We prove existence, uniqueness, and stability of solutions of the prescribed curvature problem in , , for any given and . We also develop a linear monotone iterative scheme for approximating the solution. This equation has been proposed as a model of the corneal shape in the recent paper (Okrasiński and Płociniczak in Nonlinear Anal., Real World Appl. 13:1498-1505, 2012), where a simplified version obtained by partial linearization has been investigated.
In this paper we study existence, uniqueness, stability, and approximation of classical solutions of the one-dimensional prescribed curvature problem
where and are given constants. This problem, together with its N-dimensional counterpart
has been proposed in [1–4] as a mathematical model for the geometry of the human cornea. However, in these papers a simplified version of (2) has been investigated, where the mean curvature operator has been replaced by its linearization around 0. In particular, it has been proved in  that, if , then the problem
has a unique solution which is the limit of a sequence of successive approximations. The above limitations on the parameters have recently been removed in .
Unlike all these works we tackle here the fully nonlinear problem (1) and we prove the existence of a unique solution for the whole range of positive parameters a, b. The study of problem (1) requires some care because, even if pairs of constant lower and upper solutions can easily be exhibited, the presence of the curvature term rules out in general the possibility of applying the standard existence results, due to the possible occurrence of derivative blow-up phenomena (see, e.g., ). On the other hand, the non-variational structure of (1) puts the problem, as it stands, out of the scope of the methods developed in [6–8] for the curvature equation. Nevertheless, we show that an a priori bound in for all possible solutions can be obtained by an elementary, but delicate, argument which exploits the qualitative properties - positivity, monotonicity, and concavity - of the solutions themselves. These estimates eventually enable us to use a degree argument in order to prove the existence of solutions. The proof of the uniqueness is then based on suitable fixed point index calculations, which are performed via linearization. A similar approach, applied to an associated evolutionary problem, is exploited for detecting the linear stability of the solution.
Next, taking inspiration from [9, 10], we develop a linear iterative scheme for approximating the solution by two monotone sequences of strict lower and upper solutions, starting from an explicit pair of constant lower and upper solutions. These two sequences, besides providing accurate two-sided bounds on the solution, yield the strict order stability and hence, according to , the (Lyapunov) asymptotic stability of the solution itself, yielding as well an explicitly computable estimate of its basin of attractivity. We finally illustrate the use of this approximation scheme in order to compute numerically the solution u of (1) for the same choice of the parameters a and b as the one considered in .
We finally mention that part of our results extends to the general N-dimensional problem (2); this topic will be discussed elsewhere.
In this section we are concerned with the study of the existence, the qualitative properties and the approximation of classical solutions, i.e., belonging to , of problem (1), where and are fixed constants. Clearly, problem (1) can be written in the equivalent form
Let us set for convenience, for all ,
It is obvious that, due to the symmetry properties of the function f, the mixed problem (4) is equivalent to the Dirichlet problem
Notations As usual, for functions , we write in if for all and in if and . We also write in if for all and, if , , where , denote the left Dini derivatives; this is equivalent to requiring that there exists such that for all . Whenever no confusion occurs, we omit the indication of the interval.
Existence, uniqueness, and linear stability
We start with a preliminary result, where some properties of the possible solutions of problem (1) are highlighted.
Lemma 2.1The following assertions hold.
Any solutionuof (1) satisfiesfor allandfor all .
Any solutionuof (1) is such thatandfor all .
Any solutionuof (1) is such thatfor all , where .
Proof In the following steps u denotes a solution of (1), or equivalently of (4). From the equation in (4) it follows that, if is such that and , then
Step 1. Proof of (i). Let us first show that in . Assume by contradiction that . The boundary condition implies that . Suppose that . We have and . Condition (6) yields . Hence there exists such that for all and therefore for all , which is a contradiction. Now suppose that . We have and and condition (6) yields again a contradiction. Hence we conclude that in . In a completely similar way we prove that in .
