Positive solutions of BVPs on the half-line involving functional BCs
aa r X i v : . [ m a t h . C A ] D ec POSITIVE SOLUTIONS OF BVPS ON THE HALF-LINE INVOLVINGFUNCTIONAL BCS
GENNARO INFANTE AND SERENA MATUCCI
Abstract.
We study the existence of positive solutions on the half-line of a second orderordinary differential equation subject to functional boundary conditions. Our approachrelies on a combination between the fixed point index for operators on compact intervals,a fixed point result for operators on noncompact sets, and some comparison results forprincipal and nonprincipal solutions of suitable auxiliary linear equations. Introduction
In this manuscript we discuss the existence of multiple non-negative solutions of the bound-ary value problem (BVP)(1.1) ( p ( t ) u ′ ( t )) ′ + f ( t, u ( t )) = 0 , t ≥ ,αu (0) − βu ′ (0) = B [ u ] , u (+ ∞ ) = 0 , where p, f are continuous functions on their domains, α > β ≥ B is a suitablefunctional with support in [0 , R ]. The functional formulation of the boundary conditions(BCs) covers, as special cases , the interesting setting of (not necessarily linear) multi-pointand integral BCs; there exists a wide literature on this topic, we refer the reader to the recentpaper [11] and references therein.The approach that we use to solve the BVP (1.1), in the line of the papers [7, 8, 10, 19, 20],consists in considering two auxiliary BVPs separately, the first one on the compact interval[0 , R ], where B has support and f is nonnegative, and the second one on the half-line[ R, ∞ ), where f is allowed to change its sign. Unlike the above cited articles, in which theproblem of gluing the solutions is solved with some continuity arguments and an analysisin the phase space, here both the auxiliary problems have the same slope condition in thejunction point (namely the condition u ′ ( R ) = 0), which simplifies the arguments. This kindof decomposition is some sort of an analogue of one employed by Boucherif and Precup [2]utilized for equations with nonlocal initial conditions, where the associated nonlinear integral Mathematics Subject Classification.
Primary 34B40, secondary 34B10, 34B18.
Key words and phrases.
Fixed point index, cone, positive global solution, functional boundary condition. perator is decomposed into two parts, one of Fredholm-type (that takes into account thefunctional conditions) and another one of Volterra-type.We make the following assumptions on the terms that occur in (1.1). • p : [0 , + ∞ ) → (0 , + ∞ ) is continuous, with(1.2) P = Z + ∞ R p ( t ) dt < + ∞ . • f : [0 , + ∞ ) × [0 , + ∞ ) → R is continuous, with f ( t, v ) ≥ t, v ) ∈ [0 , R ] × [0 , + ∞ ), f ( t,
0) = 0 for t ≥ • There exist two continuous functions b , b : [ R, + ∞ ) → R , with b ≥
0, and twonondecreasing C -functions F , F : [0 , + ∞ ) → [0 , + ∞ ), with F j (0) = 0, F j ( v ) > v > j = 1 ,
2, such that(1.3) b ( t ) F ( v ) ≤ f ( t, v ) ≤ b ( t ) F ( v ) , for all ( t, v ) ∈ [ R, + ∞ ) × [0 , + ∞ ) . • For j = 1 , v → + F j ( v ) v < + ∞ . • Let b − , b +1 be the negative and the positive part of b , i.e. b − ( t ) = max { , − b ( t ) } , b +1 ( t ) = max { , b ( t ) } . Then B − = Z + ∞ R b − ( t ) dt < + ∞ , (1.5) Z + ∞ R p ( t ) Z tR b +1 ( s ) dsdt = + ∞ , (1.6)Notice that (1.2), (1.6) imply Z ∞ R b +1 ( t ) dt = + ∞ , and, in particular, the function b cannot be negative in a neighbourhood of infinity.As we will show in Section 3 (see also Theorem 4.1), if the condition (1.6) is not satisfied,then our approach leads to the existence of a bounded non-negative solution of the differentialequation in (1.1), namely a solution of the problem(1.7) ( p ( t ) u ′ ( t )) ′ + f ( t, u ( t )) = 0 , t ≥ ,αu (0) − βu ′ (0) = B [ u ] , u ≥ , ∞ ) . The problem of the existence and multiplicity of the solutions for the equation in (1.1),which are non-negative in the interval [0 , R ] and satisfy the functional BCs and the additionalassumptions at u ′ ( R ) = 0, is considered in Section 2 and is solved by means of the classicalfixed point index for compact maps. A BVP on [ R, + ∞ ) is examined in Section 3, where e deal