Existence Of Solutions For Integer With Integral Boundary Conditions And Fractional Differential Equation

nonlinear differential equation has been one of the most important research fields in modern mathematics.The theory of singular boundary value problems has become an important area of investigation in recent years.The existence of sym-metric positive solution of boundary value problems has been studied by several authors in the literature; for example,see [17-21] and the reference therein.This article mainly use the Lower and upper solution method and Monotone iterative technique of nonlinear function analysis, researching a necessary and sufficient condition for existence of symmetric positive solutions of integral boundary value problems and the solution of Riemann-Liouville fractional derivatives.The main contents are as follows:Chapter 1 gives some preliminary knowledge of nonlinear functional analysis, including some lemmas which play an important role in the proving of following theorem.If you wang to understand other knowledge of nonlinear functional anal-ysis,you can see.Chapter 2 considers the existence of Symmetric positive solution of fourth-order differential equation with integral boundary conditions. [φp(x"(t))]"=f(t,x(t),x'(t),-x"(t)), t∈(0,1), whereφp(t)=|t|p-2t, p>1,0<ζ<η<1, are constants, a andβare right continuous on [ζ,η), left continuous at t=η, and nondecreasing on [ζ,η], with denote the Riemann-Stieltjes integrals of x andφp(x")with respect to a andβ,respectively,and f satisfy the following non-monotonic hypothesis. for (t,x,y,z)∈(0,1)×(0,∞)×(-∞,+∞)×(0,∞),f(t,x,y,z)is symmetric in t and even in y,i.e.,f satisfies f(1-t,x,y,z)=f(t,x,y,z)and f(t,x,-y,z)=f(t,x,y,z) and f(t,x,y,z)is nondecreasing in x,and there exist constantsιi,μi,(ιi> 0,μi≥0,i=0,1,2,μ0+μ1+μ2<p-1,)such that for t∈(0,1),y∈(-∞,+∞),x,z∈(0,∞) cμ0 f(t,x,y,z)≤f(t,cx,y,z)≤c-ι0f(t,x,y,z), if 0<c≤1ï¼›cμ1f(t,x,y,z)≤f(t,x,cy,z)≤c-ι1f(t,x,y,z), if 0<c≤1ï¼›cμ2f(t,x,y,z)≤f(t,x,y,cz)≤c-ι2f(t,x,y,cz),if 0<c≤1ï¼›Chapter 3 by means of the lower and upper solution method and monotone iter-ative technique,investigate Riemann-Liouville fractional derivatives其中,0<T<∞,Da-ky(x)|x=0=limxâ†'0+Dα-ky(x),(k=1,2,…,n-1) Dα-ny(x)|x=0=limxâ†'0+In-αy(x)其中n=[(?)(α)]+1...