Skip to main content

Method of quasilinearization for a nonlocal singular boundary value problem in weighted spaces

Abstract

This paper studies the existence and uniqueness of solutions for a nonlocal singular boundary value problem of second-order integro-differential equations in weighted spaces. The method of quasilinearization is applied to obtain monotone sequences of approximate solutions converging uniformly and quadratically to a unique solution of the problem at hand. An illustrative example is presented.

MSC:34A45, 34B16, 45J05.

1 Introduction

Boundary value problems (BVPs) for nonlinear differential equations arise in a variety of areas of applied mathematics, physics and in variational problems of control theory. A point of central importance in the study of nonlinear boundary value problems is to understand how the properties of nonlinearity in a problem influence the nature of the solutions to the boundary value problems. During the last two decades, the theory of singular boundary value problems has also been extensively developed. This is largely due to the fact that the mathematical models in the study of nonlinear phenomena give rise to singular boundary value problems. Examples include energy analysis problems [1, 2], plasma and electric potential theory [3], circular membrane theory [4], membrane response of a spherical cap [5, 6], deformation of membrane cap [7], theory of colloids [8], flow and heat transfer [9], draining flow [10, 11], flow of a gas through porous media [12], Homann flow [13], boundary layer problems [14], pseudoplastic fluids [15], etc. For the theoretical background of singular boundary value problems, we refer the reader to the references [16, 17].

Integro-differential equations arise in many engineering and scientific disciplines, often as an approximation to partial differential equations, which represent much of the continuum phenomena. Many forms of these equations are possible. Some of the applications are unsteady aerodynamics and aero-elastic phenomena, visco-elasticity, visco-elastic panel in super sonic gas flow, fluid dynamics, electro-dynamics of complex medium, many models of population growth, polymer rheology, neural network modeling, sandwich system identification, materials with fading memory, mathematical modeling of the diffusion of discrete particles in a turbulent fluid, heat conduction in materials with memory, theory of lossless transmission lines, theory of population dynamics, compartmental systems, nuclear reactors and mathematical modeling of hereditary phenomena. Also, the governing equations in the problems of biological sciences, such as spreading of disease by the dispersal of infectious individuals, the reaction-diffusion models in ecology to estimate the speed of invasion etc., are integro-differential equations. Thus, it is important to study singular boundary value problems for nonlinear integro-differential equations. It is worth mentioning that most of the results on singular boundary value problems deal with the existence and uniqueness of the solution of the problems under certain conditions. There are only a few papers which develop some constructive methods for the solution of the nonlinear singular problems.

The study of singular nonlocal boundary value problems for nonlinear differential equations was initiated by Kiguradze and Lomtatidze [18] and Lomtatidze [19, 20]. Since then, more general nonlinear singular nonlocal boundary value problems have been studied extensively. Some results concerning the positive solutions of singular boundary value problems can be found in [2125] and references therein. A great deal of the work on singular boundary value problems is mainly concerned with the existence of the solution. It is equally important to construct the solution of the problem once its existence is proved.

The monotone iterative technique is one of the efficient analytic methods for solving nonlinear boundary value problems. This technique coupled with the method of upper and lower solutions [26] manifests itself as an effective and flexible mechanism that offers theoretical as well as constructive existence results in a closed set, generated by the lower and upper solutions. In general, the convergence of the sequence of approximate solutions given by the monotone iterative technique is at most linear. To obtain a sequence of approximate solutions converging quadratically, we use the method of quasilinearization (QSL). The origin of the quasilinearization lies in the theory of dynamic programming [27]. In fact, the quasilinearization technique is a variant of Newton’s method. 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. In view of its diverse applications, the quasilinearization approach is quite elegant and easier for application algorithms. A detailed description of the QSL method can be found in the monograph [28] and a series of papers [2948].

Devi and Vatsala [49] discussed the QSL method for second-order singular boundary value problems with solutions in weighted spaces. Eloe [50] developed the QSL method for singular boundary value problems on an unbounded domain. Ramos [51] discussed piecewise quasilinearization techniques for singular boundary value problems. El-Gebeily and O’Regan [52] studied the QSL method for second-order singular nonlinear differential equations with nonlinear boundary conditions. For some existence results on singular initial and boundary value problems, see [53, 54]. The theoretical background of integro-differential equations can be found in the text by Lakshmikantham and Rao [55]. To the best of our knowledge, the QSL method has not been discussed for singular nonlocal second-order boundary value problems involving nonlinear integro-differential equations on a bounded domain.

In this paper, we consider the following singular boundary value problem (SBVP):

{ ( t n u ) = f ( t , t n 1 u ) + 0 t K ( t , s , s n 1 u ( s ) ) d s , 0 < t < 1 , n > 2 , lim t 0 + t n 1 u ( t ) = u 0 , u ( 1 ) = g ( u ( σ ) ) , 0 < σ < 1 ,
(1.1)

where f:[0,1]×RR, K:[0,1]×[0,1]×RR, and g:RR. In the forthcoming analysis, g(u(σ)) will be written as g(u) for the sake of convenience.

Here we remark that a solution to a singular problem may not lie in the space C[0,1], but it should depend on the singularity in some sense. For example, the solution u=( u 0 + u 1 )+ u 0 / t 2 to the singular problem

( t 3 u ) =0,0<t<1; lim t 0 + t 2 u(t)= u 0 ,u(1)= u 1

( u 0 , u 1 are constants) is not continuous at t=0 but t 2 uC[0,1]. Such a situation provides motivation to consider the singular boundary value problems in a weighted Banach space.

2 Preliminaries and some existence results

In this section, we follow the terminology introduced in [53].

Definition 2.1 A function u C 2 (0,1) with t n 1 uC[0,1], t n u C[0,1] and ( t n 1 u ) L 1 [0,1] is a solution of (1.1) if it satisfies singular boundary value problem (1.1).

Remark 2.2 Whenever we say u(t) is a solution of (1.1), it means that t n 1 u(t) is a solution of (1.1). Similar terminology will be used for lower and upper solutions.

Throughout the forthcoming analysis, we work in a Banach space of functions defined by

E= { u C 1 ( 0 , 1 ) : t n 1 u C [ 0 , 1 ] , t n u C [ 0 , 1 ] , t [ 0 , 1 ] }

with the norm

u=max { sup t [ 0 , 1 ] | t n 1 u ( t ) | , sup t [ 0 , 1 ] | t n u ( t ) | } .

Lemma 2.3 Singular boundary value problem (1.1) is equivalent to the singular integral equation

u ( t ) = g ( u ) u 0 ( 1 1 t n 1 ) + 1 t n 1 ( n 1 ) t n 1 0 t ( f ( s , s n 1 u ( s ) ) + 0 s K ( s , p , p n 1 u ( p ) ) d p ) d s + 1 ( n 1 ) t 1 1 s n 1 s n 1 ( f ( s , s n 1 u ( s ) ) + 0 s K ( s , p , p n 1 u ( p ) ) d p ) d s .
(2.1)

Proof Differentiating (2.1) and rearranging the terms, we have

t n u (t)=(n1) u 0 0 t ( f ( s , s n 1 u ( s ) ) + 0 s K ( s , p , p n 1 u ( p ) ) d p ) ds.

Again differentiating, it follows that

( t n u ) =f ( t , t n 1 u ) + 0 t K ( t , s , s n 1 u ( s ) ) ds.

