Stationary Kirchhoff equations and systems with reaction terms

Abstract

In this paper, the operator approach based on the fixed point principles of Banach and Schaefer is used to establish the existence of solutions to stationary Kirchhoff equations with reaction terms. Next, for a coupled system of Kirchhoff equations, it is proved that under suitable assumptions, there exists a unique solution which is a Nash equilibrium with respect to the energy functionals associated to the equations of the system. Both global Nash equilibrium, in the whole space, and local Nash equilibrium, in balls are established. The solution is obtained by using an iterative process based on Ekeland’s variational principle and whose development simulates a noncooperative game..

Authors

Radu Precup
Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Andrei Stan
Department of Mathematics Babes-Bolyai University, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Keywords

Kirchhoff equation; fixed point principle; Nash equilibrium; Ekeland variational principle.

Paper coordinates

R. Precup, A. Stan, Stationary Kirchhoff equations and systems with reaction terms, AIMS Mathematics, 7 (2022) no. 8, pp.15258-15281. https://doi.org/10.3934/math.2022836

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AIMS Mathematics

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AIMS Mathematics

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[1] Be ldzinski, M., Galewski, M.,Nash–type equilibria for systems of non-potentialequations, Appl. Math. Comput.385(2020), 125456.

[2] Benedetti, I., Cardinali, T., Precup, R.,Fixed point-critical point hybrid theo-rems and applications to systems with partial variational structure, submitted.

[3] Cournot, A.,The mathematical principles of the theory of wealth,EconomicJ.,1838.

[4] Mawhin, J., Willem, M.,Critical Point Theory and Hamiltonian Systems,Springer, Berlin, 1989.

[5] Nash, J.,Non-cooperative games,Ann. of Math.54(1951), 286-295.

[6] Precup, R.,Methods in Nonlinear Integral Equations, Springer, Amsterdam,2002.

[7] Precup, R.,Nash-type equilibria and periodic solutions to nonvariational sys-tems, Adv. Nonlinear Anal.4(2014), 197-207

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Stationary Kirchhoff equations and systems with reaction terms

Stationary Kirchhoff equations and systems with reaction terms

Radu Precup Andrei Stan 1 2 \addr11affiliationmark: Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania & Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, 400110 Cluj-Napoca, Romania \addr22affiliationmark: Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania & Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, 400110 Cluj-Napoca, Romania
Abstract

In this paper, the operator approach based on the fixed point principles of Banach and Schaefer is used to establish the existence of solutions to stationary Kirchhoff equations with reaction terms. Next, for a coupled system of Kirchhoff equations, it is proved that under suitable assumptions, there exists a unique solution which is a Nash equilibrium with respect to the energy functionals associated to the equations of the system. Both global Nash equilibrium, in the whole space, and local Nash equilibrium, in balls are established. The solution is obtained by using an iterative process based on Ekeland’s variational principle and whose development simulates a noncooperative game.

keywords:
Kirchhoff equation, Fixed point principle, Nash equilibrium, Ekeland variational principle.
Mathematics Subject Classification: 47J25, 35J65

1 Introduction

The famous Kirchhoff equation [1]

utt(a+bΩu2𝑑x)u′′=h(t,x)

(a,b>0) is an extension of the classical D’Alembert’s wave equation for vibrations of elastic strings, which takes into account the changes in mass density and/or tension force of the string produced by transverse vibrations. In higher dimensions, the equation reads as follows

utt(a+bΩ|u|2𝑑x)Δu=h(t,x).

One can also consider the parabolic type equation

ut(a+bΩ|u|2𝑑x)Δu=h(t,x)

which models diffusion processes with a diffusion coefficient globally dependent on gradient.

Several authors (see, e.g., [2], [3], [4], [5], [6], [7], [8]) have considered a more general Kirchhoff type equation, by replacing the integral factor a+b|u|L22 with an expression of the form η(|u|L2), where η is an increasing and nonnegative function.

Kirchhoff type problems are referred to be nonlocal due to the presence of the integral over the entire Ω, and due to this specificity, some difficulties arise in their investigation.

The study of such equations and systems have been made using variational and topological methods, as well as upper and lower solution techniques (see, e.g., [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20] and the references therein).

In this paper, we first study the Dirichlet problem for a stationary integro-differential equation of Kirchhoff type with a reaction external force term, on a bounded domain Ωn,

{(a+bΩ|u|2𝑑x)Δu=f+g(x,u,u)in Ωu|Ω=0,

and next we focus on the Dirichlet problem for a coupled system of Kirchhoff equations

{(a+b|u|H012)Δu=f1+g1(x,u,v)(a+b|v|H012)Δv=f2+g2(x,u,v)u|Ω=v|Ω=0 (1.1)

for which the solution is a Nash type equilibrium.

To our knowledge, Nash equilibria of system (1.1) have not been considered so far, and our objective is to provide sufficient conditions for such solutions to exist. To this aim we use the approach initiated in [21] (see also [22], [23], [24], [25], [26] and [27]). The idea is to put system (1.1) in an operator form, as a fixed point system,

{N1(u,v)=uN2(u,v)=v, (1.2)

where the operators N1 and N2 admit a variational structure, i.e., there exist (energy) functionals E1(u,v) and E2(u,v) such that system (1.2) is equivalent with

{E11(u,v)=0E22(u,v)=0. (1.3)

where E11 is the partial Fréchet derivative of E1 with respect to the first variable and E22 is the partial Fréchet derivative of E2 with respect to the second variable. A solution (u,v) of (1.1) is a Nash equilibrium if

{E1(u,v)=infE1(,v)E2(u,v)=infE2(u,).

The notion of a Nash equilibrium originated in game theory and economics, where a number of players or traders with their own costing criteria are in competition and each aims to optimize its cost in relation to the others. When no one can further improve his criterion, it means that the system has reached a Nash equilibrium state. Such kind of situations also hold for systems modeling real processes from physics, biology etc., when stationary states are Nash equilibria for the associated energy functionals.

Non-cooperative games in which the players move alternately suggest an iterative method based on Ekeland’s variational principle for finding and approximating Nash equilibria. The convergence of the iterative process is established by using unilateral Lipschitz conditions on the reaction terms and working techniques with inverse-positive matrices.


The outline of this paper is as follows: Section 3 provides a comprehensive picture of the theoretical aspects of the Kirchhoff solution operator for the Dirichlet problem. Section 4 is dedicated to the Dirichlet problem for the stationary Kirchhoff equation with a reaction force term; the existence of solutions is established via Banach contraction principle and Schaefer’s fixed point theorem. Finally in Section 5 there are provided sufficient conditions for a system of two Kirchhoff equations to admit a Nash equilibrium.

