The role of concavity in applications of avery type fixed point theorems to higher order differential equations

Abstract

In this article we apply an extension of an Avery type fixed point theorem to a family of boundary value problems for higher order ordinary differential equations. The theorem employs concave and convex functionals defined on a cone in a Banach space. We begin by extending a known application to a right focal boundary value problem for a second order problem to a conjugate boundary value problem for a second order problem. We then extend inductively to a two point boundary value problem for a higher order equation. Concavity of differentiable functions plays a key role in the application to second order equations. A concept of generalized concavity plays the same key role in the application to the higher order equation. (2010): 34B15, 34B27, 47H10. Mathematics subject classification

Concavity in fractional calculus

“…Using this property, it is shown that if D α 0 + u ∈ C[0, b], u(0) = 0, and −D α 0 + u(t) ≥ 0 for all t ∈ [0, b], then u also satisfies this concavity like property. This property gives a geometric meaning to sign properties of fractional derivatives of order α ∈ (1,2], similar to the geometric meaning of sign properties of the first and second derivative. Interestingly, this property provides a geometric link between monotonicity and concavity.…”

“…The inequality (1) and related inequalities are useful when applying Avery type fixed point theorems to prove the existence of positive solutions of second order boundary value problems satisfying Dirichlet, right focal, periodic, and other boundary conditions. For some examples, see [2,4,5,10].…”

“…There has been limited work done on concavity properties of Caputo and Riemann Liouville fractional derivatives. In [1], Al-Refai shows that if f ∈ C (2) [0, 1] attains its minimum at t 0 ∈ (0, 1) and f (0) ≤ 0, D δ C f (t 0 ) ≥ 0 for all δ ∈ (1, 2). Here D δ C is the Caputo fractional derivative.…”

Concavity of solutions of a 2n-th order problem with symmetry

“…Recently, Al Twaty and Eloe [3] applied these types of theorems to a two point conjugate type boundary value problem for a second order ordinary differential equation. Concavity was employed as in [6] and symmetry of functions was employed to construct appropriate concave or convex functionals.…”

“…Symmetry will be employed as in [3] and [7]. In [7] a new inequality representing concavity was obtained for functions satisfying a fourth order differential inequality (and more importantly, a new inequality will be obtained for an associated Green's function).…”