Study of Nonlinear Second-Order Differential Inclusion Driven by a <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.4619007pt" id="M1" height="11.8613pt" version="1.1" viewBox="-0.0657574 -11.3994 23.2294 11.8613" width="23.2294pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-150" d="M488 658H270V630C328 626 335 619 335 579C217 574 44 509 44 330C44 147 221 88 335 86C335 38 328 32 265 28V0H488V28C425 32 418 38 418 86C534 88 710 153 710 337C710 517 538 575 418 579C418 619 425 626 488 630V658ZM335 118C249 124 136 180 136 337C136 490 251 543 335 547V118ZM418 547C502 543 618 491 618 333S503 124 418 118V547Z"/></g><g transform="matrix(.017,0,0,-0.017,12.939,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g></svg>Laplacian Operator Using the Lower and Upper Solutions Method

Béhi, Droh Arsène; Adjé, Assohoun; Etienne Goli, Konan Charles

doi:https://doi.org/10.1155/2024/2258546

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Research Article | Open Access

Volume 2024 | Article ID 2258546 | https://doi.org/10.1155/2024/2258546

Study of Nonlinear Second-Order Differential Inclusion Driven by a Laplacian Operator Using the Lower and Upper Solutions Method

Droh Arsène Béhi,¹Assohoun Adjé,²and Konan Charles Etienne Goli³

Academic Editor: Xiaolong Qin

Received03 Jul 2023

Revised04 Jan 2024

Accepted21 Feb 2024

Published14 Mar 2024

Abstract

In this paper, we study a second-order differential inclusion under boundary conditions governed by maximal monotone multivalued operators. These boundary conditions incorporate the classical Dirichlet, Neumann, and Sturm–Liouville problems. Our method of study combines the method of lower and upper solutions, the analysis of multivalued functions, and the theory of monotone operators. We show the existence of solutions when the lower solution and the upper solution are well ordered. Next, we show how our arguments of proof can be easily exploited to establish the existence of extremal solutions in the functional interval . We also show that our method can be applied to the periodic case.

1. Introduction

We consider the nonlinear second-order problemwhere is a subdifferential of a lower semicontinuous, proper, and convex function which are not identically equal to , is Caratheodory multifunction, , , is a maximal monotone operator, is a not necessarily continuous map, is a monotone homeomorphism, and is a continuous and positive function.

Introduced by Picard in 1890 in [1, 2], the method of lower and upper solutions is still intensely used today to establish existence and multiplicity results for boundary problems of second or other order. For example, in [3–5], it was combined, respectively, with the topological degree theory, a fixed point theorem for ordered Banach spaces and fixed point index theory to establish existence and multiplicity results for nonlinear second-order differential equations while in [6], it is used to study the fractional evolution equation with order , . Also, recently in [7, 8], the method has been extended, respectively, to semilinear and generalized second-order random impulsive differential problem driven, respectively, by the scalar Laplacian operator and with linear boundary conditions. Much earlier, Frigon [9, 10] generalized this method to differential inclusions but her study was only focused on a semilinear problem with linear boundary conditions. It was followed by other authors such as Bader-Papageorgiou [11] and Staicu-Papageorgiou [12]who worked only with a nonlinear homogeneous differential operator, the p-Laplacian operator with nonlinear and multivalued boundary conditions that encompass the Dirichlet, Neumann, and Sturm–Liouville problems. They also show that their method stay true for the periodic problems but like the references given above, their work does not include variational inequalities.

The aim of this paper is to extend the aforementioned works to a large class of problems incorporating the operators used and variational inequalities with nonlinear and multivalued boundary conditions. At this end, we deal with a nonhomogeneous and nonlinear differential operator, the Laplacian operator , in a problem that incorporates variational inequalities. Our proof is based on a fixed point theorem for ordered Banach spaces due to Heikilla and Hu [13].

