Existence of solutions for weighted p(r)-Laplacian impulsive system mixed type boundary value problems

This paper investigates the existence of solutions for weighted p(r)-Laplacian impulsive system mixed type boundary value problems. The proof of our main result is based upon Gaines and Mawhin's coincidence degree theory. Moreover, we obtain the existence of nonnegative solutions.

In this paper, we mainly consider the existence of solutions for the weighted p(r)-Laplacian system

where u: [0, T] → ℝ^N, with the following impulsive boundary conditions

where p ∈ C ([0, T], ℝ) and p(r) > 1, -Δ_p(r) u:= -(w(r) |u'|^p(r)-2 u'(r))' is called weighted p(r)-Laplacian; 0 < r₁ < r₂ < ⋯ < r_k < T; A_i, B_i ∈ C(ℝ^N × ℝ^N, ℝ^N); a, b, c, d ∈ [0, +∞), ad + bc > 0.

Throughout the paper, o(1) means functions which uniformly convergent to 0 (as n → +∞); for any v ∈ ℝ^N, v^j will denote the j-th component of v; the inner product in ℝ^N will be denoted by 〈·,·〉; |·| will denote the absolute value and the Euclidean norm on ℝ^N. Denote J = [0, T], J' = [0, T]\{r₀, r₁,..., r_k+1}, J₀ = [r₀, r₁], J_i = (r_i, r_i+1], i = 1, ..., k, where r₀ = 0, r_k+1 = T. Denote the interior of J_i, i = 0, 1,..., k. Let PC(J, ℝ^N) = {x: J → ℝ^N | x ∈ C(J_i, ℝ^N), i = 0, 1,..., k, and exists for i = 1,..., k}; w ∈ PC(J, ℝ) satisfies 0 < w(r), ∀r ∈ J', and ; , and exists for i = 0, 1,..., k}. For any x = (x¹,..., x^N) ∈ PC(J, ℝ^N), denote |xⁱ|₀ = sup_r∈J' |xⁱ(r)|. Obviously, PC(J, ℝ^N) is a Banach space with the norm , PC¹(J, ℝ^N) is a Banach space with the norm . In the following, PC(J, ℝ^N) and PC¹(J, ℝ^N) will be simply denoted by PC and PC¹, respectively. Let L¹ = L¹(J, ℝ^N) with the norm , ∀x ∈ L¹, where . We will denote

The study of differential equations and variational problems with nonstandard p(r)-growth conditions is a new and interesting topic. It arises from nonlinear elasticity theory, electro-rheological fluids, image processing, etc. (see [1-4]). Many results have been obtained on this problems, for example [1-25]. If p(r) ≡ p (a constant), (1) is the well-known p-Laplacian system. If p(r) is a general function, -Δ_p(r) represents a nonhomogeneity and possesses more nonlinearity, thus -Δ_p(r) is more complicated than -Δ_p; for example, if Ω ⊂ ℝ^N is a bounded domain, the Rayleigh quotient

is zero in general, and only under some special conditions λ_p(·) > 0 (see [8,17-19]), but the property of λ_p > 0 is very important in the study of p-Laplacian problems.

Impulsive differential equations have been studied extensively in recent years. Such equations arise in many applications such as spacecraft control, impact mechanics, chemical engineering and inspection process in operations research (see [26-28] and the references therein). It is interesting to note that p(r)-Laplacian impulsive boundary problems are about comparatively new applications like ecological competition, respiratory dynamics and vaccination strategies. On the Laplacian impulsive differential equation boundary value problems, there are many results (see [29-37]). There are many methods to deal with this problem, e.g., subsupersolution method, fixed point theorem, monotone iterative method and coincidence degree. Because of the nonlinearity of -Δ_p, results on the existence of solutions for p-Laplacian impulsive differential equation boundary value problems are rare (see [38,39]). On the Laplacian (p(x) ≡ 2) impulsive differential equations mixed type boundary value problems, we refer to [30,32,34].

