EXISTENCE AND UNIQUENESS THEOREMS ON CERTAIN DIFFERENCE-DIFFERENTIAL EQUATIONS

Elecronic Journal of Differenial Equaions, Vol. 29(29), No. 49, pp. 2. ISSN: 72-669. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu fp ejde.mah.xsae.edu EXISTENCE AND UNIQUENESS THEOREMS ON CERTAIN DIFFERENCE-DIFFERENTIAL EQUATIONS BABURAO G. PACHPATTE Absrac. In he presen paper, exisence and uniqueness heorems for he soluions of cerain nonlinear difference-differenial equaions are esablished. The main ools employed in he analysis are based on he applicaions of he Leray-Schauder alernaive and he well known Bihari s inegral inequaliy.. Inroducion Le R n denoe he real n-dimensional Euclidean space wih appropriae norm denoed by and J [, T ] (T > ), R [, ) be he given subses of R, he se of real numbers. In his paper, we consider he difference-differenial equaion of he form x () f(, x(), x( )), (.) for J under he iniial condiions x( ) φ() ( < ), x() x, (.2) f C(J R n R n, R n ) and φ() is a coninuous funcion for <, lim φ() exiss, for which we denoe by φ( ) c. If we consider he soluions of (.) for J, we obain a funcion x( ) which is unable o define as a soluion for <. Hence, we have o impose some condiion, for example he firs condiion in (.2). We noe ha, if T is less han, he problem is reduced o ordinary differenial equaion x () f(, x(), φ()), (.3) for <, wih he second condiion in (.2). Here, i is essenial o obain he soluions of equaion (.) for T, so ha, we suppose, in he sequel, T is no less han. I is easy o observe ha he inegral equaions which are equivalen o (.)-(.2) are x() x f(s, x(s), φ(s)), (.4) 2 Mahemaics Subjec Classificaion. 34K5, 34K, 34K4. Key wor and phrases. Exisence and uniqueness; difference-differenial equaions; Leray-Schauder alernaive; Bihari s inegral inequaliy; Winner ype; Osgood ype. c 29 Texas Sae Universiy - San Marcos. Submied June 9, 28. Published April 7, 29.

2 B. G. PACHPATTE EJDE-29/49 for < and x() x f(s, x(s), φ(s)) f(s, x(s), x(s )), (.5) for T. Problems of exisence and uniqueness of soluions of equaions of he form (.) and is more general versions, under various iniial condiions have been sudied by many auhors by using differen echniques. The fundamenal ools used in he exisence proofs, are essenially, he mehod of successive approximaion, Schauder-Tychonoff s fixed poin heorem, Banach conracion mapping principle and comparison mehod, see [2,6,9,2,4,5] and he references cied herein. Our main objecive here is o invesigae he global exisence of soluion o (.)-(.2) by using simple and classical applicaion of he opological ransversaliy heorem of Granas [5, p. 6], also known as Leray-Schauder alernaive. Osgood ype uniqueness resul for he soluions of (.)-(.2) is esablished by using he well known Bihari s inegral inequaliy. Exisence and uniqueness heorems for cerain perurbed difference-differenial equaion are also given. 2. Main Resuls In proving exisence of soluion of (.)-(.2), we use he following opological ransversaliy heorem given by Granas [5, p.6]. Lemma 2.. Le B be a convex subse of a normed linear space E and assume B. Le F : B B be a compleely coninuous operaor and le U(F ) {x : x λf x} for some < λ <. Then eiher U(F ) is unbounded or F has a fixed poin. We also need he following inegral inequaliy, ofen referred o as Bihari s inequaliy [8, p. 7]. Lemma 2.2. Le u(), p() C(R, R ). Le w(u) be a coninuous, nondecreasing funcion defined on R, w(u) > for u > and w(). If u() c p(s)w(u(s)), for R, c is a consan, hen for, u() W [ ] W (c) p(s), W (r) r r w(s), r >, r >, W is he inverse funcion of W and R be chosen so ha W (c) for all R lying in he inerval. p(s) Dom(W ), The following heorem deals wih he Winner ype global exisence resul for he soluion of (.)-(.2).

