Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 64, pp. 1–19. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu SENSITIVITY OF A NONLINEAR ORDINARY BVP WITH FRACTIONAL DIRICHLET-LAPLACE OPERATOR DARIUSZ IDCZAK Abstract. In this article, we derive a sensitivity result for a nonlinear frac- tional ordinary elliptic system on a bounded interval with Dirichlet boundary conditions. More precisely, using a global implicit function theorem, we show that for each functional parameter there exists a unique solution, and that its dependence on the functional parameters is continuously differentiable. 1. Introduction In this article, we study a nonlinear ordinary boundary value problem on the interval (0, π), involving a Dirichlet-Laplace operator (−∆)β of order β > 1/2, (−∆)βx(t) = f(t, x(t), u(t)), a.e. t ∈ (0, π), (1.1) where (−∆) : H1 0∩H2 → L2 is the Dirichlet-Laplace operator, H1 0 = H1 0 ((0, π),Rm) and H2 = H2((0, π),Rm) are classical Sobolev spaces, L2 = L2((0, π),Rm) is the classical Lebesgue space, f : (0, π)× Rm × Rr → Rm (m, n ∈ N), x : (0, π)→ Rm is an unknown function and u : (0, π)→ Rr is a functional parameter. Problems involving fractional Laplacians are extensively investigated in resent years because of their numerous applications, among others in probability, fluid mechanics, hydrodynamics; see, for example, [3, 4, 8, 9, 15] and references therein. The definition of the fractional Laplacian adopted in our paper comes from the Stone-von Neumann operator calculus and is based on the spectral integral representation theorem for a self-adjoint operator in Hilbert space. It reduces to a series form which is taken by other authors as a definition [3, 6, 8]. Our more general approach allows us to obtain useful properties of this fractional operator in a smart way. This approach has also been used in [12]. In the first part of this paper, we recall some facts from the theory of spectral integral and Stone-von Neumann operator calculus. Next, we derive some proper- ties of positive powers of the ordinary Dirichlet-Laplace operator and their domains (among others some embedding theorems). In the second part, we use a global im- plicit function theorem [10, 11] to prove existence and uniqueness of a solution to problem (1.1) as well as its sensitivity. By sensitivity we mean continuous differen- tiability of the mapping u 7→ xu 2010 Mathematics Subject Classification. 34B15, 34A08, 34B08, 34L05. Key words and phrases. Fractional Dirichlet-Laplace operator; Palais-Smale condition; Stone-von Neumann operator calculus; global implicit function theorem. c©2021. This work is licensed under a CC BY 4.0 license. Submitted July 19, 2020. Published July 12, 2021. 1 2 D. IDCZAK EJDE-2021/64 where xu is a unique solution to the problem, corresponding to a parameter u. This property can be used to study optimal control problems associated with system (1.1). Similar method but based on a global diffeomorphism theorem [13] and applied to a nonlinear integral Hammerstein equation is presented in [5]. An application of the obtained results to the problem λ(−∆)σ/2x(t) + h(t, x(t)) = (−∆)σ/2u(t), t ∈ (0, 1), where λ ∈ R, σ ∈ (1, 2], h : [−1, 1]× Rn → Rn (n ∈ N), with the exterior Dirichlet boundary condition x(t) = 0, t ∈ (−∞,−1] ∪ [1,∞). In [6], a problem of type (1.1) on a bounded Lipschitzian domain Ω ⊂ Rn (n ≥ 2) and with an exterior Dirichlet boundary condition, is studied. Continuous depen- dence of solutions on parameters (stability) is investigated therein. In [12], using a variational method, we derive an existence result for the so-called bipolynomial fractional Dirichlet-Laplace problem k∑ i,j=0 αiαj [(−∆)ω]βi+βju(x) = DuF (x, u(x)), a.e. x ∈ Ω, where αi > 0 for i = 0, . . . , k (k ∈ N ∪ {0}) and 0 ≤ β0 < β1 < · · · < βk, (−∆)ω : D((−∆)ω) ⊂ L2 → L2 is a weak Dirichlet-Laplace operator, Ω ⊂ RN (N ∈ N) is a bounded open set, F : Ω×R→ R, DuF is the partial derivative of F with respect to u. 2. Integral representation of a self-adjoint operator Results presented in this section can be found, in the case of complex Hilbert space, for example, in [1, 14]. Their proofs can be moved