ACTA issues

Higher order functional boundary value problems: existence and location results

John R. Graef, Lingju Kong, Feliz M. Minhós

Acta Sci. Math. (Szeged) 77:1-2(2011), 87-100

Abstract. This paper considers the $n^{th}$-order boundary value problem consisting of the equation $$ -(\phi(u^{(n-1)}(x)))^{\prime }=f(x,u(x),\ldots,u^{(n-1)}(x)), x\in(0,1), $$ together with the boundary conditions $$\eqalign{ g_{i}(u,u^{\prime },\ldots,u^{(n-2) },u^{(i)}(0))&=0,\hbox{ }i=0,\ldots,n-3, \cr g_{n-2}(u,u^{\prime },\ldots,u^{(n-2)},u^{(n-2)}(0),u^{(n-1)}(0))&=0, \cr g_{n-1}(u,u^{\prime },\ldots,u^{(n-2)},u^{(n-2)}(1),u^{(n-1)}(1))&=0, }$$ where $\phi $ is an increasing homeomorphism such that $\phi(0)=0$, $n\geq2$ is an integer, $I:=[0,1]$, and $f\colon I\times{\msbm R}^n\rightarrow{\msbm R}$ is an $L^1$-Carathéodory function. Here, $g_{i}\colon(C(I))^{n-1}\times{\msbm R}\rightarrow{\msbm R}$, $i=0,\ldots,n-3$, and $g_{n-2}$, $g_{n-1}\colon( C(I))^{n-1}\times{\msbm R}^2\rightarrow{\msbm R}$ are continuous functions satisfying certain monotonicity assumptions. We present sufficient conditions on the nonlinearity and the boundary conditions to ensure the existence of solutions. Moreover, from the lower and upper solutions method, some information is given about the location of the solution and its qualitative properties. Due to the functional dependence in the boundary conditions, this work generalizes several results for higher order problems with many types of boundary conditions. The main results are illustrated with examples.

AMS Subject Classification (1991): 34B15, 34B10

Keyword(s): boundary value problems, increasing homeomorphism, functional boundary conditions, Nagumo condition, lower and upper solutions

Received October 23, 2009. (Registered under 5476/2009.)