Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 39, pp. 1–17. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
POSITIVE SOLUTIONS OF THE p-LAPLACIAN DYNAMIC EQUATIONS ON TIME SCALES WITH SIGN CHANGING NONLINEARITY ABDULKADIR DOGAN Communicated by Mokhtar Kirane
Abstract. This article concerns the existence of positive solutions for pLaplacian boundary value problem on time scales. By applying fixed point index we obtain the existence of solutions. Emphasis is put on the fact that the nonlinear term is allowed to change sign. An example illustrates our results.
1. Introduction The development of the theory of time scales was initiated by Stefan Hilger in his Ph.D. thesis in 1988 [12] as a means of a unifying structure for the study of differential equations in the continuous case and the study of finite difference equations in the discrete case. The study of time scales has led to several important applications, e.g., in the study of insect population models, heat transfer, neural networks, phytoremediation of metals, wound healing, and epidemic models [5, 14, 19, 24]. In the last few years, there is much attention focused on the existence of positive solutions for second-order boundary value problems (BVPs) on time scales [2, 6, 7, 8, 11, 16, 17, 18, 21, 22, 23, 25, 27]. But for the existence of positive solutions for p-Laplacian BVPs with sign changing nonlinearity on time scales [20, 28], few works were done as far as we know. Agarwal, L¨ u and O’Regan [1] studied the singular BVP (ϕp (y 0 ))0 + q(t)f (t, y) = 0,
0 < t < 1,
y(0) = y(1) = 0, by means of the upper and lower solution method, where the nonlinearity f is allowed to change sign. Anderson, Avery and Henderson[3] studied the BVP (g(u∆ ))∇ + c(t)f (u) = 0, ∆
u(a) − B0 (u (ν)) = 0,
a < t < b, ∆
u (b) = 0,
2010 Mathematics Subject Classification. 34B15, 34B18, 34N05. Key words and phrases. Time scales; p-Laplacian; positive solutions; fixed point index; cone. c
2018 Texas State University. Submitted November 17, 2017. Published January 30, 2018. 1
2
A. DOGAN
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where g(z) = |z|p−2 z, p > 1, ν ∈ (a, b), f ∈ Cld ([0, +∞), [0, +∞)), c(t) ∈ Cld ([a, b], [0, +∞)) and Km x ≤ B0 (x) ≤ KM x for some positive constants Km , KM . They proved the existence of at least one positive solution by using a fixed point theorem of cone expansion and compression of functional type. Ji, Feng and Ge [13] considered the existence of multiple positive solutions to BVP for the one-dimensional p-Laplacian equation (φp (u0 (t)))0 + q(t)f (t, u(t)) = 0, u(0) =
n X
αi u(ξi ),
u(1) =
i=1
t ∈ (0, 1),
n X
βi u(ξi ),
i=1
Pn Pn where 0 < ξ1 < ξ2 < · · · < ξn < 1, αi , βi ∈ [0, ∞) satisfy 0 < i=1 αi , i=1 βi < 1. The nonlinear term f may change sign. By applying a fixed-point theorem for operators in a cone, they provided sufficient conditions for the existence of multiple positive solutions to the BVP. In [18], Sang, Su and Xu studied the existence of positive solutions of the following dynamic equations on time scales: (φ(u∆ ))∇ + a(t)f (t, u(t)) = 0, φ(u∆ (0)) =
m−2 X
ai φ(u∆ (ξi )),
t ∈ [0, T ],
u(T ) =
i=1
m−2 X
bi u(ξi ),
i=1
where φ : R → R is an increasing homeomorphism and homomorphism and φ(0) = 0. They established several existence theorems of positive solutions for nonlinear m-point BVP. The nonlinearity f is allowed to change sign. Su, Li and Sun [20] investigated the following singular m-point p-Laplacian BVP with the sign changing nonlinearity on time scales: (ϕp (u∆ (t)))∇ + q(t)f (t, u(t)) = 0, u(0) = 0,
u(T ) −
m−2 X
t ∈ (0, T )T ,
ψi (u(ξi )) = 0,
i=1
where ϕp (u) = |u|p−2 u, p > 1, 0 < ξ1 < ξ2 < · · · < ξm−2 < ρ(T ), q(t) ∈ Cld ((0, T ), (0, +∞)), f ∈ Cld ((0, T ) × (0, +∞), (−∞, +∞)). They presented some new existence criteria for positive solutions of the problem by using the well-known Schauder fixed point theorem and upper and lower solutions method. Wang and Hou [26] considered the multipoint BVP (φp (u0 (t)))0 + f (t, u(t)) = 0, φp (u0 (0)) =
n−2 X i=1
ai φp (u0 (ξi )),
t ∈ (0, 1),
u(1) =
n−2 X
βi u(ξi ),
i=1
where φp (u) = |u|p−2 u, p > 1, ξi ∈ (0, 1) with 0 < ξ1 < ξ2 < · · · < ξn−2 < 1, and Pn−2 Pn−2 ai , bi satisfy ai , bi ∈ [0, ∞), 0 < i=1 ai < 1, and 0 < i=1 bi < 1. Using a fixed point theorem for operators on a cone, they provided sufficient conditions for the existence of at least three positive solutions to the above BVP. Motivated by works mentioned above, in this paper, we are concerned with the existence of multiple positive solutions for p-Laplacian multipoint BVP on time
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3
scales (φp (u∆ (t)))∇ + w(t)f (t, u(t)) = 0, m−2 X
u(0) =
φp (u∆ (T )) =
ai u(ξi ),
t ∈ [0, T ]T ,
(1.1)
bi φp (u∆ (ξi )),
(1.2)
m−2 X
i=1
i=1
where φp (u) is p-Laplacian operator, i.e., φp (u) = |u|p−2 u, for p > 1 with (φp )−1 = φq and p1 + 1q = 1, 0 < ξ1 < · · · < ξm−2 < ρ(T ). The usual notation and terminology for time scales as can be found in [4, 5, 9], will be used here. We need the following hypotheses: Pm−2 Pm−2 (H1) ai , bi ∈ [0, +∞) satisfy i=1 ai < 1 and 0 < i=1 bi < 1; Rξ (H2) w ∈ Cld ([0, T ]T , [0, +∞)) with 0 1 w(τ )∇τ > 0; (H3) f : [0, T ]T × [0, +∞) → R is continuous. We find some new results on the existence of at least two positive solutions to the BVP (1.1) and (1.2) by using fixed point index. The interesting point of this article is that the nonlinear term f is allowed to change sign. 2. Preliminaries Let the Banach space E = Cld ([0, T ]T , R) with norm kuk = supt∈[0,T ]T |u(t)|, and define two cones: P = {u : u ∈ E, u(t) ≥ 0, t ∈ [0, T ]T }, 0
P = {u : u ∈ E, u is concave, nonnegative and increasing on [0, T ]T }. Pm−2 Pm−2 Lemma 2.1. If 1− i=1 ai 6= 0 and 1− i=1 bi 6= 0, then for h ∈ Cld ([0, T ]T , R), the BVP (φp (u∆ (t)))∇ + h(t) = 0, u(0) =
m−2 X
ai u(ξi ),
φp (u∆ (T )) =
m−2 X
i=1
t ∈ (0, T )T ,
(2.1)
bi φp (u∆ (ξi ))
(2.2)
i=1
has a unique solution for which the following representation holds Z T Z s u(t) = φq h(τ )∇τ − C ∆s t
+
0
1− Z T
−
1 Pm−2 i=1
φq
Z
0
h m−2 X ai
s
Z
T
ai
φq ξi
i=1
s
Z
h(τ )∇τ − C ∆s
0
i h(τ )∇τ − C ∆s ,
0
where C=
1−
−1 Pm−2 i=1
h m−2 X bi
Z
ξi
Z h(τ )∇τ −
bi 0
i=1
T
i h(τ )∇τ .