without any or with small changes to the case of real Hilbert space. We will continue to deal only with real Hilbert spaces. Such a preliminary section has also been included in [12]. Let H be a real Hilbert space with a scalar product 〈·, ·〉 : H ×H → R. Let us denote by Π(H) the set of all projections of H on closed linear subspaces, and by B the σ-algebra of Borel subsets of R. By the spectral measure in R we mean a set function E : B → Π(H) that satisfies the following conditions: • for each x ∈ H, the function B 3 P 7→ E(P )x ∈ H (2.1) is a vector measure • E(R) = I • E(P ∩Q) = E(P ) ◦ E(Q) for P,Q ∈ B. By a support of a spectral measure E we mean the complement of the sum of all open subsets of R with zero spectral measure. If b : R→ R is a bounded Borel measurable function, defined a.e. in E, then the integral ∫∞ −∞ b(λ)E(dλ) is defined by(∫ ∞ −∞ b(λ)E(dλ) ) x = ∫ ∞ −∞ b(λ)E(dλ)x EJDE-2021/64 SENSITIVITY OF A NONLINEAR ORDINARY BVP 3 for each x ∈ H where the integral ∫∞ −∞ b(λ)E(dλ)x (with respect to the vector measure) is defined in a standard way, namely, with the aid of the sequence of simple functions converging a.e. in E(dλ)x to b (see [1]). If b : R→ R is an unbounded Borel measurable function defined a.e. in E, then, for each x ∈ H such that ∫ ∞ −∞ |b(λ)|2‖E(dλ)x‖2 <∞ (2.2) (the above integral is taken with respect to the nonnegative measure B 3P 7→ ‖E(P )x‖2 ∈ R+ 0 ), there exists the limit lim ∫ ∞ −∞ bn(λ)E(dλ)x of integrals (with respect to the vector measure (2.1)) where bn : R 3 λ 7→ { b(λ) if |b(λ)| ≤ n, 0 if |b(λ)| > n. Let us denote the set of all points x with property (2.2) by D. One proves that D is dense linear subspace of H, and by ∫∞ −∞ b(λ)E(dλ) one denotes the operator∫ ∞ −∞ b(λ)E(dλ) : D ⊂ H → H given by (∫ ∞ −∞ b(λ)E(dλ) ) x = lim ∫ ∞ −∞ bn(λ)E(dλ)x. Of course, D = H and lim ∫ ∞ −∞ bn(λ)E(dλ)x = ∫ ∞ −∞ b(λ)E(dλ)x when b : R→ R is a bounded Borel measurable function, defined a.e. in E. For x ∈ D, we have ‖ (∫ ∞ −∞ b(λ)E(dλ) ) x‖2 = ∫ ∞ −∞ |b(λ)|2‖E(dλ)x‖2. Moreover, ( ∫ ∞ −∞ b(λ)E(dλ) )∗ = ∫ ∞ −∞ b(λ)E(dλ), (2.3) i.e., the operator ∫∞ −∞ b(λ)E(dλ) is self-adjoint. Remark 2.1. To integrate a Borel measurable function b : B → R where B is a Borel set containing the support of the measure E, it is sufficient to extend b on R to a whichever Borel measurable function (putting, for example, b(λ) = 0 for λ /∈ B). If b : R→ R is Borel measurable and σ ∈ B, then by the integral∫ σ b(λ)E(dλ) we mean the integral ∫ ∞ −∞ χσ(λ)b(λ)E(dλ), 4 D. IDCZAK EJDE-2021/64 where χσ is the characteristic function of the set σ. The integral ∫ σ b(λ)E(dλ) can be also defined with the aid of the restriction of E to the set σ. The next theorem plays the fundamental role in the spectral theory of self-adjoint operators. Theorem 2.2. If A : D(A) ⊂ H → H is self-adjoint and the resolvent set ρ(A) is non-empty, then there exists a unique spectral measure E with the closed support Λ = σ(A), such that A = ∫ ∞ −∞ λE(dλ) = ∫ σ(A) λE(dλ). The basic notion in the Stone-von Neumann operator calculus is a function of a self-adjoint operator. Namely, if A : D(A) ⊂ H → H is self-adjoint and E is the spectral measure determined according to the above theorem, then, for each Borel measurable function b : R→ R, one defines the operator b(A) by b(A) = ∫ ∞ −∞ b(λ)E(dλ) = ∫ σ(A) b(λ)E(dλ). It is known that the spectrum σ(b(A)) of b(A) is given by σ(b(A)) = b(σ(A)) (2.4) provided that b is continuous (it is sufficient to assume that b is continuous on σ(A)). We have the following general result. Proposition 2.3. If b, d : R → R are Borel measurable functions and E is the spectral measure for a self-adjoint operator A : D(A) ⊂ H → H with non-empty resolvent set, then (b · d)(A) ⊃ b(A) ◦ d(A) and (b · d)(A) = b(A) ◦ d(A) (2.5) if and only if D((b · d)(A)) ⊂ D(d(A)). Using the above proposition one can deduce that for each n ∈ N with n ≥ 2, and a Borel measurable function b : R→ R, (b(A))n = bn(A). (2.6) When b(λ) = λ, equality (2.6) gives An = ∫ ∞ −∞ λnE(dλ). (2.7) If n = 1, then (2.7) follows from Theorem 2.2. Since E(R) = I, therefore the identity operator I can be written as I = ∫ ∞ −∞ 1E(dλ). If