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Multiple positive solutions of resonant and non-resonant non-local fourth-order boundary value problems. (English) Zbl 1250.34019

The differential equation \[ u^{(4)}(t) - \omega^4 u(t) = f(t,u(t)), \quad 0 < t < 1 \] with \(0 < \omega < \pi\) is considered together with the non-local functional conditions \[ u(0) = \beta_1[u], \quad u''(0) + \beta_2[u] = 0, \quad u(1) = \beta_3[u], \quad u''(1) + \beta_4[u] = 0. \] The functionals \(\beta_i\) are linear functionals in \(C[0,1]\) determined by \[ \beta_i[u] = \int_0^1u(s) \, dB_i(s), \] where \(B_i\) are functions of bounded variations.
The problem is considered in both resonant and non-resonant scenarios. The resonance occurs if \(\lambda =0\) is the eigenvalue of the problem \(u^{(4)} - \omega^4 u = \lambda u\) subject to the functional conditions above. In a non-resonant case it is supposed that \(f(t,u)+k^4 u \geq 0\) for \(u \geq 0\) while \(f(t,u)\) is not positive for all \(u \geq 0\). Here, \(k\) is a suitably chosen constant.
Positive solutions of the problem at resonance are studied using perturbations of the linear operator in the left side of the differential equation. That is, the equivalent problem \[ u^{(4)}(t) - \tilde{\omega}^4 u(t) = \tilde{f}(t,u(t)), \] where \(\tilde{\omega}^4 = \omega^4 - k^4\), \(0 < k < \omega\), \(\tilde{f}(t,u) = f(t,u) +k^4 u\), is non-resonant.
The results are obtained using cone-theoretic theorems and the spectral theory.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34L05 General spectral theory of ordinary differential operators
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