2 Preliminaries

In this section we collect a number of notions and results that will be used in the following.

First we recall the weak form of Ekeland’s variational principle (see, e.g., [28, Corollary 8.1]).

Theorem 1 (Ekeland)

Let (X,d) be a complete metric space and E:X a lower semicontinuous functional bounded from below. For each ε>0, there is an element xX such that the following two properties hold:

E(x) infyXE(y)+ε,
E(x) E(y)+εd(x,y)for all yX.

Next we recall Perov’s fixed point theorem (see, e.g., [28, pp 151-154]) for mappings defined on the Cartesian product of two metric spaces.

Theorem 2 (Perov)

Let (Xi,di), i=1,2 be complete metric spaces and Ni:X1×X2Xi be two mappings for which there exists a square matrix M of size two with nonnegative entries and the spectral radius ρ(M)<1 such that the following vector inequality

(d1(N1(x,y),N1(u,v))d2(N2(x,y),N2(u,v)))M(d1(x,y)d2(u,v))

holds for all (x,y),(u,v)X1×X2. Then there exists a unique point (x,y)X1×X2 with x=N1(x,y) and y=N2(x,y). Moreover, the point (x,y) can be obtained by the method of successive approximations starting from any initial point (x0,y0), and

(d1(N1k(x0,y0),x)d2(N2k(x0,y0),y))Mk(IM)1(d1(x0,N1(x0,y0))d2(y0,N2(x0,y0)))

for every k.

Here I stands for the unit matrix of size two. Note that the property of a square matrix M with nonnegative entries of having the spectral radius ρ(M) less than 1 is equivalent to each one of the properties: (a) Mk tends to the zero matrix as k+; (b) The matrix IM is nonsingular and the entries of its inverse (IM)1 are nonnegative.

For our Kirchhoff system (1.1) both fixed point and critical point formulations ((1.2) and (1.3)) being available, both Perov approach and Ekeland variational approach can be used. The first approach offers the approximation procedure for the solution given by the method of successive approximations, while by the second approach, an approximation procedure more appropriate to the concept of Nash equilibrium can be established.

We conclude this preliminary section by some notations and results related to Laplacian. For details we refer the reader to the book [29]. We consider the well-known Sobolev space H01(Ω) whose scalar product and norm are

(u,v)H01=Ωuvdx,|u|H01=|u|L2=(Ω|u|2𝑑x)1/2.

The notation H1(Ω) stands for the dual of H01(Ω) and for any fH1(Ω),uH01(Ω), by (f,u) we mean the value at u of the continuous linear functional f. One has the continuous embeddings H01(Ω)L2(Ω)H1(Ω) and the Poincaré inequalities

|u|L2 1λ1|u|H01(uH01(Ω)),
|u|H1 1λ1|u|L2(uL2(Ω)),

where λ1 is the first eigenvalue of the Dirichlet problem for the operator Δ. We use the notation (Δ)1 for the inverse of the Laplacian with respect to the homogeneous Dirichlet boundary condition. More exactly, (Δ)1f=u, where u is the unique function in H01(Ω) satisfying (u,v)H01=(f,v) for all vH01(Ω), i.e., u is the weak solution of the Dirichlet problem Δu=fin Ω;u=0 on Ω. Recall that (Δ)1 is an isometry between H1(Ω) and H01(Ω).

3 Stationary Kirchhoff equations

3.1 The Kirchhoff solution operator

First we focus on the stationary equation

(a+bΩ|u|2𝑑x)Δu=h

under the Dirichlet condition u=0 on Ω.

Theorem 3

(The solution operator) For each hH1(Ω), the Dirichlet problem has a unique weak solution uH01(Ω), i.e.,

(a+bΩ|u|2𝑑x)(u,v)H01=(h,v),vH01(Ω), (3.1)

and the solution operator S:H1(Ω)H01(Ω), hu, is continuous and

|S(h)|H011a|h|H1. (3.2)
Proof 3.4.

(a) Existence: Let hH1(Ω) be fixed and consider the operator Sh:H01(Ω)H01(Ω) defined by

Sh(v)=1a+b|v|H012(Δ)1h.

Clearly, Sh is completely continuous. In addition,

|Sh(v)|H011a|h|H1,vH01(Ω). (3.3)

Hence, if we denote B={vH01(Ω):|v|H011a|h|H1}, then Sh(B)B and according to Schauder’s fixed point theorem, there exists at least one u such that Sh(u)=u. Clearly u is a solution of the Dirichlet problem.

(b) Uniqueness: Assume that u1,u2 are two solutions of (3.1). Then

(a+b|u1|H012)|u1|H012 = (h,u1),
(a+b|u2|H012)(u1,u2)H01 = (h,u1).

It follows that

(a+b|u1|H012)|u1|H012 = (a+b|u2|H012)(u1,u2)H01
(a+b|u2|H012)|u1|H01|u2|H01.

Hence

(a+b|u1|H012)|u1|H01(a+b|u2|H012)|u2|H01.

The function (a+bx2)x being strictly increasing on +, we have that |u1|H01|u2|H01. By symmetry the converse inequality also holds. Thus |u1|H01=|u2|H01. Now the uniqueness of solution for the Dirichlet problem related to Δ yields u1=u2.

(c) Continuity: Let hkh in H1(Ω) and let uk:=S(hk). Using (3.3) we have that the sequence (uk) is bounded. Hence, passing if necessary to a subsequence, we may assume that the sequence of real numbers (|uk|) is convergent. We now prove that the sequence (uk) is Cauchy. From

Δuk=1a+b|uk|H012hk,

we have

Δ(ukup)=1a+b|uk|H012hk1a+b|up|H012hp

in the weak sense. Consequently

|ukup|H012 = (1a+b|uk|H012hk1a+b|up|H012hp,ukup)
= 1a+b|uk|H012(hkhp,ukup)
+(1a+b|uk|H0121a+b|up|H012)(hp,ukup).

Furthermore

|ukup|H0121a|hkhp|H1|ukup|H01+ba2||uk|H012|up|H012||hp|H1|ukup|H01,

whence

|ukup|H011a|hkhp|H1+ba2||uk|H012|up|H012||hp|H1.

Since |hp|H1 is bounded, (hk) and (|uk|H012) are convergent, one immediately obtain that (uk) is Cauchy. Hence there is u with uku and passing to the limit we see that u=S(h). Finally the uniqueness of the solution implies that the whole sequence (uk) converges to S(h), that is S(hk)S(h).

Theorem 3.5.

