Monotonic Solutions of a Class of Quadratic Singular Integral Equations of Volterra type

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1 In. J. Conemp. Mah. Sci., Vol. 2, 27, no. 2, 89-2 Monoonic Soluions of a Class of Quadraic Singular Inegral Equaions of Volerra ype Mahmoud M. El Borai Deparmen of Mahemaics, Faculy of Science, Alexandria Universiy, Alexandria, Egyp m m elborai@yahoo.com Wagdy G. El-sayed Deparmen of Mahemaics, Faculy of Science, Alexandria Universiy, Alexandria, Egyp wagdygomaa@yahoo.com Mohamed I. Abbas Deparmen of Mahemaics, Faculy of Science, Alexandria Universiy, Alexandria, Egyp m i abbas77@yahoo.com Absrac We sudy nonlinear singular inegral equaion of Volerra ype in Banach space of real funcions defined and coninuous on a bounded and closed inerval. Using a suiable measure of noncompacness we prove he exisence of monoonic soluions. Also a generalized resul is aken in he consideraion. Mahemaics Subjec Classificaion: 32A55, D9 Keywords: Measure of noncompacness, Fixed-poin heorem, Monoonic soluions, Quadraic singular inegral equaion Preliminaries and Inroducion In his paper, we are going o sudy he solvabiliy of a nonlinear singular inegral equaion of Volerra ype of he form:

2 9 M. M. El-Borai, W. G. El-Sayed and M. I. Abbas x () =a ()x(), () where I =,M, M < and <α. We look for soluions of ha equaion in he Banach space of real funcions being defined and coninuous on a bounded and closed inerval. The main ool used in our invesigaions is a special measure of noncompacness consruced in such a way enable us o sudy he solvabiliy of considered equaions in he class of monoonic funcions. For furher purposes, we collec a few auxiliary resuls which will be needed in he sequel. Assume ha (E,. ) is an infinie-dimensional Banach space wih he zero elemen θ. Denoe by B (x, r) he closed ball cenered a x and wih radius r. The symbol B r sands for he ball B (θ, r). If X is a subse of E, hen X, Conv X denoe he closure and convex closure of X, respecively. We use he symbols λx and X Y o denoe he algebraic operaions on ses. The family of all nonempy and bounded subses of E will be denoed by M E and is subfamily consising of all relaively compac ses is denoed by N E. Throughou his secion, we accep he following definiion of he noion of a measure of noncompacness. Definiion. A funcion μ : M E R =, ) is said o be a measure of noncompacness in E if i saisfies he following condiions: he family ker μ = {X M E : μ (X) =} is nonempy and ker μ N E ; 2 X Y = μ (X) μ (Y ); 3 μ ( X ) = μ (Conv X) =μ (X); 4 μ (λx ( λ) Y ) λμ (X)( λ) μ (Y ),for x, ; 5 if (X n ) is a sequence of closed ses from M E such ha X n X n,for n =, 2,..., and if lim μ (X n)=,hen he se X = X n is nonempy. n n=

3 Quadraic Singular Inegral Equaions 9 The family ker μ described in noncompacness μ. is called he kernel of he measure of Furher facs concerning measures of noncompacness and is properies may be found in 4. For our furher purposes, we shall only need he following fixed-poin heorem 8. Theorem.2 Le Q be a nonempy bounded closed convex subse of he space E and le F : Q Q be a coninuous ransformaion such ha μ (FX) Kμ(X) for any nonempy subse X of Q, where K, ) is a consan. Then F has a fixed poin in he se Q. Remark.3 Under he assumpions of he above heorem, i can be shown ha he se F ixf of fixed poins of F belonging o Q is a member he family ker μ. This fac permis us o characerize soluions of considered operaor equaions. In wha follows, we shall work in he classical Banach space C,M consising of all real funcions defined and coninuous on he inerval,m. For convenience, we wrie I =,M and C (I) = C,M. The space C (I) is furnished by he sandard norm x = max { x () : I}. Now, we recall he definiion of a measure of noncompacness in C (I) which will be used in our furher invesigaions. Tha measure was inroduced and sudied in 4. To do his, le us fix a nonempy and bounded subse X of C (I). For x X and ε denoed by ω (x, ε), he modulus of coninuiy of he funcion x, i.e., Furher, le us pu ω (x, ε) = sup { x () x () :, I, ε}. ω (X, ε) = sup {ω (x, ε) :x X}, ω o (X) = lim ω (X, ε). ε Nex, le us define he following quaniies: d (x) = sup { x () x () x () x () :, I, }, i (x) = sup { x () x () x () x () :, I, }, d (X) = sup {d (x) :x X}, i (X) = sup {i (x) :x X}.