Multiplying (2.1) by t n 1 and taking the limit t 0 + , we get lim t 0 + t n 1 u(t)= u 0 , and t=1 in (2.1) yields u(1)=g(u). Thus, u(t) defined by (2.1) satisfies (1.1). A straightforward computation shows that (1.1) implies (2.1). This completes the proof. □

Now we prove a general existence principle for singular boundary value problem (1.1). The proof of this principle is based on the Schauder fixed point theorem.

Lemma 2.4 Assume that f:[0,1]×RR, K:[0,1]×[0,1]×RR and g:RR are continuous functions. Furthermore, there exist positive constants M f , M K , M g such that |f(t,u)| M f for (t,u)[0,1]×R, |K(t,s,u)| M K for (t,s,u)[0,1]×[0,1]×R, and |g(u)| M g for uR. Then problem (1.1) has a solution.

Proof We reduce the problem of finding a solution uE of (1.1) to a fixed point problem u=Fu, where F:EE is given by

F u ( t ) = g ( u ) u 0 ( 1 1 t n 1 ) + 1 t n 1 ( n 1 ) t n 1 0 t ( f ( s , s n 1 u ( s ) ) + 0 s K ( s , p , p n 1 u ( p ) ) d p ) d s + 1 ( n 1 ) t 1 1 s n 1 s n 1 ( f ( s , s n 1 u ( s ) ) + 0 s K ( s , p , p n 1 u ( p ) ) d p ) d s .

If u m u in , then

| t n 1 F u m ( t ) t n 1 F u ( t ) | t n 1 | g ( u m ) g ( u ) | + ( 1 t n 1 ) ( n 1 ) 0 t ( | f ( s , s n 1 u m ( s ) ) f ( s , s n 1 u ( s ) ) | + 0 s | K ( s , p , p n 1 u m ( p ) ) K ( s , p , p n 1 u ( p ) ) | d p ) d s + t n 1 ( n 1 ) t 1 ( | f ( s , s n 1 u m ( s ) ) f ( s , s n 1 u ( s ) ) | + 0 s | K ( s , p , p n 1 u m ( p ) ) K ( s , p , p n 1 u ( p ) ) | d p ) d s ,

and

| t n ( F u m ) ( t ) t n ( F u ) ( t ) | 0 t ( | f ( s , s n 1 u m ( s ) ) f ( s , s n 1 u ( s ) ) | + 0 s | K ( s , p , p n 1 u m ( p ) ) K ( s , p , p n 1 u ( p ) ) | d p ) d s .

In view of the continuity of f, K, g, it follows that F u m Fu in and so F:EE is continuous.

Now, for r,t[0,1] with r<t, we have

| t n 1 F u ( t ) r n 1 F u ( r ) | | t n 1 r n 1 | ( M g + | u 0 | ) + 1 ( n 1 ) | t n 1 r n 1 | 0 r ( M f + 0 s M K d p ) d s + ( 1 t n 1 ) ( n 1 ) r t ( M f + 0 s M K d p ) d s + 1 ( n 1 ) | t n 1 r n 1 | t 1 1 s n 1 s n 1 ( M f + 0 s M K d p ) d s + ( 1 t n 1 ) ( n 1 ) r t ( M f + 0 s M K d p ) d s + r n 1 ( n 1 ) r t 1 s n 1 s n 1 ( M f + 0 s M K d p ) d s

and

| r n ( F u ) ( r ) t n ( F u ) ( t ) | r t ( M f + 0 s M K d p ) ds,

which are independent of u. So F:EE is relatively compact. Hence, by the Arzela-Ascoli theorem, is compact on . Thus, by Schauder’s fixed point theorem, has a fixed point in . This completes the proof. □

Definition 2.5 A function β C 2 (0,1) such that t n 1 βC[0,1], t n β C[0,1], ( t n 1 β ) L 1 [0,1] is called an upper solution for (1.1) if

( t n β ) f ( t , t n 1 β ) + 0 t K ( t , s , s n 1 β ( s ) ) d s , 0 < t < 1 , lim t 0 + t n 1 β ( t ) u 0 , β ( 1 ) g ( β ) .

Similarly, a function α C 2 (0,1) such that t n 1 αC[0,1], t n α C[0,1], ( t n 1 α ) L 1 [0,1] is called a lower solution for (1.1) if

( t n α ) f ( t , t n 1 α ) + 0 t K ( t , s , s n 1 α ( s ) ) d s , 0 < t < 1 , lim t 0 + t n 1 α ( t ) u 0 , α ( 1 ) g ( α ) .

Lemma 2.6 Assume that f:[0,1]×RR and K:[0,1]×[0,1]×RR are continuous functions, and g:RR is a Lipschitz function. Suppose that there exist an upper solution β and a lower solution α for (1.1) such that αβ on (0,1), and lim t 0 + t n 1 α(t) u 0 lim t 0 + t n 1 β(t). Then (1.1) has a solution u satisfying the relation t n 1 α(t) t n 1 u(t) t n 1 β(t) for t[0,1].

Proof Consider the modified problem

( t n u ) = F ( t , t n 1 u ) , 0 < t < 1 , lim t 0 + t n 1 u ( t ) = u 0 , u ( 1 ) = g ( u ) ,
(2.2)

where

F(t,v)= { f ( t , t n 1 β ) + 0 t K ( t , s , s n 1 β ( s ) ) d s ( v t n 1 β ) if  v t n 1 β ( t ) , f ( t , v ) + 0 t K ( t , s , v ) d s if  t n 1 α ( t ) v t n 1 β ( t ) , f ( t , t n 1 α ) + 0 t K ( t , s , s n 1 α ( s ) ) d s ( v t n 1 α ) if  v t n 1 α ( t ) .

By Lemma 2.4, problem (2.2) has a solution u. Let us claim that

t n 1 α(t) t n 1 u(t) t n 1 β(t),t[0,1].
(2.3)

If (2.3) is true, then we are done, that is, u is a solution of (1.1). On the contrary, assume that t n 1 u(t) t n 1 β(t) for t[0,1]. Then there exists t 0 (0,1) such that ( t n 1 u(t) t n 1 β(t)) has a positive maximum at t 0 . Thus, ( t n 1 u ( t ) t n 1 β ( t ) ) ( t 0 )=0 and ( t n 1 u ( t ) t n 1 β ( t ) ) ( t 0 )0. On the other hand,

( t n 1 u ( t ) t n 1 β ( t ) ) ( t 0 ) = 1 t 0 ( t 0 n u ( t 0 ) ) + ( n 2 ) t 0 n 2 u ( t 0 ) + ( n 1 ) ( n 2 ) t 0 n 3 u ( t 0 ) 1 t 0 ( t 0 n β ( t 0 ) ) ( n 2 ) t 0 n 2 β ( t 0 ) ( n 1 ) ( n 2 ) t 0 n 3 β ( t 0 ) = 1 t 0 ( t 0 n u ( t 0 ) ) 1 t 0 ( t 0 n β ( t 0 ) ) 1 t 0 ( t 0 n u ( t 0 ) ) + 1 t 0 ( f ( t 0 , t 0 n 1 β ( t 0 ) ) + 0 t 0 K ( t 0 , s , s n 1 β ( s ) ) d s ) = 1 t 0 ( t 0 n 1 u ( t 0 ) t 0 n 1 β ( t 0 ) ) > 0 ,

which is a contradiction. Hence t n 1 u(t) t n 1 β(t) for t(0,1). Similarly, it can be shown that t n 1 α(t) t n 1 u(t) for t(0,1). In view of assumptions that g is a Lipschitz function and αβ on (0,1), it follows that α(1)β(1). Thus we conclude that

t n 1 α(t) t n 1 u(t) t n 1 β(t)for t[0,1].