(Monotonicity) If 0h1h2, then |S(h1)|H01|S(h2)|H01.

Proof 3.6.

Denote u:=S(h1) and v=S(h2). Since h1,h20, one has u,v0.Then

(1+|u|H012)|u|H012=(h1,u)(h2,u)=(1+|v|H012)(u,v)(1+|v|H012)|u|H01|v|H01

which gives

(1+|u|H012)|u|H01(1+|v|H012)|v|H01,

whence |u|H01|v|H01.

Theorem 3.7.

(The energy functional) A function uH01(Ω) is the weak solution of the Dirichlet problem if and only if it is a critical point of the C1 functional E:H01(Ω),

E(v)=14(2a+b|v|H012)|v|H012(h,v). (3.4)
Proof 3.8.

One has

|v+λw|H012|v|H012 = 2λ(v,w)H01+λ2|w|H012,
|v+λw|H014|v|H014 = (|v+λw|H012|v|H012)(|v+λw|H012+|v|H012)
= (2λ(v,w)H01+λ2|w|H012)(|v+λw|H012+|v|H012).

Consequently

limλ0E(v+λw)E(v)λ=(a+b|v|H012)(v,w)H01(h,w).

Hence

(E(v),w)=(a+b|v|H012)(v,w)H01(h,w). (3.5)
Theorem 3.9.

Function uH01(Ω) solves the Dirichlet problem if and only if it minimizes the energy functional (3.4).

Proof 3.10.

If u is a minimum point of E, then E(u)=0 and according to (3.5) it solves the problem. Assume now that u is a solution. Then for every v, by direct computation, we have

E(u+v) = E(u)+(a+b|u|H012)(u,v)H01(h,v)
+a2|v|2+b4(|v|4+2|u|2|v|2+4(u,v)2+4|v|2(u,v)2)
= E(u)+a2|v|2+b4(|v|4+2|u|2|v|2+4(u,v)2+4|v|2(u,v)2)
E(u)+a2|v|2+b4((|v|2+2(u,v))2+2|u|2|v|2)>0

for every v0. Hence u is the unique minimum point of E.

4 Kirchhoff equations with reaction terms

Consider the Dirichlet problem

{(a+bΩ|u|2𝑑x)Δu=f+g(x,u,u)in Ωu=0on Ω. (4.1)

Here Ωn is open bounded, fH1(Ω), g:Ω××n satisfies the Carathéodory conditions and g(,0,0)=0.

We look for weak solutions to (4.1), namely uH01(Ω) with g(,u,u)H1(Ω) and

(a+bΩ|u|2𝑑x)(u,v)H01=(f+g(,u,u),v)for all vH01(Ω).

A function uH01(Ω) is a weak solution of (4.1) if

u=1a+bΩ|u|2𝑑x(Δ)1(f+g(,u,u)),

that is u is a fixed point of the operator

A(u)=S(f+g(,u,u)).

4.1 Existence and uniqueness of solution

We apply Banach contraction principle. Assume the Lipschitz condition

(HL)
|g(x,u,v)g(x,u¯,v¯)|L1|uu¯|+L2|vv¯|

for all u,u¯, v,v¯n and a.e. xΩ, where

θ:=1a(L1λ1+L2λ1)<1. (4.2)

Step 1: Invariance of a ball.

We prove that if L1,L2 are small, then for any large enough number R, one has |A(u)|H01R for all uH01(Ω) with |u|H01R. According with (3.2), using (HL) and Poincaré’s inequality, one has

|A(u)|H01 = |S(f+g(,u,u))|H011a|f+g(,u,u)|H1
1a(|f|H1+|g(,u,u)|H1)1a(|f|H1+1λ1|g(,u,u)|L2)
1a(|f|H1+1λ1(L1|u|L2+L2|u|L2))
1a|f|H1+1a(L1λ1+L2λ1)|u|H01.

Hence in virtue of (4.2), the invariance condition holds for any number R |f|H1/(a(1θ)).

Step 2: Contraction condition.

Fix any number R as guaranteed at the previous step. Let u,vH01(Ω) with |u|H01,|v|H01R be arbitrary and let w=S(f+g(,u,u)) and z=S(f+g(,v,v)). Assume without loss of generality that |w|H01|z|H01. Then

(a+b|w|H012)|w|H012 = (f+g(.,u,u),w),
(a+b|z|H012)(w,z)H01 = (f+g(.,v,v),w),

whence

(a+b|w|H012)|w|H012(a+b|z|H012)(w,z)H01=(g(.,u,u)g(.,v,v),w).

For the left side, one has

(a+b|w|H012)|w|H012(a+b|z|H012)(w,z)H01(a+b|w|H012)|w|H012(a+b|z|H012)|w|H01|z|H01

and for the right side

(g(.,u,u)g(.,v,v),w) |g(.,u,u)g(.,v,v)|L2|w|L2
(L1λ1+L2λ1)|uv|H01|w|H01.

Hence

(a+b|w|H012)|w|H012(a+b|z|H012)|w|H01|z|H01(L1λ1+L2λ1)|uv|H01|w|H01

and since |w|H01|z|H01,

0a(|w|H01|z|H01)(a+b|w|H012)|w|H01(a+b|z|H012)|z|H01(L1λ1+L2λ1)|uv|H01.

Consequently

0|w|H01|z|H01θ|uv|H01.

On the other hand, from

(a+b|w|H012)(w,wz)H01 = (f+g(.,u,u),wz),
(a+b|z|H012)(z,wz)H01 = (f+g(.,v,v),wz),

we deduce that

|wz|H012 = (f+g(.,u,u)a+b|w|H012f+g(.,v,v)a+b|z|H012,wz)
= 1a+b|w|H012(g(.,u,u)g(.,v,v),wz)
+(1a+b|w|H0121a+b|z|H012)(f+g(.,v,v),wz).

We have

(g(.,u,u)g(.,v,v),wz)(L1λ1+L2λ1)|uv|H01|wz|H01

and

(1a+b|w|H0121a+b|z|H012)(f+g(.,v,v),wz)
b(|w|H01|z|H01)|w|H01+|z|H01(a+b|w|H012)(a+b|z|H012)|f+g(.,v,v)|H1|wz|H01.

Since

|w|H01+|z|H01(a+b|w|H012)(a+b|z|H012)3381aab

and

|g(.,v,v)|H1aθR,

we obtain

|wz|H01θ(1+θ338aba(|f|H1+aθR))|uv|H01.

Hence if

(HC)
θ(1+θ338aba(|f|H1+aθR))<1,

then the operator A is a contraction on the ball of H01(Ω) centered at the origin and of radius R. Notice that condition (HC) is fulfilled for example if θ<1 (invariance condition for the ball of radius R) and b is small enough.

