EXISTENCE OF POSITIVE SOLUTIONS FOR NON LOCAL P -LAPLACIAN THERMISTOR PROBLEMS ON TIME SCALES

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Existence for Thermistor Problems on Time Scales Moulay Rchid Sidi Ammi

and Delfim F. M. Torres vol. 8, iss. 3, art. 69, 2007

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EXISTENCE OF POSITIVE SOLUTIONS FOR NON LOCAL P -LAPLACIAN THERMISTOR PROBLEMS

ON TIME SCALES

MOULAY RCHID SIDI AMMI AND DELFIM F. M. TORRES

Department of Mathematics University of Aveiro 3810-193 Aveiro, Portugal

EMail:{sidiammi,delfim}@mat.ua.pt

Received: 20 March, 2007

Accepted: 25 August, 2007

Communicated by: R.P. Agarwal

2000 AMS Sub. Class.: 34B18, 39A10, 93C70.

Key words: Time scales,p-Laplacian, Positive solutions, Existence.

Abstract: We make use of the Guo-Krasnoselskii fixed point theorem on cones to prove existence of positive solutions to a non localp-Laplacian boundary value problem on time scales arising in many applications.

Acknowledgements: The authors were partially supported by the Portuguese Foundation for Science and Technology (FCT) through the Centre for Research in Optimization and Con- trol (CEOC) of the University of Aveiro, cofinanced by the European Community fund FEDER/POCTI, and by the project SFRH/BPD/20934/2004.

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1. Introduction

The purpose of this paper is to prove the existence of positive solutions for the fol- lowing non localp-Laplacian dynamic equation on a time scaleT:

(1.1) − φ_p(u⁴(t))^∇

= λf(u(t)) (RT

0 f(u(τ))∇τ)^k, ∀t∈(0, T)_T =T, subject to the boundary conditions

(1.2) φ_p(u⁴(0))−βφ_p(u⁴(η)) = 0, 0< η < T, u(T)−βu(η) = 0,

whereφ_p(·)is thep-Laplacian operator defined byφ_p(s) =|s|^p−2s,p >1,(φ_p)⁻¹ = φ_q withqthe Hölder conjugate ofp, i.e. ¹_p + ¹_q = 1. The function

(H1) f : (0, T)_T→R^+∗ is assumed to be continuous

(R^+∗ denotes the positive real numbers); λ is a dimensionless parameter that can be identified with the square of the applied potential difference at the ends of a conductor; f(u) is the temperature dependent resistivity of the conductor; β is a transfer coefficient supposed to verify 0 < β < 1. Different values for p and k are connected with a variety of applications for both T = R and T = Z. When k > 1, equation (1.1) represents the thermo-electric flow in a conductor [20]. In the particular case p = k = 2, (1.1) has been used to describe the operation of thermistors, fuse wires, electric arcs and fluorescent lights [11, 12, 18, 19]. For k = 1, equation (1.1) models the phenomena associated with the occurrence of shear bands (i) in metals being deformed under high strain rates [6, 7], (ii) in the theory of gravitational equilibrium of polytropic stars [17], (iii) in the investigation of the fully turbulent behavior of real flows, using invariant measures for the Euler

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equation [10], (iv) in modelling aggregation of cells via interaction with a chemical substance (chemotaxis) [22].

The theory of dynamic equations on time scales (or, more generally, measure chains) was introduced in 1988 by Stefan Hilger in his PhD thesis (see [14, 15]).

The theory presents a structure where, once a result is established for a general time scale, then special cases include a result for differential equations (obtained by taking the time scale to be the real numbers) and a result for difference equations (obtained by taking the time scale to be the integers). A great deal of work has been done since 1988, unifying and extending the theories of differential and difference equations, and many results are now available in the general setting of time scales – see [1,2, 3,4,8,9] and the references therein. We point out, however, that results concerning p-Laplacian problems on time scales are scarce [21]. In this paper we prove the existence of positive solutions to the problem (1.1)-(1.2) on a general time scaleT.