The Laplacian operator under consideration applies to several areas such as nonlinear elasticity, non-Newtonian fluid theory, theory of capillary surfaces, and diffusion of flows in porous media (see [14]). As for differential inclusions, they arise in the mathematical modelling of certain problems in the control theory, optimisation, mathematical economics, sweeping process, stochastic analysis, and many other fields (see [15–17]). Finally, variational inequalities models many applied problems, such as differential, Nash games electrical circuits with ideal diodes, dynamic traffic networks and hybrid engineering systems with variable structures, and Coulomb friction for contacting bodies (see [18]).

2. Notations and Preliminaries

Here, we will take stock of the notations and results we will be using in the rest of the article. Our main sources are the books of Hu-Papageorgiou [19] and Zeidler [20].

We denote by a finite measure space and a separable Banach space; is the set of nonempty parts of ; is the Borel field of ; is the set of nonempty and closed parts of ; is the set of nonempty, closed, and convex parts of ; is the set of nonempty, weakly compact, and convex parts of ; is a multifunction defined on with values in ; is a positive real function defined by, for any element of , for any element of , ; is a multifunction defined on with values in , is the graph of the multifunction , i.e., the set of pairs belonging to such that belongs to ; is the set of functions belonging to such that for almost all element of , belongs to , with ; and are Hausdorff topological spaces; is a closed subset of ; is a multifunction defined on with values in ; and is the set of elements of such that .

If is closed, we say that is upper semicontinuous (usc in the abbreviated form). If has closed values and is regular, we say that has a closed graph. If is locally compact, then has a closed graph which implies that has closed values.

If for all and for all , is measurable, we say that the multifunction is measurable. If a multifunction of type is measurable, then its graph is measurable. On the other hand, the opposite is true only if is complete.

belongs to . The set may be empty. If is measurable, then the set is nonempty if only if belongs to .

and are, respectively, a reflexive Banach space and its topological dual. 〈〉 is the duality bracket between and . is the domain of . Let be a subset of , then the map is said to be(1)monotone if for all , for all , ;(2)strictly monotone if is monotone and for all , for all , leads to ;(3)maximal monotone if is monotone and for all leads to .

The maximal monotony of implies that for any element of , the set is nonempty, closed, and convex. In addition, is demiclosed. This means that if the sequence is in , either the sequence converges strongly to in and the sequence converges weakly to in or the sequence converges weakly to in and the sequence converge strongly to in , then the pair belongs to . A map defined on with values in is said to be demicontinuous, if for every sequence that converges to in , we have the sequence that converges weakly to in . If a map is monotone and demicontinuous, then it is maximal monotone. If is a map defined on with values in , with bounded or unbounded such that for , converges to as converges to , then is coercive. A maximal monotone and coercive map is surjective.

Let be Banach spaces and a map defined on with values in . is said to be completely continuous if the sequence converges weakly to in ; then, the sequence converges strongly to in and is said to be compact if it is continuous and maps bounded sets into relatively compact sets. Complete continuity is different from compactness but if is reflexive, then complete continuity leads to compactness. In addition, if is reflexive and is linear, then the complete continuity equals compactness.

Let be a reflexive Banach space, (] a proper, convex, and lower semicontinuous map. Let . The subdifferential of at is the multifunction defined bywhere is a maximal monotone map.

denotes the norm on .

denotes the duality brackets for the pair .

Theorem 1. If and are Banach spaces, is usc from into and is completely continuous and if maps bounded sets into relatively compact sets, then one of the following statements holds:(a)the set is unbounded or(b) has a fixed point.

To establish the existence of a solution for problem (1), we will need the following fixed point theorem for multifunctions in ordered Banach spaces due to Heikkila-Hu [13].

Theorem 2. Let be a separable, reflexive, and ordered Banach space and a nonempty and weakly closed set. Let be a multifunction with weakly closed values. We suppose that is bounded and(i) is nonempty;(ii)If and , then we can find such that .

Then, has a fixed point, that means there exists such that

2.1. Auxiliary Results

Our respective definitions of solutions, lower solution and upper solution, of problem (1) are as follows.

