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Positive solutions for a nonlinear system of fourth-order ordinary differential equations. (English) Zbl 1458.34051

In this paper, the authors consider the existence of positive solutions for the nonlinear system of fourth-order ordinary differential equations \[ u^{(4)}(t) +\beta_1 u''(t) -\alpha_1 u(t) =f_1 (t,u(t),v(t)),\, t\in ( 0,1), \] \[ v^{(4)}(t) +\beta_2 v''(t) -\alpha_2 v(t) =f_2(t,u (t),v(t)),\, t\in (0,1), \] \[ u(0) =u(1) =u''(0) =u''(1) =0, \] \[ v(0) =v(1) =v''(0) =v''(1) =0, \] where \(f_i\in C ([0,1] \times \mathbb{R}^+ \times \mathbb{R}^+,\mathbb{R}^+)\) and \(\alpha_i\), \(\beta_i\in \mathbb{R}(i=1,2)\) satisfy the following conditions \[ \beta_i <2\pi ^{2},\, -\beta_i^2 \leq \alpha_i,\,\alpha_i /\pi^4 +\beta_i/\pi^2<1. \] The authors define suitable cones and using fixed point theory in cones, they obtain the existence of nontrivial fixed points for the corresponding compact continuous mappings on a single cone \(P\). The first main results in the paper is as follows. Assume that \(f_1\) and \(f_2\) satisfy the condition \[ \lim_{u+v\rightarrow 0^+}\sup \max_{t\in [0,1]} \frac{f_1(t,u,v) +f_2(t,u,v)}{\lambda_1 u+\lambda_2 v}<1<\lim_{u+v\rightarrow +\infty} \inf\min_{t\in [0,1]} \frac{f_1 (t,u,v) +f_2(t,u,v)}{\lambda_1 u+\lambda_2 v}, \] where \(\lambda_i=\pi^4- \beta_i \pi^2-\alpha_i(i=1,2)\), then the considered system has at least one nonzero nonnegative solution. Moreover, if \(f_1 (t,0,v(t)) \not\equiv 0\) and \(f_2 (t,u(t),0) \not\equiv 0\) for all \((u,v) \in P\backslash \{(0,0)\}\), then the system has at least one positive solution.
The second main result in the paper is as follows. Assume that \(f_1\) and \(f_2\) satisfy the condition \[ \lim_{u+v\rightarrow 0^+} \inf \min_{t\in [ 0,1]} \frac{f_1 (t,u,v) +f_2(t,u,v)}{\lambda_1 u+\lambda_2 v}>1>\lim_{u+v\rightarrow +\infty} \sup\max_{t\in [0,1]} \frac{f_1 (t,u,v) +f_2 (t,u,v) }{\lambda_1 u+\lambda_2 v}, \] where \(\lambda_i =\pi^4 -\beta_i \pi^2 -\alpha_i(i=1,2)\), then the considered system has at least one nonzero nonnegative solution. Moreover, if \(f_1 (t,0,v (t)) \not\equiv 0\) and \(f_2 (t,u(t),0) \not\equiv 0\) for all \((u,v) \in P\backslash \{(0,0)\}\), then system has at least one positive solution.
Then, the authors investigate the existence of positive solutions for the system involving nonlinear terms in which one is uniformly superlinear or sublinear and the other is locally uniformly sublinear or superlinear. In this case, they construct a product cone \(K_1\times K_2\), which is the Cartesian product of two cones \(K_1\)and \(K_2\) and then they choose proper open sets \(D=D_1 \times D_2 \subset K_1 \times K_2\), such that the different features of nonlinearities can be exploited better. Applying the product formula for the fixed point index on product cone and the fixed point index theory in cones, they establish the existence of positive solutions for the considered system.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47H11 Degree theory for nonlinear operators
47N20 Applications of operator theory to differential and integral equations
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