Multiple non-negative solutions of systems with coupled nonlinear BCs
aa r X i v : . [ m a t h . C A ] J un MULTIPLE POSITIVE SOLUTIONS OF SYSTEMS WITH COUPLEDNONLINEAR BCS
GENNARO INFANTE AND PAOLAMARIA PIETRAMALA
Abstract.
Using the theory of fixed point index, we discuss the existence and multi-plicity of non-negative solutions of a wide class of boundary value problems with couplednonlinear boundary conditions. Our approach is fairly general and covers a variety ofsituations. We illustrate our theory in an example all the constants that occur in ourtheory. Introduction
The aim of this paper is to present a theory for the existence of positive solution fora fairly general class of systems of ordinary differential equations subject to nonlinear,nonlocal boundary conditions. In particular we are interested in systems that present acoupling in the boundary conditions (BCs); this type of problems have been studied in[5, 6, 7, 10, 11, 32, 36, 47] and often occur in applications, for example when modellingthe displacement of a suspension bridge subject to nonlinear controllers.In [33], Lu and co-authors, by means of the Krasnosel’ski˘ı-Guo Theorem on cone com-pressions and cone expansions, studied existence of positive solutions of the system ofordinary differential equations (ODEs)(1.1) u ′′ ( t ) + f ( t, v ( t )) = 0 , t ∈ (0 , ,v (4) ( t ) = f ( t, u ( t )) , t ∈ (0 , , subject to the BCs(1.2) u (0) = u (1) = v (0) = v (1) = v ′′ (0) = v ′′ (1) = 0 . The motivation, given in [33], for studying the BVP (1.1)-(1.2) is that it can be seenas the stationary case of a model for the oscillations of the center-span of a suspensionbridge, where the forth order equation represents the road-bed (seen as an elastic beam)and second order equation models the main cable (seen as a vibrating string). The BCsin this case illustrate the fact that the beam is simply supported and that the two endsof the cable are supposed to be immovable, see also, for example, [29, 35].
Mathematics Subject Classification.
Primary 45G15, secondary 34B10, 34B18, 47H30.
Key words and phrases.
Fixed point index, cone, non-negative solution, nonlinear boundary condi-tions, coupled boundary conditions.
The existence of positive solutions of a coupled system with an elastic beam equationof the type(1.3) u ′′ ( t ) + f ( t, v ( t )) = 0 , t ∈ (0 , ,v (4) ( t ) = f ( t, u ( t ) , v ( t )) , t ∈ (0 , , has been studied by Sun in [41], by monotone iterative techniques, under the BCs(1.4) u (0) = u (1) = v (0) = v (1) = v ′ (0) = v ′′ (1) = 0 . A common feature of the systems (1.3)-(1.4) and (1.1)-(1.2) is that the BCs underconsideration are local and homogeneous.In [22], Infante and co-authors, by means of classical fixed point index theory, provideda fairly general theory suitable to study the existence of non-negative solutions of a varietyof systems of ODEs subject to linear , nonlocal conditions, one example being the system(1.5) u ′′ ( t ) + g ( t ) f ( t, u ( t ) , v ( t )) = 0 , t ∈ (0 , ,v (4) ( t ) = g ( t ) f ( t, u ( t ) , v ( t )) , t ∈ (0 , , with the BCs(1.6) u (0) = β [ u ] , u (1) = δ [ v ] , v (0) = β [ v ] , v ′′ (0) = 0 , v (1) = 0 , v ′′ (1) + δ [ u ] = 0 , where β ij [ · ], δ ij [ · ] are bounded linear functionals given by Riemann-Stieltjes integrals,namely β ij [ w ] = Z w ( s ) dB ij ( s ) , δ ij [ w ] = Z w ( s ) dC ij ( s ) . This type of formulation includes, as special cases, multi-point or integral conditions,when α ij [ w ] = m X l =1 α ijl w ( η ijl ) and α ij [ w ] = Z α ij ( s ) w ( s ) ds, see for example [17, 16, 24, 25, 27, 28, 34, 37, 42, 44].In the case of the system (1.5)-(1.6), the BCs u (0) = βu ( ξ ) , u (1) = v (1) = v ′′ (0) = v (0) = 0 , v ′′ (1) + δu ( η ) = 0 , can be interpreted as a cable-beam model with two devices of feedback control, wherethe displacement of the left end of cable is related to displacement of another point ξ of the cable and the bending moment in the right end of the beam depends upon thedisplacement registered in a point η of the string. We point out that not necessarilythe response of the controllers needs to be of linear type, for example this happens withconditions of the type u (0) = H ( u ( ξ )) , u (1) = v (1) = v ′′ (0) = v (0) = 0 , v ′′ (1) + L ( u ( η )) = 0;we refer to [20] for more details regarding the illustration of nonlinear controllers on abeam. YSTEMS WITH COUPLED NONLINEAR BCS 3
Our approach allows us to deal with a larger class of nonlinear nonlocal BCs, oneexample given by the BCs u (0) = H ( β [ u ]) + L ( δ [ v ]) , u (1) = H ( β [ u ]) + L ( δ [ v ]) ,v (0) = H ( β [ v ]) + L ( δ [ u ]) , v ′′ (0) = 0 , v (1) = 0 ,v ′′ (1) + H ( β [ v ]) + L ( δ [ u ]) = 0 , (1.7)where H ij , L ij are continuous functions. For earlier contributions on problems withnonlinear BCs we refer the reader to [1, 2, 3, 9, 12, 13, 14, 18, 20, 38] and referencestherein.Here we develop an existence theory for multiple positive solutions of the perturbedHammerstein integral equations of the type u ( t ) = X i =1 , γ i ( t ) (cid:16) H i ( β i [ u ]) + L i ( δ i [ v ]) (cid:17) + Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds,v ( t ) = X i =1 , γ i ( t ) (cid:16) L i ( δ i [ u ]) + H i ( β i [ v ]) (cid:17) + Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds. Similar systems of perturbed Hammerstein integral equations were studied in [8, 10, 11,19, 21, 22, 26, 46]. Our theory covers, as a special case, the system (1.5)-(1.7) and weshow in an example that all the constants that occur in our theory can be computed.We make use of the classical fixed point index theory (see for example [4, 15]) and alsobenefit of ideas from the papers [18, 21, 22, 23, 43].2.
Positive solutions for systems of perturbed integral equations
We begin with stating some assumptions on the terms that occur in the system ofperturbed Hammerstein integral equations u ( t ) = X i =1 , γ i ( t ) (cid:16) H i ( β i [ u ]) + L i ( δ i [ v ]) (cid:17) + F ( u, v )( t ) ,v ( t ) = X i =1 , γ i ( t ) (cid:16) L i ( δ i [ u ]) + H i ( β i [ v ]) (cid:17) + F ( u, v )( t ) , (2.1)where F i ( u, v )( t ) := Z k i ( t, s ) g i ( s ) f i ( s, u ( s ) , v ( s )) ds, namely: • For every i = 1 , f i : [0 , × [0 , ∞ ) × [0 , ∞ ) → [0 , ∞ ) satisfies Carath´eodoryconditions, that is, f i ( · , u, v ) is measurable for each fixed ( u, v ) and f i ( t, · , · ) iscontinuous for almost every (a.e.) t ∈ [0 , r > φ i,r ∈ L ∞ [0 ,
