EXISTENCE OF POSITIVE SOLUTIONS FOR NON LOCAL p-laplacian THERMISTOR PROBLEMS ON TIME SCALES

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1 Volume 8 27, Issue 3, Article 69, 1 pp. EXISENCE OF POSIIVE SOLUIONS FOR NON LOCAL p-laplacian HERMISOR PROBLEMS ON IME SCALES MOULAY RCHID SIDI AMMI AND DELFIM F. M. ORRES DEPARMEN OF MAHEMAICS UNIVERSIY OF AVEIRO AVEIRO, PORUGAL sidiammi@mat.ua.pt delfim@mat.ua.pt URL: Received 2 March, 27; accepted 25 August, 27 Communicated by R.P. Agarwal ABSRAC. 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. Key words and phrases: ime scales, p-laplacian, Positive solutions, Existence. 2 Mathematics Subject Classification. 34B18, 39A1, 93C7. 1. INRODUCION he purpose of this paper is to prove the existence of positive solutions for the following non local p-laplacian dynamic equation on a time scale : 1.1 φ p u t = λfut fuτ τk, t, =, subject to the boundary conditions 1.2 φ p u φ p u η =, < η <, u uη =, where φ p is the p-laplacian operator defined by φ p s = s p 2 s, p > 1, φ p 1 = with q the Hölder conjugate of p, i.e = 1. he function p q H1 f :, R + is assumed to be continuous he authors were partially supported by the Portuguese Foundation for Science and echnology FC through the Centre for Research in Optimization and Control CEOC of the University of Aveiro, cofinanced by the European Community fund FEDER/POCI, and by the project SFRH/BPD/2934/

2 2 MOULAY RCHID SIDI AMMI AND DELFIM F. M. ORRES 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; fu is the temperature dependent resistivity of the conductor; is a transfer coefficient supposed to verify < < 1. Different values for p and k are connected with a variety of applications for both = R and = Z. When k > 1, equation 1.1 represents the thermo-electric flow in a conductor [2]. 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 equation [1], iv in modelling aggregation of cells via interaction with a chemical substance chemotaxis [22]. he 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]. he 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 on a general time scale. 2. PRELIMINARIES Our main tool to prove the existence of positive solutions heorem 3.5 is the Guo-Krasnoselskii fixed point theorem on cones. heorem 2.1 Guo-Krasnoselskii fixed point theorem on cones [13, 16]. Let X be a Banach space and K E be a cone in X. Assume that Ω 1 and Ω 2 are bounded open subsets of K with Ω 1 Ω 1 Ω 2 and that G : K K is a completely continuous operator such that i either Gw w, w Ω 1, and Gw w, w Ω 2 ; or ii Gw w, w Ω 1, and Gw w, w Ω 2. hen, G has a fixed point in Ω 2 \Ω 1. Using the properties of f on a bounded set,, we construct an operator an integral equation whose fixed points are solutions to the problem 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 scale is an arbitrary nonempty closed subset of R. he forward jump operator σ and the backward jump operator ρ, both from to, are defined in [14]: σt = inf{τ : τ > t}, ρt = sup{τ : τ < t}. A point t is left-dense, left-scattered, right-dense, or right-scattered if ρt = t, ρt < t, σt = t, or σt > t, respectively. If has a right scattered minimum m, define k = {m}; otherwise set k =. If has a left scattered maximum M, define k = {M}; otherwise set k =. Let f : R and t k assume t is not left-scattered if t = sup, then the delta derivative of f at the point t is defined to be the number f t provided it exists with the J. Inequal. Pure and Appl. Math., 83 27, Art. 69, 1 pp.

3 EXISENCE FOR HERMISOR PROBLEMS ON IME SCALES 3 property that for each ɛ > there is a neighborhood U of t such that fσt fs f tσt s σt s, for all s U. Similarly, for t assume t is not right-scattered if t = inf, 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 ɛ > there is a neighborhood U of t such that fρt fs f tρt s ρt s, for all s U. If = R, then x t = x t = x t. If = Z, then x t = xt + 1 xt is the forward difference operator while x t = xt xt 1 is the backward difference operator. A function f is left-dense continuous ld-continuous if f is continuous at each left-dense point in and its right-sided limit exists at each right-dense point in. Let f be ld-continuous. If F t = ft, then the nabla integral is defined by b a ft t = F b F a ; if F t = ft, then the delta integral is defined by b a ft t = F b F a. In the remainder of this article is a closed subset of R with k, k ; E = C ld [, ], R, which is a Banach space with the maximum norm u = max [, ] ut. 3. MAIN RESULS By a positive solution of we understand a function ut which is positive on, and satisfies 1.1 and 1.2. Lemma 3.1. Assume that hypothesis H1 is satisfied. hen, ut is a solution of if and only if ut E is solution of the integral equation where ut = gs = t gs s + B, λhur r A, A = φ p u = λ hur r, λfut hut = k, fuτ τ B = u = 1 { gs s } gs s. Proof. We begin by proving necessity. Integrating the equation 1.1 we have φ p u s = φ p u On the other hand, by the boundary condition 1.2 φ p u = φ p u η = φ p u λhur r. λhur r. J. Inequal. Pure and Appl. Math., 83 27, Art. 69, 1 pp.

