Lower and Upper Solutions for System of Differential Equations Involving Homeomorphism and Nonlinear Boundary Conditions

Results Math (2024) 79:181 c 2024 The Author(s) 1422-6383/24/050001-12 published online June 14, 2024 https://doi.org/10.1007/s00025-024-02213-4 Results in Mathematics Lower and Upper Solutions for System of Differential Equations Involving Homeomorphism and Nonlinear Boundary Conditions Jorge Rodr´ıguez–L´opez , Katarzyna Szyma´nska-D¸ebowska ,and Miroslawa Zima Abstract. We study the existence of solution to the system of differential equations (φ(u))=f(t, u, u) with nonlinear boundary conditions g(u(0),u,u )=0,h(u(1),u,u )=0, where f:[0,1]×Rn×Rn→Rn,g,h :Rn×C([0,1],Rn)×C([0,1],Rn)→ Rnare continuous, φ:n i=1(−ai,a i)→Rn,0<a i≤+∞,φ(s)= (φ1(s1),...,φ n(sn)) and φi:(−ai,a i)→Ris a one dimensional regular or singular homeomorphism. Our proofs are based on the concept of the lower and upper solutions. Mathematics Subject Classification. Primary 34B10; Secondary 34B15, 47H11. Keywords. Nonlinear boundary value problem, nonlinear boundary conditions, homeomorphism, lower solution, upper solution, Schauder fixed point theorem. 1. Introduction We consider the following system of differential equations (φ(u))=f(t, u, u),(1) subject to nonlinear boundary conditions of the following type g(u(0),u,u )=0,h(u(1),u,u )=0,(2) 181 Page 2 of 12 J. Rodr´ıguez–L´opez et al. Results Math where f:[0,1] ×Rn×Rn→Rn,g,h :Rn×C([0,1],Rn)×C([0,1],Rn)→Rn are continuous and φ:n i=1(−ai,a i)→Rngiven by φ(s)=(φ1(s1),...,φ n(sn)) is such that: φi:(−ai,a i)→Ris a one dimensional increasing homeomorphism with φi(0) = 0 and 0 <a i≤+∞,i=1,...,n.Ifai=+∞then φi:R→R is a regular homeomorphism and if ai<+∞,φiis said to be a singular homeomorphism, cf. [1,5,10]. Lower and upper solutions are an effective tool in proving the existence of solutions to differential equations. The idea of lower solution comes from Picard [19]. More than twenty years later, Perron [18] and then M¨uller [16] proved the existence of solutions to the first order Cauchy problem together with their localization between functions αand β, which are ordered α≤β. Papers of Scorza Dragoni [21,22], for second order Dirichlet boundary value problem, indicate the important role played by the functions αand β,which we call today lower and upper solutions, and form the core of the method. Later, Nagumo [17] enlarged the class of functions to be considered as lower and upper solutions: the differential inequalities must be satisfied between the functions αand βand not, as in Scorza Dragoni’s papers, for larger sets of values of u. Since then, the theory of lower and upper solutions has been constantly developing, see, for instance [6]. There are many results regarding the existence of solutions, using lower and upper solutions, for two-points boundary value problems for second-order differential equations. We refer to monographs [7,20] and the literature included therein for a detailed description of these results. Since then, using the method of lower and upper solutions, the mentioned results have been generalized for problems with non-local boundary conditions [4,8,13], non-linear second-order coupled systems [15], non-linear boundary conditions