Georgian Mathematical Journal Volume 10 (2003), Number 3, 595–602

ON TWO-POINT BOUNDARY VALUE PROBLEMS FOR TWO-DIMENSIONAL DIFFERENTIAL SYSTEMS WITH SINGULARITIES

S. MUKHIGULASHVILI

Abstract. For a differential system

\frac{du_{1}}{dt} = h_{0}(t, u_{1}, u_{2})u_{2}, \frac{du_{2}}{dt} = −h_{1}(t, u_{1}, u_{2})u_{1}^{−λ} − h_{2}(t, u_{1}, u_{2}),

where λ ∈]0, 1[ and h_{i} :]a, b[×]0, +∞[×ℝ → [0, +∞[ (i = 0, 1, 2) are continuous functions, we have established sufficient conditions for the existence of at least one solution satisfying one of the two boundary conditions

lim_{t→a} u_{1}(t) = 0, lim_{t→b} u_{1}(t) = 0

and

lim_{t→a} u_{1}(t) = 0, lim_{t→b} u_{2}(t) = 0.

2000 Mathematics Subject Classification: 34B16.

Key words and phrases: Two-dimensional nonlinear differential system with singularities, two-point boundary value problem, existence theorem.

Two-point boundary value problems for second order ordinary differential equations with singularities with respect to one of the phase variable frequently occur in applications (see, for example, [1, 3, 4, 10, 11, 12]) and are investigated with sufficient thoroughness in the works of S. Taliafero [13], J. E. Bouillet and S. M. Gomes [2], Yu. A. Klokov and A. I. Lomakina [7], A. G. Lomtatidze [8, 9], I. Kiguradze and B. Shekhter [6] and others. However analogous problems for two-dimensional singular differential systems still remain little studied. In this paper we investigate the first and second boundary value problems for nonlinear differential system with singularities with respect to both independent and/or one of the phase variables.

We consider the differential system

\frac{du_{1}}{dt} = h_{0}(t, u_{1}, u_{2})u_{2}, \frac{du_{2}}{dt} = −h_{1}(t, u_{1}, u_{2})u_{1}^{−λ} − h_{2}(t, u_{1}, u_{2}),  (1)

with the boundary conditions

lim_{t→a} u_{1}(t) = 0, lim_{t→b} u_{1}(t) = 0  (2)

or

lim_{t→a} u_{1}(t) = 0, lim_{t→b} u_{2}(t) = 0.  (3)

Here λ ∈]0, 1[, h_{i} :]a, b[×]0, +∞[×ℝ → [0, +∞[ (i = 0, 1, 2) are continuous functions. Moreover, there exist a continuous function h :]a, b[→ [0, +∞[ and

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

positive numbers l, l_{0} such that 0 < ∫_{a}^{b} h(t) dt < ∞ and the inequalities

h(t) ≤ h_{0}(t, x, y) ≤ l h(t), h_{1}(t, x, y) + h_{2}(t, x, y) ≥ l_{0}h(t)  (4)

hold on the set ]a, b[×]0, +∞[×ℝ.

Solutions of problems (1), (2) and (1), (3) are sought for in the class of continuously differentiable vector functions (u_{1}, u_{2}) :]a, b[→]0, +∞[×ℝ.

We set

δ_{1}(t) = ∫_{a}^{t} h(s) ds ∫_{t}^{b} h(s) ds, δ_{2}(t) = ∫_{a}^{t} h(s) ds,
δ(t) = l δ_{1}(t)( ∫_{a}^{b} h(s) ds)^{−1}.

Theorem 1. Let, along with (4) the conditions

δ_{1}(t) > 0 for a < t < b  (5)

and

lim sup_{ρ→+∞} [(2 l l_{0}^{−1})^{λ}ρ^{λ^{2}−1} ∫_{a}^{b} δ_{1}^{1−λ}(t) h_{1}^{∗}(t, ρ) dt + ρ^{−1} ∫_{a}^{b} δ_{1}(t) h_{2}^{∗}(t, ρ) dt]

< l^{−2} ∫_{a}^{b} h(t) dt  (6)

be satisfied, where

h_{i}^{∗}(t, ρ) = sup {h_{i}(t, x, y) : 0 < x < l, δ(t)|y| < ρ} (i = 1, 2).  (7)

Then problem (1), (2) is solvable.

