Solvability of multipoint BVPs at resonance for various kernels

In this paper, the Mawhin’s continuation theorem in the theory of coincidence degree has been used to investigate the existence of solutions for a class of nonlinear second-order differential systems of equations in ℝ

Journal of Science Technology and Food 20 (1) (2020) 3-16 SOLVABILITY OF MULTIPOINT BVPs AT RESONANCE FOR VARIOUS KERNELS Phan Dinh Phung Ho Chi Minh City University of Food Industry Email: pdphungvn@gmail.com Received 20 December 2019; Accepted 21 February 2020 ABSTRACT In this paper, the Mawhin’s continuation theorem in the theory of coincidence degree has been used to investigate the existence of solutions for a class of nonlinear second-order differential systems of equations in ℝ𝑛 associated with a multipoint boundary conditions at resonance This is the first time that a resonant boundary condition of multipoint with large dimension of the kernel has been considered An example has also been provided to illustrate the result Keywords: Coincidence degree, Fredholm operator of index zero, multipoint boundary value problem, resonance INTRODUCTION In the theory of partial differential equations such as the method of separation of variables, we encounter differential equations for several parameters with some requirement of solutions which is called multi-point boundary condition This then leads to an extensive development of spectral theory with multi-parameter [1] Many multi-point boundary value problems (for short, BVPs) are established when looking for solutions to free-boundary problems [2] Multipoint BVPs can also arise in other ways like physics and mechanics [3, 4] In recent decades, the nonlinear multi-point BVPs especially at resonance have received much attention of many mathematicians, for instance, with the results of higher order BVPs [5, 6], the fractional order BVPs [7, 8], the positive solutions [9] In particular, Phung P.D et al also had some contributions on this topic [10-12] This note is to study the existence of solutions to the m-point BVPs in ℝ𝑛 u (t ) = f (t , u (t ), u (t )), t  (0,1),  m−2   u (  ) = 0, u (1) = Ai u (i ),   i =1  (1.1) where 𝜃 is zero element in ℝ𝑛 , 𝑓: ℝ2𝑛 → ℝ𝑛 satisfies the Carathéodory condition, that is, (a) 𝑓(⋅, 𝑢, 𝑣) is Lebesgue measurable for every (𝑢, 𝑣) ∈ ℝ𝑛 × ℝ𝑛 , (b) 𝑓(𝑡,⋅,⋅) is continuous on ℝ𝑛 × ℝ𝑛 for almost every 𝑡 ∈ [0, 1], (c) for each compact set 𝐾 ⊆ ℝ2𝑛 , the function ℎ𝐾 (𝑡) = sup {|𝑓(𝑡, 𝑢, 𝑣)|: (𝑢, 𝑣) ∈ 𝐾} is Lebesgue integrable on [0, 1], where | ⋅ | is the max-norm in ℝ𝑛 , and 𝜂1 , 𝜂2 , … , 𝜂𝑚−2 ∈ (0,1), 𝑚 ≥ 3, and 𝐴1 , 𝐴2 , … , 𝐴𝑚−2 are square matrices of order 𝑛 satisfying (G1) The matrix I −  i =1 i Ai is invertible, m− Phan Dinh Phung ( (G3) (  (G2) m−2 i =1 m− i =1  2i Ai )(  ) =  i =1 Ai ) ( A or (  m−2 i =1 m− Ai = i m−2 i =1 m−2 i =1 )(   A ) , A ) = I , here I stands for the identity matrix of order m−2 Ai i =1 i i i n In most of boundary conditions, the operator 𝐿𝑢 = 𝑢′′ defined on some Banach spaces is invertible Such a case is the so-called non-resonant; otherwise, the more complicated one called resonant In [13], Gupta first studied the existence of solutions for 𝑚-point BVP of the form u (t ) = f (t , u (t ), u (t )), t  (0,1),  m−2   u (  ) = 0, u (1) =  i u (i ),   i =1  at resonance with the resonant condition ∑𝑚−2 𝑖=1 𝛼𝑖 = After that, Feng [14] and Ma [15] also achieved these similar results with some improvement on the nonlinear term The main tool is Mawhin continuation theorem