Journl of Applied Mthemtics Volume 2011, Article ID 743923, 7 pges doi:10.1155/2011/743923 Reserch Article On Existence nd Uniqueness of Solutions of Nonliner Integrl Eqution M. Eshghi Gordji, 1 H. Bghni, 1 nd O. Bghni 2 1 Center of Excellence in Nonliner Anlysis nd Applictions (CENAA), Semnn University, Semnn, Irn 2 Deprtment of Applied Mthemtics, Fculty of Mthemticl Sciences, Ferdowsi University of Mshhd, Mshhd, Irn Correspondence should be ddressed to M. Eshghi Gordji, mdjid.eshghi@gmil.com Received 26 My 2011; Revised 10 August 2011; Accepted 27 August 2011 Acdemic Editor: Kupplplle Vjrvelu Copyright q 2011 M. Eshghi Gordji et l. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. The purpose of this pper is to study the existence of fixed point for nonliner integrl opertor in the frmework of Bnch spce X : C, b, R n. Lter on, we give some exmples of pplictions of this type of results. 1. Introduction In this pper, we intend to prove the existence nd uniqueness of the solutions of the following nonhomogeneous nonliner Volterr integrl eqution: u x f x ϕ ) F x, t, u t dt Tu, u X, 1.1 where x, t, b, <<b<, f :, b R n is mpping, nd F is continuous function on the domin D : { x, t, u : x, b,t, x,u X}. The solutions of integrl equtions hve mjor role in the fields of science nd engineering 1, 2. A physicl event cn be modeled by the differentil eqution, n integrl eqution, n integrodifferentil eqution, or system of these 3, 4. Investigtion on existence theorems for diverse nonliner functionl-integrl equtions hs been presented in other references such s 5 10.
2 Journl of Applied Mthemtics In this study, we will use n itertive method to prove tht 1.1 hs the mentioned cses under some pproprite conditions. Finlly, we offer some exmples tht verify the ppliction of this kind of nonliner functionl-integrl equtions. 2. Bsic Concepts In this section, we recll bsic result which we will need in this pper. Consider the nonhomogeneous nonliner Volterr integrl eqution 1.1. Through this rticle, we consider the complete metric spce X, d, which d f, g mx x,b f x g x, for ll f, g X nd ssume tht ϕ is bounded liner trnsformtion on X. Note tht the liner mpping ϕ : X X is clled bounded, if there exists M>0 such tht ϕx M x ; for ll x X. In this cse, we define ϕ sup{ ϕx / x ; x / 0, x X}. Thus, ϕ is bounded if nd only if ϕ <, 11. Note 1. As ϕ is bounded liner mpping on X, then ϕ x λx, where λ does not depend on x X. Definition 2.1. Let S denote the clss of those functions α : 0, 0, 1 stisfying the condition lim sup α s < 1, t 0,. 2.1 s t Definition 2.2. Let B denoted the clss of those functions φ : 0, 0, which stisfies the following conditions: i φ is incresing, ii for ech x>0,φ x <x, iii α x φ x /x S, x / 0. For exmple, φ t μt, where 0 μ<1, φ t t/ t 1 nd φ t ln 1 t re in B. 3. Existence nd Uniqueness of the Solution of Nonliner Integrl Equtions In this section, we will study the existence nd uniqueness of the nonliner functionl-integrl eqution 1.1 on X. Theorem 3.1. Consider the integrl eqution 1.1 such tht i ϕ : X X is bounded liner trnsformtion, ii F : D R n nd f :, b R n re continuous,
