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

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.

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 [21–25] 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 [29–48].

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):

where f:[0,1]\times \mathbb{R}\to \mathbb{R}, K:[0,1]\times [0,1]\times \mathbb{R}\to \mathbb{R}, and g:\mathbb{R}\to \mathbb{R}. In the forthcoming analysis, g(u(\sigma )) 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

(u_0, u_1 are constants) is not continuous at t=0 but t^{2}u\in C[0,1]. Such a situation provides motivation to consider the singular boundary value problems in a weighted Banach space.

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

Definition 2.1 A function u\in C^2(0,1) with t^{n-1}u\in C[0,1], t^n{u}^{\prime }\in C[0,1] and {(t^{n-1}u)}^{\prime }\in 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

with the norm

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

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

Again differentiating, it follows that

Multiplying (2.1) by t^{n-1} and taking the limit t\to 0^{+}, we get {lim}_{t\to 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]\times \mathbb{R}\to \mathbb{R}, K:[0,1]\times [0,1]\times \mathbb{R}\to \mathbb{R} and g:\mathbb{R}\to \mathbb{R} are continuous functions. Furthermore, there exist positive constants M_f, M_K, M_g such that |f(t,u)|\le M_f for (t,u)\in [0,1]\times \mathbb{R}, |K(t,s,u)|\le M_K for (t,s,u)\in [0,1]\times [0,1]\times \mathbb{R}, and |g(u)|\le M_g for u\in \mathbb{R}. Then problem (1.1) has a solution.

Proof We reduce the problem of finding a solution u\in \mathcal{E} of (1.1) to a fixed point problem u=\mathcal{F}u, where \mathcal{F}:\mathcal{E}\to \mathcal{E} is given by

If u_m\to u in ℰ, then

and

In view of the continuity of f, K, g, it follows that \mathcal{F}{u}_m\to \mathcal{F}u in ℰ and so \mathcal{F}:\mathcal{E}\to \mathcal{E} is continuous.

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

and

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

Definition 2.5 A function \beta \in C^2(0,1) such that t^{n-1}\beta \in C[0,1], t^n{\beta }^{\prime }\in C[0,1], {(t^{n-1}\beta )}^{\prime }\in L^1[0,1] is called an upper solution for (1.1) if

Similarly, a function \alpha \in C^2(0,1) such that t^{n-1}\alpha \in C[0,1], t^n{\alpha }^{\prime }\in C[0,1], {(t^{n-1}\alpha )}^{\prime }\in L^1[0,1] is called a lower solution for (1.1) if

Lemma 2.6 Assume that f:[0,1]\times \mathbb{R}\to \mathbb{R} and K:[0,1]\times [0,1]\times \mathbb{R}\to \mathbb{R} are continuous functions, and g:\mathbb{R}\to \mathbb{R} is a Lipschitz function. Suppose that there exist an upper solution β and a lower solution α for (1.1) such that \alpha \le \beta on (0,1), and {lim}_{t\to 0^{+}}{t}^{n-1}\alpha (t)\le u_0\le {lim}_{t\to 0^{+}}{t}^{n-1}\beta (t). Then (1.1) has a solution u satisfying the relation t^{n-1}\alpha (t)\le t^{n-1}u(t)\le t^{n-1}\beta (t) for t\in [0,1].

Proof Consider the modified problem

where

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

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)\nleqq t^{n-1}\beta (t) for t\in [0,1]. Then there exists t_0\in (0,1) such that (t^{n-1}u(t)-t^{n-1}\beta (t)) has a positive maximum at t_0. Thus, {(t^{n-1}u(t)-t^{n-1}\beta (t))}^{\prime }(t_0)=0 and {(t^{n-1}u(t)-t^{n-1}\beta (t))}^{″}(t_0)\le 0. On the other hand,

which is a contradiction. Hence t^{n-1}u(t)\le t^{n-1}\beta (t) for t\in (0,1). Similarly, it can be shown that t^{n-1}\alpha (t)\le t^{n-1}u(t) for t\in (0,1). In view of assumptions that g is a Lipschitz function and \alpha \le \beta on (0,1), it follows that \alpha (1)\le \beta (1). Thus we conclude that

This completes the proof. □

Lemma 2.7 Let \alpha ,\beta \in \mathcal{E} be lower and upper solutions of (1.1) respectively with

Further, suppose that f:[0,1]\times \mathbb{R}\to \mathbb{R}, K:[0,1]\times [0,1]\times \mathbb{R}\to \mathbb{R} and g:\mathbb{R}\to \mathbb{R} are continuous functions and satisfy

whenever x\ge y. Then t^{n-1}\alpha (t)\le t^{n-1}\beta (t) on [0,1].

