Existence And Uniqueness Of Several Classes Solutions For Fractional Differential Equations

In recent years, with the rapid development of science and technology, and researchis increasingly complex. fractional calculus theory not only has important applications insuch fields as Quantum mechanics, solid mechanics, viscous fluid mechanics, rheology,viscoelasticity and image processing, etc., but also has played a positive role in promotingother fields of scientific research as biomechanics, medical ultrasound detection, signalprocessing, electrochemical, electrical engineering, biological engineering, control theoryetc. Fractional calculus is the extension of the calculus with integer order, which is of highvalue in fields of mathematics. Therefore, the existence and uniqueness of solutions forfractional differential equations is of great theoretical significance and practical value.In this thesis, we discuss the existence of several class of solutions for fractionaldifferential equations are studied, which based on the theory of fractional differentialequation. And individual examples are presented to illustrate the main resultsrespectively.Firstly, using Krasnosel’skii fixed point theorem、Banach contraction mappingprinciple and the significance of the-norm discuss the existence and uniqueness ofmild solutions for semilinear impulsive neutral fractional integrodifferential equationswith nonlocal conditions in the-norm，some sufficient conditions for the existence anduniqueness of solution and at least one or three solutions for the nonlocal boundary valueproblem are established.Secondly, using Banach contraction mapping principle and fixed point theoremconsider existence and uniqueness of weighted S-asymptotically ω-periodic solutionsfor fractional integro-differential neutral equations and existence and uniqueness ofweighted S-asymptotically ω-periodic solutions for fractional integro-differential neutralequations with nonlocal conditions, we give the uniqueness of solution and at leastone solution for boundary value problems.Finally, by Banach contraction mapping principle, we study the existence anduniqueness of weighted pseudo-almost periodic solutions for semilinear fractionaldifferential equations with nonlocal initial conditions. we obtain the uniqueness of solutionand at least one solution for nonlocal initial boundary value problems.