Theorem 2. Let, along with (4), the conditions

δ_{2}(t) > 0 for a < t ≤ b

and

lim sup_{ρ→+∞} [(\frac{2}{l_{0} δ_{2}(b)})^{λ}ρ^{λ^{2}−1} ∫_{a}^{b} δ_{2}^{1−λ}(t) h_{1}^{∗}(t, ρ) dt + ρ^{−1} ∫_{a}^{b} δ_{2}(t) h_{2}^{∗}(t, ρ) dt] < \frac{1}{l}

hold, where

h_{i}^{∗}(t, ρ) = sup {h_{i}(t, x, y) : 0 < x < ρ, δ_{2}(t)|y| < ρ} (i = 1, 2).

Then problem (1), (3) is solvable.

As an example, consider the differential system

\begin{aligned}\frac{du_{1}}{dt} & = (t − a)^{α} (b − t)^{β} ρ_{0}(t, u_{1}, u_{2})u_{2}, \\ \frac{du_{2}}{dt} & = −\frac{p_{1}(t, u_{1}, u_{2})}{(t − a)^{α_{1}} (b − t)^{β_{1}}} u_{1}^{−λ} − \frac{p_{2}(t, u_{1}, u_{2})}{(t − a)^{α_{2}} (b − t)^{β_{2}}} , \end{aligned}  (8)

where λ ∈]0, 1[, and

α > −1, β > −1, α_{i} ≥ −α, β_{i} ≥ −β (i = 1, 2),

and p_{i} :]a, b[×]0, +∞[×ℝ →]r_{1}, r_{2}[ (i = 0, 1, 2) are continuous functions, r_{i} = const > 0 (i = 1, 2).

Theorems 1 and 2 give rise to

Corollary 1. The inequalities

α_{1} < (α + 1)(1 − λ) + 1, β_{1} < (β + 1)(1 − λ) + 1, α_{2} < α + 2, β_{2} < β + 2

guarantee the solvability of problem (1), (2), and the inequalities

α_{1} < (α + 1)(1 − λ) + 1, β_{1} < 1, α_{2} < α + 2, β_{2} < 1

guarantee the solvability of problem (1), (3).

The above example shows that Theorem 1 (Theorem 2) covers the case where the functions h_{1} and h_{2} have singularities of arbitrary order at t = a and t = b (at t = a).

To prove the formulated theorems, we need two lemmas from [5] on the representation of a solution of the differential system

\frac{dv_{1}}{dt} = p(t) v_{2}, \frac{dv_{2}}{dt} = q(t)  (9)

under the boundary conditions

v_{1}(a) = 0, v_{1}(b) = 0  (10)

or

v_{1}(a) = 0, v_{2}(b) = 0.  (11)

Lemma 1. Let p, q :]a, b[→ ℝ be continuous functions such that

∫_{a}^{b} |p(t)| dt < +∞, ∫_{a}^{b} ( ∫_{a}^{t} |p(τ)| dτ ∫_{t}^{b} |p(τ)| dτ)|q(t)| dt < +∞ (i = 1, 2),

γ_{0} =^{def} ∫_{a}^{b} p(t) dt ≠ 0.  (12)

Then problem (9), (10) has a unique solution v = (v_{1}, v_{2}) and the representations

v_{i}(t) = ∫_{a}^{b} g_{i}(t, s) q(s) ds (i = 1, 2)  (13)

are valid, where

g_{1}(t, s) = \begin{cases}−\frac{1}{γ_{0}} ∫_{a}^{s} p(τ) dτ ∫_{t}^{b} p(τ) dτ & for s ≤ t, \\ −\frac{1}{γ_{0}} ∫_{a}^{t} p(τ) dτ ∫_{s}^{b} p(τ) dτ & for s > t, \end{cases}

g_{2}(t, s) = \begin{cases}\frac{1}{γ_{0}} ∫_{a}^{s} p(τ) dτ & for s ≤ t, \\ −\frac{1}{γ_{0}} ∫_{s}^{b} p(τ) dτ & for s > t. \end{cases}

Lemma 2. Let p, q ∈]a, b[→ ℝ be continuous functions such that conditions (12) and

∫_{a}^{b} p(t) dt < +∞, ∫_{a}^{b} ( ∫_{a}^{t} |p(τ)| dτ)q(t) dt < +∞

are fulfilled. Then problem (9), (11) has a unique solution v = (v_{1}, v_{2}) and representations (13) are valid, where

g_{1}(t, s) = \begin{cases}− ∫_{a}^{s} p(τ) dτ & for s ≤ t, \\ − ∫_{a}^{t} p(τ) dτ & for s > t, \end{cases} g_{2}(t, s) = \begin{cases}0 & for s ≤ t, \\ −1 & for s > t. \end{cases}

