28 research outputs found
Exact boundary behavior of the unique positive solution for singular second-order differential equations
Existence and Global Asymptotic Behavior of Positive Solutions for Nonlinear Fractional Dirichlet Problems on the Half-Line
We are interested in the following fractional boundary value problem: DÎąu(t)+atuĎ=0, tâ(0,â), limtâ0âĄt2-Îąu(t)=0, limtâââĄt1-Îąu(t)=0, where 1<Îą<2, Ďâ(-1,1), DÎą is the standard Riemann-Liouville fractional derivative, and a is a nonnegative continuous function on (0,â) satisfying some appropriate assumptions related to Karamata regular variation theory. Using the Schauder fixed point theorem, we prove the existence and the uniqueness of a positive solution. We also give a global behavior of such solution
Existence of Positive Solutions for Some Superlinear Fourth-Order Boundary Value Problems
We are concerned with the following superlinear fourth-order equation u4t+utĎt,âut=0,  ââtâ0, 1; âu0=u1=0,âââuâ˛0=a,âââ uâ˛1=-b, where a,âb are nonnegative constants such that a+b>0 and Ďt,âs is a nonnegative continuous function that is required to satisfy some appropriate conditions related to a class K satisfying suitable integrability condition. Our purpose is to prove the existence, uniqueness, and global behavior of a classical positive solution to the above problem by using a method based on estimates on the Green function and perturbation arguments
On Some Quasimetrics and Their Applications
We aim at giving a rich class of quasi-metrics from which we obtain as an application an interesting inequality for the Greens function of the fractional Laplacian in a smooth domain in ℝn
Existence and Uniqueness of Positive Solution for a Fractional Dirichlet Problem with Combined Nonlinear Effects in Bounded Domains
We prove the existence and uniqueness of a positive continuous solution to the following singular semilinear fractional Dirichlet problem (-Î)Îą/2u=a1(x)uĎ1+a2(x)uĎ2, in âDââlimxâzââD(δ(x))1-(Îą/2)u(x)=0, where 0<Îą<2,âĎ1,ââĎ2â(-1,1),âD is a bounded C1,1-domain in ân,nâĽ2, and δ(x) denotes the Euclidian distance from x to the boundary of D. The nonnegative weight functions a1,ââa2 are required to satisfy certain hypotheses related to the Karamata class. We also investigate the global behavior of such solution