Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations

Alikhani, Robab; Bahrami, Fariba; Jabbari, Adel

doi:10.1016/j.na.2011.09.021

Cited by 51 publications

(29 citation statements)

References 17 publications

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“…Since the inception of the fuzzy set theory, Soliman and Mantawy [5] showed that five major strongly connected branches have been developed, including fuzzy mathematics, fuzzy logic and artificial intelligence, fuzzy systems, uncertainty and information, and fuzzy decision-making. Their subbranches have also been established; for example, fuzzy differential equations [6][7][8][9][10][11][12][13][14] and fuzzy integrodifferential equations [15][16][17][18][19][20][21][22] are of fuzzy mathematics while fuzzy-number ranking, the focus of this paper, is of fuzzy decision-making. Specifically, based on its feasible mathematical capacity for representing the imprecise information in practice, we have observed many successful cases spreading in disparate disciplines, such as robot selection [23], supplier selection [24], logistics center allocation [25], facility location determination [26], choosing mining methods [27], manufacturing process monitoring [1,2,[28][29][30][31], cutting force prediction [32], firm-environmental knowledge management [33,34], green supply-chain operation [35], and weapon procurement decision [36].…”

Section: Introductionmentioning

confidence: 99%

Methods in Ranking Fuzzy Numbers: A Unified Index and Comparative Reviews

Nguyễn¹

2017

Complexity

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Fuzzy set theory, extensively applied in abundant disciplines, has been recognized as a plausible tool in dealing with uncertain and vague information due to its prowess in mathematically manipulating the knowledge of imprecision. In fuzzy-data comparisons, exploring the general ranking measure that is capable of consistently differentiating the magnitude of fuzzy numbers has widely captivated academics' attention. To date, numerous indices have been established; however, counterintuition, less discrimination, and/or inconsistency on their fuzzy-number rating outcomes have prohibited their comprehensive implementation. To ameliorate their manifested ranking weaknesses, this paper proposes a unified index that multiplies weighted-mean and weighted-area discriminatory components of a fuzzy number, respectively, called centroid value and attitude-incorporated left-and-right area. From theoretical proof of consistency property and comparative studies for triangular, triangular-and-trapezoidal mixed, and nonlinear fuzzy numbers, the unified index demonstrates conspicuous ranking gains in terms of intuition support, consistency, reliability, and computational simplicity capability. More importantly, the unified index possesses the consistency property for ranking fuzzy numbers and their images as well as for symmetric fuzzy numbers with an identical altitude which is a rather critical property for accurate matching and/or retrieval of information in the field of computer vision and image pattern recognition.

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Section: Introductionmentioning

confidence: 99%

Methods in Ranking Fuzzy Numbers: A Unified Index and Comparative Reviews

Nguyễn¹

2017

Complexity

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“…[5,6,13,17]). Recently, there are some papers dealing with the existence of solution for nonlinear set valued and fuzzy differential equations whose methods are based on the monotone method, the method of upper and lower solutions and fixed point theorems [13,7,1,3,2]. Some works also have been done on 154 R. Alikhani and F. Bahrami the existence and uniqueness results of solutions for interval-valued second-order differential equations by contraction principle and successive approximations [9,10].…”

Section: Introductionmentioning

confidence: 99%

“…Some works also have been done on 154 R. Alikhani and F. Bahrami the existence and uniqueness results of solutions for interval-valued second-order differential equations by contraction principle and successive approximations [9,10]. Moreover, author in [3], has proved the existence and uniqueness of global solutions for fuzzy integro-differential equation of Volterra type by means of the fixed point theory, the successive iteration method and Gronwall inequality. Among of them, we can find results on existence of solution for fuzzy differential equations in presence of both lower and upper solutions relative to the problem considered.…”

Section: Introductionmentioning

confidence: 99%

Global solutions to nonlinear second order interval integrodifferential equations by fixed point in partially ordered sets

Alikhani

Bahrani

2018

B Soc Paran Mat

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In this paper, we prove the existence and uniqueness of global solution for second order interval valued integrodifferential equation with initial conditions admitting only the existence of a lower solution or an upper solution. In this study, in order to make the global solution on entire [a, b], we use a fixed point in partially ordered sets on the subintervals of [a, b] and obtain local solutions. Also, under weak conditions we show being well-defined a special kind of H-difference involved in this work. Moreover, we compare the results of existence and uniqueness under consideration of two kind of partial ordering on fuzzy numbers.

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“…Alikharni et al ( [1]) proved the existence of global solutions to nonlinear fuzzy Volterra integrodifferential equations. Balachandran and Duar ( [3]) established the existence of perturbed fuzzy integral equations and fuzzy delay differential equations with nonlocal conditions.…”

Section: Introductionmentioning

confidence: 99%