Vacková, Jitka

Miskolc Mathematical Notes

Jitka Vackova

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On the solvability of boundary value problems for some nonlinear systems of ordinary differential equations on $\mathbb{R}_+$

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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 18 (2017), No. 1, pp. 479–488 DOI: 10.18514/MMN.2017.2096ON THE SOLVABILITY OF BOUNDARY VALUE PROBLEMS FORSOME NONLINEAR SYSTEMS OF ORDINARY DIFFERENTIALEQUATIONS ON RCJITKA VACKOVA´Received 22 September, 2016Abstract. Sufficient conditions guaranteeing the unique existence of a bounded solution of quasi-linear boundary value problems are established and a successive approximation method for con-structing the solution is described, together with the proof of the stability of the used method.2010 Mathematics Subject Classification: 34B10; 34B15Keywords: boundary value problems, existence, uniqueness and construction of solution,Carathe´odory conditions, stability1. INTRODUCTIONOn RC we consider the boundary value problem for the quasi-linear differentialsystem:x0.t/D P.t/x.t/Cf .t;x.t//; (1.1)`.x/D h.x/; (1.2)where P 2 Lloc.RC;Rnn/, ` W Cloc.RC;Rn/! Rn is linear bounded functional,h W Cloc.RC;Rn/! Rn is continuous and generally nonlinear operator and f W RCRn! Rn is a function satisfying the local Caratheodory conditions :(1) f .;x/ W RC! Rn is a measurable function for an arbitrary x 2 Rn,(2) f .t; / W Rn! Rn is a continuous function for almost every t 2 RC,(3) 8r > 0 9kr 2L.RCIRC/ such that kf .t;x/k kr.t/ for almost every t 2RCand x 2 Rn such that kxk  r .If we put f .t;x.t// q.t/ and h.x/ c0 we get from the problem (1.1), (1.2) thelinear boundary value problem for ordinary differential equations on RC which wasstudied in [3].The solvability of boundary value problems for ordinary differential systems onRC is studied in much bibliography, for example in monograph [1] and publications[3, 5, 7] and in references mentioned there. The construction methods of the solutionc 2017 Miskolc University Press480 JITKA VACKOVA´of boundary value problems for systems of ordinary linear differential equations onclosed interval is studied for example in 4.5. of monograph [4].In Miskolc Mathematical Notes there are authors studying similar problems, thesolutions of quasilinear differential equations on RC in publication [2] and problemsof integral-differential equations in publication [8].In this work Theorems 1 and 2 provide the criteria of solvability and uniquenesssolvability including the construction method of successive aproximations for gener-ally nonlinear boundary value problems on RC. Theorem 3 proves the stability of themethod used for constructing the solution.By a solution of system (1.1) on RC it is uderstood a local absolutely continuousvector function x 2 Cloc.RC;Rn/ which is bounded on RC, i.e.supfkx.t/k W t 2 RCg<C1;and satisfies the equation (1.1) for almost every t 2 RC. A solution of system (1.1)satisfying condition (1.2) is called a solution of problem (1.1), (1.2) on RC.Basic notation:RD . 