Next, in order to prove that
for all , it is sufficient to note that, if for some , then (6) would yield , which is impossible. Moreover, as the constant function is a solution of the equation in (4), the uniqueness of solutions for any Cauchy problem associated with this equation implies that
for all .
Step 2. Proof of (ii). As , assertion (i) implies that there exists such that for all and for all . Let us show that
for all . If this is not the case, set . Then, by (6), we have and hence there exists such that , for all , which contradicts the definition of .
Let us now prove that
in . By contradiction, assume that there exists such that . As , there exist and such that and for all . Since we have for all , the function is decreasing in . We also know that the function au is decreasing in . Hence the function is decreasing in . On the other hand, as , from the equation in (4) we get
for all . Then the equation in (4) yields for all , which is a contradiction.
Step 3. Proof of (iii). Since by assertion (i) in , we get in , and then, from the equation in (4), we conclude that
in . Multiplying by , where in by assertion (ii), and integrating between 0 and 1, we obtain
On the one hand, using the boundary condition we have
On the other hand, the boundary condition and assertion (i) imply
In conclusion, setting
we get . Since is non-increasing, we conclude
for all . □
We are now in position to prove the existence of a unique solution of problem (1), which is linearly stable.
Theorem 2.2Letandbe given. Then there exists a unique solutionuof (1) and it satisfies the conditions (i), (ii), and (iii). Further, uis linearly stable as an equilibrium of the parabolic problem
Proof The proof is divided into three steps.
Step 1. Existence. Let us prove that there exists at least one solution of (1), or equivalenty of (4). Let be the operator which associates with any the unique solution w of
Clearly, is completely continuous. Moreover, let be the Nemitski operator associated with f, i.e., for any . The operator is continuous and maps bounded sets into bounded sets. Introduce the open bounded subset of
Finally, define a completely continuous operator by . The fixed points of are precisely the solutions of (4).
An inspection of the assertions of Lemma 2.1 shows that, if satisfies, for some ,
then . The invariance property of the degree under homotopy implies that
where ℑ stands for the identity operator. Therefore there exists a fixed point of , which is a solution of (4).
By Lemma 2.1, u satisfies the conditions (i), (ii), and (iii).
Step 2. Uniqueness. Set . As the function is of class , the operators and, hence, are of class , with Fréchet differentials
for any given and all .
Observe that, for any , is invertible. Indeed, let us fix and assume that for some . This means that w is the solution of
for all , the maximum principle [, Appendix, Theorem 5.2] implies that . Hence the local inversion theorem applies to Φ at every point and thus any fixed point of is isolated. The compactness in of the set of all fixed points of then implies that is finite, i.e., for some positive integer N.
Denote by the open ball in centered at u and having radius r. Pick so small that for all , and for all , with . The excision and the additivity properties of the degree yield
where, for each ,
denotes the fixed point index of . Using again (15), we see as above that, for any given and all , the problem
has no non-trivial solution w. Accordingly, for any given , the operator does not have any eigenvalue . Therefore, we infer from [, Theorem 3.20] that
for all . Finally, by (13), (16), and (17) we conclude that , i.e., there is a unique solution u of problem (4).
Step 3. Linear stability. The solution u of (4) is an equilibrium of the parabolic problem (11), in particular, it is a 1-periodic solution of (11). In order to show that u is linearly stable, and hence locally exponentially asymptotically stable, it is enough, after a standard cut-off argument, to show that the eigenvalue problem
does not have any eigenvalue (see, e.g., [, Chapter III.23]), or [, Chapter V.22]). Indeed, if w is a solution of (18) for some , then using again condition (15), together with the interior form of the parabolic maximum principle and the Hopf boundary point lemma (see, e.g., [, Chapter III.13]), we conclude that . □
Monotone approximation and order stability
In this section we discuss approximation and stability of the solution of (1), or equivalently of (4). To this end, we define a linear iterative scheme that allows one to construct an increasing sequence of strict lower solutions and a decreasing sequence of strict upper solutions of (4) which converge in to the unique solution u of (4), that is, of (1). Then, according to [11, 13], we see that u is strictly order stable from above and from below and hence it is (Lyapunov) asymptotically stable as an equilibrium of the parabolic problem (11). In addition, the converging sequences of lower and upper solutions provide explicitly computable estimates of the basin of attractivity of the solution.