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2. Preliminaries

Our main tool to prove the existence of positive solutions (Theorem3.5) is the Guo- Krasnoselskii fixed point theorem on cones.

Theorem 2.1 (Guo-Krasnoselskii fixed point theorem on cones [13, 16]). LetX be a Banach space andK ⊂Ebe a cone inX. Assume thatΩ₁andΩ₂ are bounded open subsets ofK with0 ∈ Ω₁ ⊂ Ω₁ ⊂ Ω₂ and thatG : K → K is a completely continuous operator such that

(i) eitherkGwk ≤ kwk,w∈∂Ω₁, andkGwk ≥ kwk,w∈∂Ω₂; or (ii) kGwk ≥ kwk,w∈∂Ω₁, andkGwk ≤ kwk,w∈∂Ω₂.

Then,Ghas a fixed point inΩ₂\Ω₁.

Using the properties off on a bounded set(0, T)_T, we construct an operator (an integral equation) whose fixed points are solutions to the problem (1.1)-(1.2).

Now we introduce some basic concepts of time scales that are needed in the sequel. For deeper details, the reader can see, for instance, [1,5,8]. A time scaleT is an arbitrary nonempty closed subset ofR. The forward jump operatorσ and the backward jump operatorρ, both fromTtoT, are defined in [14]:

σ(t) = inf{τ ∈T:τ > t} ∈T, ρ(t) = sup{τ ∈T:τ < t} ∈T.

A point t ∈ T is left-dense, left-scattered, right-dense, or right-scattered if ρ(t) = t, ρ(t) < t, σ(t) = t, orσ(t) > t, respectively. IfThas a right scattered minimum m, defineTk=T− {m}; otherwise setTk=T. IfThas a left scattered maximum M, defineT^k =T− {M}; otherwise setT^k =T.

Let f : T → R and t ∈ T^k (assume t is not left-scattered if t = supT), then the delta derivative off at the pointtis defined to be the numberf^∆(t)(provided it

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exists) with the property that for each >0there is a neighborhoodU oftsuch that f(σ(t))−f(s)−f^∆(t)(σ(t)−s)

≤ |σ(t)−s|, for alls∈U .

Similarly, fort∈T(assumetis not right-scattered ift = infT), the nabla derivative of f at the point t is defined to be the number f^∇(t) (provided it exists) with the property that for each >0there is a neighborhoodU oftsuch that

|f(ρ(t))−f(s)−f^∇(t)(ρ(t)−s)| ≤ |ρ(t)−s|, for alls∈U .

IfT = R, thenx^∆(t) = x^∇(t) = x⁰(t). If T = Z, then x^∆(t) = x(t+ 1)−x(t) is the forward difference operator while x^∇(t) = x(t)−x(t−1)is the backward difference operator.

A function f is left-dense continuous (ld-continuous) iff is continuous at each left-dense point inT and its right-sided limit exists at each right-dense point inT. Letf beld-continuous. IfF^∇(t) =f(t), then the nabla integral is defined by

Z b

a

f(t)∇t=F(b)−F(a) ; ifF^∆(t) =f(t), then the delta integral is defined by

Z b

a

f(t)∆t=F(b)−F(a).

In the remainder of this article T is a closed subset of R with 0 ∈ Tk, T ∈ T^k; E = Cld([0, T],R), which is a Banach space with the maximum norm kuk = max_[0,T_]_T|u(t)|.

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3. Main Results

By a positive solution of (1.1)-(1.2) we understand a functionu(t)which is positive on(0, T)_Tand satisfies (1.1) and (1.2).