This strongly depends on the dimension of ker 𝐿 and most of results consider only the case that dim ker 𝐿 = 1, because of which constructing projections P and Q (in Mawhin’s method) is quite simple Therefore, the larger dimension of ker 𝐿 is, the more difficult the resonant problem is In this paper, we aim to generalize these works so that considering the multi-point BVPs (1.1) with the difficulty of resonance that ≤ dim ker 𝐿 ≤ 𝑛 by using the Mawhin's continuation theorem In addition, an example to illustrate the main result, especially the resonant conditions, was provided PRELIMINARIES We begin this section by recalling some definitions and abstract results from the coincidence degree theory For more details on the Mawhin’s theory, we refer to [16, 17] Let X and Z be two Banach spaces Definition 2.1 ([Ch III-16, 17]) Let L : domL  X → Z be a linear operator Then one says that L is a Fredholm operator provided that (i) (ii) ker L is finite dimensional, Im L is closed and has finite codimension Then the index of L is defined by ind L = dim ker L − codim Im L It follows from Definition 2.1 that if L is a Fredholm operator of index zero then there exist continuous projections P : X → X and Q : Z → Z such that Im P = ker L, ker Q = Im L, X = ker L  ker P, Z = Im L  Im Q Furthermore, the restriction of L on domL ker P, LP : domL ker P, → Im L, is invertible We will denote its inverse by K P The generalized inverse of L denoted by K P ,Q = K P ( I − Q ) On the other hand, for every isomorphism J : Im Q → ker L, the mapping JQ + K P ,Q : Z → domL is an isomorphism, and Solvability of multipoint BVPs at resonance for various kernels ( JQ + K ) P ,Q −1 u = ( L + J −1 P ) u, u  domL Now let  be an open bounded subset of X such that domL     Definition 2.2 ([Ch III-16, 17]) Let L be a Fredholm operator of index zero The operator N : X → Z is said to be L-compact in  if ( ) • the map QN :  → Z is continuous and QN  is bounded in Z, • the map KP,Q N :  → X is completely continuous Moreover, we say that N is L-completely continuous if it is L-compact on every bounded set in X Note that if L is a Fredholm operator of index zero and N is L-compact in  then the existence of a solution to equation Lu = Nu , u  is equivalent to the existence of a fixed point of  in  , where  = P + ( JQ + K P,Q ) N This can be guaranteed by the following theorem due to Mawhin [16] Theorem 2.1 Let   X be open and bounded, L be a Fredholm mapping of index zero and N be L-compact on  Assume that the following conditions are satisfied: i) Lu   Nu for every ( u,  )  (( domL \ ker L ) )  ( 0,1) ; ii) QNu  for every u  ker L  ; iii) for some isomorphism J : Im Q → ker L we have deg B ( JQN |ker L ;   ker L, )  0, where Q : Z → Z is a projection given as above Then the equation 𝐿𝑢 = 𝑁𝑢has at least one solution in domL   Next, to achieve the existence of problem (1.1) by applying Theorem 2.1, we introduce the spaces X = C1 [0,1];R n endowed with the norm ( ) u = max  u (  , u   ) where  stands for the sup-norm and Z = L1 [0,1];R n endowed with the Lebesgue norm denoted by Further, we use the Sobolev space defined by X = u  X : u   Z   X Then we define the operator L : domL  X → Z by Lu = u , where m−2   domL = u  X : u  ( ) =  , u (1) =  Ai (i ) i =1   It is easy to see that u  X  u ( t ) = u ( ) + u  ( ) t + I 0+ Lu ( t ) , where I 0k+ z ( t ) =  ( t − s ) t k −1 z ( s ) ds, for k  1, 2 Thus, by substituting the boundary conditions, dom L is easily written by Phan Dinh Phung   