with the existence of positive global solutions which have zero initial slope and arebounded or tend to zero at infinity. This second problem is solved by using a fixed pointtheorem for operators defined in a Frech´et space, by a Schauder’s linearization device, see [5,Theorem 1.3], and does not require the explicit form of the fixed point operator, but onlysome a-priori bounds. These estimates are obtained using some properties of principal andnonprincipal solutions of auxiliary second-order linear equations, see [15, Chapter 11] and [9].Finally, the existence and multiplicity of solutions for the BVP (1.1) and (1.7) is obtained inSection 4, thanks to the fact that the problem in [ R, + ∞ ) has at least a solution for everyinitial value u ( R ) sufficiently small. An example completes the paper.2. An auxiliary BVP on the compact interval [0 , R ]In this Section we investigate the existence of multiple positive solutions of the BVP(2.1) ( p ( t ) u ′ ( t )) ′ + f ( t, u ( t )) = 0 , t ∈ [0 , R ] ,αu (0) − βu ′ (0) = B [ u ] , u ′ ( R ) = 0 . First of all we recall some results regarding the linear BVP(2.2) − ( p ( t ) u ′ ( t )) ′ = 0 , t ∈ [0 , R ] ,αu (0) − βu ′ (0) = 0 , u ′ ( R ) = 0 . It is known, see for example [18], that the Green’s function k for the BVP (2.2) is given by k ( t, s ) := 1 α βp (0) + α R s p ( µ ) dµ, s ≤ t, βp (0) + α R t p ( µ ) dµ, s ≥ t. and satisfies the inequality (see [18, Lemma 2.1]) c ( t )Φ( s ) ≤ k ( t, s ) ≤ Φ( s ) , ( t, s ) ∈ [0 , R ] , where(2.3) Φ( s ) := βαp (0) + Z s p ( µ ) dµ, and c ( t ) := βp (0) + α R t p ( µ ) dµ βp (0) + α R R p ( µ ) dµ . Note that the constant function 1 α solves the BVP − ( p ( t ) u ′ ( t )) ′ = 0 , t ∈ [0 , R ] ,αu (0) − βu ′ (0) = 1 , u ′ ( R ) = 0 . e associate to the BVP (2.1) the perturbed Hammerstein integral equation(2.4) u ( t ) = F u ( t ) + H [ u ] α := T u ( t ) , where F u ( t ) := Z R k ( t, s ) f ( s, u ( s )) ds. We seek fixed points of the operator T in a suitable cone of the space of continuousfunctions C [0 , R ], endowed with the usual norm k w k := max {| w ( t ) | , t ∈ [0 , R ] } .We recall that a cone K in a Banach space X is a closed convex set such that λ x ∈ K for x ∈ K and λ ≥ K ∩ ( − K ) = { } . In the following Proposition we recall the mainproperties of the classical fixed point index for compact maps, for more details see [1, 12].In what follows the closure and the boundary of subsets of a cone K are understood to berelative to K . Proposition 2.1.
Let X be a real Banach space and let K ⊂ X be a cone. Let D be an openbounded set of X with ∈ D K and D K = K , where D K = D ∩ K . Assume that T : D K → K is a compact operator such that x = T x for x ∈ ∂D K . Then the fixed point index i K ( T, D K ) has the following properties: ( i ) If there exists e ∈ K \ { } such that x = T x + λe for all x ∈ ∂D K and all λ > ,then i K ( T, D K ) = 0 . ( iii ) If T x = λx for all x ∈ ∂D K and all λ > , then i K ( T, D K ) = 1 . (iv) Let D be open bounded in X such that D K ⊂ D K . If i K ( T, D K ) = 1 and i K ( T, D K ) =0 , then T has a fixed point in D K \ D K . The same holds if i K ( T, D K ) = 0 and i K ( T, D K ) = 1 . The assumptions above allow us to work in the cone(2.5) K := { u ∈ C [0 , R ] : u ≥ , min t ∈ [ a,b ] u ( t ) ≥ c k u k} , a type of cone firstly used by Krasnosel’ski˘ı, see [16], and D. Guo, see e.g. [12]. In (2.5)[ a, b ] is a suitable subinterval of [0 , R ] and c := min t ∈ [ a,b ] c ( t ), with c ( t ) given by (2.3). Wehave freedom of choice of [ a, b ], with the restriction a > β = 0. Note also that theconstant function equal to r ≥ r ) belongs to K ,so K = { } .Regarding the functional H we assume that • H : K → [0 , + ∞ ) is continuous and maps bounded sets in bounded sets. ith these ingredients it is routine to show that T leaves K invariant and is compact. Wemake use of the following open bounded set (relative to K ) K ρ := { u ∈ K : k u k < ρ } . We now employ some local upper and lower estimates for the functional H , in the spiritof [13, 14]. We begin with a condition which implies that the index is 1. Lemma 2.2.
