In this paper, we study the following second order scalar differential inclusion: under nonlinear general boundary conditions incorporating a large number of boundary problems including Dirichlet, Neumann, Neumann–Steklov, Sturm–Liouville, and periodic problems. By means of approximate problems, we succeed in establishing existence results for the main problem. We establish the existence of solutions when the approximate problem admits two sign conditions or a single sign condition and a single lower solution or a single sign condition and a single upper solution. Our proofs combine the method of lower and upper solutions, sign conditions, the analysis of multifunctions, and Yosida’s approximation.

1. Introduction

This paper is devoted to the study of the following problem:

()

where

, with 0 < a < +∞, is an increasing homeomorphism,

is maximal monotone map,

is L^p−Caratheodory multifunction, p > 1, with nonempty, convex, and closed values,

is a strictly positive continuous,

are continuous functions.

The presence of the operator A in (1) incorporates into our framework variational inequalities. The variational inequalities models many applied problems, such as electrical circuits with ideal diodes, differential Nash games, Coulomb friction for contacting bodies, dynamic traffic networks and hybrid engineering systems with variable structures (see [1]).

Differential inclusions are a generalization of single valued differential equations. They are mathematical models of several physical phenomena such as control theory, optimization, mathematical economics, biology, physics, chemistry, economics, ecology, sweeping process, stochastic analysis, and in other fields. For these reasons, they have been of great interest to a large number of experts and scholars during the past decades. For examples, see [2–9]. In [4, 5], the authors prove existence results of a control problems by using some fixed point theorems and some techniques for nonautonomous second order differential inclusions. In [7, 8], the authors use a method that combines lower and upper solutions method, fixed point techniques, and monotonic iterative method to prove existence results. By using Leray–Schauder nonlinear alternative for multivalued maps and a fixed point theorem, the authors establish some existence results in [9]. This paper is inspired by [2, 3, 6], where the authors study single valued problem by using sign conditions and lower and upper solutions method.

This work extends the results of [2, 3, 6] to differential inclusion, to variational inequalities, to the nonlinear differential operator and to more general boundary conditions which incorporate Dirichlet, Neumann, Neumann–Steklov, periodic, and Sturm–Liouville problems. To establish the various proofs, we consider an approximation of the original problem. By combining the theory of topological degree with that of sign conditions and lower- and upper-solutions, we obtain three existence results for the approximate problem. If the problem admits two sign conditions or a single sign condition and a single lower solution or a single sign condition and a single upper solution, we show that the approximate problem admits at least one solution. Then, for each of the three cases, we show that any sequence of solutions of a family of approximate problems converges to a solution of the main problem.

The Φ−Laplacian operator used is a relativity operator. Consequently, the problem models problems related to relativity. For example, the dynamics of a charged particle in electric and magnetic fields when the velocities of the particles are relativistic; the dynamics of the acceleration of a particle of mass one at rest moving in a straight line at the speed of light normalised to one; the forced pendulum with relativistic effect, mean curvature in Minkowski space.

The rest of the article is organized as follows: In Section 2, we define the notions and notations that we will use in the sequel. In Section 3, we consider an approximate problem for which we establish three existence results. We deduce three existence results for the main problem in Section 4. In Section 5, we give an example of application of our results. Finally, in Section 6, we give a conclusion.

2. Notations and Preliminaries

This section is devoted to the notations and results that we will use to establish our results. Our main sources are the books of Hu-Papageorgiou [10] and Zeidler [11].

Let (Λ, Σ, μ) be a finite measure space and X be a separable Banach space.

is the family of nonempty subsets of X and B(X) denotes the Borel σ−field of X. We will also use the following notations:

()

The multifunction

is said to be measurable if for all x ∈ X, the

valued function

()

is Σ−measurable. A multifunction

is said to be graph measurable, if its graph

()

belongs to Σ × B(X). For

valued multifunctions, measurability implies graph measurability, while the converse is true if Σ is μ−complete. Given a multifunction

and 2 ≤ p ≤ +∞, we define the set

()

This set may be empty. For a graph measurable function , the set is nonempty if only if w⟼ inf{‖u‖ : u ∈ F(w)} belongs to L^p(Λ)₊. Let Y and Z be Hausdorff topological spaces. A multifunction has a closed graph, if GrG = {(y, z) ∈ Y × Z : z ∈ G(y)} is a closed subsets of Y × Z.

