Positive solutions of nonlinear multi-point boundary value problems

This paper deals with the existence of positive solutions of nonlinear differential equation

$$\begin{aligned} u^{\prime \prime }(t)+ a(t) f(u(t) )=0,\quad 0<t <1, \end{aligned}$$

subject to the boundary conditions

$$\begin{aligned} u(0)=\sum _{i=1}^{m-2} a_i u (\xi _i) ,\quad u^{\prime } (1) = \sum _{i=1}^{m-2} b_i u^{\prime } (\xi _i), \end{aligned}$$

where \( \xi _i \in (0,1) \) with \( 0< \xi _1<\xi _2< \cdots<\xi _{m-2} < 1,\) and \(a_i,b_i \) satisfy   \(a_i,b_i\in [0,\infty ),~~ 0< \sum _{i=1}^{m-2} a_i <1,\) and \( \sum _{i=1}^{m-2} b_i <1. \) By using Schauder’s fixed point theorem, we show that it has at least one positive solution if f is nonnegative and continuous. Positive solutions of the above boundary value problem satisfy the Harnack inequality

$$\begin{aligned} \displaystyle \inf _{0 \le t \le 1} u(t) \ge \gamma \Vert u\Vert _\infty . \end{aligned}$$

     

Multiplicity of nodal solutions to the Yamabe problem

Given a compact Riemannian manifold (M, g) without boundary of dimension \(m\ge 3\) and under some symmetry assumptions, we establish existence of one positive and multiple nodal solutions to the Yamabe-type equation

$$\begin{aligned} -\text {div}_{g}(a\nabla u)+bu=c|u|^{2^{*}-2}u\quad \text { on }M, \end{aligned}$$

where \(a,b,c\in \mathcal {C}^{\infty }(M), a\) and c are positive, ? div\(_{g}(a\nabla )+b\) is coercive, and \(2^{*}=\frac{2m}{m-2}\) is the critical Sobolev exponent. In particular, if \(R_{g}\) denotes the scalar curvature of (M, g), we give conditions which guarantee that the Yamabe problem

$$\begin{aligned} \Delta _{g}u+\frac{m-2}{4(m-1)}R_{g}u=\kappa u^{2^{*}-2}\quad \text { on }M \end{aligned}$$

admits a prescribed number of nodal solutions.

     

A new hyperstability result for the Apollonius equation on a restricted domain and some applications

Let \((G,+)\) be an abelian group equipped with a complete ultrametric d that is invariant (i.e., \(d(x + z, y + z)= d(x, y\)) for \(x, y, z \in G\)), X be a normed space and \(U\subset X\setminus \{0\} \) be a nonempty subset. Under some weak natural assumptions on U and on the function \(\chi :U^3\rightarrow [0,\infty )\), we study new hyperstability results when \(f:U\rightarrow G\) satisfy the following Apollonius inequality

$$\begin{aligned}&d\Big (4f\Big (z-\frac{x+y}{2}\Big )+f(x-y),2f(x-z)+2f(y-z)\Big )\leqslant \chi (x,y,z),\\ {}&\quad x, y, z\in U,\;\;x-z,y-z,x-y,z-\frac{x+y}{2}\in U. \end{aligned}$$

Moreover, we derive some consequences from our main results.

     

Existence results to positive solutions of fractional BVP with $${\varvec{}}$$-derivatives

This paper investigates the existence and uniqueness of positive and nondecreasing solution for nonlinear boundary value problem with fractional q-derivative

$$\begin{aligned}&D_{q}^{\alpha }u(t)+f(t,u(t))=0, \quad {0<t<1, ~3<\alpha \le 4,} \\&u(0)= D_{q}u(0)=D_{q}^{2}u(0)=0, \quad D_{q}^{2}u(1)=\beta D_{q}^{2}u(\eta ), \end{aligned}$$

where \(D_{q}^{\alpha }\) denotes the Riemann–Liouville q-derivative of order \(\alpha \), \(0<\eta <1\) and \(1-\beta \eta ^{\alpha -3}>0\). Our analysis relies a fixed point theorem in partially ordered sets. An example to illustrate our results is given.

