A topological approach to periodic oscillations related to the Liebau phenomenon
José Ángel Cid, Gennaro Infante, Milan Tvrdý, Mirosława Zima
AA TOPOLOGICAL APPROACH TO PERIODIC OSCILLATIONSRELATED TO THE LIEBAU PHENOMENON
J. A. CID, G. INFANTE, M. TVRD ´Y, AND M. ZIMA
Abstract.
We give some sufficient conditions for existence, non-existence and localizationof positive solutions for a periodic boundary value problem related to the Liebau phenome-non. Our approach is of topological nature and relies on the Krasnosel’ski˘ı-Guo theorem oncone expansion and compression. Our results improve and complement earlier ones in theliterature. Introduction
In the 1950’s the physician G. Liebau developed some experiments dealing with a valvelesspumping phenomenon arising on blood circulation and that has been known for a longtime: roughly speaking, Liebau showed experimentally that a periodic compression made onan asymmetric part of a fluid-mechanical model could produce the circulation of the fluidwithout the necessity of a valve to ensure a preferential direction of the flow [1, 12, 13]. Afterhis pioneering work this effect has been known as the Liebau phenomenon.In [14] G. Propst, with the aim of contributing to the theoretical understanding of theLiebau phenomenon, presented some differential equations modeling a periodically forcedflow through different pipe-tank configurations. He was able to prove the presence of pump-ing effects in some of them, but the apparently simplest model, the “one pipe-one tank”configuration, skipped his efforts due to a singularity in the corresponding differential equa-tion model, namely(1) u (cid:48)(cid:48) ( t ) + a u (cid:48) ( t ) = 1 u (cid:0) e ( t ) − b ( u (cid:48) ( t )) (cid:1) − c, t ∈ [0 , T ] ,u (0) = u ( T ) , u (cid:48) (0) = u (cid:48) ( T ) , being a ≥ , b > , c > e ( t ) continuous and T -periodic on R . The singular periodic problem (1) was studied in [4], where the authors gave general resultsfor the existence and asymptotic stability of positive solutions by performing the change ofvariables u = x µ , where µ = b + 1 , which transforms the singular problem (1) into the regular Mathematics Subject Classification.
Primary 34B18, secondary 34B27, 34B60.
Key words and phrases.
Valveless pumping; Periodic boundary value problem; Krasnosel’ski˘ı-Guo fixedpoint theorem; Green’s function. a r X i v : . [ m a t h . C A ] A ug ne(2) x (cid:48)(cid:48) ( t ) + a x (cid:48) ( t ) = e ( t ) µ x − µ ( t ) − cµ x − µ ( t ) , t ∈ [0 , T ] ,x (0) = x ( T ) , x (cid:48) (0) = x (cid:48) ( T ) . Then the existence and stability of positive solutions for (2) were obtained by means of thelower and upper solution technique [5] and using tricks analogous to those used in [15].In this paper we deal with the existence of positive solutions for the following generalizationof the problem (2)(3) (cid:40) x (cid:48)(cid:48) ( t ) + ax (cid:48) ( t ) = r ( t ) x α ( t ) − s ( t ) x β ( t ) , t ∈ [0 , T ] ,x (0) = x ( T ) , x (cid:48) (0) = x (cid:48) ( T ) , where we assume(H0) a ≥ r, s ∈ C [0 , T ], 0 < α < β < . Note that, by defining r ( t ) = e ( t ) µ , s ( t ) = cµ , α = 1 − µ and β = 1 − µ , the problem (2) fitswithin (3).Our approach is essentially of topological nature: in Section 2 we rewrite problem (3) asan equivalent fixed point problem suitable to be treated by means of the Krasnosel’ski˘ı-Guocone expansion/compression fixed point theorem. A careful analysis of the related Green’sfunction, necessary in our approach, is postponed to a final Appendix. In Section 3 we presentour main results: existence, non-existence and localization criteria for positive solutions ofthe problem (3). Some corollaries with more ready-to-use results are also addressed. Wepoint out that our results are valid not only for the more general problem (3), but also whenapplied to the singular model problem (1) we improve previous results of [4].2. A fixed point formulation