Proof of Theorem 1. By virtue of condition (6), the constant ρ can be chosen so that the inequalities

ρ > 1 + \frac{l_{0}}{2 l}( ∫_{a}^{b} h(s) ds)^{2},  (14)
(\frac{2 l}{l_{0}})^{λ}ρ^{λ^{2}} ∫_{a}^{b} δ_{1}^{1−λ}(t) h_{1}^{∗}(t, ρ) dt + ∫_{a}^{b} δ_{1}(t) h_{1}^{∗}(t, ρ) dt < \frac{ρ}{l^{2}} ∫_{a}^{b} h(t) dt (15)

be fulfilled.

Let 𝔹 be a Banach space of two-dimensional continuous vector functions v = (v_{1}, v_{2}) : [a, b] → ℝ^{2} with the norm

∥v∥ = max {|v_{1}(t)| + |v_{2}(t)| : a ≤ t ≤ b}

and 𝔹_{ρ} be a set of all v = (v_{1}, v_{2}) ∈ 𝔹 satisfying the conditions

\frac{l_{0}}{2 l} ρ^{−λ}δ_{1}(t) ≤ v_{1}(t) < ρ, |v_{2}(t)| ≤ ρ for a ≤ t ≤ b.

In view of inequality (14), 𝔹_{ρ} is a nonempty closed convex subset of the space 𝔹.

For arbitrary v = (v_{1}, v_{2}) ∈ 𝔹_{ρ} we set

γ(v_{1}, v_{2}) = ∫_{a}^{b} h_{0}(τ, v_{1}(τ), \frac{v_{2}(τ)}{σ(τ)}) dτ,
q(v_{1}, v_{2})(t) = −h_{1}(t, v_{1}(t), \frac{v_{2}(t)}{σ(t)})v_{1}^{−λ}(t) − h_{2}(t, v_{1}(t), \frac{v_{2}(t)}{σ(t)}),

g_{1}(v_{1}, v_{2})(t, s) · γ(v_{1}, v_{2}) = \begin{cases}− ∫_{a}^{s} h_{0}(τ, v_{1}(τ), \frac{v_{2}(τ)}{σ(τ)}) dτ \\ × ∫_{t}^{b} h_{0}(τ, v_{1}(τ), \frac{v_{2}(τ)}{σ(τ)}) dτ & for s ≤ t, \\ − ∫_{a}^{t} h_{0}(τ, v_{1}(τ), \frac{v_{2}(τ)}{σ(τ)}) dτ \\ × ∫_{s}^{b} h_{0}(τ, v_{1}(τ), \frac{v_{2}(τ)}{σ(τ)}) dτ & for s > t, \end{cases}

g_{2}(v_{1}, v_{2})(t, s) · γ(v_{1}, v_{2}) = \begin{cases}∫_{a}^{s} h_{0}(τ, v_{1}(τ), \frac{v_{2}(τ)}{σ(τ)}) dτ & for s ≤ t, \\ − ∫_{s}^{b} h_{0}(τ, v_{1}(τ), \frac{v_{2}(τ)}{σ(τ)}) dτ & for s > t. \end{cases}

Let us now define the continuous operators W_{i} : 𝔹 → C([a, b]) (i = 1, 2) and W : 𝔹 → C([a, b]) × C([a, b]) by the equalities

W_{i}(v_{1}, v_{2})(t) = (σ(t))^{i−1} ∫_{a}^{b} g_{i}(v_{1}, v_{2})(t, s) q(v_{1}, v_{2})(s) ds (i = 1, 2)

and

W (v_{1}, v_{2})(t) = (W_{1}(v_{1}, v_{2})(t), W_{2}(v_{1}, v_{2})(t)).

Then if problem (1), (2) has a solution u = (u_{1}, u_{2}) satisfying the conditions

\frac{l_{0}}{2 l} ρ^{−λ} δ_{1}(t) ≤ u_{1}(t) ≤ ρ, σ(t)|u_{2}(t)| ≤ ρ for a ≤ t ≤ b,  (16)
lim_{t→a} σ_{1}(t) u_{2}(t) = 0, lim_{t→b} σ_{1}(t) u_{2}(t) = 0,  (17)

the vector function v = (v_{1}, v_{2}) with the components

v_{1}(t) = u_{1}(t), v_{2}(t) = σ(t) u_{2}(t)  (18)

belongs to the set 𝔹_{ρ} and, by virtue of Lemma 1, is a solution of the operator equation

v(t) = W (v)(t).  (19)

Conversely, if equation (19) has a solution v = (v_{1}, v_{2}) ∈ 𝔹_{ρ}, then, again by Lemma 1, the function u = (u_{1}, u_{2}), the components of which are defined from equalities (18), is a solution of problem (1), (2) satisfying conditions (17). Thus to prove the theorem it is sufficient to establish that the operator W has a fixed point on the set 𝔹_{ρ}.

