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Zbl 0807.34021
Brykalov, S.A.
A second-order nonlinear problem with two-point and integral boundary conditions. (English)
[J] Georgian Math. J. 1, No.3, 243-249 (1994). ISSN 1072-947X; ISSN 1572-9176/e

Let $f: [a,b]\times \bbfR\times \bbfR\to \bbfR$ satisfy Carathéodory conditions. Moreover, let be $\vert f(t,x\sb 0,x\sb 1)\vert\le M$ for almost all $t$ and all $x\sb 0$, $x\sb 1$, where $M>0$. Let be $g\in \bbfR$ fixed, $w: \bbfR\times \bbfR\to \bbfR$, $\varphi[0,\infty)\to \bbfR$ continuous, $w(s\sb 1,s\sb 2)$ nondecreasing in each of the arguments $s\sb 1$, $s\sb 2$ and strictly increasing at least in one of the arguments $s\sb 1$, $s\sb 2$, the set of pairs $s\sb 1$, $s\sb 2$ that satisfy equality $w(s\sb 1,s\sb 2)= 0$ be nonempty, $\varphi(z)$ strictly increasing and $\lim\varphi(z)= \infty$ as $z\to\infty$. Under these conditions the author proves the theorem: If $g\ge A\sb \varphi$ then every solution of the boundary value problem $$(1)\quad \ddot x= f(t,x,\dot x),\ t\in [a,b],\quad(2)\quad w(x(a),x(b))= 0,\quad(3)\quad \int\sp b\sb a \varphi(\vert \dot x(t)\vert)dt= g$$ is strictly monotone and there exist at least one increasing solution and at least one decresing solution.
[M.Švec (Bratislava)]

MSC 2000:: *34B15 Nonlinear boundary value problems of ODE
34K10 Boundary value problems for functional-differential equations

Keywords: nonlinear boundary value problem; two-point and integral boundary conditions; increasing solution; decresing solution

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