First of all, by means of a shifting argument, we rewrite the problem (3) in the equivalentform(4) (cid:40) x (cid:48)(cid:48) ( t ) + ax (cid:48) ( t ) + m x ( t ) = r ( t ) x α ( t ) − s ( t ) x β ( t ) + m x ( t ) := f m ( t, x ( t )) , t ∈ [0 , T ] ,x (0) = x ( T ) , x (cid:48) (0) = x (cid:48) ( T ) , with m ∈ R . A similar approach has been used, under a variety of boundary conditions in[7, 9, 16, 17].We say that problem (4) is non-resonant if zero is the unique solution of the homogeneouslinear problem (cid:40) x (cid:48)(cid:48) ( t ) + ax (cid:48) ( t ) + m x ( t ) = 0 , t ∈ [0 , T ] ,x (0) = x ( T ) , x (cid:48) (0) = x (cid:48) ( T ) . n this case the non-homogeneous linear problem(5) (cid:40) x (cid:48)(cid:48) ( t ) + ax (cid:48) ( t ) + m x ( t ) = h ( t ) , t ∈ [0 , T ] ,x (0) = x ( T ) , x (cid:48) (0) = x (cid:48) ( T ) , is also uniquely solvable and its unique solution is given by K h ( t ) = (cid:90) T G m ( t, s ) h ( s ) ds, where G m ( t, s ) is the related Green’s function (see [2, 3]).In particular, if m > m < (cid:16) πT (cid:17) + (cid:16) a (cid:17) then the problem (4) is non-resonant andmoreover G m ( t, s ) satisfies the following properties (see the Appendix for the details):i) G m ( t, s ) > , for all ( t, s ) ∈ [0 , T ] × [0 , T ].ii) (cid:90) T G m ( t, s ) ds = 1 m . iii) There exists a constant c m ∈ (0 ,
1) such that G m ( t, s ) ≥ G m ( s, s ) ≥ c m G m ( t, s ) , for all ( t, s ) ∈ [0 , T ] × [0 , T ] . A cone in a Banach space X is a closed, convex subset of X such that λ x ∈ K for x ∈ K and λ ≥ K ∩ ( − K ) = { } . Here we work in the space X = C [0 , T ] endowed with theusual maximum norm (cid:107) x (cid:107) = max {| x ( t ) | : t ∈ [0 , T ] } and use the cone P = { x ∈ X : x ( t ) ≥ c m (cid:107) x (cid:107) on [0 , T ] } , a type of cone firstly used by Krasnosel’ski˘ı, see for example [10], and D. Guo, see for example[6]. The cone P is particularly useful when dealing with singular nonlinearities (see [11]), orfor localizing the solutions (see for example [8]).Let us define the operator F : P → X (6) F x ( t ) = (cid:90) T G m ( t, s ) f m ( s, x ( s )) ds. Note that a fixed point of F in P is a non-negative solution of the problem (4). In or-der to get such a fixed point we employ the following well-known Krasnosel’ski˘ı-Guo conecompression/expansion theorem. Theorem 2.1. [6]
Let P be a cone in X and suppose that Ω and Ω are bounded open setsin X such that ∈ Ω and Ω ⊂ Ω . Let F : P ∩ (Ω \ Ω ) → P be a completely continuousoperator such that one of the following conditions holds: (i) (cid:107) F x (cid:107) ≥ (cid:107) x (cid:107) for x ∈ P ∩ ∂ Ω and (cid:107) F x (cid:107) ≤ (cid:107) x (cid:107) for x ∈ P ∩ ∂ Ω , (ii) (cid:107) F x (cid:107) ≤ (cid:107) x (cid:107) for x ∈ P ∩ ∂ Ω and (cid:107) F x (cid:107) ≥ (cid:107) x (cid:107) for x ∈ P ∩ ∂ Ω . Then F has a fixed point in the set P ∩ (Ω \ Ω ) . n the sequel, for a given continuous function h : [0 , T ] → R we use the notation h ∗ = min { h ( t ) : t ∈ [0 , T ] } and h ∗ = max { h ( t ) : t ∈ [0 , T ] } . Main results
Integrating on [0 , T ] the differential equation in the problem (3), and taking into accountthe boundary conditions, we get a necessary condition for the existence of positive solutions,namely 0 = (cid:90) T [ r ( t ) x α ( t ) − s ( t ) x β ( t )] dt. So it is easy to arrive to the following non-existence result.
