# Picard Type Iterative Scheme with Initial Iterates in Reverse Order for a Class of Nonlinear Three Point BVPs

Authors: Mandeep Singh; Amit K. Verma
Journal: International Journal of Differential Equations
 (2013)

---

## Abstract

We consider the following class of three point boundary value problemy′′(t)+f(t,y)=0, 0<t<1,y′(0)=0,y(1)=δy(η), where δ>0, 0<η<1, the source term f(t,y) is Lipschitz and continuous. We use monotone iterative technique in the presence of upper and lower solutions for both well-order and reverse order cases. Under some sufficient conditions, we prove some new existence results. We use examples and figures to demonstrate that monotone iterative method can efficiently be used for computation of solutions of nonlinear BVPs.


---

## Body

## 1. Introduction

In recent years, multipoint boundary value problems have been extensively studied by many authors ([1–5] and the references there in). Multipoint BVPs have lots of applications in various branches of science and engineering; for example, Webb [6] studied a second-order nonlinear boundary value problem subject to some nonlocal boundary conditions, which models a thermostat, and Zou et al. [7] studied the design of a large size bridge with multipoint supports.It is well known that one of the most important tools for dealing with existence results for nonlinear problems is the method of upper and lower solutions. The method of upper and lower solutions has a long history and some of its ideas can be traced back to Picard [8]. Later, it was extensively studied by Dragoni [9].Recently, there have been numerous results in the presence of an upper solutionu0 and a lower solution v0 with u0≥v0. But, in many cases, the upper and lower solutions may occur in the reversed order also, that is, u0≤v0. Cabada et al. [10] considered the monotone iterative method for the following BVP:
(1)y′′=f(t,y),y′(a)=0=y′(b)
with reversed ordered upper and lower solutions. So far, there have been some results in the presence of reverse ordered upper and lower solutions [10–13]. Xian et al. [14] considered the following second-order three point BVP:
(2)y′′(t)+f(t,y)=0,0<t<1,y(0)=0,y(1)-δy(η)=0,
where f(I×R,R), I=[0,1], 0<η<1, 0<δ<1. They used the fixed point index theory with non-well-ordered upper and lower solutions.Recently, Li et al. [15] studied the existence and uniqueness of solutions of second-order three point BVP
(3)y′′+f(t,y)=0,0<t<1,y′(0)=0,y(1)=δy(η),0<η<1,δ>0
with upper and lower solutions in the reversed order via the monotone iterative method in Banach space.The present work proves some new existing results for three point BVPs. Our technique is based on Picard-type iterative scheme and is quite simple and efficient from computational point of view. We believe that it can be very well adapted for this type of problem. In this paper we consider the following three point BVP:(4)y′′(t)+f(t,y)=0,0<t<1,y′(0)=0,y(1)=δy(η),
where f(I×R,R), I=[0,1], 0<η<1, δ>0. We have allowed sup(∂f/∂y) to take both negative and positive values.The paper is divided into 4 sections. In Section2, we construct Green's function and establish maximum and antimaximum principle. In Section 3, we generate monotone sequences by using results of Section 2 with upper and lower solutions as initial iterates ordered in one way or the other. We prove our final result of existence. In Section 4, we show that the monotone iterative scheme is a powerful technique. For that by using iterative scheme proposed in this paper we have computed the members of sequences in both cases (well-ordered and non-well-ordered case).


