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The authors prove the existence of at least one nontrivial solution to an anisotropic discrete boundary value problem of p(k)-Kirchhoff type and its associated parametric version. They provide an e...

The authors give sufficient conditions for the existence of at least three classical solutions to the nonlinear impulsive fractional boundary value problem with a p-Laplacian and Dirichlet conditions $$\begin{aligned} {\left\{ \begin{array}{ll} D^{\alpha }_{T^-}\varPhi _p(^cD^{\alpha }_{0^+}u(t))+|u(t)|^{p-2}u(t)= \lambda f(t,u(t)), &{}t\ne t_j,\ \ t\in (0,T),\\ \varDelta (D^{\alpha -1}_{T^-}\varPhi _p(^cD^{\alpha }_{0^+}u))(t_j)=I_j(u(t_j)),\\ u(0)=u(T)=0, \end{array}\right. } \end{aligned}$$ where $$\alpha \in (\frac{1}{p}, 1]$$ and $$p > 1$$ . Their approach is based on variational methods. The main result is illustrated with an example.

The authors establish sufficient conditions for the existence of solutions to a boundary value problem for fractional differential inclusions involving the Caputo-Hadamard type derivative of order $ r \in (1, 2] $ on infinite intervals. Both cases of convex and nonconvex valued right hand sides are considered. The technique of proof involves fixed point theorems combined with a diagonalization method.

The research by J. R. Graef was supported in part by a University of Tennessee at Chattanooga SimCenter - Center of Excellence in Applied Computational Science and Engineering (CEACSE) grant. In this paper, the authors examine the existence of positive solutions to a third-order boundary value problem having a sign changing nonlinearity. The proof makes use of fixed point index theory. An example is included to illustrate the applicability of the results. University of Tennessee at Chattanooga SimCenter - Center of Excellence in Applied Computational Science and Engineering (CEACSE) grant

This paper deals with existence and uniqueness of solutions to a class of impulsive boundary value problem for nonlinear implicit fractional differential equations involving the Caputo-exponential fractional derivative. The existence results are based on Schaefer's fixed point theorem and the uniqueness result is established via Banach's contraction principle. Two examples are given to illustrate the main results.

In this paper, a Banach space and an associated functional are presented to study the existence of solutions to a second order discrete boundary value problem with mixed periodic boundary conditions. The proofs make use of critical point theory. Some illustrative examples are provided.

Criteria are established for the existence of at least two nontrivial solutions to the discrete fourth order boundary value problem Δ4u(t-2)-αΔ2u(t-1)+βu(t)=f(t,u(t)),t∈[1,N]Z,u(-1)=Δu(-1)=0,u(N+1)...

The authors establish sufficient conditions for the existence of solutions to boundary value problems for fractional differential inclusions involving the Hadamard type fractional derivative of order $\alpha \in (1,2]$ in Banach spaces. Their approach uses M\"onch's fixed point theorem and the Kuratowski measure of noncompacteness.

In this paper, the authors study the forward and inverse problems for a fractional boundary value problem with Dirichlet boundary conditions. The existence and uniqueness of solutions for the forward problem is first proved. Then an inverse source problem is considered.

This manuscript is devoted to proving some results concerning the existence of solutions to a class of boundary value problems for nonlinear implicit fractional differential equations with non-instantaneous impulses and generalized Hilfer fractional derivatives. The results are based on Banach&rsquo, s contraction principle and Krasnosel&rsquo, skii&rsquo, s fixed point theorem. To illustrate the results, an example is provided.

We obtain a new upper estimate for the Green’s function associated with a higher order fractional boundary value problem. As an application of this result, criteria for the existence of positive solutions of the problem are then established.

This paper is concerned with the existence of multiple anti-periodic solutions to a nonlinear fourth order difference equation. The analysis is based on variational methods and critical point theory. Clark’s critical point theorem is used to prove the main results. An example illustrates the applicability of the results.

This paper is concerned with periodic boundary value problems for a fourth order nonlinear difference equation. Via variational methods and critical point theory, sufficient conditions are given fo...

In this paper, the authors discuss the existence of multiple solutions to a class of second-order Sturm–Liouville boundary value systems. Their proofs are based on variational methods and critical point theory.

