EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 4, 2019, 1662-1675 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Generalized Quasilinearization using coupled lower and upper solutions for periodic boundary value problem of an integro differential equation Ch. V. Sreedhar1, J. Vasundhara Devi1,∗ 1 Department of Mathematics, GVP-Prof.V.Lakshmikantham Institute for Advanced Studies, Gayatri Vidya Parishad College of Engineering (Autonomous), Visakhapatnam 530048, India Abstract. In this paper we first develop the method of generalized quasilinearization for initial value problem of an integro differential equation and then use it to develop quasilinearization for the periodic boundary value problem of the integro differential equation by using the coupled lower and upper solutions of Type-I. 2010 Mathematics Subject Classifications: 45J05, 47G20 Key Words and Phrases: Periodic boundary value problem(PBVP), integro differential equa- tion, coupled lower and upper solutions, existence, quasilinearization. 1. Introduction Integro differential equations [1] arise quite frequently as mathematical models in vari- ous disciplines of physical, social and biological sciences and engineering. Models involving integro differential equations can be found in unsteady aerodynamics and aero-elastic phe- nomena etc. The qualitative theory of integro differential equations deals with existence and uniqueness of solutions, stability of solutions etc. The existence and uniqueness results are studied using various approaches like fixed point theory and iterative techniques. There are various iterative techniques for solving integro differential equations. Some of the iterative methods are monotone iterative technique, quasilinearization and their generalizations. The monotone iterative technique and quasilinearization are two iterative techniques that are widely used to obtain existence and uniqueness results of various types of differential equations. Both monotone iterative technique and quasilinearization [2, 3, 4, 5] along with the method of upper and lower solutions yield monotone iterates which are solutions of certain linear differential equations obtained from the hypothesis of the given problem. These iterates converge to a solution of the original problem. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3529 Email address: jvdevi@gmail.com (J. Vasundhara Devi) http://www.ejpam.com 1662 c© 2019 EJPAM All rights reserved. Ch. V. Sreedhar, J. Vasundhara Devi, / Eur. J. Pure Appl. Math, 12 (4) (2019), 1662-1675 1663 The monotone iterative technique had undergone various extensions and generaliza- tions. The right hand side of the problem was considered as a sum of a nondecreasing and nonincreasing function. This gave rise to various notions of coupled solutions and much work has been done in this setup for various types of differential equations. In [6, 7] the authors obtained the existence of solutions for an integro differential equation with