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A nonlocal boundary value problem with singularities in phase variables. (English) Zbl 1068.34019
Sufficient conditions for the solvability are obtained for the second-order nonlinear boundary value problem \[ (g(x'(t)))'=f(t,x(t)),\;x'(t)),\quad x(0)=x(T)=-\gamma\min\{x(t): t\in[0,T]\}. \] Here, \(g\) is a continuous increasing odd function. The right-hand side \(f(t,x,x')>0\) may be singular both at \(x=0\) and \(x'=0\). The number \(\gamma\) is positive. The operation of minimum makes the boundary value problem nonlocal. Boundary conditions that fix the minimal and maximal values of a solution to a second-order differential equation were considered by S. A. Brykalov [Differ. Equations 29, No. 6, 802–805 (1993; Zbl 0826.34020)].

MSC:
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
Citations:
Zbl 0826.34020
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References:
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