Existence And Multiplicity Of Positive Solutions For Hadamard-Type Fractional Differential Equations With Nonlocal Conditions On Infinite Intervals

Fractional-order differential equation is a relatively classical research field,which can make a better description of some objective phenomena.Experts and scholars have made many valuable new achievements in the study of nonlinear fractional-order differential equation.Fractional differential equations have been widely applied in various fields such as medicine,physics,biology,control theory,and so on.Hadamard-type fractional equations with nonlocal boundary conditions on an unbounded or bounded domain have evolved as an interesting subject of research.In this paper,we discuss the existence and multiplicity of positive solutions to nonlocal boundary value problems for several classes of Hadamard-type fractional equations on an infinite interval.In the first chapter,we consider the existence and multiplicity of positive solutions to nonlocal boundary value problems of Hadamard-type fractional-order differential equations on infinite intervals when the parameter λ takes different values:where 2<α<3;1<η<ξ1<ξ2<…<ξn<+∞;λ>0;b,αi,σj≥0(i=1,2,…,m;j=1,2,…,n);f∈C([1,+∞)×[0,+∞),[0,+∞));a(t):[1,+∞)→[0,+∞).By using the Guo-Krasnoselskii fixed point theorem,we obtain the existence,nonexistence and multiplicity of positive solutions under different conditions at 0 or ∞ in different ranges of parameterλ.In the second chapter,we study a class of nonlocal boundary value problems of Hadamardtype fractional differential equations with nonlinear terms and lower derivative terms:where 2<α≤3;βi>0(i=1,2,…,m);1<η<ξ1<ξ2<…<ξn<+∞;b,αi,σj≥ 0(i=1,2,…,m;j=1,2,…,n);f∈C([1,+∞)×[0,+∞))3,[0,+∞));a(t):[1,+∞)→[0,+∞).In the new function space,we overcome the difficulties caused by infinite interval and nonlinear term containing lower derivative term,and obtain the existence of three positive solutions by using the fixed point theorem.