Layer structures for the solutions to the perturbed simple pendulum problems

Size: px
Start display at page:

Download "Layer structures for the solutions to the perturbed simple pendulum problems"

Transcription

1 J. Math. Anal. Appl. 35 (26) Layer structures for the solutions to the perturbed simple pendulum problems Tetsutaro Shibata Applied Mathematics Research Group, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, , Japan Received 28 July 24 Available online 2 August 25 Submitted by R.E. O Malley, Jr. Abstract We consider the perturbed simple pendulum equation u (t) = µf ( u(t) ) + λ sin u(t), t := ( T,T), u(t) >, t, u(±t)=, where λ>andµ R are parameters. The typical example of f is f(u)= u p u(p>). The purpose of this paper is to study the shape of the solutions when λ. More precisely, by using a variational approach, we show that there exist two types of solutions: one is almost flat inside and another is like a step function with two steps. 25 Elsevier nc. All rights reserved. Keywords: nterior layer; Boundary layer; Simple pendulum problems. ntroduction We consider the perturbed simple pendulum equation u (t) = µf ( u(t) ) + λ sin u(t), t := ( T,T), (.) u(t) >, t, (.2) address: shibata@amath.hiroshima-u.ac.jp X/$ see front matter 25 Elsevier nc. All rights reserved. doi:.6/j.jmaa

2 726 T. Shibata / J. Math. Anal. Appl. 35 (26) u(±t)=, (.3) where T>isaconstant and λ>, µ R are parameters. We assume that f satisfies the following conditions: (A) f C (R), f ( u) = f(u)for u R and f(u)>foru>. (A2) f () =. (A3) f(u)/uis increasing for <u<π. The typical example of f(u)is f(u)= u p u(p>). The purpose of this paper is to study the shape of the solutions of (.) (.3) when λ. More precisely, by using a variational approach, we show that (.) (.3) has two types of solutions: one is almost flat inside and another is like a step function with two steps. Therefore, it is shown that the structure of the solutions (.) (.3) is rich. Linear and nonlinear multiparameter problems have been investigated intensively by many authors. We refer to [ 4,6 9] and the references therein. n particular, one of the main topics for the nonlinear problems is to study the equations which develop layer type solutions. ndeed, concerning the layer structure of the solutions, a possible layer structure was brought out in [6,7] for one-parameter singular perturbation problems; and it is known that the solutions with layers appear for two-parameter problems, which are different from (.) (.3) (cf. [ 2,4]). Recently, Shibata [3] considered (.) (.3) for the case µ< by means of the following constrained minimization method. Let } U β := {u H (): ( ) cos u(t) dt = β, where <β<4t is a fixed constant and H () is the usual real Sobolev space. Regarding µ< as a given parameter, consider the minimizing problem, which depends on µ: Minimize 2 u 2 2 µ F ( u(t) ) dt under the constraint u U β, (.4) where F(u) := u f(s)ds. Then by Lagrange multiplier theorem, for a given µ<, a unique solution triple (µ, λ(µ), u(µ)) R 2 + U β was obtained, where λ(µ) is the Lagrange multiplier. Then the following result was obtained in [3]: Theorem. [3]. Let <θ β <π satisfies cos θ β = β/(2t). Then u(µ) θ β uniformly on any compact subset in and λ(µ) as µ. We see from Theorem. that u(µ) is almost flat inside and develops boundary layer as µ. We emphasize that this asymptotic behavior of u(µ) is the most characteristic feature of the solution of two-parameter problem (.) (.3) in the following sense. Let µ = µ < be fixed in (.) (.3) and consider a one-parameter problem

3 T. Shibata / J. Math. Anal. Appl. 35 (26) v (t) = µ f ( v(t) ) + λ sin v(t), t, (.5) v(t) >, t, (.6) v(±t)=. (.7) Then for a given λ, there exists a unique solution (λ, v λ ) R + C 2 ( ). Moreover, if λ, then v λ π locally uniformly in (cf. [5]). Therefore, we do not have any solutions {v λ } of a one-parameter problem (.5) (.7) such that v λ θ β (< π) as λ. t should be pointed out that only a flat solution has been obtained in [3], since only the case where µ< has been considered. ndeed, if µ< is assumed, then we see from [5] that the maximum norm of the solution is less than π. Therefore, by the variational method (.4), it is impossible to treat the solution of (.) (.3) with maximum norm larger than π. To treat both cases mentioned above at the same time, we adopt here another sort of variational approach. Namely, we regard λ> as a given parameter here and using different type of variational approach from (.4), we show that (.) (.3) has both flat and step function type solutions. t is shown that the maximum norm of step function type solution is bigger than π. We now explain the variational framework used here. Let M α := {v H (): Q(v) := F ( v(t) ) } dt = 2TF(α), (.8) where α> is a constant. Then consider the minimization problem, which depends on λ>: Minimize K λ (v) := 2 v 2 2 λ ( ) cos v(t) dt Let under the constraint v M α. (.9) β(λ,α) := min K λ (v). v M α Then by Lagrange multiplier theorem, for a given λ>, there exists (λ, µ(λ), u λ ) R 2 M α which satisfies (.) (.3) with K λ (u λ ) = β(λ,α), where µ(λ), which is called the variational eigenvalue, is the Lagrange multiplier. Now we state our results. Theorem.. Let <α<π be fixed. Then (a) µ(λ) < for λ. (b) u λ α locally uniformly on as λ. (c) µ(λ) = C λ + o(λ) as λ, where C = sin α/f (α). The following Theorem.2 is our main result in this paper.

4 728 T. Shibata / J. Math. Anal. Appl. 35 (26) Theorem.2. Let π<α<3π be fixed. Then (a) µ(λ) >. (b) µ(λ) as λ. More precisely, as λ, λ exp ( ( ) ) ( ( ) ) t α + o() λ <µ(λ)<λexp tα o() λ, where t α := (F (α) F(π))T/(F(3π) F(π)), which is positive by the condition π<α<3π and (A). (c) Assume 3F(α)<F(3π)+ 2F(π). (.) Then, as λ, u λ 3π locally uniformly on ( t α,t α ), u λ π locally uniformly on ( T, t α ) (t α,t). Remark.3. (i) Theorem.(b) implies that for <α<π, u λ is almost flat inside and develops boundary layer as λ. On the other hand, Theorem.2(c) implies that for π<α<3π satisfying (.), u λ has both boundary layers and interior layers. Moreover, u λ is almost flat in ( t α,t α ) and ( T, t α ) (t α,t). n other words, u λ is almost a step function with two steps in this case. Therefore, the structures of u λ for <α<π and π<α<3π are totally different. (ii) The rough idea of the proof of Theorem.2(c) is as follows. We first show that u λ has boundary layers at t =±T. Secondly, we show that u λ has a interior layer in (,T) and is almost equal to π and 3π. nequality (.) is a technical condition to obtain the estimate of u λ from above. Then the position of the interior layer is established automatically. f f(u)= u p u (p >), then (.) implies π<α<((3 p+ + 2)/3) /(p+) π. For instance, if p = 7, then (.) is equivalent to π<α<(6563/3) /8 π = π. (iii) t is certainly important to consider the asymptotic shape of u λ as λ for the case α = π. Clearly, u λ is almost equal to π in ( T,) (,T). By Theorem.(b), we see that if α<π and α is very close to π, then u λ is almost flat and equal to π inside when λ. On the other hand, by Theorem.2(c), if α>π and α is very close to π, then u λ is almost flat and equal to π in ( T, t α ) (t α,t), and u λ is almost equal to 3π in ( t α,t α ). Since α>π and α is nearly equal to π, we see that t α is very small by definition of t α and t α asα π. Therefore, if α = π, then the asymptotic shape of u λ when λ is expected to be a box with spike at t =. Therefore, it is quite interesting to determine whether the asymptotic shape of u λ is like a box with spike at t = asλ when α = π. However, it is difficult to treat this problem by our methods here. The reason whyisasfollows.weregardα as a parameter and denote the minimizer by u λ = u λ,α if u λ M α. Then it is quite natural to consider a sequence of minimizer {u λ,α } for a fixed λ, and observe a limit function u λ,π = lim α π u λ,α. Then it is not so difficult to show that u λ,π is also a minimizer of (.9) for α = π, and satisfies (.) (.3). Therefore, it is expected that u λ,π 3π as λ. However, to show this, the uniform estimate

