Existence and multiplicity of solutions of Stieltjes differential equations via topological methods

J. Fixed Point Theory Appl. (2024) 26:9 https://doi.org/10.1007/s11784-024-01098-8 Published online February 23, 2024 c The Author(s) 2024 Journal of Fixed Point Theory and Applications Existence and multiplicity of solutions of Stieltjes differential equations via topological methods Vˇera Krajˇsˇc´akov´a and F. Adri´an F. Tojo Abstract. In this work, we use techniques from Stieltjes calculus and fixed point index theory to show the existence and multiplicity of solution of a first order non-linear boundary value problem with linear boundary conditions that extend the periodic case. We also provide the Green’s function associated to the problem as well as an example of application. Mathematics Subject Classification. 34B15, 26A42. Keywords. Stieltjes derivative, green’s functions, differential equations, fixed point index theory. 1. Introduction In recent years, there has been a surge regarding the interest in Stieltjes Calculus, that is, the study of differentiability, integration and differential and integral equations where a derivator is involved, that is, a nondecreasing left continuous function g:[0,T]→R. The reason behind this interest is the fact that Stieltjes differential equations include ordinary, difference, impulsive and time scale equations, providing a context where stationary and impulsive behavior is allowed with great generality and the powerful tools of measure theory can be applied. Many previous works in this area deal with the basic properties of the Stieltjes derivative (which sometimes behaves in unexpected ways), see, for instance, [5–7,16,23], whereas others focus their attention on Stieltjes differential equations and systems [1,6,8,11,17,18]. Despite all of these articles, there is much less work concerning the application of topological methods to Stieltjes differential equations, the exceptions being [20,22,24,25]. Topological methods provide a series of powerful tools to study the existence, nonexistence and multiplicity of solutions of differential problems of all types. In regards to their use for Stieltjes differential equations, in [25]the 9Page 2 of 23 V. Krajˇsˇc´akov´a, F. A. F. Tojo JFPTA authors applied Schaeffer’s fixed point theorem to study Stieltjes differential problems, whereas in [24] they used Bohnenblust-Karlin set-valued fixedpoint theorem for differential inclusions. In [20] they use the upper and lower solution method. Some of the methods yet to be explored are those that rely on the fixed point index theory for cones in Banach spaces—see [4,9,10,13– 15,19,26,27] for the case or ordinary differential equations—which provide simple criteria to study the qualitative properties of a given problem. As said before, in [25] the authors applied Schaeffer’s fixed point theorem to