Theorem 3.1.
If one of the following conditions holds, (1) r ∗ ≥ and s ∗ < , (2) r ∗ > and s ∗ ≤ , (3) r ∗ ≤ and s ∗ > , (4) r ∗ < and s ∗ ≥ ,then problem (3) does not have positive solutions. Our main result, concerning not only the existence but also the localization of positivesolutions, is the following one.
Theorem 3.2.
Assume that (H0) and the following condition hold: (H1)
There exist m > and < R < R such that m < (cid:16) πT (cid:17) + (cid:16) a (cid:17) , (7) f m ( t, x ) = r ( t ) x α − s ( t ) x β + m x ≥ for t ∈ [0 , T ] and x ∈ [ c m R , R ] , (8) f m ( t, x ) ≥ m R for t ∈ [0 , T ] and x ∈ [ c m R , R ] and (9) f m ( t, x ) ≤ m R for t ∈ [0 , T ] and x ∈ [ c m R , R ] . Then, the problem (3) has a positive solution x ( t ) such that c m R ≤ x ( t ) ≤ R . Proof.
The problem (4) with the value of m given by (H1) is non-resonant and its Green’sfunction G m ( t, s ) satisfies the properties i), ii) and iii) stated in the Introduction. Let usdefine Ω j = { x ∈ X : (cid:107) x (cid:107) < R j } j = 1 , . hen by (7) we have f m ( t, x ( t )) ≥ x ∈ P ∩ (Ω \ Ω ). Now, for every x ∈ P ∩ (Ω \ Ω )and for every t ∈ [0 , T ], we have that (cid:90) T G m ( s, s ) f m ( s, x ( s )) ds ≤ F x ( t ) = (cid:90) T G m ( t, s ) f m ( s, x ( s )) ds ≤ (cid:90) T G m ( s, s ) c m f m ( s, x ( s )) ds, and therefore F ( P ∩ (Ω \ Ω )) ⊂ P . A standard argument shows that F is a completelycontinuous operator.Now, we check that condition (i) in Theorem 2.1 is fulfilled. Claim 1. (cid:107)
F x (cid:107) ≥ (cid:107) x (cid:107) for x ∈ P ∩ ∂ Ω . Let x ∈ P ∩ ∂ Ω , that is, x ∈ P and (cid:107) x (cid:107) = R . Then we have that c m R ≤ x ( t ) ≤ R forall t ∈ [0 , T ] and hence F x ( t ) = (cid:90) T G m ( t, s ) f m ( s, x ( s )) ds ≥ (cid:90) T G m ( t, s ) m R ds = m R (cid:90) T G m ( t, s ) ds = m R m = R = (cid:107) x (cid:107) . Claim 2. (cid:107)
F x (cid:107) ≤ (cid:107) x (cid:107) for x ∈ P ∩ ∂ Ω . Let x ∈ P ∩ ∂ Ω , that is, x ∈ P and (cid:107) x (cid:107) = R . Then c m R ≤ x ( t ) ≤ R for all t ∈ [0 , T ]and hence F x ( t ) = (cid:90) T G m ( t, s ) f m ( s, x ( s )) ds ≤ (cid:90) T G m ( t, s ) m R ds = m R (cid:90) T G m ( t, s ) ds = m R m = R = (cid:107) x (cid:107) . Thefore by Theorem 2.1 the existence of a solution of the problem (3) with the desiredlocalization property immediately follows. (cid:3)
The following result is a consequence of Theorem 3.2.
Theorem 3.3.