β > 0, then formula (2.6) with b : R 3 λ→ { 0, λ < 0 λβ/2, λ ≥ 0 and n = 2 implies the following proposition (cf. Remark 2.1). EJDE-2021/64 SENSITIVITY OF A NONLINEAR ORDINARY BVP 5 Proposition 2.4. If σ(A) ⊂ [0,∞), then Aβ/2 ◦Aβ/2 = Aβ . (2.8) 3. Fractional Dirichlet-Laplace operator Consider the one-dimensional Dirichlet-Laplace operator on the interval (0, π), −∆ : H1 0 ∩H2 ⊂ L2 → L2 given by −∆x(t) = −x′′(t). In an elementary way, one can check that this operator is self-adjoint, σ(−∆) = σp(−∆) = {j2 : j ∈ N} (σp(−∆) is the pointwise spectrum of (−∆)) and the eigenspace N(j2) correspond- ing to the eigenvalue λj = j2 is the set {c sin jt : c ∈ Rm}. The system of functions ej,i = (0, . . . , 0, √ 2 π sin jt︸ ︷︷ ︸ i-th entry , 0, . . . , 0), j = 1, 2, . . . , i = 1, . . . ,m, is the Hilbertian basis (complete orthonormal system) in L2. Now, let us fix any β > 0 and consider the operator (−∆)β : D((−∆)β) ⊂ L2 → L2 where D((−∆)β) = {x(t) ∈ L2 : ∫ σ(−∆) |λβ |2‖E(dλ)x‖2 = ∞∑ j=1 ((j2)β)2|aj |2 <∞, (3.1) where x(t) = (∫ σ(−∆) 1E(dλ)x ) (t) = ∞∑ j=1 aj √ 2 π sin jt . Here E is the spectral measure given by Theorem 2.2 for the operator (−∆), aj √ 2 π sin jt is the projection of x on the m-dimensional eigenspace N(j2) of the operator (−∆), and (−∆)βx(t) = ((∫ σ(−∆) λβE(dλ) ) x ) (t) = ( lim ∫ σ(−∆) (λβ)nE(dλ)x ) (t) = ∞∑ j=1 (j2)βaj √ 2 π sin jt for x(t) = ∞∑ j=1 aj √ 2 π sin jt ∈ D((−∆)β) . The series is meant in L2 but from the Carleson theorem it follows that x(t) =∑∞ j=1 aj √ 2 π sin jt a.e. on (0, π) (cf. [7, Theorem 5.17]). Equality (2.4) and the fact that isolated points of the spectrum of a self-adjoint operator are the eigenvalues imply that σ((−∆)β) = σp((−∆)β) = {(j2)β : j ∈ N}. 6 D. IDCZAK EJDE-2021/64 The corresponding eigenspaces for (−∆) and (−∆)β are the same (it follows from a general result concerning the power of any self-adjoint operator). The operator (−∆)β will be called the Dirichlet-Laplace operator of order β, and the function (−∆)βx - the Dirichlet-Laplacian of order β of x. Lemma 3.1. D((−∆)β) with the scalar product 〈x, y〉β = 〈x, y〉L2 + 〈(−∆)βx, (−∆)βy〉L2 is a Hilbert space. Proof. The assertion follows from the operator (−∆)β being self-adjoint is closed (cf. (2.3)). � The scalar product 〈·, ·〉β and the scalar product 〈x, y〉∼β = 〈(−∆)βx, (−∆)βy〉L2 generate equivalent norms in D((−∆)β). Indeed, it is sufficient to observe that the following Poincare inequality holds: ‖x‖2L2 = ∞∑ j=1 a2 j ≤ ∞∑ j=1 ((j2)β)2a2 j = ‖(−∆)βx‖2L2 = ‖x‖2∼β (3.2) for each x(t) = ∞∑ j=1 aj √ 2 π sin jt ∈ D((−∆)β). Next, we shall consider D((−∆)β) with the norm ‖ · ‖∼β . 3.1. Embeddings. From the description of the domain D((−∆)β) it follows that D((−∆)β2) ⊂ D((−∆)β1) (3.3) for each 0 < β1 < β2. Using this relation and equality (2.7) with A = (−∆) we assert that C∞c ⊂ D((−∆)β) for each β > 0 (C∞c = C∞c ((0, π),Rm) is the set of smooth functions with the supports contained in (0, π)). Lemma 3.2. If β > 1/4, then D((−∆)β) ⊂ L∞m = L∞((0, π),Rm) and this embedding is continuous, more precisely, ‖x‖L∞m ≤ √ 2 π ζ(4β)‖x‖∼β for x ∈ D((−∆)β), where ζ(4β) is the value of the Riemann zeta function ζ(γ) =∑∞ j=1 1/jγ at γ = 4β. Proof. Let x(t) = ∞∑ j=1 aj √ 2 π sin jt ∈ D((−∆)β). EJDE-2021/64 SENSITIVITY OF A NONLINEAR ORDINARY BVP 7 Since ∑∞ j=1((j2)β)2a2 j <∞ and β > 1/4, for t ∈ (0, π) a.e., we have |x(t)|2 = | ∞∑ j=1 aj √ 2 π sin jt|2 ≤ 2 π ( ∞∑ j=1 |aj | )2 = 2 π ( ∞∑ j=1 (j2)β |aj | (j2)β )2 ≤ 2 π ( ∞∑ j=1 ((j2)β)2|aj |2 )( ∞∑ j=1 1( (j2)β )2) = 2 π ‖x‖2∼βζ(4β) <∞ and the proof is complete. � Lemma 3.3. If β ≥ 1/2, then D((−∆)β) ⊂ H1 0 , and consequently D((−∆)β) ⊂ C = C([0, π],Rm). Proof. Of course it is sufficient to show that D((−∆)1/2) ⊂ H1 0 (cf. (3.3)). Indeed, let x(t) = ∑∞ j=1 aj √ 2 π sin jt ∈ D((−∆)1/2) and consider this series on the interval [0, π]. The sequence (Sn) of partial sums converges in L2 to x. From the convergence of the series ∑∞ j=1 j 2a2 j it follows that the sequence (S′n) of derivatives converges in L2 to a function. So (cf. [7]), one can choose a subsequence (S′nk) convergent a.e. on [0, π] to this function and bounded pointwise a.e. on [0, π] by a function