Unbounded solutions for three-point boundary value problems with nonlinear boundary conditions on \([0,+\infty )\)
The paper presents existence and multiplicity results for the second-order differential equation \[ x''(t)+\phi(t)f(t,x(t),x'(t))=0, \quad t\in [0,+\infty) \] with the boundary conditions \[ x(0)-ax'(0)=b\text{ and }\lim_{t\rightarrow+\infty}x'(t)=g(x(\eta)). \] Here, \(a\) and \(\eta\) are positive constants, \(b\in\mathbb{R}\), \(\phi,f\) and \(g\) are continuous functions that satisfy the following assumptions. The function \(f\) satisfies the Nagumo condition with respect to a suitable pair of lower and upper solutions \(\alpha_1(t)\) and \(\beta_1(t)\), \(\alpha_1(t)\leq\beta_1(t)\), i.e., \(|f(t,u,v)|\leq\psi(t)h(|v|)\) for all \(t\geq0\) and \(\alpha_1(t)\leq u\leq\beta_1(t)\) where \(\psi\) is a non-negative continuous function and \(h\) is a Nagumo function. Next, \(g:\mathbb{R}\rightarrow\mathbb{R}\) is assumed to be increasing and the function \(\phi:(0,+\infty)\rightarrow(0,+\infty)\) is in the class \(L^1(0,+\infty)\) and such that \(\sup_{t\in(0,+\infty)}(1+t)^k\phi(t)\psi(t)<+\infty\) for a constant \(k>1\). The first main result states that there is at least one solution of the problem located between the lower and upper solutions \(\alpha_1(t)\) and \(\beta_1(t)\). Furthermore, if in addition there are lower and upper solutions \(\alpha_2\) and \(\beta_2\), respectively, such that they are not themselves solutions of the boundary value problem, \(\alpha_1\leq\alpha_2\leq\beta_1\), \(\alpha_1\leq\beta_2\leq\beta_1\), and \(\alpha_2\nleq\beta_2\) on the interval \([0,+\infty)\), then the problem has at least three distinct solutions \(x_1\), \(x_2\) and \(x_3\) such that \(\alpha_1\leq x_1\leq\beta_2\), \(\alpha_2\leq x_2\leq\beta_1\), \(x_3\nleq\beta_2\) and \(x_3\ngeq\alpha_2\). An example demonstrates the applicability of the main results.
lower and upper solutions on unbounded interval
scientific article; zbMATH DE number 5782218
scientific article