A NOTE ON A FOURTH ORDER DISCRETE BOUNDARY VALUE PROBLEM. Marek Galewski and Joanna Smejda

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1 Opuscula Mathematica Vol. 3 No A NOE ON A FOURH ORDER DISCREE BOUNDARY VALUE PROBLEM Marek Galewski and Joanna Smejda Abstract. Using variational methods we investigate the existence of solutions and their dependence on parameters for certain fourth order difference equations. Keywords: discrete boundary value problem, variational method, coercivity, continuous dependence on paremeters. Mathematics Subject Classification: 39A10, 39A INRODUCION In this note we will study a Dirichlet boundary value problem for a fourth order discrete equation ( p(k) x(k ) ) ( x(k 1)) f (k, x(k)) = g(k), k Z[, ], x(0) = x(1) = x( 1) = x( ) = 0. (1.1) For fixed a, b N we define Z[a, b] = {a, a 1,..., b 1, b} as the so called discrete interval. is the forward difference operator x(k) = x(k 1) x(k). By a solution of problem (1.1) we mean such a function x : Z[0, ] R which satisfies the difference equation on Z[, ] and the given boundary conditions. We note that since we do not assume anything about the sign condition of f near 0 our results may apply for both positone (i.e. when f(k, 0) 0, k Z[, ]) and non-positone (i.e. when f(k, 0) < 0, k Z[, ]) problems within one approach. his is not common within the boundary value problems, compare with [8]. Solutions are obtained in the space E of functions x : {0, } R such that x(0) = x(1) = x( 1) = x( ) = 0 considered with a norm x = ( x(k )). 115

2 116 Marek Galewski and Joanna Smejda All functions from E are defined on a finite set, and therefore these are continuous. he space E can be also considered with the following norms x 1 = 1 ( x(k 1)) and x 0 = x (k). Since E has finite dimension these norms are equivalent, thus β x x 1 β 1 x, x 0 γ x (1.) for a certain constants β, β 1, γ > 0 which do not depend on x. We assume that: (A1) f C(Z [, ] R, R), p C(Z [, 3], R), q C(Z [, ], R), g C(Z [, ], R); (A) there exists a constant α > 0 such that xf(t, x) 0 for x α; (A3) M < Nβ, where M = sup t {,3,..., 3} p(t), N = inf t {,3,..., } q(t). Problems such as (1.1) arise when fourth order Dirichlet problems are being discretization and may be viewed as a discrete version of a simply supported elastic beam equation, see for example [, 5]. he approach through symmetric Green s function is used in [6, 7], the Krein-Rutman heorem is applied in [10], while in [9] the critical point theory is used with some other growth conditions. In fact the variational framework for problem (1.1) which we follow is descried in [9]. However, in the sources mentioned, the approach is somewhat different and with different set of assumptions. While in the literature mainly the problem of the existence of solutions and their multiplicity is considered we are going to go further and investigate also the dependence on a functional parameter. he paper is organized as follows. We are going first to apply a variational approach based on the so called direct variational method in order to get the existence result and next investigate the dependence of the solution on a functional parameter. We think that for our problem, as far as the existence is concerned, a lower-upper solution method introduced in [4] could also be applied. However the latter approach does not seem to allow for the investigations of the dependence on parameters due to the non-uniqueness of solutions and therefore we do not apply it. For the sake of convenience, we now recall same basic tools used in our note, see []. A mapping J of a real Banach X space to R will be called a functional. A point x 0 where J (x 0 ) = θ is called a critical point of J, assuming that J is Gâteaux differentiable and that J denotes the Gâteaux derivative. J is weakly lower semi-continuous at x X if x n x lim inf n J(x n) J(x)

