- Email: [email protected]
A note on the multiplicity of solutions for a second-order difference equation with a parameter
Applied Mathematics and Computation 218 (2011) 3954–3956
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A note on the multiplicity of solutions for a second-order difference equation with a parameter Marek Galewski Technical University of Lodz, Institute of Mathematics, Wolczanska 215, 90-924 Lodz, Poland
a r t i c l e
i n f o
a b s t r a c t We undertake the question of the multiplicity of solutions to the discrete Emden–Fowler BVP by correcting some results contained in [2]. Ó 2011 Elsevier Inc. All rights reserved.
Keywords: Discrete boundary value problem Variational method Multiplicity of solutions
1. Introduction Functional J 2 C1(E, R), where E is a real Banach space, satisfies (PS) condition when any sequence {un} E such that {J(un)} is bounded and J 0 ðun Þ ! 0 as n ? 1, has a convergent subsequence. It is known that when E has finite dimension, the coercivity of J implies that J satisfies (PS) condition. In the investigation of the existence of multiple solutions for discrete equations the following Clark’s Theorem appears a useful tool, see for example [1–4]. Theorem 1 [5]. Let E be a real Banach space, let J 2 C1(E, R) be an even functional, bounded from below and satisfying (PS) condition. Suppose J(0E) = 0, there is a set K closed and symmetric with respect to zero and such that K is a homeomorphic to Sj1 (i.e. j 1 dimension unit sphere) by an odd map, and supx2KJ(x) < 0. Then J has at least j distinct pairs of nonzero critical points. In [2] the Author considers the multiplicity of solutions for the second order difference equations known as the generalized Emden–Fowler equation
Dðpðk 1ÞDxðk 1ÞÞ þ qðkÞxðkÞ þ kf ðk; xðkÞÞ ¼ 0
ð1Þ
subject to a parameter k > 0 and with boundary conditions
xð0Þ ¼ xðTÞ;
pð0ÞDxð0Þ ¼ pðTÞDxðTÞ:
ð2Þ
Here T P 3 is a fixed natural number; [a, b], where a 6 b are integers, is the discrete interval {a, a + 1, . . . , b 1, b} ; D is the forward difference operator, i.e. Dx(k 1) = x(k) x(k 1) ; f 2 C([1, T] R R), p 2 C([0, T + 1], R), q 2 C([1, T], R). Let us denote as in [3]
2 6 6 6 6 6 M¼6 6 6 6 4
pð0Þ þ pð1Þ
pð1Þ
pð1Þ
pð1Þ þ pð2Þ
0 .. .
pð2Þ .. .
0 pð0Þ
0 0
0
...
0
pð0Þ
pð2Þ
...
0
0
pð2Þ þ pð3Þ . . . .. .. . .
0 .. .
0 .. .
0 0
. . . pðT 2Þ þ pðT 1Þ pðT 1Þ ... pðT 1Þ pðT 1Þ þ pð0Þ
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.09.004
3 7 7 7 7 7 7 7 7 7 5
M. Galewski / Applied Mathematics and Computation 218 (2011) 3954–3956
3955
and
2
qð1Þ
0
0
...
0
0
0
qð2Þ
0
...
0
0
0 .. .
0 .. .
qð3Þ . . . .. .. . .
0 .. .
0 .. .
0
0
0
. . . qðT 1Þ
0
0
0
...
6 6 6 6 6 Q ¼6 6 6 6 4
0
0
3 7 7 7 7 7 7: 7 7 7 5
qðTÞ
Let us assume that (H1) (H2) (H3) (H4)
p(T) – 0; f 2 C([1, T] R ,R); f is odd in u for all k 2 [1, T]; M + Q is positive definite.
