p < p (5.4 Proof In Theorem 5.3, let φ(x, y =θ( x p + y p. Then we can choose L = j( p and we get the desired result. 6. Stability of the cubic type fuzzy set-valued functional equation In this section, we will prove the Hyers-Ulam-Rassias stability of the cubic type fuzzy set-valued functional equation by using the same method as applied in Section 6. Definition 6. Let X be a cone with the vertex 0 and let f :X KC (Y be a fuzzy set valued mapping. The cubic type fuzzy set-valued functional equation is defined by f (x + y f (x y =f (x + y f (x y f (x (6. for all x, y X. Every solution of (6. is called a cubic type fuzzy set-valued mapping. Remark 6. Obviously, it can be checked that for the triangular fuzzy set-valued mapping f as defined in Example 3., f (t =tu 3 0 is a solution of (6. for any t R+ and any fixed u 0 KC (Y. Page 5 of

16 Mursaleen et al., Cogent Mathematics (07, 4: Theorem 6.3 Let j {, } be fixed and let φ:x X [0, + be a function such that there exists a positive constant L < φ(x,0 8 j Lφ( j x,0 Moreover, assume that φ satisfies lim 8 jn φ( jn x, jn y=0 (6. (6.3 for all x, y X. If a mapping f :X KC (Y satisfies f (0 =I 0 and the inequality ( f (x + y f (x y, f (x + y f (x y f (x φ(x, y (6.4 for all x, y X, then A(x =lim 8 jn f ( jn x (6.5 exists for each x X and defines a unique cubic type fuzzy set-valued mapping C:X KC Y such that (f (x, A(x 8( L L 8( L φ(x,0 if j = φ(x,0 if j = (6.6 Proof Letting y = 0 in (6.4. Since u I 0 = u for any u KC (Y, we get ( f (x, 8 f (x 8 φ(x,0 (6.7 Furthermore, it follows from (6. that ( f (x,8f ( x L 8 φ(x,0 (6.8 Consider the set E ={g g:x KC (Y, g(0 =I 0 } and introduce the generalized metric D on E, which is defined by D(g, h =inf{μ (0, ( g(x, h(x μφ(x,0, x X} (6.9 where, as usual, inf =. It can be easily verified that (E, D is a complete generalized metric space (see Radu, 003. Now, we consider the linear mapping J:E E such that Jg(x =8 j g( j x (6.0 Moreover, we can infer from (6.7 and (6.8 that D ( f, Jf { 8 L 8 if j = if j =. (6. The rest of the proof is similar to the proof of Theorem 3.5. Page 6 of

17 Mursaleen et al., Cogent Mathematics (07, 4: Corollary 6.4 Let j {, } be fixed and let p, θ be positive real numbers with p 3, and let X be a cone with the vertex 0 contained in a real normed space. Suppose that f :X KC (Y is a mapping ( f (x + y f (x y,f (x + y f (x y f (x θ( x p + y p (6. for all x, y X. Then A(x =lim 8 jn f ( jn x (6.3 exists for each x X and defines a unique cubic type fuzzy set-valued mapping C:X KC (Y such that (f (x, C(x { θ x p, if j =, 0 < p < 3 8 p, if j =, p > 3 θ x p p 8 (6.4 Proof In Theorem 6.3, let φ(x, y =θ( x p + y p. Then we can choose L = j(p 3 and we get the desired result. 7. Stability of the Jensen type cubic fuzzy set-valued functional equation In this section, we will prove the Hyers-Ulam-Rassias stability of the Jensen type cubic fuzzy setvalued functional equation by using the same method as applied in Section 6. Definition 7. Let X be a cone with the vertex 0 and let f :X KC (Y be a fuzzy set valued mapping. The Jensen type cubic fuzzy set-valued functional equation is defined by ( 3x + y f f ( x + 3y = f ( x + y f (x f (y for all x, y X. Every solution of (7. is called a Jensen type cubic fuzzy set-valued mapping. Remark 7. Obviously, it can be checked that for the triangular fuzzy set-valued mapping f as defined in Example 3., f (t =tu 3 0 is a solution of (4. for any t R+ and any fixed u 0 KC (Y. (7. Theorem 7.3 Let j {, } be fixed and let φ:x X [0, + be a function such that there exists a positive constant L < φ(x, y 8 j Lφ( j x, j y Moreover, assume that φ satisfies lim 8jn φ( jn x, jn y=0 for all x, y X. If a mapping f :X KC (Y satisfies f (0 =I 0 and the inequality ( ( 3x + y x + 3y ( x + y (f f, f f (x f (y φ(x, y for all x, y X, then ( A(x =lim 8 jn x f jn (7. (7.3 (7.4 (7.5 Page 7 of