3 A note on a fourth order discrete boundary value problem 117 and J is coercive on X if lim J(x) =, x where x stands for a norm in X and denotes weak convergence in X. heorem 1.1. Let E be a reflexive Banach space, D E be weakly closed, and J : E R be weakly lower semi-continuous and coercive, then J has a minimum over D.. HE EXISENCE OF SOLUIONS Let F (k, y(k)) = y(k) 0 f(k, t)dt, y E. he action functional J : E R corresponding to our problem is J(y) = ( p(k) ( y(k )) ) ( y(k 1)) ( F (k, y(k)) g(k)y(k)). Lemma.1. J is a Gâteaux differentiable functional; y E is a critical point of J if and only if it is a solution to (1.1). Proof. We denote by ϕ : R R the function ϕ(ε) = J(y εh) for y, h E and ε R; here y, h E are fixed. hen ϕ(ε) = ( p(k) ( (y εh)(k )) ) ( F (k, y(k) εh(k)) g(k) (y(k) εh(k))) and since ϕ is differentiable we get ( (y εh)(k 1)) ϕ (0) = ( p(k) y(k ) h(k )) y(k 1) h(k 1) ( f(k, y(k))h(k) g(k)h(k)) = ( ( p(k) y(k ) ) h(k) ( y(k 1) ) h(k) f(k, y(k))h(k) g(k)h(k) ). hus y E is critical point of J if and only if y satisfies equation (1.1).

4 118 Marek Galewski and Joanna Smejda heorem.. Assume that (A1), (A), (A3) hold. hen functional J is weakly lower semi-continuous and coercive on E. Proof. Since J is continuous it is lower semi-continuous and since E is finite dimensional it is weakly lower semi-continuous. We have just demonstrated that J is Gâteaux differentiable. We show that J is coercive on E. o do this first notice that by (A) F (k, y(k)) = y(k) 0 f(k, t)dt α α Further from (1.), (.1) and (A3) for any sequence {y n } E J(y n ) = ( p(k) ( y n (k )) ) M M ( M So J(y n ) as y n. ( y n(k 1)) y n N y n 1 C f(k, t) dt C. (.1) (g(k)y n (k) F (k, y n (k))) g(k) y n (k) y n N β y n C g (k) y n 0 N β ) y n C γ g (k) y n. he main result of this section is contained in the next theorem. heorem.3. Assume that (A1), (A), (A3) hold. hen problem (1.1) has at least one solution v E such that J (v) = inf y E J(y). Proof. We use heorem 1.1 and Lemma.1. Let D = E. hen D as a closed and convex set is weakly closed. By heorem. J is weakly lower semi-continuous and coercive on D. So by heorem 1.1 it has at least one argument for a minimum. Let us denote it by v. Since J is differentiable in the sense of Gâteaux, it follows that J (v) = 0 and the assertion follows by Lemma HE DEPENDENCE ON PARAMEERS he usage of a variational method allows us to consider a boundary value problem which is subject to some functional parameter and later to investigate the dependence

5 A note on a fourth order discrete boundary value problem 119 of the solution on the parameter as it varies. We do not need to have uniqueness of solutions in order to investigate their dependence on parameters. In this section we will investigate the following Dirichlet problem ( p(k) x(k ) ) ( x(k 1)) f (k, x(k), u(k)) = g(k), k Z[, ], (3.1) x(0) = x(1) = x( 1) = x( ) = 0 subject to parameter u L D = {u C(Z[, ], R) : u C D}, where D > 0 is fixed and u C denotes classical maximum norm u C = max k Z[, ] u(k). Now we assume that: (A4) f C(Z [, ] R, R), p C(Z [, 3], R), q C(Z [, ], R), g C(Z [, ], R); (A5) there exists α > 0 such that xf(t, x, u) 0 for x α, u D. With these assumptions and with (A3) the action functional J u : E R corresponding to (3.1) with a fixed function u L D reads J u (y) = ( p(k) ( y(k )) ) ( y(k 1)) (g(k)y(k) F (k, y(k), u(k))), where F (k, y(k), u(k)) = y(k) 0 f(k, t, u(k))dt, y E. Reasoning as in the proof of heorem. we obtain heorem 3.1. Assume that (A3), (A4), (A5) hold. hen for any fixed u L D the problem (3.1) has at least one solution in V u. Let for any fixed u L D V u = {y E : J u (y) = inf v E J u(v) and J u(y) = 0} be the set which consists of the arguments of a minimum to J u. Due to heorem 3.1 V u. We will investigate the behavior of the sequence {y n } of solutions to (3.1) depending on convergence of the sequence of parameters {u n }. heorem 3.. Assume that (A3), (A4), (A5) hold. For any fixed u L D there exists at least one solution y V u to problem (3.1). Let {u n } L D be a convergent sequence of parameters, where lim n u n = ū L D. For any sequence {y n } of solutions y n V n to the problem (3.1) corresponding to u n, there exist a subsequence {y ni } E and an element y E such that lim i y ni = y and Jū(y) = inf J ū(y). y E Moreover y Vū, i.e. y satisfies ( p(k) y(k ) ) ( y(k 1) ) f ( k, y(k), ū(k) ) = g(k), y(0) = y(1) = y( 1) = y( ) = 0.

