Theory of fuzzy integrals and its applications

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

Some integral inequalities for fuzzy-interval-valued functions

In this note the concept of convexity and Jensen’s Inequality are reviewed. Jensen’s Inequality plays a central role in the derivation of the Expectation Maximization algorithm [1] and the proof of consistency of maximum likelihood estimators. Definition Let f(x) be a real valued function defined on the interval I = [a, b]. f is said to be convex if for every x1, x2 ∈ [a, b] and 0 ≥ λ ≥ 1, f(λx1 + (1− λ)x2) ≥ λf(x1) + (1− λ)f(x2) A function is said to be strictly convex if the equality is strict for x1 6= x2. Definition f(x) is said to be concave (strictly concave) if −f(x) is convex (strictly convex). Intuitively, the definition of convexity states that function falls below never above the straight line between the points (x1, f(x1)) to (x2, f(x2)) (see Fig. 1). Theorem 0.1 If f ′′(x) exists on [a, b] and f ′′(x) ≥ 0 on [a, b] then f(x) is convex on [a, b]. Theorem 0.2 (Jensen’s Inequality) Let f(x) be a convex function defined on an interval I. If x1, x2, . . . , xN ∈ I and λ1, λ2, . . . , λN ≥ 0 with ∑N i=1 λi,

Jensen's inequality

In this paper, we use a new method to obtain the necessary and sufficient condition guaranteeing the validity of the Minkowski-H\"{o}lder type inequality for the generalized upper Sugeno integral in the case of functions belonging to a wider class than the comonotone functions. As a by-product, we show that the Minkowski type inequality for seminormed fuzzy integral presented by Daraby and Ghadimi in General Minkowski type and related inequalities for seminormed fuzzy integrals, Sahand Communications in Mathematical Analysis 1 (2014) 9--20 is not true. Next, we study the Minkowski-H\"{o}lder inequality for the lower Sugeno integral and the class of $\mu$-subadditive functions introduced in On Chebyshev type inequalities for generalized Sugeno integrals, Fuzzy Sets and Systems 244 (2014) 51--62. The results are applied to derive new metrics on the space of measurable functions in the setting of nonadditive measure theory. We also give a partial answer to the open problem 2.22 posed by Borzov\'a-Moln\'arov\'a and et al in The smallest semicopula-based universal integrals I: Properties and characterizations, Fuzzy Sets and Systems 271 (2015) 1--17.

On the Minkowski-H\"{o}lder type inequalities for generalized Sugeno integrals with an application

In this paper, we introduce a new iterative scheme to approximate fixed point of generalized α-nonexpansive mappings and then, we prove that the proposed iteration process is faster than all of Picard, Mann, Ishikawa, Noor, Agarwal, Abbas, and Thakur processes for contractive mappings We also obtain some weak and strong convergence theorems for generalized α-nonexpansive mappings At the end, by using an example for generalized α-nonexpansive mappings, we compare the convergence behavior of new iterative process with other iterative processes

Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces by new faster iteration process

Generalization of the Stolarsky type inequality for pseudo-integrals

In this note, we present some properties of Felbin-type fuzzy inner product spaces and fuzzy bounded linear operators on the same spaces with some operator norms. At the first, we study representation of functionals on fuzzy Hilbert spaces. We introduce the fuzzy Hilbert-adjoint operator of a bounded linear operator on a fuzzy Hilbert space and define fuzzy adjoint operator. By a counterexample, it is shown that uniqueness of fuzzy Hilbert-adjoint operator is not valid for this fuzzy setting. Also, we prove that the fuzzy Hilbert-adjoint operator has the same norm as the operator itself and it holds similarly for fuzzy adjoint operator. Finally we study the relation between the fuzzy adjoint operator and the fuzzy Hilbert-adjoint operator.

Some Properties of Fuzzy Hilbert Spaces

General related inequalities to Carlson-type inequality for the Sugeno integral

Recently, Hasankhani et al. proved that any Felbin-fuzzy inner product space can be imbedded in a complete Felbin-fuzzy inner product space or Felbin-fuzzy Hilbert space. In this paper, it is showed a general result that any classical Hilbert space is a Felbin-fuzzy Hilbert space, so it shows that all results in classical Hilbert spaces are immediate consequences of the corresponding results for Felbin-fuzzy Hilbert spaces. Moreover by an example, it is showed that the spectrum of the category of Felbin-fuzzy Hilbert spaces is broader than the category of classical Hilbert spaces. Finally the authors are able to state a transformation theorem from an ascending family of crisp inner product into Felbin-fuzzy inner product that shows how to formulate the recent results.

Bayaz Daraby

Papers

Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces by new faster iteration process

Generalization of the Stolarsky type inequality for pseudo-integrals

Some Properties of Fuzzy Hilbert Spaces

General related inequalities to Carlson-type inequality for the Sugeno integral

A note on fuzzy Hilbert spaces