4 4 MOULAY RCHID SIDI AMMI AND DELFIM F. M. ORRES hen, It follows that A = φ p u = λ hur r. u s = λ Integrating the last equation, we obtain 3.1 ut = u hur r + A = gs. t gs s. Moreover, by 3.1 and the boundary condition 1.2, we have u = u + = uη + = u gs s gs s gs s + gs s. hen, u = B = 1 gs s + gs s. Sufficiency follows by a simple calculation, taking the delta derivative of ut. Lemma 3.2. Suppose H1 holds. hen, a solution u of satisfies ut for all t,. Proof. We have A = λ hur r. hen, gs = λ hur A. It follows 1 that φ p gs. Since < < 1, we also have u = B = 1 { } gs s gs s and u = u = 1 { gs s gs s gs s + 1 } gs s gs s = gs s + gs s = { } gs s gs s. gs s J. Inequal. Pure and Appl. Math., 83 27, Art. 69, 1 pp.

5 EXISENCE FOR HERMISOR PROBLEMS ON IME SCALES 5 If t,, ut = u t gs s gs s + u = u. Lemma 3.3. If H1 holds, then u ρu, where ρ = η η. Proof. We have φ p u s = φ p u λhur r. Since A = φ pu, then u. his means that u = u, inf t, ut = u. Moreover, φ p u s is non increasing which implies, with the monotonicity of φ p, that u is a non increasing function on,. It follows from the concavity of ut that each point on the chord between, u and, u is below the graph of ut. We have u uη u u +. η Alternatively, uη ηu ηu. Using the boundary condition 1.2, it follows that η u ηu. hen, u η η u. In order to apply heorem 2.1, we define the cone K by { } K = u E, u is concave on, and inf ut ρ u t, It is easy to see that has a solution u = ut if and only if u is a fixed point of the operator G : K E defined by 3.2 Gut = where g and B are defined as in Lemma 3.1. t Lemma 3.4. Let G be defined by 3.2. hen, i GK K; ii G : K K is completely continuous. gs s + B, Proof. Condition i holds from previous lemmas. We now prove ii. Suppose that D K is a bounded set. Let u D. We have: t Gut = gs s + B t s = λfur r A s + B fuτ τk. J. Inequal. Pure and Appl. Math., 83 27, Art. 69, 1 pp.

6 6 MOULAY RCHID SIDI AMMI AND DELFIM F. M. ORRES λ sup u D fu r A s + B, inf u D k In the same way, we have A = λ λ = λ hur r fur r fuτ rk sup u D fu inf u D k η. It follows that B 1 1 Gut As a consequence, we get Gu 2 gs s λ supu D fu inf u D k λ supu D fu inf u D k 2 λ supu D fu inf u D k λ supu D fu inf u D k s + η s. s + η s + B. s + s + η η s. We conclude that GD is bounded. Item ii follows by a standard application of the Arzela- Ascoli and Lebesgue dominated theorems. heorem 3.5 Existence result on cones. Suppose that H1 holds. Assume furthermore that there exist two positive numbers a and b such that H2 H3 where and A 1 = 2 φ p B 1 = max fu φ paa 1, u a min fu φ pbb 1, u b 1 inf u a fu k + η η φ λ pηφ p sup u b fu k. hen, there exists < λ < 1 such that the non local p-laplacian problem has at least one positive solution u, a u b, for any λ, λ. J. Inequal. Pure and Appl. Math., 83 27, Art. 69, 1 pp.

