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In this paper, we study a nonlinear fractional differential equation involving two mixed fractional orders with nonlocal boundary conditions. By using some new techniques, we introduce a formula of solutions for above problem, which can be regarded as a novelty item. Moreover, under the weak assumptions and using Leray–Schauder degree theory, we obtain the existence result of solutions for above problem. Furthermore, we discuss the Ulam–Hyers stability of the above fractional differential equation. Three examples illustrate our results.

In this paper, we investigate the existence of positive solutions for a system of nonlinear fractional differential equations nonlocal boundary value problems with parameters and p-Laplacian operator. Under different combinations of superlinearity and sublinearity of the nonlinearities, various existence results for positive solutions are derived in terms of different values of parameters via the Guo-Krasnosel’skii fixed point theorem.

A couple ($\sigma ,\tau $) of lower and upper slopes for the resonant second order boundary value problem $$ x'' = f(t,x,x'), \quad x(0) = 0,\quad x'(1) = \int _0^1 x'(s) {\rm d}g(s), $$ with $g$ increasing on $[0,1]$ such that $\int _0^1 dg = 1$, is a couple of functions $\sigma , \tau \in C^1([0,1])$ such that $\sigma (t) \leq \tau (t)$ for all $t \in [0,1]$, \begin {gather} \sigma '(t) \geq f(t,x,\sigma (t)), \quad \sigma (1) \leq \int _0^1 \sigma (s) {\rm d}g(s),\nonumber \\ \tau '(t) \leq f(t,x,\tau (t)), \quad \tau (1) \geq \int _0^1 \tau (s) {\rm d}g(s),\nonumber \end {gather} in the stripe $\int _0^t\sigma (s) {\rm d}s \leq x \leq \int _0^t \tau (s) {\rm d}s$ and $t \in [0,1]$. It is proved that the existence of such a couple $(\sigma ,\tau )$ implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained.

An m-point nonlocal boundary value problem is posed for quasi- linear differential equations of first order on the plane. Nonlocal boundary value problems are investigated using the algorithm of reducing nonlocal boundary value problems to a sequence of Riemann-Hilbert problems for a generalized analytic function. The conditions for the existence and uniqueness of a generalized solution in the space are considered.

This paper focuses on the maximum and minimum solutions for a fractional order differential system, involving a p-Laplacian operator and nonlocal boundary conditions, which arises from many complex processes such as ecological economy phenomena and diffusive interaction. By introducing new type growth conditions and using the monotone iterative technique, some new results about the existence of maximal and minimal solutions for a fractional order differential system is established, and the estimation of the lower and upper bounds of the maximum and minimum solutions is also derived. In addition, the iterative schemes starting from some explicit initial values and converging to the exact maximum and minimum solutions are also constructed.

In the light of the fixed point theorems, we analytically establish the conditions for the uniqueness of solutions as well as the existence of at least one solution in the nonlocal boundary value problem for a specific kind of nonlinear fractional differential equation. Furthermore, we provide a representative example to illustrate a possible application of the established analytical results.

Nonlocal boundary value problems at resonance for a higher order nonlinear differential equation with a p-Laplacian are considered in this paper. By using a new continuation theorem, some existence results are obtained for such boundary value problems. An explicit example is also given in this paper to illustrate the main results.

The paper is concerned with the delay differential equation u ″ + λ b ( t ) f ( u ( t − τ ) ) = 0 satisfying u ( t ) = 0 for − τ ⩽ t ⩽ 0 and u ( 1 ) = g ( ∫ 0 1 u ( t ) d β ( t ) ) , where ∫ 0 1 u ( t ) d β ( t ) denotes the Riemann–Stieltjes integral. By applying the fixed point theorem in cones, we show the relationship between the asymptotic behaviors of the quotient f ( u ) u (at zero and infinity) and the open intervals (eigenvalue intervals) of the parameter λ such that the problem has zero, one and two positive solution(s). If g ( t ) = t , by using a property of the Riemann–Stieltjes integral, the above nonlocal boundary value problem educes a three-point boundary value problem with delay, for which some similar results are established.

International Conference on Advancements in Mathematical Sciences (AMS) -- NOV 05-07, 2015 -- Antalya, TURKEY WOS: 000371818700081 The nonlocal boundary value problem for elliptic-Schrodinger equations is considered. The stability estimates for the solution of this problem are presented. In applications, the stability estimates of the mixed type boundary value problems for elliptic-Schrodinger equations are obtained. Fatih Univ, Badji Mokhtar Annaba Univ, Inst Math & Math Modeling

In this article, we prove an existence result for a nonlocal boundary value problem at resonance concerning a second order differential equation. Our method is based upon the coincidence degree theory of Mawhin.

Criteria for the existence and uniqueness of a solution of the boundary value problem are established, where ƒ :]a, b[×R 2 → R satisfies the local Carathéodory conditions, and μ : [a, b] → R is the function of bounded variation. These criteria apply to the case where the function ƒ has nonintegrable singularities in the first argument at the points a and b.

In the paper, the problem on the existence and uniqueness of a solution to the nonlocal problem u'(t)=l(u)(t)+q(t), u(a)=h(u)+c is studied. They are found conditions guaranteeing the unique solvability of considered problem. Je studována problematika existence a jednoznačnosti nelokální okrajové úlohy u'(t)=l(u)(t)+q(t), u(a)=h(u)+c. Jsou nalezeny podmínky zaručující jednoznačnou řešitelnost zvažované úlohy.

We consider a second order nonlinear differential equation with nonlocal (integral) condition. The spectrum of it differs essentially from the known ones. First Published Online:14 Oct 2010

In this paper we consider the following nth order nonlocal boundary value problem at resonance case x(t) = f(t, x(t), x′(t), . . . , x(n−1)(t)), t ∈ (0, 1), x(0) = 0, i = 0, 1, . . . , n − 2, x(n−1)(1) = ∫ 1 0 x(n−1)(s) dg(s), where f : [0, 1]×Rn → R is a continuous function, g : [0, 1] → [0,∞) is a nondecreasing function with g(0) = 0. Under the resonance condition g(1) = 1, by applying the coincidence degree theory of Mawhin, we obtain some existence results for the boundary value problems. We also give an example to illustrate our results.

In this paper, we investigate a second-order nonlinear difference equation with sign-changing nonlinearity subject to two different sets of nonlocal boundary conditions. The explicit expressions of the associated Green's functions are presented. By using a recently developed fixed point theorem, we establish sufficient conditions for the existence of multiple positive solutions of the boundary value problem.

In this paper we investigate the existence of multiple nontrivial solutions of a nonlinear heat flow problem with nonlocal boundary conditions. Our approach relies on the properties of a vector field on the phase plane and utilizes Sperner’s Lemma, combined with the continuum property of the solutions funnel.