study a Stieltjes differential problem. In particular, they considered the problem u g(t)+b(t)u(t)=f(t, u(t)),t∈[0,T], u(0) = u(T). In this paper we will consider the more general case u g(t)+b(t)u(t)=f(t, u(t)),t∈[0,T], u(0) = u(T)+kB(u),(1.1) where kis a constant and Bis a linear functional. To this end, we will use fixed point index methods on cones and, in particular, we will adapt the results given in [3] for the n-th order non-local linear boundary value problem with parameter dependence: Tn[M]u(t)=σ(t),t∈[a, b], Bi(u)=δiCi(u),i=1,...,n, (1.2) where Tn[M]u(t):=Lnu(t)+Mu(t),t∈[a, b], with Lnu(t):=u(n)(t)+a1(t)u(n−1) (t)+···+an(t)u(t),t∈[a, b]. Here, σand akare continuous functions for all k=0,...,n−1, M∈R and δi∈Rfor all i=1,...,n.TheCi:C([a, b]) →Rare linear continuous operators and the Biare such that they cover the general two-point linear boundary conditions, i.e.,: Bi(u)= n−1  j=0 αi ju(j)(a)+βi ju(j)(b),i=1,...,n, where αi j,β i jare real constants for all i=1,...,n,j=0,...,n−1. In [3], the authors described the spectrum of the non-local problem (1.2) by assuming that the local problem (δi=0,i=1,...,n) is uniquely solvable and they obtained the expression of its related Green’s function in terms of the one associated to the local problem. Furthermore, in the particular case of the problem ⎧ ⎪ ⎨ ⎪ ⎩ u(t)+Mu(t)=σ(t),t∈[0,1], u(0) −u(1) = k1 0 u(s)ds, Vol. 26 (2024) Existence and multiplicity... Page 3 of 23 9 the authors characterize the values of Mand kthat ensure the constant sign of its Green’s function G, and, as a direct consequence of [3, Theorem 2], they prove that it is given by the expression G(t, s, k, M)=gM(t, s)+ kgM(t, 0) 1−k1 0gM(t, 0) d t1 0 gM(t, s)dt, where gMis the Green’s function of the homogeneous periodic problem u(t)+Mu(t)=σ(t),t∈[0,1], u(0) −u(1) = 0. Here we will take the same approach to study problem (1.1). 2. Preliminaries Through this work we will consider a nondecreasing left continuous function g:[0,T]→Rcalled a derivator, and we will denote by μgthe Lebesgue– Stieltjes measure associated to g, which is given by μg([a, b)) = g(b)−g(a)a, b ∈[0,T],a<b, for intervals of the form [a, b) and extended to a complete measure through Carath´eodory’s extension theorem. We will consider L1 g([a, b),R), the set of functions f:[a, b)→Rwhich are μg–measurable and [a,b)|f|dμg<∞. We will also take into account some special sets in [0,T] related to g such as Cg:= {t∈[0,T]:gis constant on(t−ε, t +ε) for some ε>0} =n∈Λ(an,b n), where (al,b l)∩(aj,b j)=∅for l=jand Λ is countable; and Dg={t∈[0,T]:Δ +g(t)>0}, where Δ+g(t):=g(t+)−g(t), t∈R,andg(t+) denotes the right hand side limit of gat t. Now, we introduce the concept of g-continuity—see e.g., [6,21]. Definition 2.1. A function f:[a, b]→Ris g–continuous at a point t∈[a, b], or continuous with respect to gat t, if for every >0, there exists δ>0 such that |f(t)−f(s)|<, for all s∈[a, b] such that |g(t)−g(s)|<δ. If fis g–continuous at every point t∈[a, b], we say that fis g-continuous on [a, b]. We denote by Cg([a, b],R) the set of g-continuous functions. We denote by BC g([a, b],R) the set of g-continuous functions which are also bounded. BC g([a, b],R) is Banach space with the sup-norm f∞=sup t∈[a,b] |f(t)|. 