1;C1/, RC D Œ0;1/,Rn – the space of n-dimensional column vectors x D .xi /niD1 with the normkx k DnXiD1jxi j and jxj D .jxi j/niD1;RnC D fx D .xi /niD1 2 Rn W xi 2 RC.i D 1; : : : ;n/g;Rnn – the space of nn matrices X D .xij /ni;jD1 with the normkX k DnXi;jD1jxij j and jX j D .jxij j/ni;jD1;RnnC D fX D .xij /ni;jD1 2 Rnn W xij 2 RC .i;j D 1; : : : ;n/g:if x;y 2 Rn and X;Y 2 Rnn, thenx  y” y x 2 RnC; X  Y” Y  X 2 RnnC :For the matrix X D .xij /ni;jD1 2 Rnn by the notation Xd and Xd we meanXd D0BB@x1 0 : : : 00 x2 : : : 0: : : : : : : : : : : : : : : :0 0 : : : xn1CCA ; Xd D0BB@0 x12 : : : x1nx21 0 : : : x2n: : : : : : : : : : : : : : : : : : :xn1 xn2 : : : 01CCA ;det.X/ – the determinant of the matrix X ,r.X/ – the spectral radius of the matrix X ,X 1 – the inverse matrix to X ,E – the unit matrix,NONLINEAR BOUNDARY VALUE PROBLEMS ON RC 481C.RC;Rn/ – the space of continuous and bounded vector functions1 x W RC! Rnwith the normkxkC D supfkx.t/k W t 2 RCg and jxjC D .kxikC /niD1;L.RC;Rn/ – the space of integrable vector functions x W RC! Rn with the normkxkL DZ C10kx.t/kdt and jxjL D .jxi jL/niD1;L.RC;Rnn/ – the space of integrable matrix functions X W RC ! Rnn with thenormkXkL DZ C10kX.t/kdt and jX jL D .jxij jL/ni;jD1;if X 2 C.RC;Rnn/ is a matrix function with columns x1; : : : ;xn and` W C.RC;Rn/! Rn is a linear functional, then `.X/ stands for the matrix functionwith columns `.x1/; : : : ;`.xn/.2. RESULTSTo formulate the Theorem on solvability of the problem (1.1), (1.2) we need toformulate the following estimates 8x;y 2 C0.RC;Rn/ :jf .t;x/ f .t;y/j  NL.t/jx yjCwhere NL.t/ 2 L.RC;Rnn/,jh.x/ h.y/j H jx yjCwhere H 2 Rnn is constant matrix andj`.x/j  N`.jxj/;where N` W C0.RC;Rn/! Rn is linear positive bounded functional. Moreover we usethe notations:P d D supZ tt0Pd .s/ds W t 2 RC;P  D supˇˇˇˇZ tt0expZ tsPd ./djPd.s/jdsˇˇˇˇW t 2 RC;f  D supˇˇˇˇZ tt0expZ tsPd ./dj NL.s/jdsˇˇˇˇW t 2 RC;where t0 2 RC.1A vector or matrix function is said to be continuous, bounded, integrable etc., if such are itselements482 JITKA VACKOVA´Theorem 1. Assume that P d;P  and N`.E/, defined previously, satisfy the condi-tionkP d kCkP kCk N`.E/k<C1; (2.1)the matrixD `.expZ :t0Pd .s/ds/ is nonsingular (2.2)andr.S/ < 1 (2.3)whereS D P Cf Cd j 1jH C N`.E/.P Cf /; (2.4)andd D supexpZ tt0Pd ./d W t 2 RC:Then the problem (1.1), (1.2) has just one solution.To prove Theorem 1 we shall need to formulate the following statement that cor-responds with the methods used in publications of I.Kiguradze (see [4, 5, 7] and thebibliography mentioned there). To get the effective criteria of solvability of the con-sidered problem we use one of the methods presented in 1.3. of [6].Proposition 1. Every solution of the problem (1.1), (1.2) is also a solution of anoperator equationx D T x (2.5)whereT x DZ tt0expZ tsPd ./dPd.s/x.s/Cf .s;x.s//dsC expZ tt0Pd ./d 1h.x/ `Z :t0expZ :sPd ./dPd.s/x.s/Cf .s;x.s//ds(2.6)where t0 2 RC is an arbitrary fixed point. And from the other side, every solution ofthe operator equation (2.5) is also a solution of the problem (1.1), (1.2).Proof. Let x.t/ be a solution of the problem (1.1), (1.2). Then by integration of(1.1) we getx.t/D x.t0/expZ tt0Pd ./dCZ tt0expZ tsPd ./dhPd.s/x.s/Cf .s;x.s//ids: (2.7)NONLINEAR