We make use of the Guo-Krasnoselskii fixed point theorem on cones to prove existence of positive solutions to a non local p-Laplacian boundary value problem on time scales arising in many applications. © 2007 Victoria University. All rights reserved.CEOCFCTFEDER/POCTISFRH/BPD/20934/200

Ammi, M.R.S.

Torres, D.F.M.

English

Repositório Institucional da Universidade de Aveiro

Volume 8 (2007), Issue 3, Article 69, 10 pp.EXISTENCE OF POSITIVE SOLUTIONS FOR NON LOCAL p-LAPLACIANTHERMISTOR PROBLEMS ON TIME SCALESMOULAY RCHID SIDI AMMI AND DELFIM F. M. TORRESDEPARTMENT OF MATHEMATICSUNIVERSITY OF AVEIRO3810-193 AVEIRO, PORTUGALsidiammi@mat.ua.ptdelfim@mat.ua.ptURL: http://www.mat.ua.pt/delfimReceived 20 March, 2007; accepted 25 August, 2007Communicated by R.P. AgarwalABSTRACT. We make use of the Guo-Krasnoselskii fixed point theorem on cones to prove ex-istence of positive solutions to a non local p-Laplacian boundary value problem on time scalesarising in many applications.Key words and phrases: Time scales, p-Laplacian, Positive solutions, Existence.2000 Mathematics Subject Classification. 34B18, 39A10, 93C70.1. INTRODUCTIONThe purpose of this paper is to prove the existence of positive solutions for the following nonlocal p-Laplacian dynamic equation on a time scale T:(1.1) − (φp(u4(t)))∇ = λf(u(t))(∫ T0f(u(τ))∇τ)k, ∀t ∈ (0, T )T = T ,subject to the boundary conditions(1.2) φp(u4(0))− βφp(u4(η)) = 0, 0 < η < T,u(T )− βu(η) = 0,where φp(·) is the p-Laplacian operator defined by φp(s) = |s|p−2s, p > 1, (φp)−1 = φq with qthe Hölder conjugate of p, i.e. 1p+ 1q= 1. The function(H1) f : (0, T )T → R+∗ is assumed to be continuousThe authors were partially supported by the Portuguese Foundation for Science and Technology (FCT) through the Centre for Researchin Optimization and Control (CEOC) of the University of Aveiro, cofinanced by the European Community fund FEDER/POCTI, and by theproject SFRH/BPD/20934/2004.085-072 MOULAY RCHID SIDI AMMI AND DELFIM F. M. TORRES(R+∗ denotes the positive real numbers); λ is a dimensionless parameter that can be identifiedwith the square of the applied potential difference at the ends of a conductor; f(u) is the tem-perature dependent resistivity of the conductor; β is a transfer coefficient supposed to verify0 < β < 1. Different values for p and k are connected with a variety of applications for bothT = R and T = Z. When k > 1, equation (1.1) represents the thermo-electric flow in a con-ductor [20]. In the particular case p = k = 2, (1.1) has been used to describe the operation ofthermistors, fuse wires, electric arcs and fluorescent lights [11, 12, 18, 19]. For k = 1, equa-tion (1.1) models the phenomena associated with the occurrence of shear bands (i) in metalsbeing deformed under high strain rates [6, 7], (ii) in