or equations involving homeomorphisms [2,11,20]. The aim of this paper is to give sufficient conditions for the existence of solutions to the problem (1)–(2), using the method of lower and upper solutions. To the best of our knowledge this approach has never been used before in the context of the systems of differential equations involving homeomorphism and nonlinear boundary conditions (although there are known results for this type of systems of equations using other methods, cf. [9,12]). Firstly, in Sect. 2, we consider an auxiliary problem, which from the Schauder fixed point theorem has a solution, under certain additional assumptions (Theorem 2.1). The concept of lower and upper solutions for a system of equations is described in Sect. 3(Definition 3.1), cf. [14]. In Sect. 4, we obtain the existence and localization of a solution to the problem (1)–(2) (Theorem 4.1). In our approach we used some ideas from the paper [3]. Our results cover some recent ones derived in [23]. The scalar equation with the nonlinear BCs considered in [23] are a special case of the problem (1)–(2) (see Sect. 4). Despite the fact that the main results in [23] are based on the extension of Mawhin’s Vol. 79 (2024) Lower and Upper Solutions for System Page 3 of 12 181 continuation theorem for quasi-linear operators, the assumptions made there allow us to find the lower and upper solutions for this particular case of the problem (1)–(2) (Corollary 4.1). In Sect. 4, we also present an example which illustrates an application of Theorem 4.1. 2. Auxiliary Problem Denote by C1([0,1],Rn) the Banach space of all continuously differentiable functions u:[0,1] →Rnendowed with the norm u= max{u∞,u∞}. The following assumptions will be needed throughout the paper: (A1) f:[0,1] ×Rn×Rn→Rnis continuous; (A2) φ=(φ1,...,φ n):n i=1(−ai,a i)→Rnand φiis one dimensional increasing homeomorphism with φi(0) = 0 and 0 <a i≤+∞,i=1,...,n; (A3) g,h :Rn×C([0,1],Rn)×C([0,1],Rn)→Rnare continuous. For the purposes of this section, we introduce the following assumptions: (S1) fis bounded; (S2) A,B:C1([0,1],Rn)→Rnare continuous and bounded not necessarily linear functionals with BC1([0,1],Rn)⊂n i=1(−ai,a i). Consider the following auxiliary problem (φ(u))=f(t, u, u),u(0) = A(u),u (1) = B(u).(3) Note that the solutions to (3) are exactly the fixed points of the operator T:C1([0,1],Rn)→C1([0,1],Rn) defined as T(u)(t)=A(u)+t 0 φ−1φ(B(u)) −1 s f(z,u(z),u (z)) dzds. (4) We have (T(u))(t)=φ−1φ(B(u)) −1 t f(s, u(s),u (s)) ds. If A, B, M are bounds for A,B,f, respectively, then for any u∈C1([0,1],Rn) and t∈[0,1] we obtain |T(u)(t)|≤A+φ−1(φ(B)+M):=R, (5) and |(T(u))(t)|≤φ−1(φ(B)+M):=R≤R. (6) Clearly, the functions T(u), (T(u))are continuous and, from (5) and (6), the operator Tis well defined. Moreover, from (5) and (6), the fact that φis a homeomorphism and the Lebesgue Dominated Convergence Theorem, Tis continuous. 