9Page 4 of 23 V. Krajˇsˇc´akov´a, F. A. F. Tojo JFPTA Let us now define the Stieltjes derivative,org-derivative (we follow here [6] for this form of the definition), of a function f:[a, b]→Rat a point t∈[a, b]as f g(t)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ lim s→t f(s)−f(t) g(s)−g(t),t∈ Dg∪Cg, lim s→t+ f(s)−f(t) g(s)−g(t),t∈Dg, lim s→b+ n f(s)−f(bn) g(s)−g(bn),t∈(an,b n)⊂Cg, provided the corresponding limits exist (when considering the function inside the limit to be defined wherever the expression makes sense). In that case, we say that fis g-differentiable at t. For Stieltjes derivatives we have the following result—see [6,21]. Proposition 2.2. Let f1,f 2:[a, b]→Rbe two g-differentiable functions at a point t. Then: 1. The function λ1f1+λ2f2is g-differentiable at tfor any λ1,λ 2∈Rand (λ1f1+λ2f2) g(t)=λ1(f1) g(t)+λ2(f2) g(t). 2. The product f1f2is g-differentiable at tand (f1f2) g(t)=(f1) g(t)f2(t)+(f2) g(t)f1(t)+(f1) g(t)(f2) g(t)Δ+g(t). 3. If f2(t)(f2(t)+(f2) g(t)Δ+g(t)) =0,the quotient f1/f2is g-differentiable at tand f1 f2 g (t)= (f1) g(t)f2(t)−(f2) g(t)f1(t) f2(t)(f2(t)+(f2) g(t)Δ+g(t)). The next result connects the definition of g-absolutely continuous function and the Fundamental Theorem of Calculus for the Lebesgue–Stieltjes integral. Theorem 2.3. [21, Theorem 2.7 - Fundamental Theorem of Calculus for the Lebesgue–Stieltjes integral] Let F:[a, b]→R. The following conditions are equivalent: 1. The function Fis g-absolutely continuous on [a, b], that is, for every >0, there exists δ>0such that for every open pairwise disjoint family of subintervals {(an,b n)}m n=1 satisfying m  n=1 (g(bn)−g(an)) <δ, we have that m  n=1 |F(bn)−F(an)|<. 2. (i) there exists F g(t)for μg–a.a. t∈[a, b); (ii) F g(t)∈L1 g([a, b),R); Vol. 26 (2024) Existence and multiplicity... Page 5 of 23 9 (iii) for each t∈[a, b], F(t)=F(a)+[a,t) F g(s)dμg(s). Theorem 2.4. [11, Theorem 3.26 - Fundamental Theorem of Calculus for the Stieltjes derivative] Let f∈L1 g([a, b),R).Then the function F:[a, b]→R, F(t)=[a,t) f(s)dμg(s) is well-defined, g-absolutely continuous on [a, b]and F g(t)=f(t)for μg−a.a. t∈[a, b). Finally we present the definitions and results regarding fixed point index in cones that we will use. Definition 2.5. [9, Definition 2.1] Let (N,·) be a real normed space. A cone Kin Nis a closed set such that 1. u+v∈Kfor all u, v ∈K; 2. λu ∈Kfor all u∈K,λ ∈[0,∞); 3. K∩(−K)={0}. Consider a cone Pin the Banach space BC g([0,T],R). Let Ω be a bounded open subset of BC g([0,T],R). We denote by Ω its closure and by ∂Ω its boundary. We will denote ΩP=Ω∩P. Lemma 2.6. [2, Theorem 11.1] Let Ωbe an open bounded set with 0∈ΩP. Assume that F:ΩP→Pis a continuous compact map such that x=Fx for all x∈∂ΩP. Then the fixed point index iP(F, ΩP)has the following properties. 