# 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

/lp/springer_journal/coercive-nonlocal-elements-in-fractional-differential-equations-rqi4xjic8X
Publisher
Springer Journals
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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations