Impulsive coupled systems with regular and singular φ -Laplacians and generalized jump conditions

This work contains suﬃcient conditions for the solvability of a third-order coupled system with two diﬀerential equations involving diﬀerent Laplacians, fully discontinuous nonlinearities, two-point boundary conditions, and two sets of impulsive eﬀects. The ﬁrst existing result is obtained from Schauder’s ﬁxed point theorem, and the second one provides also the localization of a solution via the lower and upper solutions technique. We point out that it is the ﬁrst time that impulsive coupled systems with strongly nonlinear fully diﬀerential equations and generalized impulse eﬀects are considered simultaneously. Moreover, the singular case is applied to a special relativity model in classical electrodynamics.

Usually φ and ψ are known as φ, ψ-Laplacian as they generalize the one-dimensional Laplacian and the p-Laplacian, and they were used by many authors in a broad range of problems.Some examples: [34] to obtain a positive periodic solution for a ϕ-Laplacian Liénard equation with a singularity; [21] proving the multiplicity of solutions of p-Laplacian Dirichlet boundary value problem with discontinuous nonlinearities; [35] giving sufficient conditions for the existence of at least three positive solutions of one-dimensional p-Laplacian boundary value problem; [7,31] to obtain positive solutions for some p-Laplacian problem in superlinear cases; and [29] based on nonnegative nonlinearities under a version of the Krasnosel'skii expansion and compression cone theory.
Nonlinear coupled systems, where the unknown functions and their derivatives can interact, have been considered in several works in recent years, such as, among others: [30] via Schauder's fixed point theorem; [11] for fractional differential equations at resonance applying coincidence degree theory; [24] including the study of different types of differential and integral equations; [36] via lower and upper solutions technique; and [18] applied to reaction-diffusion Robin problems.
Impulsive differential equations model many real phenomena in which the nonlinearities have sudden discontinuous jumps in their values.These types of events can occur in population dynamics, control, and optimization theory, ecology, biology and biotechnology, economics, pharmacokinetics, and other physics and mechanics problems.For some examples of the approach to impulsive differential equations, we refer to: [22] for a general theory; [19] via fixed point index; [23,27] applied to functional impulsive problems; and [38] with a monotone iterative technique for approximating the solution.The study of φ-Laplacian impulsive problems can be seen, for instance, in: [17] in periodic problems applying a continuation theorem; [9,10] for bounded and unbounded intervals; [33] for fractional equations with p-Laplacian; and [1,8,28] for Brownian motion.
Combining all these areas and results, we consider, to the best of our knowledge, for the first time the methods and techniques suggested in, for example, [6,14] to an impulsive coupled system with fully differential equations including different regular and singular Laplacians and generalized impulsive conditions, whose jumps depend on both variables and some of its derivatives.
The paper is organized as follows: Sect. 2 contains the functional framework and some preliminary results, namely the explicit solution for the associated impulsive linear problem, Nagumo-type growth conditions, and a priori bounds for the second derivatives.In Sect.3, we present an existence theorem for the general case.Section 4 contains an existence and localization result applied to a particular case of the initial impulsive conditions and a concrete example to show the applicability of the localization tool.Section 5 applies our method to the singular case and to special relativity theory.

Definitions and preliminary results
This section introduces some preliminary results and the functional framework. Define and consider the sets of piecewise continuous functions: , be the usual Banach space equipped with the norm • ∞ , defined by where and 2), and impulsive effects (1.3).
The next lemma gives the unique solution for the homogeneous problem related to (1.1)-(1.3).Lemma 2.2 Let φ, ψ : R → R be increasing homeomorphisms and p, q The problem composed by the differential system and conditions (1.2) and (1.3) has a unique solution given by u Proof Integrating the first equation of (2.1), for x ∈ (x n , b], we have, by (1.2), x n ], integrating (2.1), by (1.3) and (2.2), we obtain So, by mathematical induction, for x ∈ [a, b], and therefore By a new integration of (2.3) from a to x, when x ∈ [a, x 1 ], (2.4) According to (1.3), when x → x + 1 , we have Similarly, Likewise, for the second equation, we have The Nagumo condition, introduced in [25], is an important tool for controlling the second derivatives.We consider here a Nagumo-type condition given by the following definition.
Proof Let (u(x), v(x)) be a solution of (1.1) on the set (S).By the mean value theorem, there are x ∈ (x i , x i+1 ) and x ∈ (τ j , τ j+1 ) such that ), we obtain by (2.8), (S), and (2.5) the contradiction (2.9) the contradiction is similar.Assume now that there are x, By the continuity of u (x), there exists Making the change of variable φ(u (x)) = s and using (2.6) and (2.9), and by (2.10) Therefore u (x * ) < N 1 , and as x * is taken arbitrarily, then u (x) < N 1 for the values of x whenever u (x) > μ 1 .
The case for x > x * follows similar arguments.The other possible case where u (x) > -μ 1 and u x * < -μ 1 can be proved by the previous techniques.Therefore u ∞ ≤ N 1 .
By a similar method as above, it can be shown that and, with the same type of arguments, obtaining that v ∞ ≤ N 2 .
The arguments forward will require the following lemma of [32].
Then, for each u ∈ C 1 (I), the next two properties hold: Schauder's fixed point theorem will be the key existence tool.
Theorem 2.7 [37] Let Y be a nonempty, closed, bounded, and convex subset of a Banach space X, and suppose that P : Y → Y is a compact operator.Then P has at least one fixed point in Y .