Assume that (I ρ ) there exists ρ > , such that the following algebraic inequality holds: (2.6) 1 m f ρ + 1 α H ρ < ρ, where f ρ := max ( t,u ) ∈ [0 ,R ] × [0 ,ρ ] f ( t, u ) , H ρ := sup u ∈ ∂K ρ H [ u ] and m := sup t ∈ [0 ,R ] Z R k ( t, s ) ds. Then i K ( T, K ρ ) is .Proof. Note that if u ∈ ∂K ρ then we have 0 ≤ u ( t ) ≤ ρ for every t ∈ [0 , R ]. We prove that µ u = T u for every µ ≥ u ∈ ∂K ρ . In fact, if this does not happen, there exist µ ≥ u ∈ ∂K ρ such that, for every t ∈ [0 , R ], we have µ u ( t ) = T u ( t ) = F u ( t ) + 1 α H [ u ] . Then we obtain, for t ∈ [0 , R ],(2.7) µ u ( t ) ≤ Z R k ( t, s ) f ρ ds + 1 α H ρ ≤ m f ρ + 1 α H ρ . Taking the supremum for t ∈ [0 , R ] in (2.7) and using the inequality (2.6) we obtain µρ < ρ ,a contradiction that proves the result. (cid:3) Now we give a condition which implies that the index is 0 on the set K ρ . Lemma 2.3.
Assume that (I ρ ) there exists ρ > such that the following algebraic inequality holds: (2.8) 1 M f ρ + 1 α H ρ [ u ] > ρ, where f ρ := min ( t,u ) ∈ [ a,b ] × [ cρ,ρ ] f ( t, u ) , H ρ := inf u ∈ ∂K ρ H [ u ] and M := inf t ∈ [ a,b ] Z ba k ( t, s ) ds. Then i K ( T, K ρ ) is . roof. Note that the constant function 1 belongs to K . We prove that u = T u + λ u ∈ ∂K ρ and for every λ ≥
0. If this is false, there exist u ∈ ∂K ρ and λ ≥ u = T u + λ
1. Then we have, for t ∈ [ a, b ],(2.9) ρ ≥ u ( t ) = F u ( t ) + 1 α H [ u ] + λ ≥ F u ( t ) + 1 α H [ u ] ≥ Z ba k ( t, s ) f ( s, u ( s )) ds + 1 α H [ u ] ≥ Z ba k ( t, s ) f ρ ds + 1 α H ρ [ u ] ≥ M f ρ + 1 α H ρ [ u ] . Using the inequality (2.8) in (2.9) we obtain ρ > ρ , a contradiction that proves the result. (cid:3)
In view of the Lemmas above, we may state our result regarding the existence of one ormore nontrivial solutions. Here, for brevity, we provide sufficient conditions for the existenceof one, two or three solutions. It is possible to obtain more solutions, by adding moreconditions of the same type, see for example [17].
Theorem 2.4.