domL = u  X : u (t ) = u ( 0) + I 02+ Lu (t ) with u ( 0) =  ( Lu ) , (2.1) where • m−2  = I −  Ai i =1 •  : Z → R n is a continuous linear mapping defined by m−2  ( z ) =  Ai I o2 z (i ) − I o2 z (1) , z  Z + + (2.2) i =1 Hence, it is not difficult to show that ker L = u  X : u (t ) = c, t [0,1], c  ker M   ker M Moreover we have Im L = z  Z :  ( z )  Im M  Indeed, let z  Im L so that z = Lu for some u  domL From (2.1) we have Mu ( ) =  ( z ) which implies  ( z )  Im M Conversely, if z  Z such that  ( z ) = M   Im M then it is easy to see that z = Lu , where u  domL, defined by u ( t ) =  + I o2+ z ( t ) This shows that z  Im L Now we prove some useful lemmas The methods of the proofs are similar to some previous works [6, 10-12] Lemma 2.1 Assume that (G1)-(G3) hold Then the operator L is a Fredholm operator of index zero Proof Since  is continuous and Im M is closed in R n it is clear that Im L is a closed subspace of Z Further, we have dim ker L = dim ker M  n   It remains to show that codim Im L = dim ker L For this we consider the continuous linear mapping Q : Z → Z defined by, for z  Z , Qz ( t ) = ( I −  M ) D ( z ) , t [0,1] (2.3) where   m−2  m−2 1, if G holds , that is , A  ( )1   i  =  Ai , i =1   i =1   =  m−2  1  , if ( G3)2 holds, that is,   Ai  = I ,  i =1   and −1  m−2  D =   i2 Ai − I   i =1  Since (G1) holds, the matrix D exists It's necessary to note that if z ( t ) = h  R n , t [0,1], then Solvability of multipoint BVPs at resonance for various kernels m−2 i i =1 0  ( z ) =  Ai  (i − s ) hds −  (1 − s ) hds = D −1 h (2.4) From (G3), it's not difficult to show that  M and I −  M are two projections on Rn Moreover, one can prove, for two cases, that ( I −  M ) D −1 = D −1 ( I −  M ) (2.5) m−2 Indeed, if ( G 3)1 holds then I −  M =  Ai By (G2) we get (2.5) Otherwise, we have i =1 1  I −  M =  I +  Ai  Therefore 2 i =1  m−2 m−2 m−2 m−2    m−2  m−2      I +  Ai   i Ai − I  = i Ai  I +  Ai  −  I +  Ai  i =1 i =1 i =1    i =1  i =1     m−2  m−2   =  i2 Ai − I   I +  Ai  i =1  i =1   Hence (2.5) is proved This follows from (2.4) – (2.5) that Q z ( t ) = ( I −  M ) D ( Qz ) = ( I −  M ) DD −1Qz = ( I −  M ) Qz = ( I −  M ) z (t ) = ( I −  M ) z (t ) = Qz ( t ) , t  [0,1] Thus, the map Q is idempotent and consequently Q is a continuous projection Now we prove the following three assertions i) ker Q = Im L, ii) Z = Im L  Im Q, iii) Im Q = ker L, which allow us to complete the proof of the lemma In order to get i) we note that D−1 M = MD−1 , due to (2.5), which implies DM = MD So   Im M if and only if D  Im M Hence, for z  Z , z  ker Q  D ( z )  ker ( I −  M )  D ( z )  Im ( M )   ( z )  Im ( M )  z  Im L, which shows that ker Q = Im L Hence, we also obtain ii), that is, Z = ker Q  Im Q = Im L  Im Q Now, let z  Im Q Assume that z = Qz1 , for z1  Z Then we have  Mz ( t ) =  M ( I −  M ) D ( z1 ) =  , t [0,1], due to  M is a projection This implies Qz1  ker ( M )  ker M Therefore z  ker L Conversely, for each z  ker L, there exists   ker M such that z ( t ) =  for all t [0,1] Then we have Qz ( t ) = ( I −  M ) D ( z ) = ( I −  M ) D ( D−1  ) = ( I −  M )  =  = z (t ) , t [0,1], Hence z  Im Q and so we get Im Q = ker L Then Lemma 2.4 has proved Phan Dinh Phung Now we define an operator P : X → X by setting Pu ( t ) = ( I −  M ) u ( ) , t [0,1] (2.6) Lemma 2.2 We have i) The mapping P defined by (2.6) is a continuous projection satisfying the identities Im P = ker L, X = ker L  ker P ii) The linear operator KP : Im L → domL ker P can