EJDE-29/49 EXISTENCE AND UNIQUENESS THEOREMS 3 Theorem 2.3. Suppose ha he funcion f in (.) saisfies he condiion f(, x, y) h()[g( x ) g( y )], (2.) h C(J, R) and g : R (, ) is coninuous and nondecreasing funcion. Then (.)-(.2) has a soluion x() defined on J provided T saisfies T [h(s) h(s )] < α x α g(s), (2.2) h(s)g( φ(s) ). (2.3) Proof. The proof will be given in hree seps. Sep I. To use Lemma, we esablish he priori boun on he soluions of he problem x () λf(, x(), x( )), (2.4) under he iniial condiions (.2) for λ (, ). Le x() be a soluion of (2.4)-(.2), hen we consider he following wo cases. Case : <. From he hypoheses, we have x() x x α λf(s, x(s), φ(s)) h(s)[g( x(s) ) g( φ(s) )] h(s)g( x(s) ). (2.5) Le u() be defined by he righ hand side of (2.5), hen u() α, x() u() and ha is, u () h()g( x() ) h()g(u()); u () h(). (2.6) g(u()) Inegraion of (2.6) from o ( < ), he change of variable, and he condiion (2.2) gives u() α g(s) h(s) h(s) < α g(s). (2.7) From his inequaliy, we conclude ha, here is a consan Q independen of λ (, ) such ha u() Q for < and hence x() Q.

4 B. G. PACHPATTE EJDE-29/49 Case 2: T. From he hypoheses, we have x() x x x α λf(s, x(s), φ(s)) h(s)[g( x(s) ) g( φ(s) )] h(s)[g( x(s) ) g( x(s ) )] h(s)g( φ(s) ) h(s)g( x(s) ) h(s)g( x(s) ) I I λf(s, x(s), x(s )) h(s)g( x(s) ) h(s)g( x(s ) ) By he change of variable, from (2.9), we observe ha I h(σ )g( x(σ) )dσ Using his inequaliy in (2.8), we obain x() α (2.8) h(s)g( x(s ) ). (2.9) h(σ )g( x(σ) )dσ. (2.) [h(s) h(s )]g( x(s) ). (2.) Le v() be defined by he righ hand side of (2.), hen v() α, x() v() and ha is, v () [h() h( )]g( x() ) [h() h( )]g(v()); v () [h() h( )]. (2.2) g(v()) Inegraion of (2.2) from o, T, he change of variable, and he condiion (2.2) give v() α g(s) [h(s) h(s )] T [h(s) h(s )] < α g(s). (2.3) From (2.3) we conclude ha here is a consan Q 2 independen of λ (, ) such ha v() Q 2 and hence x() Q 2 for T. Le Q max{q, Q 2 }. Obviously, x() Q for J and consequenly, x sup{ x() : J} Q. Sep II. We define B C(J, R n ) o be he Banach space of all coninuous funcions from J ino R n endowed wih sup norm defined above. We rewrie he problem (.)-(.2) as follows. If y B and x() y() x, i is easy o see ha y() f(s, y(s) x, φ(s)),

EJDE-29/49 EXISTENCE AND UNIQUENESS THEOREMS 5 for < and y() f(s, y(s) x, φ(s)) f(s, y(s) x, y(s ) x ), for T, y() y if and only if x() saisfies (.)-(.2). Le B {y B : y } and define F : B B by for < and F y() F y() f(s, y(s) x, φ(s)) f(s, y(s) x, φ(s)), (2.4) f(s, y(s) x, y(s ) x ), (2.5) for T. Then F is clearly coninuous. Now, we shall prove ha F is uniformly bounded. Le {b m } be a bounded sequence in B, ha is, b m b for all m, b > is a consan. Le N sup{h() : J}. We have o consider he wo cases. Case : <. From (2.4) and hypoheses, we have F b m () γ f(s, b m (s) x, φ(s)) h(s)[g( b m (s) x ) g( φ(s) )] h(s)g( φ(s) ) Ng(b x ) γ Ng(b x ). γ h(s)g(b x ) (2.6) h(s)g( φ(s) ). (2.7) Case 2: T. From (2.5) and he hypoheses, we have F b m () γ f(s, b m (s) x, φ(s)) h(s)[g( b m (s) x ) g( φ(s) )] f(s, b m (s) x, b m (s ) x ) h(s)[g( b m (s) x ) g( b m (s ) x )] h(s)g( φ(s) ) h(s)g( b m (s) x ) h(s)g( b m (s) x ) h(s)g( b m (s) x ) I 2, h(s)g( b m (s ) x ) (2.8)

