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Nonlinear discrete Sturm-Liouville problems at resonance. (English) Zbl 1129.39006
Nonlinear boundary value problems of the form $$\Delta [p(t-1)\Delta y(t-1)]+q(t)y(t)+\lambda _{k}y(t)=f(t,y(t))+h(t),$$ $$a_{11}y(a)+a_{12}\Delta y(a)=0,\quad a_{21}y(b+1)+a_{22}\Delta y(b+1)=0,$$ are studied. Here $\lambda _{k}$ is an eigenvalue of the associated linear problem, and $f$ is subject to the sublinear growth condition $\vert f(t,s)\vert \leq A\vert s\vert ^{\alpha }+B,$ $t\in \{a+1,\dots,b+1\},s\in R$ for some $0\leq \alpha <1$ and $A,B\in (0,\infty )$. The existence and multiplicity of solutions are proved by using the connectivity properties of solution sets of parameterized families of compact vector fields.

39A12Discrete version of topics in analysis
34B15Nonlinear boundary value problems for ODE
39A10Additive difference equations
Full Text: DOI
[1] Costa, D. G.; Gonçalves, J. V. A.: Existence and multiplicity results for a class of nonlinear elliptic boundary value problems at resonance. J. math. Anal. appl. 84, No. 2, 328-337 (1981) · Zbl 0479.35037
[2] Kelley, W. G.; Peterson, A. C.: Difference equations. (1991) · Zbl 0733.39001
[3] Ma, R.: Multiplicity results for a third order boundary value problem at resonance. Nonlinear anal. 32, No. 4, 493-499 (1998) · Zbl 0932.34014
[4] Rodriguez, J.: Nonlinear discrete Sturm--Liouville problems. J. math. Anal. appl. 308, No. 1, 380-391 (2005) · Zbl 1076.39016