1] such that f i ( t, u, v ) ≤ φ i,r ( t ) for u, v ∈ [0 , r ] and a. e. t ∈ [0 , . GENNARO INFANTE AND PAOLAMARIA PIETRAMALA • For every i = 1 , k i : [0 , × [0 , → [0 , ∞ ) is measurable, and for every τ ∈ [0 , t → τ | k i ( t, s ) − k i ( τ, s ) | = 0 for a. e. s ∈ [0 , . • For every i = 1 ,
2, there exist a subinterval [ a i , b i ] ⊆ [0 , i ∈ L ∞ [0 , c i ∈ (0 , k i ( t, s ) ≤ Φ i ( s ) for t ∈ [0 ,
1] and a. e. s ∈ [0 , ,k i ( t, s ) ≥ c i Φ i ( s ) for t ∈ [ a i , b i ] and a. e. s ∈ [0 , . • For every i = 1 , g i Φ i ∈ L [0 , g i ≥ R b i a i Φ i ( s ) g i ( s ) ds > • For every i, j = 1 , β ij [ · ] and δ ij [ · ] are linear functionals given by β ij [ w ] = Z w ( s ) dB ij ( s ) , δ ij [ w ] = Z w ( s ) dC ij ( s ) , involving Riemann-Stieltjes integrals; B ij and C ij are of bounded variation and dB ij , dC ij are positive measure. • H ij , L ij : [0 , ∞ ) → [0 , ∞ ) are continuous functions such that there exist h ij , h ij , l ij ∈ [0 , ∞ ), i, j = 1 ,
2, with h ij w ≤ H ij ( w ) ≤ h ij w, L ij ( w ) ≤ l ij w, for every w ≥ • γ ij ∈ C [0 , , γ ij ( t ) ≥ t ∈ [0 , , h ij β ij [ γ ij ] < c ij ∈ (0 ,
1] such that γ ij ( t ) ≥ c ij k γ ij k ∞ for every t ∈ [ a i , b i ] , where k w k ∞ := max {| w ( t ) | , t ∈ [0 , } . • D i := (1 − h i β i [ γ i ])(1 − h i β i [ γ i ]) − h i h i β i [ γ i ] β i [ γ i ] > , i = 1 , D i > D i := (1 − h i β i [ γ i ])(1 − h i β i [ γ i ]) − h i h i β i [ γ i ] β i [ γ i ] > . We work in the space C [0 , × C [0 ,
1] endowed with the norm k ( u, v ) k := max {k u k ∞ , k v k ∞ } . Let ˜ K i := { w ∈ C [0 ,
1] : w ( t ) ≥ t ∈ [0 ,
1] and min t ∈ [ a i ,b i ] w ( t ) ≥ ˜ c i k w k ∞ } , where ˜ c i = min { c i , c i , c i } , and consider the cone K in C [0 , × C [0 ,
1] defined by K := { ( u, v ) ∈ ˜ K × ˜ K } . For a positive solution of the system (2.1) we mean a solution ( u, v ) ∈ K of (2.1) suchthat k ( u, v ) k > YSTEMS WITH COUPLED NONLINEAR BCS 5
Under our assumptions, it is routine to show that the integral operator T ( u, v )( t ) := P i =1 , γ i ( t ) (cid:16) H i ( β i [ u ]) + L i ( δ i [ v ]) (cid:17) + F ( u, v )( t ) P i =1 , γ i ( t ) (cid:16) L i ( δ i [ u ]) + H i ( β i [ v ]) (cid:17) + F ( u, v )( t ) := T ( u, v )( t ) T ( u, v )( t ) ! , leaves the cone K invariant and is compact, see for example Lemma 1 of [22].We use the following (relative) open bounded sets in K : K ρ = { ( u, v ) ∈ K : k ( u, v ) k < ρ } , and V ρ = { ( u, v ) ∈ K : min t ∈ [ a ,b ] u ( t ) < ρ and min t ∈ [ a ,b ] v ( t ) < ρ } . The set V ρ (in the context of systems) was introduced by the authors in [19] and is equalto the set called Ω ρ/c in [8]. Ω ρ/c is an extension to the case of systems of a set given byLan [31]. For our index calculations we make use of the fact that K ρ ⊂ V ρ ⊂ K ρ/c , where c = min { ˜ c , ˜ c } . We denote by ∂K ρ and ∂V ρ the boundary of K ρ and V ρ relativeto K .We utilize the following results of [43] regarding order preserving matrices: Definition 2.1.
A 2 × Q is said to be order preserving (or positive) if p ≥ p , q ≥ q imply Q p q ! ≥ Q p q ! , in the sense of components. Lemma 2.2. [43]
Let Q = a − b − c d ! with a, b, c, d ≥ and det Q > . Then Q − is order preserving. Remark 2.3.