6 B. G. PACHPATTE EJDE-29/49 γ is given by (2.7), and I 2 By he change of variable, we have I 2 h(σ )g( b m (σ) x )dσ Using (2.2) in (2.8), we have h(s)g( b m (s ) x ). (2.9) h(σ )g( b m (σ) x )dσ. (2.2) F b m () γ γ T [h(s) h(s )]g( b m (s) x ) 2Ng(b x ) γ 2NT g(b x ). (2.2) From (2.6) and (2.2), i follows ha {F b m } is uniformly bounded. Sep III. We shall show ha he sequence {F b m } is equiconinuous. Le {b m } and N be as in Sep II. We mus consider hree cases. Case : and are conained in <. From (2.4), i follows ha F b m () F b m ( ) f(s, b m (s) x, φ(s)) f(s, b m (s) x, φ(s)) f(s, b m (s) x, φ(s)). From he above equaliy and hypoheses, we have (2.22) F b m () F b m ( ) f(s, b m (s) x, φ(s)) h(s)[g( b m (s) x ) g( φ(s) )] N[g(b x ) g( c )] N[g(b x ) g( c )]. (2.23) Case 2: and are conained in T. From (2.5), i follows ha F b m () F b m ( ) f(s, b m (s) x, φ(s)) f(s, b m (s) x, φ(s)) f(s, b m (s) x, b m (s ) x ). f(s, b m (s) x, b m (s ) x ) f(s, b m (s) x, b m (s ) x ) (2.24)

EJDE-29/49 EXISTENCE AND UNIQUENESS THEOREMS 7 From his equaliy and he hypoheses, we have F b m () F b m ( ) f(s, b m (s) x, b m (s ) x ) h(s)[g( b m (s) x ) g( b m (s ) x )] Ng( b m (s) x ) I 3 (2.25) I 3 Ng( b m (s ) x ). (2.26) By he change of variable, we have I 3 Using he above inequaliy in (2.25), we obain Ng( b m (σ) x )dσ Ng(b x )( ). (2.27) F b m () F b m ( ) 2Ng(b x ). (2.28) Case 3: and are respecively conained in [, ) and [, T ]. From (2.4) and (2.5), i follows ha F b m () F b m ( ) f(s, b m (s) x, φ(s)) f(s, b m (s) x, φ(s)) f(s, b m (s) x, φ(s)) f(s, b m (s) x, b m (s ) x ). From (2.29) and using he hypoheses, we have F b m () F b m ( ) f(s, b m (s) x, φ(s)) f(s, b m (s) x, b m (s ) x ) h(s)[g( b m (s) x ) g( φ(s) )] f(s, b m (s) x, b m (s ) x ) f(s, b m (s) x, φ(s)) h(s)[g( b m (s) x ) g( b m (s ) x )] N[g(b x ) g( c )] M( ), N[g(b x ) g(b x )] (2.29) (2.3)

8 B. G. PACHPATTE EJDE-29/49 M max{n[g(b x ) g( c )], 2Ng(b x )}. From (2.23), (2.28), (2.3), we conclude ha {F b m } is equiconinuous. By Arzela- Ascoli heorem (see [4,7]), he operaor F is compleely coninuous. Moreover, he se U(F ) {y B : y λf y, λ (, )} is bounded, since for every y in U(F ) he funcion x() y() x is a soluion of (2.4)-(.2), for which we have proved x Q and hence y Q x. Now, an applicaion of Lemma, he operaor F has a fixed poin in B. This means ha (.)-(.2) has a soluion. The proof is complee. Remark. We noe ha he advanage of our approach here is ha, i yiel simuaneously he exisence of soluion of (.)-(.2) and maximal inerval of exisence. In he special case, if we ake h() in (2.2) and he inegral on he righ hand side in (2.2) is assumed o diverge, hen he soluion of (.)-(.2) exiss for every T < ; ha is, on he enire inerval R. Our resul in Theorem yiel exisence of soluion of (.)-(.2) on R, if he inegral on he righ hand side in (2.2) is divergen i.e., α g(s). Thus Theorem can be considered as a furher exension of he well known heorem on global exisence of soluion of ordinary differenial equaion due o Winner given in [6]. The nex heorem deals wih he Osgood ype uniqueness resul for he soluions of (.)-(.2). Theorem 2.4. Consider (.) wih f C(R R n R n, R n ), under he iniial condiions in (.2). Suppose ha: (i) he funcion f saisfies (ii) le f(, x, y) f(, x, ȳ) h()[g( x x ) g( y ȳ )], (2.3) h C(R, R ), g(u) is a coninuous, nondecreasing funcion for u, g() ; G(r) r r g(s), ( < r r), wih G being he inverse funcion of G and assume ha lim r G(r), for any fixed r. Then (.)-(.2) has a mos one soluion on R. Proof. Le x(), y() be wo soluions of equaion (.), under he iniial condiions x( ) y( ) φ(), ( < ), x() y() x, (2.32) and le u() x() y(), R. We consider he following wo cases. Case : <. From he hypoheses, we have u() ε f(s, x(s), φ(s)) f(s, y(s), φ(s)) h(s)g( x(s) y(s) ) h(s)g(u(s)), (2.33)