0
Proof. Integrating (2.1) from 0 to t, one gets ∆
∆
Z
φp (u (t)) − φp (u (0)) = −
t
h(τ )∇τ. 0
(2.3)
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Let C = φp (u∆ (0)), then we can write φp (u∆ (t)) = −
t
Z
h(τ )∇τ + C,
(2.4)
h(τ )∇τ + C .
(2.5)
0
i.e.
t
Z
∆
u (t) = φq − 0
Integrating the dynamic equation (2.5) from t to T , we have Z T Z s h(τ )∇τ + C ∆s. φq − u(T ) − u(t) = 0
t
Then we obtain Z u(t) = u(T ) +
T
φq
s
Z
t
h(τ )∇τ − C ∆s.
(2.6)
0
Applying the first boundary condition, one gets Z T Z s u(0) = u(T ) + φq h(τ )∇τ − C ∆s 0
=
m−2 X
ai u(ξi ) =
i=1
0 m−2 X
T
Z h ai u(T ) +
φq
ξi
i=1
s
Z
i h(τ )∇τ − C ∆s ,
0
i.e., u(T ) =
h m−2 X
1 Pm−2
Z ai
T
φq
Z
s
1 − i=1 ai i=1 ξi 0 Z T Z s i − φq h(τ )∇τ − C ∆s . 0
h(τ )∇τ − C ∆s (2.7)
0
Therefore, by (2.6) and (2.7), we have h m−2 X Z T Z s 1 ai φq h(τ )∇τ − C ∆s u(t) = Pm−2 1 − i=1 ai i=1 ξi 0 Z T Z s i Z T Z s − φq h(τ )∇τ − C ∆s + φq h(τ )∇τ − C ∆s. 0
0
t
0
Applying the second boundary condition in (2.4), we have Z T φp (u∆ (T )) = − h(τ )∇τ + C =
0 m−2 X i=1
∆
bi φp (u (ξi )) =
m−2 X i=1
Z
bi −
(2.8)
ξi
h(τ )∇τ + C . 0
Solving the above equation, we obtain Z T h m−2 i X Z ξi −1 bi h(τ )∇τ − h(τ )∇τ . C= Pm−2 1 − i=1 bi i=1 0 0 Let u1 and u2 be two solutions of problem (2.1), (2.2). Then, using representation (2.3), we obtain u1 (t) − u2 (t) = 0, t ∈ [0, T ]. Thus u in (2.3) is the unique solution of (2.1) and (2.2). The proof is complete.
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p-LAPLACIAN DYNAMIC EQUATIONS
5
Lemma 2.2. Suppose (H1) holds, for h ∈ Cld ([0, T ]T , R) and h(t) ≥ 0, t ∈ [0, T ]. Then, the unique solution u of (2.1) and (2.2) satisfies u(t) ≥ 0, for t ∈ [0, T ]T . Proof. Let Z
s
h(τ )∇τ − C .
ϕ0 (s) = φq 0
Since Z s
RT h(τ )∇τ − 0 h(τ )∇τ h(τ )∇τ + Pm−2 1 − i=1 bi 0 R Rξ Pm−2 R T T Z s h(τ )∇τ + T i h(τ )∇τ − 0 h(τ )∇τ i=1 bi 0 = h(τ )∇τ + Pm−2 1 − i=1 bi 0 R Pm−2 R T Pm−2 R T T Z s h(τ )∇τ − i=1 bi 0 h(τ )∇τ 0 i=1 bi ξi h(τ )∇τ h(τ )∇τ − = − Pm−2 Pm−2 1 − i=1 bi 1 − i=1 bi 0 Pm−2 R T Pm−2 R T Z s 1 − i=1 bi 0 h(τ )∇τ i=1 bi ξi h(τ )∇τ = h(τ )∇τ − − Pm−2 Pm−2 1 − i=1 bi 1 − i=1 bi 0 Pm−2 R T Z s Z T i=1 bi ξi h(τ )∇τ = h(τ )∇τ − h(τ )∇τ − Pm−2 1 − i=1 bi 0 0 Pm−2 R T Z T i=1 bi ξi h(τ )∇τ ≤ 0, =− h(τ )∇τ − Pm−2 1 − i=1 bi s Pm−2 i=1
bi
R ξi 0
it follows that ϕ0 (s) ≤ 0. According to Lemma 2.1, we obtain Z T Z T h m−2 i X Z T 1 u(0) = a ϕ (s)∆s − ϕ0 (s)∆s + ϕ0 (s)∆s Pm−2 i 0 1 − i=1 ai i=1 0 ξi 0 Pm−2 R T Pm−2 R T − i=1 ai 0 ϕ0 (s)∆s + i=1 ai ξi ϕ0 (s)∆s = Pm−2 1 − i=1 ai h i P Rξ RT RT Pm−2 m−2 − i=1 ai 0 i ϕ0 (s)∆s + ξi ϕ0 (s)∆s + i=1 ai ξi ϕ0 (s)∆s = Pm−2 1 − i=1 ai Pm−2 R ξ − i=1 ai 0 i ϕ0 (s)∆s = ≥0 Pm−2 1 − i=1 ai and u(T ) =
1 Pm−2
h m−2 X
Z ai
T
Z ϕ0 (s)∆s −
T
i ϕ0 (s)∆s