1. If there exists e∈P\{0}such that x=Fx+λe for all x∈∂ΩPand all λ>0,then iP(F,ΩP)=0. 2. If λx =Fx for all x∈∂ΩPand for every λ≥1,then iP(F,ΩP)=1. 3. If iP(F,ΩP)=0,then Fhas a fixed point in ΩP. 4. Let Ω1be open in BC g([0,T],R)with Ω1 P⊂ΩP.IfiP(F,ΩP)=1and iPF,Ω1 P=0, then Fhas a fixed point in ΩP\Ω1 P.Thesameresult holds if iP(F,ΩP)=0and iPF,Ω1 P=1. 3. The Green’s function In this section, we obtain a Green’s function of the linear boundary value problem for the Stieltjes differential equation u g(t)+b(t)u(t)=f(t),for μg−a.a. t∈[0,T], u(0) −u(T)=kB(u),(3.1) where f∈L1 g([0,T),R),(3.2) b∈L1 g([0,T),R),such that 1 −b(t)Δ+g(t)=0 ∀t∈[0,T),(3.3) 9Page 6 of 23 V. Krajˇsˇc´akov´a, F. A. F. Tojo JFPTA and B:BC g([0,T],R)→Ris a linear bounded functional. In order to obtain this function, we will make use of the results in [21]. First, we introduce the g-exponential function. Theorem 3.1. [21, Theorem 3.4] Let c∈L1 g([t0,t 0+T),R)be such that 1+c(t)Δ+g(t)=0for all t∈[t0,t 0+T)∩Dg. Then the set T− c={t∈[t0,t 0+T)∩Dg1+c(t)Δ+g(t)<0} has a finite cardinality. Furthermore, if T− c={t1,...,t k},t0≤t1<t 2< ···<t k<t k+1 =t0+T, then the map c:[t0,t 0+T)→R, defined as c(t)=⎧ ⎨ ⎩ c(t)if t∈[t0,t 0+T)\Dg, log |1+c(t)Δ+g(t)| Δ+g(t)if t∈[t0,t 0+T)∩Dg, belongs to L1 g([t0,t 0+T),R); the map ec(·,t 0):[t0,t 0+T]→R\{0}, given by ec(t, t0)=⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ exp [t0,t)c(s)dμg(s)if t0≤t≤t1, (−1)jexp [t0,t)c(s)dμg(s)if tj<t≤tj+1,j=1,...,k, is well-defined and g-absolutely continuous on [t0,t 0+T]; and the map x: [t0,t 0+T]→R, given by x(t)=x0ec(t, t0),t∈[t0,t 0+T], solves the initial value problem x g(t)=c(t)x(t),μ g-a.e. t∈[t0,t 0+T),x(t0)=x0. We will call the map ecthe g-exponential function associated to c. Now, from Theorem 3.1, the corresponding result for the nonlinear case follows. Theorem 3.2. [21, Theorem 4.7] Let d, h ∈L1 g([t0,t 0+T),R)be such that 1−d(t)Δ+g(t)=0for all t∈[t0,t 0+T)∩Dg. Then the unique g-absolutely continuous solution of an initial value problem x g(t)+d(t)x(t)=h(t),μ g-a.e. t∈[t0,t 0+T),x(t0)=x0, is given by the map x(t) =e−d(t, t0)x0+[t0,t) h(s) e−d(s, t0)(1 −d(s)Δ+g(s)) dμg(s),t∈[t0,t 0+T]. Remark 3.3. In this paper we will use the map ecwith t0=0,c=−band b satisfying (3.3). The function e−b(·,0) will be denoted as e−b. Let us return our attention to the original boundary value problem (3.1). First, we investigate the periodic problem u g(t)+b(t)u(t)=f(t),μ g-a.e. t∈[0,T], u(0) −u(T)=0,(3.4) Vol. 26 (2024) Existence and multiplicity... Page 7 of 23 9 The thesis of next Lemma is essentially that of [25, Theorem 