Thus Banach’s contraction principle applied to operator A in the ball of radius R yields the following existence and uniqueness result.

Theorem 4.11.

Assume that conditions (HL) and (HC) hold. Then problem (4.1) has a unique solution u such that

|u|H01|f|H1/(a(1θ)).
Example 4.12.

Consider the Dirichlet problem,

{(4+|u|2𝑑x)Δu=2|x|+λ1u+λ1sin|u|on u|=0, (4.3)

where Ω= and is the open ball centered at the origin of n and of radius ρ whose measure equals 1. Here

a=4, b=1, f(x)=2|x| and g(x,u,v)=λ1u+λ1sin|v|,

for u and vn. Note that fH1() with |f|H1=1. Indeed, the function u0(x)=|x|1 is the weak solution of Dirichlet problem Δu=f in , u|=0 and consequently

|f|H1=|u0|H01=|u0|L2=|x|x||L2=1.

Clearly, g is a Carathéodory function, g(,0,0)=0 and satisfies condition (HL) with L1=λ1 and L2=λ1 and θ=2/a=1/2.

For R=|f|H1/(a(1θ))=1/ 2, the condition (HC) is fulfilled, since

θ(1+θ338aba(|f|H1+aθR))=12(1+3364)<1.

Therefore, the problem (4.3) has a unique solution uH01() with |u|H011/2.

4.2 Existence via Schaefer’s fixed point theorem

Step 1: Complete continuity of the operator A:H01(Ω)H01(Ω).

Recall that (Δ)1:H1(Ω)H01(Ω) is an isometry between H1(Ω) and H01(Ω). This implies that the operator A is completely continuous if the operator

uB(u):=g(,u,u)

is well-defined and completely continuous from H01(Ω) to H1(Ω).

Assume that n3. Then the embedding H01(Ω)Lp(Ω) is continuous for 1p2=2n/(n2), and compact for 1p<2, and consequently the embedding Lq(Ω)H1(Ω) holds and is compact for q>(2)=2n/(n+2).

We would like to represent B as a composition of three operators: B=JNP, where

P : H01(Ω)L2(Ω)×L2(Ω;n),P(u)=(u,u),
N : L2(Ω)×L2(Ω;n)Lq(Ω),N(w1,w2)=g(,w1,w2),
J : Lq(Ω)H1(Ω),J(v)=v.

Clearly, since the embedding H01(Ω)L2(Ω) is continuous, P is a bounded linear operator. Also, if q>(2), then J is completely continuous. It remains to clarify the case of Nemytskii’s operator N. It suffices that N is well-defined, continuous and bounded (maps bounded sets into bounded sets). To this aim, recall the main result about Nemytskii’s operator (see, e.g., [29, Section 9.1]). According to this result, we need a growth condition on g, namely

|g(x,w1,w2)|c1|w1|2q+c2|w2|2q+h(x)(w1,w2n,a.a. xΩ)

where c1,c2+ are constants and hLq(Ω). Notice that instead of the exponents 2/q, 2/q one may have smaller exponents, let they be α and β, hence a growth condition like

|g(x,w1,w2)|c1|w1|α+c2|w2|β+h(x) (4.4)

with 1α2q, 1β2q. These give some conditions on q:

q2α,q2β.

Thus we can take

q=min{2α,2β}.

Finally, the condition q>(2)   holds if

α<2(2),β<2(2).

Note that

2(2)=n+2n2,2(2)=n+2n.

Therefore, the operator N is as desired provided that g satisfies the growth condition (4.4) for

1α<n+2n2, 1β<n+2n

and hL2(Ω)

Step 2: A priori boundedness of solutions.

Let uH01(Ω) be any solution of the equation λA(u)=u for some λ(0,1). Then u is a weak solution of the problem

{(a+bλ2Ω|u|2𝑑x)Δu=λf+λg(x,u,u)in Ωu=0on Ω.

Hence

(a+bλ2Ω|u|2𝑑x)(u,v)H01=(λf+λg(,u,u),v),vH01(Ω).

Letting v=u gives

(a+bλ2|u|H012)|u|H012=λ(f,u)+λ(g(,u,u),u).

Since g(,u,u)Lq(Ω), one has (g(,u,u),u)=Ωug(x,u,u). Assume that g satisfies the sign condition

ug(x,u,v)0 for all u,vn,a.a. xΩ. (4.5)

Then (g(,u,u),u)0 and so

(a+bλ2|u|H012)|u|H012λ(f,u)|f|H1|u|H01.

Thus

a|u|H01(a+bλ2|u|H012)|u|H01|f|H1,

that is the solutions are bounded independently of λ, namely |u|H01|f|H1/a.

Therefore, based on Schaefer’s fixed point theorem, we have the following existence result.

Theorem 4.13.

Assume that g satisfies the growth condition (4.4) for some numbers 1α<(n+2)/(n2), 1β<(n+2)/n and function hL2(Ω). Also assume that g has the sign property (4.5). Then problem (4.1) has at least one weak solution uH01(Ω) with |u|H01|f|H1/a.

Example 4.14.

Consider the Dirichlet problem,

{(1+|u|2𝑑x)Δu=2|x|u3u2+1uu2+1|u|on u|=0, (4.6)

where is as in Example 4.12. We apply Theorem 4.13. Here

f(x)=2|x|,g(,u,v)=u3u2+1uu2+1|v|

for u and vn. Similarly to Example 4.12, one has fH1(Ω) and |f|H1=1. Moreover, g satisfies the growth condition (4.4) with α=β=1 and the sign condition (4.5) since

|g(x,u,v)||u|+12|v|

and

ug(x,u,v)=u4u2+1u2u2+1|v|0,

for all u and vn. Consequently, problem (4.6) has at least one weak solution in H01() with |u|H011.

5 Nash equilibrium for Kirchhoff systems

In this section our focus is on system (1.1), where we look for a solution which is a Nash equilibrium.

5.1 Global Nash equilibrium

We start by an existence and uniqueness result in the whole space H01(Ω)×H01(Ω).

Each equation of system (1.1) has a variational structure given respectively by the energy functionals E1,E2:H01(Ω)×H01(Ω),

E1(u,v) = 14(2a+b|u|H012)|u|H012(f1,u)ΩG1(x,u(x),v(x))𝑑x,
E2(u,v) = 14(2a+b|v|H012)|v|H012(f2,v)ΩG2(x,u(x),v(x))𝑑x,

where G1(x,u,v)=0ug1(x,s,v)𝑑s and G2(x,u,v)=0vg2(x,u,s)𝑑s. Using (3.5), we easily see that

E11(u,v) = (a+b|u|H012)u(Δ)1(f1+g1(,u,v)),
E22(u,v) = (a+b|v|H012)v(Δ)1(f2+g2(,u,v)),

for every u,vH01(Ω).

Before stating the main result of this section we introduce the following notion: A function H:Ω× is said to be of coercive-type if the functional ϕ:H01(Ω),

ϕ(v)=14(2a+b|v|H012)|v|H012(f2,v)ΩH(x,v)𝑑x (5.1)

is coercive, i.e., ϕ(v)+ as |v|H01+.

We have the following result on the existence of a Nash equilibrium under unilateral Lipschitz (monotonicity type) conditions.

Theorem 5.15.

Assume that for i=1,2, fiH1(Ω), gi:Ω×2 is a Carathéodory function and gi(,0,0)=0. In addition assume that the following conditions are satisfied:

(h1)

There exist constants aij+(i,j=1,2) such that

aii<λ1a,i=1,2,
a12a21<(λ1aa11)(λ1aa22) (5.2)

and

(g1(x,u,v)g1(x,u¯,v¯))(uu¯) a11|uu¯|2+a12|uu¯||vv¯|, (5.3)
(g2(x,u,v)g2(x,u¯,v¯))(vv¯) a21|uu¯||vv¯|+a22|vv¯|2

for all u,v,u¯,v¯ and a.e. xΩ.

(h2)

There exist   two functions H1,H2:Ω× of coercive-type such that

H1(x,v)G2(x,u,v)H2(x,v)

for all u,v, a.e. xΩ.

Then system (1.1) has a unique solution which is a Nash equilibrium for the pair of functionals (E1,E2).

Proof 5.16.

The proof follows the idea from [22]. For a clear comprehending, we structure our proof in six steps.

Step 1: The functionals E1(,v) and E2(u,) are bounded from below. First let us remark that from (5.3), for every u,v, there exist θ(0,1) such that

G1(x,u,v) = 0ug1(x,s,v)𝑑s=ug1(x,θu,v)
= 1θg1(x,θu,v)θu1θ(a11|θu|2+a12|θu||v|)
= a11θu2+a12|u||v|a11u2+a12|u||v|.

Similarly

G2(x,u,v)a21|u||v|+a22v2.

Now let vH01(Ω) be fixed. For any uH01(Ω), one has

E1(u,v)= 14(2a+b|u|H012)|u|H012(f1,u)ΩG1(x,u(x),v(x))𝑑x (5.4)
14(2a+b|u|H012)|u|H012|f1|H1|u|H01(a11|u|L22+a12|u|L2|v|L2)
14(2a+b|u|H012)|u|H012a111λ1|u|H012a121λ1|u|H01|v|H01|f1|H1|u|H01
b4|u|H014+(a2a11λ1)|u|H012(|f1|H1+a12λ1|v|H01)|u|H01, (5.5)

which is bounded from below since the coefficient of the term of forth degree of the quartic expression in |u|H01 is positive. Similarly the functional E2(u,) is bounded from below for each u.

Step 2: Construction of an approximation sequence (uk,vk).

Now, similarly to [21], starting with an arbitrary v0 and using Ekeland’s variational principle, we recursively construct a sequence (uk,vk)H01(Ω)×H01(Ω) such that

E1(uk,vk1)infH01(Ω)E1(,vk1)+1k , E2(uk,vk)infH01(Ω)E2(uk,)+1k,
|E11(uk,vk1)|H011k, |E22(uk,vk)|H011k. (5.6)

Step 3: Boundedness of the sequence (vk).

Let ϕ1,ϕ2 be the functionals of type (5.1) with ϕ replaced by ϕ1 and ϕ2, respectively. As coercive functionals they are bounded from below.

Obviously, for every u,v, one has

ϕ1(v)E2(u,v)ϕ2(v).

The coerciveness of ϕ2 implies that there is R>0 with

ϕ2(v)infH01(Ω)ϕ1+1,|v|H01>R.

Since infH01(Ω)ϕ1infH01(Ω)E2(u,) for all u, we obtain

E2(u,v)infH01(Ω)E2(u,)+1for all u,vH01(Ω),|v|H01>R. (5.7)

Since for k2,

E2(uk,vk)infH01(Ω)E2(uk,)+1k<infH01(Ω)E2(uk,)+1,

in view of (5.7) we must have |vk|H01R, that is the boundedness of the sequence (vk).

Step 4: Convergence of the sequences (uk) and (vk).

For every u,u¯,v,v¯H01(Ω), we have

(E11(u,v)E11(u¯,v¯),uu¯)H01 = ((a+b|u|H012)u(a+b|u¯|H012)u¯,uu¯)H01
(g1(,u,v)g1(,u¯,v¯),uu¯)L2
= a|uu¯|H012+b(|u|H012u|u¯|H012u¯,uu¯)H01
(g1(,u,v)g1(,u¯,v¯),uu¯)L2.

Since

(|u|H012u|u¯|H012u¯,uu¯)H01 = |u|H014+|u¯|H014(|u|H012+|u¯|H012)(u,u¯)H01
|u|H014+|u¯|H014(|u|H012+|u¯|H012)|u|H01|u¯|H01
= (|u|H012+|u¯|H012+|u|H01|u¯|H01)(|u|H01|u¯|H01)20

we obtain

(E11(u,v)E11(u¯,v¯),uu¯)H01 a|uu¯|H012(g1(,u,v)g1(,u¯,v¯),uu¯)L2
a|uu¯|H012a11|uu¯|L22a12|uu¯|L2|vv¯|L2
(aa11λ1)|uu¯|H012a12λ1|uu¯|H01|vv¯|H01. (5.8)

Similarly

(E22(u,v)E22(u¯,v¯),vv¯)H01(aa22λ1)|vv¯|H012a21λ1|uu¯|H01|vv¯|H01. (5.9)

On the other hand, from (5.6) we obtain

(E11(uk+p,vk+p1)E11(uk,vk1),uk+puk)H01(1k+p+1k)|uk+puk|H01,
(E22(uk+p,vk+p)E11(uk,vk),vk+pvk)H01(1k+p+1k)|vk+pvk|H01.

Consequently, if we denote mii=aaiiλ1(i=1,2),m12=a12λ1 and m21=a21λ1, then

m11|uk+puk|H01m12|vk+p1vk1|H012k,m21|uk+puk|H01+m22|vk+pvk|H012k. (5.10)

Under the notations xk,p:=|uk+puk|H01 and yk,p=|vk+pvk|H01, relations (5.10) can be put under the matrix form

M[xk,pyk,p]2k[11](MM)[xk1,pyk1,p],

where

M=[m11m12m21m22],M=[m110m21m22].