     

Jump and Variational Inequalities for Rough Operators

In this paper, we systematically study jump and variational inequalities for rough operators, whose research have been initiated by Jones et al. More precisely, we show some jump and variational inequalities for the families \(\mathcal T:=\{T_\varepsilon \}_{\varepsilon >0}\) of truncated singular integrals and \(\mathcal M:=\{M_t\}_{t>0}\) of averaging operators with rough kernels, which are defined respectively by

$$\begin{aligned} T_\varepsilon f(x)=\int _{|y|>\varepsilon }\frac{\Omega (y')}{|y|^n}f(x-y)dy \end{aligned}$$

and

$$\begin{aligned} M_t f(x)=\frac{1}{t^n}\int _{|y|<t}\Omega (y')f(x-y)dy, \end{aligned}$$

where the kernel \(\Omega \) belongs to \(L\log ^+\!\!L(\mathbf S^{n-1})\) or \(H^1(\mathbf S^{n-1})\) or \(\mathcal {G}_\alpha (\mathbf S^{n-1})\) (the condition introduced by Grafakos and Stefanov). Some of our results are sharp in the sense that the underlying assumptions are the best known conditions for the boundedness of corresponding maximal operators.

     

New stability results for the radical sextic functional equation related to quadratic mappings in $$(2,\beta )$$-Banach spaces

We first introduce basic concepts of \((2,\beta )\)-Banach spaces and we will reformulate the fixed point theorem [8, Theorem 1] in this space. After that, we achieve the general solution of the radical sextic functional equation

$$\begin{aligned} f\left( \root 3 \of {x^3+y^3}\right) +f\left( \root 3 \of {x^3-y^3}\right) = 2f(x)+2f(y),\quad x,y\in {\mathbb {R}}, \end{aligned}$$

for f a mapping from the set of all real numbers \({\mathbb {R}}\) into a vector space. In addition, we prove a new type of stability results for radical sextic functional equation in \((2,\beta )\)-Banach spaces. The method is based on a fixed point Theorem 3.1 in some functions spaces.

     

Large conformal metrics with prescribed sign-changing Gauss curvature

Let \((M,g)\) be a two dimensional compact Riemannian manifold of genus \(g(M)>1\). Let \(f\) be a smooth function on \(M\) such that

$$\begin{aligned} f \ge 0, \quad f\not \equiv 0, \quad \min _M f = 0. \end{aligned}$$

Let \(p_1,\ldots ,p_n\) be any set of points at which \(f(p_i)=0\) and \(D^2f(p_i)\) is non-singular. We prove that for all sufficiently small \(\lambda >0\) there exists a family of “bubbling” conformal metrics \(g_\lambda =e^{u_\lambda }g\) such that their Gauss curvature is given by the sign-changing function \(K_{g_\lambda }=-f+\lambda ^2\). Moreover, the family \(u_\lambda \) satisfies

$$\begin{aligned} u_\lambda (p_j) = -4\log \lambda -2\log \left( \frac{1}{\sqrt{2}} \log \frac{1}{\lambda }\right) +O(1) \end{aligned}$$

and

$$\begin{aligned} \lambda ^2e^{u_\lambda }\rightharpoonup 8\pi \sum _{i=1}^{n}\delta _{p_i},\quad \text{ as } \lambda \rightarrow 0, \end{aligned}$$

where \(\delta _{p}\) designates Dirac mass at the point \(p\).

     

On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

Let \(\alpha ,\beta \) be orientation-preserving diffeomorphism (shifts) of \(\mathbb {R}_+=(0,\infty )\) onto itself with the only fixed points \(0\) and \(\infty \) and \(U_\alpha ,U_\beta \) be the isometric shift operators on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\), \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\), and \(P_2^\pm =(I\pm S_2)/2\) where

$$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$

is the weighted Cauchy singular integral operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \), and

$$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|<1, \quad \limsup _{t\rightarrow s}|d(t)|<1, \quad s\in \{0,\infty \}, \end{aligned}$$

then the operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.

     

On the stability of the generalized Cauchy–Jensen set-valued functional equations

In this paper, we prove the Hyers–Ulam–Rassias stability of the generalized Cauchy–Jensen set-valued functional equation defined by

$$\begin{aligned} \alpha f\left( \frac{x+y}{\alpha } + z\right) = f(x) \oplus f(y)\oplus \alpha f(z) \end{aligned}$$

for all \(x,y,z \in X\) and \(\alpha \ge 2\) on a Banach space by using the fixed point alternative theorem.