Journl of Applied Mthemtics 3 iii there exists integrble function p :, b, b R such tht F x, t, u F x, t, v p x, t φ u v, 3.1 for ech x, t, b nd u, v R n. iv sup x,b b p2 x, t dt 1/ ϕ 2 b. Then, the integrl eqution 1.1 hs unique fixed point u in X. Proof. Consider the itertive scheme u n 1 x f x ϕ ) F x, t, u n t dt Tu n, n 0, 1,... 3.2 ) Tu n x Tu n 1 x ϕ F x, t, u n t dt ϕ F x, t, u n 1 t dt) ϕ F x, t, u n t F x, t, u n 1 t dt) ϕ x F x, t, u n t F x, t, u n 1 t dt x ϕ F x, t, u n t F x, t, u n 1 t dt 3.3 b ϕ p x, t φ u n t u n 1 t dt ( b ) 1/2 ( b 1/2 ϕ p 2 x, t dt φ 2 u n t u n 1 t dt). As the function φ is incresing then φ u n t u n 1 t φ d u n,u n 1, 3.4 so, we obtin d 2 u n 1,u n ϕ 2 ( b sup x,b φ 2 d u n,u n 1. )( b ) p 2 x, t dt φ 2 d u n,u n 1 dt 3.5 Therefore, d u n 1,u n φ d u n,u n 1 φ d u n,u n 1 d u n,u n 1 d u n,u n 1 α d u n,u n 1 d u n,u n 1, 3.6
4 Journl of Applied Mthemtics nd so the sequence {d u n 1,u n } is nonincresing nd bounded below. Thus, there exists τ 0 such tht lim n d u n 1,u n τ. Since lim sup s τ α s < 1ndα τ < 1, then there exist r 0, 1 nd ɛ>0 such tht α s <rfor ll s τ, τ ɛ. We cn tke ν N such tht τ d u n 1,u n τ ɛ for ll n N with n ν. On the other hnd, we hve d u n 2,u n 1 α d u n 1,u n d u n 1,u n rd u n 1,u n, 3.7 for ll n N with n ν. It follows tht ν d u n 1,u n d u n 1,u n r n d u ν 1,u ν <, n 1 n 1 n 1 3.8 nd hence, {u n } is Cuchy sequence. Since X, d is complete metric spce, then there exists u X such tht lim n u n u. Now, by tking the limit of both sides of 3.2, we hve ( u lim u n 1 x lim f x ϕ n n f x ϕ f x ϕ )) F x, t, u n t dt ( F x, t, lim u n t n ) F x, t, u t dt. ) ) dt 3.9 So, there exists solution u X such tht Tu u. It is cler tht the fixed point of T is unique. Note 2. Theorem 3.1 ws proved with the condition i, but there exist some nonliner exmples ϕ, such tht by the nlogue method mentioned in this theorem, the existence, nd uniqueness cn be proved for those. For exmple ϕ x sin x. 4. Applictions In this section, for efficiency of our theorem, some exmples re introduced. For Exmples 4.1 nd 4.2, 5 is used. Mleknejd et l. presented some exmples tht the existence of their solutions cn be estblished using their theorem. Generlly, Exmples 4.1 nd 4.2 re introduced for the first time in this work. On the other hnd, for Exmple 4.3, 12 is pplied. In Chpter 6 of this reference, the existence theorems for Volterr integrl equtions with wekly singulr kernels is discussed. Exmple 4.1 is extrcted from this chpter. Exmple 4.1. Consider the following liner Volterr integrl eqution: ( u x ln x 2) x 0 txu t dt, x, t 0, 1. 4.1
Journl of Applied Mthemtics 5 We hve F x, t, u F x, t, v tx u tx v tx u v ( ) tx u v tx (μ u ) v, μ 4.2 where 3/3 μ < 1. Now, we put p x, t tx /μ nd φ t μt. Becuse sup x 0,1 1 0 p2 x, t dt 1/3μ 2 1, then by pplying the result obtined in Theorem 3.1, we deduce tht 4.1 hs unique solution in Bnch spce C 0, 1, R. Exmple 4.2. Consider the following nonliner Volterr integrl eqution: ( ) 1 u x sin x x cos ( x 2 t ) rctn u t dt, x, t 0, 1. 4.3 1 x 9 2 0 1 xt We write x cos ( x 2 t ) F x, t, u F x, t, v rctn u rctn v 2 9 1 xt x cos ( x 2 t ) 1 xt u v 2 9. 4.4 Tke p x, t x cos x 2 t / 1 xt 2 nd φ t t/9. Since sup x 0,1 1 0 p2 x, t dt 1, then 4.3 hs unique solution in C 0, 1, R. Exmple 4.3 see 12. Consider the following singulr Volterr integrl eqution u x f x λ where 0 λ<1nd0<α<1/2. Then, x 0 x t α u t dt, x, t 0,T, 4.5 F x, t, u F x, t, v λ u v x t α λ u v x t α. 4.6 Put p x, t x t α nd φ t λt. We hve T T sup p 2 x, t dt sup x t 2α dt T 1 2α x 0,T 0 x 0,T 0 1 2α. 4.7 It follows tht if T 1 α 1 2α 1/2, then 4.5 hs unique solution in complete metric spce C 0,T, R.
