---

In the present paper, we write out the eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal parity condition of the first kind. We analyze the  completeness,  the basis  property,  and the minimality of the eigenfunctions  in the space `overline(W)_p^(2l) (0,pi)`,  where  `overline(W)_p^(2l) (0,pi)` be the set of functions `f in  W_p^(2l) (0,pi)`, satisfying of the following conditions: `f^{(2k-1)}(0)=0, k=1,2,...,l`.

---

In this work, the common T approximate strict fixed point property for multi-valued mappings F,G : X -> P_{cl,bd}(X) is introduced to prove necessary and sufficient condition for existence of a common strict xed point of multi-valued mappings involved therein. Our results extend and unify comparable results in the existing literature. We also provide examples to support our results.

 

---

In this paper, some basic results under various conditions for $varphi$-convex functions are investigated.

 

---

In this paper we introduce two new mapping in connection to Hermite-Hadamard type inequality. Some results concerning these mappings associated to the celebrated Hermite-Hadamard integral inequality for preinvex functions are given.