Lower and upper solutions Let us consider the problem
where is locally Lipschitz continuous. A lower solution of (19) is a function which satisfies
Similarly an upper solution of (19) is a function which satisfies
Remark 2.1 The Lipschitz character of g implies (see [, Chapter 3, Proposition 1.7, Proposition 2.7]) that a lower solution α of (19), which is not a solution, is a strict lower solution, that is, any solution u of (19), such that , satisfies in . Similarly, an upper solution β of (19), which is not a solution, is a strict upper solution, that is, any solution u of (19), such that , satisfies in .
Remark 2.2 Any constant is a strict lower solution of (4) and any constant is a strict upper solution of (4). In particular, one can choose and . We wish to point out that, with this choice of lower and upper solutions, the existence of at least one solution u of problem (4) between α and β can be alternatively achieved by applying [, Chapter 2, Theorem 3.1]; the relevant observations being here the facts that , and f satisfies the one-sided Nagumo condition
for all such that . We point out that one-sided Nagumo conditions were introduced for the first time by Kiguradze in .
Let us consider the following modified problem:
Here is defined as follows. We first introduce an auxiliary function by setting, for all ,
where R is defined in (10). Then we set, for all ,
The function is locally Lipschitz continuous and satisfies the following conditions:
(h1) there existssuch that
holds for all, with;
(h2) there existssuch that
holds for all.
We can choose in (h1) and in (h2).
Remark 2.3 Any constant and any constant are, respectively, a strict lower and a strict upper solution of (21) too.
Lemma 2.3A functionis a solution of (21) if and only if it is a solution of (1).
Proof Let u be a solution of (21). In order to prove that u is also a solution of (1), or equivalently of (4), it is sufficient to show that u satisfies and in .
The function satisfies the following conditions:
It can be verified, proceeding as in the proof of Lemma 2.1 and using the maximum principle, that and hence in .
Next we prove that in . The proof of assertion (ii) in Lemma 2.1 can be repeated verbatim in order to show that for all and in . Assume now that there exists such that . In particular, we have in and in . By definition of , u satisfies
in . As and in , we easily get from (23)
From (24), using again and in , we obtain
in . Since
in and in , we finally get from (26)
Combining (25) and (27) yields
which is precisely (9) in Step 3 of the proof of Lemma 2.1. As there we conclude that in . Accordingly, u is a solution of (4).
Conversely, the definition of implies that any solution of (1), or equivalently of (4), is a solution of (21) as well. □
Let us consider the following auxiliary linear problem:
Here, is a continuous function and and are given constants. Notice that problem (28) has a unique solution . The following result is inspired from  and [, Chapter 5].
Lemma 2.4There existssuch that for all , for all , within , and for all , the solutionwof (28) satisfies
In addition, ifinor , then
Proof Let us denote by and the respective solutions of
Step 1. The functions and satisfy
A simple computation yields
The conclusion then easily follows by direct calculations.
Step 2. There exists such that, for any , the following inequalities hold:
Let us first show that (37) holds. We have, for all ,
Since and , we can conclude that, for any sufficiently large, for all . Namely, if we set
and we take , we have
Moreover, since , the inequality
holds as well. This yields the validity of (37).
As for (38), by the sign properties of and , we see that for all provided that : this condition holds as .
