We extend Zajíček’s theorem from [Za] about points of singlevaluedness of monotone operators on Asplund spaces. Namely we prove that every monotone operator on a subspace of a Banach space containing densely a continuous image of an Asplund space (these spaces are called GSG spaces) is singlevalued on the whole space except a $\sigma$-cone supported set.

This work is devoted to the existence of solutions for a class of singular third-order boundary value problem associated with a $\phi$-Laplacian operator and posed on the positive half-line; the nonlinearity also depends on the first derivative. The upper and lower solution method combined with the fixed point theory guarantee the existence of positive solutions when the nonlinearity is monotonic with respect to its arguments and may have a space singularity; however no Nagumo type condition is assumed....

Iterative methods based on small functions are used both to show local surjectivity of certain operators and a fixed point property of mappings on scales of complete metric spaces.

We give an affirmative answer to Schauder's fixed point question.

We present sufficient conditions for the existence of solutions of Fredholm integral inclusion equations using new sort of contractions, named as multivalued almost F -contractions and multivalued almost F -contraction pairs under ı-distance, defined in b-metric spaces. We give its relevance to fixed point results in orbitally complete b-metric spaces. To rationalize the notions and outcome, we illustrate the appropriate examples.

Les solutions d’équations d’évolution $\frac{du}{dt}+Au\ni f$ où $A$ est un opérateur maximal monotone d’un espace de Hilbert $H$, et $f\in L^1(0,T,H)$ sont étudiées dans le cas général en introduisant une notion de solution faible. Des résultats particuliers sont donnés lorsque $H$ est de dimension finie ou plus généralement lorsque l’intérieur de $D\left(A\right)$ est non vide.

In this paper, we are concerned with the existence of solutions of the following multi-point boundary value problem consisting of the higher-order differential equation $$x^{\left(n\right)}\left(t\right)=f(t,x\left(t\right),x^{\text{'}}\left(t\right),\cdots ,x^{(n-1)}\left(t\right))+e\left(t\right),0<t<1,(*)$$ and the following multi-point boundary value conditions $$\begin{array}{cc}\hfill 1*-1x^{\left(i\right)}\left(0\right)& =0fori=0,1,\cdots ,n-3,\hfill \\ \hfill x^{(n-1)}\left(0\right)& =\alpha x^{(n-1)}\left(\xi \right),x^{(n-2)}\left(1\right)=\sum _{i=1}^m{\beta }_i{x}^{(n-2)}\left({\eta }_i\right).**\hfill \end{array}$$ Sufficient conditions for the existence of at least one solution of the BVP $(*)$ and $(**)$ at resonance are established. The results obtained generalize and complement those in [13, 14]. This paper is directly motivated by Liu and Yu [J. Pure Appl. Math. 33 (4)(2002), 475–494...

2000 Mathematics Subject Classification: 45G15, 26A33, 32A55, 46E15.Schauder's fixed point theorem is used to establish an existence result for an infinite system of singular integral equations in the form: (1) xi(t) = ai(t)+ ∫t0 (t − s)− α (s, x1(s), x2(s), …) ds, where i = 1,2,…, α ∈ (0,1) and t ∈ I = [0,T]. The result obtained is applied to show the solvability of an infinite system of differential equation of fractional orders.