Existence of extremal solutions for quadratic fuzzy equations
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Some results on the existence of solution for certain fuzzy equations are revised and extended. In this paper, we establish the existence of a solution for the fuzzy equation Ex2 + Fx + G = x, where E, F, G, and x are positive fuzzy numbers satisfying certain conditions. To this purpose, we use fixed point theory, applying results such as the wellknown fixed point theorem of Tarski, presenting some results regarding the existence of extremal solutions to the above equation. 1. Preliminaries In [1], it is studied the existence of extremal fixed points for a map defined in a subset of the set E1 of fuzzy real numbers, that is, the family of elements x : R → [0,1] with the properties: (i) x is normal: there exists t0 ∈ R with x(t0 ) = 1. (ii) x is upper semicontinuous. (iii) x is fuzzy convex,
x λt1 + (1 − λ)t2 ≥ min x t1 ,x t2 ,
∀t1 , t2 ∈ R,λ ∈ [0,1].
(1.1)
(iv) The support of x, supp(x) = cl({t ∈ R : x(t) > 0}) is a bounded subset of R. In the following, for a fuzzy number x ∈ E1 , we denote the α-level set
[x]α = t ∈ R : x(t) ≥ α
(1.2)
by the interval [xαl ,xαr ], for each α ∈ (0,1], and
[x]0 = cl ∪α∈(0,1] [x]α = x0l ,x0r .
(1.3)
Note that this notation is possible, since the properties of the fuzzy number x guarantee that [x]α is a nonempty compact convex subset of R, for each α ∈ [0,1]. Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:3 (2005) 321–342 DOI: 10.1155/FPTA.2005.321
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Existence of extremal solutions for quadratic fuzzy equations
We consider the partial ordering ≤ in E1 given by x, y ∈ E1 ,
x ≤ y ⇐⇒ xαl ≤ yαl ,
xαr ≤ yαr ,
∀α ∈ (0,1],
(1.4)
and the distance that provides E1 the structure of complete metric space is given by
d∞ (x, y) = sup dH [x]α ,[y]α , α∈[0,1]
for x, y ∈ E1 ,
(1.5)
being dH the Hausdorff distance between nonempty compact convex subsets of R (that is, compact intervals). For each fuzzy number x ∈ E1 , we define the functions xL : [0,1] → R, xR : [0,1] → R given by xL (α) = xαl and xR (α) = xαr , for each α ∈ [0,1]. Theorem 1.1 [1, Theorem 2.3]. Let u0 , v0 ∈ E1 , u0 < v0 . Let
B ⊂ u0 ,v0 = x ∈ E1 : u0 ≤ x ≤ v0
(1.6)
be a closed set of E1 such that u0 ,v0 ∈ B. Suppose that A : B → B is an increasing operator such that u0 ≤ Au0 ,
Av0 ≤ v0 ,
(1.7)
and A is condensing, that is, A is continuous, bounded and r(A(S)) < r(S) for any bounded set S ⊂ B with r(S) > 0, where r(S) denotes the measure of noncompactness of S. Then A has a maximal fixed point x∗ and a minimal fixed point x∗ in B, moreover x∗ = lim vn ,
x∗ = lim un ,
n−→+∞
n−→+∞
(1.8)
where vn = Avn−1 and un = Aun−1 , n = 1,2,... and u0 ≤ u1 ≤ · · · ≤ un ≤ · · · ≤ vn ≤ · · · ≤ v1 ≤ v0 .
(1.9)
Corollary 1.2 [1, Corollary 2.4]. In the hypotheses of Theorem 1.1, if A has a unique fixed point x¯ in B, then, for any x0 ∈ B, the successive iterates xn = Axn−1 ,
n = 1,2,...
(1.10)
¯ that is, d∞ (xn , x) ¯ → 0 as n → +∞. converge to x, Theorem 1.1 is used in [1] to solve the fuzzy equation Ex2 + Fx + G = x,
(1.11)
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