5 T. Shibata / J. Math. Anal. Appl. 35 (26) u λ,α 3π δ for all λ>λ and π<α<π+ δ for some <δ is necessary. Since this estimate is quite difficult to show, it is so hard to show whether u λ 3π or π as λ. From these points of view, it is not easy to study the case where α = π by the simple calculation. The future direction of this study is certainly to extend our investigation to the case where α = π,3π,... The remainder of this paper is organized as follows. n Section 2, we prove Theorem.2. Section 3 is devoted to the proof of Theorem.. For the sake of completeness, we show the existence of (λ, µ(λ), u λ ) R 2 M α in Appendix A. 2. Proof of Theorem.2 n what follows, we denote by C the various constants which are independent of λ. n particular, the several characters C, which appear in an equality or an inequality repeatedly, may imply the different constants each other. Proof of Theorem.2(a). Assume that µ(λ). Then u λ > satisfies u (t) ( ) µ(λ) f u(t) = λ sin u(t), t, u(±t)=. Then it follows from [5] that u λ <π. ndeed, let <m λ <π satisfy µ(λ) f(m λ ) = λ sin m λ. Then we know from [5] that u λ <m λ. This is impossible, since u λ M α and α>π. We next prove Theorem.2(c). To do this, we need some lemmas. For a given γ>, let t γ,λ [,T] satisfy u λ (t γ,λ ) = γ, which is unique if it exists, since u λ (t) <, <t T. (2.) Lemma 2.. Let <δ be fixed. Then t π δ,λ T as λ. Proof. Since u λ M α (π<α<3π), we see that u λ >π. Therefore, there exists unique t π,λ. By (.), we have { u λ (t) + µ(λ)f ( u λ (t) ) + λ sin u λ (t) } u λ (t) =. This implies that for t [,T] 2 u λ (t)2 + µ(λ)f ( u λ (t) ) + λ ( cos u λ (t) ) constant = µ(λ)f ( ) ( ) u λ + λ cos uλ (put t = ) = 2 u λ (t π,λ) 2 + µ(λ)f (π) + 2λ (put t = t π,λ ). (2.2) By this and (2.), we see that for t [,T]

6 73 T. Shibata / J. Math. Anal. Appl. 35 (26) u λ (t) = 2λ ( + cos u λ (t) ) + 2µ(λ) ( F(π) F ( u λ (t) )) + u λ (t π,λ) 2. (2.3) By this and Theorem.2(a), for t π δ,λ t T, u λ (t) 2λ( cos δ). (2.4) By this, we obtain π δ = T t π δ,λ u λ This implies our conclusion. (t) dt 2λ( cos δ)(t tπ δ,λ ). Lemma 2.2. Assume that u λ 3π for λ. Let an arbitrary <δ be fixed. Then t π+δ,λ t 3π δ,λ as λ. Lemma 2.2 can be proved by the same argument as that in Lemma 2.. So we omit the proof. Lemma 2.3. Let an arbitrary <δ be fixed. Then for λ t π,λ t π+δ,λ >t π δ,λ t π,λ. (2.5) Proof. By (2.3) and putting θ := π u λ (t), we obtain t π δ,λ t π,λ = Similarly, by (2.3) t π δ,λ t π,λ u λ (t) dt 2λ( + cos u λ (t)) + 2µ(λ)(F (π) F(u λ (t))) + u λ (t π,λ) 2 δ dθ =. (2.6) 2λ( cos θ)+ 2µ(λ)(F (π) F(π θ))+ u λ (t π,λ) 2 δ dθ t π,λ t π+δ,λ =. (2.7) 2λ( cos θ) 2µ(λ)(F (θ + π) F(π))+ u λ (t π,λ) 2 By this and (2.6), we obtain (2.5). Lemma 2.4. Assume that there is a constant <δ satisfying lim sup λ u λ > 3π + δ. Then for <δ δ and λ t 3π,λ t 3π+δ,λ >t π,λ t π+δ,λ. (2.8) Proof. Let <δ δ be fixed. Put t = t 3π,λ in (2.2). Then we obtain 2 u λ (t π,λ) 2 + µ(λ)f (π) + 2λ = 2 u λ (t 3π,λ) 2 + µ(λ)f (3π)+ 2λ. (2.9)

7 T. Shibata / J. Math. Anal. Appl. 35 (26) By this and (A), we see that u λ (t 3π,λ) 2 <u λ (t π,λ) 2. (2.) By this and (2.2), for t [t 3π+δ,λ,t 3π,λ ],wehave 2 u λ (t)2 = λ ( + cos u λ (t) ) µ(λ) ( F ( u λ (t) ) F(3π) ) + 2 u λ (t 3π,λ) 2 <λ ( + cos u λ (t) ) µ(λ) ( F ( u λ (t) ) F(3π) ) + 2 u λ (t π,λ) 2. (2.) This along with (2.) and the same argument as that to obtain (2.6) implies that t 3π,λ t 3π+δ,λ δ dθ >. (2.2) 2λ( cos θ) 2µ(λ)(F (θ + 3π) F(3π))+ u λ (t π,λ) 2 By (A3), it is easy to see that for θ δ, F(θ + 3π) F(3π)>F(θ + π) F(π). (2.3) Then (2.7), (2.2) and (2.3) imply (2.8). Thus the proof is complete. Proof of Theorem.2(c). We first show that lim sup u λ 3π. (2.4) λ To do this, we assume that there exists a constant <δ and a subsequence of { u λ }, which is denoted by { u λ } again, such that u λ > 3π + δ and derive a contradiction. For <δ<δ and λ, we put π,δ,λ := t π δ,λ t π+δ,λ, 3π,δ,λ := t 3π,λ t 3π+δ,λ. Then we see from Lemmas 2.3 and 2.4 that π,δ,λ < 2 3π,δ,λ, π,δ,λ + 3π,δ,λ <T. (2.5) By (2.5), for λ, we obtain π,δ,λ < 2T/3. Since u λ M α and u λ (t) = u λ ( t) for t [,T], by this and Lemmas 2. and 2.2, we obtain T TF(α)= F ( u λ (t) ) dt F(π δ) π,δ,λ + F(3π δ)(t π,δ,λ ) + o() F(3π)T ( F(3π) F(π) ) π,δ,λ + O(δ) + o() F(3π)T 2 3 T ( F(3π) F(π) ) + O(δ)+ o() = 3 TF(3π)+ 2 TF(π)+ O(δ) + o(). (2.6) 3

8 732 T. Shibata / J. Math. Anal. Appl. 35 (26) Since <δ is arbitrary, this contradicts (.). Therefore, we obtain (2.4). Then there are two possibilities: (i) lim λ u λ = π or (ii) lim λ u λ = 3π. However, (i) is impossible, since u λ M α (π < α < 3π). Hence, we obtain (ii). Then we see from Lemmas 2. and 2.2 that for any t [,T), we have only two possibilities: (i) lim λ u λ (t) = π or (ii) lim λ u λ (t) = 3π. Then by (2.) and u λ M α, obviously, Theorem.2(c) holds, since u λ (t) = u λ ( t) for t [,T]. Thus the proof is complete. Proof of Theorem.2(b). The proof is divided into three steps. Step. We first show that for λ u λ 3π. (2.7) ndeed, if there exists a subsequence of { u λ }, which is denoted by { u λ } again, such that u λ > 3π, then by putting θ = u λ (t) 3π and δ λ := u λ 3π >, we see from the same calculation as that to obtain (2.2) that t 3π,λ > δ λ dθ. (2.8) 2λ( cos θ) 2µ(λ)(F (θ + 3π) F(3π))+ u λ (t π,λ) 2 Further, by putting δ = δ λ in (2.6) and (2.7), we obtain t π δλ,λ t π,λ = δ λ t π,λ t π+δλ,λ = δ λ dθ, (2.9) 2λ( cos θ)+ 2µ(λ)(F (π) F(π θ))+ u λ (t π,λ) 2 dθ. (2.2) 2λ( cos θ) 2µ(λ)(F (θ + π) F(π))+ u λ (t π,λ) 2 By (2.3) and (2.8) (2.2), we obtain t 3π,λ >t π,λ t π+δλ,λ >t π δλ,λ t π,λ. (2.2) By this and the same argument to obtain (2.6), we obtain a contradiction. Therefore, we obtain (2.7). Step 2. Assume that there exists a subsequence of {µ(λ)}, denoted by {µ(λ)} again, such that µ(λ) δ >. By (2.2), we have 2 u λ (t)2 = µ(λ) ( F ( ) ( u λ F uλ (t) )) + λ ( ) cos u λ (t) cos u λ. (2.22) Let <δ be fixed. By mean value theorem and (A.3), for t [,t 3π 2δ,λ ] and λ, we obtain

