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Abstract The novelty of this paper is that, based on Mawhin’s continuation theorem, we present some sufficient conditions that ensure that there is at least one solution to a particular kind of a boundary value problem with the p-Laplacian and generalized fractional Caputo derivative.

Abstract In this paper, we study the existence and uniqueness of a periodic solution for a third-order neutral delay differential equation (NDDE) by applying Mawhin’s continuation theorem of coincidence degree and analysis techniques. An illustrative example is given as an application to support our results. To confirm the accuracy of our results, we also present a plot of the behavior of the periodic solution.

Abstract In this paper, the existence of positive periodic solutions is studied for super-linear neutral Liénard equation with a singularity of attractive type ( x ( t ) − c x ( t − σ ) ) ″ + f ( x ( t ) ) x ′ ( t ) − φ ( t ) x μ ( t ) + α ( t ) x γ ( t ) = e ( t ) , $$ \bigl(x(t)-cx(t-\sigma)\bigr)''+f\bigl(x(t) \bigr)x'(t)-\varphi(t)x^{\mu}(t)+ \frac{\alpha(t)}{x^{\gamma}(t)}=e(t), $$ where f : ( 0 , + ∞ ) → R $f:(0,+\infty)\rightarrow R$ , φ ( t ) > 0 $\varphi(t)>0$ and α ( t ) > 0 $\alpha(t)>0$ are continuous functions with T-periodicity in the t variable, c, γ are constants with | c | < 1 $|c| 1 $\mu>1$ by using Mawhin’s continuation theorem as well as the techniques of a priori estimates. At last, an example is given to show applications of the theorem.

Abstract By defining the Banach spaces endowed with the appropriate norm, constructing a suitable projection scheme, and using the coincidence degree theory due to Mawhin, we study the existence of solutions for functional boundary value problems at resonance on the half-line with dim Ker L = 1 $\operatorname{dim}\operatorname{Ker}L = 1$ . And an example is given to show that our result here is valid.

Abstract By constructing a suitable projection scheme and using the coincidence degree theory of Mawhin, we study the existence of solutions for conjugate boundary value problems with functional boundary conditions at resonance with dimKerL=1 $\operatorname{dim}\operatorname{Ker}L = 1$. Examples are given to illustrate our main results.

Abstract In this paper, by using Mawhin’s continuation theorem, we establish some sufficient conditions for the existence of at least one solution for a class of fractional infinite point boundary value problem at resonance. Moreover, an example is given to illustrate our results.

Abstract In this paper, by using Mawhin’s continuation theorem, we investigate the existence of solutions for a class of fractional differential equations with multi-point boundary value problems at resonance, and the dimension of the kernel for a fractional differential operator is three. An example is given to show our main result.