This completes the proof. □

Lemma 2.7 Let α,βE be lower and upper solutions of (1.1) respectively with

lim t 0 + t n 1 α(t) u 0 lim t 0 + t n 1 β(t).

Further, suppose that f:[0,1]×RR, K:[0,1]×[0,1]×RR and g:RR are continuous functions and satisfy

f ( t , t n 1 x ) f ( t , t n 1 y ) L 1 ( x y ) , L 1 > 0 , K ( t , s , s n 1 x ) K ( t , s , s n 1 y ) L 2 ( x y ) , L 2 > 0 , g ( x ) g ( y ) L 3 ( x y ) , 0 < L 3 < 1 ,

whenever xy. Then t n 1 α(t) t n 1 β(t) on [0,1].

Proof For the sake of contradiction, suppose that the conclusion does not hold, that is, t n 1 α(t)> t n 1 β(t) at some t(0,1]. Then, by continuity, there exists a local maximum at t 0 (0,1). Thus, at t= t 0 , we have

( t n 1 ( α β ) ) =0,
(2.4)
( t n 1 ( α β ) ) 0.
(2.5)

In view of (2.4), (2.5) takes the form

0 [ t n 1 ( α β ) ] = [ ( n 1 ) t n 2 ( α β ) + t n 1 ( α β ) ] = ( n 1 ) ( n 2 ) t n 3 ( α β ) + ( n 1 ) t n 2 ( α β ) + ( n 1 ) t n 2 ( α β ) + t n 1 ( α β ) = ( n 1 ) ( n 2 ) t n 3 ( α β ) + 2 ( n 1 ) t n 2 ( α β ) + t n 1 ( α β ) = ( n 1 ) ( n 2 ) t n 3 ( α β ) + ( 2 n 2 ) t n 2 ( α β ) + t n 1 ( α β ) = ( n 1 ) ( n 2 ) t n 3 ( α β ) + ( n 2 ) t n 2 ( α β ) + n t n 2 α n t n 2 β + t n 1 ( α β ) = n 2 t [ ( n 1 ) t n 2 ( α β ) + t n 1 ( α β ) ] + n t n 2 α n t n 2 β + t n 1 α t n 1 β = n 2 t [ t n 1 ( α β ) ] + n t n 2 α + t n 1 α n t n 2 β t n 1 β = n 2 t ( 0 ) + n t n 2 α + t n 1 α n t n 2 β t n 1 β = 1 t ( t n α ) 1 t ( t n β ) ,

which can alternatively be written as t 1 ( t n α ) t 1 ( t n β ) at t= t 0 (0,1). Using the definition of lower and upper solutions together with (2.5), we obtain

f ( t 0 , t 0 n 1 α ) + 0 t 0 K ( t 0 , s , s n 1 α ( s ) ) d s ( t 0 n α ) ( t 0 n β ) f ( t , t 0 n 1 β ) + 0 t 0 K ( t 0 , s , s n 1 β ( s ) ) d s ,

which yields a contradiction:

0 f ( t 0 , t 0 n 1 α ) f ( t , t 0 n 1 β ) + 0 t 0 [ K ( t 0 , s , s n 1 α ( s ) ) K ( t 0 , s , s n 1 β ( s ) ) ] d s ( L 1 + t 0 L 2 ) t 0 n 1 ( α β ) < 0 .

Hence we obtain t n 1 α(t) t n 1 β(t) on (0,1). As 0<σ<1, therefore, in view of the foregoing arguments, at t 0 =1, we have

0 < α ( 1 ) β ( 1 ) g ( α ) g ( β ) L 3 σ n 1 ( σ n 1 α ( σ ) σ n 1 β ( σ ) ) 0 , 0 < σ < 1 ,

which is a contradiction. This completes the proof. □

3 Main result

Theorem 3.1 Assume that

(A1) α 0 , β 0 E are lower and upper solutions of (1.1) respectively such that t n 1 α 0 (t) t n 1 β 0 (t) for t(0,1] with lim t 0 + t n 1 α 0 (t) u 0 lim t 0 + t n 1 β 0 (t);

(A2) f(t,v) is convex, that is, f v v (t,v) exists, continuous with f v v (t,v)0 for each (t,v)(0,1]×R, and f v (t, t n 1 u)<0, where t n 1 α 0 v t n 1 β 0 ;

(A3) K(t,s,v) is convex, that is, K v v (t,s,v) exists, continuous with K v v (t,s,v)0 for each (t,s,v)(0,1]×(0,1]×R, and K v (t,s, s n 1 u)<0, where t n 1 α 0 v t n 1 β 0 ;

(A4) g is continuous on such that g , g exist and 0 g <1, g 0.

Then there exist monotone sequences { t n 1 α k (t)} and { t n 1 β k (t)} that converge uniformly and quadratically in the space of continuous functions on [0,1] to a unique solution t n 1 u(t) of (1.1).

Proof Using the generalized mean value theorem together with (A2), (A3), and (A4), we obtain

f ( t , t n 1 u ) f ( t , t n 1 w ) + f v ( t , t n 1 w ) (uw) t n 1 ,uw,
(3.1)
K ( t , s , s n 1 u ) K ( t , s , s n 1 w ) + k v ( t , s , s n 1 w ) (uw) s n 1 ,uw,
(3.2)
g(u)g(w)+ g (w)(uw).
(3.3)

Now, we set

F ( t , t n 1 α 1 ; α 0 ) = f ( t , t n 1 α 0 ) + f v ( t , t n 1 α 0 ) ( α 1 α 0 ) t n 1 , F ¯ ( t , t n 1 β 1 ; α 0 , β 0 ) = f ( t , t n 1 β 0 ) + f v ( t , t n 1 α 0 ) ( β 1 β 0 ) t n 1 , K ( t , s , s n 1 α 1 ; α 0 ) = K ( t , s , s n 1 α 0 ) + K v ( t , s , s n 1 α 0 ) ( α 1 α 0 ) s n 1 , K ( t , s , s n 1 β 1 ; α 0 , β 0 ) = K ( t , s , s n 1 β 0 ) + K v ( t , s , s n 1 α 0 ) ( β 1 β 0 ) s n 1 , h 1 ( α 1 ; α 0 , β 0 ) = g ( α 0 ) + g ( β 0 ) ( α 1 α 0 ) , h 2 ( β 1 ; β 0 ) = g ( β 0 ) + g ( β 0 ) ( β 1 β 0 ) .