It is a consequence of Lemma 2.2 that if N = − a − b − c − d ! , satisfies the hypotheses of Lemma 2.2, p ≥ , q ≥ µ > N − µ pq ! ≤ N − pq ! , GENNARO INFANTE AND PAOLAMARIA PIETRAMALA where N µ = µ − a − b − c µ − d ! . In the sequel of the paper we use the following notation. K ij ( s ) := Z k i ( t, s ) dB ij ( t ) , Q i = X l =1 , β i [ γ il ] l il δ il [1] , S i = X l =1 , β i [ γ il ] l il δ il [1] ,θ i = 1 − h i β i [ γ i ] D i , θ i = h i β i [ γ i ] D i , θ i = h i β i [ γ i ] D i , θ i = 1 − h i β i [ γ i ] D i , We are now able to prove a result concerning the fixed point index on the set K ρ . Lemma 2.4. (I ρ ) there exists ρ > such that for every i = 1 , f ,ρi (cid:16)(cid:16) k γ i k ∞ h i θ i + k γ i k ∞ h i θ i (cid:17) Z K i ( s ) g i ( s ) ds + (cid:16) k γ i k ∞ h i θ i + k γ i k ∞ h i θ i (cid:17) Z K i ( s ) g i ( s ) ds + 1 m i (cid:17) + k γ i k ∞ h i ( θ i Q i + θ i S i ) + k γ i k ∞ h i ( θ i Q i + θ i S i ) + X j =1 , k γ ij k ∞ l ij δ ij [1] < where f ,ρi = sup n f i ( t, u, v ) ρ : ( t, u, v ) ∈ [0 , × [0 , ρ ] × [0 , ρ ] o and m i = sup t ∈ [0 , Z k i ( t, s ) g i ( s ) ds. Then the fixed point index, i K ( T, K ρ ) , is equal to 1.Proof. We show that µ ( u, v ) = T ( u, v ) for every ( u, v ) ∈ ∂K ρ and for every µ ≥
1; thisensures that the index is 1 on K ρ . In fact, if this does not happen, there exist µ ≥ u, v ) ∈ ∂K ρ such that µ ( u, v ) = T ( u, v ). Assume, without loss of generality, that k u k ∞ = ρ and k v k ∞ ≤ ρ . Then µu ( t ) = X i =1 , γ i ( t ) (cid:16) H i ( β i [ u ]) + L i ( δ i [ v ]) (cid:17) + F ( u, v )( t )and therefore, since v ( t ) ≤ ρ, for all t ∈ [0 , µu ( t ) ≤ X i =1 , γ i ( t ) h i β i [ u ] + X i =1 , γ i ( t ) l i δ i [ ρ ] + F ( u, v )( t )(2.3) = X i =1 , γ i ( t ) h i β i [ u ] + ρ X i =1 , γ i ( t ) l i δ i [1] + F ( u, v )( t ) . Applying β and β to both sides of (2.3) gives YSTEMS WITH COUPLED NONLINEAR BCS 7 µβ [ u ] ≤ X i =1 , β [ γ i ] h i β i [ u ] + ρ X i =1 , β [ γ i ] l i δ i [1] + β [ F ( u, v )] ,µβ [ u ] ≤ X i =1 , β [ γ i ] h i β i [ u ] + ρ X i =1 , β [ γ i ] l i δ i [1] + β [ F ( u, v )] . Thus we have( µ − h β [ γ ]) β [ u ] − h β [ γ ] β [ u ] ≤ ρ X i =1 , β [ γ i ] l i δ i [1] + β [ F ( u, v )] , − h β [ γ ] β [ u ] + ( µ − h β [ γ ]) β [ u ] ≤ ρ X i =1 , β [ γ i ] l i δ i [1] + β [ F ( u, v )] , that is µ − h β [ γ ] − h β [ γ ] − h β [ γ ] µ − h β [ γ ] ! β [ u ] β [ u ] ! ≤ ρ P i =1 , β [ γ i ] l i δ i [1] + β [ F ( u, v )] ρ P i =1 , β [ γ i ] l i δ i [1] + β [ F ( u, v )] ! . (2.4)The matrix M µ = µ − h β [ γ ] − h β [ γ ] − h β [ γ ] µ − h β [ γ ] ! , satisfies the hypotheses of Lemma 2.2, thus ( M µ ) − is order preserving. If we apply( M µ ) − to both sides of the inequality (2.4) we obtain β [ u ] β [ u ] ! ≤ M µ ) µ − h β [ γ ] h β [ γ ] h β [ γ ] µ − h β [ γ ] ! × ρ P i =1 , β [ γ i ] l i δ i [1] + β [ F ( u, v )] ρ P i =1 , β [ γ i ] l i δ i [1] + β [ F ( u, v )] ! , and by Remark 2.3, we have β [ u ] β [ u ] ! ≤ D − h β [ γ ] h β [ γ ] h β [ γ ] 1 − h β [ γ ] ! × ρ P i =1 , β [ γ i ] l i δ i [1] + β [ F ( u, v )] ρ P i =1 , β [ γ i ] l i δ i [1] + β [ F ( u, v )] ! , that is β [ u ] β [ u ] ! ≤ θ θ θ θ ! ρQ + β [ F ( u, v )] ρS + β [ F ( u, v )] ! . Thus β [ u ] β [ u ] ! ≤ ρ ( θ Q + θ S ) + θ β [ F ( u, v )] + θ β [ F ( u, v )] ρ ( θ Q + θ S ) + θ β [ F ( u, v )] + θ β [ F ( u, v )] ! . GENNARO INFANTE AND PAOLAMARIA PIETRAMALA