g ∈ L2. Consequently, the sequence (S′nk) is equiabsolutely integrable on [0, π]. So, the sequence (Snk) is equiabsolutely continuous on [0, π]. Of course, Snk(0) = 0, thus |Snk(t)| = ∣∣Snk(0) + ∫ t 0 S′nk(s)ds ∣∣ ≤ ∫ π 0 g(s)ds <∞ for t ∈ [0, π]. It means that elements of the sequence (Snk) satisfy the assumptions of the Ascoli-Arzela theorem for absolutely continuous functions and, in conse- quence, there exists a subsequence (Snki ) converging uniformly on [0, π] to an abso- lutely continuous function x. Clearly, (Snki ) converges to x in L2. The uniqueness of the limit in L2 means that x = x a.e. on (0, π). So, x has a representative which is absolutely continuous on [0, π] and satisfies Dirichlet boundary conditions, i.e. x ∈ W 1,1 0 ((0, π),Rm) (the classical Sobolev space). Consequently, there exists a function g ∈ L1 such that∫ π 0 x(t)ϕ′(t) dt = − ∫ π 0 g(t)ϕ(t) dt for each ϕ ∈ C∞c . But∫ π 0 x(t)ϕ′(t) dt = ∫ π 0 ( ∞∑ j=1 aj √ 2 π sin jt ) ϕ′(t) dt = ∫ π 0 lim n→∞ Sn(t)ϕ′(t) dt = ∞∑ j=1 ∫ π 0 aj √ 2 π sin jtϕ′(t) dt 8 D. IDCZAK EJDE-2021/64 = ∞∑ j=1 − ∫ π 0 jaj √ 2 π cos jtϕ(t) dt = − ∫ π 0 ∞∑ j=1 jaj √ 2 π cos jtϕ(t) dt for ϕ ∈ C∞c . The last equality follows from ∑∞ j=1 j 2a2 j < ∞, and consequently,∑∞ n=1 jaj √ 2 π cos jt ∈ L2. Thus, g(t) = ∞∑ n=1 jaj √ 2 π cos jt ∈ L2 and, finally, x ∈ H1 0 . The second part of the theorem follows from a known property of Sobolev space W 1,1((0, π),Rm). � Lemma 3.4. If β > 3/4, then any bounded the set B ⊂ D((−∆)β) is equicontinu- ous on [0, π]. Proof. Similarly as in the proof of Lemma 3.2 we obtain |x(t1)− x(t2)|2 = ∣∣∣ ∞∑ j=1 aj √ 2 π (sin jt1 − sin jt2) ∣∣∣2 ≤ ( ∞∑ j=1 |aj | √ 2 π 2| sin j(t1 − t2) 2 | )2 ≤ 2 π |t1 − t2|2 ( ∞∑ j=1 |aj |j )2 = 2 π |t1 − t2|2 ( ∞∑ j=1 (j2)β |aj | (j2)βj−1 )2 ≤ 2 π |t1 − t2|2 ( ∞∑ j=1 ((j2)β)2|aj |2 )( ∞∑ j=1 1 (j2β−1)2 ) = 2 π |t1 − t2|2‖x‖2∼βζ(4β − 2) <∞ for t1, t2 ∈ (0, π) a.e., where x(t) = ∑∞ j=1 aj √ 2 π sin jt ∈ D((−∆)β). Identifying x with its absolutely continuous representative on [0, π] we assert that the above estimation holds for all t1, t2 ∈ [0, π]. � Using Lemmas 3.2, 3.3, 3.4 we obtain the following result. Corollary 3.5. If β > 3/4, then the embedding D((−∆)β) ⊂ C is compact. EJDE-2021/64 SENSITIVITY OF A NONLINEAR ORDINARY BVP 9 3.2. Equivalence of equations. Fact that the operator (−∆)β (β > 0) is self- adjoint means that its domain satisfies the equality D((−∆)β) = { x ∈ L2 : there exists z ∈ L2 such that∫ π 0 x(t)(−∆)βy(t) dt = ∫ π 0 z(t)y(t) dt for each y ∈ D((−∆)β) } (3.4) and (−∆)βx = z (3.5) for x ∈ D((−∆)β). From (2.8) it follows that x ∈ D((−∆)β) if and only if x ∈ D((−∆)β/2), (−∆)β/2x ∈ D((−∆)β/2), and this case (−∆)β/2((−∆)β/2x) = (−∆)βx. (3.6) Using this fact and (3.4), (3.5), we obtain the following lemma. Lemma 3.6. If β > 0 and g ∈ L2, then x ∈ D((−∆)β) and (−∆)βx = g if and only if x ∈ D((−∆)β/2) and∫ π 0 (−∆)β/2x(t)(−∆)β/2y(t) dt = ∫ π 0 g(t)y(t) dt for each y ∈ D((−∆)β/2). 4. Global implicit function theorem Let X be a real Banach space and I : X → R be a functional of class C1. We say that I satisfies Palais-Smale (PS) condition if any sequence (xk) such that • |I(xk)| ≤M for all k ∈ N and some M > 0, • I ′(xk)→ 0, admits a convergent subsequence. Here I ′(xk) denotes the Frechet differential of I at xk. A sequence (xk) satisfying the above conditions is called the (PS) sequence for I. From [10, 11] we have the following result. Theorem 4.1. Let X, U be real Banach spaces, H be a real Hilbert space. If F : X × U → H is continuously differentiable with respect to (x, u) ∈ X × U and • for each u ∈ U , the functional ϕ : X 3 x 7→ 1 2 ‖F (x, u)‖2 ∈ R satisfies (PS) condition • F ′x(x, u) : X → H is bijective for each (x, u) ∈ X × U , then there exists a unique function λ : U → X such that F (λ(u), u) = 0 for each u ∈ U and this function is of class C1 with differential λ′(u) at u given by λ′(u) = −[F ′x(λ(u), u)]−1 ◦ F ′u(λ(u), u). (4.1) 10 D. IDCZAK EJDE-2021/64 5. A boundary value problem Let us consider boundary value problem (1.1). Using the global implicit func- tion theorem, we shall show (under suitable assumptions) that, for each fixed u ∈ L∞r = L∞((0, π),Rr), problem (1.1) has a unique solution xu ∈ D((−∆)β) and the mapping L∞r 3 u 7→ xu ∈ D((−∆)β) is continuously differentiable. Consider the mapping F : D((−∆)β)× L∞r 3 (x, u) 7→ (−∆)βx(t)− f(t, x(t), u(t)) ∈ L2. We shall formulate conditions guaranteeing that • F is of class C1, • differential Fx(x, u) : D((−∆)β)→ L2 is bijective for each (x, u) in D((−∆)β)× L∞r , • for each u ∈ L∞r , functional ϕ : D((−∆)β) 3 x 7→ 1 2 ‖F (x, u)‖2L2 ∈ R satisfies the (PS) condition. 