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22
From Fractal Image Compression to Fractalbased Methods in Mathematics
"... Introduction In keeping with the philosophy of this workshop, the aim of this presentation is to provide an overview of the research done over the years at Waterloo on fractalbased methods of approximation and associated inverse problems. Near the end, some new and encouraging results regarding \f ..."
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Introduction In keeping with the philosophy of this workshop, the aim of this presentation is to provide an overview of the research done over the years at Waterloo on fractalbased methods of approximation and associated inverse problems. Near the end, some new and encouraging results regarding \fractal enhancement" will be presented. The paper concludes with thoughts and challenges on how the mathematical methods that underlie fractal image compression could be used in other areas of mathematics. Let us go back to rst principles for a moment in order to recall some of the early thinking behind fractal image compression (FIC). In fact, since the early work of Barnsley, Jacquin et al., there has been very little change in the basic idea of FIC. Most eorts have focussed on developing strategies to perform \collage coding" as eectively as possible { whether it be in the pixel or wavelet domain. This includes the the competition between employing the largest possible domain pools and
ITERATIVE ALGORITHM FOR A NEW SYSTEM OF NONLINEAR SETVALUED VARIATIONAL INCLUSIONS INVOLVING (H, η)MONOTONE MAPPINGS
"... ABSTRACT. In this paper, a new system of nonlinear setvalued variational inclusions involving (H, η)monotone mappings in Hilbert spaces is introduced and studied. By using the resolvent operator method associated with (H, η)monotone mappings, an existence theorem of solutions for this kind of sys ..."
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ABSTRACT. In this paper, a new system of nonlinear setvalued variational inclusions involving (H, η)monotone mappings in Hilbert spaces is introduced and studied. By using the resolvent operator method associated with (H, η)monotone mappings, an existence theorem of solutions for this kind of system of nonlinear setvalued variational inclusion is established and a new iterative algorithm is suggested and discussed. The results presented in this paper improve and generalize some recent results in this field. Key words and phrases: (H, η)monotone mapping; System of nonlinear setvalued variational inclusions; Resolvent operator method; Iterative algorithm. 2000 Mathematics Subject Classification. 49J40; 47H10. 1.
SetValued Maps With Fixed And Coincidence Points
, 1996
"... . In this paper, we prove the existence of solutions of functional equations f i x 2 Sx " Tx and x = f i x 2 Sx " Tx under certain nonlinear hybrid contraction and asymptotic regularity conditions, generalize and improve a recent result due to V. Popa concerning common fixed points of multivalued ..."
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. In this paper, we prove the existence of solutions of functional equations f i x 2 Sx " Tx and x = f i x 2 Sx " Tx under certain nonlinear hybrid contraction and asymptotic regularity conditions, generalize and improve a recent result due to V. Popa concerning common fixed points of multivalued mappings weakly commuting with a singlevalued mapping and satisfying a generalized contraction type condition. In the sequel, we derive several independent results. (1991) A.M.S. (MOS) Subject Classification Codes. 54H25, 47H10. Keywords and Phrases. Coincidence and common fixed points, Hausdorffmetric, orbital complete, proximinal subset & weakly commuting mappings. I. Introduction In [16], Nadler proved a fixed point theorem for multivalued contraction, commonly known as Nadler's contraction principle. Subsequently, a large number of generalizations of Nadler's contraction principle appeared in Ciric [1], Khan [10], Kubiak [12], [13], Kaneko [8], [9], Sessa [22][24], MeadeSingh [14]...
Convergence of Iterative Schemes for Multivalued QuasiVariational Inclusions
"... Relying on the resolvent operator method and using Nadler's theorem, we suggest and analyze a class of iterative schemes for solving multivalued quasivariational inclusions. In fact, by considering problems involving composition of mutivalued operators and by replacing the usual compactness conditi ..."
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Relying on the resolvent operator method and using Nadler's theorem, we suggest and analyze a class of iterative schemes for solving multivalued quasivariational inclusions. In fact, by considering problems involving composition of mutivalued operators and by replacing the usual compactness condition by a weaker one, our result can be considered as an improvement and a signicant extension of previously known results in this eld.
ERROR BOUNDS FOR DEGENERATE CONE INCLUSION PROBLEMS
"... Abstract. Error bounds for cone inclusion problems in Banach spaces are established under conditions weaker than Robinson’s constraint qualification. The results allow the cone to be more general than the origin, therefore also generalize a classical error bound result concerning equalityconstrained ..."
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Abstract. Error bounds for cone inclusion problems in Banach spaces are established under conditions weaker than Robinson’s constraint qualification. The results allow the cone to be more general than the origin, therefore also generalize a classical error bound result concerning equalityconstrained sets in optimization. Key words. cone inclusion problems, error bounds, Robinson’s constraint qualification, tangent cones.