1 − i=1 ai i=1 ξi 0 R ξi RT Pm−2 R T i=1 ai ξi ϕ0 (s)∆s − 0 ϕ0 (s)∆s − ξi ϕ0 (s)∆s = Pm−2 1 − i=1 ai Rξ Pm−2 R T − 1 − i=1 ai ξi ϕ0 (s)∆s − 0 i ϕ0 (s)∆s = Pm−2 1 − i=1 ai
6
A. DOGAN
Z
R ξi
T
=−
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ϕ0 (s)∆s − ξi
ϕ0 (s)∆s Pm−2 ≥ 0. 1 − i=1 ai 0
If t ∈ (0, T )T , we have u(t) Z =
T
t
=
1− +
1 Pm−2 i=1
m−2 X
1− +
i=1
i=1
=
ai
T
T
i ϕ0 (s)∆s
0
t
i ϕ0 (s)∆s
ϕ0 (s)∆s −
0
Z
T
ξi
T
ai
ϕ0 (s)∆s 0
i=1
Z
1 Pm−2
m−2 X
T
ϕ0 (s)∆s −
ai
Z
0
hZ T
Z
T
ϕ0 (s)∆s −
i=1
Z
ξi
1 Pm−2
m−2 X
t
ϕ0 (s)∆s −
ai
i=1
≥
ai
Z ai
1 − i=1 ai i=1 ξi Z Z m−2 T h T X ϕ0 (s)∆s − ai ϕ0 (s)∆s
T
Z
h m−2 X
1 Pm−2
ϕ0 (s)∆s +
i ϕ0 (s)∆s
0
h
−
m−2 X
ai
Z
ξi
Z
m−2 X Z ai ϕ0 (s)∆s +
T
ϕ0 (s)∆s +
1 − i=1 ai 0 i=1 Pm−2 R ξi − i=1 ai 0 ϕ0 (s)∆s ≥ 0. = Pm−2 1 − i=1 ai
ξi
i=1
T
i ϕ0 (s)∆s
ξi
Therefore u(t) ≥ 0, t ∈ [0, T ]T .
Lemma 2.3. If u ∈ P 0 and it satisfies (1.2), then inf t∈[0,T ]T
where γ1 =
Pm−2 i=1
u(t) ≥ γ1 kuk,
ai ξTi , kuk = maxt∈[0,T ]T |u(t)|.
Proof. Since u∆∇ (t) ≤ 0, it follows that u∆ (t) is nonincreasing. Thus, for 0 < t < T, Z t u(t) − u(0) = u∆ (s)∆s ≥ tu∆ (t) 0
and Z
T
u(T ) − u(t) =
u∆ (s)∆s ≤ (T − t)u∆ (t)
t
from which we have u(t) ≥
tu(T ) + (T − t)u(0) t t ≥ u(T ) = kuk, T T T
and so u(ξi ) ≥ ξTi kuk. Pm−2 With the boundary condition u(0) = i=1 ai u(ξi ), we obtain inf t∈[0,T ]T
u(t) = u(0) =
m−2 X i=1
ai u(ξi ) ≥
m−2 X i=1
ai
ξi kuk = γ1 kuk. T
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p-LAPLACIAN DYNAMIC EQUATIONS
7
This completes the proof.
Let n K = u : u ∈ E, u is nonnegative and increasing on [0, T ]T ,
o min u(t) ≥ γkuk , t∈[0,T ]T
where γ = γ1 γ2 , Pm−2 i=1
γ2 =
R ξi
ai
RT
0
R
φq
0
R
T s
φq T s
Pm−2
R bi ξT w(τ )∇τ i Pm−2 ∆s 1− i=1 bi . Pm−2 R T i=1 bi ξi w(τ )∇τ Pm−2 ∆s 1− i=1 bi i=1
w(τ )∇τ +
w(τ )∇τ +
Note that u is a solution of BVP (1.1) and (1.2) if and only if Z s Z T w(τ )f (τ, u(τ ))∇τ − C ∆s φq u(t) = 0
t
+
h m−2 X
1 Pm−2
1− Z T
−
i=1
ai
Z
s
φq 0
Z
T
ai
s
Z
w(τ )f (τ, u(τ ))∇τ − C ∆s
φq ξi
i=1
0
i w(τ )f (τ, u(τ ))∇τ − C ∆s ,
0
where C=
1−
−1 Pm−2 i=1
h m−2 X bi
ξi
Z
Z
T
w(τ )f (τ, u(τ ))∇τ −
bi 0
i=1
i w(τ )f (τ, u(τ ))∇τ .
0
We define the operators A : P → E and F : K → E as follows Z s Z T (Au)(t) = φq w(τ )f (τ, u(τ ))∇τ − C ∆s t
+
0
1 Pm−2
1− Z T
i=1
−
φq
h m−2 X ai
s
Z
Z
T
Z
ai ξi
i=1
s
φq
w(τ )f (τ, u(τ ))∇τ − C ∆s (2.9)
0
i w(τ )f (τ, u(τ ))∇τ − C ∆s ,
0
0
−1 Pm−2
h m−2 X
where C=
1−
i=1 T
Z (F u)(t) =
Z
T
w(τ )f (τ, u(τ ))∇τ −
bi i=1 0 Z s φq w(τ )f + (τ, u(τ ))∇τ − C˜ ∆s
t
+
ξi
Z bi
i w(τ )f (τ, u(τ ))∇τ ,
0
0
1− Z T
−
1 Pm−2
h m−2 X
i=1
ai
Z
s
φq 0
i=1
Z
T
ai
φq ξi
Z
s
w(τ )f + (τ, u(τ ))∇τ − C˜ ∆s
0
i w(τ )f + (τ, u(τ ))∇τ − C˜ ∆s ,
0
(2.10) where C˜ =
1−
−1 Pm−2 i=1
h m−2 X bi
i=1
Z
ξi +
Z
w(τ )f (τ, u(τ ))∇τ −
bi 0
0
T
i w(τ )f + (τ, u(τ ))∇τ ,
8
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f + (t, u(t)) = max{f (t, u(t)), 0},
t ∈ [0, T ]T .