Substituting into (2.3) gives µu ( t ) ≤ ρ (cid:16) γ ( t ) h ( θ Q + θ S ) + γ ( t ) h ( θ Q + θ S ) + X i =1 , γ i ( t ) l i δ i [1] (cid:17) + (cid:16) γ ( t ) h θ + γ ( t ) h θ (cid:17) β [ F ( u, v )]+ (cid:16) γ ( t ) h θ + γ ( t ) h θ (cid:17) β [ F ( u, v )]+ F ( u, v )( t )= ρ (cid:16) γ ( t ) h ( θ Q + θ S ) + γ ( t ) h ( θ Q + θ S ) + X i =1 , γ i ( t ) l i δ i [1] (cid:17) + (cid:16) γ ( t ) h θ + γ ( t ) h θ (cid:17) Z K ( s ) g ( s ) f ( s, u ( s ) , v ( s )) ds + (cid:16) γ ( t ) h θ + γ ( t ) h θ (cid:17) Z K ( s ) g ( s ) f ( s, u ( s ) , v ( s )) ds + F ( u, v )( t ) . Taking the supremum over [0 ,
1] gives µρ ≤ ρ (cid:16) k γ k ∞ h ( θ Q + θ S ) + k γ k ∞ h ( θ Q + θ S ) + X i =1 , k γ i k ∞ l i δ i [1] (cid:17) + ρf ,ρ (cid:16) k γ k ∞ h θ + k γ k ∞ h θ (cid:17) Z K ( s ) g ( s ) ds + ρf ,ρ (cid:16) k γ k ∞ h θ + k γ k ∞ h θ (cid:17) Z K ( s ) g ( s ) ds + ρf ,ρ m . Using the hypothesis (2.2) we obtain µρ < ρ.
This contradicts the fact that µ ≥ (cid:3) We give a first Lemma that shows that the index is 0 on a set V ρ . Lemma 2.5.
Assume that (I ρ ) there exist ρ > such that for every i = 1 , f i, ( ρ,ρ/c ) (cid:16)(cid:0) c i k γ i k h i D i (1 − h i β i [ γ i ]) + c i k γ i k h i D i h i β i [ γ i ] (cid:1) Z b i a i K i ( s ) g i ( s ) ds + (cid:0) c i k γ i k h i D i h i β i [ γ i ]) + c i k γ i k h i D i (1 − h i β i [ γ i ]) (cid:1) Z b i a i K i ( s ) g i ( s ) ds + 1 M i (cid:17) > , where f , ( ρ,ρ/c ) = inf n f ( t, u, v ) ρ : ( t, u, v ) ∈ [ a , b ] × [ ρ, ρ/c ] × [0 , ρ/c ] o ,f , ( ρ,ρ/c ) = inf n f ( t, u, v ) ρ : ( t, u, v ) ∈ [ a , b ] × [0 , ρ/c ] × [ ρ, ρ/c ] o and M i = inf t ∈ [ a i ,b i ] Z b i a i k i ( t, s ) g i ( s ) ds. YSTEMS WITH COUPLED NONLINEAR BCS 9
Then i K ( T, V ρ ) = 0 .Proof. Let e ( t ) ≡ t ∈ [0 , e, e ) ∈ K . We prove that( u, v ) = T ( u, v ) + µ ( e, e ) for ( u, v ) ∈ ∂V ρ and µ ≥ . In fact, if this does not happen, there exist ( u, v ) ∈ ∂V ρ and µ ≥ u, v ) = T ( u, v ) + µ ( e, e ). Without loss of generality, we can assume that for all t ∈ [ a , b ] wehave ρ ≤ u ( t ) ≤ ρ/c, min u ( t ) = ρ and 0 ≤ v ( t ) ≤ ρ/c. Then, for t ∈ [ a , b ], we obtain u ( t ) = X i =1 , γ i ( t ) (cid:16) H i ( β i [ u ]) + L i ( δ i [ v ]) (cid:17) + F ( u, v )( t ) + µe and therefore(2.6) u ( t ) ≥ X i =1 , γ i ( t ) H i ( β i [ u ]) + F ( u, v )( t ) + µe ≥ X i =1 , γ i ( t ) h i β i [ u ] + F ( u, v )( t ) + µe. Applying β and β to both sides of (2.6) gives β [ u ] ≥ h β [ γ ] β [ u ] + h β [ γ ] β [ u ] + β [ F ( u, v )] + µβ [ e ] ,β [ u ] ≥ h β [ γ ] β [ u ] + h β [ γ ] β [ u ] + β [ F ( u, v )] + µβ [ e ] . Thus we have(1 − h β [ γ ]) β [ u ] − h β [ γ ] β [ u ] ≥ β [ F ( u, v )] + µβ [ e ] , − h β [ γ ] β [ u ] + (1 − h β [ γ ]) β [ u ] ≥ β [ F ( u, v )] + µβ [ e ] , that is − h β [ γ ] − h β [ γ ] − h β [ γ ] 1 − h β [ γ ] ! β [ u ] β [ u ] ! ≥ β [ F ( u, v )] + µβ [ e ] β [ F ( u, v )] + µβ [ e ] ! ≥ β [ F ( u, v )] β [ F ( u, v )] ! . The matrix M = − h β [ γ ] − h β [ γ ] − h β [ γ ] 1 − h β [ γ ] ! satisfies the hypotheses of Lemma 2.2, thus ( M ) − is order preserving. If we apply( M ) − to both sides of the last inequality we obtain β [ u ] β [ u ] ! ≥ D − h β [ γ ] h β [ γ ] h β [ γ ] 1 − h β [ γ ] ! β [ F ( u, v )] β [ F ( u, v )] ! and therefore u ( t ) ≥ (cid:16) γ ( t ) D h (1 − h β [ γ ]) + γ ( t ) D h h β [ γ ] (cid:17) × Z K ( s ) g ( s ) f ( s, u ( s ) , v ( s )) ds + (cid:16) γ ( t ) D h h β [ γ ] + γ ( t ) D (1 − h β [ γ ]) h (cid:17) × Z K ( s ) g ( s ) f ( s, u ( s ) , v ( s )) ds + Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds + µ. Then we have, for t ∈ [ a , b ], u ( t ) ≥ (cid:16) c k γ k D h (1 − h β [ γ ]) + c k γ k D h h β [ γ ] (cid:17) × Z b a K ( s ) g ( s ) f ( s, u ( s ) , v ( s )) ds + (cid:16) c k γ k D h h β [ γ ] + c k γ k D (1 − h β [ γ ]) h (cid:17) × Z b a K ( s ) g ( s ) f ( s, u ( s ) , v ( s )) ds + Z b a k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds + µ. Taking the minimum over [ a , b ] gives ρ = min t ∈ [ a ,b ] u ( t ) ≥ ρf , ( ρ,ρ/c ) (cid:16) c k γ k D h (1 − h β [ γ ]) + c k γ k D h h β [ γ ] (cid:17) × Z b a K ( s ) g ( s ) ds + ρf , ( ρ,ρ/c ) (cid:16) c k γ k D h h β [ γ ] + c k γ k D (1 − h β [ γ ]) h (cid:17) × Z b a K ( s ) g ( s ) ds + ρf , ( ρ,ρ/c ) M + µ. Using the hypothesis (2.5) we obtain ρ > ρ + µ , a contradiction. (cid:3) The following Lemma provides a result of index 0 on V ρ of a different flavour; the idea isto control the growth of just one nonlinearity f i , at the cost of having to deal with a largerdomain. The proof is omitted as it follows from the previous proof, for details see [21, 22].We mention that nonlinearities with different growth were studied also in [39, 40, 46]. Lemma 2.6.
Assume that
YSTEMS WITH COUPLED NONLINEAR BCS 11 (I ρ ) ⋆ there exist ρ > such that for some i = 1 , f ∗ i, (0 ,ρ/c ) (cid:16)(cid:0) c i k γ i k h i D i (1 − h i β i [ γ i ]) + c i k γ i k h i D i h i β i [ γ i ] (cid:1) Z b i a i K i ( s ) g i ( s ) ds + (cid:0) c i k γ i k h i D i h i β i [ γ i ])+ c i k γ i k h i D i (1 − h i β i [ γ i ]) (cid:1) Z b i a i K i ( s ) g i ( s ) ds + 1 M i (cid:17) > . where f ∗ i, (0 ,ρ/c ) = inf n f i ( t, u, v ) ρ : ( t, u, v ) ∈ [ a i , b i ] × [0 , ρ/c ] × [0 , ρ/c ] o . Then i K ( T, V ρ ) = 0 . The above Lemmas can be combined to prove the following Theorem, here we dealwith the existence of at least one, two or three solutions. We stress that, by expandingthe lists in conditions ( S ) , ( S ) below, it is possible, in a similar way as in [30], to stateresults for four or more positive solutions. We omit the proof which follows from theproperties of fixed point index. Theorem 2.7.