Existence result
The next theorem will guarantee the existence of a solution of (1.1)-(1.3)through the existence of fixed points of a convenient operator.Proof Define the operators T 1 : with Define L > 0 and M > 0 such that and Since f and g are L 1 -Carathéodory functions and a nonnegative function The proof will follow several steps that, for clarity, are detailed for the T 1 (u, v) operator.The technique for the T 2 (u, v) operator is similar.
Step 1: T is well defined, continuous, and uniformly bounded.
So, TB is equiconvergent at each impulsive point.
Step 4: T : X 2 → X 2 has a fixed point.
with L > 0 given by (3.2) and (3.5), and M > 0 defined in (3.3) such that B ⊂ .According to Step 1, we have So, T ⊂ , and by Theorem 2.7, the operator T(u, v) = (T 1 (u, v), T 2 (u, v)) has a fixed point (u * , v * ).By standard techniques and Lemma 2.2, it can be shown that this fixed point is a solution of problem (1.1)-(1.3).
Example 3.2 Consider the following system of coupled differential equations: with the boundary conditions and the impulsive effects given by with x i = i 5 for i = 1, 2, 3, 4 and τ j = j 2 10 for j = 1, 2, 3.
This problem is a particular case of (1.1)- It is clear that the functions in (3.9) verify assumption (H1) and f and g satisfy a Nagumotype condition in sets such as, for some piecewise continuous functions γ (l)  k (x), (l)

Existence and localization results
In addition to the existence of a solution, it is possible to obtain an existence and localization theorem, that is, not only it guarantees the existence of at least a solution, but provides also a strip where this solution is localized.
However, the localization part is obtained for a particular case of impulsive conditions (1.4), applying lower and upper functions, defined as follows.
To obtain this goal, we consider local monotone assumptions: and The existence and localization theorem is given as follows.
Proof Define the truncate functions δ im : [a, b] × R → R for κ = 1, 2 and l = 0, 1 given by Consider the following modified coupled system composed by the truncated and perturbed differential equations x, δ 10 (x, u(x)), δ 11 (x, u (x)), u (x), with the truncated impulsive conditions for i = 1, 2, . . ., m, j = 1, 2, . . ., n, and boundary conditions (1.2).It is clear that the functions F and G, given by satisfy the Nagumo type conditions, as in Definition 2.3, relative to the set S * with Therefore, applying the same arguments as in Theorem 3.1, it can be proved that problem (4.3), (1.2), (4.4) has at least a solution (u(x), v(x)).
To prove that this solution is also a solution to the initial problem (1.1), (1.2), (1.4), it will be enough to show that For the second inequality, assume, by contradiction, that there is x ∈ [a, b] such that u (x) > β 1 (x), and define As, by boundary conditions (1.2) and Definition 4.1, u (a)β 1 (a) ≤ 0, then x = a.In the same way, u (b -)β 1 (b -) ≤ 0, therefore x = b.
(ii) Suppose now that there is p * ∈ {1, 2, . . ., n} such that x = x p * .That is, As u, β 1 ∈ X, by (i), we obtain the contradiction By (4.4), (H3), and Definition 4.1, we obtain the contradiction By similar arguments, the remaining inequality can be proved, and therefore The other inequalities follow similar steps.
By integration of (4.9) for x ∈ [a, x 1 ], and for x ∈ (x 1 , x 2 ] we have, by (H3), By recurrence, it can be shown, analogously, that Analogously, the remaining inequality can be proved, and therefore Analogously, it can be proved that To illustrate the importance of the location arguments, we consider the following example.
The functions f and g satisfy the Nagumo condition relative to the sets Consider a constant K k > 0, k = 1, 2, and μ k as defined in (2.5), then, in S 1 , So, by Theorem 4.2, there is at least one pair of functions (u(x), v(x)) ∈ X 2 , a solution of problem (4.11)-(4.13);moreover, as shown in Fig. 2 and Fig. 3, and with N 1 and N 2 given by Lemma 2.4.

Singular φ-Laplacian equations in special relativity
Relativity implies that physical laws do not depend on the chosen reference frame.In special relativity, the speed of light c is recognized as the maximum speed with which information can travel in free space from one frame of reference to another [4].Let us consider two frames of reference P 0 and P in uniform relative motion to each other, that is, moving with relative speed v. Taking into account the upper limit c of the speed of information propagation, the space-time coordinates of the frames P 0 and P must be related by Lorentz transformations [13].The Lorentz factor depends nonlinearly on the relative velocity v and is defined by The theory of special relativity is fundamental in the development of the modern theory of classical electrodynamics.The fact that an electric charge q generates an electric field E and in motion generates a magnetic field B is intuitively compatible with the statement that the electric and magnetic fields are covariant under a Lorentz transformation from one inertial system to another [20].

Conclusion
This work shows, mainly, that local monotonies on the nonlinearities and the impulsive functions are sufficient conditions for the solvability of a third-order impulsive coupled system with two differential equations involving different Laplacians, fully discontinuous nonlinearities, and two-point boundary conditions.The localization information given by the lower and upper solutions had been underused to obtain qualitative data on the solutions, such as growth type, sign, and estimation of the unknown function and its derivatives, as it is illustrated in both examples.To the best of our knowledge, it is the first time where impulsive coupled systems with strongly nonlinear fully differential equations and generalized impulse effects are considered simultaneously.There remain arguments and techniques to be used, to obtain the localization part for coupled systems with jumps on the Laplacians.

Figure 2 Figure 3
Figure 2 At least one solution (u(x), v(x)) of problem (4.11)-(4.13) is located in the colored region, when x ∈ [0, 1] are, respectively, lower and upper solutions of problem (5.1)-(5.3)according to Definition 4.1.In fact, the differential inequalities are verified in the interval [-1, 1], as shown in Fig. 4. and impulsive conditions verify the inequalities of Definition 4.1, as shown in Table

Table 1
Impulsive conditions for functions α 1 and β 1

Table 2
Impulsive conditions for functions α 2 and β 2

Table 3
Impulsive conditions for functions α 1 and β 1