the theory of gravitational equilibrium ofpolytropic stars [17], (iii) in the investigation of the fully turbulent behavior of real flows, us-ing invariant measures for the Euler equation [10], (iv) in modelling aggregation of cells viainteraction with a chemical substance (chemotaxis) [22].The theory of dynamic equations on time scales (or, more generally, measure chains) wasintroduced in 1988 by Stefan Hilger in his PhD thesis (see [14, 15]). The theory presents astructure where, once a result is established for a general time scale, then special cases includea result for differential equations (obtained by taking the time scale to be the real numbers) anda result for difference equations (obtained by taking the time scale to be the integers). A greatdeal of work has been done since 1988, unifying and extending the theories of differential anddifference 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 concerningp-Laplacian problems on time scales are scarce [21]. In this paper we prove the existence ofpositive solutions to the problem (1.1)-(1.2) on a general time scale T.2. PRELIMINARIESOur main tool to prove the existence of positive solutions (Theorem 3.5) is the Guo-Krasnoselskiifixed point theorem on cones.Theorem 2.1 (Guo-Krasnoselskii fixed point theorem on cones [13, 16]). Let X be a Banachspace and K ⊂ E be a cone in X . Assume that Ω1 and Ω2 are bounded open subsets of K with0 ∈ Ω1 ⊂ Ω1 ⊂ Ω2 and that G : K → K is a completely continuous operator such that(i) either ‖Gw‖ ≤ ‖w‖, w ∈ ∂Ω1, and ‖Gw‖ ≥ ‖w‖, w ∈ ∂Ω2; or(ii) ‖Gw‖ ≥ ‖w‖, w ∈ ∂Ω1, and ‖Gw‖ ≤ ‖w‖, w ∈ ∂Ω2.Then, G has a fixed point in Ω2\Ω1.Using the properties of f on a bounded set (0, T )T, we construct an operator (an integralequation) 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. Fordeeper details, the reader can see, for instance, [1, 5, 8]. A time scale T is an arbitrary nonemptyclosed subset of R. The forward jump operator σ and the backward jump operator ρ, both fromT to T, 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, or σ(t) > t, respectively. If T has a right scattered minimumm, define Tk = T−{m};otherwise set Tk = T. If T has a left scattered maximum M , define Tk = T− {M}; otherwiseset Tk = T.Let f : T → R and t ∈ Tk (assume t is not left-scattered if t = supT), then the deltaderivative of f at the point t is defined to be the number f∆(t) (provided it exists) with theJ. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 69, 10 pp. http://jipam.vu.edu.au/EXISTENCE FOR THERMISTOR PROBLEMS ON TIME SCALES 3property that for each  > 0 there is a neighborhood U of t such that∣∣f(σ(t))− f(s)− f∆(t)(σ(t)− s)∣∣ ≤ |σ(t)− s|, for all s ∈ U .Similarly, for t ∈ T (assume t is not right-scattered if t = inf T), the nabla derivative of f atthe point t is defined to be the number f∇(t) (provided it exists) with the property that for each > 0 there is a neighborhood U of t such that|f(ρ(t))− f(s)− f∇(t)(ρ(t)− s)| ≤ |ρ(t)− s|, for all s ∈ U .If T = R, then x∆(t) = x∇(t) = x′(t). If T = Z, then x∆(t) = x(t + 1)− x(t) is the forwarddifference operator while x∇(t) = x(t)− x(t− 1) is the backward difference operator.A function f is left-dense continuous (ld-continuous) if f is continuous at each left-densepoint in T and its right-sided limit exists at each right-dense point in T. Let f be ld-continuous.If F∇(t) = f(t), then the nabla integral is defined by∫ baf(t)∇t = F (b)− F (a) ;if F∆(t) = f(t), then the delta integral is defined by∫ baf(t)∆t = F (b)− F (a) .In the remainder of this article T is a closed subset of R with 0 ∈ Tk, T ∈ Tk; E =Cld([0, T ],R), which is a Banach space with the maximum norm ‖u‖ = max[0,T ]T |u(t)|.3. MAIN RESULTSBy a positive solution of (1.1)-(1.2) we understand a function u(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) ifand only if u(t) ∈ E is solution of the integral equationu(t) = −∫ t0φq (g(s))4s+B,whereg(s) =∫ s0λh(u(r))∇r − A,A = φp(u4(0)) = − λβ1− β∫ η0h(u(r))∇r,h(u(t)) =λf(u(t))(∫ T0f(u(τ))∇τ)k ,B = u(0) =11− β{∫ T0φq(g(s))4s− β∫ η0φq(g(s))4s}.Proof. We begin by proving necessity. Integrating the equation (1.1) we haveφp(u4(s)) = φp(u4(0))−∫ s0λh(u(r))∇r.On the other hand, by the boundary condition (1.2)φp(u4(0)) = βφp(u4(η)) = β(φp(u4(0))−∫ η0λh(u(r))∇r).J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 69, 10 pp. http://jipam.vu.edu.au/4 MOULAY RCHID SIDI AMMI AND DELFIM F. M. TORRESThen,A = φp(u4(0)) =−λβ1− β∫ η0h(u(r))∇r.It follows thatu4(s) = φq(−λ∫ s0h(u(r))∇r + A)= −φq(g(s)).Integrating the last equation, we obtain(3.1) u(t) = u(0)−∫ t0φq(g(s))4s.Moreover, by (3.1) and the boundary condition (1.2), we haveu(0) = u(T ) +∫ T0φq(g(s))4s= βu(η) +∫ T0φq(g(s))4s= β(u(0)−∫ η0φq(g(s))4s)+∫ T0φq(g(s))4s.Then,u(0) = B =11− β(−β∫ η0φq(g(s))4s+∫ T0φq(g(s))4s).Sufficiency follows by a simple calculation, taking the delta derivative of u(t). Lemma 3.2. Suppose (H1) holds. Then, a solution u of (1.1)-(1.2) satisfies u(t) ≥ 0 for allt ∈ (0, T )T.Proof. We have A = −λβ1−β∫ η0h(u(r))∇r ≤ 0. Then, g(s) = λ ∫ s0h(u(r)) − A ≥ 0. It followsthat φp(g(s)) ≥ 0. Since 0 < β < 1, we also haveu(0) = B =11− β{∫ T0φq(g(s))4s− β∫ η0φq(g(s))4s}≥ 11− β{β∫ T0φq(g(s))4s− β∫ η0φq(g(s))4s}≥ 0andu(T ) = u(0)−∫ T0φq(g(s))4s=−β1− β∫ η0φq(g(s))4s+ 11− β∫ T0φq(g(s))4s−∫ T0φq(g(s))4s=−β1− β∫ η0φq(g(s))4s+ β1− β∫ T0φq(g(s))4s=β1− β{∫ T0φq(g(s))4s−∫ η0φq(g(s))4s}≥ 0.J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 69, 10 pp. http://jipam.vu.edu.au/EXISTENCE FOR THERMISTOR PROBLEMS ON TIME SCALES 5If t ∈ (0, T )T,u(t) = u(0)−∫ t0φq(g(s))4s≥ −∫ T0φq(g(s))4s+ u(0) = u(T ) ≥ 0 .Lemma 3.3. If (H1) holds, then u(T ) ≥ ρu(0), where ρ = β T−ηT−βη ≥ 0.Proof. We have φp(u4(s)) = φp(u4(0)) −∫ s0λh(u(r))∇r ≤ 0. Since A = φp(u4(0)) ≤ 0,then u4 ≤ 0. This means that ‖u‖ = u(0), inft∈(0,T )T u(t) = u(T ). Moreover, φp(u4(s)) isnon increasing which implies, with the monotonicity of φp, that u4 is a non increasing functionon (0, T )T. It follows from the concavity of u(t) that each point on the chord between (0, u(0))and (T, u(T )) is below the graph of u(t). We haveu(T ) ≥ u(0) + T u(T )− u(η)T − η .Alternatively,Tu(η)− ηu(T ) ≥ (T − η)u(0).Using the boundary condition (1.2), it follows that(Tβ− η)u(T ) ≥ (T − η)u(0).Then,u(T ) ≥ β T − ηT − βηu(0).In order to apply Theorem 2.1, we define the cone K byK ={u ∈ E, u is concave on (0, T )T and inft∈(0,T )Tu(t) ≥ ρ‖u‖}.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 theoperator G : K → E defined by(3.2) Gu(t) = −∫ t0φq (g(s))4s+B,where g and B are defined as in Lemma 3.1.Lemma 3.4. Let G be 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 ⊆ K isa bounded set. Let u ∈ D. We have:|Gu(t)| =∣∣∣∣−∫ t0φq (g(s))4s+B∣∣∣∣=∣∣∣∣∣−∫ t0φq(∫ s0λf(u(r))(∫ T0f(u(τ))∇τ)k∇r − A)4s+B∣∣∣∣∣J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 69, 10 pp. http://jipam.vu.edu.au/6 MOULAY RCHID SIDI AMMI AND DELFIM F. M. TORRES≤∫ T0φq(∫ s0λ supu∈D f(u)(T infu∈D)k∇r − A)4s+ |B|,|A| =∣∣∣∣ λβ1− β∫ η0h(u(r))∇r∣∣∣∣=∣∣∣∣∣ λβ1− β∫ η0f(u(r))(∫ T0f(u(τ))∇r)k∇r∣∣∣∣∣≤ λβ1− βsupu∈D f(u)(T infu∈D)kη.In the same way, we have|B| ≤ 11− β∫ T0φq(g(s))4s≤ 11− β∫ T0φq(λ supu∈D f(u)(T infu∈D)k(s+β1− βη))4s .It follows that|Gu(t)| ≤∫ T0φq(λ supu∈D f(u)(T infu∈D)k(s+βη1− β))4s+ |B|.As a consequence, we get‖Gu‖ ≤ 2− β1− β∫ T0φq(λ supu∈D f(u)(T infu∈D)k(s+βη1− β))≤ 21− βφq(λ supu∈D f(u)(T infu∈D)k)∫ T0φq(s+βη1− β)4s .We conclude that G(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 furthermore thatthere exist two positive numbers a and b such thatmax0≤u≤af(u) ≤ φp(aA1),(H2)min0≤u≤bf(u) ≥ φp(bB1),(H3)whereA1 =1− βT (2− β)φp(1(T inf0≤u≤a f(u))k(T +βη1− β))andB1 =1− ββ(T − η)φp(η)φp(λ(T sup0≤u≤b f(u))k).Then, there exists 0 < λ∗ < 1 such that the non local p-Laplacian problem (1.1)-(1.2) has atleast one positive solution u, a ≤ u ≤ b, for any λ ∈ (0, λ∗).J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 69, 10 pp. http://jipam.vu.edu.au/EXISTENCE FOR THERMISTOR PROBLEMS ON TIME SCALES 7Proof. Let Ωr = {u ∈ K, ‖u‖ ≤ r}, ∂Ωr = {u ∈ K, ‖u‖ = r}. If u ∈ ∂Ωa, then 0 ≤ u ≤ a,t ∈ (0, T )T. This implies f(u(t)) ≤ max0≤u≤a f(u) ≤ φp(aA). We can write that‖Gu‖ ≤∫ T0φq(g(s))4s+B≤∫ T0φq(∫ s0λf(u(r))(∫ T0f(u(τ))∇τ)k∇r − A)4s+B ,|A| = λβ1− β∫ η0f(u(r))(∫ T0f(u(τ))∇τ)k∇r ≤ λβ1− β(aA1)p−1(T inf0≤u≤a f(u))kη ,g(s) ≤ λ(aA1)p−1(T inf0≤u≤a f(u))k(T +βη1− β).Then,∫ T0φq(g(s))4s ≤ φq(λ(aA1)p−1(T inf0≤u≤a f(u))k(T +βη1− β))T= aA1Tφq(λ(T inf0≤u≤a f(u))k(T +βη1− β)).Moreover,B =11− β(∫ T0φq(g(s))4s− β∫ η0φq(g(s))4s)≤ 11− β(∫ T0φq(g(s))4s)≤ aA1 T1− βφq(λ(T inf0≤u≤a f(u))k(T +βη1− β)).For A1 as in the statement of the theorem, it follows that‖Gu‖ ≤ aA1T 2− β1− βφq(λ(T inf0≤u≤a f(u))k(T +βη1− β))≤ φq(λ)aA1T 2− β1− βφq(1(T inf0≤u≤a f(u))k(T +βη1− β))≤ φq(λ∗)aA1T 2− β1− βφq(1(T inf0≤u≤a f(u))k(T +βη1− β))≤ φq(λ∗)a≤ a = ‖u‖.J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 69, 10 pp. http://jipam.vu.edu.au/8 MOULAY RCHID SIDI AMMI AND DELFIM F. M. TORRESIf u ∈ ∂Ωb, we have‖Gu‖ ≥ −∫ T0φq (g(s))4s+B≥ −∫ T0φq (g(s))4s+ 11− β∫ T0φq (g(s))4s− β1− β∫ η0φq (g(s))4s≥ β1− β∫ T0φq (g(s))4s− β1− β∫ η0φq (g(s))4s≥ β1− β∫ Tηφq (g(s))4s.Since A ≤ 0, we haveg(s) = λ∫ s0h(u(r))∇r − A ≥ λ∫ s0h(u(r))∇r≥ λ∫ s0f(u)(T sup0≤u≤b f(u))k≥ λ (bB1)p−1(T sup0≤u≤b)ks.Using the fact that φq is nondecreasing we getφq(g(s)) ≥ φq(λ(bB1)p−1(T sup0≤u≤b)ks)≥ bB1φq(λ(T sup f(u))k)φq(s).Then, using the expression of B1,‖Gu‖ ≥ β1− β bB1φq(λ(T sup f(u))k)∫ Tηφq(s)4s≥ bB1 β1− βφq(λ(T sup f(u))k)φq(η)(T − η)≥ b = ‖u‖.As a consequence of Lemma 3.4 and Theorem 2.1, G has a fixed point theorem u such thata ≤ u ≤ b. 4. AN EXAMPLEWe consider a function f which arises with the negative coefficient thermistor (NTC-thermistor).For this example the electrical resistivity decreases with the temperature.Corollary 4.1. Assume (H1) holds. Iff0 = limu→0f(u)φp(u)= 0 , f∞ = limu→∞f(u)φp(u)= +∞,orf0 = +∞ , f∞ = 0 ,then problem (1.1)-(1.2) has at least one positive solution.J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 69, 10 pp. http://jipam.vu.edu.au/EXISTENCE FOR THERMISTOR PROBLEMS ON TIME SCALES 9Proof. If f0 = 0 then ∀ A1 > 0 ∃ a such that f(u) ≤ (A1u)p−1, 0 ≤ u ≤ a. Similarly asabove, we can prove that ‖Gu‖ ≤ ‖u‖, ∀ u ∈ ∂Ωa. On the other hand, if f∞ = +∞, then ∀B1 > 0, ∃ b > 0 such that f(u) ≥ (B1u)p−1, u ≥ b. As in the proof of Theorem 3.5, we have‖Gu‖ ≥ ‖u‖, ∀ u ∈ ∂Ωb. By Theorem 2.1, G has a fixed point. For the NTC-thermistor, the dependence of the resistivity to the temperature can be expressedby(4.1) f(s) = 1(1 + s)k, k ≥ 2 .For p = 2, we havef0 = limu→0f(u)φp(u)= +∞ , f∞ = limu→∞f(u)φp(u)= 0 .It follows from Corollary 4.1 that the boundary value problem (1.1)-(1.2) with p = 2 and f asin (4.1) has at least one positive solution.REFERENCES[1] R.P. AGARWAL AND M. BOHNER, Basic calculus on time scales and some of its applications,Result. Math., 35 (1999), 3–22.[2] R.P. AGARWAL, M. BOHNER, D. O’REGAN AND A. 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Existence of positive solutions for non local p-Laplacian thermistor problems on time scales

Existence of positive solutions for non local p-Laplacian thermistor problems on time scales

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