The convergence of normal S-iterative method to solution of a nonlinearFredholm integral equation with modied argument is established. The correspondingdata dependence result has also been proved. An example in support of the established results is included in our analysis

Atalan, Yunus

Gursoy, Faik

Rahim Khan, Abdul

English

University of Niš: Facta Universitatis (E-Journals) / Универзитет у Нишу

FACTA UNIVERSITATIS (NIŠ)Ser. Math. Inform. Vol. 36, No 4 (2021), 685–694https://doi.org/10.22190/FUMI190116051AOriginal Scientific PaperCONVERGENCE OF S-ITERATIVE METHOD TO A SOLUTIONOF FREDHOLM INTEGRAL EQUATION AND DATADEPENDENCYYunus Atalan1, Faik Gürsoy2 and Abdul Rahim Khan31 Faculty of Science and Arts, Department of MathematicsAksaray University, 68100 Aksaray, Turkey2 Faculty of Science and Arts, Department of MathematicsAdiyaman University, 02040, Adiyaman, Turkey3 Department of Mathematics and Statistics,King Fahd University of Petroleum and Minerals, 31261, Dhahran, Saudi ArabiaAbstract. The convergence of normal S-iterative method to solution of a nonlinearFredholm integral equation with modified argument is established. The correspondingdata dependence result has also been proved. An example in support of the establishedresults is included in our analysis.Key words: Fredholm equation, data dependency, Fixed-point theorem.1. Introduction and PreliminariesThe past few decades have witnessed substantial developments in the field ofintegral equations and their applications have arisen in many areas, ranging fromeconomics to engineering. Now it is an unquestionable fact that the theory ofiterative approximation of fixed points plays a significant role in recent progressof integral equations and their applications. In this context, fixed point iterativemethods for solving integral equations have already gained a splendid boost overthe past few years (see, for example [1],[2],[4],[5],[7],[8],[16],[17],[19],[20]).Received January 16, 2019, accepted: June 23, 2019Communicated by Marko PetkovićCorresponding Author: Yunus Atalan, Faculty of Science and Arts, Department of Mathematics,Aksaray University, 68100 Aksaray, Turkey | E-mail: yunusatalan@aksaray.edu.tr2010 Mathematics Subject Classification. Primary 47H10; Secondary 54H25, 45G10© 2021 by University of Nǐs, Serbia | Creative Commons License: CC BY-NC-ND686 Y. Atalan, F. Gürsoy and A. R. KhanIn 2011, Sahu [23] introduced a normal S-iterative method as follows: x0 ∈ X,xn+1 = Tyn,yn = (1− ξn)xn + ξnTxn, n ∈ N(1.1)where X is an ambient space, T is a self-map of X and {ξn}∞n=0 is a real sequencein [0, 1] satisfying certain control condition(s).It has been shown both analytically and numerically in [23] and [12] that the it-erative method (1.1) converges faster than Picard [22], Mann [21], and Ishikawa [10]iterative processes in the sense of Berinde [3] for the class of contraction mappings.This iterative method, due to its simplicity and fastness, has attracted the atten-tion of many researchers and has been examined in various settings (see [9],[11],[13],[14],[15],[18],[24]).In this paper, inspired by the above mentioned achievements of normal S-iterative method (1.1), we will use it to show that normal S-iterative method (1.1)converges strongly to the solution of the following integral equation which has beenconsidered in [6]:x (t) =b∫aK (t, s) · h (s, x (s) , x (a) , x (b)) ds+ f (t) , t ∈ [a, b] ,(1.2)where K : [a, b] × [a, b]→ R, h : [a, b]× R3 → R and f , x : [a, b]→ R.Also we give a data dependence result for the solution of integral equation (1.2)with the help of normal S-iterative method (1.1).We need the following pair of known results:Theorem 1.1. [6] Assume that the following conditions are satisfied:(A1) K ∈ C ([a, b]× [a, b]);(A2) h ∈ C([a, b]× R3);(A3) f , x ∈ C [a, b];(A4) there exist constants α, β, γ > 0 such that|h (s, u1, u2, u3)− h (s, v1, v2, v3)| ≤ α |u1 − v1|+ β |u2 − v2|+ γ |u3 − v3| ,for all s ∈ [a, b], ui, vi ∈ R, i = 1, 2, 3;(A5) MK (α+ β + γ) (b− a) < 1,where MK denotes a positive constant such that for all t, s ∈ [a, b]|K (t, s)| ≤MK .Then the equation (1.2) has a unique solution x∗ ∈ C [a, b], which can be obtainedby the successive approximations method starting with any element x0 ∈ C [a, b].Moreover, if xn is the n-th successive approximation, then one has:|xn − x∗| ≤[MK (α+ β + γ) (b− a)]n1−MK (α+ β + γ) (b− a)· |x0 − x1| .Iterative Method and Fredholm Integral Equation 687Lemma 1.1. [25] Let {βn}∞n=0 be a sequence of non negative numbers for whichone assumes there exists n0 ∈ N (set of natural numbers), such that for all n ≥ n0βn+1 ≤ (1− µn)βn + µnγn,where µn ∈ (0, 1) , for all n ∈ N,∞∑n=0µn = ∞ and γn ≥ 0, ∀n ∈ N. Then thefollowing inequality holds:0 ≤ lim supn→∞βn ≤ lim supn→∞γn.2. Main ResultsTheorem 2.1. Assume that all the conditions (A1) − (A5) in Theorem 1.1 arefulfilled. Let {ξn}∞n=0 be a real sequence in [0, 1] satisfying∞∑n=0ξn = ∞. Thenequation (1.2) has a unique solution x∗ ∈ C [a, b] and normal S-iterative method(1.1) converges to x∗ with the following estimate:‖xn+1 − x∗‖ ≤[MK (α+ β + γ) (b− a)]n+1e(1−MK(α+β+γ)(b−a))∑nk=0 ξk‖x0 − x∗‖ .Proof. We consider the Banach space B = (C [a, b] , ‖·‖C), where ‖·‖C is the Cheby-shev’s norm on C [a, b], defined by ‖·‖C = {sup |x (t)| : t ∈ [a, b]} . Let {xn}∞n=0 beiterative sequence generated by Normal-S iteration method (1.1) for the operatorT : B → B defined byT (x (t)) =b∫aK (t, s) · h (s, x (s) , x (a) , x (b)) ds+ f (t) , t ∈ [a, b] .