Let X be a reflexive Banach space and X^′ the topological dual of X. A map is said to be monotone, if for all x, y ∈ D(A) and for all x^′ ∈ A(x), y^′ ∈ A(y), we have 〈x^′ − y^′, x − y〉 ≥ 0. By 〈.〉, we denote the duality brackets for the pair (X, X^′). If additionally, the fact that 〈x^′ − y^′, x − y〉 = 0 implies that x = y, then we say that A is strictly monotone. The map A is said to be maximal monotone, if it is monotone and for all x ∈ D(A), x^′ ∈ A(x), the fact that 〈x^′ − y^′, x − y〉 ≥ 0 implies that y ∈ D(A) and y^′ ∈ A(y). It is clear from this definition that A is maximal monotone if and only if its graph GrA = {(x, x^′) ∈ X × X^′ : x^′ ∈ A(x)} is maximal with respect to inclusion among the graphs of monotone maps. If A is maximal monotone, for any x ∈ D(A), the set A(x) is nonempty, closed, and convex. Moreover, GrA is semiclosed, i.e., if , either x_n⟶x in X and in X^′, or x_n⇀x in X and in X^′, then (x, x^′) ∈ GrA.

is a maximal monotone operator if and only if the operator I + ηA is surjective, for all η > 0. We now define the following operators:

()

J_η is nonexpansive, i.e., ;

A_η is monotone, continuous and Lipschitz with constant 1/η (therefore, it is maximal monotone);

A_η(x) ∈ A(J_η(x)) for all x ∈ X;

for all x ∈ D(A) with A⁰(x) = proj_A(x){0} (recall that );

A_η(x)⟶A⁰(x) when η↘0 for all x ∈ D(A) and, when η↘0 for all x not in A(x);

when η↘0 for all x ∈ X (recall that ).

and

are some continuous operators defined by

()

We will use the following notations:

()

In the sequel, we will need the following Yankov–Von Neumann–Aumann’s selection theorem. For the proof of this theorem, see Hu-Papageorgiou ([12], p. 158).

Theorem 1. If (Λ, Σ, μ) is a complete, σ−finite measure space, X is a separable complete metric space and is a multifunction such that GrG = {(w, x) ∈ Λ × X : x ∈ G(w)} ∈ Σ × B(X) (i.e. G has a measurable graph), then there exists a Σ−measurable function g : Λ⟶X such that g(w) ∈ G(w) for all w ∈ Λ.

We will also need the following proposition which gives information about the pointwise behavior of a weakly convergent sequence in the Lebesgue–Bochner space L^p(Λ), 1 < p < ∞. For a proof of this result, see the book of Hu-Papageorgiou ([12], p. 694).

Proposition 1. If (Λ, Σ, μ) is a finite measure space, X is a Banach space, in L^p(Λ) and for μ−almost all w ∈ Λ there exists a nonempty, weakly compact set G(w) such that g(w) ∈ G(w) for all n ≥ 1, then

()

We consider the following assumptions on the data of (1).

•
(H_b): is a continuous strictly positive function such that there exist m, M > 0 such that
()
•
(H_Φ): , 0 < a < ∞, is an increasing homeomorphism such that Φ(0) = 0.
•
(H_A): is a maximal monotone multivalued map such that 0 ∈ A(0).
•
(H_G) is a multifunction such that:
- (i)
  For all is graph measurable.
- (ii)
  For almost all x ∈ Λ, (u, v)⟶G(x, u, v) has a closed graph.
- (iii)
  For every r > 0, there exists γ_r ∈ L^q(Λ) such that for almost all x ∈ Λ and for all with |u|, |v| ≤ r and for all l ∈ G(x, u, v) we have : |l| ≤ γ_r(x).

Let us define what we mean by solution of (1).

Definition 1. A function z ∈ C¹(Λ) such that b(z(.))Φ(z^′(.)) ∈ W^1,q(0, α), with (1/p) + (1/q) = 1 and p ≥ 2, is said to be a solution of problem (1), if and there exist and such that

()

3. Approximate Problem

For all η > 0 fixed, the map

is the Yosida approximation of A. Consider the following approximate problem:

()

Let us define what we mean by solution, lower solution and upper solution of (12).