The BVP (2.1) has at least one non-negative solution u , with ρ < u ( R ) <ρ if either of the following conditions holds. ( S ) There exist ρ , ρ ∈ (0 , + ∞ ) with ρ < ρ such that (I ρ ) and (I ρ ) hold. ( S ) There exist ρ , ρ ∈ (0 , + ∞ ) with ρ < ρ such that (I ρ ) and (I ρ ) hold.The BVP (2.1) has at least two non-negative solutions u and u , with ρ < u ( R ) < ρ t, nd therefore u ′ ≥ , R ], which, in turn, implies u ( R ) = k u k .Assume now that condition ( S ) holds, then we obtain in addition that i K ( T, K ρ ) = 0. ByProposition 2.1 we obtain the existence of a second solution u of the integral equation (2.4)in K ρ \ K ρ . A similar argument as above holds for the monotonicity of u .The remaining cases are dealt with in a similar way. (cid:3) An auxiliary BVP on the half-line [ R, ∞ ]In this section we state sufficient conditions for the existence of solutions of the follow-ing BVP ( p ( t ) u ′ ( t )) ′ + f ( t, u ( t )) = 0 , t ≥ R, (3.1) u ′ ( R ) = 0 , u ( R ) = u , u ( t ) > , lim t →∞ u ( t ) = 0 , (3.2)where u > (cid:0) p ( t ) u ′ ( t ) (cid:1) ′ + g ( t ) | u ( t ) | β sgn( u ( t )) = 0 , if β > g is allowed to take negative values, has solutions which tend to infinity in finitetime, see [4, 3]. Moreover, again in the superlinear case β >
1, if g is non-negative withisolated zeros, then (3.3) may have solutions which change sign infinitely many times in theleft neighborhood of some ¯ t > R , and so these solutions are not continuable to infinity, see[6]. Further, even if global solutions exist (that is, solutions which are defined in the wholehalf-line [ R, + ∞ )), their positivity is not guaranteed in general. Indeed, (3.1) may exhibitthe coexistence of nonoscillatory and oscillatory solutions; further, nonoscillatory solutionsmay have an arbitrary large number of zeros.The problem (3.1), (3.2) has been consider in [9] for nonlinear equations with p -laplacianoperator and nonlinear term f ( t, u ( t )) = b ( t ) F ( u ( t )). We address the reader to such a paperfor a complete discussion on the issues related to the BVP (3.1), (3.2) and for a review ofthe existing literature on related problems. The approach used in [9] to solve the BVP wasbased on a combination of the Schauder’s (half)-linearization device, a fixed point result inthe Fr´echet space of continuous functions on [ R, + ∞ ), and comparison results for principaland nonprincipal solutions of suitable auxiliary half-linear equations, which allow to findgood upper and lower bounds for the solutions of the (half)-linearized problem. The sameapproach, with minor modifications, allow us to treat also the present case with a general onlinearity f ( t, u ( t )), under the assumptions (1.3), (1.4). In the following Proposition werecall the fixed-point result [9, Theorem 1], based on [5, Theorem 1.3], in the form suitablefor the present problem. Proposition 3.1.
Let J = [ t , ∞ ) . Consider the BVP (3.4) ( p ( t ) u ′ ) ′ + f ( t, u ) = 0 , t ∈ J,u ∈ S, where f is a continuous function on J × R and S is a subset of C ( J, R ) . Let g be a continuousfunction on J × R such that g ( t, c, c ) = f ( t, c ) for all ( t, c ) ∈ J × R , and assume that there exist a closed convex subset Ω of C ( J, R ) and a bounded closed subset S of S ∩ Ω which make the problem (3.5) ( p ( t ) y ′ ) ′ + g ( t, y, q ) = 0 , t ∈ J,y ∈ S uniquely solvable for all q ∈ Ω . Then the BVP (3.4) has at least one solution in Ω . In view of this result, no topological properties of the fixed-point operator are needed tobe checked, since they are a direct consequence of a-priori bounds for the solutions of the“linearized” problem (3.5).We state here the existence results in the form which will be used in the next section,addressing to [9, Theorem 2] for the general result, in case of a factored nonlinearity. Forreader’s convenience we provide a short proof, focusing only on those points which requiresome adjustments due to the more general nonlinearity. We point out that the present resultsare obtained by using the Euler equation(3.6) t y ′′ + nty ′ + (cid:18) n − (cid:19) y = 0 , n > , as a Sturm majorant of the linearized auxiliary equation (see also [9, Corollary 3]), butany other linear equation having a positive decreasing solution can be used as a majorantequation, obtaining different conditions. The first result states sufficient conditions for theexistence of a global positive solution of (3.1), bounded on [ R, + ∞ ). Theorem 3.2.
