Second order nonlinear multivalued boundary problems in Hilbert spaces

Abstract

In this paper we consider second order differential inclusions in real Hilbert space, namely p(t) · x (t) + r(t) · x (t) ∈ Ax(t) + F (t, x(t)), a.e. on [0, T ], under the nonlinear boundary conditions. Using techniques from multivalued analysis and the theory of operators of monotone type, we prove the existence of solutions for both the 'convex' and 'nonconvex' problems. Finally, we present a special case of interest, which fit into our framework, illustrating the generality of our results.  2004 Elsevier Inc. All rights reserved.

A note on the continuous dependence on data for second-order difference inclusions

“…This result complements the main theorem from [1], where a similar study was done on finite sets of integers i. Other properties of second-order evolution equations associated to maximal monotone operators were studied in [5,7,8], while their discrete variants are treated in [6,8,9]. A detailed study can be found in [4].…”

Continuous dependence on data for bilocal difference equations

“…In this paper, we prove the continuous dependence on data for the solution of the boundary value problem Such problems, together with some generalizations, are studied in [1,11,15,16]. The existence of the solution, asymptotic behaviour for problems on semi-axis and applications to singularly perturbed problems are presented in [12] (Chapter 5) and in [9].…”

Existence and Relaxation Results for Second Order Multivalued Systems

“…In the literature such problems are usually approached by using some variant of the Hartman or Nagumo-Hartman condition, which leads to an a priori uniform bound for the solutions. We refer to the works of Frigon-Montoki [4], Halidias-Papageorgiou [10], Kandilakis-Papageorgiou [12], Kyritsi-Matzakos-Papageorgiou [13], Ma-Xue [15], Pruszko [25], Zhang-Li [26] and the recent work of Gasiński-Papageorgiou [9] for first order systems. Here instead we employ a condition which involves the principal eigenvalue of the corresponding eigenvalue problem for the vector p-Laplacian.…”