BOUNDARY VALUE PROBLEMS ON RC 483To calculate x.t0/ we shall substitute x(t) in the left side of the boundary condition(1.2) and with use of the linearity of the functional ` we geth.x/D `expZ t0Pd ./dx.t0/C`Z t0expZ sPd ./dhPd.s/x.s/Cf .s;x.s//ids:Then according to (2.2) we obtainx.t0/D 1h.x/ `Z t0expZ sPd ./dhPd.s/x.s/Cf .s;x.s//ids:By substituting x.t0/ back in (2.7) we get (2.6).From the other side, if x.t/ is a solution of (2.5) then it satisfies the condition (1.2),because by substituting t0 for t in (2.6) we get:x.t0/D 1h.x/ `x./ x.t0/expZ tt0Pd ./d;and then with use of linearity of `we get (1.2). By derivating of (2.5) we get equation(1.1). It means that the solution x.t/ is also the solution of the problem (1.1), (1.2).Proof of Theorem 1. To deduce the sufficient criterias of the solvability of theequation (2.5) we have to prove the contractivity of the operator T W C0.RC;Rn/!C0.RC;Rn/, where C0.RC;Rn/ is a subspace of the space Cloc.RC;Rn/ withbounded functions on RC. This space is a Banach space with a supreme norm andthus the operator T displays a Banach space into itself. Let’s make the followingestimation:jT x TyjC ˇˇˇˇZ tt0expZ tsPd ./dPd.s/ Œx.s/ y.s/ˇˇˇˇCCˇˇˇˇZ tt0expZ tsPd ./dŒf .s;x.s// f .s;y.s//dsˇˇˇˇCCˇˇˇexpZ tt0Pd ./d 1 Œh.x/ h.y/  expZ tt0Pd ./d 1`Z tt0expZ tsPd ./dPd.s/ Œx.s/ y.s/dsCZ tt0expZ tsPd ./dŒf .s;x.s// f .s;y.s//dsˇˇˇC .P Cf /jx yjCCd j 1jhH jx yjC C N`.E/.P Cf /jx yjCi484 JITKA VACKOVA´D S jx yjCwhereS D P Cf Cd j 1jH C N`.E/.P Cf /:Thus, for x;y 2 C0.RC;Rn/, we getkT x TykC  kSkkx ykC (2.8)and since for S is valid r.S/ < 1we get kSk<1. Thus we proved that T is a contrac-tion. The existence of the unique solution of the operator equation (2.5) follows fromBanach Theorem and by Lemma 1 this solution is also the solution of the problem(1.1), (1.2). The construction of the solution of problem (1.1), (1.2) follows from Banach The-orem.Theorem 2. Suppose that all assumptions of Theorem 1 are fulfilled. Then theproblem (1.1), (1.2) has the solution x on RC which is a limit of the sequence of thesolutions of ancillary system:x0mC1.t/D Pd .t/xmC1.t/CPd .t/xm.t/Cf .t;xm.t//; (2.9)`.xmC1/D h.xm/; (2.10)where m 2N and x1.t/ 2 C0.RC;Rn/ is arbitrary.Proof. It is obvious that the solution to the ancillary system (2.9), (2.10) is equi-valent with the solution to the operator equationxmC1 D T xm (2.11)where operator T is defined by (2.6), m 2N and t0 2 RC is an arbitrary given point.Let x1.t/ be arbitrary. With respect to the contractivity of operator T and to Theorem1 we shall prove that x2D T x1;x3D T x2; : : : ;xmD T xm 1;xmC1D T xm : : : is theCauchy sequence:kxmC1 xmkC D kT xm T xm 1kC  kSkkxm xm 1kCD kSkkT xm 1 T xm 2kC  kSk2kxm 1 xm 2kC : : : kSkm 1kx2 x1kCwhere m 2N is arbitrary. By virtue of this we get for n > m;n 2Nkxn xmkC  kxmC1 xmkC C Ckxn xn 1kC kSkm 1kx2 x1kC CkSkmkx2 x1kC C CkSkn 2kx2 x1kCD kx2 x1kC kSkm 1.1CkSkC   CkSkn m 1/ kx2 x1kC kSkm 1.1CkSkC : : : /D kSkm 11 kSk kx2 x1kCNONLINEAR BOUNDARY VALUE PROBLEMS ON RC 485Let  > 0 be arbitrary. Since r.S/ < 1 then kSkm 1! 