In the first place, we will show that

W (𝔹_{ρ}) ⊂ 𝔹_{ρ}.  (20)

Let (v_{1}, v_{2}) ∈ 𝔹_{ρ}. Then by virtue of (7)

|q(v_{1}, v_{2})(t)| ≤ φ(t) for a < t < b,

φ(t) = (\frac{2 l}{l_{0}})^{λ} ρ^{λ^{2}} δ_{1}^{−λ}(t) h_{1}^{∗}(t, ρ) + h_{2}^{∗}(t, ρ),

and, also, by virtue of the second of equalities (4)

|q(v_{1}, v_{2})(t)| ≥ ρ^{−λ} l_{0} h(t) for a < t < b.

Taking into account the latter estimates and the first of equalities (4), we obtain

W_{1}(v_{1}, v_{2})(t) ≥ \frac{l_{0}}{l} ρ^{−λ}( ∫_{a}^{b} h(τ) dτ)^{−1}[ ∫_{a}^{b} h(τ) dτ ∫_{a}^{t} ( ∫_{a}^{s} h(τ) dτ)h(s) ds
+ ∫_{a}^{t} h(τ) dτ ∫_{t}^{b} ( ∫_{s}^{b} h(τ) dτ)h(s) ds]

= \frac{l_{0}}{2 l} ρ^{−λ} δ_{1}(t) for a ≤ t ≤ b,  (21)
W_{1}(v_{1}, v_{2})(t) ≤ l^{2}( ∫_{a}^{b} h(τ) dτ)^{−1}[ ∫_{t}^{b} h(τ) dτ ∫_{a}^{t} ( ∫_{a}^{s} h(τ) dτ)φ(s) ds
+ ∫_{a}^{t} h(τ) dτ ∫_{t}^{b} ( ∫_{t}^{b} h(τ) dτ)φ(s) ds] for a ≤ t ≤ b  (22)

and

W_{2}(v_{1}, v_{2})(t) ≤ l^{2}( ∫_{a}^{b} h(τ) dτ)^{−2}δ_{1}(t)[ ∫_{a}^{t} ( ∫_{a}^{s} h(τ) dτ)φ(s) ds
+ ∫_{t}^{b} ( ∫_{s}^{b} h(τ) dτ)φ(s) ds] for a ≤ t ≤ b.  (23)

From equalities (21), (23) with (15) taken into account we obtained

W_{i}(v_{1}, v_{2})(t) ≤ l^{2}( ∫_{a}^{b} h(τ) dτ)^{−1} ∫_{a}^{b} δ_{1}(τ) φ(τ) dτ < ρ (i = 1, 2)  (24)
for a ≤ t ≤ b.

This and (21) imply that inclusion (20) is valid.

Now note that from the definition of the operators W_{i} (i = 1, 2) and Lemma 1 it follows that the functions W_{i}(v_{1}, v_{2})(t) (i = 1, 2) are continuously differentiable on the interval ]a, b[ for any v = (v_{1}, v_{2}) ∈ 𝔹_{ρ} and

\frac{d}{dt} W_{1}(v_{1}, v_{2})(t) = h_{0}(t, v_{1}, \frac{v_{2}}{σ} ) δ^{−1}(t) W_{2}(v_{1}, v_{2})(t),
\frac{d}{dt} W_{2}(v_{1}, v_{2})(t) = −(h_{1}(t, v_{1}, \frac{v_{2}}{σ} ) W_{1}^{−λ}(v_{1}, v_{2})(t) + h_{2}(t, v_{1}, v_{2}))δ(t)

+ \frac{δ′(t)}{δ(t)} W_{2}(v_{1}, v_{2})(t).

From this, taking into account estimates (21) and (24), we obtain

|\frac{d}{dt} W_{i}(v_{1}, v_{2})| ≤ η_{i}(t) (i = 1, 2) for a < t < b,  (25)

where η_{i} :]a, b[→ [0, +∞[ are continuous functions defined by the equalities

η_{1}(t) = ρ \frac{h(t)}{δ(t)}, η_{2}(t) = δ(t) φ(t) + ρ \frac{h(t)}{δ_{1}(t)} ∫_{a}^{b} h(τ) dτ.