Definition 3. A function such that , with and , is said to be a solution of problem (1) if it verifies

Definition 4. (a)A function such that is said to be a lower solution of problem (1) if there exist such that(b)A function such that is said to be an upper solution of problem (1) if there exist such that

Our hypotheses on the data of (1) are the following:

: problem (1) admits a pair of well-ordered lower and upper solutions and .

: is a continuous positive function such that there exist satisfying

is an increasing homeomorphism map such that(a);(b)there exist such that , for all .

Remark 5. Any increasing homeomorphism of the form, for all , with a continuous map and , for all , satisfies hypotheses .
: is a maximal monotone multivalued map defined bywhere is a function such that(i)for all is measurable;(ii)for almost all is a proper, convex, and lower semicontinuous function.(iii)for every , there exists such that for a.e and for all with and for all , we have .

Remark 6. There exists a nondecreasing function such thatwhere

is a multifunction such that(i)for all is a graph measurable;(ii)for almost all has a closed graph;(iii)for almost all , for all , we can find in such a manner that where , and a nondecreasing function that can be measured in the Borel sense in such a manner that with and (iv)for all , we can find such that for almost all and for all with and for every , .

Remark 7. is measurable. In addition, Remark 1.2 and hypothesis lead to. The hypothesis shows that the derivatives of the solution functions of (1) are uniformly bounded. This is the Bernstein–Nagumo–Wintner growth condition.

: for , the map : is a maximal monotone and .

Remark 8. We can find an increasing positive function such that , where and , .

: is a not necessarily continuous function such that we can find and in such a manner that is decreasing. Also, it maps bounded sets to bounded sets.

Lemma 9. Suppose that and hypotheses and are satisfied andwith . Ifthen there exists that depends to such that , for all .

Proof. We setBy hypothesis (iii) of , we can find such thatAs in the proof of Lemma 1 of [3] or Lemma 5 of [12], we show that for all . In fact, if we reason by the absurd, we arrive at the following contradiction:Let us introduce, respectively, the truncation map , the penalty function , and the map defined bywhere ;We set . For and all , we have . Moreover, for almost all , all , and all , we have with . For every , we setwhere is the Nemitsky operator corresponding to . Then, we define by

Proposition 10. If hypothesis hold, then is usc from into (by , we denote the Lebesgue space furnished with the weak topology).

Proof. See the proof of Proposition 3.7 of Bader-Papageorgiou [11].

Let us introduce the set and the operators and defined, respectively, by

Let be the restriction of to the set . We have,where is lower semicontinuous, proper, and convex (see Barbu [21]) and is a reflexive Banach space. Then, is a maximal monotone map.

Proposition 11. Suppose that hypotheses , , , , and are satisfied. Then, is maximal monotone.

Proof. Let . Let us consider the following nonlinear boundary value problem:The problem (23) has a single solution . To demonstrate this, consider the following problem:where . Let us set . We have, and . Then, let be the function defined by . That means that . By replacing by its expression as a function of in (24), we obtain the following problem:To study (25), let us consider the nonlinear operator defined bywhere is strictly monotone and demicontinuous (see the proof of Proposition 3.10 of Behi-Adjé-Goli [4]). Whence, is maximal monotone.
For all , we haveUsing the hypotheses , it followsTherefore, there exist some such that for all such thatSo, is coercive.
Let be the restriction of to the set . is a proper, semicontinuous, convex, functional map and is a maximal monotone map. Moreover, for all and all,and we have,Then, from (29) and (31), we obtainTherefore, is weakly coercive. Also, since and are maximal monotone maps, with defined on all , is maximal monotone. Moreover, is surjective (because is maximal monotone and weakly coercive). Then, is surjective. Whence, there exists such that . Since is strictly monotone, is unique. It follows that is the unique solution of problem (25). Then, is the unique solution of problem (24). We can define the solution map which assigns to each pair the unique solution of the problem (24). Let be defined byAs in the proof of the Proposition 3.10 of Béhi-Adjé-Goli [4] or Proposition 13 of [3], we can show that is monotone, continuous, and weakly coercive. We infer that is surjective. Set and . We define by , for all . Let be defined byBy Corollary 2.7, p.36 of [22], previous arguments on and hypothesis , we deduce that is surjective. Then, there exists such that . Whence, . Thus, is the single solution of problem (23).
We consider as the operator defined bywhere is maximal monotone because is continuous and monotone. Given any choice of in , arguments above show that is surjective. We deduce that is maximal monotone.