17], but with simpler assumptions. A similar result can be found in [24, Section 3]. Lemma 3.4. Let (3.2), (3.3)and 1−e−b(T)= 0 (3.5) hold. Then there exists a unique solution of the problem (3.4), which is in the form v(t)=[0,T ) hb(t, s)f(s) 1−b(s)Δ+g(s)dμg(s), where hb(t, s)= e−b(t) e−b(s)(1−e−b(T)) ·1,0≤s<t≤T, e−b(T),0≤t≤s≤T. (3.6) Proof. According to Theorem 3.2 and Remark 3.3, each solution of the Stieltjes differential equation in problem (3.4) can be written in the form v(t)=e−b(t)v(0) + [0,t) f(s) e−b(s)(1−b(s)Δ+g(s)) dμg(s).(3.7) Taking t=Twe get v(T)=e−b(T)v(0) + [0,T ) f(s) e−b(s)(1−b(s)Δ+g(s)) dμg(s). Therefore, vsatisfies the boundary condition in (3.4) if and only if (1 −e−b(T)) v(0) = e−b(T)[0,T ) f(s) e−b(s)(1−b(s)Δ+g(s)) dμg(s). By (3.5) we arrive at the identity v(0) = e−b(T) 1−e−b(T)[0,T ) f(s) e−b(s)(1−b(s)Δ+g(s)) dμg(s). Using this expression to substitute v(0) in (3.7) we get (after some calculation) the desired formula for the solution of (3.4).  Lemma 3.5. Let (3.2), (3.3)and(3.5) hold and hbbe the Green’s function of problem (3.4) given by expression (3.6). Then the function hb(·,0) is a solution of the problem u g(t)+b(t)u(t)=0,μ g-a.e. t∈[0,T], u(0) −u(T)=1. Proof. Since hb(t, 0) = e−b(t) 1−e−b(T),t∈[0,T), it is direct to check that the equation and the boundary conditions hold.  Now we are ready to write and prove the main theorem of this section. Its proof was inspired by a similar result [3, Theorem 2]. 9Page 8 of 23 V. Krajˇsˇc´akov´a, F. A. F. Tojo JFPTA Theorem 3.6. Let (3.2), (3.3), (3.5) hold, k∈R,andB:BC g([0,T],R)→ Rbe a linear bounded operator such that 1=kB(hb(·,0)).(3.8) Then there exists a unique solution of the problem (3.1) given by u(t)=[0,T ) H(t, s, k, b)f(s) 1−b(s)Δ+g(s)dμg(s) (3.9) where H(t, s, k, b)=hb(t, s)+ khb(t, 0) 1−kB(hb(·,0))B(hb(·,s)),(t, s)∈[0,T)2. (3.10) Proof. Existence: Let us write u(t)=v(t)+hb(t, 0) ·kB(v) 1−kB(hb(·,0)),t∈[0,T),(3.11) where vis the unique solution of (3.4)andhbis given by (3.6). Then, according to Lemmas 3.4 and 3.5 we have, for t∈[0,T), u g(t)=v g(t)+(hb(·,0)) g(t)·kB(v) 1−kB(hb(·,0)) =−b(t)v(t)+f(t)−b(t)hb(t, 0) ·kB(v) 1−kB(hb(·,0)) =−b(t)v(t)+hb(t, 0) kB(v) 1−kB(hb(·,0))+f(t) =−b(t)u(t)+f(t). Since u(0) −u(T) =v(0) + hb(0,0) ·kB(v) 1−kB(hb(·,0)) −v(T)−hb(T,0) ·kB(v) 1−kB(hb(·,0)) =(hb(0,0) −hb(T,0)) kB(v) 1−kB(hb(·,0)) =kB(v) 1−kB(hb(·,0)), and kB(u)=kB(v)+kB(hb(·,0)) ·kB(v) 1−kB(hb(·,0)) =kB(v)(1−kB(hb(·,0)) + kB(hb(·,0))) 1−kB(hb(·,0)) =kB(v) 1−kB(hb(·,0)), ualso satisfies the boundary condition in (3.1). Substituting vgiven by (3.6) into (3.11), we arrive at formula (3.9). Uniqueness: Let u, v be two solutions of (3.1). Let us write w=u−v. Then wis a solution of the boundary value problem w g(t)+b(t)w(t)=0,t∈[0,T], w(0) −w(T)=kB(w).