Definition 2. A function z ∈ C¹(Λ) such that b(z(.))Φ(z^′(.)) ∈ W^1,q(0, α), with (1/p) + (1/q) = 1 and p ≥ 2, is said to be a solution of problem (1), if and there exists such that

()

Definition 3.

(a)
A function σ ∈ C¹(Λ) such that Φ(b(z(.)σ^′(.))) ∈ W^1,p(Λ), for all z ∈ C¹(Λ), is said to be a lower solution of problem (12), if and there exists such that:
()
(b)
A function ρ ∈ C¹(Λ) such that b(z(.))Φ(ρ^′(.)) ∈ W^1,p(Λ), for all z ∈ C¹(Λ), is said to be an upper solution of problem (12), if and there exists such that:

()

3.1. Existence of Solution Under Two Sign Conditions

Lemma 1. For each (h, d, u) ∈ C(Λ)×] − aα, aα[×C¹(Λ) there exists a unique ϱ = Q_Φ(h, d) such that

()

Moreover, the function is continuous.

Proof 1. Let (h, d, u) ∈ C(Λ)×] − aα, aα[×C¹, the function defined by

()

is well defined, continuous and decreasing. We have

()

From where there exists a unique solution ϱ = Q_Φ(h, d) of (4). To show that Q_Φ(h, d) is continuous on C(Λ)×] − aα, aα[, let (h_n, d_n) ⊂ C(Λ)×] − aα, aα[ be such that (h_n, d_n)⟶(h₀, d₀) in C(Λ)×] − aα, aα[. Without loss of generality, passing if necessary to a subsequence, we may assume that Q_Φ(h_n, d_n)⟶ϱ₀. Using the dominated convergence theorem we deduce that , so we have that ϱ₀ = Q_Φ(h₀, d₀).

Let N : C¹(Λ)⟶L^q(Λ) be the operator defined by

()

where

and N_g : C¹(Λ)⟶L^q(Λ) are some operators associated to A_η and

Lemma 2. N is bounded and continuous.

Proof 2. Let us show that N is bounded.

Let z be a bounded function in C¹(Λ). Then z is bounded in C(Λ). Since A_η is continuous, for all x ∈ Λ, A_η(z(x)) is bounded in . It follows that is bounded in L^q(Λ). Let N_g ∈ G(., z(.), z^′(.)). By hypothesis (H_G)(iii), N_g is bounded in L^q(Λ). So, N is bounded in L^q(Λ).

Let us show that N is continuous.

We have to show that, for all sequence of C¹(Λ) which converges to z in C¹(Λ), that the sequence converges to N(z) in L^q(Λ). We have , for all n ≥ 1. Since A_η is continuous on , is continuous in L^q(Λ), i.e., in L^q(Λ). It remains to show that N_g(z_n)⟶N_g(z) in L^q(Λ). Let the sequence , for almost all x ∈ Λ. Then, by hypothesis (H_G)(ii), (z(x), z^′(x), N_g(z(x))) ∈ GrG, with for almost all x ∈ Λ, z_n(x)⟶z(x), and N_g(z_n(x))⟶N_g(z(x)) in . By hypothesis (H_G)(iii), we can find γ ∈ L^q(Λ) such that |N_g(z_n(x))| < γ(x), for all n ≥ 1. Then, by Lebesgue’s dominated convergence theorem, the sequence converges to N_g(z) in L^q(Λ). Thus, N is continuous in L^q(Λ).

Let us consider the following family of boundary value problems (P_λ), λ ∈ [0, 1],

()

For each ∈[0, 1], problem (P_λ) can be written equivalently

()

For each λ ∈ [0, 1], we associate with (P_λ) the nonlinear operator M(λ.) where M is defined on [0, 1] × C¹(Λ) by

()

where

is Nemitsky’s operator associate to b and for all x ∈ Λ,

()

Lemma 3. M is completely continuous on C¹(Λ).

Proof 3. We adopt the method used to prove Lemma 3 of [13].