As a result, is surjective and strictly monotone. Thus, is well-defined, single-valued, and maximal monotone.

Proposition 12. If hypotheses and hold, then is completely continuous.

Proof. Let be a sequence which converge weakly to in . As in the proof of Proposition 14 of [3], we can show that in . Therefore, the operator is completely continuous.

2.2. Existence Results

We introduce the functional interval

We consider the operator defined by

We see that is bounded and is continuous.

Let . We consider the following auxiliary boundary problem:

Proposition 13. If the hypotheses , and hold, then problem (38) has a solution .

Proof. Let be the nonlinear operator defined bywhere . From the Proposition 10 and the continuity of the operators and , we infer that is usc from into . Let be the set defined bySuppose that . Then, . It follows thatWhence, for some , we havewhere denotes the duality brackets between and . By integration by parts, we obtainwhere implies that . and are maximal monotone, is monotone, and . We deduce thatUsing hypotheses , , (43), and (44), we obtainFor all ,because . Also, for all ,Furthermore, hypotheses on and implyUsing (44)–(48) in (42), we deduce thatThen,Whence, the set is bounded. As a result, it follows that maps bounded sets into relatively compact sets. Thus, by Theorem 2, we obtain such that . Then, we havewith and . Also, by definition, a function is said to be a lower solution of problem (1), if there exist such thatThen, as in the proof of the Proposition 4.1 of [4] or Proposition 9 of [12], we show that any solution of (1) belongs to .

Theorem 14. If the hypotheses , and hold, then problem (1) has a solution .

Proof. We will use Theorem 2 to establish the proof of this theorem. Let us set . is a separable, reflexive, ordered Banach space. is the solution multifunction for the auxiliary problem . Then, for every , is subset of solutions of problem . From Proposition 13, we know that and . Then, . Moreover, for all , it follows from the proof of the Proposition 13 that is a weakly closed part of and is bounded part of . Now, let us check points (i) and (ii) in Theorem 2. Suppose that . Then, by Proposition 13, and . It follows that if , . So, is verified.
It remains to verify statement of Theorem 2. Let , , and with . implies that , for almost all and . is the map defined byWe have, . Then, for all . Whence, for all , the auxiliary problem becomesSince , by hypothesis , we haveUsing (55) in (54), we obtainfor some and some . It follows that is the lower solution of the boundary value problem.Furthermore, we recall that is an upper solution of (1). Then, by definition,for some and some . Since , we use hypothesis , and we obtainUsing (59) in (57), we obtainIt follows is an upper solution of (57). So, and are ordered lower and upper solutions of (57), respectively. Then, using the same arguments as in the auxiliary problem (38), we obtain a solution of (1) such that .

2.3. Existence of Extremal Solutions

Theorem 15. If the hypotheses , and hold, then problem (1) has some extremal solutions in the order interval .

Proof. By making a few modifications to the proof of Theorem 13 of [12] in relation to the above arguments, we can easily establish the existence of extremal solutions of problem (1) in the functional interval .

2.4. Example and the Periodic Problem

2.4.1. Example

Let us consider the following problem:

Here,

We have, , , where Suppose that for , where is the indicator function of , a closed interval of real numbers containing 0. Then, and are maximal monotone maps such that . If , then (61) becomes a homogeneous Dirichlet problem. If , then (61) becomes a homogeneous Neumann problem. If for , then (61) becomes a Sturm–Liouvile problem.

