## Existence of one-signed solutions of nonlinear four-point boundary value problems.(English)Zbl 1265.34053

The paper deals with the existence of one-sided solutions to boundary value problems of the form $\mu \,u^{\prime \prime }+M\,u=r\,g(t)\,f(u),\quad u(0)=u(\varepsilon ),\;u(1)=u(1-\varepsilon ),$ where $$\mu \in \{-1,1\},$$ $$M,r\in (0,\infty ),$$ $$\varepsilon \in (0,1),$$ $$f:\mathbb {R}\to \mathbb {R}$$ and $$g\:[0,1]\to (0,\infty )$$ are continuous, $$x\,f(x)>0$$ for $$x\neq 0,$$ $$f_0=\lim _{| x| \to 0} f(x)/x\in ( 0,\infty )$$ and $$f_\infty =\lim _{| x| \to \infty } f(x)/x\in (0,\infty ).$$ The authors states two theorems depending on whether $$\mu =1$$ or $$\mu =-1$$ which ensure the existence of a pair of solutions, one of them being positive and the latter one being negative. The proofs rely on the bifurcation principle due to P. H. Rabinowitz [J. Funct. Anal. 7, 487–513 (1971; Zbl 0212.16504)].

### MSC:

 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations

Zbl 0212.16504
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### References:

 [1] J. Chu, Y. Sun, H. Chen: Positive solutions of Neumann problems with singularities. J. Math. Anal. Appl. 337 (2008), 1267–1272. · Zbl 1142.34315 [2] K. Deimling: Nonlinear Functional Analysis. Springer, Berlin, 1985. [3] D. Jiang, H. Liu: Existence of positive solutions to second order Neumann boundary value problems. J. Math. Res. Expo. 20 (2000), 360–364. · Zbl 0963.34019 [4] X. Li, D. Jiang: Optimal existence theory for single and multiple positive solutions to second order Neumann boundary value problems. Indian J. Pure Appl. Math. 35 (2004), 573–586. · Zbl 1070.34038 [5] Z. Li: Positive solutions of singular second-order Neumann boundary value problem. Ann. Differ. Equations 21 (2005), 321–326. · Zbl 1090.34524 [6] R. Ma, B. Thompson: Nodal solutions for nonlinear eigenvalue problems. Nonlinear Anal., Theory Methods Appl. 59 (2004), 707–718. · Zbl 1059.34013 [7] A.R. Miciano, R. Shivaji: Multiple positive solutions for a class of semipositone Neumann two-point boundary value problems. J. Math. Anal. Appl. 178 (1993), 102–115. · Zbl 0783.34016 [8] P.H. Rabinowitz: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7 (1971), 487–513. · Zbl 0212.16504 [9] I. Rachunková, S. Staněk, M. Tvrdý: Solvability of Nonlinear Singular Problems for Ordinary Differential Equations. Hindawi Publishing Corporation, New York, 2008. [10] J. Sun, W. Li: Multiple positive solutions to second-order Neumann boundary value problems. Appl. Math. Comput. 146 (2003), 187–194. · Zbl 1041.34013 [11] J. Sun, W. Li, S. Cheng: Three positive solutions for second-order Neumann boundary value problems. Appl. Math. Lett. 17 (2004), 1079–1084. · Zbl 1061.34014 [12] Y. Sun, Y. J. Cho, D. O’Regan: Positive solution for singular second order Neumann boundary value problems via a cone fixed point theorem. Appl. Math. Comput. 210 (2009), 80–86. · Zbl 1167.34313
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