9 T. Shibata / J. Math. Anal. Appl. 35 (26) F ( u λ ) F ( uλ (t) ) f(3π 2δ) ( u λ u λ (t) ) ( f(3π) Cδ )( u λ u λ (t) ). (2.23) By (2.7) and the fact that u λ 3π as λ,fort [,t 3π 2δ,λ ] and λ, cos u λ (t) cos u λ = sin u λ ( uλ (t) u λ ) 2 cos( θ u λ + ( θ)u λ (t) )( u λ u λ (t) ) 2 2( Cδ o() )( uλ u λ (t) ) 2, (2.24) where <θ<. By (2.22) (2.24), for λ u λ (t) 2 ( f(3π) Cδ ) µ(λ) ( u λ u λ (t) ) + λ ( Cδ o() )( u λ u λ (t) ) 2 ( := A λ uλ u λ (t) ) 2 ( + Bλ uλ u λ (t) ), (2.25) where A λ = λ ( Cδ o() ), By this, for λ, t 3π 2δ,λ = = t 3π 2δ,λ dt u λ 3π+2δ < Aλ 3δ t 3π 2δ,λ B λ = 2 ( f(3π) Cδ ) µ(λ). u λ (t) Aλ ( u λ u λ (t)) 2 + B λ ( u λ u λ (t)) dt Aλ θ 2 + B λ θ dθ dθ (θ + B λ /(2A λ )) 2 Bλ 2/(4A2 λ ) = [log ( Aλ 3δ + B λ + 3δ + B ) ] 2 λ B2 λ 2A λ 2A λ log B λ. (2.26) 2A λ Step 3. There are three cases to consider. Case (i). Assume that there exists a subsequence of {µ(λ)} satisfying µ(λ)/λ as λ. Then since µ(λ) δ,asλ, t 3π 2δ,λ C ( C + C log λ ) C ) λ (C + C log λδ. λ µ(λ) This along with Lemmas 2. and 2.2 implies that u λ (t) π (t \{}) asλ.this is a contradiction, since u λ M α and π<α. Case (ii). Assume that there is a subsequence of {µ(λ)} satisfying C µ(λ)/λ C. Then by this and (2.26), as λ, 4A 2 λ

10 734 T. Shibata / J. Math. Anal. Appl. 35 (26) t 3π 2δ,λ C λ. (2.27) By the same reason as that of case (i), this is a contradiction. Case (iii). Assume that there exists a subsequence of {µ(λ)} satisfying µ(λ)/λ. Then by this and (2.26), as λ, ( ) t 3π 2δ,λ o λ. (2.28) By the same reason as that of case (i), this is a contradiction. Therefore, we see that µ(λ) asλ. Finally, we show the decay rate of µ(λ) as λ. Since t 3π 2δ,λ t α as λ,by (2.26), L := Aλ < Aλ 2δ 3δ Aλ θ 2 + B λ θ dθ < t 3π 2δ,λ Aλ θ 2 + B λ θ dθ := L 2. (2.29) Since µ(λ) asλ, L = ( ) ( + o() log ( 3δ + o() ) ) λ ( ) + log <t α + o(). (2.3) λ Cµ(λ) This implies that log λ µ(λ) <t ( ) α + o() λ. (2.3) By this, we obtain µ(λ) > λ exp ( ( ) ) t α + o() λ. (2.32) By the same argument as above, we also obtain µ(λ) < λ exp ( ( ) ) t α o() λ. (2.33) Thus the proof is complete. 3. Proof of Theorem. To prove Theorem., we follows the idea of the proof of [3, Theorem 2]. Proof of Theorem.(a). We assume that µ(λ) and derive a contradiction. There are three cases to consider. Case. Assume that there exists a subsequence of { u λ }, denoted by { u λ } again, such that u λ π.let<δ be fixed. Then by (2.2), for t [t π δ,λ,t]

11 T. Shibata / J. Math. Anal. Appl. 35 (26) u λ (t)2 λ ( + cos u λ (t) ) λ( cos δ). By this and (2.), π δ = T π δ,λ u λ (t) dt 2λ( cos δ)(t t π δ,λ ). This implies that t π δ,λ T as λ. This is a contradiction, since u λ M α and < α<π. Case 2. Assume that there exists a subsequence of { u λ }, denoted by { u λ } again, such that u λ π as λ. Then we see that α + δ< u λ <π for <δ. Then it is clear that t α+δ,λ T as λ. ndeed, if t α+δ,λ T as λ, then 2TF(α)= lim sup Q(u λ ) 2TF(α+ δ). λ This is a contradiction. Therefore, there exists a constant <ɛ such that <t α+δ,λ T ɛ for λ. We choose φ C () satisfying supp φ (T ɛ,t). Since u λ (t) α + δ for t (T ɛ,t)by (2.), we see that for t (T ɛ,t)and λ sin u λ (t) δ >. (3.) u λ (t) Then for λ µ(λ) = inf v H () v v 2 2 λ sin u λ (t) u λ (t) f(u λ (t)) u λ (t) v 2 dt v 2 dt φ 2 2 λδ f(u λ (t)) u λ (t) φ2 dt φ 2 dt <. (3.2) Case 3. Assume that there exists a subsequence of { u λ }, denoted by { u λ } again, such that lim λ u λ <π. Since (3.) holds for any t [,T] in this case, we choose φ C () and obtain (3.2). Thus the proof is complete. For simplicity, we put µ(λ) := µ(λ) >. Then (.) is equivalent to u (t) + µ(λ)f ( u(t) ) = λ sin u(t), t. (3.3) Therefore, in what follows, we consider (3.3) with conditions (.2) and (.3). Lemma 3.. µ(λ) Cλ for λ. Proof. Let θ λ ={t (,π): µ(λ)f (θ) = λ sin θ}. Then by [5], we see that u λ <θ λ <π. Furthermore, since u λ M α, we see that for λ u λ >δ >. By the same calculation as that to obtain (2.2), we have (3.4) (3.5)

12 736 T. Shibata / J. Math. Anal. Appl. 35 (26) u λ (t)2 λ cos u λ (t) µ(λ)f ( u λ (t) ) = λcos u λ µ(λ)f ( ) u λ By (A3), (3.5) and (3.6), we obtain = 2 u λ (T )2 λ. (3.6) µ(λ)f (δ ) µ(λ)f ( ) u λ = 2 u λ (T )2 + λ ( ) cos u λ 2λ. Thus the proof is complete. We put g λ (u) := Then by (3.3), sin u µ(λ)f (u)/λ. (3.7) u u λ (t) = λg( u λ (t) ) u λ (t), t. (3.8) Lemma 3.2. g λ (u λ (t)) locally uniformly on as λ. Proof. We assume that there exists a constant δ>, t [,T)and a subsequence of {λ}, denoted by {λ} again, such that g λ (u λ (t )) δ for λ. Since g λ (u) is decreasing for <u<π by (A3), we see from (2.) that for any t [t,t)and λ ( g λ uλ (t) ) ( g λ uλ (t ) ) δ. (3.9) We choose φ C () with supp φ (t,t). Then by (3.8), we obtain λ = inf v H () v v 2 2 g λ(u λ (t))v 2 dt φ 2 2 δ φ 2. 2 This is a contradiction. Thus the proof is complete. Lemma 3.3. λ C µ(λ) for λ. Proof. We assume that there exists a subsequence of {λ/ µ(λ)}, denoted by {λ/ µ(λ)} again, such that λ/ µ(λ) as λ, and derive a contradiction. Multiply (3.3) by u λ. Then integration by parts along with (3.4), (3.8) and Lemma 3.2 implies that, as λ, 2 2 = λ u λ (t) sin u λ (t) dt µ(λ) f ( u λ (t) ) u λ (t) dt u λ ( = λ g λ uλ (t) ) u λ (t) 2 dt = o(λ). (3.) By (3.4) and the assumption, µ(λ) f ( u λ (t) ) u λ (t) dt = o(λ).

