Boundary value problem for differential inclusions in Fréchet spaces with multiple solutions of the homogeneous problem

Irene Benedetti; Luisa Malaguti; Valentina Taddei

Mathematica Bohemica (2011)

  • Volume: 136, Issue: 4, page 367-375
  • ISSN: 0862-7959

Abstract

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The paper deals with the multivalued boundary value problem x ' ∈ A ( t , x ) x + F ( t , x ) for a.a. t ∈ [ a , b ] , M x ( a ) + N x ( b ) = 0 , in a separable, reflexive Banach space E . The nonlinearity F is weakly upper semicontinuous in x . We prove the existence of global solutions in the Sobolev space W 1 , p ( [ a , b ] , E ) with 1 < p < ∞ endowed with the weak topology. We consider the case of multiple solutions of the associated homogeneous linearized problem. An example completes the discussion.

How to cite

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Benedetti, Irene, Malaguti, Luisa, and Taddei, Valentina. "Boundary value problem for differential inclusions in Fréchet spaces with multiple solutions of the homogeneous problem." Mathematica Bohemica 136.4 (2011): 367-375. <http://eudml.org/doc/196940>.

@article{Benedetti2011,
abstract = {The paper deals with the multivalued boundary value problem $x^\{\prime \}\in A(t,x)x+F(t,x)$ for a.a. $t \in [a,b]$, $Mx(a)+Nx(b) =0$, in a separable, reflexive Banach space $E$. The nonlinearity $F$ is weakly upper semicontinuous in $x$. We prove the existence of global solutions in the Sobolev space $W^\{1,p\}([a,b], E)$ with $1<p<\infty $ endowed with the weak topology. We consider the case of multiple solutions of the associated homogeneous linearized problem. An example completes the discussion.},
author = {Benedetti, Irene, Malaguti, Luisa, Taddei, Valentina},
journal = {Mathematica Bohemica},
keywords = {multivalued boundary value problem; differential inclusion in Banach space; compact operator; fixed point theorem; multivalued boundary value problem; differential inclusion in Banach space; compact operator; fixed point theorem},
language = {eng},
number = {4},
pages = {367-375},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Boundary value problem for differential inclusions in Fréchet spaces with multiple solutions of the homogeneous problem},
url = {http://eudml.org/doc/196940},
volume = {136},
year = {2011},
}

TY - JOUR
AU - Benedetti, Irene
AU - Malaguti, Luisa
AU - Taddei, Valentina
TI - Boundary value problem for differential inclusions in Fréchet spaces with multiple solutions of the homogeneous problem
JO - Mathematica Bohemica
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 136
IS - 4
SP - 367
EP - 375
AB - The paper deals with the multivalued boundary value problem $x^{\prime }\in A(t,x)x+F(t,x)$ for a.a. $t \in [a,b]$, $Mx(a)+Nx(b) =0$, in a separable, reflexive Banach space $E$. The nonlinearity $F$ is weakly upper semicontinuous in $x$. We prove the existence of global solutions in the Sobolev space $W^{1,p}([a,b], E)$ with $1<p<\infty $ endowed with the weak topology. We consider the case of multiple solutions of the associated homogeneous linearized problem. An example completes the discussion.
LA - eng
KW - multivalued boundary value problem; differential inclusion in Banach space; compact operator; fixed point theorem; multivalued boundary value problem; differential inclusion in Banach space; compact operator; fixed point theorem
UR - http://eudml.org/doc/196940
ER -

References

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  1. Andres, J., Malaguti, L., Taddei, V., On boundary value problems in Banach spaces, Dyn. Syst. Appl. 18 (2009), 275-301. (2009) Zbl1195.34091MR2543232
  2. Basova, M. M., Obukhovski, V. V., 10.1007/s10958-008-0071-7, J. Math. Sci. 149 (2008), 1376-1384. (2008) MR2336427DOI10.1007/s10958-008-0071-7
  3. Benedetti, I., Malaguti, L., Taddei, V., 10.1016/j.jmaa.2010.03.002, J. Math. Anal. Appl. 368 (2010), 90-102. (2010) Zbl1198.34109MR2609261DOI10.1016/j.jmaa.2010.03.002
  4. Benedetti, I., Malaguti, L., Taddei, V., Two-point b.v.p. for multivalued equations with weakly regular r.h.s, Nonlinear Analysis, Theory Methods Appl. 74 (2011), 3657-3670. (2011) Zbl1221.34161MR2803092
  5. Castaing, C., Valadier, V., 10.1007/BFb0087688, Lect. Notes Math. 580, Springer, Berlin (1977). (1977) Zbl0346.46038MR0467310DOI10.1007/BFb0087688
  6. Daleckiĭ, Ju. L., Kreĭn, M. G., Stability of Solutions of Differential Equations in Banach Spaces, Translation of Mathematical Monographs, American Mathematical Society, Providence, R. I. (1974). (1974) MR0352639
  7. Edwards, R. E., Functional Analysis: Theory and Applications, Holt Rinehart and Winston, New York (1965). (1965) Zbl0182.16101MR0221256
  8. Kamenskii, M. I., Obukhovskii, V. V., Zecca, P., Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Space, W. de Gruyter, Berlin (2001). (2001) MR1831201
  9. Marino, G., 10.1016/0362-546X(90)90061-K, Nonlinear Anal., Theory Methods Appl. 14 (1990), 545-558. (1990) MR1044285DOI10.1016/0362-546X(90)90061-K

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