Research Article Applications of Normal S-Iterative Method to a Nonlinear Integral Equation

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1 e Scientific World Journal Volue 2014, Article ID , 5 pages Research Article Applications of Noral S-Iterative Method to a Nonlinear Integral Equation Faik Gürsoy Departent of Matheatics, Faculty of Science and Letters, Yildiz Technical University, Davutpasa Capus, Esenler, Istanbul, Turkey Correspondence should be addressed to Faik Gürsoy; faikgursoy02@hotail.co Received 4 June 2014; Accepted 22 July 2014; Published 10 August 2014 Acadeic Editor: M. Mursaleen Copyright 2014 Faik Gürsoy. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. It has been shown that a noral S-iterative ethod converges to the solution of ixed type Volterra-Fredhol functional nonlinear integral equation. Furtherore, a data dependence result for the solution of this integral equation has been proven. 1. Introduction The scientists working in alost every field of science are faced with nonlinear probles, because nature itself is intrinsically nonlinear. Such probles can be odelled as nonlinear atheatical equations. Solving nonlinear equations is, of course, considered to be atter of the utterost iportance in atheatics and its anifold applications. There are nuerous systeatic approaches which are classified as direct and iterative ethods to solve such equations in the existing literature. Indeed, by using direct ethods, finding solutions to a coplicated nonlinear equation can be an alost insurountable challenge. In this context, iterative ethods have becoe very iportant atheatical tools for finding solutions to a nonlinear equation. For a coprehensive review and references to the extensive literature on the iterative ethods, the interested reader ay refer to soe recent works [1 8]. Recently, Sahu [9] and Khan[10], who was probably unaware of Sahu s work, introduced the following iterative process which has been called noral S-iterative ethod and Picard-Mann hybrid iterative process by Sahu and Khan, respectively, and hereinafter referred to as the noral S- iterative ethod. Definition 1. Let X be an abient space and let T be a selfap of X. A noral S-iterative ethod is defined by x 0 X, x n+1 =Ty n, y n =(1 ξ n )x n +ξ n Tx n, n N, where {ξ n } n=0 is a real sequence in [0, 1] satisfying certain control condition(s). It has been shown both analytically and nuerically in [9, 10]thatiterativeethod(1)convergesataratefasterthan all Picard [11], Mann [12], and Ishikawa [13] iterative processes in the sense of Berinde[14] for the class of contraction appings. This iterative ethod, due to its siplicity and fastness, has attracted the attention of any researchers and has been exainedinvariousaspects;see[15 20]. In this paper, inspired by the perforance and achieveents of noral S-iterative ethod (1), we will give soe of its applications. We will show that noral S-iterative ethod (1) converges strongly to the solution of the following ixed type Volterra-Fredhol functional nonlinear integral equation which was considered in [21]: x (t) =F(t,x(t), K (t, s, x (s)) ds, H (t, s, x (s)) ds), (1) (2)

2 2 The Scientific World Journal where [ ; ] [ ; ] is an interval in R, K, H : [ ; ] [ ; ] [ ; ] [ ; ] R R continuous functions, and F:[ ; ] [ ; ] R 3 R. Also we give a data dependence result for the solution of integral equation (2) with the help of noral S-iterative ethod (1). We end this section with soe known results which will be useful in proving our ain results. Theore 2 (see [21]). We suppose that the following conditions are satisfied: (A 1 ) K, H C([ ; ] [ ; ] [ ; ] [ ; ] R); (A 2 ) F C([ ; ] [ ; ] R 3 ); (A 3 ) there exist nonnegative constants α, β,andγ such that F(t,u 1, V 1,w 1 ) F(t,u 2, V 2,w 2 ) α u 1 u 2 +β V 1 V 2 +γ w 1 w 2, (3) for all t [ ; ] [ ; ], u i, V i, w i R,,2; (A 4 ) thereexistnonnegativeconstantsl K and L H such that K (t, s, u) K(t, s, V) L K u V, H (t, s, u) H(t, s, V) L H u V, for all t, s [ ; ] [ ; ], u, V R; (A 5 ) α + (βl K +γl H )( ) ( )<1. Then (2) has a unique solution x C([ ; ] [ ; ]). Lea 3 (see [22]). Let {β n } n=0 be a nonnegative sequence for which one assues there exists n 0 N,suchthatforalln n 0 one has satisfied the inequality (4) β n+1 (1 μ n ) β n +μ n γ n, (5) where μ n (0, 1),foralln N, n=0 μ n =,andγ n 0,for all n N. Then the following inequality holds: 2. Main Results 0 li sup β n li sup γ n. (6) n n Theore 4. One opines that all conditions (A 1 ) (A 5 ) in Theore 2 are perfored. Let {ξ n } n=0 [0, 1] be a real sequence satisfying n=0 ξ n =.Then(2) has a unique solution, say x,inc([ ; ] [ ; ]) and noral S- iterative ethod (1) converges to x. Proof. We consider the Banach space B=C([ ; ] [ ; ], C ),where C is Chebyshev s nor. Let {x n } n=0 be an iterative sequence generated by noral S-iterative ethod (1)for the operator A:B Bdefined by A (x)(t) = F (t, x (t), K (t, s, x (s)) ds, H (t, s, x (s)) ds). We will show that x n x as n. Fro (1), (2), and assuptions (A 1 ) (A 4 ), we have that x n+1 x = Ay n x = A(y n) (t) A(x ) (t) = F (t, y n (t), K(t,s,y n (s))ds, H(t,s,y n (s))ds) F (t, x (t), K(t,s,x (s))ds, α y n (t) x (t) +β +γ t 1 t H(t,s,x (s))ds) K(t,s,y n (s))ds K(t,s,x (s))ds H(t,s,y n (s))ds α y n (t) x (t) H(t,s,x (s))ds +β K(t,s,y n (s)) K(t,s,x (s)) a ds +γ H(t,s,y n (s)) H(t,s,x (s)) a ds (7)