Consider the singular BVP

{ ( t n α 1 ) = F ( t , t n 1 α 1 ; α 0 ) + 0 t K ( t , s , s n 1 α 1 ; α 0 ) d s , 0 < t < 1 , lim t 0 + t n 1 α 1 ( t ) = u 0 , α 1 ( 1 ) = h 1 ( α 1 ; α 0 , β 0 ) .
(3.4)

Using assumption (A1), (3.1) and (3.2), we obtain

( t n α 0 ) f ( t , t n 1 α 0 ) + 0 t K ( t , s , s n 1 α 0 ( s ) ) d s = F ( t , t n 1 α 0 ; α 0 ) + 0 t K ( t , s , s n 1 α 0 ; α 0 ) d s

and

( t n β 0 ) f ( t , t n 1 β 0 ) + 0 t K ( t , s , s n 1 β 0 ) d s f ( t , t n 1 α 0 ) + f v ( t , t n 1 α 0 ) ( β 0 α 0 ) t n 1 + 0 t [ K ( t , s , s n 1 α 0 ) + k v ( t , s , s n 1 α 0 ) ( β 0 α 0 ) s n 1 ] d s = F ( t , t n 1 β 0 ; α 0 ) + 0 t K ( t , s , s n 1 β 0 ; α 0 ) d s .

Further, we note that α 0 (1)g( α 0 )= h 1 ( α 0 ; α 0 , β 0 ) and using assumption (A4), for c 0 ( α 0 , β 0 ), we find that

g ( β 0 ) h 1 ( β 0 ; α 0 , β 0 ) = g ( β 0 ) g ( α 0 ) g ( β 0 ) ( β 0 α 0 ) = [ g ( c 0 ) g ( β 0 ) ] ( β 0 α 0 ) 0 ,

which implies that β 0 (1)g( β 0 ) h 1 ( β 0 ; α 0 , β 0 ). Thus, it follows that α 0 and β 0 are respectively lower and upper solutions of (3.4). Since h i = g (i=1,2), therefore it follows by Lemmas 2.6 and 2.7 that there exists a unique solution α 1 of (3.4) satisfying

t n 1 α 0 (t) t n 1 α 1 (t) t n 1 β 0 (t),t[0,1].

Observe that the hypotheses of Lemma 2.7 hold in view of the conditions f v <0, K v <0, and 0 g <1, demanded in (A2), (A3), and (A4).

Next, consider the singular BVP

{ ( t n β 1 ) = F ¯ ( t , t n 1 β 1 ; α 0 , β 0 ) + 0 t K ( t , s , s n 1 β 1 ; α 0 , β 0 ) d s , 0 < t < 1 , lim t 0 + t n 1 β 1 ( t ) = u 0 , β 1 ( 1 ) = h 2 ( β 1 ; β 0 ) .
(3.5)

Using the definition of the lower and upper solutions, (3.1) and (3.2), we get

( t n α 0 ) f ( t , t n 1 α 0 ) + 0 t K ( t , s , s n 1 α 0 ( s ) ) d s f ( t , t n 1 β 0 ) + f v ( t , t n 1 α 0 ) ( α 0 β 0 ) t n 1 + 0 t [ K ( t , s , s n 1 β 0 ) + K v ( t , s , s n 1 α 0 ) ( α 0 β 0 ) s n 1 ] d s = F ¯ ( t , t n 1 α 0 ; α 0 , β 0 ) + 0 t K ( t , s , s n 1 α 0 ; α 0 , β 0 ) d s , ( t n β 0 ) f ( t , t n 1 β 0 ) + 0 t K ( t , s , s n 1 β 0 ) d s = F ¯ ( t , t n 1 β 0 ; α 0 , β 0 ) + 0 t K ( t , s , s n 1 β 0 ; α 0 , β 0 ) d s .

By assumption (A4), we have α 0 (1) h 2 ( α 0 ; β 0 ) as

g ( α 0 ) h 2 ( α 0 ; β 0 ) = g ( α 0 ) g ( β 0 ) + g ( β 0 ) ( β 0 α 0 ) = [ g ( c 1 ) + g ( β 0 ) ] ( β 0 α 0 ) 0 , c 1 ( α 0 , β 0 ) ,

and β 0 (1)g( β 0 )= h 2 ( β 0 ; β 0 ). In view of the above inequalities, we find that α 0 and β 0 are respectively lower and upper solutions of (3.5). Therefore it follows by Lemmas 2.6 and 2.7 that there exists a unique solution β 1 of (3.5) satisfying

t n 1 α 0 (t) t n 1 β 1 (t) t n 1 β 0 (t),t[0,1].

Now we show that t n 1 α 1 (t) t n 1 β 1 (t). Using (3.1), (3.2), and assumption (A1), we obtain

( t n α 1 ) = F ( t , t n 1 α 1 ; α 0 ) + 0 t K ( t , s , s n 1 α 1 ; α 0 ) d s = f ( t , t n 1 α 0 ) + f v ( t , t n 1 α 0 ) ( α 1 α 0 ) t n 1 + 0 t [ K ( t , s , s n 1 α 0 ) + K v ( t , s , s n 1 α 0 ) ( α 1 α 0 ) s n 1 ] d s f ( t , t n 1 β 0 ) + f v ( t , t n 1 α 0 ) ( α 0 β 0 ) t n 1 + f v ( t , t n 1 α 0 ) ( α 1 α 0 ) t n 1 + 0 t [ K ( t , s , s n 1 β 0 ) + K v ( t , s , s n 1 α 0 ) ( α 0 β 0 ) s n 1 + K v ( t , s , s n 1 α 0 ) ( α 1 α 0 ) s n 1 ] d s = f ( t , t n 1 β 0 ) + f v ( t , t n 1 α 0 ) ( α 1 β 0 ) t n 1 + 0 t [ K ( t , s , s n 1 β 0 ) + K v ( t , s , s n 1 α 0 ) ( α 1 β 0 ) s n 1 ] d s = F ¯ ( t , t n 1 α 1 ; α 0 , β 0 ) + 0 t K ( t , s , s n 1 α 1 ; α 0 , β 0 ) d s ,

and in view of (A4), we get g( α 1 ) h 2 ( α 1 ; β 0 )0, which implies that α 1 (1) h 2 ( α 1 ; β 0 ). Clearly, the above inequalities and singular boundary value problem (3.5) satisfy the hypotheses of Lemma 2.7. Therefore, by the conclusion of Lemma 2.7, we have t n 1 α 1 (t) t n 1 β 1 (t), t[0,1]. Consequently, we obtain

t n 1 α 0 (t) t n 1 α 1 (t) t n 1 β 1 (t) t n 1 β 0 (t),t[0,1].

As a next step, we prove that

t n 1 α k (t) t n 1 α k + 1 (t) t n 1 β k + 1 (t) t n 1 β k (t),t[0,1],

for k>1. For that, we consider the following SBVP:

{ ( t n α k + 1 ) = F ( t , t n 1 α k + 1 ; α k ) + 0 t K ( t , s , s n 1 α k + 1 ; α k ) d s , 0 < t < 1 , lim t 0 + t n 1 α k + 1 ( t ) = u 0 , α k + 1 ( 1 ) = h 1 ( α k + 1 ; α k , β 0 ) .
(3.6)

Using (3.1), (3.2), (3.3), and the inequality t n 1 α k 1 t n 1 α k , we get

( t n α k ) = F ( t , t n 1 α k ; α k 1 ) + 0 t K ( t , s , s n 1 α k ; α k 1 ) d s = f ( t , t n 1 α k 1 ) + f v ( t , t n 1 α k 1 ) ( α k α k 1 ) t n 1 + 0 t [ K ( t , s , s n 1 α k 1 ) + K v ( t , s , s n 1 α k 1 ) ( α k α k 1 ) s n 1 ] d s f ( t , t n 1 α k ) f v ( t , t n 1 α k 1 ) ( α k α k 1 ) t n 1 + f v ( t , t n 1 α k 1 ) ( α k α k 1 ) t n 1 + 0 t [ K ( t , s , s n 1 α k ) K v ( t , s , s n 1 α k 1 ) ( α k α k 1 ) s n 1 + K v ( t , s , s n 1 α k 1 ) ( α k α k 1 ) s n 1 ] d s = f ( t , t n 1 α k ) + 0 t K ( t , s , s n 1 α k ) d s = F ( t , t n 1 α k ; α k ) + 0 t K ( t , s , s n 1 α k ; α k ) d s , α k ( 1 ) g ( α k ) = h 1 ( α k , α k ; β 0 ) .