(3.12) Vol. 26 (2024) Existence and multiplicity... Page 9 of 23 9 Let us prove that problem (3.12) has only a trivial solution. From (3.12)and Theorem 3.2 we get that there exists w0∈Rsuch that w(t)=w0·e−b(t),t∈[0,T). This function satisfies the boundary condition if and only if w0−w0·e−b(T)=kB(w0·e−b). The linearity of Bimplies that this equation is equivalent to w0(1 −e−b(T)−kB(e−b)) = 0. From the Assumption (3.5)weget w01−kB e−b 1−e−b(T)=0, which is w0(1 −kB(hb(·,0))) = 0. From (3.8) we get that w0= 0. Therefore, the problem (3.1) has only one solution.  4. The integral operator We will consider the operator Tx(t)=[0,T ) H(t, s, k, b)f(s, x(s)) 1−b(s)Δ+g(s)dμg(s),(4.1) where His defined in (3.10). Let us recall that B:BC g([0,T],R)→Ris a linear bounded functional, i.e., there exists c∈R+such that |B(x)|≤c·x∞(4.2) for every x∈BC g([0,T],R). Now, we aim to prove that the operator Tmaps BC g([0,T],R)into BC g([0,T],R) and that it is continuous and compact. Let us recall some theorems and definitions. Definition 4.1. [11, Definition 7.1] Let g:R→Rbe a nondecreasing and left-continuous function, J⊂Rbe an interval and X⊂Rm. A function f:J×X→Rnis said to be g–Carath´eodory if the following properties are satisfied: 1. f(·,x)isμg–measurable for all x∈X, 2. f(t, ·) is continuous for μg−a.a. t∈J, 3. for all r>0 there exists pr∈L1 g(J, [0,∞)) such that f(t, x)≤pr(t),μ g−a.a. t∈J, x ∈X, x≤r. Lemma 4.2. [12, Lemma 8.2] Let Xbe a nonempty subset of Rnand f: [0,T]×X→Rnag–Carath´eodory function. Then, for every x∈BC g([0,T], X), the map f(·,x(·)) is in L1 g([0,T),Rn). Now we are ready to prove next theorem. 9Page 16 of 23 V. Krajˇsˇc´akov´a, F. A. F. Tojo JFPTA Now, according to Lemma 2.6, we give conditions that ensure the fixed point index is 1 or 0 in a certain open subset of P. First, for some ρ>0, we define the sets Mρ:= u∈BC g([0,T],R): inf t∈[0,T )u(t)<ρ ∩P, Nρ:= u∈BC g([0,T],R): sup t∈[0,T ) u(t)<ρ ∩P. We also define a new function n:R→Rby n(ρ):= ρ e−b(T),ρ>0. We are ready to deduce the following inclusions. Lemma 5.6. For every ρ>0we have Nρ⊂Mρ⊂Nn(ρ).(5.9) Proof. Let u∈Nρi.e., supt∈[0,T )u(t)<ρ. Therefore, inft∈[0,T )u(t)<ρand first inclusion is valid. Now we take u∈Mρi.e., inf t∈[0,T )u(t)≥e−b(T)sup t∈[0,T ) u(t) and inft∈[0,T )u(t)<ρ. Thus, supt∈[0,T )u(t)<ρ e−b(T)=n(ρ) and (5.9) holds.  Remark 5.7. Notice that, from previous result, we deduce that for any ρ>0 given, the sets Mρand Nρare bounded. Moreover, from the definition of Mρ and Nρit directly follows that, for every ρ1<ρ 2,weget Mρ1⊂Mρ2,(5.10) Nρ1⊂Nρ2.(5.11) Lemma 5.8. Assume that there exists ρ>0such that fρ·[0,T ) 1 1−e−b(T)1+ k 1−kB (hb(·,0))B(hb(·,s))dμg(s)<1, (5.12) where fρ=1 ρsup f(t, u(t)) 1−b(t)Δ+g(t);t∈[0,T),u∈P, inf t∈[0,T )u(t)=ρ. Then iP(T,Mρ)=1. Proof. In the proof we will use Lemma 2.6. Assume that there exist u∈∂Mρ and λ≥1 such that λu =Tu,thatis, λu(t)=[0,T ) H(t, s, k, b)f(s, u(s)) 1−b(s)Δ+g(s)dμg(s). Vol. 26 (2024) Existence and multiplicity... Page 17 of 23 9 Taking the infimum on [0,T) on both sides of previous equality, and using upper estimate of H,fromTheorem5.4, we deduce that λρ =λinf t∈[0,T )u(t) ≤[0,T ) 1 1−e−b(T) ×1+ k 1−kB (hb(·,0))B(hb(·,s))f(s, u(s)) 1−b(s)Δ+g(s)dμg(s) ≤ρfρ[0,T ) 1 1−e−b(T) ×1+ k 1−kB (hb(·,0))B(hb(·,s))dμg(s)<ρ. This is a contradiction.  