Existence,Stability And Controllability Of Solutions For Some Classes Of Differential Equations

The theory of differential equations has a very wide applications,such as in physics,medicine,engineering and other fields.Using the fixed point theory,topological degree theory,and the semigroups theory,this paper studies the existence,stability,and controllability of solutions for several kinds of differential equations.Chapter 1 mainly introduces the research background,research status,and some necessary preparatory knowledge of several kinds of differential equations.Chapter 2 considers the following singular semipositone boundary value problem of fourth-order differential systems with parameters#12 where βi,αi∈R(i=1,2)satisfies a two-parameter non-resonance condition,name1y,βi<2π2,-βi/4≤αi,αi/π4+βi/π2<1,f1,f2 ∈C(0,1)×R0+×R,R],R0+=(0,+∞).Firstly,two transformations are used to overcome the technical difficulties arising from singularity and semipositone.Then,by constructing a special cone and applying fixed point index theory,some existence results of multiple solutions for the considered systems are obtained under some suitable assumptions.Chapter 3 investigates the implicit-type Caputo fractional coupled system with integral boundary conditions#12Firstly,the Green function is obtained,and then some transformations are used to convert the considered system into the corresponding explicit form and integral form,respectively.Under the condition that one"certain variable" satisfies the Lipschitz condition in nonlinear terms,the existence result of solutions for the considered system is obtained by means of topological degree theory.Secondly.under some suitable conditions,we introduce the definition of Ulam-Hyers stability for theconsidered system.The obtained stability results enrich the application of Ulam-Hyers st ability theory in Caputo fractional order systems.Chapter 4 investigates aclass of impulsive ψ-Caputo fractional evolution equations wit.h nonlocal conditions#12 where △x|t=ti=x(ti+0)-x(ti-0),0<α<1,l<+∞.A is an infinitesimal generat.or of a,C0-semigroup ｛T(t)}t≥0 on X.The control function u is given in L2[J,V].Here,V is a Banach space.B is a linear bounded operator from V to X.The function g:PC[J.X]→X.0<t1<t2<…<tk<tk+1=l.Ii:X→X(i=1,2,…,k)are impulsive functions.The Volterra integral operator Gx(t):∫0K(t,s)x(s)ds is equipped with integral kernel K ∈C[Ω,R+],Ω:={(t,s):0≤t≤s≤l}.Firstly,by generalized Laplace transforms,the definition of mild solutions and the controllability are introduced for considered problems.Secondly,by the semigroups theory,the Hausdorff measure of non-compactness,and the M?nch fixed point theorem,the exact controllability result is obtained under some suitable assumptions.