LEADER 07105nam 2200553 450 001 996490344203316 005 20231110213200.0 010 $a3-031-11619-4 035 $a(CKB)5840000000091653 035 $a(MiAaPQ)EBC7102040 035 $a(Au-PeEL)EBL7102040 035 $a(PPN)26495355X 035 $a(EXLCZ)995840000000091653 100 $a20230226d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aCombined measure and shift invariance theory of time scales and applications /$fChao Wang, Ravi P. Agarwal 210 1$aCham, Switzerland :$cSpringer,$d[2022] 210 4$d©2022 215 $a1 online resource (443 pages) 225 1 $aDevelopments in Mathematics ;$vv.77 311 $a3-031-11618-6 320 $aIncludes bibliographical references and index. 327 $aIntro -- Preface -- Contents -- 1 Riemann Integration, Stochastic Calculus, and Shift Operators on Time Scales -- 1.1 Riemann Integration on Time Scales -- 1.1.1 Riemann Delta and Nabla Integration on Time Scales -- 1.1.2 Some Fundamental Results of the Riemann Integral -- 1.1.3 Fundamental Theorems of Calculus -- 1.2 Stochastic Calculus and Dynamic Equations on Time Scales -- 1.2.1 Relation Between Classical Integrable Functions and ?-Integrable Functions -- 1.2.2 Stochastic Calculus -- 1.2.3 Stochastic Dynamic Equations on Time Scales -- 1.3 Shift Operators on Time Scales -- 1.3.1 Shift Operators -- 1.3.2 Periodicity of Time Scales -- 1.4 Combined Dynamic Derivatives on Time Scales -- 1.4.1 Delta and Nabla Derivatives -- 1.4.2 Diamond-? Dynamic Derivative -- 1.4.3 Some Basic Properties of Combined Dynamic Derivatives -- 1.4.4 Combined Dynamic Integrations -- 1.5 Combined Liouville Formula and ?-Matrix Exponential Solutions -- 1.5.1 Liouville Formula for ?-Dynamic Equations -- 1.5.2 Liouville Formula for -Dynamic Equations and Some Lemmas -- 1.5.3 Liouville Formula for Diamond-? Dynamic Equations -- 1.6 Quaternion Combined Impulsive Matrix Dynamic Equation on Time Scales -- 1.6.1 Quaternion Delta and Nabla Exponential Function -- 1.6.2 Quaternion Matrix Diamond-Exponential Function and Quaternion Combined Matrix Dynamic Equation -- 1.6.3 Quaternion Impulsive Nonhomogeneous Matrix Combined Dynamic Equation -- 2 ?-Measurability and Combined Measure Theory on Time Scales -- 2.1 Lebesgue Measurable and Lebesgue ?-Measurable Sets -- 2.1.1 ?-Measure -- 2.1.2 Lebesgue Measurable and Lebesgue ?-Measurable Sets -- 2.2 ?-Measurable Functions and Lebesgue Measurable Functions -- 2.2.1 ?-Measurable Functions -- 2.2.2 Lebesgue Measurable and Lebesgue ?-Measurable Functions -- 2.3 ?-Integral on Time Scales -- 2.3.1 Riemann ?-Integral. 327 $a2.3.2 Lebesgue ?-Integral on Time Scales -- 2.3.3 Relation Between Lebesgue ?-Integral and Riemann ?-Integral -- 2.4 Lebesgue-Stieltjes Combined ?-Measure and Integral on Time Scales -- 2.4.1 Lebesgue-Stieltjes ?-Outer Measure and Related Properties -- 2.4.2 Lebesgue-Stieltjes ?-Measure and Related Properties -- 2.4.3 Relationships, Extensions, and Composition Theorems for Lebesgue-Stieltjes ?-Measure -- 2.5 Lebesgue-Stieltjes ?-Measurable Function and ?-Integral -- 2.5.1 Lebesgue-Stieltjes ?-Measurable Function and Convergence Theorems -- 2.5.2 Lebesgue-Stieltjes ?-Integral -- 2.5.3 Relationships, Extensions, and Composition Theorems of Lebesgue-Stieltjes ?