## 2. Preliminaries

### 2.1. Construction of Green's Function

To investigate (4), we consider the following linear three point BVP:
(5)-y′′(t)-λy(t)=h(t),0<t<1,y′(0)=0,y(1)=δy(η)+b,
where h∈C(I) and b is any constant. In this section, we construct the Green's function. We divide it into two cases.Case I (λ>0).
Let us assume that(H0)
0<λ≤π2/4, sinλ-δsinλη≥0, δcosλη-cosλ>0.It is easy to see that(H0) can be satisfied.Lemma 1.
Green's function for the following linear three point BVP(6)y′′(t)+λy(t)=0,0<t<1,y′(0)=0,y(1)=δy(η),
is given by
(7)G(t,s)=1λ(δcosλη-cosλ)×{[sinλ(1-s)+δsinλ(s-η)]cosλt,0≤t≤s≤η,cosλssinλ(1-t)+δcosλssinλ(t-η),s≤t,s≤η,sinλ(1-s)cosλt,t≤s,η≤s,cosλssinλ(1-t)+δcosληsinλ(t-s),η≤s≤t≤1,
and if H0 holds then G(t,s)≥0.Proof.
See proof of Lemma 2.1 in [15].Lemma 2.
Whenλ>0, y∈C2(I) is a solution of boundary value problem (5) and is given by
(8)y(t)=bcosλtcosλ-δcosλη-∫01G(t,s)h(s)ds.Proof.
See proof of Lemma 2.3 in [15].Remark 3.
Particularyy∈C2(I) is a solution of the boundary value problem (5) if and only if y∈C(I) is a solution of the integral equation
(9)y(t)=bcosλtcosλ-δcosλη-∫01G(t,s)h(s)ds.Case II (λ<0).
Assume that(H0′)
λ<0, δcosh|λ|η-cosh|λ|<0, sinh|λ|-δsinh|λ|η≥0.It is easy to see that(H0′) can be satisfied.Lemma 4.
Green's function for the following three point BVP(10)y′′(t)+λy(t)=0,0<t<1,y′(0)=0,y(1)=δy(η),
for λ<0 is given by
(11)G(t,s)=1Dλ{[sinh|λ|(1-s)+δsinh|λ|(s-η)]cosh|λ|t,0≤t≤s≤η,cosh|λ|ssinh|λ|(1-t)+δcosh|λ|ssinh|λ|(t-η),s≤t,s≤η,sinh|λ|(1-s)cosh|λ|t,t≤s,η≤s,cosh|λ|ssinh|λ|(1-t)+δcosh|λ|ηsin|λ|(t-s),η≤s≤t≤1,
where Dλ=|λ|(δcosh|λ|η-cosh|λ|) and if H0′ holds then G(t,s)≤0.Proof.
Proof is same as given in Lemma1.Lemma 5.
Whenλ<0, y∈C2(I) is a solution of boundary value problem (5) and is given by
(12)y(t)=bcosh|λ|tcosh|λ|-δcosh|λ|η-∫01G(t,s)h(s)ds.Proof.
Proof is same as given in Lemma2.


### 2.2. Maximum and Antimaximum Principle

Proposition 6 (antimaximum principle).
Let0<λ≤π2/4, sinλ-δsinλη≥0, δcosλη-cosλ>0, b≥0, and h(t)∈C[0,1] is such that h(t)≥0; then y(t) is nonpositive on I.Proposition 7 (maximum principle).
Letλ<0, sinh|λ|-δsinh|λ|η≥0, δcosh|λ|η-cosh|λ|<0, b≥0 and h(t)∈C[0,1] is such that h(t)≥0; then y(t) is nonnegative on I.


## 2.1. Construction of Green's Function

To investigate (4), we consider the following linear three point BVP:
(5)-y′′(t)-λy(t)=h(t),0<t<1,y′(0)=0,y(1)=δy(η)+b,
where h∈C(I) and b is any constant. In this section, we construct the Green's function. We divide it into two cases.Case I (λ>0).
Let us assume that(H0)
0<λ≤π2/4, sinλ-δsinλη≥0, δcosλη-cosλ>0.It is easy to see that(H0) can be satisfied.Lemma 1.
Green's function for the following linear three point BVP(6)y′′(t)+λy(t)=0,0<t<1,y′(0)=0,y(1)=δy(η),
is given by
(7)G(t,s)=1λ(δcosλη-cosλ)×{[sinλ(1-s)+δsinλ(s-η)]cosλt,0≤t≤s≤η,cosλssinλ(1-t)+δcosλssinλ(t-η),s≤t,s≤η,sinλ(1-s)cosλt,t≤s,η≤s,cosλssinλ(1-t)+δcosληsinλ(t-s),η≤s≤t≤1,
and if H0 holds then G(t,s)≥0.Proof.
See proof of Lemma 2.1 in [15].Lemma 2.
Whenλ>0, y∈C2(I) is a solution of boundary value problem (5) and is given by
(8)y(t)=bcosλtcosλ-δcosλη-∫01G(t,s)h(s)ds.Proof.
See proof of Lemma 2.3 in [15].Remark 3.
Particularyy∈C2(I) is a solution of the boundary value problem (5) if and only if y∈C(I) is a solution of the integral equation
(9)y(t)=bcosλtcosλ-δcosλη-∫01G(t,s)h(s)ds.Case II (λ<0).
Assume that(H0′)
λ<0, δcosh|λ|η-cosh|λ|<0, sinh|λ|-δsinh|λ|η≥0.It is easy to see that(H0′) can be satisfied.Lemma 4.
Green's function for the following three point BVP(10)y′′(t)+λy(t)=0,0<t<1,y′(0)=0,y(1)=δy(η),
for λ<0 is given by
(11)G(t,s)=1Dλ{[sinh|λ|(1-s)+δsinh|λ|(s-η)]cosh|λ|t,0≤t≤s≤η,cosh|λ|ssinh|λ|(1-t)+δcosh|λ|ssinh|λ|(t-η),s≤t,s≤η,sinh|λ|(1-s)cosh|λ|t,t≤s,η≤s,cosh|λ|ssinh|λ|(1-t)+δcosh|λ|ηsin|λ|(t-s),η≤s≤t≤1,
where Dλ=|λ|(δcosh|λ|η-cosh|λ|) and if H0′ holds then G(t,s)≤0.Proof.
Proof is same as given in Lemma1.Lemma 5.
Whenλ<0, y∈C2(I) is a solution of boundary value problem (5) and is given by
(12)y(t)=bcosh|λ|tcosh|λ|-δcosh|λ|η-∫01G(t,s)h(s)ds.Proof.
Proof is same as given in Lemma2.