The system (2.1) has at least one positive solution in K if either of thefollowing conditions hold. ( S ) There exist ρ , ρ ∈ (0 , ∞ ) with ρ /c < ρ such that (I ρ ) [ or (I ρ ) ⋆ ] , (I ρ ) hold. ( S ) There exist ρ , ρ ∈ (0 , ∞ ) with ρ < ρ such that (I ρ ) , (I ρ ) hold.The system (2.1) has at least two positive solutions in K if one of the following condi-tions hold. ( S ) There exist ρ , ρ , ρ ∈ (0 , ∞ ) with ρ /c < ρ < ρ such that (I ρ ) [ or (I ρ ) ⋆ ] , (I ρ ) and (I ρ ) hold. ( S ) There exist ρ , ρ , ρ ∈ (0 , ∞ ) with ρ < ρ and ρ /c < ρ such that (I ρ ) , (I ρ ) and (I ρ ) hold.The system (2.1) has at least three positive solutions in K if one of the followingconditions hold. ( S ) There exist ρ , ρ , ρ , ρ ∈ (0 , ∞ ) with ρ /c < ρ < ρ and ρ /c < ρ such that (I ρ ) [ or (I ρ ) ⋆ ] , (I ρ ) , (I ρ ) and (I ρ ) hold. ( S ) There exist ρ , ρ , ρ , ρ ∈ (0 , ∞ ) with ρ < ρ and ρ /c < ρ < ρ such that (I ρ ) , (I ρ ) , (I ρ ) and (I ρ ) hold. Remark 2.8.
If the nonlinearities f and f have some extra positivity properties, forexample if the condition ( S ) holds and moreover we assume that f ( t, , v ) > a , b ] ×{ } × [0 , ρ ] and f ( t, u, > a , b ] × [0 , ρ ] × { } , then the solution ( u, v ) of thesystem (2.1) is such that k u k ∞ and k v k ∞ are strictly positive. An application to coupled systems of BVPs
We study the existence of positive solutions for the system of second order ODEs(3.1) u ′′ ( t ) + g ( t ) f ( t, u ( t ) , v ( t )) = 0 , t ∈ (0 , ,v (4) ( t ) = g ( t ) f ( t, u ( t ) , v ( t )) , t ∈ (0 , , with the nonlocal nonlinear BCs u (0) = H ( β [ u ]) + L ( δ [ v ]) , u (1) = H ( β [ u ]) + L ( δ [ v ]) ,v (0) = H ( β [ v ]) + L ( δ [ u ]) , v ′′ (0) = 0 , v (1) = 0 ,v ′′ (1) + H ( β [ v ]) + L ( δ [ u ]) = 0 , (3.2)This differential system can be rewritten in the integral form u ( t ) =(1 − t )( H ( β [ u ]) + L ( δ [ v ])) + t ( H ( β [ u ]) + L ( δ [ v ]))+ Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds,v ( t ) =(1 − t )( H ( β [ v ]) + L ( δ [ u ])) + 16 t (1 − t )( H ( β [ v ]) + L ( δ [ u ]))+ Z k ( t, s ) g ( s ) f ( s, u ( s ) , v ( s )) ds, where k ( t, s ) = s (1 − t ) , s ≤ t,t (1 − s ) , s > t, and k ( t, s ) = s (1 − t )(2 t − s − t ) , s ≤ t, t (1 − s )(2 s − t − s ) , s > t, are non-negative continuous functions on [0 , × [0 , a , b ] and [ a , b ] may be chosen arbitrarily in (0 , k ( t, s ) ≤ s (1 − s ) := Φ ( s ) , min t ∈ [ a ,b ] k ( t, s ) ≥ c s (1 − s ) , where c = min { − b , a } . Furthermore, see [45], we have that k ( t, s ) ≤ Φ ( s ) := √ s (1 − s ) , for 0 ≤ s ≤ , √ (1 − s ) s (2 − s ) , for < s ≤ , and k ( t, s ) ≥ c ( t )Φ ( s ) , where c ( t ) = √ t (1 − t ) , for t ∈ [0 , / , √ t (1 − t )(2 − t ) , for t ∈ (1 / , , so that c = min t ∈ [ a ,b ] c ( t ) > . The existence of multiple solutions of the system (3.1)-(3.2) follows from Theorem 2.7.