3 The Scientific World Journal 3 α y n (t) x t (t) +β 1 t L K a y n (s) x (s) ds +γ L H a y n (s) x (s) ds [α+(βl K +γl H ) y n x (1 ξ n ) x n (t) x (t) +ξ n A(x n) (t) A(x ) (t) (b i a i )] y n x, (8) or, fro assuption (A 5 ), x n+1 x x n x. Thus, by induction, we get x n+1 x x 0 x n k=0 {1 ξ k (1 [α + (βl K +γl H ) (11) =(1 ξ n ) x n (t) x (t) +ξ n F(t,x n (t), K(t,s,x n (s))ds, H(t,s,x n (s))ds) F (t, x (t), K(t,s,x (s))ds, H(t,s,x (s))ds) (1 ξ n ) x n (t) x (t) +ξ nα x n (t) x (t) +ξ n β L K a x n (s) x (s) ds +ξ n γ L H a x n (s) x (s) ds x n x. Cobining (8)with(9), we obtain (9). Since ξ k [0, 1] for all k N, assuption (A 5 ) yields 1 ξ k (1 [α + (βl K +γl H ) (12) (b i a i )]) < 1. (13) Having regard to the fact that e x 1 xfor all x [0,1],we can rewrite (12)as x n+1 x x 0 x e (1 [α+(βl K+γL H ) (b i a i )]) n k=0 ξ k, (14) which yields li n x n x =0. We now prove the data dependence of the solution for integral equation (2) with the help of the noral S-iterative ethod (1). Let B be as in the proof of Theore 4 and T, T :B B two operators defined by T (x)(t) = F (t, x (t), K (t, s, x (s)) ds, H (t, s, x (s)) ds), (15) x n+1 x [α+(βl K +γl H ) (b i a i )] t 1 T (x)(t) = F (t, x (t), t K (t, s, x (s)) ds, H (t, s, x (s)) ds), (16) {1 ξ n (1 [α + (βl K +γl H ) x n x, (10) where K, K, H, H C([ ; ] [ ; ] [ ; ] [ ; ] R). Theore 5. Let F, K, andh be defined as in Theore 2 and let {x n } n=0 be an iterative sequence defined by noral

4 4 The Scientific World Journal S-iterative ethod (1) associated with T. Let{ x n } n=0 iterative sequence generated by x 0 B, be an α y n (t) y n (t) +β (L K a y n (s) y n (s) +ε 1)ds x n+1 = T y n, y n =(1 ξ n ) x n +ξ n T x n, n N, (17) where B is defined as in the proof of Theore 4 and {ξ n } n=0 is a real sequence in [0, 1] satisfying (i) 1/2 ξ n,foralln N, and (ii) n=0 ξ n =. One supposes further that (iii) there exist nonnegative constants ε 1 and ε 2 such that K(t, s, u) K(t, s, u) ε 1 and H(t, s, u) H(t, s, u) ε 2,forallu R and for all t, s [ ; ] [ ; ]. If x and x are solutions of corresponding equations (15) and (16), respectively, then one has that x x 3(βε 1 +γε 2 ) (b i a i ) 1 [α+(βl K +γl H ) (b i a i )]. (18) Proof. Using (1), (15), (16), (17), and assuptions (A 1 ) (A 4 ) and (iii), we obtain x n+1 x n+1 = Ty n T y n +γ (L H a y n (s) y n (s) +ε 2)ds α y n y n +β(l K y n y n +ε 1) +γ(l H y n y n +ε 2) [α+(βl K +γl H ) +(βε 1 +γε 2 ) y n y n (b i a i ), (1 ξ n ) x n (t) x n (t) +ξ n T(x n) (t) T( x n ) (t) (b i a i ) (b i a i )] y n y n (b i a i ) (19) = F (t, y n (t), K(t,s,y n (s))ds, H(t,s,y n (s))ds) t 1 F(t, y n (t), t K(t,s, y n (s))ds, H(t,s, y n (s))ds) α y n (t) y n (t) +β K(t,s,y n (s)) K(t,s, y n (s)) a ds +γ H(t,s,y n (s)) H(t,s, y n (s)) a ds α y n (t) y n (t) +β ( a K(t,s,y n (s)) K(t,s, y n (s)) +γ + K(t,s, y n (s)) K(t,s, y n (s)) )ds ( a H(t,s,y n (s)) H(t,s, y n (s)) + H(t,s, y n (s)) H(t,s, y n (s)) )ds (1 ξ n ) x n (t) x n (t) +ξ n {α x n (t) x n (t) +β {L K a x n (s) x n (s) +ε 1}ds +γ {L H a x n (s) x n (s) +ε 2}ds} x n x n +ξ n (βε 1 +γε 2 ) (b i a i ). (20) Cobining (19) with(20) and using assuptions (A 5 )and 1/2 ξ n in the resulting inequality, we get x n+1 x n+1 x n x n

