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Positive solutions to singular system with four-point coupled boundary conditions. (English) Zbl 1232.34034

This paper is devoted to the solvability of the system

-x '' (t)=ft , x ( t ) , y ( t ),t(0,1),
-y '' (t)=gt , x ( t ) , y ( t ),t(0,1),
x(0)=0,x(1)=αy(ξ),y(0)=0,y(1)=βx(η),

where f,g:(0,1)×[0,)×[0,)[0,) are continuous and unbounded at t=0 and t=1, and the parameters α,β,ξ,η are such that ξ,η(0,1) and 0<αβξη<1·

It is assumed in addition that the function f is such that

f(t,1,1)C((0,1),(0,)), 0 1 t(1-t)f(t,1,1)dt<+

and there exist constants α i ,β i ,i=1,2, and c with the properties 0α i β i <1,i=1,2,β 1 +β 2 <1 and for t(0,1) and x,y[0,)

c β 1 f(t,x,y)f(t,cx,y)c α 1 f(t,x,y),0<c1,c α 1 f(t,x,y)f(t,cx,y)c β 1 f(t,x,y),c1,c β 2 f(t,x,y)f(t,x,cy)c α 2 f(t,x,y),0<c1,c α 2 f(t,x,y)f(t,x,cy)c β 2 f(t,x,y),c1·

It is assumed also that similar conditions hold for g.

Using the Guo-Krasnosel’skii fixed point theorem, the authors establish the existence of a solution (x,y) such that x,yC[0,1]C 2 (0,1) and x and y are positive on (0,1)·

MSC:
34B16Singular nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
34B10Nonlocal and multipoint boundary value problems for ODE
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