0 for m!1. Thus thereexists n0 2N such that for m n0 the following holds:kSkm 1kx2 x1kC 11 kSk < :This implies kxn xmkC <  for m;n  n0. Thus we proved that fxmg is a Cauchysequence. Since C0 is a complete space fxmg is convergent. That means xm! x0,x0 2 C0. With respect to the Banach Theorem x0 is the only one fixed point of theoperator T and also the solution of the operator equation (2.11). This solution is alsothe solution of the problem (2.9); (2.10). The following Theorem proves that the process of constructing the solution ofproblem (2.9), (2.10) described above is stable in some sense, analogously to theassertions published by I.Kiguradze in [4].Theorem 3. Consider a sequence of vectors m 2 Rn .m D 1;2; : : : / and a se-quence of vector-valued functions m 2 L.RC;Rn/ .m D 1;2; : : : /. Let P1mD1 mbe convergent wherem DkmkC supZ tt0km.s/kds W t 2 Rand assume that all the conditions of Theorem 1 are satisfied. Then the solution xon RC of problem (1.1), (1.2) is the limit of the sequence of solutions of ancillarysystem:x0mC1.t/D Pd .t/xmC1.t/CPd .t/xm.t/Cf .t;xm.t//Cm.t/; (2.12)`.xmC1/D h.xm/Cm; (2.13)where m 2N and x1.t/ 2 C0.RC;Rn/ is arbitrary.Proof. To prove the theorem we have to show that the operator equation (2.11)enriched with error function doesn’t breach the Cauchy property of the sequenceoccurred in Theorem 2. Therefore we consider the operator equationxmC1 D T xmCbT .m;m/where operator T is defined by (2.6) andbT .m;m/D Z tt0expZ tsPd ./dm.s/dsC expZ tt0Pd ./d 1m `Z :t0expZ :sPd ./dm.s/dswhere m 2 N and t0 2 RC is an arbitrary given point. Now, to prove the stabilityof the solution we show that also the sequence x2 D T x1CbT .1;1/; : : : ;xm DT xm 1CbT .m 1;m 1/;xmC1 D T xmCbT .m;m/ : : : is a Cauchy sequence,486 JITKA VACKOVA´with respect to Theorem 1, 2 and to the contractivity of operator .T CbT /, analog-ously to the proof of Theorem 2. In addition to (2.8) we need to make the followingestimation:kbT .m;m/ bT .m 1;m 1/kZ tt0expZ tsPd ./dm.s/ds m 1.s/dsCexpZ tt0Pd ./d 1m m 1CexpZ tt0Pd ./d 1`Z :t0expZ :sPd ./dm.s/ m 1ds d j 1jm m 1Cd .1Cj 1j N`.E/d /Z tt0m.s/ds m 1.s/ds kckkm m 1kCkSkZ tt0km.s/ds m 1.s/kdswherec Dd j 1j; S Dd .1Cj 1j N`.E/d /:ThenkxmC1 xmkC D kT xmCbT .m;m/  .T xm 1CbT .m 1;m 1//kC kTxm  T xm 1kC CkbT .m;m/ bT .m 1;m 1/k kSkkxm xm 1kCkckkm m 1kCkSkZ tt0km.s/ds m 1.s/kds kSkkTxm 1  T xm 2kC CkbT .m 1;m 1/ bT .m 2;m 2/kCkckkm m 1kCkSkZ tt0km.s/ds m 1.s/kds  : : : kSkm 1kx2 x1kCkckmXiD2ki  i 1kCkSkmXiD2Z tt0ki .s/ i 1.s/kdswhere m 2N is arbitrary. For n > m;n 2N we getkxn xmkC D kSkm 11 kSk kx2 x1kC CkckmXiD2ki  i 1kNONLINEAR BOUNDARY VALUE PROBLEMS ON RC 487CkSkmXiD2Z tt0ki .s/ i 1.s/kdsCkckmC1XiD2ki  i 1kCkSkmC1XiD2Z tt0ki .s/ i 1.s/kdsC:::Ckckn 1XiD2ki  i 1kCkSkn 1XiD2Z tt0ki .s/ i 1.s/kdsCkcknXiD2ki  i 1kCkSknXiD2Z tt0ki .s/ i 1.s/kds kSkm 11 kSk kx2 x1kCCnXjDmkckkj kCkSksupZ tt0kj .s/kds W t 2 R:Convergency ofPnjD1 j implies the convergency of the series of both addends ofm and their linear combinations as well. Then by virtue of Theorem 2 the sequencefxmg D fT xmCbT .m;m/g is Cauchy and xm D T xmCbT .m;m/! x0 2 C0which is the solution of the problem (2.12), (2.13). Remark 1. If m D 0 and m  0 then from Theorem 3 we get immediatelyTheorem 2. Thus Theorem 2 could be perceived as