periodic boundary condition using monotone iterative technique. This was a very interesting result because of two reasons: 1) no additional lemmas were needed to prove the result 2) no extra conditions were needed for uniqueness, as the uniqueness of solution was generated by the method itself. This led to a spurt of publications in monotone iterative technique for various types of differential equations [8, 9]. This idea has been extended to quasilinearization and generalized quasilinearisation had been developed for periodic boundary value problem of a graph differential equation and a matrix differential equation through natural upper and lower solutions [10]. In [11] it was observed that quasilinearization for periodic boundary value problem through coupled lower and upper solutions of the initial value problem can be obtained with certain restrictions. In this paper, using the approach given in [11] we develop quasilinearization technique, using coupled lower and upper solutions, for initial value problem of an integro differential equation and using this result to obtain existence and uniqueness of solutions for periodic boundary value problem of an integro differential equation. 2. Preliminaries Consider the periodic boundary value problem of an integro differential equation given by x′ = f1(t, x, Sx) + f2(t, x, Sx), (1) x(0) = x(T ). (2) To develop the method of the quasilinearization technique (1) and (2), we first develop quasilinearization technique for the corresponding initial value problem of an integro dif- ferential equation given by x′ = f1(t, x, Sx) + f2(t, x, Sx), (3) x(0) = x0, (4) where f1, f2 ∈ C[I × Rn × Rn, Rn], Sx(t) = t∫ 0 K(t, s)x(s)ds, with K ∈ C[I × I,R+] and I=[0,T]. To do so we first define the various types of lower and upper solution for (3) and (4), Definition 1. Let α0, β0 ∈ C1[I,Rn]. Then α0, β0 are said to be (a) natural lower and upper solutions of (3) and (4) if α′0 ≤ f1(t, α0, Sα0) + f2(t, α0, Sα0), α0(0) ≤ x0, β′0 ≥ f1(t, β0, Sβ0) + f2(t, β0, Sβ0), β0(0) ≥ x0, t ∈ I; } (5) Ch. V. Sreedhar, J. Vasundhara Devi, / Eur. J. Pure Appl. Math, 12 (4) (2019), 1662-1675 1664 (b) coupled lower and upper solutions of Type I of (3) and (4) if α′0 ≤ f1(t, α0, Sα0) + f2(t, β0, Sβ0), α0(0) ≤ x0, β′0 ≥ f1(t, β0, Sβ0) + f2(t, α0, Sα0), β0(0) ≥ x0, t ∈ I; } (6) (c) coupled lower and upper solutions of Type II of (3) and (4) if α′0 ≤ f1(t, β0, Sβ0) + f2(t, α0, Sα0), α0(0) ≤ x0, β′0 ≥ f1(t, α0, Sα0) + f2(t, β0, Sβ0), β0(0) ≥ x0, t ∈ I; } (7) (d) coupled lower and upper solutions of Type III of (3) and (4) if α′0 ≤ f1(t, β0, Sβ0) + f2(t, β0, Sβ0), α0(0) ≤ x0, β′0 ≥ f1(t, α0, Sα0) + f2(t, α0, Sα0), β0(0) ≥ x0, t ∈ I. } (8) We observe that whenever we have α(t) ≤ β(t), t ∈ I, f1(t, x, Sx) is nondecreasing in x and y and f2(t, x, Sx) is nonincreasing in x and y for each t ∈ I, the lower and upper solutions defined by (5) and (8) reduce to (6) and (7) consequently, hence it is sufficient to investigate the cases (6) and (7). 