be defined by K P z ( t ) =  M  ( z ) + I 02+ z ( t ) , t  [0,1], (2.7) Moreover K P satisfies K P = ( L |domL ker P ) −1 and K P z  C z ,  m−2  where C = +  M * 1 + i Ai *  ( * is the maximum absolute column sum norm of 1=1   matrices) Proof i) It is clear that P is a continuous projection Further we have Im P = ker L Indeed, if v  Im P then there exists u  X such that v ( t ) = Pu ( t ) = ( I −  M ) u ( ) , t [0,1] (2.8) Thus  Mv ( t ) =  M ( I −  M ) u ( ) =  which implies that v  ker L, by the definition of ker L Conversely if v  ker L then v ( t ) =   ker M , t  [0,1] Then we deduce that Pv ( t ) = ( I −  M ) v ( ) = ( I −  M )  =  = v ( t ) , t [0,1] This shows that v  Im P Therefore we can conclude that Im P = ker L and consequently X = ker L  ker P ii) Let z  Im L Then we have  ( z )  Im M which implies that  ( z ) = M  , where   Rn It follows from (2.6) and (2.7) that PK P z ( t ) = ( I −  M ) K P z ( ) =  ( I −  M ) M  ( z ) =  , t [0,1] Thus KP z  ker P In addition, clearly  ( z )  Im M and  M is the projection, implying M  ( z ) =  M  ( z ) =  ( z ) Then, it is easy to show that K P z  domL So K P is well defined On the other hand, if u  domL  ker P then u ( t ) = u ( ) + I 02+ Lu ( t ) , with  Mu ( ) =  ( Lu ) ,  u ( )  Im ( M ) Thus Solvability of multipoint BVPs at resonance for various kernels K P Lu ( t ) =  M  ( Lu ) + I 02+ Lu ( t ) = ( M ) u ( 0) + I 02+ Lu ( t ) = u ( ) + I 02+ Lu ( t ) = u ( t ) by  M is a projection This deduces that K P = ( L |domL ker P ) −1 by LK P z ( t ) = z ( t ) , t [0,1], for all z  Im L Finally, from the definition of K P we have ( K P z ) ( t ) = I 01 z ( t ) , t [0,1] (2.9) + Combining (2.2), (2.7) and (2.9) we have • • • KP z  2 M *  ( z) + z ,  m−2   i =1   ( z )  1 +  i Ai *  z , ( K P z )   z These show that K P z  C z The lemma is proved Lemma 2.3 The operator N : X → Z defined by Nu (t ) = f (t , u (t ) , u  (t )) , a.e., t [0,1] is L-completely continuous Proof Let  be a bounded set in X Put R = sup  u : u  From the assumptions of the function f there exists a function mR  Z such that, for all u   we have Nu ( t ) = f (t , u ( t ) , u  (t ) )  mR (t ) , a.e., t [0,1] (2.10) It follows from (2.2), (2.13) and the identity QNu ( t ) = ( I −  M ) D ( Nu ) ( ) that QN  (2.11) is bounded and QN is continuous by using the Lebesgue's dominated convergence theorem We now prove that K P ,Q N is completely continuous Note that, for every u  , we have K P ,Q Nu ( t ) = K P ( I − Q ) Nu ( t ) = K P ( Nu − QNu )( t ) = KP  Nu − ( I −  M ) D ( Nu ) (t ) = I 02+ Nu ( t ) − t2 ( I −  M ) D ( Nu ) +  M  ( Nu ) , (2.12) and (K P ,Q Nu ) ( t ) = I 01+ Nu ( s ) ds − t ( I −  M ) D ( Nu ) (2.13) Further, it follows from (2.13) and the definition of  that  m−2   m−2   i =1   i =1   ( z )  1 + i Ai *  z  1 + i Ai *  mR , (2.14) Phan Dinh Phung Combining (2.10) and (2.12) – (2.14) we can find two positive constants C1 , C2 such that K P,Q Nu ( t )  C1 mR , ( K P,Q Nu ) ( t )  C2 mR , (2.15) for all t [0,1] and for all u  This shows that KP,Q Nu  max C1 , C2  mR , that is, K P ,Q N (  ) is uniformly bounded in X On the other hand, for t1 , t2 [0,1] with t1  t2, we have t2 a t1 K P,Q Nu ( t2 ) − K P,Q Nu ( t1 )   ds  Nu ( ) d + ( t2 − t1 )( I −  M ) D ( Nu )  C3 mR t2 − t1 , and t2 ( KP,Q Nu ) (t2 ) − ( KP,Q Nu ) (t1 )   mR ( s ) ds + C4 mR t2 − t1 , t1 which prove that the family K P ,Q N (  ) is equi-continuous in X Thanks to Arzelà-Ascoli theorem, K P ,Q N (  ) is a relatively compact subset in X Lastly, it