If , for the cases of Dirichlet and Neumann problems, we show that and defined by and are well-ordered lower and upper solutions of (61). It follows that the problem admits a solution and extremal solutions in the functional interval .

In a general view, if hypothesis is satisfied, the problem (61) admits a solution and extremal solutions in the functional interval because hypotheses , , , and are satisfied.

Suppose that Then, problem (1) becomes the following variational inequality:where . Thus, our results stay true for this kind of problems.

2.4.2. Periodic Problem

Our method stays true for the following periodic problem:

Indeed, setand consider the nonlinear operator is defined by

To establish that the operator is maximal monotone, consider the following auxiliary problem:

We replace the auxiliary problem (23) by the following nonhomogeneous Dirichlet problem:where . Setting , problem (68) becomes the following homogeneous Dirichlet problem:

Then, the nonlinear operator is defined by

Arguing as in the proof of Proposition 11, we show that is strictly monotone, demicontinuous, and coercive. Whence, is strictly monotone and surjective. Then, there exists a unique such that which is the unique solution of problem (68). Then, is the unique solution of the problem (69). We can define the solution map which assigns to each pair the unique solution of the problem (24). Let be defined bywhere . Arguing as in the proof of Proposition 11, we show that is monotone, continuous, and coercive, and is the unique solution of (67). We deduce that is well-defined, single-valued, maximal monotone and completely continuous (from into ).

Finally, with slight modifications to the rest of the arguments, we can establishe the existence of solutions and extremal solutions for the periodic problem.

3. Conclusion

In this paper, we have studied two second-order nonlinear differential inclusions containing a nonhomogeneous Laplacian operator and variational inequalities. One is subject to multivalued boundary conditions encompassing the classical Dirichlet, Neumann, and Sturm–Liouville boundary conditions, and the other is subject to periodic boundary conditions. To study these problems, we have used a method that combine the lower and upper solutions methods, the analysis of multifunctions, the theory of monotone operators, and a fixed point theorem for reflexive Banach spaces. We have obtained results showing the existence of solutions and extremal solutions when the lower and upper solutions are well ordered. We have also demonstrated the applications of our results using some examples [23, 24].

Data Availability

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

Conflicts of Interest

The authors declare that Droh Arsène Béhi is the main author of this article. Each of the authors Assohoun Adjé and Konan Charles Etiennes Goli contributed to the revision of the article at the following three levels: (1) correction of typing errors in the manuscript and provision of more recent bibliographical references; (2) help in providing more details in certain proofs, in particular, concerning the proofs of equation (29) and the periodic case; and (3) advice on improving the introduction to the referees’ reports.