(2.1)We will show that xn → x∗ as n→∞.From (1.1), (2.1), and assumptions (A1)− (A4), we have that|xn+1 (t)− x∗ (t)| = |T (yn (t))− T (x∗ (t))|=∣∣∣∣∣∣b∫aK (t, s) ·[h (s, yn (s) , yn (a) , yn (b))−h (s, x∗ (s) , x∗ (a) , x∗ (b))]ds∣∣∣∣∣∣≤b∫a|K (t, s)| ·∣∣∣∣ h (s, yn (s) , yn (a) , yn (b))−h (s, x∗ (s) , x∗ (a) , x∗ (b))∣∣∣∣ ds≤ MKb∫a[α |yn (s)− x∗ (s)|+ β |yn (a)− x∗ (a)|+γ |yn (b)− x∗ (b)|]ds,688 Y. Atalan, F. Gürsoy and A. R. Khan|yn (t)− x∗ (t)| ≤ (1− ξn) |xn (t)− x∗ (t)|+ ξn |T (xn) (t)− T (x∗) (t)|= (1− ξn) |xn (t)− x∗ (t)|+ξn∣∣∣∣∣∣b∫aK (t, s) ·[h (s, xn (s) , xn (a) , yn (b))−h (s, x∗ (s) , x∗ (a) , x∗ (b))]ds∣∣∣∣∣∣≤ (1− ξn) |xn (t)− x∗ (t)|+ξnMKb∫a[α |xn (s)− x∗ (s)|+ β |xn (a)− x∗ (a)|+γ |xn (b)− x∗ (b)|]dsNow, by taking supremum in the above inequalities, we get‖xn+1 − x∗‖ ≤MK (α+ β + γ) (b− a) ‖yn − x∗‖ ,(2.2)and‖yn − x∗‖ ≤ [1− ξn (1−MK (α+ β + γ) (b− a))] ‖xn − x∗‖ ,(2.3)respectively.Combining (2.2) with (2.3), we obtain‖xn+1 − x∗‖(2.4)≤ MK (α+ β + γ) (b− a) [1− ξn (1−MK (α+ β + γ) (b− a))] ‖xn − x∗‖ .Thus, by induction, we get‖xn+1 − x∗‖ ≤ ‖x0 − x∗‖ [MK (α+ β + γ) (b− a)]n+1×∏nk=0[1− ξk (1−MK (α+ β + γ) (b− a))] .(2.5)Since ξk ∈ [0, 1] for all k ∈ N, the assumption (A5) yieldsξk (1−MK (α+ β + γ) (b− a)) < 1.(2.6)From the classical analysis, we know that 1 − x ≤ e−x for all x ∈ [0, 1]. Hence byutilizing this fact with (2.6) in (2.5), we obtain‖xn+1 − x∗‖ ≤ ‖x0 − x∗‖ [MK (α+ β + γ) (b− a)]n+1(2.7)×e−(1−MK(α+β+γ)(b−a))∑nk=0 ξk ,which yields limn→∞ ‖xn − x∗‖ = 0.We now prove a closeness of solutions of integral equation (1.2) with the help of thenormal-S iterative method (1.1).We consider the following equation:T̃ (x̃ (t)) =b∫aK (t, s) · h̃ (s, x̃ (s) , x̃ (a) , x̃ (b)) ds+ g (t) , t ∈ [a, b] ,(2.8)Iterative Method and Fredholm Integral Equation 689where K : [a, b]× [a, b]→ R, h̃ : [a, b]× R3 → R and g : [a, b]→ R.Now, we define the following normal-S iterative methods associated with T in(2.1) and T̃ in (2.8), respectively:x0 ∈ C [a, b] ,xn+1 =b∫aK (t, s) · h (s, yn (s) , yn (a) , yn (b)) ds+ f (t) ,yn = (1− ξn)xn+ξnb∫aK (t, s) · h (s, xn (s) , xn (a) , xn (b)) ds+ f (t) , t ∈ [a, b] , n ∈ N,(2.9)andx̃0 ∈ C [a, b] ,x̃n+1 =b∫aK (t, s) · h̃ (s, ỹn (s) , ỹn (a) , ỹn (b)) ds+ g (t) ,ỹn = (1− ξn) x̃n+ξnb∫aK (t, s) · h̃ (s, x̃n (s) , x̃n (a) , x̃n (b)) ds+ g (t) , t ∈ [a, b] , n ∈ N,(2.10)where {ξn}∞n=0 is a real sequence in [0, 1], K : [a, b]×[a, b]→ R, h, h̃ : [a, b]×R3 → Rand f ,g : [a, b]→ R.Theorem 2.2. Consider the sequences {xn}n=0 and {x̃n}∞n=0 generated by (2.9)and (2.10), respectively, with the real sequence {ξn}∞n=0in [0, 1] satisfying12 ≤ ξn forall n ∈ N. Assume that:(i) all the conditions of Theorem 2.1 hold and x∗ and x̃∗ are solutions of equa-tions (2.1) and (2.8), respectively;(ii) there exist non negative constants ε1 and ε2 such that∣∣∣h (s, u, v, w)− h̃ (s, u, v, w)∣∣∣ ≤ ε1 and |f (t)− g (t)| ≤ ε2, for all t, s ∈ [a, b], u, v, w ∈R.If the sequence {x̃n}∞n=0 converge to x̃∗, then we have‖x∗ − x̃∗‖ ≤ 3 [MK (b− a) ε1 + ε2]1−MK (α+ β + γ) (b− a).