181 Page 4 of 12 J. Rodr´ıguez–L´opez et al. Results Math Denote by BRthe ball of radius Rcentered at the origin in the space C1([0,1],Rn). To prove that Tis completely continuous it is sufficient to observe that the image of BRunder Tis relatively compact. From (5) and (6), T(BR) is uniformly bounded in C1([0,1],Rn). Observe that, for every u∈BR, we have t1 t2 f(r, u(r),u (r)) dr ≤M|t1−t2|. As φis a homeomorphism, one can see that {(T(u)):u∈BR}is equicontinuous on [0,1]. The fact that {(T(u)):u∈BR}is uniformly bounded implies that {T(u):u∈BR}is equicontinuous on [0,1]. Hence Tis completely continuous. Consequently, from the Schauder fixed point theorem, the operator T: BR→BRdefined in (4) has a fixed point and the following theorem holds true. Theorem 2.1. Let Assumptions (A1),(A2),(S1) and (S2) be satisfied. Then the problem (3)has at least one solution. Remark 2.1. Note that in the case where φis a singular homomorphism, then Theorem 2.1 holds under assumptions (A1), (A2) and (S2), i.e. the condition (S1) is not necessary to prove this Theorem. Remark 2.2. Theorem 2.1 holds true if (A1) is replaced by the assumption that fis a Carath´eodory function. In case when fis an L1–Carath´eodory function, one can also skip condition (S1). 3. Lower and Upper Solutions By a solution to the problem (1)–(2) we mean a function u:[0,1] →Rn, u=(u1,...,u n)∈C1([0,1],Rn) with |u i(t)|<a ion [0,1] for each iand φ(u)∈C1([0,1],Rn), which satisfies (1) and (2)on[0,1]. Let α, β ∈C1([0,1],Rn) with |α i(t)|<a i,|β i(t)|<a ion [0,1], i= 1,2,...,n and φ(α),φ(β)∈C1([0,1],Rn). Definition 3.1. We say that αis a lower solution and βis an upper solution to problem (1)–(2), if αi(t)≤βi(t)andα i(t)≤β i(t),t∈[0,1],i∈{1,...,n}, and for each i∈{1,...,n} (φi(α i))(t)≥f(t, u1,...,u n,v 1,...,v i−1,α  i(t),v i+1 ...,v n), (φi(β i))(t)≤f(t, u1,...,u n,v 1,...,v i−1,β i(t),v i+1 ...,v n), whenever αj(t)≤uj≤βj(t), j∈{1,...,n},andα j(t)≤vj≤β j(t), j∈ {1,...,n}\{i},and gi(α(0),ν,ξ)≤0,h i(α(1),ν,ξ)≤0, Vol. 79 (2024) Lower and Upper Solutions for System Page 5 of 12 181 gi(β(0),ν,ξ)≥0,h i(β(1),ν,ξ)≥0, whenever αj(t)≤νj(t)≤βj(t), α j(t)≤ξj(t)≤β j(t), j∈{1,...,n}. Remark 3.1. In the case, where n=1andα,βare constant Definition 3.1 takes the following form. We say that αis a lower solution and βis an upper solution to problem (1)–(2), if α≤βand f(t, u, 0) = 0, whenever α≤u≤β,t∈[0,1], and g(α, ν, 0) ≤0,g(β,ν,0) ≥0,h(0,ν,0) = 0, whenever α≤ν≤β. Remark 3.2. Observe that in the case when n=1ifgand hare nonincreasing functions with respect to their second and third arguments, then the assumptions on gand hin Definition 3.1 can be written as g(α(0),α,α )≤0,h(α(1),α,α )≤0, and g(β(0),β,β)≥0,h(β(1),β,β)≥0. Remark 3.3. In the case when fis a Carath´eodory function, the definition of the lower and upper solutions can be weakened in the standard way. 4. Main Results Now we shall prove that the existence of a couple of lower and upper solutions implies the existence and localization of a solution to (1)–(2). Theorem 4.1. Let Assumptions (A1)–(A3) be satisfied. Suppose that there exist a couple (α, β)of lower and upper solutions to (1)–(2). Then problem (1)–(2) has at least one solution usuch that α≤u≤βand α≤u≤β. Proof. For each i∈{1,...,n}we define the continuous and bounded functions ζi,η i:[0,1] ×R→Rby ζi(t, x)=⎧ ⎨ ⎩ αi(t),x<α i(t) x, αi(t)≤x≤βi(t) βi(t),x>β i(t) , ηi(t, y)=⎧ ⎨ ⎩ α i(t),y<α  i(t) y, α i(t)≤y≤β i(t) β i(t),y>β  i(t) and consider the modified problem (φi(u i))=f∗ i(t, u, u),u i(0) = A∗ i(u),u  i(1) = B∗ i(u),(7) 181 Page 6 of 12 J. Rodr´ıguez–L´opez