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

Loading next page...
 
/lp/springer_journal/coercive-nonlocal-elements-in-fractional-differential-equations-rqi4xjic8X
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

References

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
discover and read the research
that matters to you.

Enjoy affordable access to
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.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off