Asymptotic integration and stability : for ordinary, functional and discrete differential equations of fractional order - Baleanu, D. (Dumitru); Mustafa, Octavian G.; World Scientific (Firm) (Singapore ; Hackensack, N.J. : World Scientific Pub. Co., 2015)
This volume presents several important and recent contributions to the emerging field of fractional differential equations in a self-contained manner. It deals with new results on existence, uniqueness and multiplicity, smoothness, asymptotic development, and stability of solutions. The new topics in the field of fractional calculus include also the Mittag-Leffler and Razumikhin stability, stability of a class of discrete fractional non-autonomous systems, asymptotic integration with a priori given coefficients, intervals of disconjugacy (non-oscillation), existence of Lp solutions for various linear, and nonlinear fractional differential equations.
1. The differential operators of order 1 + [symbol] and their integral counterparts. 1.1. The gamma function. 1.2. The Riemann-Liouville derivative. 1.3. The Abel computation. 1.4. The operators. The Caputo differential. 1.5. The integral representation of the operators. The half-line case -- 2. Existence and uniqueness of solution for the differential equations of order [symbol]. 2.1. A Lovelady-Martin uniqueness result for the equation (2.2). 2.2. A Nagumo-like uniqueness criterion for the fractional differential equations with a Riemann-Liouville derivative. 2.3. A Wintner-type existence interval for the equation (2.2) -- 3. Position of the zeros, the Bihari inequality, and the asymptotic behavior of solutions for the differential equations of order [symbol]. 3.1. A Fite-type length criterion for fractional disconjugacy. 3.2. The Bihari inequality. 3.3. Asymptotic integration of the differential equations of orders 1 and [symbol]. 3.4. The Bihari asymptotic integration theory of the differential equations of second order -- 4. Asymptotic integration for the differential equations of order 1 + [symbol]. 4.1. An asymptotic integration theory of Trench type. 4.2. Asymptotically linear solutions. 4.3. A Bihari-like result. 4.4. Convergent solutions. 4.3. Lp-solutions of the equation (4.3) -- 5. Existence and uniqueness of solution for some delay differential equations with Caputo derivatives -- 6. Existence of positive solutions for some delay fractional differential equations with a generalized N-term. 6.1. The existence theorem. 6.2. Existence and uniqueness for the solution -- 7. Stability of a class of discrete fractional nonautonomous systems -- 8. Mittag-Leffler stability of fractional nonlinear systems with delay -- 9. Razumikhin stability for fractional systems in the presence of delay -- 10. Controllability of some fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. 10.1. Preliminaries. 10.2. The problem. 10.3. A controllability result -- 11. Approximate controllability of Sobolev type nonlocal fractional stochastic dynamic systems in Hilbert spaces.
Includes bibliographical references (pages 181-194) and index.
Keywords: Fractional calculus; Fractional differential equations; Electronic books