Lemma 3.1. Assume that hypothesis (H1) is satisfied. Then, u(t) is a solution of (1.1)-(1.2) if and only ifu(t)∈Eis solution of the integral equation

u(t) =− Z t

0

φ_q(g(s))4s+B, where

g(s) = Z s

0

λh(u(r))∇r−A, A =φ_p(u⁴(0)) =− λβ

1−β Z η

0

h(u(r))∇r, h(u(t)) = λf(u(t))

RT

0 f(u(τ))∇τk, B =u(0) = 1

1−β Z T

0

φq(g(s))4s−β Z η

0

φq(g(s))4s

. Proof. We begin by proving necessity. Integrating the equation (1.1) we have

φ_p(u⁴(s)) =φ_p(u⁴(0))− Z s

0

λh(u(r))∇r.

On the other hand, by the boundary condition (1.2) φ_p(u⁴(0)) =βφ_p(u⁴(η)) =β

φ_p(u⁴(0))− Z η

0

λh(u(r))∇r

.

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Then,

A=φ_p(u⁴(0)) = −λβ 1−β

Z η

0

h(u(r))∇r.

It follows that

u⁴(s) =φ_q

−λ Z s

0

h(u(r))∇r+A

=−φ_q(g(s)).

Integrating the last equation, we obtain

(3.1) u(t) =u(0)−

Z t

0

φ_q(g(s))4s.

Moreover, by (3.1) and the boundary condition (1.2), we have u(0) =u(T) +

Z T

0

φ_q(g(s))4s

=βu(η) + Z T

0

φ_q(g(s))4s

=β

u(0)− Z η

0

φ_q(g(s))4s

+ Z T

0

φ_q(g(s))4s.

Then,

u(0) =B = 1 1−β

−β Z η

0

φ_q(g(s))4s+ Z T

0

φ_q(g(s))4s

. Sufficiency follows by a simple calculation, taking the delta derivative ofu(t).

Lemma 3.2. Suppose (H1) holds. Then, a solutionuof (1.1)-(1.2) satisfiesu(t)≥0 for allt∈(0, T)_T.

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Proof. We haveA = ^−λβ_1−β Rη

0 h(u(r))∇r ≤0. Then,g(s) = λRs

0 h(u(r))−A ≥0.

It follows thatφ_p(g(s))≥0. Since0< β < 1, we also have u(0) =B = 1

1−β Z T

0

φ_q(g(s))4s−β Z η

0

φ_q(g(s))4s

≥ 1 1−β

β

Z T

0

φ_q(g(s))4s

≥0 and

u(T) =u(0)− Z T

0

φq(g(s))4s

= −β 1−β

Z η

0

φq(g(s))4s+ 1 1−β

Z T

0

φq(g(s))4s− Z T

0

φq(g(s))4s

= −β 1−β

Z η

0

φq(g(s))4s+ β 1−β

Z T

0

φq(g(s))4s

= β

1−β Z T

0

φq(g(s))4s− Z η

0

φq(g(s))4s

≥0.

Ift ∈(0, T)_T,

u(t) = u(0)− Z t

0

φ_q(g(s))4s

≥ − Z T

0

φ_q(g(s))4s+u(0) =u(T)≥0.

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Lemma 3.3. If (H1) holds, thenu(T)≥ρu(0), whereρ=β_T^T_−βη^−η ≥0.

Proof. We have

φ_p(u⁴(s)) = φ_p(u⁴(0))− Z s

0

λh(u(r))∇r≤0.

SinceA=φ_p(u⁴(0))≤0, thenu⁴ ≤0. This means thatkuk=u(0),inf_t∈(0,T₎

Tu(t)

=u(T). Moreover,φ_p(u⁴(s))is non increasing which implies, with the monotonic- ity ofφ_p,thatu⁴ is a non increasing function on(0, T)_T. It follows from the con- cavity ofu(t)that each point on the chord between(0, u(0))and(T, u(T))is below the graph ofu(t). We have

u(T)≥u(0) +Tu(T)−u(η) T −η . Alternatively,

T u(η)−ηu(T)≥(T −η)u(0).