Let (λ_n, z_n) ⊂ [0, 1] × C¹(Λ) with is bounded. We suppose that λ_n⟶λ. For all . Let’s set v_n = M(λ_n, z_n). Then,

()

Since A_η is continuous, is continuous. Moreover, since the sequence is bounded in C(Λ), (A_η(z_n(x))) is bounded in . Then is continuous and bounded in L^q(Λ). So there exists c > 0 such that |A_η(z_n(x))| ≤ c. Using hypothesis (H_F)(iii), we obtain

()

Using (25) and (26), v_n is bounded on C(Λ). Let x₁, x₂ ∈ [0, α]. Using (26), for all , we have

()

That implies that is equicontinuous. Using Arzelà–Ascoli theorem, there exists a subsequence labeled which converges strongly to v in C(Λ). Then,

()

By (26), it follows that for all . Moreover, if x₁, x₂ ∈ [0, α], then

()

Using the uniform continuity of Φ⁻¹ on compact intervals of , it follows that is equicontinuous. Applying Arzela–Ascoli theorem, we may assume, passing to a subsequence, that converges to w in C(Λ), with ‖w‖_∞ < a. It follows that v^′ = w. So v_n⟶v in C¹(Λ). we get that M is completely continuous.

Lemma 4. Assume that there exist R > 0 and ε ∈ {−1, 1} such that

()

Then, for all sufficiently large β > 0,

()

and problem (12) has at least one solution.

Proof 4. Suppose that there exists (λ, z) ∈ [0, 1] × C¹(Λ) such that M(λ, z) = z. We have

()

It follows:

()

Since

()

we get

. If z_L ≥ R or z_M ≤ −R, by (30) and (31), we have

()

which contradict (34). Therefore, z_L < R and z_M > −R. Since z is continuous on Λ, there exist x₁, x₂ ∈ Λ such that z_L = z(x₁) and z_M = z(x₂). We have

()

Using (37), we have

()

It follows that ‖z‖_∞ < R + aα. Since and ‖z‖_∞ < R + aα, we have

()

Let M be the operator given by (33), and let β > R + (a + 1)α. Using (39) and the homotopy invariance of Leray–Schauder degree, we have

()

But the range of the mapping z⟼P(z) + QN(z) − K(z) is contained in the subspace of constant function isomorphic to , so, using, the property the reduction property of Leray–Schauder degree, we have

()

By existence property of the Leray–Schauder degree, there exist z ∈ B_β such that z = M(1, z) which is the solution of (12).

Let us decompose any z ∈ C¹(Λ) in the form , and

Lemma 5. The set of solution of problem

()

contain a continuum subset C whose projection on

and whose projection on

is contained on the ball B_a(α + 1).

Proof 5. Problem (42) is equivalent to the fixed point problem in

()

for all

. M^∗ is completely continuous on

(see the proof of Lemma 3). For all

()

From (44), for all

, all fixed point z^∗ of

is such that

()

Furthermore, for each λ ∈ [0, 1], each possible fixed point z^∗ of

()

satisfies, for the same reasons, inequality (45), which implies that

()

using (45), (47), and Lemma 2 in [14] or Theorem 1.2 in [15] imply the existence of C.

Proposition 2. Assume that there exist R > 0 and ε ∈ {−1, 1} such that

()

problem (12) has at least one solution.

Proof 6. The proof is similar to the proof of Theorem 2 in [2].

3.2. Existence of Solutions Under One Sign Condition and Only One Lower Solution or Only One Upper Solution

Let us define two functions

and

()

We introduce the following lemma (see [12], Lemma 6.3, and Corollary 6.4).

Lemma 6. For z ∈ C¹, the following three properties are true.

(a)
For i = 1, 2, (d/ dx)γ_i(x, z(x)) exists for a.e x ∈ [0, α].
(b)
and .
(c)
For i = 1, 2, if is such that z_n⟶z in C¹(Λ), then γ_i(., z_n)⟶γ_i(., z) in C¹(Λ), and for almost every x ∈ [0, α], lim_n⟶+∞(d/ dx)γ_i(x, z_n) = (d/dx)γ_i(x, z(x)).

Proposition 3. Assume that:

(i)
There exist a lower solution σ of problem (12).
(ii)
There exist R > 0 such that.

()

Then problem (12) admits at least one solution.

Proof 7. The proof will be established in several steps.

Step 1. The modified problem.

Consider the functions δ

, and

given by

()

where

. And

()

f^∗ is Carathéodory function and

and

are continuous. Consider the modified problem

()

Step 2. Any solution of problem (55) is a solution of problem (12).

