Solutions to resonant boundary value problem with boundary conditions involving Riemann-Stieltjes integrals

We study the nonlinear boundary value problem consisting of a system of second order differential equations and boundary conditions involving a Riemann-Stieltjes integrals. Our proofs are based on the generalized Miranda Theorem.

We are concerned with the boundary value problem consisting of the equations the initial conditions x (0) = 0, (2) and the non-local boundary conditions where the integrals We shall assume that 1 0 dg i (s) = 1 for i = 1, . . . , k. In this case the problem (1)- (3) is resonant, since the corresponding homogeneous linear problem has nontrivial solutions : x(t) = a ∈ R k . The existence of solutions to the problem (1)-(3) with multi-point boundary conditions have been studied extensively (see, for instance, [1,4,8,9] and the references therein).
The nonlocal resonant problem (1)- (3) was studied in particular in [19], where authors established the existence and multiplicity of positive solutions for nonperturbed boundary value problem at resonance by considering equivalent nonresonant perturbed problem with the same boundary conditions. In [5], authors, using a Leggett-Williams norm-type theorem, showed, in particular, that the problem under consideration has positive solutions. In the papers mentioned above the boundary value problems were scalar and the function f did not depend upon x . The methods used in the papers are not the same as our ones and the assumptions are of completely different kind.
In this paper, we apply the ideas from [14]. The paper is organized as follows. The next section presents some notations, definitions and the generalized Miranda Theorem (see Theorem 2.1). In Section 3 we define an auxiliary problem which allows us to apply Theorem 2.1 and prove the existence of solutions to the problem (1)-(3). We impose on the function f standard growth and sign conditions and assume that for each i = 1, . . . , k functions g i are nondecreasing. Moreover, the main theorem can also be applied to the problem (1)-(3) in the case when function f does not depend upon x (comp. Corollary 1). The paper ends with two examples, the second one concerns the application of our results to a model of thermostat.
2. Some preliminaries. In this section, we shall present some basic concepts and results for later use.
Let X and Y be topological spaces.
A space X is said to be contractible, if the identity map of X is homotopic to some constant map of X to itself [13].
A set-valued map Θ : X Y is upper semicontinuous (written usc), if, given an open set V ⊂ Y , the set {x ∈ X : Θ(x) ⊂ V } is open [6].
A compact space X is an R δ -set (we write X ∈ R δ ), if there is a decreasing sequence X n of compact contractible spaces such that X = ∞ n=1 X n [6].
We say that Θ : X Y is an R δ -map, if it is usc and, for each x ∈ X, Θ(x) ∈ R δ . We shall discuss the existence of solutions to the problem (1)-(3) by using the following theorem (comp. [14]): then there exists x such that 0 ∈ Ψ(x).
For more abstract results with using tangency conditions expressed in terms of tangent cones see [2].
3. An auxiliary problem. Consider the following auxiliary initial value problem where t ∈ [0, 1] and α ∈ R k is fixed.

SOLUTIONS TO RESONANT BOUNDARY VALUE PROBLEM 277
The following assumptions upon f will be needed throughout the paper: The following result is standard. We give its proof for completeness.
Lemma 3.1. If assumptions (i), (ii) hold, the problem (5) has at least one global solution for every fixed α ∈ R k .
Proof. Let α ∈ R k be fixed. The existence of at least one local solution to the problem (5) follows from the assumption (i). Using the theorem on a priori bounds [11], we will show that every such solution is a global one, i.e., any possible solution can be extended to the interval [0, 1]. Let x be a local solution to (5) and observe that then Now, notice that (5) is equivalent to Moreover, one has Consequently, from (8) and (ii), we obtain Putting and by Gronwall's Inequality Consequently, we have |x (t)| ≤ C α exp D < ∞. and, by (6), |x(t)| is also bounded on [0, 1], which completes the proof. Define the mapping F : and observe that by the assumptions (i) and (ii) the operator F is completely continuous.
Note that for each α ∈ R k a function x is a solution to the problem (5) if and only if x is a fixed point of the operator F (·, α).