et al. Results Math where f∗ i(t, x, y)=fi(t, ζ1(t, x1),...,ζ n(t, xn),η 1(t, y1),...,η n(t, yn)) and A∗,B∗:C1([0,1],Rn)→Rnare given by A∗ i(v)=ζi(0,v i(0) −gi(v(0),v,v)),B∗ i(v)=ηi(1,v i(1) −hi(v(1),v,v)). Clearly, f∗is bounded and B∗ i(v)∈(−ai,a i). First let us observe that Theorem 2.1 implies that problem (7) has at least one solution. Now we shall show that if uis a solution to (7), then α i(t)≤u i(t)≤β i(t),(8) for all t∈[0,1] and i∈{1,...,n}. Suppose to the contrary that for some iwe have u i≤ β i.From(7), we have u i(1) = ηi(1,u  i(1) −hi(u(1),u,u )) ∈[α i(1),β i(1)], which means that there exists t0∈(0,1] such that u i(t0)=β i(t0)andu i(t)> β i(t) for all t∈[t0−δ, t0). By the monotonicity of φi,φi(u i(t)) >φ i(β i(t)) for all t∈[t0−δ, t0). Now, knowing that uis a solution of (7) and using Definition 3.1, we reach a contradiction. Indeed, for t∈[t0−δ, t0), we obtain φi(β i(t)) −φi(u i(t)) =t0 t(φi(u i))(s)−(φi(β i))(s)ds =t0 t (f∗ i(s, u1(s),...,u n(s),u  1(s),...,β i(s),...,u  n(s)) −(φi(β i(s)))ds ≥0. The proof that α(t)≤u(t), t∈[0,1], is similar. Now, let us prove that if uis a solution to (7), then αi(t)≤ui(t)≤βi(t),(9) for all t∈[0,1] and i∈{1,...,n}.From(8), we obtain αi(t)−αi(0) ≤ui(t)−ui(0) ≤βi(t)−βi(0).(10) Moreover, we have ui(0) = ζi(0,u i(0) −gi(u(0),u,u )) ∈[αi(0),β i(0)].(11) Consequently, (10) and (11) imply (9). Now, observe that α i(1) ≤u i(1) −hi(u(1),u,u )≤β i(1),(12) Vol. 79 (2024) Lower and Upper Solutions for System Page 7 of 12 181 i∈{1,...,n}. Suppose to the contrary that u i(1) −hi(u(1),u,u )<α  i(1). Then, by the definition of ηi,u i(1) = α i(1) and thus, from Definition 3.1,one has α i(1) >u  i(1) −hi(u(1),u,u )=α i(1) −hi(α(1),u,u )≥α i(1), a contradiction. In a similar way, it can be shown that u i(1)−hi(u(1),u,u )≤ β i(1). Finally, notice that αi(0) ≤ui(0) −gi(u(0),u,u )≤βi(0),(13) i∈{1,...,n}. Indeed, if we suppose to the contrary that ui(0)−gi(u(0),u,u )< αi(0), then, by the definition of ζi,ui(0) = αi(0) and, from Definition 3.1,we reach a contradiction αi(0) >u i(0) −gi(u(0),u,u )=αi(0) −gi(α(0),u,u )≥αi(0). It is now easy to see that a solution to (7) is also a solution to (1) satisfying (9) and (8). Moreover, from (12) and (13) we obtain that this solution satisfies the boundary conditions (2).  Example 4.1. Consider the system of differential equations ⎧ ⎨ ⎩ (φ1(u 1))=(u 2−3)3eu1+u2+sinπ 3(u 1−2t),t∈[0,1] (φ2(u 2))=(u 2−3)2u2 1+u2 2+(u 1)2, (14) with φi:R→Ran increasing homeomorphism such that φi(0) = 0, subject to the nonlinear functional boundary conditions ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 3u1(0) + 1 2maxt∈[0,1] u2(t)=0, 4u1(0) −u2(0) + maxt∈[0,1] u 2(t)=0, 6u 1(1) −16 + 2 1 0u1(t)dt +1 3 0u 2(t)dt =0, u 1(1) −u 2(1) + 1 10 1 0u2 2(t)dt =0. (15) Obviously, problem (14)–(15) can be seen as a particular case of (1)–(2) with f1(t, u1,u 2,v 1,v 2)=(v2−3)3eu1+u2+sinπ 3(v1−2t), f2(t, u1,u 2,v 1,v 2)=(v2−3)2u2 1+u2 2+v2 1, and g1(z1,z 2,ν 1,ν 2,ξ 1,ξ 2)=3z1+1 2max t∈[0,1] ν2(t), g2(z1,z 2,ν 1,ν 2,ξ 1,ξ 2)=4z1−z2+ max t∈[0,1] ξ2(t), h1(z1,z 2,ν 1,ν 2,ξ 1,ξ 2)=6z1−16 + 2 1 0 ν1(t)dt +1 3 0 ξ2(t)dt, 181 Page 8 of 12 J. Rodr´ıguez–L´opez et al. Results Math h2(z1,z 2,ν 1,ν 2,ξ 1,ξ 2)=z1−z2+1 10 1 0 ν2 2(t)dt. Observe that the