In [39], Tian and Ge have studied nonlinear IBVP

where Φ_p(x) = |x|^p-2 x, p > 1, ρ, s ∈ L^∞ [a, b] with essin f_{[a, b]} ρ > 0, and essin f_[a,b] s > 0, 0 < ρ(a), p(b) <∞, σ₁ ≤ 0, σ₂ ≥ 0, α, β, γ, σ > 0, a = t₀ < t₁ < ⋯ < t_l < t_l+1 = b, I_i ∈ C([0, +∞), [0, ∞)), i = 1,..., l, f ∈ C ([a, b] × [0, +∞), [0, ∞)), f(·, 0) is nontrivial. By using variational methods, the existence of at least two positive solutions was obtained.

In [24,25], the present author investigates the existence of solutions of p(r)-Laplacian impulsive differential equation (1-3) with periodic-like or multi-point boundary value conditions.

In this paper, we consider the existence of solutions for the weighted p(r)-Laplacian impulsive differential system mixed type boundary value condition problems, when p(r) is a general function. The proof of our main result is based upon Gaines and Mawhin's coincidence degree theory. Since the nonlinear term f in (5) is independent on the first-order derivative, and the impulsive conditions are simpler than (2), our main results partly generalized the results of [30,32,34,39]. Since the mixed type boundary value problems are different from periodic-like or multi-point boundary value conditions, and this paper gives two kinds of mixed type boundary value conditions (linear and nonlinear), our discussions are different from [24,25] and have more difficulties. Moreover, we obtain the existence of nonnegative solutions. This paper was motivated by [24-26,38,40].

Let N ≥ 1, the function f: J × ℝ^N × ℝ^N → ℝ^N is assumed to be Caratheodory; by this, we mean:

(i) for almost every t ∈ J, the function f(t, ·, ·) is continuous;

(ii) for each (x, y) ∈ ℝ^N × ℝ^N, the function f(·, x, y) is measurable on J;

(iii) for each R > 0, there is a α_R ∈ L¹ (J, ℝ), such that, for almost every t ∈ J and every (x, y) ∈ ℝ^N × ℝ^N with |x| ≤ R, |y| ≤ R, one has

We say a function u: J → ℝ^N is a solution of (1) if u ∈ PC¹ with w(·) |u'|^p(·)-2 u'(·) absolutely continuous on every , i = 0, 1,..., k, which satisfies (1) a.e. on J.

This paper is divided into three sections; in the second section, we present some preliminary. Finally, in the third section, we give the existence of solutions and nonnegative solutions of system (1)-(4).

Let X and Y be two Banach spaces and L: D(L) ⊂ X → Y be a linear operator, where D(L) denotes the domain of L. L will be a Fredholm operator of index 0, i.e., ImL is closed in Y and the linear spaces KerL and coImL have the same dimension which is finite. We define X₁ = KerL and Y₁ = coImL, so we have the decompositions X = X₁ ⊕ coKerL and Y = Y₁ ⊕ ImL. Now, we have the linear isomorphism Λ: X₁ → Y₁ and the continuous linear projectors P: X → X₁ and Q: Y → Y₁ with KerQ = ImL and ImP = X₁.

Let Ω be an open bounded subset of X with Ω ∩ D(L) ≠ ∅. Operator be a continuous operator. In order to define the coincidence degree of (L, S) in Ω, as in [40,41], denoted by d(L - S, Ω), we assume that

It is easy to see that the operator , M = (L + ΛP)^-1 (S + ΛP) is well defined, and

If M is continuous and compact, then S is called L-compact, and the Leray-Schauder degree of I_X - M (where I_X is the identity mapping of X) is well defined in Ω, and we will denote it by d_LS (I_X - M, Ω, 0). This number is independent of the choice of P, Q and Λ (up to a sign) and we can define

Definition 2.1. (see [40,41]) The coincidence degree of (L, S) in Ω, denoted by d(L - S, Ω), is defined as d(L - S, Ω) = d_LS (I_X - M, Ω, 0).

There are many papers about coincidence degree and its applications (see [40-43]).

Proposition 2.2. (see [40]) (i) (Existence property). If d(L - S, Ω) ≠ 0, then there exists x ∈ Ω such that Lx = Sx.