is obvious that K P ,Q N is continuous Therefore, the operator N is L-completely continuous The proof of the theorem is completed MAIN RESULTS In this section we employ Theorem 2.1 to prove the existence of the solutions of problem (1.1) For this aim we assume that the following conditions hold: (B1) there exist the positive functions a, b, c  Z with ( I −M * + C )( a + b )  such that f ( t, u, v )  a (t ) u + b (t ) v + c (t ) , (3.1) for all t [0,1] and u, v  R n where C is the constant given in Lemma 2.2; (B2) there exists a positive 1 constant u ( t )  1 , t [0,1], then m−2 s i such that  A  ds  f ( , u ( ) , u  ( ) ) d  Im M ; i =1 i for each u  domL, if (3.2) (B3) there exists a positive constant  such that for any   Rn with   ker M and    , either  , QN ( )  or  , QN ( )  0, where , stand for the scalar product in Rn 10 (3.3) Solvability of multipoint BVPs at resonance for various kernels Lemma 3.1 Let 1 = u  domL \ ker L : Lu =  Nu,  [0,1] Then 1 is bounded in X Proof Let u 1 Assume that Lu =  Nu for    Then it is clear that Nu  Im L = ker Q, which implies  ( Nu )  Im M by the definition of Im L On the other hand, we have m−2 s s i 0  A  ds  f ( t , u ( ) , u  ( ) ) d = − ( Nu ) − M  ds  f (t , u ( ) , u  ( ) ) d i =1 i Hence we deduce that m−2 s i =1 i  Ai  ds  f ( , u ( ) , u  ( ) ) d  Im M By assumption (B2), there exists t0 [0,1] such that u ( t0 )  1 Then we get t t t0 u ( t ) = u ( t0 ) +  u  ( s ) ds  1 + u   and u  ( t )   u  ( s ) ds  u   Nu , (3.4) for all t [0,1] These give us Pu = ( I −  M ) u ( 0)  I −  M On the other hand, it is noted that ( Id X − P ) u  domL Then ( Id X * ( + Nu ) (3.5) ker P since P is a projection on X − P ) u = KP L ( Id X − P ) u  KP Lu  C Nu , (3.6) where the constant C is defined as in Lemma 2.5 and Id X is the identity operator on X Combining (3.5) and (3.6) obtains u = Pu + ( Id X − P ) u  Pu + ( Id X − P ) u  1 I −  M * + ( I − M * + C ) Nu (3.7) u + c (3.8) By (B1) and the definition of N we have Nu   f ( s, u ( s ) , u  ( s ) ) ds  a u  + b u  + c  ( a + b ) Combining (3.7) and (3.8) gives us Nu  1 I −  M 1− ( I −M (a * * + b 1)+ c + C )( a + b ) The last inequality and (3.4) deduce that sup u = sup max  u u1 u1  , u     + Therefore 1 is bounded in X The lemma is proved Lemma 3.2 The set 2 = u  ker L : Nu  Im L is a bounded subset in X 11 Phan Dinh Phung Proof Let u 2 Assume that u ( t ) = c, t [0,1], where c  ker M Since Nu  Im L we have  ( Nu )  Im M By the same arguments as in the proof of Lemma 3.1 we can point out that there exists t0 [0,1] such that u ( t0 )  1 Therefore u = u  = u ( t0 ) = c  1 So  is bounded in X The lemma is proved Lemma 3.3 The sets 3− = u  ker L : −u + (1 −  ) QNu =  ,  [0,1] and 3+ = u  ker L : u + (1 −  ) QNu =  ,  [0,1] are bounded in X provided that the first and the second part of (3.3) is satisfied, respectively Proof Case 1:  , QN   Let u 3− Then there exists   ker M such that u ( t ) =  , t  [0,1], and (1 −  ) QN =  (3.9) If  = then it follows from (3.9) that N  ker Q = Im L, which means u 2 Using Lemma 3.2 we deduce that u  1 On the other hand, if  [0,1] and    then, by assumption (B3), we get a contradiction    = (1 −  )  , QN  Thus u =    or  3− is bounded in X Case 2:  , QN   In this case, using the similar arguments as in Case we show that  3+ is also bounded in X Theorem 3.1 Let the assumptions (B1)-(B3) hold Then the problem (1.1) has at least one solution in X Proof We prove that all of the conditions of Theorem 2.1 are satisfied, where  be