13 T. Shibata / J. Math. Anal. Appl. 35 (26) By this and (3.), as λ, u λ (t) sin u λ (t) dt. (3.) Since u λ M α, by (3.4) and (3.), we see that u λ (t) π, t [,t ), (3.2) u λ (t), t (t,t], (3.3) where t := F(α)T/F(π). Then by (3.4), (3.6), (3.2) and (3.3), for t (t,t] and λ 2 u λ (t)2 = λ ( ) ( ( cos u λ (t) cos u λ + µ(λ) F uλ (t) ) F ( )) u λ = 2 ( + o() ) λ. (3.4) Let <δ be fixed. Then by (2.) and (3.4), for λ π>u λ (t + δ) u λ (t + 2δ) = t +2δ t +δ This is a contradiction. Thus the proof is complete. u λ (t) dt 2 λδ ( + o() ). Proof of Theorem.(b). Let an arbitrary <t 2 <T be fixed. We first prove that u λ (t 2 ) δ for λ. ndeed, if there exists a subsequence of {u λ (t 2 )} λ, denoted by {u λ (t 2 )} λ again, such that u λ (t 2 ) asλ, then by (A2) and Lemmas , as λ, sin u λ (t 2 ) u λ (t 2 ) ( = g λ uλ (t 2 ) ) + µ(λ) f(u λ (t 2 )). (3.5) λ u λ (t 2 ) This is a contradiction, since the left-hand side of (3.5) tends to as λ. This implies that u λ (t) δ for any t t 2 and λ. Now let ξ>beanarbitrary accumulation point of { µ(λ)/λ}. We see that θ λ π as λ. ndeed, if θ λ π as λ, then µ(λ) λ = sin θ λ f(θ λ ). This contradicts Lemma 3.3. Therefore, we see from (3.4), (3.5) and the argument above that for t [,t 2 ] and λ δ u λ (t) π δ. By this and Lemma 3.2, for t t 2,asλ, ( µ(λ) sin ξ = lim λ λ = lim uλ (t) λ f(u λ (t)) g ) λ(u λ (t))u λ (t) sin u λ (t) = lim f(u λ (t)) λ f(u λ (t)). (3.6) Since sin u λ (t)/f (u λ (t)) is increasing for t [,T], and u λ M α, this implies that u λ α locally uniformly as λ and ξ = C. Now our assertion follows from a standard compactness argument. Thus the proof is complete.

14 738 T. Shibata / J. Math. Anal. Appl. 35 (26) Acknowledgment The author thanks the referee for helpful comments that improved the manuscript. Appendix A n this section, we show the existence of (λ, µ(λ), u λ ) R 2 M α, where u λ is the minimizer of the problem (.9). Let λ> and α> be fixed. Since K λ (v) 4Tλfor any v M α, we can choose a minimizing sequence {u n } n= M α such that, as n, K λ (u n ) β(λ,α) 4Tλ. (A.) Since K λ (u n ) = K λ ( u n ) and u n M α by assumption (A), without loss of generality, we may assume that u n forn N. By(A.),forn N, u 2 n 2 2 K λ(u n ) + 4Tλ<C. Therefore, we can choose a subsequence of {u n } n=, denoted by {u n} n= again, such that, as n, u n u λ weakly in H (), (A.2) u n u λ in C( ). (A.3) By (A.3), we see that u λ M α. n particular, u λ in. Furthermore, by (A.2) and (A.3), K λ (u λ ) = u 2 λ 2 2 λ ( cos uλ (t) ) dt lim inf n lim inf n u 2 2 n 2 lim λ ( cos un (t) ) dt n ( u 2 2 n 2 λ ( cos un (t) ) dt This implies that u λ is a minimizer of (.9). Then Q (u λ )u λ = f ( u λ (t) ) u λ (t) dt >, ) = β(λ,α). where the prime denotes the Fréchet derivative of Q. Now we apply the Lagrange multiplier theorem to our situation and obtain (λ, µ(λ), u λ ) R 2 M α, which satisfies (.) and (.3) in a weak sense. Here µ(λ) is the Lagrange multiplier. Then by a standard regularity theorem, we see that u λ C 2 ( ) and it follows from the strong maximum principle that u λ > in. Thus the proof is complete.

15 T. Shibata / J. Math. Anal. Appl. 35 (26) References [] P.A. Binding, P.J. Browne, Asymptotics of eigencurves for second order differential equations,, J. Differential Equations 88 (99) [2] P.A. Binding, Y.X. Huang, The principal eigencurves for the p-laplacian, Differential ntegral Equations 8 (995) [3] P.A. Binding, Y.X. Huang, Bifurcation from eigencurves of the p-laplacian, Differential ntegral Equations 8 (995) [4] M. Faierman, Two-Parameter Eigenvalue Problems in Ordinary Differential Equations, Pitman Res. Notes Math. Ser., vol. 25, Longman, Harlow, 99. [5] D.G. Figueiredo, On the uniqueness of positive solutions of the Dirichlet problem u = λ sin u, in: Pitman Res. Notes Math. Ser., vol. 22, Longman, Harlow, 985, pp [6] R.E. O Malley Jr., Phase-plane solutions to some singular perturbation problems, J. Math. Anal. Appl. 54 (976) [7] R.E. O Malley Jr., Singular Perturbation Methods for Ordinary Differential Equations, Springer, New York, 989. [8] B.P. Rynne, Bifurcation from eigenvalues in nonlinear multiparameter problems, Nonlinear Anal. 5 (99) [9] B.P. Rynne, Uniform convergence of multiparameter eigenfunction expansions, J. Math. Anal. Appl. 47 (99) [] T. Shibata, Asymptotic behavior of eigenvalues of two-parameter nonlinear Sturm Liouville problems, J. Anal. Math. 66 (995) [] T. Shibata, nterior transition layers of solutions to perturbed Sine-Gordon equation on an interval, J. Anal. Math. 83 (2) 9 2. [2] T. Shibata, Multiple interior layers of solutions to perturbed Sine-Gordon equation on an interval, Topol. Methods Nonlinear Anal. 5 (2) [3] T. Shibata, Asymptotic behavior of eigenvalues for two-parameter perturbed elliptic Sine-Gordon type equations, Results Math. 39 (2) [4] T. Shibata, Precise spectral asymptotics for the Dirichlet problem u (t) + g(u(t)) = λ sin u(t), J. Math. Anal. Appl. 267 (22)

Layer structures for the solutions to the perturbed simple pendulum problems

Layer structures for the solutions to the perturbed simple pendulum problems Layer structures for the solutions to the perturbed simple pendulum problems Tetsutaro Shibata Applied Mathematics Research Group, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima,

More information

Two dimensional exterior mixed problem for semilinear damped wave equations

Two dimensional exterior mixed problem for semilinear damped wave equations J. Math. Anal. Appl. 31 (25) 366 377 www.elsevier.com/locate/jmaa Two dimensional exterior mixed problem for semilinear damped wave equations Ryo Ikehata 1 Department of Mathematics, Graduate School of

More information

Existence of Multiple Positive Solutions of Quasilinear Elliptic Problems in R N

Existence of Multiple Positive Solutions of Quasilinear Elliptic Problems in R N Advances in Dynamical Systems and Applications. ISSN 0973-5321 Volume 2 Number 1 (2007), pp. 1 11 c Research India Publications http://www.ripublication.com/adsa.htm Existence of Multiple Positive Solutions