YSTEMS WITH COUPLED NONLINEAR BCS 13
The nonlinearities that occurs in the next example, taken from [22], are used to illus-trate, under a mathematical point of view, the constants that occur in our theory.
Example 3.1.
Consider the system u ′′ + (1 / u + t v ) + 2 =0 , t ∈ (0 , ,v (4) = √ tu + 13 v , t ∈ (0 , ,u (0) = H ( u (1 / L ( v (1 / , u (1) = H ( u (3 / L ( v (3 / ,v (0) = H ( v (1 / L ( u (1 / , v ′′ (0) = 0 , v (1) = 0 ,v ′′ (1) + H ( v (2 / L ( u (2 / , (3.3)where the nonlocal conditions are given by the functionals β ij [ w ] = δ ij [ w ] = w ( η ij ) andthe functions H ij and L ij satisfy the condition h ij w ≤ H ij ( w ) ≤ h ij w, L ij ( w ) ≤ l ij w, with h = 16 , h = 12 , h = 19 , h = 13 , h = 16 , h = 14 , h = 12 , h = 23 l = 115 , l = 120 , l = 120 , l = 115 . The functions H ij and L ij can be built in a similar way as in [21] by choosing, for example, H ( w ) = ( w, ≤ w ≤ , w + , w ≥ , L ( w ) = 111 (cid:0) (cid:0) w − π (cid:1)(cid:1) . The choice [ a , b ] = [ a , b ] = [1 / , /
4] gives c = 1 / , c = 45 √ / , c = c = c = 1 / , c = 45 √ / ,m = 8 , M = 16 , m = 384 / , M = 768 / . We have that β [ γ ] = β [ γ ] = 34 , β [ γ ] = β [ γ ] = 14 , β [ γ ] = 23 , β [ γ ] = 481 ,β [ γ ] = 13 , β [ γ ] = 581 , δ [1] = δ [1] = δ [1] = δ [1] = 1 . Since K ij ( s ) = k i ( η ij , s ) we obtain Z K ( s ) ds = Z K ( s ) ds = 332 , Z / / K ( s ) ds = Z / / K ( s ) ds = 116 , Z K ( s ) ds = Z K ( s ) ds = 11972 , Z / / K ( s ) ds = Z / / K ( s ) ds = 3985497664 . Then, for ρ = 1 / ρ = 1 and ρ = 11, we have (the constants that follow have beenrounded to 2 decimal places unless exact)inf n f ( t, u, v ) : ( t, u, v ) ∈ [1 / , / × [0 , / × [0 , / o = f (1 / , , > . ρ , sup n f ( t, u, v ) : ( t, u, v ) ∈ [0 , × [0 , × [0 , o = f (1 , , < . ρ , sup n f ( t, u, v ) : ( t, u, v ) ∈ [0 , × [0 , × [0 , o = f (1 , , < . ρ , inf n f ( t, u, v ) : ( t, u, v ) ∈ [1 / , / × [11 , × [0 , o = f (1 / , , > . ρ , inf n f ( t, u, v ) : ( t, u, v ) ∈ [1 / , / × [0 , × [11 , o = f (1 / , , > . ρ , that is the conditions (I ρ ) ⋆ , (I ρ ) and (I ρ ) are satisfied; therefore the system (3.3) hasat least two positive solutions in K . Acknowledgments
The authors would like to thank Dr. Ing. Antonio Madeo of the Dipartimento diIngegneria Informatica, Modellistica, Elettronica e Sistemistica - DIMES, Universit`a dellaCalabria, for shedding some light on the physical interpretation of the problem.
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Gennaro Infante, Dipartimento di Matematica ed Informatica, Universit`a della Cal-abria, 87036 Arcavacata di Rende, Cosenza, Italy
E-mail address : [email protected] Paolamaria Pietramala, Dipartimento di Matematica ed Informatica, Universit`a dellaCalabria, 87036 Arcavacata di Rende, Cosenza, Italy
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