Fix now , , with in , and . Let w be the solution of problem (28). If in and , then (29) trivially follows. Therefore suppose that in or . We can express w as
for all . Inequality (29) now reads
for . The sign properties of , , , and the assumptions on h and m immediately yield (30). □
We introduce now a linear monotone iterative scheme for approximating the solution of (1); namely, we define by recurrence two sequences , and as follows:
letbe any constant, with, and, for, letbe the solution of
letbe any constant, with, and, for, letbe the solution of
Theorem 2.5Letandbe given. Then there exists , given by (39), such that for anythe sequencesandrecursively defined in (41) and (42), respectively, converge into the unique solutionuof (21) and hence of (1). In addition, for eachthe following conditions hold:
is a strict lower solution andis a strict upper solution of (21), and
Proof Let us fix , where is given by (39).
Step 1. The sequence is such that, for each , in and is a strict lower solution of (21).
The proof is done by induction. Define, for each , . The function satisfies
Notice that in . Hence the maximum principle implies that , that is, , in . Now, let us show that is a strict lower solution of (21). Using the definition of , together with conditions (h1) and (h2), we get
in . Since is a solution of (43), which is of the form of (28), with in and , Lemma 2.4 applies and yields
for all . From (44), (45) and from the boundary conditions , we conclude that is a strict lower solution of (21).
Assume now that, for some integer , is strict lower solution of (21) satisfying the boundary conditions. The function satisfies
As is strict and satisfies the boundary conditions, we have in . Hence the maximum principle yields , i.e., in . Finally, satisfies
in . Since is the solution of problem (46), which is of the form of (28), with in and , Lemma 2.4 applies and yields
for all . From (47), (48), and from the boundary conditions , we conclude that is a strict lower solution of (21), such that in .
In a similar way, one can prove the following conclusion.
Step 2. The sequence is such that, for each , in and is a strict upper solution of (21).
Step 3. We have, for each , in .
For each , let us set
where, clearly, and satisfy
By construction, we have in and
As , we conclude that for all .
Take now any and suppose that and in . From (49) we infer, using the maximum principle, that in . Let us prove that in . We easily see that
in . As is the solution of problem (49), which is of the form of (28), with in and , Lemma 2.4 applies and yields
and hence for all . The conclusion , i.e., , in for all , then follows by induction.
Step 4. There exists such that, for all ,
We know that in for all , with . Hence we get
for all . Let us set .
Suppose, by contradiction, that (51) does not hold, i.e., for every there exists such that or . Assume that the former eventuality occurs. By Step 1, using conditions (h1) and (h2), we get
in . Suppose that . Since , there exists such that . Let be such that and for all . From (53) we infer
The right-hand side of (54) diverges as , then a contradiction follows.
In a completely similar way, we achieve the conclusion if , or if .
Step 5. The sequence converges in to the solution u of (1).
It follows from the previous steps that the sequence is increasing and bounded in . Therefore there exists a function which is the pointwise limit of in ; in particular, in for all . Moreover, by the Arzelà-Ascoli theorem, any subsequence of admits a subsequence which is convergent in to u. Then the whole sequence converges in to u. From the equation in (41) we see that the convergence takes place in . Hence u is a solution of problem (21) and, by Lemma 2.3 and Theorem 2.2, it is in fact the unique solution of problem (1).
In a similar way, one can prove the following conclusion.
Step 6. The sequence converges in to the solution u of (1).
Thus the proof is completed. □
Corollary 2.6Letandbe given. Then the unique solutionuof (21) is (Lyapunov) globally asymptotically stable as an equilibrium of the parabolic problem
Proof Let us note that any lower, respectively upper, solution of (21) is a lower, respectively upper, solution of the parabolic problem
Arguing as in the proof of Theorem 2.2 we see that u is the unique solution of (56). Then Theorem 2.5 implies that u is strictly order stable from below and from above. Actually, since any constant is a strict lower solution and any constant is a strict upper solution of (56), the results in [, Section 2.6] imply that u is (Lyapunov) globally asymptotically stable as a solution of (56) and hence as an equilibrium of (55). □
Remark 2.4 The definition of implies that the solution u of (1) is strictly order stable from below and from above and (Lyapunov) asymptotically stable as an equilibrium of the parabolic problem (11).