3. Generalized quasilinearization for initial value problem of an integro differential equation. In this section we develop the method of generalized quasilinearization for the initial value problem of an integro differential equation and use it in the next section. we first state a known result from [1] corresponding to an integro differential equation which is useful in developing a sequence of iterates to be constructed while developing the quasilinearization. Lemma 1. Let p ∈ C1[I,R], where I = [0, T ] is such that and p ′(t) ≤ −Mp(t)−NSp(t) on I, p(0) ≤ 0, (9) where M > 0, N ≥ 0 are constants such that Nk1T (eMT − 1) ≤M, (10) where k1 = maxt∈I K(t, s). Then p(t) ≤ 0 on I. To prove the main theorem we need the following assumptions which are listed below for convenience. Ch. V. Sreedhar, J. Vasundhara Devi, / Eur. J. Pure Appl. Math, 12 (4) (2019), 1662-1675 1665 H1 : (i) Second order Frechet derivatives of f1(t, x, ξ), f2(t, x, ξ) with respect to all variables exist and are bounded; (ii) f1(t, x, ξ) is convex in x, ξ ; (iii) f1x(t, x, ξ) is nondecreasing in ξ for each (t, x); (iv) f1 is nondecreasing function in x, ξ for each t ∈ I and f2 is nonincreasing function in x, ξ for each t ∈ I. H2 : (i) −M1 ≤ f1x(t, x, ξ) ≤ −M, 0 < M < M1; (ii) −M2 ≤ f1ξ(t, x, ξ) ≤ −N, 0 < N < M2; (iii) Nk1T < M ; where M > 0, N ≥ 0. H3 : α0, β0 are coupled lower and upper solutions of (3) and (4). H4 : (i) f1(t, x, Sx) ≥ f1(t, y, Sy) + f1x(t, y, Sy)(x− y) + f1ξ(t, y, Sy)(Sx− Sy); (ii) |f1x(t, x, Sx)− f1x(t, y, Sy)| ≤ L1(x− y) +M1(Sx− Sy), L1,M1 ≥ 0. Theorem 1. Suppose that the assumptions H1 to H4 are satisfied. Then there exists monotone sequence {αn}, such that αn → ρ, as n → ∞ uniformly and monotonically to the unique solution ρ = u of an integro differential equation (3) and (4)on I and the convergence is quadratic. Proof: In order to construct a sequence of lower iterates that converge to the solution of the IVP we fix the upper solution β0. Now consider the following linear problem for n = 0, 1, 2, 3, ... α′n+1 = f1(t, αn, Sαn) + f1x(t, αn, Sαn)[αn+1 − αn] + f1ξ(t, αn, Sαn)[Sαn+1 − Sαn] +f2(t, β0, Sβ0), } (11) αn+1(0) = x0. (12) Since the above equation is a linear integro differential equation, it has unique solution αn+1(t) on I for each n. Now we claim that α0 ≤ α1 ≤ α2 ≤ ... ≤ αn−1 ≤ αn ≤ ... ≤ β0 (13) on I. Ch. V. Sreedhar, J. Vasundhara Devi, / Eur. J. Pure Appl. Math, 12 (4) (2019), 1662-1675 1666 We begin by setting p = α0 − α1. Then p′ = α′0 − α′1 ≤ {f1(t, α0, Sα0) + f2(t, β0, Sβ0)} −{f1(t, α0, Sα0) + f1x(t, α0, Sα0)[α1 − α0] + f1ξ(t, α0, Sα0)[Sα1 − Sα0] + f2(t, β0, Sβ0)} ≤ f1x(t, α0, Sα0)[α0 − α1] + f1ξ(t, α0, Sα0)[Sα0 − Sα1] p′(t) ≤ −Mp(t)−NSp(t). Also p(0) = α0(0) − α1(0) ≤ 0. Hence by Lemma 1 we have p(t) ≤ 0. So α0 ≤ α1 on I. Next, we show that α1 ≤ α2 on I. For this set p = α1 − α2. Then p′ = α′1 − α′2 ≤ f1(t, α1, Sα1)− {f1(t, α1, Sα1) + f1x(t, α1, Sα1)[α2 − α1] + f1ξ(t, α1, Sα1)[Sα2 − Sα1]} = f1x(t, α1, Sα1)[α1 − α2] + f1ξ(t, α1, Sα1)[Sα1 − Sα2] ≤ −Mp(t)−NSp(t). Also