An Extended Method of Quasilinearization for Nonlinear Impulsive Differential Equations with a Nonlinear Three-Point Boundary Condition

In this paper, we discuss an extended form of generalized quasilinearization technique for first order nonlinear impulsive differential equations with a nonlinear three-point boundary condition. In fact, we obtain monotone sequences of upper and lower solutions converging uniformly and quadratically to the unique solution of the problem.


Introduction
The method of quasilinearizaion provides an adequate approach for obtaining approximate solutions of nonlinear problems.The origin of the quasilinearizaion lies in the theory of dynamic programming [1][2][3].This method applies to semilinear equations with convex (concave) nonlinearities and generates a monotone scheme whose iterates converge quadratically to the solution of the problem at hand.The assumption of convexity proved to be a stumbling block for the further development of the method.The nineties brought new dimensions to this technique.The most interesting new idea was introduced by Lakshmikantham [4][5] who generalized the method of quasilinearizaion by relaxing the convexity assumption.This development proved to be quite significant and the method was studied and applied to a wide range of initial and boundary value problems for different types of differential equations, see [6][7][8][9][10][11][12][13][14][15][16][17] and references therein.Some real-world applications of the quasilinearizaion technique can be found in [18][19][20].Many evolution processes are subject to short term perturbations which act instantaneously in the form of impulses.Examples include biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics and frequency modulated systems.Thus, impulsive differential equations provide a natural description of observed evolution processes of several real world problems.Moreover, the theory of impulsive differential equations is much richer than the corresponding theory of ordinary differential equations without impulse effects since a simple impulsive differential equation may exhibit several new phenomena such as rhythmical beating, merging of solutions and noncontinuability of solutions.Thus, the theory of impulsive differential equations is quite interesting and has attracted the attention of many scientists, for example, see [21][22][23][24].In particular, Eloe and Hristova [23] discussed the method of quasilinearization for first order nonlinear impulsive differential equations with linear boundary conditions.Multi-point nonlinear boundary value problems, which refer to a different family of boundary conditions in the study of disconjugacy theory [25], have been addressed by many authors, for instance, see [26][27] and the references therein.In this paper, we develop an extended method of quasilinearization for a class of first order nonlinear impulsive differential equations involving a mixed type of nonlinearity with a nonlinear three-point boundary condition where .., p and the nonlinearity h : R −→ R is continuous.Here, it is worthmentioning that the convexity assumption on f (t, x) has been relaxed and instead f (t, x) + M 1 x 2 is taken to be convex for some M 1 > 0 while a less restrictive condition is demanded on g(t, x), namely, [g(t, x) + M 2 x 1+ ] satisfies a nondecreasing condition for some > 0 and M 2 > 0.Moreover, we also relax the concavity assumption (h (x) ≤ 0) on the nonlinearity h(x) in the boundary condition (3) by requiring h (x) + ψ (x) ≤ 0 for some continuous function ψ(x) satisfying ψ ≤ 0 on R. We construct two monotone sequences of upper and lower solutions converging uniformly and quadratically to the unique solution of the problem.Some special cases of our main result have also been recorded.

Some Basic Results
For A ⊂ R, B ⊂ R, let P C(A, B) denotes the set of all functions v : A → B which are piecewise continuous in A with points of discontinuity of first kind at the points τ k ∈ A, that is, there exist the limits lim The function β(t) ∈ P C 1 ([0, T ], R) is called an upper solution of the BVP (1)-( 3) if the inequalities are reversed in ( 4)- (6).
Let us set the following notations for the sequel.
R) be lower and upper solutions of ( 1)-( 3) respectively.Further, Moreover, h is nondecreasing on R and Proof.The method of proof is similar to the one used in proving Theorem 2.6.1 (page 87 [21]), so we omit the proof.
has a unique solution u(t) on the interval [0, T ] given by where We need the following known theorem (Theorem 1.4.1, page 32 [21]) to prove our main result.
Taking the limit n → ∞, we find that )).
Now applying Theorem 3 to the BVP ( 28)-(30) together with Lebesgue dominated convergence theorem, it follows that x(t) is the solution of the BVP (1)-(3) in S(α 0 , β 0 ).Similarly, applying Theorem 3 to the BVP (31)-(32), it can be shown that y(t) is the solution of the BVP (1)-(3) in S(α 0 , β 0 ).Therefore, by the uniqueness of the solution, x(t) = y(t).Now, we prove that the convergence of each of the two sequences is quadratic.For that, we set a n+1 (t) = x(t) − α n+1 (t), b n+1 (t) = β n+1 (t) − x(t), t ∈ [0, T ] and note that a n+1 (t) ≥ 0 and b n+1 (t) ≥ 0. We will only prove the quadratic convergence of the sequence {a n (t)} ∞ 0 as that of the sequence {b n (t)} ∞ 0 is similar one.Setting P (t, x) = f (t, x) + g(t, x) + M 1 x 2 + M 2 x 1+ , t ∈ [0, T ], t = τ k and using the mean value theorem repeatedly, we obtain where L 1 is Lipschitz constant (g x satisfies the Lipschitz condition), x ≤ β n and Similarly it can be shown that where Applying Theorem 1.4.1 (page 32 [21]) on (34)-(35), it follows that the function a n+1 (t) satisfies the estimate In view of ( 21), we have where ).Thus, we have Combining (36) and (37) yields ))a n+1 ( T )) Solving for a n+1 (0), we get )) On the same pattern, it can be proved that where ζ 1 and ζ 2 are positive constants.This establishes the quadratic convergence of the sequences.

Concluding Remarks
This paper addresses a quasilinearization method for a nonlinear impulsive first order ordinary differential equation dealing with a nonlinear function F (t, x(t)) which is a sum of two functions of different nature together with a nonlinear three-point boundary condition in contrast to a problem containing a single function and a linear boundary condition considered in [23].The condition on g(t, x(t)) in assumption (A 3 ) of Theorem 3 is motivated by the well known fact that χ(t) = t p is convex for p > 1.The following results can be recorded as a special case of this problem: EJQTDE, 2007 No. 1, p. 16 (i) If we take g(t, x(t)) = 0, h(x( T 2 )) = c (constant), we obtain the generalized quasilinearization for first order impulsive differential equations with linear boundary conditions [23].
(ii) By taking h(x( T 2 )) = u 0 , γ 1 = 1, γ 2 = 0, we can record the results of usual initial value problems with impulse.In reference [28], the authors have developed an extension of generalized quasilinearization for initial value problems without impulse.Thus our problem generalizes the results of [28] in the sense that impulsive effects have been taken into account along with a three-point nonlinear boundary condition.