Page 223 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 223
TOPOLOGICAL METHODS IN DIFFERENTIAL EQUATIONS (MS-13)
Lower and upper solutions for even order boundary value problems
Lucía López-Somoza, lucia.lopez.somoza@usc.es
Universidade de Santiago de Compostela, Spain
We will prove the existence of solutions of nonlinear boundary value problems of arbitrary even
order using the lower and upper solutions method. In particular, we will show that the existence
of a pair of lower and upper solutions of a considered problem could imply the existence of
solution of another one with different boundary conditions.
Lp-exact controllability of partial differential equations with nonlocal
terms
Luisa Malaguti, luisa.malaguti@unimore.it
University of Modena and Reggio Emilia, Italy
Coauthors: Stefania Perrotta, Valentina Taddei
This talk deals with the exact controllability of some classes of partial differential equations by
means of linear controls. The discussion takes place in infinite dimensional state spaces since
these equations are considered in their abstract formulation as semilinear equations. The linear
parts are densely defined and generate strongly continuous semigroups. The nonlinear terms
may also include a nonlocal part. The solutions satisfy suitable nonlocal constraints, which
are possibly nonlinear. Several contributions already appeared on this topic and the novelty of
our investigation lies in the introduction of an approximation solvability method which involves
a sequence of controllability problems in finite-dimensional spaces. This approach is made
possible by means of a Schauder basis in the Banach state space. The results exploit topological
methods. The exact controllability of nonlocal solutions can then be proved, with controls in
arbitrary Lp spaces, 1 < p < ∞. The results apply, in particular, to the transport equation in
arbitrary Euclidean spaces and to the nonlinear wave equation with possibly localized controls.
Boundary value problems associated with singular strongly nonlinear
equations with functional terms
Cristina Marcelli, marcelli@dipmat.univpm.it
Università Politecnica delle Marche, Italy
Coauthors: Stefano Biagi, Alessandro Calamai, Francesca Papalini
The talk concerns boundary value problems associated with singular, strongly nonlinear differ-
ential equations with functional terms of the type
(Φ(k(t)x (t))) + f (t, Gx(t))h(t, x (t)) = 0 , t ∈ [a, b]
x(a) = Ha[x], x(b) = Hb[x].
The nonlinear differential operator Φ is a general strictly increasing homeomorphism; the coef-
ficient k is non-negative and it may vanish on a set of null measure. Moreover, the differential
equation depends on a general functional term Gx. By means of a fixed point argument, we
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Lower and upper solutions for even order boundary value problems
Lucía López-Somoza, lucia.lopez.somoza@usc.es
Universidade de Santiago de Compostela, Spain
We will prove the existence of solutions of nonlinear boundary value problems of arbitrary even
order using the lower and upper solutions method. In particular, we will show that the existence
of a pair of lower and upper solutions of a considered problem could imply the existence of
solution of another one with different boundary conditions.
Lp-exact controllability of partial differential equations with nonlocal
terms
Luisa Malaguti, luisa.malaguti@unimore.it
University of Modena and Reggio Emilia, Italy
Coauthors: Stefania Perrotta, Valentina Taddei
This talk deals with the exact controllability of some classes of partial differential equations by
means of linear controls. The discussion takes place in infinite dimensional state spaces since
these equations are considered in their abstract formulation as semilinear equations. The linear
parts are densely defined and generate strongly continuous semigroups. The nonlinear terms
may also include a nonlocal part. The solutions satisfy suitable nonlocal constraints, which
are possibly nonlinear. Several contributions already appeared on this topic and the novelty of
our investigation lies in the introduction of an approximation solvability method which involves
a sequence of controllability problems in finite-dimensional spaces. This approach is made
possible by means of a Schauder basis in the Banach state space. The results exploit topological
methods. The exact controllability of nonlocal solutions can then be proved, with controls in
arbitrary Lp spaces, 1 < p < ∞. The results apply, in particular, to the transport equation in
arbitrary Euclidean spaces and to the nonlinear wave equation with possibly localized controls.
Boundary value problems associated with singular strongly nonlinear
equations with functional terms
Cristina Marcelli, marcelli@dipmat.univpm.it
Università Politecnica delle Marche, Italy
Coauthors: Stefano Biagi, Alessandro Calamai, Francesca Papalini
The talk concerns boundary value problems associated with singular, strongly nonlinear differ-
ential equations with functional terms of the type
(Φ(k(t)x (t))) + f (t, Gx(t))h(t, x (t)) = 0 , t ∈ [a, b]
x(a) = Ha[x], x(b) = Hb[x].
The nonlinear differential operator Φ is a general strictly increasing homeomorphism; the coef-
ficient k is non-negative and it may vanish on a set of null measure. Moreover, the differential
equation depends on a general functional term Gx. By means of a fixed point argument, we
221