With assumptions (H1)–(H4) and the with suitable growth conditions there are obtained in [2] the open intervals for parameter k such that for each k in such an interval problem (1) and (2) has at least T distinct pairs of solutions. The main aim of our note is to improve Theorem 3.2 from [2] which has an invalid proof. Namely, it is shown that upon assumptions of Theorem 3.2 the action functional is anti-coercive and as such it cannot be bounded from below as required by Clark’s Theorem. Moreover, one of the assumptions must be altered as well. Some minor corrections to the assumptions of Theorems 3.1 and 3.3 and Corollary 3.1 are suggested. However, these results remain valid upon our corrections. Due to (H1) solutions to (1) and (2) can be identified with vectors from RT considered with classical Euclidean norm jj; by Ry h, i we denote the scalar product. Let us define Fðk; yÞ ¼ 0 f ðk; sÞds. The action functionals J; eJ : RT ! R for (1) and (2) are given by the formulas T X 1 JðxÞ ¼ hðM þ QÞx; xi þ k Fðk; xðkÞÞ; 2 k¼1
ð3Þ
T X eJðxÞ ¼ 1 hðM þ Q Þx; xi k Fðk; xðkÞÞ 2 k¼1
ð4Þ
depending on growth conditions imposed on the nonlinear term. Both functionals eJ and J are l.s.c. and Gateaux differentiable. Critical points to both functionals correspond to in a 1 1 manner to the solutions to our problem. Thus in order to determine the existence of multiple solutions to (1) and (2) it suffice to perform these investigations for critical points to either (3) or (4). In [2] only functional J is used, while we will use functional eJ in the new version of Theorem 3.2. 2. Results We denote by k1 6 k2 6 6 k T the eigenvalues of M + Q. We shall start with the following observation concerning the correct assumptions of Theorem 3.1. in [2]. f ðk;uÞ Remark 2. In [2], Theorem 3.1., relations limu!1 f ðk;uÞ u P d and limu!0 u 6 e are assumed and it is derived that
Fðk; uÞ P 2d u2 þ c for some c 2 R. Such a conclusion can only be reached when limu!1 f ðk;uÞ u > d or else it becomes Fðk; uÞ P d2e1 u2 þ c; where e1 > 0 is as small as we please but still positive. Indeed, function f(k, u) = u + sinu satisfies that limu!1 f ðk;uÞ u ¼ 1 while it oscillates around 1. Similar comment concerns Theorem 3.3. and Corollary 3.1. where the weak inequalities in the assumptions should be replaced with the strict ones. For example, Theorem 3.1. in [2] should read: Theorem 3. Assume (H1)–(H4) and that there exist numbers d > e > 0 such that for all k 2 [1, T] (H5) kTe < k1d; > d; (H6) limu!1 f ðk;uÞ u (H7) limu!0 f ðk;uÞ < e. u Then for all k 2
k
T
d
; ke1 problem (1) and (2) has at least T pairs of distinct solutions.
3956
M. Galewski / Applied Mathematics and Computation 218 (2011) 3954–3956
Now we are concerned with Theorem 3.2. [2] which we state as follows: Theorem 4. Assume (H1)–(H4), (H6) and (H8) there exist constants e > 0 and 1 < r < 2 such that for all k 2 [1, . . . , T] it holds
lim
u!1
f ðk; uÞ < e: ur1
Then for all k > kdT problem (1) and (2) has at least T pairs of distinct solutions. Proof. Throughout the proof we fix k > kdT . f ðk;y;uÞ Let us denote limu!1 fuðk;uÞ < a þ n for all k 2 [1, T] and for r1 ¼ n. By (H8) for any a > 0 there exists B > 0 such that jyjr1 jyj P B. Since n < e we can always take a such that a + n = e. Then it follows that Fðk; yÞ 6 er jyjr for all k 2 [1, T] and for j yj P B. Denoting A = sup(k,x)2[1,T][B,B]jf(k, x)j we see that for (k, y) 2 [1, T] R by the definition of F
e
Fðk; yÞ 6 AB þ jyjr : r Since
eJðxÞ P 1 k1 2
T X
jxðkÞj2 k
k¼1
T eX
r
jxðkÞjr kTAB ! þ1;
k¼1
we note that eJ is bounded below, coercive and it satisfies the (PS) condition. Reasoning as in the above we show that (H6) implies that there exists g > 0 such that for all u 2 R with juj 6 g we have Fðk; uÞ P 2d u2 . Hence, we have on the sphere {x 2 RT : jxij = g, i = 1, . . . , T} what follows
eJðxÞ 6 1 ðkT kdÞjxj2 ¼ ðkT kdÞTjgj2 < 0: 2 Therefore by Clark’s Theorem, we reach the conclusion.
h
Remark 5. Note that in the counterpart of Theorem 4 in [2] it is assumed that
ðH9Þ lim
u!1
f ðk; uÞ ¼ 0; u
instead of (H6). However, we think that we are unable to reach Theorem 4 with (H9) since it provides no information about the behavior of the action functional near 0 and therefore, we may not construct the sphere required by the Clark’s Theorem. References [1] D. Bai, Y. Xu, Nontrivial solutions of boundary value problems of second-order difference equations, J. Math. Anal. Appl. 326 (1) (2007) 297–302. [2] L. Gao, Existence of multiple solutions for a second-order difference equation with a parameter, Appl. Math. Comput. 216 (5) (2010) 1592–1598. [3] X. He, X. Wu, Existence and multiplicity of solutions for nonlinear second order difference boundary value problems, Comput. Math. Appl. 57 (2009) 1–8. [4] F. Lian, Y. Xu, Multiple solutions for boundary value problems of a discrete generalized Emden–Fowler equation, Appl. Math. Lett. 23 (2) (2010) 8–12. [5] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in: CBMS, Registered Conference Series on Mathamatics, vol. 65, American Mathamatical Society, Providence, RI, 1986.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A note on the multiplicity of solutions for a second-order difference equation with a parameter Marek Galewski Technical University of Lodz, Institute of Mathematics, Wolczanska 215, 90-924 Lodz, Poland
a r t i c l e
i n f o
a b s t r a c t We undertake the question of the multiplicity of solutions to the discrete Emden–Fowler BVP by correcting some results contained in [2]. Ó 2011 Elsevier Inc. All rights reserved.