More information

Journal of Mathematical Analysis and Applications

Journal of Mathematical Analysis and Applications J. Math. Anal. Appl. 381 (2011) 506 512 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Inverse indefinite Sturm Liouville problems

More information

Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations

Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations J. Math. Anal. Appl. 32 26) 578 59 www.elsevier.com/locate/jmaa Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations Youming Zhou,

More information

Sturm-Liouville Problem on Unbounded Interval (joint work with Alois Kufner)

Sturm-Liouville Problem on Unbounded Interval (joint work with Alois Kufner) (joint work with Alois Kufner) Pavel Drábek Department of Mathematics, Faculty of Applied Sciences University of West Bohemia, Pilsen Workshop on Differential Equations Hejnice, September 16-20, 2007 Pavel

More information

Nonlinear Analysis 71 (2009) Contents lists available at ScienceDirect. Nonlinear Analysis. journal homepage:

Nonlinear Analysis 71 (2009) Contents lists available at ScienceDirect. Nonlinear Analysis. journal homepage: Nonlinear Analysis 71 2009 2744 2752 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na A nonlinear inequality and applications N.S. Hoang A.G. Ramm

More information

NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION

NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 210, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian

More information

SPECTRAL PROPERTIES AND NODAL SOLUTIONS FOR SECOND-ORDER, m-point, BOUNDARY VALUE PROBLEMS

SPECTRAL PROPERTIES AND NODAL SOLUTIONS FOR SECOND-ORDER, m-point, BOUNDARY VALUE PROBLEMS SPECTRAL PROPERTIES AND NODAL SOLUTIONS FOR SECOND-ORDER, m-point, BOUNDARY VALUE PROBLEMS BRYAN P. RYNNE Abstract. We consider the m-point boundary value problem consisting of the equation u = f(u), on

More information

Asymptotic analysis of the Carrier-Pearson problem

Asymptotic analysis of the Carrier-Pearson problem Fifth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 1,, pp 7 4. http://ejde.math.swt.e or http://ejde.math.unt.e

More information

Existence and Multiplicity of Solutions for a Class of Semilinear Elliptic Equations 1

Existence and Multiplicity of Solutions for a Class of Semilinear Elliptic Equations 1 Journal of Mathematical Analysis and Applications 257, 321 331 (2001) doi:10.1006/jmaa.2000.7347, available online at http://www.idealibrary.com on Existence and Multiplicity of Solutions for a Class of

More information

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply

More information

Global Solutions for a Nonlinear Wave Equation with the p-laplacian Operator

Global Solutions for a Nonlinear Wave Equation with the p-laplacian Operator Global Solutions for a Nonlinear Wave Equation with the p-laplacian Operator Hongjun Gao Institute of Applied Physics and Computational Mathematics 188 Beijing, China To Fu Ma Departamento de Matemática

More information

Scientiae Mathematicae Japonicae Online, Vol. 5, (2001), Ryo Ikehata Λ and Tokio Matsuyama y

Scientiae Mathematicae Japonicae Online, Vol. 5, (2001), Ryo Ikehata Λ and Tokio Matsuyama y Scientiae Mathematicae Japonicae Online, Vol. 5, (2), 7 26 7 L 2 -BEHAVIOUR OF SOLUTIONS TO THE LINEAR HEAT AND WAVE EQUATIONS IN EXTERIOR DOMAINS Ryo Ikehata Λ and Tokio Matsuyama y Received November

More information

A TWO PARAMETERS AMBROSETTI PRODI PROBLEM*

A TWO PARAMETERS AMBROSETTI PRODI PROBLEM* PORTUGALIAE MATHEMATICA Vol. 53 Fasc. 3 1996 A TWO PARAMETERS AMBROSETTI PRODI PROBLEM* C. De Coster** and P. Habets 1 Introduction The study of the Ambrosetti Prodi problem has started with the paper

More information

Existence and multiple solutions for a second-order difference boundary value problem via critical point theory

Existence and multiple solutions for a second-order difference boundary value problem via critical point theory J. Math. Anal. Appl. 36 (7) 511 5 www.elsevier.com/locate/jmaa Existence and multiple solutions for a second-order difference boundary value problem via critical point theory Haihua Liang a,b,, Peixuan

More information

SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction

SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.

More information

EXISTENCE OF SOLUTIONS FOR A RESONANT PROBLEM UNDER LANDESMAN-LAZER CONDITIONS

EXISTENCE OF SOLUTIONS FOR A RESONANT PROBLEM UNDER LANDESMAN-LAZER CONDITIONS Electronic Journal of Differential Equations, Vol. 2008(2008), No. 98, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE

More information

Variational eigenvalues of degenerate eigenvalue problems for the weighted p-laplacian

Variational eigenvalues of degenerate eigenvalue problems for the weighted p-laplacian Variational eigenvalues of degenerate eigenvalue problems for the weighted p-laplacian An Lê Mathematics Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720 e-mail: anle@msri.org Klaus

More information

SEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT

SEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 139, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC

More information

Sensitivity analysis for abstract equilibrium problems

Sensitivity analysis for abstract equilibrium problems J. Math. Anal. Appl. 306 (2005) 684 691 www.elsevier.com/locate/jmaa Sensitivity analysis for abstract equilibrium problems Mohamed Ait Mansour a,, Hassan Riahi b a Laco-123, Avenue Albert Thomas, Facult

More information

arxiv: v1 [math.ap] 16 Jan 2015

arxiv: v1 [math.ap] 16 Jan 2015 Three positive solutions of a nonlinear Dirichlet problem with competing power nonlinearities Vladimir Lubyshev January 19, 2015 arxiv:1501.03870v1 [math.ap] 16 Jan 2015 Abstract This paper studies a nonlinear

More information

Rolle s Theorem for Polynomials of Degree Four in a Hilbert Space 1

Rolle s Theorem for Polynomials of Degree Four in a Hilbert Space 1 Journal of Mathematical Analysis and Applications 265, 322 33 (2002) doi:0.006/jmaa.200.7708, available online at http://www.idealibrary.com on Rolle s Theorem for Polynomials of Degree Four in a Hilbert

More information

Wavelets and regularization of the Cauchy problem for the Laplace equation

Wavelets and regularization of the Cauchy problem for the Laplace equation J. Math. Anal. Appl. 338 008440 1447 www.elsevier.com/locate/jmaa Wavelets and regularization of the Cauchy problem for the Laplace equation Chun-Yu Qiu, Chu-Li Fu School of Mathematics and Statistics,

More information

The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag

The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag J. Math. Anal. Appl. 270 (2002) 143 149 www.academicpress.com The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag Hongjiong Tian Department of Mathematics,

More information

NONLINEAR DIFFERENTIAL INEQUALITY. 1. Introduction. In this paper the following nonlinear differential inequality

NONLINEAR DIFFERENTIAL INEQUALITY. 1. Introduction. In this paper the following nonlinear differential inequality M athematical Inequalities & Applications [2407] First Galley Proofs NONLINEAR DIFFERENTIAL INEQUALITY N. S. HOANG AND A. G. RAMM Abstract. A nonlinear differential inequality is formulated in the paper.

More information

EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY

EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY Electronic Journal of Differential Equations, Vol. 216 216), No. 329, pp. 1 22. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN

More information

Existence of at least two periodic solutions of the forced relativistic pendulum

Existence of at least two periodic solutions of the forced relativistic pendulum Existence of at least two periodic solutions of the forced relativistic pendulum Cristian Bereanu Institute of Mathematics Simion Stoilow, Romanian Academy 21, Calea Griviţei, RO-172-Bucharest, Sector

More information

Multiple positive solutions for a class of quasilinear elliptic boundary-value problems

Multiple positive solutions for a class of quasilinear elliptic boundary-value problems Electronic Journal of Differential Equations, Vol. 20032003), No. 07, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp) Multiple positive

More information

p-laplacian problems with critical Sobolev exponents

p-laplacian problems with critical Sobolev exponents Nonlinear Analysis 66 (2007) 454 459 www.elsevier.com/locate/na p-laplacian problems with critical Sobolev exponents Kanishka Perera a,, Elves A.B. Silva b a Department of Mathematical Sciences, Florida

More information

The Existence of Nodal Solutions for the Half-Quasilinear p-laplacian Problems

The Existence of Nodal Solutions for the Half-Quasilinear p-laplacian Problems Journal of Mathematical Research with Applications Mar., 017, Vol. 37, No., pp. 4 5 DOI:13770/j.issn:095-651.017.0.013 Http://jmre.dlut.edu.cn The Existence of Nodal Solutions for the Half-Quasilinear

More information

1 Math 241A-B Homework Problem List for F2015 and W2016

1 Math 241A-B Homework Problem List for F2015 and W2016 1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let

More information

On Positive Solutions of Boundary Value Problems on the Half-Line

On Positive Solutions of Boundary Value Problems on the Half-Line Journal of Mathematical Analysis and Applications 259, 127 136 (21) doi:1.16/jmaa.2.7399, available online at http://www.idealibrary.com on On Positive Solutions of Boundary Value Problems on the Half-Line

More information

ATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS. Emerson A. M. de Abreu Alexandre N.

ATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS. Emerson A. M. de Abreu Alexandre N. ATTRACTORS FOR SEMILINEAR PARABOLIC PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS IN VARYING DOMAINS Emerson A. M. de Abreu Alexandre N. Carvalho Abstract Under fairly general conditions one can prove that

More information

Note on the Chen-Lin Result with the Li-Zhang Method

Note on the Chen-Lin Result with the Li-Zhang Method J. Math. Sci. Univ. Tokyo 18 (2011), 429 439. Note on the Chen-Lin Result with the Li-Zhang Method By Samy Skander Bahoura Abstract. We give a new proof of the Chen-Lin result with the method of moving

More information

Research Article Almost Periodic Solutions of Prey-Predator Discrete Models with Delay

Research Article Almost Periodic Solutions of Prey-Predator Discrete Models with Delay Hindawi Publishing Corporation Advances in Difference Equations Volume 009, Article ID 976865, 19 pages doi:10.1155/009/976865 Research Article Almost Periodic Solutions of Prey-Predator Discrete Models

More information

Oscillation Criteria for Certain nth Order Differential Equations with Deviating Arguments

Oscillation Criteria for Certain nth Order Differential Equations with Deviating Arguments Journal of Mathematical Analysis Applications 6, 601 6 001) doi:10.1006/jmaa.001.7571, available online at http://www.idealibrary.com on Oscillation Criteria for Certain nth Order Differential Equations

More information

New exact multiplicity results with an application to a population model

New exact multiplicity results with an application to a population model New exact multiplicity results with an application to a population model Philip Korman Department of Mathematical Sciences University of Cincinnati Cincinnati, Ohio 45221-0025 Junping Shi Department of

More information

ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM

ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM ON WEAKLY NONLINEAR BACKWARD PARABOLIC PROBLEM OLEG ZUBELEVICH DEPARTMENT OF MATHEMATICS THE BUDGET AND TREASURY ACADEMY OF THE MINISTRY OF FINANCE OF THE RUSSIAN FEDERATION 7, ZLATOUSTINSKY MALIY PER.,

More information

SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction

SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 12, 1998, 47 59 SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS M. Grossi S. Kesavan F. Pacella M. Ramaswamy

More information

ON PARABOLIC HARNACK INEQUALITY

ON PARABOLIC HARNACK INEQUALITY ON PARABOLIC HARNACK INEQUALITY JIAXIN HU Abstract. We show that the parabolic Harnack inequality is equivalent to the near-diagonal lower bound of the Dirichlet heat kernel on any ball in a metric measure-energy

More information

Exact multiplicity of boundary blow-up solutions for a bistable problem

Exact multiplicity of boundary blow-up solutions for a bistable problem Computers and Mathematics with Applications 54 (2007) 1285 1292 www.elsevier.com/locate/camwa Exact multiplicity of boundary blow-up solutions for a bistable problem Junping Shi a,b,, Shin-Hwa Wang c a

More information

EIGENVALUE QUESTIONS ON SOME QUASILINEAR ELLIPTIC PROBLEMS. lim u(x) = 0, (1.2)

EIGENVALUE QUESTIONS ON SOME QUASILINEAR ELLIPTIC PROBLEMS. lim u(x) = 0, (1.2) Proceedings of Equadiff-11 2005, pp. 455 458 Home Page Page 1 of 7 EIGENVALUE QUESTIONS ON SOME QUASILINEAR ELLIPTIC PROBLEMS M. N. POULOU AND N. M. STAVRAKAKIS Abstract. We present resent results on some

More information

A Computational Approach to Study a Logistic Equation

A Computational Approach to Study a Logistic Equation Communications in MathematicalAnalysis Volume 1, Number 2, pp. 75 84, 2006 ISSN 0973-3841 2006 Research India Publications A Computational Approach to Study a Logistic Equation G. A. Afrouzi and S. Khademloo

More information

Positive Periodic Solutions of Systems of Second Order Ordinary Differential Equations

Positive Periodic Solutions of Systems of Second Order Ordinary Differential Equations Positivity 1 (26), 285 298 26 Birkhäuser Verlag Basel/Switzerland 1385-1292/2285-14, published online April 26, 26 DOI 1.17/s11117-5-21-2 Positivity Positive Periodic Solutions of Systems of Second Order

More information

A global solution curve for a class of free boundary value problems arising in plasma physics

A global solution curve for a class of free boundary value problems arising in plasma physics A global solution curve for a class of free boundary value problems arising in plasma physics Philip Korman epartment of Mathematical Sciences University of Cincinnati Cincinnati Ohio 4522-0025 Abstract

More information

Received 8 June 2003 Submitted by Z.-J. Ruan

Received 8 June 2003 Submitted by Z.-J. Ruan J. Math. Anal. Appl. 289 2004) 266 278 www.elsevier.com/locate/jmaa The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense

More information

Existence and uniqueness of solutions for a diffusion model of host parasite dynamics

Existence and uniqueness of solutions for a diffusion model of host parasite dynamics J. Math. Anal. Appl. 279 (23) 463 474 www.elsevier.com/locate/jmaa Existence and uniqueness of solutions for a diffusion model of host parasite dynamics Michel Langlais a and Fabio Augusto Milner b,,1

More information

EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM

EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM JENICĂ CRÎNGANU We derive existence results for operator equations having the form J ϕu = N f u, by using

More information

SYMMETRY AND REGULARITY OF AN OPTIMIZATION PROBLEM RELATED TO A NONLINEAR BVP

SYMMETRY AND REGULARITY OF AN OPTIMIZATION PROBLEM RELATED TO A NONLINEAR BVP lectronic ournal of Differential quations, Vol. 2013 2013, No. 108, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SYMMTRY AND RGULARITY

More information

Stability of a Class of Singular Difference Equations

Stability of a Class of Singular Difference Equations International Journal of Difference Equations. ISSN 0973-6069 Volume 1 Number 2 2006), pp. 181 193 c Research India Publications http://www.ripublication.com/ijde.htm Stability of a Class of Singular Difference

More information

POTENTIAL LANDESMAN-LAZER TYPE CONDITIONS AND. 1. Introduction We investigate the existence of solutions for the nonlinear boundary-value problem

POTENTIAL LANDESMAN-LAZER TYPE CONDITIONS AND. 1. Introduction We investigate the existence of solutions for the nonlinear boundary-value problem Electronic Journal of Differential Equations, Vol. 25(25), No. 94, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) POTENTIAL

More information

EXISTENCE OF NONTRIVIAL SOLUTIONS FOR A QUASILINEAR SCHRÖDINGER EQUATIONS WITH SIGN-CHANGING POTENTIAL

EXISTENCE OF NONTRIVIAL SOLUTIONS FOR A QUASILINEAR SCHRÖDINGER EQUATIONS WITH SIGN-CHANGING POTENTIAL Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 05, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF NONTRIVIAL

More information

Quasireversibility Methods for Non-Well-Posed Problems

Quasireversibility Methods for Non-Well-Posed Problems Kennesaw State University DigitalCommons@Kennesaw State University Faculty Publications 11-29-1994 Quasireversibility Methods for Non-Well-Posed Problems Gordon W. Clark Kennesaw State University Seth

