Existence And Attractivity Of Solutions For Fractional Q-difference Equations

Fractional calculus is a mathematical theory that studies the characteristics and applications of arbitrary order differential and integral operators.Its development history has been more than 300 years.The theoretical research on boundary value problems of fractional differential equations has attracted the attention of many mathematicians at home and abroad q-difference theory is an important branch of discrete mathematics.With the increasing popularity and development of information technology,q-difference is increasingly applied to natural science and engineering,especially playing an important role in mathematical physics model,dynamic system,quantum physics and economics.q-calculus(or quantum calculus)has been an important bridge between mathematics and physics.In recent years,many experts and scholars have introduced the fractional q-calculus theory into the equation and started to pay attention to the related theoretical research of fractional q-difference equation.At present,fractional q-difference equation has attracted the attention and research of scholars at home and abroad,especially on the existence and attraction of its solutions,which are the two most fundamental and important properties.It is not only the requirement of its theoretical development,but also the need of social production and life.It is expected that it can play a corresponding role in practical applicationThis paper mainly studies the existence and attractivity of solutions to the initial-boundary value problem of fractional q-difference equation,including singular equation,impulsive equation,involving the existence and uniqueness of solutions、Lyapunov inequality as well as the attractivity and obtain some new resultsThe Chapter 1 is about the background,development history and research status of fractional calculus and fractional q-difference and give the basic definition of fractional q-difference equation,lemma and the main methods are used in this paper,a brief introduction to the main content of this paper studiesSince most of the research of singular fractional q-difference equation is mostly about the Cauchy,Robin boundary value conditions.There are few papers to study Sturm-Liouville boundary value conditions.In addition,it is difficult to solve integral expression of Sturm Liouville boundary value problem.In Chapter 2,we introduce the boundary value problem of singular fractional q-difference equation with parameters.By using of the Krasnoselskii fixed point theorem,we obtain the sufficient conditions for the existence of solutionsIn Chapter 3,a class of linear and nonlinear fractional q-difference boundary value problem of Lyapunov inequality is introduced under Cauchy boundary condition.In addition,We also derive sufficient conditions for the nonexistence of boundary value problems for fractional q-difference equationsFor the boundary value problems with both the derivative of Caputo and the derivative of Riemann-Liouville in the boundary value conditions,especially the fractional q-difference equations,there are relatively few studies.In Chapter 4,the existence and uniqueness of solutions to fractional q-difference equation boundary value problems with mixed derivatives are studied.By using Guo-Krasnoselskii fixed point theorem、Banach contraction mapping principle and Scheafer fixed point theorem,the sufficient conditions for the existence and uniqueness of solutions are obtained and examples are given to illustrate the application of the main resultsMost of the researches on the initial value problem of fractional differential equation of impulse are of constant order,while the researches on the variable order,especially the fractional q-difference equation are relatively few.In Chapter 5,the existence of solutions for the initial value problem of fractional q-difference equation with impulsive is studied.By using Banach contraction mapping principle,the sufficient conditions for the existence and uniqueness of solutions are obtained and an example is given to illustrate the application of the main resultsThere are many papers about the attractivity of fractional differential equation,but few about the attractivity of fractional q-difference equation.In Chapter 6,the sufficient conditions for the global attractivity of nonlinear fractional q-difference equation is studied and the result of global attractivity of fractional q-difference equation is obtained by the improvement of Krasnoselskii fixed point theorem.Finally,several examples are given to illustrate the main resultsIn Chapter 7,we summarize the main results and the innovation in this paper.Finally,we prospect some future research work based on the paper.