     

A Characterization of Two-Weight Norm Inequality for Littlewood–Paley $$g_{\lambda }^{*}$$-Function

Let \(n\ge 2\) and \(g_{\lambda }^{*}\) be the well-known high-dimensional Littlewood–Paley function which was defined and studied by E. M. Stein,

$$\begin{aligned} g_{\lambda }^{*}(f)(x) =\bigg (\iint _{\mathbb {R}^{n+1}_{+}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda } |\nabla P_tf(y,t)|^2 \frac{\mathrm{d}y \mathrm{d}t}{t^{n-1}}\bigg )^{1/2}, \ \quad \lambda > 1, \end{aligned}$$

where \(P_tf(y,t)=p_t*f(y)\), \(p_t(y)=t^{-n}p(y/t)\), and \(p(x) = (1+|x|^2)^{-(n+1)/2}\), \(\nabla =(\frac{\partial }{\partial y_1},\ldots ,\frac{\partial }{\partial y_n},\frac{\partial }{\partial t})\). In this paper, we give a characterization of two-weight norm inequality for \(g_{\lambda }^{*}\)-function. We show that \(\big \Vert g_{\lambda }^{*}(f \sigma ) \big \Vert _{L^2(w)} \lesssim \big \Vert f \big \Vert _{L^2(\sigma )}\) if and only if the two-weight Muckenhoupt \(A_2\) condition holds, and a testing condition holds:

$$\begin{aligned} \sup _{Q : \text {cubes}~\mathrm{in} \ {\mathbb {R}^n}} \frac{1}{\sigma (Q)} \int _{{\mathbb {R}^n}} \iint _{\widehat{Q}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda }|\nabla P_t(\mathbf {1}_Q \sigma )(y,t)|^2 \frac{w \mathrm{d}x \mathrm{d}t}{t^{n-1}} \mathrm{d}y < \infty , \end{aligned}$$

where \(\widehat{Q}\) is the Carleson box over Q and \((w, \sigma )\) is a pair of weights. We actually prove this characterization for \(g_{\lambda }^{*}\)-function associated with more general fractional Poisson kernel \(p^\alpha (x) = (1+|x|^2)^{-{(n+\alpha )}/{2}}\). Moreover, the corresponding results for intrinsic \(g_{\lambda }^*\)-function are also presented.

     

Existence of multi-bump solutions for a class of elliptic problems involving the biharmonic operator

Using variational methods, we establish existence of multi-bump solutions for the following class of problems

$$\begin{aligned} \left\{ \begin{array}{l} \Delta ^2 u +(\lambda V(x)+1)u = f(u), \quad \text{ in } \quad \mathbb {R}^{N},\\ u \in H^{2}(\mathbb {R}^{N}), \end{array} \right. \end{aligned}$$

where \(N \ge 1\), \(\Delta ^2\) is the biharmonic operator, f is a continuous function with subcritical growth, \(V : \mathbb {R}^N \rightarrow \mathbb {R}\) is a continuous function verifying some conditions and \(\lambda >0\) is a real constant large enough.

     

On one-dimensional superlinear indefinite problems involving the $$\phi $$-Laplacian

Let \(\Omega := ( a,b ) \subset \mathbb {R}\), \(m\in L^{1} ( \Omega ) \) and \(\phi :\mathbb {R\rightarrow R}\) be an odd increasing homeomorphism. We consider the existence of positive solutions for problems of the form

$$\begin{aligned} \left\{ \begin{array} [c]{ll} -\phi ( u^{\prime } ) ^{\prime }=m ( x ) f ( u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on } \partial \Omega , \end{array} \right. \end{aligned}$$

where \(f: [ 0,\infty ) \rightarrow [ 0,\infty ) \) is a continuous function which is, roughly speaking, superlinear with respect to \(\phi \). Our approach combines the Guo-Krasnoselski? fixed-point theorem with some estimates on related nonlinear problems. We mention that our results are new even in the case \(m\ge 0\).

     

Uniform Bounds on the Relative Error in the Approximation of Upper Quantiles for Sums of Arbitrary Independent Random Variables

Fix any \(n\ge 1\). Let \(\tilde{X}_1,\ldots ,\tilde{X}_n\) be independent random variables. For each \(1\le j \le n\), \(\tilde{X}_j\) is transformed in a canonical manner into a random variable \(X_j\). The \(X_j\) inherit independence from the \(\tilde{X}_j\). Let \(s_y\) and \(s_y^*\) denote the upper \(\frac{1}{y}{\underline{\text{ th }}}\) quantile of \(S_n=\sum _{j=1}^nX_j\) and \(S^*_n=\sup _{1\le k\le n}S_k\), respectively. We construct a computable quantity \(\underline{Q}_y\) based on the marginal distributions of \(X_1,\ldots ,X_n\) to produce upper and lower bounds for \(s_y\) and \(s_y^*\). We prove that for \(y\ge 8\)

