Existence of solutions for quasilinear second order differential inclusions with nonlinear boundary conditions. (English) Zbl 0941.34008

Continuing their previous work [J. Differ. Equations 147, No. 1, 123-154 (1998; Zbl 0912.34020)], the authors prove the existence of a solution \(x(.)\in C^1(T,\mathbb{R}^N), \;\|x'(.)\|^{p-2}x'(.)\in W^{1,q}(T,\mathbb{R}^N)\) to a boundary value problem of the form: \[ (\|x'(t)\|^{p-2}x'(t))'\in F(t,x(t),x'(t)) \quad \text{a.e. }(T=[0,b]), \]
\[ (\|x'(0)\|^{p-2}x'(0),- \|x'(b)\|^{p-2}x'(b))\in \xi (x(0),x(b)), \] defined the convex-valued multifunction \(F(.,.,.)\), by the maximal monotone operator \(\xi (.,.)\) and by \(p\in [2,\infty)\).
The rather technical proofs of the auxiliary results and of the main one rely on the construction of some approximate selectors of Carathéodory type and on the use of the Leray-Schauder fixed point theorem for compact single-valued operators.
The authors show that their general formulation contains as particular cases the Dirichlet, Neumann, periodic solutions and some Sturm-Liouville-type problems.


34B15 Nonlinear boundary value problems for ordinary differential equations
34B24 Sturm-Liouville theory
34A60 Ordinary differential inclusions


Zbl 0912.34020
Full Text: DOI


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