Since M is invertible and its inverse

M=1[1m110m21m11m221m22]

is nonnegative, we obtain

[xk,pyk,p]M2k1[11]M[0m1200]1[xk1,pyk1,p]=[2k1m112k(m21m11m22+1m22)]+[m12m11yk1,pm21m12m11m22yn1,p]

and therefore

xk,p2km11+m12m11yk1,p, (5.11)
yk,p2k(m12m11m22+1m22)+m12m21m11m22yk1,p (5.12)

From (5.2) one has α:=m12m21m11m22<1 and hence

yk,pαyk1,p+2k(m12m11m22+1m22).

Now we use the following lemma provided in [21].

Lemma 5.17.

Let (yk,p),(zk,p) be two sequences of real numbers depending on a parameter p, such that

(y)k,p is bounded uniformly with respect to p

and

0yk,pαyk1,p+zk,p for some α(0,1).

If zk,p0 as k+ uniformly with respect to p, then yk,p0 as k+ uniformly with respect to p.

According to this result, since (vk) is bounded and then (yk,p) is bounded uniformly with respect to p, we conclude that yk,p0 as k+ uniformly with respect to p. It follows that (vk) is a Cauchy sequence. Next, the inequality (5.11) implies that (uk) is also a Cauchy sequence. Denote by u,v their limits.

Step 5: Transition to the limit.

If we pass to the limit in (5.6) we obtain

E1(u,v)=infH01(Ω)E1(,v), E2(u,v)=infH01(Ω)E2(u,), 
E11(u,v)
=E22(u,v)=0
,

i.e., (u,v) is a solution of (1.1) and also is a Nash equilibrium for the pair of functional (E1,E2).

Step 6: Uniqueness.

Assume there are two different solutions of the system (1.1), denoted with (u,v) and (u¯,v¯). Then

E11(u,v)=0 , E22(u,v)=0,
E11(u¯,v¯)=0 , E22(u¯,v¯)=0.

On the other hand, from (5.16) and (5.9), we have

0m11|uu¯|H012m12|uu¯|H01|vv¯|H01, (5.13)
0m22|vv¯|H012m21|uu¯|H01|vv¯|H01.

If u=u¯ or v=v¯ then in each case |uu¯|=0 or |vv¯|=0, concluding that u=u¯ and v=v¯. In what follows we will work under assumption that uu¯ and vv¯. From (5.13) we obtain

|uu¯|H01m12m11|vv¯|H01, (5.14)
|vv¯|H01m21m22|uu¯|H01,

whence

|vv¯|H01m12m21m11m22|vv¯|H01.

Since from (5.2) one has m12m21m11m22<1, we conclude that

|vv¯|H01<vv¯|H01,

which is impossible. Hence u=u¯ and v=v¯.

Remark 5.18 (Classical Lipschitz conditions).

Obviously the unilateral Lipschitz conditions (5.3) are satisfied if g1,g2 fulfill the classical Lipschitz conditions

|g1(x,u,v)g1(x,u¯,v¯)| a11|uu¯|+a12||vv¯|, (5.15)
|g2(x,u,v)g2(x,u¯,v¯)| a21|uu¯|+a22|vv¯|,

for all u,v,u¯,v¯ and a.e. xΩ. In this particular case considered in [21] (see also [27]), the required conditions on the coefficients aij make possible to derive the existence and uniqueness of the solution of system (1.2) directly from Perov’s fixed point theorem. We note that unilateral Lipschitz conditions for Nash equilibria of systems have been used for the first time in paper [22].

Example 5.19.

Consider the Dirichlet problem for the system of Kirchhoff type

{(1+01|u|2)u′′=usinv(1+01|v|2)v′′=v+sinuu(0)=v(0)=u(1)=v(1)=0. on (0,1) (5.16)

We apply Theorem 5.15, where

Ω=(0,1),a=b=1,g1(x,u,v)=usinv,g2(x,u,v)=sinu+v.

Note that condition (5.3) holds with aij=1 (i,j=1,2). The first eigenvalue of the Dirichlet problem u′′=λu on (0,1), u(0)=u(1)=0 is equal to π2 (see, e.g., [28, p. 72]), whence relation (5.2) is valid since 1<π2 and 1<(π21)2. In order to check (h2) we compute

G2(x,u,v)=0v(s+sinu)𝑑s=12v2+vsinu.

Consider the coercive-type functions H1(x,v)=12v2|v| and H2(x,v)=12v2+|v|. Clearly

H1(x,v)G2(x,u,v)H2(x,v).

Therefore, the Dirichlet problem (5.16) has a unique solution (u,v)H01(0,1)×H01(0,1) which is a Nash equilibrium for the corresponding energy functionals.

5.2 Local Nash equilibrium

Let R1,R2>0 and denote by BR1, BR2 two closed balls of H01(Ω), of center the origin and radius R1 and R2, respectively. Now, our interest is focused on an existence and uniqueness result of the system (1.1) on BR1×BR2.

Here an additional ingredient is given by the Leray-Schauder boundary conditions

E11(u,v)+μu 0 for all (u,v)BR1×BR2 with |u|H01=R1 and all μ>0, (5.17)
E22(u,v)+γv 0 for all (u,v)BR1×BR2 with |v|H01=R2 and all γ>0.
Theorem 5.20.

Assume that for i=1,2, fiH1(Ω), gi:Ω×2 is a Carathéodory function, gi(,0,0)=0, and that condition (h1) holds. In addition assume that

(h2’)
a11λ1R1+a12λ1R2+|f1|H1 aR1+bR13,
a21λ1R1+a22λ1R2+|f2|H1 aR2+bR23.

Then system (1.1) has in BR1×BR2 a unique solution which is a Nash equilibrium in BR1×BR2for the pair of functionals (E1,E2).

Proof 5.21.

Step 1: As at Step 1 from the proof of Theorem 5.15, the functionals E1 and E2 are bounded from below on BR1×BR2.

Step 2: E1 and E2 satisfy the Leray-Schauder boundary conditions (5.17).

Assume that there exist (u,v)BR1×BR2 with |u|H01=R1 and μ>0 such that

E11(u,v)+μu=0.