## 2.2. Maximum and Antimaximum Principle

Proposition 6 (antimaximum principle).
Let0<λ≤π2/4, sinλ-δsinλη≥0, δcosλη-cosλ>0, b≥0, and h(t)∈C[0,1] is such that h(t)≥0; then y(t) is nonpositive on I.Proposition 7 (maximum principle).
Letλ<0, sinh|λ|-δsinh|λ|η≥0, δcosh|λ|η-cosh|λ|<0, b≥0 and h(t)∈C[0,1] is such that h(t)≥0; then y(t) is nonnegative on I.


## 3. Three Point Nonlinear BVP

Based on maximum and antimaximum Principle we develop theory to solve the three point nonlinear BVP and divide it into the following two subsections.


### 3.1. Reverse Ordered Lower and Upper Solutions

Theorem 8.
Let there existv0, u0 in C2[0,1] such that u0≤v0 and satisfies
(13)-u0′′(t)≥f(t,u0),0<t<1,u0′(0)=0,u0(1)≥δu0(η),(14)-v0′′(t)≤f(t,v0),0<t<1,v0′(0)=0,v0(1)≤δv0(η).
If f:D→R is continuous on D:={(t,y)∈[0,1]×R2:u0≤y≤v0} and there exists M≥0 such that for all (t,y),(t,w)∈D,
(15)y≤w⟹f(t,w)-f(t,y)≤M(w-y),
then the boundary value problem (5) has at least one solution in the region D. Further, if ∃ a constant λ such that M-λ≤0 and (H0) is satisfied, then the sequences {un} generated by
(16)-un+1′′(t)-λun+1=F(t,un),un+1′(0)=0,un+1(1)=δun+1(η),
where F(t,un)=f(t,un)-λun, with initial iterate u0 converge monotonically (non-decreasing) and uniformly towards a solution u(t) of (5). Similarly, using v0 as an initial iterate leads to a nonincreasing sequences {vn} converging to a solution v(t). Any solution z(t) in D must satisfy
(17)u(t)≤z(t)≤v(t).Proof.
From (13) and (16) (for n=0)
(18)-(u0-u1)′′-λ(u0-u1)≥0,(u0-u1)′(0)=0,(u0-u1)(1)≥(u0-u1)(η).
Since h(t)≥0 and b≥0, by using Proposition 6, we have u0≤u1.
In view ofλ≥M, from (16) we get
(19)-un+1′′(t)≥-(M-λ)(un+1-un)+f(t,un+1)
and if un+1≥un, then
(20)-un+1′′(t)≥f(t,un+1).
Since u0≤u1, then from (20) (for n=0) and (16) (for n=1), we get
(21)-(u1-u2)′′-λ(u1-u2)≥0,(u1-u2)′(0)=0,(u1-u2)(1)≥(u1-u2)(η).
From Proposition 6 we have u1≤u2.
Now from (14) and (16) (for n=0)
(22)-(u1-v0)′′-λ(u1-v0)≥0,(u1-v0)′(0)=0,(u1-v0)(1)≥δ(u1-v0)(η).
Thus, u1≤v0 follows from Proposition 6.
Now assuming thatun+1≥un, un+1≤v0, we show that un+2≥un+1 and un+2≤v0 for all n. From (16) (for n+1) and (20), we get
(23)-(un+1-un+2)′′-λ(un+1-un+2)≥0,(un+1-un+2)′(0)=0,(un+1-un+2)(1)≥δ(un+1-un+2)(η),