Lemma 2.4. The operator F : K → K is completely continuous. Proof. Firstly, we show that F (K) ⊂ K. for all u ∈ K, it is easy to see that F u is nonnegative, concave and increasing on [0, T ]T . Thus, F u ∈ K. Moreover, we know that F u satisfies (2.2). Hence, Lemma 2.3 implies inf (F u)(t) ≥ γ1 kF uk,
for u ∈ K,
t∈[0,T ]T
i.e., F u ∈ K. Therefore, we can find that F (K) ⊂ K. Secondly, we show that F maps bounded set into itself. Suppose that c > 0 is a constant and u ∈ Kc = {u ∈ K : kuk ≤ c}. Note that the continuity of f + guarantees that there is a L > 0 such that f + (t, u(t)) ≤ φp (L) for t ∈ [0, T ]T . Therefore, kF uk = max F u(t) t∈[0,T ]T
Pm−2
RT
Rs
w(τ )f + (τ, u(τ ))∇τ − C˜ ∆s ≤ Pm−2 1 − i=1 ai RT Rs φq 0 w(τ )f + (τ, u(τ ))∇τ − C˜ ∆s − 0 Pm−2 1 − i=1 ai RT Rs φq 0 w(τ )f + (τ, u(τ ))∇τ − C˜ ∆s 0 ≤− Pm−2 1 − i=1 ai Pm−2 R T Z T Z T i=1 bi ξi w(τ )∇τ ≤L φq ∆s, w(τ )∇τ + Pm−2 1 − i=1 ai 0 s where φq f + (τ, u(τ )) ≤ L. That is, F Kc is uniformly bounded. In addition, for any t1 , t2 ∈ [0, T ]T , we have Z T Z s |F u(t1 ) − F u(t2 )| = φq w(τ )f + (τ, u(τ ))∇τ − C˜ ∆s ai
i=1
φq
ξi
t1
Z
0
T
−
φq
Z =
t2 t2
t1
0
φq
s
Z
Z
0 s
0
≤ L|t1 − t2 |φq
w(τ )f + (τ, u(τ ))∇τ − C˜ ∆s
w(τ )f + (τ, u(τ ))∇τ − C˜ ∆s
Z s
RT bi ξi w(τ )∇τ ∆s. Pm−2 1 − i=1 ai
Pm−2
T
w(τ )∇τ +
i=1
Therefore, by applying the Arzela-Ascoli theorem on time scales, we can find that F Kc is relatively compact. Finally, from the continuity of f and w(t) ∈ Cld ([0, T ]T , [0, +∞)), we can find that F is continuous. Thus, F is completely continuous. The proof is complete. The proof of our main result is based upon an application of the following fixed point theorem in a cone. Theorem 2.5 ([10]). Let K be a cone in a Banach space X. Let D be an open bounded subset of X with DK = D ∩ K 6= ∅ and DK 6= K. Suppose that F :
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p-LAPLACIAN DYNAMIC EQUATIONS
9
DK → K is a completely continuous map such that u 6= F u for u ∈ ∂DK . Then the following results hold. (i) If kF uk ≤ kuk ∀u ∈ ∂DK , then iK (F, DK ) = 1. (ii) If there exists e ∈ K \ {0} such that u 6= F u + λe ∀u ∈ ∂DK and λ > 0, then iK (F, DK ) = 0. (iii) Let U be open in X such that U ⊂ DK . If iK (F, DK ) = 1 and iK (F, UK ) = 0, then F has a fixed point in DK \ UK . The same result holds if iK (F, DK ) = 0 and iK (F, UK ) = 1, where iK (F, DK ) denotes a fixed point index. 3. Main results Lemma 3.1 ([15]). Let Kρ = {u ∈ K : kuk < ρ} and Ωρ = {u ∈ K : min u(t) < γρ}. t∈[0,T ]T
Then Ωρ has the following properties: (a) Ωρ is open relative to K; (b) Kγρ ⊂ Ωρ ⊂ Kρ ; (c) u ∈ ∂Ωρ if and only if mint∈[0,T ]T u(t) = γρ; (d) If u ∈ ∂Ωρ , then γρ ≤ u(t) ≤ ρ for t ∈ [0, T ]T . Now, for convenience, we introduce the following notation. Let Pm−2 R T Z T Z T 1 1 i=1 bi ξi w(τ )∇τ = φq w(τ )∇τ + ∆s, Pm−2 Pm−2 Λ1 1 − i=1 ai 0 1 − i=1 bi s Pm−2 R T h m−2 i X Z ξi Z T 1 1 i=1 bi ξi w(τ )∇τ = ai φq w(τ )∇τ + ∆s . Pm−2 Pm−2 Λ2 1 − i=1 ai i=1 1 − i=1 bi 0 s We remark that from (H1) it follows that 0 < Λ1 , Λ2 < +∞, Λ2 γ = Λ2 γ1 γ2 = Λ1 γ1 < Λ1 . Lemma 3.2. If f satisfies the conditions f (t, u) ≤ φp (Λ1 ρ)
and
u 6= F u, for u ∈ ∂Kρ , (t, u) ∈ [0, T ]T × [0, ρ],
then iK (F, Kρ ) = 1. Proof. If u ∈ ∂Kρ , then Z s w(τ )f + (τ, u(τ ))∇τ − C˜ 0 Z s = w(τ )f + (τ, u(τ ))∇τ 0
Pm−2 i=1
+ Z =
bi
R ξi 0
RT w(τ )f + (τ, u(τ ))∇τ − 0 w(τ )f + (τ, u(τ ))∇τ Pm−2 1 − i=1 bi
s
w(τ )f + (τ, u(τ ))∇τ
0
Pm−2 +
i=1
bi
RT 0
Rξ w(τ )f + (τ, u(τ ))∇τ + T i w(τ )f + (τ, u(τ ))∇τ Pm−2 1 − i=1 bi
10
A. DOGAN
RT 0
−
EJDE-2018/39
w(τ )f + (τ, u(τ ))∇τ Pm−2 1 − i=1 bi
s
Z
w(τ )f + (τ, u(τ ))∇τ 0 R Pm−2 R T T w(τ )f + (τ, u(τ ))∇τ − i=1 bi 0 w(τ )f + (τ, u(τ ))∇τ 0 − Pm−2 1 − i=1 bi Pm−2 R T + i=1 bi ξi w(τ )f (τ, u(τ ))∇τ − Pm−2 1 − i=1 bi Z s w(τ )f + (τ, u(τ ))∇τ = 0 Pm−2 R T 1 − i=1 bi 0 w(τ )f + (τ, u(τ ))∇τ − Pm−2 1 − i=1 bi Pm−2 R T + i=1 bi ξi w(τ )f (τ, u(τ ))∇τ − Pm−2 1 − i=1 bi Z s Z T = w(τ )f + (τ, u(τ ))∇τ − w(τ )f + (τ, u(τ ))∇τ =
0
0
Pm−2
bi
RT
+
w(τ )f (τ, u(τ ))∇τ Pm−2 1 − i=1 bi Pm−2 R T Z T + i=1 bi ξi w(τ )f (τ, u(τ ))∇τ + =− w(τ )f (τ, u(τ ))∇τ − Pm−2 1 − i=1 bi s Pm−2 R T Z h T i i=1 bi ξi w(τ )∇τ ≥ −φp (ρΛ1 ) w(τ )∇τ + Pm−2 1 − i=1 bi s i=1
+
ξi
so that ϕ(s) = φq
s
Z
w(τ )f + (τ, u(τ ))∇τ − C˜
0
≥ −ρΛ1 φq
Z
RT bi ξi w(τ )∇τ Pm−2 1 − i=1 bi
Pm−2
T
w(τ )∇τ +
i=1
s
Therefore, kF uk ≤
1−
≤−
1 Pm−2
1−
≤ ρΛ1
i=1
h m−2 X ai
1 Pm−2 i=1
1− = ρ = kuk.