References

E. Picard, “Sur l’application des méthodes d’approximations succesives à l’étude de certains équations différentielles ordinaires,” Journal of Pure and Applied Mathematics, vol. 9, pp. 217–271, 1893.
View at: Google Scholar
E. Picard, “Mémoire sur la théorie des équations aux derivés partielles et les méthode des approximations successives,” Journal of Pure and Applied Mathematics, vol. 6, pp. 145–210, 1890.
View at: Google Scholar
D. A. Béhi and A. Adjé, “Existence and multiplicity results for second-order nonlinear differential equations with multivalued boundary conditions,” Journal of Applied Mathematics and Physics, vol. 07, no. 06, pp. 1340–1368, 2019.
View at: Publisher Site | Google Scholar
D. A. Béhi and A. Adjé, “Goli: lower and upper solutions method for nonlinear second-order differential equations involving a Φ-Laplacian operator,” African Diaspora Journal of Mathematics, vol. 22, no. 1, pp. 22–41, 2019.
View at: Google Scholar
N. El Khattabi, M. Frigon, and N. Ayyadi, “Multiple solutions of boundary value problems with Φ-Laplacian operators and under a Wintner-Nagumo growth condition,” Boundary Value Problems, vol. 236, p. 21, 2013.
View at: Google Scholar
X. Shu and F. Xu, “Upper and lower solution method for fractional evolution equation with order 1<σ<2,” Journal of the Korean Mathematical Society, vol. 51, no. 6, pp. 1123–1139, 2014.
View at: Publisher Site | Google Scholar
Z. Li, X. Shu, and F. Xu, “The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem,” AIMS Mathematics, vol. 5, no. 6, pp. 6189–6210, 2020.
View at: Publisher Site | Google Scholar
Z. Li, X. Shu, and T. Miao, “The existence of solutions for Sturm–Liouville differential equation with random impulses and boundary value problems,” Boundary Value Problems, vol. 97, 2021.
View at: Publisher Site | Google Scholar
M. Frigon, “Problèmes aux limites pour des inclusions différentielles sans condition de croissance,” Annals of Polish Mathematicians, vol. 54, no. 1, pp. 69–83, 1991.
View at: Publisher Site | Google Scholar
M. Frigon, “Théorème d’existence de solutions d’inclusions differentielles, Topological Methods in Differential Equations and Inclusions,” Topological Methods in Differential Equations and Inclusions, vol. 472, pp. 51–87, 1995.
View at: Google Scholar
R. Bader and N. S. Papageorgiou, “Nonlinear multivalued boundary value problems discussiones mathematicae differential inclusions,” Control and Optimization, vol. 21, pp. 127–148, 2001.
View at: Google Scholar
V. Staicu and N. S. Papageorgiou, “The method of upper lower solutions for nonlinear second order differential inclusions,” Nonlinear Analysis, vol. 67, pp. 708–726, 2007.
View at: Google Scholar
S. Heikilla and S. Hu, “On fixed points of multifunctions in ordered spaces,” International Journal, vol. 51.
View at: Publisher Site | Google Scholar
A. Calamail, C. Marcelli, and F. Papalini, “Boundary value problems for singular second order equations,” Fixed Point Theory and Applications, vol. 2018, p. 20, 2018.
View at: Google Scholar
J. Chen, Z. Lui, F. E. Lomotsev, and V. Obuklovskii, “Optimal feedback control for a class of second-order evolution differential inclusion with Clarke’s subdifferential,” Journal Of Nonlinear and Variational Analysis, vol. 6, pp. 551–565, 2022.
View at: Google Scholar
J. R. Granada and V. A. Kovtunenko, “A shape derivative for optimal control of the nonlinear Brinkman-ForchHeimer equation,” Journal of Applied and Numerical Optimization, vol. 3, pp. 243–261, 2021.
View at: Google Scholar
C. Nuchpong, T. Schoployklang, S. K. Ntouyas, and J. Tariboon, “Boundary value problems for Hilfer-Hadamard fractional differential inclusion with nonlocal integro-multi-point boundary conditions,” Journal of Nonlinear Functional Analysis, vol. 18, p. 37, 2022.
View at: Google Scholar
L. Lu, Z. Liu, and V. Obukhovskii, “Second order differential variational inequalities involving anti-periodic boundary value conditions,” Journal of Mathematical Analysis and Applications, vol. 473, pp. 846–865, 2019.
View at: Google Scholar
S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Theory, Kluwer, Dordrecht, The Netherlands, 1997.
E. Zeidler, Nonlinear Functional Analysis and its Applications II, Springer- Verlag, New York, NY, USA, 1990.
V. Barbu and T. Precupanu, Convexity And Optimization In Banach Spaces, Sijthoff Noordoff, International Publishers, New York, NY, USA, 1978.
H. Brezis, Opérateurs Maximaux Monotones, North-Holland, Amsterdam, The Netherlands, 1973.
N. Halidias and N. S. Papageorgiou, “Existence and relaxation results for nonlinear second order multivalued boundary value problems in RN,” Journal of Differential Equations, vol. 147, pp. 123–154, 1998.
View at: Google Scholar
N. S. Papageorgiou and S. T. Kyritsi-Yiallourou, Handbook of Applied Analysis, Springer Science+Business Medi, Berlin, Germany, 2009.

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