4 92 M. M. El-Borai, W. G. El-Sayed and M. I. Abbas Observe ha d (X) = if and only if all funcions belonging o X are nondecreasing on I. In a similar way, we can characerize he se X wih i (X) =. Finally, we define he funcion μ on he family M C(I) by puing μ (X) =ω o (X)d (X). I can be shown (see 4) ha he funcion μ is a measure of noncompacness in he space C (I). The kernel ker μ of his measure conains nonempy and bounded ses X such ha funcions from X are equiconinuous and nondecreasing on he inerval I. Remark.4 The above described properies of he kernel ker μ of he measure of noncompacness μ in conjuncion wih Remark (.3) allow us o characerize soluions of he nonlinear inegal equaion considered in he nex secion. Remark.5 Observe ha, in a similar way, we can define he measure of noncompacness associaed wih he se quaniy i (X) defined above. We omi he deails concerning ha measure. 2 Main Resuls In his secion, we shall sudy he solvabiliy of nonlinear quadraic singular inegral equaion of Volerra ype( ). We shall look for soluions of ha equaion in he Banach space of real funcions being defined and coninuous on a bounded and closed inerval. The ool used in our invesigaions is a special measure of noncompacness consruced in such a way ha is use enables us o sudy he solvabiliy of considered equaion in he class of monoonic fucions. Firs, in equaion( )we noice ha he funcions a = a () and v = v (, s, x) are given while x = x () is unknown funcion. We shall invesigae equaion ( ) assuming ha he following se of hypoheses is saisfied: (i) a C (I) and he funcion a is nondecreasing and nonnegaive on I; (ii) v : I I R R is a coninuous funcion such ha v : I I R R and for arbirary fixed s I and x R he funcion v (, s, x) is nondecreasing on I; (iii) here exiss a nondecreasing funcion f : R R such ha he inequaliy v (, s, x) f ( x )

5 Quadraic Singular Inegral Equaions 93 holds for all, s I and x R; (iv) he inequaliy a r M α α f (r) r has a posiive soluion r such ha M α α f (r o). Now, we can formulae our main exisence resul. Theorem 2. Under assumpions (i) (iv), equaion( )has a leas one soluion x = x () which belonging o he space C (I) and is nondecreasing on he inerval I. Proof. Le us consider he operaor A defined on he space C (I) in he following way: (Ax)() =a ()x(). In view of assumpions (i) and (ii), i follows ha he funcion Ax is coninuous on I for any funcion x C (I), i.e., A ransforms he space C (I) ino iself. Moreover, keeping in mind assumpions (iii), we ge: (Ax)() a () x () f ( x ) a x Hence, a x f ( x ) a x α α f ( x ) a x M α α f ( x ). Ax a x M α α f ( x ). ( s) α ds Thus, aking ino accoun assumpion (iv), we infer ha here exiss r > wih M α f (r α o) and such ha he operaor A ransforms he ball B ro ino iself.