i=1
T
Z ϕ(s)∆s −
ξi
i=1 Z T
T
i ϕ(s)∆s
0
ϕ(s)∆s
ai
1 Pm−2
Z ai
0
Z ai
T
φq 0
Z s
RT bi ξi w(τ )∇τ ∆s Pm−2 1 − i=1 bi
Pm−2
T
w(τ )∇τ +
i=1
EJDE-2018/39
p-LAPLACIAN DYNAMIC EQUATIONS
11
This implies kF uk ≤ kuk, u ∈ ∂Kρ . By Theorem 2.5 (i), we have iK (F, Kρ ) = 1. Lemma 3.3. If f satisfies the conditions f (t, u) ≥ φp (Λ2 γρ)
and
u 6= F u for u ∈ ∂Ωρ , (t, u) ∈ [0, T ]T × [γρ, ρ],
then iK (F, Ωρ ) = 0. Proof. Let e(t) ≡ 1 for t ∈ [0, T ]T , then e ∈ ∂K1 . We claim that u 6= F u + λe, u ∈ ∂Ωρ , and λ > 0. In fact, if not, there exist u0 ∈ ∂Ωρ , and λ0 > 0 such that u0 = F u0 + λ0 e. Then we find that Z s w(τ )f + (τ, u0 (τ ))∇τ − C˜ 0 s
Z
w(τ )f + (τ, u0 (τ ))∇τ +
= 0
T
Z
w(τ )f + (τ, u0 (τ ))∇τ
−
1−
1 Pm−2 i=1
m−2 X bi
Z bi
i=1
ξi
w(τ )f + (τ, u0 (τ ))∇τ
0
0
Pm−2
s
Z
w(τ )f + (τ, u0 (τ ))∇τ +
= 0
Z
ξi
+
1− w(τ )f + (τ, u0 (τ ))∇τ −
T s
Z
w(τ )f + (τ, u0 (τ ))∇τ −
= 0
−
m−2 X i=1
Z
1−
T
Z T i=1 bi w(τ )f + (τ, u0 (τ ))∇τ Pm−2 0 i=1 bi RT + w(τ )f (τ, u0 (τ ))∇τ 0 Pm−2 1 − i=1 bi Z T 1 w(τ )f + (τ, u0 (τ ))∇τ Pm−2 0 i=1 bi
+
w(τ )f (τ, u0 (τ ))∇τ −
bi 0
m−2 X i=1
RT bi
ξi
w(τ )f + (τ, u0 (τ ))∇τ Pm−2 1 − i=1 bi
Pm−2 R T s 1 − i=1 bi 0 w(τ )f + (τ, u0 (τ ))∇τ = w(τ )f + (τ, u0 (τ ))∇τ − Pm−2 1 − i=1 bi 0 Pm−2 R T + i=1 bi ξi w(τ )f (τ, u0 (τ ))∇τ − Pm−2 1 − i=1 bi Z T Z s = w(τ )f + (τ, u0 (τ ))∇τ + w(τ )f + (τ, u0 (τ ))∇τ Z
0
T
Pm−2
RT
w(τ )f + (τ, u0 (τ ))∇τ Pm−2 1 − i=1 bi 0 Pm−2 R T Z T + i=1 bi ξi w(τ )f (τ, u0 (τ ))∇τ + =− w(τ )f (τ, u0 (τ ))∇τ − Pm−2 1 − i=1 bi s Pm−2 R T Z + T i=1 bi ξi w(τ )f (τ, u0 (τ ))∇τ =− w(τ )f + (τ, u0 (τ ))∇τ + Pm−2 1 − i=1 bi s Pm−2 R T Z T i=1 bi ξi w(τ )∇τ ≤ −φp (Λ2 γρ) w(τ )∇τ + Pm−2 1 − i=1 bi s Z
−
T
+
w(τ )f (τ, u0 (τ ))∇τ −
i=1
bi
ξi
12
A. DOGAN
EJDE-2018/39
so that ϕ(s) ˜ = φq
Z
s
w(τ )f + (τ, u0 (τ ))∇τ − C˜
0
≤ −Λ2 γρφq
RT bi ξi w(τ )∇τ . Pm−2 1 − i=1 bi
Pm−2
T
Z
w(τ )∇τ +
s
i=1
From (2.10), we have u0 (t) = F u0 (t) + λ0 e(t) Z T Z T h m−2 i X Z T 1 ≥ ϕ(s)∆s ˜ + ϕ(s)∆s ˜ + λ0 ai ϕ(s)∆s ˜ − Pm−2 1 − i=1 ai i=1 0 0 ξi Pm−2 R T Pm−2 R T − i=1 ai 0 ϕ(s)∆s ˜ + i=1 ai ξi ϕ(s)∆s ˜ = + λ0 Pm−2 1 − i=1 ai Rξ RT Pm−2 Pm−2 R T − i=1 ai 0 i ϕ(s)∆s ˜ + ξi ϕ(s)∆s ˜ + i=1 ai ξi ϕ(s)∆s ˜ = + λ0 Pm−2 1 − i=1 ai Pm−2 R ξ − i=1 ai 0 i ϕ(s)∆s ˜ = + λ0 Pm−2 1 − i=1 ai Pm−2 R T Pm−2 R ξi Z T i=1 bi ξi w(τ )∇τ i=1 ai 0 ∆s + λ0 w(τ )∇τ + ≥ γρΛ2 Pm−2 φq Pm−2 1 − i=1 ai 1 − i=1 bi s = γρ + λ0 , which implies λ0 ≤ 0, it is a contradiction. Hence by Theorem 2.5 (ii), it follows that iK (F, Ωρ ) = 0. Now, we set up and verify our existence results for the BVP (1.1) and (1.2). Theorem 3.4. Suppose that one of the following conditions holds. (C1) There exist ρ1 , ρ2 and ρ3 ∈ (0, +∞) with ρ1 < γρ2 , and ρ2 < ρ3 such that (i) f (t, u) ≤ φp (Λ1 ρ1 ), (t, u) ∈ [0, T ]T × [0, ρ1 ]; (ii) f (t, u) ≥ 0, (t, u) ∈ [0, T ]T ×[γρ1 , ρ3 ], in addition, f (t, u) ≥ φp (Λ2 γρ2 ), (t, u) ∈ [0, T ]T × [γρ2 , ρ2 ], u 6= F u, u ∈ ∂Ωρ2 ; (iii) f (t, u) ≤ φp (Λ1 ρ3 ), (t, u) ∈ [0, T ]T × [0, ρ3 ]. (C2) There exist ρ1 , ρ2 and ρ3 ∈ (0, +∞) with ρ1 < ρ2 < γρ3 , such that (i) f (t, u) ≥ φp (Λ2 γρ1 ), (t, u) ∈ [0, T ]T × [γ 2 ρ1 , ρ2 ]; (ii) f (t, u) ≤ φp (Λ1 ρ2 ), (t, u) ∈ [0, T ]T × [0, ρ2 ], u 6= F u, u ∈ ∂Kρ2 ; (iii) f (t, u) ≥ 0, (t, u) ∈ [0, T ]T ×[γρ2 , ρ3 ], in addition, f (t, u) ≥ φp (Λ2 γρ3 ), (t, u) ∈ [0, T ]T × [γρ3 , ρ3 ]. Then BVP (1.1) and (1.2) has at least two positive solutions u1 and u2 . Proof. Suppose (C1) holds, we verify that F has a fixed point u1 either in ∂Kρ1 or u1 in Ωρ2 \Kρ1 . If u 6= F u, u ∈ ∂Kρ1 ∪ ∂Kρ3 , by Lemmas 3.2 and 3.3, we have iK (F, Kρ1 ) = 1,
iK (F, Ωρ2 ) = 0,
iK (F, Kρ3 ) = 1.
By Lemma 3.1 (b) and ρ1 < γρ2 , we have Kρ1 ⊂ Kγρ2 ⊂ Ωρ2 . From Theorem 2.5 (iii), we obtain F has a fixed point u1 ∈ Ωρ2 \Kρ1 . Similarly, F has a fixed point
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p-LAPLACIAN DYNAMIC EQUATIONS
13
u2 ∈ Kρ3 \Ωρ2 . Obviously, ku1 k > ρ1 ,
min u1 (t) = u1 (0) ≥ γku1 k > γρ1 .
t∈[0,T ]T
Therefore, γρ1 ≤ u1 (t) ≤ ρ2 , t ∈ [0, T ]T . By (C1) (ii), we have f (t, u1 (t)) ≥ 0, t ∈ [0, T ]T , i.e. f + (t, u1 (t)) = f (t, u1 (t)). Hence, F u1 = Au1 . Consequently u1 is a fixed point of A. From u2 ∈ Kρ3 \Ωρ2 , ρ2 < ρ3 and Lemma 3.1 (b), we have Kγρ2 ⊂ Ωρ2 ⊂ Kρ3 . Clearly, ku2 k > γρ2 . This implies min u2 (t) = u2 (0) ≥ γku2 k > γ 2 ρ2 .
t∈[0,T ]T
So, γ 2 ρ2 ≤ u2 (t) ≤ ρ3 ,
t ∈ [0, T ]T .
Using that ρ1 < γρ2 and (C1) (ii), we obtain f (t, u2 (t)) ≥ 0, t ∈ [0, T ]T , i.e., f + (t, u2 (t)) = f (t, u2 (t)). Therefore u2 is another fixed point of A. Thus, we have verified that BVP (1.1) and (1.2) has at least two positive solutions u1 and u2 . The proof is similar when (C2) holds. The proof is complete. By a similar argument to that of Theorem 3.4, we can find the following new results on existence of at least one positive solution of the BVP (1.1) and (1.2). Theorem 3.5. Assume that one of the following conditions holds. (C3) There exist ρ1 , ρ2 ∈ (0, +∞) with ρ1 < γρ2 such that (i) f (t, u) ≤ φp (Λ1 ρ1 ), (t, u) ∈ [0, T ]T × [0, ρ1 ]; (ii) f (t, u) ≥ 0, (t, u) ∈ [0, T ]T ×[γρ1 , ρ2 ], in addition, f (t, u) ≥ φp (Λ2 γρ2 ), (t, u) ∈ [0, T ]T × [γρ2 , ρ2 ]. (C4) There exist ρ1 , ρ2 ∈ (0, +∞) with ρ1 < ρ2 such that (i) f (t, u) ≥ φp (Λ2 γρ1 ), (t, u) ∈ [0, T ]T × [γ 2 ρ1 , ρ2 ]; (ii) f (t, u) ≤ φp (Λ1 ρ2 ), (t, u) ∈ [0, T ]T × [0, ρ2 ]. Then BVP (1.1) and (1.2) has at least one positive solution. 4. Application {1 − ( 21 )N0 } ∪ { 13 , 1},
Let T = N0 denotes the set of all nonnegative integers. We consider the following BVP on time scales (u∆ (t))∇ + f (t, u(t)) = 0, t ∈ [0, T ]T , 2 1 1 1 1 1 1 1 u(0) = u + u , u∆ (1) = u∆ + u , 3 3 4 2 3 3 2 2
(4.1) (4.2)
where f (t, u) 5455 3 1 3 u − 41472 × 40 t , 1 3 41472 π 5455 π 40 t sin 36017 2 u − 36017 2 , 1 3 54550 25 41472 = 40 t 3078 − 41472 u + 288 u − 41472 , 3078 3078 3078 2 25 7 54550 3 288 + 7843 t u − 41472 , 25 7 54550 2 3 288 + 7843 t 10 − 41472 [1 + (u − 10)(20 − u)],
5455 (t, u) ∈ [0, T ]T × [0, 41472 ], 5455 t, u) ∈ [0, T ]T × [ 41472 , 1], 54550 (t, u) ∈ [0, T ]T × [1, 41472 ], 54550 (t, u) ∈ [0, T ]T × [ 41472 , 10], (t, u) ∈ [0, T ]T × [10, +∞).