-Integral -- 3 Shift Invariance and Matched Spaces of Time Scales -- 3.1 Periodic Time Scales with Shift Operators -- 3.2 Singularity of Time Scales Under Action of Shift Operators ? -- 3.3 Complete-Closed Time Scales Under Shifts -- 3.4 The Matched Space of Time Scales and Shift Invariance -- 3.5 Singularity of Time Scales Under Action of Shift Operators ? and ?-1 -- 3.6 Singularity Avoiding: A Shift with Direction -- 4 Almost Periodic Functions Under Matched Spaces of Time Scales -- 4.1 Periodic Functions Under Complete-Closed Time Scales in Shifts -- 4.2 ?-Almost Periodic Functions Under Matched Spaces -- 4.3 Piecewise ?-Almost Periodic Stochastic Process in Shift Operators -- 4.4 n0-Order ?-Almost Periodic Functions -- 5 Almost Automorphic Functions Under Matched Spaces of Time Scales -- 5.1 Weighted Pseudo ?-Almost Automorphic Functions Under Matched Spaces -- 5.2 n0-Order Weighted Pseudo ?-Almost Automorphic Functions -- 5.3 Discontinuous Weighted Pseudo S-Almost Automorphic Functions -- 5.3.1 S-Equipotentially Almost Automorphic Sequence Under S-CCTS -- 5.3.2 S-Almost Automorphic Functions and Weighted Pseudo S-Almost Automorphic Functions. 327 $a6 C0-Semigroup and Stepanov-Like Almost Automorphic Functions on Hybrid Time Scales -- 6.1 C0-Semigroup on a Quantum Time Scale -- 6.2 Stepanov-Like Almost Automorphic Functions on a Quantum Time Scale -- 6.3 Weak Almost Automorphy on a Quantum Time Scale -- 6.4 Shift-Semigroup Under Matched Spaces of Time Scales -- 6.5 Stepanov-Like Almost Automorphic Functions in Matched Spaces of Time Scales -- 7 Almost Periodic Dynamic Equations Under Matched Spaces -- 7.1 Basic Theory of Nonlinear Almost Periodic Dynamic Equations -- 7.2 Exponential Dichotomy of Inhomogeneous Dynamic Equations -- 7.3 ?-Almost Periodic Solutions of Dynamic Equations -- 7.4 n0-Order ?-Almost Periodic Solutions for Dynamic Equations -- 8 Almost Automorphic Dynamic Equations Under Matched Spaces -- 8.1 Weighted Pseudo ?-Almost Automorphic Solutions for Dynamic Equations -- 8.2 n0-Order Weighted Pseudo ?-Almost Automorphic Solutions for Dynamic Equations -- 8.3 Almost Automorphic Impulsive Dynamic Equations Based on Shift Operators -- 9 Applications to Dynamics Models Under Matched Spaces -- 9.1 Almost Periodic Neutral Impulsive Stochastic Lasota-Wazewska Timescale Model -- 9.1.1 Introduction and Model Description -- 9.1.2 Inverse Representation of the Neutral Term with Shift Operator -- 9.1.3 Existence of Mean-Square Almost Periodic Solution -- 9.2 Almost Periodic Impulsive Stochastic Nicholson's Blowflies Timescale Model -- 9.2.1 Model Description -- 9.2.2 Almost Periodic Stochastic Process Under Periodic Time Scales with Shifts ? -- 9.2.3 Mean-Square Positive Almost Periodic Solutions to the Model with Shift Operators -- 9.3 A Numerical Example and the Model on the Quantum Time Scale -- References -- Index. 410 0$aDevelopments in Mathematics 606 $aDifferential equations 606 $aEquacions diferencials$2thub 606 $aAnàlisi funcional$2thub 606 $aTeoria de la mesura$2thub 608 $aLlibres electrònics$2thub 615 0$aDifferential equations 615 7$aEquacions diferencials 615 7$aAnàlisi funcional 615 7$aTeoria de la mesura 676 $a381 700 $aZhao$b Bingxuan$01305898 702 $aAgarwal$b Ravi P. 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996490344203316 996 $aCombined measure and shift invariance theory of time scales and applications$93027977 997 $aUNISA