COMMON FIXED POINT OF MULTIVALUED MAPPINGS WITHOUT CONTINUITY
"... Abstract: In this paper, we prove a common fixed point theorem for singlevalued and multivalued mappings on a metric space using the minimal type commutativity condition. We show that continuity of any mapping is not necessary for the existence of a common fixed point. Key words: common fixed point ..."
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Abstract: In this paper, we prove a common fixed point theorem for singlevalued and multivalued mappings on a metric space using the minimal type commutativity condition. We show that continuity of any mapping is not necessary for the existence of a common fixed point. Key words: common fixed point, coincidence point, noncompatible maps. 1.
Abdelkrim Aliouche and Valeriu Popa COINCIDENCE AND COMMON FIXED POINT THEOREMS VIA RWEAK COMMUTATIVITY
"... Abstract. We prove common fixed point theorems for two pairs of hybrid mappings satisfying implicit relations in complete metric spaces using the concept of R−weak commutativity of type AT and we correct errors of [1], [3] and [8]. Our theorems generalize results of [13], [8], [1216] and [21]. Key ..."
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Abstract. We prove common fixed point theorems for two pairs of hybrid mappings satisfying implicit relations in complete metric spaces using the concept of R−weak commutativity of type AT and we correct errors of [1], [3] and [8]. Our theorems generalize results of [13], [8], [1216] and [21]. Key words: hybrid mappings, common fixed point, R−weakly commuting of type AT, metric space. AMS Mathematics Subject Classification: 54H25, 47H10. 1. Introduction and
A GENERALIZATION OF SOME RESULTS ON MULTIVALUED WEAKLY PICARD MAPPINGS IN b−METRIC SPACE
"... Abstract. In this paper, we establish some convergence results in a complete b−metric space for the Picard iteration associated to two multivalued weak contractions by employing the concepts of monotone and comparison functions. Our results generalize and extend those of Berinde and Berinde [8], Da ..."
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Abstract. In this paper, we establish some convergence results in a complete b−metric space for the Picard iteration associated to two multivalued weak contractions by employing the concepts of monotone and comparison functions. Our results generalize and extend those of Berinde and Berinde [8], Daffer and Kaneko [15] and Nadler [27]. Theorem 2.1 in our paper generalizes Theorem 5 of Nadler [27] and a recent result of Berinde and Berinde [8], it also extends, improves and unifies several classical results pertainning to single and multivalued contractive mappings in the fixed point theory. Also, Theorem 2.3 is a generalization and extension of
A View From Variational Analysis
, 2009
"... smm springer monographs in mathematics The implicit function theorem is one of the most important theorems in analysis and its many variants are basic tools in partial differential equations and numerical analysis. This book treats the implicit function paradigm in the classical framework and beyond ..."
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smm springer monographs in mathematics The implicit function theorem is one of the most important theorems in analysis and its many variants are basic tools in partial differential equations and numerical analysis. This book treats the implicit function paradigm in the classical framework and beyond, focusing largely on properties of solution mappings of variational problems. The purpose of this selfcontained work is to provide a reference on the topic and to provide a unified collection of a number of results which are currently scattered throughout the literature. The first chapter of the book treats the classical implicit function theorem in a way that will be useful for students and teachers of undergraduate calculus. The remaining part becomes gradually more advanced, and considers implicit mappings defined by relations other than equations, e.g., variational problems. Applications to numerical analysis and optimization are also provided. This valuable book is a major achievement and is sure to become a standard reference on the topic.
c○1995 Juliusz Schauder Center for Nonlinear Studies
"... Let Ω be a bounded domain in Euclidean space, k: Ω×Ω → R N×N a matrixvalued kernel function, and f: Ω × R N → CpCv(R N) a (multivalued) nonlinear function, where CpCv(R N) denotes the system of all nonempty compact convex subsets of R N. Consider the linear integral operator (1) Ky(s) = Ω k(s, t)y(t ..."
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Let Ω be a bounded domain in Euclidean space, k: Ω×Ω → R N×N a matrixvalued kernel function, and f: Ω × R N → CpCv(R N) a (multivalued) nonlinear function, where CpCv(R N) denotes the system of all nonempty compact convex subsets of R N. Consider the linear integral operator (1) Ky(s) = Ω k(s, t)y(t) dt generated by k, and the (multivalued) superposition operator (see e.g. [3]) (2) Nf x(t) = {y(t) : y measurable selection of f ( · , x ( ·))} generated by f. The present paper is concerned with the integral inclusion of Hammerstein type (3) x ∈ KNf x.