14
A. DOGAN
EJDE-2018/39
It is easy to see that f : [0, 1] × [0, +∞) → (−∞, +∞) is continuous. In this case p = q = 2, w(t) ≡ 1, T = 1, a1 = 32 , a2 = 14 , b1 = 13 , b2 = 12 , ξ1 = 13 , ξ2 = 12 , it follows from a direct calculation that RT RT Z T Z T b1 ξ1 ∇τ + b2 ξ2 ∇τ 1 1 φq ∇τ + ∆s = Λ1 1 − (a1 + a2 ) 0 1 − (b1 + b2 ) s 1 1 1 1 Z 1 3 1− 3 + 2 1− 3 1 1−s+ ∆s = 40, = 1 − ( 23 + 14 ) 0 1 − (1 − 13 + 21 ) RT RT h Z ξ1 Z T b1 ξ1 ∇τ + b2 ξ2 ∇τ 1 1 a1 φq ∇τ + ∆s = Λ2 1 − (a1 + a2 ) 1 − (b1 + b2 ) 0 s RT RT Z ξ2 Z T b1 ξ1 ∇τ + b2 ξ2 ∇τ i + a2 φq ∇τ + ∆s 1 − (b1 + b2 ) 0 s 1 1 1 1 h 2 Z 1/3 3 1− 3 + 2 1− 2 1 1−s+ = ∆s 1 − ( 23 + 14 ) 3 0 1 − ( 31 + 12 ) 1 1 1 1 Z i 1091 3 1− 3 + 2 1− 2 1 1/2 , + 1−s+ ∆s = 4 0 72 1− 1 + 1 3
2
2 1 1 1 25 ξ1 ξ2 γ1 = a1 + a2 = · + · = . T T 3 3 4 2 72 Let ξ1
Z
Z
T
∇τ +
b1
RT ξ1
∇τ + b2
RT ξ2
∇τ
∆s 1 − (b1 + b2 ) RT RT Z ξ2 Z T b1 ξ1 ∇τ + b2 ξ2 ∇τ + a2 φq ∇τ + ∆s 1 − (b1 + b2 ) 0 s 1 1 1 1 Z 3 1− 3 + 2 1− 2 2 1/3 = 1−s+ ∆s 3 0 1 − 13 + 21 1 1 1 1 Z 1/2 1 − + 1 − 3 3 2 2 1 1091 + 1−s+ ∆s = 1 1 4 0 864 1− +
N = a1
φq
0
s
3
2
and RT
D=
T
φq
Z
T
b1
ξ1
∇τ + b2
RT
∇τ ξ2 ∇τ + ∆s 1 − (b1 + b2 ) 0 s 1 1 1 1 Z 1 3 1− 3 + 2 1− 3 10 = 1−s+ ∆s = . 1 1 3 0 1− 3 + 2 Z
Then γ2 =
N 1091 3 1091 = · = , D 864 10 2880
γ = γ1 γ2 =
5455 , 41472
EJDE-2018/39
p-LAPLACIAN DYNAMIC EQUATIONS
15
72 5455 24 1 · = ≈ 0.00868 < = 0.025 = Λ1 . 1091 41472 2765 40
Λ2 γ =
Choose ρ1 = 1, ρ2 = 10, ρ3 = 20, it is easy to check that γρ1 < ρ1 < γρ2 < ρ2 < ρ3 . By (2.10), we obtain Z T Z s (F u)(t) = φq f + (τ, u(τ ))∇τ − C˜ ∆s t
+
0
1− Z T
−
1 Pm−2 i=1
φq
=
ai
s
Z
0 1
Z
h m−2 X
T
Z ai
φq ξi
i=1
s
Z
f + (τ, u(τ ))∇τ − C˜ ∆s
0
i f + (τ, u(τ ))∇τ − C˜ ∆s
0 s
Z
t
0 1 Z s
Z +3
1/2 Z 1
1
Z f + (τ, u(τ ))∇τ − C˜ ∆s + 8
Z
1/3
s
f + (τ, u(τ ))∇τ − C˜ ∆s
0
f + (τ, u(τ ))∇τ − C˜ ∆s
0 s
Z
− 12 0
31 ˜ f + (τ, u(τ ))∇τ − C˜ ∆s + C, 6
0
where C˜ =
h m−2 X
−1 Pm−2
1− Z = −2
bi
i=1
Z
ξi
bi
f + (τ, u(τ ))∇τ −
0
i=1
f + (τ, u(τ ))∇τ − 3
Since f (t, u) ≤
f + (τ, u(τ ))∇τ
Z
1/2
f + (τ, u(τ ))∇τ + 6
Z
0 13 80 ,
i
0
1/3
0
T
Z
1
f + (τ, u(τ ))∇τ.