5.1. Smoothness of F . Assume that function f is measurable in t ∈ (0, π), con- tinuously differentiable in (x, u) ∈ Rm × Rr and |f(t, x, u)|, |fx(t, x, u)|, |fu(t, x, u)| ≤ a(t)γ(|x|) + b(t)δ(|u|) (5.1) for (t, x, u) ∈ (0, π)×Rm ×Rr, where a, b ∈ L2 and γ, δ : R+ 0 → R+ 0 are continuous functions. Proposition 5.1. If β > 1/4, then F is of class C1 and the differential F ′(x, u) : D((−∆)β)× L∞r → L2 of F at (x, u) is given by F ′(x, u)(h, v) = (−∆)βh(t)− fx(t, x(t), u(t))h(t)− fu(t, x(t), u(t))v(t) for (h, v) ∈ D((−∆)β)× L∞r . Proof. Smoothness of the first term of F is obvious. So, let us consider the mapping G : D((−∆)β)× L∞r 3 (x, u) 7→ f(t, x(t), u(t)) ∈ L2. We shall show that the mappings Gx(x, u) : D((−∆)β) 3 h 7→ fx(t, x(t), u(t))h(t) ∈ L2, Gu(x, u) : L∞r 3 v 7→ fu(t, x(t), u(t))v(t) ∈ L2 are partial Frechet differentials of G at (x, u) and the mappings D((−∆)β)× L∞r 3 (x, u) 7→ Gx(x, u) ∈ L(D((−∆)β), L2), (5.2) D((−∆)β)× L∞r 3 (x, u) 7→ Gu(x, u) ∈ L(L∞r , L 2) (5.3) are continuous. Of course, it is sufficient to check the differentiability in Gateaux sense and continuity of the above two mappings (in such a case, the Gateaux differentials are Frechet ones). EJDE-2021/64 SENSITIVITY OF A NONLINEAR ORDINARY BVP 11 So, let us consider differentiability of G with respect to x. Linearity and conti- nuity of the mapping Gx(x, u) are obvious (in view of Lemma 3.2). To prove that Gx(x, u) is Gateaux differential of G with respect to x, we shall show that∥∥G(x+ λkh, u)−G(x, u) λk −Gx(x, u)h ∥∥2 L2 = ∫ π 0 ∣∣f(t, x(t) + λkh(t), u(t))− f(t, x(t), u(t)) λk − fx(t, x(t), u(t))h(t) ∣∣2dt→ 0 for each sequence (λk) ⊂ (−1, 1) such that λk → 0. Indeed, the sequence of functions t 7→ f(t, x(t) + λkh(t), u(t))− f(t, x(t), u(t)) λk − fx(t, x(t), u(t))h(t) converges pointwise a.e. on (0, π) to the zero function (by differentiability of f in x). Moreover, from the mean value theorem it follows that this sequence is bounded by a function from L2:∣∣f(t, x(t) + λkh(t), u(t))− f(t, x(t), u(t)) λk − fx(t, x(t), u(t))h(t) ∣∣ = |fx(t, x(t) + st,kλkh(t), u(t))h(t)− fx(t, x(t), u(t))h(t)| ≤ constx,u,h(a(t) + b(t))|h(t)|, where st,k ∈ (0, 1) and constx,u,h is a constant depending on x, u, h. Thus, using the Lebesgue dominated convergence theorem we assert that Gx(x, u) is Gateaux differential of G with respect to x. In the same way, we check that Gu(x, u) is Gateaux differential of G with respect to u. To finish the proof we shall show that the mappings (5.2), (5.3) are continuous. Let (xk, uk)→ (x0, u0) in D((−∆)β)× L∞r . Then ‖(Gx(xk, uk)−Gx(x0, u0))h‖2L2 ≤ ∫ π 0 |fx(t, xk(t), uk(t))− fx(t, x0(t), u0(t))|2|h(t)|2dt ≤ ‖h‖2∞ ∫ π 0 |fx(t, xk(t), uk(t))− fx(t, x0(t), u0(t))|2dt ≤ 2 π ζ(4β)‖h‖2∼β ∫ π 0 |fx(t, xk(t), uk(t))− fx(t, x0(t), u0(t))|2dt. Consequently, ‖Gx(xk, uk)−Gx(x0, u0)‖L(D((−∆)β),L2) ≤ √ 2 π ζ(4β)( ∫ π 0 |fx(t, xk(t), uk(t))− fx(t, x0(t), u0(t))|2dt)1/2. Using Lemma 3.2, assumption (5.1) and the Lebesgue dominated convergence the- orem we assert that Gx(xk, uk)→ Gx(x0, u0) in L(D((−∆)β), L2). In a similar way, we check the continuity of the mapping D((−∆)β)× L∞r 3 (x, u) 7→ Gu(x, u) ∈ L(L∞r , L 2). The proof is complete. � 12 D. IDCZAK EJDE-2021/64 5.2. Bijectivity of Fx(x, u). In view of the previous theorem and its proof, it is clear that if β > 1/4 and functions f , fx satisfy growth condition (5.1), then the partial differential of F with respect to x is of the form Fx(x, u) : D((−∆)β) 3 h 7→ (−∆)βh(t)− fx(t, x(t), u(t))h(t) ∈ L2. for each (x, u) ∈ D((−∆)β)× L∞r . Proposition 5.2. Assume that functions f , fx satisfy growth condition (5.1). If β > 1/2 and one of the following conditions is satisfied (a) ‖Λ‖L1 m×m < π 