On the two-point boundary value problem for quadratic second-order differential equations and inclusions on manifolds
<p>The two-point boundary value problem for second-order differential inclusions of the form <mml:math alttext="$(D/dt)dot{m}(t)in F(t,m(t),dot{m}(t))$"> <mml:mrow> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:mrow><mml:mi>D</m...
Saved in:
| Format: | Article |
|---|---|
| Language: | English |
| Published: |
Wiley
2006-01-01
|
| Series: | Abstract and Applied Analysis |
| Online Access: | http://www.hindawi.com/GetArticle.aspx?doi=10.1155/AAA/2006/30395 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | <p>The two-point boundary value problem for second-order differential inclusions of the form <mml:math alttext="$(D/dt)dot{m}(t)in F(t,m(t),dot{m}(t))$"> <mml:mrow> <mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:mrow><mml:mi>D</mml:mi><mml:mo>/</mml:mo><mml:mrow> <mml:mi>d</mml:mi><mml:mi>t</mml:mi> </mml:mrow></mml:mrow> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow><mml:mover accent='true'> <mml:mi>m</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mi>F</mml:mi><mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mrow><mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mover accent='true'> <mml:mi>m</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> on complete Riemannian manifolds is investigated for a couple of points, nonconjugate along at least one geodesic of Levi-Civitá connection, where <mml:math alttext="$D/dt$"> <mml:mrow><mml:mi>D</mml:mi><mml:mo>/</mml:mo><mml:mrow> <mml:mi>d</mml:mi><mml:mi>t</mml:mi> </mml:mrow></mml:mrow> </mml:math> is the covariant derivative of Levi-Civitá connection and <mml:math alttext="$F(t,m,X)$"> <mml:mi>F</mml:mi><mml:mrow><mml:mo>(</mml:mo> <mml:mrow> <mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi> </mml:mrow> <mml:mo>)</mml:mo></mml:mrow> </mml:math> is a set-valued vector with quadratic or less than quadratic growth in the third argument. Some interrelations between certain geometric characteristics, the distance between points, and the norm of right-hand side are found that guarantee solvability of the above problem for <mml:math alttext="$F$"> <mml:mi>F</mml:mi> </mml:math> with quadratic growth in <mml:math alttext="$X$"> <mml:mi>X</mml:mi> </mml:math>. It is shown that this interrelation holds for all inclusions with <mml:math alttext="$F$"> <mml:mi>F</mml:mi> </mml:math> having less than quadratic growth in <mml:math alttext="$X$"> <mml:mi>X</mml:mi> </mml:math>, and so for them the problem is solvable.</p> |
|---|---|
| ISSN: | 1085-3375 |