6 10 Marek Galewski and Joanna Smejda Proof. By heorem 3.1 we get for n N the existence of solution y n V un to (3.1). Notice, that for n N, By (A5) we get for some constant C > 0 y n V un {y : J un J un (0)}. F ( k, y n (k), u n (k) ) = hen (3.) and (1.) imply J u (y n ) = M M ( M y n(k) 0 α α ( p(k) ( y n (k )) ) f(k, t, u n (k))dt f(k, t, u n (k)) dt C. (g(k)y n (k) F (k, y n (k), u(k))) ( y n(k 1)) y n N y n 1 C g(k) y n (k) y n N β y n C g (k) y n 0 N β ) y n C γ g (k) y n. (3.) We also know that F (k, 0, u n (k)) = 0, so J un (0) = 0. As a consequence for y n V un we see that ( M N β ) y n γ g (k) y n C (3.3) so {y n } is bounded in E and hence it has a convergent subsequence {y ni }. We denote its limit by y. In order to demonstrate that y satisfies (3.1) corresponding to ū we follow the same steps as in the proof of heorem 1 in [3]. However, we proceed with the reasoning for the reader s convenience and slightly simplify the approach of [3]. Observe that by heorem 3.1 there exists y 0 E such that y 0 solves (3.1) with u and J u (y 0 ) = inf y E J u (y) and either J u (y 0 ) < J u (y) or J u (y 0 ) = J u (y). Suppose that J u (y 0 ) < J u (y). hen, for some constant δ > 0 we have δ < ( J uni (y ni ) J u (y 0 ) ) ( J uni (y ni ) J u (y) ). (3.4)

7 A note on a fourth order discrete boundary value problem 11 By continuity, it follows that ( lim Juni (y n i ) J u (y) ) = 0. (3.5) i Since y ni minimizes J uni over E we see that J uni (y ni ) J uni (y 0 ) for any n i. herefore, we get ( lim Juni (y n i ) J u (y 0 ) ) ( lim Juni (y 0 ) J u (y 0 ) ) = 0. (3.6) i i Using (3.5) and (3.6) we obtain δ 0 in (3.4), which is a contradiction. hus J u (y) = inf y E J u (y) and since J u is differentiable in the sense of Gâteaux we have y V u. Hence y necessarily satisfies (3.1). On the other hand, if we have J u (y 0 ) = J u (y) then the result readily follows. 4. FURHER EXISENCE RESULS AND EXAMPLES In this section we will also investigate (1.1) but our assumptions will be somewhat different. his change forces us to use a different action functional. (A6) there exists α 1 > 0 such that xf(t, x) 0 for x α 1 ; (A7) M 1 > N 1 β 1, where M 1 = inf t {,3,..., 3} p(t), N 1 = sup t {,3,..., } q(t). Now we consider the following action functional J 1 (y) = p(k) ( y(k )) ( y(k 1)) (F (k, y(k)) g(k)y(k)). heorem 4.1. Assume that (A1), (A6), (A7) hold. hen functional J 1 is weakly lower semi-continuous and coercive on E. Proof. he fact that J 1 is lower semi-continuous is obvious. We show that J 1 is coercive on E. o do this first notice that by (A6) (similar to proof of heorem.) we get for any y E F (k, y(k)) C (4.1)