(ii) (Homotopy invariant property). If is continuous, L-compact and Lx ≠ H(x, λ) for all x ∈ ∂Ω and λ ∈ [0, 1], then d(L - H (·, λ), Ω) is independent of λ.

The effect of small perturbations is negligible, as is proved in the next Proposition (see [41] Theorem III.3, page 24).

Proposition 2.3. Assume that Lx ≠ Sx for each x ∈ ∂Ω. If S_ε is such that sup_x∈∂Ω||S_εx||_Y is sufficiently small, then Lx ≠ Sx + S_εx for all x ∈ ∂Ω and d(L - S - S_ε, Ω) = d(L - S, Ω).

For any (r, x) ∈ (J × ℝ^N), denote φ_p(r)(x) = |x|^p(r)-2x. Obviously, φ has the following properties

Proposition 2.4 (see [41]) φ is a continuous function and satisfies

(i) For any r ∈ [0, T], φ_p(r)(·) is strictly monotone, i.e.,

(ii) There exists a function η: [0, +∞) → [0, +∞), η(s) → +∞ as s → +∞, such that

It is well known that φ_p(r)(·) is a homeomorphism from ℝ^N to ℝ^N for any fixed r ∈ J. Denote

It is clear that is continuous and sends bounded sets to bounded sets, and where . Let X = {(x₁, x₂) | x₁ ∈ PC, x₂ ∈ PC} with the norm ||(x₁, x₂)||_X = || x₁||₀ + ||x₂||₀, Y = L¹ × L¹ × ℝ^{2(k + 1)N}, and we define the norm on Y as

where y₁, y₂ ∈ L¹, z_m ∈ ℝ^N, m = 1,..., 2(k + 1), then X and Y are Banach spaces.

Define L: D(L) ⊂ X → Y and S: X → Y as the following

where

Obviously, the problem (1)-(4) can be written as Lx = Sx, where L: X → Y is a linear operator, S: X → Y is a nonlinear operator, and X and Y are Banach spaces.

Since

we have dimKerL = dim(Y/ImL) = 2N < +∞ is even and ImL is closed in Y, then L is a Fredholm operator of index zero. Define

at the same time the projectors P: X → X and Q: Y → Y satisfy

Since ImQ is isomorphic to KerL, there exists an isomorphism Λ: KerL → ImQ. It is easy to see that L |_D(L)∩KerP : D(L) ∩ KerP → ImL is invertible. We denote the inverse of that mapping by K_p, then K_p : ImL → D(L) ∩ KerP as

then

Proposition 2.5 (i) K_p(·) is continuous;

(ii) K_p (I - Q)S is continuous and compact.

Proof. (i) It is easy to see that K_p(·) is continuous. Moreover, the operator sends equi-integrable set of L¹ to relatively compact set of PC.

(ii) It is easy to see that K_p(I - Q)Sx ∈ X, ∀x ∈ X. Since and f is Caratheodory, it is easy to check that S is a continuous operator from X to Y, and the operators (x₁, x₂) → φ_q(r) ((w(r))^-1 x₂) and (x₁, x₂) → f (r, x₁, φ_q(r) ((w(r))^-1 x₂)) both send bounded sets of X to equi-integrable set of L¹. Obviously, A_i, B_i and QS are compact continuous. Since f is Caratheodory, by using the Ascoli-Arzela theorem, we can show that the operator is continuous and compact. This completes the proof.

Denote

where A_i, B_i are defined in (6), i = 1,..., k.

Consider

Define as M(·, ·) = (L + ΛP)^-1 (S(·, ·) + ΛP), then

Since (I - Q)S(·, 0) = 0 and K_p (0) = 0, we have

It is easy to see that all the solutions of Lx = S(x, 0) belong to KerL, then

Notice that P |_KerL = I_KerL, then

Proposition 2.6 (continuation theorem) (see [40]). Suppose that L is a Fredholm operator of index zero and S is L-compact on , where Ω is an open bounded subset of X. If the following conditions are satisfied,

(i) for each λ ∈ (0, 1), every solution x of

is such that x ∉ ∂Ω;

(ii) QS(x, 0) ≠ 0 for x ∈ ∂Ω ∩ KerL and d_B(Λ^-1 QS(·,0), Ω ∩ KerL, 0) ≠ 0, then the operator equation Lx = S(x, 1) has one solution lying in .