Assume that (1.2) – (1.5) hold, and let M j ( d ) = sup v ∈ (0 ,d ] F j ( v ) v , j = 1 , here d > is a fixed constant. If (3.7) B − ≤ log 2 M ( d ) P , and (3.8) p ( t ) ≥ t n , b ( t ) ≤ ( n − M ( d ) t n − , for all t ≥ R, for some n > , then for every u ∈ (0 , d/ the equation (3.1) has a solution u , satisfying u ( R ) = u , u ′ ( R ) = 0 , < u ( t ) ≤ u for t ≥ R .Proof. The result follows from [9, Theorem 2 and Corollary 3], with some technical adjust-ments due to the actual general form of the nonlinearity. Indeed, it is sufficient to observethat, for every continuous function q : [ R, ∞ ) → (0 , d ] fixed, the equations(3.9) ( p ( t ) w ′ ( t )) ′ − M ( d ) b − ( t ) w ( t ) = 0is a Sturm minorant of the linearized equation(3.10) ( p ( t ) u ′ ( t )) ′ + f ( t, q ( t )) q ( t ) u ( t ) = 0 , and (3.6) is a Sturm majorant, due to (3.8) (see for instance [15]). Since (3.6) is nonoscil-latory, and has the solution y ( t ) = t − ( n − / , which is positive decreasing on [ R, + ∞ ), from[9, Lemma 3] the solution of (3.10) satisfying the initial conditions u ( R ) = u , u ′ ( R ) = 0exists and is positive on [ R, + ∞ ), since it satisfies u ( t ) ≥ w ( t ) for all t ≥ R , where w isthe principal solution of (3.9). Further, double integration of (3.10) on [ R, t ], t > R , gives U ( t ) ≤ u + M ( d ) B − Z tR U ( s ) p ( s ) ds, where U ( t ) = max s ∈ [ R,t ] u ( s ) , and the Gronwall lemma, together with (3.7), gives the upper bound u ( t ) ≤ U ( t ) ≤ u .Thus, put S = { q ∈ C [ t , ∞ ) : q ( R ) = u , q ′ ( R ) = 0 , q ( t ) > t ≥ R } , Ω = (cid:8) q ∈ C [1 , ∞ ) : q ( R ) = u , q ′ ( R ) = 0 , w ( t ) ≤ q ( t ) ≤ u (cid:9) . We have S ∩ Ω = Ω and the set S = T (Ω), where T is the operator which maps every q ∈ Ωinto the unique solution of (3.10) satisfying the initial conditions u ( R ) = u , u ′ ( R ) = 0,satisfies S ⊂ S ∩ Ω = Ω and is bounded in C [ R, + ∞ ) (for the detailed proof see [9,Theorem 2]). Then Proposition 3.1 can be applied, and the existence of a solution of (3.1)in the set S ∩ Ω is proved. (cid:3) emark 3.3. Clearly, the result in Theorem 3.2 holds also if we allow a different upperbound for the solution. More precisely, if we look for a solution satisfying 0 < u ( t ) ≤ k u with k >
1, then it is sufficient to put log k instead of log 2 in (3.7), and we get the existenceof global positive solutions of the Cauchy problem associated with (3.1), for every u suchthat 0 < ku ≤ d . Remark 3.4.
Since M ( d ) is nondecreasing, condition (3.7) can be seen as an upper boundfor the values of u for which (3.1) has a global bounded solution. For instance, if F ( v ) = v β , β >
1, then M ( d ) = d β − and (3.7) can be read as2 u ≤ (cid:18) log 2 P B − (cid:19) β − if B − = 0 , while, if F ( v ) = v , then M ( d ) = 1 for all d >
0, and (3.1), (3.2) has solution for all u > Theorem 3.5.
Assume that (1.2) – (1.6) hold, and that d > exists such that (3.7) and (3.8) are satisfied for some n > . Then, for every u ∈ (0 , d/ , the BVP (3.1) , (3.2) has at leasta solution u , satisfying < u ( t ) ≤ u for t ∈ [ R, ∞ ) , u ′ ( t ) < for large t. Proof.
The proof is analogous to the second part of the proof of [9, Theorem 2], with obvi-ous modifications due to the more general form of the nonlinear term here considered. Inparticular notice that (3.1) with conditions (1.3) gives the inequality( p ( t ) u ′ ( t )) ′ F ( u ( t )) + b ( t ) ≤ u of (3.1) and all t ≥ R . Thus the arguments in the proof of [9,Theorem 2] apply also to the present case. (cid:3) We conclude this Section pointing out that the case b ( t ) ≥ t ≥ R is included in theprevious results, and in this case Theorems 3.2, 3.5 have a more simple statement. Indeed, B − = 0, and therefore (1.5) and (3.7) are trivially satisfied. Further, every solution of (3.1)is nonincreasing on the whole half-line. . The main result
Combining Theorems 2.4 and 3.2 or 2.4 and 3.5, we obtain an existence result for one ormore solutions of the BVP (3.1) and (3.2), respectively. We limit ourself to state resultsfor the existence of one or two solutions, for sake of simplicity. Clearly, as pointed out inSection 2, adding more conditions, with similar arguments it is possible to obtain sufficientconditions for the existence of three solutions (see Theorem 2.4) or more solutions.
