USA-Chile Workshop on Nonlinear Analysis, Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 101–107. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

An abstract existence result and its applications ^{∗}

Sui Sun Cheng, Bin Liu, & Jian-She Yu

Abstract By means of Borsuk’s theorem and continuation through an admissible homotopy, we establish an existence theorem for operator equation with homogeneous nonlinearity. We illustrate our theorem by considering a perturbed functional differential equation under periodic boundary conditions.

1 Introduction

Continuation theorems have been used to derive periodic solutions for differential systems with perturbations. In particular, in [1], existence criteria for ω-periodic solutions are given for the equation

x′ = g(x) + e(t, x)

by means of ‘continuation’ through an admissible homotopy carrying the given problem to the equation

x′ = g(x),

which admits only the trivial ω-periodic solution (see [1, pp. 101-103]).

In this note, we are interested in the study of a similar problem for the perturbed functional differential system

x′ = g(t, x_{t}) + h(t, x_{t}), 0 ≤ t ≤ ω,

with solutions that satisfy the periodic boundary condition

x(0) = x(ω) .

This will be achieved by first proving an abstract existence theorem utilizing Borsuk’s theorem and continuation through an admissible homotopy carrying our given problem to the equation

x′ = g(t, x_{t}),

which admits only the trivial periodic solution.

∗Mathematics Subject Classifications: 34K10, 34C20.

Key words: Borsuk’s theorem, Fredholm mapping, perturbed differential equation,

periodic solution.

\begin{matrix}c \\ ◯ \end{matrix}2001 Southwest Texas State University.

Published January 8, 2001.

2 Main Results

Let X, Y be real normed spaces with respective norms ∥·∥_{X} and ∥·∥_{Y}. Let L : dom(L) ⊆ X → Y be a linear Fredholm mapping of index zero, and let Ω be an open and bounded subset of X. It is well known [1, Section 2.2] that there exist projections P : X → X and Q : Y → Y such that Im P = ker L, ker Q = Im L and X = ker L ⊕ ker P, Y = Im L ⊕ Im Q. Suppose F : dom(L) ∩ \overline{Ω} → Y has the form F = L − N where N : \overline{Ω} → Y is L-compact on \overline{Ω} and satisfies the condition 0 ∉ F (dom(L) ∩ ∂Ω). Then a coincidence degree D_{L}(F, Ω) can be defined which satisfies the properties listed in [1, Section 2.3]. As mentioned above, we will need the following Borsuk’s Theorem: Suppose Ω is an open, bounded subset of X which is symmetric with respect to the origin and suppose further that the function F mentioned above satisfies the additional condition that F (−x) = −F (x) for every x ∈ dom(L) ∩ ∂Ω, then the coincidence degree D_{L}(F, Ω) is odd. We remark that there are a number of studies which are concerned with the existence of periodic solutions of differential equations by means of coincidence theory, see for examples [2-6].

Lemma 2.1 Let \overline{Ω} = \{x ∈ X| ∥x∥_{X} ≤ 1\}. Let N_{2} : X → Y be a continuous mapping which maps bounded sets into bounded sets and satisfies

lim_{∥x∥_{X}→∞} \begin{matrix}∥N_{2}x∥ & Y \\ ∥x∥ & {β \atop X} \end{matrix} = 0  (2.1)

for some β ∈ (0, 1]. Suppose H : \overline{Ω} × [0, 1] → Y is defined by

H(x, μ) = \{ \begin{matrix}μ^{β}N_{2}(μ^{−β}x) & if μ ∈ (0, 1] \\ 0 & if μ = 0 . \end{matrix}

Then H is continuous and bounded on \overline{Ω} × [0, 1].

Proof. To show that H is continuous, it suffices to show that H is continuous at (x, 0) where x ∈ \overline{Ω}. For any ε ∈ (0, 1), in view of assumption (2.1), we see that there exists a constant ρ > 0 such that for arbitrary x ∈ X which satisfies ∥x∥_{X} > ρ, ∥N_{2}x∥_{Y} ≤ ε ∥x∥_{X}^{β}. Since N_{2} maps bounded sets into bounded sets, hence

M = sup {∥N_{2}x∥_{Y} : ∥x∥_{X} ≤ ρ < ∞} > 0.

Let μ_{0} = (\frac{ε}{M+1})^{1/β}. Clearly,

0 < μ_{0} < (\frac{1}{M + 1})^{1/β} .

For every positive μ ≤ μ_{0} and every x ∈ \overline{Ω}, we assert that ∥H(x, μ)∥_{Y} < ε. In fact, if μ^{−β} ∥x∥_{X} > ρ, then

∥H(x, μ)∥_{Y} ≤ μ^{β} ∥N_{2}(μ^{−β}x)∥_{Y}

≤ μ^{β}ε ∥μ^{−β}x∥_{X}^{β}
≤ μ^{β}εμ^{−β^{2}} ∥x∥_{X}^{β}
≤ μ_{0}^{β(1−β)}ε
< (\frac{1}{M + 1})^{1−β}ε < ε,

and if μ^{−β} ∥x∥_{X} ≤ ρ, then

∥H(x, μ)∥_{Y} ≤ μ^{β} ∥N_{2}(μ^{−β}x)∥_{Y} ≤ μ^{β}M ≤ \frac{ε}{M + 1}M < ε.