The importance of the above result is that it gives sufficient conditions for being able to calculate the coincidence degree as the Brouwer degree (denoted with d_B) of a related finite dimensional mapping. It is known that the degree of finite dimensional mappings is easier to calculate. The idea of the proof is the use of the homotopy of the problem Lx = S(x, 1) with the finite dimensional one Lx = S(x, 0).

Let us now consider the following simple impulsive problem

where J' = [0, T]\{r₀, r₁, ..., r_k+1}, a_i, b_i ∈ ℝ^N; g ∈ L¹.

If u is a solution of (7), then we have

Denote ρ₀ = w(0)_φp(0) (u'(0)). Obviously, ρ₀ is dependent on g, a_i, b_i. Define F: L¹ → PC as

By (8), we have

If a ≠ 0, then the boundary condition implies that

The boundary condition implies that

Denote H = L¹ × ℝ^2kN with the norm

then H is a Banach space. For fixed h ∈ H, we denote

Lemma 2.7 The mapping Θ_h(·) has the following properties

(i) For any fixed h ∈ H, the equation

has a unique solution ρ(h) ∈ ℝ^N.

(ii) The mapping ρ: H → ℝ^N, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, , where h = (g, a_i, b_i) ∈ H, , the notation p^# means .

Proof. (i) From Proposition 2.4, it is immediate that

and hence, if (10) has a solution, then it is unique.

Let . Since and F(g) ∈ PC, if |ρ| > R₀, it is easy to see that there exists a j₀ such that, the j₀-th component of ρ satisfies

Obviously,

then

and

By (11) and (12), the j₀-th component of keeps the same sign of on J and

Combining (13) and (14), the j₀-th component of satisfies

From the definition , we have , then , and

Without loss of generality, we may assume that , then we have

Therefore, the j₀-th component of keeps the same sign of . Since the j₀-th component of keeps the same sign of , a, b, c, d ∈ [0, +∞) and ad + bc > 0, we can easily see that the j₀-th component of Θ_h(ρ) keeps the same sign of , and thus

Let us consider the equation

According to the above discussion, all the solutions of (15) belong to b(R₀ + 1) = {x ∈ ℝ^N| |x| < R₀ + 1}. So, we have

It means the existence of solutions of Θ_h(ρ) = 0.

In this way, we define a mapping ρ(h): H → ℝ^N, which satisfies

(ii) By the proof of (i), we also obtain ρ sends bounded set to bounded set, and

It only remains to prove the continuity of ρ. Let {u_n} is a convergent sequence in H and u_n → u, as n → +∞. Since {ρ(u_n)} is a bounded sequence, it contains a convergent subsequence satisfies as j → +∞. Since Θ_h(ρ) consists of continuous functions, and

Letting j → +∞, we have

from (i) we get ρ_* = ρ(u), it means that ρ is continuous.

This completes the proof.

If a = 0, the boundary condition implies that

Since ad + bc > 0, we have c > 0. Thus,

the boundary condition implies that

Denote G: H → ℝ^N as

It is easy to see that

Lemma 2.8 The function G(·) is continuous and sends bounded sets to bounded sets. Moreover, , where , the notation p* means .

In this section, we will apply coincidence degree to deal with the existence of solutions for (1)-(4). In the following, we always use C and C_i to denote positive constants, if it cannot lead to confusion.

Theorem 3.1 Assume that Ω is an open bounded set in X such that the following conditions hold.

(1⁰) For each λ ∈ (0, 1) the problem

has no solution on ∂Ω.

(2⁰) (0, 0) ∈ Ω.

Then, problem (1)-(4) has a solution u satisfies , where v = w(r)φ_p(r)(u'(r)), ∀r ∈ J'.

Proof. Let us consider the following operator equation

It is easy to see that x = (x₁, x₂) is a solution of Lx = S(x, 1) if and only if x₁(r) is a solution of (1)-(4) and , ∀r ∈ J'.

