## Differential and Integral Equations

- Differential Integral Equations
- Volume 17, Number 3-4 (2004), 443-480.

### Strongly nonlinear multivalued, periodic problems with maximal monotone terms

Evgenia H. Papageorgiou and Nikolaos S. Papageorgiou

#### Abstract

In this paper we study periodic, nonlinear, second-order differential inclusions, driven by the differential operator $$ x\rightarrow (\alpha(x)\|x'\|^{p-2}x')' $$ and involving a maximal monotone term $A$ and a multivalued nonlinearity $F(t,x)$ which satisfies the Hartman condition. We do not assume that $domA$ is all of $\mathbb{R}^{N}$, and so our formulation incorporates variational inequalities. Then we obtain partial generalizations. First, we allow $F$ to depend on $x'$, but for $p=2$ and for the scalar problem ($N=1$). Second, we assume a general multivalued, nonlinear differential operator $x\rightarrow \alpha(x,x')'$; the nonlinearity $F$ depends also on $x'$, but the boundary conditions are Dirichlet. Our methods are based on notions and techniques from multivalued analysis and from the theory of operators of monotone type.

#### Article information

**Source**

Differential Integral Equations, Volume 17, Number 3-4 (2004), 443-480.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060440

**Mathematical Reviews number (MathSciNet)**

MR2037985

**Zentralblatt MATH identifier**

1224.34024

**Subjects**

Primary: 34G25: Evolution inclusions

Secondary: 34A60: Differential inclusions [See also 49J21, 49K21] 34B15: Nonlinear boundary value problems

#### Citation

Papageorgiou, Evgenia H.; Papageorgiou, Nikolaos S. Strongly nonlinear multivalued, periodic problems with maximal monotone terms. Differential Integral Equations 17 (2004), no. 3-4, 443--480. https://projecteuclid.org/euclid.die/1356060440