Assume that (H0) and the following conditions hold: (H2)
There exists m > such that m < (cid:16) πT (cid:17) + (cid:16) a (cid:17) and f m ( t, x ) = r ( t ) x α − s ( t ) x β + m x ≥ for t ∈ [0 , T ] and x ≥ . (H3) r ∗ > and s ∗ > .Then the problem (3) has a positive solution.Proof. We are going to prove that assumption (H1) in Theorem 3.2 is satisfied. By (H0),(H3) and since c m ∈ (0 ,
1) we can choose 0 < R < R such that(10) (1 − c m ) m R − α + s ∗ R β − α ≤ r ∗ c αm nd(11) r ∗ ≤ s ∗ c βm R β − α . Note that the inequality (7) follows from (H2) so it only remains to check the inequalities(8) and (9).If t ∈ [0 , T ] and x ∈ [ c m R , R ] then, taking into account (H3) and (10), we have that f m ( t, x ) ≥ r ∗ ( c m R ) α − s ∗ R β + m c m R ≥ m R , so the inequality (8) is fulfilled.If t ∈ [0 , T ] and x ∈ [ c m R , R ] then, taking into account (H3) and (11), we have that f m ( t, x ) ≤ r ∗ R α − s ∗ ( c m R ) β + m R ≤ m R , so the inequality (9) is also fulfilled. (cid:3) Remark 3.4.
Note that (H0) and (H2) imply that r ∗ >
0, thus the first part of condition(H3) is redundant but we have included it for the sake of clarity. On the other hand, noticethat condition (7) in Theorem 3.2 could be satisfied even if r ( t ) assumes negative values.Next, we present an explicit condition sufficient in order to get the condition (H2) inTheorem 3.3. Corollary 3.5.
Assume that (H0) and (H3) hold and, moreover, that (12) s ∗ < min { (cid:16) πT (cid:17) + (cid:16) a (cid:17) , r ∗ } . Then the problem (3) has a positive solution.Proof.
It is enough to show that condition (12) implies (H2). Indeed, by (12) we can choose m > s ∗ < m < (cid:16) πT (cid:17) + (cid:16) a (cid:17) . Now, we are going to prove that for such m we have that f m ( t, x ) ≥ t ∈ [0 , T ] andall x ≥ Step 1. We show that f m ( t, x ) ≥ for all t ∈ [0 , T ] and all x ∈ [0 , . Since 0 < α < β < x β ≤ x α for 0 ≤ x ≤
1. Moreover s ∗ < r ∗ by (12) and then f m ( t, x ) = r ( t ) x α − s ( t ) x β + m x ≥ r ∗ x α − s ∗ x β + m x ≥ r ∗ x β − s ∗ x β + m x = ( r ∗ − s ∗ ) x β + m x > . Step 2. We show that f m ( t, x ) ≥ for all t ∈ [0 , T ] and all x ≥ . Since 0 < β < x β ≤ x for x ≥ f m ( t, x ) = r ( t ) x α − s ( t ) x β + m x ≥ r ∗ x α − s ∗ x β + m x ≥ r ∗ x α − s ∗ x β + m x β = r ∗ x α + ( m − s ∗ ) x β ≥ . In the particular case of problem (2) we recover [4, Theorem 1.8].
Corollary 3.6.
Assume that a ≥ , b > , c > and e ∗ > . Then problem (2) has a positivesolution provided that (13) ( b + 1) c e ∗ < (cid:16) πT (cid:17) + a . Proof.
We recall that the problem (2) is of the form (3) with µ = 1 b + 1 , r ( t ) = e ( t ) µ , s ( t ) = cµ , α = 1 − µ, β = 1 − µ. From our assumptions it follows immediately that the conditions (H0) and (H3) of Theorem3.3 hold. Thus it only remains to check the condition (H2). By (13) we can choose m > b + 1) c e ∗ < m < (cid:16) πT (cid:17) + a . Then for all t ∈ [0 , T ] and x ≥ f m ( t, x ) = e ( t ) µ x − µ − cµ x − µ + m x ≥ e ∗ µ x − µ − cµ x − µ + m x = x − µ µ (cid:0) e ∗ − cx µ + m µ x µ (cid:1) ≥ , because e ∗ − cx µ + m µ x µ > x µ which has by (14) negativediscriminant c − m µe ∗ <
0, so it has constant sign, and positive independent term e ∗ > (cid:3) The following example shows that Theorem 3.2 is in fact more general than Corollary 3.6.Therefore, even in the case of the model problem (2), Theorem 3.2 is a true extension of [4,Theorem 1.8]. Moreover, notice that we also obtain information about the localization of thesolution which was not provided in [4]. Some computations here were made with MAPLE.