In this paper, the authors establish the existence and uniqueness of solutions for a class of boundary value problem for nonlinear implicit fractional differential equations with impulses and Caputo–Hadamard type fractional derivatives. The stability of this problem is also studied. The arguments are based on the Banach contraction principle and Schaefer’s fixed point theorem. Two examples are presented to show the applicability of the results.

In this paper the authors study the Hammerstein generalized integral equation $$\begin{aligned} u(t)=\int _{0}^{1}k(t,s)\text { }g(s)\text { }f(s,u(s),u^{\prime }(s),\dots ,u^{(m)}(s))\,ds, \end{aligned}$$ where $$k:[0,1]^{2}\rightarrow {\mathbb {R}}$$ are kernel functions, $$m\ge 1$$ , $$g:[0,1] \rightarrow [0,\infty )$$ , and $$f:[0,1]\times {\mathbb {R}}^{m+1} \rightarrow [0,\infty )$$ is a $$L^{\infty }-$$ Caratheodory function. The existence of solutions of integral equations has been studied in concrete and abstract cases, by different methods and techniques. However, in the existing literature, the nonlinearity depends only on the unknown function. This paper is one of a very few to consider equations having discontinuous nonlinearities that depend on the derivatives of the unknown function and having discontinuous kernels functions that have discontinuities in the partial derivatives with respect to their first variable. Our approach is based on the Krasnosel’skiĭ–Guo compression/expansion theorem on cones and it can be applied to boundary value problems of arbitrary order $$n>m$$ . The last two sections of the paper contain an application to a third order nonlinear boundary value problem and a concrete example.

For the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

The authors discuss the existence of multiple solutions of the system of fractional boundary value problems of the formwhere , is a parameter, , and are, respectively, the left and right Riemann–Liouville fractional integrals of order , and is a given function. Using critical point theory, sufficient conditions are found under which the above problem has at least two or infinitely many nontrivial solutions.

In this paper, the authors obtain the existence of infinitely many classical solutions to the boundary value system with Sturm–Liouville boundary conditions $$\left\{\begin{array}{ll}-(\phi_{p_i}(u_{i}^\prime))^\prime = \lambda F_{u_{i}}(x,u_{1},\ldots,u_{n})h_{i}(u^\prime_i)\quad {\rm in} \, (a,b), \\ \alpha_iu_{i}(a)-\beta_iu^ \prime_{i}(a)=0, \quad \gamma_iu_{i}(b)+\sigma_iu^\prime_{i}(b)=0, \end{array}\quad{i = 1, \ldots , n.} \right.$$ Critical point theory and Ricceri’s variational principle are used in the proofs.

In this paper, the authors consider a nonlinear fractional boundary value problem defined on a star graph. By using a transformation, an equivalent system of fractional boundary value problems with mixed boundary conditions is obtained. Then the existence and uniqueness of solutions are investigated by fixed point theory.

Sufficient conditions are established for the existence of at least one nontrivial classical solution to the boundary value system with Sturm-Liouville boundary conditions {−(φpi(u′i(x)))′ = λFui (x,u1 , . . . ,un)hi(u′i(x)) in (a,b), αiui(a)−βiu′i(a) = 0, γiui(b)+σiu′i(b) = 0, i = 1, . . . ,n. The analysis is based on variational methods and critical point theory. Mathematics subject classification (2010): 34B15, 47J10.

In this article, we study a type of nonlinear fractional boundary value problem with integral boundary conditions. By constructing an associated Green's function, applying spectral theory and using fixed point theory on cones, we obtain criteria for the existence, multiplicity and nonexistence of positive solutions.

We study a class of nonlinear discrete fourth-order Lidstone boundary value problems with dependence on two parameters. The existence, uniqueness and dependence of positive solutions on the parameters are discussed. Two sequences are constructed so that they converge uniformly to the unique solution of the problems. One example is included in the paper. Numerical computations of the example confirm our theoretical results. Recent results in the literature are extended and improved.