According to Proposition 2.5, we can conclude that S(·, ·) is L-compact from X × [0, 1] to Y. We assume that for λ = 1, (16) does not have a solution on ∂Ω, otherwise we complete the proof. Now from hypothesis (1⁰), it follows that (16) has no solutions for (x, λ) ∈ ∂Ω × (0, 1]. For λ = 0, (17) is equivalent to Lx = S(x, 0), namely the following usual problem

The problem (??) is a usual differential equation. Hence,

where c₁, c₂ ∈ ℝ^N are constants. The boundary value condition of (??) holds,

Since (ad + bc) > 0, we have

which together with hypothesis (2⁰), implies that (0, 0)∈ Ω. Thus, we have proved that (16) has no solution on ∂Ω × [0, 1]. It means that the coincidence degree d[L - S(·, λ), Ω] is well defined for each λ ∈ [0, 1]. From the homotopy invariant property of that degree, we have

Now, it is clear that the following problem

is equivalent to problem (1)-(4), and (18) tells us that problem (19) will have a solution if we can show that

Since by hypothesis (2⁰), this last degree

where ω_*(c₁, c₂) = (ac₁ - bφ_q(0)(c₂), cc₁ + dc₂). This completes the proof.

Our next theorem is a consequence of Theorem 3.1. Denote

Theorem 3.2 Assume that the following conditions hold

(1⁰) a > 0;

(2⁰) lim_{|u| + |v| → +∞} (f(r, u, v)/(|u| + |v|)^{β(r) -1}) = 0, for r ∈ J uniformly, where β(r) ∈ C(J, ℝ), and 1<β ^- ≤ β ⁺ < p ^-;

(3⁰) when |u| + |v| is large enough, where ;

(4⁰) when |u| + |v| is large enough, where 0 ≤ ε < β ⁺ - 1.

Then, problem (1)-(4) has at least one solution.

Proof. Now, we consider the following operator equation

For any λ ∈ (0, 1], x = (x₁, x₂) = (u, v) is a solution of (20) if and only if and u(r) is a solution of the following

We claim that all the solutions of (21) are uniformly bounded for λ ∈ (0, 1]. In fact, if it is false, we can find a sequence (u_n, λ_n) of solutions for (21), such that ||u_n||₁ > 1 and ||u_n||₁ → +∞ when n → +∞, λ_n ∈ (0, 1]. Since (u_n, λ_n) are solutions of (21), we have

for any r ∈ J', where and

By computation, we have

Denote

We claim that

If it is false, without loss of generality, we may assume that

then for any n = 1, 2, ..., there is a j_n ∈ {1, ..., N} such that the j_n-th component of ρ_n satisfies

Thus, when n is large enough, the j_n-th component of Γ_n(r) keeps the same sign as and satisfies

When n is large enough, we can conclude that the j_n-th component of F{φ_q(r)[Γ_n(r)]} (T) keeps the same sign as and satisfies

Since

from (22) and (24), we can see that keeps the same sign as , when n is large enough.

But the boundary value conditions (4) mean that

It is a contradiction. Thus (23) is valid. Therefore,

It means that

From (22), (23) and (25), for any r ∈ J, we have

then

From (25) and (26), we get that all the solutions of (20) are uniformly bounded for any λ ∈ (0, 1].

When λ = 0, if (x₁, x₂) is a solution of (20), then (x₁, x₂) is a solution of the following usual equation

we have

Thus, there exists a large enough R₀ > 0 such that all the solutions of (20) belong to B(R₀) = {x ∈ X | || x ||_X < R₀}. Thus, (20) has no solution on ∂B (R₀). From theorem 3.1, we obtain that (1)-(4) has at least one solution. This completes the proof.

Theorem 3.3 Assume that the following conditions hold

(1⁰) a = 0;

(2⁰) lim_{|u| + |v| → +∞} f(r, u, v)/(|u| + |v|)^ε = 0 for r ∈ J uniformly, where 0 ≤ ε min(1, p^- - 1);

(3⁰) when |u| + |v| is large enough, where 0 < θ < 1;

(4⁰) when |u| + |v| is large enough, where 0 ≤ ε < min(1, p^- - 1).

