A topological approach to nonlocal elliptic partial differential equations on an annulus

Abstract

For q≥1 we consider the nonlocal ordinary differential equation −a∫01false|yfalse|q0.33emdsy′′false(tfalse)=λffalse(t,y(t)false),0<t<1,subject to the Dirichlet boundary conditions y(0)=0=y(1). Due to the term a∫01false|yfalse|q0.33emds appearing in the equation, this is a class of nonlocal differential equations. By using a novel order cone we are able to establish existence of a positive solution to this problem by means of topological fixed point theory. The preceding problem is really a special case of a more general problem that we consider – namely, the existence of a positive radially symmetric solution to the nonlocal elliptic partial differential equation −a∫Ωfalse|ufalse|q0.33emdbold-italicsΔufalse(bold-italicxfalse)=λg(ufalse(bold-italicxfalse)),x∈Ω,subject to u(x)≡0, for x∈∂Ω, where Ω is an annular region when n≥3.

A Partial Differential Equation-Based Image Restoration Method in Environmental Art Design

“…Image restoration techniques are widely used in the field of film and television, to restore the deteriorated film and also to remove unwanted dates from pictures and to eliminate mosaics, among other things. As early as the Renaissance [2], image restoration techniques emerged as an important branch of image processing, where professional restorers restored damaged oil painting articles by hand painting, but there were many shortcomings in using manual restoration, experts specializing in the restoration of artifacts were very rare, and manual restoration was time-consuming and risky.…”

Nontrivial Solutions of Systems of Perturbed Hammerstein Integral Equations with Functional Terms

“…In the context of ODEs and radial solutions of PDEs in annular domains, a recent and very interesting paper is the one by Goodrich [3]. Goodrich studied the existence of one positive solution of the nonlocal ODE − A 1 0 |u(s)| q ds u (t) = λ f (t, u(t)), t ∈ (0, 1),…”

“…The approach in [3] relies on classical fixed point index theory applied in the cone of positive continuous functionŝ…”

“…where w ∞ := sup t∈[0,1] |w(t)|. Goodrich also studied in [3] the following generalization of (1), namely…”

Asymptotic properties of PDEs in compact spaces

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