Theorem 4.1.
Suppose that (1.2) – (1.5) are satisfied, and that there exist ρ , ρ ∈ (0 , + ∞ ) ,with ρ < ρ , such that either ( S ) or ( S ) holds. If n > exists, such that (3.7) , (3.8) aresatisfied with d = 2 ρ , then the BVP (1.7) has at least one solution u , with u ( t ) ≤ ρ forall t ≥ .If, in addition, there exists ρ ∈ (0 , + ∞ ) , with ρ < ρ , such that either ( S ) or ( S ) holds,and (3.7) , (3.8) are satisfied with d = 2 ρ , then the BVP (1.7) has an additional solution u , with u ( t ) ≤ ρ . Notice that, from Theorem 2.4, the solutions u , u satisfy ρ < u ( R ) < ρ < u ( R ) < ρ and therefore they are distinct solutions. Further, since solutions on [0 , R ] are increasing,then u ( t ) < ρ , u ( t ) < ρ , for all t ∈ [0 , R ].In case also assumption (1.6) is satisfied, from the above Theorem we obtain sufficientconditions for the existence of solutions of the BVP (1.1). Theorem 4.2.
Suppose that (1.2) – (1.6) are satisfied, and that there exist ρ , ρ ∈ (0 , + ∞ ) ,with ρ < ρ , such that either ( S ) or ( S ) holds. If n > exists, such that (3.7) , (3.8) aresatisfied with d = 2 ρ , then the BVP (1.1) has at least one solution u , with u ( t ) ≤ ρ forall t ≥ .If, in addition, there exists ρ ∈ (0 , + ∞ ) , with ρ < ρ , such that either ( S ) or ( S ) holds,and (3.7) , (3.8) are satisfied with d = 2 ρ , then the BVP (1.1) has an additional solution u , with u ( t ) ≤ ρ . We conclude this Section with the following illustration of our results.
Example 4.3.
Let us consider the BVP(4.1) ( p ( t ) u ′ ( t )) ′ + b ( t ) u ( t ) = 0 , t ≥ ,u (0) = B [ u ] , u (+ ∞ ) = 0 , where p ( t ) = , ≤ t ≤ ,t n , t > , b ( t ) = , ≤ t ≤ , sin + ( π t ) − µt sin − ( π t ) , t > , or some n > µ >
0, and B [ u ] = 12 sZ u ( t ) dt. The definition of the operator B leads to the natural choice [0 , R ] = [0 , m = 2. The choice of [ a, b ] = [1 / ,
1] leads to c = 1 / M = 4.Furthermore note that f ρ = ρ , f ρ = ρ and12 √ ρ ≥ B [ u ] ≥ sZ / u ( t ) dt ≥ √ ρ, for every u ∈ ∂K ρ . Observe that the inequalities14 f ρ + H ρ [ u ] ≥ (cid:0) ρ + √ ρ (cid:1) > ρ , f ρ + H ρ ≤ (cid:0) ρ + √ ρ (cid:1) < ρ , f ρ + H ρ [ u ] ≥ (cid:0) ρ + √ ρ (cid:1) > ρ . are satisfied for ρ = 1 / ρ = 3 / ρ = 15. Thus ( S ) and ( S ) hold for ρ = 1 / ρ = 3 / ρ = 15. Note that (1.2)–(1.5) are satisfied, with P = 1 / ( n − F ( v ) = F ( v ) = v , b ( t ) = b ( t ) , b ( t ) = 1, and B − < µ holds. Straightforward calculations showthat also (1.6) is satisfied. Since M ( d ) = M ( d ) = d , applying Theorem 4.2 we get thefollowing result: • If ( n − ≥ u , with 0 ≤ u ( t ) ≤ /
2, for every µ ≤ n −
1) log 2 / • If ( n − ≥
120 then (4.1) has at least two distinct solution u , u , with 0 ≤ u ( t ) ≤ /
2, 0 ≤ u ( t ) ≤
30, for every µ ≤ ( n −
1) log 2 / Acknowledgments
G. Infante and S. Matucci were partially supported by G.N.A.M.P.A. - INdAM (Italy).
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Gennaro Infante, Dipartimento di Matematica e Informatica, Universit`a della Calabria,87036 Arcavacata di Rende, Cosenza, Italy
Email address : [email protected] Serena Matucci, Department of Mathematics and Computer Science “Ulisse Dini”, Uni-versity of Florence, 50139 Florence, Italy
Email address : [email protected]@unifi.it