(2.11)Proof. Using (1.1), (2.1), (2.8)-(2.10), and assumptions (A1)-(A4) and (ii), we ob-tain|xn+1 (t)− x̃n+1 (t)| =∣∣∣T (yn) (t)− T̃ (ỹn) (t)∣∣∣=∣∣∣∣∣∣b∫aK (t, s) · h (s, yn (s) , yn (a) , yn (b)) ds+ f (t)690 Y. Atalan, F. Gürsoy and A. R. Khan−b∫aK (t, s) · h̃ (s, ỹn (s) , ỹn (a) , ỹn (b)) ds− g (t)∣∣∣∣∣∣≤∣∣∣∣∣∣b∫aK (t, s) ·[h (s, yn (s) , yn (a) , yn (b))−h̃ (s, ỹn (s) , ỹn (a) , ỹn (b))]ds∣∣∣∣∣∣+ |f (t)− g (t)|≤ MKb∫a∣∣∣∣ h (s, yn (s) , yn (a) , yn (b))−h (s, ỹn (s) , ỹn (a) , ỹn (b))∣∣∣∣+∣∣∣∣ h (s, ỹn (s) , ỹn (a) , ỹn (b))−h̃ (s, ỹn (s) , ỹn (a) , ỹn (b))∣∣∣∣ ds+ |f (t)− g (t)|≤ MKb∫a(α |yn (s)− ỹn (s)|+β |yn (a)− ỹn (a)|+ γ |yn (b)− ỹn (b)|+ ε1)ds+ε2≤ MKb∫a(α |yn (s)− ỹn (s)|+β |yn (a)− ỹn (a)|+ γ |yn (b)− ỹn (b)|)ds+MKb∫aε1ds+ ε2,|yn (t)− ỹn (t)| ≤ (1− ξn) |xn (t)− x̃n (t)|+ ξn∣∣∣T (xn) (t)− T̃ (x̃n) (t)∣∣∣≤ (1− ξn) |xn (t)− x̃n (t)|+ξnMKb∫a∣∣∣∣ h (s, xn (s) , xn (a) , xn (b))−h (s, x̃n (s) , x̃n (a) , x̃n (b))∣∣∣∣+∣∣∣∣ h (s, x̃n (s) , x̃n (a) , x̃n (b))−h̃ (s, x̃n (s) , x̃n (a) , x̃n (b))∣∣∣∣ ds+ξn |f (t)− g (t)|≤ (1− ξn) |xn (t)− x̃n (t)|+ξnMKb∫a(α |xn (s)− x̃n (s)|+β |xn (a)− x̃n (a)|+ γ |xn (b)− x̃n (b)|+ ε1)ds+ξnε2.Now, by taking supremum in the above inequalities, we get‖xn+1 − x̃n+1‖ ≤ MK (α+ β + γ) (b− a) ‖yn − ỹn‖(2.12)+MK (b− a) ε1 + ε2,Iterative Method and Fredholm Integral Equation 691and‖yn − ỹn‖ ≤ [1− ξn (1−MK (α+ β + γ) (b− a))] ‖xn − x̃n‖(2.13)+ξnMK (b− a) ε1 + ξnε2,respectively.Combining (2.12) with (2.13) and using assumptions (A5) and12 ≤ ξn for alln ∈ N in the resulting inequality, we get‖xn+1 − x̃n+1‖ ≤ [1− ξn (1−MK (α+ β + γ) (b− a))] ‖xn − x̃n‖+ξnMK (b− a) ε1 + ξnε2 + 2ξnMK (b− a) ε1 + 2ξnε2= [1− ξn (1−MK (α+ β + γ) (b− a))] ‖xn − x̃n‖+ξn (1−MK (α+ β + γ) (b− a))× 3 [MK (b− a) ε1 + ε2]1−MK (α+ β + γ) (b− a).(2.14)Denote byβn = ‖xn − x̃n‖ ,µn = ξn (1−MK (α+ β + γ) (b− a)) ∈ (0, 1) ,γn =3 [MK (b− a) ε1 + ε2]1−MK (α+ β + γ) (b− a)≥ 0.The assumption 12 ≤ ξn for all n ∈ N implies∞∑n=0ξn = ∞. Now it can be easilyseen that (2.14) satisfies all the conditions of Lemma 1.1. Hence it follows by itsconclusion that0 ≤ lim supn→∞‖xn − x̃n‖ ≤ lim supn→∞3 [MK (b− a) ε1 + ε2]1−MK (α+ β + γ) (b− a).By (i), we have that limn→∞ xn = x∗. Using this fact and the assumptionlimn→∞ x̃n = x̃∗, we get‖x∗ − x̃∗‖ ≤ 3 [MK (b− a) ε1 + ε2]1−MK (α+ β + γ) (b− a).Remark 2.1. The result given in Theorem 2.2 relate the solutions of equations (2.1) and(2.8) in the sense that if f is close to g and h is close to h̃, then