More information

The Navier-Stokes Equations with Time Delay. Werner Varnhorn. Faculty of Mathematics University of Kassel, Germany

The Navier-Stokes Equations with Time Delay. Werner Varnhorn. Faculty of Mathematics University of Kassel, Germany The Navier-Stokes Equations with Time Delay Werner Varnhorn Faculty of Mathematics University of Kassel, Germany AMS: 35 (A 35, D 5, K 55, Q 1), 65 M 1, 76 D 5 Abstract In the present paper we use a time

More information

GORDON TYPE THEOREM FOR MEASURE PERTURBATION

GORDON TYPE THEOREM FOR MEASURE PERTURBATION Electronic Journal of Differential Equations, Vol. 211 211, No. 111, pp. 1 9. ISSN: 172-6691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu GODON TYPE THEOEM FO

More information

Nonlinear Analysis. Global solution curves for boundary value problems, with linear part at resonance

Nonlinear Analysis. Global solution curves for boundary value problems, with linear part at resonance Nonlinear Analysis 71 (29) 2456 2467 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Global solution curves for boundary value problems, with linear

More information

Blow-up solutions for critical Trudinger-Moser equations in R 2

Blow-up solutions for critical Trudinger-Moser equations in R 2 Blow-up solutions for critical Trudinger-Moser equations in R 2 Bernhard Ruf Università degli Studi di Milano The classical Sobolev embeddings We have the following well-known Sobolev inequalities: let

More information

The Hausdorff measure of a class of Sierpinski carpets

The Hausdorff measure of a class of Sierpinski carpets J. Math. Anal. Appl. 305 (005) 11 19 www.elsevier.com/locate/jmaa The Hausdorff measure of a class of Sierpinski carpets Yahan Xiong, Ji Zhou Department of Mathematics, Sichuan Normal University, Chengdu

More information

University of Bristol - Explore Bristol Research. Publisher's PDF, also known as Version of record

University of Bristol - Explore Bristol Research. Publisher's PDF, also known as Version of record Netrusov, Y., & Safarov, Y. (2010). Estimates for the counting function of the laplace operator on domains with rough boundaries. In A. Laptev (Ed.), Around the Research of Vladimir Maz'ya, III: Analysis

More information

Blow up points of solution curves for a semilinear problem

Blow up points of solution curves for a semilinear problem Blow up points of solution curves for a semilinear problem Junping Shi Department of Mathematics, Tulane University New Orleans, LA 70118 Email: shij@math.tulane.edu 1 Introduction Consider a semilinear

More information

MULTIPLE POSITIVE SOLUTIONS FOR P-LAPLACIAN EQUATION WITH WEAK ALLEE EFFECT GROWTH RATE

MULTIPLE POSITIVE SOLUTIONS FOR P-LAPLACIAN EQUATION WITH WEAK ALLEE EFFECT GROWTH RATE Differential and Integral Equations Volume xx, Number xxx,, Pages xx xx MULTIPLE POSITIVE SOLUTIONS FOR P-LAPLACIAN EQUATION WITH WEAK ALLEE EFFECT GROWTH RATE Chan-Gyun Kim 1 and Junping Shi 2 Department

More information

NONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS. Shouchuan Hu Nikolas S. Papageorgiou. 1. Introduction

NONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS. Shouchuan Hu Nikolas S. Papageorgiou. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 34, 29, 327 338 NONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS Shouchuan Hu Nikolas S. Papageorgiou

More information

On the simplest expression of the perturbed Moore Penrose metric generalized inverse

On the simplest expression of the perturbed Moore Penrose metric generalized inverse Annals of the University of Bucharest (mathematical series) 4 (LXII) (2013), 433 446 On the simplest expression of the perturbed Moore Penrose metric generalized inverse Jianbing Cao and Yifeng Xue Communicated

More information

Positive Solutions of a Third-Order Three-Point Boundary-Value Problem

Positive Solutions of a Third-Order Three-Point Boundary-Value Problem Kennesaw State University DigitalCommons@Kennesaw State University Faculty Publications 7-28 Positive Solutions of a Third-Order Three-Point Boundary-Value Problem Bo Yang Kennesaw State University, byang@kennesaw.edu

More information

Sufficient conditions for functions to form Riesz bases in L 2 and applications to nonlinear boundary-value problems

Sufficient conditions for functions to form Riesz bases in L 2 and applications to nonlinear boundary-value problems Electronic Journal of Differential Equations, Vol. 200(200), No. 74, pp. 0. ISSN: 072-669. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Sufficient conditions

More information

Research Article Existence and Uniqueness of Homoclinic Solution for a Class of Nonlinear Second-Order Differential Equations

Research Article Existence and Uniqueness of Homoclinic Solution for a Class of Nonlinear Second-Order Differential Equations Applied Mathematics Volume 2012, Article ID 615303, 13 pages doi:10.1155/2012/615303 Research Article Existence and Uniqueness of Homoclinic Solution for a Class of Nonlinear Second-Order Differential

More information

PERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS

PERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 207 222 PERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS Fumi-Yuki Maeda and Takayori Ono Hiroshima Institute of Technology, Miyake,

More information

Internal Stabilizability of Some Diffusive Models

Internal Stabilizability of Some Diffusive Models Journal of Mathematical Analysis and Applications 265, 91 12 (22) doi:1.16/jmaa.21.7694, available online at http://www.idealibrary.com on Internal Stabilizability of Some Diffusive Models Bedr Eddine

More information

Existence of homoclinic solutions for Duffing type differential equation with deviating argument

Existence of homoclinic solutions for Duffing type differential equation with deviating argument 2014 9 «28 «3 Sept. 2014 Communication on Applied Mathematics and Computation Vol.28 No.3 DOI 10.3969/j.issn.1006-6330.2014.03.007 Existence of homoclinic solutions for Duffing type differential equation

More information

Persistence and global stability in discrete models of Lotka Volterra type

Persistence and global stability in discrete models of Lotka Volterra type J. Math. Anal. Appl. 330 2007 24 33 www.elsevier.com/locate/jmaa Persistence global stability in discrete models of Lotka Volterra type Yoshiaki Muroya 1 Department of Mathematical Sciences, Waseda University,

More information

LIFE SPAN OF BLOW-UP SOLUTIONS FOR HIGHER-ORDER SEMILINEAR PARABOLIC EQUATIONS

LIFE SPAN OF BLOW-UP SOLUTIONS FOR HIGHER-ORDER SEMILINEAR PARABOLIC EQUATIONS Electronic Journal of Differential Equations, Vol. 21(21), No. 17, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu LIFE SPAN OF BLOW-UP

More information

Upper and lower solutions method and a fractional differential equation boundary value problem.

Upper and lower solutions method and a fractional differential equation boundary value problem. Electronic Journal of Qualitative Theory of Differential Equations 9, No. 3, -3; http://www.math.u-szeged.hu/ejqtde/ Upper and lower solutions method and a fractional differential equation boundary value

More information

Spectrum of one dimensional p-laplacian Operator with indefinite weight

Spectrum of one dimensional p-laplacian Operator with indefinite weight Spectrum of one dimensional p-laplacian Operator with indefinite weight A. Anane, O. Chakrone and M. Moussa 2 Département de mathématiques, Faculté des Sciences, Université Mohamed I er, Oujda. Maroc.

