Báo cáo hóa học: Research Article General Fritz Carlson’s Type Inequality for Sugeno Integrals

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article General Fritz Carlson’s Type Inequality for Sugeno Integrals
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Báo cáo hóa học: " Research Article General Fritz Carlson’s Type Inequality for Sugeno Integrals"Hindawi Publishing CorporationJournal of Inequalities and ApplicationsVolume 2011, Article ID 761430, 9 pagesdoi:10.1155/2011/761430Research ArticleGeneral Fritz Carlson’s Type Inequality forSugeno Integrals Xiaojing Wang and Chuanzhi Bai Department of Mathematics, Huaiyin Normal University, Huaian, Jiangsu 223300, China Correspondence should be addressed to Chuanzhi Bai, czbai8@sohu.com Received 18 August 2010; Revised 23 November 2010; Accepted 20 January 2011 Academic Editor: L´ szlo Losonczi a´ Copyright q 2011 X. Wang and C. Bai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Fritz Carlson’s type inequality for fuzzy integrals is studied in a rather general form. The main results of this paper generalize some previous results.1. Introduction and PreliminariesRecently, the study of fuzzy integral inequalities has gained much attention. The most popu-lar method is using the Sugeno integral 1 . The study of inequalities for Sugeno integral wasinitiated by Rom´ n-Flores et al. 2, 3 and then followed by the others 4–11 . a Now, we introduce some basic notation and properties. For details, we refer the readerto 1, 12 . Suppose that Σ is a σ -algebra of subsets of X , and let μ : Σ → 0, ∞ be a nonnegative,extended real-valued set function. We say that μ is a fuzzy measure if it satisfies 1 μ∅ 0, 2 E, F ∈ Σ and E ⊂ F imply μ E ≤ μ F monotonicity ; ∞ 3 {En } ⊂ Σ, E1 ⊂ E2 ⊂ · · · imply limn → ∞ μ En μ En continuity from below , n1 ∞ 4 {En } ⊂ Σ, E1 ⊃ E2 ⊃ · · · , μ E1 < ∞, imply limn → ∞ μ En μ En continuity n1 from above . If f is a nonnegative real-valued function defined on X , we will denote by Lα f {x ∈X : f x ≥ α} {f ≥ α} the α-level of f for α > 0, and L0 f {x ∈ : f x > 0} supp f isthe support of f . Note that if α ≤ β, then {f ≥ β} ⊂ {f ≥ α}. μ Let X, Σ, μ be a fuzzy measure space; by F X we denote the set of all nonnegativeμ-measurable functions with respect to Σ.2 Journal of Inequalities and Applications μDefinition 1.1 see 1 . Let X, Σ, μ be a fuzzy measure space, with f ∈ F X , and A ∈ Σ,then the Sugeno integral or fuzzy integral of f on A with respect to the fuzzy measure μ isdefined by α∧μ A∩ f ≥α fdμ , 1.1 A α≥0where ∨ and ∧ denote the operations sup and inf on 0, ∞ , respectively. It is well known that the Sugeno integral is a type of nonlinear integral; that is, forgeneral cases, af bg dμ a f dμ b g dμ 1.2does not hold. The following properties of the fuzzy integral are well known and can be found in 12 . μProposition 1.2. Let X, Σ, μ be a fuzzy measure space, with A, B ∈ Σ and f, g ∈ F X ; then fdμ ≤ μ A , 1 A k ∧ μ A , for k a nonnegative constant, 2 kdμ A 3 if f ≤ g on A then fdμ ≤ gdμ, A A 4 if A ⊂ B then fdμ ≤ fdμ, A A 5 μ A ∩ {f ≥ α} ≥ α ⇒ fdμ ≥ α, A 6 μ A ∩ {f ≥ α} ≤ α ⇒ f dμ ≤ α, A fdμ < α ⇔ there exists γ < α such that μ A ∩ {f ≥ γ } < α, 7 A fdμ > α ⇔ there exists γ > α such that μ A ∩ {f ≥ γ } > α. 8 A μ A∩Remark 1.3. Let F be the distribution function associated with f on A, that is, F α{f ≥ α} . By 5 and 6 of Proposition 1.2 α⇒ Fα fdμ α. 1.3 AThus, from a numerical point of view, the Sugeno integral can be calculated by solving theequation F α α. Fritz Carlson’s integral inequality states 13, 14 that ¢ ¢ ¢ ∞ ∞ ∞ √ 1/4 1/4 f x dx ≤ ...