7 EXISENCE FOR HERMISOR PROBLEMS ON IME SCALES 7 Proof. Let Ω r = {u K, u r}, Ω r = {u K, u = r}. If u Ω a, then u a, t,. his implies fut max u a fu φ p aa. We can write that Gu gs s + B s λfur r A s + B, fuτ τk hen, A = λ gs fur λ r fuτ τk λaa 1 p 1 + η. inf u a fu k aa 1 p 1 inf u a fu k η, λaa 1 p 1 gs s + η inf u a fu k λ = aa 1 + η. inf u a fu k Moreover, B = 1 1 aa 1 gs s gs s λ inf u a fu k For A 1 as in the statement of the theorem, it follows that gs s + η. Gu aa 1 2 φ λ q + η inf u a fu k λaa 1 2 φ 1 q + inf u a fu k λ aa 1 2 φ 1 q + inf u a fu k λ a a = u. η η J. Inequal. Pure and Appl. Math., 83 27, Art. 69, 1 pp.

8 8 MOULAY RCHID SIDI AMMI AND DELFIM F. M. ORRES If u Ω b, we have Gu Since A, we have gs s + B gs s + 1 η gs s gs s. gs = λ λ gs s hur r A λ fu sup u b fu k bb 1 p 1 λ sup u b s. k gs s gs s hur r Using the fact that is nondecreasing we get bb 1 gs λ p 1 sup u b s k λ bb 1 φ sup fu k q s. hen, using the expression of B 1, Gu bb 1 bb 1 b = u. λ φ sup fu k q s s η λ φ sup fu k q η η As a consequence of Lemma 3.4 and heorem 2.1, G has a fixed point theorem u such that a u b. 4. AN EXAMPLE We consider a function f which arises with the negative coefficient thermistor NC-thermistor. For this example the electrical resistivity decreases with the temperature. Corollary 4.1. Assume H1 holds. If or f = lim u fu φ p u =, f = lim u fu φ p u = +, f = +, f =, then problem has at least one positive solution. J. Inequal. Pure and Appl. Math., 83 27, Art. 69, 1 pp.

9 EXISENCE FOR HERMISOR PROBLEMS ON IME SCALES 9 Proof. If f = then A 1 > a such that fu A 1 u p 1, u a. Similarly as above, we can prove that Gu u, u Ω a. On the other hand, if f = +, then B 1 >, b > such that fu B 1 u p 1, u b. As in the proof of heorem 3.5, we have Gu u, u Ω b. By heorem 2.1, G has a fixed point. For the NC-thermistor, the dependence of the resistivity to the temperature can be expressed by fs = 1 + s, k 2. k For p = 2, we have f = lim u fu φ p u = +, f = lim u fu φ p u =. It follows from Corollary 4.1 that the boundary value problem with p = 2 and f as in 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., , [2] R.P. AGARWAL, M. BOHNER, D. O REGAN AND A. PEERSON, Dynamic equations on time scales: a survey, J. Comput. Appl. Math., , [3] R.P. AGARWAL, M. BOHNER AND P. WONG, Sturm-Liouville eigenvalue problem on time scales, Appl. Math. Comput., , [4] R.P. AGARWAL AND D. O REGAN, Nonlinear boundary value problems on time scales, Nonlinear Anal., 44 21, [5] F.M. AICI AND G.Sh. GUSEINOV, On Green s functions and positive solutions for boundaryvalue problems on time scales, J. Comput. Appl. Math., , [6] J.W. BEBERNES AND A.A. LACEY, Global existence and finite-time blow-up for a class of nonlocal parabolic problems, Adv. Diff. Eqns., , [7] J. W. BEBERNES, C. LI AND P. ALAGA, Single-point blow-up for non-local parabolic problems, Physica D, , [8] M. BOHNER AND A. PEERSON, Dynamic Equations on ime Scales An Introduction with Applications, Birkhäuser, Boston, 21. [9] M. BOHNER AND A. PEERSON, Advances in Dynamic Equations on ime Scales, Birkhäuser Boston, Cambridge, MA, 23. [1] E. CAGLIOI, P-L. LIONS, C. MARCHIORO AND M. PULVIRENI, A special class of stationary flows for two-dimensinal Euler equations: a statistical mechanics description, Comm. Math. Phys., , [11] A. EL HACHIMI AND M.R. SIDI AMMI, Semidiscretization for a nonlocal parabolic problem, Int. J. Math. Math. Sci., , [12] A. EL HACHIMI, M.R. SIDI AMMI AND D.F.M. ORRES, A dual mesh method for a nonlocal thermistor problem, SIGMA Symmetry Integrability Geom. Methods Appl., 2 26, Paper 58, 1 pp. electronic. [ONLINE: Paper58/index.html] [13] D. GUO AND V. LAKSHMIKANHAM, Nonlinear Problems in Abstract Cones, Academic press, Boston, J. Inequal. Pure and Appl. Math., 83 27, Art. 69, 1 pp.

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