2ζ(2β) , (b) Λ(t) ≤ 0, i.e. matrix Λ(t) is nonpositive, for a.e. t ∈ (0, π), (c) Λ ∈ L∞m×m and ‖Λ‖∞ < 1, where Λ(t) := fx(t, x(t), u(t)), Lpm×m = Lp((0, π),Rm×m) for p = 1,∞, then dif- ferential Fx(x, u) : D((−∆)β)→ L2 is bijective. By the norm of a matrix C = [ci,j ] ∈ Rm×m we mean the value ( ∑m i,j=1 |ci,j |2)1/2. Remark 5.3. In Part (c) one can assume that β > 1/4. In such a case the proof of coercivity of a (see the proof of Proposition 5.2) remains unchanged and to show its continuity one estimates |a(h, y)| ≤ ‖h‖∼β/2‖y‖∼β/2 + ‖Λ‖∞‖h‖L2‖y‖L2 ≤ (1 + ‖Λ‖∞)‖h‖∼β/2‖y‖∼β/2. Proof of Proposition 5.2. We shall show that, for each function g ∈ L2, equation (−∆)βh(t)− Λ(t)h(t) = g(t) (5.4) has a unique solution in D((−∆)β). Using Lemma 3.6, we see that it is equivalent to show that there exists a unique function h ∈ D((−∆)β/2) such that∫ π 0 (−∆)β/2h(t)(−∆)β/2y(t) dt = ∫ π 0 (Λ(t)h(t) + g(t))y(t) dt for each y ∈ D((−∆)β/2). So, let us define a bilinear form a : D((−∆)β/2) × D((−∆)β/2)→ R by a(h, y) = ∫ π 0 (−∆)β/2h(t)(−∆)β/2y(t) dt− ∫ π 0 Λ(t)h(t)y(t) dt. This form is continuous. Indeed (cf. Lemma 3.2), |a(h, y)| ≤ ‖h‖∼β/2‖y‖∼β/2 + ‖Λ‖L1‖h‖∞‖y‖∞ ≤ (1 + ‖Λ‖L1 2 π ς(2β))‖h‖∼β/2‖y‖∼β/2 for h, y ∈ D((−∆)β/2). We have the following three parts Part a. |a(h, h)| = ∣∣ ∫ π 0 (−∆)β/2h(t)(−∆)β/2h(t) dt− ∫ π 0 Λ(t)h(t)h(t) dt ∣∣ ≥ ‖h‖2∼β/2 − ‖Λ‖L1‖h‖2∞ ≥ (1− ‖Λ‖L1 2 π ς(2β))‖h‖2∼β/2 . EJDE-2021/64 SENSITIVITY OF A NONLINEAR ORDINARY BVP 13 Part b. |a(h, h)| ≥ ∫ π 0 (−∆)β/2h(t)(−∆)β/2h(t) dt− ∫ π 0 Λ(t)h(t)h(t) dt ≥ ‖h‖2∼β/2 . Part c. |a(h, h)| = ∣∣ ∫ π 0 (−∆)β/2h(t)(−∆)β/2h(t) dt− ∫ π 0 Λ(t)h(t)h(t) dt ∣∣ ≥ ‖h‖2∼β/2 − ‖Λ‖∞‖h‖ 2 L2 ≥ (1− ‖Λ‖∞)‖h‖2∼β/2 . So, a is coercive. From Lax-Milgram theorem it follows that for each linear continuous functional l : D((−∆)β/2) → R there exists a unique h ∈ D((−∆)β/2) such that a(h, y) = l(y) for each y ∈ D((−∆)β/2). Since the functional D((−∆)β/2) 3 y 7→ ∫ π 0 g(t)y(t) dt ∈ R is linear and continuous, therefore there exists a unique h ∈ D((−∆)β/2) such that∫ π 0 (−∆)β/2h(t)(−∆)β/2y(t) dt− ∫ π 0 Λ(t)h(t)y(t) dt = ∫ π 0 g(t)y(t) dt for each y ∈ D((−∆)β/2). The proof is complete. � 5.3. (PS) condition. As in the proof of Proposition 5.2, one can show that, for each β > 0 and any function g ∈ L2, there exists a unique function xg ∈ D((−∆)β/2) such that∫ π 0 (−∆)β/2xg(t)(−∆)β/2y(t) dt = ∫ π 0 g(t)y(t) dt for each y ∈ D((−∆)β/2). It means, in view of Lemma 3.6, that the following lemma holds. Lemma 5.4. For any β > 0 and g ∈ L2, there exists a unique solution xg ∈ D((−∆)β) of the equation (−∆)βx = g. Lemma 5.5. If β > 1/2, then the operator [(−∆)β ]−1 : L2 3 g 7→ xg ∈ L2 is compact, i.e. the image of any bounded set in L2 is relatively compact in L2. Proof. Since x(g1,...,gm) = (xg1 , . . . , xgm) for each (g1, . . . , gm) ∈ L2, one can assume that m = 1. Let us recall the Kolmogorov-Frechet-Riesz theorem [7]: if F is a bounded set in Lp(Rn) (1 ≤ p <∞) and ∀ ε>0 ∃ δ>0 ∀ |h|<δ ∀ f∈F ‖τhf − f‖Lp(Rn) < ε (5.5) (where, τhf(x) = f(x + h)), then F |Ω is relatively compact in Lp(Ω) for each measurable set Ω ⊂ Rn with finite Lebesgue measure. 14 D. IDCZAK EJDE-2021/64 Let G ⊂ L2((0, π),R) be a set bounded by a constant C. Consider the functions g(t) = ∞∑ j=1 bgj √ 2 π sin jt ∈ G, xg(t) = ∞∑ j=1 agj √ 2 π sin jt (both series are convergent in L2 and, in view of the Carleson theorem, a.e. on (0, π)). Since (−∆)βxg(t) = g(t), i.e. ∞∑ j=1 (j2)βagj √ 2 π sin jt = ∞∑ j=1 bgj √ 2 π sin jt, it follows that agj = bgj (j2)β for j ∈ N. Now, we shall show that the set of functions {x̃g; g ∈ G}, where x̃g : R 3 t 7→ { xg(t) t ∈ (0, π), 0 otherwise, satisfies condition (5.5) (of course, it is bounded in L2(R,R)). Let us fix 0 < h < π and consider the integral∫ ∞ −∞ |x̃g(t+ h)− x̃g(t)|2dt = ∫ 0 −h |x̃g(t+ h)|2dt+ ∫ π−h 0 |x̃g(t+ h)− x̃g(t)|2dt + ∫ π π−h |x̃g(t+ h)− x̃g(t)|2dt = ∫ h 0 |xg(t)|2dt+ ∫ π−h 0 |xg(t+ h)− xg(t)|2dt+ ∫ π π−h |xg(t)|2dt. (5.6) The first term of the above expression can be estimated as follows (to obtain third inequality we use Hölder inequality for series)∫ h 0 |xg(t)|2dt = ∫ h 0 ∣∣∣ ∞∑ j=1 bgj (j2)β √ 2 π sin jt ∣∣∣2dt ≤ 2 π ∫ h 0 ( ∞∑ j=1 |bgj | (j2)β )2 dt ≤ 2 