Existence Theory of Second Order Random Differential Equations
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1 Global Journal of Mahemaical Sciences: Theory and Pracical. ISSN Volume 4, Number 3 (22), pp Inernaional Research Publicaion House hp:// Exisence Theory of Second Order Random Differenial Equaions D.S. Palimkar Deparmen of Mahemaics, Vasanrao Naik College, Nanded- 4363, M.S., India palimkar@rediffmail.com Absrac In his paper, an exisence resul for nonlinear second order ordinary random differenial equaions is proved under a Carah eodory condiion. My invesigaions is placed in he Banach space of coninuous real valued funcions on closed and bounded inervals of he real line ogeher wih an applicaion of he random version of he Leray-Schauder principle. Keywor: Nonlinear random differenial equaion, muli-valued funcion, local araciviy, global araciviy, boundary value problem. Mahemaics Subjec Classificaions: 6H25, 47H4, 47N2. Saemen of he Problem Le R denoe he real line and le J = [, T ] be a closed and bounded inerval in R. le C ( J, R) denoe he class of realvalued funcions defined and coninuously differeniable on J. Given a measurable space ( Ω, A) and for a given measurable funcion x : Ω C ( J, R), consider he boundary value problem of second order ordinary random differenial equaions (RDE), x"(, ω) = f(, x(, ω), ω) a. e. J, (.) x(, ω) = q ( ω), x'(, ω) = q ( ω), for all ω Ω, where f : J R Ω R, q, q: Ω R. By a random soluion of he RDE. (.), I mean a measurable funcion x : Ω AC ( J, R) ha saisfies he equaions in (.) where AC ( J, R) is he space of
2 34 D.S. Palimkar real valued funcions defined and absoluely coninuously differeniable on J. The RDE (.) is no new o he heory random differenial equaions. When he random parameer ω is absen, he RDE (.) reduces o he classical RDE of second order ordinary differenial equaions (ODE), x"( ) = f (, x( )) a. e. J, x() = x, x'() = x, (.2) where f : J R R. The classical ODE (.2) has been sudied in he lieraure by several auhors for differen aspecs of he soluions. See for example, Heikkilä and Lakshikanham [9] and he references herein. In his paper, I discuss he RDE (.) for exisence of random soluions, under suiable condiions of he nonlineariy f which hereby generalize several exisence resuls of he RDE (.2) proved in he above papers. My analyses on he random of he nonlinear alernaive of Leray-Schauder ype(dhage[4, 5] and an algebraic random fixed poin heorem of Dhage[4]. Auxiliary Resuls Theorem 2. (Dhage [4, 5] ) Le E be a separable Banach space and le Q: Ω E E be a compleely coninuous random operaor. Then, eiher. he random equaion Q( ω ) x= xhas a random soluion, i.e. here is a measurable funcion ξ : Ω E such ha Q( ω) ξω ( ) = ξω ( ) for all ω Ω, or 2. he se ε = { x : Ω E is measurable / λω ( ) Q( ω) x = x} is unbounded for some measurable λ : Ω R wih < λ( ω) < on Ω. An immediae corollary o above heorem in applicable form is Corollary 2. Le E be a separable Banach space and le Q: Ω E E be a compleely coninuous random operaor. Then, eiher. he random equaion Q( ω ) x= x has a random soluion, i.e. here is a measurable funcion ξ : Ω E such ha Q( ω) ξω ( ) = ξω ( ) for all ω Ω, or 2. hese ε = { x : Ω E is measurable / λq( ω) x = x} is unbounded for some ω Ω saisfying < λ <. The following heorem is used in he sudy of nonlinear disconinuous random differenial equaions. Theorem 2.2 (Carah eodory) Le Q: Ω E E be a mapping such ha Q(., x ) is measurable for all x E and Q( ω,.) is coninuous for allω Ω. Then he map ( ω, x ) Q( ω, x ) is joinly measurable. The following lemma is useful in he sudy of second order iniial value problems of ordinary random differenial equaions via fixed poin echniques.
