Mah. Sci. Le., No., 9-7 (3) 9 Mahemaical Sciences Leers An Inernaional Journal @ 3 NSP Naural Sciences Publishing Cor. Posiive coninuous soluion of a quadraic inegral equaion of fracional orders A. M. A. El-Sayed, H. H. G Hashem, and Y. M. Y. Omar 3 Faculy of Science, Alexandria Universiy, Alexandria, Egyp Faculy of Science, Qassim Universiy, Buraidah, Saudi Arabia 3 Faculy of Science, Omar Al-Mukhar Universiy, Al-Beida, Libya E.mails : amasayed@homail.com,hendhghashem@yahoo.com, yasmin9843@yahoo.com Received: 9 Jul., Revised Oc., Acceped Oc. Published online: Jan. 3 Absrac: We are concerned here wih he exisence of a unique posiive coninuous soluion for he quadraic inegral of fracional orders ( ( x( ) a( ) f ( s, x( ). f ( s, x( ), I ( ) ( ) where f and f are Carahéodory funcions. As an applicaion he Cauchy problems of fracional order differenial equaion D x( ) f (, x( )), wih one of he wo iniial values x() or I x( ) will be sudied. Some examples are considered as applicaions of our resuls. Keywor: Quadraic inegral equaion of fracional-order, Banach fixed poin heorem, Cauchy problem.. Inroducion Fracional differenial and inegral equaions have received increasing aenion during recen years due o is applicaions in numerous diverse fiel of science and engineering. In fac, fracional differenial and inegral equaions are considered as models alernaive o nonlinear differenial equaions []. here has been a significan developmen in fracional differenial equaions. We refer readers o he monographs of Kilbas e al. [7] and he papers []- [8]. Quadraic inegral equaions are ofen applicable in he heory of radiaive ransfer, he kineic heory of gases, he heory of neuron ranspor, he queuing heory and he raffic heory. Many auhors sudied he exisence of soluions for several classes of nonlinear quadraic inegral equaions (see e.g. [], [7] and [9]-[5]. However, in mos of he above lieraure, he main resuls are realized wih he help of he echnique associaed wih he measure of noncompacness. Insead of using he echnique of measure of noncompacness we use he Banach conracion fixed poin heorem. Le I [, ], C C[, ] be he space of coninuous funcions on I, and space of Lebesgue inegrable funcions on I. L L [, ] be he Firsly, we deal wih he quadraic inegral equaion of fracional order ( ( x( ) a( ) f ( s, x( ). f ( s, x( ), I ( ) ( ) Where is a real number. Also, Eqn. () can be wrien in operaor form as: ()
El-Sayed, Hashem, Omar: Quadraic inegral equaion x( ) a( ) ( I f (, x( ))) ( I f (, x( ))), I where, (,]. We prove he exisence and uniqueness of posiive coninuous soluion of (). As an applicaion of our resuls, he exisence of a unique soluion of he iniial value problems D x( ) f (, x( )), () x()=. and D x( ) f (, x( )), (3) I x( ). will be sudied. Le, be wo posiive real numbers, hen he definiion of he fracional (arbirary) order inegraion is given by : Definiion. Le f ( ) L, R. he fracional (arbirary) order inegral of he funcion of order is defined as (see [9], and []) ( Ia f ( ) f (. a ( ) When a we can wrie I f ( ) I f ( ). Definiion..he Riemann-Liouville fracional-order derivaive of f() of order (,) is defined as (see [], [9]) d ( Da f ( ) f ( d a ( ) or d Da f ( ) I a f ( ). d Main heorem Consider he following assumpions: (i) a : I is coninuous on I. R (ii) fi : I R R are measurable in for all x R, and saisfy he Lipschiz condiion wih respec o he second argumen x for almos all I. i.e f (, x) f (, y) L x y, L i,. (4) i i i i for each (, x),(, y) I R. (iii) here exis wo funcions mi L such ha
El-Sayed, Hashem, Omar: Quadraic inegral equaion f (, x) m ( ), I, i,. i Le max{, }, and M max{ I m ( ) : I, }, i,. i i Now, for he exisence of a unique coninuous posiive soluion of he quadraic inegral equaion () we have he following heorem. heorem. Le he assumpions (i)-(iii) be saisfied. Moreover, if LM LM. ( ) ( ) ( ) ( ) hen he quadraic inegral equaion () has a unique posiive coninuous soluion x C. Proof. Equaion () can be wrien as x( ) a( ) I ( I f (, x( ))) I ( I f (, x( ))). Define he operaor F by: ( ( Fx( ) a( ) f ( s, x( ) f ( s, x( ) ( ) ( ) he operaor maps ino iself. For his we have, le, I, such ha, hen Fx( ) Fx( ) a( ) a( ) ( ( f ( s, x( ) f ( s, x( ) ( ) ( ) ( ( f ( s, x( ) f ( s, x( ) ( ) ( ) a( ) a( ) ( ( ( f ( s, x( ) f ( s, x( ) ) ( ) ( ) ( ( ( f ( s, x( ) f( s, x( ) ) ( ) ( ) ( ( f ( s, x( ) f ( s, x( ) ( ) ( ) a ( ) a ( )