Example 3.7.
Consider the problem (2), that is, x (cid:48)(cid:48) ( t ) + a x (cid:48) ( t ) = e ( t ) µ x − µ − cµ x − µ , t ∈ [0 , T ] ,x (0) = x ( T ) , x (cid:48) (0) = x (cid:48) ( T ) , with the parameter values a = 1 . e ( t ) ≡ . µ = 0 . c = 1 .
49 and T = 1.Therefore we have b = 99 (recall that µ := b +1 ) , r ( t ) := e ( t ) µ ≡ , s ( t ) := cµ ≡ ,α := 1 − µ = 0 .
98 and β := 1 − µ = 0 . . hus, condition (13) in Corollary 3.6 is not satisfied because( b + 1) c e ∗ ≈ . > . ≈ (cid:16) πT (cid:17) + a . Notice that neither condition (12) in Corollary 3.5 is satisfied since s ∗ = 149 > . ≈ (cid:16) πT (cid:17) + a . However, the condition (H0) in Theorem 3.2 is clearly satisfied and if we define m := 0 . R := 27 and R := 29 then (H1) is also fulfilled (notice that c m ≈ .
10 20 30 40 50 (cid:45) (cid:45) f m ( t, x ) on [0 , f m ( t, x ) on [ c m R , R ] f m ( t, x ) − m R on [ c m R , R ] (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) f m ( t, x ) − m R on [ c m R , R ]Thus, Theorem 3.2 ensures the existence of a solution x ( t ) of this problem such that25 . ≈ c m R ≤ x ( t ) ≤ R = 29 . Observe that f m is not positive for all x ≥ emark 3.8. From a careful reading of the proof of [4, Theorem 1.8] it follows that theexistence of a positive periodic solution to problem (2) is ensured if condition (13) in Corollary3.6 is replaced by the more general one:(H) There is δ > f ( t, x ) < t ∈ [0 , T ] and all x ∈ (0 , δ ) and λ ∗ x − f ( t, x ) > t ∈ [0 , T ] and all x >
0, where λ ∗ = (cid:0) πT (cid:1) + a .However, condition (H) is neither satisfied for the problem presented in Example 3.7because in that case λ ∗ x − f ( t, x ) ≈ . x − x . + 154 x . , which is a sing-changing nonlinearity on the positive real line. Indeed, dividing by x . andmaking the change y = x . , it is clear that λ ∗ x − f ( t, x ) is negative on ( r , r ) and positiveon (0 , r ) ∪ ( r , ∞ ), being r ≈ . r ≈ . ∗ .4. Appendix: The Green’s function
The case < m < a . The Green’s function related to problem (5) is given by(15) G m ( t, s ) = 1 λ − λ e λ ( t − s ) − e λ T − e λ ( t − s ) − e λ T , ≤ s ≤ t ≤ T,e λ ( t − s + T ) − e λ T − e λ ( t − s + T ) − e λ T , ≤ t ≤ s ≤ T, where λ and λ are the characteristic roots of the homogeneous equation x (cid:48)(cid:48) ( t ) + ax (cid:48) ( t ) + m x ( t ) = 0 , that is, λ = − a − √ a − m , λ = − a + √ a − m . Note that λ < λ < Remark 4.1. G m ( s, s ) is constant for s ∈ [0 , T ], G m ( s, s ) = 1 λ − λ (cid:18) − e λ T − − e λ T (cid:19) , and(16) (cid:90) T G m ( t, s ) ds = 1 λ λ = 1 m . The Green’s function (15) satisfies(17) G m ( t, s ) ≥ G m ( s, s ) > , and(18) G m ( s, s ) ≥ c m G m ( t, s ) , or all ( t, s ) ∈ [0 , T ] × [0 , T ], where(19) c m = G m ( s, s )max { G m ( t, s ) : t, s ∈ [0 , T ] } , that is,(20) c m = − λ λ − λ e λ T − e λ T (1 − e λ T ) (cid:16) (1 − e λ T ) λ (1 − e λ T ) λ (cid:17) λ λ − λ . The case m = a . The