6 94 M. M. El-Borai, W. G. El-Sayed and M. I. Abbas In wha follows, we shall consider he operaor A on he subse B r o defined in he following way: B ro of he ball B r o = {x B ro : x (), for I}. Obviously, he se B r o is nonempy, bounded, closed and convex. In view of hese facs and assumpions (i) and (ii), we deduce ha A ransforms he se B r o ino iself. Now, we shall show ha A is coniguous on he se B r o. To do his, le us fix ε> and ake arbirary x, y B r o such ha x y ε. Then, for I, we derive he following esimaes: (Ax)() (Ay)() x () v (, s, y (s)) y () x () y () y () v (, s, y (s)) ( s) α ds y () ( s) α ds x y εf (r o ) where we denoed ( s) α ds y ( s) α ds r o β ro (ε) εf (r o ) M α α r oβ ro (ε) M α α, v (, s, y (s)) ( s) α ds ( s) α ds β ro (ε) = sup { v (, s, x) v (, s, y) :, s I, x, y,r o, x y ε}. Obviously, β ro (ε) asε which is a simple consequence of he uniform coninuiy of he funcion v on he se I I,r o. From he above esimae, we derive he following inequaliy: Ax Ay εf (r o ) M α α r oβ ro (ε) M α α, which implies he coninuiy of he operaor A on he se B ro. In wha follows, le us ake a nonempy se X B r o. Furher, fix arbirary number ε> and choose x X and,,m such ha ε. Wihou loss of generaliy, we may assume ha. Then, in view of our assumpions we obain (Ax)() (Ax)() a () a () x () ( s) α ds x ()

7 Quadraic Singular Inegral Equaions 95 where we have denoed w (a, ε) x () ( s) α ds x () ( s) α ds x () ( s) α ds x () ( s) α ds x () ( s) α ds x () x () x () ω (a, ε) x () x () ( s) α ds x () ( s) α ds x () ( s) α ( s) α ds x () ( s) α ds ω (a, ε)ω (x, ε) f (r o ) ( s) α ds r o γ ro (ε) ( s) α ds r o f (r o ) ( s) α ( s) α ds r o f (r o ) ( s) α ds ω (a, ε)ω (x, ε) f (r o ) α α r oγ ro (ε) α α α ( ) α r o f (r o ) α ( ) α r o f (r o ) α α α α ω (a, ε)ω (x, ε) f (r o ) M α α r oγ ro (ε) M α α α r o f (r o ) α α, α γ ro (ε) = sup { v (, s, x) v (, s, x) :, I, ε, x,r o }. α α α α By applying The Mean Value Theorem on he bracke ge α α α ( ) = < ε α δ α δ, α for all <δ<. Then we ge (Ax)() (Ax)() ω (a, ε)ω (x, ε) f (r o ) M α, we α r oγ ro (ε) M α α r of (r o ) ε δ. α (2)

8 96 M. M. El-Borai, W. G. El-Sayed and M. I. Abbas Noice ha, in view of he uniform coninuiy of he funcion v on he se I I,r o, we have γ ro (ε) asε. Now, fix arbirary x X and, I such ha. Then we have he following chain of esimaes: (Ax)() (Ax)() (Ax)() (Ax)() = a ()x() ( s) α ds a () x () a ()x() ( s) α ds a () x () { a () a () a() a ()} x () ( s) α ds x () x () ( s) α ds x (), and since a () is nondecreasing, we can deduce, according o he definiion of d (x), ha: d (C (I)) = sup {d (a) :a C (I)} =. Then, (Ax)() (Ax)() (Ax)() (Ax)() x () ( s) α ds x () x () ( s) α ds x () x () ( s) α ds x () ( s) α ds x () ( s) α ds x () x () ( s) α ds x () ( s) α ds x () ( s) α ds x () x () x () ( s) α ds x () ( s) α ds x () x () ( s) α ds x () ( s) α ds { x () x () x() x ()} ( s) α ds { x () ( s) α ( s) α ds ( s) α ds ( s) α ds