not only the solutionsof equations (2.1) and (2.8) are close to each other, but also depend continuously on thefunctions involved therein. Further, if ε1 → 0 and ε2 → 0, then the solution x∗ of equation(2.1) tends the solution x̃∗ of the equation (2.8).692 Y. Atalan, F. Gürsoy and A. R. KhanExample 2.1. Consider the following integral equationx (t) =1∫03t− 2s5[s− sinx (s)2+x (0) + x (1)3]ds+t+ e−t3, t ∈ [0, 1] .where K ∈ C ([0, 1]× [0, 1]), K (t, s) = 3t−2s5, h ∈ C([0, 1]× R3), h (s, u, v, w) = s−sinu2+v+w3, f ∈ C [0, 1], f (t) = t+e−t3, x ∈ C [0, 1] and its perturbed integral equationx̃ (t) =1∫03t− 2s5[s− sin x̃ (s)2+x̃ (0) + x̃ (1)3− s+ 17]ds+t+ 2e−t3, t ∈ [0, 1] ,where K ∈ C ([0, 1]× [0, 1]), K (t, s) = 3t−2s5, k ∈ C([0, 1]× R3), k (s, u, v, w) = s−sinu2+v+w3− s+ 17, g ∈ C [0, 1], g (t) = t+2e−t3, x̃ ∈ C [0, 1].Define the operator T : C [0, 1]→ C [0, 1] byT (x (t)) =1∫03t− 2s5[s− sinx (s)2+x (0) + x (1)3]ds+t+ e−t3, t ∈ [0, 1] .We now show that the operator T is a contraction with contractivity factor 710. Indeed,|T (x1 (t))− T (x2 (t))|=∣∣∣∣∣∣1∫03t− 2s5[s− sinx1 (s)2+x1 (0) + x1 (1)3− s− sinx2 (s)2− x2 (0) + x2 (1)3]ds∣∣∣∣∣∣≤∣∣∣∣∣∣1∫0∣∣∣∣3t− 2s5∣∣∣∣ ∣∣∣∣s− sinx1 (s)2 + x1 (0) + x1 (1)3 − s− sinx2 (s)2 − x2 (0) + x2 (1)3∣∣∣∣ ds∣∣∣∣∣∣≤∣∣∣∣∣∣1∫0∣∣∣∣3t− 2s5∣∣∣∣ [12 |sinx1 (s)− sinx2 (s)|+ 13 |x1 (0)− x2 (0)|+ 13 |x1 (1)− x2 (1)|]ds∣∣∣∣∣∣ .Now using the Chebyshev norm, we obtain‖Tx1 − Tx2‖ ≤ supt,s∈[0,1]∣∣∣∣3t− 2s5∣∣∣∣ (12 + 13 + 13)(1− 0) ‖x1 − x2‖=710‖x1 − x2‖ .One can easily show on the same lines as above that the mapping T̃ : C [0, 1] → C [0, 1]defined byT̃ (x̃ (t)) =1∫03t− 2s5[s− sin x̃ (s)2+x̃ (0) + x̃ (1)3− s+ 17]ds+t+ 2e−t3, t ∈ [0, 1] ,is also a contraction with contractivity factor 710.Iterative Method and Fredholm Integral Equation 693Since all the conditions of Theorem 2.1 are satisfied by the integral equations (2.1) and(2.8) so by its conclusion, normal S-iterative method (1.1) converges to unique solution x∗and x̃∗, respectively in C [0, 1].Now we have the following estimates:|K (t, s)| =∣∣∣∣3t− 2s5∣∣∣∣ ≤ 35 = MK , t, s ∈ [0, 1] ,|h (s, u, v, w)− k (s, u, v, w)| =∣∣∣∣s− 17∣∣∣∣ ≤ 17 = ε1, foralls ∈ [0, 1] , u, v, w ∈ R,|f (t)− g (t)| =∣∣∣∣ t+ e−t − t− 2e−t3∣∣∣∣ = e−t3 ≤ 13 = ε2, s ∈ [0, 1] .In view of the above estimates, all the conditions of Theorem 2.2 are satisfied and hencefrom (2.11), we have‖x∗ − x̃∗‖ ≤ 8821.REFERENCES1. 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CONVERGENCE OF S-ITERATIVE METHOD TO A SOLUTION OF FREDHOLM INTEGRAL EQUATION AND DATA DEPENDENCY

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CONVERGENCE OF S-ITERATIVE METHOD TO A SOLUTION OF FREDHOLM INTEGRAL EQUATION AND DATA DEPENDENCY

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