QUADRATICALLY CONVERGING ITERATIVE SCHEMES FOR NONLINEAR VOLTERRA INTEGRAL EQUATIONS AND AN APPLICATION SUDHAKAR

A generalized quasilinear technique is employed to derive iterative schemes 
for nonlinear Volterra integral equations under various monotonicity and 
convexity (concavity) conditions on the kernels. The iterates in the 
schemes are linear, and converge monotonically, uniformly and quadratically to the unique solution. An application to a boundary-layer theory problem and examples illustrating the results are presented.


Introduction
For the nonlinear Volterra integral equation (t)-h(t)+ / K(t,,())d, (1.1) 0 when K is nondecreasing in u and satisfies a Lipschitz condition, the successive appro- ximations method [9] yields a monotonic sequence converging uniformly to the unique solution of (1.1).On the other hand, if K is nonincreasing in u and satisfies a Lip- schitz condition, then there is an alternating sequence of successive approximations converging to the unique solution of (1.1) in a closed set bounded by lower and upper functions [8].The iterates defining the above sequences are nonlinear and the rate of convergence is linear.The iterates employed in the monotone iterative technique [3,5] are linear, and so is their convergence rate.Based on the quaslinearization idea [1], the generalized quasilinear technique initiated in [6] and later extended in [4] offers two monotonic sequences of linear iterates converging uniformly and quadratically (and hence more rapidly) to the unique solution of the initial value problem u' f(t, u), u u(t).
(1.2) 170 SUDHAKAR G. PANDIT The generalized quasilinear technique has recently found its way [2] into the initial- boundary value problem for the two dimensional analog of (1.2), namely Uxu f(x,y,U, Ux, Uu) u u(x,y), (1.3) whereas it is still open for the (more difficult) periodic-boundary value problem asso- ciated with (1.3).
The purpose of this paper is to develop linearly defined and quadratically conver- gent iterative schemes for (1.1) under various monotonicity and convexity (concavity)   conditions on K. Of special interest is the case when K is nonincreasing and convex in u.The boundary-layer theory problem [10], when transformed into a Volterra inte- gral equation has a nonincreasing and convex kernel (see Example 4.2).It is apt to note that the method of successive approximations when applied to nonlinear pro- blems is especially useful, even for numerical computations, when the nonlinearities are nonincreasing [7,8].In this case, we employ coupled lower, and upper solutions in the development of our schemes (see Theorem 3.2).We present examples illustrat- ing the results obtained.
A function v C[J,N] is called an upper solution of (2.1) on J if v(t) >_ h(t) + / g(t,s, v(s))ds, t e J, 0 and a lower solution, if the reversed inequalities hold.If v(t) <_ + f >_ + ] 0 0 tEJ, then v and w are said to be coupled lower and upper solutions of (2.1) on J.
Utilizing the ideas of Theorem 2.1 we can prove the existence of solutions of a finite system of Volterra integral equations in a closed set.(See [3] for a similar re- sult for systems of ordinary differential equations.)In what follows, vectorial inequali- ties mean that the same inequalities hold for their corresponding components.Recall that a vector function is continuous or nondecreasing (nonincreasing) when all its components are such.For Vo, W o G c[g, Nn], such that Vo(t <_ wo(t on J, let fn {(t,s, u) D x Rn'Vo(t) <_ u <_ w0(t)}.Consider the system h(t) + ] n(t, If v0, w o E C[J, Rn], Vo(t _ wo(t on J, are lower and upper solutions of (2.4) on J respectively, and R is nondecreasing in u for each fixed pair of s and t, then there exists a solution u of (2.4) such that vo(t _ u(t) _ wo(t on J. Furthermore, if R satisfies Lipschitz condition n Rj(t,s,u)-Rj(t,s,v)l <_LIE ui-vil' l<_j_n, (2.6) --1 then, u is unique; (b) wo(t on J, are coupled lower and upper solu- tions of (2.4) on g and R is nonincreasing in u for each fixed pair of s and t, then there exists a solution (unique solution, if R satisfies (2.6)) u of (2.4) satisfying (2.5).
The following vector integral inequality of Gronwall type is required in establish- ing the quadratic convergence of iterates.
Lemma 2.1: Let u(t) _ h(t) + f Au(s)ds, t e g where u,h G c[g,n] and 0 A (aij) is an n x n constant matrix with aij >_ O.Then, u(t) _ h(t) + / A ezp(A(t-s))h(s)ds, t J. 0 3. Generalized Quasilinearizaion In the result below, we employ upper and lower solutions to develop a quadratically convergent iterative scheme for (2.1) when the kernel K is nondecreasing and convex in u.For Vo, W oec[J,R] and vo(t)<u<wo(t on J, let ft {(t, s, u) eDxR; vo(t _ u _ wo(t), t J}.Let ]l u I I max u(t) I, t J.
In our next result, we employ coupled lower and upper solutions to make use of the predominant effect of the nonincreasing character of K.
Then, there exist monotone sequences {vn(t)} and {wn(t)} converging uniformly and quadratically to the unique solution of (2.1) on J.