couple α(t)=(t2−1,3t), β(t)=(3t, 3t+ 1), t∈[0,1], are a lower and an upper solution for problem (14)–(15). Indeed, let t∈[0,1] and αj(t)≤uj≤βj(t), j=1,2. Since α 2(t)=β 2(t)=3,forα 2(t)≤v2≤β 2(t) we have (φ1(α 1))(t)≥0=(3−3)3eu1+u2+sinπ 3(2t−2t)=f1(t, u1,u 2,α  1(t),v 2), and (φ1(β 1))(t)=0=(3−3)3eu1+u2+sinπ 3(3 −2t)=f1(t, u1,u 2,β 1(t),v 2), while for α 1(t)≤v1≤β 1(t)weget (φ2(α 2))(t)≥0=(3−3)2(u2 1+u2 2+v2 1)=f2(t, u1,u 2,v 1,α  2(t)), and (φ2(β 2))(t)=0=(3−3)2(u2 1+u2 2+v2 1)=f2(t, u1,u 2,v 1,β 2(t)). Moreover, if αj(t)≤νj(t)≤βj(t), α j(t)≤ξj(t)≤β j(t), j=1,2, then g1(α1(0),α 2(0),ν 1,ν 2,ξ 1,ξ 2)=−3+1 2max t∈[0,1] ν2(t)=−3+2=−1≤0, g2(α1(0),α 2(0),ν 1,ν 2,ξ 1,ξ 2)=−4 + max t∈[0,1] ξ2(t)=−4+3=−1≤0, g1(β1(0),β 2(0),ν 1,ν 2,ξ 1,ξ 2)=1 2max t∈[0,1] ν2(t)=2≥0, g2(β1(0),β 2(0),ν 1,ν 2,ξ 1,ξ 2)=−1 + max t∈[0,1] ξ2(t)=−1+3=2≥0, h1(α 1(1),α  2(1),ν 1,ν 2,ξ 1,ξ 2)=12−16 + 2 1 0 ν1(t)dt +1 3 0 ξ2(t)dt ≤0, h2(α 1(1),α  2(1),ν 1,ν 2,ξ 1,ξ 2)=2−3+ 1 10 1 0 ν2 2(t)dt ≤− 13 45 <0, h1(β 1(1),β 2(1),ν 1,ν 2,ξ 1,ξ 2)=18−16 + 2 1 0 ν1(t)dt +1≥5 3>0, h2(β 1(1),β 2(1),ν 1,ν 2,ξ 1,ξ 2)= 1 10 1 0 ν2 2(t)dt ≥0. Therefore, Theorem 4.1 ensures that problem (14)–(15) has at least one solution (u1,u 2) such that for t∈[0,1] t2−1≤u1(t)≤3t, 3t≤u2(t)≤3t+1, and 2t≤u 1(t)≤3,u  2(t)=3. Vol. 79 (2024) Lower and Upper Solutions for System Page 9 of 12 181 It is worth mentioning that the couple α(t)=(t2−1,3t), β(t)=(3t, 3t+ 1), are a lower and an upper solution to problem (14)–(15) involving any homeomorphism with ai>3. We finish this paper by considering the scalar case of the differential equation (1) with the homeomorphism φ:(−a, a)→R,0<a≤+∞, and the boundary conditions g(u(0),G(u)) = 0,h(u(1),H(u)) = 0,(16) where the functions g,h :R2→Rare continuous and G, H :C([0,1]) →R are continuous functionals and map bounded sets into bounded sets, cf. [23]. Moreover, assume that the following conditions hold: (H1) there exist M1,M 2∈(−a, a), M1≤0≤M2, such that for all y∈Rwe have h(M1,y)≤0,h(M2,y)≥0; (H2) there exist m1,m 2∈R,m1≤m2, such that for all y∈Rwe have g(m1,y)≤0,g(m2,y)≥0; (H3) for each t∈[0,1] and each x∈[m1+M1,m 2+M2], we have f(t, x, M1)≤0,f(t, x, M2)≥0. Now, we are in a position to establish an existence result in the line of those in [23], as a consequence of Theorem 4.1. In this case, the lower and upper solutions are straight lines. Corollary 4.1. Assume that conditions (A1),(A2) and (H1)–(H3) hold. Then problem (1)–(16)has at least one solution usuch that m1+M1≤u(t)≤m2+M2,M 1≤u(t)≤M2,t∈[0,1]. Proof. Note that problem (1)–(16) is a particular case of problem (1)–(2) and, under assumptions (H1), (H2) and (H3), the functions αand βdefined as α(t)=M1t+m1,β(t)=M2t+m2, are a couple of lower and upper solutions for problem (1)–(2). Therefore the conclusion follows from Theorem 4.1. Author contributions All authors contributed equally to this manuscript. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. M. Zima was partially supported by the Centre for Innovation and Transfer of Natural Science and Engineering Knowledge of University of Rzesz´ow. J. Rodr´ıguez-L´opez was partially supported by Agencia Estatal de Investigaci´on, Spain, Project PID2020-113275GBI00, and Xunta de Galicia, Spain, Project ED431C 2023/12.