$$\begin{aligned} 6^{-1} \gamma _{3y/16}\underline{Q}_{3y/16}\le s^*_{y}\le \underline{Q}_y \end{aligned}$$

where

$$\begin{aligned} \gamma _y=\frac{1}{2w_y+1} \end{aligned}$$

and \(w_y\) is the unique solution of

$$\begin{aligned} \Big (\frac{w_y}{e\ln (\frac{y}{y-2})}\Big )^{w_y}=2y-4 \end{aligned}$$

for \(w_y>\ln (\frac{y}{y-2})\), and for \(y\ge 37\)

$$\begin{aligned} \frac{1}{9}\gamma _{u(y)}\underline{Q}_{u(y)}<s_y \le \underline{Q}_y \end{aligned}$$

where

$$\begin{aligned} u(y)=\frac{3y}{32} \left( 1+\sqrt{1-\frac{64}{3y}}\right) . \end{aligned}$$

The distribution of \(S_n\) is approximately centered around zero in that \(P(S_n\ge 0) \ge \frac{1}{18}\) and \(P(S_n\le 0)\ge \frac{1}{65}\). The results extend to \(n=\infty \) if and only if for some (hence all) \(a>0\)

$$\begin{aligned} \sum _{j=1}^{\infty }E\{(\tilde{X}_j-m_j)^2\wedge a^2\}<\infty . \end{aligned}$$

(1)

     

Zeta functions and asymptotic additive bases with some unusual sets of primes

Fix \(\delta \in (0,1]\), \(\sigma _0\in [0,1)\) and a real-valued function \(\varepsilon (x)\) for which \(\varlimsup _{x\rightarrow \infty }\varepsilon (x)\leqslant 0\). For every set of primes \(\mathcal {P}\) whose counting function \(\pi _\mathcal {P}(x)\) satisfies an estimate of the form

$$\begin{aligned} \pi _\mathcal {P}(x)=\delta \,\pi (x)+O\bigl (x^{\sigma _0+\varepsilon (x)}\bigr ), \end{aligned}$$

we define a zeta function \(\zeta _\mathcal {P}(s)\) that is closely related to the Riemann zeta function \(\zeta (s)\). For \(\sigma _0\leqslant \frac{1}{2}\), we show that the Riemann hypothesis is equivalent to the non-vanishing of \(\zeta _\mathcal {P}(s)\) in the region \(\{\sigma >\frac{1}{2}\}\).

For every set of primes \(\mathcal {P}\) that contains the prime 2 and whose counting function satisfies an estimate of the form

$$\begin{aligned} \pi _\mathcal {P}(x)=\delta \,\pi (x)+O\bigl ((\log \log x)^{\varepsilon (x)}\bigr ), \end{aligned}$$

we show that \(\mathcal {P}\) is an exact asymptotic additive basis for \(\mathbb {N}\), i.e. for some integer \(h=h(\mathcal {P})>0\) the sumset \(h\mathcal {P}\) contains all but finitely many natural numbers. For example, an exact asymptotic additive basis for \(\mathbb {N}\) is provided by the set

$$\begin{aligned} \{2,547,1229,1993,2749,3581,4421,5281\ldots \}, \end{aligned}$$

which consists of 2 and every hundredth prime thereafter.

     

A Sharp Weak-Type $$(\infty ,\infty )$$ Inequality for the Hilbert Transform

The paper is devoted to sharp weak type \((\infty ,\infty )\) estimates for \({\mathcal {H}}^{\mathbb {T}}\) and \({\mathcal {H}}^{\mathbb {R}}\), the Hilbert transforms on the circle and real line, respectively. Specifically, it is proved that

$$\begin{aligned} \left\| {\mathcal {H}}^{\mathbb {T}}f\right\| _{W({\mathbb {T}})}\le \Vert f\Vert _{L^\infty ({\mathbb {T}})} \end{aligned}$$

and

$$\begin{aligned} \left\| {\mathcal {H}}^{\mathbb {R}}f\right\| _{W({\mathbb {R}})}\le \Vert f\Vert _{L^\infty ({\mathbb {R}})}, \end{aligned}$$

where \(W({\mathbb {T}})\) and \(W({\mathbb {R}})\) stand for the weak-\(L^\infty \) spaces introduced by Bennett, DeVore and Sharpley. In both estimates, the constant \(1\) on the right is shown to be the best possible.