6 Journl of Applied Mthemtics Remrk 4.4. The unique solution u C 0, 1, R of the Volterr integrl 4.5 is given by u x E 1 α ( λγ 1 α x 1 α ) u 0, x 0, 1, 4.8 where f x u 0 nd E β z : z k k 0Γ ( 1 kβ ), ( ) β>0 4.9 denotes the Mittg-Leffler function. The Mittg-Leffler function ws introduced erly in the 20th century by the Swedish mthemticin whose nme it bers. Additionl properties nd pplictions cn be found, for exmple, in Erdélyi 13 nd, especilly, in the survey pper by Minrdi nd Gorenflo 14. Acknowledgments The uthors thnk the referees for their pprecition, vluble comments, nd suggestions. References 1 S. R. Mnm, Multiple integrl equtions rising in the theory of wter wves, Applied Mthemtics Letters, vol. 24, no. 8, pp. 1369 1373, 2011. 2 S. Hussin, M. A. Ltif, nd M. Alomri, Generlized double-integrl Ostrowski type inequlities on time scles, Applied Mthemtics Letters, vol. 24, no. 8, pp. 1461 1467, 2011. 3 C. Bozky nd M. Tezer-Sezgin, Boundry element solution of unstedy mgnetohydrodynmic duct flow with differentil qudrture time integrtion scheme, Interntionl Journl for Numericl Methods in Fluids, vol. 51, no. 5, pp. 567 584, 2006. 4 M. Dehghn nd D. Mirzei, Numericl solution to the unstedy two-dimensionl Schrödinger eqution using meshless locl boundry integrl eqution method, Interntionl Journl for Numericl Methods in Engineering, vol. 76, no. 4, pp. 501 520, 2008. 5 K. Mleknejd, K. Nouri, nd R. Mollpoursl, Existence of solutions for some nonliner integrl equtions, Communictions in Nonliner Science nd Numericl Simultion, vol. 14, no. 6, pp. 2559 2564, 2009. 6 M. Meehn nd D. O Regn, Existence theory for nonliner Volterr integrodifferentil nd integrl equtions, Nonliner Anlysis. Theory, Methods nd Applictions, vol. 31, no. 3-4, pp. 317 341, 1998. 7 J. Bnś nd B. Rzepk, On existence nd symptotic stbility of solutions of nonliner integrl eqution, Journl of Mthemticl Anlysis nd Applictions, vol. 284, no. 1, pp. 165 173, 2003. 8 R. P. Agrwl, D. O Regn, nd P. J. Y. Wong, Positive Solutions of Differentil, Difference nd Integrl Equtions, Kluwer Acdemic Publishers, Dordrecht, The Netherlnds, 1999. 9 R. P. Agrwl nd D. O Regn, Existence of solutions to singulr integrl equtions, Computers nd Mthemtics with Applictions, vol. 37, no. 9, pp. 25 29, 1999. 10 D. O Regn nd M. Meehn, Existence Theory for Nonliner Integrl nd Integrodifferentil Equtions, vol. 445 of Mthemtics nd Its Applictions, Kluwer Acdemic Publishers, Dordrecht, The Netherlnds, 1998. 11 G. B. Follnd, Rel Anlysis, John Wiley nd Sons, New York, NY, USA, 1984, Modern Techniques nd Their Applictions. 12 H. Brunner, Colloction Methods for Volterr Integrl nd Relted Functionl Differentil Equtions, vol. 15 of Cmbridge Monogrphs on Applied nd Computtionl Mthemtics, Cmbridge University Press, Cmbridge, UK, 2004.
Journl of Applied Mthemtics 7 13 A. Erdélyi, W. Mgnus, F. Oberhettinger, nd F. G. Tricomi, Higher Trnscendentl Functions, vol. 3, McGrw-Hill Book Compny, New York, NY, USA, 1955. 14 F. Minrdi nd R. Gorenflo, On Mittg-Leffler-type functions in frctionl evolution processes, Journl of Computtionl nd Applied Mthemtics, vol. 118, no. 1-2, pp. 283 299, 2000.
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