EJDE-29/49 EXISTENCE AND UNIQUENESS THEOREMS 9 ε > is sufficienly small consan. Now, an applicaion of Lemma 2 o (2.33) yiel x() y() G [ G(ε ) Case 2: <. From he hypoheses, we have u() f(s, x(s), φ(s)) f(s, y(s), φ(s)) f(s, x(s), x(s )) f(s, y(s), y(s )) h(s)g(u(s)) h(s)g(u(s)) h(s)g(u(s)) I 4, I 4 By he change of variable, we observe ha I 4 Using (2.37) in (2.35), we obain u() h(s)[g(u(s)) g(u(s ))] h(s)g(u(s)) [h(s) h(s )]g(u(s)) ε 2 h(s) ]. (2.34) h(s)g(u(s )) (2.35) h(s)g(u(s )). (2.36) h(s )g(u(s)). (2.37) [h(s) h(s )]g(u(s)), (2.38) ε 2 > is sufficienly small consan. Now, an applicaion of Lemma 2 o (2.38) yiel x() y() G [G(ε 2 ) [h(s) h(s )]]. (2.39) To apply he esimaions in (2.34), (2.39) o he uniqueness problem, we use he noaion G(r, r ) insead of G(r) and impose he assumpion lim r G(r, r ), for fixed r, hen we obain lim r G (r, r ), see [5, p. 77]. From (2.34), (2.39), i follows ha x() y() for R and hence x() y() on R. Thus, here is a mos one soluion o (.)-(.2) on R. Remark. We noe ha he condiion (2.3) correspon o he Osgood ype condiion concerning he uniqueness of soluions in he heory of differenial equaions (see [4, p. 35]). 3. Perurbed equaions In his secion, we consider he difference-differenial equaion of he form x () A()x() B()x( ) f(, x(), x( )), (3.)

B. G. PACHPATTE EJDE-29/49 for J wih he iniial condiions (.2). perurbaion of he linear sysem The equaion (3.) is reaed as a x () A()x() B()x( ), (3.2) for J wih he iniial condiions (.2), A(), B() are coninuous funcions on J and f, φ are he funcions as in (.), (.2). Following Sugiyama [4], le K(, s) be a marix soluion of equaions: K(, s) A()K(, s) B()K(, s) ( s < ), K(, s) A()K(, s) ( < < s <, s < < ), K(, ), K(, s) ( < ). The funcion K(, s) saisfying he above properies is called he kernel funcion for equaion (3.2). For more deails concerning he kernel funcion and is use in he sudy of various properies of soluions of difference-differenial equaions wih perurbed erms, see Bellman and Cooke [,2]. By means of he kernel funcion K(, s), i follows ha he soluion of (3.) wih (.2) considered as a perurbaion of (3.2) wih (.2) is represened by (see [3, p.457]) x() x () K(, s)f(s, x(s), x(s )), (3.3) x () is a unique soluion of (3.2) wih (.2). I is easy o observe ha (see [3]) he inegral equaions which are equivalen o (3.3) are for < and x() x () x() x () K(, s)f(s, x(s), φ(s)) K(, s)f(s, x(s), φ(s)), (3.4) K(, s)f(s, x(s), x(s )), (3.5) for T. The following heorems concerning he exisence and uniqueness of soluions of (3.)-(.2) hold. Theorem 3.. Suppose ha: (i) he funcion f in (3.) saisfies (2.), (ii) he unique soluion x () of (3.2)-(.2) is bounded, ha is x () c, (3.6) for J, c is a posiive consan, (iii) he kernel funcion K(, s) for (3.2) is bounded; ha is, K(, s) L, (3.7) for s T, L is a consan, and for each J, lim is saisfied for J. T K(, s) K(, s), (3.8)