p(0) = α1(0) − α2(0) = 0. Hence by Lemma 1 we have p(t) ≤ 0. Thus α1 ≤ α2 on I. Now we show α1 ≤ β0 on I by setting p = α1 − β0. Then, p′ = α′1 − β′0 ≤ {f1(t, α0, Sα0)− f1(t, β0, Sβ0)}+ {f1x(t, α0, Sα0)[α1 − α0] + f1ξ(t, α0, Sα0)[Sα1 − Sα0]}+ {f2(t, β0, Sβ0)− f2(t, α0, Sα0)} ≤ f1x(t, α0, Sα0)[α1 − β0] + f1ξ(t, α0, Sα0)[Sα1 − Sβ0] ≤ −Mp(t)−NSp(t). Also p(0) = α1(0) − β0(0) ≤ 0. Hence by Lemma 1 we have p(t) ≤ 0. Which means that α1 ≤ β0 on I. Similarly we can show α2 ≤ β0 on I. Thus α0 ≤ α1 ≤ α2 ≤ β0, on I. Now we assume that the result holds for n = k and prove it for n = k+1. We now consider the following linear integro differential equation, α′k+1 = f1(t, αk, Sαk)+f1x(t, αk, Sαk)[αk+1−αk]+f1ξ(t, αk, Sαk)[Sαk+1−Sαk]+f2(t, β0, Sβ0), αk+1(0) = x0. Ch. V. Sreedhar, J. Vasundhara Devi, / Eur. J. Pure Appl. Math, 12 (4) (2019), 1662-1675 1667 The above linear integro differential equation has the unique solution αk+1 where αk and β0 are known lower and upper solutions of (3) and (4). Further αk is the solution of the linear integro differential equation α′k = f1(t, αk−1, Sαk−1) + f1x(t, αk−1, Sαk−1)[αk − αk−1] +f1ξ(t, αk−1, Sαk−1)[Sαk − Sαk−1] + f2(t, β0, Sβ0), αk(0) = x0. We now consider p = αk − αk+1 p′ = α′k − α′k+1 ≤ {f1x(t, αk, Sαk)[αk − αk+1] + f1ξ(t, αk, Sαk)[Sαk − Sαk+1] ≤ −Mp(t)−NSp(t). Also p(0) = αk(0)−αk+1(0) = 0. Hence by Lemma 1 we have p(t) ≤ 0. Thus αk ≤ αk+1 on I. To show αk+1 ≤ β0 on I. Set p = αk+1 − β0 p′ = α′k+1 − β′0 ≤ {f1x(t, αk, Sαk)[αk − β0] + f1ξ(t, αk, Sαk)[Sαk − Sβ0]}] + {f1x(t, αk, Sαk)[αk+1 − αk] + f1ξ(t, αk, Sαk)[Sαk+1 − Sαk]} + {f2(t, β0, Sβ0)− f2(t, α0, Sα0)} ≤ {f1x(t, αk, Sαk)[αk+1 − αk] + f1ξ(t, αk, Sαk)[Sαk+1 − Sαk]} ≤ −Mp(t)−NSp(t). Also p(0) = αk+1(0) − β0(0) ≤ 0. Hence by Lemma 1 we have p(t) ≤ 0 which implies that αk+1 ≤ β0 on I. Thus αk ≤ αk+1 ≤ β0, on I. Now using the principle of mathematical induction, we deduce the relation (13) and our claim holds. Also from relation (13), we can seen that the sequences are uniformly bounded. Since f1, f2 are uniformly bounded, the sequence {αn} is equicontinuous on [0, T ] and therefore by using Ascoli-Arzela Theorem, there exists a subsequence {αnk } that converges uniformly on [0, T ]. In view of (13) it also follows that the entire sequence {αn} converges uniformly to ρ. Since f1x exists and is bounded on [0, T ], we obtain that f1 is Lipschitz and hence the solution u is unique. To show that the convergence is quadratic, we begin by writing pn+1 = u− αn+1 and consider p′n+1 = u′ − α′n+1 Ch. V. Sreedhar, J. Vasundhara Devi, / Eur. J. Pure Appl. Math, 12 (4) (2019), 1662-1675 1668 = [f1(t, u, Su) + f2(t, u, Su)] − [{f1(t, αn, Sαn) + f1x(t, αn, Sαn)[αn+1 − αn] + f1ξ(t, αn, Sαn)[Sαn+1 − Sαn] + f2(t, β0, Sβ0)}] p′n+1 ≤ A+B + f1x(t, αn, Sαn)pn+1 + f1ξ(t, αn, Sαn)Spn+1 (14) where A = f1(t, u, Su)− f1(t, αn, Su)− f1x(t, αn, Sαn)[u− αn]; B = f1(t, αn, Su)− f1(t, αn, Sαn)− f1ξ(t, αn, Sαn)[Su− Sαn]. Our aim is to simplify each of the term A,B and substitute in (14). In this