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: Dαu(t)+atuσ=0, t∈(0,∞), limt→0⁡t2-αu(t)=0, limt→∞⁡t1-αu(t)=0, where 1<α<2, σ∈(-1,1), Dα is the standard Riemann-Liouville fractional derivative, and a is a nonnegative continuous function on (0,∞) 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+utφt,−ut=0,    t∈0, 1; −u0=u1=0,  −u′0=a,  − u′1=-b, where a,−b are nonnegative constants such that a+b>0 and φt,−s 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 &#x211d;n

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 (-Δ)α/2u=a1(x)uσ1+a2(x)uσ2, in  D  limx→z∈∂D(δ(x))1-(α/2)u(x)=0, where 0<α<2, σ1,  σ2∈(-1,1), D is a bounded C1,1-domain in ℝn,n≥2, and δ(x) denotes the Euclidian distance from x to the boundary of D. The nonnegative weight functions a1,  a2 are required to satisfy certain hypotheses related to the Karamata class. We also investigate the global behavior of such solution