Let us prove that every solution z of (55) is such that σ(x) ≤ z(x) for all x ∈ Λ. We will argue by contradiction. Suppose that the set {x ∈ Λ : σ(x) − z(x) > 0} is nonempty. Then, there exists x₀ ∈ Λ such that

()

If x₀∈]0, α[, then σ^′(x₀) = z^′(x₀). Then, b(z(x₀))Φ(σ^′(x₀)) = b(z(x₀))Φ(z^′(x₀)). We can find ϵ > 0 such that for all x∈]x₀, x₀ + ϵ[, σ(x) > z(x). We have

()

Then, for all x∈]x₀, x₀ + ϵ[,

()

b(z(x))Φ(σ^′(x)) − b(z(x))Φ(z^′(x)) > 0⟹σ^′(x) − z^′(α) > 0. This contradicts the fact that σ − u reach its maximum at x₀.

If x₀ = 0, then

()

contradiction with the definition of a lower solution.

If x₀ = α, then

()

contradiction with the definition of a lower solution. Therefore, σ(x) ≤ z(x) for all x ∈ Λ. So, z is a solution of (12).

Step 3. Existence of solutions of problem (12).

Let Δ = [σ_L, σ_M]×] − a, a[. Since G is L^p−Caratheodory, there exists φ ∈ L^p(Λ) such that for all w ∈ G(x, u, v), |w| ≤ φ(x) for a.e x ∈ Λ and for all (u, v) ∈ Δ. Furthermore, since for all η > 0 fixed, A_η is continuous on the compact interval [σ_L, σ_M], there exists c₁ > 0 such that |A_η(u)| < c₁. Let

()

By (51), if z ∈ C¹(Λ) is such that ‖z‖_∞ < a and z_L ≥ R₁, then

()

Furthermore, if z ∈ C¹(Λ) is such that ‖z‖_∞ < a and z_M < −R₁, then

()

for all x ∈ Λ. It follows:

()

Using (62), (64), and Proposition 2, we deduce that problem (55) has at least one solution, which is also a solution of problem (12) by Step 2.

Proposition 4. Assume that:

(i)
There exists a lower solution σ of problem (12).
(ii)
There exists R > 0 such that.

()

Then problem (12) admits at least one solution.

Proof 8. The proof is similar to that of Proposition 3.

4. Main Results

4.1. Existence of Solution Under Two Sign Conditions

Theorem 2. If hypotheses (H_b), (H_Φ), (H_G), (H_A), and (H_g) are satisfied and there exist R > 0 and ε ∈ {−1, 1} such that

()

then problem (1) admits at least a solution in C¹(Λ).

Proof 9. We adopt the same approach as the proof of the Theorem 2 of [16].

Since (66) and (67) are satisfied, by Proposition 2, the approximate problem (12) admits at least one solution. Let η_n⟶0, η_n > 0 and

be a sequence of solutions of the approximate problem (12). For all n ≥ 1 and all

. It follows that, for all n ≥ 1 and all x ∈ Λ, |z_n(x)| < R + aα. We deduce that

is bounded in W^1,p(Λ). So, we can find a subsequence labeled

which converge weakly to z in W^1,p(Λ). Because of compact embedding of W^1,p(Λ) in C(Λ), this subsequence converge strongly to z in C(Λ). We have

for all x ∈ Λ. Since A⁰ is bounded on the compacts subsets of

, there exists c > 0 such that

. Therefore, for all x ∈ Λ,

. As a result the sequence

is bounded in L^q(Λ). Then, it admits a subsequence which converge weakly to w in L^q(Λ). Furthermore, by previous arguments and hypothesis (H_G)(iii), the sequence

is bounded in L^q(Λ). By integration, the sequence

is bounded in L^q(Λ). Where the sequence

is bounded in W^1,p(Λ). Then, we can find a subsequence labeled

which converge weakly to b(z)Φ(z^′) in W^1,p(Λ). Because of compact embedding of W^1,p(Λ) in C(Λ), this subsequence converge strongly to b(z)Φ(z^′) in C(Λ). We consider hypothesis (H_G)(i). Applying Theorem 1, we obtain a sequence

. By hypothesis (H_G)(iii), the sequence

is bounded in L^q(Λ). So, it admits a subsequence labeled

which converge weakly to g in L^q(Λ). The Proposition 1 and hypothesis (H_G)(ii) imply that

. Thus, when n⟶+∞, we obtain:

()

The continuity of fonction g_i, i = 1, 2, leads to the convergence of sequence

, to (g_i(z(0), z(x), z^′(0), z^′(α)). Thus, we obtain

()

It remains to be shown that w(x) ∈ A(z(x)). To this end, let

defined by

()

Let us show that is maximal monotone.