open and bounded such that 3i=1 i   It is clear that the conditions (1) and (2) of Theorem 2.1 are fulfilled by using Lemma 3.1 and Lemma 3.2 So, it remains to verify that the third condition holds For this, we apply the degree property of invariance under a homotopy Let us define H ( u,  ) =  u + (1 −  ) QNu, where we choose the isomorphism J : Im Q → ker L is the identity operator By Lemma 3.3, we have H ( u,  )   for all ( u,  )  ( ker L   )  [0,1], so that deg (QN |ker L ;  ker L, ) = deg ( H (., 0) ,  = deg (  Id ,  ker L,  ) = deg ( H (.,1) ,  ker L,  ) ker L,  ) = 1  Hence, Theorem 3.1 is proved In order to end this paper, we provide an example dealing with the solvability of a second order system of differential equations associated with four-point boundary conditions by applying the above results Example 3.1 Consider the following boundary value problem 12 Solvability of multipoint BVPs at resonance for various kernels  x ( t ) = f1 ( t , x ( t ) , y ( t ) , x ( t ) , y  ( t ) ) , t  ( 0,1) ,   y ( t ) = f ( t , x ( t ) , y ( t ) , x ( t ) , y  ( t ) ) , t  ( 0,1) ,    x ( ) = y  ( ) = 0,   x (1) = −4 x (1 / 3) + y (1 / 3) + x (1 / ) − y (1 / ) ,  y (1) = − x (1 / 3) − y (1 / 3) + x (1 / ) ,   (3.10) where the functions fi :[0,1]  R4 → R ( i = 1, ) are given by f1 ( t , x1 , y1 , x2 , y2 ) = f ( t , x1 , y1 , x2 , y2 ) = t+2 180 t+2 180 ( x1 + y1 ) + (x + y1 ) + ( ) t5 ln + x22 + y22 , 60 t5 x22 + y22 , 60 (3.11) (3.12) for all t [0,1] and ( x1 , y1 ) , ( x2 , y2 )  R In what follows we prove that problem (3.10) has at least one solution by using Theorem 3.1 First we put  −4  6 −4 , A2 =     −1 −1 2  1 = / 3, 2 = / 2, A1 =  and define the function f :[0,1]  R  R → R by f ( t , u1 , u2 ) = ( f1 (t , u1 , u2 ) , f2 (t , u1 , u2 )) , (3.13) for all t [0,1] and u1 = ( x1 , y1 ) , u2 = ( x2 , y2 )  R Then problem (3.10) has one solution if and only if problem (1.1) (with m = 4, 1 , 2 , A1 , A2 and f defined as above) has one solution So we need only show that the conditions of Theorem 3.1 hold ( ) Indeed, it is clear that I − 12 A1 + 22 A2 is invertible, (  / −4 / 3 A1 + 22 A2 ) ( A1 + A2 ) = ( A1 + A2 ) (12 A1 + 22 A2 ) =  ,  / −2 / 3 and ( A1 + A2 )  −2 = A1 + A2 =   1 −1 On the other hand, from (3.11) - (3.13), the function f satisfies Carathéodory condition Next we verify conditions (B1) - (B3) It follows from (3.11), (3.12) and (3.13) that f (t, u, v )  a (t ) u + b (t ) v , for all t [0,1] and u, v  R2 , where a (t ) = ( Since a, b  L1 [0,1];R + ) and ( I −M t+2 t5 , b (t ) = 90 30 * + C )( a + b ) = 41 / 90  1, condition (B1) is satisfied In order to check (B2) we note that 13 Phan Dinh Phung f1 (t , u (t ) , u  (t ))  f2 (t , u (t ) , u  (t )) , t [0,1], for all u = ( x, y )  dom ( L ) This implies that m−2 s m−2 s i =1 i i =1 i  Ai  ds  f1 ( t , u ( t ) , u  ( t ) ) dt   Ai  ds  f ( t , u ( t ) , u  ( t ) ) dt Therefore m−2 s i  A  ds  f ( t , u ( t ) , u  ( t ) ) dt  Im ( M ) i i =1 due to Im ( M ) = ( q, q ) : q  R It means that (B2) holds Finally, we note that  −120 / 13 84 / 13 D = DG =    −42 / 13 / 13  Then  −12 12 Qz ( t ) = ( I −  M ) D ( z ) =   ( z),  −6  ( ) for all z  L1 [0,1];R , where 1/3 s 1/2 0  ( z ) = A1  ds  z ( ) d + A2  s s 0 ds  z ( ) d −  ds  z ( ) d Let  = ( 2a, a )  ker ( M ) , we have N  = ( f1 ( t , a, ) , f ( t , a, ) ) = t+2 60 ( a, a ) , and  −7 55 55 13  a− a, a− a  77760 7776 15552 1215    ( N ) =  So we obtain Q ( N ) = 47 2 ( a − a ) , a − a   2160  Thus,  , QN = 47 ( a2 − a a ) 432 This