Existence of symmetric positive solutions for a multipoint boundary value problem with sign-changing nonlinearity on tim
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RESEARCH
Open Access
Existence of symmetric positive solutions for a multipoint boundary value problem with sign-changing nonlinearity on time scales Fatma Tokmak and Ilkay Yaslan Karaca* *
Correspondence: [email protected] Department of Mathematics, Ege University, Bornova, Izmir, 35100, Turkey
Abstract In this paper, we make use of the four functionals fixed point theorem to verify the existence of at least one symmetric positive solution of a second-order m-point boundary value problem on time scales such that the considered equation admits a nonlinear term f whose sign is allowed to change. The discussed problem involves both an increasing homeomorphism and homomorphism, which generalizes the p-Laplacian operator. An example which supports our theoretical results is also indicated. MSC: 34B10; 39A10 Keywords: symmetric positive solution; fixed-point theorem; time scales; m-point boundary value problem; increasing homeomorphism and homomorphism
1 Introduction The theory of time scales was introduced by Stefan Hilger [] in his PhD thesis in in order to unify continuous and discrete analysis. This theory was developed by Agarwal, Bohner, Peterson, Henderson, Avery, etc. [–]. Some preliminary definitions and theorems on time scales can be found in books [, ] which are excellent references for calculus of time scales. There have been extensive studies on a boundary value problem (BVP) with signchanging nonlinearity on time scales by using the fixed point theorem on cones. See [, ] and references therein. In [], Feng, Pang and Ge discussed the existence of triple symmetric positive solutions by applying the fixed point theorem of functional type in a cone. In [], Ji, Bai and Ge studied the following singular multipoint boundary value problem: φp u (t) + a(t)f u(t) = , u () –
m– i=
αi u(ξi ) = ,
t ∈ (, ),
u () +
m–
αi u(ηi ) = ,
i=
where < ξ < ξ < · · · < ξm– < , < η < η < · · · < ηm– < , ξi < ηi , αi > for i = , , . . . , m – . By using fixed point index theory [] and the Legget-Williams fixed point theorem [], sufficient conditions for the existence of countably many positive solutions are established. © 2013 Tokmak and Karaca; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Tokmak and Karaca Boundary Value Problems 2013, 2013:52 http://www.boundaryvalueproblems.com/content/2013/1/52
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Sun, Wang and Fan [] studied the nonlocal boundary value problem with p-Laplacian of the form ∇ φp u (t) + h(t)f t, u(t) = , u (t ) –
n
θj u (ηj ) –
j=
m–
t ∈ [t , tm )T , u (tm ) = ,
i u(ξi ) = ,
i=
where ≤ t < ξ < ξ < · · · < ξm– < tm and t < η < η < · · · < ηn < tm < +∞ and i > , θj ≥ for i = , , . . . , m – and j = , , . . . , n. By using the fou
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