3 Exisence Theory of Second Order Random Differenial Equaions 35 Lemma2.. For any funcion h: J L( J, R), a funcion x : J C ( J, R) is a soluion o he differenial equaion, () () ( ) ( ) x" = h a. e. J, x = q, x = q, (2.) if and only if i is a soluion of he inegral equaion. () = + + () x q q ( s) h s. (2.2) Exisence Resuls Some basic definiions are needed summarise as follows Definiion 3. A Funcion f : J R Ω Ris called random Carah eodory if he following condiions are saisfied:-. he map(, ω) (, x, ω) is joinly measurable for all x R, and 2. he map x f(, x, ω) is coninuous for all J and ω Ω. Definiion 3.2 A Carah eodory funcion f : J R Ω Ris called random L - Caraheodory if for each real number r > here is a measurable and bounded funcion h : Ω L( J, R) such ha r f (, x, ω) h (, ω) a.e. J r for all ω Ω and x R wih x r. Similarly, a Carah eodory funcion f is called random L R -Carah eodory if here is a measurable and bounded funcion h: Ω L( J, R) such ha f (, x, ω) h(, ω) a.e. J for all ω Ω and x R. I consider he following se of hypoheses in wha follows: (H ) The funcions q, q : Ω Rare measurable and bounded wih Q ( ) = Sup q ω Ω ω and Q ( ) = Sup q ω Ω ω. (H ) The funcions f is random Carah eodory on J R Ω. (H 2 ) There exiss a measurable and bounded funcion γ : Ω L( J, R) and a coninuous and nondecreasing funcion : R (, ) such ha γ +
4 36 D.S. Palimkar f (, x, ω) γ(, ω) ψ( x) a. e. J dr for all ω Ω and x R.Moreover, assume ha = for all C. ψ () r C Main Exisence Resul Theorem 3. Assume ha he hypoheses (H ) (H 2 ) hold. Suppose ha C dr > γ ( ω) L ψ ( r) (3.) for all ω Ω, where C = Q + QT. Then he RDE (.) has a random soluion defined on J. Proof. Se E = CJR (, ) and define a mapping Q: Ω E E by Q( ω) x( ) = q( ω) + q( ω) + ( s) f ( s, x( s, ω), ω) (3.2) for all J and ω Ω. Now he map q( ω) + q( ω) is coninuous for all ω Ω. Again, as he indefinie inegral is coninuous on J, Q( ω) defines a mapping Q: Ω E E. we show ha Q saisfies all he condiions of Corollary 2. on E. Firs, I show ha Q is random operaor on E. Since f (, xω, ) is random Carah eodory, he map ω f(, x, ω) is measurable in view of Theorem 2.2 Similarly, he produc ( s) f( s, x( s, ω), ω) of a coninuous and a measurable funcion is again measurable. Furher, he inegral is a limi of a finie sum of measurable funcions, herefore, he map ω q ( ω) + q ( ω) + ( s) f ( s, x( s, ω), ω) = Q( ω) x( ) is measurable. As a resul, Q is a random operaor on Ω E ino E. Le B be a bounded subse of E. hen, here is real number r > such ha x r for all x B. Nex, I show ha he random operaor Q( ω ) is coninuous on B. le { x n } be a sequence of poins in B converging o he poin x B. Then i is enough o prove ha Q( ) x () = Q( ) x() for all J and ω Ω. By he dominaed lim ω ω n n convergence heorem, hen,
5 Exisence Theory of Second Order Random Differenial Equaions 37 lim Q( ω ) x ( ) n n = lim q ( ω) + q ( ω) + ( s) f ( s, x ( s, ω), ω) n lim [ n ] = q ( ω) + q ( ω) + ( s) f ( s, x ( s, ω), ω) n [ ] = q ( ω) + q ( ω) + ( s) f ( s, x( s, ω), ω) = Q( ω) x( ) for all J and ω Ω. This shows ha Q( ω ) is a coninuous random operaor on E. Now, I show ha Q( ω ) is a oally bounded random operaor on E. I prove ha Q( ω)( B) is a oally bounded subse of E for each bounded subse B of E. To finish, i is enough o prove ha Q( ω)( B) is a uniformly bounded and equi-coninuous se in E for each ω Ω. Since he map ω γ(, ω) is bounded, by hypohesis (H ), here is a consan c such ha γω ( ) L c for all ω Ω. Leω Ω be fixed. Then for any x : Ω B, one has, Q( ω) x( ) q ( ω) + q ( ω) + ( s) f ( s, x( s, ω), ω) q ( ω ) + q ( ω ) + ( s) γ ( s, ω ) ψ ( x( s, ω ) ) Q + Q + ( s) γ ( s, ω ) ψ ( x( ω ) ) T ( Q + Q T ) + T γ ( ω) ψ ( r ) K, L n for all J, where K = ( Q + Q T ) + ctψ ( r ). This shows ha Q( ω )( B) is a uniformly bounded subse of E for each ω Ω. Nex, I show ha Q( ω)( B) is an equi-coninuous se in E. Le x B be arbirary. Then, for any, 2 J, one has Q( ω) x( ) Q( ω) x( ) q ( ω) q ( ω) ( s) f ( s, x( s, ω), ω) ( s) f ( s, x( s, ω), ω) 2