and hence from Proposition 6 we have un+1≤un+2.
From (16) (for n+1) and (14) we get
(24)-(un+2-v0)′′-λ(un+2-v0)≥0,(un+2-v0)′(0)=0,(un+2-v0)(1)≥δ(un+2-v0)(η).
Then, from Proposition 6, un+2≤v0 and hence we have
(25)u1≤u2≤⋯≤un≤un+1≤⋯≤v0,
and starting with v0 it is easy to get
(26)v1≥v2≥⋯≥vn≥vn+1≥⋯≥u0.
Finally, we show that un≤vn for all n. For this, by assuming un≤vn, we show that un+1≤vn+1. From (16) it is easy to get
(27)-(un+1-vn+1)′′-λ(un+1-vn+1)≥0,(un+1-vn+1)′(0)=0,(un+1-vn+1)(1)≥δ(un+1-vn+1)(η).
Hence, from Proposition 6, un+1≤vn+1. Thus we have
(28)v0≥v1≥v2≥⋯≥vn≥vn+1≥⋯≥un+1≥un≥⋯≥u2≥u1≥u0.
So, the sequences un and vn are monotonically nondecreasing and nonincreasing, respectively, and are bounded by u0 and v0. Hence by Dini's theorem they converge uniformly. Let u(t)=limn→∞un(t) and v(t)=limn→∞vn(t).
Using Lemma2, the solution un of (16) is given by
(29)un=bcosλtcosλ-δcosλη-∫01G(t,s)(f(t,un)-λun)ds.
Then, by Lebesgue's dominated convergence theorem, taking the limit as n approaches to ∞, we get
(30)u(t)=bcosλtcosλ-δcosλη-∫01G(t,s)(f(t,u)-λu)ds,
which is the solution of boundary value problem (5).
Any solutionz(t) in D can play the role of u0(t); hence, z(t)≤v(t) and similarly one concludes that z(t)≥u(t).


### 3.2. Well-Ordered Lower and Upper Solutions

Theorem 9.
Let there existv0, u0 in C2[0,1] such that u0≥v0 and satisfies
(31)-u0′′(t)≥f(t,u0),0<t<1,u0′(0)=0,u0(1)≥δu0(η),-v0′′(t)≤f(t,v0),0<t<1,v0′(0)=0,v0(1)≤δv0(η).
If f:D0→R is continuous on D0:={(t,y)∈[0,1]×R2:v0≤y≤u0} and there exists M≥0 such that for all (t,y~),(t,w~)∈D0(32)y~≤w~⟹f(t,w~)-f(t,y~)≥-M(w~-y~),
then the boundary value problem (5) has at least one solution in the region D0. If ∃ a constant λ<0, such that λ+M≤0 and (H0′) is satisfied, then the sequences {un} generated by
(33)-un+1′′(t)-λun+1=F(t,un),un+1′(0)=0,-un+1′′(t)-λun+1(1)=δun+1(η),
where F(t,un)=f(t,un)-λun, with initial iterate u0, converge monotonically (nonincreasing) and uniformly towards a solution u~(t) of (5). Similarly, using v0 as an initial iterate leads to a nondecreasing sequences {vn} converging to a solution v~(t). Any solution z~(t) in D0 must satisfy
(34)v~(t)≤z~(t)≤u~(t).Proof.
Proof follows from the analysis of Theorem8.