Since f v (t,v), k v (t,s,v) are increasing in v by assumptions (A2) and (A3), therefore, f v (t, t n 1 α 0 ) f v (t, t n 1 α k ), K v (t,s, s n 1 α 0 ) K v (t,s, s n 1 α k ) for t n 1 α 0 t n 1 α k . Consequently, in view of (3.1), (3.2), (3.3), we obtain the inequality

( t n β k ) = F ¯ ( t , t n 1 β k ; α 0 , β k 1 ) + 0 t K ( t , s , s n 1 β k ; α 0 , β k 1 ) d s = f ( t , t n 1 β k 1 ) + f v ( t , t n 1 α 0 ) ( β k β k 1 ) t n 1 + 0 t [ K ( t , s , s n 1 β k 1 ) + K v ( t , s , s n 1 α 0 ) ( β k β k 1 ) s n 1 ] d s f ( t , t n 1 α k ) + f v ( t , t n 1 α k ) ( β k 1 α k ) t n 1 f v ( t , t n 1 α 0 ) ( β k 1 β k ) t n 1 + 0 t [ K ( t , s , s n 1 α k ) + K v ( t , s , s n 1 α k ) ( β k 1 α k ) s n 1 K v ( t , s , s n 1 α 0 ) ( β k 1 β k ) s n 1 ] d s f ( t , t n 1 α k ) + f v ( t , t n 1 α k ) ( β k 1 α k ) t n 1 f v ( t , t n 1 α k ) ( β k 1 β k ) t n 1 + 0 t [ K ( t , s , s n 1 α k ) + K v ( t , s , s n 1 α k ) ( β k 1 α k ) s n 1 K v ( t , s , s n 1 α k ) ( β k 1 β k ) s n 1 ] d s = f ( t , t n 1 α k ) + f v ( t , t n 1 α k ) ( β k α k ) t n 1 + 0 t [ K ( t , s , s n 1 α k ) + K v ( t , s , s n 1 α k ) ( β k α k ) s n 1 ] d s = F ( t , t n 1 β k ; α k ) + 0 t K ( t , s , s n 1 β k ; α k ) d s ,

and by virtue of the inequality

g ( β k ) h 1 ( β k , α k ; β 0 ) = g ( β k ) g ( α k ) g ( β 0 ) ( β k α k ) = [ g ( c 2 ) g ( β 0 ) ] ( β k α k ) 0 , α k c 2 β k β 0 ,

we get β k (1)g( β k )= h 1 ( β k , α k ; β 0 ). Thus, as argued earlier, there exists a unique solution α k + 1 of (3.6) such that

t n 1 α k (t) t n 1 α k + 1 (t) t n 1 β k (t),t[0,1].

Now let us consider the following SBVP:

{ ( t n β k + 1 ) = F ¯ ( t , t n 1 β k + 1 ; α 0 , β k ) ( t n β k + 1 ) = + 0 t K ( t , s , s n 1 β k + 1 ; α 0 , β k ) d s , 0 < t < 1 , lim t 0 + t n 1 β k + 1 ( t ) = u 0 , β k + 1 ( 1 ) = h 2 ( β k + 1 , β k ) .
(3.7)

Again, using assumptions (A2)-(A4), (3.1), (3.2), (3.3), and the inequality

t n 1 α 0 t n 1 α k 1 t n 1 α k t n 1 β k t n 1 β k 1 t n 1 β 0 ,

we obtain

( t n β k ) = F ¯ ( t , t n 1 β k ; α 0 , β k 1 ) + 0 t K ( t , s , s n 1 β k ; α 0 , β k 1 ) d s = f ( t , t n 1 β k 1 ) + f v ( t , t n 1 α 0 ) ( β k β k 1 ) t n 1 + 0 t [ K ( t , s , s n 1 β k 1 ) + K v ( t , s , s n 1 α 0 ) ( β k β k 1 ) s n 1 ] d s f ( t , t n 1 β k ) + f v ( t , t n 1 β k ) ( β k 1 β k ) t n 1 f v ( t , t n 1 α 0 ) ( β k 1 β k ) t n 1 + 0 t [ K ( t , s , s n 1 β k ) + K v ( t , s , s n 1 β k ) ( β k 1 β k ) s n 1 K v ( t , s , s n 1 α 0 ) ( β k 1 β k ) s n 1 ] d s f ( t , t n 1 β k ) + 0 t K ( t , s , s n 1 β k ) d s = F ¯ ( t , t n 1 β k ; α 0 , β k ) + 0 t K ( t , s , s n 1 β k ; α 0 , β k ) d s , β k ( 1 ) g ( β k ) = h 2 ( β k , β k ) , ( t n α k ) = F ( t , t n 1 α k ; α k 1 ) + 0 t K ( t , s , s n 1 α k ; α k 1 ) d s = f ( t , t n 1 α k 1 ) + f v ( t , t n 1 α k 1 ) ( α k α k 1 ) t n 1 + 0 t [ K ( t , s , s n 1 α k 1 ) + K v ( t , s , s n 1 α k 1 ) ( α k α k 1 ) s n 1 ] d s f ( t , t n 1 β k ) f v ( t , t n 1 α k 1 ) ( β k α k 1 ) t n 1 + f v ( t , t n 1 α k 1 ) ( α k α k 1 ) t n 1 + 0 t [ K ( t , s , s n 1 β k ) K v ( t , s , s n 1 α k 1 ) ( β k α k 1 ) s n 1 + K v ( t , s , s n 1 α k 1 ) ( α k α k 1 ) s n 1 ] d s = f ( t , t n 1 β k ) f v ( t , t n 1 α k 1 ) ( β k α k ) t n 1 + 0 t [ K ( t , s , s n 1 β k ) K v ( t , s , s n 1 α k 1 ) ( β k α k ) s n 1 ] d s f ( t , t n 1 β k ) + f v ( t , t n 1 α 0 ) ( α k β k ) t n 1 + 0 t [ K ( t , s , s n 1 β k ) + K v ( t , s , s n 1 α 0 ) ( α k β k ) s n 1 ] d s = F ¯ ( t , t n 1 α k ; α 0 , β k ) + 0 t K ( t , s , s n 1 α k ; α 0 , β k ) d s , α k ( 1 ) = g ( α k ) h 2 ( α k , α k 1 ) .

Applying the earlier arguments, it follows that there exists a unique solution β k + 1 of (3.7) such that

t n 1 α k (t) t n 1 β k + 1 (t) t n 1 β k (t),t[0,1].