9 Quadraic Singular Inegral Equaions 97 x () ( s) α ds { ( s) α ( s) α ds } ( s) α ds } ( s) α ds ( s) α ds. Again,as above, le us applying The Mean Value Theorem on he bracke ( s) α ( s) α, we ge ( s) α ( s) α ( ) = α δ α < α ε δ α, for all <δ <.Then we obain he following inequaliy: (Ax)() (Ax)() (Ax)() (Ax)() f (r o ) { x () x () x() x ()} ( s) α ds 2α ε x() ds x () δ α { } ( s) α ds. Taking ino accoun ha he las erm in he above inequaliy will be vanished (noice ha he funcion v (, s, x) is nondecreasing on I). Finally we ge (Ax)() (Ax)() (Ax)() (Ax)() { x() x() x() x()} M α α f(r o)2α ε δ α x() v(, s, x(s))ds. (3) By adding Eq.( 2) and Eq.( 3) and aking he supremum of he resulan inequaliy hen le ε, keeping in mind he definiion of he measure of noncompacness μ(x) =ω o (X)d(X), herefore we obain μ (AX) M α α f (r o) μ (X). Now, aking ino accoun he above inequaliy and he fac ha M α f (r α o) < and applying Theorem(.2), we complee he proof. Remark 2.2 Taking ino accoun Remarks (.3) and (.4) and he descripion of he kernel of he measure of noncompacness μ given in secion, we deduce easily from he proof of Theorem ( 2.) ha he soluions of he inegral equaion ( ) belonging o he se B r o are nondecreasing and coninuous on he inerval I. Moreover, hose soluions are also posiive provided a () > for I.

10 98 M. M. El-Borai, W. G. El-Sayed and M. I. Abbas 3 Generalized Resuls The resuls in his secion generalize and complee he resuls in secion ( 2). We consider he following nonlinear singular inegral equaion of Volerra ype: x () =a ()(Bx)(), (4) where ( he operaor B saisfies he following se of condiions: ) i The operaor B : C (I) C (I) is coninuous and saisfies he condiions of Theorem (.2) for he measure of noncompacness μ wih a consan K and, ( moreover, B is a posiive operaor, i.e., Bx ifx. ) ii There exis nonnegaive consans b and c such ha: (Bx)() b c x, for each x C (I) and I. We replace he assumpion (iv) in Secion ( 2) wih he following assumpion: (iii ) The inequaliy a (b cr) M α α f (r) r has a posiive soluion r o such ha K M α α f (r o) <. By connecion beween he assumpions (i) (iii), in Secion ( 2), and he assumpions ( i ) ( iii ) we can formulae he following exisence resul. Theorem 3. Under assumpions (i) (iii) and ( i ) ( iii ), he equaion ( 4) has a leas one soluion x = x () which belongs o he space C (I) and is nondecreasing on he inerval I. Proof. Le us consider he operaor V defined on he space C (I) in he following way: (Vx)() =a ()(Bx)(). In a similar way as in Proof of Theorem ( 2.), we ge he following esimaes:. (Vx)() a (b c x ) M α α f ( x ),

11 Quadraic Singular Inegral Equaions 99 which proves ha V ransforms he space C (I) ino iself. 2. (Vx)() (Vy)() Bx By f (r o ) M α α (b cr o) β ro (ε) M α α, and from he uniform coninuiy of he funcion v on he se I I,r o and he coninuiy of V, he las inequaliy implies he coninuiy of he operaor V on he se B r o. 3. (Vx)() (Vy)() a () a () (Bx)() ( s) α ds (Bx)() w (a, ε) (Bx)() ( s) α ds (Bx)() ( s) α ds (Bx)() ( s) α ds (Bx)() ( s) α ds (Bx)() ( s) α ds (Bx)() (Bx)() (Bx)() ω (a, ε) (Bx)() (Bx)() ( s) α ds (Bx)() ( s) α ds (Bx)() ( s) α ( s) α ds (Bx)() ( s) α ds ω (a, ε)ω (Bx,ε) f (r o ) ( s) α ds (b cr o ) γ ro (ε) ( s) α ds (b cr o ) f (r o ) ( s) α ( s) α ds (b cr o ) f (r o ) ( s) α ds ω (a, ε)ω (Bx,ε) f (r o ) M α Hence, we ge (Vx)() (Vy)() ω (a, ε)ω (Bx,ε) f (r o ) M α α (b cr o) γ ro (ε) M α α (b cr o) f (r o ) ε δ α. α (b cr o) γ ro (ε) M α α (b cr o) f (r o ) ε δ. α (5)