## 3.1. Reverse Ordered Lower and Upper Solutions

Theorem 8.
Let there existv0, u0 in C2[0,1] such that u0≤v0 and satisfies
(13)-u0′′(t)≥f(t,u0),0<t<1,u0′(0)=0,u0(1)≥δu0(η),(14)-v0′′(t)≤f(t,v0),0<t<1,v0′(0)=0,v0(1)≤δv0(η).
If f:D→R is continuous on D:={(t,y)∈[0,1]×R2:u0≤y≤v0} and there exists M≥0 such that for all (t,y),(t,w)∈D,
(15)y≤w⟹f(t,w)-f(t,y)≤M(w-y),
then the boundary value problem (5) has at least one solution in the region D. Further, if ∃ a constant λ such that M-λ≤0 and (H0) is satisfied, then the sequences {un} generated by
(16)-un+1′′(t)-λun+1=F(t,un),un+1′(0)=0,un+1(1)=δun+1(η),
where F(t,un)=f(t,un)-λun, with initial iterate u0 converge monotonically (non-decreasing) and uniformly towards a solution u(t) of (5). Similarly, using v0 as an initial iterate leads to a nonincreasing sequences {vn} converging to a solution v(t). Any solution z(t) in D must satisfy
(17)u(t)≤z(t)≤v(t).Proof.
From (13) and (16) (for n=0)
(18)-(u0-u1)′′-λ(u0-u1)≥0,(u0-u1)′(0)=0,(u0-u1)(1)≥(u0-u1)(η).
Since h(t)≥0 and b≥0, by using Proposition 6, we have u0≤u1.
In view ofλ≥M, from (16) we get
(19)-un+1′′(t)≥-(M-λ)(un+1-un)+f(t,un+1)
and if un+1≥un, then
(20)-un+1′′(t)≥f(t,un+1).
Since u0≤u1, then from (20) (for n=0) and (16) (for n=1), we get
(21)-(u1-u2)′′-λ(u1-u2)≥0,(u1-u2)′(0)=0,(u1-u2)(1)≥(u1-u2)(η).
From Proposition 6 we have u1≤u2.
Now from (14) and (16) (for n=0)
(22)-(u1-v0)′′-λ(u1-v0)≥0,(u1-v0)′(0)=0,(u1-v0)(1)≥δ(u1-v0)(η).
Thus, u1≤v0 follows from Proposition 6.
Now assuming thatun+1≥un, un+1≤v0, we show that un+2≥un+1 and un+2≤v0 for all n. From (16) (for n+1) and (20), we get
(23)-(un+1-un+2)′′-λ(un+1-un+2)≥0,(un+1-un+2)′(0)=0,(un+1-un+2)(1)≥δ(un+1-un+2)(η),
and hence from Proposition 6 we have un+1≤un+2.
From (16) (for n+1) and (14) we get
(24)-(un+2-v0)′′-λ(un+2-v0)≥0,(un+2-v0)′(0)=0,(un+2-v0)(1)≥δ(un+2-v0)(η).
Then, from Proposition 6, un+2≤v0 and hence we have
(25)u1≤u2≤⋯≤un≤un+1≤⋯≤v0,
and starting with v0 it is easy to get
(26)v1≥v2≥⋯≥vn≥vn+1≥⋯≥u0.
Finally, we show that un≤vn for all n. For this, by assuming un≤vn, we show that un+1≤vn+1. From (16) it is easy to get
(27)-(un+1-vn+1)′′-λ(un+1-vn+1)≥0,(un+1-vn+1)′(0)=0,(un+1-vn+1)(1)≥δ(un+1-vn+1)(η).
Hence, from Proposition 6, un+1≤vn+1. Thus we have
(28)v0≥v1≥v2≥⋯≥vn≥vn+1≥⋯≥un+1≥un≥⋯≥u2≥u1≥u0.
So, the sequences un and vn are monotonically nondecreasing and nonincreasing, respectively, and are bounded by u0 and v0. Hence by Dini's theorem they converge uniformly. Let u(t)=limn→∞un(t) and v(t)=limn→∞vn(t).
Using Lemma2, the solution un of (16) is given by
(29)un=bcosλtcosλ-δcosλη-∫01G(t,s)(f(t,un)-λun)ds.
Then, by Lebesgue's dominated convergence theorem, taking the limit as n approaches to ∞, we get
(30)u(t)=bcosλtcosλ-δcosλη-∫01G(t,s)(f(t,u)-λu)ds,
which is the solution of boundary value problem (5).
Any solutionz(t) in D can play the role of u0(t); hence, z(t)≤v(t) and similarly one concludes that z(t)≥u(t).