shows  , QN  for all a  R, that means (B3) is satisfied Thanks to Theorem 3.4, the problem (3.10) has at least one solution CONCLUSION This note is dedicated to deal with the existence of a multi-point BVP for second-order differential systems at resonance in the case of various kernel spaces This provides a technique 14 Solvability of multipoint BVPs at resonance for various kernels to construct the two projections in the method of Mawhin’s coincidence degree when dimension of the kernel is large as well as to prove an operator with complicated boundary condition is Fredholm of index zero In a forthcoming research, it is possible to extend to a wider class of resonant conditions on the matrices 𝐴𝑖 and with a more general boundary condition REFERENCES Gregus M., Neuman F., Arscott F M - Three-point boundary value problems in differential equations, Journal of the London Mathematical Society s2-3 (3) (1971) 429436 Berger M S., Fraenkel L E - Nonlinear desingularization in certain free-boundary problems, Communications in Mathematical Physics 77 (2) (1980) 149-172 Moshiinsky M - Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas, Boletin De La Sociedad Matematica Mexicana (1950) 1-25 Timoshenko S - Theory of Elastic Stability, McGraw-Hill, New York (1961) Chang S K., Pei M - Solvability for some higher order multi-point 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three-point boundary-value problems at resonance, Electronic Journal of Differential Equations 115 (2016) 1-13 12 Phung P D., Minh H B - Existence of solutions to fractional boundary value problems at resonance in Hilbert spaces, Boundary Value Problems (2017) 2017:105 13 Gupta C P - Existence theorems for a second order m-point boundary value problem at resonance, International Journal of Mathematics and Mathematical Sciences 18 (4) (1995) 705-710 14 Feng W - On an m-point boundary value problems, Nonlinear Analysis 30 (1997) 53695374 15 Ma R - Existence results of a m-point boundary value problem at resonance, Journal of Mathematical Analysis and Applications 294 (2004) 147-157 16 Gaines R E., Mawhin J - Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math., vol 568, Springer-Verlag, Berlin, 1977 15 Phan Dinh Phung 17 Mawhin J - Theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, Journal of Differential Equations 12 (1972) 610-636 TĨM TẮT TÍNH GIẢI ĐƯỢC CỦA BÀI TOÁN BIÊN ĐA ĐIỂM CỘNG HƯỞNG VỚI NHÂN THAY ĐỔI Phan Đình Phùng Trường Đại học Công nghiệp Thực phẩm TP.HCM Email: pdphungvn@gmail.com Trong báo tác giả sử dụng định lý liên tục Mawhin lý thuyết bậc coincidence để nghiên cứu tồn nghiệm cho lớp hệ phương trình vi phân cấp hai phi tuyến ℝ𝑛 kết hợp với biên đa điểm điều kiện cộng hưởng Đây kết nghiên cứu điều kiện biên cộng hưởng loại đa điểm với số chiều hạt nhân lớn Tác giả xây dựng ví dụ để minh họa kết Từ khóa: Bậc coincidence, tốn tử Fredholm số 0, toán biên đa điểm, cộng hưởng 16 ... multi-point BVP for second-order differential systems at resonance in the case of various kernel spaces This provides a technique 14 Solvability of multipoint BVPs at resonance for various kernels to... (3.3) Solvability of multipoint BVPs at resonance for various kernels Lemma 3.1 Let 1 = u  domL ker L : Lu =  Nu,  [0,1] Then 1 is bounded in X Proof Let u 1 Assume that Lu =  Nu for. .. isomorphism, and Solvability of multipoint BVPs at resonance for various kernels ( JQ + K ) P ,Q −1 u = ( L + J −1 P ) u, u  domL Now let  be an open bounded subset of X such that domL   