El-Sayed, Hashem, Omar: Quadraic inegral equaion ( ( f ( s, x( ) f( s, x( ) ( ) ( ) ( ( f( s, x( ) f( s, x( ) ( ) ( ) ( ( f( s, x( ) f( ( ) s, x( ) ( ) a( ) a( ) ( ( m ( m( ( ) ( ) ( ( m( m( ( ) ( ) ( ( m( m( ( ) ( ) a ( ) a ( ) ( ( M ( ) ( ) ( ( M ( ) ( ) ( ( M ( ) ( ) a( ) a( ) M [ ( ) ( ) ][ ] ( ) ( ) ( ) ( ) ( ) ( ) M [ ][ ] ( ) ( ) ( ) ( ) M [ ][ ]. Which proves ha F : C C. Now, o show ha F is conracion. Le x, y C, hen we have ( ( Fx( ) Fy( ) f ( s, x( ) f ( s, x( ) ( ) ( ) ( ( f ( s, y( ) f ( s, y( ) ( ) ( ) ( ( f ( s, x( ) f ( s, x( ) ( ) ( )
El-Sayed, Hashem, Omar: Quadraic inegral equaion 3 ( ( f ( s, y( ) f ( s, y( ) ( ) ( ) ( ( f ( s, x( ) f ( s, y( ) ( ) ( ) ( ( f ( s, x( ) f ( s, y( ) ( ) ( ) ( ( f ( s, x( ) ( ) f ( s, x( ) f ( s, y( ) ( ) ( ( f ( s, y( ) f ( s, x( ) f( s, x( ) ( ) ( ) ( ( L m ( x( y( ( ) ( ) L ( ( ) ( m ( x( y( ( ) ( LM x y ( ) ( ) ( LM x y ( ) ( ) LM x y ( ) ( ) LM x y ( ) ( ) L M L M [ ( ] x y ) ( ) ( ) ( ) Since LM LM. ( ) ( ) ( ) ( ) hen F is conracion. herefore, by he Banach conracion fixed poin heorem [6], he operaor F has a unique fixed poin x C (i.e. he quadraic inegral equaion () has a unique soluion x C ). which complees he proof. As paricular cases of heorem. we have he following corollaries. Corollary. Le he assumpions (i) and (iii) be saisfied. If fi :[, ] R R, i, are coninuous and saisfy Lipschiz condiion (4), hen he quadraic inegral equaion () has a unique coninuous soluion x C.
4 El-Sayed, Hashem, Omar: Quadraic inegral equaion Corollary.3 Le he assumpions of heorem. be saisfied (wih, and, ). If LM, L max{ L, L }, hen he quadraic inegral equaion has a unique coninuous soluion x C, which is he same equaion sudied in []. x( ) a( ) f ( s, x( ). f ( s, x( ) Corollary.4 Le he assumpions of heorem. be saisfied (wih f f f and ), hen he quadraic inegral equaion x( ) ( I f (, x( ))) (5) has a unique coninuous soluion x C. Corollary.5 Le he assumpions of heorem. be saisfied (wih, f, f f ), hen he inegral equaion x( ) a( ) I f (, x( )), I (6) has a unique coninuous soluion x C. Proof. Le f, f f in (), and aking he limi as, we have lim x( ) a( ) lim( I ) ( I f (, x( ))), I x( ) a( ) I f (, x( )), I. 3 Fracional order differenial equaions Lemma 3. Le ( ) f : I R R, be measurable in I for any x R, and coninuous in x R for almos all I. ( ) here exiss an inegrable funcion m L, such ha f (, x) m( ), hen I f (, x). Proof: Le, le M max I m( ). Now I f (, x) I ( I f (, x)), hen I f (, x) I M M ( ) his implies ha I f (, x). We shall prove he following corollary.
El-Sayed, Hashem, Omar: Quadraic inegral equaion 5 Corollary 3. Le he assumpions ( i ) and ( ii ) be saisfied. If f (, x) f (, y) L x y, L, (, x) I R. hen he iniial value problem () has a unique coninuous soluion x C. Proof. Equaion () can be wrien as By inegraing boh sides, we obain operaing by I on boh sides, we ge d I x f x d ( ) (, ( )). I x If x c ( ) (, ( )), I x( ) I f (, x( )) c, ( ) (7) differeniaing equaion (7), we ge x( ) I f (, x( )) c, ( ) leing =, hen by Lemma 3. we deduce ha c=, hen and we obain equaion (5). Conversely, le hen x x( ) ( I f (, x( ))) ( ) ( I f (, x( ))), x( ) I f (, x( )) operaing by and Finally I o boh sides, we ge I x( ) If (, x( )) x d I x f x d ( ) (, ( )). (8) () ( I f (, x( )) ). (9) hen, he iniial value problem () and he quadraic inegral equaion (5) are equivalen. Consequenly, from Corollary.4 we deduce ha he iniial value problem () has a unique coninuous soluion x C.. Also, he following corollary can be proved (see [3]). Corollary 3.3 Le he assumpions of Corollary 3.3, hen he iniial value problem of equaion (3) has a unique coninuous soluion x C.