Following the procedure employed to prove t n 1 α 1 (t) t n 1 β 1 (t), it can be shown that t n 1 α k + 1 (t) t n 1 β k + 1 (t). Hence we have

t n 1 α 0 ( t ) t n 1 α 1 ( t ) t n 1 α k + 1 ( t ) t n 1 β k + 1 ( t ) t n 1 β 1 ( t ) t n 1 β 0 ( t ) , t [ 0 , 1 ] .
(3.8)

As the monotone sequences { t n 1 α k + 1 (t)} and { t n 1 β k + 1 (t)} are both bounded, therefore, they converge to the limit functions t n 1 ρ(t) and t n 1 σ(t) pointwise respectively.

Now we show that the convergence of the sequences { t n 1 α k + 1 (t)} and { t n 1 β k + 1 (t)} to the limit functions { t n 1 ρ(t)} and { t n 1 σ(t)} respectively is indeed uniform. Using SBVP (3.6), integral equation (2.1), and relation (3.8), we observe that { t n 1 α k + 1 (t)}, { ( t n 1 α k + 1 ( t ) ) }, { t n α k + 1 (t)}, { ( t n α k + 1 ( t ) ) } are uniformly bounded sequences. Thus, by the Arzela-Ascoli theorem, the sequences { t n 1 α k + 1 (t)}, { t n α k + 1 (t)} have uniformly convergent subsequences. Hence, by the monotonicity of the sequence { t n 1 α k + 1 (t)} in , it follows that the sequence converges uniformly to the limit function t n 1 ρ(t) in . In a similar manner, the sequence { t n 1 β k + 1 (t)} in converges uniformly to the limit function t n 1 σ(t) in .

Finally, we show that the convergence of the sequences is quadratic. We only prove the quadratic convergence for the sequence { t n 1 α k + 1 (t)} as that of the sequence { t n 1 β k + 1 (t)} follows a similar procedure.

Let us define

e k + 1 (t)=ρ(t) α k + 1 (t).

Then, for t n 1 ζ 1 ( t n 1 α k , t n 1 ρ), s n 1 ζ 2 ( s n 1 α k , s n 1 ρ), t n 1 ζ 3 ( t n 1 α k , t n 1 ζ 1 ), s n 1 ζ 4 ( s n 1 α k , s n 1 ζ 2 ), we have

( t n α k + 1 ( t ) ) = ( t n ρ ( t ) ) + ( t n α k + 1 ( t ) ) = f ( t , t n 1 ρ ( t ) ) + 0 t K ( t , s , s n 1 ρ ( s ) ) d s f ( t , t n 1 α k ) f v ( t , t n 1 α k ) ( α k + 1 α k ) t n 1 0 t [ K ( t , s , s n 1 α k ) + K v ( t , s , s n 1 α k ) ( α k + 1 α k ) s n 1 ] d s = f v ( t , t n 1 ζ 1 ) ( ρ α k ) t n 1 f v ( t , t n 1 α k ) ( α k + 1 α k ) t n 1 + 0 t [ K v ( t , s , s n 1 ζ 2 ) ( ρ α k ) s n 1 K v ( t , s , s n 1 α k ) ( α k + 1 α k ) s n 1 ] d s = [ f v ( t , t n 1 ζ 1 ) f v ( t , t n 1 α k ) ] ( ρ α k ) t n 1 + f v ( t , t n 1 α k ) ( ρ α k + 1 ) t n 1 + 0 t [ { K v ( t , s , s n 1 ζ 2 ) K v ( t , s , s n 1 α k ) } ( ρ α k ) s n 1 + K v ( t , s , s n 1 α k ) ( ρ α k + 1 ) s n 1 ] d s = f v v ( t , t n 1 ζ 3 ) ( ζ 1 α k ) ( ρ α k ) ( t n 1 ) 2 + f v ( t , t n 1 α k ) ( ρ α k + 1 ) t n 1 + 0 t [ K v v ( t , s , s n 1 ζ 4 ) ( ζ 2 α k ) ( ρ α k ) ( s n 1 ) 2 + K v ( t , s , s n 1 α k ) ( ρ α k + 1 ) s n 1 ] d s f v v ( t , t n 1 ζ 3 ) ( ( ρ α k ) t n 1 ) 2 + f v ( t , t n 1 α k ) ( ρ α k + 1 ) t n 1 + 0 t [ K v v ( t , s , s n 1 ζ 4 ) ( ( ρ α k ) s n 1 ) 2 + K v ( t , s , s n 1 α k ) ( ρ α k + 1 ) s n 1 ] d s .

Setting N= N 1 + N 2 , M= M 1 + M 2 , where N 1 and N 2 provide bounds for f v v (t,) ( t n 1 ) 2 and 0 t [ K v v (t,s,) ( s n 1 ) 2 ]ds respectively and f v (t,) M 1 , 0 t K v (t,s,)ds M 2 , the above inequality takes the form

( t n α k + 1 ( t ) ) N | e k | 2 M e k + 1 ( t ) t n 1 .
(3.9)

Given M, N and | e k | 2 , there exists μ R + such that

0=N | e k | 2 Mμ t n 1 .
(3.10)

By a comparison theorem [56], it follows from (3.9) and (3.10) that

e k + 1 (t) t n 1 μ t n 1 = N M | e k | 2 ,t[0,1),

which implies that

| e k + 1 | N M | e k | 2 .
(3.11)

Now, we consider

e k + 1 ( 1 ) = ρ ( 1 ) α k + 1 ( 1 ) = g ( ρ ) h 1 ( α k + 1 , α k , β 0 ) = g ( ρ ) g ( α k ) g ( β 0 ) ( α k + 1 α k ) = [ g ( η 1 ) g ( β 0 ) ] ( ρ α k ) + g ( β 0 ) ( ρ α k + 1 ) [ g ( α k ) g ( ρ ) ] ( ρ α k ) + g ( β 0 ) ( ρ α k + 1 ) = g ( η 2 ) | e k | 2 + g ( β 0 ) e k + 1 ,

where η 1 , η 2 ( α k ,ρ). Letting g () λ 2 on ( α k ,ρ) and g () λ 1 <1, we find that

| e k + 1 | λ 2 1 λ 1 | e k | 2 .
(3.12)

From (3.11) and (3.12), we conclude that the sequence { t n 1 α k + 1 (t)} converges to the unique solution of SBVP (1.1) quadratically. Similarly, we can prove the quadratic convergence of the sequence { t n 1 β k + 1 (t)}. This completes the proof. □

Example Consider the following singular nonlocal boundary value problem:

{ ( t 3 u ) = A ( 1 + t ) e t 2 u ( t ) + 1 ( t 3 u ) = + 0 t [ B ( 2 t ) s 2 e s 2 u ( s ) 1 C s 2 u ( s ) ] d s , 0 < t < 1 , lim t 0 + t 2 u ( t ) = 0 , u ( 1 ) = 1 + 1 2 u ( 1 2 ) ,
(3.13)

where f(t, t 2 u(t))=A(1+t) e t 2 u ( t ) + 1 , K(t,s,u(s))=[B(2t) s 2 e s 2 u ( s ) 1 C s 2 u(s)], n=3, g(u(1/2))=1+ 1 2 u( 1 2 ), and A, B, C are suitable positive constants. Let α(t)=0 and β(t)=2t be respectively lower and upper solutions of (3.13). Clearly, α(t) and β(t) are not the solutions of (3.13) and α(t)β(t), t[0,1]. Moreover, assumptions (A1), (A2), (A3), and (A4) of Theorem 3.1 are satisfied. Thus, the conclusion of Theorem 3.1 applies to problem (3.13).

