# Coercive nonlocal elements in fractional differential equations

Coercive nonlocal elements in fractional differential equations We consider the fractional boundary problem \begin{aligned} -\left[ D_{0^+}^{\nu }y\right] (t)= & {} \lambda f\big (t,y(t)\big )\text {, }0<t<1\\ y^{(i)}(0)= & {} 0\text {, }0\le i\le n-2\\ \left[ D_{0^+}^{\alpha }y\right] (1)= & {} H\big (\varphi (y)\big ),\nonumber \end{aligned} - 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 $$n\in \mathbb {N}_4$$ n ∈ N 4 , $$n-1<\nu \le n$$ n - 1 < ν ≤ n , $$\alpha \in [1,n-2]$$ α ∈ [ 1 , n - 2 ] , and $$\lambda >0$$ λ > 0 is a parameter. Here the element $$\varphi$$ 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, $$\nu$$ ν , of the fractional derivative. Finally, the methods we develop are applicable to a range of fractional-order boundary value problems. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Coercive nonlocal elements in fractional differential equations

, Volume 21 (1) – May 21, 2016
18 pages
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Publisher
Springer International Publishing
Copyright
Copyright © 2016 by Springer International Publishing
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-016-0427-z
Publisher site
See Article on Publisher Site

### Abstract

We consider the fractional boundary problem \begin{aligned} -\left[ D_{0^+}^{\nu }y\right] (t)= & {} \lambda f\big (t,y(t)\big )\text {, }0<t<1\\ y^{(i)}(0)= & {} 0\text {, }0\le i\le n-2\\ \left[ D_{0^+}^{\alpha }y\right] (1)= & {} H\big (\varphi (y)\big ),\nonumber \end{aligned} - 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 $$n\in \mathbb {N}_4$$ n ∈ N 4 , $$n-1<\nu \le n$$ n - 1 < ν ≤ n , $$\alpha \in [1,n-2]$$ α ∈ [ 1 , n - 2 ] , and $$\lambda >0$$ λ > 0 is a parameter. Here the element $$\varphi$$ 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, $$\nu$$ ν , of the fractional derivative. Finally, the methods we develop are applicable to a range of fractional-order boundary value problems.

### Journal

PositivitySpringer Journals

Published: May 21, 2016

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