8 1 Marek Galewski and Joanna Smejda for some constant C > 0. Further from (1.), (4.1) for any sequence {y n } E J 1 (y n ) = p(k) ( y n (k )) ( y n(k 1)) (F (k, y n (k)) g(k)y n (k)) M1 y n N1 y n 1 C g(k) y n (k) M1 y n N1 β 1 y n C g (k) y n 0 ( M1 N1 β 1) y n C γ g (k) y n. So J 1 (y n ) as y n. he existence of a minimum for J 1 follows from heorem 1.1 (with D = E) and heorem 4.1. So we get the following theorem. heorem 4.. Assume that (A1), (A6), (A7) hold. hen the problem (1.1) has at least one solution. Example 4.3. Let l be any natural number and let r C(R, R) be bounded. Define function f (t, x) = r(t)h(x) with { x l, x < 0, h(x) = x l, x 0. For r C(R, R ) function f satisfy (A), but it does not satisfy (A6). For r C(R, R ) it does not satisfy (A), but it satisfies (A6). Example 4.4. Let r C(R, R) be bounded. Define function f(t, x) = r(t)h(x) with h(x) = arctan x. For r C(R, R ) function f satisfies (A6), but it does not satisfy (A). For r C(R, R ) it does not satisfy (A6), but it satisfies (A).

9 A note on a fourth order discrete boundary value problem 13 REFERENCES [1] J. Mawhin, Problèmes de Dirichlet Variationnels Non Linéaires, Les Presses de l Université de Montréal, [] R. Ma, J. Li, C. Gao, Existence of positive solutions of a discrete elastic beam equation, Discrete Dyn. Nat. Soc. 010, Art. ID 58919, 15 pp. [3] M. Galewski, Dependence on parameters for discrete second order boundary value problems, arxiv:091.54v1, to appear in J. Difference Equ. Appl. [4] I. Rachùnková, L. Rachùnek, Solvability of discrete Dirichlet problem via lower and upper functions method, J. Difference Equ. Appl. 13 (007) 5, [5] S. Huang, Z. Zhou, On the nonexistence and existence of solutions for a fourth-order discrete boundary value problem, Adv. Difference Equ. 009, Art. ID 38964, 18 pp. [6]. He, Y. Su, On discrete fourth-order boundary value problems with three parameters, J. Comput. Appl. Math. 33 (010) 10, [7] C. Yuan, D. Jiang, D. O Regan, Existence and uniqueness of positive solutions for fourth-order nonlinear singular continuous and discrete boundary value problems, Appl. Math. Comput. 03 (008) 1, [8] F.J.S.A. Corrca, On multiple positive solutions of positone and non-positone problems, Abstr. Appl. Anal. 4 (1999), [9] X. Cai, Z. Guo, Existence of solutions of nonlinear fourth order discrete boundary value problem, J. Difference Equ. Appl. 1 (006) 5, [10] R. Ma, C. Gao, Y. Chang, Existence of solutions of a discrete fourth-order boundary value problem, Discrete Dyn. Nat. Soc. 010, Article ID , 19 pp. Marek Galewski marek.galewski@p.lodz.pl echnical University of Lodz Institute of Mathematics Wolczanska 15, Lodz, Poland Joanna Smejda jsmejda@math.uni.lodz.pl Lodz University Faculty of Mathematics and Computer Science Banacha, Lodz, Poland Received: January 4, 011. Revised: March 11, 011. Accepted: March 15, 011.