4 Conclusions

In this paper, we have presented monotone sequences of approximate solutions converging uniformly and quadratically to a unique solution of a nonlocal singular boundary value problem involving second-order integro-differential equations in weighted spaces. The results established in this project are new and contribute to the present theory of singular boundary value problems of integro-differential equations. The present work provides a guideline to extend it further by relaxing the convexity assumptions on the nonlinear functions f(t,v) and K(t,s,v) in (1.1). In fact, we can find a continuous function ϕ(t,v) such that f v v (t,v)+ ϕ v v (t,v)0, where ϕ v v (t,v) exists, continuous with ϕ v v (t,v)0. In a similar manner, the convexity assumption on K(t,s,v) and the concavity assumption on g(u) can be relaxed. Further, for g(u)= u 1 and K(t,s,v)0 in (1.1), our results become the existence results obtained in [49]. The results for a nonlocal singular boundary value problem of second-order integro-differential equations involving a purely integral type of nonlinearity follow by taking f(t,v)0 in (1.1). Thus, the work presented in this paper takes care of numerous interesting situations.

References

  1. Castro A, Kurepa A: Energy analysis of a nonlinear singular equation and applications. Rev. Colomb. Mat. 1988, 21: 155-166.

    MathSciNet  Google Scholar 

  2. Kurepa A: Existence and uniqueness theorem for singular initial value problem and applications. Publ. Inst. Math. (Belgr.) 1990, 45: 89-93.

    MathSciNet  Google Scholar 

  3. Agarwal RP, O’Regan D: Infinite interval problems modeling phenomena which arise in the theory of plasma and electrical potential theory. Stud. Appl. Math. 2003, 111: 339-358. 10.1111/1467-9590.t01-1-00237

    Article  MathSciNet  Google Scholar 

  4. Shin JY: A singular nonlinear boundary value problem in the nonlinear circular membrane under normal pressure. J. Korean Math. Soc. 1995, 32: 761-773.

    MathSciNet  Google Scholar 

  5. Baxley J: A singular nonlinear boundary value problem: membrane response of a spherical cap. SIAM J. Appl. Math. 1988, 48: 497-505. 10.1137/0148028

    Article  MathSciNet  Google Scholar 

  6. Agarwal RP, O’Regan D, Lakshmikantham V: Existence criteria for singular boundary value problems modelling the membrane response of a spherical cap. Nonlinear Anal., Real World Appl. 2003, 4: 223-244. 10.1016/S1468-1218(02)00007-X

    Article  MathSciNet  Google Scholar 

  7. Johnson KN: Circularly symmetric deformations of shallow elastic membrane caps. Q. Appl. Math. 1997, 55: 537-550.

    Google Scholar 

  8. Agarwal RP, O’Regan D: Boundary value problems on the half line in the theory of colloids. Math. Probl. Eng. 2002, 8: 143-150. 10.1080/10241230212905

    Article  MathSciNet  Google Scholar 

  9. Soewona E, Vajravelu K, Mohapatra RN: Existence of solutions of a nonlinear boundary value problem arising in the flow and heat transfer over a stretching sheet. Nonlinear Anal. 1992, 18: 93-98. 10.1016/0362-546X(92)90049-K

    Article  MathSciNet  Google Scholar 

  10. Bernis F, Peletier LA: Two problems from draining flows involving third order ordinary differential equations. SIAM J. Math. Anal. 1996, 27: 515-527. 10.1137/S0036141093260847

    Article  MathSciNet  Google Scholar 

  11. Wang JY, Zhang ZX: A boundary value problem from draining and coating flows involving a third order differential equation. Z. Angew. Math. Phys. 1998, 49: 506-513. 10.1007/s000000050104

    Article  MathSciNet  Google Scholar 

  12. Agarwal RP, O’Regan D: Infinite interval problems modeling the flow of a gas through a semi-infinite porous medium. Stud. Appl. Math. 2002, 108: 245-257. 10.1111/1467-9590.01411

    Article  MathSciNet  Google Scholar 

  13. Agarwal RP, O’Regan D: A singular Homann differential equation. Z. Angew. Math. Mech. 2003, 83: 344-350. 10.1002/zamm.200310048

    Article  Google Scholar 

  14. Wang J, Gao W, Zhang Z: Singular nonlinear boundary value problems arising in boundary layer theory. J. Math. Anal. Appl. 1999, 233: 246-256. 10.1006/jmaa.1999.6290

    Article  MathSciNet  Google Scholar 

  15. Nachman A, Callegari A: A nonlinear singular boundary value problem in the theory of pseudoplastic fluids. SIAM J. Appl. Math. 1980, 38: 275-282. 10.1137/0138024

    Article  MathSciNet  Google Scholar 

  16. O’Regan D: Theory of Singular Boundary Value Problems. World Scientific, Singapore; 1994.

    Book  Google Scholar 

  17. O’Regan D: Existence Theory for Nonlinear Ordinary Differential Equations. Kluwer Academic, Dordrecht; 1997.

    Book  Google Scholar 

  18. Kiguradze IT, Lomtatidze AG: On certain boundary value problems for second-order linear ordinary differential equations with singularities. J. Math. Anal. Appl. 1984, 101: 325-347. 10.1016/0022-247X(84)90107-0

    Article  MathSciNet  Google Scholar 

  19. Lomtatidze AG: A boundary value problem for second-order nonlinear ordinary differential equations with singularities. Differ. Equ. 1986, 22: 416-426.

    MathSciNet  Google Scholar 

  20. Lomtatidze AG: Positive solutions of boundary value problems for second-order ordinary differential equations with singularities. Differ. Equ. 1987, 23: 1146-1152.

    MathSciNet  Google Scholar 

  21. Agarwal RP, O’Regan D: Positive solutions to superlinear singular boundary value problems. J. Comput. Appl. Math. 1998, 88: 129-147. 10.1016/S0377-0427(97)00205-7

    Article  MathSciNet  Google Scholar 

  22. O’Regan D: Singular second order boundary value problems. Nonlinear Anal. 2000, 15: 1097-1109.

    Article  Google Scholar 

  23. Hao Z-C, Liang J, Xiao T-J: Positive solutions of operator equations on half-line. J. Math. Anal. Appl. 2006, 314: 423-435. 10.1016/j.jmaa.2005.04.004

    Article  MathSciNet  Google Scholar 

  24. Liu L, Zhang X, Wu Y: Existence of positive solutions for singular higher-order differential equations. Nonlinear Anal. 2008, 68: 3948-3961. 10.1016/j.na.2007.04.032

    Article  MathSciNet  Google Scholar 

  25. Wang Y, Liu L, Wu Y: Positive solutions of singular boundary value problems on the half-line. Appl. Math. Comput. 2008, 197: 789-796. 10.1016/j.amc.2007.08.013

    Article  MathSciNet  Google Scholar 

  26. Ladde GS, Lakshmikantham V, Vatsala AS: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston; 1985.

    Google Scholar 

  27. Bellman R, Kalaba R: Quasilinearization and Nonlinear Boundary Value Problems. Am. Elsevier, New York; 1965.

    Google Scholar 

  28. Lakshmikantham V, Vatsala AS Mathematics and Its Applications 440. In Generalized Quasilinearization for Nonlinear Problems. Kluwer Academic, Dordrecht; 1998.