direction, consider A = f1(t, u, Su)− f1(t, αn, Su)− f1x(t, αn, Sαn)[u− αn]; = [f1x(t, η1, Su)(u− αn) − f1x(t, αn, Sαn)](u− αn) = [f1x(t, η1, Su) − f1x(t, αn, Sαn)](u− αn) = [f1x(t, η1, Su)− f1x(t, αn, Su) +f1x(t, αn, Su)− [f1x(t, αn, Sαn)]pn(t) = f1xx(t, τ1, Su)pn[η1 − αn] + 1∫ 0 f1xξ(t, αn, sSu+ (1− s)Sαn)[Su− Sαn]pnds ≤ f1xx(t, τ1, Su)pn 2 + 1∫ 0 f1xξ(t, αn, sSu+ (1− s)Sαn)[Spn]pnds ≤ l1|pn|2 + l2k1T |pn||pn| 1∫ 0 ds ≤ l1|pn|2 + l2k1T |pn|2 Next consider B = f1(t, αn, Su)− f1(t, αn, Sαn)− f1ξ(t, αn, Sαn)[Su− Sαn]; = 1∫ 0 [f1ξ(t, αn, s(Su) + (1− s)Sαn)− [f1ξ(t, αn, Sαn)](Su− Sαn)ds. Let η2(s) = s(Su) + (1− s)Sαn. Then B = 1∫ 0 [f1ξ(t, αn, η2(s))− f1ξ(t, αn, Sαn)](Su− Sαn)ds = 1∫ 0 1∫ 0 f1ξξ(t, αn, σ η2(s) + (1− σ)Sαn)s(Spn)(Su− Sαn)dsdσ = 1∫ 0 1∫ 0 f1ξξ(t, αn, σ η2(s) + (1− σ)Sαn)s(Spn)(Spn)dsdσ Ch. V. Sreedhar, J. Vasundhara Devi, / Eur. J. Pure Appl. Math, 12 (4) (2019), 1662-1675 1669 = 1∫ 0 1∫ 0 f1ξξ(t, αn, σ η2(s) + (1− σ)Sαn)s(Spn)2dsdσ ≤ l3k21T 2|p2n| 1∫ 0 1∫ 0 s dsdσ ≤ l3k21T 2|p2n| p′n+1 ≤ {l1|pn|2 + l2k1T |pn|2}+ {l3k21T 2|p2n|} −Mpn+1(t)−NSpn+1(t) ≤ l −Mpn+1(t)−NSpn+1(t) where l = {l1|pn|2 + l2k1T |pn|2} + {l3k21T 2|p2n|}. Now multiplying throughout by eMt and setting p̃n+1 = eMtpn+1, we get (pn+1(t)e Mt)′ ≤ Nk1 h2∫ 0 pn+1(s)e Mtds+ leMt ≤ Nk1 ∫ t 0 p̃n+1(s)e M(t−s)ds+ leMt = w′(t) (say) Then choosing w(0)=0, we get p̃n+1 ≤ w(t) on I. Clearly w′(t) ≥ 0, which means that w(t) is nondecreasing on I. Now for t ∈ I, w(t) ≤ Nk1 t∫ 0 z∫ 0 p̃n+1(s)e M(z−s)dsdz + t∫ 0 leMudu ≤ Nk1 t∫ 0 z∫ 0 w(z)eM(z−s)dsdz + l[max[0,T ]{ e Mt M − 1 M }] ≤ Nk1 t∫ 0 z∫ 0 w(z)eM(z−s)dsdz + l[max[0,T ]{ e Mt M }] ≤ Nk1 t∫ 0 z∫ 0 w(z)eM(z−s)dsdz + l{ eMT M } by setting c2 = l{ eMT M } and c1 = N M k1e Mt we get w(t) ≤ c1 t∫ 0 w(z)dz + c2. Now by using Gronwall’s inequality we get w(t) ≤ c2e t∫ 0 c1ds Ch. V. Sreedhar, J. Vasundhara Devi, / Eur. J. Pure Appl. Math, 12 (4) (2019), 1662-1675 1670 ≤ c2ec1t ≤ c2ec1T . Hence p̃n+1(t) ≤ w(t) ≤ max[0,T ]l(t)[eMt]ec1T p̃n+1(t) ≤ w(t) ≤ max[0,T ]l(t)[e(M+c1)t] |pn+1| ≤ (e(N1+c1)T )[M |pn|2], Therefore the sequence {αn} converges quadratically on I. Hence the theorem. 4. Generalized quasilinearization for periodic boundary value problem In this section an existence and uniqueness result is obtained for an PBVP of an integro differential equation using the method of generalized quasilinearization. For this first we define the various types of lower and upper solutions for the periodic boundary value problem of an integro differential equation given by x′ = f1(t, x, Sx) + f2(t, x, Sx), (15) x(0) = x(T ), (16) where f1, f2 ∈ C[I×Rn×Rn, Rn], Sx(t) = t∫ 0 K(t, s)x(s)ds, and K ∈ C[I×I,R+], I=[0,T]. Definition 2. Let α0, β0 ∈ C1[I,Rn]. Then α0, β0 are said to be (a) natural lower and upper solutions of (15) and (16) if α′0 ≤ f1(t, α0, Sα0) + f2(t, α0, Sα0), α0(0) ≤ α0(T ), β′0 ≥ f1(t, β0, Sβ0) + f2(t, β0, Sβ0), β0(0) ≥ β0(T ), t ∈ I; } (17) (b) coupled lower and upper solutions of Type I of (15) and (16) if α′0 ≤ f1(t, α0, Sα0) + f2(t, β0, Sβ0), α0(0) ≤ α0(T ), β′0 ≥ f1(t, β0, Sβ0) + f2(t, α0, Sα0), β0(0) ≥ β0(T ), t ∈ I; } (18) (c) coupled lower and upper solutions of Type II of (15) and (16) if α′0 ≤ f1(t, β0, Sβ0) + f2(t, α0, Sα0), α0(0) ≤ α0(T ), β′0 ≥ f1(t, α0, Sα0) + f2(t, β0, Sβ0), β0(0) ≥ β0(T ), t ∈ I; } (19) (d) coupled lower and upper solutions of Type III of (15) and (16) if α′0 ≤ f1(t, β0, Sβ0) + f2(t, β0, Sβ0), α0(0) ≤ α0(T ), β′0 ≥ f1(t, α0, Sα0) + f2(t, α0, Sα0), β0(0) ≥ β0(T ), t ∈ I. } (20) Ch. V. Sreedhar, J. Vasundhara Devi, / Eur. J. Pure Appl. Math, 12 (4) (2019), 1662-1675 1671 Now we will prove the following theorem related to coupled lower and upper solutions of Type I and we develop the generalized quasilinearization method for the periodic boundary value problem of an integro diffential equation via the initial value problem approach. Theorem 2. Suppose that the assumptions of Theorem 1 are satisfied. Then there exists monotone sequence {αn}, such that αn → ρ, as n → ∞ uniformly and monotonically to the unique solution ρ = u for PBVP of an integro differential equation (15) and (16)on I and the convergence is quadratic. Proof: In order to construct a sequence of lower and upper iterates that converge to the solution of the PBVP we fix the upper solution β0. Now consider the following linear problem for n= 0,1,2,3... α′n+1 = f1(t, αn, Sαn)+f1x(t, αn, Sαn)[αn+1−αn]+f1ξ(t, αn, Sαn)[Sαn+1−Sαn]+f2(t, β0, Sβ0), αn+1(0) = αn(T ). Since the above equation is a linear integro differential equation, so it has unique solution αn+1(t) on I. Now we claim that α0 ≤ α1 ≤ α2 ≤ ... ≤ αn−1 ≤ αn ≤ ... ≤ β0 (21) on I. We begin by setting p = α0 − α1. Then p′ = α′0 − α′1 = f1x(t, α0, Sα0)p(t) + f1ξ(t, α0, Sα0)Sp(t) p′(t) ≤ −Mp(t)−NSp(t) Also p(0) = α0(0) − α1(0) ≤ 0. Hence by Lemma 1 we have p(t) ≤ 0. Thus α0 ≤ α1 on I. Now we show α1 ≤ β0 on I by setting p = α1 − β0. Then, p′ = α′1 − β′0 ≤ {f1(t, α0, Sα0)− f1(t, β0, Sβ0)}+ {f1x(t, α0, Sα0)[α1 − α0] + f1ξ(t, α0, Sα0)[Sα1 − Sα0]}+ {f2(t, β0, Sβ0)− f2(t, α0, Sα0)} ≤ f1x(t, α0, Sα0)[α1 − β0] + f1ξ(t, α0, Sα0)[Sα1 − Sβ0] ≤ −Mp(t)−NSp(t). Also p(0) = α1(0) − β0(0) ≤ 0. Hence by Lemma 1 we have p(t) ≤ 0. Hence α1 ≤ β0 on I. Thus α0 ≤ α1 ≤ β0 Ch. V. Sreedhar, J. Vasundhara Devi, / Eur. J. Pure Appl. Math, 12 (4) (2019), 1662-1675 1672 on I. Now assuming that the result is true for n = k and prove it for n = k + 1. In order to prove our claim we consider the following linear integro differential equation. α′k+1 = f1(t, αk, Sαk)+f1x(t, αk, Sαk)[αk+1−αk]+f1ξ(t, αk, Sαk)[Sαk+1−Sαk]+f2(t, β0, Sβ0), αk+1(0) = αk(T ). The above linear integro differential equation has unique solution αk+1, where αk and β0 are known lower and upper solutions of (15) and (16). Further αk is the solution of the linear integro differential equation α′k = f1(t, αk−1, Sαk−1)+f1x(t, αk−1, Sαk−1)[αk−αk−1]+f1ξ(t, αk−1, Sαk−1)[Sαk−Sαk−1] +f2(t, β0, Sβ0), αk(0) = αk−1(T ). We now claim that αk ≤ αk+1 on I. For this set p = αk − αk+1 p′ = α′k − α′k+1 ≤ {f1x(t, αk, Sαk)[αk − αk+1] + f1ξ(t, αk, Sαk)[Sαk − Sαk+1] ≤ −Mp(t)−NSp(t) Also p(0) = αk(0)− αk+1(0) ≤ 0. Then by Lemma 1 Thus p(t) ≤ 0. So αk ≤ αk+1 on I. Next to show αk+1 ≤ β0 on I, Set p = αk+1 − β0 p′ = α′k+1 − β′0 ≤ {f1x(t, αk, Sαk)[αk+1 − αk] + f1ξ(t, αk, Sαk)[Sαk+1 − Sαk]} ≤ −Mp(t)−NSp(t) Also p(0) = αk+1(0) − β0(0) ≤ 0. Using Lemma 1 we get p(t) ≤ 0. Which means that αk+1 ≤ β0 on I. Now using the principle of mathematical induction, we deduce the relation (21) and our claim holds. Also from relation (21), we can seen that the sequences are uniformly bounded. Since f1, f2 are uniformly bounded so the sequence {αn} equicontinuous on [0, T ] and therefore by using Ascoli-Arzela Theorem, there exists subsequence {αnk } that converges uniformly on [0, T ]. In view of (21) it also follows that the entire sequence {αn} converges uniformly to ρ. Since f1x exists and is bounded on [0, T ], we obtain that f1 is Lipschitz and hence the solution u is unique. Ch. V. Sreedhar, J. Vasundhara Devi, / Eur. J. Pure Appl. Math, 12 (4) (2019), 1662-1675 1673 To show that the convergence is quadratic, we begin by writing pn+1 = u−αn+1. Then p′n+1 = u′ − α′n+1 ≤ [f1(t, u, Su) + f2(t, u, Su)] − [{f1(t, αn, Sαn) + f1x(t, αn, Sαn)[αn+1 − αn] + f1ξ(t, αn, Sαn)[Sαn+1 − Sαn] + f2(t, β0, Sβ0)}] = A+B + fx(t, αn, Sαn)pn+1 +fξ(t, αn, Sαn, αnt, α t n)Spn+1 = A+B + f1x(t, αn, Sαn)pn+1 + f1ξ(t, αn, Sαn)Spn+1 (22) where A = f1(t, u, Su)− f1(t, αn, Su)− f1x(t, αn, Sαn)[u− αn]; B = f1(t, αn, Su)− f1(t, αn, Sαn)− f1ξ(t, αn)[Su− Sαn]. Our aim is to simplify each of the term A,B and substitute in (22). In this direction, consider A = f1(t, u, Su)− f1(t, αn, Su)− f1x(t, αn, Sαn)[u− αn]; ≤ f1xx(t, τ1, Su)pn 2 + 1∫ 0 f1xξ(t, αn, sSu+ (1− s)Sαn)[Spn]pnds ≤ l1|pn|2 + l2k1T |pn||pn| 1∫ 0 ds ≤ l1|pn|2 + l2k1T |pn|2 Next consider B = f1(t, αn, Su)− f1(t, αn, Sαn)− f1ξ(t, αn)[Su− Sαn]; = 1∫ 0 [f1ξ(t, αn, s(Su) + (1− s)Sαn)− [f1ξ(t, αn, Sαn)](Su− Sαn)ds. Let η2(s) = s(Su) + (1− s)Sαn. Then B = 1∫ 0 [f1ξ(t, αn, η2(s))− f1ξ(t, αn, Sαn)](Su− Sαn)ds ≤ l3k21T 2|p2n| 1∫ 0 1∫ 0 s dsdσ ≤ l3k21T 2|p2n| p′n+1 ≤ {l1|pn|2 + l2k1T |pn|2}+ {l3k21T 2|p2n|} −Mpn+1(t)−NSpn+1(t) ≤ l −Mpn+1(t)−NSpn+1(t) REFERENCES 1674 where l = {l1|pn|2 + l2k1T |pn|2} + {l3k21T 2|p2n|}. Now multiplying throughout by eMt and setting p̃n+1 = eMtpn+1, we get (pn+1(t)e Mt)′ ≤ Nk1 h2∫ 0 pn+1(s)e Mtds+ leMt ≤ Nk1 ∫ t 0 p̃n+1(s)e M(t−s)ds+ leMt = w′(t) (say) Then choosing w(0)=0, we get p̃n+1 ≤ w(t) on I. Clearly w′(t) ≥ 0, which means that w(t) is nondecreasing on I. Now for t ∈ I, w(t) ≤ Nk1 t∫ 0 z∫ 0 p̃n+1(s)e M(z−s)dsdz + t∫ 0 leMudu by setting c2 = l{ eMT M } and c1 = N M k1e Mt we get w(t) ≤ c1 t∫ 0 w(z)dz + c2. Now by using Gronwall’s inequality we get w(t) ≤ c2e t∫ 0 c1ds ≤ c2ec1t ≤ c2ec1T Hence p̃n+1(t) ≤ w(t) ≤ max[0,T ]l(t)[eMt]ec1T p̃n+1(t) ≤ w(t) ≤ max[0,T ]l(t)[e(M+c1)t] |pn+1| ≤ (e(N1+c1)T )[M |pn|2], Therefore the sequence {αn} converges quadratically on I. Hence the theorem. 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