The authors study the singular boundary value problem with fractional q-derivatives $$\begin{gathered} - (D_q^\nu u)(t) = f(t,u),t \in (0,1), \hfill \\ (D_q^i u)(0) = 0,i = 0,...,n - 2,(D_q u)(1) = \sum\limits_{j = 1}^m {a_j (D_q u)(t_j ) + \lambda ,} \hfill \\ \end{gathered} $$ , where q ∈ (0, 1), m ≥ 1 and n ≥ 2 are integers, n − 1 < ν ≤ n, λ ≥ 0 is a parameter, f: (0, 1] × (0,∞) → [0,∞) is continuous, ai ≥ 0 and ti ∈ (0, 1) for i = 1, …,m, and Dqν is the q-derivative of Riemann-Liouville type of order ν. Sufficient conditions are obtained for the existence of positive solutions. Their analysis is mainly based on a nonlinear alternative of Leray-Schauder.

The authors consider a nonlinear fractional boundary value problem with the Dirichlet boundary condition. An associated Green's function is constructed as a series of functions by applying spectral theory. Criteria for the existence and uniqueness of solutions are obtained based on it.

In this paper, we establish the existence of at least three solutions of the multi-point boundary value system \[\left\{\begin{array}{ll} -(\phi_{p_i}(u'_{i}))'=\lambda F_{u_{i}}(x,u_{1},\ldots,u_{n}),\ t\in(0,1),\\ u_{i}(0)=\sum_{j=1}^m a_ju_i(x_j),\ u_{i}(1)=\sum_{j=1}^m b_ju_i(x_j), \end{array}\right. i=1,\ldots,n.\] The approaches used are based on variational methods and critical point theory.

The authors study a class of nonlinear higher order boundary value problem with fractional $q$-erivativesand dependence on a positive parameter $\lm$.The existence, uniqueness, and dependence of positive solutions on $\lm$ are discussed.Two sequences are constructed so that they converge uniformly to the unique solution of the problems.Two examples are included in the paper. Numerical computations of the examples confirm their theoretical results.

A second-order impulsive differential inclusion with Sturm-Liouville boundary conditions is studied. By using a nonsmooth version of a three critical point theorem of Ricceri, the existence of three solutions is obtained.

The authors consider a higher order boundary value problem that originates from the study of boundary value problems for the beam equation. Upper and lower estimates for positive solutions of the problem are obtained.

The authors study a type of nonlinear fractional boundary value problem with non-homogeneous integral boundary conditions. The existence and uniqueness of positive solutions are discussed. An example is given as the application of the results.

The authors study two related boundary value problems for discrete and continuous beam equations. Some upper and lower estimates for positive solutions of the problems are obtained. Sufficient conditions for the existence and nonexistence of positive solutions are established.

The authors study the boundary value problem with fractional q -derivatives - ( D q ν u ) ( t ) = f ( t , u ) , t ∈ ( 0 , 1 ) , ( D q i u ) ( 0 ) = 0 , i = 0 , … , n - 2 , ( D q u ) ( 1 ) = ∑ j = 1 m a j ( D q u ) ( t j ) + λ , where q ∈ ( 0 , 1 ) , m ⩾ 1 and n ⩾ 2 are integers, n - 1 ν ⩽ n , λ ⩾ 0 is a parameter, f : [ 0 , 1 ] × R → [ 0 , ∞ ) is continuous, a i ⩾ 0 and t i ∈ ( 0 , 1 ) for i = 1 , … , m , and D q ν is the q -derivative of Riemann–Liouville type of order ν . The uniqueness, existence, and nonexistence of positive solutions are investigated in terms of different ranges of λ .

The authors use the mixed monotone operator method to study the fractional boundary value problem −D ν+ u(t )= λ f (t, u), t ∈ (0,1), u (j) (0 )= 0, j = 0,...,n −2, (D α+ u(t))t=1 = 0. Here, m 1a ndn 3 are integers, n −1 < ν n ,1 α n −2, u(t)=(u1(t),...,um(t)) T , λ =(λ1,...,λm), λ f (t, u)=(λ1 f1(t, u),...,λm fm(t, u)) T ,a ndD ν+ is the Riemann-Liouville fractional derivative of order ν . Existence, uniqueness, and dependence of positive solutions on the parameter λ are discussed. An application to a special problem is also presented.

The authors obtain sufficient conditions for the existence of at least one and two positive solutions of a higher order semipositone fractional boundary value problem with a forcing term in the differential equation. Examples are included to illustrate the results.