Thus we have shown that H is continuous at (x, 0) ∈ \overline{Ω} × [0, 1].

By arguments similar to those just described, we may show by means of the continuity of H at (x, 0) ∈ \overline{Ω} × [0, 1] that there exists a constant δ > 0 and a real number M_{1} such that for (x, μ) ∈ \overline{Ω} × [0, δ], ∥H(x, μ)∥_{Y} ≤ M_{1}. Since N_{2} maps bounded sets into bounded sets, there exists a number M_{2} such that ∥H(x, μ)∥_{Y} ≤ M_{2} for (x, μ) ∈ \overline{Ω} × [δ, 1]. Thus H is bounded on \overline{Ω} × [0, 1]. The proof is complete. Let us now consider the operator equation

Lx = N_{1}x + N_{2}x, x ∈ X,  (2.2)

where

H1) L is a linear Fredholm mapping of index zero,

H2) N_{1} : X → Y is a continuous mapping which satisfies N_{1}(λx) = λN(x) for λ ∈ (−∞, ∞) and x ∈ X,

H3) N_{2} : X → Y is a continuous mapping which maps bounded sets into bounded sets and satisfies (2.1) for some β ∈ (0, 1],

H4) N_{1}, N_{2} are L-completely continuous.

Theorem 2.2 Suppose the conditions H1-H4 hold. Suppose further that

Lx = N_{1}x  (2.3)

admits only the trivial solution. Then (2.2) has a nontrivial solution in dom L∩ \overline{Ω}.

Proof. Let Ω = \{x ∈ X| ∥x∥_{X} ≤ 1\}. Let T : \overline{Ω} × [0, 1] → Y be defined by

T (x, μ) = \{ \begin{matrix}N_{1}x + μ^{β}N_{2}(μ^{−β}x) & if μ ∈ (0, 1] \\ N_{1}x & if μ = 0 . \end{matrix}  (2.4)

Then

T (x, 1) = N_{1}x + N_{2}x, x ∈ \overline{Ω},

furthermore, in view of Lemma 2.1, T is continuous and bounded on \overline{Ω} × [0, 1]. Since N_{1} and N_{2} are L-completely continuous, it is also easy to see that T is L-compact on \overline{Ω} × [0, 1].

Note that, in view of the assumption that (2.3) admits only the trivial solution, for any x ∈ ∂Ω, (x, 0) cannot be a solution of

Lx = T (x, μ).  (2.5)

Note further that if (x, μ) ∈ ∂Ω × (0, 1] is a nontrivial solution of (2.5), then in view of (2.4) and (H2), μ^{−β}x will be a nontrivial solution of (2.2).

Let F̃ = L − T. Suppose to the contrary that the operator equation (2.2) does not have any nontrivial solutions, then in view of the above discussions, 0 ∉ F̃((dom(L) ∩ ∂Ω) × [0, 1]). Thus the degree D_{L}(F̃(·, μ), Ω) can be defined for arbitrary μ ∈ [0, 1], and it takes constant on [0, 1]. But since

F̃(−x, 0) = −Lx − T (−x, 0) = −Lx − N_{1}(−x)
= −Lx + N_{1}x = −Lx + T (x, 0) = −F̃(x, 0)

for all x ∈ X, by Borsuk’s Theorem stated above, we see that D_{L}(F̃(·, 0), Ω), and (hence) D_{L}(F̃(·, 1), Ω) are odd. But this is contrary to the existence property of the coincidence degree. The proof is complete.

Let us now turn back to the perturbed functional differential equation

x′ = g(t, x_{t}) + h(t, x_{t}), 0 ≤ t ≤ ω,  (2.6)

under the periodic boundary condition

x(0) = x(ω),  (2.7)

where x(t) ∈ C(R, R^{n}), x_{t} ∈ BC (R, R^{n}) are given by x_{t}(s) = x(t + s), and g, h : [0, ω]×BC(R, R^{n}) → R^{n} are continuous mappings that take bounded sets into bounded sets. Here BC(R, R^{n}) is the linear normed space of all continuous and bounded functions from R into R^{n} endowed with the usual supremum norm.

Theorem 2.3 Assume that

g(t, λx) = λg(t, x), λ, t ∈ R; x ∈ BC(R, R^{n}),  (2.8)

and there exists β ∈ (0, 1] such that

lim_{∥x∥→∞} \frac{|h(t, x)|}{∥x∥^{β}} = 0 uniformly in t ∈ [0, ω].  (2.9)

Suppose further that the boundary value problem

x′ = g(t, x_{t}) t ∈ [0, ω]
x(0) = x(ω)  (2.10)
x(t) = x(0) t ∈ (−∞, 0] ∪ [ω, ∞)

admits only the trivial solution. Then (2.6) has a nontrivial solution x that satisfies (2.7).