6 El-Sayed, Hashem, Omar: Quadraic inegral equaion Example: Consider he following quadraic inegral equaion 3 x x( ) I [ 5 ( log( x( ) 3) )]. I [ e ], I [,]. () 3 Se f x log x I (, ) 5 ( ( ( ) 3) ), x f(, x) e. 3 hen we have: ( i) f(, z) f(, y) 5 ( log( z( ) 3) ) 5 ( log( y( ) 3) ) ( log( z( ) 3) ) ( log( y( ) 3) ) z y z x ( ii) f(, z) f(, x) e. e. 3 3 e z e x 3 x z x z x z. 3 3 Example: Consider he following Cauchy problem D x( ) x( ), I [,] () 3 wih he iniial condiion x()= Se f (, x) x( ), I 3 hen easily we can deduce ha: f (, z) f (, y) z y. 3 Example:3 Consider he following Cauchy problem D x( ) sin x( ), I [,] 4 () wih he iniial condiion I x( ) Se
El-Sayed, Hashem, Omar: Quadraic inegral equaion 7 f (, x) sin x( ), I 4 hen easily we ge f (, z) f (, x) sin z( ) sin x( ) 4 4 sin z ( ) sin x ( ) 4 zx. 4 References [] I. K. Argyros, Quadraic equaions and applicaions o Chandrasekhar s and relaed equaions, Bull. Ausral. Mah. Soc. 3 (985) 75-9. [] I. K. Argyros, On a class of quadraic inegral equaions wih perurbaions, Func. Approx. (99) 5-63. [3] J. Banas, M. Lecko, W.G. El-Sayed, Eixsence heorems of some quadraic inegral equaion, J. Mah. Anal. Appl. 7 (998)76-79. [4] J. Banas, A. Marinon, Monoonic soluions of a quadraic inegral equaion of Volerra ype, Compu. Mah. Appl. 47 (4) 7-79. [5] J. Banas, J. Caballero, J. Rocha and K. Sadarangani, Monoonic soluions of a class of quadraic inegral equaion of Volerra ype, Comp. and Mah. wih Applicaions, 49 (5) 943-95. [6] J. Banas, J. Rocha Marin, K. Sadarangani, On he soluions of a quadraic inegral equaion of Hemmersien ype, Mahemaecal and Compuer Modelling. vol. 43 (6) 97-4. [7] J. Banas, B. Rzepka, Monoonic soluions of a quadraic inegral equaions of fracional -order J. Mah. Anal. Appl. 33 (7) 37-378. [8] J. Banas, B. Rzepka, Nondecreasing soluions of a quadraic singular Volerra inegral equaion, Mah. Compu. Modelling 49 (9) 488-496. [9] M. A. Darwish, On monoonic soluions of a singular quadraic inegral equaion wih supremum, Dynam. Sys. Appl., 7 (8), 539-55. [] A. M. A. El-Sayed, M. M. Saleh and E. A. A. Ziada, Numerical and Analyic Soluion for a nonlinear quadraic Inegral Equaion, Mah. Sci. Res. J., (8) (8), 83-9. [] A. M. A. El-Sayed, H. H. G. Hashem, Carahèodory ype heorem for a nonlinear quadraic inegral equaion, MAH. SCI. RES. J. (4), (8)7-95. [] A. M. A. El-Sayed, H. H. G. Hashem, Inegrable and coninuous soluions of a nonlinear quadraic inegral equaion, Elecronic Journal of Qualiaive heory of Differenial Equaions 5 (8), -. [3] A. M. A. El-Sayed, M. SH. Mohamed, F. F. S. Mohamed, Exisence of posiive coninuous soluion of a quadraic inegral equaion of fracional orders, Journal of Fracional Calculus and Applicaions Vol. (), No. 9, pp. -7. [4] A. M. A. El-Sayed, H. H. G. Hashem, and E. A. A. Ziada, Picard and Adomian meho for quadraic inegral equaion, Compuaional and Applied Mahemaics, Vol 9. No 3, pp 447-463, (). [5] W. G. El-Sayed, B. Rzepka, Nondecreasing Soluions of a quadraic Inegral Equaion of Urysohn ype, Compuers and Mahemaics wih Applicaions 5 (6) 65-74. [6] Goebel, K. and Kirk W. A., opics in Meric Fixed poin heory, Cambridge Universiy Press, Cambridge (99). [7] A. A. Kilbas, H. M. Srivasava, J. J. rujillo, heory and Applicaions of Fracional Differenial Equaions, Elsevier, Amserdam, 6. [8] V. Lakshmikanham, heory of fracional funcional differenial equaions, Nonlinear Anal., 69 (8), 3337-3343. [9] I. Podlubny and A. M. A. El-Sayed, On wo definiions of fracional calculus. Preprin UEF. 3-96 (ISBN 8-799-5-) Solvak Academy of scince Insiue of Experimenal phys. (996). [] I. Podlubny, Fracional Differenial Equaions. Acad. Press, San Diego-New York-London (999).