0
t ∈ [0, T ]T , u ∈ [0, 10], for u ∈ ∂K10 , we have
kF uk = max F u(t) t∈[0,T ]T
Z
1
s
Z
=8 1/3
0 1
Z − 12 1
=8 1/3
Z −3
1
f + (τ, u(τ ))∇τ ∆s + 3
f + (τ, u(τ ))∇τ ∆s
0
31 ˜ C 6 Z 1 Z 1/2
Z
0 1/2
s
f + (τ, u(τ ))∇τ ∆s +
0
Z − 12
0 s
Z
Z
1/2 s
Z
0
Z
1
Z f + (τ, u(τ ))∇τ ∆s + 3
s
+
f (τ, u(τ ))∇τ ∆s +
0
f + (τ, u(τ ))∇τ + 6
0
Z
31 h 6
Z
1/3
f + (τ, u(τ ))∇τ
−2 0
1
f + (τ, u(τ ))∇τ
0
This implies F u 6= u, for u ∈ ∂K10 . As a result, f satisfies the following conditions: 1 40 ,
f + (τ, u(τ ))∇τ ∆s
0
13 ≤ < 10 = kuk. 4
(i) f (t, u) ≤ φp (Λ1 ρ1 ) =
s
(t, u) ∈ [0, T ]T × [0, 1];
i
16
A. DOGAN
EJDE-2018/39
5455 (ii) f (t, u) ≥ 0, (t, u) ∈ [0, T ]T × [ 41472 , 20]; in addition, f (t, u) ≥ φp (Λ2 γρ2 ) = 5455 5 54550 72 · · 4 = , (t, u) ∈ [0, T ] T × [ 41472 , 10]; 1091 41472 144 7 (iii) f (t, u) ≤ φp (Λ1 ρ3 ) = 40 , (t, u) ∈ [0, T ]T × [0, 20].
Therefore, (C1) of Theorem 3.4 is satisfied, then problem (4.1) and (4.2) has two positive solutions u1 , u2 satisfying ku1 k ≤ 10,
ku2 k > 10.
Acknowledgments. The author would like to thank the anonymous referees and editor for their valuable comments and suggestions. References [1] R. P. Agarwal, H. L¨ u, D. O’Regan; Existence theorems for the one-dimensional singular p-Laplacian equation with sign changing nonlinearities, Appl. Math. Comput., 143 (2003) 15-38. [2] D. R. Anderson; Solutions to second-order three-point problems on time scales, J. Difference Equ. Appl., 8 (2002) 673-688. [3] D. R. Anderson, R. Avery, J. Henderson; Existence of solutions for a one-dimensional pLaplacian on time scales, J. Difference Equ. Appl., 10 (2004) 889-896. [4] M. Bohner, A. Peterson; Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, Cambridge, MA 2001. [5] M. Bohner, A. Peterson; Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, Cambridge, MA 2003. [6] A. Dogan; Existence of multiple positive solutions for p-Laplacian multipoint boundary value problems on time scales, Adv. Differ. Equ. 2013, 238 (2013), 1-23. [7] A. Dogan; Triple positive solutions for m-point boundary-value problems of dynamic equations on time scales with p-Laplacian, Electron. J. Differential Equations, 131 (2015) 1-12. [8] A. Dogan; On the existence of positive solutions of the p-Laplacian dynamic equations on time scales, Math. Meth. Appl. Sci., 40 (2017), 4385-4399. [9] S. Georgiev; Integral Equations on Time Scales, Atlantis Press, Paris, 2016. [10] D. Guo, V. Lakshmikantham; Nonlinear Problems in Abstract Cones, Academic Press, San Diego, 1988. [11] Z. He, Z. Long; Three positive solutions of three-point boundary value problems for pLaplacian dynamic equations on time scales, Nonlinear Anal., 69 (2008) 569-578. [12] S. Hilger; Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18-56. [13] D. Ji, M. Feng, W. Ge; Multiple positive solutions for multipoint boundary value problems with sign changing nonlinearity, Appl. Math. Comput., 196 (2008), 511-520. [14] M. A. Jones, B. Song, D. M. Thomas; Controlling wound healing through debridement, Math. Comput. Modelling, 40 (2004), 1057-1064. [15] K. Q. Lan; Multiple positive solutions of semilinear differential equations with singularities, J. London Math. Soc. 63, (2001) 690-704. [16] H. Luo; Positive solutions to singular multi-point dynamic eigenvalue problems with mixed derivatives, Nonlinear Anal., 70 (2009), 1679-1691. [17] A. C. Peterson, Y. N. Raffoul, C. C. Tisdell; Three point boundary value problems on time scales, J. Difference Equ. Appl., 10 (2004), 843-849. [18] Y. Sang, H. Su, F. Xu; Positive solutions of nonlinear m-point BVP for an increasing homeomorphism and homomorphism with sign changing nonlinearity on time scales, Comput. Math. Appl., 58 (2009), 216-226. [19] V. Spedding; Taming nature’s numbers, New Scientist: The Global Science and Technology Weekly, 2404 (2003), 28-31. [20] Y. H. Su, W. T. Li, H. R. Sun; Positive solutions of singular p-Laplacian BVP’s with sign changing nonlinearity on time scales, Math. Comput. Model., 48 (2008), 845-858. [21] Y. H. Su, W. T. Li; Triple positive solutions of m-point BVPs for p-Laplacian dynamic equations on time scales, Nonlinear Anal., 69 (2008), 3811-3820.
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[22] H. R. Sun; Triple positive solutions for p-Laplacian m-point boundary value problem on time scales, Comput. Math. Appl., 58 (2009), 1736-1741. [23] H. R. Sun, W. T. Li; Existence theory for positive solutions to one-dimensional p-Laplacian boundary value problems on time scales, J. Differential Equations, 240 (2007) 217-248. [24] D. M. Thomas, L. Vandemuelebroeke, K. Yamaguchi; A mathematical evolution model for phytoremediation of metals, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 411-422. [25] D. B. Wang; Three positive solutions of three-point boundary value problems for p-Laplacian dynamic equations on time scales, Nonlinear Anal., 68 (2008), 2172-2180. [26] Y. Wang, C. Hou; Existence of multiple positive solutions for one-dimensional for pLaplacian, J. Math. Anal. Appl., 315 (2006), 144-153. [27] Y. Yang, F. Meng; Positive solutions of the singular semipositone boundary value problem on time scales, Math. Comput. Model., 52 (2010), 481-489. [28] Y. Zhu, J. Zhu; The multiple solutions for p-Laplacian multipoint BVP with sign changing nonlinearity on time scales, J. Math. Anal. Appl., 344 (2008), 616-626. Abdulkadir Dogan Department of Applied Mathematics, Faculty of Computer Sciences, Abdullah Gul University, Kayseri, 38039 Turkey. Tel +90 352 224 88 00, Fax +90 352 338 88 28 E-mail address:
[email protected] 