Then, problem (1)-(4) has at least one solution.

Proof Now, we consider the following operator equation

If (x₁, x₂) is a solution of (27) when λ = 0, then (x₁, x₂) is a solution of the following usual equation

Then, we have

For any λ ∈ (0, 1], x = (x₁, x₂) = (u, v) is a solution of (27) if and only if and u(r) is a solution of the following

We only need to prove that all the solutions of (28) are uniformly bounded for λ ∈ (0, 1].

In fact, if it is false, we can find a sequence (u_n, λ_n) of solutions for (28), such that ||u_n||₁ > 1 and ||u_n||₁ → +∞ when n → +∞. Since (u_n, λ_n) are solutions of (28), we have

where .

From conditions (2⁰) and (4⁰), we have

Thus,

Denote

By solving u_n(r), we have

where .

From condition (3⁰), we have

The boundary value condition implies

From (31) and conditions (2⁰), (3⁰) and (4⁰), we have

From (30) and (32), we have

From (29) and (33), we can conclude that {||u_n||₁} is uniformly bounded for λ ∈ (0, 1]. This completes the proof.

Now, let us consider the following mixed type boundary value condition

Theorem 3.4 Assume that the following conditions hold

(1⁰) , for r ∈ J uniformly, where β(r) ∈ C(J, ℝ), and 1 < β ^- ≤ β ⁺< p^-;

(2⁰) when |u| + |v| is large enough, where ;

(3⁰) when |u| + |v| is large enough, where 0 ≤ ε < β ⁺ - 1.

Then, problem (1) with (2), (3) and (34) has at least one solution.

Proof. It is similar to the proof of Theorem 3.2 and Theorem 3.3, we omit it.

Denote

where f_* (r, u, v) is Caratheodory.

Let us consider

Theorem 3.5 Under the conditions of Theorem 3.2, Theorem 3.3 or Theorem 3.4, then (35) with (2), (3) and (4) or (34) has at least a solution when δ is small enough.

Proof We only need to prove the existence of solutions under the conditions of Theorem 3.2, the rest is similar. If δ = 0, the proof of Theorem 3.2 means that all the solutions of (35) with (2), (3) and (4) are bounded and belong to U(R₀) = {u ∈ PC¹| ||u||₁ < R₀}. Define S_δ : X → Y as

where , .

Since f_* (r, u, v) is a Caratheodory function, we have ||S_δx - S₀x||_Y → 0 as δ → 0, for uniformly. According to Proposition 2.3, we get the existence of solutions.

In the following, we will consider the existence of nonnegative solutions. For any x = (x¹, ..., x^N) ∈ ℝ^N, the notation x ≥ 0 means xⁱ ≥ 0 for any i = 1, ..., N.

Theorem 3.6 We assume

(i) f(r, u, v) ≤ 0, ∀(r, u, v) ∈ J × ℝ^N × ℝ^N;

(ii) for any i = 1, ..., k, B_i(u, v) ≤ 0,∀(u, v) ∈ ℝ^N × ℝ^N;

(iii) for any i = 1, ..., k, j = 1, ..., N, ∀(u, v) ∈ ℝ^N × ℝ^N.

Then, the solution u in Theorem 3.2, Theorem 3.3 or Theorem 3.4 is nonnegative.

Proof We only need to prove that the solution u in Theorem 3.2 is nonnegative, and the rest is similar. Denote

where

Similar to (8) and (9), we have

where

and ρ is the solution (unique) of

Denote

From (i), (ii) and (36), we can see that Φ(r) is decreasing, namely

We claim that

If it is false, then there exists some j₀ ∈ {1, ..., N}, such that the j₀-th component of ρ satisfies

Combining (i), (ii), (iii) and (40), we can see that the j₀-th component of D is negative. It is a contradiction to (37). Thus, (39) is valid. So, we have

We claim that

If it is false. Then, there exists some j₁ ∈ {1,..., N}, such that the j₁-th component of Φ(T) satisfies

From (38) and (42), we have

Combining (i), (ii), (iii) and (42), we can see that the j₁-th component of D is positive. It is a contradiction to (37). Thus, (41) is valid.