4.
The existence of solutions. In this section, considering the family of all initial value problems (5), we shall show that there is an α ∈ R k such that for the α a solution to the problem (5) is also a solution to the problem (1)  Proof. Since the operator Fix F (·, α) is completely continuous, one can show that Θ is usc with compact values ( [14], Lemma 2). The fact that Θ is an R δ -map follows from the assumption (ii). Indeed, it is well known that if f has a linear growth then the set of all solutions of the problem (5) is an R δ -set ( [7], p. 162). Now, let us introduce the following assumption: (iii) for every i = 1, . . . , k the function g i is nondecreasing,  Proof. The linearity and continuity of θ is obvious. We shall show that θ is surjective. Let d = (d 1 , . . . , d k ) ∈ R k . Observe that one can always find a function x i (s) dg i (s) and set Consequently, for each d there is an x such that θ(x) = d.
The lemma below follows from Lemmas 4.1 and 4.2. Now, the following assumption will be needed: (iv) there exist M i > 0 such that for every t ∈ [0, 1], x ∈ R k and y ∈ R k we have Proof. To prove the existence of solutions to the problem (1)-(3), since Ψ is admissible, it is sufficient to show that the condition (4) of Theorem 2.1 holds. We shall show that for any i = 1, . . . , k and every y ∈ Ψ(α), we have α i · y i > 0 for |α i | = M i , where M i is as in the assumption (iv). First let us consider the case when α i = M i . Let x be a solution to the problem (5) with α i = M i , i = 1, . . . , k, and observe that x ∈ C 2 ([0, 1], R k ). First, we shall prove that x i (t) < x i (1) for t ∈ [0, 1), i = 1, . . . , k.
Observe that x i (0) = α i and x i (0) = 0. Moreover, by (iv), we have Consequently, x i has a local minimum at 0 and there exists an ε > 0 such that x i (t) > α i for t ∈ (0, ε). Now, assume that for some t ∈ (ε, 1] we have x i (t) < α i . Then there exists t := inf{t : x i (t) < α i } such that x(t) = α i and x i (t) ≥ α i for t ≤ t. Hence, there exists t 0 ∈ (0, t) such that x i has a local maximum at t 0 greater than α i . By the assumption (iv), we reach a contradiction. Indeed, we have We have proved that x i (t) ≥ α i for t ∈ [0, 1]. Hence, by (iv), we obtain Consequently, x i is increasing on (0, 1) and x i (t) > 0 for t ∈ (0, 1), since x i (0) = 0. Hence, one has t ∈ [0, 1). Integrating this inequality over [0, 1] with respect to g i and using the assumption (iii), we obtain which means that in this case α i · y i > 0 with y ∈ Ψ(α).
By similar arguments, one can show that when Finally, we get that there is an α ∈ R k such that 0 ∈ Ψ(α). This finishes the proof. Now, consider the case when the function f does not depend upon x and assume that the following conditions hold: ( where a 1 , a 2 > 0; (iv) for every i = 1, . . . , k there exist M i > 0 such that for every t ∈ [0, 1] and   Observe, that the problem (1)-(3) with the function f given above has at least one nontrivial solutions. Indeed, note that f satisfies the assumptions (i) and (ii). Moreover, we have . Hence, setting M 2 > 3, we obtain that x 2 f 2 (t, x, y) for |x 2 | ≥ M 2 . Let M 1 > 2. By similar arguments one can show that x 1 f 1 (t, x, y) > 0 when |x 1 | ≥ M 1 . Consequently, the assumption (iv) of Theorem 4.4 is satisfied and the problem (1)-(3) with f defined above has at least one nontrivial solution.
Example 4.2. In [15], a three-point second-order boundary value problem has been used to model the temperature on a heated bar with a controller at the right end of the bar which adds or eliminates heat depending on the temperature detected by sensors at intermediate points. Solutions of such problem are stationary solutions for a one-dimensional heat equation. Recently, the study of nonlocal boundary value problems for second-order equations has been shown to be effective to the modeling of a thermostat with sensors expressed as linear functionals, see [18].
The problem (1)-(3) can also be interpreted as a thermostat model. In our case sensors give feedback to the endpoints where controllers add or remove heat according to that feedback. In addition, we generalize the equation considering the heat equation with a nonlinear gradient source terms that vary in time (for the heat equation with a gradient source see, for instance, [3], [12] and papers cited therein). Moreover, now, the heated bar with a controller at 1 adds or removes heat depending on the temperature detected by a sensors put at any points of the bar (it depends on the function g). One can control the heat at 0 and 1, depending on what happens over the entire length of the bar. Theorems presented in this paper provide information on the existence of solutions to such problems.