To do this, we will show that

with J the p−Laplacian operator. Let h ∈ L^q(Λ) and let

()

where r(x) = |h(x)|^{(1/p − 1)} + 1. By Theorem III.3.3 p. 334 of [17], A + J is maximal monotone. By the proof of Theorem III.6.28, p. 371 of [17], Υ(x) ≠ ∅ a.e on Λ. Moreover,

()

with

Lebesgue’s σ−algèbre on Λ (for more details see the proof of Theorem 2 of [16]). Then, by Theorem 1, we obtain measurable maps

such that (z(x), a(x)) ∈ Υ(x) a.e on Λ. Then z ∈ L^p(Λ) and a ∈ L^q(Λ). Therefore,

Let us show that the surjectivity of leads to the maximal monotonicity of the operator .

Suppose that (y, v) ∈ L^p(Λ) × L^q(Λ) and:

()

Since

, we can find

such that

. Then

. Since J is strictly monotone,

, i.e.,

. This prove that

is maximal monotone. Let

be the resolvant. Then, ∀x ∈ Λ,

()

Recall that

. We have

()

and

in L^q(Λ). Since

is maximal monotone, passing to the limit in relation (31), we have

, i.e., w(x) ∈ A(z(x)) a.e Λ. That prove that z ∈ C¹(Λ) is a solution of (1).

4.2. Existence of Solutions Under One Sign Condition and Only One Lower Solution or Only One Upper Solution

Theorem 3. If hypotheses (H_b), (H_Φ), (H_G), (H_A), and (H_g) are satisfied and

(i)
There exists a lower solution σ of problem (3).
(ii)
There exist R > 0 such that.

()

then problem (1) admits at least a solution in C¹(Λ).

Proof 10. Since (i) and (76) are satisfied, by Proposition 3 problem (12) admits at least one solution. Finally, by arguing as in the proof of Theorem 2, we complete the proof.

Theorem 4. If hypotheses (H_b), (H_Φ), (H_G), (H_A), and (H_g) are satisfied and

(i)
There exist an upper solution ρ of problem (12).
(ii)
There exist R > 0 such that.

()

then problem (1) admits at least a solution in C¹(Λ).

Proof 11. The proof is similar to the proof of Theorem 2.

Corollary 1. Assume that:

(a)
There exists such that N(z)(x) ≥ A for a.e x ∈ Λ and all z ∈ C¹(Λ).
(b)
lim_u⟶+∞ g₁(u, u + l, v, w) − g₂(u, u + l, v, w) = +∞, for all .
(c)
There exist a lower solution σ of problem (12); then problem (1) admits at least a solution.

Proof 12. By (a), we have

()

By (b) there exists R > 0 such that (32) is true. By Proposition 3, problem (12) admits at least one solution. Finally, by arguing as in the proof of Theorem 2, we complete the proof.

Corollary 2. Assume that:

(a)
There exists such that N(z)(x) ≤ A for a.e x ∈ Λ and all z ∈ C¹(Λ).
(b)
lim_u⟶−∞ g₁(u, u + l, v, w) − g₂(u, u + l, v, w) = −∞, for all .
(c)
There exist a upper solution ρ of problem (12); then problem (1) admits at least a solution.

Proof 13. The proof is similar to that of Corollary 1.

5. Examples

()

where ∂|.| is the subdifferential of the absolute value. ∂|.| is maximal monotone and 0 ∈ ∂|0|. We have

()

For all ψ ∈ G(x, u, v), |ψ| < 32. For all η > 0, |A_η(u)| ≤ |A⁰(u)| with

()

Then, A_η(0) = 0. The function σ, defined by σ(u) = 0 for all

, is a lower solution of approximate problem of (34) and

()

Then, by Corollary 1, problem (79) admits at least one solution.

Suppose that

()

Then, problem (79) becomes one of variational inequality. Consequently, our results apply to it.

Now, let us consider some classical problems of Dirichlet, Neumann, Sturm–Liouville, and periodic boundary conditions, respectively.

()

Here, g₁(u, u + l, v, w) = b(u)Φ(v) + 2u − 1, g₂(u, u + l, v, w) = b(u + l)Φ(w) + u + 1 with b(u) = 2. The function σ, defined by σ(u) = 0 for all

, is a lower solution of approximate problem of (86) and

()

Then, by Corollary 1, problem (86), admits at least one solution.