    Chapter  Google Scholar 

  29. Mandelzweig VB, Tabakin F: Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs. Comput. Phys. Commun. 2001, 141: 268-281. 10.1016/S0010-4655(01)00415-5

    Article  MathSciNet  Google Scholar 

  30. Eloe P, Gao Y: The method of quasilinearization and a three-point boundary value problem. J. Korean Math. Soc. 2002, 39: 319-330.

    Article  MathSciNet  Google Scholar 

  31. Cabada A, Nieto JJ: Quasilinearization and rate of convergence for higher order nonlinear periodic boundary value problems. J. Optim. Theory Appl. 2001, 108: 97-107. 10.1023/A:1026413921997

    Article  MathSciNet  Google Scholar 

  32. Nikolov S, Stoytchev S, Torres A, Nieto JJ: Biomathematical modeling and analysis of blood flow in an intracranial aneurysms. Neurol. Res. 2003, 25: 497-504. 10.1179/016164103101201724

    Article  Google Scholar 

  33. Ahmad B: A quasilinearization method for a class of integro-differential equations with mixed nonlinearities. Nonlinear Anal., Real World Appl. 2006, 7: 997-1004. 10.1016/j.nonrwa.2005.09.003

    Article  MathSciNet  Google Scholar 

  34. Krivec R, Mandelzweig VB: Quasilinearization method and WKB. Comput. Phys. Commun. 2006, 174: 119-126. 10.1016/j.cpc.2004.12.017

    Article  MathSciNet  Google Scholar 

  35. Akyildiz FT, Vajravelu K: Existence, uniqueness, and quasilinearization results for nonlinear differential equations arising in viscoelastic fluid flow. Differ. Equ. Nonlinear Mech. 2006., 2006: Article ID 71717

    Google Scholar 

  36. Amster P, De Napoli P: A quasilinearization method for elliptic problems with a nonlinear boundary condition. Nonlinear Anal. 2007, 66: 2255-2263. 10.1016/j.na.2006.03.016

    Article  MathSciNet  Google Scholar 

  37. Pei M, Chang SK: A quasilinearization method for second-order four-point boundary value problems. Appl. Math. Comput. 2008, 202: 54-66. 10.1016/j.amc.2008.01.026

    Article  MathSciNet  Google Scholar 

  38. Pei M, Chang SK: A quasilinearization method for second-order four-point boundary value problems. Appl. Math. Comput. 2008, 202: 54-66. 10.1016/j.amc.2008.01.026

    Article  MathSciNet  Google Scholar 

  39. Ahmad B, Nieto J: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions. Nonlinear Anal. 2008, 69: 3291-3298. 10.1016/j.na.2007.09.018

    Article  MathSciNet  Google Scholar 

  40. Ahmad B, Alghamdi B: Approximation of solutions of the nonlinear Duffing equation involving both integral and non-integral forcing terms with separated boundary conditions. Comput. Phys. Commun. 2008, 179: 409-416. 10.1016/j.cpc.2008.04.008

    Article  MathSciNet  Google Scholar 

  41. Ahmad B, Alsaedi A: Existence of approximate solutions of the forced Duffing equation with discontinuous type integral boundary conditions. Nonlinear Anal., Real World Appl. 2009, 10: 358-367. 10.1016/j.nonrwa.2007.09.004

    Article  MathSciNet  Google Scholar 

  42. Nieto JJ, Ahmad B: Approximation of solutions for an initial and terminal value problem for the forced Duffing equation with non-viscous damping. Appl. Math. Comput. 2010, 216: 2129-2136. 10.1016/j.amc.2010.03.046

    Article  MathSciNet  Google Scholar 

  43. Sun L, Zhou M, Wang G: Generalized quasilinearization method for nonlinear boundary value problems with integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2010, 66: 1-14.

    MathSciNet  Google Scholar 

  44. Alsaedi A: Approximation of solutions for second-order m -point nonlocal boundary value problems via the method of generalized quasilinearization. Bound. Value Probl. 2011., 2011: Article ID 929061

    Google Scholar 

  45. Wang P, Gao W: Quasilinearization of an initial value problem for a set valued integro-differential equation. Comput. Math. Appl. 2011, 61: 2111-2115. 10.1016/j.camwa.2010.08.084

    Article  MathSciNet  Google Scholar 

  46. Motsa SS: A new piecewise-quasilinearization method for solving chaotic systems of initial value problems. Cent. Eur. J. Phys. 2012, 10: 936-946. 10.2478/s11534-011-0124-2

    Google Scholar 

  47. Hartung F: Parameter estimation by quasilinearization in differential equations with state-dependent delays. Discrete Contin. Dyn. Syst., Ser. B 2013, 18: 1611-1631.

    Article  MathSciNet  Google Scholar 

  48. Motsa SS, Sibanda P: Some modifications of the quasilinearization method with higher-order convergence for solving nonlinear BVPs. Numer. Algorithms 2013, 63: 399-417. 10.1007/s11075-012-9629-z

    Article  MathSciNet  Google Scholar 

  49. Devi JV, Vatsala AS: Quasilinearization for second order singular boundary value problems with solutions in weighted spaces. J. Korean Math. Soc. 2000, 37: 823-832.

    MathSciNet  Google Scholar 

  50. Eloe PW: The quasilinearization method on an unbounded domain. Proc. Am. Math. Soc. 2002, 131: 1481-1488.

    Article  MathSciNet  Google Scholar 

  51. Ramos JI: Piecewise quasilinearization techniques for singular boundary-value problems. Comput. Phys. Commun. 2004, 158: 12-25. 10.1016/j.comphy.2003.11.003

    Article  Google Scholar 

  52. El-Gebeily M, O’Regan D: A quasilinearization method for a class of second order singular nonlinear differential equations with nonlinear boundary conditions. Nonlinear Anal., Real World Appl. 2007, 8: 174-186. 10.1016/j.nonrwa.2005.06.008

    Article  MathSciNet  Google Scholar 

  53. Kannan R, O’Regan D: A note on singular boundary value problems with solutions in weighted spaces. Nonlinear Anal. 1999, 37: 791-796. 10.1016/S0362-546X(98)00072-8

    Article  MathSciNet  Google Scholar 

  54. Agarwal RP, O’Regan D, Lakshmikantham V, Leela S: Existence of positive solutions for singular initial and boundary value problems via the classical upper and lower solution approach. Nonlinear Anal. 2002, 50: 215-222. 10.1016/S0362-546X(01)00747-7

    Article  MathSciNet  Google Scholar 

  55. Lakshmikantham V, Rao MRM: Theory of Integro-Differential Equations. Gordon & Breach, London; 1995.

    Google Scholar 

  56. Lakshmikantham V, Leela S I. In Differential and Integral Inequalities. Academic Press, New York; 1969.

    Google Scholar 

Download references

Acknowledgements

This work was supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bashir Ahmad.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors, RPA, BA and AA contributed to each part of this work equally and read and approved the final version of the manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Agarwal, R.P., Ahmad, B. & Alsaedi, A. Method of quasilinearization for a nonlocal singular boundary value problem in weighted spaces. Bound Value Probl 2013, 261 (2013). https://doi.org/10.1186/1687-2770-2013-261

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-2770-2013-261

Keywords