Lemma 5.9. Assume that there exists ρ>0such that fρ·[0,T )e−b(T) 1−e−b(T)1+ k 1−kB (hb(·,0))B(hb(·,s))dμg(s)>1, (5.13) where fρ=1 ρinf f(t, u(t)) 1−b(t)Δ+g(t);t∈[0,T),u∈P, sup t∈[0,T ) u(t)=ρ. Then iP(T,Nρ)=0. Proof. In the proof we will again use Lemma 2.6.Letp∈P, such that p=0. Assume that there exist u∈∂Nρand λ>0 such that u=Tu+λp,thatis, u(t)=[0,T ) H(t, s, k, b)f(s, u(s)) 1−b(s)Δ+g(s)dμg(s)+λp(t). Taking the supremum on [0,T) on both sides of previous equality, and using the lower estimate of H,fromTheorem5.4 we arrive at the following contradiction ρ=sup t∈[0,T ) u(t) ≥[0,T )e−b(T) 1−e−b(T)1+ k 1−kB (hb(·,0))B(hb(·,s)) ×f(s, u(s)) 1−b(s)Δ+g(s)dμg(s) ≥ρfρ[0,T )e−b(T) 1−e−b(T)1+ k 1−kB (hb(·,0))B(hb(·,s))dμg(s) >ρ.  9Page 18 of 23 V. Krajˇsˇc´akov´a, F. A. F. Tojo JFPTA Due to the properties of the fixed point index from Lemma 2.6 and the above results we get the following theorem which is the main result of the paper. Theorem 5.10. Let g:R→Rbe a nondecreasing left-continuous function. Let b∈L1 g([0,T),R)and (5.2), (5.3), (5.4), (5.5) hold. Then, problem (1.1) has at least one nontrivial solution in Pif one of the following conditions hold: (S1)there exist ρ1,ρ 2>0with ρ2>ρ 1such that (5.13) holds for ρ1and (5.12) holds for ρ2, (S2)there exist ρ1,ρ 2>0with ρ2>n(ρ1)such that (5.12) holds for ρ1 and (5.13) holds for ρ2. Problem (1.1) has at least two nontrivial solutions in Pif one of the following conditions hold: (S3)there exist ρ1,ρ 2,ρ 3>0with ρ2>ρ 1and ρ3>n(ρ2)such that (5.13) holds for ρ1,(5.12) holds for ρ2and (5.13) holds for ρ3, (S4)there exist ρ1,ρ 2,ρ 3>0with ρ2>n(ρ1)and ρ3>ρ 2such that (5.12) holds for ρ1,(5.13) holds for ρ2and (5.12) holds for ρ3. Proof. First we point out that uis a solution of the problem (1.1) if and only if it is a fixed point of the operator T. We give the proof for case (S3). Proofs of the other cases are similar. Let (S3)hold.From(5.9) and (5.11)weget Mρ2⊂Nn(ρ2)⊂Nρ3. From (5.9) and (5.10) it follows Nρ1⊂Mρ1⊂Mρ2. So Nρ1⊂Mρ2⊂Nρ3. Therefore, from Lemma 5.9, it follows that iP(T,Nρ1)=0andiP(T,Nρ3)= 0 and, from Lemma 5.8, it follows that iP(T,Mρ2)=1. We obtain the existence of solution by applying Lemma 2.6. 