A STUDY OF SOME FUNCTIONAL DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

Abstract

Evolution equation arises in di erent elds of science and engineering, such as the propagation of waves, generic repression, laser optics, whaling control, control theory, climate models, reaction di usion equations, coupled oscillators, population ecology, viscosity materials, food webs, structured population models, enzyme kinetics, neural networks, modern physics and biology, can be modelled as abstract functional and neutral functional di erential equations in Hilbert spaces or more generally, in Banach spaces, where the space variable is suppressed and the equation looks like an ordinary di erential equation in time variable in an in nite dimensional space. If we are concerned with the invariant properties of certain classes of problems, then the best way to study such problems is to consider the abstract formulations of such problems. By an abstract formulation, we indicate a functional analytic representation of the problems. The evolution equations are the abstract formulation of the prototype of many problems. In in nite dimensional space, ordinary di erential equations can be regarded as evolution equations. Functional di erential equations may give a mathematical model for a physical system in which the rate of progress of the system may rely on its past history, that is, the future state of the system depends on the present as well as a part of its past history. The functional di erential equation with nonlocal conditions is considered to be a valuable tool in the modeling of many phenomena in various areas of sciences, engineering and economics due to their practical application to many physical v problems(population dynamics, pollution process in river and sea caused by sewage and many others) and hence they have picked up impressive consideration. Mathematical formulation of nonlocal problems also arises naturally in various engineering models, such as, heat conduction, semiconductor modelling, nonlocal reactive transport in underground waterows in porous media and biotechnology. The quantity of physical phenomena displayed by partial di erential equations with nonlocal conditions which haves abstract formulation as a functional di erential equation is continually expanding. These facts emphasize the importance of examining the nonlocal problems being considered as a model of a physical system for the existence, uniqueness and regularity results. The fractional calculus, which provides the di erentiation and integration of arbitrary order, has been a productive eld of research in many area of sciences and engineering such as, in viscoelasticity and damping, electromagnetism, di usion and wave propagation, chaos and fractals, biology, electronics, signal processing, robotics, tra c systems, genetic algorithms, percolation, chemistry, irreversibility, physics, control systems as well as economy, nance and many others. In fact, fractional di erential equations are considered as an alternative model to nonlinear di erential equations. The most important advantage of utilizing them is their nonlocal property. It is well realized that the fractional order di erential operator is non-local but the integer order di erential operator is a local operator. Di erential inclusions are used for describing systems with hysteresis. All the problems considered for di erential equation, i.e. existence of mild solutions, continuation of solutions, dependence on initial conditions and parameters, are present in the theory of di erential inclusion. Since a di erential inclusion usually has many solutions starting at a given point, new issues may appear, for example, study of topological properties of the set of solutions, optimal control theory, selection of vi solutions with given properties evaluation of the reachability sets and so on. However, the di erential inclusions are not only models for many dynamical processes but also, they present a powerful tool for various branches of mathematical analysis. Di erential inclusion is used to derive su cient conditions of optimality, play an essential role in the theory of control under conditions of uncertainty and in di erential game theory. A stochastic di erential equation of fractional order is a fractional di erential equation containing random term/s (noise term). The investigation of stochastic fractional di erential equations is an energizing topic which unites procedures from probability hypothesis, functional analysis, and the hypothesis of fractional partial di erential equations. In the recent decades, the investigation of stochastic partial di erential equations has turned into a standout amongst the most quickly extending ranges in probability theory. In addition to applications to various issues in mathematical physics and life sciences, enthusiasm for such studies is motivated by a desire to understand and control the behavior of complex systems that show up in numerous zones of natural and social sciences. Thus, there has been a great deal of interest in optimal control systems described by stochastic and partial di erential equations. These optimal control problems lead to stochastic and partial di erential inclusions. Several evolutionary operations from areas as diverse as population dynamics, orbital transfer of satellites, sampled-data systems and engineering are portrayed by the fact that they undergo abrupt changes. The total length of such changes is negligible in comparison with the entire duration of the process and thus the abrupt changes can