()

Here,

, for all t∈] − 2; 2[, g₁(u, u + l, v, w) = b(u), g₂(u, u + l, v, w) = b(u + l) with l a strictly negative constant and b(u) = u² + 1 for all

. The function σ, defined by σ(u) = 0 for all

, is a lower solution of approximate problem of (86) and

()

Then, by Corollary 1, problem (86), admits at least one solution.

()

Here, g₁(u, u + l, v, w) = b(u)Φ(v) + 2u + v, g₂(u, u + l, v, w) = b(u)Φ(w) + u + w with b(u) = 2 for all

. The function σ, defined by σ(u) = 0 for all

, is a lower solution of approximate problem of (86) and

()

Then, by Corollary 1, problem (88) admits at least one solution.

()

Here, g₁(u, t, v, w) = b(u)Φ(v) + u − t, g₂(u, t, v, w) = b(u)Φ(w) + v − w with b(u) = 1 for all . By Theorem 2, problem (90) admits at least one solution (see example 1 of [3]).

6. Conclusion

In this paper, we have studied a second order nonlinear differential inclusion driven by the nonlinear differential operator that incorporates variational inequality problem, with nonlinear general boundary conditions that encompass classicals Dirichlet, Neuman, periodic and Sturm–Liouville type problems. We have obtained results for the existence of solutions for the main problem when the approximate problem admits two sign conditions or a single sign condition and a single lower solution or a single sign condition and a single upper solution. To achieve this, we used a method that combines the method of lower and upper solutions, sign conditions, the analysis of multifunctions, the Yosida approximation and the theory of monotone maximal operators. Ours work ends with an example of the application of our results. The problem studied does not include the classical Φ–Laplacian operator and the nonsuperjective Φ–Laplacian operator. In the future, a study could be carried out in this direction. The results could also be extended to .

Conflicts of Interest

The authors declare no conflicts of interest.

Funding

This study was not funded.

Open Research

Data Availability Statement

The data used to support the findings of this study are included within the article.