Then

(a+b|u|H012)|u|H012+μ|u|H012((Δ)1(f1+g1(,u,v)),u)H01=0,

which gives

(a+bR12)R12+μR12 = ((Δ)1(f1+g1(,u,v)),u)H01
= (f1+g1(,u,v),u)L2
R1|f1|H1+a11|u|L22+a12|u|L2|v|L2
R1|f1|H1+a11λ1R12+a12λ1R1R2,

whence

(a+bR12)R1+μR1|f1|H1+a11λ1R1+a12λ1R2,

which contradicts the first relation in (h2’). An analog reasoning applies for E2.

Step 3: Construction of an approximation sequence.

As in the proof of Lemma 2.1 in [24], starting from an arbitrarily initial point v0BR2 and applying recursively Ekeland’s variational principle, we obtain a sequence (uk,vk)BR1×BR2 such that

E1(uk,vk1)infBR1E1(,vk1)+1k , E2(uk,vk)infBR2E2(uk,)+1k,
|E11(uk,vk1)+μkuk|H011k, |E22(uk,vk)+γkvk|H011k,

where

μk={1R12(E11(uk,vk1),uk)H01, if |uk|H01=R1 and (E11(uk,vk1),uk)H01<00, otherwise  (5.18)

and

γk={1R22(E22(uk,vk),vk)H01, if |vk|H01=R2 and (E22(uk,vk),vk)H01<00, otherwise.  (5.19)

Step 4: Convergence to zero of the sequences (μk) and (γk).

Assume the contrary. Then, passing eventually to subsequences, we may assume that μkμ>0 or γkγ>0. Using the expressions of E11 and E22 and denoting

αk:=E11(uk,vk1)+μkuk,βk:=E22(uk,vk)+γkvk, (5.20)

we have

uk = S(f1+g1(,uk,vk1))+μka+b|uk|H012, (5.21)
vk = S(f2+g2(,uk,vk))+γka+b|vk|H012.

The sequences (uk),(vk) being bounded and the operators S(f1+g1(,u,v)),S(f2+g2(,u,v)) being compact, we have that the two sequences from the right-hand sides in (5.21) are compact; thus (uk) and (vk) have convergent subsequences (ukj),(vkj). The same reasoning applied to the second formula in (5.21) with kj1 instead of k allows us, passing again to subsequence, to assume that the sequences (ukj),(vkj) and (vkj1) are convergent. Let u,v,v¯ be their limits. If we take the limit in (5.20)

E11(u,v¯)+μu=0,E22(u,v)+γv=0,

where |u|H01=R1 if μ>0 and |v|H01=R2 if γ>0. In each case, one of the two Leray-Schauder conditions (5.17) is contradicted. Therefore μk0 and γk0 as k+.

Step 5: Estimations for (uk) and (vk).

We can proceed similarly to Theorem 5.15, Step 4, to obtain inequalities (5.16) and (5.9). Under the notations from Step 4 in the proof of the previous theorem, and the additional notations ck,p=|μk+pμk|, dk,p=|γk+pγk|, we arrive to the matrix inequality

Mk[xk,pyk,p]2k[11](MkMk)[xk1,pyk1,p]+[ck,pR1dk,pR2],

where now

Mk=[m11+μkm12m21m22+μk],Mk=[m11+μk0m21m22+μk].

Since for any k, Mk is invertible and

Mk=1[1m11+μk0m21(m11+μk)(m22+γk)1m22+γk]

we obtain

[xk,pyk,p]2kM[11]k1M[0m1200]k1[xk1,pyk1,p]+M[ck,pR1dk,pR2]k1.

Thus

xk,p2k(m11+μk)+m12m11+μkyk1,p+ck,pR11m11+μk
2km11+m12m11yk1,p+ck,pR11m11,
yk,p2k(1m22+γk)
+1(m11+μk)(m22+γk)[2km12+m12m21yk1,p+m21ck,pR1]+1m22+γkdk,pR2
2k(m12m11m22+1m22)+m12m21m11m12yk1,p+m21m11m22ck,pR1+1m22dk,pR2.

Hence

yk,pαyk1,p+2k(m12m11m22+1m22)+m21m11m22ck,pR1+1m22dk,pR2,

where α:=m12m21m11<1 and ck,p,dk,p converge to zero uniformly with respect to p. Now the conclusion follows as in the proof of Theorem 5.15 with the limits u and v of the sequences (uk) and (vk) satisfying

E11(u,v)=0, E22(u,v)=0

and

E1(u,v)=infBR1E1(,v),E22(u,v)=infBR2E2(u,).

Step 6: Uniqueness.

Similar to the proof in Theorem 5.15.

Example 5.22.

Consider the Dirichlet problem for the system of Kirchhoff type

{(2+01|u|2)u′′=u3+usinv+π2sin(πx)(2+01|v|2)v′′=v3+v+sinuu(0)=v(0)=u(1)=v(1)=0. on (0,1) (5.22)

For R1=R2=1, we apply Theorem 5.20, where

Ω=(0,1),a=2,b=1,f1(x)=π2sin(πx),f20,
g1(x,u,v)=u3+usinv,g2(x,u,v)=v3+v+sinu.

Since

(u3+u¯3)(uu¯)=(uu¯)2(u2+uu¯+u¯2)0,

one has

(g1(x,u,v)g1(x,u¯,v¯))(uu¯) (usinvu¯+sinv¯)(uu¯)
|uu¯|2+|uu¯||vv¯|.

Similarly

(g2(x,u,v)g2(x,u¯,v¯))(vv¯)|uu¯||vv¯|+|vv¯|2.

Hence, condition (5.3) holds with aij=1 for i,j=1,2. In addition, since λ1=π2, condition (5.2) also holds. Thus assumption (h1) is satisfied. Next we check condition (h2’). We have |f2|H1=0 and that the function u0(x)=sin(πx) is the solution of the Dirichlet problem u′′=f1 in (0,1), u(0)=u(1)=0. Then

|f1|H1=|u0|H01=|u0|L2=(01π2cos2(πx))12=π2.

Now, condition (h2’) is verified since both 2/π2+π/2 and 2/π2 are less than 3.

Therefore, the Dirichlet problem (5.22) has a unique solution

(u,v){uH01(0,1):|u|H011}×{vH01(0,1):|v|H011},

which is a Nash equilibrium for the corresponding energy functionals.

6 Conclusions

In this paper, we have studied the existence, uniqueness, localization and variational properties of solutions for some equations and systems of Kirchhoff type. First we have defined the solution operator associated to nonhomogeneous equations subjected to the Dirichlet boundary condition and we have made the connexion with the corresponding energy functional. Next, we have considered equations with a reaction term and using Banach contraction principle and Schaefer’s fixed point theorem we have established sufficient conditions so that a solution exist and be localized in some bounded sets. For a system of two Kirchhoff equations, under appropriate conditions, we have proved the existence of a unique solution which appears as a Nash equilibrium for the associated energy functionals. Both global Nash equilibrium, in the whole space, and local Nash equilibrium, in balls, have been obtained by using an iterative procedure simulating a noncooperative game and based on Ekeland’s variational principle.