12 M. M. El-Borai, W. G. El-Sayed and M. I. Abbas Noice ha, in view of he uniform coninuiy of he funcion v on he se I I,r o, we have γ ro (ε) asε. 4. (Vx)() (Vx)() (Vx)() (Vx)() = a ()(Bx)() ( s) α ds a () (Bx)() a ()(Bx)() ( s) α ds a () (Bx)() { a () a () a() a ()} (Bx)() ( s) α ds (Bx)() (Bx)() ( s) α ds (Bx)() (Bx)() ( s) α ds (Bx)() (Bx)() ( s) α ds (Bx)() (Bx)() ( s) α ds (Bx)() ( s) α ds (Bx)() ( s) α ds (Bx)() (Bx)() ( s) α ds (Bx)() ( s) α ds (Bx)() ( s) α ds (Bx)() { (Bx)() (Bx)() (Bx)() (Bx)()} ( s) α ds { (Bx)() ( s) α ( s) α ds } ds ( s) α ds { (Bx)() ( s) α ( s) α ds ( s) α ds { (Bx)() (Bx)() (Bx)() (Bx)()} 2α ε δ α (Bx)() (Bx)() v(, s, x(s))ds ( s) α ( s) α f (r o ) ( s) α ds } ds ( s) α ds { } ( s) α ds = { (Bx)() (Bx)() (Bx)() (Bx)()} M α α f (r o)

13 Quadraic Singular Inegral Equaions 2α ε δ α Hence, we ge (Bx)() ds. (Vx)() (Vx)() (Vx)() (Vx)() { (Bx)() (Bx)() (Bx)() (Bx)()} M α α f (r o) 2α ε δ α (Bx)() ds. (6) Finally (as in he pervious secion), by adding Eq.( 5) and Eq.( 6)and keeping in mind he definiion of he measure of noncompacness μ, we obain μ (VX) M α α f (r o) μ (BX) M α α f (r o) Kμ(X). Now, aking ino accoun he above inequaliy and he fac ha M α α f (r o) K< and applying Theorem (.2),we complee he proof. ACKNOWLEDGMENTS. The auhors would like o hank Professor Dr. Emil Minchev for his suggesions, correcions and valuable remarks. References R. P. Agarwal, D. O Regan, Infinie Inerval Problems for Differenial, Difference and Inegral Equaions, Kluwer Academic Publishers, Dordrech, 2. 2 I. K. Argyros, Quadraic equaions and applicaions o Chandrasekhar s and relaed equaions, Bull. Ausral. Mah. Soc. 32 (985), J. Banaś, M. Lecko and W. G. El-Sayed, Exisence heorems for some quadraic inegral equaions, J. Mah. Anal. Appl. 222 (998), J. Banaś, K. Geobel, Measure of Noncompacness in Banach Spaces, in:lecure Noes in Pure and Applied Mahemaics, Vol.6, Dekker, New York, M. M. El-Borai, On some fracional differenial equaions in Hilber space, Disc. Con. Dynam. Sys.(25), M. M. El-Borai, The fundamenal soluions for fracional evoluion equaions of parabolic ype, J. Applied Mah. and S.A.,H.P.C. (24).

14 2 M. M. El-Borai, W. G. El-Sayed and M. I. Abbas 7 W. G. El-Sayed, Nonlinear funcional inegral equaions of convoluion ype, Porugaliae Mah. J. Vol. 54 fasc. 4 (997). 8 G. Darbo, Puni unii in ransformazioni a condominio non compao, Rend. Sem. Ma. Univ. Padova 24, (955), Received: February 8, 26