6. Example In this section we apply the theory developed in the previous ones to study whether a given problem has a nontrivial solution. We will be considering the following problem u g(t)+bu(t)=|u(t)|α,t∈[0,2], u(0) = u(2) + k[0,2) u(t)dμg(t),(6.1) where b∈(0,1), α>1, g(t)=t, t ∈[0,1], t+1,t∈(1,2] Vol. 26 (2024) Existence and multiplicity... Page 19 of 23 9 and k∈[0,k 0), where k0=1−(1 −b)e−2b [0,2) e−b(s)dμg(s). Note that e−b(t)=e−bt,t∈[0,1], (1 −b)e−bt,t∈(1,2]. It is clear that gis a nondecreasing and left continuous function such that Dg={1}and Δ+g(1) = 1. We will prove that there exists at least one nontrivial solution of (6.1). We start by verifying the Assumptions (5.1)–(5.5). Evidently, |u|α≥0 whenever u≥0, so (5.1) holds. Note that b(t)=bis a constant function with b∈(0,1), so (5.2) holds as well. Also, we have that 1−b(t)Δ+g(t)=1,t∈[0,2] \{1}, 1−b, t ∈{1}. Therefore, (5.3) is satisfied. Operator Bis defined by B(u)=[0,2) u(t)dμg(t),u∈BC g([0,2],R). It is obvious that (5.4) holds. We also have that k∈[0,k 0), so 1>k[0,2) e−b(s)dμg(s) 1−(1 −b)e−2b. Therefore (5.5) is satisfied. Now we work with the cone P:= {u∈BC g([0,2],R):u(t)≥(1 −b)e−2bu∞,t∈[0,2]}. We take a constant ρ1such that 0<ρ 1<(1 −b)α+1 e−2αb I1 α−1 , where I=[0,2) 1 1−(1 −b)e −2b ×1+ k 1−k[0,2) hb(t, 0) d μg(t)[0,2) hb(t, s)dμg(t)dμg(s). Observe that I>0. Such ρ1meets Assumption (5.12). Observe as well that, due to the definition of the cone P,weget u(t)≤infs∈[0,2] u(s) (1 −b)e−2b. 9Page 20 of 23 V. Krajˇsˇc´akov´a, F. A. F. Tojo JFPTA As f(t, u(t)) 1−b(tΔ+g(t)) =|u(t)|αt∈[0,2] \{1}, |u(t)|α 1−b,t∈{1}, we have that fρ1≤max ρα−1 1 ((1 −b)e−2b)α,ρα−1 1 ((1 −b)e−2b)α·1 1−b=ρα−1 1 ((1 −b)e−2b)α·1 1−b. Therefore, Assumption (5.12) is valid for ρ1. Now we take a constant ρ2such that ρ2>1 (1 −b)α+1 e−2b(α+1) ·I1 α−1 . The constant ρ2then meets Assumption (5.13). Indeed, due to the definition of the cone P,wehavethat u(t)≥(1 −b)e−2bsup s∈[0,2] u(s). In this case we get fρ2≥min (1 −b)e−2bαρα−1 2,(1 −b)e−2bαρα−1 2 1−b=(1 −b)e−2bαρα−1 2, so (5.13) holds for ρ2. Besides, n(ρ1)= ρ1 (1 −b)e−2b<(1 −b)α+1 e−2αb I1 α−1 · 1 (1 −b)e−2b<1 (1 −b)α+1 e−2b(α+1) ·I1 α−1 <ρ 2. Thus, all assumptions in (S2) in Theorem 5.10 hold and problem (6.1)has at least one nontrivial solution in P. Acknowledgements The authors greatly appreciate the ideas, comments and suggestions of Profs. Alberto Cabada and Jan Tomeˇcek. We also would like to acknowledge the careful revision by the anonymous referee. These contributions have greatly helped improve the manuscript. F. Adri´an F. Tojo was partially supported by Grant PID2020-113275GB-I00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” of the “European Union”. Vˇera Krajˇsˇc´akov´a was supported by Palack´y University in Olomouc (grant no. IGA PrF 2023 009) Author contributions Vera Krajˇsˇc´akov´a and F. Adrian F. Tojo wrote the main manuscript text and both reviewed the manuscript. Vol. 26 (2024) Existence and multiplicity... Page 21 of 23 9 Funding Open access publishing supported by the National Technical Library in Prague. Declarations Conflict of interest The authors declare no competing interests. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. 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