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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.

##### MSC:
 39A12 Discrete version of topics in analysis 34B15 Nonlinear boundary value problems for ODE 39A10 Additive difference equations
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##### References:
 [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