The authors study the first-order singular boundary value problem One criterion is established for the existence of positive solutions. Two examples applying the result are also given. The existence result extends some recent work in the literature. Their analysis is mainly based on a nonlinear alternative of Leray–Schauder.

We study the existence of solutions of the boundary value problem (@f(u^(^n^-^1^)(t)))^'+f(t,u(t),u^'(t),...,u^(^n^-^1^)(t))=0,[email protected]?(0,1),g"i(u,u^',...,u^(^n^-^1^),u^(^i^)(0))=0,i=0,...,n-2,g"n"-"1(u,u^',...,u^(^n^-^1^),u^(^n^-^2^)(1))=0, where n>=2, @f and g"i, i=0,...,n-1, are continuous, and f is a Caratheodory function. We obtain an existence criterion based on the existence of a pair of coupled lower and upper solutions. We also apply our existence theorem to derive some explicit conditions for the existence of a solution of a special case of the above problem. In our problem, both the differential equation and the boundary conditions may have dependence on all lower order derivatives of the unknown function, and many boundary value problems with various boundary conditions, studied extensively in the literature, are special cases of our problem. Consequently, our results improve and cover a number of known results in the literature. Examples are given to illustrate the applicability of our theorems.

The authors consider the boundary value problem with a two-parameter nonhomogeneous multi-point boundary condition \gather u''+g(t)f(t,u)=0, \quad t\in (0,1),\nonumber \\ u(0)=\alpha u(\xi )+\lambda ,\quad u(1)=\beta u(\eta )+\mu .\nonumber \endgather Criteria for the existence of nontrivial solutions of the problem are established. The nonlinear term $f(t,x)$ may take negative values and may be unbounded from below. Conditions are determined by the relationship between the behavior of $f(t, x)/x$ for $x$ near $0$ and $\pm \infty $, and the smallest positive characteristic value of an associated linear integral operator. The analysis mainly relies on topological degree theory. This work complements some recent results in the literature. The results are illustrated with examples.

We employ fixed point index theory to establish existence results for positive solutions to the singular boundary value problem � −(au � ) � (t )= b(t)f(t,u(t)), t ∈ (0,1)

The authors consider the nth-order differential equation ?(?(u(n?1)(x)))?= f(x, u(x), ..., u(n?1)(x)), for 2?(0, 1), where ?: R? R is an increasing homeomorphism such that ?(0) = 0, n?2, I:= [0,1], and f : I ?Rn ? R is a L1-Carath?odory function, together with the boundary conditions gi(u, u?, ..., u(n?2), u(i)(1)) = 0, i = 0, ..., n? 3, gn?2 (u, u?, ..., u(n?2), u(n?2)(0), u(n?1)(0)) = 0, gn?1 (u, u?, ..., u(n?2), u(n?2)(1), u(n?1)(1)) = 0, where gi : (C(I))n?1?R ? R, i = 0, ..., n?3, and gn?2, gn?1 : (C(I))n?1?R2 ? R are continuous functions satisfying certain monotonicity assumptions. The main result establishes sufficient conditions for the existence of solutions and some location sets for the solution and its derivatives up to order (n?1). Moreover, it is shown how the monotone properties of the nonlinearity and the boundary functions depend on n and upon the relation between lower and upper solutions and their derivatives.

We consider the boundary value problem where n ⩾ 2 and m ⩾ 1 are integers, tj ∈ [0, 1] for j = 1, …, m, and f and gi, i = 0, …, n − 1, are continuous. We obtain sufficient conditions for the existence of a solution of the above problem based on the existence of lower and upper solutions. Explicit conditions are also found for the existence of a solution of the problem. The differential equation has dependence on all lower order derivatives of the unknown function, and the boundary conditions cover many multi-point boundary conditions studied in the literature. Schauder’s fixed point theorem and appropriate Nagumo conditions are employed in the analysis. Examples are given to illustrate the results. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The authors consider a fourth order two-point boundary value problem. Some a priori estimates to positive solutions for the problem are obtained. Explicit sufficient conditions for existence and nonexistence of positive solutions to the problem are established. An example to illustrate the results is included.