5 The Scientific World Journal 5 +ξ n (1 [α + (βl K +γl H ) (b i a i )]) 3(βε 1 +γε 2 ) (b i a i ) 1 [α+(βl K +γl H ) (b i a i )]. (21) Denote that β n = x n x n, μ n =ξ n (1 [α + (βl K +γl H ) (b i a i )]) (0, 1), 3(βε γ n = 1 +γε 2 ) (b i a i ) 1 [α+(βl K +γl H ) (b i a i )] 0. (22) It is clear that inequality (21) satisfies all conditions in Lea 3, and hence it follows that x x 3(βε 1 +γε 2 ) (b i a i ) 1 [α+(βl K +γl H ) (b i a i )]. (23) Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper. Acknowledgent Theauthorwouldliketothanktheanonyousreviewers for their valuable coents and suggestions to iprove the quality of the paper. References [1] C. Chidue, Geoetric Properties of Banach Spaces and Nonlinear Iterations, vol. 1965, Springer, London, UK, [2] F. Gürsoy and V. Karakaya, Soe convergence and stability results for two new Kirk type hybrid fixed point iterative algoriths, Function Spaces, vol.2014,articleid , 8 pages, [3] F. Gürsoy, V. Karakaya, and B. E. Rhoades, Data dependence results of new ulti-step and S-iterative schees for contractive-like operators, Fixed Point Theory and Applications, vol. 2013, artcile 76, 12 pages, [4]H.KiziltuncandS.Teir, Convergencetheoresbyanew iteration process for a finite faily of nonself asyptotically nonexpansive appings with errors in Banach spaces, Coputers & Matheatics with Applications,vol.61,no.9,pp , [5] M. Basarir and A. Sahin, On the strong and Δ convergence of new ulti-step and S-iteration processes in a CAT (0) space, Inequalities and Applications, vol.2013,article482, [6] M. O. Olatinwo, Convergence and stability results for soe iterative schees, Acta Universitatis Apulensis, no. 26, pp , [7]S.Alezel,Q.H.Ansari,andM.A.Khasi,Eds.,Topics in Fixed Point Theory, Springer, [8] V. Berinde, Iterative Approxiation of Fixed Points, Springer, Berlin,Gerany,2007. [9] D. R. Sahu, Applications of the S-iteration process to constrained iniization probles and split feasibility probles, Fixed Point Theory,vol.12,no.1,pp ,2011. [10] S. H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Theory and Applications,vol.2013,article69,2013. [11] E. Picard, Méoire sur la théorie des équations aux dérivées partielles et léthode des approxiations successives, Journal de Matéatiques Pures et Appliquées, vol.6,pp , [12] W. R. Mann, Mean value ethods in iteration, Proceedings of the Aerican Matheatical Society,vol.4,pp ,1953. [13] S. Ishikawa, Fixed points by a new iteration ethod, Proceedings of the Aerican Matheatical Society,vol.44,pp , [14] V. Berinde, Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators, Fixed Point Theory and Applications, vol. 2004, no. 2, pp , [15] D. R. Sahu and A. Petruşel, Strong convergence of iterative ethods by strictly pseudocontractive appings in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications,vol. 74,no.17,pp ,2011. [16] N.Hussain,V.Kuar,andM.A.Kutbi, Onrateofconvergence of Jungck-type iterative schees, Abstract and Applied Analysis, vol. 2013, Article ID , 15 pages, [17] S. H. Khan, Fixed points of contractive-like operators by a faster iterative process, WASET International Matheatical, Coputational Science and Engineering, vol.7, pp , [18] S.M.Kang,A.Rafiq,andS.Lee, Convergenceanalysisofan iterative schee for Lipschitzian heicontractive appings in Hilbert spaces, Inequalities and Applications, vol. 2013,article312,5pages,2013. [19]S.M.Kang,A.Rafiq,andS.Lee, Strongconvergenceofan iplicit S -iterative process for Lipschitzian heicontractive appings, Abstract and Applied Analysis, vol. 2012, ArticleID , 7 pages, [20] V. Kuar, A. Latif, A. Rafiq, and N. Hussain, S-iteration process for quasi-contractive appings, JournalofInequalitiesand Applications,vol.2013,article206,15pages,2013. [21] C. Crăciun and M. Şerban, A nonlinear integral equation via Picard operators, Fixed Point Theory, vol. 12, no. 1, pp , [22] Ş. M. Şoltuz and T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive-like operators, Fixed Point Theory and Applications, vol.2008,articleid242916,7 pages, 2008.

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