Combined measure and shift invariance theory of time scales and applications / / Chao Wang, Ravi P. Agarwal

• Autore: Zhao Bingxuan
• Pubblicazione: Cham, Switzerland : , : Springer, , [2022]; ©2022
• Descrizione fisica: 1 online resource (443 pages)
• Disciplina: 381
• Soggetto topico: Differential equations; Equacions diferencials; Anàlisi funcional; Teoria de la mesura
• Soggetto genere / forma: Llibres electrònics
• Persona (resp. second.): AgarwalRavi P.
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- 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 Δ-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-α Dynamic Derivative -- 1.4.3 Some Basic Properties of Combined Dynamic Derivatives -- 1.4.4 Combined Dynamic Integrations -- 1.5 Combined Liouville Formula and α-Matrix Exponential Solutions -- 1.5.1 Liouville Formula for Δ-Dynamic Equations -- 1.5.2 Liouville Formula for -Dynamic Equations and Some Lemmas -- 1.5.3 Liouville Formula for Diamond-α 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 α-Measurability and Combined Measure Theory on Time Scales -- 2.1 Lebesgue Measurable and Lebesgue α-Measurable Sets -- 2.1.1 α-Measure -- 2.1.2 Lebesgue Measurable and Lebesgue α-Measurable Sets -- 2.2 α-Measurable Functions and Lebesgue Measurable Functions -- 2.2.1 α-Measurable Functions -- 2.2.2 Lebesgue Measurable and Lebesgue α-Measurable Functions -- 2.3 α-Integral on Time Scales -- 2.3.1 Riemann α-Integral. ; 2.3.2 Lebesgue α-Integral on Time Scales -- 2.3.3 Relation Between Lebesgue α-Integral and Riemann α-Integral -- 2.4 Lebesgue-Stieltjes Combined α-Measure and Integral on Time Scales -- 2.4.1 Lebesgue-Stieltjes α-Outer Measure and Related Properties -- 2.4.2 Lebesgue-Stieltjes α-Measure and Related Properties -- 2.4.3 Relationships, Extensions, and Composition Theorems for Lebesgue-Stieltjes α-Measure -- 2.5 Lebesgue-Stieltjes α-Measurable Function and α-Integral -- 2.5.1 Lebesgue-Stieltjes α-Measurable Function and Convergence Theorems -- 2.5.2 Lebesgue-Stieltjes α-Integral -- 2.5.3 Relationships, Extensions, and Composition Theorems of Lebesgue-Stieltjes α-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 δ -- 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 δ and δ-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 δ-Almost Periodic Functions Under Matched Spaces -- 4.3 Piecewise δ-Almost Periodic Stochastic Process in Shift Operators -- 4.4 n0-Order Δ-Almost Periodic Functions -- 5 Almost Automorphic Functions Under Matched Spaces of Time Scales -- 5.1 Weighted Pseudo δ-Almost Automorphic Functions Under Matched Spaces -- 5.2 n0-Order Weighted Pseudo Δ-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. ; 6 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 δ-Almost Periodic Solutions of Dynamic Equations -- 7.4 n0-Order Δ-Almost Periodic Solutions for Dynamic Equations -- 8 Almost Automorphic Dynamic Equations Under Matched Spaces -- 8.1 Weighted Pseudo δ-Almost Automorphic Solutions for Dynamic Equations -- 8.2 n0-Order Weighted Pseudo Δ-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 δ -- 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.
• Titolo autorizzato: Combined measure and shift invariance theory of time scales and applications
• ISBN: 3-031-11619-4
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996490344203316
• Lo trovi qui: Univ. di Salerno
• Serie: Developments in Mathematics