Keywords: Discrete boundary value problem Variational method Multiplicity of solutions
1. Introduction Functional J 2 C1(E, R), where E is a real Banach space, satisfies (PS) condition when any sequence {un} E such that {J(un)} is bounded and J 0 ðun Þ ! 0 as n ? 1, has a convergent subsequence. It is known that when E has finite dimension, the coercivity of J implies that J satisfies (PS) condition. In the investigation of the existence of multiple solutions for discrete equations the following Clark’s Theorem appears a useful tool, see for example [1–4]. Theorem 1 [5]. Let E be a real Banach space, let J 2 C1(E, R) be an even functional, bounded from below and satisfying (PS) condition. Suppose J(0E) = 0, there is a set K closed and symmetric with respect to zero and such that K is a homeomorphic to Sj1 (i.e. j 1 dimension unit sphere) by an odd map, and supx2KJ(x) < 0. Then J has at least j distinct pairs of nonzero critical points. In [2] the Author considers the multiplicity of solutions for the second order difference equations known as the generalized Emden–Fowler equation
Dðpðk 1ÞDxðk 1ÞÞ þ qðkÞxðkÞ þ kf ðk; xðkÞÞ ¼ 0
ð1Þ
subject to a parameter k > 0 and with boundary conditions
xð0Þ ¼ xðTÞ;
pð0ÞDxð0Þ ¼ pðTÞDxðTÞ:
ð2Þ
Here T P 3 is a fixed natural number; [a, b], where a 6 b are integers, is the discrete interval {a, a + 1, . . . , b 1, b} ; D is the forward difference operator, i.e. Dx(k 1) = x(k) x(k 1) ; f 2 C([1, T] R R), p 2 C([0, T + 1], R), q 2 C([1, T], R). Let us denote as in [3]
2 6 6 6 6 6 M¼6 6 6 6 4
pð0Þ þ pð1Þ
pð1Þ
pð1Þ
pð1Þ þ pð2Þ
0 .. .
pð2Þ .. .
0 pð0Þ
0 0
0
...
0
pð0Þ
pð2Þ
...
0
0
pð2Þ þ pð3Þ . . . .. .. . .
0 .. .
0 .. .
0 0
. . . pðT 2Þ þ pðT 1Þ pðT 1Þ ... pðT 1Þ pðT 1Þ þ pð0Þ
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.09.004
3 7 7 7 7 7 7 7 7 7 5
M. Galewski / Applied Mathematics and Computation 218 (2011) 3954–3956
3955
and
2
qð1Þ
0
0
...
0
0
0
qð2Þ
0
...
0
0
0 .. .
0 .. .
qð3Þ . . . .. .. . .
0 .. .
0 .. .
0
0
0
. . . qðT 1Þ
0
0
0
...
6 6 6 6 6 Q ¼6 6 6 6 4
0
0
3 7 7 7 7 7 7: 7 7 7 5
qðTÞ
Let us assume that (H1) (H2) (H3) (H4)
p(T) – 0; f 2 C([1, T] R ,R); f is odd in u for all k 2 [1, T]; M + Q is positive definite.
