Monotone iterative technique for the initial value problem for differential equations with non-instantaneous impulses

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Abstract

An algorithm for constructing two monotone sequences of upper and lower solutions of the initial value problem for a scalar nonnlinear differential equation with non-instantaneous impulses is given. The impulses start abruptly at some points and their action continue on given finite intervals. We prove that the functional sequences are convergent and their limits are minimal and maximal solutions of the considered problem. An example is given to illustrate the results.

Introduction

In the real world life there are many processes and phenomena that are characterized by rapid changes in their state. In the literature there are two popular types of impulses:

  • instantaneous impulses – the duration of these changes is relatively short compared to the overall duration of the whole process. The model is given by impulsive differential equations (see, for example, [5], [11], the monographs [10], [14], [19] and the cited references therein);

  • noninstantaneous impulses – an impulsive action, which starts at an arbitrary fixed point and remains active on a finite time interval. Hernandez and O’Regan [9] introduced this new class of abstract differential equations where the impulses are not instantaneous and they investigated the existence of mild and classical solutions. For recent work we refer the reader to [6], [16], [17], [18], [20], [21], [22].

In this paper the impulses start abruptly at some points and their action continue on given finite intervals. As a motivation for the study of these systems we consider the following simplified situation concerning the hemodynamical equilibrium of a person. In the case of a decompensation (for example, high or low levels of glucose) one can prescribe some intravenous drugs (insulin). Since the introduction of the drugs in the bloodstream and the consequent absorption for the body are gradual and continuous processes, we can interpret the situation as an impulsive action which starts abruptly and stays active on a finite time interval. The model of this situation is the so called noninstanteneous impulsive differential equation.

The present paper deals with an initial value problem for a nonlinear scalar noninstantaneous impulsive differential equation on a closed interval. The monotone iterative technique combined with the method of lower and upper solutions is applied to find approximately the solution of the given problem. A procedure for constructing two monotone functional sequences is given. The elements of these sequences are solutions of suitably chosen initial value problems for scalar linear noninstantaneous impulsive differential equations for which there is an explicit formula. Also, the elements of these sequences are lower/upper solutions of the given problem. We prove that both sequences converge and their limits are minimal and maximal solutions of the studied problem. An example, generalizing the logistic equation, is given to illustrate the procedure.

Note iterative techniques combined with lower and upper solutions are applied to approximately solve various problems for ordinary differential equations (see the classical monograph [13]), for second order periodic boundary value problems [4], for differential equations with maxima [1], [7], for difference equations with maxima [2], for impulsive differential equations [3], [5], for impulsive integro-differential equations [8], for impulsive differential equations with supremum [11], and for differential equations of mixed type [12].

Section snippets

Noninstantaneous impulses in differential equations

In this paper we will assume two increasing finite sequences of points {ti}i=1p+1 and {si}i=0p are given such that s0=0<tisi<ti+1 , i=1,2,,p, and points t0,TR+ are given such that s0=0<t0<t1,tp<Ttp+1, p is a natural number.

Consider the initial value problem (IVP) for the nonlinear noninstantaneous impulsive differential equation (NIDE) x=fk(t,x)fort(sk,tk+1][t0,T],k=0,1,,p,x(t)=ϕk(t,x(t),x(tk0))fort(tk,sk][t0,T],k=1,2,,p,x(t0)=x0,where x,x0R,fk:[sk,tk+1][t0,T]×RR,(k=0,1,,p),ϕk:[t

Lower and upper solutions of NIDE

Definition 1

We will say that the function v(t) ∈ D is a minimal (maximal) solution of the IVP for NIDE (1) in the set D if it is a solution of (1) and for any solution u(t) ∈ D of (1) the inequality v(t) ≤ u(t) (v(t) ≥ u(t)) holds on [t0, T] where D ⊂ PC1([t0, T]).

Definition 2

We will say that function v(t) is a lower (upper) solution of the IVP for NIDE (1) if v()fk(t,v)fort(sk,tk+1][t0,T],k=0,1,2,,p,v(t)()ϕk(t,v(t),v(tk0))fort(tk,sk][t0,T],k=1,2,,p,v(t0)()x0.

Example 2

Consider the IVP for the scalar NIDE (1)

Main results

We give an algorithm for constructing two sequences of successive approximations.

Theorem 1

Let the following conditions be fulfilled:

  • 1.

    The functions v, wPC1([t0, T]) are lower and upper solutions of the IVP for NIDE (1), respectively, and v(t) ≤ w(t) for t ∈ [t0, T].

  • 2.

    The functionsfkC([sk,tk+1][t0,T],R),(k=0,1,,p) and there exist constants Mk > 0, k=0,1,2,,p such that for anyt[sk,tk+1][t0,T],k=0,1,2,,p, and x, yΩk(t, v, w): xy the inequality fk(t,x)fk(t,y)Mk(xy)holds.

  • 3.

    The functionsϕkC([t

Applications

Let the finite sequences of points {ti}i=1p+1 and {si}i=0p:s0=0<tisi<ti+1 , i=1,2,,p, be given, t0=0,T(tp,tp+1], p is a natural number.

Consider the following nonlinear NIDE which is a generalization of the logistic equation x=ak(t)x(xb)fort(sk,tk+1][0,T],k=0,1,,p,x(t)=dkex(t)+ckx(tk0)dkfort(tk,sk][0,T],k=1,2,,p,x(0)=x0,where x,x0R,akC[[sk,tk+1],(0,)](k=0,1,,p), b > 0.5, ck, dk, (k=1,2,3,,p), are nonnegative constants such that dk<0.5e0.51 and ck12dk(e0.51).

The NIDE (32)

Conclusions

This paper considers initial value problems for a nonlinear scalar differential equation with noninstantaneous impulses. The monotone iterative technique combined with the method of lower and upper solutions is applied to find approximately the solution of the given problem. An algorithm for constructing two monotone functional sequences of upper and lower solutions is suggested. The elements of these sequences are solutions of suitably chosen initial value problems for scalar linear

Acknowledgments

Research was partially supported by the Fund NPD, Plovdiv University, No. MU15-FMIIT-008.

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