π h ∞∑ j=1 |bgj | 2 ∞∑ j=1 1 (j2)2β = 2 π h‖g‖2L2ζ(4β) ≤ 2 π Cζ(4β)h. In the same way one can estimate third term of (5.6). EJDE-2021/64 SENSITIVITY OF A NONLINEAR ORDINARY BVP 15 For the second term, we have∫ π−h 0 |xg(t+ h)− xg(t)|2dt = ∫ π−h 0 ∣∣∣ ∞∑ j=1 bgj (j2)β √ 2 π (sin j(t+ h)− sin jt) ∣∣∣2dt ≤ ∫ π−h 0 ( ∞∑ j=1 |bgj | (j2)β √ 2 π |2 sin jh 2 cos(jt+ jh 2 )| )2 dt ≤ 8 π ∫ π−h 0 ( ∞∑ j=1 |bgj | (j2)β | sin jh 2 | )2 dt ≤ 8 π (π − h) ∞∑ j=1 |bgj | 2 ∞∑ j=1 sin2 jh 2 (j2)2β ≤ 8 π (π − h)C ∞∑ j=1 jh 2(j2)2β ≤ 4Ch ∞∑ j=1 1 j4β−1 = 4Cζ(4β − 1)h. If −π < h < 0, we proceed in the same way. Finally, ‖τhf − f‖Lp(Rn) ≤ const|h| for |h| < π. So, the set {x̃g ∣∣ (0,π) : g ∈ G} = {xg : g ∈ G} is relatively compact in L2. The proof is complete. � Using the above lemma we obtain the following result. Lemma 5.6. If β > 1/2 and xk ⇀ x0 weakly in D((−∆)β), then xk → x0 strongly in L2 and (−∆)βxk ⇀ (−∆)βx0 weakly in L2. Proof. From the continuity of the linear operators D((−∆)β) 3 x 7→ x ∈ L2, D((−∆)β) 3 x 7→ (−∆)βx ∈ L2, it follows that xk ⇀ x0 weakly in L2 and (−∆)βxk ⇀ (−∆)βx0 weakly in L2. Lemma 5.5 implies that the sequence (xk) contains a subsequence (xki) converging strongly in L2 to a limit. Of course, this limit is the function x0, i.e. xki → x0 strongly in L2. Supposing contrary and repeating the above argumentation we assert that xk → x0 strongly in L2. � Remark 5.7. Lemmas 5.5 and Lemma 5.6 are valid for each β > 0. The proofs of such stronger results, in the case of bounded open set Ω ⊂ Rn (n ≥ 1), can be found in [12]. We give here weaker theorems for two reasons. First, to prove more general results (in fact, a counterpart of Lemma 5.5 because the proof of Lemma 5.6 remains unchanged) some additional considerations, concerning the spectral representation of the inverse operator, are needed. Second, due to the other assumptions (cf. Proposition 5.2) assumption β > 1/2 in Theorem 6.1 can not be omitted. The main tool for proving that ϕ satisfies the (PS) condition is the following lemma. 16 D. IDCZAK EJDE-2021/64 Lemma 5.8. If β > 1/4, f satisfies the growth condition |f(t, x, u)| ≤ a(t)|x|+ b(t)δ(|u|) for (t, x, u) ∈ (0, π) × Rm × Rr, where a, b ∈ L2, δ : R+ 0 → R+ 0 is a continuous function and √ 2 π ζ(4β)‖a‖L2 < 1, (5.7) then, for each u ∈ L∞r , the functional ϕ : D((−∆)β) 3 x 7→ 1 2 ‖F (x, u)‖2L2 ∈ R is coercive, i.e. ‖x‖∼β →∞ implies ϕ(x)→∞. Proof. We have ‖F (x, u)‖L2 = ‖(−∆)βx(t)− f(t, x(t), u(t))‖L2 ≥ ‖(−∆)βx(t)‖L2 − ‖f(t, x(t), u(t))‖L2 . But ‖f(t, x(t), u(t))‖L2 ≤ (∫ π 0 (a(t)|x(t)|+ b(t)δ(|u(t)|))2dt )1/2 ≤ (∫ π 0 |a(t)|2|x(t)|2dt )1/2 +D ≤ ‖x‖∞‖a‖L2 +D ≤ √ 2 π ζ(4β)‖a‖L2‖(−∆)βx‖L2 +D where D = ( ∫ π 0 |b(t)|2(δ(|u(t)|))2dt)1/2. Thus, ‖F (x, u)‖L2 ≥ ‖(−∆)βx‖L2 − √ 2 π ζ(4β)‖a‖L2‖(−∆)βx‖L2 −D = (1− √ 2 π ζ(4β)‖a‖L2)‖x‖∼β −D. It means that ϕ is coercive. � Now, we are in a position to prove that ϕ satisfies the (PS) condition. Proposition 5.9. If β > 1/2, f and fx satisfy the growth conditions |f(t, x, u)| ≤ a(t)|x|+ b(t)δ(|u|), |fx(t, x, u)| ≤ a(t)γ(|x|) + b(t)δ(|u|) for (t, x, u) ∈ (0, π)×Rm×Rr, where a, b ∈ L2 and γ, δ : R+ 0 → R+ 0 are continuous functions, and (5.7) holds true, then ϕ (with any fixed u ∈ L∞r ) satisfies the (PS) condition. Proof. From Proposition 5.1 it follows that ϕ is of class C1 and its differential ϕ′(x) : D((−∆)β)→ R is ϕ′(x)h = ∫ π 0 ((−∆)βx(t)− f(t, x(t), u(t)))((−∆)βh(t)− fx(t, x(t), u(t))h(t)) dt EJDE-2021/64 SENSITIVITY OF A NONLINEAR ORDINARY BVP 17 for h ∈ D((−∆)β). Consequently, for xk, x0 ∈ D((−∆)β), we have (ϕ′(xk)− ϕ′(x0))(xk − x0) = ‖xk − x0‖2∼β/2 + 5∑ i=1 ψi(xk) where ψ1(xk) = ∫ π 0 (−∆)βxk(t)fx(t, xk(t), u(t))(x0(t)− xk(t)) dt, ψ2(xk) = ∫ π 0 (−∆)βx0(t)fx(t, x0(t), u(t))(xk(t)− x0(t)) dt, ψ3(xk) = ∫ π 0 f(t, xk(t), u(t))fx(t, xk(t), u(t))(xk(t)− x0(t)) dt, ψ4(xk) = ∫ π 0 f(t, x0(t), u(t))fx(t, x0(t), u(t))(x0(t)− xk(t)) dt, ψ5(xk) = ∫ π 0 (f(t, x0(t), u(t))− f(t, xk(t), u(t)))((−∆)βxk(t)− (−∆)βx0(t)) dt. Now, let (xk) be a (PS) sequence for ϕ. Since ϕ is coercive, therefore (xk) is bounded in D((−∆)β). So, one can choose a subsequence (xki) weakly converging in D((−∆)β) to some x0. From Lemma 5.6 it follows that xki → x0 