Proof. Let

X = {x ∈ C(R, R^{n})| x(0) = x(ω), x(t) = x(0), t ∈ (−∞, 0] ∪ [ω, ∞)} ,

and Y = C ([0, ω], R^{n}). Then X is a closed subset in BC (R, R^{n}) , and therefore it is a Banach space. Let dom(L) = {x ∈ X| x′ is continuous on [0, ω]} , let L : dom(L)∩X → Y be defined by (Lx)(t) = x′(t) for t ∈ R, and let N : X → Y be defined by

(Nx)(t) = (N_{1}x)(t) + (N_{2}x)(t), t ∈ R,

where (N_{1}x)(t) = g(t, x_{t}), (N_{2}x)(t) = h(t, x_{t}) for t ∈ R. Then it is easy to show that the kernel of L is

ker L = {x ∈ X| x = c ∈ R^{n}} ,

the image of L is

Im L = \{y ∈ Y | \frac{1}{ω} ∫_{0}^{ω} y(s)ds = 0\} ,

and dim ker L = codim Im L = n. Furthermore, if we define the projections P : X → X and Q : Y → Y by

(P x)(t) = x(0), t ∈ R,

and

(Qy)(t) = \frac{1}{ω} ∫_{0}^{ω} y(s)ds, t ∈ R,

respectively, then ker L = Im P and ker Q = Im L. Thus, L is a Fredholm operator with index zero, and the generalized inverse K_{P} : Im L → ker P ∩ dom(L) of L is given by

(K_{P} y)(t) = \{ \begin{matrix}∫_{0}^{t} y(s)ds & if 0 ≤ t ≤ ω \\ 0 & if t ∈ (−∞, 0] ∪ [ω, ∞) , \end{matrix}

and is compact. Since

(QN)(x) = \frac{1}{ω} ∫_{0}^{ω}(g(s, x_{s}) + h(s, x_{s}))ds,

we easily see that QN(\overline{Ω}) is bounded, furthermore, by the Arzela-Ascoli theorem, it is also easily seen that K_{P} (I − Q)N : \overline{Ω} → X is compact. As a consequence, N is L-compact on \overline{Ω}.

Note that the conditions (H2) and (H3) follow (2.8) and (2.9) respectively, and that Lx = N_{1}x admits only the trivial solution. By Theorem 2.2, (2.6) will have a nontrivial solution which satisfies (2.7). The proof is complete.

As an example, consider the boundary value problem

x′ = p(t)x(t − τ) + p(t) (−x^{1/2}(t − τ) + a) , 0 ≤ t ≤ ω,
x(0) = x(ω),

where a, τ, ω are real numbers which satisfy 0 < ω < τ and a ≤ 1/4. The function p ∈ C(R, R) is bounded and

∫_{0}^{ω} p(s)ds ≠ 0.

Let β = 3/4. Then

lim_{|x|→∞} \frac{|p(t) (−x^{1/2} + a)|}{|x|^{β}} ≤ lim_{|x|→∞} \begin{matrix}max & |p(t)| & (|x|^{1/2} + |a|) \\ |x| & 3/4 \end{matrix} = 0.

Furthermore, since x(t − τ) = x(0) for 0 ≤ t ≤ ω, x ≡ 0 is the unique solution of the periodic boundary problem

x′ = p(t)x(t − τ) t ∈ [0, ω]
x(0) = x(ω)
x(t) = x(0) − τ ≤ t ≤ 0

By Theorem 2.3, there will be a nontrivial solution of our boundary value problem. In fact,

x(t) = (\frac{1 + \sqrt{1 − 4a}}{2})^{1/2}, −τ ≤ t ≤ ω,

is one of its nontrivial solutions.

We remark that similar results can be obtained for boundary-value problems involving infinite delay, or problems of the form

x^{(m)}(t) = g (t, x′_{t}, ..., x_{t}^{(m−1)}) + h (t, x′_{t}, ..., x_{t}^{(m−1)}) , 0 ≤ t ≤ T,
x^{(i)}(0) = x^{(i)}(T), i = 0, 1, ..., m − 1.

References

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[2] F. Zanolin, Periodic solutions for differential systems of Rayleigh type, Tend. Istit. Mat. Univ. Trieste, 12(1980), no. 1-2, 69-77.

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[4] G. Q. Wang and S. S. Cheng, A priori bounds for periodic solutions of a delay Rayleigh equation, Applied Math, Lett., 12(1999), 41-44.

[5] S. W. Ma, J. S. Yu and Z. C. Wang, The periodic solutions of functional differential equations with perturbation, Chinese Sci. Bull., 43(1998), 1386-Sui Sun Cheng Department of Mathematics, Tsing Hua University Taiwan 30043, R. O. China e-mail: sscheng@math.nthu.edu.tw Bin Liu Department of Mathematics, Hubei Normal Univeristy Huangshi, Hubei 435002, P. R. China Jian-She Yu Department of Applied Mathematics, Hunan University Changsha, Hunan 410082, P. R. China