Positive Solutions of a Three-Point Boundary-Value Problem for the p -Laplacian Dynamic Equation on Time Scales
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POSITIVE SOLUTIONS OF A THREE-POINT BOUNDARY-VALUE PROBLEM FOR THE p-LAPLACIAN DYNAMIC EQUATION ON TIME SCALES A. Dogan
UDC 517.9
We consider a three-point boundary-value problem for a p-Laplacian dynamic equation on time scales. We prove the existence of at least three positive solutions of the boundary-value problem by using the Avery and Peterson fixed-point theorem. The conditions used in this case differ from the conditions used in the major part of available papers. As an interesting point, we can mention the fact that the nonlinear term f involves the first derivative of the unknown function. As an application, an example is given to illustrate our results.
1. Introduction The present paper deals with the existence of positive solutions of the p-Laplacian dynamic equation on time scales �r � � � φp (u∆ (t)) + g(t)f t, u(t), u∆ (t) = 0, or
� � u(0) − B0 u∆ (⌫) = 0, u∆ (0) = 0,
t 2 [0, T ]T ,
u∆ (T ) = 0,
� � u(T ) + B1 u∆ (⌫) = 0,
(1.1) (1.2)
(1.3)
where φp (s) is the p-Laplacian operator, i.e., φp (s) = |s|p−2 s for p > 1, with (φp )−1 = φq and 1/p + 1/q = 1, ⌫ 2 (0, ⇢(T ))T . Some basic knowledge about time scales and definitions can be found in [7, 8]. As far as we know, if the nonlinear term f is involved in the first-order derivative, then we immediately encounter serious difficulties. In the present work, we use a fixed-point theorem proposed by Avery and Peterson to overcome the difficulties. Throughout the paper, we suppose that the following conditions are satisfied: (H 1 ) T is a time scale with 0, T 2 T, ⌫ 2 (0, ⇢(T ))T ; � ⇢ T (H 2 ) let ⇣ ≥ min t 2 T : t ≥ ; there exists ⌧ 2 T such that ⇣ < ⌧ < T holds; 2 (H 3 ) f : [0, T ]T ⇥ R+ ⇥ R ! R+ is continuous and does not identically vanish on any closed subinterval of [0, T ]T ; � � (H 4 ) g : T ! R+ is left dense continuous i.e., g 2 Cld (T, R+ ) , and does not identically vanish on any closed subinterval of [0, T ]T ; Department of Applied Mathematics, A. Gul University, Kayseri, Turkey; e-mail: [email protected]. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 6, pp. 790–805, June, 2020. Ukrainian DOI: 10.37863/umzh.v72i6.646. Original article submitted April 10, 2017; revision submitted June 18, 2019. 0041-5995/20/7206–0917
c 2020
Springer Science+Business Media, LLC
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A. D OGAN
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(H 5 ) both B0 (υ) and B1 (υ) are continuous odd functions defined on R and there exist A, B > 0 such that Bυ Bj (υ) Aυ,
υ ≥ 0,
j = 0, 1.
In [3], Anderson established the existence of multiple positive solutions to the nonlinear second-order threepoint boundary-value problem (BVP) on the time scale T given by u∆r (t) + f (t, u(t)) = 0, u(0) = 0,
t 2 (0, T ) ⇢ T,
au(⌘) = u(T ).
He employed the Leggett–Williams fixed-point theorem in an appropriate cone to guarantee the existence of at least three positive solutions to this nonlinear problem. Anderson, et al. [4] studied the time scale delta-nabla dynamic equation (g(u∆ ))r + c(t)f (u) = 0
for a < t < b
with boundary conditions � � u(a) − B0 u∆ (⌫) = 0
and
u∆ (b) =
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