Green’s function related to the problem (5) is given by(21) G m ( t, s ) = 1 e mT − e m ( t − s ) (cid:20) T e mT e mT − s − t (cid:21) , ≤ s ≤ t ≤ T,e m ( T − s + t ) (cid:20) Te mT − s − t (cid:21) , ≤ t ≤ s ≤ T. Then we have (cid:90) T G m ( t, s ) ds = 1 m ,G m ( t, s ) ≥ G m ( s, s ) > , and G m ( s, s ) ≥ c m G m ( t, s ) , for all ( t, s ) ∈ [0 , T ] × [0 , T ], where c m = G m ( s, s )max { G m ( t, s ) : t, s ∈ [0 , T ] } , that is, c m = κe − κ , with κ = mTe mT − . The case m > a and m < (cid:16) πT (cid:17) + (cid:16) a (cid:17) . We use the following notation: γ = − a , δ = √ m − a , D = 1 − e γT cos( δT ) + e γT . The Green’s function related to the problem (5) is given by(22) G m ( t, s ) = 1 δD e γ ( T + t − s ) sin( δ ( T + s − t )) + e γ ( t − s ) sin( δ ( t − s )) , ≤ s ≤ t ≤ T,e γ ( T + t − s ) sin( δ ( T + t − s )) + e γ (2 T + t − s ) sin( δ ( s − t )) , ≤ t ≤ s ≤ T. hen we have (cid:90) T G m ( t, s ) ds = 1 m ,G m ( t, s ) ≥ G m ( s, s ) > , and G m ( s, s ) ≥ c m G m ( t, s ) , for all ( t, s ) ∈ [0 , T ] × [0 , T ], where c m = G m ( s, s )max { G m ( t, s ) : t, s ∈ [0 , T ] } = 1 δD e γT sin( δT )max { G m ( t, s ) : t, s ∈ [0 , T ] } . If a = 0 then γ = 0 , δ = m, D = 2(1 − cos mT ) and it is known that max { G m ( t, s ) : t, s ∈ [0 , T ] } = m sin mT , see [16]. Therefore we have c m = cos mT .On the other hand, if a > G m ( t, s ), butwe are able to get an upper bound in the following way: since 0 < δT < π we have for0 ≤ s ≤ t ≤ T (the case 0 ≤ t ≤ s ≤ T is analogous) e γ ( T + t − s ) sin( δ ( T + s − t )) + e γ ( t − s ) sin( δ ( t − s )) ≤ sin( δ ( T + s − t )) + sin( δ ( t − s ))= 2 sin δT (cid:18) δ ( T − ( t − s )) (cid:19) ≤ δT . So, max { G m ( t, s ) : t, s ∈ [0 , T ] } ≤ δD sin δT which yields c m ≥ e γT cos δT . Remark 4.2. If m = (cid:16) πT (cid:17) + (cid:16) a (cid:17) then G m ( s, s ) = 0. Acknowledgements
J. A. Cid was partially supported by Ministerio de Educaci´on y Ciencia, Spain, andFEDER, Project MTM2010-15314, G. Infante was partially supported by G.N.A.M.P.A.- INdAM (Italy), M. Tvrd´y was supported by GA ˇCR Grant P201/14-06958S and RVO:67985840 and M. Zima was partially supported by the Centre for Innovation and Transferof Natural Science and Engineering Knowledge of University of Rzesz´ow.
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E-mail address : [email protected] Gennaro Infante, Dipartimento di Matematica e Informatica, Universit`a della Calabria,87036 Arcavacata di Rende, Cosenza, Italy
E-mail address : [email protected] Milan Tvrd´y, Mathematical Institute, Academy of Sciences of the Czech Republic, CZ 115 67Praha 1, ˇZitn´a 25, Czech Republic
E-mail address : [email protected] Miros(cid:32)lawa Zima, Department of Differential Equations and Statistics, Faculty of Math-ematics and Natural Sciences, University of Rzesz´ow, Pigonia 1, 35-959 Rzesz´ow, Poland.
E-mail address : [email protected]@ur.edu.pl