EJDE-29/49 EXISTENCE AND UNIQUENESS THEOREMS Then (3.)-(.2) has a soluion x() defined on J provided T saisfies T L[h(s) h(s )] < β c β g(s), (3.9) Lh(s)g( φ(s) ). (3.) Theorem 3.2. Consider (3.) wih f C(R R n R n, R n ) and he condiions in (.2), as a perurbaion of (3.2) for R wih (.2). Suppose ha: (i) he condiion (i) of Theorem 3 hol and he kernel funcion K(, s) saisfies he condiion (3.7), (ii) he condiions (i)-(ii) of Theorem 2 hold. Then (3.)-(.2) has a mos one soluion on R. The proofs of Theorems 3 and 4 can be compleed by following he proofs of Theorems and 2 given above, wih suiable modificaions and closely looking a he proofs of exisence resuls given in [9,]. We omi he deails here. Remark. We noe ha our approach o he exisence sudy of (.)-(.2) and (3.)- (.2) is differen from hose used in [-5] and we believe ha he resuls given here are of independen ineres. For he sudy of numerical soluion of general Volerra inegral equaion wih delay argumens of he form (3.3), see [3]. Acknowledgemens. The auhor wishes o express his sincere hanks o he anonymous referee and Professor J. G. Dix for helpful commens and suggesions. References [] R. Bellman and K. L. Cooke; Sabiliy heory and adjoin operaors for linear differenialdifference equaions, Trans.Amer.Mah.Soc. 92(959), 47-5. [2] R. Bellman and K. L. Cooke; Differenial-Difference Equaions, Academic Press, New York, 963. [3] B. Cahlon, L. J. Nachman and D. Schmid; Numerical soluion of Volerra inegral equaions wih delay argumens, J. Inegral Equaions 7(984), 9-28. [4] C. Corduneanu; Principles of Differenial and Inegral Equaions, Chelsea, New York, 97. [5] J. Dugundji and A. Granas; Fixed poin Theory, Vol. I, Monografie Mahemayczne, PWN, Warsaw, 982. [6] J. K. Hale; Theory of Funcional Differenial Equaions, Springer-Verlag, New York, 977. [7] M. A. Krasnoselskii; Topological Meho in he Theory of Nonlinear Inegral Equaions, Pergamon Press, Oxford, 964. [8] B. G. Pachpae; Inequaliies for Differenial and Inegral Equaions, Academic Press, New York, 998. [9] B. G. Pachpae; On higher order differenial equaion wih rearded argumen, In: Differenial Equaions and Applicaions, Vol. 4, 27, Nova Science Publishers, Inc., New York, 27, Ediors: Yeol Je Cho e.al., pp. 93-3. [] B. G. Pachpae; On a perurbed sysem of Volerra inegral equaions, Numer. Func. Anal. Opim. 29(28), 97-22. [] B. G. Pachpae; Some basic heorems on difference-differenial equaions, Elec. J. Diff. Eqs. Vol. 28(28), No. 75, pp. -. [2] S. Sugiyama; On he exisence and uniqueness heorems of difference-differenial equaions, Kodi Mah.Sem.Rep. 2(96), 79-9. [3] S. Sugiyama; On he boundedness of soluions of difference-differenial equaions, Proc. Japan Acad. 36(96), 456-46. [4] S. Sugiyama; Exisence heorems on difference-differenial equaions, Proc. Japan. Acad. 38(962), 45-49.

2 B. G. PACHPATTE EJDE-29/49 [5] S. Sugiyama; Dependence properies of soluions of he reardaion and iniial values in he heory of difference-differenial equaions, Kodai Mah. Sem. Rep. 5(963), 67-78. [6] A. Winner; The nonlocal exisence problem for ordinary differenial equaions, Amer. J. Mah. 67(945), 277-284. Baburao G. Pachpae 57 Shri Nikean Colony, Near Abhinay Talkies, Aurangabad 43 (Maharashra), India E-mail address: bgpachpae@gmail.com