Using the boundary condition (1.2), it follows that T

β −η

u(T)≥(T −η)u(0).

Then,

u(T)≥β T −η T −βηu(0).

In order to apply Theorem2.1, we define the coneKby K =

u∈E, uis concave on(0, T)_Tand inf

t∈(0,T)_Tu(t)≥ρkuk

.

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It is easy to see that (1.1)-(1.2) has a solution u = u(t) if and only if u is a fixed point of the operatorG:K →E defined by

(3.2) Gu(t) =−

Z t

0

φ_q(g(s))4s+B, whereg andB are defined as in Lemma3.1.

Lemma 3.4. LetGbe defined by (3.2). Then, (i) G(K)⊆K;

(ii) G:K →K is completely continuous.

Proof. Condition(i)holds from previous lemmas. We now prove(ii). Suppose that D⊆Kis a bounded set. Letu∈D. We have:

|Gu(t)|=

− Z t

0

φq(g(s))4s+B

=

− Z t

0

φ_q Z s

0

λf(u(r)) (RT

0 f(u(τ))∇τ)^k∇r−A

!

4s+B

≤ Z T

0

φ_q Z s

0

λsup_u∈Df(u)

(T inf_u∈D)^k ∇r−A

4s+|B|,

|A|=

λβ 1−β

Z η

0

h(u(r))∇r

=

λβ 1−β

Z η

0

f(u(r)) (RT

0 f(u(τ))∇r)^k∇r

≤ λβ 1−β

sup_u∈Df(u) (T infu∈D)^k η.

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In the same way, we have

|B| ≤ 1 1−β

Z T

0

φ_q(g(s))4s

≤ 1 1−β

Z T

0

φ_q

λsup_u∈Df(u) (T infu∈D)^k

s+ β 1−βη

4s .

It follows that

|Gu(t)| ≤ Z T

0

φq

s+ βη 1−β

4s+|B|.

As a consequence, we get kGuk ≤ 2−β

1−β Z T

0

φ_q

s+ βη 1−β

≤ 2 1−βφ_q

Z T

0

φ_q

s+ βη 1−β

4s .

We conclude thatG(D)is bounded. Item (ii)follows by a standard application of the Arzela-Ascoli and Lebesgue dominated theorems.

Theorem 3.5 (Existence result on cones). Suppose that (H1) holds. Assume fur- thermore that there exist two positive numbersaandbsuch that

0≤u≤amax f(u)≤φ_p(aA₁), (H2)

0≤u≤bmin f(u)≥φ_p(bB₁), (H3)

where

A₁ = 1−β T(2−β)φ_p

1

(T inf_0≤u≤af(u))^k

T + βη 1−β

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and

B₁ = 1−β

β(T −η)φ_p(η)φ_p λ

T sup_0≤u≤bf(u)k

! .

Then, there exists0 < λ∗ < 1such that the non local p-Laplacian problem (1.1)- (1.2) has at least one positive solutionu,a≤u≤b, for anyλ∈(0, λ_∗).

Proof. LetΩ_r = {u ∈ K,kuk ≤ r},∂Ω_r = {u ∈ K,kuk = r}. Ifu ∈ ∂Ω_a, then 0≤u≤a,t ∈(0, T)_T. This impliesf(u(t))≤max0≤u≤af(u)≤φ_p(aA). We can write that

kGuk ≤ Z T

0

φq(g(s))4s+B

≤ Z T

0

φ_q Z s

0

λf(u(r)) (RT

0 f(u(τ))∇τ)^k∇r−A

!

4s+B ,

|A|= λβ 1−β

Z η

0

f(u(r)) (RT

0 f(u(τ))∇τ)^k∇r ≤ λβ 1−β

(aA₁)^p−1

(Tinf0≤u≤af(u))^kη , g(s)≤ λ(aA₁)^p−1

(T inf0≤u≤af(u))^k

T + βη 1−β

. Then,

Z T

0

φ_q(g(s))4s≤φ_q

λ(aA₁)^p−1 (T inf0≤u≤af(u))^k

T + βη 1−β

T

=aA₁T φ_q

λ

T + βη 1−β

.