References

1 Lu L., Liu Z., and Obukhovskii V., Second Order Differential Variational Inequalities Involving Anti-Periodic Boundary Value Conditions, Journal of Mathematical Analysis and Applications. (2019) 473, no. 2, 846–865, https://doi.org/10.1016/j.jmaa.2018.12.072, 2-s2.0-85060086165.
10.1016/j.jmaa.2018.12.072
Web of Science® Google Scholar
2 Bereanu C. and Mawhin J., Nonhomogeneous Boundary Value Problems for Some Nonlinear Equations With Singular, Φ-Laplacian, Journal of Mathematical Analysis and Applications. (2009) 352, no. 1, 218–233, https://doi.org/10.1016/j.jmaa.2008.04.025, 2-s2.0-57749100460.
10.1016/j.jmaa.2008.04.025
Web of Science® Google Scholar
3 Bereanu C. and Mawhin J., Existence and Multiplicity Results for Some Nonlinear Problems With Singular Φ-Laplacian, Journal of Differential Equations. (2007) 243, no. 2, 536–557, https://doi.org/10.1016/j.jde.2007.05.014, 2-s2.0-36048953390.
10.1016/j.jde.2007.05.014
Web of Science® Google Scholar
4 Cardinali T. and De Angelis E., Non-Autonomous Second Order Differential Inclusions With a Stabilizing Effect, Results in Mathematics. (2021) 76, no. 1, https://doi.org/10.1007/s00025-020-01305-1.
10.1007/s00025-020-01305-1
PubMed Web of Science® Google Scholar
5 Chen J., Lui Z., Lomotsev F. E., and Obuklovskii V., Optimal Feedback Control for a Class of Second-Order Evolution Differential Inclusion With Clarke’s Subdifferential, Journal of Nonlinear and Variational Analysis. (2022) 6, 551–565.
Web of Science® Google Scholar
6 Goli K. C. E. and Adjé A., New Existence Results for Some Periodic and Neumann-Steklov Boundary Value Problems With ϕ-Laplacian, Boundary Value Problems. (2016) 2016, no. 1, https://doi.org/10.1186/s13661-016-0676-6, 2-s2.0-84988663696.
10.1186/s13661-016-0676-6
Google Scholar
7 Li Z., Shu X., and Xu F., The Existence of Upper and Lower Solutions to Second Order Random Impulsive Differential Equation with Boundary Value Problem, AIMS Mathematics. (2020) 5, no. 6, 6189–6210, https://doi.org/10.3934/math.2020398.
10.3934/math.2020398
Web of Science® Google Scholar
8 Li Z., Shu X., and Miao T., The Existence of Solutions for Sturm–Liouville Differential Equation with Random Impulses and Boundary Value Problems, Boundary Value Problems. (2021) 2021, no. 1, https://doi.org/10.1186/s13661-021-01574-x.
10.1186/s13661-021-01574-x
PubMed Web of Science® Google Scholar
9 Nuchpong C., Schoployklang T., Ntouyas S. K., and Tariboon J., Boundary Value Problems for Hilfer-Hadamard Fractional Differential Inclusion With Nonlocal Integro-Multi-Point Boundary Conditions, Journal of Nonlinear Functional Analysis. (2022) .
Google Scholar
10 Hu S. and Papageorgiou N. S., Handbook of Multivalued Analysis. Volume I: Theory, 1997, Kluwer, Dordrecht, The Netherlands.
10.1007/978-1-4615-6359-4
Google Scholar
11 Zeidler E., Nonlinear Functional Analysis and its Applications II, 1990, Springer- Verlag, New York.
Google Scholar
12 De Coster C. and Habets P., Two-Point Boundary Value Problems: Lower and Upper Solutions, 2006, Elsevier, Amsterdam.
Google Scholar
13 Bereanu C. and Mawhin J., Nonlinear Neumann Boundary Value Problem With φ Laplacian Operators, Analele stiintifice ale Universitatii Ovidius Constanta. (2004) 12, no. 2, 73–82.
Google Scholar
14 Mawhin J., Rebello C., and Zanolin F., Continuation Theorems for Ambrosetti–Prodi Type Periodic Problems, Communications in Contemporary Mathematics. (2000) 2, 87–126.
10.1142/S0219199700000074
Web of Science® Google Scholar
15 Massabo I. and Pejsachowicz J., On the Connectivity Properties of the Solution Set of Parametrized Families of Compact Vector Fields, Journal of Functional Analysis. (1984) 59, no. 2, 151–166, https://doi.org/10.1016/0022-1236(84)90069-7, 2-s2.0-26844560839.
10.1016/0022-1236(84)90069-7
Web of Science® Google Scholar
16 Gasinski L. and PapageorgiouA N. S., Nonlinear Second-Order Multivalued Boundary Value Problems, Proceedings of the Indian Academy of Sciences-Section A. (2003) 113, no. 3, 293–319.
Google Scholar
17 Hu S. and Papageorgiou N. S., Handbook of Multivalued Analysis. Volume I: Theory, 1997, Kluwer, Dordrecht, The Netherlands.
10.1007/978-1-4615-6359-4
Google Scholar

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Nonlinear Second Order Scalar Differential Inclusion Involving a Singular Φ−Laplacian Operator, With Nonlinear General Boundary Conditions

Abstract

1. Introduction

2. Notations and Preliminaries

3. Approximate Problem

3.1. Existence of Solution Under Two Sign Conditions

3.2. Existence of Solutions Under One Sign Condition and Only One Lower Solution or Only One Upper Solution

4. Main Results

4.1. Existence of Solution Under Two Sign Conditions

4.2. Existence of Solutions Under One Sign Condition and Only One Lower Solution or Only One Upper Solution

5. Examples

6. Conclusion

Conflicts of Interest

Funding

Open Research

Data Availability Statement

References

References

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Nonlinear Second Order Scalar Differential Inclusion Involving a Singular Φ−Laplacian Operator, With Nonlinear General Boundary Conditions

Abstract

1. Introduction

2. Notations and Preliminaries

3. Approximate Problem

3.1. Existence of Solution Under Two Sign Conditions

3.2. Existence of Solutions Under One Sign Condition and Only One Lower Solution or Only One Upper Solution

4. Main Results

4.1. Existence of Solution Under Two Sign Conditions

4.2. Existence of Solutions Under One Sign Condition and Only One Lower Solution or Only One Upper Solution

5. Examples

6. Conclusion

Conflicts of Interest

Funding

Open Research

Data Availability Statement

References

References

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