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Positive solutions for boundary value problems of fourth-order differential equations with delay. (Chinese. English summary) Zbl 1313.34190

Summary: Using the fixed point index in cones, the existence of positive solutions is studied for the fourth-order functional differential equations with delay \[ \begin{aligned} & u^{(4)}(t)+au''(t)-bu(t)=f(t, u_t), \;t \in[0, 1],\\ & u(t)=\phi(t), \;t \in [-\tau, 0],\\ & u(0)=u(1)=u''(0)=u''(1)=0, \end{aligned} \] where \(f: [0, 1]\times C^+\to [0, +\infty)\) is continuous, \(C^+=\{\varphi\in C| \varphi(\theta)\geqslant 0,\;\theta\in [-\tau, 0]\},\;\phi(t) \in C([-\tau, 0],\;[0, +\infty)),\;\phi(0)=0\), for every \(t\in [0, 1],\;u_t(\theta)=u(t+\theta),\;\theta\in [-\tau, 0],\;0\leqslant \tau<\frac 12\), and \(a, b\in \mathbb {R}, \;a<2\pi^2,\;b>-\frac{a^2}4,\frac b{\pi^4}+\frac a{\pi^2}<1\). The results in this paper improve and generalize some known ones.

MSC:

34K10 Boundary value problems for functional-differential equations
47N20 Applications of operator theory to differential and integral equations
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