Proof For the sake of contradiction, suppose that the conclusion does not hold, that is, t^{n-1}\alpha (t)>t^{n-1}\beta (t) at some t\in (0,1]. Then, by continuity, there exists a local maximum at t_0\in (0,1). Thus, at t=t_0, we have

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

which can alternatively be written as -t^{-1}{(t^n{\alpha }^{\prime })}^{\prime }\ge -t^{-1}{(t^n{\beta }^{\prime })}^{\prime } at t=t_0\in (0,1). Using the definition of lower and upper solutions together with (2.5), we obtain

which yields a contradiction:

Hence we obtain t^{n-1}\alpha (t)\le t^{n-1}\beta (t) on (0,1). As , therefore, in view of the foregoing arguments, at t_0=1, we have

which is a contradiction. This completes the proof. □

Theorem 3.1 Assume that

(A₁) {\alpha }_0,{\beta }_0\in \mathcal{E} are lower and upper solutions of (1.1) respectively such that t^{n-1}{\alpha }_0(t)\le t^{n-1}{\beta }_0(t) for t\in (0,1] with {lim}_{t\to 0^{+}}{t}^{n-1}{\alpha }_0(t)\le u_0\le {lim}_{t\to 0^{+}}{t}^{n-1}{\beta }_0(t);

(A₂) f(t,v) is convex, that is, f_{vv}(t,v) exists, continuous with f_{vv}(t,v)\ge 0 for each (t,v)\in (0,1]\times \mathbb{R}, and , where t^{n-1}{\alpha }_0\le v\le t^{n-1}{\beta }_0;

(A₃) K(t,s,v) is convex, that is, K_{vv}(t,s,v) exists, continuous with K_{vv}(t,s,v)\ge 0 for each (t,s,v)\in (0,1]\times (0,1]\times \mathbb{R}, and , where t^{n-1}{\alpha }_0\le v\le t^{n-1}{\beta }_0;

(A₄) g is continuous on ℝ such that g^{\prime }, g^{″} exist and , g^{″}\le 0.

Then there exist monotone sequences \{t^{n-1}{\alpha }_k(t)\} and \{t^{n-1}{\beta }_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

Now, we set

Consider the singular BVP

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

and

Further, we note that {\alpha }_0(1)\le g({\alpha }_0)=h_1({\alpha }_0;{\alpha }_0,{\beta }_0) and using assumption (A₄), for c_0\in ({\alpha }_0,{\beta }_0), we find that

which implies that {\beta }_0(1)\ge g({\beta }_0)\ge h_1({\beta }_0;{\alpha }_0,{\beta }_0). Thus, it follows that {\alpha }_0 and {\beta }_0 are respectively lower and upper solutions of (3.4). Since h_i^{\prime }=g^{\prime } (i=1,2), therefore it follows by Lemmas 2.6 and 2.7 that there exists a unique solution {\alpha }_1 of (3.4) satisfying

Observe that the hypotheses of Lemma 2.7 hold in view of the conditions , , and , demanded in (A₂), (A₃), and (A₄).

Next, consider the singular BVP

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

By assumption (A₄), we have {\alpha }_0(1)\le h_2({\alpha }_0;{\beta }_0) as

and {\beta }_0(1)\ge g({\beta }_0)=h_2({\beta }_0;{\beta }_0). In view of the above inequalities, we find that {\alpha }_0 and {\beta }_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 {\beta }_1 of (3.5) satisfying

Now we show that t^{n-1}{\alpha }_1(t)\le t^{n-1}{\beta }_1(t). Using (3.1), (3.2), and assumption (A₁), we obtain