Coercive nonlocal elements in fractional differential equations

• Published in: Positivity : an international journal devoted to the theory and applications of positivity in analysis Vol. 21; no. 1; pp. 377 - 394
• Main Author: Goodrich, Christopher S.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.03.2017; Springer Nature B.V
• Subjects: Boundary conditions; Boundary value problems; Calculus of Variations and Optimal Control; Optimization; Differential equations; Econometrics; Fourier Analysis; Mathematics; Mathematics and Statistics; Operator Theory; Potential Theory; Studies; Primary 26A33; Riemann–Liouville fractional derivative; 34B10; coercivity; 34A08; Secondary 34B18; 45G10; positive solution; 47H30; nonlocal boundary condition; Hammerstein integral equation; 45M20; 47H14
• ISSN: 1385-1292; 1572-9281
• DOI: 10.1007/s11117-016-0427-z

We consider the fractional boundary problem - D 0 + ν y ( t ) = λ f ( t , y ( t ) ) , 0 < t < 1 y ( i ) ( 0 ) = 0 , 0 ≤ i ≤ n - 2 D 0 + α y ( 1 ) = H ( φ ( y ) ) , where n ∈ N 4 , n - 1 < ν ≤ n , α ∈ [ 1 , n - 2 ] , and λ > 0 is a parameter. Here the element φ is a linear functional that represents a nonlocal boundary condition. We show that by introducing a new order cone, we can ensure that this functional is coercive, which is of importance in proving existence results for the above boundary value problem under minimal assumptions on the functions f and H . We also develop a new open set attendant to the cone. By means of examples we investigate both the usefulness of the new set as well as the strength of the coercivity condition and its dependence on the order, ν , of the fractional derivative. Finally, the methods we develop are applicable to a range of fractional-order boundary value problems.