With assumptions (H1)–(H4) and the with suitable growth conditions there are obtained in [2] the open intervals for parameter k such that for each k in such an interval problem (1) and (2) has at least T distinct pairs of solutions. The main aim of our note is to improve Theorem 3.2 from [2] which has an invalid proof. Namely, it is shown that upon assumptions of Theorem 3.2 the action functional is anti-coercive and as such it cannot be bounded from below as required by Clark’s Theorem. Moreover, one of the assumptions must be altered as well. Some minor corrections to the assumptions of Theorems 3.1 and 3.3 and Corollary 3.1 are suggested. However, these results remain valid upon our corrections. Due to (H1) solutions to (1) and (2) can be identified with vectors from RT considered with classical Euclidean norm jj; by Ry h, i we denote the scalar product. Let us define Fðk; yÞ ¼ 0 f ðk; sÞds. The action functionals J; eJ : RT ! R for (1) and (2) are given by the formulas T X 1 JðxÞ ¼ hðM þ QÞx; xi þ k Fðk; xðkÞÞ; 2 k¼1
ð3Þ
T X eJðxÞ ¼ 1 hðM þ Q Þx; xi k Fðk; xðkÞÞ 2 k¼1
ð4Þ
depending on growth conditions imposed on the nonlinear term. Both functionals eJ and J are l.s.c. and Gateaux differentiable. Critical points to both functionals correspond to in a 1 1 manner to the solutions to our problem. Thus in order to determine the existence of multiple solutions to (1) and (2) it suffice to perform these investigations for critical points to either (3) or (4). In [2] only functional J is used, while we will use functional eJ in the new version of Theorem 3.2. 2. Results We denote by k1 6 k2 6 6 k T the eigenvalues of M + Q. We shall start with the following observation concerning the correct assumptions of Theorem 3.1. in [2]. f ðk;uÞ Remark 2. In [2], Theorem 3.1., relations limu!1 f ðk;uÞ u P d and limu!0 u 6 e are assumed and it is derived that
Fðk; uÞ P 2d u2 þ c for some c 2 R. Such a conclusion can only be reached when limu!1 f ðk;uÞ u > d or else it becomes Fðk; uÞ P d2e1 u2 þ c; where e1 > 0 is as small as we please but still positive. Indeed, function f(k, u) = u + sinu satisfies that limu!1 f ðk;uÞ u ¼ 1 while it oscillates around 1. Similar comment concerns Theorem 3.3. and Corollary 3.1. where the weak inequalities in the assumptions should be replaced with the strict ones. For example, Theorem 3.1. in [2] should read: Theorem 3. Assume (H1)–(H4) and that there exist numbers d > e > 0 such that for all k 2 [1, T] (H5) kTe < k1d; > d; (H6) limu!1 f ðk;uÞ u (H7) limu!0 f ðk;uÞ < e. u Then for all k 2
k
T
d
; ke1 problem (1) and (2) has at least T pairs of distinct solutions.
3956
M. Galewski / Applied Mathematics and Computation 218 (2011) 3954–3956
Now we are concerned with Theorem 3.2. [2] which we state as follows: Theorem 4. Assume (H1)–(H4), (H6) and (H8) there exist constants e > 0 and 1 < r < 2 such that for all k 2 [1, . . . , T] it holds
lim
u!1
f ðk; uÞ < e: ur1
Then for all k > kdT problem (1) and (2) has at least T pairs of distinct solutions. Proof. Throughout the proof we fix k > kdT . f ðk;y;uÞ Let us denote limu!1 fuðk;uÞ < a þ n for all k 2 [1, T] and for r1 ¼ n. By (H8) for any a > 0 there exists B > 0 such that jyjr1 jyj P B. Since n < e we can always take a such that a + n = e. Then it follows that Fðk; yÞ 6 er jyjr for all k 2 [1, T] and for j yj P B. Denoting A = sup(k,x)2[1,T][B,B]jf(k, x)j we see that for (k, y) 2 [1, T] R by the definition of F
e
Fðk; yÞ 6 AB þ jyjr : r Since
eJðxÞ P 1 k1 2
T X
jxðkÞj2 k
k¼1
T eX
r
jxðkÞjr kTAB ! þ1;
k¼1
we note that eJ is bounded below, coercive and it satisfies the (PS) condition. Reasoning as in the above we show that (H6) implies that there exists g > 0 such that for all u 2 R with juj 6 g we have Fðk; uÞ P 2d u2 . Hence, we have on the sphere {x 2 RT : jxij = g, i = 1, . . . , T} what follows
eJðxÞ 6 1 ðkT kdÞjxj2 ¼ ðkT kdÞTjgj2 < 0: 2 Therefore by Clark’s Theorem, we reach the conclusion.
h
Remark 5. Note that in the counterpart of Theorem 4 in [2] it is assumed that
ðH9Þ lim
u!1
f ðk; uÞ ¼ 0; u
instead of (H6). However, we think that we are unable to reach Theorem 4 with (H9) since it provides no information about the behavior of the action functional near 0 and therefore, we may not construct the sphere required by the Clark’s Theorem. References [1] D. Bai, Y. Xu, Nontrivial solutions of boundary value problems of second-order difference equations, J. Math. Anal. Appl. 326 (1) (2007) 297–302. [2] L. Gao, Existence of multiple solutions for a second-order difference equation with a parameter, Appl. Math. Comput. 216 (5) (2010) 1592–1598. [3] X. He, X. Wu, Existence and multiplicity of solutions for nonlinear second order difference boundary value problems, Comput. Math. Appl. 57 (2009) 1–8. [4] F. Lian, Y. Xu, Multiple solutions for boundary value problems of a discrete generalized Emden–Fowler equation, Appl. Math. Lett. 23 (2) (2010) 8–12. [5] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in: CBMS, Registered Conference Series on Mathamatics, vol. 65, American Mathamatical Society, Providence, RI, 1986.