## 3.2. Well-Ordered Lower and Upper Solutions

Theorem 9.
Let there existv0, u0 in C2[0,1] such that u0≥v0 and satisfies
(31)-u0′′(t)≥f(t,u0),0<t<1,u0′(0)=0,u0(1)≥δu0(η),-v0′′(t)≤f(t,v0),0<t<1,v0′(0)=0,v0(1)≤δv0(η).
If f:D0→R is continuous on D0:={(t,y)∈[0,1]×R2:v0≤y≤u0} and there exists M≥0 such that for all (t,y~),(t,w~)∈D0(32)y~≤w~⟹f(t,w~)-f(t,y~)≥-M(w~-y~),
then the boundary value problem (5) has at least one solution in the region D0. If ∃ a constant λ<0, such that λ+M≤0 and (H0′) is satisfied, then the sequences {un} generated by
(33)-un+1′′(t)-λun+1=F(t,un),un+1′(0)=0,-un+1′′(t)-λun+1(1)=δun+1(η),
where F(t,un)=f(t,un)-λun, with initial iterate u0, converge monotonically (nonincreasing) and uniformly towards a solution u~(t) of (5). Similarly, using v0 as an initial iterate leads to a nondecreasing sequences {vn} converging to a solution v~(t). Any solution z~(t) in D0 must satisfy
(34)v~(t)≤z~(t)≤u~(t).Proof.
Proof follows from the analysis of Theorem8.


## 4. Numerical Illustration

To verify our results, we consider examples and show that there exists at least one value ofλ∈ℝ∖{0} such that iterative scheme generates monotone sequences which converge to solutions of nonlinear problem. Thus, these examples validate sufficient conditions derived in this paper.Example 10 (reverse order).
Consider the boundary value problem(35)-y′′(t)=ey32-164,0<t<1,y′(0)=0,y(1)=2y(13).
Here, f(t,y)=(ey/32)-(1/64), δ=2, η=1/3. This problem has v0=1 and u0=-1 as lower and upper solutions; that is, it is non-well-ordered case. The nonlinear term is Lipschitz in y and continuous for all value of y, and Lipschitz constants are M=e/32. For 0.0849463≤λ≤π2/4 we can see that (H0) will be true. To verify (H0), in Figure 1 we plot inequalities assumed in (H0). From Figures 2, 3, and 4, we plot members of monotone sequences un, vn for different values of λ.Figure 1
Plot ofsinλ-δsinλη and δcosλη-cosλ.Figure 2
Plot ofun (red) and vn (blue), n=0,1,2,3,4 for λ=0.9.Figure 3
Plot ofun (red) and vn (blue), n=0,1,2,3,4 for λ=1.Figure 4
Plot ofun (red) and vn (blue), n=0,1,2,3,4 for λ=2.3.Example 11 (well order).
Consider the boundary value problem(36)-y′′(t)=132[e24-sint4-2(y(t))3],y′(0)=0,y(1)=13y(12).
Here, f(t,y)=(1/32)[(e2/4)-(sint/4)-2(y(t))3], δ=1/3, η=1/2. This problem has v0=-1 and u0=1 as lower and upper solutions; that is, it is well-ordered case. The nonlinear term is Lipschitz in y and continuous for all value of y, and Lipschitz constants are M=3/16. For λ≤-0.1875, we can see that (H0′) will be true. To verify (H0′), in Figure 5 we plot inequalities assumed in (H0′). From Figures 6, 7, and 8, we plot members of monotone sequences un, vn for different values of λ.Figure 5
Plot ofδcosh|λ|η-cosh|λ|and sinh|λ|-δsinh|λ|η.Figure 6
Plot ofun (blue) and vn (red), n=1,2,3,4,5 for λ=-0.2.Figure 7
Plot ofun  (blue) and vn (red), n=1,2,3,4,5 for λ=-1.Figure 8
Plot ofun  (blue) and  vn (red), n=0,1,2,…,20 for λ=-20.


## 5. Conclusion

The monotone iterative technique coupled with upper and lower solutions is a powerful tool for computation of solutions of nonlinear three point boundary value problems. It proves the existence of solutions analytically and gives us a tool so that numerical solutions can also be computed and then some real-life problems, for example, bridge design problem, thermostat problem, and so forth, can be solved. We have plotted sequences for bothλ>0 and λ<0. The plots are quite encouraging and will motivate researchers to explore further possibilities. Employing this technique, Mathematica/Maple/MATLAB user-friendly packages can be developed (see [16]).



---

*Source: 728149-2013-07-24.xml*