Conflict of interest

The authors declare no conflict of interest.

References

  • 1 G. Kirchhoff, Vorlesungen über Mechanik, Leipzig: Teubner, 1883.
  • 2 G. Autuori, P. Pucci, M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Rational Mech. Anal., 196 (2010), 489-516. \doilinkhttps://doi.org/10.1007/s00205-009-0241-x
  • 3 M. Dreher, The Kirchhoff equation for the p-Laplacian, Rend. Sem. Mat. Univ. Pol. Torino, 64 (2006), 217-238.
  • 4 J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284-346. \doilinkhttps://doi.org/10.1016/S0304-0208(08)70870-3
  • 5 T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), e1967-e1977. \doilinkhttps://doi.org/10.1016/j.na.2005.03.021
  • 6 T. F. Ma, J. E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett., 16 (2003), 243-248. \doilinkhttps://doi.org/10.1016/S0893-9659(03)80038-1
  • 7 S. I. Pokhozhaev, A quasilinear hyperbolic Kirchhoff equation (Russian), Differ. Uravn., 21(1985), 101-108.  
  • 8 C. F. Vasconcellos, On a nonlinear stationary problem in unbounded domains, Rev. Mat. Univ. Complut. Madrid, 5 (1992), 309-318. \doilinkhttps://doi.org/10.5209/revREMA.1992.v5.n2.17919
  • 9 C. O. Alves, F. J. S. A. Corrêa, T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93. \doilinkhttps://doi.org/10.1016/j.camwa.2005.01.008
  • 10 G. Che, H. Chen, Infinitely many solutions of systems of Kirchhoff-type equations with general potentials, Rocky Mountain J. Math., 48 (2018), 2187-2209. \doilink10.1216/RMJ-2018-48-7-2187
  • 11 P. Chen, X. Liu, Positive solutions for Kirchhoff equation in exterior domains, J. Math. Phys., 62 (2021). \doilinkhttps://doi.org/10.1063/5.0014373
  • 12 M. Chipot, V. Valente, G. V. Caffarelli, Remarks on a nonlocal problem involving the Dirichlet energy, Rend. Semin. Mat. Univ. Padova, 110 (2003), 199-220. \doilinkhttp://dx.doi.org/10.5167/uzh-21865
  • 13 N. T. Chung, An existence result for a class of Kirchhoff type systems via sub and supersolutions method, Appl. Math. Lett., 35 (2014), 95-101. \doilinkhttps://doi.org/10.1016/j.aml.2013.11.005
  • 14 K. Perera, Z. Zhang, Nontrival solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations, 221 (2006), 246-255. \doilinkhttp://dx.doi.org/10.1016/j.jde.2005.03.006
  • 15 P. Pucci, V. D. Rădulescu, Progress in nonlinear Kirchhoff problems, Nonlinear Anal., 186 (2019), 1-5. \doilinkhttp://dx.doi.org/10.1016/j.na.2019.02.022
  • 16 B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, J. Glob. Optim., 46 (2010), 543-549. \doilinkhttps://doi.org/10.1007/s10898-009-9438-7
  • 17 Z. T. Zhang, Y. M. Sun, Existence and multiplicity of solutions for nonlocal systems with Kirchhoff type, Acta Math. Appl. Sin. Engl. Ser., 32 (2016), 35-54. \doilinkhttp://dx.doi.org/10.13140/2.1.3805.4403
  • 18 A. Deep, Deepmala, C. Tunç, On the existence of solutions of some non-linear functional integral equations in Banach algebra with applications, Arab J. Basic Appl. Sci., 27 (2020), 279-286. \doilinkhttps://doi.org/10.1080/25765299.2020.1796199
  • 19 S. Islam, M. N. Alam, M. F. Al-Asad, C. Tunç, An analytical technique for solving new computational solutions of the modified Zakharov-Kuznetsov equation arising in electrical engineering, J. Appl. Comput. Mech., 7 (2021), 715-726. \doilinkhttps://dx.doi.org/10.22055/jacm.2020.35571.2687
  • 20 M. N. Alam, C. Tunç, An analytical method for solving exact solutions of the nonlinear Bogoyavlenskii equation and the nonlinear diffusive predator–prey system, Alexandria Eng. J., 55(2016), 1855–1865. \doilinkhttps://doi.org/10.1016/j.aej.2016.04.024
  • 21 R. Precup, Nash-type equilibria and periodic solutions to nonvariational systems, Adv. Nonlinear Anal., 4 (2014), 197-207. \doilinkhttps://doi.org/10.1515/anona-2014-0006
  • 22 R. Precup, Nash-type equilibria for systems of Szulkin functionals, Set-Valued Var. Anal., 24 (2016), 471-482. \doilinkhttps://doi.org/10.1007/s11228-015-0356-1
  • 23 A. Budescu, R. Precup, Variational properties of the solutions of singular second-order differential equations and systems, J. Fixed Point Theor. Appl., 18 (2016), 505–518. \doilinkhttps://doi.org/10.1007/s11784-016-0284-1
  • 24 R. Precup, A critical point theorem in bounded convex sets and localization of Nash-type equilibria of nonvariational systems, J. Math. Anal. Appl., 463 (2018), 412-431. \doilinkhttps://doi.org/10.1016/j.jmaa.2018.03.035
  • 25 M. Beldinski, M. Galewski, Nash type equilibria for systems of non-potential equations, Appl. Math. Comput., 385 (2020). \doilinkhttps://doi.org/10.1016/j.amc.2020.125456
  • 26 I. Benedetti, T. Cardinali, R. Precup, Fixed point-critical point hybrid theorems and applications to systems with partial variational structure, J. Fixed Point Theory Appl., 23 (2021), 1-19. \doilinkdoi.org/10.1007/s11784-021-00852-6
  • 27 A. Stan, Nonlinear systems with a partial Nash type equilibrium, Stud. Univ. Babeş-Bolyai Math., 66 (2021), 397-408. \doilinkdoi:10.24193/subbmath.2021.2.14
  • 28 R. Precup, Methods in Nonlinear Integral Equations, Amsterdam: Springer, 2002. \doilinkhttps://doi.org/10.1007/978-94-015-9986-3
  • 29 R. Precup, Linear and Semilinear Partial Differential Equations, Berlin: De Gruyter, 2013. \doilinkhttps://doi.org/10.1515/9783110269055
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