be well-approximated in terms of instantaneous changes of the state i.e. impulses. Such process can be modeled by impulsive di erential equations allowing for discontinuous in the evolution of the state. Impulsive di erential equations are vii usually de ned by a pair of equations to be satis ed during the continuous portions of evolution and a di erence equation de ning the discrete impulsive action. If some spatio-temporal relation satis es, then impulses occur. Particular impulsive di erential equations simulating the work of concrete systems have been investigated by many authors. Also, some biological events can be appropriately described by impulsive di erential equations [31], [214]. The main object of thesis is to study of numerous type of di erential equations of integer and fractional order, functional integro-di erential equations of the rst order, neutral fractional di erential equations with nite delay or in nite delay, nonlocal stochastic fractional integro-di erential inclusion with impulses, and investigate the existence of mild solution such considered systems utilizing di erent techniques. Chapter 1 provides some introductory matter and literature survey of the problems considered in the subsequent chapters giving an inspiration to the study provided in the thesis. Some elementary de nitions and results, which are used in subsequent chapters, are considered in Chapter 2. Chapter 3 contains two problems and deals with the semi-linear di erential equations in a Banach space involving non-instantaneous impulses. By using xed point theorems for convex-power condensing, the su cient conditions for the existence of the mild solution is obtained via the techniques of the measure of noncompactness and analytic semigroup without assuming Lipschitz continuity of nonlinear function. For the second problem, we also obtain the su cient condition to prove the existence of positive mild solution. For illustrating the obtained theory, we consider the examples. With the help of Hausdor measure of noncompactness and fractional operators, viii Chapter 4 studies an impulsive neutral integro-di erential equation with in nite delay in Banach space. Utilizing Darbo-Sadovskii xed point theorem, the existence of the mild solution is established without assuming Lipschitz continuity of nonlinear function. In order to prove required result, we need only equicontinuity of resolvent operator. That's why, we consider the analytic resolvent operator. An application is also provided at the end of the chapter. Chapter 5 investigates the existence of the mild solution for the fractional di erential equation with state-dependent delay in Banach space. The su cient conditions, providing existence of a mild solution are given by means of resolvent operator and some xed point theorems. Finally, some examples are considered to illustrate the discussed theory. An impulsive di erential equation of fractional order with a deviated argument is under consideration in Chapter 6. We study the approximate solution of considered impulsive di erential equation with deviating argument via the technique of Faedo-Galerkin approximations and analytic semigroup method. The Faedo- Galerkin method will result in a nite-dimensional approximation of the evolution equation providing more regular solutions under weaker assumptions. First of all we provide preliminaries and assumptions required in this chapter. We study an associated integral equation and then consider a sequence of approximate integral equations obtained by applying the projection operator on the associated integral equation. Utilizing Banach xed point theorem and semigroup theory, we study the existence and uniqueness of each approximate integral equation. Then, we show the convergence of approximate integral equation to limiting function that satis es the associated integral equation. Next, we consider the Faedo-Galerkin approximation and demonstrate some convergence results with the help of it. Finally, we give an ix example showing the e ectiveness of obtained abstract results. Following the idea of Chapter 6, Chapter 7 studies the Faedo-Galerkin approximations of solutions to a nonlocal neutral di erential equations of fractional order in a separable Hilbert space. We establish the existence and uniqueness of the solution to every approximate integral equation obtained by applying the projection operator on associate integral equation via xed point technique and semigroup theory. To illustrate the discussed abstract theory, we give an example at the end of the Chapter. An impulsive neutral stochastic integro-di erential equation of fractional order with in nite delay is studied in the Chapter 8. This chapter discusses two existence results. The su cient condition to prove the rst existence result providing existence and uniqueness of the mild solution is derived via the technique of Banach xed point theorem and resolvent operator method. The second result is obtained by utilizing Krasnoselskii-Schaefer xed point theorem with compact resolvent operator. An example is also considered illustrating the given theory. The closing Chapter 9 is devoted to study a nonlocal stochastic fractional integro-di erential inclusion with impulsive conditions in a real separable Hilbert space. The existence of a mild solution is established by utilizing a xed point theorem for multi-valued operators due to Dhage and resolvent operator with fractional power operator theory. An example is also provided to show the applicability of obtained results. The relevant references are appended at the end.