     

Algebraic independence of certain Mahler functions

A monotonicity-type result for functions \(f\ : \ \mathbb {N}_a\rightarrow \mathbb {R}\) satisfying the sequential fractional difference inequality

$$\begin{aligned} \Delta _{1+a-\mu }^{\nu }\Delta _{a}^{\mu }f(t)\ge 0, \end{aligned}$$

for \(t\in \mathbb {N}_{2+a-\mu -\nu }\), where \(0<\mu <1\), \(0<\nu <1\), and \(1<\mu +\nu <2\), is proved, subject to the restriction that

$$\begin{aligned} \mu <2(1-\nu ). \end{aligned}$$

We demonstrate that this result is sharp in the sense that the restriction \(\mu <2(1-\nu )\) cannot be improved.

     

Existence of Multiple Positive Solutions for Nonlinear Fractional Boundary Value Problems on the Half-Line

In this paper, we deal with the following nonlinear fractional differential problem in the half-line \({\mathbb{R}^{+}=(0,+ \infty)}\)

$$\left\{\begin{array}{l}D^{\alpha }u(x)+f(x,u(x),D^{p}u(x))=0,\quad x \in \mathbb{R}^{+},\\ u(0)=u^{\prime } \left( 0\right) = \cdots =u^{\left( m-2\right) }(0)=0,\end{array}\right.$$

where \({m\in \mathbb{N}, m \geq 2, m-1 < \alpha \leq m, 0 < p \leq \alpha -1}\), the differential operator is taken in the Riemann–Liouville sense and f is a Borel measurable function in \({\mathbb{R}^{+} \times \mathbb{R}^{+} \times \mathbb{R} ^{+}}\) satisfying certain conditions. More precisely, we show the existence of multiple unbounded positive solutions, by means of Schäuder fixed point theorem. Some examples illustrating our main result are also given.

     

Fixed points and additive $${\rho}$$-functional equations

In this paper, we solve the additive \({\rho}\)-functional equations

$$\begin{aligned} f(x+y)-f(x)-f(y)= & {} \rho(2f(\frac{x+y}{2})-f(x)-f(y)), \\ 2f(\frac{x+y}{2})-f(x)-f(y)= & {} \rho(f(x+y)-f(x)-f(y)), \end{aligned}$$

where \({\rho}\) is a fixed non-Archimedean number or a fixed real or complex number with \({\rho \neq 1}\). Using the fixed point method, we prove the Hyers–Ulam stability of the above additive \({\rho}\)-functional equations in non-Archimedean Banach spaces and in Banach spaces.

     

Multiplicity and bifurcation of positive solutions for nonhomogeneous semilinear fractional Laplacian problems

In this paper we consider the following nonhomogeneous semilinear fractional Laplacian problem

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u+u=\lambda (f(x,u)+h(x)) \,\, \text {in}\,\, \mathbb {R}^N,\\ u\in H^s(\mathbb {R}^N), u>0\,\, \text {in}\,\, \mathbb {R}^N, \end{array}\right. } \end{aligned}$$

where \(\lambda >0\) and \(\lim _{|x|\rightarrow \infty }f(x,u)=\overline{f}(u)\) uniformly on any compact subset of \([0,\infty )\). We prove that under suitable conditions on f and h, there exists \(0<\lambda ^*<+\infty \) such that the problem has at least two positive solutions if \(\lambda \in (0,\lambda ^*)\), a unique positive solution if \(\lambda =\lambda ^*\), and no solution if \(\lambda >\lambda ^*\). We also obtain the bifurcation of positive solutions for the problem at \((\lambda ^*,u^*)\) and further analyse the set of positive solutions.

     

Perpetual Integrals for Lévy Processes

Given a Lévy process \(\xi \), we find necessary and sufficient conditions for almost sure finiteness of the perpetual integral \(\int _0^\infty f(\xi _s)\hbox {d}s\), where \(f\) is a positive locally integrable function. If \(\mu =\mathbb {E}[\xi _1]\in (0,\infty )\) and \(\xi \) has local times we prove the 0–1 law

$$\begin{aligned} \mathbb {P}\Big (\int _0^\infty f(\xi _s)\,\hbox {d}s<\infty \Big )\in \{0,1\} \end{aligned}$$

with the exact characterization

$$\begin{aligned} \mathbb {P}\Big (\int _0^\infty f(\xi _s)\,\hbox {d}s<\infty \Big )=0\qquad \Longleftrightarrow \qquad \int ^\infty f(x)\,\hbox {d}x=\infty . \end{aligned}$$

The proof uses spatially stationary Lévy processes, local time calculations, Jeulin’s lemma and the Hewitt–Savage 0–1 law.