6 38 D.S. Palimkar q ( ω ) 2 + ( s) f ( s, x( s, ω), ω) ( s) f ( s, x( s, ω), ω) ( s) f ( s, x( s, ω), ω) ( s) f ( s, x( s, ω), ω) 2 2 q ( ω ) 2 T + ( ) f ( s, x( s, ω), ω) + ( s) f ( s, x( s, ω), ω) Q + ( ) f ( s, x( s, ω), ω) + T f ( s, x( s, ω), ω) T Q + f ( s, x( s, ω), ω) T T f ( s, x( s, ω), ω) Q + γ ( s, ω) ψ ( x( s, ω) T Tγ ( s, ω) ψ ( x( s, ω) Q + γ ( ω) ψ ( r) p(, ω) p(, ω) 2 L [ Q + cψ ( r)] + p(, ω) p(, ω) (3.3) for all ω Ω, where 2 2 p(, ω) = Tγ( s, ωψ ) ( r). 2 Hence, for all, J 2, Q( ω) x( ) Q( ω) x( 2) as 2, uniformly for all x B and ω Ω. Therefore, Q( ω)( B) is an equi-coninuous se in E. As Q( ω )( B) is uniformly bounded and equi-coninuous, i is compac by he Arzela-Ascolli heorem for each ω Ω. Consequenly, Q( ω) is a compleely coninuous random operaor on B. Finally, I prove ha he se ε given in conclusion (ii) of corollary 2. does no hold. Le u ε be arbirary and le ω Ω be fixed. Then u (, ω) = λq( ω) u ( ) for all J and ω Ω, where < λ <. Then, one has u (, ω) λ Q( ω) u ()
7 Exisence Theory of Second Order Random Differenial Equaions 39 q ( ω ) + q ( ω ) + ( s) f ( s, u( s, ω ), ω ) Q + Q T + ( s) γ( s, ω) ψ ( u( s, ω) ) C + T + γ ( s, ω) ψ ( u( s, ω) ) for all J andω Ω, wherec = Q + QT. m Le mω s [, ] * (, ω) u (, ω) (, ) = sup us (, ω). Then, here is a =. Then from he inequaliy (3.3) i follows ha * [, ] such ha m * (, ω) = u (, ω) * = + + Q Q T T γ ( s, ω) ψ ( u( s, ω) ) C + T + T γ ( s, ω) ψ ( m( s, ω )). ω γ ω ψ ω Pu, w(, ) = C + T ( s, ) ( m( s, )) for J. Now differeniaing his wih respec o, we obain w (, ω) = Tγ(, ω) ψ ( m(, ω) w(, ω) = C, for J. Fom he above inequaliy, I obain w (, ω) Tγ(, ω) ψ ( w(, ω), w(, ω) = C, or w (, ω ) ψ ( w (, ω)) w(, ω ) = C. Tγ (, ω ), Inegraing from o, we ge w ( s, ω ) T γ( s, ω ) ψ ( ws (, ω )). By change of variable,
8 3 D.S. Palimkar w(, ω ) C dr ψ dr T γ ( ω ) < = L ( r) ψ ( r) C. Now an applicaion of he mean value heorem for inegral calculus, here exiss a consan M > such ha u (, ω) m (, ω) w (, ω) M for all J andω Ω. Hence he condiion (ii) of Corollary 2. does no hold. As a resul, he condiion(i) hol and he operaor equaion Q( ω ) x = x has a random soluion. This furher implies ha he random differenial equaion(.)has a random soluion defined on Ω J. This complees he proof. Acknowledgemen This aricle is an some oupu resul of Minor Research Projec funded by UGC, New Delhi, India. References [] T. Bharucha-Reid, On he heory of random equaions, Proc. Symp.Appl.6 h, (963), 4-69, Ame.Soc., Providence, Rhode, Island, (964). [2] A.T. Bharucha-Reid, Random Inegral Equaions, Academic Press, New York, 972. [3] K. Deimling, Muli-valued Differenial Equaions, De Gruyer, Berlin, 998. [4] B. C. Dhage, Some algebraic and opological random fixed poin heorems wih applicaions o nonlinear random inegral equaions, Tamkang J. Mah. 35(24), [5] B. C. Dhage, A random version of a Schaefer ype fixed poin heorem wih applicaions o funcional random inegral equaions, Nonlinear Func. Anal. Appl 9 (24), [6] B. C. Dhage, Monoone ieraive echnique for Carah eodory heory of nonlinear funcional random inegral equaions, Tamkang J. Mah. 35 (24), [7] K. P. Gupa, Measure Theory, Springer-Verlag, New York, Berlin, 97. [8] P.Hans, Random Fixed Poin Theorem, Transacions of Firs Prague Conference on Informaion Theory, Saisical Decision Funcions, Pure and Applied Mahemaics, Marcel Dekker, New York, 994. [9] S. Heikkila and V. Lakshmikanham, Monoone ieraive echnique for disconinuous onlinear differenial equaions, Pure and Applied Mahs., Marcel Dekker, New York, 994. [] J. Heimmelberg, Measurable relaions, Fund. Mah. 87 (975), [] S. Ioh, Random fixed poin heorems wih applicaions o random differenial equaions in Banach spaces, J. Mah. Anal. Appl. 67 (979),
9 Exisence Theory of Second Order Random Differenial Equaions 3 [2] G. S. Ladde and V. Lakshmikanham, Random Di erenial Inequaliies, Academic Press, New York, 98. [3] V. Lakshmikanham and S. Leela, Remarks on firs and second order periodic boundary valueproblem, Nonlinear Anal. 8 (984), [4] E.Zeidler, Nonlinear Funcional analysis and Applicaions:Par-I, SpringerVerlag, 985
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