LEADER 05082nam 2200685 450 001 9910830294703321 005 20210209153116.0 010 $a1-118-99854-5 010 $a1-118-99851-0 035 $a(CKB)2670000000617964 035 $a(EBL)1895199 035 $a(SSID)ssj0001538474 035 $a(PQKBManifestationID)11848812 035 $a(PQKBTitleCode)TC0001538474 035 $a(PQKBWorkID)11528270 035 $a(PQKB)10248389 035 $a(SSID)ssj0001517902 035 $a(PQKBManifestationID)16114672 035 $a(PQKBTitleCode)TC0001517902 035 $a(PQKBWorkID)11506417 035 $a(PQKB)22099512 035 $a(MiAaPQ)EBC4040414 035 $a(MiAaPQ)EBC1895199 035 $a(Au-PeEL)EBL1895199 035 $a(OCoLC)910802665 035 $a(PPN)192893491 035 $a(EXLCZ)992670000000617964 100 $a20150601h20152015 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aI-Smooth analysis $etheory and applications /$fA. V. Kim 205 $a1st ed. 210 1$aSalem, Massachusetts :$cScrivener Publishing,$d2015. 210 4$d©2015 215 $a1 online resource (294 p.) 300 $aDescription based upon print version of record. 311 $a1-118-99852-9 311 $a1-118-99836-7 320 $aIncludes bibliographical references and index. 327 $aCover; Title Page; Copyright Page; Contents; Preface; Part I Invariant derivatives of functionals and numerical methods for functional differential equations; 1 The invariant derivative of functionals; 1 Functional derivatives; 1.1 The Frechet derivative; 1.2 The Gateaux derivative; 2 Classifi cation of functionals on C[a, b]; 2.1 Regular functionals; 2.2 Singular functionals; 3 Calculation of a functional along a line; 3.1 Shift operators; 3.2 Superposition of a functional and a function; 3.3 Dini derivatives; 4 Discussion of two examples; 4.1 Derivative of a function along a curve 327 $a4.2 Derivative of a functional along a curve5 The invariant derivative; 5.1 The invariant derivative; 5.2 The invariant derivative in the class B[a, b]; 5.3 Examples; 6 Properties of the invariant derivative; 6.1 Principles of calculating invariant derivatives; 6.2 The invariant differentiability and invariant continuity; 6.3 High order invariant derivatives; 6.4 Series expansion; 7 Several variables; 7.1 Notation; 7.2 Shift operator; 7.3 Partial invariant derivative; 8 Generalized derivatives of nonlinear functionals; 8.1 Introduction; 8.2 Distributions (generalized functions) 327 $a13.1 Functional Differential Equations13.2 FDE types; 13.3 Modeling by FDE; 13.4 Phase space and FDE conditional representation; 14 Existence and uniqueness of FDE solutions; 14.1 The classic solutions; 14.2 Caratheodory solutions; 14.3 The step method for systems with discrete delays; 15 Smoothness of solutions and expansion into the Taylor series; 15.1 Density of special initial functions; 15.2 Expansion of FDE solutions into Taylor series; 16 The sewing procedure; 16.1 General case; 16.2 Sewing (modification) by polynomials; 16.3 The sewing procedure of the second order 327 $a16.4 Sewing procedure of the second order for linear delay differential equation 330 $aThe edition introduces a new class of invariant derivatives  and shows their relationships with other derivatives, such as the Sobolev generalized derivative and the generalized derivative of the distribution theory. This is a new direction in mathematics.   i-Smooth analysis is the branch of functional analysis that considers the theory and applications of the invariant derivatives of functions and functionals. The important direction of i-smooth analysis is the investigation of the relation of invariant derivatives with the Sobolev generalized derivative and the generalized derivative of distribution theory.   Until now, i-smooth analysis has been developed mainly to apply to the theory of functional differential equations, and the goal of this book is to present i-smooth analysis as a branch of functional analysis.  The notion of the invariant derivative (i-derivative) of nonlinear functionals has been introduced in mathematics, and this in turn developed the corresponding i-smooth calculus of functionals and showed that for linear continuous functionals the invariant derivative coincides with the generalized derivative of the distribution theory.  This book intends to introduce this theory to the general mathematics, engineering, and physicist communities.. 606 $aFunctional differential equations$xNumerical solutions 606 $aFunctional analysis$xResearch 615 0$aFunctional differential equations$xNumerical solutions. 615 0$aFunctional analysis$xResearch. 676 $a515 700 $aKim$b A. V.$0890977 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910830294703321 996 $aI-Smooth analysis$91990213 997 $aUNINA