If c > 0. We have

Since Φ(r) is decreasing, Φ(0) = ρ ≥ 0 and Φ(T) ≤ 0, for any j = 1, ..., N, there exists ξ_j ∈ J such that

Combining condition (iii), we can conclude that u^j(r) is increasing on [0, ξ_j], and u^j(r) is decreasing on (ξ_j, T]. Notice that u(0) ≥ 0 and u(T) ≥ 0, then we have u(r) ≥ 0,∀r ∈ [0, T].

If c = 0, boundary condition (4) means that Φ(T) = 0. Since Φ(r) is decreasing, we get that Φ(r) ≥ 0. Combining condition (iii), we can conclude that u(r) is increasing on J, namely u(t₂) - u(t₁) ≥ 0,∀t₂, t₁ ∈ J, t₂ > t₁. Notice that u(0) ≥ 0, then we have u(r) ≥ 0,∀t ∈ J. This completes the proof.

Corollary 3.7 We assume

(i) f(r, u, v) ≤ 0,∀(r, u, v) ∈ J × ℝ^N × ℝ^N with u ≥ 0;

(ii) for any i = 1, ..., k, B_i(u, v) ≤ 0,∀(u, v) ∈ ℝ^N × ℝ^N with u ≥ 0;

(iii) for any i = 1, ..., k, j = 1, ..., N, ,∀(u, v) ∈ ℝ^N × ℝ^N with u ≥ 0.

Then, we have

(1₀) Under the conditions of Theorem 3.2 or Theorem 3.3, (1)-(4) has a nonnegative solution.

(2₀) Under the conditions of Theorem 3.4, (1) with (2), (3) and (34) has a nonnegative solution.

Proof We only need to prove that (1)-(4) has a nonnegative solution under the conditions of Theorem 3.2, and the rest is similar. Define

where

Denote

then satisfies Caratheodory condition, and for any (r, u, v) ∈ J × ℝ^N × ℝ^N.

For any i = 1, ..., k, we denote

then and are continuous and satisfy

Obviously, we have

(2⁰)' , for r ∈ J uniformly, where β(r) ∈ C(J, ℝ), and1 < β ^- ≤ β ⁺< p^- ;

(3⁰)' when |u| + |v| is large enough, where ;

(4⁰)' when |u| + |v| is large enough, where 0 ≤ ε < β ⁺ - 1.

Let us consider

From Theorem 3.2 and Theorem 3.6, we can see that (43) has a nonnegative solution u. Since u ≥ 0, we have ϕ(u) = u, and then

Thus, u is a nonnegative solution of (1)-(4). This completes the proof.

Example 4.1. Consider the following problem

where p(r) = 5 + cos 3r, , 0 ≤ g(r) ∈ L¹, e₀, e₁ ∈ ℝ^N, w(r) = 3 + sin r, σ is a nonnegative parameter.

Obviously, is Caratheodory, q(r) ≤ 3.5 < 4 ≤ p (r) ≤ 6, then the conditions of Theorem 3.5 are satisfied, then (P₁) has a solution when δ > 0 is small enough. Moreover, when σ = 0, the conditions of Corollary 3.7 are satisfied, then (P₁) has a nonnegative solution.

Example 4.2. Consider the following problem

where p(r) = 5 + cos 3r, , 0 ≤ g(r) ∈ L¹, e₀, e₁ ∈ ℝ^N, w(r) = 3+sin r, σ is a nonnegative parameter.

Obviously, is Caratheodory, 1 < q(r) < 2 < 4 ≤ p (r) ≤ 6, then conditions of Theorem 3.5 are satisfied, then (P₂) has a solution when δ > 0 is small enough. Moreover, when σ = 0, the conditions of Corollary 3.7 are satisfied, and (P₂) has a nonnegative solution.

The authors declare that they have no competing interests.

All authors typed, read and approved the final manuscript.

The authors would like to thank the referees very much for their helpful comments and suggestions. This study was partly supported by the National Science Foundation of China (10701066 & 10926075 & 10971087) and China Postdoctoral Science Foundation funded project (20090460969) and the Natural Science Foundation of Henan Education Committee (2008-755-65).

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