We present here some experiments concerning the numerical approximation of the solution of problem (1), for the same choice of the parameters as in .
The iterative scheme in case We have computed various approximations, at different precision levels, of the unique solution u of problem (1) by implementing in MatLab the linear iterative scheme defined by (41) and (42); at each step of the iteration the resulting linear equations have been solved using the bvp4c routine with a 100-point grid. We have chosen , with given by (39), and . Theorem 2.5 guarantees that the approximating sequences and are constituted by lower and upper solutions and monotonically converge to u, in an increasing or decreasing fashion, respectively; thus, for each n, the couple , brackets the solution u, thus providing lower and upper estimates. In what follows the -norm of a given function is intended to have been computed as the -norm of its discretization on the given grid. We have denoted by the minimum number of iterations needed in order that for ; the corresponding values are , and . In Table 1 we have tabulated , , for , at the mesh points ; the graphs of , are displayed in Figure 1; whereas Figure 2 describes the rate of decay of , as well as of the errors and , plotted against the number n of iterations. Here u denotes a reference approximation of the solution of (1), calculated using the same scheme up to a precision of 10−5. Although the lower solutions converge slightly faster than the upper solutions , it is evident that the monotone iterative scheme defined by (41) and (42) turns out to be extremely slow.
Values of the approximations,, defined by (41), (42) with, such thatfor
t = 0
t = 0.2
t = 0.4
t = 0.6
t = 0.8
t = 1
The iterative scheme in case We start from the obvious observation that the iterative scheme given by (41) and (42) is well defined for any fixed ; hence it is clear that, if the resulting sequences and are Cauchy sequences in , then, by the uniqueness of the solution of (1), they converge in to u. Of course, if we cannot anymore guarantee that either is a lower solution, or is an upper solution, or the sequences and enjoy any monotonicity property. Let us take in (41) and let be the sequence of iterates obtained for some given . The numerical experiments, we have performed for several different choices of , show that the sequence converges to u, but the magnitude of L strongly affects the speed of convergence; namely, as L decreases, the required number of iterations n in order that goes beneath a prescribed threshold, decreases. In particular, the speed of convergence significantly increases as L approximates 1 and, for this choice of L, it becomes comparable even with the speed of Newton’s method. Indeed, if we fix an error tolerance of 10−3, the iterative scheme defined by (41), with and , converges in 4 iterations, whereas Newton’s method, starting from too, converges in 2 iterations: these results are displayed in Tables 2 and 3. This computational remark suggests the possibility of using the iterative scheme also in case the condition fails; however, its convergence properties should be theoretically analyzed.
Values of the approximations, defined by (41) with, for
t = 0
t = 0.2
t = 0.4
t = 0.6
t = 0.8
t = 1
Values of the Newton approximationsfor
t = 0
t = 0.2
t = 0.4
t = 0.6
t = 0.8
t = 1
A comparison between the solutions of (1) and (3) Here we present a numerical comparison between the solution u of the fully nonlinear problem (1) and the solution of the partially linearized problem (3) investigated in . We have approximated u by the lower solution obtained by implementing the monotone iterative scheme given by (41), with , and stopping criterion . An approximation of , matching the one obtained in , has been calculated using the bvp4c routine of MatLab with a 100-point grid. Table 4 reports the values of u and at the mesh points and Figure 3 displays the graphs of u and .
This paper was written under the auspices of INdAM-GNAMPA. The first named author has been supported by Fundação para a Ciência e a Tecnologia (SFRH/BD/61484/2009). The last two named authors have been supported by Università di Trieste, in the frame of the FRA projects ‘Equazioni differenziali ordinarie: aspetti qualitativi e numerici’ and ‘Nonlinear Ordinary Differential Equations: Qualitative Theory, Numerics and Applications’. They also wish to thank Igor Moret for some useful discussions.
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