More information

Upper and lower solution method for fourth-order four-point boundary value problems

Upper and lower solution method for fourth-order four-point boundary value problems Journal of Computational and Applied Mathematics 196 (26) 387 393 www.elsevier.com/locate/cam Upper and lower solution method for fourth-order four-point boundary value problems Qin Zhang a, Shihua Chen

More information

Blow-up of solutions for the sixth-order thin film equation with positive initial energy

Blow-up of solutions for the sixth-order thin film equation with positive initial energy PRAMANA c Indian Academy of Sciences Vol. 85, No. 4 journal of October 05 physics pp. 577 58 Blow-up of solutions for the sixth-order thin film equation with positive initial energy WENJUN LIU and KEWANG

More information

arxiv:math/ v1 [math.fa] 20 Feb 2001

arxiv:math/ v1 [math.fa] 20 Feb 2001 arxiv:math/12161v1 [math.fa] 2 Feb 21 Critical sets of nonlinear Sturm-Liouville operators of Ambrosetti-Prodi type H. Bueno and C. Tomei Dep. de Matemática - ICEx, UFMG, Caixa Postal 72, Belo Horizonte,

More information

Existence of global solutions of some ordinary differential equations

Existence of global solutions of some ordinary differential equations J. Math. Anal. Appl. 340 (2008) 739 745 www.elsevier.com/locate/jmaa Existence of global solutions of some ordinary differential equations U. Elias Department of Mathematics, Technion IIT, Haifa 32000,

More information

ON SINGULAR PERTURBATION OF THE STOKES PROBLEM

ON SINGULAR PERTURBATION OF THE STOKES PROBLEM NUMERICAL ANALYSIS AND MATHEMATICAL MODELLING BANACH CENTER PUBLICATIONS, VOLUME 9 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 994 ON SINGULAR PERTURBATION OF THE STOKES PROBLEM G. M.

More information

Eigenvalues and eigenfunctions of the Laplacian. Andrew Hassell

Eigenvalues and eigenfunctions of the Laplacian. Andrew Hassell Eigenvalues and eigenfunctions of the Laplacian Andrew Hassell 1 2 The setting In this talk I will consider the Laplace operator,, on various geometric spaces M. Here, M will be either a bounded Euclidean

More information

Viscosity approximation methods for nonexpansive nonself-mappings

Viscosity approximation methods for nonexpansive nonself-mappings J. Math. Anal. Appl. 321 (2006) 316 326 www.elsevier.com/locate/jmaa Viscosity approximation methods for nonexpansive nonself-mappings Yisheng Song, Rudong Chen Department of Mathematics, Tianjin Polytechnic

More information

How large is the class of operator equations solvable by a DSM Newton-type method?

How large is the class of operator equations solvable by a DSM Newton-type method? This is the author s final, peer-reviewed manuscript as accepted for publication. The publisher-formatted version may be available through the publisher s web site or your institution s library. How large

More information

On the validity of the Euler Lagrange equation

On the validity of the Euler Lagrange equation J. Math. Anal. Appl. 304 (2005) 356 369 www.elsevier.com/locate/jmaa On the validity of the Euler Lagrange equation A. Ferriero, E.M. Marchini Dipartimento di Matematica e Applicazioni, Università degli

More information

A Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators

A Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators A Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators Lei Ni And I cherish more than anything else the Analogies, my most trustworthy masters. They know all the secrets of

More information

This article was published in an Elsevier journal. The attached copy is furnished to the author for non-commercial research and education use, including for instruction at the author s institution, sharing

More information

Positive solutions for a class of fractional boundary value problems

Positive solutions for a class of fractional boundary value problems Nonlinear Analysis: Modelling and Control, Vol. 21, No. 1, 1 17 ISSN 1392-5113 http://dx.doi.org/1.15388/na.216.1.1 Positive solutions for a class of fractional boundary value problems Jiafa Xu a, Zhongli

More information

Positive eigenfunctions for the p-laplace operator revisited

Positive eigenfunctions for the p-laplace operator revisited Positive eigenfunctions for the p-laplace operator revisited B. Kawohl & P. Lindqvist Sept. 2006 Abstract: We give a short proof that positive eigenfunctions for the p-laplacian are necessarily associated

More information

HOMOCLINIC SOLUTIONS FOR SECOND-ORDER NON-AUTONOMOUS HAMILTONIAN SYSTEMS WITHOUT GLOBAL AMBROSETTI-RABINOWITZ CONDITIONS

HOMOCLINIC SOLUTIONS FOR SECOND-ORDER NON-AUTONOMOUS HAMILTONIAN SYSTEMS WITHOUT GLOBAL AMBROSETTI-RABINOWITZ CONDITIONS Electronic Journal of Differential Equations, Vol. 010010, No. 9, pp. 1 10. ISSN: 107-6691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu HOMOCLINIC SOLUTIONS FO

More information

The stability of travelling fronts for general scalar viscous balance law

The stability of travelling fronts for general scalar viscous balance law J. Math. Anal. Appl. 35 25) 698 711 www.elsevier.com/locate/jmaa The stability of travelling fronts for general scalar viscous balance law Yaping Wu, Xiuxia Xing Department of Mathematics, Capital Normal

More information

BIHARMONIC WAVE MAPS INTO SPHERES

BIHARMONIC WAVE MAPS INTO SPHERES BIHARMONIC WAVE MAPS INTO SPHERES SEBASTIAN HERR, TOBIAS LAMM, AND ROLAND SCHNAUBELT Abstract. A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed.

More information

PERIODIC SOLUTIONS OF THE FORCED PENDULUM : CLASSICAL VS RELATIVISTIC

PERIODIC SOLUTIONS OF THE FORCED PENDULUM : CLASSICAL VS RELATIVISTIC LE MATEMATICHE Vol. LXV 21) Fasc. II, pp. 97 17 doi: 1.4418/21.65.2.11 PERIODIC SOLUTIONS OF THE FORCED PENDULUM : CLASSICAL VS RELATIVISTIC JEAN MAWHIN The paper surveys and compares some results on the

More information

HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM

HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM Georgian Mathematical Journal Volume 9 (2002), Number 3, 591 600 NONEXPANSIVE MAPPINGS AND ITERATIVE METHODS IN UNIFORMLY CONVEX BANACH SPACES HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM

More information

SIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP AND UNIFORM BLOW-UP PROFILES FOR REACTION-DIFFUSION SYSTEM

SIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP AND UNIFORM BLOW-UP PROFILES FOR REACTION-DIFFUSION SYSTEM Electronic Journal of Differential Euations, Vol. 22 (22), No. 26, pp. 9. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SIMULTANEOUS AND NON-SIMULTANEOUS

More information

INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES. Note on the fast decay property of steady Navier-Stokes flows in the whole space

INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES. Note on the fast decay property of steady Navier-Stokes flows in the whole space INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES Note on the fast decay property of stea Navier-Stokes flows in the whole space Tomoyuki Nakatsuka Preprint No. 15-017 PRAHA 017 Note on the fast

More information

Junhong Ha and Shin-ichi Nakagiri

Junhong Ha and Shin-ichi Nakagiri J. Korean Math. Soc. 39 (22), No. 4, pp. 59 524 IDENTIFICATION PROBLEMS OF DAMPED SINE-GORDON EQUATIONS WITH CONSTANT PARAMETERS Junhong Ha and Shin-ichi Nakagiri Abstract. We study the problems of identification

More information

Existence and Multiplicity of Positive Solutions to a Quasilinear Elliptic Equation with Strong Allee Effect Growth Rate

Existence and Multiplicity of Positive Solutions to a Quasilinear Elliptic Equation with Strong Allee Effect Growth Rate Results. Math. 64 (213), 165 173 c 213 Springer Basel 1422-6383/13/1165-9 published online February 14, 213 DOI 1.17/s25-13-36-x Results in Mathematics Existence and Multiplicity of Positive Solutions

More information

Author(s) Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 41(1)

Author(s) Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 41(1) Title On the stability of contact Navier-Stokes equations with discont free b Authors Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 4 Issue 4-3 Date Text Version publisher URL

More information

Minimization problems on the Hardy-Sobolev inequality

Minimization problems on the Hardy-Sobolev inequality manuscript No. (will be inserted by the editor) Minimization problems on the Hardy-Sobolev inequality Masato Hashizume Received: date / Accepted: date Abstract We study minimization problems on Hardy-Sobolev

More information

An Inverse Problem for the Matrix Schrödinger Equation

An Inverse Problem for the Matrix Schrödinger Equation Journal of Mathematical Analysis and Applications 267, 564 575 (22) doi:1.16/jmaa.21.7792, available online at http://www.idealibrary.com on An Inverse Problem for the Matrix Schrödinger Equation Robert

More information

Lecture Notes on PDEs

Lecture Notes on PDEs Lecture Notes on PDEs Alberto Bressan February 26, 2012 1 Elliptic equations Let IR n be a bounded open set Given measurable functions a ij, b i, c : IR, consider the linear, second order differential

More information