18 Mursaleen et al., Cogent Mathematics (07, 4: exists for each x X and defines a unique cubic type fuzzy set-valued mapping C:X KC (Y such that (f (x, A(x L 8( L 8( L φ(x, x if j = φ(x, x if j = (7.6 Proof Letting y = x in (7.4. Since u I 0 = u for any u KC (Y, we get ( f (x, 8 f (x φ(x, x 8 (7.7 Furthermore, it follows from (7. that ( x (f (x,8f L φ(x, x 8 (7.8 Consider the set E ={g g:x KC (Y, g(0 =I 0 } and introduce the generalized metric D on E, which is defined by D(g, h =inf{μ (0, ( g(x, h(x μφ(x, x, x X} (7.9 where, as usual, inf =. It can be easily verified that (E, D is a complete generalized metric space (see Radu, 003. Now, we consider the linear mapping J:E E such that ( Jg(x =8 j x g j (7.0 Moreover, we can infer from (7.7 and (7.8 that D ( f, Jf { L 8 8 if j = if j =. (7. The rest of the proof is similar to the proof of Theorem 3.5. Corollary 7.4 Let j {, } be fixed and let p, θ be positive real numbers with p 3, and let X be a cone with the vertex 0 contained in a real normed space. Suppose that f :X KC (Y is a mapping ( 3x + y (f f ( x + 3y, f ( x + y f (x f (y θ( x p + y p (7. for all x, y X. Then A(x =lim 8 jn f ( jn x (7.3 exists for each x X and defines a unique cubic-jensen type fuzzy set-valued mapping C:X KC (Y such that Page 8 of

19 Mursaleen et al., Cogent Mathematics (07, 4: (f (x, C(x { θ x p p 8, if j =, p > 3 θ x p 8 p, if j =, 0 < p < 3 (7.4 Proof In Theorem 7.3, let φ(x, y =θ( x p + y p. Then we can choose L = j(3 p and we get the desired result. Corollary 7.5 Let j {, } be fixed and let p, θ be positive real numbers with p 3, and let X be a cone with the vertex 0 contained in a real normed space. Suppose that f :X KC (Y is a mapping ( 3x + y (f f for all x, y X. Then ( x + 3y, f ( x + y f (x f (y θ x p y p (7.5 A(x =lim 8 jn f ( jn x (7.6 exists for each x X and defines a unique cubic-jensen type fuzzy set-valued mapping C:X KC (Y such that (f (x, C(x { θ x p 4 p 8, if j =, p > 3 θ x p 8 4 p, if j =, 0 < p < 3 (7.7 Proof result. In Theorem 7.3, let φ(x, y =θ x p y p. Then we can choose L = j(3 p and we get the desired Corollary 7.6 Let j {, } be fixed and let p, θ be positive real numbers with p + q 3, and let X be a cone with the vertex 0 contained in a real normed space. Suppose that f :X KC (Y is a mapping ( 3x + y (f f for all x, y X. Then ( x + 3y, f ( x + y f (x f (y θ ( x p y q + x p+q + y p+q (7.8 A(x =lim 8 jn f ( jn x (7.9 exists for each x X and defines a unique cubic-jensen type fuzzy set-valued mapping C:X KC (Y such that (f (x, C(x { 3θ x p+q p+q 8, if j =, p + q > 3 3θ x p+q 8 p+q, if j =, 0 < p + q < 3 (7.0 Proof In Theorem 7.3, let φ(x, y =θ ( x p y q + x p+q + y p+q. then we can choose L = j(3 p q and we get the desired result. Page 9 of

20 Mursaleen et al., Cogent Mathematics (07, 4: Funding The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia. Author details M. Mursaleen, S.A. Mohiuddine Khursheed J. Ansari 3 ansari.jkhursheed@gmail.com Department of Mathematics, Aligarh Muslim University, Aligarh 000, India. Operator Theory and Applications Research Group, Faculty of Science, Department of Mathematics, King Abdulaziz University, P.O. Box 8003, Jeddah 589, Saudi Arabia. 3 Department of Mathematics, King Khalid University, Abha, Saudi Arabia. Citation information Cite this article as: On the stability of fuzzy set-valued functional equations, M. Mursaleen, S.A. Mohiuddine & Khursheed J. Ansari, Cogent Mathematics (07, 4: References Al-Fhaid, A. S., & Mohiuddine, S. A. (03. On the Ulam stability of mixed type QA mappings in IFN-spaces. Advances in Difference Equations, 03, Article ID 03. Alotaibi, A., & Mohiuddine, S. A. (0. 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