strongly in L2 and (−∆)βxki(t) ⇀ (−∆)βx0(t) weakly in L2. Since the sequence (xki) is bounded in D((−∆)β), therefore it is bounded in L∞m and, consequently (β > 1/2), in C. Moreover, there exists a subsequence of the sequence (xki) (let us denote it by (xki)) converging to x0 pointwise a.e. on (0, π). Term ψ1(xki) tends to zero. Indeed, fx(t, xki(t), u(t)), k ∈ N, are equibounded on (0, π) by a square integrable function. Functions fx(t, xki(t), u(t))(x0(t)− xmk(t)) belong to L2 and converge pointwise (a.e. on (0, π)) to zero function. More- over, they are equibounded on (0, π) by a square integrable function. So, from the Lebesgue dominated convergence theorem it follows that the sequence (fx(t, xki(t), u(t))(x0(t)− xmk(t))) converges in L2 to the zero function. Thus, in view of the weak convergence of the sequence ((−∆)βxk) to (−∆)βx0 in L2, ψ1(xki)→ 0. Similarly, ψl(xki)→ 0 for remaining l. Finally, since ϕ′(xki)(xki − x0)→ 0 and ϕ′(x0)(xki − x0)→ 0, it follows that ‖xki − x0‖2∼β/2 → 0, i.e. ϕ satisfies the (PS) condition. � 6. Final result Thus, we have proved the following result. Theorem 6.1. Assume that β > 1/2, function f is measurable in t ∈ (0, π), continuously differentiable in (x, u) ∈ Rm × Rr and |f(t, x, u)| ≤ a(t)|x|+ b(t)δ(|u|), |fx(t, x, u)|, |fu(t, x, u)| ≤ a(t)γ(|x|) + b(t)δ(|u|) 18 D. IDCZAK EJDE-2021/64 for (t, x, u) ∈ (0, π) × Rm × Rr, where a, b ∈ L2, γ, δ : R+ 0 → R+ 0 are continuous functions and √ 2 π ζ(4β)‖a‖L2 < 1. If, for each pair (x, u) ∈ D((−∆)β) × L∞r one of the following assumptions is satisfied (a) ‖fx(t, x(t), u(t))‖L1 m×m < π 2ζ(2β) , (b) fx(t, x(t), u(t)) ≤ 0 for a.e. t ∈ (0, π), (c) fx(t, x(t), u(t)) ∈ L∞m×m and ‖fx(t, x(t), u(t))‖∞ < 1, then, for each u ∈ L∞r , there exists a unique solution xu ∈ D((−∆)β) of problem (1.1) and the mapping λ : L∞r 3 u 7→ xu ∈ D((−∆)β) is continuously differentiable with the differential λ′(u) at u ∈ L∞r such that, for each v ∈ L∞r , (−∆)β(λ′(u)v)(t)− fx(t, xu(t), u(t))(λ′(u)v)(t) = fu(t, xu(t), u(t))v(t) for t ∈ (0, π) a.e. Remark 6.2. Thus, for each u ∈ L∞r , v ∈ D((−∆)β) the function λ′(u)v ∈ D((−∆)β) is a solution to the equation (−∆)βy(t)− fx(t, xu(t), u(t))y(t) = fu(t, xu(t), u(t))v(t), a.e.t ∈ (0, π) . Example 6.3. Let β > 1/2,,m = 2, and r = 2. It is easy to see that the function f(t, x, u) = ( f1(t, x1, x2, u1, u2), f2(t, x1, x2, u1, u2) ) = (a sin(x2) + t−1/3eu1 , b cos(x1) + tu2) satisfies assumptions of Theorem 6.1 with a(t) = √ a2 + b2, γ(s) = √ 2, b(t) = t−1/3 + t+ |b|, δ(s) = es, where a, b ∈ R are such that √ a2 + b2 ≤ 1 2 √ 2ζ(2β) . Consequently, for each u = (u1, u2) ∈ L∞2 , there exists a unique solution xu ∈ D((−∆)β) of the problem (−∆)βx1(t) = a sin(x2(t)) + t−1/3eu1(t) (−∆)βx2(t) = b cos(x1(t)) + tu2(t) for t ∈ (0, π) a.e., and the mapping λ(u) = (x1 u, x 2 u) is continuously differentiable with the differential λ′(u) : L∞2 → D((−∆)β) such that (−∆)β(λ′(u)v)(t)− [ 0 a cos((xu)2(t)) −b sin((xu)1(t)) 0 ] (λ′(u)v)(t) = [ t−1/3eu1(t) 0 0 tu2(t) ] v(t), a.e. t ∈ (0, π), for each v ∈ L∞2 , i.e. (−∆)β((λ′(u)v)1)(t) = a cos((xu)2(t)) (λ′(u)v)2 (t) + t− 1 3 eu1(t)v1(t) EJDE-2021/64 SENSITIVITY OF A NONLINEAR ORDINARY BVP 19 (−∆)β((λ′(u)v)1)(t) = −b sin((xu)1(t)) (λ′(u)v)1 (t) + tu2(t)v2(t) for a.e. t ∈ (0, π) and every v = (v1, v2) ∈ L∞2 . References [1] A. Alexiewicz; Functional Analysis, PWN, Warsaw, 1969 (in Polish). [2] H. Attouch, G. Buttazzo, G. Michaille; Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and Optimization, SIAM-MPS, Philadelphia, 2006. [3] B. Barrios, E. 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Vazquez; Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, vol. 7 of Abel Symposia (2012), 271-298. Dariusz Idczak Faculty of Mathematics and Computer Science, University of Lodz, Banacha 22 90-238 Lodz, Poland Email address: dariusz.idczak@wmii.uni.lodz.pl 1. Introduction 2. Integral representation of a self-adjoint operator 3. Fractional Dirichlet-Laplace operator 3.1. Embeddings 3.2. Equivalence of equations 4. Global implicit function theorem 5. A boundary value problem 5.1. Smoothness of F 5.2. Bijectivity of Fx(x,u) 5.3. (PS) condition 6. Final result References