Analogously, from estimates (22) and (23) we obtain

|W_{1}(v_{1}, v_{2})(t)| + |W_{2}(v_{1}, v_{2})(t)| ≤ ε_{0}(t) for a ≤ t ≤ b,  (26)

where the continuous function ε_{0} : [a, b] → [0, +∞[ is defined by the equality

ε_{0}(t) = 2 l^{2}( ∫_{a}^{b} h(τ) dτ)^{−1}[ ∫_{t}^{b} h(τ) dτ ∫_{a}^{t} ( ∫_{a}^{s} h(τ) dτ)φ(τ) dτ
+ ∫_{a}^{t} h(τ) dτ ∫_{t}^{b} ( ∫_{s}^{b} h(τ) dτ)φ(τ) dτ],

and therefore

ε_{0}(a) = 0, ε_{0}(b) = 0.  (27)

From (25)–(27) we conclude that the set of functions W (𝔹_{ρ}) is equicontinuous and uniformly bounded, i.e., the continuous operator W transforms the bounded convex set 𝔹_{ρ} into its compact subset and, by virtue of the Shauder’s principle of fixed point, equation (19) has at least one solution. □

Theorem 2 is proved in an analogous manner with the only difference that we use Lemma 2.

Acknowledgement

The work was supported by a fellowship from the INTAS YS 2001-2/80 as for a visiting professor at the University of Udine, Italy.

References

1. J. A. Ackroyd, On the laminar compressible boundary layer with stationary origin on a moving flat wall. Proc. Cambridge Phil. Soc. 63(1967), 871–888.

2. J. E. Bouillet and S. M. Gomes, An equation with a singular nonlinearity related to diffusion problems in one dimension. Quart. Appl. Math. 42(1985), No. 4, 395–402.

3. A. J. Callegary and M. B. Friedman, An analytical solution of a nonlinear singular boundary value problem in the theory of viscous fluids. J. Math. Anal. Appl. 21(1968), No. 3, 510–529.

4. A. J. Callegary and A. Nachman, Some singular, nonlinear equations arising in boundary-layer theory. J. Math. Anal. Appl. 64(1978), No. 1, 96–105.

5. I. T. Kiguradze, On boundary value problems for linear differential systems with singularities. (Russian) Differentsial’nye Uravneniya 39(2002), No. 2, 198–209.

6. I. T. Kiguradze and B. L. Shekhter, Singular boundary value problems for secondorder ordinary differential equations. (Russian) Itogi Nauki i Tekhniki, Current problems in mathematics. Newest results, Vol. 30 (Russian), 105–201, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987; English translation: J. Sov. Math. 43(1988), No. 2, 2340–2417.

7. Yu. A. Klokov and A. I. Lomakina, A certain boundary value problem with singularities at the endpoints of the interval. (Russian) Latvian mathematical yearbook, 17 (Russian), 179–186. Izdat. “Zinatne”, Riga, 1976.

8. A. G. Lomtatidze, Solvability of boundary value problems for second-order nonlinear ordinary differential equations with singularities. (Russian) Reports of the extended sessions of a seminar of the I. N. Vekua Institute of Applied Mathematics, Vol. I, No. 3 (Russian) (Tbilisi, 1985), 85–92, 170–171, Tbilis. Gos. Univ., Tbilisi, 1985.

9. A. G. Lomtatidze, On positive solutions of singular boundary value problems for nonlinear second order ordinary differential equations. (Russian) Differentsial’nye Uravneniya 22(1986), No. 6, p. 1092.

10. N. F. Morozov, On the analytic structure of the solution of the membrane equation. (Russian) Dokl. Akad. Nauk SSSR 152(1963), No. 1, 78–80.

11. N. P. Morozov and L. S. Srubshchik, Application of Chaplygin’s method to the study of the membrane equation. (Russian) Differentsial’nye Uravneniya 2(1966), No. 3, 425–427.

12. L. S. Srubshchik and V. I. Yudovich, The asymptotics of the equation for large bending of a round symmetrically loaded lamina. (Russian) Sibirsk. Mat. Zh. 4(1963), No. 3, 657–672.

13. S. Taliaferro, A nonlinear singular boundary value problem. Nonlinear Anal. 3(1979), No. 6, 897–904.

(Received 1.05.200)

Author’s address:

A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 0193 Georgia E-mail: mukhig@rmi.acnet.ge