the corollary of Theorem 3. Forclearer view on calculations in proofs of both theorems it is convenient to formulatefirst Theorem 2 separately and then Theorem 3 including the error function.All consequences for linear boundary value problems published in [3] and analog-ous statements for solutions of linear boundary value problems for nonlinear systemsof ordinary differential equations follow directly from Theorem 1 and Theorem 2.The following example illustrates the case of nonlinear boundary value problemfor nonlinear system of ordinary differential equations:x0i .t/D xi .t/Cgi .t/sinxi .t/Cqi .t/; .i D 1; : : : ;n/;xi .0/D ci cosxi .1/C c0i ; .i D 1; : : : ;n/;where gi ;qi 2 Lloc.RC;R/;ci ; c0i 2 R .i D 1; : : : ;n/.Then we haveP.t/ E  Pd .t/; Pd .t/ 0;f .t;x/D .gi .t/sinxi .t/Cqi .t//niD1;488 JITKA VACKOVA´l.x/D .xi .0//niD1; h.x/D .ci cosxi .1/C c0i /niD1and thenjf .t;x/j  .jgi .t/jC jqi .t/j/niD1; jh.x/j  .jci jC jc0i j/niD1; NL.t/ 0; NhD 0:Because P dD 0;P  D 0; N`.E/ D E then it follows that condition (2.1) is fulfilledon assumptionkgkLCkqkL <C1:Then S D 0 which means that the differential equation has a unique solution.REFERENCES[1] R. P. Agarwal and D. O’Regan, Infinite interval problems for differential, difference and integralequations. Kluwer Academic Publishers, Dordrecht, 2001. doi: 10.1007/978-94-010-0718-4.[2] V. M. Evtukhov and L. I. Kusick, “Asymptotic behaviour of solutions of some systems of quasilin-ear differential equations,” Math. Notes (Miskolc), vol. 4, no. 1, pp. 3–24, 2003.[3] J. Jandova´ and B. Pu˚zˇa, “On the solvability of a linear boundary value problems for system ofordinary differential equations on RC,” in Colloquium on Differential and Difference Equations—CDDE 2006, ser. Folia Fac. Sci. Natur. Univ. Masaryk. Brun. Math. Masaryk Univ., Brno, 2007,vol. 16, pp. 75–83.[4] I. Kiguradze, Nachalnaya i kraevye zadachi dlya sistem obyknovennykh differentsialnykh uravnenii.Tom I. “Metsniereba”, Tbilisi, 1997, lineinaya teoriya. [Linear theory].[5] I. Kiguradze, “On some boundary value problems with conditions at infinity for nonlinear differen-tial systems,” Bull. Georgian Natl. Acad. Sci., vol. 175, no. 1, pp. 27–33, 2007.[6] I. Kiguradze and B. Pu˚zˇa, Boundary value problems for systems of linear functional differentialequations, ser. Folia Facultatis Scientiarium Naturalium Universitatis Masarykianae Brunensis.Mathematica. Masaryk University, Brno, 2003, vol. 12.[7] I. Kiguradze and Z. Sokhadze, “A priori estimates of solutions of systems of functional differentialinequalities and some of their applications,” Mem. Differential Equations Math. Phys., vol. 41, pp.43–67, 2007.[8] N. A. Perestyuk and V. Y. Slusarchuk, “Oscillation of nonlinear differential-integral equation in aBanach space with respect to its subspace,” Math. Notes (Miskolc), vol. 4, no. 1, pp. 53–64, 2003.Author’s addressJitka Vackova´Department of Mathematics and Statistics, Masaryk University Brno, Czech RepublicE-mail address: jitka@finnsub.com

On the solvability of boundary value problems for some nonlinear systems of ordinary differential equations on R+

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