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Moreover,

B = 1 1−β

Z T

0

φ_q(g(s))4s

≤ 1 1−β

Z T

0

φ_q(g(s))4s

≤aA₁ T 1−βφ_q

λ

T + βη 1−β

. ForA₁ as in the statement of the theorem, it follows that

kGuk ≤aA₁T2−β 1−βφ_q

λ

T + βη 1−β

≤φ_q(λ)aA₁T2−β 1−βφ_q

1

T + βη 1−β

≤φ_q(λ∗)aA₁T2−β 1−βφ_q

1

T + βη 1−β

≤φ_q(λ_∗)a

≤a =kuk.

Ifu∈∂Ω_b, we have kGuk ≥ −

Z T

0

φ_q(g(s))4s+B

≥ − Z T

0

φ_q(g(s))4s+ 1 1−β

Z T

0

φ_q(g(s))4s− β 1−β

Z η

0

φ_q(g(s))4s

≥ β 1−β

Z T

0

φ_q(g(s))4s− β 1−β

Z η

0

φ_q(g(s))4s

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≥ β 1−β

Z T

η

φ_q(g(s))4s.

SinceA≤0, we have g(s) =λ

Z s

0

h(u(r))∇r−A≥λ Z s

0

h(u(r))∇r

≥λ Z s

0

f(u)

(T sup_0≤u≤bf(u))^k

≥λ (bB1)^p−1 (T sup_0≤u≤b)^ks.

Using the fact thatφq is nondecreasing we get φ_q(g(s))≥φ_q

λ (bB1)^p−1 (Tsup_0≤u≤b)^ks

≥bB₁φ_q

λ (T supf(u))^k

φ_q(s).

Then, using the expression ofB₁, kGuk ≥ β

1−βbB₁φ_q

λ (T supf(u))^k

Z T

η

φ_q(s)4s

≥bB₁ β 1−βφ_q

λ (T supf(u))^k

φ_q(η)(T −η)

≥b=kuk.

As a consequence of Lemma3.4 and Theorem 2.1, Ghas a fixed point theorem u such thata≤u≤b.

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4. An Example

We consider a functionfwhich arises with the negative coefficient thermistor (NTC- thermistor). For this example the electrical resistivity decreases with the temperature.

Corollary 4.1. Assume (H1) holds. If f₀ = lim

u→0

f(u)

φ_p(u) = 0, f_∞ = lim

u→∞

f(u)

φ_p(u) = +∞, or

f₀ = +∞, f∞= 0, then problem (1.1)-(1.2) has at least one positive solution.

Proof. If f0 = 0 then ∀ A1 > 0 ∃ a such that f(u) ≤ (A1u)^p−1, 0 ≤ u ≤ a.

Similarly as above, we can prove thatkGuk ≤ kuk,∀u∈∂Ω_a. On the other hand, iff_∞ = +∞, then∀B₁ > 0, ∃b >0such thatf(u)≥(B₁u)^p−1,u ≥b. As in the proof of Theorem3.5, we havekGuk ≥ kuk,∀u ∈∂Ω_b. By Theorem2.1, Ghas a fixed point.

For the NTC-thermistor, the dependence of the resistivity to the temperature can be expressed by

(4.1) f(s) = 1

(1 +s)^k , k ≥2. Forp= 2,we have

f₀ = lim

u→0

f(u)

φ_p(u) = +∞, f∞ = lim

u→∞

f(u) φ_p(u) = 0.

It follows from Corollary4.1that the boundary value problem (1.1)-(1.2) withp= 2 andf as in (4.1) has at least one positive solution.

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